IHTKZ@KDIOTK @KCFOHICS GA @OTKZIOLS Skcghd Kditigh
@icfinoh Tkcfhglgnicol [hivkrsity
KVROHDIHN KD[COTIGHOL FGZI]GHS, LLC
Fgunftgh, @icfinoh `odfuvojlk.grn :2?4
Cgpyrinfts C :2?4 Kxpohdihn Kducotighol Fgrizghs, LLC `odfuvojlk.grn
Oll rinfts rkskrvkd. Hg port ga tfis pujlicotigh `oy jk rkprgduckd, stgrkd ih o rktrikvol systk`, gr trohs`ittkd, ih ohy agr` gr jy ohy `kohs, klkctrghic, `kcfohicol, pfgtgcgpyihn, rkcgrdihn, gr gtfkrwisk, witfgut prigr pkr`issigh ga Kxpohdihn Kducotighol Fgrizghs.
Ropkr cgpy= k-jgge=
ISJH=9>6-2-99?:441-2-> ISJH=9>6-2-99?:441-?-4
Tg `y arikhd ohd `y lgvk Rrgakssgr Rusfpolotfo @urtfy
CGHTKHTS
CGHTKHTS RZKAOCK TG TFK AIZST KDITIGH
V
RZKAOCK TG TFK SKCGHD KDITIGH
V
O HGTK TG ST[DKHTS
VI
O HGTK TG TFK IHSTZ[CTGZ
VI
OCEHG^LKDNK@KHTS
VII
CFORT KZ ?
STZKSS OHD STZOIH
?
?.?
Strkss gh o Suraock
:
?.:.? ?.:.: ?.:.3 ?.:.4 ?.:.; ?.:.1
Strkss ot o Rgiht Sinh Cghvkhtigh agr Strkss Strkss Klk`khts Cghstructigh ga o Strkss Cujk Strkss Klk`kht ih Cylihdricol Cggrdihotks Strkss Klk`kht ih Spfkricol Cggrdihotks Rlohk Strkss
3 4 4 4 ; 1 1
?.3.? ?.3.: ?.3.3
Strkss Trohsagr`otigh ih Twg Di`khsighs @otrix @ktfgd ih Twg Di`khsighs Rrihcipol Strkssks @oxi`u` Sfkor Strkss
> 6 9 ?2
?.4.? ?.4.: ?.4.3 ?.4.4
Strkss Trohsagr`otigh ih Tfrkk Di`khsighs Rrihcipol Strkssks Rrihcipol Strkss Cghvkhtigh Cforoctkristic Kquotigh ohd Strkss Ihvoriohts Gctofkdrol Strkssks
?? ?: ?: ?3 ?3
?.;.?
Ovkronk Hgr`ol ohd Sfkor Stroihs [hits ga Ovkronk Stroihs
?4 ?;
?.1.?
Stroih ot o Rgiht ohd Stroih–Displock`kht Kquotighs Rlohk Stroih
?1 ?>
?.>
Aihitk Diaakrkhck Opprgxi`otigh ga Stroihs
?>
?.6
Stroih Trohsagr`otigh Rrihcipol Stroihs
?6 ?9
?.9
Clgsurk
:2
CFORTKZ :
@OTKZIOL DKSCZIRTIGH
:?
:.?
Lihkor @otkriol @gdkl @ghgclihic @otkriols Grtfgtrgpic @otkriols Trohsvkrskly Isgtrgpic @otkriols Isgtrgpic @otkriols
:? :: :3 :4 :4
:.:
Rlohk Strkss ohd Rlohk Stroih
:;
:.3
Aoilurk Tfkgriks @oxi`u` Sfkor Strkss Tfkgry @oxi`u` Gctofkdrol Sfkor Strkss Tfkgry @oxi`u` Hgr`ol Strkss Tfkgry @gdiaikd @gfr ’s Tfkgry
:1 :> :> :6 :6
:.4
Soiht-Qkhoht’s Rrihciplk
:9
:.;
Strkss Cghckhtrotigh Aoctgr
32
:.1
Strkss Ihtkhsity Aoctgr
3?
?.:
?.3
?.4
?.; ?.1
?.6.?
:.?.? :.?.: :.?.3 :.?.4
:.3.? :.3.: :.3.3 :.3.4
Q
:.1.?
Oholysis Rrgckdurk
33
:.>
Aotinuk
33
:.6
Clgsurk
3;
CFORTKZ 3
JOSIC STZ[CT[ZOL @K@JKZS
31
3.?
Lgnic ih Structurol Oholysis
3>
3.: 3.:.? 3.:.: 3.:.3 3.:.4
Josic Tfkgriks ga Ghk-Di`khsighol Structurol @k`jkrs Li`itotighs Dkaihitigh ga voriojlks ohd sinh cghvkhtighs Sfkor Strkss ih Tfih Sy``ktric Jko`s Strkssks ohd Stroihs
36 39 39 43 4;
3.3.? 3.3.: 3.3.3
Jguhdory Qoluk Rrgjlk`s Oxiol Displock`kht Tgrsighol Zgtotigh Jko` Dkalkctigh
41 4> 4> 46
3.4.? 3.4.: 3.4.3 3.4.4
Discghtihuity Auhctighs Dkaihitigh ga Discghtihuity Auhctighs Oxiol displock`kht Tgrsighol Zgtotigh Jko` Dkalkctigh
;2 ;? ;: ;3 ;3
3.;.? 3.;.: 3.;.3 3.;.4 3.;.; 3.;.1
Sy``ktric Jkhdihn ga Curvkd Jko`s Eihk`otics @otkriol @gdkl Stotic Kquivolkhcy Lgcotigh ga tfk Hkutrol Oxis Strkss Agr`ulo Cg`jihkd Lgodihn
;; ;; ;1 ;> ;> ;> ;6
3.1
Clgsurk
;9
CFORTKZ 4
CG@RGSITK STZ[CT[ZOL @K@JKZS
12
4.?
Cg`pgsitk Oxiol @k`jkrs
1?
4.:
Cg`pgsitk Sfoats
13
4.3.? 4.3.: 4.3.3
Cg`pgsitk Sy``ktric Jko`s Hgr`ol Jkhdihn Strkss ih Cg`pgsitk Jko`s Lgcotigh ga tfk Hkutrol Oxis (grinih) ih Cg`pgsitk Jko`s Jkhdihn Sfkor Strkss ih Cg`pgsitk Jko`s
1; 1; 11 11
4.4.? 4.4.: 4.4.3 4.4.4 4.4.; 4.4.1
@icrg`kcfohics ga Cg`pgsitks Grtfgtrgpic Cg`pgsitk Lo`ihok [hit Ckll Opprgxi`otigh Lghnitudihol @gdulus ga Klosticity Kx Rgissgh’s Zotig hxy Trohsvkrsk @gdulus ga Klosticity Ky Sfkor @gdulus Nxy
19 19 19 >2 >2 >? >?
4.;
Clgsurk
>:
CFORTKZ ;
IHKLOSTIC STZ[CT[ZOL JKFOQIGZ
>3
;.?
Kaakcts ga Tk`pkroturk
>4
;.:
Ihitiol Strkss gr Stroih ih Oxiol @k`jkrs
>;
;.3
Tk`pkroturk ohd Ihitiol stroih Kaakcts ih Oxiol @k`jkrs
>1
3.3
3.4
3.;
4.3
4.4
CGHTKHTS ;.4
QII
Hghlihkor @otkriol @gdkls Klostic–Rkrakctly Rlostic @otkriol @gdkls Lihkor Stroih-Fordkhihn @otkriol @gdkl Rgwkr Low @otkriol @gdkl
>> >6 >6 >9
Klostic–Rkrakctly Rlostic Oxiol @k`jkrs
62
Klostic–Rkrakctly Rlostic Circulor Sfoats Zksiduol Sfkor Strkss
62 6:
;.>
Klostic–Rkrakctly Rlostic Jko`s
63
;.6
@otkriol Hghlihkority ih Structurol @k`jkrs
61
;.9
Qiscgklosticity @oxwkll @gdkl Eklvih @gdkl Crkkp Tkst Zkloxotigh Tkst Nkhkrolizkd Qiscgklostic Lihkor @gdkls
6> 66 66 69 92 92
;.?2
Clgsurk
9?
CFORT KZ 1
TFIH-^OLLKD STZ[CT[ZOL @K@JKZS
93
1.? 1.?.? 1.?.: 1.?.3 1.?.4 1.?.; 1.?.1 1.?.> 1.?.6 1.?.9 1.?.?2 1.?.?? 1.?.?:
Tfkgry ga [hsy``ktric Jkhdihn ga Jko`s Dkagr`otigh Stroih Distrijutigh @otkriol @gdkl Ihtkrhol Agrcks ohd @g`khts Sinh Cghvkhtigh Jkhdihn Agr`ulos Lgcotigh ga Grinih Skcghd Orko @g`kht ga Ihkrtios Rrihcipol @g`khts ga Ihkrtio Hkutrol Oxis Kquilijriu` Kquotighs Jguhdory Qoluk Rrgjlk`
94 9; 91 91 9> 96 96 96 99 99 ?2? ?2: ?2:
1.:.?
Sfkor Strkss ih Tfih Gpkh Skctighs Sfkor Ckhtkr ga Tfih Gpkh Skctighs
?23 ?24
1.3.?
Sfkor Strkssks ih Tfih Clgskd Skctighs Sfkor Ckhtkr ga Tfih Clgskd Skctighs
?26 ?29
1.4.? 1.4.:
Tgrsigh ga Tfih-^ollkd Tujks Tgrsighol Dkagr`otigh Tgrsigh ga @ultickll Tujks
?29 ??? ??3
1.;
Cg`jihkd Lgodihn
??4
1.1
Clgsurk
??1
CFORT KZ >
KHKZNY @KTFGDS
??>
>.?
Stroih Khkrny
??>
>.:
Lihkor Stroih Khkrny Dkhsity Oxiol Stroih Khkrny Tgrsighol Stroih Khkrny ga Circulor Sfoats Jkhdihn Stroih Khkrny
??6 ??9 ?:2 ?:2
^gre
?:?
Qirtuol ^gre Typks ga Jguhdory Cghditighs Eihk`oticolly Od`issijlk Auhctighs Stoticolly Od`issijlk Auhctighs
?:: ?:3 ?:3 ?:4
;.4.? ;.4.: ;.4.3 ;.; ;.1 ;.1.?
;.9.? ;.9.: ;.9.3 ;.9.4 ;.9.;
1.: 1.3 1.4
>.:.? >.:.: >.:.3 >.3 >.4 >.4.? >.4.: >.4.3
>.4.4 >.4.;
Qirtuol Displock`kht @ktfgd Qirtuol Agrck @ktfgd
?:; ?:;
>.;.? >.;.: >.;.3
Du``y [hit Lgod @ktfgd Oxiol @k`jkrs Tgrsigh ga Circulor Sfoats Sy``ktric Jkhdihn ga Jko`s
?:1 ?:1 ?:1 ?:>
>.1
Costinliohg’s Tfkgrk`s
?:6
>.>
Dkalkctigh ga Curvkd Jko`s
?3?
>.6
@ihi`u` Rgtkhtiol Khkrny
?3:
Zoylkinf–Zitz @ktfgd Oxiol @k`jkrs Tgrsigh ga Circulor Sfoats Sy``ktric Jkhdihn ga Jko`s
?33 ?33 ?3; ?3;
>.?2
Clgsurk
?31
CFORTKZ 6
KLOSTICITY OHD TFK @KCFOHICS GA @OTKZIOLS
?3>
6.? 6.?.? 6.?.: 6.?.3 6.?.4
Klosticity Kquotighs Cg`potijility Kquotighs Rlohk Strkss ohd Rlohk Stroih Kquilijriu` Kquotighs Jguhdory Cghditighs
?3> ?36 ?36 ?39 ?42
6.:.? 6.:.:
Oxisy``ktric Rrgjlk`s Oxisy``ktric Rlohk Stroih Oxisy``ktric Rlohk Strkss
?42 ?42 ?4?
6.3
Zgtotihn Dises
?4?
6.4 6.4.? 6.4.:
Tfice Fgllgw Cylihdkrs Ihtkrhol Rrkssurk Ghly Kxtkrhol Rrkssurk Ghly
?4: ?43 ?43
6.;.? 6.;.: 6.;.3
Tfih Dises Ihtkrhol Rrkssurk Ghly Kxtkrhol Rrkssurk Ghly Sfrihe Aittihn
?43 ?44 ?44 ?44
Oiry Strkss Auhctigh
?4;
6.>.? 6.>.: 6.>.3
Sglutigh jy Rglyhg`iols _uodrotic Rglyhg`iols Cujic Rglyhg`iols Agurtf-Grdkr Rglyhg`iols
?41 ?41 ?41 ?4>
6.6.? 6.6.: 6.6.3
Tgrsigh ga Hghcirculor Sfoats Soiht-Qkhoht’s @ktfgd Rrohdtl’s @ktfgd Rrgckdurk agr Sglvihn Rrgjlk`s ga Tgrsigh ga Hg H ghcirculor Sf S foats
?4> ?46 ?46 ?;2
6.9.?
@k`jrohk oholgny agr tgrsigh ga hgh-circulor sfoats @k`jrohk oholgny agr crgss skctigh witf fglks
?;? ?;:
6.?2.?
Tgrsigh ga tfih-wollkd gpkh skctigh Khd kaakcts gh rkctohnulor crgss skctighs
?;3 ?;4
6.??
Clgsurk
?;;
ORRKHDIV O
IHAGZ@OTIGH CFOZTS OHD TOJLKS
?;1
O.?
Strkss Cghckhtrotigh Aoctgr agr O Aihitk Rlotk witf o Ckhtrol Fglk
?;1
O.:
Strkss Cghckhtrotigh Aoctgr agr Stkppkd Oxiol Circulor Jors witf Aillkt?
?;>
O.3
Strkss Cghckhtrotigh Aoctgr agr Stkppkd Circulor Sfoats witf Aillkt?
?;>
>.;
>.9 >.9.? >.9.: >.9.3
6.:
6.;
6.1 6.>
6.6
6.9 6.?2
CGHTKHTS O.4
Strkss Cghckhtrotigh Aoctgr agr O Stkppkd Circulor Jko` witf Aillkt?
?;6
O.;
Orkos, Ckhtrgids, ohd Skcghd orko `g`khts ga ihkrtio
?;9
O.1
Dkalkctighs ohd Slgpks ga jko`s
?12
O.>
Rrgpkrtiks ga TYpicol Aijkr ohd @otrix @otkriols
?12
O.6
Rrgpkrtiks ga Troditighol `otkriols (ARS uhits)
?1?
O.9
Rrgpkrtiks ga Troditighol @otkriols (`ktric [hits)
?1?
O RRKHDIV J
JIJLIGNZORFY
?1:
IHDKV
?13
[HIT CGHQKZSIGH
?1;
SI RZKAIVKS
?1;
NZKKE OLRFOJKT
?1;
AGZ@[LO SFKKT
?11
IV
RZKAOCK TG TG TFK AIZST KDITIGH Jig`kdicol khnihkkrihn is o nrgwihn hkw disciplihk, ohd plostics khnihkkrihn is jkcg`ihn o pgpulor hkw `ihgr5 ih jgtf aiklds, strkss oholysis rkquirks oh uhdkrstohdihn ga tfk `kcfohics ga ihklostic ohd hghlihkor `otkriol jkfovigr. Opplicotighs ga ihklostic ohd hghlihkor `otkriol ork olsg nrgwihn os `ktols cg`pktk witf hkw `otkriols ih khnihkkrihn dksinh jy gpkrotihn ih tfk plostic rknigh tfrgunf prkstrkssihn. Tfk nrgwihn usk ga `ktol `otrix cg`pgsitks, pgly`kr cg`pgsitks, rkihagrckd cghcrktk, ohd wggdkh jko`s stiaakhkd witf stkkl strips ohd gtfkr lo`ihotkd structurks k`pfosizks tfk hkkd tg kxpgsk st udkhts tg tfk oholysis ga ghkdi`khsighol cg`pgsitk structurol `k`jkrs. Tfk ujiquity ga tfk aihitk klk`kht `ktfgd ih khnihkkrihn oholysis ohd dksinh k`pfosizks tfk kducotighol i`pgrtohck ga khkrny prihciplks. Kquolly sinhiaicoht, tfk kvoluotigh ohd usk ga tfk rksults prgduckd jy cg``krciol aihitk klk`kht cg`putkr sgatwork rkquirks oh uhdkrstohdihn ga strkss ohd stroih trohsagr`otigh ih tfrkk di`khsighs ohd ga aoilurk tfkgriks. Copstghk skhigr dksinh prgmkcts fovk oddkd tg tfk i`pgrtohck ga uhdkrstohdihn uhsy``ktric jkhdihn ohd tfk cghckpt ohd usk ga sfkor ckhtkrs ih dksinh. Jko` ohd sfoat vijrotighs, jko`s gh klostic aguhdotighs, Ti`gsfkheg jko`s, ohd sg gh ork o`ghn tfk `ohy tgpics ih kxistihn okrgspock, civil, ohd `kcfohicol khnihkkrihn cgursks tfot usk tfk prihciplks ga `kcfohics ga `otkriols. Ia o studkht is tg jk tounft tfk `kcfohics ga tfk tgpics dkscrijkd witfgut jkihn gvkrwfkl`kd jy oll tfkir ihfkrkht cg`plkxitiks, tfkh tfk prkskhtotigh ga tfk `otkriol `ust fovk cgfkrkhck ohd cg`pocthkss tfot cghsglidotks wfot fk gr sfk fos olrkody lkorhkd ih tfk ihtrgductgry cgursk ohd juilds gh it. Tfis is tfk uhdkrlyihn dksinh ga tfis jgge, ohd tfk Hgtk tg tfk Ihstructgr klojgrotks gh it. Oh ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols is rkquirkd ih `gst khnihkkrihn disciplihks. Tfk kducotighol pfilgsgpfy jkfihd tfk rkquirk`kht is tg tkocf cg``gh `kcfohics cghckpts ohd prihciplks ih o sihnlk cgursk ohd tg prkskht tfk kxtkhsighs ohd opplicotighs ga tfk `kcfohics cghckpts ohd prihciplks ih ihdividuol disciplihks. Tfis kducotighol pfilgsgpfy oddrkssks tfk hkkd ga ihtkrdisciplihory kducotigh wfilk rkolizihn curriculu` kaaicikhcy jy rkducihn duplicotigh ga cgursk cghtkht—oh i`pgrtoht cghsidkrotigh os kducotgrs ottk`pt tg `gdkrhizk khnihkkrihn kducotigh jy ihcgrpgrotihn rkskorcf ihtg o jurnkghihn curriculu`. Tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols fos skrvkd tfk khnihkkrihn cg``uhity wkll, jut tfk trk`khdgus nrgwtf ih tfk opplicotighs ga `kcfohics ga `otkriols is jrihnihn oddkd i`pgrtohck tg o skcghd cgursk, Ihtkr`kdiotk @kcfohics ga @otkriols. Tfk ihtkr`kdiotk cgursk is gatkh tounft jy ihstructgrs wfg usk tfkir gwh hgtks ih cghmuhctigh witf kitfkr gptighol tgpics arg` o tkxtjgge dksinhkd pri`orily tg tkocf tfk ihtrgductgry cgursk gr sklkctkd tgpics arg` o jgge pri`orily dksinhkd agr o nroduotk cgursk gh odvohckd `kcfohics ga `otkriols. Tfk pkdongnicol hkkds ga tfk studkhts witf rkspkct tg tfkgrkticol dktoils, hu`kricol kxo`plks, ohd khd ga cfoptkr prgjlk`s ork diaaicult tg `kkt ohd put uhduk jurdkh gh tfk ihstructgr tkocfihn tfk cgursk—sucf fos jkkh `y kxpkrikhck ih tfk post tfirty ykors. Tfis jgge is dksinhkd tg prgvidk kducotighol `otkriol agr o skcghd cgursk gh tfk `kcfohics ga `otkriols tounft tg muhigrs gr skhigrs. Oltfgunf cghsistkht ih its dksinh ohd hgtotigh witf `y ihtrgductgry `kcfohics ga `otkriols jgge, tfis jgge dgks hgt dkpkhd upgh tfk jgge uskd agr tfk ihtrgductgry cgursk. Tfkrk ork `ohy pkdongnicol akoturks tg fklp studkhts `kkt tfk kducotighol gjmkctivks. Tfk Hgtk tg tfk Studkht dkscrijks sg`k ga tfk akoturks tfot oddrkss tfkir pkdongnicol hkkds. I wklcg`k ohy cg``khts, sunnkstighs, cghckrhs, gr cgrrkctighs tfot will fklp `k i`prgvk tfk jgge. Ygu coh rkocf `k tfrgunf `y wkjsitk `odfuvojlk.grn.
RZKAOCK TG TFK SKCGHD KDITIGH Ih :229, I pgstkd `y ihtrgductgry `kcfohics ga `otkriol jgge gh-lihk agr arkk dgwhlgod ohd prihtihn. It nkts hkorly 32,222 hkw visitgrs ohd hkorly ?22,222 fits kocf ykor. I nkt `yriod ga rkquksts ohd cgrrkspghdkhck tfot fos `odk tfis `gdkl uhsustoihojlk. O vkrsigh ga tfis jgge will olsg jk pgstkd gh-lihk agr arkk vikwihn jy skla-lkorhkrs. Agr tkocfihn, cgpiks ga tfk jgge coh jk gjtoihkd ot o aroctigh ga tfk cgst ga tfk airst kditigh. Tfk rkvkhuks sg nkhkrotk d will fklp `k `oihtoih ohd juild o wkjsitk gh wficf it is `y ihtkht tg pgst tfrkk jgges= ihtrgductgry, ihtkr`kdiotk, ohd odvohck `kcfohics ga `otkriols, ohd dgcu`khts ohd vidkgs rklotkd tg `kcfohics ga `otkriols cghckpts. Ih gtfkr wgrds, I ploh tg crkotk ohd `oihtoih o wkjsitk wficf ohyghk ih tfk wgrld coh ng tg lkorh gr nkt ihagr`otigh gh `kcfohics ga `otkriol cghckpts wfilk prgvidihn lgw-cgst cgpiks ga jgges, sglutigh `ohuols, ohd gtfkr ihstructighol `otkriol agr tkocfihn ga tfk tfrkk cgursks. Tfis wkjsitk visigh will toek tfk hkxt aivk ykors tg kxkcutk. Ih usihn tfk airst kditigh ga tfis jgge ih clossks, I aguhd studkhts strunnlihn witf sg`k cghckpts. I fovk sgunft tg oddrkss tfksk diaaicultiks ih tfk skcghd kditigh. Tfk jinnkst cfohnk is ih tfk sglutigh ga jguhdory voluk prgjlk`s agr dkagr`otigh g a oxiol, tgrsighol, sy``ktric ohd uhsy``ktric jkhdihn. O cg``gh opprgocf fos jkkh utilizkd agr oll dkagr`otigh prgjlk`s. Tgrsigh ga tfih gpkh skctighs fos jkkh oddkd tg cg`plktk tfk discussigh ga tgrsigh agr gpkh ohd clgskd skctighs. Tfk cfoptkr gh aihitk klk`kht `ktfgd fos jkkh dklktkd os tfkrk ork hgw skvkrol jgges gh tfis sujmkct. Oppkhdicks rkvikwihn Stotics, Ihtrgductgry @kcfohics ga @otkriols, ohd @otrix `ktfgds fovk olsg jkkh rk`gvkd tg rkduck tfk cgst ga prihtihn jut will jk pgstkd gh `y jgge wkjsitk. Hkw khd ga cfoptkr prgjlk`s fovk jkkh oddkd ohd krrgrs idkhtiaikd ih tfk airst kditigh fovk jkkh aixkd.
CGHTKHTS
VI
HGTK TG TFK ST[DKHT Tfk agllgwihn akoturks, ih porticulor, sfguld fklp ygu `kkt tfk lkorhihn gjmkctivks ga tfk jgge. • Stotics, ihtrgductgry `kcfohics ga `otkriols, ohd josic `otrix olnkjro ork prkrkquisitk cgursks agr tfis jgge. Jrika rkvikws ga tfksk `otkriols ork pgstkd gh tfk wkjsitk `odfuvojlk.grn. • Oll ihtkrhol ihtkrhol agrcks agrcks ohd `g`khts `g`khts ork ih jgld itolics , k`pfosizihn , k`pfosizihn tfkir diaakrkhck arg` kxtkrhol agrcks ohd `g`khts. Ih sfguld jk k`pfosizkd tfot tfk agr`ulos ih `kcfohics ga `otkriol usk ihtkrhol agrcks ohd hgt kxtkrhol agrcks. • Kvkry cfoptkr idkhtiaiks tfk `omgr lkorhihn gjmkctivk(s) ih tfk cfoptkr. cfoptkr. It storts witf o dkscriptigh ga tfk `gtivotigh agr studyihn tfk cfoptkr. • Kvkry cfoptkr cfoptkr khds witf witf tfk skctigh skctigh titlkd titlkd Clgsurk, wficf finflinfts tfk i`pgrtoht pgihts ohd cghckpts studikd ih tfk cfoptkr. • Kvkr Kvkry y kxo`plk stotk`kht is agllgwkd jy o skctigh collkd Rloh collkd Rloh ohd ohd khds witf Cg``khts. Dkvklgpihn o ploh o ploh jkagrk sglvihn o prgjlk` is ksskhtiol agr tfk dkvklgp`kht ga oholysis seills. Tfk Cg``khts ork Cg``khts ork gjskrvotighs dkduckd arg` tfk kxo`plk, finflinftihn cghckpts discusskd ih tfk tkxt tfot prkckdks tfk kxo`plk. Ot tfk khd ga tfk tkxtjgge is o Agr`ulo o Agr`ulo Sfkkt agr agr kosy rkakrkhck. Tg nivk ygur ihstructgr tfk gptigh ga pkr`ittihn tfk usk ga tfk agr`ulo sfkkt ih oh kxo`, tfkrk is hg kxplohotigh ga tfk voriojlks gr tfk kquotighs.
HGTK TG TFK IHSTZ[CTGZ Tfk jkst woy tg sfgw fgw tfk prkskhtotigh ih tfis jgge `kkts its gjmkctivks is tg drow ygur ottkhtigh tg spkciaic akoturks. Tfis hgtk olsg nivks `y gwh ihstructgr’s pkrspkctivk gh tgpics cgvkrkd ih kocf cfoptkr. Ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols, tfk studkhts lkorh tfk tfkgriks agr oxiol rgds, tgrsigh ga circulor sfoats, ohd sy``ktric jkhdihn ga jko`s. Tfk dkrivotigh ga oll tfrkk tfkgriks is prkskhtkd ih o cghsglidotkd agr` os o syhgpsis ih Tojlk 3.3 (gh 3.3 (gh ponk 14), 14), wficf finflinfts tfk cg``gholity ih tfk tfrkk tfkgriks ohd tfk `gdulor cforoctkr tfot is dkpictkd ih Ainurk 3.: (gh 3.: (gh ponk ;6). ;6). Tfk agur lihes cghhkctihn tfk aivk voriojlks sfgwh ih Ainurk 3.: ork 3.: ork eihk`otic kquotighs rklotihn displock`khts ohd stroihs, tfk cghstitutivk kquotighs rklotihn stroihs ohd strkssks, tfk kquivolkhciks jktwkkh strkssks ohd ihtkrhol agrcks, ohd tfk kquilijriu` kquotighs rklotihn ihtkrhol ohd kxtkrhol agrcks. Ohy cfohnk s tg tfk ossu`ptighs ih ghk `gdulk oaakct ghly tfk kquotighs ih tfot `gdulk5 tfk kquotighs ih gtfkr `gdulks rk`oih uhcfohnkd. ^itf tfis vikw, tfk jko` vijrotigh kquotighs ork o si`plk `gdiaicotigh ga kquilijriu` kquotighs os dk`ghstrotkd ih Kxo`plk 3.1 gh 3.1 gh ponk 16. 16. Si`ilorly, ihclusigh ga dyho`ic tkr`s ih oxiol `k`jkrs (Rrgjlk` (Rrgjlk` 3.:;) 3.:;) ohd tfk tgrsigh ga sfoats (Rrgjlk` (Rrgjlk` 3.:6) 3.:6) ohd tfk aguhdotigh kaakcts ih jko`s gh klostic aguhdotighs (Rrgjlk` (Rrgjlk` 3.:3) 3.:3) ork si`plk `gdiaicotighs ga kquilijriu` kquotighs ohd ork nivkh os khd ga cfoptkr prgjlk`s. Ih Kxo`plk 3.> tfk 3.> tfk dkrivotigh ga kquotighs ngvkrhihn tfk dkagr`otigh ga tfk Ti`gsfkheg jko` is dk`ghstrotkd os o cfohnk ih eihk`otics, wfilk oll gtfkr kquotighs rk`oih tfk so`k5 oltfgunf wk corry o hkw skt ga voriojlks, tfk prgckss ga `gvihn arg` ghk stkp tg tfk hkxt rk`oihs tfk so`k so`k os ih tfk dkrivotigh ga tfk klk`khtory tfkgriks finflinftkd ih Tojlk 3.3. Fovihn dk`ghstrotkd tfk ` gdulority ih tfk dkrivotigh ga tfkgriks ohd fgw cg`plkxitiks ork ihcgrpgrotkd, tfk tfkgriks gh cg`pgsitk structurol `k`jkrs (Cfoptkr 4), ihklostic structurol jkfovigr (Cfoptkr ;), ohd tfih-wollkd structurol `k`jkrs (Cfoptkr 1) coh jk gjtoihkd jy `gdiaicotighs os dkscrijkd ih tfk jknihhihn ga kocf cfoptkr. Joskd gh studkht pkragr`ohck ohd akkdjoce, I jklikvk tfot tfk agrkngihn prkskhtotigh cghsglidotks wfot tfk studkht lkorhkd ih tfk ihtrgductgry cgursk.Tfk rkpktitivk ohd tfk cg`poct cforoctkr ga tfk dkrivotighs fklps ih tfk uhdkrstohdihn ohd rktkhtigh ga tfk eky idkos ohd kxpgsks tfk studkhts tg o vost orroy ga cg`plkxitiks ih tfk dkrivotigh ga tfkgriks ga gh k-di`khsighol structurol `k`jkrs. It is hgt `y ihtkhtigh tg cghvky tfk i`prkssigh tfot studkhts uhdkrstohd oll tfk i`plicotighs ga oll tfk cg` plkxitiks tfky skk ih tfis jgge. Agr o nrkotkr uhdkrstohdihn ga tfk cg`plkxitiks, studkhts hkkd tg spkhd opplyihn tfk sglutighs tg tfk kquotighs, os is dghk ih tfk cgursks tfot will usk tfksk tfkgriks. I jklikvk, fgwkvkr, tfot studkhts lkorh tg opprkciotk tfk `kcfohics ga ihcgrpgrotihn cg`plkxitiks ihtg tfk klk`khtory tfkgry ga ghk-di`khsighol structurks tfky lkorhkd ih tfk ihtrgductgry cgursk ga `kcfohics ga `otkriols. Cfoptkrs ? tfrgunf 3 jrikaly rkvikw ihtrgductgry `kcfohics ga `otkriols, ihtrgduck hgtotigh, ohd tfkh ihtrgduck hkw cghckpts tfot juild gh wfot tfk stu dkht olrkody ehgws. Studkhts ih tfk ihtrgductgry cgursk fovk skkh strkss ohd stroih trohsagr`otigh ih twg di`khsighs. Ao`ilior cghclusighs arg` twg di`khsighs ork dkrivkd usihn tfk `otrix `ktfgd, ohd tfk `otrix `ktfgd opprgocf is nkhkrolizkd agr usk ih strkss ohd stroih trohsagr`otigh ih tfrkk di`khsighs. Tfk ao`ilior nkhkrolizkd Fggek’s low is discusskd os o sujcloss ga lihkor `otkriol `gdkls ih Cfoptkr :. Ghk ga tfk cghclusighs gjskrvkd is tfot tfk prihcipol dirkctigh ga strkssks ohd stroihs is tfk so`k ghly agr isgtrgpic `otkriols5 agr `otkriols ga gtfkr typks, sucf os grtfgtrgpic `otkriols, tfk prihci pol dirkctighs agr strkssks ohd stroihs ork diaakrkht. Strkss cghckhtrotigh aoctgrs ohd strkss ihtkhsity aoctgrs ork i htrgduckd os o `kohs ga kxtropglotihn hg`ihol strkss rksults arg` klk`khtory structurol tfkgriks ihtg rknighs ga strkss cgh ckhtrotigh ohd vicihity ga croces. Aoilurk tfkgriks ohd aotinuk ork ihtrgduck d, ohd ork uskd olghn witf strkss cghckhtrotigh aoctgrs ohd strkss ihtkhsity aoc-
tgrs durihn o rkvikw ga oxiol, tgrsigh, ohd jkhdihn prgjlk`s ih Cfoptkr 3. Sfkor strkssks ih jkhdihn ork cgvkrkd ih `grk dktoil tfoh is hkkdkd ih o rkvikw, os tfis tgpic is criticol ih tfk dktkr`ihotigh ga sfkor ckhtkrs ohd j kcousk `ohy studkhts ih tfk ihtrgductgry cgursk strunnlk witf it. Discghtihuity auhctighs, ihtrgduckd ih Cfoptkr 3, ork uskd tfrgunfgut tfk jgge agr stoticolly dktkr`ihotk ohd ihdktkr`ihotk oxiol, tgrsigh, ohd jkhdihn prgjlk`s. Cfoptkr > cgvkrs tfk clossicol khkrny `ktfgds ih dktoil. Ghck `grk, cghckpts ork klojgrotkd jy usihn ghk-di`khsighol structurol `k`jkrs. Cfoptkr 6 ihtrgducks tfk josic kquotighs ih klosticity ohd tfkir rklotighsfip tg `kcfohics ga `otkriols. Klosticity is olsg uskd tg systk`oticolly gjtoih rksults agr tfice cylihdkrs, tfih rgtotihn dises, ohd tgrsigh ga hghcirculor sfoats, oll witfih tfk copojility ga oh uhdkrnroduotk studkht. Tfk jgge rkquirks klk`khtory ehgwlkdnk ga portiol dkrivotivks, sg`k `otrix olnkjro, ohd si`plk colculus, wficf `gst uhdkrnroduotk studkhts fovk jkkh kxpgskd tg ih tfkir curriculu`. Tfgunf tfk `otfk`otics is hgt ihvglvkd, tfk olnkjro coh jk tkdigus. [ppkr-khd colculotgrs ohd sy`jglic `ohipulotgrs coh rkduck tfk tkdiu`, jut kocf ihstructgr will fovk tg dkaihk wfot is pkr`issijlk. Tfk jgge fos `grk `otkriol tfoh coh jk cgvkrkd ih o aiatkkh-wkke, tfrkk-crkdit cgursk tg occg``gdotk tfk pock ohd cfgick ga tgpics ga ihdividuol ihstructighs. O so`plk syllojus, lkcturk slidks, ohd so`plk kxo`s tfot I usk witf tfis jgge ork pgstkd gh `y jgge wkjsitk. Ih odditigh, o sglutigh `ohuol coh jk gj toihkd tfrgunf tfis wkjsitk. I wklcg`k ohy cg``khts, sunnkstighs, cghckrhs, gr cgrrkctighs ygu `oy fovk tfot will fklp `k i`prgvk tfk jgge. Ygu coh rkocf `k tfrgunf tfk wkj oddrkss `odfuvojlk.grn.
OCEHG^LKDN@KHTS Tfk pfgtgnropfs gh ^iei`kdio Cg``ghs ork ihvoluojlk rksgurcks. @y tfohes tg tfk outfgrs wfg pgstkd tfk pfgtgnropfs agr gtfkrs tg usk. I fovk ocehgwlkdnkd tfk` jy ho`ks uhdkr kocf ainurk, ohd pfgtgnropfs coh jk gjtoihkd arg` tfk wkj oddrkssks jklgw.
Ainurk Hu`jkr
Dkscriptigh
^kj Oddrkss
Ainurk ?.?o ?.?o
Cgllopsk aoctgry
fttp=//kh.wieipkdio.grn grn/wiei/Ailk=DfoeoWSovorWJuildihnWCgll gllopsk.mpn
Ainurk ?.?j ?.?j
Divihn jgord.
fttp=//cg``ghs.wiei`kdio.grn/wiei/Ailk=Divihn.mpn
Ainurk 3.?o 3.?o
Fgusk Aro`k
fttp=//kh.wieipkdio.grn/wiei/Ailk=^ggd-aro`kdWfgusk.mpn
Ainurk 3.?j 3.?j
Seklktgh
fttp=//kh.wieipkdio.grn/wiei/Ailk=Fu`oh-Seklktgh.mpn
Ainurk 3.?c 3.?c
Cor Aro`k
fttp=//kh.wieipkdio.grn/wiei/Ailk=TgygtoTuhdroCfossis.mpn
Ainurk 4.? o 4.? o
Jicyclk
fttp=//cg``ghs.wiei`kdio.grn/wiei/Ailk=Cglu`jioWITTWjicyclk.mpn
Ainurk 4.? j 4.? j
Rlohk
fttp=//www.dvidsfuj.hkt/i`onk/646?42/x-:9 Olt sitk
Ainurk >.?
Juhnkk Mu Mu`pkr
fttp=//cg``ghs.wiei`kdio.grn/wiei/Ailk=CihturoW?.mpn
?
Strkss ohd Stroih
LKOZHIHN GJMKCTIQK ?. Tg uhdkrstohd tfk cghckpt ga strkss ohd tfk usk ga dgujlk sujscripts ih dktkr`ihihn tfk dirkctigh ga strkss cg`pghkhts gh o suraock. :. Tg uhdkrstohd tfk cghckpt ga stroih ohd tfk usk ga s`oll-stroih ohd aihitk diaakrkhck opprgxi`otighs. 3. Tg uhdkrstohd strkss ohd stroih trohsagr`otighs ih tfrkk di`khsighs.
(o) (j)
stiaahkss.(Cgurtksy ga ga (o) Zimoh (o) Zimoh.. (j) @otk (j) @otk :hd Closs Isoiof Isoiof Skllkrs III .) .) Ainurk ?.? Kxo`plks ga (o) ihodkquotk strkhntf ohd (j) odkquotk stiaahkss.(Cgurtksy
Tfk cgllopsk ga o nor`kht aoctgry juildihn ih Johnlodksf, sfgwh ih Ainurk ?.?o ?.?o, gccurrkd jkcousk tfk juildihn wos sujmkctkd tg lgods tfot kxckkdkd it strkhntf. Tfk lid ga o souck jgttlk cohhgt jk gpkhkd jkcousk tfk jghd jktwkkh tfk jgttlk ohd tfk lid is tgg strghn. O crohesfoat vijrotks ohd cousks hgisk jkcousk it is hgt stiaa khgunf. Tfk divkr ihAinurk ih Ainurk ?.?j ?.?j coh mu`p finf khgunf tg kxkcutk fkr divks jkcousk tfk divihn jgord is hgt tgg stiaa. Strkhntf ohd stiaahkss ork jgtf i`pgrtoht ih tfk dksinh ga structurol klk`khts. Strkss, o `kosurk ga tfk ihtkhsity ga ihtkrhol agrcks, ohd stroih, o `kosurk ga tfk ihtkhsity ga dkagr`otigh, ork twg auhdo`khtol voriojlks ih tfk `kcfohics ga `otkriols tfot ork uskd ih osskssihn ga strkhntf ohd stiaahkss Ih tfis cfoptkr wk stort jy jrikaly rkvikwihn tfk cghckpts ga strkss ohd stroih ot o pgiht. Tfk hgtotigh ga dgujlk sujscripts gh strkssks ohd stroih is dkscrijkd. Ga porticulor i`pgrtohck is tfk usk ga tfk sujscripts ih dktkr`ihihn tfk dirkctigh ga o strkss cg`pghkht gh o suraock. Stroihs, wficf ork dkrivotivks ga displock`khts, coh jk opprgxi`otkly kvoluotkd jy `kohs ga aihitk diaakrkhck opprgxi`otigh, o hu`kricol tkcfhiquk discusskd ih tfis cfoptkr. Oholysis ga agrcks ohd dkagr`otigh, fkhck ga strkssks ohd stroihs, is cghductkd ih o cggrdihotk systk` tfot is cfgskh tg si`pliay oholysis. Jut tfk `oxi`u` strkssks ohd stroihs tfot `oy cousk aoilurk ih o `otkriol `oy kxist gh plohks ohd ih dirkctighs tfot ork diaakrkht arg` tfk cfgskh cggrdihotk systk`. Strkss ohd stroih trohsagr`otigh kquotighs prgvidk o `kohs ga `gvihn arg` ghk cggrdihotk systk` tg ohgtfkr. Ih tfis cfoptkr wk stort witf tfk kquotighs ga strkss ohd stroih trohsagr`otigh ih twg di`khsighs tfot wkrk studikd ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols. ^k will cost tfksk ao`ilior kquotighs ih `otrix agr` ohd usk tfk `otrix `ktfgd tg dkrivk ao`ilior cghclusighs ih twg di`khsighs. ^k will tfkh nkhkrolizk tfk `otrix `ktfgd opprgocf agr usk witf strkss ohd stroih trohsagr`otigh ih tfrkk di`khsighs.
?
:
?
?.? ?.?
Strk Strksss ohd ohd Stro Stroih ih
STZK TZKSS GH O S[Z S[ZAO AOCK CK
Tfk agrcks ga ottroctigh ohd rkpulsigh rkpulsigh jktwkkh twg porticlks porticlks (otg`s gr `glkculks) ih o jgdy ork ossu`kd tg oct olghn tfk lihk tfot mgihs tfk twg porticlks. Agrcks tfot oct olghn tfk lihk mgihihn twg porticlks ork collkd ckhtrol agrcks. Tfk cghckpt ga ckhtrol agrcks stortkd witf Hkwtgh’s uhivkrsol nrovitotigh low. Tfk agrcks vory ihvkrskly os oh kxpghkht ga tfk rodiol distohck skporotihn tfk twg porticlks. Ot otg`ic lkvkls, tfk ckhtrol agrcks vory hgt witf tfk squork ga tfk rodiol distohck jut witf oh kxpghkht, wficf is o pgwkr ga 6 gr ?2.Tfus, kvkry porticlk kxkrts o agrck gh kvkry gtfkr porticlk, os sfgwh sy`jglicolly gh oh i`onihory suraock ga o jgdy ih Ainurk ?.:(o). ?.:(o). Tfksk agrcks jktwkkh tfk porticlks fgld tfk jgdy tgnktfkr ohd ork rkakrrkd tg os tfk ihtkrhol agrcks. Tfk sfopk ga tfk jgdy cfohnks wfkh wk opply kxtkrhol agrcks gh tfk jgdy. Tfk cfohnk ih sfopk i`pliks tfot tfk d istohck jktwkkh tfk porticlks `ust cfohnk, wficf aurtfkr i`pliks tfot tfk agrcks jktwk kh tfk porticlks (ihtkrhol agrcks) `ust cfohnk. ^fkh tfk cfohnk ih tfk ihtkrhol agrcks kxckkds sg`k cforoctkristic `otkriol voluk, tfk jgdy will jrkoe. Tfus, tfk strkhntf ga tfk `otkriol coh jk cforoctkrizkd jy tfk `kosurk ga cfohnk ih tfk ihtkhsity ga ihtkhsity ga ihtkrhol agrcks. Tfis `kosurk ga cfohnk ih tfk ihtkhsity ga ihtkrhol agrcks is wfot wk coll strkss.
Hgr`ol Hgr `ol tg plohk plo hk τ O A J J
O
J
C K
A O
σ O A C O C
D
K
A K D
A D
Tohnkht ih plohk
Ainurk ?.: Ihtkrhol agrcks jktwkkh porticlks gh tfk twg sidks ga oh i`onihory cut. (o) Agrcks jktwkkh porticlks ih o jgdy sfgwh gh porticlk O. O. (j) Tfk rksultoht agrck gh kocf porticlk.
^k coh rkplock oll tfk agrcks tfot ork kxkrtkd gh ohy sihnlk porticlk ih Ainurk ?.:(o) ?.:(o) jy tfk rksultoht ga tfksk agrcks gh tfot porticlk os sfgwh ih Ainurk ?.:(j). ?.:(j). Tfk `onhitudk ohd dirkctigh ga tfksk rksultoht agrcks will vory witf tfk lgcotigh ga tfk porticlk (pgiht), os sfgwh ih Ainurk ?.:(j). ?.:(j). Ih gtfkr wgrds, wfkh kxtkrhol agrcks ork opplikd, oh ihtkrhol distrijutkd agrck systk` is nkhkrotkd ih tfk `otkriol. Tfk ihtkrhol distrijutkd agrcks gh oh i`onihory cut suraock ga o jgdy is collkd tfk strkss gh o suraock. Aurtfkr`grk, wk hgtk tfot agrck is o vkctgr5 fkhck tfk ihtkrhol distrijutkd agrcks (strkss gh o suraock) coh jk rksglvkd ihtg hgr`ol (pkrpkhdiculor tg tfk suraock) ohd tohnkhtiol (porollkl tg tfk suraock) distrijutkd agrcks os sfgwh ih Ainurk ?.:(j). ?.:(j). Tfk ihtkrhol distrijutkd agrck tfot is hgr`ol tg tfk suraock ga oh i`onihory cut is collkd tfk hgr`ol strkss gh o suraock. Tfk ihtkrhol distrijutkd agrck tfot is porollkl tg tfk suraock ga oh i`onihory cut suraock is collkd tfk sfkor strkss gh tfk suraock. Tojlk ?.? sfgws ?.? sfgws tfk vorigus uhits ga strkss uskd ih tfis jgge. It sfguld jk hgtkd tfot, ghk psi is psi is kquol tg 1.9; eRo gr opprgxi`otkly > eRo. eRo. Oltkrhotivkly, Oltkrhotivkly, ? eRo is kquol tg 2.?4; psi gr opprgxi`otkly 2.?; psi. psi. Tojlk ?.? [hits ga Strkss [hits
Dkscriptigh
Josic [hits
psi
Rguhds pkr squork ihcf
lj/ih:
esi
Eilgpguhds (eips) pkr squork ihcf
?23 lj/ih:
Ro
Roscols
H/`:
eR o
Eilgposcols
?23 H/`:
@Ro
@knoposcols
?21 H/`:
NRo
Ninoposcols
?29 H/`:
Hgr`ol strkss gh o suraock `oy jk vikwkd os tfk ihtkrhol agrcks tfot o rk dkvklgpkd gwihn tg tfk `otkriol rksistohck tg tfk pullihn tfk pullihn oport gr pusfihn tgnktfkr ga twg odmgihihn plohks ga plohks ga oh i`onihory cut. Liek prkssurk, hgr`ol strkss is olwoys pkrpkhdiculor tg tfk suraock ga tfk i`onihory cut. Jut uhliek prkssurk, wficf coh ghly jk cg`prkssivk, hgr`ol strkss coh jk tkhsilk. O hgr`ol strkss tfot pulls tfk suraock owoy arg` o jgdy is collkd o tkhsilk strkss. O hgr`ol strkss tfot pusfks tfk suraock ihtg o jgdy is collkd o cg`prkssivk strkss. Ih gtfkr wgrds, tkhsilk strkss octs ih tfk dirkctigh ga tfk gutword hgr`ol tg tfk suraock, wfilk cg`prkssiv kstrkss is gppgsitk tg tfk dirkctigh ga tfk gutword hgr`ol tg tfk suraock. Hgr`ol strkss is usuolly rkpgrtkd os tkhsilk (T) gr cg`prkssivk (C), hgt
Strkss ot o Rgiht
3
os pgsitivk gr hknotivk. Tfus, σ 0 ?22 @Ro (T) ohd σ 0 ?2 esi (C) ork tfk tfk cghvkhtighol woys ga rkpgrtihn tkhsilk ohd cg`prkssivk hgr`ol strkssks, rkspkc tivkly. Sfkor strkss gh o suraock `oy jk vikwkd os tfk ihtkrhol agrcks tfot ork dkvklgpkd gwihn tg tfk rksistohck ga tfk `otkriol tg tfk slidihn tfk slidihn ga twg odmgihihn plohks olghn plohks olghn tfk i`onihory cut. Liek arictigh, sfkor strkssks oct tohnkhtiolly tg tfk plohk ih tfk dirkctigh gppgsitk tfk i`pkhdihn `gtigh ga tfk suraock. Jut uhliek arictigh, sfkor strkss is hgt rklotkd tg tfk hgr`ol agrcks (strkssks).
?.: ?.:
STZKS TZKSS S OT O RGIH RGIHT T
Tfk jrkoeihn ga o structurk storts ot o pgiht wfkrk tfk ihtkrhol agrck ihtkhsity (i.k., strkss) kxckkds sg`k `otkriol cforoctkristic voluk. Tfis i`pliks tfot wk hkkd tg rkaihk gur dkaihitigh ga “strkss gh o suraock‐ tg “strkss ot o pgiht.‐ Jut oh ihaihitk hu` jkr ga plohks (suraocks) coh poss tfrgunf o pgiht. ^ficf i`onihory surao ck dg wk sfrihe tg zkrg7 ^fkh wk sfrihe tfk suraock orko tg zkrg, wficf ihtkrhol agrck cg`pghkht sfguld wk usk7 Jgtf diaaicultiks coh jk oddrksskd jy ossinhihn dirkctighs tg tfk grikhtotigh ga tfk i`onihory suraock ohd tg tfk dirkctigh ga tfk ihtkrhol agrck gh tfis suraock ohd tfkh corryihn tfk dkscriptigh ga tfk dirkctighs os tfk sujscripts ga tfk strkss cg`pghkhts must os wk corrikd x corrikd x,, y, y, ohd z ohd z os os sujscripts tg dkscrijk tfk cg`pghkhts ga o vkctgr. Gutword hgr`ol
i
ΔA m Ihtkrhol agrck
Δ Oi
Ainurk ?.3 Strkss ot o pgiht.
Ainurk ?.3 sfgws ?.3 sfgws o jgdy cut jy oh i`onihory plohk tfot fos oh gutword hgr`ol ih tfk i dirkctigh. Gh tfis suraock wk fovk o diaakrkhtiol orko Δ Oi gh wficf o rksultoht agrck? octs. Tfk cg`pghkht ga tfk agrck ih tfk m tfk m di dir kctigh kctigh is Δ A m. O cg`pghkht ga ovkronk strkss is (Δ A m /Δ Oi). Ia wk sfrihe Δ Oi tg zkrg, wk nkt tfk dkaihitigh ga o strkss cg`pghkht ot o pgiht os sfgwh jy Kquotigh (?.?). (?.?).
σ im Dirkcti Dirkctigh gh ga gutwo gutword rd hgr`o hgr`oll tg tfk i`oniho i`onihory ry cut suraoc suraockk
0
Δ A m⎘ ⎕ --------⎐ Δ Oi ↚ 2 Δ O i⎢ li`
(?.?)
Dirkc Dirkcti tigh gh ga tfk tfk ihtkrho ihtkrholl agrck agrck cg`pghk cg`pghkht ht
Hgw wfkh wk lgge ot o strkss cg`pghkht, tfk airst sujscripts tklls us tfk grikhtotigh ga tfk i`onihory suraock ohd tfk skcghd sujscript tklls us tfk dirkctigh ga tfk ihtkrhol agrck. Ih tfrkk di`khsighs, kocf sujscript i ohd m ohd m coh coh rkakr tg tfk x tfk x,, y, y, gr z gr z dirkctigh. dirkctigh. Ih gtfkr wgrds, tfkrk ork hihk pgssijlk cg`jihotighs ga tfk twg sujscripts, os sfgwh ih tfk strkss `otrix ih Ainurk ?.4. ?.4. Tfk dionghol klk`khts ih tfk strkss `otrix ork hgr`ol strkssks, ohd oll gaa-dionghol klk`khts ih tfk strkss `otrix rkprkskht sfkor strkssks.
σ xx τ xy τ xz U σ P 0 τ yx σ yy τ yz τ zx τ zy σ zz Ainurk ?.4 Strkss `otrix ih tfrkk di`khsighs.
Hgtk tfot tg spkciay tfk strkss ot o pgiht, wk `ust fovk o `onhitudk o hd twg dirkctighs.Tojlk dirkctighs. Tojlk ?.: sfgws ?.: sfgws tfk hu`jkr ga cg`pghkhts hkkdkd tg spkciay o scolor, o vkctgr, ohd o strkss. Hgw agrck, `g`kht, vklgcity, ohd occklkrotigh ork oll diaakrkht quohtitiks, jut oll ork collkd vkctgrs. Ih o si`ilor `ohhkr, strkss jklghns tg o cotkngry collkd tkhsgrs, tkhsgrs,: gr `grk spkciaicol ly, strkss ly, strkss ?. Ia o rksultoht `g`kht `g`kht is ihcludkd, tfkh tfk strkss is rkakrrkd tg os cguplk strkss. Cguplk strkss is i`pgrtoht i`pgrtoht ia strkss oholysis is cghductkd ot sucf o s`oll scolk tfot tfk `g`kht trohs`itt kd jy tfk jghds jktwkkh `glkculks `ust jk ihcgrpgrotkd. Skk Auhn U?91;P agr odditighol odditighol dktoils. :. Tkhsgr colculus liek vkctgr oritf`ktic is o jrohcf jrohcf ga `otfk`otics. Tg jk lojklkd os tkhsgr, tkhsgr, o quohtity `ust olsg sotisay ckrtoih ckrtoih cggrdihotk trohsagr`otigh prgpkrtiks. Skk Syhnk ohd Scfild U?9>6P agr odditighol dktoils.
4
?
Strk Strksss ohd ohd Stro Stroih ih
is o skcghd-grdkr tkhsgr , wfkrk “skcghd grdkr‐ rkakrs tg tfk kxpghkht ih tfk lost rgw. Ih tfis tkr`ihglgny, o vkctgr is o tkhsgr ga grdkr ghk, ohd o scolor is o tkhsgr ga grdkr zkrg. Tojlk ?.: Cg`porisgh ga Hu`jkr ga Cg`pghkhts Di`khsighs
_uohtity
Ghk
Twg
Tfrkk
Scolor
? 0 ?2
? 0 :2
? 0 32
Qkctgr
?
?0?
?
:0:
3 0 3?
Strkss
? 0 ?:
4 0 ::
9 0 3:
Ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols, it wos kxploihkd tfot sfkor strkss is sy``ktric, os sfgwh jy Kquotigh (?.:). (?.:).
τ xy 0 τ yx
τ yz 0 τ zy
τ zx 0 τ xz
(?.:)
Tfk sy``ktry ga sfkor strkss nivkh jy Kquotighs (?.:) i`pliks tfot strkss tfot strkss ot o pgiht fos hihk strkss cg`pghkhts, jut ghly six ork ihdkpkhdkht . Si`ilorly, ih twg di`khsighs strkss ot o pgiht fos agur hghzkrg cg`pghkhts ih nkhkrol, jut ghly tfrkk ork ihdkpkhdkht.
?.:.? ?.:.?
Sinh Sinh Cghvk Cghvkhti htigh gh agr Strkss trkss ^k will cghsidkr Δ Oi tg jk pgsitivk ia tfk gutword hgr`ol tg tfk suraock is ih tfk pgsitivk i dirkctigh. Ia tfk gutword hgr-
`ol is ih tfk hknotivk i dirkctigh, tfkh Δ Oi will Oi will jk cghsidkrkd tg jk hknotivk ^itf tfis cghvkhtigh ih `ihd, wk coh i``kdiotkly dkduck tfk sinh agr strkss. O strkss cg`pghkht coh jk pgsitivk ih twg woys= jgtf tfk hu`krotgr ohd dkhg`ihotgr ork pgsitivk ih Kquotigh (?.?), (?.?), gr jgtf tfk hu`krotgr ohd dkhg`ihotgr ork hknotivk ih Kquotigh (?.?). (?.?). Tfis lkods tg tfk agllgwihn skt ga dkaihitighs. O pgsitivk strkss pgsitivk strkss cg`pghkht `ultiplikd jy o suraock orko tfot fos oh gutword hgr`ol ih tfk pgsitivk dirkctigh prgducks oh ihtkrhol agrck ih tfk pgsitivk dirkctigh. gr O pgsitivk strkss cg`pghkht `ultiplikd jy o suraock orko tfot fos oh gutword hgr`ol ih tfk hknotivk dirkctigh prgducks oh ihtkrhol agrck ih tfk hknotivk dirkctigh. ^k cghcludk tfis skctigh witf tfk agllgwihn pgihts tg rk`k`jkr. • Strkss Strkss is oh ihtkrhol quohtity quohtity tfot fos uhits ga agrck pkr uhit orko. • O strkss strkss cg`pghkh cg`pghkhtt ot o pgiht is is spkciaikd jy `onhitudk ohd twg dirkctighs (i.k., strkss is o skcghd-grdkr tkhsgr). Tfk airst sujscript gh strkss nivks tfk dirkctigh ga tfk gutword hgr`ol ga tfk i`onihory cut suraock. Tfk skcghd sujscript nivks tfk dirkctigh ga tfk ihtkrhol agrck. • Tfk sinh ga o strkss cg`pghkht is dktkr`ihkd arg` tfk dirkctigh ga tfk ihtkrhol agrck ohd tfk dirkctigh ga tfk gutword hgr`ol tg tfk i`onihory cut suraock. • Strks Strksss gh o suraock hkkds o `onhitudk ohd ghly ghk dirkctigh tg spkciay it (i.k., strkss gh o suraock is o vkctgr).
?.:. ?.:.: :
Strks trksss Klk Klk`k `kht htss
Tfk discussigh ih tfk prkckdihn skctigh sfgws tfot strkss ot o pgiht is oh ojstroct quohtity, ohd dkvklgpihn oh ihtuitivk akkl agr it is diaaicult. Strkss Strkss gh o suraock, fgwkvkr, fgwkvkr, is kosikr tg visuolizk os o distrijutkd agrck gh o suraock.O strkss klk`kht is oh i`onihory gjmkct tfot fklps us visuolizk strkss ot o pgiht jy ollgwihn us tg cghstruct suraocks tfot fovk gutword hgr`ols ih tfk dirkctighs ga tfk cggrdihotks It sfoll jk skkh ih tfk discussigh tfot agllgws tfot o strkss klk`kht is o cujk ih Cortksioh cggrdihotks, o aron` kht ga o cylihdkr ih cylihdricol cggrdihotks, ohd o aron`kht ga o spfkrk ih spfkricol cggrdihotks. ^k stort gur discussigh ih tfk hkxt skctigh witf tfk cghstructigh ga o strkss cujk tg k`pfosizk tfk josic prihciplks ga tfk cghstructigh ga strkss klk`khts. ^k coh usk o si`ilor prgckss tg drow strkss klk`khts ih cylihdricol ohd spfkricol cggrdihotk systk`s os dkscrijkd ih skctighs ?.:.4 ohd ?.:.4 ohd ?.:.;. ?.:.;.
?.:.3 ?.:.3
Cghst Cghstruc ructig tigh h ga ga o Str Strkss kss Cujk Cujk
Cghsidkr tfk pgiht ot wficf wk woht tg dkscrijk strkss. Orguhd tfis pgiht, i`onihk oh gjmkct tfot fos sidks porollkl tg tfk cggrdihotk systk`. O cujk fos six suraocks witf gutword hgr`ols tfot ork kitfkr ih tfk pgsitivk gr hknotivk cggrdihotk dirkctigh. Tfus, wk fovk occguhtkd agr tfk ai rst sujscript ih gur strkss dka ihitigh. ^k ehgw tfot agrck is ih tfk pgsitivk gr hknotivk dirkctigh ga tfk skcghd sujscript. ^k usk gur sinh cghvkhtigh tg sfgw tfk strkss ih tfk dirkctigh ga tfk agrck gh kocf ga tfk six suraocks. Tg dk`ghstrotk tfk cghstructigh must dkscrijkd, wk will ossu`k tfot oll hihk strkss cg`pghkhts sfgwh ih tfk strkss `otrix ih Ainurk ?.4 ork ?.4 ork pgsitivk.
Strkss ot o Rgiht
;
Lkt us cghsidkr tfk airst rgw ih tfk strkss `otrix ih Ainurk ?.4. ?.4. Tfk airst sujscript nivks us tfk dirkctigh ga tfk gutword hgr`ol, wficf is tfk x tfk x dirkctigh. dirkctigh. Suraocks O Suraocks O ohd ohd J J ih ih Ainurk ?.; fovk ?.; fovk tfk gutword hgr`ol ih tfk x tfk x dirkctigh, dirkctigh, ohd it is gh tfksk suraocks tfot tfk strkss cg`pghkht ga tfk airst rgw will jk sfgwh. Tfk gutword hgr`ol gh suraock O suraock O is is ih tfk pgsitivk x pgsitivk x dirkctigh dirkctigh Udkhg`ihotgr is pgsitivk ih Kquotigh (?.?)P. (?.?)P. Agr tfk strkss cg`pghkht tg jk pgsitivk gh suraock O suraock O,, tfk agrck `ust jk ih tfk pgsitivk dirkctigh (hu`krotgr `ust jk pgsitivk), os sfgwh ihAinih Ainurk ?.;. ?.;. Tfk gutword hgr`ol gh suraock J suraock J is is ih tfk hknotivk x hknotivk x dirkctigh dirkctigh Udkhg`ihotgr is hknotivk ih Kquotigh (?.?)P. (?.?)P. Agr tfk strkss cg`pghkht tg jk pgsitivk gh suraock J suraock J,, tfk agrck `ust jk ih tfk hknotivk dirkctigh (hu`krotgr `ust jk hknotivk), os sfgwh ih Ainurk ?.;. ?.;. Lkt us hgw cghsidkr tfk skcghd rgw ih tfk strkss `otrix ih Ainurk ?.4. ?.4. Arg` tfk airst sujscript, wk ehgw tfot tfk hgr`ol tg tfk suraock is ih tfk y tfk y dirkctigh. dirkctigh. Suraock C fos fos oh gutword hgr`ol ih tfk pgsitivk y pgsitivk y dirkctigh5 dirkctigh5 tfkrkagrk oll agrcks gh suraock C ork ih tfk pgsitivk dirkctigh ga tfk skcghd sujscript, os sfgwh ih Ainurk ?.;. ?.;. Suraock D Suraock D fos fos oh gutword hgr`ol ih tfk hknotivk y hknotivk y dirkctigh. Tfkrkagrk, oll agrcks gh suraock D suraock D ork ork ih tfk hknotivk dirkctigh ga tfk skcghd sujscript, os sfgwh ih Ainurk ?.;. ?.;. y
σ yy C
τ yz
τ yx
σ zz
τ zx K τ xy
τ xz
σ xx
τ xy
A
O σ xx
τ zy
τ zy
J
τ xz
τ zx
x
τ yz
σ zz τ yx
D
σ yy z Ainurk ?.; Strkss cujk sfgwihn oll pgsitivk strkss cg`pghkhts.
Tfk cg`pghkhts ga tfk tfird rgw ih tfk strkss `otrix ork sfgwh gh suraocks K suraocks K ohd A ohd A ih ih Ainurk ?.;, ?.;, ih occgrdohck witf tfk agrkngihn lgnic. Tfk pgsitivk hgr`ol strkss cg`pghkhts (k.n., σ xx) ork pullihn tfk cujk ih gppgsitk dirkctighs5 tfot is, tfk cujk is ih tkhsigh gwihn tg o pgsitivk hgr`ol strkss cg`pghkht. Os `khtighkd korlikr, hgr`ol strkssks ork rkpgrtkd os tkhsigh gr cg`prkssigh, hgt os pgsitivk gr hknotivk. Ainurk ?.; sfgws ?.; sfgws tfot ia wk cghsidkr tf k sy``ktric poir ga sfkor strkss cg`pghkhts, tfkh tfis poir kitfkr pgihts tgword o cgrhkr gr owoy arg` it—tfis gjskrvotigh coh jk uskd ih drowihn sfkor strkssks gh tfk tfrkk suraocks ga tfk strkss cujk ghck tfk sfkor strkss gh ghk ga its suraocks fos jkkh drowh.
?.:.4 ?.:.4
Strk trkss ss Klk`kh Klk`khtt ih ih Cylih Cylihdric dricol ol Cggrdih Cggrdihotks otks
Ainurk ?.1o ?.1o sfgws tfk strkss `otrix ih cylihdricol cggrdihotks. Tfk strkss klk`kht is o aron`kht ga o cylihdkr ih cylihdricol cggrdihotks systk` os sfgwh ih Ainurk ?.1j ?.1j. ^k ossu`k oll tfk cg`pghkhts ih tfk st rkss `otrix ork pgsitivk ork pgsitivk.. Tfk airst rgw ga tfk strkss `otrix will jk sfgwh gh suraocks O suraocks O ohd J ohd J witf gutword hgr`ols ih pgsitivk ohd hknotivk x hknotivk x dirkctigh, rkspkctivkly. rkspkctivkly. Tfk skcghd rgw ga tfk strkss `otrix will jk sfgwh gh suraocks C ohd D ohd D witf gutword hgr`ols ih pgsitivk ohd hknotivk r hknotivk r dirkctigh, rkspkctivkly. rkspkctivkly. Tfk tfird rgw ga tfk strkss `otrix will jk sfgwh gh suraocks K suraocks K ohd Y ohd Y witf gutword hgr`ols ih pgsitivk ohd hknotivk ζ dirkctigh, rkspkctivkly. rkspkctivkly. Fovihn aixkd tfk sinh ga tfk dkhg`ihotgr ih Kquotigh (?.?) wk (?.?) wk sfgw tfk dirkctigh ga strkss cg`pghkhts —ih tfk pgsitivk dirkctigh ga tfk agrcks wfkh gutword hgr`ol is ih tfk pgs itivk dirkctigh. —ohd ih tfk hknotivk dirkctigh ga tfk agrcks wfkh gutwo rd hgr`ol is ih tfk hknotivk dirkctigh. Tfk skctigh k`pfosizks tfot tfk usk ga Kquotigh (?.?) tg (?.?) tg cghstruct tfk strkss klk`kht ih cylihdricol cggrdihotks is si`ilor tg tfot ga cghstructigh ga strkss cujk ih cortksioh cggrdihotks.
1
?
Strk Strksss ohd ohd Stro Stroih ih r
σ rr
σ xx τ xr τ x ζ τrx
σ rr
τζ x
τζ r σζζ
σ xx
τr ζ
τ rx
C
τ x ζ
τ r ζ
J
τ ζ x
τ ζ r
x
τ xr O
τ ζ r τ rx
A
σ ζζ
ζ
K
τ xr
(o)
r
σ ζζ
τ ζ x
τ r ζ D
σ xx
τ x ζ
(j) x
σ rr
ζ
( o) strkss `otrix. (j ( j) strkss klk`kht. Ainurk ?.1 Cylihdricol cggrdihotks= (o
?.:.;
Strk trkss ss Klk`kht Klk`kht ih Spfk Spfkrico ricoll Cggr Cggrdiho dihotks tks
Ainurk ?.?4o ?.?4o sfgws tfk strkss `otrix ih spfkricol cggrdihotks. Ossu`ihn oll strkss cg`pghkhts ih tfk strkss `otrix ork pgsitivk, tfk strkss klk`kht coh jk drowh os sfgwh ih Ainurk ?.?4j ?.?4j. Tfk cghstructigh ga tfk strkss klk`kht ih spfkricol cggrdihotks agllgw tfk so`k rkosghihn os dkscrijkd agr cortksioh ohd cylihdricol cggrdihotks ih tfk prkvigus twg skctighs. Tfk rkodkrs ork khcguronkd tg cfkce tfk rksults agr tfk`sklvks. r
ζ
σ ρρ τ ρζ
σrr
τr ζ
τr ρ
τζ r σ ζζ τρ r τρζ
τζρ
τ ζ r r
τ r ρ
D
C
τ r ρ ρ
τ ζ r τ r ζ ζ
σ ρρ
τ r ζ
O
τ ρ r r
τ ζρ σζζ
σ rr
K
r
σ ζζ τ ζρ
J
τρ r
σ rr
A
τρζ σ ρρ
ρ
z
(o)
ρ (j) y
ζ x (o) strkss `otrix. (j (j) strkss klk`kht. Ainurk ?.> Spfkricol Cggrdihotks= (o
?.:.1 .:.1
Rloh lohk Str Strks ksss
Rlohk strkss is ghk ga tfk twg typks ga twg-di`khsighol si`pliaicotigh ih tfk `kcfohics ga `otkriols. Ih Skctigh ?.1.?, ?.1.?, wk will study plohk stroih, tfk gtfkr typk ga twg-di`khsighol si`pliaicotigh, ohd ih Skctigh :.:, :.:, wk will skk tfk diaakrkhck jktwkkh tfk twg typks ga si`pliaicotigh. Jy “twg-di`khsighol‐ wk i`p ly tfot ghk ga tfk cggrdihotks dgks hgt ploy o rglk ih tfk dkscriptigh ga tfk prgjlk`. Ia wk cfggsk z cfggsk z tg tg jk tfk cggrdihotk, wk skt oll strkssks witf sujscript z sujscript z tg tg jk zkrg. Ainurk ?.6o ?.6o sfgws
Strkss Trohsagr`otigh ih Twg Di`khsighs
>
tfk strkss `otrix ih plohk strkss, Ainurk ?.6j ?.6j tfk ossgciotkd strkss cujk agr tfk pgiht, ohd Ainurk ?.6c ?.6c tfk si`pliaikd twg-di`khsighol rkprkskhtotigh ga tfk strkss cujk os vikwkd arg` tfk z tfk z oxis. oxis. ^k hgtk tfot tfk plohk witf gutword hgr`ol ih tfk z tfk z dirkctigh dirkctigh is strkss arkk. Strkss-arkk suraocks Strkss-arkk suraocks ork olsg collkd arkk collkd arkk suraocks. suraocks. y σ σ yy yy τ τ yx yx τ xy dy σ xx τ xy 2 τ xy σ σ xx xx τ yx σ yy 2 dy σ x σ xx xx 2 2 2 τ dx xy τ dz xy τ yx G τ yx dx z σ σ yy yy Ainurk ?.6 Strkss cujk ih plohk strkss.
KVO@RLK ?.? Sfgw tfk hghzkrg strkss cg`pghkhts gh suraocks O suraocks O,, J, J, ohd C ga ga twg cujks rkprkskhtkd ih Ainurk ?.9 agr ?.9 agr tfk agllgwihn cggrdihotk systk`s= z
σ xx 0 62 @Ro ( T ) τ xy 0 32 τ yx 0 32 @Ro σ yy 0 2 τ zx 0 – >2 @R o τ zy 0 2
@Ro
C
τ xz 0 – >2 @R o τ yz 0 2 σ zz 0 42 @Ro ( C )
C J
O
y
Cujk ? z
x
y
J O Cujk :
x
Ainurk ?.9 Cujks ih diaakrkht cggrdihotk systk`s.
SGL[TIGH z 62 32 C >2 J
>2 O 42
y
32
C
>2 y
>2 O 62
z
x
42
32
J 32
x
(o)
(j)
?.? (o) Cujk ?. (j) Cujk :. Ainurk ?.?2 Sglutigh ga Kxo`plk ?.? (o)
?.3
STZKSS TZKSS TZOHSAG ZOHSAGZ@O Z@OTIG TIGH H IH T^G DI@KHS DI@KHSIGH IGHS S
Ih tfis skctigh tfk strkss trohsagr`otigh kquotighs arg` tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols ork jrikaly rkvikwkd ohd cost ih `otrix agr`. Tfis nivks ohgtfkr pkrspkctivk gh ao`ilior cghckpts. Ih Skctigh ?.4, ?.4, tfis hkw pkrspkctivk will jk kxtkhdkd tg strkss trohsagr`otigh ih tfrkk di`khsighs. Tfk rklotighsfip ga strkssks ot o ot o pgiht ih ih diaakrkht cggrdihotk systk`s, tfot is, trohsagr`otigh ga strkss cg`pghkhts witf cggrdihotk systk`s, is collkd strkss trohsagr`otigh. Strkss trohsagr`otigh coh olsg jk vikwkd os rklotihn strkssks gh diaakrkht plohks tfot poss tfrgunf o pgiht . Tfk hgr`ols ga tfk plohk dkaihk tfk oxis ga o cggrdihotk systk` tg wficf wk ork trohsagr`ihn tfk strkss cg`pghkhts. Tfk aixkd rkakrkhck cggrdihotk systk` ih wficf tfk khtirk prgjlk` is dkscrijkd is collkd tfk nlgjol cggrdihotk systk`. O cggrdihotk systk` tfot coh jk aixkd ot ohy pgiht gh tfk jgdy ohd fos oh grikhtotigh tfot is dkaihkd witf rkspkct tg tfk nlgjol cggrdihotk systk` is collkd tfk lgcol cggrdihotk systk`. Ih tfis jgge tfk nlgjol cggrdihotk systk` will `gst gatkh jk tfk x tfk x-- y y- z z Cortksioh Cortksioh cggrdihotk systk`. Zklotihn ihtkrhol agrcks ohd `g`khts tg kxtkrhol agrcks ohd `g`khts is usuolly dghk ih o nlgjol cggrdihotk systk`. Tfk ihtkrhol quohtitiks ork tfkh uskd tg gjtoih strkssks ih tfk nlgjol cggrdihotk systk`.
6
?
Strk Strksss ohd ohd Stro Stroih ih y t
k h o l R l o c i t r k Q
ζ
I h c l i h k d R l o h k
y
h Gutword hgr`ol tg ihclihkd plohk t ζ
h ζ
x
ζ
ε
Fgrizghtol Rlohk
x
z z
(o)
(j)
Ainurk ?.?? Lgcol ohd nlgjol cggrdihotk systk`s.
^k ossu`k tfot tfk pgiht is ih plohk strkss, ohd wk ehgw tfk strkssks ih tfk x tfk x-- y y- z z cggrdihotk cggrdihotk systk`, tfot is, gh tfk fgrizghtol ohd vkrticol plohks sfgwh ih Ainurk ?.??o ?.??o. ^k skke tg aihd strkssks gh ihclihkd plohks, wficf coh jk gjtoihkd jy rgtotigh ojgut tfk z tfk z oxis. oxis. Oltkrhotivkly, wk skke tg aihd strkssks ih tfk h-t - z z cggrdihotk cggrdihotk systk`, wfkrk h is tfk hgr`ol dirkctigh tg o plohk tfrgunf tfot pgiht os sfgwh ih Ainurk ?.??o ?.??o ohd j, ohd t is is tfk tohnkht dirkctigh tg tfk plohk sucf tfot h-t - z z is is o rinft-fohdkd cggrdihotk systk` os sfgwh ih Ainurk ?.??j ?.??j. Ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols, tfk strkssks ih tfk h-t - z z cggrdihotk cggrdihotk systk` wkrk rklotkd tg strkssks ih tfk Cortksioh cggrdihotk systk` jy `kohs ga tfk wkdnk `ktfgd, wficf is jrikaly klojgrotkd fkrk.Ainurk fkrk. Ainurk ?.?:o ?.?:o sfgws tfk strkssks octihn gh tfk vkrticol, fgrizghtol, ohd ihclihkd plohks tfot cg`pgsk tfk strkss wkdnk. Tfk strkssks ork cghvkrtkd tg agrcks jy `ultiplyihn jy tfk orkos ga kocf plohk gh wficf tfk strkssks oct, os sfgwh gh tfk agrck wkdnk ih Ainurk ?.?:j ?.?:j. Jy kquilijriu` ga agrcks ih tfk h ohd t dirkctighs, dirkctighs, wk gjtoih Kquotigh (?.3j) ohd (?.3j) ohd (?.3j). (?.3j).
σhh 0 σ xx cgs :ζ + σ yy sih :ζ + : τ xy sih ζ τ ht 0 – σ xx cgs ζ sih ζ + σ yy sih ζ
cgs
cgs
ζ
(?.3o)
ζ + τ xy ( cgs :ζ – sih :ζ )
(?.3j)
dz
σhh
τht
τht (dt dz )
σhh (dt dz )
τ xy σ xx
ζ
σ xx(dt cgsζ dz )
dt t
ζ
τ xy (dt cgsζ dz )
h
ζ
τ yx (dt sihζ dz )
τ yx σ yy (dt sihζ dz )
σ yy (o)
(j)
(j) Agrck wkdnk. Ainurk ?.?: (o) Strkss wkdnk. (j
Jy sujstitutihn 92g + ζ agr ζ ih Kquotigh (?.3j), (?.3j), wk gjtoih tfk hgr`ol strkss ih tfk t dirkctigh dirkctigh os
σ tt 0 σ xx sih :ζ + σ yy cgs :ζ – : τ xy cgs ζ sih ζ
(?.3c)
Gh oddihn Kquotigh (?.3j) ohd (?.3j) ohd (?.3c), (?.3c), wk gjtoih
σhh + σtt 0 σ xx + σ yy
(?.4)
Kquotigh (?.4) i`pliks (?.4) i`pliks tfot tfk su` ga hgr`ol strkssks ih oh grtfgnghol systk` is ihdkpkhdkht ga cggrdihotk trohsagr`otigh.
?.3.? ?.3.?
@otri @otrix x @ktf @ktfgd gd ih ih Twg Twg Di` Di`khs khsigh ighss
Ainurk ?.??c ?.??c sfgws tfk hgr`ol (h (h) ohd tohnkht (t (t ) dirkctighs ga o plohk. Tfk dirkctigh cgsihks ga tfk uhit vkctgrs ih tfk h ohd t dirkctighs coh jk writtkh os ih Kquotigh (?.;o). (?.;o).
Strkss Trohsagr`otigh ih Twg Di`khsighs h x Hgtihn tfot
ε 0
0
cgs
ζ
h y
0 sih ζ
t x
0
t y
0 h x
cgs
ε
t y
0 sih ε
9
(?.;o)
92 ² + ζ , wk gjtoih Kquotigh (?.;j). (?.;j). t x
0 – h y
(?.;j)
Tg dkvklgp strkss trohsagr`otigh kquotighs ih `otrix agr`, wk ihtrgduck tfk `otrix hgtotigh nivkh ih Kquotigh (?.;c). (?.;c).
⎮ h x ⎥ {h} 0 ⎦ ⎬ ⎩ h y ⎠
⎮ t x ⎥ { t } 0 ⎦ ⎬ ⎩ t y ⎠
UσP 0
σ xx τ xy τ yx σ yy
(?.;c)
Tfk sy``ktry ga sfkor strkssks i`pliks tfot tfk strkss `otrix is sy``ktric. Ih `otrix hgtotigh, tfis sy``ktry is kxprksskd os
U σ P T 0 U σ P wfkrk
U P
T
(?.1)
i`pliks tfk trohspgsk ga tfk `otrix. Ih `otrix hgtotigh, hgtotig h, Kquotigh (?.3j), (?.3j), (?.3j), (?.3j), ohd (?.3c) coh (?.3c) coh jk writtkh os
σhh 0 { h } T U σ P { h }
τht 0 { t } T U σ P { h }
σ tt 0 { t } T U σ P { t }
(?.>)
wfkrk { } T i`pliks tfk trohspgsk ga tfk cglu`h. Kquotigh (?.>) sfgw (?.>) sfgw tfot wk coh gjtoih o strkss cg`pghkht jy prk- ohd pgst`ultiplyihn tfk strkss `otrix jy tfk uhit vkctgrs ih tfk dirkctigh ga tfk sujscripts. sujscripts. Jkcousk tfk sfkor sfkor strkss is sy``ktric, wk cguld fovk prk`ultiplikd jy tfk uhit hgr`ol ohd pgst`ultiplikd jy tfk uhit tohnkht. Ih gtfkr wgrds, tfk grdkr ga `ultiplicotigh is i``otkriol ih tfk colculotigh ga sfkor strkss. Strkss gh o suraock is collkd troctigh gr o strkss vkctgr. Hgtk tfot strkss ot o pgiht is o skcghd-grdkr tkhsgr. ^fkh wk spkciay o suraock, tfk grikhtotigh ga tfk suraock is dkaihkd5 fkhck tfk strkss gh tfk suraock hkkds ghly ghk dirkctigh tg spkciay, wficf i`pliks tfot strkss gh o suraock is o vkctgr quohtity. @otfk`oticolly tfk strkss vkctgr {S {S } is dkaihkd os ih Kquotigh (?.6). (?.6).
{ S } 0 U σ P { h }
(?.6)
Kquotigh (?.6) ih (?.6) ih lghn hgtotigh coh jk writtkh os S x
0 σ xx h x + τ xy h y
S y
0 τ yx h x + σ yy h y
(?.9)
Sihck dirkctigh cgsihks dg hgt fovk uhits, tfk strkss vkctgr fos tfk uhits ga strkss. Tfus, wk hgw ehgw tfrkk diaakrkht quohtitiks tfot fovk uhits ga agrck pkr uhit orko= • prkssurk, prkssurk, wficf is o scolor scolor quohtity, quohtity, • troctigh, troctigh, wficf is o vkctgr vkctgr quohtity quohtity,, • strkss, wficf wficf is o skcghd-grdkr skcghd-grdkr tkhsgr tkhsgr.. Tg jkttkr opprkciotk tfk cghckpt ga troctigh, wk ghck `grk cghsidkr tfk agrck wkdnk sfgwh ih Ainurk ?.?:j ?.?:j. ^k rkprkskht tfk ihclihk orko dt dz 0 0 dO ohd dO ohd usk tfk dirkctigh cgsihk hgtotigh ga Kquotigh (?.;o) tg (?.;o) tg sfgw tfk twg-di`khsighol rkprkskhtotigh ga tfk agrck wkdnk, os ih Ainurk ?.?3o ?.?3o. Tfk agrcks sfgwh gh tfk ihclihkd plohk cguld jk rksglvkd ihtg agrcks ih tfk x tfk x ohd ohd y y cggrdihotks, os sfgwh ih Ainurk ?.?3j ?.?3j. Jy kquilijriu` ga agrcks ih Ainurk ?.?3j ?.?3j, wk gjtoih Kquotigh (?.9). (?.9). Oltkrhotivkly, tfk strkss vkctgr {S {S } cguld jk writtkh os
{ S } 0 σ hh { h } + τ ht { t }
(?.?2)
Kquotigh (?.6) ohd (?.6) ohd (?.?2) ork (?.?2) ork rkprkskhtotighs ga tfk vkctgr {S {S } ih twg diaakrkht cggrdihotk systk`s, os sfgwh ih Ainurk ?.?3c ?.?3c. S y (dO) dO)
y
dO) τht (dO)
σhh{h} dO) σhh (dO)
dO) σxx ( h x dO)
S x (dO) dO)
dO) σ xx ( h x dO)
{S}
Sy
τht{t} Sx dO) τxy (h x dO)
τ xy (h x dO) x
τyx σyy
τyx
(h y dO) dO)
σ yy
(h y dO) dO) (o)
(h y dO) dO)
(h y dO) dO) (j)
(c)
Ainurk ?.?3 Stoticolly kquivolkht agrck wkdnks.
?.3. ?.3.: :
Rrih Rrihci cipo poll Strks trkssk skss
Zkcoll tfk agllgwihn dkaihitighs arg` tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols. Rlohks gh wficf tfk sfkor strkssks ork zkrg ork collkd tfk prihcipol plohks. Tfk hgr`ol dirkctigh tg tfk prihcipol plohks is rkakrrkd tg os tfk prihcipol dirkctigh, gr
?2
?
Strk Strksss ohd ohd Stroi troih h
tfk prihcipol oxis agr strkssks. Tfk ohnlks tfk prihcipol oxis `oeks witf tfk nlgjol cggrdihotk systk` ork collkd prihcipol ohnlks. Hgr`ol strkss gh o prihcipol plohk is collkd prihcipol strkss. Tfk nrkotkst prihcipol strkss is collkd prihcipol strkss ?. Ih plohk strkss, tfk plohk witf gutword hgr`ol ih tfk z tfk z dirkctigh dirkctigh fos zkrg sfkor strkss. Tfkrkagrk jy dkaihitigh, tfk z tfk z dirkctigh is o prihcipol dirkctigh. Sih ck tfkrk ork tfrkk prihcipol plohks, t fkrk ork tfrkk prihcipol strkssks ot o pgiht. Tfk tfrkk prihcipol strkssks ork lojklkd σ?, σ:, ohd σ3. Lkt tfk uhit hgr`ol vkctgr ga o prihcipol plohk jk nivkh jy { p} p} ohd tfk cgrrkspghdihn prihcipol strkss jk σ p. Jy dkaihitigh, tfk plohk witf hgr`ol { p} p} will fovk zkrg sfkor strkss ohd tfk hgr`ol strkss will jk tfk prihcipol strkss ohd Kquotighs (?.6) ohd (?.?2) coh (?.?2) coh jk writtkh os
{ S } 0 U σ P { p } 0 σ p { p }
(?.??)
Kquotigh (?.??) ih (?.??) ih lghn hgtotigh coh jk writtkh os agllgws=
{ S } 0
σ xx τ xy ⎮ p x ⎥ σ 2 ⎮ p x ⎥ ⎦ ⎬ 0 p ⎦ ⎬ 2 σ p ⎩ p y ⎠ τ yx σ yy ⎩ p y ⎠
(?.?:o)
Oltkrhotivkly, Kquotigh (?.?:o) coh (?.?:o) coh jk writtkh os Kquotigh (?.?:j). (?.?:j).
( σ xx – σ p ) τ xy ⎮ p x ⎥ ⎦ ⎬0 τ yx ( σ yy – σ p ) ⎩ p y ⎠
2
(?.?:j)
Kquotigh (?.?:j) rkprkskhts (?.?:j) rkprkskhts twg kquotighs ih tfk twg uhehgwhs p uhehgwhs p x ohd p ohd p y. Agr o hghtriviol (hghzkrg (hghzkrg)) sglutigh tg kxist, tfk dktkr`ihoht ga tfk `otrix `ust jk zkrg. Tfis is o clossic stotk`kht ga oh kinkhvoluk prgjlk`. Tg sfgw tfot tfk kinkhvoluks ga tfk strkss `otrix ork tfk prihcipol strkssks, wk skt tfk dktkr`ihoht ga tfk `otrix tg jk zkrg tg gjtoih : σ p: – σ p ( σ xx + σ yy ) + ( σ xx σ yy – τ xy ) 0
2
(?.?3)
Kquotigh (?.?3) is (?.?3) is collkd tfk cforoctkristic kquotigh ih oh kinkhvoluk prgjlk`. Tfk ojgvk kquotigh is o quodrotic wfgsk rggts ork
σ?,:
: U ( σ xx + σ yy ) ¿ ( σ xx + σ yy ): – 4 ( σ xx σ yy – τ xy ) P ⁄ :
0
(?.?4o)
Tfk tkr`s ih Kquotigh (?.?4o) coh (?.?4o) coh jk rkorrohnkd tg gjtoih
σ?,:
0
σ xx + σ yy⎘ σ xx – σ yy⎘ : : ⎕ --------------------⎕ --------------------¿ ⎐ : ⎢ ⎐ : ⎢ + τ xy
(?.?4j)
Kquotigh (?.?4j) nivks (?.?4j) nivks tfk agr`ulos agr prihcipol strkssks wk sow ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols. ^k cghcludk tfk kinkhvoluks ga tfk strkss `otrix ork tfk prihcipol strkssks ohd tfk kinkhvkctgrs ga tfk strkss `otrix ork tfk prihcipol dirkctighs. Tg dktkr`ihk tfk kinkhvkctgrs, wk coh usk kitfkr ga tfk twg kquotighs rkprkskhtkd ih (?.?:j), (?.?:j), olghn witf tfk aoct tfot tfk su` ga tfk squork ga dirkctigh cgsihks is ?. Kxo`plk ?.: dk`ghstrotks ?.: dk`ghstrotks tfk prgckss ga dktkr`ihihn tfk kinkhvkctgrs.
?.3. ?.3.3 3
@oxi @oxi`u `u` ` Sfk Sfkor or Strks trksss
Ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols, wk sow tfot tfk `oxi`u` sfkor strkss kxists gh o plohk tfot is 4;² tg tfk prihcipol plohks. Tfk `onhitudk ga tfk `oxi`u` sfkor strkss is nivkh jy
τ`ox
σ? – σ : σ: – σ3 , σ3 – σ? ⎘ 0 `ox ⎕ --------------------------------- ⎢ ⎐ : , ----------------: :
(?.?;)
wfkrk tfk tkr` gh tfk rinft-fohd sidk is cfgskh tg n ivk tfk `oxi`u` `onhitudk ga diaakrkhck jktwkkh tfk twg pri hcipol strkssks. Ainurks ?.?4 tfrgunf ?.?4 tfrgunf ?.?1 sfgw ?.?1 sfgw tfk vorigus plohks gh wficf tfk `oxi`u` sfkor strkss will kxist, dkpkhdihn upgh tfk rklotivk voluks ga tfk prihcipol strkssks. p p: :
τ:3 p?
p3
p?
p3
Ainurk ?.?4 Rlohks ga `oxi`u` sfkor strkss tfot ork 4; ² tg prihcipol plohks : ohd 3.
σ: – σ3 0 – τ 3: 0 ----------------:
Strkss Trohsagr`otigh ih Tfrkk Di`khsighs
??
p:
p:
p?
p3
σ 3 – σ? 0 – τ ?3 0 ----------------:
τ3?
p?
p3
Ainurk ?.?; Rlohks ga `oxi`u` sfkor strkss tfot ork 4; ² tg prihcipol plohks ? ohd 3.
p:
p:
p?
p?
p3
σ? – σ : 0 – τ :? 0 ----------------:
τ ?:
p3
Ainurk ?.?1 Rlohks ga `oxi`u` sfkor strkss tfot ork 4; ² tg prihcipol plohks ? ohd :.
KVO@RLK ?.: Tfk agllgwihn strkssks wkrk aguhd ot o pgiht ih plohk strkss gh o ausklonk ga oh oircroat=
σ xx
0 32 @Ro ( T )
σ yy
0 12 @Ro ( C )
τ xy
0 ;2 @Ro
[sk tfk `otrix `ktfgd tg dktkr`ihk (o) tfk hgr`ol ohd sfkor strkssks gh o plohk possihn tfrgunf tfk pgiht tfot is ot 32g cguhtkrclgcewisk tg tfk x tfk x oxis, oxis, (j) tfk prihcipol strkssks, ohd (c) prihcipol dirkctigh ?. 62.6 @Ro ( C ) σ ? σhh 0 62.6 5 τht 0 ?4.2 @Ro σ:
OHS.
0 ;:.3 ;:.3 @Ro ( T ) 0 6:.3 6:.3 @Ro ( C )
5
⎮ ζ x ⎦ ⎩ ζ y
0 :4 ² ⎥ 0
⎬ 11 ² ⎠
gr
⎮ ζ x ⎦ ⎩ ζ y
0 ?;1 ² ⎥
⎬
0 ??4 ² ⎠
KVO@RLK ?.3 Rrgvk tfot tfk cgkaaicikhts ga tfk cforoctkristic kquotighs ork ihdkpkhdkht ga tfk cggrdihotks.
?.4
STZKSS TZKSS TZOH TZOHSAG SAGZ@O Z@OTIGH TIGH IH TFZKK TFZKK DI@KHS DI@KHSIGH IGHS S
Tfk cghckpts ohd agr`ulos dkvklgpkd ih twg di`khsighs coh jk kxtkhdkd tg tfrkk di`khsighs jy hgtihn tfot tfk `otrix rklotighsfips dkscrijkd ih Skctigh ?.3 dg hgt dkpkhd upgh tfk sizk ga tfk `otrix. Ainurk ?.?>(o) ?.?>(o) sfgws tfk dirkctigh cgsihks ga tfk uhit hgr`ol ih tfrkk di`khsighs, wficf coh jk writtkh os agllgws= h x 0 cgs
ζ x
h y 0 cgs
ζ y
h z 0 cgs
ζ z
y
y
σhh{h}
h
O O x ζ y
{S }
O z ζ x
ζ z
x
x
O y
z
τht {t }
(o)
z
( j) Kquilijrotihn sfkor strkss. Ainurk ?.?> (o) Dirkctigh cgsihks ga o uhit hgr`ol. (j
^k dkaihk tfk agllgwihn quohtitiks ih `otrix hgtotigh
(j)
?:
?
Strk Strksss ohd ohd Stroi troih h
⎮ h x ⎥ ⎭ ⎭ { h } 0 ⎦ h y ⎬ ⎭ ⎭ ⎩ h z ⎠
⎮ S x ⎥ ⎭ ⎭ { S } 0 ⎦ S y ⎬ ⎭ ⎭ ⎩ S z ⎠
UσP 0
σ xx τ xy τ xz τ yx σ yy τ yz τ zx τ zy σ zz
(?.?1)
Tfk hgr`ol strkss coh jk aguhd arg` Kquotigh (?.>). (?.>). Ih twg di`khsighs, tfk hgr`ol ohd tohnkht dirkctighs wkrk rklotkd. Ih tfrkk di`khsighs, fgwkvkr, tfkrk ork oh ihaihitk hu`jkr ga tohnkhts ih tfk ihclihkd plohk sfgwh ih Ainurk ?.?>o ?.?>o. Tfk colculotigh ga sfkor strkss ih tfrkk di`khsighs khtoils prgjlk`s ga twg typks. ?. Tfk tohnkht tohnkht dirkctigh is ehgwh. ehgwh. Tfk sfkor strkss ih tfot dirkctigh dirkctigh `oy jk aguhd arg`Kquotigh arg`Kquotigh (?.>). (?.>). Sfkor strkss aguhd arg` Kquotigh (?.>) rkprkskhts (?.>) rkprkskhts tfk cg`pghkht ga tfk strkss vkctgr ih tfk nivkh tohnkht dirkctigh5 tfis `oy hgt jk tfk sfkor strkss hkckssory agr kquilijriu` ga tfk wkdnk sfgwh ih Ainurk ?.?>o ?.?>o. :. Tfk sfkor strkss strkss tfot is hkckssory hkckssory agr kquilijriu` kquilijriu` `oy jk aguhd arg` Kquotigh (?.?2), (?.?2), os sfgwh ih Ainurk ?.?>j ?.?>j. ^k coh aihd tfk strkss vkctgr {S {S } arg` Kquotigh (?.6) ohd (?.6) ohd σhh arg` Kquotigh (?.>). (?.>). Arg` Kquotigh (?.?2) wk gjtoih τ ht { t } 0 { S } – σ hh { h } . Tfk `onhitudk ohd dirkctigh ga tfk sfkor strkss coh jk dktkr`ihkd arg` tfk vkctgr τht {t }. }.
KVO@RLK ?.4 Tfk strkssks ot o pgiht gh o aro`k ga o cor ork nivkh jy tfk agllgwihn strkss `otrix= 32
2
:2
32 – ?2 @R o
2
:2 – ?2
2
Dktkr`ihk (o) tfk hgr`ol strkss gh o plohk tfot fos gutword hgr`ols ot 12g, 12g, ohd 4;g tg tfk x tfk x,, y, y, ohd z ohd z dirkctighs, dirkctighs, rkspkctivkly5 (j) tfk sfkor strkss `onhitudk ohd dirkctigh gh tfk plohk ih port (o) tfot is ih kquilijriu` witf tfk nivkh stotk ga strkss5 ohd (c) tfk voluk ga tfk sfkor strkss gh tfk plohk witf gutword hgr`ols nivkh ih port (o) ohd tohnkht dirkctigh tfot `oeks tfk ohnlks 4;g, ?3;g, ohd 92g tg tfk x tfk x,, y, y, ohd z ohd z dirkctighs, dirkctighs, rkspkcti vkly.
OHS.
?.4. ?.4.? ?
(o)
σ hh
0 ::.? ::.? @Ro ( T ) 5 (j)
τht
0 :?.:? @Ro 5 { t k } 0
⎮ 2.6;3> ⎥ ⎭ ⎭ ⎦ – 2.?41; ⎬ 5 (c) τht ⎭ – 2.; ⎭ ⎩ ⎠
0 ?; @Ro
Rrih Rrihci cipo poll Strks trkssk skss
Tfk kquivolkht agr` ga Kquotigh (?.?:o) ohd (?.?:o) ohd (?.?:j) ih (?.?:j) ih tfrkk di`khsighs coh jk writtkh os agllgws=
σ xx τ xy τ xz ⎮ p x ⎥ ⎭ ⎭ τ yx σ yy τ yz ⎦ p y ⎬ ⎭ ⎭ τ zx τ zy σ zz ⎩ p z ⎠
0
σ p
2
2
σ p
2
2
⎮ p x ⎥ ⎭ ⎭ 2 ⎦ p y ⎬ ⎭ ⎭ σ p ⎩ p z ⎠
( σ xx – σ p ) τ yx τ zx
2
gr
τ xy ( σ yy – σ p ) τ zy
⎮ p x ⎥ ⎭ ⎭ ⎦ p y ⎬ ⎭ ⎭ ( σ zz – σ p ) ⎩ p z ⎠ τ xz τ yz
0 2
(?.?>)
Tfk `otrix Kquotigh (?.?>) rkprkskhts (?.?>) rkprkskhts tfrkk kquotighs ih tfk tfrkk uhehgwh dirkctigh cgsihks. Agr o hghtriviol sglutigh, tfk dktkr`ihoht ga tfk `otrix `ust jk zkrg. Ghck `grk wk gjskrvk tfot tfk kinkhvoluks ga tfk strkss `otrix ork tfk prihcipol strkssks, ohd tfk kinkhvkctgrs ork tfk prihcipol dirkctighs. Tg dktkr`ihk tfk kinkhvkctgrs cgrrkspghdihn tg oh kinkhvoluk, wk coh usk ohy twg ga tfk tfrkk kquotighs ih Kquotigh (?.?>), (?.?>), olghn witf tfk ehgwlkdnk tfot tfk squork ga tfk dirkctigh cgsihks is ?, os sfgwh ih Kquotigh (?.?6). (?.?6). :
:
:
p x + p y + p z 0 ?
?.4.: ?.4.:
(?.?6)
Rrihci Rrihcipol pol Strkss trkss Cghvk Cghvkhti htigh gh
Rrihcipol strkssks `oy jk grdkrkd gr uhgrdkrkd. Nkhkrolly spkoeihn, tfk agllgwihn cghvkhtigh is gjskrvkd agr grdkrihn ga prihci pol strkssks. Ih tfrkk di`khsighs tfk prihcipol strkssks ork rkpgrtkd sucf tfot σ ? 8 σ : 8 σ 3 . Ih twg di`khsighs tfk prihcipol strkssks ork rkpgrtkd sucf tfot σ ? 8 σ: , jut σ 3 is hgt ngvkrhkd jy ohy grdkr. Tfis is jkcousk σ3 is kquol tg σ zz , wficf dkpkhds upgh wfktfkr tfk stotk ga strkss is plohk strkss gr plohk stroih. Jy cghv khtigh, colculotkd prihcipol strkssks ork rkpgrtkd ih grdkrkd agr`. Ih discussigh ga aoilurk tfkgriks (skk, k.n., Skctigh :.3), tfk prihcipol strkssks ork usuolly cghsidkrkd tg jk uhgrdkrkd, sg tfot tfk aoilurk critkrigh is uhoaakctkd jy tfk grdkrihn cghvkhtigh. Tfk prihcipol ohnlks ork colculotkd arg` tfk dirkctigh cgsihks ga tfk prihcipol dirkctighs. Jy cghvkhtigh, tfk ohnlks rkpgrtkd ork jktwkkh zkrg ohd ?62², tfot is, 2 ² ≤ ζ x, ζ y, ζ z ≤ ?62 ²
(?.?9)
Strkss Trohsagr`otigh ih Tfrkk Di`khsighs
?3
KVO@RLK ?.; Sfgw tfot tfk prihcipol dirkctighs ork grtfgnghol ia tfk prihcipol strkssks ork hgt kquol.
KVO@RLK ?.1 Ot o pgiht gh o suj`orihk full t fk strkss `otrix is os nivkh. Dktkr`ihk tfk prihcipol strkssks ohd tfk prihcipol dirkctighs ot tfk pgiht. 2
1
2
;
2 @R o
1
2 – 6
σ : 0 ; @Ro ( T ) 5 σ 3 0 ?2 @Ro ( C ) Rrihcipol dirkctigh ? ζ x 0 ?6.4 ² 5ζ y 0 92 ² 5ζ z 0 >?.1 ² gr ζ x 0 ?1?.1 ² 5ζ y 0 92 ² 5ζ z 0 ?26.4 ² Rrihcipol dirkctigh : ζ x 0 92 ² 5ζ y 0 2 5ζ z 0 92 ² gr ζ x 0 92 ² 5ζ y 0 ?62 ² 5ζ z 0 92 ² Rrihcipol dirkctigh 3 ζ x 0 >?.1 ² 5ζ y 0 92 ² 5ζ z 0 ?1?.1 ² gr ζ x 0 ?26.4 ² 5ζ y 0 92 ² 5ζ z 0 ?6.4 ² OHS.
?.4.3 ?.4.3
σ?
6
0 ?2 @Ro ( T ) 5
Cforoct Cforoctkris kristic tic Kqu Kquotig otigh h ohd ohd Strk Strkss ss Ihvor Ihvorioht iohtss
^k skt tfk dktkr`ihoht ga tfk `otrix ih Kquotigh (?.?>) tg (?.?>) tg zkrg tg gjtoih tfk cforoctkristic kquotigh, wficf is writtkh ih tfk agl lgwihn agr`.
σ p3 √ I ? σ p: + I : σ p √ I 3
0 2
(?.:2)
Tfk tfrkk rggts3 ga tfk cforoctkristic kquotigh yikld tfk prihcipol strkssks, os gjskrvkd ih Skctigh ?.3.:. ?.3.:. Tfk cgkaaicikhts I cgkaaicikhts I ?, I :, ohd I ohd I 3 ork collkd strkss ihvoriohts jkcousk tfk voluk ga tfksk quohtitiks ork ihdkpkhdkht ga cggrdihotk trohsagr`otigh, os wos sfgwh agr plohk strkss prgjlk` ih Kxo`plk ?.3. ?.3. Hgtick tfot tfk sinh ih tfk tkr`s cghtoihihn gdd strkss ihvoriohts is hknotivk. Tfk tfrkk strkss ihvoriohts coh jk sfgwh (skk Rrgjlk` ?.34) tg jk nivkh jy Kquotigh (?.:?). (?.:?). I ? 0
σ xx + σ yy + σ zz
I :
σ τ 0 xx xy τ yx σ yy
+
σ yy τ yz τ zy σ zz
+
σ xx τ xz τ zx σ zz
I 3
σ xx τ xy 0 τ yx σ yy τ zx τ zy
τ xz τ yz σ zz
(?.:?)
wfkrk tfk dktkr`ihoh dktkr`ihohtt ga tfk quohtity quohtity sfgwh is khclgskd khclgskd ih vkrticols= vkrticols= | |. • Tfk airst airst strkss strkss ihvorioht ihvorioht I I ? is tfk su` ga dionghol tkr`s ga tfk strkss `otrix. • Tfk skcghd skcghd strkss ihvorioht ihvorioht I I : is tfk su` ga `ihgr dktkr`ihohts ga tfk strkss `otrix. • Tfk tfird tfird strkss strkss ihvorioht ihvorioht I I 3 is tfk dktkr`ihoht ga tfk khtirk strkss `otrix. Sihck strkss ihvoriohts ork ihdkpkhdkht ga tfk cggrdihotk systk`, tfk voluk ga tfks k quohtitiks will will jk tfk so`k so`k ih o prihcipol cggrdihotk cggrdihotk systk`. Tfk strkss `otrix `otrix ih o prihcipol cggrdihotk systk` coh jk writtkh os
σ?
2
2
2
σ:
2
2
2
σ3
(?.::)
Tfk strkss ihvoriohts coh jk writtkh ih tkr`s ga prihcipol strkss os I ? 0
σ? + σ: + σ3
I : 0
σ?
2
2
σ:
+
σ:
2
2
σ3
+
σ?
2
2
σ3
σ? 0
σ ? σ: + σ: σ3 + σ 3 σ?
I 3 0 2 2
2
2
σ:
2
2
σ3
0
σ? σ: σ3
(?.:3)
Tfk strkss ihvoriohts, porticulorly tfk airst ohd tfk tfird, prgvidk o quice ohd kosy cfkce gh tfk lojgrigusly colculotkd prihcipol strkssks. Tfk `oxi`u` sfkor strkss coh jk aguhd arg`Kquotigh arg` Kquotigh (?.?;) ohd (?.?;) ohd will jk gh o plohk tfot is 4;g tg tfk prihcipol plohks os discusskd ih Skctigh ?.3.3. ?.3.3.
?.4. ?.4.4 4
Gcto Gc tofk fkdr drol ol Strks trkssk skss
O plohk tfot `oeks kquol ohnlks witf tfk prihcipol plohks is collkd oh gctofkdrol plohk. Ih Skctigh :.3, :.3, wk will skk tfot ghk ga tfk tfkgriks ga aoilurk is joskd gh tfk strkssks gh o gctofkdrol plohk. 3.
Tfk rggts rggts ga ga kquo kquotigh tigh x x3 √ I ? x: + I : x √ I 3 0 2 ork x ork x? 0 : O cgs O cgs ο + I ?/3, ohd x ohd x:,3 0 –: O cgs O cgs (ο ¿ 12²) + I ?/3, wfkrk O 0 3
cgs 3 ο 0 U : ( I ? ⁄ 3 ) – ( I ? ⁄ 3 ) I : + I 3 P ⁄ ( : O
3
).
( I ? ⁄ 3 ): – I : ⁄ 3 ohd
?4
?
Strk Strksss ohd ohd Stroi troih h :
σ?
σ3 Ainurk ?.?6 Tfk kinft gctofkdrol plohks.
Ainurk ?.?6 sfgws ?.?6 sfgws kinft (gctol) plohks tfot `oek kquol ohnlks witf tfk prihcipol plohks. Tfk strkssks gh gctofkdrol plohks ork rkakrrkd tg os gctofkdrol strkssks. ^k usk h?, h:, ohd h3 tg rkprkskht tfk dirkctigh cgsihks ga tfk gutword hgr`ol ga o plohk `kosurkd arg` tfk prihcipol oxis.Tfk hgr`ol strkss ohd sfkor strkss gh o plohk witf dirkctigh cgsihksh cgsihks h?, h:, ohd h3 coh jk sfgwh (skk Rrgjlk` ?.:9) tg jk nivkh jy Kquotigh (?.:4o) ohd (?.:4o) ohd (?.:4j). (?.:4j).
σhh τht
0
σ? h ?: + σ: h :: + σ3 h 3:
(?.:4o)
: ( σ?: h?: + σ:: h :: + σ3: h 3: ) – σhh
(?.:4j)
0
Tfgunf tfk sinh ga tfk dirkctigh cgsihks cfohnks witf kocf ga tfk kinft gctofkdrol plohks, tfkir `onhitudk is tfk so`k agr oll kinft plohks, tfot is,
h ? 0 h : 0 h 3 0 ? ⁄ 3
. Sujstitutihn tfk dirkctigh cgsihks agr tfk gctofkdrol plohks ihKquotigh ih Kquotigh
(?.:4o) wk (?.:4o) wk gjtoih tfk gctofkdrol hgr`ol strkss os
σgc t Kquotigh (?.:4c) sfgws
σgc t
0
( σ? + σ: + σ3 ) ⁄ 3
(?.:4c)
0 I ? ⁄ 3 , wfkrk I wfkrk I ? is tfk airst strkss ihvorioh ihvorioht. t. O `grk ihtkrkstihn ihtkrpr ktotigh ga gcto-
fkdrol hgr`ol strkss is gjtoihkd jy cgh sidkrihn o pgiht ih aluid. Agr o pgih t ih aluid, tfk sfkor strkssks ork zkrg ohd oll hgr`ol strkss ks (prihcipol (prih cipol strkssks) ork cg`prkssivk ohd kquol tg tfk prkssurk ot tfot pgiht, tfot is, σ ? 0 fydrgstotic prkssurk. Arg` Kquotigh (?.:4c) wk gjtoih
σgc t
σ:
0
σ3
0 – p , wfkrk p is p is tfk
0 – p 5 ih gtfkr wgrds, tfk gctofkdrol hgr`ol strkss is si`ilor tg tfk
fydrgstotic prkssurk. Sujstitutihn tfk dirkctigh cgsihks agr tfk gctofkdrol plohks ih Kquotigh (?.:4j) wk (?.:4j) wk gjtoih tfk `onhitudk `onhitudk ga sfkor sfkor strkss, wficf coh jk si`pliaikd os sfgwh ih Kquotigh (?.:4d). (?.:4d).
τ gc t 0
( σ?: + σ:: + σ 3: ) ( σ? + σ: + σ 3 ) :
------------------------------------ – ------------------------------------3 9
τ gc t 0
? -3
0
? -3
( : σ ?: + : σ:: + : σ3: ) – : ( σ? σ: + σ: σ3 + σ 3 σ? )
( σ ? – σ: ) : + ( σ: – σ3 ): + ( σ3 – σ ? ) :
gr (?.:4d)
Ih Skctigh :.3 it :.3 it will jk skkh tfot ductilk `otkriols coh witfstohd vkry lornk fydrgstotic prkssurks ohd yikldihn is ngvkrhkd jy τgct rkocfihn o criticol voluk.
KVO@RLK ?.> Tfk strkssks ot o pgiht gh tfk aro`k ga o cor ork nivkh jy tfk strkss `otrix sfgwh korlikr, ihKxo`plk ih Kxo`plk ?.4. ?.4. Dktkr`ihk (o) tfk prihcipol strkssks, (j) tfk tfird prihcipol dirkctigh, (c) tfk `oxi`u` sfkor strkss, ohd (d) tfk gctofkdrol hgr`ol ohd sfkor strkssks. OHS. σ ? 0 4?.9 4?.9 @Ro ( T ) 5 σ : 0 32 @Ro ( T ) 5 σ 3 0 ??.9 ??.9 @Ro ( C ) 5 ζ x 0 ??4.9 ² 5ζ y 0 >>.6 ² 5ζ z 0 ζ z 0 :6.? ² 5 gr
ζ x 0 1;.? ² 5ζ y 0 ?2:.: ² 5ζ z 0 ?;?.9 ² 5 τ`o x ?.;
0 :1.9 @Ro 5
σgc t
0 :2 @Ro ( T ) 5
τgc t
0 :3.? @Ro 5
OQKZO QKZONK NK HGZ@O HGZ@OL L OHD SFK SFKOZ OZ STZ STZOI OIHS HS
Stroih is o `kosurk ga tfk ihtkhsity ga cfohnk ih tfk sfopk ga o jgdy. Tg `kosurk cfohnks, wk `ust cghsidkr tfk `gvk`kht ga pgihts gh tfk jgdy, wficf rkquirks o clkor dklihkotigh ga `gvk`kht duk tg rinid jgdy `gtigh arg` `gvk`kht ossgciotkd witf cfohnks ih tfk sfopk g a o jgdy. Tfk tgtol `gvk`kht ga ga o pgiht witf rkspkct tg o aixkd rkakrkhck cggrdihotks is collkd displock`kht. Tfk rklotivk `gvk`kht ga ga o pgiht witf rkspkct tg ohgtfkr pgiht gh tfk jgdy is collkd dkagr`otigh. Tfus displock`kht ga o pgiht rkprkskhts cfohnks duk tg dkagr`otigh plus rinid jgdy `gtig h. Ih dkscrijihn cfohnks, wk `ust spkciay tfk rkakrkhck voluk arg` wficf wk ork `kosurihn cfohnk. Tfis lkods tg tfk agllgwihn twg dkaihitighs. Lonrohnioh stroih is cg`putkd arg` dkagr`otigh jy usihn tfk grinihol uhdkagr`kd nkg`ktry os nkg`ktry os tfk rkakrkhck nkg`ktry. Kulkrioh stroih is cg`putkd arg` dkagr`otigh jy usihn tfk aihol tfk aihol dkagr`kd nkg`ktry nkg`ktry os tfk rkakrkhck nkg`ktry. Tfk Lonrohnioh dkscriptigh is usuolly uskd ih sglid `kcfohics. Tfk Kulkrioh dkscriptigh is usuolly uskd ih aluid `kcfohics. ^fkh o `otkriol uhdkrngks vkry lornk dkagr`otighs, sucf os ih o sgat rujjkr gr ih prgmkctilk pkhktrotigh ga `ktols,
Ovkronk Hgr`ol ohd Sfkor Stroihs
?;
kitfkr dkscriptigh `oy jk uskd, dkpkhdihn upgh tfk oholysis. Ih tfis jgge, wk will usk Lonrohnioh stroih kxckpt agr o akw prgjlk`s. Zkcoll tfot tfkrk ork twg eihds ga stroih—hgr`ol stroihs ohd sfkor stroihs. Hgr`ol stroihs ork `kosurks ga cfohnk ga lkhntf ohd sfkor stroihs ork `kosurks ga cfohnk ga ohnlks. Hgr`ol stroih is usuolly dksinhotkd jy tfk Nrkke lkttkr kpsilgh (λ). Tfk cfohnk ga lkhntf, olsg collkd dkagr`otigh, is dksinhotkd jy tfk lgwkrcosk Nrkke lkttkr dklto (δ). Tfk ovkronk hgr`ol Lonrohnioh stroih ga o lihk is tfk rotig ga dkagr`otigh δ ga tfk lihk tg tfk grinihol lkhntf L lkhntf L2 os agllgws=
λ
δ
0 ---- L 2
(?.:;)
rksultt ih pgsitivk hgr`ol stroihs. Cghtroctighs (δ < 2) rksul rksultt ih hknotivk hgr`ol stroihs. Klghnotighs (δ 8 2) rksul Ih s`oll-stroih opprgxi`otighs (λ < 2.2?), tfk dkagr`otigh δ coh jk opprgxi`otkd jy tfk dkagr`otigh cg`pghkht cg`pghkht ih tfk dirkctigh ga tfk uhdkagr`kd lihk klk`kht. Ainurk ?.?9 sfgws ?.?9 sfgws o jor OR jor OR tfot tfot dkagr`s tg tfk pgsitigh OR pgsitigh OR ?. Agr s`oll-stroih colculotighs, tfk dkagr`kd lkhntf ga OR ga OR ? is opprgxi`otkd jy lkhntf OR lkhntf OR :, wficf is gjtoihkd jy drowihn o pkrpkhdiculor arg` R arg` R ? tg tfk grinihol dirkctigh OR dirkctigh OR . Ohgtfkr pkrspkctivk gh s`oll-stroih opprgxi`otigh is tfot tfk dkagr`otigh δ is tfk cg`pghkht ga dkagr`otigh vkctgr RR RR ? ih tfk grinihol dirkctigh. Tfk s`oll-stroih opprgxi`otigh rksults ih lihkor oholysis ohd nrkotly si`pliaiks colculotighs. R ?
g h i t i s p g d R ` k r a g g h D k i t it g s i O
R : δ
p g h h o l i n n i G r
Ainurk ?.?9 Opprgxi`otigh ga o s`oll hgr`ol stroih.
Sfkor stroih is usuolly dksinhotkd jy tfk Nrkke lkttkr no``o (γ). Tfk ovkronk Lonrohnioh sfkor stroih is dkaihkd os o cfohnk ga ohnlk arg` tfk rinft ohnlk os
γ
ς
0 -- – ο :
(?.:1)
wfkrk, tfk Nrkke lkttkr olpfo (ο) is tfk aihol ohnlk `kosurkd ih rodiohs (rod), ohd tfk Nrkke lkttkr pi (ς) 0 3.?4?;9 rod. Dkcrkosks rod. Dkcrkosks ih tfk ohnlk (ο < ς /:) rksult ih pgsitivk sfkor stroih. Ihcrkosks stroih. Ihcrkosks ih ih tfk ohnlk ( ο 8 ς /:) rksult ih hknotivk sfkor stroih Agr s`oll sfkor stroih (γ < 2.2?), tfk agllgwihn opprgxi`otigh ga tfk tringhg`ktric auhctighs `oy jk uskd= toh γ ≆ γ
?.;.? ?.;.?
sih γ ≆ γ
cgs γ ≆ ?
[hits [hits ga Ovkron vkronkk Stro Stroihs ihs
Kquotigh (?.:;) sfgws (?.:;) sfgws tfot hgr`ol stroih is di`khsighlkss, fkhck sfguld fovk hg uhits. Fgwkvkr, tg diaakrkhtiotk ovkronk stroih ohd stroih ot o pgiht (discusskd ih Skctigh ?.1), ?.1), uhits ga lkhntf pkr uhit lkhntf ork uskd agr ovkronk hgr`ol stroihs. Tfus, ovkronk hgr`ol stroihs ork rkpgrtkd witf tfk uhits ga ihcf/ihcf (gr c`/c`, `/`, ktc.). Ovkronk sfkor stroihs ork rkpgrtkd ih rodiohs. Ih rkpgrtihn kxpkri`khtol rksults ohd agr dkscrijihn vkry lornk dkagr`otighs, stroihs ork nivkh jy `kohs ga o pkrckhtonk cfohnk= tfot is, tfk rinft-fohd sidk ga Kquotigh (?.:;) ohd (?.:;) ohd (?.:1) is (?.:1) is `ultiplikd jy ?22 jkagrk tfk rksults ork rkpgrtkd. Tfus, o hgr`ol stroih ga 2.;% is kquol tg o stroih ga 2.22;. Tfk Nrkke lkttkr `u (μ) is gatkh uskd ih rkpgrtihn s`oll stroihs jkcousk it stohds agr `icrg ( `icrg (μ 0 ?2√1). Tfus, o stroih ga ?222 μih/ih rkprkskhts o hgr`ol stroih ga 2.22? ihcf pkr ihcf.
KVO@RLK ?.6 Twg jors ork cghhkctkd tg o rgllkr tfot slidks ih o slgt os sfgwh ih Ainurk ?.:2. ?.:2. Dktkr`ihk tfk stroih ih jor OR jor OR jy jy aihdihn tfk dkagr`kd lkhntf ga OR ga OR (o) (o) witfgut s`oll-stroih opprgxi`otigh ohd (j) witf s`oll-stroih opprgxi`otigh. J
:22 :22 `` 3;g
O ?.6. Ainurk ?.:2 Nkg`ktry ih Kxo`plk ?.6.
OHS.
(o)
λ OR
0 1>?.:
μ ` ` ⁄ ` `
(j)
λ OR
0 1>?.2?
μ ` ` ⁄ ` `
2.: `` δ p 0 2.: R
?1
?.1
?
Strk Strksss ohd ohd Stroi troih h
STZOIH TZOIH OT O RGIHT RGIHT OHD OHD STZOI STZOIH–D H–DISRL ISRLOCK OCK@KH @KHT T K_[OTIG K_[OTIGHS HS
Lkt u, v, ohd w jk displock`khts ih tfk x tfk x,, y, y, ohd z ohd z dirkctighs, dirkctighs, rkspkctivkly. Tfk khnihkkrihn stroih ot o pgiht coh jk visuolizkd jy cghsidkrihn tfk dkagr`otigh ga o cujk ga ihaihitksi`ol di`khsighs Δ x, x, Δ y, y, ohd Δ z , sfgwh ih Ainurk ?.:?. ?.:?. y
Δu
y
Δv Δ y Δ y
Δu
Δx
Δv
Δ x
x
x
Δ z
Δw
z
z
(o)
⎕ ς-- – γ ⎘ ⎐ : xy⎢
(j) y
y
⎕ ς-- – γ ⎘ ⎐ : zx⎢
Δw
Δ z
Δ y x Δv
Δ z
⎕ ς-- – γ ⎘ ⎐ : yz ⎢
Δ x Δw
x
Δu z
z
(d )
(c) (o) hgr`ol stroihs, (j (j) sfkor stroih γ xy, (c) sfkor stroih γ yz , ohd (d) sfkor stroih γ zx. Ainurk ?.:? Stroih cg`pghkhts= (o
Os tfk di`khsighs ga tfk cujk tkhd tg zkrg, wk nkt tfk `otfk`oticol dkaihitigh ga stroih ot o pgiht. Jkcousk tfk li` itihn gpkrotigh is ih o nivkh dirkctigh, wk gjtoih portiol dkrivotivks, hgt tfk tgtol dkrivotivks. Tfk dgujlk sujscript ih tfk cosk ga sfkor stroih ihdicotks tfk cfohnk ga ohnlk jktwkkh twg cggrdihotk lihks dkaihkd jy tfk sujscripts. ^k olsg rkcgrd dgujlk sujscripts agr hgr`ol stroih agr soek ga cghsistkhcy, os wkll os agr gur sujskqukht `otrix dkaihitigh ga stroih. Tfk `otfk`oticol dkaihitighs ga khnihkkrihn stroihs ot o pgiht ork os nivkh ih Kquotigh (?.:>o) tfrgunf (?.:>o) tfrgunf (?.:>d). (?.:>d).
λ xx
⎕ -Δ----u--⎘ Δ x ↚ 2⎐ Δ x⎢
0 li`
0
∊u ∊ x γ xy
λ yy 0
γ yx
0
γ xz
λ zz
Δw ∊w 0 li` ⎕ --------⎘ 0 ⎐ ⎢ z Δ Δ z ↚ 2 ∊ z
(?.:>o)
Δ u- + -Δ----v--⎘ ⎕ -----Δ x⎢
0
∊u + ∊v ∊ y ∊ x
(?.:>j)
Δ v- + -Δ----w ⎕ --------⎘ Δ y ⎢
0
∊v + ∊w ∊ z ∊ y
(?.:>c)
Δ w- + -Δ----u--⎘ ⎕ ------⎐ Δ x ↚ 2 Δ x Δ z ⎢
0
∊w + ∊u ∊ x ∊ z
(?.:>d)
0 li`
Δ x ↚ 2⎐ Δ y Δ y ↚ 2
γ yz 0 γ zy 0
γ zx
Δv ∊v 0 li` ⎕ -------⎘ 0 ⎐ ⎢ y Δ Δ y ↚ 2 ∊ y
li`
Δ y ↚ 2⎐ Δ z Δ z ↚ 2
0 li`
Δ z ↚ 2
Kquotigh (?.:>j) tfrgunf (?.:>j) tfrgunf (?.:>d) sfgw (?.:>d) sfgw tfot sfkor stroih is sy``ktric ohd tfot tfk grdkr ga sujscripts is i``otkriol. Tfk sy``ktry ga sfkor stroih `oeks ihtuitivk skhsk. Tfk cfohnk ga ohnlk jktwkkh tfk x tfk x ohd ohd y y dirkctighs dirkctighs is gjvigusly tfk so`k os jktwkkh tfk y tfk y ohd ohd x x dirkctighs. dirkctighs. Hgtk tfot tfk sinh ga sfkor stroih dkpkhds ghly upgh wfktfkr tfk ohnlk ihcrkosks gr dkcrkosks, hgt upgh tfk grdkr ih wficf wk cghsidkr x cghsidkr x ohd ohd y y.. Ainurk ?.:: sfgws ?.:: sfgws tfk khnihkkrihn stroih cg`pghkhts os oh khnihkkrihn stroih `otrix. Tfk `otrix is sy``ktric jkcousk ga tfk sy``ktry ga sfkor stroih. Tfus, ih tfrkk-di`khsighol prgjlk`s stroih fos hihk cg`pghkhts, jut ghly six ga tfk hihk cg`pghkhts ork ihdkpkhdkht jkcousk ga sy``ktry ga sfkor stroih.
λ xx γ xy γ xz γ yx λ yy γ yz γ zx γ zy λ zz Ainurk ?.:: Khnihkkrihn stroih `otrix.
Aihitk Diaakrkhck Opprgxi`otigh ga Stroihs
?>
^k skk tfot liek strkss, khnihkkrihn stroih fos twg sujscripts, wficf wguld skk` tg sunnkst tfot khnihkkrihn stroih is olsg o skcghd-grdkr tkhsgr liek strkss. Fg wkvkr uhliek strkss, khnihkkrihn stroih dgks hgt sotisa y ckrtoih cggrdihotk trohsagr`otigh lows, wficf wk will study ih Skctigh ?.6. ?.6. Fkhck it is hgt o skcghd-grdkr tkhsgr jut is rklotkd tg it, os sfgwh ih Kquotigh (?.:6). (?.:6). tkhs tkhsgr gr hgr` hgr`ol ol stro stroih ih 0 khni khnihk hkkr krihn ihn hgr hgr`o `oll stroi stroih h
tkhs tkhsgr gr sfk sfkor or stro stroih ih 0 khni khnihkk hkkri rihn hn sfk sfkor or str stroi oih/: h/:
(?.:6)
Ih Skctigh ?.6, ?.6, wk sfoll skk tfot tfk aoctgr ga ?/: tfot cfohnks tfk khnihkkrihn sfkor stroih t g tkhsgr sfkor stroih4 pkr`its tfk kxtkhsigh ga strkss trohsagr`otigh rksults tg stroih trohsagr`otigh. tfk displock`kht u is ghly o auhctigh ga x ga x Ui.k., Ui.k., u( x)P, x)P, tfkh tfk portiol dkrivotivk ih Kquotigh (?.:>o) jkcg`ks (?.:>o) jkcg`ks oh grdi Ia tfk hory dkrivotivk rksultihn ih tfk agllgwihn kquotigh= du (?.:9) λ xx 0 -----(x) dx
?.1.? .1.?
Rloh lohk Stro troih
Rlohk stroih is tfk gtfkr typk ga twg-di`khsighol prgjlk`s ih tfk study ga tfk `kcfohics ga `otkriols. Tfk diaakrkhcks jktwkkh plohk strkss ohd plohk stroih ork discusskd ih Skctigh :.:. :.:. Jy “twg-di`khsighol,‐ wk i`ply tfot ghk ga tfk cggrdihotks dgks hgt ploy o rglk ih tfk dkscriptigh ga tfk prgjlk`, tfus wk skt oll stroihs witf sujscript z sujscript z tg tg jk zkrg os sfgwh ih tfk stroih `otrix ih Ainurk ?.:3. ?.:3. Oltfgunf agur cg`pghkhts ga stroih ork hkkdkd ih plohk stroih, jkcousk ga tfk sy``ktry ga sfkor stroih, ghly tfrkk ga tfk agur cg`pghkhts ork ihdkpkhdkht.
λ xx γ xy γ yx λ yy 2
2 2
2
2
Ainurk ?.:3 Rlohk stroih `otrix.
Tfk ossu`ptigh ga plohk stroih is gatkh `odk ih oholyzihn vkry tfice jgdiks sucf os pgihts orguhd tuhhkls gr `ihk sfoats, gr o pgiht ih tfk `iddlk ga o tfice cylihd kr sucf os o suj`orihk full. Ih tfice jgdiks, it is o rnukd tfot tg `gvk, o pgiht `ust pusf o lgt ga `otkriol ih tfk tficehkss dirkctigh5 fkhck tfk stroihs coh jk kxpkctkd tg jk s`oll ih tfk tficehkss dirkctigh. Tfus, tfk stroih ih tfk tficehkss dirkctigh is hgt zkrg jut is s`oll khgunf tg hknlkct. Rlohk stroih is o `otfk`oticol opprgxi`otigh `odk tg si`pliay tfk oholysis.
KVO@RLK ?.9 O jgdy uhdkr opplikd lgods ohd jgdy agrcks prgduck tfk displock`kht aikld u02
:
:
v 0 Ex ( y – z ) + Eoxz
w0
√ : Ex yz – Eo xy
wfkrk u, v, ohd w ork displock`khts ih tfk x tfk x,, y, y, ohd z ohd z dirkctighs, dirkctighs, rkspkctivkly. Dktkr`ihk oll tfk stroih cg`pghkhts ot x ot x 0 :o :o ih tkr`s ga E ga E , o, y, y, ohd z ohd z . OHS.
?.>
λ xx
0 2 5
λ yy
0 4 Eo y 5
λ zz
0 – 4 Eo y 5
γ xy
:
:
0 E ( y – z ) + Eo z 5
γ yz
0 – 6 Eo z 5
γ xz
0 – Ey U : z + o P
AIHITK AIHITK DIAAKZ DIAAKZKHC KHCK K ORRZ ORRZGVI GVI@OTI @OTIGH GH GA GA STZO STZOIHS IHS
^fkh o dkrivotivk is opprgxi`otkd jy usihn o aihitk lkhntf ih plock ga ihaihitksi`ol lkhntf, wk coll it o aihitk diaakrkhck opprgxi`otigh. Arg` colculus wk ehgw tfot tfk dkrivotivk ga o auhctigh is tfk slgpk ga tfk tohnkht tg tfk curvk dkaihkd jy tfk auhctigh. Ih colculotihn tfis slgpk ga tfk tohnkht jy `kohs ga aihitk lkhntfs, dg wk • [sk tfk lkhntf lkhntf ih arght ga ga tfk pgiht7 pgiht7 • [sk tfk lkhntf lkhntf jkfihd jkfihd tfk tfk pgiht7 • [sk tfk lkhntf lkhntf ih arght ohd jkfihd jkfihd tg colculotk colculotk oh ovkronk voluk7 voluk7 Tfk cfgick `odk ih ohswkrihn tfksk tfrkk qukstighs lkods tg tfrkk `ktfgds= agrword diaakrkhck, joceword diaakrkhck, ohd ckhtrol diaakrkhck. Tg distihnuisf o`ghn tfk tfrkk `ktfgds, cghsidkr o plgt ga u vs. x, vs. x, wficf wficf wk usk tg aihd tfk opprgxi`otk voluk ga stroih ?.:4. Tfk stroih ot tfk pgiht is tfk slgpk ga tfk tohnkht tg tfk octuol auhctigh ot pgiht i. λ xx ot pgiht i ih Ainurk ?.:4. • Agrword diaakrkhck opprgxi`otks tfk slgpk ga tfk tohnkht jy usihn tfk pgiht ofkod ga pgiht i os
( λ xx ) i
u i + ? – u i 0 ------------------ x i + ? – x i
(?.32o)
• Joceword diaakrkhck opprgxi`otks tfk slgpk ga tfk tohnkht jy usihn tfk pgiht jkfihd i os 4. O cg`poct hgtotigh sfgwihn tfk rklotigh ga tkhsgr stroihs ohd tfk dkrivotivks is
λ im 0 ( ∊ ui ⁄ ∊ x m + ∊ u m ⁄ ∊ x i ) ⁄ :
x?, x:, ohd x ohd x3 ork tfk x tfk x,, y, y, ohd z ohd z cggrdihotks, cggrdihotks, ohd u?, u:, ohd u3 ork tfk displock`khts u, v, ohd w, rkspkctivkly.
, wfkrk i ohd m ohd m coh coh jk ?, :, gr 3,
?
?6
Strk Strksss ohd ohd Stroi troih h
( λ xx ) i
ui – u i √ ? 0 ------------------ xi – x i √ ?
(?.32j)
• Ckhtrol diaakrkhck toeks tfk ovkronk voluk jy usihn tfk pgiht ofkod ohd jkfihd os ? u i + ? – u i u i – u i √ ? 0 -- ------------------- + ------------------: x i + ? – x i x i – x i √ ?
( λ xx )i
(?.32c)
Tfk ckhtrol diaakrkhck opprgxi`otigh jiskcts tfk slgpk ga tfk lihks tfot yikld tfk agrword ohd joceword diaakrkhcks. Os coh jk skkh arg` Ainurk ?.:4, ?.:4, tfk jkst opprgxi`otigh is arg` tfk ckhtrol diaakrkhck. Jut ia tfk pgiht is gh tfk lkat-`gst jguhdory, tfkrk is hg pgiht jkagrk it ohd ghly o agrword diaakrkhck opprgxi`otigh coh jk usk d. Si`ilorly, ia tfk pgiht is rinft-`gst jguhdory, jguhdory, ghly o joceword diaakrkhck opprgxi`otigh coh jk uskd.
u
Ckhtrol Diaakrkhck opprgxi`otigh ga tohnkht
ui+? Octuol auhctigh ui
Agrword Diaakrkhck opprgxi`otigh ga tohnkht
Octuol tohnkht Joceword Diaakrkhck opprgxi`otigh ga tohnkht
ui-?
xi-?
xi
x
xi+?
Ainurk ?.:4 Tfk tfrkk aihitk diaakrkhck `ktfgds.
KVO@RLK ?.?2 Tfk displock`khts u ohd v ih tfk x tfk x ohd ohd y y dirkctighs, dirkctighs, rkspkctivkly, wkrk `kosurkd jy @giré ihtkrakrg`ktryo. Ainurk ?.:; nivks ?.:; nivks tfk displock`khts ga hihk pgihts gh tfk jgdy. Tfk nrid spocihn is f 0 2.222; ``. Dktkr`ihk Dktkr`ihk tfk tfk stroihs stroihs λ xx, λ yy, ohd γ xy ot pgiht ; usihn (o) agrword diaakrkhck, (j) joceword diaakrkhck, ohd (c) ckhtrol diaakrkhck ohd tfk doto ih Tojlk ?.3. ?.3. Tojlk ?.3 Displock`khts agr Kxo`plk ?.?2
y
>
6
9
f 4
;
1
f ?
: f
3
x
f
?.?2. Ainurk ?.:; (o) Nrid agr Kxo`plk ?.?2.
Rgiht
u (μ``)
v (μ``)
?
2.222
2.2222
:
2.1:;
– 2.3?:;
3
?.;22
– 2.;222
4
– 2.;22
– 2.;1:;
;
2.:;2
– ?.?:;2
1
?.:;2
– ?.;1:;
>
– ?.:;2
– ?.:;2
6
– 2.3>;
– :.21:;
9
2.>;2
– :.>;22
OHS.
λ xx 0 :222 μ ` ` ⁄ ` ` 5 λ yy 0 – ?6>; μ ` ` ⁄ ` ` 5 γ xy (j) λ xx 0 ?;22 μ ` ` ⁄ ` ` 5 λ yy 0 – ?1:; μ ` ` ⁄ ` ` (c) λ xx 0 ?>;2 μ ` ` ⁄ ` ` 5 λ yy 0 – ?>;2 μ ` ` ⁄ ` ` 5 γ xy (o)
0 – :?:; 5
γ xy
μ ro d
0 – ?6>;
0 – :222
μ ro d
μ ro d
o. @giré `ktfgds `kosurk displock`khts displock`khts ga pgihts gh o nrid. Ih `giré ihtkrakrg`ktry, ihtkrakrg`ktry, displock`khts ga nrid pgihts ork `kosurkd jy `kohs ga tfk ihtkrakrkhck ga linft. Skk Rgst U?993P agr odditighol dktoils.
?.6 ?.6
STZOI TZOIH H TZOHS ZOHSAG AGZ@ Z@OT OTIG IGH H
Ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols, tfk stroihs (λhh, λtt , ohd γht ) ih tfk cggrdihotk systk` h-t - z z sfgwh sfgwh Ainurk ?.??(c), ?.??(c), wkrk rklotkd tg tfk stroihs (λ xx, λ yy, ohd γ xy) ih tfk Cortksioh cggrdihotk systk` os sfgwh jy Kquotighs (?.39). Tfk ohnlk tfk x oxis oxis ohd is cghsidkrkd tg jk pgsitivk ih tfk cguhtkrclgcewisk dirkctigh. ζ is `kosurkd arg` tfk x
λhh
0
λ xx cgs :ζ + λ yy sih :ζ + γ xy sih ζ
cgs
ζ
(?.3?o)
Stroih Trohsagr`otigh
λtt γ ht
λ xx sih :ζ + λ yy cgs :ζ √ γ xy sih ζ
0
0 – : λ xx sih
ζ
cgs
ζ + : λ yy sih ζ
cgs
?9
ζ
(?.3?j)
ζ + γ xy ( cgs :ζ – sih :ζ )
(?.3?c)
cgs
Tfk ojgvk kquotighs ork si`ilor tg tfk strkss trohsagr`otigh rklotighs UKquotighs UKquotighs (?.3o) tfrgunf (?.3o) tfrgunf (?.3c)P (?.3c)P witf tfk diaakrkhck tfot tfk cgkaaicikht ga sfkor strkss tkr` is twick tfk cgkaaicikht ga tfk sfkor stroih tkr`. Tfis diaakrkhck kxists jkcousk wk or k usihn khnihkkrihn stroih ihstkod ga tkhsgr stroih. Ia wk cghvkrt tfk khnihkkrihn stroihs tg tkhsgr stroihs, jy dividihn oll khnihkkrihn sfkor stroihs jy :, tfkh tfk rksults agr tfk `otrix `ktfgd ih strkss trohsagr`otigh coh jk odoptkd agr stroih trohsagr`otigh. Ainurk ?.:; sfgws ?.:; sfgws tfk tkhsgr stroih `otrix tfot is gjtoihkd arg` tfk khnihkkrihn stroihs.
Uλ P
0
λ xx λ yx 0 γ yx ⁄ : λ zx 0 γ zx ⁄ :
λ xy 0 γ xy ⁄ : λ yy λ zy 0 γ zy ⁄ :
λ xz 0 γ xz ⁄ : λ yz 0 γ yz ⁄ : λ zz
Ainurk ?.:; Tkhsgr stroih `otrix gjtoihkd arg` oh khnihkkrihn stroih `otrix.
Cghsidkr twg pkrpkhdiculor dirkctighs h ohd t witf witf {h {h} ohd {t {t } rkprkskhtihn tfk uhit vkctgrs ih tfksk dirkctighs, rkspkctivkly. Oholgngus tg Kquotigh (?.>) wk (?.>) wk gjtoih tfk stroih trohsagr`otigh kquotighs agr tfk hgr`ol ohd sfkor stroih ih tfk h-t dirkc dirkctighs os sfgwh jklgw.
λ hh
0
{ h } T U λ P { h }
λht
0
{ t } T U λ P { h }
λtt
0
{ t } T U λ P { t }
(?.3:)
It sfguld jk hgtkd tfot tfk sfkor stroih λht ih Kquotigh (?.3:) is (?.3:) is tkhsgr sfkor stroih, wficf `ust jk `ultiplikd jy : tg gjtoih tfk khnihkkrihn sfkor stroih γht . Tfk stroih vkctgr ih tfk dirkctigh ga h is oholgngus tg tfk strkss vkctgr ohd coh jk dkaihkd os sfgwh ih Kquotigh (?.33). (?.33).
{ K } ?.6. ?.6.? ?
Uλ P{ h }
0
(?.33)
Rrih Rrihci cipo poll Stro Stroih ihss
^k rkcoll tfk agllgwihn dkaihitighs arg` tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriol.Tfk prihcipol cggrdihotk dirkctighs agr stroih ork tfk oxks ih wficf tfk sfkor tfk sfkor stroih is zkrg. zkrg. Hgr`ol stroihs ih tfk prihcipol dirkctighs ork collkd prihcipol stroihs.Tfk nrkotkst .Tfk nrkotkst prihcipol prihcipol stroih is collkd prihcipol stroih ?. Tfk ohnlks tfk prihcipol oxis `oeks witf tfk nlgjol cggrdihotk systk` ork collkd prihcipol ohnlks. Ih plohk stroih, tfk sfkor stroihs witf sujscripts z sujscripts z ork ork zkrg, wficf i`pliks tfot ghk prihcipol dirkctigh liks olghn tfk z tfk z oxis. Ih plohk stroih prgjlk`s wk will lojkl tfk z tfk z dirkctigh dirkctigh os prihcipol dirkctigh 3. Tfkrk ork tfrkk prihcipol stroihs ot o pgiht, wficf ork lojklkd os λ?, λ:, ohd λ3. Tfk prihcipol stroihs λ p ork tfk kinkhvoluks ga tfk tkhsgr stroih `otrix, ohd tfk prihcipol dirkctighs { p} p} ork tfk kinkhvkctgrs ga tfk stroih `otrix, wficf ork aguhd ih kxoctly tfk so`k `ohhkr os tfk prihcipol strkssks ohd tfk ossgciotkd prihcipol dirkctighs. Tfk cforoctkristic kquotigh oholgngus tg Kquotigh (?.:2) is (?.:2) is
λ p3 – I ? λ p: + I : λ p – I 3
0 2
(?.34)
ohd its rggts ga tfk cforoctkristic yikld tfk prihcipol stroihs. Tfk cgkaaicikhts I cgkaaicikhts I ?, I :, ohd I ohd I 3 ork collkd tfk stroih ihvoriohts jkcousk tfkir voluk is ihdkpkhdkht ga cggrdihotk trohsagr`otigh. Hgtick tfot tfk sinh ih tkr`s cghtoihihn tfk gdd strkss ihvoriohts is hknotivk. Tfk stroih ihvoriohts I ihvoriohts I ?, I :, ohd I ohd I 3, oholgngus tg strkss ihvoriohts ih Kquotigh (?.:?) ork (?.:?) ork I ? 0
λ xx + λ yy + λ zz
I : 0
λ xx λ xy λ yx λ yy
+
λ yy λ yz λ zy λ zz
+
λ xx λ xz λ zx λ zz
I 3 0
λ xx λ xy λ xz λ yx λ yy λ yz λ zx λ zy λ zz
(?.3;)
Tfk stroih ihvoriohts ih tkr`s ga prihcipol stroihs ork I ? 0
λ ? + λ: + λ3
I : 0
λ?
2
2
λ:
+
λ:
2
2
σ3
+
λ?
2
2
λ3
0
λ? λ : + λ: λ3 + λ 3 λ?
I 3 0
λ?
2
2
2
λ:
2
2
2
λ3
0
λ?λ: λ3
(?.31)
Tfk stroih ihvoriohts, porticulorly tfk airst ohd tfk tfird, prgvidk o quice ohd kosy cfkce gh tfk lojgrigusly colculotkd prihcipol stroihs. Tfk `oxi`u` khnihkkrihn sfkor stroih is rklotkd tg tfk diaakrkhck ih prihcipol stroihs ohd coh jk kxprksskd os
γ `ox
λ? – λ : λ: – λ 3 , λ3 – λ? ⎘ --------- 0 `ox⎕ ----------------------------- ⎢ ⎐ : , --------------: : :
(?.3>)
:2
?
Strk Strksss ohd ohd Stroi troih h
KVO@RLK ?.?? Tfk khnihkkrihn stroihs ot o pgiht gh o juildihn aro`k wkrk dktkr`ihkd jy `kohs ga tfk aihitk klk`kht `ktfgdo os λ xx 0 122 μ,
λ yy 0 422 μ, λ zz 0 :22 μ, γ xy 0 622 μ, γ yz 0 422 μ, γ zx 0 122 μ. Dktkr`ihk (o) tfk hgr`ol stroih olghn o lihk grikhtkd ot 42g, 12g, ohd 11.:g witf tfk x tfk x,, y, ohd z oxks, oxks, rkspkctivkly, (j) tfk prihcipol stroihs ohd tfk `oxi`u` khnihkkrihn sfkor stroih, ohd (c) y, ohd z prihcipol stroih dirkctigh ?. OHS.
λhh 0 ?2;> μ 5 λ ? 0 ?21: μ 5 λ : ζ x 0 4:.6 ² 55ζ y 0 ;1 ² 5ζ z 0 1>.4 ²
μ 5 λ 3 0 3>.1 μ 5 γ`ox 0 ?2:4.> μ 5 gr ζ x 0 ?3>.: ² 5ζ y 0 ζ y 0 ?:4 ² 5ζ z 0 ??:.1 ²
0 ?22
o. Aihitk klk`kht `ktfgd is o hu`kricol hu`kricol `ktfgd agr sglvihn `kcfohics ohd gtfkr khnihkkrihn prgjlk`s. prgjlk`s. Skk Zkddy U?993P agr odditighol dktoils.
?.9
CLGS[ZK
Ih tfis cfoptkr wk sow tfot strkss gh o suraock is oh ihtkrhol distrijutkd agrck systk` tfot is o vkctgr quohtity, wfilk strkss ot o pgiht is o skcghd-grdkr tkhsgr wfgsk sinh is dktkr`ihkd arg` tfk dirkctigh ga tfk gutword hgr`ol ga oh i`onihory suraock tfrgunf tfk pgiht ohd tfk dirkctigh ga tfk ihtkrhol agrck. ^k olsg sow tfot khnihkkrihn stroih, wficf is hgt o skcghd-grdkr tkhsgr, `oy jk trohsagr`kd ihtg o skcghd-grdkr tkhsgr jy dividihn oll sfkor stroihs ot tfot pgiht jy o aoctgr ga :. Rrihcipol strkssks ohd stroihs ork tfk kinkhvoluks, ohd prihcipol dirkctighs ork tfk kinkhvkctgrs, ga tfk strkss ohd tkhsgr stroih `otricks. ^k sow tfot tfk prihcipol strkssks ohd stroihs ot o pgiht ork uhiquk, wficf i`pliks tfot tfk cgkaaicikhts cgkaaicikhts ga tfk cforoctkristic cforoctkristic kquotigh ork uhiquk. uhiquk. Tfk cgkaaicikhts ga tfk cforoctkristic kquotighs ork tfk strkss ohd stroih ihvoriohts tfot dg hgt dkpkhd upgh tfk cggrdihotk systk`—oh gjskrvotigh wk uskd tg cfkce gur colculotighs ga prihcipol strkssks ohd stroihs. ^k sow tfot s`oll-stroih opprgxi`otigh is o lihkor rklotighsfip jktwkkh displock`khts ohd stroihs tfot is volid agr stroihs lkss tfoh ?%. S`oll-stroih opprgxi`otighs ollgw sinhiaicoht si`pliaicotigh, ihcludihn colculotigh ga hgr`ol stroih, ih wficf wk opprgxi`otk tfk octuol dkagr`otigh jy tfk cg`pghkht ga tfk dkagr`otigh ih tfk grinihol dirkctigh ga tfk lihk klk`kht, ohd sfkor stroih, ih wficf tfk sihk ohd tohnkht auhctighs ga sfkor stroih ork opprgxi`otkd jy tfkir ornu`khts. Tfk dkrivotivks ga displock`khts ih s`oll stroihs coh jk opprgxi`otkd jy tfk aihitk diaakrkhck `ktfgd. Ckhtrol diaakrkhck sfguld jk uskd wfkh pgssijlk jkcousk tfis `ktfgd prgducks `grk occurotk rksults. Fgwkvkr agrword ohd joceword diaakrkhcks `oy fovk tg jk uskd hkor jguhdoriks.
:
@otkriol Dkscriptigh
LKOZHIHN GJMKCTIQK ?. Tg uhdkrstohd tfk dkaihitighs ih ohd diaakrkhcks jktwkkh vorigus lihkor `otkriol `gdkls. :. Tg uhdkrstohd uhdkrstohd tfk stotk`khts ohd tfk opplicotig opplicotighs hs ga aoilurk tfkgriks. tfkgriks. 3. Tg uhdkrstohd tfk cghckpts ohd opplicotighs ga strkss cghckhtrotigh aoctgr ohd strkss ihtkhsity aoctgr aoctgr ih oholysis ohd dksinh.
( o) tkhsigh ohd (j (j) tgrsigh. (Cgurtksy (Cgurtksy ga Rrgakssgr I. @iseignlu.) @iseignlu. ) Ainurk :.? @ocfihks agr tkstihn (o
O `otkriol coh jk dkscrijkd quolitotivkly o hd quohtitotivkly. _uolitotivk dkscriptigh jy odmkctivks sucf os klostic, lihkor, ductilk, ohd tgunf agr`s tfk khnihkkrihn lohnuonk ga `otkriol dkscriptigh. _uohtitotivk dkscriptighs ga o `otkriol ork tfk kquotighs rklotihn strkssks ohd stroihs tfot ork kstojlisfkd kxpkri`khtolly, usuolly arg` o tkhsigh gr tgrsigh tkst `odk gh `ocfihks sucf os tfgsk sfgwh ih Ainurk :.?. :.?. O `otkriol `gdkl, olsg collkd o cghstitutivk `gdkl, is tfk quolitotivk ohd quohtitotivk opprgxi`otigh ga `otkriol jkfovigr. Tfk hu`jkr ga poro`ktkrs gr `otkriol cghstohts tg jk kxpkri`khtolly dktkr`ihkd dkpkhds upgh tfk `otkriol `gdkl cfgskh tg rklotk strkssks ohd stroihs. Ih tfis cfoptkr wk discuss skvkrol lihkor `otkriol `gdkls. Sg`k ihklostic ohd hghlihkor `otkriol `gdkls ork discusskd ih Cfoptkr ;. Ih o tkhsigh tkst, tfkrk is ghly ghk strkss cg`pghkht. Fgw dg wk opply tfk voluk ga aoilurk strkss arg` tfk tkhsigh tkst tg o stotk ga strkss ih wficf oll hihk strkss c g`pghkhts ork prkskht7 Tfk ohswkr tg tfis qukstigh is prgvidkd jy aoilurk tfkgriks, sg`k ga wficf wk will stud y ih tfis cfoptkr. Tfk klk`khtory structurol tfkgriks studikd ih oh ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols nivk agr`ulos gjtoihkd jy `oeihn `ohy si`pliayihn si`pliayihn ossu`ptighs. ossu`ptighs. Ih rknighs rknighs ga suddkh suddkh cfohnks cfohnks ih nkg`ktriks nkg`ktriks ohd ohd croces, tfk josic ossu`ptighs ossu`ptighs ga tfk klk`khtory tfkgriks ork hg lghnkr volid. Coh wk kxtropglotk rksults arg` gur agr`ulos ihtg tfksk rknighs7 ^k will study twg cghckpts jy wficf wficf wk coh kxtropglotk kxtropglotk tfk rksults rksults ga klk`khtory klk`khtory tfkgry agr usk ih tfksk rknighs hkor suddkh cfohnks ih nkg`ktry ohd croces. Tfksk twg cghckpts ork strkss ork strkss cghckhtrotigh aoctgr ohd strkss ohd strkss ihtkhsity aoctgr. aoctgr.
:.? :.?
LIHK LIHKOZ OZ @OTK @OTKZI ZIOL OL @GDK @GDKL L
Ih tfk `kcfohics ga `otkriols, tfk kxpkri`khtol rklotighsfip jktwkkh strkssks ohd stroihs is ihcgrpgrotkd ihtg tfk lgnicol aro`kwgre ga `kcfohics tg prgduck agr`ulos agr tfk oholysis ohd dksinh ga structurol `k`jkrs. Ia o `otkriol `gdkl dgks hgt ait tfk kxpkri`khtol doto wkll, tfkrk will jk o finf dknrkk ga krrgr ih tfk tfkgrkticol prkdictighs. Ykt o `otkriol `gdkl tfot aits tfk kxpkri`khtol doto vkry occurotkly `oy jk sg cg`plkx tfot hg oholyticol `gdkl (tfkgry) coh jk juilt. Tfus, tfk cfgick ga `otkriol `gdkl is dictotkd jgtf jy tfk kxpkri`khtol doto ohd jy tfk occurocy hkkds ga tfk oholysis. :?
::
:
@otkriol Dkscriptigh
Tfk si`plkst `otkriol `gdkl is o lihkor rklotighsfip jktwkkh strkssks ohd stroihs. ^itf hg odditighol ossu`ptighs, tfk lihkor rklotighsfip ga tfk six stroih cg`pghkhts tg six strkss cg`pghkhts coh jk writtkh os
λ xx λ yy λ zz γ yz γ zx γ xy
0 0 0 0 0 0
C ?? σ xx
+ C ?: σ yy + C ?3 σ zz + C ?4 τ yz + C ?; τ zx + C ?1 τ xy C :? σ xx + C :: σ yy + C :3 σ zz + C :4 τ yz + C :; τ zx + C :1 τ xy C 3? σ xx + C 3: σ yy + C 33 σ zz + C 34 τ yz + C 3; τ zx + C 31 τ xy C 4? σ xx + C 4: σ yy + C 43 σ zz + C 44 τ yz + C 4; τ zx + C 41 τ xy C ;? σ xx + C ;: σ yy + C ;3 σ zz + C ;4 τ yz + C ;; τ zx + C ;1 τ xy C 1? σ xx + C 1: σ yy + C 13 σ zz + C 14 τ yz + C 1; τ zx + C 11 τ xy
(:.?o)
Tfk ojgvk kquotigh coh jk writtkh ih `otrix agr` os
⎮ λ xx ⎥ ⎭ ⎭ ⎭ λ yy ⎭ ⎭ ⎭ ⎭ λ zz ⎭ 0 ⎦ ⎬ ⎭ γ yz ⎭ ⎭γ ⎭ ⎭ zx ⎭ ⎭γ ⎭ ⎩ xy ⎠
C ?? C ?: C ?3 C ?4 C ?; C ?1 C ?: C :: C :3 C :4 C :; C :1 C ?3 C :3 C 33 C 34 C 3; C 31 C 4? C 4: C 43 C 44 C 4; C 41 C ;? C ;: C ;3 C ;4 C ;; C ;1 C 1? C 1: C 13 C 14 C 1; C 11
⎮ σ xx ⎥ ⎭ ⎭ ⎭ σ yy ⎭ ⎭ ⎭ ⎭ σ zz ⎭ ⎦ ⎬ ⎭ τ yz ⎭ ⎭τ ⎭ ⎭ zx ⎭ ⎭τ ⎭ ⎩ xy ⎠
(:.?j)
Tfk `otrix C im ih Kquotigh (:.?j) is (:.?j) is collkd tfk cg`pliohck `otrix. Kquotigh (:.?j) i`pliks (:.?j) i`pliks tfot wk hkkd 31 `otkriol cghstohts tg dkscrijk tfk `gst nkhkrol lihkor rklotighsfip jktwkkh strkssks ohd stroihs. Fgwkvkr, it coh jk sfgwh tfot tfk `otrix agr`kd jy tfk cghstohts C im is o sy``ktric `otrix5 tfot is, C im 0 C ohd m coh coh jk ohy hu`jkr arg` ? tg 1. Tfis sy``ktry mi, wfkrk i ohd m coh jk prgvkd jy usihn tfk rkquirk`kht tfot t fk stroih khkrny—khkrny stgrkd ih o `otkriol gwihn tg dkagr`otigh, `ust olwoys jk pgsitivk. Tfk prgga ga sy``ktry is jkyghd tfk scgpk ga tfis jgge. Stroih khkrny is discusskd ih Skctigh >.?. >.?. Tfk sy``ktry rkquirk`kht rkducks tfk hu`jkr ga ihdkpkhdkht cghstohts tg :? agr tfk `gst nkhkrol lihkor rklotighsfip jktwkkh strkss ohd stroih. Kquotigh (:.?j) prksuppgsks (:.?j) prksuppgsks tfot tfk rklotigh jktwkkh tfk strkss ohd tfk stroih ih tfk x tfk x dirkctigh is diaakrkht arg` tfk rklotigh ga strkss ohd stroih ih tfk y tfk y gr gr z z dirkctighs. dirkctighs. Oltkrhotivkly stotkd, Kquotigh (:.?j) i`pliks (:.?j) i`pliks tfot ia wk opply o agrck (strkss) ih tfk x tfk x dirkctigh dirkctigh ohd gjskrvk tfk dkagr`otigh (stroih), tfis dkagr`otigh will jk diaakrkht arg` tfk dkagr`otigh tfot wguld jk prgduckd ia wk opplikd tfk so`k agrck ih tfk y tfk y dirkctigh. dirkctigh. Tfis pfkhg`khgh is hgt gjskrvojlk jy tfk hoekd kyk agr `gst `ktols, jut agr `ktols ot tfk crystol-sizk lkvkl, tfk hu`jkr ga cghstohts hkkdkd tg dkscrijk tfk strkss–stroih rklotighsfip dgks dkpkhd upgh tfk crystol structurk. Tfus wk `ust ose wfktfkr wk ork cghductihn tfk oholysis ot kyk lkvkl gr ot crystol-sizk lkvkl. Agr oholysis ot kyk lkvkl, wk ovkronk tfk i`poct ga tfk crystol structurk, tg gjtoih tfk si`plkst `otkriol dkscriptigh. Oh isgtrgpic `otkriol fos strkss–stroih rklotighsfips tfot ork ihdkpkhdkht ga tfk grikhtotigh ga tfk cggrdihotk systk` ot o pgiht. Oh ohisgtrgpic `otkriol is o `otkriol tfot is hgt isgtrgpic. Tfk `gst nkhkrol ohisgtrgpic `otkriol rkquirks :? ihdkpkhdkht `otkriol cghstohts tg cghstohts tg dkscrijk its lihkor strkss–stroih rklotighsfips. Oh isgtrgpic j isgtrgpic jgdy gdy rkquirks ghly twg ihdkpkhdkht `otkriol cghstohts tg dkscrijk its lihkor strkss–stroih rklotighsfips. Sg`k ga tfk aoctgrs tfot ihalukhck wfktfkr wk trkot o `otkriol os isgtrgpic gr ohisgtrgpic ork tfk dknrkk ga diaakrkhck ih `otkriol prgpkrtiks witf grikhtotigh, tfk scolk ot wficf tfk oholysis is jkihn cghductkd, ohd tfk eihd ga ihagr`otigh tfot is dksirkd arg` tfk oholysis. Fg`gnkhkity is ohgtfkr opprgxi `otigh tfot is gatkh uskd tg dkscrijk o `otkriol jkfovigr. O `otkriol is soid tg jk fg`gnkhkgus ia tfk `otkriol prgpkrtiks ork tfk so`k ot oll pgihts gh tfk jgdy. Oltkrhotivkly, ia tfk `otkriol cghstohts C im ork auhctighs ga tfk cggrdihotks x cggrdihotks x,, y, y, gr z gr z , tfk `otkriol is collkd hghfg`gnkhkgus. @gst `otkriols ot tfk otg`ic lkvkl, crystollihk lkvkl, gr nroih-sizk lkvkl ork hghfg`gnkhkgus. Tfk trkot`kht ga o `otkriol os fg`gnkhkgus gr hghfg`gnkhkgus dkpkhds ghck `grk gh tfk typk ga ihagr`otigh tfot is rkquirkd arg` tfk oholysis. Fg`gnkhizotigh ga `otkriol prgpkrtiks is tfk prgckss ga ovkronihn diaakrkht `otkriol prgpkrtiks jy oh gvkroll `otkriol prgpkrty. Ohy jgdy coh jk trkotkd os o fg`gnkhkgus jgdy ia tfk scolk ot wficf oholysis is cghductkd is `odk suaaicikhtly lornk Isgtrgpic–fg`gnkhkg Isgtrgpic–fg`gnkhkgus, us, ohisgtrgpic–fg`gnkh ohisgtrgpic–fg`gnkhkgus, kgus, isgtrgpic–hghfg`gnk isgtrgpic–hghfg`gnkhkgus, hkgus, ohd ohisgtrgpic–hghfg`gnkhkgus ork oll pgssijlk dkscriptighs ga `otkriol jkfovigr. Tfkrk ork 3? typks ga crystol. Crystollihk jgdiks coh jk nrgupkd ihtg clossks agr tfk purpgsk ga dkaihihn tfk ihdkpkhdkht `otkriol cghstohts hkkdkd ih tfk lihkor strkss–stroih rklotighsfip. Ih jktwkkh tfk isgtrgpic `otkriol ohd tfk `gst nkhkrol ohisgtrg pic `otkriol, tfkrk ork `otkriol nrgups ga skvkrol typks, sg`k ga wficf ork dis cusskd jrikaly ih Skctighs:.?.? Skctighs :.?.? tfrgunf tfrgunf :.?.4. :.?.4.
:.?. :.?.? ?
@ghg @ghgcl clih ihic ic @otk @otkri riol olss
Tfk xy Tfk xy plohk plohk is tfk plohk ga sy``ktry ih `ghgclihic `otkriols. Tfis i`pliks tfot tfk strkss–stroih rklotighsfip is tfk so`k ih tfk pgsitivk z pgsitivk z ohd ohd hknotivk z hknotivk z dirkctighs. dirkctighs. Ih `otrix agr`, tfk strkss–stroih rklotighsfip agr `ghgclihic `otkriols is nivkh jy Kquotigh (:.:). (:.:). Tfk kquotigh sfgws tfot o `ghgclihic `otkriol rkquirks ?3 ihdkpkhdkht `otkriol cghstohts ih cghstohts ih tfrkk di`khsighs.
Lihkor @otkriol @gdkl
⎮ λ xx ⎥ ⎭ ⎭ ⎭ λ yy ⎭ ⎭ ⎭ ⎭ λ zz ⎭ 0 ⎦ ⎬ ⎭ γ yz ⎭ ⎭γ ⎭ ⎭ zx ⎭ ⎭γ ⎭ ⎩ xy ⎠ :.?. :.?.: :
C ??
C ?:
C ?3
2
2
C ?1
C ?:
C ::
C :3
2
2
C :1
C ?3
C :3
C 33
2
2
C 31
2
2
2
C 44
C 4;
2
2
2
2
C 4;
C ;;
2
C ?1 ?1
C :1 :1
C 31
2
2
C 11
⎮ σ xx ⎥ ⎭ ⎭ ⎭ σ yy ⎭ ⎭ ⎭ ⎭ σ zz ⎭ ⎦ ⎬ ⎭ τ yz ⎭ ⎭τ ⎭ ⎭ zx ⎭ ⎭τ ⎭ ⎩ xy ⎠
:3
(:.:)
Grtf Gr tfgt gtrg rgpi picc @otk @otkri riol olss
Grtfgtrgpic `otkriols fovk twg oxis ga sy``ktry. Tfot is, ia wk rgtotk o so`plk jy 92g ojgut tfk x tfk x gr gr tfk y tfk y oxis, oxis, wk will gjtoih tfk so`k strkss–stroih rklotigh. Ih `otrix agr`, tfk strkss–stroih rklotighsfip agr grtfgtrgpic `otkriols is nivkh jy Kquotigh (:.3). (:.3).
⎮ λ xx ⎥ ⎭ ⎭ ⎭ λ yy ⎭ ⎭ ⎭ ⎭ λ zz ⎭ ⎦ ⎬ ⎭ γ yz ⎭ ⎭γ ⎭ ⎭ zx ⎭ ⎭γ ⎭ ⎩ xy ⎠
0
C ??
C ?:
C ?3
2
2
2
C ?:
C ::
C :3
2
2
2
C ?3
C :3
C 33
2
2
2
2
2
2
C 44
2
2
2
2
2
2
C ;;
2
2
2
2
2
2
C 11
⎮ σ xx ⎥ ⎭ ⎭ ⎭ σ yy ⎭ ⎭ ⎭ ⎭ σ zz ⎭ ⎦ ⎬ ⎭ τ yz ⎭ ⎭τ ⎭ ⎭ zx ⎭ ⎭τ ⎭ ⎩ xy ⎠
(:.3)
Kquotigh (:.3) sfgws (:.3) sfgws tfot oh grtfgtrgpic `otkriol rkquirks hihk ihdkpkhdkht cghstohts ih cghstohts ih tfrkk di`khsighs. Agr grtfgtrg pic `otkriols, tfk hgr`ol stroihs ork hgt oaakctkd jy tfk sfkor strkssks, ohd tfk sfkor stroihs ork h gt oaakctkd jy tfk hgr`ol strkssks, wficf is hgt tfk cosk agr `ghgclihic oh d nkhkrol ohisgtrgpic `otkriols. Ia wk trohsagr` tfk cggrd ihotk systk` arg` tfk x x y y- z z cggrdihotk cggrdihotk systk`, tfk agr` ga Kquotigh (:.3) will (:.3) will cfohnk. Ih tfk trohsagr`kd cggrdihotk systk`, `ohy ga tfk zkrgs ih tfk cg`pliohck `otrix `oy jk cg`k hghzkrg, tfk hgr`ol stroihs `oy jk oaakctkd jy tfk sfkor strkssks, ohd tf k sfkor stroihs `oy `oy jk oaakctkd jy tfk hgr`ol strkssks. Tfus tfk agr` ga kquotighs ih Kquotigh (:.3) is (:.3) is volid agr o spkciaic cggrdihotk systk`, tfk `otkriol cggrdihotk systk`.
Lo`iho y
x
(o)
(j)
(j) Cghstructigh ga o lo`ihotkd cg`pgsitk. Ainurk :.: (o) O cg`pgsitk lo`iho. (j
Tg dk`ghstrotk tfk pfysicol sinhiaicohck ga `otkriol cggrdihotk systk`s, cghsidkr tfk lo`iho ga o cg`pgsitk `otkriol sfgwh ih Ainurk :.:o :.:o. Sucf o lo`iho wguld jk prgduckd jy loyihn lghn aijkrs ih o nivkh dirkctigh ohd jihdihn tfk` tgnktfkr witf kpgxy. Aijkrs ork ihfkrkhtly stiaakr ohd strghnkr tfoh tfk jule `otkriol. Tfk ihcrkosk ih strkhntf ohd stiaahkss is duk tg tfk rkductigh ga dkakcts ohd olinh`kht ga crystols olghn tfk aijkr oxis. Clkorly tfk `kcfohicol prgpkrtiks will jk diaakrkht ih tfk dirkctigh ga tfk aijkr ohd ih tfk dirkctigh pkrpkhdiculor tg tfk aijkr. Ia tfk prgpkrtiks ga tfk aijkr ohd tfk kpgxiks ork ovkronkd gr fg`gnkhizkd (Skk Skctigh 4.4), 4.4), kocf lo`iho coh jk rknordkd os oh grtfgtrgpic `otkriol, ohd tfk dirkctighs porollkl ohd pkrpkhdiculor tg tfk aijkr ork tfk `otkriol oxis dirkctighs.
:4
:
@otkriol Dkscriptigh
Lo`ihok witf diaakrkht aijkr grikhtotighs coh jk put tgnktfkr tg crkotk o cg`pgsitk lo`ihotk, os sfgwh ih Ainurk :.:j. :.:j. Tfk gvkroll prgpkrtiks ga tfk lo`ihotk coh jk cghtrgllkd jy tfk grikhtotigh ga tfk aijkrs ohd tfk stoceihn skqukhck ga tfk lo`ihok. Agr ckrtoih stoceihn skqukhcks, tfk lo`ihotk will rkspghd liek oh grtfgtrgpic `otkriol. Agr plohk Agr plohk strkss strkss prgjlk`s, prgjlk`s, tfk grtfgtrgpic strkss–stroih strkss–stroih rklotighsfips rklotighsfips usihn khnihkkrihn khnihkkrihn cghstohts ork os writtkh ih Kquotigh (:.4)
λ xx 0
ΰ yx
σ
xx ------ – ------- σ yy K x K y
λ yy 0
ΰ xy
σ
yy ------ – ------- σ xx K y K x
γ xy 0
ΰ
τ
--- yx ---
xy -------N xy
K y
0
ΰ
--- xy ---
K x
(:.4)
wfkrk K wfkrk K ohd K tfk x ohd ohd y y (`otkriol (`otkriol oxis) dirkctighs, ΰ xy ohd ΰ yx ork tfk Rgissgh rotigs gjtoihkd x ohd K y ork tfk `gduli ga klosticity ih tfk x wfkh tfk tkhsigh tkst is cghductkd ih tfk x tfk x ohd ohd y y dirkctighs, dirkctighs, rkspkctivkly, ohd N xy is tfk sfkor `gdulus ga klosticity. Hgtk ghly agur ga tfk aivk `otkriol cghstohts ork ihdkpkhdkht. Tfus, ih twg-di`khsigh oh grtfgrtrgpic `otkriol fos agur fos agur ihdkpkhdkht ihdkp khdkht cghstohts tg dkscrijk tfk lihkor strkss-stroih rklotighsfips.
:.?.3 :.?.3
Trohsvk rohsvkrs rskly kly Isgtr Isgtrgpi gpicc @otkrio @otkriols ls
Trohsvkrskly isgtrgpic `otkriols ork isgtr gpic ih o plohk. Tfot is, rgtotigh jy oh orjitrory ohnlk ojgut tfk z oxis dgks hgt cfohnk tfk strkss–stroih rklotighs, ohd tfk `otkriol is isgtrgpic ih tfk x tfk x-- y plohk. y plohk. Ih `otrix agr`, tfk strkss–stroih rklotighsfip agr trohsvkrskly isgtrgpic `otkriol is nivkh jy Kquotigh (:.;). (:.;).
⎮ λ xx ⎥ ⎭ ⎭ ⎭ λ yy ⎭ ⎭ ⎭ ⎭ λ zz ⎭ 0 ⎦ ⎬ ⎭ γ yz ⎭ ⎭γ ⎭ ⎭ zx ⎭ ⎭γ ⎭ ⎩ xy ⎠
C ??
C ?:
C ?3
2
2
C ?:
C ??
C ?3
2
2
C ?3
C ?3
C 33
2
2
2
2
2
C 44
2
2
2
2
2
C 44
2
2
2
2
2
⎮ σ xx ⎥ ⎭ ⎭ 2 ⎭ σ yy ⎭ ⎭ ⎭ 2 ⎭ σ zz ⎭ ⎦ ⎬ 2 ⎭ τ yz ⎭ ⎭τ ⎭ 2 ⎭ zx ⎭ : ( C ?? – C ?: ) ⎭ ⎭ ⎩ τ xy ⎠ 2
(:.;)
Kquotigh (:.;) i`pliks (:.;) i`pliks tfot trohsvkrskly isgtrgpic `otkriol rkquirks aivk rkquirks aivk ihdkpkhdkht `otkriol cghstohts ih cghstohts ih tfrkk di`khsighs. Cg``gh trohsvkrskly isgtrgpic `otkriols ork aijkrnloss pohkls ohd gtfkr sfgrt-aijkr cg`pgsitks. Ghk woy ga prgducihn sfgrt-aijkr cg`pgsitks is tg sproy aijkrs ghtg kpgxy ohd curk tfk rksultihn `ixturk. Tfk rohdg` grikhtotigh ga tfk aijkrs rksu lts ih gvkroll (fg`gnkhizkd) isgtrgpic `otkriol jkfovigr witfih tfk plohk ga tfk pohkl. Rkrpkhdi culor tg tfk aijkrs, wk kxpkct o diaakrkht `otkriol rkspghsk, ohd fkhck tfk strkss–stroih rklotigh pkrpkhdiculor tg tfk plohk ga tfk pohkl (aijkrs) is diaakrkht, rksultihn ih oh gvkroll trohsvkrskly isgtrgpic `otkriol rkspghsk.
:.?. :.?.4 4
Isgt Isgtrg rgpi picc @otk @otkri riol olss
Oh isgtrgpic `otkriol rkquirks ghly twg ihdkpkhdkht `otkriol cghstohts. cghstohts. Tfkrk ork `ohy cghstohts uskd tg dkscrijk isgtrgpic `otkriol, fgwkvkr, ghly twg ork ihdkpkhdkht cghstohts. Zgtotigh ojgut tfk x tfk x,, y, y, gr z gr z oxis oxis jy ohy orjitrory ohnlk rksults ih tfk so `k strkss–stroih rklotighsfip. Ih `otrix hgtotigh, tfk strkss–stroih rklotighsfip agr isgtrgpic `otkriol is nivkh jy Kquotigh (:.1). (:.1).
⎮ λ xx ⎥ ⎭ ⎭ ⎭ λ yy ⎭ ⎭ ⎭ ⎭ λ zz ⎭ 0 ⎦ ⎬ ⎭ γ yz ⎭ ⎭γ ⎭ ⎭ zx ⎭ ⎭γ ⎭ ⎩ xy ⎠
C ??
C ?:
C ?:
C ?:
C ??
C ?:
C ?:
C ?:
C ??
2
2
2
2
2
2
2
2
2
⎮ σ xx ⎥ ⎭ ⎭ ⎭ σ yy ⎭ 2 2 2 ⎭ ⎭ 2 2 2 ⎭ σ zz ⎭ ⎦ ⎬ : ( C ?? – C ?: ) 2 2 ⎭ τ yz ⎭ ⎭τ ⎭ : ( C ?? – C ?: ) 2 2 ⎭ zx ⎭ : ( C ?? – C ?: ) ⎭ τ ⎭ 2 2 ⎩ xy ⎠ 2
2
2
(:.1)
Kquotigh (:.1) coh (:.1) coh jk writtkh usihn tfrkk cghstohts= tfk `gdulus ga klosticity K klosticity K , tfk Rgissgh rotig ΰ, ohd tfk sfkor `gdulus ga klosticity N os sfgwh jklgw. λ xx 0 U σ xx – ΰ ( σ yy + σ zz ) P ⁄ K λ yy 0 U σ yy – ΰ ( σ zz + σ xx ) P ⁄ K λ zz 0 U σ zz – ΰ ( σ xx + σ yy ) P ⁄ K (:.>) γ xy 0 τ xy ⁄ N γ yz 0 τ yz ⁄ N γ zx 0 τ zx ⁄ N Cg`porihn Kquotigh (:.>) witf (:.>) witf Kquotigh (:.1), (:.1), wk gjtoih tfk agllgwihn= C ?? 0 ?/ K K
ΰ/ K C ?: 0 – ΰ/ K
:(C :(C ?? – C ?:) 0 ?/N
Sujstitutihn C ?? ohd C ?: ih tfk lost rklotighsfip, wk gjtoih N
0
K -------------------:( ? + ΰ )
(:.6)
Kquotigh (:.>) olghn (:.>) olghn witf Kquotigh (:.6), (:.6), ork collkd tfk Nkhkrolizkd Fggek’s low. It is volid ghly agr lihkor, klostic, isgtrgpic `otkriols ot ohy pgiht, ihcludihn o pgiht ih o hghfg`gnkhkgus `otkri ol. Oh oltkrhotivk agr` agrKquotigh agrKquotigh (:.>) tfot (:.>) tfot `oy jk kosikr tg rk`k`jkr is tfk `otrix agr` nivkh jkl gw.
Rlohk Strkss ohd Rlohk Stroih
⎮ λ xx ⎥ ⎭ ⎭ ? ? --- – ΰ ⎦ λ yy ⎬ 0 K ⎭ ⎭ – ΰ ⎩ λ zz ⎠
⎮ σ xx ⎥ ⎭ ⎭ ? – ΰ ⎦ σ yy ⎬ ⎭ ⎭ – ΰ ? ⎩ σ zz ⎠
:;
– ΰ – ΰ
(:.9)
Ohgtfkr oltkrhotivk is λ ii 0 U ( ? + ΰ )σ ii – ΰ I ? P ⁄ K wfkrk I ? 0 σ xx + σ yy + σ zz Tfk Nkhkrolizkd Fggek’s low agr isgtrgpic `otkriol is volid agr ohy grtfgnghol cggrdihotk systk`. systk`. Tfus, wk cguld writk kquivolkht agr`s agr Kquotigh (:.>) agr (:.>) agr cylihdricol (pglor) cggrdihotks (r (r , ζ, z ) ohd agr spfkricol cggrdihotks (r (r , ζ, ρ). Tfk prihcipol cggrdihotk systk`s agr strkssks ohd stroihs ork olsg grtfgnghol. Agr ghly isgtrgpic `otkriols tfk `otkriols tfk prihcipol dirkctighs agr strkssks ohd stroihs ork idkhticol, fkhck wk coh writk tfk nkhkrolizkd Fggek’s low ih prihcipol cggrdihotk agr` os sfgwh jklgw.
λ? 0 U σ? – ΰ ( σ : + σ3 ) P ⁄ K
λ: 0 U σ: – ΰ ( σ 3 + σ? ) P ⁄ K
λ3 0 U σ3 – ΰ ( σ ? + σ : ) P ⁄ K
(:.?2)
It sfguld jk k`pfosizkd tfot tfk ojgvk kquotighs rklotihn prihcipol strkssks ohd prihcipol stroihs ork hgt volid agr grtfgtrgpic ohd gtfkr ohisgtrgpic `otkriols jkcousk tfk prihcipol dirkctighs agr strkssks ohd stroihs ork diaakrkht. Tfk hgr`ol strkssks coh jk sglvkd ih tkr`s ga hgr`ol stroihs jy usihn tfk Kquotigh (:.>) tg (:.>) tg gjtoih. K σ xx 0 U ( ? – ΰ )λ xx + ΰλ yy + ΰλ zz P -------------------------------------( ? – : ΰ ) ( ? + ΰ ) K σ yy 0 U ( ? – ΰ )λ yy + ΰλ zz + ΰλ xx P -------------------------------------( ? – : ΰ ) ( ? + ΰ )
(:.??)
K σ zz 0 U ( ? – ΰ )λ zz + ΰλ xx + ΰλ yy P -------------------------------------( ? – : ΰ ) ( ? + ΰ ) Oh oltkrhotivk agr` tfot is kosikr tg rk`k`jkr is σ ii 0 : N λ ii + ε ( I ? ) , wfkrk i coh jk x jk x,, y, y, gr z gr z 5 I ? 0 λ xx + λ yy + λ zz 5 N is tfk sfkor `gdulus5 ohd ε 0 ( : N ΰ ) ⁄ ( ? – : ΰ ) is collkd Lo`é’s cghstoht, ho`kd oatkr N. Lo`é (?>9;–?6>2). Skk Ti`gsfkheg U?963P agr odditighol dktoils
:.:
RLOH RLOHK K STZ STZKS KSS S OHD OHD RLOH RLOHK K STZ STZOI OIH H
Ia wk toek tfk dkaihitighs ga plohk strkss ohd plohk stroih ohd opply tfk` tg Kquotigh (:.>), (:.>), wk will gjtoih tfk rinft-`gst `otricks ih Ainurk :.3. :.3. Tfk diaakrkhcks jktwkkh tfk poir ga twg-di`khsighol idkolizotighs ga `otkriol jkfovigr ork ih tfk zkrg ohd hghzkrg voluks ga tfk hgr`ol stroih ohd hgr`ol strkss ih tfk z tfk z dirkctigh. dirkctigh. Ih plohk strkss, σ zz 0 2 , wficf i`pliks tfot tfk hgr`ol stroih ih tfk z tfk z dirkctigh dirkctigh is λ zz 0 – ΰ ⁄ K ( σ xx + σ yy ) . Ih plohk stroih, λ zz 0 2 , wficf i`pliks tfot tfk hgr`ol strkss ih tfk z tfk z dirkctigh dirkctigh is σ zz 0 ΰ ( σ xx + σ yy ) .
σ xx τ xy τ yx σ yy
Rlohk strkss
Rlohk stroih
2 2
2
2
2
λ xx γ yx
γ xy λ yy
2
2
2
2
2
Nkhkrolizkd Fggek’s low
λ xx γ xy γ yx λ yy 2
Nkhkrolizkd Fggek’s low
2
σ xx τ xy τ yx σ yy 2
2
2 2
λ zz 0 – -ΰ-- ( σ xx + σ yy ) K
2 2
σ zz 0 ΰ ( σ xx + σ yy )
Ainurk :.3 Strkss ohd stroih `otricks ih plohk strkss ohd plohk stroih.
Arkk Suraock
Dkagr`otigh
( σ zz 0 2 )
( λ zz ≢ 2 )
Zinid Suraock Arkk Suraock
( σ zz 0 2 )
Dkagr`otigh
( j) Rlohk stroih. Ainurk :.4 (o) Rlohk strkss. (j
( λ zz ≢ 2 )
Zkoctigh Agrck
( σ zz ≢ 2 ) Zinid Suraock
( λ zz 0 2 )
( λ zz 0 2 ) Zkoctigh Agrck
( σ zz ≢ 2 )
:1
@otkriol Dkscriptigh
:
Tg jkttkr opprkciotk tfk diaakrkhck jktwkkh plohk strkss ohd plohk stroih, cghsidkr tfk twg plotks sfgwh ihAinurk ih Ainurk :.4, :.4, gh wficf ghly cg`prkssivk hgr`ol strkssks ih tfk x tfk x ohd y ohd y dirkctigh dirkctigh ork opplikd. Tfk tgp ohd jgttg` suraocks gh tfk plotk ih Ainurk :.4o :.4o ork arkk suraocks (plohk strkss)5 jut jkcousk tfk plotk is arkk tg kxpohd, tfk dkagr`otigh (stroih) ih tfk z tfk z dirkctigh dirkctigh is hgt zkrg. Tfk plotk ih Ainurk :.4j :.4j is cghstroihkd arg` kxpohdihn ih tfk z tfk z dirkctigh dirkctigh jy tfk rinid plotks. Os tfk `otkriol pusfks gh tfk plotk, o rkoctigh agrck dkvklgps, ohd tfis rkoctigh agrck rksults ih o hghzkrg voluk ga hgr`ol strkss ih tfk z tfk z dirkctigh. dirkctigh. Tfgunf tfk kxo`plk ih Ainurk :.4 fklps :.4 fklps kxploih tfk diaakrkhck, it sfguld jk k`pfosizkd tfot plohk strkss gr plohk stroih stroih gatkh is oh opprgxi`otigh opprgxi`otigh `odk `odk tg si`pliay oholysis. oholysis. Tfk opprgxi`otigh ga plohk st rkss is gatkh `odk ih tfih plotks ohd sfklls, sucf os tfk seih ga oh oircroat gr tfk wolls ga prkssurk vksskls. Tfk opprgxi`otigh ga plohk stroih is gatkh `odk ih tfice plotks ohd sfklls, sucf os suj`orihk fulls gr pgihts jkhkotf tfk kortf’s suraock. O diaakrkhck ga oh grdkr ga `onhitudk gr `grk jktwkkh tfk tficehkss ohd gtfkr di`khsighs is nkhkrolly uskd tg clossiay o jgdy os “tfih “tfih.‐ .‐ It sfguld jk rkcgnhizkd tfot ih plohk stroih ohd plohk strkss tfkrk ork ghly tfrkk ihdkpkhdkht quohtitiks—kvkh tfgunf tfk hu`jkr ga hghzkrg quohtitiks is `grk tfoh tfrkk. Agr kxo`plk, ia wk ehgw σ xx , σ yy, ohd τ xy, wk coh colculotk λ xx , λ yy, γ xy, λ zz , ohd σ zz agr plohk strkss ohd pl ohk stroih. Si`ilorly, ia wk ehgw λ xx , λ yy, ohd γ xy, wk coh colculotk σ xx , σ yy, τ xy, σ zz , ohd λ zz agr plohk strkss ohd plohk stroih. Tfus, ih jgtf plohk strkss ohd plohk stroih, tfk hu`jkr ga ihdkpkhdkht ihdkpkhdkht strkss gr stroih cg`pghkhts is tfrkk, oltfgunf tfk hu`jkr ga hghzkrg cg`pghkhts is nrkotkr tfoh tfrkk. Kxo`plk :.? klojgrotks :.? klojgrotks gh tfk diaakrkhcks jktwkkh plohk strkss ohd plohk stroih ohd tfk diaakrkhcks jktwkkh hghzkrg ohd ihdkpkhdkht quohtitiks.
KVO@RLK :.? Tfk strkssks ot o pgiht gh stkkl wkrk aguhd tg jk σ xx 0 ?; esi (T), σ yy 0 32 esi (C), ohd τ xy 0 :; esi. [sihn K [sihn K 0 0 32,222 esi ohd N 0 ?:,222 esi, dktkr`ihk dktkr`ihk tfk stroihs stroihs λ xx , λ yy, γ xy, λ zz , ohd strkss σ zz uhdkr twg ossu`ptighs= (o) tfk pgiht is ih plohk strkss ohd (j) tfk pgiht is ih plohk stroih. OHS.
λ xx (j) λ xx (o)
μ 5 λ yy 0 –??:; μ 5 λ zz 0 ?:; μ 5 γ xy 0 :263 μ >6?.: μ 5 λ yy 0 –?294 μ 5 σ zz 0 3.>; 3.>; esi ( C )5 γ xy 0
0 >;2 0
:263
μ
KVO@RLK :.: Tfk strkssks
σ xx
0 4 esi ( T ), σ yy 0 ?2 esi ( C ), ohd
τ xy
0 4 esi wkrk colculotkd ot o pgiht gh o arkk suraock ga oh grtfgtrg-
pic cg`pgsitk `otkriol. Dktkr`ihk (o) prihcipol strkssks ? ohd : ohd prihcipol dirkctigh ? agr strkssks ohd (j) prihcipol stroihs ? ohd : ohd prihcipol dirkctigh ? agr stroihs. [sk tfk agllgwihn voluks agr tfk `otkriol cghstohts= K cghstohts= K x 0 >;22 esi, K esi, K y 0 :;22 :;22 esi, esi, N xy 0 ?:;2 ?:;2 esi, esi, ohd ohd ΰ xy 0 2.3. 2.3. OHS. ;.21: esi ( T ) 5 σ : 0 ??.21: ??.21: esi ( C ) σ ? 0 ;.21: gr ζ x 0 ?4.6>² 5ζ y 0 >;.?3² ζ x 0 ?1;.?² 5ζ y 0 ?24.6>² λ? 0 ?393.93 μ 5 λ: 0 –41:2.93 μ gr ζ x 0 ?1.2> ² 5ζ y 0 >3.93 ² ζ x 0 ?13.93 ² 5ζ y 0 ?21.2> ²
KVO@RLK :.3 O jgdy uhdkr opplikd lgods ohd jgdy agrcks prgducks tfk agllgwihn displock`kht aikld u 0 2
v 0 Ex( Ex( y y: √ z :) + Eoxz
w 0 – : Exyz √ Eoxy
wfkrk u, v, ohd w ork displock`khts ih tfk x tfk x,, y, y, ohd z ohd z dirkctighs, dirkctighs, rkspkctivkly. Ossu`k o lihkor, klostic, isgtrgpic, fg`gnkhkgus `otkriol witf `gdulus ga klosticity K klosticity K ohd ohd Rgissgh rotig ΰ. Dktkr`ihk oll tfk strkss cg`pghkhts ot x ot x 0 :o ih tkr`s ga E ga E , o, K , ΰ, y, y, ohd z ohd z . OHS. 0 25
τ xz
KE y 0 – --------------------( : z + o ) :( ? + ΰ )
:.3 :.3
σ yy
4 KE oy 0 ----------------- 5 (? + ΰ)
σ xx
σ zz
4 KE oy 0 – ----------------- 5 (? + ΰ)
τ xy
KE : : 0 --------------------U ( y – z ) + oz P 5 :( ? + ΰ)
τ yz
4 EK oz 0 – ----------------- 5 (? + ΰ)
AOIL AOIL[Z [ZK K TF TFKG KGZ ZIKS IKS
Tfk `oxi`u` strkhntf ga o `otkriol is its otg`ic strkhntf. Ih jule `otkriols, fgwkvkr, tfk distrijutigh ga alows (i`puritiks, `icrg fglks, `icrgcroces, ktc.) crkotks lornk strkss nrodikhts. Tfksk lornk strkss nrodikhts cousk tfk jule strkhntf ga tfk `otkriol tg jk grdkrs ga `onhitudk lgwkr tfoh tfk otg`ic strkhntf ga tfk `otkriol. Aoilurk tfkgriks ossu`k o fg`gnkhkgus `otkriol ih
Aoilurk Tfkgriks
:>
wficf tfk kaakcts ga alows fovk jkkh ovkronkd ih sg`k `ohhkr. Tfis ossu`ptigh ga fg`gnkhkity rksults ih ovkronk `otkriol strkhntf voluks tfot ork odkquotk agr `gst prgjlk`s ih khnihkkrihn dksinh ohd oholysis. Agr fg`gnkhkgus–isgtrgpic-`otkriols, tfk cforoctkristic aoilurk strkss is kitfkr tfk yikld strkss gr tfk ulti`otk strkss, usuolly gjtoihkd arg` o uhioxiol tkhsilk tkst, o tkst ih wficf tfkr k is ghly ghk hghzkrg strkss cg`pg hkht. Fgw dg wk rklotk tfk ghk strkss cg`pghkht ga o uhioxiol strkss stotk tg tfk strkss cg`pghkhts ih twg- ohd tfrkk-di`khsighol stotks ga strkss7 O aoilurk tfkgry is o stotk`kht ojgut tfk rklotighsfip ga strkss cg`pghkhts tg `otkriol aoilurk cforoctkristics voluks. Hg ghk tfkgry is opplico jlk tg oll `otkriols. ^k sfoll cghsidkr agur tfkgriks (skk Tojlk :.?). :.?). @oxi`u` sfkor strkss tfkgry ohd tfk `oxi`u` gctofkdrol sfkor strkss tfkgry ork nkhkrolly uskd agr ductilk `otkriols. Ih ductilk `otkriols, aoilurk is cforoctkrizkd jy yikld strkss. Tfk `oxi`u` hgr`ol strkss tfkgry ohd `gdiaikd @gfr’s tfkgry ork nkhkrolly uskd agr jrittlk `otkriols. Ih jrittlk `otkriols, aoilurk is cforoctkrizkd jy ulti`otk strkss. Tojlk :.? Syhgpsis ga Aoilurk Tfkgriks
Ductilk @otkriol
:.3.? :.3.?
Jrittlk @otkriol
Cforoctkristic aoilurk strkss
Yikld strkss
[lti`otk strkss
Tfkgriks
?. @oxi`u` sfkor strkss :. @oxi`u` gctofkdrol sfkor strkss
?. @oxi`u` hgr`ol strkss :. @gdiaikd @gfr
@oxi` @oxi`u` u` Sf Sfkor kor Strkss trkss Tf Tfkgr kgry y
Agr ductilk `otkriols, tfk tfkgry stotks= O stotks= O `otkriol will aoil wfkh tfk `oxi`u` sfkor strkss kxckkds tfk sfkor strkss ot tfk yikld gjtoihkd arg` o uhioxiol tkhsilk tkst. Tfk tkst. Tfk aoilurk critkrigh dkscrijkd is writtkh os
τ`ox ≤ τyikld
(:.?:)
wfkrk τyikld is tfk `oxi`u` sfkor strkss ot tfk yikld pgiht aguhd ih o uhioxiol tkhsigh tkst. Tfk `oxi`u` sfkor strkss is nivkh jy tfk Kquotigh (?.?;). (?.?;). Sujstitutihn tfk strkssks ot tfk yikld pgiht ih o uhioxiol tkhsigh tkst os σ? 0 σyikld, σ: 0 2, ohd σ3 0 2 ih Kquotigh (?.?;) ohd (?.?;) ohd kquotihn tfk rksult tg τyikld, wk gjtoih tfk `oxi`u` sfkor strkss ot yikld pgiht os agllgws=
τ yikld
σyikld ⁄ :
0
(:.?3)
Trksco’s yikld critkrigh nivkh jklgw. Sujstitutihn Kquotighs (:.?:) ohd (:.?:) ohd (?.?;) ihtg (?.?;) ihtg Kquotigh (:.?3), (:.?3), wk gjtoih tfk Trksco’s
`ox (
σ? – σ : , σ: – σ 3 , σ3 – σ? ) ≤ σyikld
(:.?4)
σ: σyikld
@oxi`u` gctofkdrol sfkor strkss.
√σyikld
σyikld
σ?
@oxi`u` sfkor strkss
√σyikld Ainurk :.; Aoilurk khvklgpks agr ductilk `otkriols ih plohk strkss.
:.3.: :.3.:
@oxi`u` @oxi`u` Gct Gctofkd ofkdrol rol Sfkor Sfkor Strk trkss ss Tfkgry Tfkgry
Tfk `oxi`u` gctofkdrol sfkor strkss tfkgry agr ductilk `otkriols is= O is= O `otkriol will aoil wfkh tfk `oxi`u` gctofkdrol sfkor strkss kxckkds tfk gctofkdrol sfkor strkss ot tfk yikld gjtoihkd arg` o uhioxiol tkhsilk tkst . Tfk aoilurk critkrigh is
τgct ≤ τyikld
(:.?;)
wfkrk τ yikld is tfk gctofkdrol sfkor strkss ot yikld pgiht ih o uhioxiol tkhsilk tkst. Sujstitutihn σ? 0 σyikld, σ: 0 2, ohd σ3 0 2, tfk strkssks ot yikld pgiht ih o uhioxiol tkhsigh tkst, ihtg tfk kxprkssigh ga gctofkdrol sfkor strkss ih Kquotigh (?.:4d), (?.:4d), wk gjtoih (?.:4d) ihtg Kquotigh (:.?;), (:.?;), wk gjtoih τyikld 0 : σ yikld ⁄ 3 . Sujstitutihn tfis ohd Kquotigh (?.:4d) ihtg ? ------- ( σ ? – σ : ) : + ( σ : – σ 3 ) : + ( σ 3 – σ ? ) : ≤ σ yikld (:.?1) : Tfk lkat-fohd sidk ga tfk ojgvk kquotigh is rkakrrkd tg os vgh @isks strkss. strkss. Jkcousk vgh @isks strkss σvgh is kxtkhsivkly uskd ih tfk dksinh ga structurks ohd `ocfihks, wk agr`olly dkaihk it os agllgws=
σ vgh
? 0 ------:
( σ ? – σ: ) : + ( σ: – σ 3 ) : + ( σ3 – σ? ):
(:.?>)
:6
:
@otkriol Dkscriptigh
(:.?1) is sg`kti`ks rkakrrkd tg os tfk vgh @isks yikld critkrigh ohd is Τfk aoilurk critkrigh rkprkskhtkd jy Kquotigh (:.?1) is stotkd os agllgws=
σvgh ≤ σ yikld
(:.?6o)
Hgtk tfot ih tfis tfkgry wk ossu`kd tfot tfk hgr`ol strkss gh tfk gctofkdrol plohk fos hg ihalukhck gh aoilurk. Ih Skctigh ?.4.4 it wos sfgwh tfot tfk gctofkdrol hgr`ol strkss cgrrkspghds tg tfk fydrgstotic stotk ga strkss5 tfus ih tfis tfkgry wk ork ossu`ihn tfot fydrgstotic prkssurk fos o hknlinijlk kaakct gh tfk yikldihn ga ductilk `otkriol—o cghclusigh tfot is cghair`kd jy kxpkri`khtol gjskrvotigh agr vkry ductilk `otkriols liek olu`ihu`. Kquotighs (:.?4) ohd (:.?4) ohd (:.?1) ork (:.?1) ork aoilurk khvklgpks ih o spock ih wficf tfk oxks ork prihcipol strkssks. Aoilurk khvklgpks skporotk tfk dksinh spock arg` aoilurk spock. Ih drowihn aoilurk khvklgpks, tfk cghvkhtigh tfot σ? 8 σ: is inhgrkd. Ia tfk cghvkhtigh wkrk khagrckd, tfkrk wguld jk hg khvklgpk ih tfk skcghd quodroht ohd ghly oh khvklgpk jklgw tfk 4;g lihk wguld jk od`issijlk ih tfk tfird quodroht, rksultihn ih o vkry strohnk-lggeihn khvklgpks. Agr o plohk strkss (σ3 0 2) prgjlk`, wk coh sfgw tfksk aoilurk khvklgpks os ih Ainurk :.;. :.;. Hgtick tfot tfk `oxi`u` gctofkdrol sfkor strkss khvklgpk khcg`possks tfk `oxi`u` sfkor strkss khvklgpk. Kxpkri`khts sfgw tfot agr `gst ductilk `otkriols, tfk `oxi`u` gctofkdrol sfkor strkss tfkgry nivks jkttkr rksults tfoh tfk `oxi`u` sfkor strkss tfkgry, jut `oxi`u` sfkor strkss tfkgry is si`plkr tg usk. Tfis tfkgry is olsg collkd tfk `oxi`u` distgrtigh stroih khkrny tfkgry. Skk Kxo`plk >.3 agr >.3 agr odditighol dktoils.
:.3.3 :.3.3
@oxi` @oxi`u` u` Hgr` Hgr`ol ol Str Strkss kss Tf Tfkg kgry ry
Tfk `oxi`u` hgr`ol strkss tfkgry agr jrittlk `otkriols is= O is= O `otkriol will aoil wfkh tfk `oxi`u` hgr`ol strkss ot o pgiht kxckkds tfk ulti`otk hgr`ol strkss ( strkss (σult) gjtoihkd arg` o uhioxiol tkhsigh tkst . Tfk tfkgry nivks nggd rksults agr jrittlk `otkriols prgvidkd tfk airst prihcipol strkss is tkhsilk gr tfk tkhsilk yikld strkss fos tfk s o`k `onhitudk os tfk yikld strkss ih cg`prkssigh. Tfk aoilurk critkrigh is `ox ( σ ?, σ :, σ 3 )
≤ σult
(:.?9)
Agr cghcrktk ohd `ohy gtfkr jrittlk `otkriols, tfk ulti`otk strkss ih tkhsigh is aor lkss tfoh tfk ulti`otk strkss ih cg` prkssigh jkcousk `icrgcroces tkhd tg nrgw ih tkhsigh ohd tg clgsk ih cg`prkssi gh. Jut tfk si`plicity ga tfk aoilurk critkrigh `oeks tfk tfkgry ottroctivk, ohd it coh jk uskd ia prihcipol strkss ? is tkhsilk ohd tkhsilk ohd is tfk dg`ihoht prihcipol dg`ihoht prihcipol strkss.
:.3. :.3.4 4
@gdi @gdiai aikd kd @gf @gfr’ r’ss Tfkg Tfkgry ry
Tfk @gfr’s tfkgry agr jrittlk `otkriols is= O is= O `otkriol will aoil ia o strkss strkss stotk is gh tfk khvklgpk tfot is tohnkht tg tfk tfrkk tfrkk @gfr’s @gfr’s circlks cgrrkspghdihn tg uhioxiol ulti`otk strkss ih tkhsigh, uhioxiol ulti`otk strkss ih cg`prkssigh, ohd purk sfkor. sfkor. C^ τ Aoilurk khvklgpk σT
σC
Tohnkht pgihts
τS
σC (C)
τS
σT
τS
σ (T)
Tohnkht pgihts
CC^ (o)
(j)
( j) @gfr’s aoilurk khvklgpk. Ainurk :.1 (o) Strkss cujks. (j
^k coh cghduct tfrkk kxpkri`khts ohd dktkr`ihk tfk ulti`otk strkss ih tkhsigh σT, tfk ulti`otk strkss ih cg`prkssigh strkss stotks ork sfgwh gh tfk strkss cujks cujks ih Ainurk :.1o :.1o. O @gfr’s σC, ohd tfk ulti`otk sfkor strkss ih purk sfkor τS. Tfk tfrkk strkss circlk agr kocf ga tfk tfrkk strkss stotks is tfkh drowh. Aiholly oh khvklgpk tfot is tohnkht tg tfk tfrkk circlks is drowh, wficf rkprkskhts tfk aoilurk khvklgpk (Ainurk (Ainurk :.1j :.1j). Ia o @gfr’s circlk cgrrkspghdihn tg o strkss stotk must tgucfks tfk khvklgpk ot ohy pgiht, tfk `otkriol is ot ihcipikht aoilurk. Ia ohy port ga tfk @gfr’s circlk agr o strkss stotk is gutsidk tfk khvklgpk, tfk `otkriol fos aoilkd ot tfot pgiht. ^k coh olsg plgt tfk aoilurk khvklgpk ga Ainurk :.1 jy :.1 jy usihn tfk prihcipol strkssks os tfk cggrdihotk oxis. Ih plohk strkss tfis khvklgpk is rkprkskhtkd jy tfk sglid lihk ih Ainurk :.>. :.>. Agr `gst jrittlk `otkriols, tfk purk sfkor tkst is gatkh inhgrkd. Ih sucf o cosk tfk lihk tohnkht tg tfk circlks ga uhioxiol cg`prkssigh ohd tkhsigh wguld jk o stroinft lihk ih Ainurk :.1. :.1. Tfk rksultihn si` pliaicotigh agr plohk strkss is sfgwh os o dosfkd lihk ih Ainurk :.> ohd :.> ohd dkscrijks wfot is collkd @gdiaikd collkd @gdiaikd @gfr’s @gfr’s tfkgry. Ainurk :.> k`pfosizks :.> k`pfosizks tfk agllgwihn pgihts.
Soiht-Qkhoht’s Rrihciplk
:9
?. Ia jgtf prihcipol strkssks ork tkhsilk, tfkh tfk `oxi`u` hgr`ol strkss `ust jk lkss tfoh tfk ulti`otk tkhsilk strkhntf. :. Ia jgtf prihcipol prihcipol strkssks strkssks ork hknotivk, tfkh tfkh tfk `oxi`u` hgr`ol hgr`ol strkss `ust jk lkss tfoh tfk ulti`otk ulti`otk cg` prkssivk strkhntf. 3. Ia tfk prihcipol strkssks ork ga diaakrkht sinhs, tfkh agr tfk `gdiaikd @gfr’s tfkgry tfk aoilurk is ngvkrhkd jy Kquotigh (:.:2). (:.:2).
σ σ ≤? σ C σT
------: – -----?-
(:.:2)
σ: σ σ ------: – -----?σC σT
(-τS,τS) 0 ?
σT σT
σC
σ?
@gfr’s Tfkgry (τS,-τS) @gdiaikd @gfr’s Tfkgry
σ σ σT σC
-----?- – ------: 0 ?
σC Ainurk :.> Aoilurk khvklgpk agr plohk strkss occgrdihn tg @gfr’s tfkgry.
KVO@RLK :.4 Ot o criticol pgiht gh o `ocfihk port `odk ga stkkl, tfk strkss cg`pghkhts ork σ xx 0 ?22 @Ro (T), (T), σ yy 0 ;2 @Ro (C), ohd τ xy 0 32 @Ro. Ossu`ihn tfot tfk pgiht is ih plohk strkss ohd tfk yikld strkss ih tkhsigh is ::2 @Ro, dktkr`ihk tfk aoctgr ga soaktyo jy usihn (o) tfk `oxi`u` sfkor strkss tfkgry ohd (j) tfk `oxi`u` gctofkdrol sfkor strkss tfkgry. OHS. e τ 0 ?.31 5 e σ 0 ?.;;
o. Aoctgr ga soakty soakty 0 aoilurk-prgducihn aoilurk-prgducihn voluk dividkd dividkd jy cg`putkd cg`putkd (ollgwojlk) (ollgwojlk) voluk.
KVO@RLK :.; Nivkh o lgod R, tfk strkssks ot o pgiht gh o arkk suraock wkrk aguhd tg jk σ xx 0 3 R esi esi (C), σ yy 0 ; R esi esi (T), ohd τ xy 0 –: R esi, esi, wfkrk R wfkrk R is is `kosurkd ih eips. Tfk jrittlk `otkriol fos o tkhsilk tkhsilk strkhntf ga ?6 esi ohd o cg`prkssivk strkhntf ga ga 31 esi. Dktkr`ihk tfk `oxi`u` voluk ga tfk lgod R lgod R tfot tfot coh jk opplikd gh tfk structurk jy usihn tfk `gdiaikd @gfr ’s tfkgry. tfkgry. OHS. R `ox 0 :.46 :.46 eips eips
:.4
SOIH SOIHTT-Q QKHOH KHOHT’ T’S S RZIH RZIHCIR CIRLK LK
Tfkgriks ih tfk `kcfohics ga `otkriols ork cghstructkd jy `oeihn ossu`ptighs rknord ihn lgod, nkg`ktry, ohd `otkriol voriotighs. Tfksk ossu`ptighs usuolly ork hgt volid hkor cghckhtrotkd agrcks gr `g`khts sucf os tfgsk tfot `oy jk prkskht hkor suppgrts, hkor cgrhkrs gr/ohd fglks, hkor ihtkraocks ga twg `otkriols, ohd ot alows sucf os croces. Disturjohcks ih tfk strkss ohd displock`kht aiklds, fgwkvkr, dissipotk ropidly os ghk `gvks owoy arg` tfk rknighs ih wficf tfk ossu`ptighs ga tfk tfkgry ork viglotkd. Tfk stotk`kht ojgut ga dissipotigh ga o disturjohck witf distohck is stotkd ih Soiht-Qkhoht’s Soiht-Qkhoht’s prihciplk os agllgws= • Twg stoticolly kquivolkht lgod systk`s systk`s prgduck hkorly tfk so`k strkss ih rknighs ot o distohck tfot is ot lkost kquol tg tfk lornkst di`khsigh ih tfk lgodkd rknigh. Cghsidkr tfk twg stoticolly kquivolkht lgod systk`s sfgwh ih Ainurk :.6. :.6. Ia wk drow o strkss cujk uhdkr tfk cghckhtrotkd agrck R agrck R ih ih tfk Ainurk :.6o :.6o, wk kxpkct tfot tfk strkssks will tkhd tgword ihaihity jkcousk ga tfk ihaihitksi`ol orko ga tfk strkss cujk. Fgwkvkr, ot o crgss skctigh tfot is ot o distohck ^ jklgw jklgw tfk opplikd lgod, tfk strkss distrijutigh will jk clgskr tg uhiagr` distri jutigh, witf o voluk kquol tg R tg R dividkd dividkd jy tfk crgss-skctighol orko. Ih Ainurk :.6j :.6j wk fovk o distrijutkd agrck tfot prgducks o tgtol agrck ga R ga R . Tfgunf tfk distrijutkd agrck is sfgwh uhiagr`, ohy distrijutigh tfot dgks hgt prgduck o `g`kht cguld jk uskd. Tfus,
32
:
@otkriol Dkscriptigh R t
t O ^
^
x
x ---σ ≆ ---R
^t
R -- O
ot x ot x 8 ^
(j)
(o) Ainurk :.6 Strkss duk tg twg stoticolly kquivolkht lgod systk`s.
tfis distrijutkd agrck is stoticolly kquivolkht tg tf k ghk gh tfk lkat. Jy Soiht-Qkhoht’s Soiht-Qkhoht’s prihciplk, tfk strkss ot o distohck^ distohck^ ih ih Ainurk :.6j :.6j will jk hkorly uhiagr`. Ih tfk rknigh ot o distohck lkss tfoh ^ , tfk strkss distrijutigh will jk diaakrkht ohd it is pgssijlk tfot sfkor strkss cg`pghkhts will jk prkskht os wkll. Ih o si`ilor `ohhkr, cfohnks ih nkg`ktry ohd `otkriols fovk lgcol kaakcts tfot coh jk inhgrkd ot distohcks. ^k will cghsidkr tfk kaakct ga cfohnks ih nkg`ktry ohd oh khnihkkrihn sglutigh tg tfk prgjlk` ih Skctigh :.;, gh strkss cghckhtrotigh. Tfk i`pgrtohck ga Soih t-Qkhoht’s t-Qkhoht’s prihciplk is tfot it ollgws us tg dkvklgp gur tfkgriks witf rkosghojlk cghaidkhck owoy arg` tfk rknighs ga strkss cghckhtrotigh. ^k coh tfkh usk tfk strkss cghckhtrotigh aoctgr discusskd ih tfk hkxt skctigh tg gjtoih `oxi`u` strkssks ih rknighs ga strkss cghckhtrotigh.
:.;
STZKS TZKSS S CGH CGHCK CKHT HTZO ZOTI TIGH GH AOCT AOCTGZ GZ
O lornk strkss nrodikhts ih o s`oll rknigh is collkd strkss cghckhtrotigh. Tfksk lornk nrodikhts cguld jk duk tg suddkh cfohnks ih nkg`ktry, `otkriol prgpkrtiks, gr lgodihn, os `khtighkd korlikr. ^k ^k ehgw arg` S oiht-Qkhoht’s oiht-Qkhoht’s prihciplk, discusskd ihSkctigh ihSkctigh :.4, :.4, tfot wk coh usk gur tfkgrkticol `gdkls tg colculotk strkss owoy arg` tfk rknighs ga lornk strkss cghckhtrotigh. Tfksk strkss voluks prkdictkd jy tfk tfkgrkticol `gdkls owoy arg` tfk rknighs ga strkss cghckhtrotigh ork lojklk d hg`ihol strkss. Ainurk :.9 sfgws :.9 sfgws ? pfgtgklostic picturks ga twg structurol `k`jkrs uhdkr uhioxiol tkhsigh. Lornk strkss nrodikhts hkor tfk circulor cutgut jguhdoriks cousk arihnks tg jk agr`kd. Tfk arihnk grdkr coh jk uskd ih colculotihn tfk strkssks.
σhg`ihol
σhg`ihol
σhg`ihol
σhg`ihol
( Cgurtksy ga Rrgakssgr I. @iseignlu.) @iseignlu. ) Ainurk :.9 Rfgtgklostic picturks sfgwihn strkss cghckhtrotigh. (Cgurtksy
Strkss cghckhtrotigh aoctgr is is oh khnihkkrihn cghckpt tfot pkr`its us tg kxtropglotk tfk rksults ga gur klk`khtory tfkgry ihtg tfk rknighs ga lornk strkss cghckhtrotigh, wfkrk tfk ossu`ptighs gh wficf tfk t fkgry is joskd ork viglotkd. Tfk strkss cghckhtrotigh aoctgr E aoctgr E cghc is dkaihkd os `oxi`u` strkss E cghc 0 --------------------------------------hg`ihol strkss
(:.:?)
Tfk strkss cghckhtrotigh aoctgr E aoctgr E cghc is aguhd arg` cforts, tojlks, gr agr`ulos tfot fovk jkkh dktkr`ihkd kxpkri`khtolly, hu`kricolly, oholyticolly gr jy o cg`jihotigh ga tfk tfrkk. Oppkhdix O sfgws skvkrol nropfs, wficf coh jk uskd ih tfk colculotigh ga strkss cghckhtrotigh aoctgrs agr prgjlk`s ih tfis jgge. Odditighol nropfs coh jk aguhd ih fohdjgges (Skk Rileky U?99>P) dkscrijihn diaakrkht situotighs. ^fkh wk ehgw tfk hg` ihol strkss ohd tfk strkss cghckhtrotigh aoctgr, wk coh ksti`otk tfk ?. Rfgtgklostic `otkriols ork trohsporkht `otkriols tfot, oaakct tfk spkkd ot wficf linft possks tfrgunf tfrgunf tfk` wfkh strksskd ohd tfus coh uskd ih tfk `kosurk`kht ga strkss. Skk Jurnkr U?993P agr odditighol dktoi ls.
Strkss Ihtkhsity Aoctgr
3?
`oxi`u` strkss ohd usk it ih dksinh gr tg ksti`otk tfk aoctgr ga soakty. Kxo`plk :.1 dk`ghstrotks :.1 dk`ghstrotks tfk usk ga tfk strkss cghckhtrotigh aoctgr.
KVO@RLK :.1 Aihitk klk`kht oholysis sfgws tfot o lghn structurol cg`pghkht corriks o uhiagr` oxiol strkss ga 3; @Ro (T). O fglk ih tfk ckhtkr hkkds tg jk drillkd agr possihn cojlks tfrgunf tfk structurol cg`pghkhts. Tfk yikld strkss ga tfk `otkriol is :22 @Ro. Dktkr`ihk tfk `oxi`u` dio`ktkr ga tfk fglk tfot coh jk drillkd, usihn o aoctgr ga soakty ga ?.1, ia aoilurk duk tg yikldihn is tg jk ovgidkd. ?2 ``
F 0 F 0 ?22 ``
d :.1. Ainurk :.?2 Cg`pghkht nkg`ktry ih Kxo`plk :.1.
OHS.
d `ox 0 31 ``
:.1 :.1
STZKS TZKSS S IHTK IHTKHS HSIT ITY Y AOCT AOCTGZ GZ
Strkss ihtkhsity ihtkhsity aoctgr is ohgtfkr cghckpt tfot pkr`its us tg kxtkhd gur klk`khtory tfkgriks ohd agr`ulos tg `otkriols cghtoihihn alows sucf os s`oll croces. Arocturk croces. Arocturk `kcfohics, tfk `kcfohics, tfk oholysis ga strkssks ohd dkagr`otigh ga `otkriols cghtoihihn croces, is jkyghd tfk scgpk ga tfis jgge (skk Zglak ohd Jorsg` U?9>>P). Jut tfk cghckpt ga strkss ihtkhsity aoctgr ohd its opplicotigh ih dksinh coh jk dkscrijkd jy usihn tfk cghclusighs dkrivkd `otfk`oticolly ih arocturk `kcfohics.
σ
σ σ
σ O :j
σ
:o
σ
O
σ
σ
σ
σ
Ainurk :.?? Oh kllipticol fglk ih oh ihaihitk plotk.
Cghsidkr o s`oll kllipticol fglk ih oh ihaihitk `otkriol os sfgwh ih Ainurk :.??. :.??. Tfk klosticity sglutigh sfgws tfot tfk `oxi`u` strkss ot tfk tip ga tfk `omgr oxis (pgiht O (pgiht O)) coh jk aguhd jy usihn tfk strkss cghckhtrotigh aoctgr E cghc 0 ( ? + : o ⁄ j ) , wfkrk j ohd o ork fola tfk dio`ktkr ga tfk `ihgr ohd tfk `omgr oxks, rkspkctivkly. Ia wk wkrk tg lkt tfk `ihgr oxis dio`ktkr j ng tg zkrg, tfk kllipsk wguld jkcg`k o croce5 jut gur strkss cghckhtrotigh aoctgr wguld jkcg`k ihaihitk—tfk `oxi`u` strkss ot pgiht O pgiht O wguld wguld jkcg`k ihaihitk. Ih gtfkr wgrds, tfk usk ga tfk strkss cghckhtrotigh cghckpt agr croces i`pliks tfot tfk `g`kht o tihy croce is agr`kd, tfk khtirk jgdy sfguld jrkoe jkcousk hg `otkriol coh sustoih oh ihaihitk strkss! ^k ehgw tfis cohhgt jk cgrrkct jkcousk wk fovk gatkh skkh tihy croces ih `otkriols ohd structurks tfot cghtihuk tg auhctigh wkll. Tfk kxplohotigh is tfot tfk tip ga tfk croce jkcg`ks slinftly rguhdkd, jut tfk jluhtihn ga tfk croce tip cohhgt jk occurotkly prkdictkd ohd occguhtkd agr ih tfk tfkgrkticol klostic `gdkls. Aurtfkr`grk, wk olsg ehgw tfot sg`kti`ks croces nrgw ohd `ocfihk cg`pghkhts (gr structurol `k`jkrs) jrkoe. Strkss ihtkhsity aoctgr is o cghckpt tfot fklps rkcghcilk tfk ihaihitk strkss prkdictigh ga tfkgrkticol `gdkls tg tfk rkolity ga stojlk gr uhstojlk croces ih `otkriols. Tfis is occg`plisfkd jy usihn rksults arg` klosticity tfot sfgw tfot tfk strkss cg`pghkhts ih tfk i``kdiotk vicihity ga o croce tip coh jk writtkh os
σim
0 E ihtkh a im
(:.::)
wfkrk a wfkrk a im ork auhctighs tfot dkpkhd upgh tfk lgcotigh ga tfk pgiht ot wficf strkss is aguhd rklotivk tg tfk lgcotigh ga tfk croce tip. Tfk auhctighs a auhctighs a im jkcg`k ihaihitk ot tfk croce tip, rkalkctihn tfk jkfovigr ga tfk tfkgrkticol `gdkl, wficf is tg inhgrk jluhtihn ga tfk croce tip. Tfk aoctgr E aoctgr E ihtkh is tfk strkss ihtkhsity aoctg r tfot dkpkhds upgh tfk strkss lkvkl ohd croce lkhntf lkhntf agr s`oll croces croces ih
3:
:
@otkriol Dkscriptigh
vkry lornk jgdiks (`gdklkd (`gdklkd os ihaihitk jgdiks). jgdiks). Ih aihitk jgdiks witf croces, tfk strkss ihtkhsity aoctgr olsg dkpkhd upgh nkg`ktric voriojlks tfot ork dkaihkd ih fohdjgges (skk @uroeo`iU?96>P).
σ
σ
Zupturk strkss
Strghnkr `otkriol
@gdulus ga tgunfhkss
Tgunfkr `otkriol
λ
λ (o)
(j)
Ainurk :.?: Twg rkprkskhtotighs ga `gdulus ga tgunfhkss.
^fkh tfk voluk ga E ga E ihtkh rkocfks o criticol voluk E voluk E crit, tfk croce storts tg nrgw. Ductilk aoilurk rkakrs tg croce nrgwtf tfot rkquirks khgunf khkrny tg crkotk hkw croce suraocks ohd tfk nrgwtf ga croce is slgw. Ih Jrittlk aoilurk croces nrgws ot hkorly tfk spkkd ga sguhd ghck croce rkocfks o criticol croce lkhntf ohd is gatkh cotostrgpfic. ^k will agcus gh tfk issuk ga croce rkocfihn criticol croce lkhntf ohd hgt cghckrh gurskla witf tfk issuk ga wfot foppkhs witf tfk croce nrgwtf. Tfk criticol voluk ga tfk strkss ihtkhsity aoctgr E aoctgr E crit is o `otkriol prgpkrty. Tfk finfkr tfk voluk ga E ga E crit, tfk nrkotkr tfk rksistohck ga tfk `otkriol tg croce nrgwtf ohd tfk tgunfkr tfk `otkriol. Zkcgllkct arg` tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols tfot tfk “`gdulus ga tgunfhkss‐ is kquol tg tfk stroih khkrny dkhsity ot rupturk, os sfgwh ih Ainurk :.?:o :.?:o. Criticol strkss ihtkhsity aoctgrs ork rklotkd tg tfk `gdulus ga tgunfhkss. ^k olsg hgtk tfk distihctigh jktwkkh strghn `otkriols, wficf ork cforoctkrizkd jy finf ulti`otk strkss, ohd tgunf `otkriols, wficf ork cforoctkrizkd jy o lornk `gdulus ga tgunfhkss, os sfgwh ih Ainurk :.?:j :.?:j. ^k rkcgrd tfk agllgwihn gjskrvotighs. • Tfk strkss ihtkhsity aoctgr dkpkhds upgh tfk strkss lkvkl ohd tfk lkhntf ga tfk croce. • Tfk criticol strkss ihtkhsity aoctgr is o `otkriol prgpkrty tfot is ihdkpkhdkht ga strkss lkvkl gr croce lkhntf. • O croce jkcg`ks uhstojlk (tfk `otkriol jrkoes) wfkh tfk strkss ihtkhsity aoctgr kxckkds tfk criticol strkss ihtkhsity aoctgr.
Croce tip
Croce tip Croce tip
@gdk I
@gdk II
@gdk III
Ainurk :.?3 Tfrkk `gdks ga rklotivk croce suraock `gvk`kht.
^k hkkd ghk `grk cghckpt rklotkd tg croce nrgwtf jkagrk wk coh usk tfk strkss ihtkhsity aoctgr ih dksinh.Ainurk dksinh. Ainurk :.?3 sfgws tfrkk pgssijlk `gdks ih wficf tfk twg suraocks gh kitfkr sidk ga o croce coh `gvk rklotivk tg kocf gtfkr. @gdk I is oh gpkhihn oport ih wficf tkhsilk strkssks cousk tfk croce tg n rgw. @gdk II is o slidihn `gdk duk tg ih-plohk sfkor strkss. @gdk III is o tkorihn `gdk, kxk`pliaikd wfkh wk tkor o pikck ga popkr witf jgtf fohds. @gdk I is prkdg`ihohtly aguhd ih `gst aoilurks ohd fkhck is kxtkhsivkly studikd. @ixkd-`gdk aoilurks ork o sujmkct ga ghngihn rkskorcf. ^k will cghsidkr o si`plk `gdkl ga `ixkd `gdks I ohd II ih tfis jgge. Ainurk :.?4 sfgws :.?4 sfgws croces agr `gdk I ohd `gdk II ih tfih ihaihitk plohks. Arocturk `kcfohics sfgws tfot tfk strkss ihtkhsity aoctgrs agr `gdks l ohd II agr o croce ih o tfih ihaihitk plohk ork os nivkh jklgw. E I 0
σ hg` ς o
(:.:3o)
E II 0
τ hg` ς o
(:.:3j)
Aotinuk
:o
σhg`
σhg`
:o
τhg`
33
τhg`
(j)
(o) Ainurk :.?4 Strkss ihtkhsity aoctgrs agr `gdk I ohd `gdk II.
wfkrk σhg` ohd τhg` ork tfk hg`ihol hgr`ol ohd sfkor strkss gjtoihkd arg` klk`khtory tfkgriks. ^fkh jgtf `gdks ork prkskht, wk will usk tfk agllgwihn kquivolkht strkss ihtkhsity aoctgr= E kquiv 0
:
:
E I + E II
(:.:4)
^fkh tfk kquivolkht strkss ihtkhsity aoctgr kxckkds tfk criticol strkss ihtkhsity aoctgr, tfk croce jkcg`ks uhstojlk. Fohdjgges (skk @uroeo`iU?96>P) supply agr`ulos ohd nropfs tfot coh jk uskd tg gjtoih strkss ihtkhsity aoctgrs ih o vorikty ga situotighs.
:.1. :.1.? ?
Ohol Oholys ysis is Rrgc Rrgckd kdur urkk
Agllgwihn ork tfk stkps agr ihcgrpgrotihn oh kxistihn croce ihtg oh oholysis. Stkp ?. Dktkr`ihk tfk stotk ga strkss ot tfk pgiht ga tfk croce= usk klk`khtory tfkgry tfot dgks hgt ihcludk o croce. Stkp :. Tg aihd tfk hgr`ol ohd sfkor strkssks gh tfk ihclihkd croce suraock. Tfksk hgr`ol ohd sfkor strkssks gh tfk croce suraock ork tfk hg`ihol strkssks. (:.:3o) ohd (:.:3j) tg (:.:3j) tg aihd tfk strkss ihtkhsity aoctgrs E aoctgrs E I ohd E ohd E II. Stkp 3. [sk Kquotighs (:.:3o) ohd (:.:4) tg aihd tfk kquivolk ht strkss ihtkhsity aoctgr. Stkp 4. [sk Kquotighs (:.:4) tg Stkp ;. Cg`pork tfk kquivolkht strkss ihtkhsity aoctgr witf tfk criticol strkss ihtkhsity aoctgr agr o `otkriol ohd dkcidk wfktfkr tfk croce is gr is hgt stojlk. Ia hg kxistihn croce grikhtotigh is spkciaikd, wk sfoll ossu`k `icrgcroces tfot will nrgw ih `gdk I gh o plohk ga `oxi`u` tkhsilk hgr`ol strkss (i.k., gh prihcipol plohk ?). ^k rkcgrd tfis gjskrvotigh agr auturk usk. • @icrgcroces will jk ossu`kd tg nrgw ih `gdk I gwihn tg prihcipol strkss ? ia tfk plohk is ih tkhsigh.
KVO@RLK :.> Tfk prgpkllkr sfoat ga o suj`orihk is sujmkctkd tg o tkhsilk oxiol strkss ohd o tgrsighol sfkor strkss wfkh tfk vksskl rkvkrsks its dirkctigh. Gh disploy, tfk prgpkllkr sfoat sfgwkd o croce ot oh ohnlk ga :>² tg tfk oxis ga tfk sfoat. Ot tfk pgiht wfkrk croce wos skkh, tfk strkssks ork ksti`otkd os sfgwh ih Ainurk :.?;. :.?;. Tfk sfoat `otkriol fos o criticol strkss ihtkhsity aoctgr ga ?42 e si si i h . Ia tfk suj`orihk wos still ih gpkrotigh, ot wfot croce lkhntf wguld ygu rkcg``khd tfot it jk pullkd gut ga wotkr agr rkpoirs, ossu`ihn (o) tfot tfk dktkctkd croce cguld nrgw ohd (j) tfot tfkrk is hg prkkxistihn croce.
>; esi
?:2 ?:2 esi esi C r o oc e G r i k h t o ot i ig h g
:>
:.>. Ainurk :.?; Croce ih o prgpkllkr agr Kxo`plk :.>.
OHS.
(o) : o cr 0 2.>61 ih (j) : o I 0 2.;?: ih
:.>
AOTIN[K
Tg opprkciotk tfk diaakrkhck jktwkkh stotic ulti`otk strkhntf ohd aotinuk strkhntf, try tg jrkoe o pikck ga wirk (k.n., o popkr clip) jy pullihn gh it witf jgtf fohds—ygu will hgt jk ojlk tg jrkoe it jkcousk ygu cohhg t kxckkd tfk ulti`otk stotic strkss ga tfk `otkriol. Hkxt toek tfk so`k pikck ga wi rk ohd jkhd it airst ghk woy, tfkh tfk gtfkr o akw ti`ks. Ygu will aihd tfot it jrkoes kosily. Tfk kxplohotigh ga tfis pfkhg`khgh is klojgrotkd ih tfk hkxt poronropf.
34
:
@otkriol Dkscriptigh
Oll `otkriols ork ossu`kd tg tg fovk `icrgcroces. Tfksk croces ork hgt criticol jkcousk tfkir lkhntf is vkry s`oll, ohd fkhck ih stotic prgjlk`s, tfk jule strkhntf ga tfk `otkriol cgrrkspghds tg tfk ulti`otk strkss dktkr`ihkd jy o tkhsigh tkst.Fgwkvkr, ia tfk `otkriol is sujmkctkd tg cyclic lgodihn, `icrgcroces coh nrgw till ghk gr `grk rkocfks o criticol lkhntf, ot wficf ti`k tfk rk`oihihn `otkriol jrkoes. Tfk strkss voluk ot rupturk ih cyclic lgodihn is sinhiaicohtly lgwkr tfkh tfk ulti`otk strkss ga tfk `otkriol. Aoilurk duk tg cyclic lgodihn ot strkss lkvkls sinhiaicohtly lgwkr tfoh tfk stotic ulti`otk strkss is collkd aotinuk.
Striotigh `ores
(Cgurtksy Rrgakssgr I. @iseignlu). @iseignlu). Ainurk :.?1 Aoilurk duk tg aotinuk. (Cgurtksy
Aoilurk duk tg aotinuk is liek o jrittlk aoilurk irrkspkctivk ga wfktfkr tfk `otkriol is jrittlk gr ductilk. ^k skk tfot tfkrk ork twg pfosks ga aoilurk. Ih tfk airst pfosk tfk `icrgcroces nrgw, ohd rknighs ga croce nrgwtf coh jk idkhtiaikd jy tfk striotigh `ores, olsg collkd jkocf `ores, os sfgwh ih Ainurk :.?1. :.?1. Gh kxo`ihotigh ga o arocturkd suraock, tfk rknigh ga `icrgcroce nrgwtf sfgws ghly s`oll dkagr`otigh. Ih tfk skcghd pfosk, wficf jknihs ghck tfk criticol croce lkhntf fos jkkh rkocfkd, tfk aoilurk suraock ga tfk rknigh sfgws sinhiaicoht dkagr`otigh, os sfgwh ih Ainurk :.?1. :.?1.
H curvks. Ainurk :.?> S - H
Tfk agllgwihn strotkny is uskd ih dksinh tg occguht agr aotinuk aoilurk. Kxpkri`khts ork cg hductkd ot diaakrkht `onhitudk lkvkls ga cyclic strkss, ohd tfk hu`jkr ga cyclks ot wficf tfk `otkriol aoils is rkcgrdkd. O plgt is `odk ga strkss ((S S ) vs. hu`jkr ga cyclks ( H ) tg aoilurk. Tfkrk is olwoys sinhiaicoht scottkr ih tfk doto. O curvk tfrgunf tfk doto is cghstructkd. Tfis curvk is collkd H tfk S - H H curvk. curvk. Ot lgw lkvkls ga strkss tfk aoilurk `oy gccur ghly oatkr `illighs gr kvkh jillighs ga cyclks. Tg occg``gdotk tfis lornk scolk, o lgn scolk is uskd agr tfk hu`jkr ga cyclks. Ainurk :.?> sfgws :.?> sfgws sg`k typicol S - H H curvks. curvks. Hgtick tfot tfk curvks opprgocf o strkss lkvkl osy`ptgticolly, i`plyihn tfot ia strkssks ork ekpt jklgw tfis lkvkl, tfk `otkriol will hgt aoil uhdkr cyclic lgodihn. Tfis finfkst (osy`ptgtic) lkvkl agr wficf o `otkriol will hgt aoil uhdkr cyclic lgodihn is collkd tfk khdurohck li`it gr otinuk strkhntf. a otinuk It sfguld jk k`pfosizkd tfot o porticulor S - H H curvk curvk agr o `otkriol dkpkhds upgh `ohy aoctgrs sucf os `ohuaocturihn prgckss, `ocfihihn prgckss, suraock prkporotigh, ohd gpkrotihn k hvirgh`kht. Tfus twg spkci`khs `odk arg` so`k stkkl ollgy jut witf diaakrkht fistgriks will yikld diaakrkht S - H curvks. curvks. Cork `ust jk toekh tg usk oh S-H oh S-H curvk tfot cgrrkspghds os clgskly os pgssijlk tg tfk octuol situotigh. Ih o typicol prkli`ihory dksinh, stotic strkss oholysis wguld jk cghductkd jy usihn tfk pkoe lgod ga tfk cycli c lgodihn. Ih gtfkr wgrds, wk wguld dktkr`ihk tfk finfkst strkss lkvkl tfot `inft gccur ih o cyclic lgodihn. [sihn oh opprgpriotk opprgpriotkS S - H H curvk, curvk, tfk hu`jkr ga cyclks tg aoilurk agr tfk cg`putkd strkss voluk is tfkh colculotkd. Tfis hu`jkr ga cyclks tg aoilurk is tfk prkdictkd liak ga tfk structurol cg`pghkht. Ia tfk prkdictkd liak is uhocckptojlk, tfkh tfk cg`pghkht wguld jk rkdksinhkd tg lgwkr tfk strkss lkvkl ohd fkhck ihcrkosk tfk hu`jkr ga cyclks tg aoilurk.
Clgsurk
3;
KVO@RLK :.6 Tfk stkkl plotk rkprkskhtkd ih Ainurk :.?6 fos :.?6 fos oh S - H H liek liek tfk tgp curvk sfgwh ih Ainurk :.?>. :.?>. (o) Dktkr`ihk tfk `oxi`u` dio`ktkr ga tfk fglk tg tfk hkorkst `illi`ktkr ia tfk prkdictkd liak ga 2.; `illigh cyclks is dksirkd agr o uhiagr` aor aikld strkss gccur ia tfk prkdictkd liak is tg σ 0 >; @Ro. (j) Agr tfk fglk rodius ih port (o), wfot pkrckhtonk rkductigh ih aor aikld strkss `ust gccur jk ihcrkoskd tg ? `illigh cyclks7
σ
σ
σ
σ
?>2 ?>2 ``
σ
σ
d
σ σ
:.6. Ainurk :.?6 Stkkl plotk agr Kxo`plk :.6.
OHS.
d 0 13 `` 5 Tfk rkquirk rkquirkd d pkrckhtonk pkrckhtonk rkductigh rkductigh is ;.?3%.
: .6
CLGS[ZK
Ih tfis cfoptkr wk sow tfot o lihkor `gdkl rkquirks :? ihdkpkhdkht `otkriol cghstohts agr `otkriols tfot ork aully ohisgtrgpic5 ?3 agr `ghgclihic5 9 ohd 4 agr grtfgtrgpic ih tfrkk ohd twg di`khsighs, rkspkctivkly5 ohd ghly : agr isgtrgpic `otkriols. Tfk strkss– stroih rklotighsfip agr isgtrgpic `otkriols collkd tfk nkhkrolizkd Fggek’s low is volid agr ohy grtfgnghol systk` ohd is ihdkpkhdkht ga tfk grikhtotigh ga cggrdihotk systk`. Agr oll `gdkls gtfkr tfoh isgtrgpic, tfk strkss–stroih rklotighsfip is dkscrijkd ih o `otkriol cggrdihotk systk` ohd tfk prihcipol cggrdihotks agr strkssks ohd stroihs fovk diaakrkht grikhtotighs. ^k olsg sow tfot o `otkriol coh aoil wfkh tfk `otkriol cforoctkristic strkhntf is kxckkdkd os dkscrijkd jy aoilurk tfkgriks5 wfkh o croce nrgws ih `gdk I gr `ixkd `gdk5 ohd wfkh cyclic lgodihn cousks aotinuk. ^k sow tfot suddkh cfohnks ih nkg`ktry, lgodihn, gr `otkriol prgpkrtiks cousk o strkss cghckhtrotigh, wficf quicely dissipotks witf distohck occgrdihn tg SoihtQkhoht’s prihciplk. Ih tfk hkxt cfoptkr wk will study tfk strkss oholysis ga ghk-di`khsighol structurol klk`khts ih o woy tfot occguhts agr tfksk vorigus agr`s ga `otkriol aoilurk.
3
Josic Structurol @k`jkrs
LKOZHIHN GJMKCTIQK ?. *Tg uhdkrstohd tfk li`itotighs ga josic tfkgry ohd fgw cg`plkxitiks `oy jk oddkd tg tfk josic tfkgriks ga oxiol `k`jkrs, tgrsigh ga circulor sfoats, ohd sy``ktric jkhdihn ga jko`s. :. Tg uhdkrstohd tfk cghckpt ohd usk ga discghtihuity auhctighs ih tfk oholysis ga structurol `k`jkrs sujmkctkd tg discghtihugus lgods.
(o)
(j)
(c)
`k`jkrs. Cgurtksy (o) (o) Moes`oto Moes`oto (j) Sel`oto (c) mrge (c) mrge Ainurk 3.? Ossk`jliks ga ghk-di`khsighol structurol `k`jkrs.Cgurtksy
Ih Ainurk 3.?, 3.?, tfk aro`k ga o juildihn, tfk seklktgh ga o fu`oh jkihn, ohd tfk aro`k ga o cor cghtoih `ohy ghk-di`khsighol structurol `k`jkrs. Cfoir lkns, utility pglks, drills, jggesfklvks, ohd divihn jgords ork o`ghn tfk cguhtlkss opplicotighs ga ghk-di`khsighol structurol `k`jkrs. Tfk tfkgriks ohd agr`ulos agr ghk-di`khsighol structurol `k`jkrs vory ih cg`plkxity, dkpkhdihn upgh tfk cg`plkxitiks ih lgodihn, `otkriol prgpkrtiks, ohd crgss-skctighol nkg`ktriks. Ih tfis cfoptkr wk jrikaly rkvikw tfk si`plkst tfkgriks ih o aro`kwgre tfot pkr`its rkody ihcgrpgrotigh ga cg`plkxitiks, sg`k ga wficf ork studikd ih tfis cfoptkr ohd gtfkrs ih sujskqukht cfoptkrs. Ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols, tfk josic tfkgriks ga oxiol `k`jkrs, tgrsigh ga circulor sfoats, ohd sy``ktric jkhdihn ga jko`s wkrk dkvklgpkd ohd studikd. Ih tfis cfoptkr o syhgpsis ga oll tfrkk tfkgriks is prkskhtkd ih tojulor agr` tg k`pfosizk tfk cg``gholitiks tfk cg``gholitiks ohd diaakrkhcks ih tfk dkrivotigh ga strkss ohd dkagr`otigh agr`ulos. Tfk li`itotighs ohd ossu`ptighs tfot ork `odk tg nkt tfk si`plkst pgssijlk agr`ulos ork idkhtiaikd. Ih tfis ohd lotkr cfoptkrs it is sfgwh tfot cg` plkxitiks ga ihtkrkst coh jk oddkd jy drgppihn o li`itotigh gr oh ossu`ptigh. It wi ll jk skkh tfot `gst cg`plkxitiks ih oll tfrkk structurol `k`jkrs (oxiol rgds, circulor sfoats, ohd sy``ktric jko`s) khtkr tfk dkrivotigh ot tfk so`k pgiht ohd ork occguhtkd agr ih hkorly tfk so`k `ohhkr. Discghtihugus lgodihn gh o `k`jkr is aoirly cg``gh. Ih tfk ihtrgductgry cgursk tfksk discghtihuitiks ih lgodihn wkrk occguhtkd agr jy skn`khtihn tfk structurol `k`jkr sucf tfot ih kocf skn`kht tfk lgodihn wos cghtihugus. Zklotivk d kagr`otighs ga skn`kht khds wkrk aguhd, ohd tfk gvkroll dkagr`otigh wos gjtoihkd jy usihn tfk gjs krvotigh tfot dkagr`otigh wos cghtihugus ot tfk pgiht ga tfk lgodihn discghtihuity. Tfis opprgocf is olnkjroicolly tkdigus, porticulorly agr jko`-jkhdihn prgjlk`s ohd stoticolly ihdktkr`ihotk structurol `k`jkrs. Ih tfis cfoptkr tfk cghckpt ga discghtihuity auhctighs is ihtrgduckd ohd kaakctivkly uskd agr oholyzihn tfk dkagr`otigh ga stoticolly dktkr`ihotk ohd ihdktkr`ihotk structurol `k`jkrs tfot ork sujmkctkd tg discghtihugus lgods.
31
3>
3.?
3
Josic Structurol @k`jkrs
LGNI LG NIC C IH IH STZ STZ[C [CT[ T[ZO ZOL L OHO OHOLY LYSIS SIS
Displock`khts, stroihs, strkssks, ihtkrhol agrcks ohd `g`khts, ohd kxtkrhol agrcks ohd `g`khts ork tfk auhdo`khtol voriojlks ih `kcfohics ga `otkriols. Tfksk auhdo`khtol voriojlks ork lgni colly rklotkd os sy`jglicolly sfgwh ihAinurk ih Ainurk 3.:. 3.:. Tfis lgnic is uskd agr dkvklgpihn jgtf tfk si`plk tfkgriks discusskd ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols ohd tfk odvohckd tfkgriks khcguhtkrkd ih nroduotk cgursks. It is pgssijlk tg stort ot ohy pgiht ih tfk lgnic ohd `gvk kitfkr clgcewisk (sglid orrgws) gr cguhtkrclgcewisk (fgllgw orrgws). It is hgt pgssijlk tg rklotk displock`kht dirkctly tg kxtkrhol agrcks witfgut i`pgsihn li`itotighs ohd `oeihn ossu`ptighs rknordihn tfk nkg`ktry ga tfk jgdy, `otkriol jkfovigr, ohd kxtkrhol lgodihn. Tfis is k`pfosizkd jy tfk ojskhck ga orrgws dirkctly liheihn displock`khts ohd kxtkrhol agrcks ihAinurk ih Ainurk 3.:.l 3.:.l
Displock`khts
? E i h k ` o t i c s Kxtkrhol agrcks ohd `g`khts
Stroihs
l s g d k l @ r i o : o t k @
K K h k q r n y u i li @ j r i 4 k t u ` f g d s
Ihtkrhol agrcks ohd `g`khts
Stotic Kquivolkhcy
Strkssks
3
Ainurk 3.: Lgnic ih structurol oholysis.
Tfk lgnic sfgwh ih Ainurk 3.: is 3.: is ihtrihsicolly vkry `gdulor5 tfot is, tfk kquotighs (eihk`otics, `otkriol `gdkls, stotic kquivolkhcy, ohd kquilijriu`) liheihn ohy twg auhdo`khtol voriojlks ork `utuolly ihdkpkhdkht . Tfus, ia cg`plkxity is oddkd ih ghk port, tfkh tfk tfkgry coh jk rkdkrivkd witf `ihi`u` cfohnk. Tfis will jk dk`ghstrotkd `ohy ti`ks ih tfis jgge tfrgunf kxo`plks, ih pgsttkxt prgjlk`s, ohd ih Cfoptkrs 4, ;, ohd 1. [hdkrstohdihn tfk `gdulor hoturk ga tfis lgnic ohd lkorhihn fgw cg`plkxitiks coh jk ihcgrpgrotkd tg rkaihk josic tfkgriks is oh i`pgrtoht lkorhihn gjmkctivk ih tfis jgge. Ih tfis skctigh tfrkk kxo`plks dk`ghstrotk tfk opplicotigh ga tfis lgnic. Ih Cfoptkr > wk will skk tfot khkrny `ktfgds ork oh oltkrhotivk tg Hkwtghioh `k tfgds ga kquilijriu` kquotighs ohd coh rkplock tfk lost lihe ih tfk lgnic. Tfk `gdulor cforoctkr ga tfk lgnic fgwkvkr rk`oihs uhcfohnkd ohd cg`plkxitiks coh os kosily ihcgrpgrotkd witf khkrny `ktfgds os witf kquilijriu` kquotighs.
KVO@RLK 3.? Twg tfih jors ork skcurkly ottocfkd tg o rinid plotk, os sfgwh ih Ainurk 3.3. 3.3. Tfk crgss-skctighol crgss-skctighol orko ga kocf kocf jor is :2 ``:. Tfk agrck A agrck A kxt is tg jk plockd sucf tfot tfk rinid plotk `gvks ghly fgrizghtolly jy 2.2; `` witfgut rgtotihn. Dktkr`ihk tfk agrck A agrck A kxt ohd its lgcotigh f agr tfk agllgwihn cosks. ga K 0 0 :22 :22 NRo. NRo. Cosk ?= Jgtf jors ork `odk arg` stkkl witf `gdulus ga klosticity ga K Cosk := Jor ? is `odk ga stkkl ( K 0 K 0 :22 NRo), ohd jor : is `odk ga olu`ihu` olu`ihu` K 0 (K 0 >2 NRo) NRo).. O
Jor ?
J A
x Jor :
:2 ``
f
:22 :22 ``
Ainurk 3.3 Oxiol jors ih Kxo`plk 3.?. OHS. Cosk ?= A kxt 0 :222 H 5 f 0 ?2 `` 5 Cosk := A kxt 0 ?3;2 H 5 f 0 ?4.6? ?4.6? ` `
Josic Tfkgriks ga Ghk-Di`khsighol Structurol @k`jkrs
36
KVO@RLK 3.: Twg tfih jors ga ford rujjkr fovk o crgss-skctighol orko ga :2 ``:. Tfk jors ork ottocfkd tg o rinid dise ga rodius :2 ``, os sfgwh ih Ainurk 3.4. 3.4. Tfk opplikd tgrquk T kxt, cousks tfk rinid dise tg rgtotk jy oh ohnlk ga 2.24 rod ojgut tfk oxis ga tfk dise. Dktkr`ihk tfk opplikd tgrquk agr tfk agllgwihn cosks= Cosk ?= Ossu`k tfot tfk ford rujjkr is ngvkrhkd jy tfk lihkor vkrsigh ga Fggek’s low τ
0
Cosk := Ossu`k tfot tfk ford rujjkr is ngvkrhkd jy tfk hghlihkor strkss–stroih rklotigh τ
0
N ? γ , wfkrkN wfkrk N? 0 :62 :62 @Ro. @Ro. N: γ
2.6
, wfkrkN wfkrk N: 0 322 322 @Ro. @Ro.
T kxt
` ` : 2
J
:22 :22 ``
C
O
Ainurk 3.4 Nkg`ktry agr Kxo`plk 3.:. 3.:. OHS. Cosk ?= T kxt
0
2.691 H ⋋ ` 5 Cosk := T kxt
0
:.9 H ⋋ `
KVO@RLK 3.3 O cohgk is tikd dgwh gh tgp ga o cor jy `kohs ga strktcf cgrds os sfgwh ih Ainurk 3.;. 3.;. Tfk uhdkagr`kd lkhntf ga tfk strktcf cgrd is 42 ih, tfk ihitiol ihitiol dio`ktkr is d 0 d 0 ?/: ih, ohd tfk `gdulus `gdulus ga klosticity klosticity K K is is ;?2 psi. @ores @ores wkrk `odk `odk gh tfk cgrd kvkry kvkry : ihcfks tg prgduck o tgtol ga :2 skn`khts. Tfk strktcf cgrd is sy``ktric witf rkspkct tg tfk tgp ga tfk cohgk. Tfk stortihn pgiht ga tfk airst skn`kht is gh tfk corrikr roil ga tfk cor, ohd tfk khd pgiht ga tfk tkhtf skn`kht is gh tfk tgp ga tfk cohgk. Tfk `kosurkd lkhntf ga kocf skn`kht is sfgwh ih Tojlk 3.?. 3.?. ^k fovk twg cosks. Cosk ?= Ossu`k s`oll stroih= tfot is,
λ s
Cosk := Ossu`k lornk stroih, nivkh jy
0
λ L
δ ⁄ L 2 . 0
? :
δ ⁄ L 2 + -- ( δ ⁄ L 2 ) : .
wfkrk δ is tfk dkagr`otigh ohd ohd L L2 is tfk grinihol lkhntf ga o skn`kht. Dktkr`ihk (o) tfk tkhsigh ih tfk cgrd ga kocf skn`kht ohd (j) tfk agrck kxkrtkd jy tfk cgrd gh tfk corrikr ga tfk cor. 3.3. Tojlk 3.? Cgrd Divisigh agr Kxo`plk 3.3. Skn`kht Hu`jkr
Dkagr`kd Lkhntf (ih)
Skn`kht Hu`jkr
Dkagr`kd Lkhntf (ih)
?
3.4
1
3.4
:
3.4
>
3.?
3
3.4
6
:.>
4
3.4
9
:.3
;
3.4
?2
:.:
Ainurk 3.; Cohgk ottocfkd tg cor agr Kxo`plk 3.3 OHS. Cosk ?= Z 0 >2.? lj Cosk := Z 0 94.1 lj
3.:
JOSIC JOSIC TFKGZI TFKGZIKS KS GA GHK-D GHK-DI@KH I@KHSIG SIGHOL HOL STZ[ STZ[CT[ CT[ZOL ZOL @K@JKZ @K@JKZS S
Tfis skctigh prkskhts tfk josic tfkgry gh oxiol `k`jkrs, sy``ktric jkhdihn ga jko`s, ohd tgrsigh ga circulor sfoats ih tojulor agr` tg k`pfosizk tfk cg``gholitiks tfk cg``gholitiks ohd diaakrkhcks ih tfk dkrivotigh ga vorigus agr`ulos. ^k will stort jy rkvikwihn tfk si` plkst pgssijlk tfkgriks, discusskd ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols. Tfk ossu`ptighs tfot lkod tg si`pliaicotighs will jk idkhtiaikd. ^k will drgp sg`k ga tfksk ossu`ptighs ohd ihcgrpgrotk cg`plkxitiks tfot ork ga procticol ihtkrkst ih lotkr skctighs ohd cfoptkrs. It will jk skkh tfot tfksk cg`plkxitiks khtkr tfk tfkgrkticol dkrivotigh ot tfk so`k lkvkl ohd ork occguhtkd agr ih hkorly tfk so`k `ohhkr agr oll tfrkk structurol `k`jkrs= oxiol rgds, sy``ktric jko`s, ohd circulor sfoats. Tojlk 3.: is 3.: is o syhgpsis ga tfk dkrivotigh tfk klk`khtory tfkgriks. Ossu`ptighs ohd kquotighs witf suaaixks -O, -J, ohd T rkakr tg oxiol, jkhdihn, ohd tgrsigh, rkspkctivkly. Oh ossu`ptigh witfgut ohy suaaix is opplicojlk tg oll tfrkk tfkgriks. Tfis hgto-
39
3
Josic Structurol @k`jkrs
tigh is uskd tg aocilitotk tfk dk`ghstroti gh, ih sujskqukht skctighs ohd cfoptkrs, tfot cg`plkxitiks khtkr tfk dkrivotigh ot tfk s o`k lkvkl ohd ork occguhtkd agr ih hkorly o si`ilor `ohhkr agr oll tfrkk structurol `k`jkrs.
3.:.? .:.?
Li`i i`itot totigh ighs
Tfk agllgwihn li`itotighs ork cg``gh tg tfkgriks ojgut oxiol `k`jkrs, tgrsigh ga circulor sfoats, ohd sy``ktric jkhdihn ga jko`s. ?. Tfk lkhntf ga tfk `k`jkr is sinhiaicohtly nrkotkr (opprgxi`otkly ?2 ti`ks) tfoh tfk nrkotkst di`khsigh ih tfk crgss skctigh. Opprgxi`otigh ocrgss tfk crgss skctigh is hgw pgssijlk jkcousk tfk rknigh ga opprgxi`otigh is s`oll. :. ^k ork owoy arg` rknighs ga strkss cghckhtrotigh, cghckhtrotigh, wfkrk displock`khts ohd strkssks coh jk tfrkk-di`khsighol. Tfk rksults arg` tfk si`pliaikd tfkgriks coh jk kxtropglotkd ihtg rknighs ga strkss cghckhtrotigh os dkscrijkd ih Skctigh :.;. :.;. Ih o si`ilor `ohhkr, rksults coh jk kxtropglotkd ihtg rknighs hkor croces jy `kohs ga strkss ihtkhsity aoctgrs, dkscrijkd ih Skctigh :.1. :.1. 3. Tfk voriotigh ga kxtkrhol lgods gr cfohnks ih tfk crgss-skctighol crgss-skctighol orko is nroduol kxckpt ih rknighs ga strkss cghckhtrotigh. Tfk tfkgry ga klosticity sfgws tfot tfis li`itotigh is hkckssory5 gtfkrwisk opprgxi`otighs ocrgss tfk crgss skctigh wguld jk uhtkhojlk. 4. Tfk kxtkrhol lgods ork sucf tfot tfk oxiol, tgrsigh, ohd jkhdihn prgjlk`s coh jk studikd ihdividuolly. ihdividuolly. Tfis rkquirks hgt ghly tfot tfk opplikd lgods jk ih o nivkh dirkctigh, jut olsg tfot tfk lgods poss tfrgunf spkciaic pgihts ossgciotkd witf tfk crgss skctigh. Tfk lgcotigh ga tfksk pgihts is dktkr`ihkd witfih tfk dkvklgp`kht ga tfk tfkgriks. Agr tgrsigh ohd jkhdihn, wk i`pgsk tfk agllgwihn odditighol li`itotighs tfot ih lotkr cfoptkrs will jk drgppkd. • Tgrsigh= ^k li`it gursklvks tg circulor crgss skctighs. Tfis pkr`its us tg usk ornu`khts ga oxisy``ktry ih dkducihn dkagr`otigh. • Jkhdihn= ^k li`it gursklvks tg jko` crgss skctighs tfot fovk o plohk ga sy``ktry tg wficf tfk lgodihn is rkstrictkd. Tfis li`itotigh khsurks sy``ktric jkhdihn. It is ossu`kd tfot tfk x tfk x-- y plohk y plohk is tfk plohk ga sy``ktry ohd tfot jkhdihn is gccurrihn ojgut tfk z tfk z oxis oxis (i.k., crgss skctighs rgtotk ojgut tfk z tfk z oxis) oxis)
3.:.:
Dkaihiti Dkaihitigh gh ga voriojlk voriojlkss ohd ohd sinh cghvkht cghvkhtighs ighs
Ainurk 3.1 sfgws 3.1 sfgws o slkhdkr `k`jkr witf tfk x tfk x oxis oxis olghn tfk oxiol dirkctigh ohd y ohd y ohd z ohd z ih ih tfk plohk ga tfk crgss skctigh. ^k will usk tfis cggrdihotk systk` agr oxiol ohd jkhdihn prgjlk`s. Agr tgrsigh prgjlk`s wk will usk o pglor cggrdihotk systk` (Ϗ, ζ, x). x).
ζ (ρ) y (v) y (v)
Ϗ
z (w (w)
x
( u )
Ainurk 3.1 Cggrdihotk systk`s ohd dkagr`otigh voriojlks.
Tfk voriojlks uskd ih Tojlk 3.:, 3.:, o syhgpsis ga tfk dkrivotigh ga tfk klk`khtory tfkgriks studikd ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols, ork dkaihkd os agllgws.
?. Tfk displo displock`k ck`khts hts u, v, ohd w will jk cghsidkrkd tg jk pgsitivk ih tfk pgsitivk x, y, ohd z dirkctigh, dirkctigh, rkspkctivkly. :. Tfk Tfk rgtot rgtotig igh h ρ ga tfk crgss skctigh will jk cghsidkrkd tg jk pgsitivk cguhtkr cl gcewisk witf rkspkct t g tfk x oxis. x oxis. 3. Tfk kxtkrhol kxtkrhol distrijutkd distrijutkd tgrquk tgrquk pkr uhit uhit lkhntf lkhntf t ( x) x) is pgsitivk cguhtkrclgcewisk witf rkspkct tg tfk x tfk x oxis. oxis. 4. Tfk kxtkrhol kxtkrhol distrijutkd distrijutkd agrcks agrcks pkr uhit uhit lkhntf p lkhntf p x( x) x) ohd p ohd p y( x) x) ork cghsidkrkd tg jk pgsitivk ih tfk pgsitivk x pgsitivk x ohd ohd y dirkctigh, y dirkctigh, rkspkctivkl y. ;. Gh o arkk jgdy dionro`, dionro`, tfk pgsitivk pgsitivk ihtkrhol ihtkrhol oxiol agrck agrck H is is sfgwh os tkhsilk. 1. Gh o arkk jgdy dionro`, dionro`, tfk pgsitivk pgsitivk ihtkrhol ihtkrhol sfkor agrck agrck Q y is sfgwh ih tfk so`k dirkctigh os pgsitivk τ xy . @ >. Gh o arkk jgdy jgdy dionro`, tfk tfk pgsitivk ihtkrhol ihtkrhol jkhdihn jkhdihn `g`kht @ z is sfgwh tg put tfk pgsitivk y aock ih cg`prkssigh.
6. Gh o arkk jgdy jgdy dionro`, dionro`, pgsitivk pgsitivk ihtkrhol ihtkrhol tgrquk T is is sfgwh cguhtkrclgcewisk witf rkspkct tg t fk gutword hgr`ol.
Josic Tfkgriks ga Ghk-Di`khsighol Structurol @k`jkrs
42
Ia tfk sinh cghvkhtighs agr ihtkrhol agrcks ohd `g`khts ork agllgwkd ih drowihn arkk jgdy dionro`s, tfk agr`ulos ih Tojlk 3.: will 3.: will nivk tfk rinft sinh agr displock`kht ohd strkss cg`pghkhts. Tfk displock`kht dirkctigh coh tfkh jk ihtkrprktkd ih occgrdohck witf tfk agrkngihn dkaihitighs, ohd tfk dirkctigh ga tfk strkss cg`pghkhts coh jk dktkr`ihkd jy usihn sujscripts os discusskd ih Cfoptkr ?. Ia tfk ihtkrhol agrcks ohd `g`khts ork drowh ih o `ohhkr tfot skrvks t g kquilijrotk tfk kxtkrhol agrcks ohd `g `khts, tfk agr`ulos ih Tojlk 3.: sfguld 3.: sfguld jk uskd ghly tg dktkr`ihk tfk `onhitudk ga tfk displock`kht ohd strkss cg`pghkhts5 tfk dirkctigh ga tfksk cg`pghkhts `ust jk dghk jy ihspkctigh. Tfk dkrivotigh ga sfkor strkss ih jkhdihn is o jkoutiaul dk`ghstrotigh ga khnihkkrihn opprgxi`otigh tg gvkrcg`k tfk li`itotighs ga cghsistkht lgnic. Tfis is discusskd skporotkly ih Skctigh 3.:.3. 3.:.3. Tfk colculotigh ga jko` dkalkctigh is olsg discusskd skporotkly, sihck kxtkrhol lgods nkhkrolly vory witf x witf x ohd ohd rkquirk jguhdory cghditighs tg dktkr`ihk tfk jko` dkalkctigh. Tojlk 3.: Syhgpsis ga Klk`khtory Tfkgriks ga Ghk-Di`khsighol Structurol @k`jkrs Oxiol
Jkhdihn
Tgrsigh
Dkagr`otighs Ossu`ptigh ?
Dkagr`otighs ork hgt o auhctighs ga ti`k.
Ossu`ptigh :
:-O= Rlohk skctighs rk`oih plohk ohd porollkl.
u 0 u2 ( x )
:o-J= Squosfihn dkagr`otigh is sinhiaicohtly s`ollkr tfoh dkagr`otigh duk tg jkhdihn. :j-J= Rlohk skctighs jkagrk dkagr`otigh rk`oih plohk oatkr dkagr`otigh. :c-J= O plohk pkrpkhdiculor tg tfk jko` oxis rk`oihs hkorly pkrpkhdiculor hkorly pkrpkhdiculor oatkr dkagr`otigh.
v 0 v(x)
(3.?o-J)
d v u 0 – y -----dx
(3.?j-J)
(3.?-O)
:o-T= Rlohk skctighs pkrpkhdiculor tg tfk oxis rk`oih plohk durihn dkagr`otigh. :j-T= Oll rodiol lihks rgtotk jy kquol ohnlk durihn dkagr`otigh gh o crgss skctigh. :c-T= Zodiol lihks rk`oih stroinft durihn dkagr`otigh.
ρ
0
ρ(x)
(3.?-T)
Stroihs Ossu`ptigh 3
Tfk stroihs ork s`oll. du 2 0 -------- ( x ) dx
λ xx
:
(3.:-O)
λ xx
d v 0 – y --------: ( x ) dx
(3.:-J)
γ x ζ
d Ϗ -----ρ- ( x )
0
dx
(3.:-T)
Strkssks Ossu`ptigh Ossu`ptigh Ossu`ptigh Ossu`ptigh
4 ; 1 >
[sihn Fggek’s low
Tfk `otkriol is isgtrgpic. Tfkrk ork hg ihklostic stroihs Tfk `otkriol is klostic. Strkss ohd stroih ork lihkorly rklotkd.
σ xx
du 2 0 K -------- ( x ) dx
:
(3.3-O)
σ xx
d v 0 – Ky --------: ( x ) dx
(3.3-J)
τ x ζ
d ρ 0 N Ϗ ------ ( x ) dx
(3.3-T)
∥ Ϗ τ ζ d O
(3.4-T)
Ihtkrhol Agrcks ohd @g`khts Stotic kquivolkhcy
H 0
∥σ
xx d O
(3.4o-O)
H 0
O
∥σ
xx d O
0 2
(3.4o-J)
O
∥
@ z 0 – y σ xx d O 0 2
(3.4j-O)
x
O
∥
@ z 0 – y σ xx d O
O
T 0
(3.4j-J)
O
∥
@ y 0 – z σ xx d O 0 2
(3.4c-O)
Q y 0
O
∥τ
xy d O
(3.4c-J)
O
Sinh cghvkhtigh
+σ xx
Cg`prkssivk pgsitivk y pgsitivk y aock aock
+τ xy
σ xx Distrijutigh
+T
+ H
+Q y
+τ xζ + @ z
4?
3
Josic Structurol @k`jkrs
Tojlk 3.: Syhgpsis ga Klk`khtory Tfkgriks ga Ghk-Di`khsighol Structurol @k`jkrs Oxiol
Jkhdihn
du
Tgrsigh :
d ρ dx
d v
2 Sujstitutihn strkssks ihtg ihtkrhol agrcks ohd `g`khts ohd hgtihn -------- , -------- , ohd ------ ork auhctighs ga x ga x ghly, ghly, wfilk tfk ihtknrotigh is
dx
dx
:
witf rkspkct tg y tg y ohd ohd z z Grinih lgcotigh
∥ yK d O 0 2
∥ yK d O 0 2
(3.;-O)
O
O
d u2 H 0 -------- K d O dx
∥
:
d v : @ z 0 --------: Ky d O d x O
(3.1-O)
O
Ossu`ptigh 6 Tfk grinih is ot tfk ckhtrgid ga tfk crgss skctigh.
(3.;-J)
∥
d ρ : T 0 ------ N Ϗ d O dx
(3.1-T)
T d ρ ------ 0 ------dx NM
(3.6-T)
∥
(3.1-J)
O
Tfk `otkriol is fg`gnkhkgus ocrgss tfk crgss skctigh.
∥ y d O 0 2
∥ y d O 0 2
(3.>-O)
O
(3.>-J)
O
du H --------2 0 ------dx KO
: @ z d v -------- 0 --------: KI zz dx
(3.6-O)
O 0 crgss skctigho skctigholl orko KO 0 oxiol rinidity rinidity
(3.6-J)
I zz 0 skcghd orko `g`kht ga ihkrtio ojgut z ojgut z KI zz 0 jkhdihn jkhdihn rinidity rinidity
M 0 M 0 pglor `g`kht `g`kht ga tfk orko NM 0 NM 0 tgrsighol tgrsighol rinidity rinidity
Strkss Agr`ulos
Sujstitutihn Kquotighs (3.6-O), (3.6-O), (3.6-J), (3.6-J), ohd (3.6-T) ihtg (3.6-T) ihtg Kquotighs (3.3-O), (3.3-O), (3.3-J), (3.3-J), ohd (3.3-T)
σ xx
H 0 --- O
σ xx
(3.9-O)
@ z y 0 – ⎕ ---------- ⎘ ⎐ I zz ⎢
(3.9-J)
τ x ζ
T Ϗ 0 ------ M
(3.9-T)
Skk Skctigh 3.:.3 agr sfkor strkssks ih jkhdihn. Dkagr`otigh Agr`ulos Ossu`ptigh 9 Ossu`ptigh ?2 Ossu`ptigh ??
Tfk `otkriol is fg`gnkhkgus jktwkkh x jktwkkh x? ohd x ohd x:. Tfk structurol `k`jkr is hgt topkrkd jktwkkh x jktwkkh x? ohd x ohd x: Tfk kxtkrhol lgods dg hgt cfohnk witf x witf x jktwkkh jktwkkh x x? ohd x ohd x:. Ihtknrotihn Kquotighs (3.6-O) ohd (3.6-O) ohd (3.6-T)
H ( x : – x ? ) u : – u ? 0 ------------------------ KO
Skk Skctigh 3.4.3 agr jko` dkalkctigh.
ρ : – ρ ?
(3.?2-O)
T ( x : – x ? ) 0 -----------------------NM
(3.?2-T)
Kquilijriu` Kquotighs
H
p x( x) x) Δ x
H +Δ H
Δ x
T @ z + Δ @ z
@ z
t ( x) x) Δ x
Δ x Q y
p y( x) x) Δ x
Q y + ΔQ y
Δ x T+ΔT
d H ------- 0 – p x ( x ) dx
(3.??-O)
Q d Q -------- y- 0 – p y ( x ) dx
(3.??o-J)
d @ ---------- z 0 – Q y dx
(3.??j-J)
T d T ------- 0 – t ( x ) dx
(3.??-T)
Diaakrkhtiol Kquotighs
Sujstitutihn Kquotighs (3.6-O), (3.6-O), (3.6-J), (3.6-J), ohd (3.6-T) ihtg (3.6-T) ihtg Kquotighs (3.??-O), (3.??o-J), (3.??o-J) , (3.??j-J) , ohd (3.??-T)
du d ------ ⎕ KO --------2⎘ 0 – p x ( x ) ⎐ dx dx ⎢
(3.?:-O)
d ⎕ d v⎘ -------- ⎜ KI zz -------- ⎗ 0 p y ( x ) : : dx ⎐ dx ⎢ :
:
(3.?:-J)
d d ρ ------ ⎕ NM ------⎘ 0 – t ( x ) dx ⎐ dx ⎢
(3.?:-T)
Josic Tfkgriks ga Ghk-Di`khsighol Structurol @k`jkrs
4:
KVO@RLK 3.4 O stkppkd circulor jko` ih o turjg `ocfihk is `gdklkd os sfgwh ih Ainurk 3.>. 3.>. Ia tfk ollgwojlk jkhdihn jkhdihn hgr`ol strkss strkss is :22 @Ro, dktkr`ihk tfk s`ollkst aillkt rodius tfot coh jk uskd ot skctigh J skctigh J agr agr o `oxi`u` jkhdihn lgod ksti`otkd tg jk R jk R 0 0 :22 H. ; 2 ` ` O
: ; ` ` R R
J
? .; `
C
? ` 3.4. Ainurk 3.> Stkppkd jko` agr Kxo`plk 3.4.
OHS.
r 0 3.>; ``
KVO@RLK 3.; Tfk fgllgw sfoat ga o sfip’s prgpkllkr is `gdklkd os sfgwh ih Ainurk 3.6. 3.6. Tfk sfoat fos oh gutsidk dio`ktkr ga ?22 `` ohd oh ihsidk dio`ktkr dio`ktkr ga >; ``. Dktkr`ihk Dktkr`ihk tfk `oxi`u` voluk voluk ga R ga R agr agr tfk agllgwihn cosks. Cosk I= [sk tfk `oxi`u` gctofkdrol sfkor strkss tfkgry ohd o yikld strkhntf ga σyikld 0 :62 :62 @Ro. @Ro. Cosk II= [sk o criticol strkss ihtkhsity aoctgr ga E crit 0 33 @ R o ` , ohd hgtk tfot tfk s`oll croce sfgwh ot pgiht O ih O ih Ainurk 3.6 fos 3.6 fos o lkhntf ga 4 ``. y y z
x
O
C r o oc e
32g
2.? R
x R
Ainurk 3.6 Fgllgw cylihdkr agr Kxo`plk 3.;. 3.;. OHS. Cosk I= R `o x 0 :??.1 :??.1 e H 5 Cosk II= R `o x 0 412 eH
KVO@RLK 3.1 Sfgw tfot tfk si`plkst diaakrkhtiol kquotigh agr jko` vijrotighs uhdkr o dyho`ic distrijutkd agrck p y ( x, t ) is 4
∊:v : ∊ v -------- + c -------∊ : x ∊ : t wfkrk Ϗ` is tfk `otkriol `oss dkhsity ohd c 0
p y ( x, t ) 0 ----------------Ϗ` O
(3.?3-J)
KI zz ⁄ Ϗ ` O . Hknlkct rgtotighol ihkrtio.
KVO@RLK 3.> Ih Ti`gsfkheg jko`so, tfk ossu`ptigh ga plohks rk`oihihn pkrpkhdiculor tg tfk oxis ga tfk jko` is drgppkd tg occguht agr sfkor. Tfk dkrivotigh storts witf tfk displock`kht aikld (3.?4o-J) u 0 – y φ ( x ) v 0 v( x ) wfkrk φ is tfk ohnlk ga rgtotigh ga tfk plohk ga crgss skctigh arg` tfk vkrticol. Sfgw tfot tfk si`plkst diaakrkhtiol kquotighs agr Ti`gsfkheg jko`s is nivkh jy tfk agllgwihn kquotighs=
d v ------ NO ⎕ -d ----- – φ⎘ ⎐ ⎢ dx dx o. Skk Auhn Auhn U?91;P U?91;P gr Zkddy Zkddy U?993P. U?993P.
0 – p y
d d φ d v ------ ⎕ KI zz -------⎘ 0 – NO ⎕ ----- – φ⎘ ⎐ ⎢ ⎐ ⎢ dx dx dx
(3.?4j-J)
43
3
3.:.3 3.:.3
Josic Structurol @k`jkrs
Sfkor Sfk or Str Strkss kss ih ih Tfih Tfih Sy``k Sy``ktri tricc Jko` Jko`ss
Ossu`ptigh :c-J i`pliks tfot tfk sfkor stroih γ xy `ust jk sg s`oll tfot wk coh hknlkct it ih eihk`otic kquotighs. Jy Fggek’s low tfis i`pliks tfot τ xy sfguld jk s`oll. Ih jko` jkhdihn, o cfkce gh tfk volidity ga tfk oholysis is tg cg`pork tfk `oxi`u` sfkor strkss τ xy witf tfk `oxi`u` hgr`ol strkss σ xx agr tfk khtirk jko`. khtirk jko`. Ia tfk twg strkss cg`pghkhts ork cg`porojlk, tfkh tfk sfkor stroih cohhgt jk hknlkctkd ih eihk`otic cghsidkrotighs ohd wk sfguld usk Ti`gsfkheg jko` tfkgry ga Kxo`plk 3.>. 3.>. Tfk `oxi`u` jkhdihn hgr`ol strkss σ xx ih tfk jko` sfguld jk hkorly oh grdkr ga `onhitudk nrkotkr tfoh tfk `oxi`u` jkhdihn sfkor strkss τ xy. Ainurk 3.9o 3.9o sfgws tfk jkhdihn ga agur skporotk wggdkh strips, wfilk Ainurk 3.9j 3.9j sfgws tfk jkhdihn ga tfk so`k agur strips oatkr tfky fod jkkh nlukd tgnktfkr. Ih tfk Ainurk 3.9o 3.9o kocf wggdkh strip slidks rklotivk tg tfk gtfkrs ih tfk oxiol dirkctigh, jut ih tfk Ainurk 3.9j 3.9j tfk rklotivk slidihn is prkvkhtkd jy tfk sfkor rksistohck ga tfk nluk5 tfot is, sfkor strkss ih tfk nluk is dkvklgpkd tg prkvkht tfk rklotivk slidihn jktwkkh tfk strips. Ghk `oy tfus fypgtfksizk tfot ih ohy jko` tfkrk will jk sfkor strkss gh o i`onihory suraock porollkl tg tfk oxis ga tfk jko`.
Zklotivk slidihn
Hg rklotivk slidihn
(o)
(j)
( o) skporotk jko`s ohd (j ( j) nlukd jko`s. Ainurk 3.9 Kaakcts ga sfkor strkss ih jkhdihn= (o
Cghsidkr o circulor crgss skctigh tfot is nlukd tgnktfkr arg` six wggdkh strips os sfgwh ih Ainurk 3.?2. 3.?2. Tfk jkhdihn lgod R lgod R , cousks tfk `g`kht @ (3.9-J) wk ehgw tfot tfk `onhitudk ga z tg vory olghn tfk lkhntf ga tfk jko`. Tfus arg` Kquotigh (3.9-J) wk tfk hgr`ol strkss σ xx will cfohnk olghn tfk lkhntf ga tfk jko`. Arg` tfk kvidkhck ga tfk pfgtgnropfs ih Ainurk 3.9, 3.9, wk ehgw tfot sfkor strkss will kxist ot kocf nlukd suraock tg rksist rklotivk slidihn jy tfk wggd strips. Tfk sfkor strkss ih jkhdihn `ust jolohck tfk voriotigh ga hgr`ol strkss σ xx olghn tfk lkhntf ga tfk jko`. t Arkk Suraock
Arkk Suraock
H
y Arkk Suraock
Q 0τ xs(tdx) tdx)
?
s z
dx
?
1
: +d H H +d
:
; 4
3
s x
(o)
(j)
R Ainurk 3.?2 Sfkor strkss ih o circulor crgss skctigh.
Tfk gutword hgr`ol ga tfk suraock will jk ih o diaakrkht dirkctigh agr kocf nluk suraock gh wficf wk cghsidkr tfk sfkor strkss. Ia wk dkaihk o tohnkhtiol cggrdihotk “ s‐ s‐ tfot is ih tfk dirkctigh ga tfk tohnkht tg tfk ckhtkrlihk ga tfk crgss skctigh, tfk gutword hgr`ol tg tfk nlukd suraock will jk ih tfk s tfk s dirkctigh dirkctigh ohd tfk sfkor strkss will jk τ sx. Ghck `grk, jy sy``ktry ga sfkor strkssks, τ xs 0 τ sx. Ot o pgiht, ia tfk s tfk s dirkctigh dirkctigh ohd tfk y tfk y dirkctigh dirkctigh ork tfk so`k, tfkh τ xs will kquol plus gr `ihus τ xy. tfk s dirkc dirkc xy. Ia tfk s tigh ohd tfk z tfk z dirkctigh dirkctigh ork tfk so`k ot o pgiht, tfkh τ xs will kquol plus gr `ihus τ xz . Ainurk 3.?2j 3.?2j sfgws o diaakrkhtiol klk`kht ga tfk jko`. Tfk sfkor agrck tfot jolohcks tfk cfohnk ih oxiol agrck H s is sfgwh gh ghly ghk suraock. Tfk suraock gh tfk gtfkr khd ga tfk arkk jgdy dionro` is ossu`kd tg jk o arkk suraock5 tfot is, tfk sfkor strkss is zkrg gh tfksk gtfkr suraocks. Tfk dirkctigh ga tfk s cggrdihotk is arg` tfk arkk suraock tgword tfk pgiht ot wficf tfk sfkor strkss is jkihn colculotkd. Ih o jko` crgss skctigh, tfk tgp, jgttg`, ohd sidk suraocks ork olwoys ossu`kd tg jk suraocks gh wficf tfk sfkor strkss is zkrg. ^k aurtfkr ossu`k tfot tfk jko` is tfih pkrpkhdiculor tg tfk ckhtkrlihk ga tfk crgss skctigh (i.k., skctigh (i.k., t is is s`oll). Tfk ossu`ptighs i`ply
Josic Tfkgriks ga Ghk-Di`khsighol Structurol @k`jkrs
44
• tfk sfkor strkss hgr`ol tg tfk ckhtkrlihk is s`oll jkcousk tfk suraocks gh kitfkr sidk ga tfk ckhtkrlihk ork arkk suraocks, ohd fkhck coh jk hknlkctkd. • tfk sfko sfkorr strkss strkss τ sx is uhiagr` ih tfk dirkctigh hgr`ol tg tfk ckhtkrlihk. H H Jy kquilijriu` ga agrcks ih tfk x tfk x dirkctigh dirkctigh ih Ainurk 3.?2j 3.?2j, wk gjtoih ( H s + d s) √ H s + τ sx t dx 0 2, gr
d H
d 0 – -----dx dx
∥σ
τ sx t 0 – --------- s
xx d O
(3.?;-J)
O s
Ih tfk ojgvk kquotigh, tfk idkhtiaicotigh ga tfk orko O orko O s is criticol. Orko criticol. Orko O s is tfk orko gh tfk crgss skctigh tfot is jktwkkh tfk arkk suraock ohd tfk pgiht ot wficf tfk sfkor strkss is jkihn kvoluotkd. Sujstitutihn Kquotigh (3.9-J) ihtg (3.9-J) ihtg Kquotigh (3.?;-J), (3.?;-J), ohd hgtihn tfot tfk `g`kht @ ihkrtio I z ohd tfk orko `g`kht ga ihkrtio I zz dg hgt vory gvkr tfk crgss skctigh, wk gjtoih
τ sx t 0
d @ z _ z 0 ------ ------------dx I zz
d @ ------ ------ z - y d O dx I zz
∥
O s
(3.?1-J)
wfkrk _ z is rkakrrkd tg os tfk airst `g`kht ga tfk orko O orko O s ohd is dkaihkd os _ z 0
∥ y d O
(3.?>-J)
O s
^k ossu`k tfk ossu`k tfk jko` is hgt topkrkd . Tfis i`pliks tfot I tfot I ga x,, ohd tfksk quohtitiks coh jk toekh zz ohd _ z ork hgt auhctighs ga x gutsidk tfk dkrivotivk sinh, wk gjtoih τ sx t 0 ( _ z ⁄ I zz ) ( d @ d x . Sujstitutihn Kquotigh (3.??j-J), (3.??j-J) , tfk rklotighsfip jktwkkh ⁄ ) z sfkor agrck Q y ohd `g`kht @ z , wk gjtoih _ z Q y⎘ τ sx t 0 – ⎕⎐ -----------⎢
gr
I zz
O s
τ sx
0
y
τ xs
Q y _ z 0 – ⎕ ------------ ⎘ ⎐ I zz t ⎢
Ckhtrgid ga O ga O s
Lihk olghn wficf sfkor strkss is jkihn aguhd.
y s S
(3.?6-J)
_ z 0 O s y s
S Hkutrol Oxis
z
O s*
y s*
_ z ∞ 0 O s∞ y s∞
Ckhtrgid ga O ga O s* s*
Ainurk 3.?? Colculotigh ga _ z .
Arg` Kquotigh (3.?>-J), (3.?>-J), wk hgtk tfot _ z is tfk airst `g`kht ga tfk orko O orko O s ojgut tfk z tfk z oxis. oxis. Ainurk 3.?? sfgws 3.?? sfgws tfk orko O s jktwkkh tfk tgp arkk suraock ohd tfk pgiht ot wficf sfkor strkss is jkihn aguhd (tfk S – S S lihk). lihk). Tfk ihtknrol ih Kquotigh (3.?>-J) is tfk hu`krotgr ih tfk dkaihitigh ga tfk ckhtrgid ga tfk orko O orko O s. Oholgngus tg tfk `g`kht duk tg o agrck, tfk airst `g`kht ga oh orko coh jk aguhd jy plocihn tfk orko O orko O s ot its ckhtrgid ohd aihdihn tfk `g`kht ojgut tfk hkutrol oxis. Tfot is, _ z is tfk prgduct ga orko O s ohd tfk distohck ga t fk ckhtrgid ga tfk orko O s arg` tfk hkutrol oxis, oxis, os sfgwh ih Ainurk 3.??. 3.??. Oltkrhotivkly, _ z coh jk aguhd jy usihn tfk jgttg` suraock os tfk arkk suraock, sfgwh os _ z * ih Ainurk 3.??. 3.??. Ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols it wos sfgwh tfot _ z is `oxi`u` ot tfk hkutrol oxis5 oxis5 fkhck jkhdihn sfkor strkss is `oxi`u` ot tfk hkutrol oxis ga o crgss skctigh. ^k hgtk tfot sihck y sihck y is is `kosurkd arg` tfk ckhtrgid, tfk airst `g`kht ga tfk khtirk crgss-skctighol orko is zkrg5 fkhck _ z + _ z ∞ 0 2 . Tfk kquotigh i`pliks tfot _ z ohd _ z * will fovk tfk so`k `onhitudk jut gppgsitk sinhs. Tfus, wk will n kt tfk so`k `onhitudk ga tfk sfkor strkss wfktfkr wk usk _ z gr _ z * ih Kquotigh (3.?6-J). (3.?6-J). Jut wficf nivks tfk cgrrkct sinh (dirkctigh)7 Tfk ohswkr is tfot jgtf will nivk tfk cgrrkct sinh, prgvidkd tfk agllgwihn pgiht is rk`k`jkrkd. Tfk s Tfk s dirkctigh dirkctigh ih Kquotigh (3.?6-J) is (3.?6-J) is `kosurkd arg` tfk arkk suraock uskd ih tfk colculotigh ga _ z.. Tfk prgduct ga tfk sfkor strkss ohd tfk tficehkss is collkd tfk sfkor tfk sfkor algw, algw, dksinhotkd jy tfk sy`jgl q, os sfgwh jklgw. q 0
τ xs t
(3.?9-J)
Tfk uhits ga sfkor algw q ork agrck ork agrck pkr uhit lkhntf. lkhntf. Tfk tkr`ihglgny is arg` aluid algw ih cfohhkls, jut tfk tkr` is uskd kxtkhsivkly agr discussihn sfkor strkssks ih tfih crgss skctighs, prgjojly jkcousk ga tfk visuol i`onk ga o algw it cghvkys ih tfk
4;
3
Josic Structurol @k`jkrs
discussigh ga sfkor strkss dirkctigh. Tfk dirkctigh ga sfkor algw (sfkor strkss) coh jk dktkr`ihkd jy ihspkctigh os klojgrotkd hkxt ohd ih Kxo`plk 3.6. 3.6. Drow tfk sfkor algw olghn tfk ckhtkrlihk ga tfk crgss skctigh ih o dirkctigh tfot will sotisay tfk agllgwihn rulks= ?. Tfk rksul rksultoh tohtt agrck agrck ih tfk y tfk y dirkctigh dirkctigh is ih tfk so`k dirkctigh os Q y . :. Tfk rksul rksultoh tohtt agrck agrck ih tfk z tfk z dirkctigh dirkctigh is zkrg. 3. It is sy``k sy``ktri tricc ojgut ojgut tfk y tfk y oxis. oxis. Tfis rkquirks sfkor algw tg cfohnk dirkctigh os ghk crgssks tfk y tfk y oxis oxis gh tfk ckhtkrlihk. Sg`kti`ks tfis will i`ply tfot sfkor strkss is zkrg ot tfk pgiht(s) wfkrk ckhtkrlihk ihtkrskcts tfk y tfk y oxis. ^k coh aihd tfk `onhitudk ohd tfk dirkctigh ga tfk jkhdihn sfkor strkss ih twg woys= ?. [sk Kquotigh (3.?6-J) tg (3.?6-J) tg aihd tfk `onhitudk ga tfk sfkor strkss. [sk tfk rulks must listkd tg dktkr`ihk tfk dirkctigh ga sfkor strkss. :. Agllgw tfk sinh sinh cghvkhtigh cghvkhtigh agr dktkr`ihihn dktkr`ihihn tfk sfkor sfkor agrck Q y . Tfk sfkor strkss is aguhd arg` Kquotigh (3.?6J), J), ohd tfk dirkctigh ga sfkor strkss is dktkr`ihkd arg` tfk sujscripts.
3.:. 3.:.4 4
Strks trkssk skss ohd ohd Stroi troihs hs
Tojlk 3.3 sfgws 3.3 sfgws tfk strkssks ohd stroihs ih tfk tfrkk structurol klk`khts discusskd sg aor. Tojlk 3.3 Strkssks ohd Stroihs Oxiol Strkssks
Sy``ktric Jkhdihn Stroihs
Strkssks Ojgut tfk z tfk z Ox Oxis
σ
Tgrsigh Stroihs
σ
σ xx 0
2
λ xx 0
2
2
ΰ σ ⎘ --λ yy 0 – ⎕⎐ -------- xx K ⎢
σ yy 0
2
λ yy 0
2
2
ΰ σ ⎘ --λ zz 0 – ⎕⎐ -------- xx K ⎢
σ zz 0
2
λ zz 0
2
--- xs --N
τ x ζ 0
T Ϗ ------ M
γ x ζ 0
--- x----ζ N
2
τ yz 0
2
γ yz 0
2
@ z σ xx 0 – ⎕⎐ ------ y----⎘⎢ I yy
ΰ σ ⎘ --λ yy 0 – ⎕⎐ -------- xx K ⎢
σ yy 0
2
σ yy 0
2
ΰ σ ⎘ --λ zz 0 – ⎕⎐ -------- xx K ⎢
σ zz 0
2
σ zz 0
2
γ xy 0 γ xz 0 γ yz 0
2
Q _ τ xs 0 – ⎕⎐ ---- y-------- z ⎘⎢ I zz t
Q _ τ xs 0 – ⎕⎐ ---- z -------- y⎘⎢ I yy t
γ xs 0
2
τ yz 0
τ yz 0
γ yz 0
H --- O
σ yy 0
2
σ zz 0 τ xy 0 τ xz 0 τ yz 0
2 2
λ xx 0
xx ------ K
2
2
Stroihs
Ojgut tfk y Oxis y Oxis
@ y σ xx 0 – ⎕⎐ ------ z ---- ⎘⎢ I zz
σ xx 0
Strkssks
λ xx 0
2
---- xx -- K
τ
τ
Strkss agr`ulos agr jkhdihn ojgut tfk y tfk y oxis oxis coh jk gjtoihkd arg` tfk agr`ulos agr jkhdihn ojgut tfk z tfk z oxis oxis jy ihtkrcfohnihn y cfohnihn y ohd ohd z z . Tfk nkhkrolizkd Fggek’s low coh jk uskd tg gjtoih tfk str oihs arg` tfk agr`ulos dkrivkd agr st rkssks. Tfk jkhdihn sfkor strkss τ xs ohd tfk tgrsighol sfkor strkss τ xζ cguld jk τ xy gr τ xz gr hkitfkr, dkpkhdihn upgh tfk lgcotigh ga tfk pgiht. Fgwkvkr, gur strkss ohd stroih trohsagr`otigh agr`ulos ork ih Cortksioh cggrdihotk systk`s. Tfus, wk hkkd tg trohsagr` tfk sfkor strkssks ihtg strkssks ih Cortksioh cggrdihotks. Tfis coh jk dghk jy airst sfgwihn tfk strkssks arg` tfk agr`ulo gh o strkss cujk ohd tfkh kxo`ihihn tfk strkssks gh tfk strkss cujk ih o Cortksioh cggrdihotk systk`.
KVO@RLK 3.6 Ossu`ihn o pgsitivk sfkor agrck Q y , sektcf tfk dirkctigh ga tfk sfkor algw olghn tfk ckhtkrlihk gh tfk tfih crgss skctighs sfgwh ih Ainurk 3.?:. 3.?:. y
y O
y
y O
O O
z
z
z
z
J
(o) 3.6. Ainurk 3.?: Crgss skctighs agr Kxo`plk 3.6.
(j)
(c)
(d )
Jguhdory Qoluk Rrgjlk`s
41
OHS. D
O
K
O
K
D
K
(o)
J
K
D
D
C
A
O C
C
O
C
(j)
(c)
(d )
3.6. Ainurk 3.?3 Sfkor algw agr Kxo`plk 3.6.
KVO@RLK 3.9 O pgsitivk sfkor agrck ga Q 3.?4. Dktkr`ihk tfk sfkor algw olghn tfk y 0 32 H octs gh tfk tfih crgss skctighs sfgwh ih Ainurk 3.?4. ckhtkrlihks ohd sektcf it.
y y
?22 ?22 `` O
D 1
I zz 0 :.3; ( ?2 ) ``
z
4
K
K
?2``
I zz 0 1.6 ( ?2 ) ``
G
?22 ?22 ``
>>.; >>.; ``
?2`` 1
z
G
A
O
4
?22 ?22 ``
>2.1 >2.1 `` C
C
D
?22 `` ?2 `` ?2 `` ?22
?2 ``
(j) Cosk II
(o) Cosk I 3.9 (hgt drowh tg scolk). Ainurk 3.?4 Crgss skctighs agr Kxo`plk 3.9 (hgt
OHS. Ihtkhsity ga Sfkor Algw
Ihtkhsity ga 63.3 63.3 H/` Sfkor Algw
?>;.; ?>;.; H/` D
63.3 63.3 H/` K
K
63.3 3.3 H`
O
63.3 H/` A 63.
O 3;? 3;? H/` H/`
G
Ihtkhsity ga Sfkor Algw
?29.9 ?29.9 H/` 363.4 363.4 H/`
Ihtkhsity ga Sfkor Algw
C
(o)
Ihtkhsity ga Sfkor Algw
G
?29.9 ?29.9 H/`
D
C
(j)
3.9. Ainurk 3.?; Sfkor algw gh tfk crgss skctighs ga Kxo`plk 3.9.
3.3
JG[H JG [HDO DOZY ZY QOL[K OL[K RZGJ RZGJLK@ LK@S S
O jguhdory voluk prgjlk` is o `otfk`oticol listihn ga tfk diaakrkhtiol kquotighs, tfk jguhdory cghditighs, ohd ohy gtfkr cghditighs hkckssory tg sglvk tfk diaakrkhtiol kquotighs. Tfk diaakrkhtiol kquotighs agr tfk oxiol, jkhdihn, ohd tgrsigh cosks ork nivkh jy Kquotighs (3.?:-O), (3.?:-O), (3.?:-J), (3.?:-J), ohd (3.?:T), T), rkspkctivkly ih Tojlk 3.:. 3.:. Tfk distrijutkd lgods p lgods p x( x), x), p p y( x), x), ohd t ( x) x) ih tfk diaakrkhtiol kquotighs ork ehgwh auhctighs. Tfk ihtknrotigh ga tfksk distrijutkd lgods rksults ih tfk porticulor sglutigh. Tfk fg`gnkhgus sglutigh cgrrkspghds tg tfk rinft fohd sidk ga tfk diaakrkhtiol kquotigh skt tg zkrg, i.k., wk skt p skt p x( x), x), p p y( x), x), ohd t ( x) x) tg zkrg ih tfk diaakrkhtiol kquotigh tg gjtoih tfk fg`gnkhgus sglutigh jy ihtknrotigh. Tfk su` ga tfk porticulor sglutigh ohd fg`gnkhgus sglutigh rkprkskhts tfk cg`plktk sglutigh ga tfk diaakrkhtiol kquotigh. Tfk fg`gnkhgus sglutigh olwoys cghtoih uhehgwh cghstohts. Tfksk cghstohts ork dktkr`ihkd arg` tfk jguhdory cghditighs gh tfk structurol `k`jkrs. Tfk jguhdory cghditighs, wficf ork cghditighs gh tfk ihtkrhol agrcks/`g`khts gr displock`khts/rgtotighs ot tfk twg khds (jguhdory) ga tfk `k`jkr, coh jk writtkh kitfkr jy ihspkctigh gr jy drowihn o arkk jgdy dionro` ga oh ihaihitksi`ol klk`kht ot tfk khd, os discusskd ih tfk skctighs tfot agllgw. Tg uhdkrstohd tfk prihciplk ga fgw jguhdory cghditighs ork dktkr`ihkd, wk nkhkrolizk o prihciplk discusskd ih stotics agr dktkr`ihihn tfk rkoctigh agrck ohd/gr `g`khts. Zkcoll tfot ih drowihn arkk jgdy dionro`s, wk uskd tfk agllgwihn prihciplks agr dktkr`ihihn rkoctigh agrcks ohd `g`kht ot tfk suppgrts.
4>
3
Josic Structurol @k`jkrs ?. Ia o pgiht cohhgt `gvk ih o nivkh dirkctigh, tfkh o rkoctigh agrck gppgsitk tg tfk dirkctigh octs ot tfot suppgrt pgiht. :. Ia o lihk cohhgt rgtotk ojgut oh oxis ih o nivkh dirkctigh, tfkh o rkoctigh `g`kht gppgsitk tg tfk dirkctigh octs ot tfot suppgrt.
Ia wk wkrk tg `oek oh i `onihory cut vkry clgsk tg tfk suppgrt ohd tfkh drow o arkk jgdy dionro`, wk wguld aihd tfot tfk ihtkrhol agrck will kquol tg tfk rkoctigh agrck ohd tfk ihtkrhol `g`kht will kquol tfk rkoctigh `g`kht. Tfus, tfk airst gjskrvotigh i`pliks tfot ia o pgiht cohhgt `gvk (displock`kht is zkrg), tfkh tfk ihtkrhol agrck is hgt ehgwh jkcousk tfk rkoctigh agrck is hgt ehgwh. Si`ilorly, tfk skcghd gjskrvotigh i`pliks tfot ia o lihk cohhgt rgtotk (rgtotigh is zkrg) orguhd oh oxis ot tfk suppgrt, tfkh tfk ihtkrhol `g`kht is hgt ehgwh jkcousk tfk rkoctigh `g`kht is hgt ehgwh. Tfk rkvkrsk is kquolly truk. Ia structurol `k`jkr’s khd is arkk, tfkh tfk ihtkrhol agrck ohd ihtkrhol `g`kht will jk zkrg5 jut tfk arkk khd coh displock ohd rgtotk jy ohy o`guht tfot is dictotkd jy tfk distrijutkd lgodihn. Tfus, wfkh wk spkciay o voluk ihtkrhol agrck, tfkh wk cohhgt spkciay displock`kht5 ohd wfkh wk spkciay tfk voluk ga oh ihtkrhol `g`kht ot o pgiht, tfkh wk cohhgt spkciay rgtotigh. ^k will skk tfis hoturk ga jguhdory cghditighs agr tfk tfrkk structurol `k`jkrs discusskd hkxt.
3.3. 3.3.? ?
Oxio Oxioll Disp Displo lock ck`k `kht ht
Tfk oxiol displock`kht u2( x) x) coh jk aguhd arg` tfk diaakrkhtiol kquotigh (3.?:-O) wficf (3.?:-O) wficf wk rkwritk jklgw agr cghvkhikhck. • Diaakrkhtio Diaakrkhtioll kquotigh kquotigh du d ------ ⎕ KO --------2⎘ 0 – p x ( x ) dx ⎐ dx ⎢
(3.:2-O)
Ihtknrotihn tfk diaakrkhtiol kquotigh twick wk gjtoih= du 2 KO -------- 0 I ? ( x ) + C ? dx KO u 2 0 I : ( x ) + C ? x + C :
I ? ( x ) 0 – p x ( x ) d x I : ( x ) 0
∥ I ( x )d x ?
(3.:?-T) (3.::-T)
wfkrk, tfk ihdkaihitk ihtknrols coh jk aguhd os p os p x( x) x) is ehgwh5 C ? ohd C : ork cghstohts tg jk dktkr`ihkd arg` tfk jguhdory cghditighs. Tfk porticulor sglutigh is I : ( x ) ⁄ ( KO ) ohd tfk fg`gnkhgus sglutigh is ( C ? x + C : ) ⁄ ( KO ) . Ih tfk ojgvk ihtknrotigh, wk ossu`kd KO ossu`kd KO is is o cghstoht, jut ia it is hgt, tfkh it coh jk kosily occguhtkd agr jy dividihn Kquotigh (3.:?-T) jy (3.:?-T) jy KO KO jkagrk jkagrk ihtknrotihn. ^k hgtk tfot tfk ihtkrhol oxiol agrck is= H 0 K O ( d u2 ⁄ dx ) . Os discusskd korlikr, it is tfk hoturk ga jguhdory cghditighs tfot ot kocf khd (jguhdory) ga tfk oxiol `k`jkr, kitfkr tfk oxiol displock`kht u2 gr tfk ihtkrhol oxiol agrck H `ust `ust jk spkciaikd os sfgwh jklgw. • Jguhdory Jguhdory cghditigh= cghditigh= Ot kocf khd khd spkciay spkciay u2
gr
H
(3.:3-O) p ( x )λ x O
H O
λ
A kxt
Ainurk 3.?1 Jguhdory cghditigh gh tfk ihtkrhol oxiol agrck.
Tg dktkr`ihk tfk jguhdory cghditigh gh tfk ihtkrhol oxiol agrck H , wk `oek oh i`onihory cut oh ihaihitksi`ol distohck (λ) arg` tfk khd (soy pgiht O pgiht O)) ohd drow o arkk jgdy dionro` ga tfk ihaihitksi`ol klk`kht os ih Ainurk 3.?1. 3.?1. Tfk ihtkrhol oxiol agrck is drowh ih tkhsigh ih occgrdohck witf gur sinh cghvkhtigh (skk Tojlk 3.:). 3.:). Jy kquilijriu` ga agrcks ohd jy toeihn tfk li`it λ ↚ 2 , wk gjtoih tfk agllgwihn jguhdory cghditigh= li` ( A kxt – H O – p x ( x O )λ ) 0 2
λ↚2
gr
H O 0 A kxt
Tfk ojgvk kquotigh sfgws tfot tfk distrijutkd oxiol agrck dgks hgt oaakct tfk jguhdory cghditigh gh tfk ihtkrhol oxiol agrck. Tfk `onhitudk ga tfk ihtkrhol oxiol agrck H ot ot tfk khd ga oh oxiol jor is kquol tg tfk cghckhtrotkd kxtkrhol oxiol agrck opplikd ot tfk khd, jut nkttihn tfk rinft tfk rinft sinh rkquirks sinh rkquirks drowihn tfk arkk jgdy dionro` witf tfk ihtkrhol agrck H drowh drowh ih occgrdohck witf tfk sinh cghvkhtigh. Ia A Ia A kxt is zkrg, wk coh writk tfk jguhdory cghditigh ga zkrg voluk ga H jy jy ihspkctigh.
3.3. 3.3.: :
Tgrsi grsigh ghol ol Zgto Zgtoti tigh gh
Tfk tgrsighol rgtotigh ρ( x) x) coh jk aguhd arg` tfk diaakrkhtiol kquotigh (3.?:-T) wficf (3.?:-T) wficf wk rkwritk jklgw agr cghvkhikhck. • Diaakrkhtio Diaakrkhtioll kquotigh kquotigh d d ρ ------ ⎕ NM ------⎘ 0 – t ( x ) ⎐ dx dx ⎢
(3.:4-T)
Jguhdory Qoluk Rrgjlk`s
46
Ihtknrotihn tfk diaakrkhtiol kquotigh twick wk gjtoih= d ρ NM ------ 0 I ? ( x ) + C ? dx
I ? ( x ) 0 – t ( x ) d x
(3.:;-T)
NM ρ 0 I : ( x ) + C ? x + C :
I : ( x ) 0
∥ I ( x )d x
(3.:1-T)
∥
?
wfkrk, tfk ihdkaihitk ihtknrols coh jk aguhd os t ( x) x) is ehgwh5 C ? ohd C : ork cghstohts tg jk dktkr`ihkd arg` jguhdory cghditighs. Tfk porticulor sglutigh is I : ( x ) ⁄ ( NM ) ohd tfk fg`gnkhgus sglutigh is ( C ? x + C : ) ⁄ ( NM ) . Ih tfk ojgvk ihtknrotigh, wk ossu`kdNM ossu`kdNM is o cghstoht, jut ia it is hgt, tfkh it coh jk kosily occguhtkd agr jy dividihn Kquotigh (3.:;-T) jy (3.:;-T) jy NM jkagrk jkagrk ihtknrotihn. ^k hgtk tfot ihtkrhol tgrquk is= T 0 NM ( d ρ / dx ) . Os discusskd korlikr, it is tfk hoturk ga jguhdory cghditighs tfot ot kocf khd (jguhdory) ga tfk circulor sfoat, kitfkr tfk rgtotigh ρ gr tfk ihtkrhol tgrquk T `ust `ust jk spkciaikd os sfgwh jklgw. • Jguhdory Jguhdory cghditigh= cghditigh= Ot kocf kocf khd spkciay spkciay
ρ
gr
T
(3.:>-T)
Tg dktkr`ihk tfk jguhdory cghditigh gh ihtkrhol tgrquk T , wk `oek oh i`onihory cut oh ihaihitksi`ol distohck (λ) arg` tfk khd (soy pgiht O pgiht O)) ohd drow o arkk jgdy dionro` ga tfk ihaihitksi`ol klk`kht os sfgwh ih Ainurk 3.?>. 3.?>. Tfk ihtkrhol tgrquk T is is drowh ih occgrdohck witf gur sinh cghvkhtigh (skk Tojlk 3.:).. 3.:).. T O
t ( x O )λ λ
T kxt
Ainurk 3.?> Jguhdory cghditigh gh tfk ihtkrhol tgrquk.
Jy kquilijriu` ga `g`khts ohd jy toeihn tfk li`it λ ↚ 2 , wk gjtoih tfk jguhdory cghditigh li` ( T kxt – T O – t ( x O )λ ) 0 2
λ↚2
gr
T O 0 T kxt
Tfk ojgvk kquotigh sfgws tfot tfk distrijutkd tgrquk dgks hgt oaakct tfk jguhdory cghditigh gh tfk ihtkrhol tgrquk. Tfk `onhitudk ga tfk ihtkrhol tgrquk T ot ot tfk khd ga o sfoat is kquol tg tfk cghckhtrotkd kxtkrhol tgrquk opplikd ot tfk khd, jut nkttihn tfk rinft tfk rinft sinh rkquirks sinh rkquirks drowihn tfk arkk jgdy dionro` witf tfk ihtkrhol tgrquk tgrquk T drowh ih occgrdohck witf tfk sinh cghvkhtigh. Ia T kxt is zkrg, wk coh writk tfk jguhdory cghditigh ga zkrg voluk ga T jy jy ihspkctigh.
3.3. 3.3.3 3
Jko` Jk o` Dkal Dkalkc kcti tigh gh
Tfk jko` dkalkctigh v( x) x) coh jk aguhd arg` tfk diaakrkhtiol kquotigh (3.?:-J) wficf (3.?:-J) wficf is writtkh jklgw agr cghvkhikhck. • Diaakrkhtio Diaakrkhtioll kquotigh kquotigh :
:
d d v -------- ⎕ KI zz --------⎘ 0 p y ( x ) :⎐ : dx dx ⎢
(3.:6-J)
Ihtknrotihn tfk diaakrkhtiol kquotigh agur ti`ks wk gjtoih= :
d d v ------ ⎕ KI zz --------⎘ 0 I ? ( x ) + C ? : dx ⎐ dx ⎢ :
d v KI zz --------: 0 I : ( x ) + C ? x + C : dx d v : KI zz ------ 0 I 3 ( x ) + C ? ( x ⁄ : ) + C : x + C 3 dx KI zz v 0 I 4 ( x ) + C ? ( x
3
⁄ 1 ) + C : ( x : ⁄ : ) + C 3 x + C 4
I ? ( x ) 0
∥ p ( x ) d x
(3.:9o-J)
I : ( x ) 0
∥ I ( x )d x
(3.:9j-J)
I 3 ( x ) 0
I : ( x ) d x
(3.:9c-J)
I 4 ( x ) 0
∥ I ( x )d x
(3.:9d-J)
y
?
3
wfkrk, tfk ihdkaihitk ihtknrols coh jk aguhd os p os p y( x) x) is ehgwh5 C ?, C :, C 3, C 4 ork cghstohts tg jk dktkr`ihkd arg` jguhdory cgh3 : ditighs. Tfk porticulor sglutigh is I 4 ( x ) ⁄ ( KI zz ) ohd tfk fg`gnkhgus sglutigh is U C ? ( x ⁄ 1 ) + C : ( x ⁄ : ) + C 3 x + C 4 P ⁄ ( KI zz ) . Ih tfk ojgvk ihtknrotigh, wk ossu`kd KI zz is o cghstoht, jut ia it is hgt, tfkh it coh jk kosily occguhtkd agr jy dividihnKquotigh dividihnKquotigh (3.:9cJ) jy KI zz jkagrk ihtknrotihn. ^k writk tfk agllgwihn kxprkssighs agr tfk ihtkrhol `g`kht ohd sfkor agrck arg` Kquotighs (3.6-J) ohd (3.6-J) ohd (3.??j-J) :
d v @ z ( x ) 0 KI zz --------: dx
(3.32o-J)
49
3
Josic Structurol @k`jkrs
⎕ Q y ( x ) 0 –
d @ z ⎘
⎐ d x ⎢
:
d d v 0 – ------ ⎕ KI zz --------:⎘ dx ⎐ dx ⎢
(3.32j-J)
@ Tfk ihtknrotigh ga tfk diaakrkhtiol kquotigh agur ti`ks rksultkd ih v, d v ⁄ d x , @ z , gr Q y. Tfk agur quohtitiks agr` twg nrgups agr tfk purpgsk ga dktkr`ihihn tfk jguhdory cghditighs.
• Nrgup ?= ?= Ot o jguhdory pgiht kitfkr tfk dkalkctigh v gr tfk ihtkrhol sfkor agrck Q y coh jk spkciaikd, jut hgt jgtf. • Nrgup := := Ot o jguhdory pgiht kitfkr tfk slgpk d v ⁄ dx gr tfk ihtkrhol jkhdihn `g`kht @ z coh jk spkciaikd, jut hgt jgtf. Tg nkhkrotk agur jguhdory cghditighs, twg cghditig hs ork spkciaikd ot kocf khd ga tfk jko`. ^k cfggsk ghk cghditigh arg` nrgup ? ohd ghk arg` nrgup :5 oltkrhotivkly wk coh `oek tfk agllgwihn stotk`kht. • Jguhdory Jguhdory cghditighs= cghditighs= ot kocf kocf khd spkciay spkciay v
gr
Q y
oh d
(3.32c-J) d v -----@ z gr dx Tg dktkr`ihk tfk jguhdory cghditigh gh tfk ihtkrhol `g`kht @ z , ohd tfk sfkor agrck Q y, wk `oek oh i`onihory cut oh ihaihitksi`ol distohck (λ) arg` tfk khd (soy pgiht O pgiht O)) ohd drow o arkk jgdy dionro` ga tfk ihaihitksi`ol klk`kht os sfgwh ih Ainurk 3.?1. 3.?1. Tfk ihtkrhol sfkor agrck ohd ihtkrhol jkhdihn `g`kht ork drowh ih occgrdohck witf gur sinh cghvkhtigh (skk Tojlk 3.:) 3.:) Q O
p y ( x O )λ
y @ O
z
A kxt
x @ kxt λ
Ainurk 3.?6 Jguhdory cghditigh gh tfk ihtkrhol sfkor agrck ohd jkhdihn `g`kht.
Jy kquilijriu` ga agrcks ohd `g`kht ohd jy toeihn tfk li`it λ ↚ 2 , wk gjtoih tfk jguhdory cghditighs jklgw. li` ( A kxt + Q O – p y ( x O )λ ) 0 2
λ↚2
gr
Q O 0 – A kxt
(3.3?o-J)
λ:
@ O 0 @ kxt (3.3?j-J) li` ⎕ @ O – @ kxt + λ A kx t + p y ( x O ) ----⎘ 0 2 gr :⎢ λ ↚ 2⎐ Tfk ojgvk kquotighs sfgw tfot tfk distrijutkd agrck dgks hgt oaakct tfk jguhdory cghditigh gh kitfkr tfk ihtkrhol sfkor agrck gr tfk jkhdihn `g`kht. Gh o jguhdory pgiht, tfk voluk ga tfk ihtkrhol sfkor agrck Q y ohd tfk jkhdihn `g`kht @ z is kquol ih `onhitudk tg tfk cghckhtrotkd kxtkrhol agrck ohd tfk cghckhtrotkd kxtkrhol `g`kht, rkspkctivkly5 jut nkttihn tfk tfk rinft rinft sinh rkquirks drowihn tfk arkk jgdy dionro` witf tfk ihtkrhol sfkor agrck Q y ohd tfk ihtkrhol jkhdihn `g`kht @ z drowh ih occgrdohck witf tfk sinh cghvkhtigh. Ia A Ia A kxt ohd @ ohd @ kxt ork zkrg, wk coh writk tfk jguhdory cghditigh ga zkrg voluk agr sfkor agrck ohd `g`kht jy ihspkctigh. O syhgpsis ga tfk jguhdory voluk prgjlk`s agr oxiol, tgrsigh, ohd sy``ktric jkhdihn is prkskhtkd ih Tojlk 3.4. 3.4. Qoluk Rrgjlk`s Tojlk 3.4 Syhgpsis ga Jguhdory Qoluk Oxiol
Tgrsigh
Diaakrkhtiol kquotigh
du d ------ ⎕ KO --------2⎘ 0 – p x ( x ) ⎐ dx dx ⎢
d d ρ ------ ⎕ NM ------⎘ 0 – t ( x ) dx⎐ dx ⎢
d ⎕ d v⎘ -------- ⎜ KI zz -------- ⎗ 0 p y ( x ) : : dx ⎐ dx ⎢
Jguhdory cghditighs ot kocf khd
u2
ρ
v
gr
H
gr
Sy``ktric Jkhdihn
T
:
:
gr ohd
d v -----dx Ihtkrhol agrcks/`g`khts
Q y
d u2 H 0 KO -------dx
d ρ T 0 NM -----dx
gr
@ z
:
@ z
0
d v KI zz --------: dx
Q y
: d ⎕ d v⎘ 0 – ------ ⎜ KI zz --------:⎗ dx⎐ dx ⎢
Discghtihuity Auhctighs
;2
KVO@RLK 3.?2 Tfk kxtkrhol tgrquk gh o drill jit voriks lihkorly tg o `oxi`u` ihtkhsity ga q ih ⋋ lj/ih os sfgwh ih Ainurk 3.?9. 3.?9. Ia tfk drill jit dio`ktkr is d , its lkhntf L lkhntf L,, ohd tfk `gdulus ga rinidity N, dktkr`ihk tfk rklotivk rgtotigh ga tfk khd ga tfk drill jit wit f rkspkct tg tfk cfuce. L x J
O
q ( x ⁄ L ) ih.-lj/ih.
3.?2. Ainurk 3.?9 Distrijutkd tgrquk gh tfk drill jit ga Kxo`plk 3.?2.
OHS. :
3:qL 3: qL ρ J – ρ O 0 ---------------3 ς Nd 4
ccw
KVO@RLK 3.?? O linft pglk is sujmkctkd tg wihd prkssurk tfot voriks quodroticolly os sfgwh ih Ainurk 3.:2. 3.:2. Ih tkr`s ga K ga K , I , w, L, L, ohd x ohd x,, dktkr`ihk (o) tfk dkalkctigh ot tfk arkk khd ohd (j) tfk nrguhd rkoctighs. w
J
:
x w ⎕ -----:⎘ ⎐ L ⎢
L
x y O 3.??. Ainurk 3.:2 Jko` ohd lgodihn agr Kxo`plk 3.??.
OHS. 4
v( L)
?3wL ?3 wL ⎘ - 5 0 – ⎕⎐ -----------------⎢ ?62 KI zz
Z O
0
wL ⁄ 3
@ O
0
wL
:
⁄ 4
KVO@RLK 3.?: Ih tkr`s ga K ga K , I , w, L, L, ohd x ohd x,, dktkr`ihk (o) tfk klostic curvk ohd (j) tfk rkoctigh agrck ot O ot O ih Ainurk 3.:?. 3.:?. y
x
O
J
L 3.?:. Ainurk 3.:? Jko` ohd lgodihn agr Kxo`plk 3.?:.
OHS.
v( x )
3.4 3.4
4 3 : : w - ( : x – ; Lx + 3 L x ) 5 0 – ---------------
46 KI zz
Z O
0
3 wL ⁄ 6
DISC DISCGH GHTI TIH[ H[IT ITY Y A[HCT A[HCTIG IGHS HS
O structurol `k`jkr usuolly fos distrijutkd lgods ohd pgiht lgods opplikd tg it. Rgiht agrcks cousk mu`ps ih tfk ihtkrhol agrcks, ohd pgiht `g`khts cousk mu`ps ih ihtkrhol `g`khts os ghk crgssks tfk pgiht lgods arg` ghk sidk tg tfk gtfkr. Tfk discghtihuity auhctighs pkr`it us tg `gdkl tfksk mu`ps sucf tfot o sihnlk kxprkssigh agr distrijutkd lgod coh jk uskd gvkr tfk khtirk lkhntf ga tfk structurol `k`jkr. Tfk usk ga must ghk kxprkssigh fos tfrkk odvohtonks=
;?
3
Josic Structurol @k`jkrs ?. [hliek tfk ihtkrhol agrcks ohd `g`khts, wficf ork discghtihugus ot pgihts ga cghckhtrotkd agrcks ohd `g`khts, tfk displock`khts ohd rgtotighs ork cghtihugus. Ih tfk ihtrgductgry cgursk gh `kcfohics ga `otkriols, tfk dis plock`khts ohd tfk rgtotigh colculotighs f od tg jk cghduct kd agr kocf skn`k ht skporotkl y, ohd tfk cghtihuity cghditighs fod tg jk i`pgskd kxplicitly tg kxplicitly tg sglvk tfk prgjlk`. ^itf discghtihuity auhctighs, tfk sihnlk kxprkssigh agr distrijutkd lgod coh jk sujstitutkd ihtg tfk diaakrkhtiol kquotighs (3.?:-O), (3.?:-O), (3.?:-J), (3.?:-J), ohd (3.?:-T) ohd (3.?:-T) ohd tfk jguhdory voluk prgjlk` sglvkd os wk did ih Skctigh 3.3. 3.3. Tfk cghtihuity cghditighs ork i`plicitly sotisaikd i`plicitly sotisaikd rkducihn tfk olnkjroic kaagrt sinhiaicohtly os sfoll jk skkh ih sujskqukht skctighs. :. Agr stoticolly ihdktkr`ihotk prgjlk`s tfk jguhdory voluk prgjlk` coh jk sglvkd os dkscrijkd ojgvk. Agr stoticolly dktkr`ihotk prgjlk`s, tfk sihnlk kxprkssigh agr distrijutkd lgods coh jk sujstitutkd ihtg tfk kquilijriu` kquotighs (3.??-O), (3.??-O), (3.??o-J), (3.??o-J), (3.??j-J), (3.??j-J), ohd (3.??-T) ohd (3.??-T) ohd ihtknrotkd tg yikld o sihnlk kxprkssigh agr tfk ihtkrhol agrcks ohd `g`khts gvkr tfk khtirk structurol `k`jkr. Ot ohy pgiht gh tfk `k`jkr, tfk voluks ga tfk ihtkrhol agrcks ohd `g`khts coh jk gjtoihkd jy sujstitutihn tfk opprgpriotk voluk ga x ga x.. Tfksk ihtkrhol agrcks ohd `g`khts coh jk uskd agr aihdihn strkssks ot ohy pgiht gh structurol `k`jkr. 3. Tfk kquotighs agr ihtkrhol agrcks ohd `g`khts coh jk uskd tg dktkr`ihk tfk `oxi`u` voluks ga tfk ihtkrhol quohtitiks kitfkr jy colculus, gr nropficolly jy plgttihn tfk kquotighs. Tfk `oxi`u` strkssks coh tfkh jk dktkr`ihk arg` tfk `oxi`u` voluks ga tfk ihtkrhol agrcks ohd `g`khts.
3.4.? 3.4.?
Dkaih Dkaihiti itigh gh ga Discg Discghti htihu huity ity Auh Auhcti ctigh ghss
Cghsidkr o distrijutkd lgod p lgod p ohd ohd oh kquivolkht lgod R lgod R 0 3.::. Suppgsk wk hgw lkt tfk ihtkhsity ga tfk dis 0 pλ os sfgwh ih Ainurk 3.::. trijutkd lgod cghtihugusly ihcrkosk tg ihaihity, wfilk wk dkcrkosk tfk lkhntf gvkr wficf tfk distrijutkd agrck is opplikd tg zkrg ih sucf o `ohhkr tfot tfk orko p orko pλ rk`oihs o aihitk quohtity5 tfkh wk gjtoih o cghckhtrotkd agrck R agrck R opplikd opplikd ot x ot x 0 o. o. @otfk`oticolly, tfis is stotkd os agllgws= R 0 li`
p ↚ ∘
li` ( p λ )
(3.3:)
λ↚2
Zotfkr tfoh writk tfk li`it gpkrotighs tg rkprkskht o cghckhtrotkd agrck, tfk agllgwihn hgtotigh is uskd agr rkprkskhtihn – ? tfk cghckhtrotkd agrck= R 〈 x – o⌭ . x 0 o
λ ⁄ :
〈 x – o⌭ – ?
λ ⁄ :
∘
x
p
x
x 0 o
x x 0 o
R0pλ
λ ⁄ : λ ⁄ :
(o)
(j)
Ainurk 3.:: Tfk dklto auhctigh. – ?
Tfk auhctigh 〈 x – o⌭ is collkd tfk Diroc dklto auhctigh, gr must tfk dklto auhctigh. Tfk dklto auhctigh is zkrg kxckpt ih oh ihaihitksi`ol rknigh hkor o. Os x Os x tkhds tkhds tgword o, tfk dklto auhctigh tkhds tg ihaihity, jut tfk orko uhdkr t fk auhctigh is kquol tg ghk. @otfk`oticolly, tfk dklto auhctigh is dkaihkd os agllgws= – ?
〈 x – o⌭
0
⎮2 ⎦ ⎩∘
o
x≢o x↚o x
λ
∥
〈 x – o⌭ – ? d x 0
?
(3.33o)
( o √λ )
∥
– ?
Hgw cghsidkr tfk ihtknrol ga tfk dklto auhctigh= 〈 x – o⌭ d x . Tfk lgwkr li`it ga `ihus ihaihity k`pfosizks tfot tfk – ∘ pgiht is jkagrk o. Ia x < o , tfkh ih tfk ihtkrvol ga ihtknrotigh, tfk dklto auhctigh is zkrg ot oll pgihts, ohd fkhck tfk ihtknrol voluk is zkrg. Ia x 8 o , tfkh wk coh writk tfk ihtknrol os tfk su` ga tfrkk ihtknrols= o + λ)
( o √λ )
∥ – ∘
– ?
〈 x – o⌭
d x +
∥ ( o √λ )
x – ?
〈 x – o⌭
d x +
∥
〈 x – o⌭ – ? d x
(o +λ )
Tfk airst ohd tfk tfird ihtknrols ork zkrg jkcousk tfk dklto auhctigh is zkrg ot oll pgihts ih tfk ihtkrvol ga ihtknrotigh, wfilk tfk skcghd ihtknrol is kquol tg ?, pkr Kquotigh (3.33o). (3.33o). Tfus, tfk ihtknrol 〈 x – o⌭ – ? d x is zkrg jkagrk o ohd ghk oatkr o. – ∘
Discghtihuity Auhctighs
;:
2
Tfk ihtknrol, rkprkskhtkd jy tfk hgtotigh 〈 x – o⌭ , is collkd tfk stkp auhctigh. It is dkaihkd ih Kquotigh (3.33j) ohd (3.33j) ohd sfgwh ih Ainurk 3.:3o 3.:3o. x
〈 x – o⌭
2
∥
0
〈 x – o⌭ – ? d x
0
– ∘
x < o⎥
⎮2 ⎦ ⎩?
⎬
(3.33j)
x 8 o⎠
〈 x – o⌭ ?
〈 x – o⌭ 2
〈 x – o⌭ : Zo`p auhctigh
Stkp auhctigh
x
x x 0 o
x
x0o
x0o (c)
(j)
(o) Ainurk 3.:3 Discghtihuity auhctighs. x
Hgw cghsidkr tfk ihtknrol ga tfk stkp auhctigh,
∥ 〈 x – o⌭ d x 2
– ∘
Ia x < o , tfkh ih tfk ihtkrvol ga ihtknrotigh tfk stkp auhctigh is zkrg ot oll pgihts, ohd fkhck tfk ihtknrol voluk is zkrg. Ia x 8 o , tfkh wk coh writk tfk ihtknrol os su` ga twg ihtknrols= o
x
∥ 〈 x – o⌭ d x + ∥ 〈 x – o⌭ d x 2
2
– ∘
o
Tfk airst ihtknrol is zkrg jkcousk tfk stkp auhctigh is zkrg ot oll pgihts ih tfk ihtkrvol ga ihtknrotigh, wfilk tfk skcghd x ihtknrol voluk is ( x – x – o). o). Tfk ihtknrol, ihtknrol, 〈 x – o⌭ 2 d x , collkd tfk ro`p auhctigh, is rkprkskhtkd jy tfk hgtotigh 〈 x – o⌭ ? . It is – ∘ sfgwh ih Ainurk 3.:3j 3.:3j. Rrgckkdihn ih tfis `o hhkr, wk coh dkaihk oh khtirk closs ga auhctighs tfot ork `otfk`oticolly rkprkskhtkd os agllgws=
∥
〈 x – o⌭ h
0
⎮2 ⎦ h ⎩ ( x – o )
x ≤ o ⎥
h≫2
⎬
x 8 o ⎠
(3.34)
^k coh olsg nkhkrotk tfk ihtknrol agr`ulo ga Kquotigh (3.3;) arg` (3.3;) arg` Kquotigh (3.34)= (3.34)= x
∥
〈 x – o⌭ h + ? 〈 x – o⌭ h d x 0 -------------------------
h≫2
h+?
(3.3;)
– ∘ – :
Ghk `grk auhctigh, collkd tfk dgujlkt auhctigh , rkprkskhtkd jy tfk hgtotigh 〈 x – o⌭
〈 x – o⌭ – :
0
⎮2 ⎦ ⎩∘
, is dkaihkd os
x
x≢o
⎥ ⎬ x ↚ o⎠
∥ 〈 x – o⌭
– :
d x 0
〈 x – o⌭ – ?
(3.31)
– ∘
– ?
– :
Tfk dklto auhctigh 〈 x – o⌭ ohd tfk dgujlkt auhctigh 〈 x – o⌭ jkcg`k ihaihitk ot x ot x 0 o. o. Oltkrhotivkly stotkd, tfksk h auhctighs ork sihnulor ot x ot x 0 o ohd o ohd ork rkakrrkd tg os sihnulority auhctighs. Tfk khtirk closs ga auhctighs 〈 x – o⌭ agr pgsitivk ohd hknotivk h ork collkd tfk discghtihuity auhctighs. Hgtk tfot tfk discghtihuity auhctighs ork zkrg ia tfk ornu`kht is hknotivk. hknotivk. Jy diaakrkhtiotihn Kquotighs (3.33j), (3.33j), (3.3;), (3.3;), ohd (3.31), (3.31), wk coh gjtoih tfk agr`ulos agr diaakrkhtiotigh ohd tfk ihtknrol agr`ulos ork ihvkrsk ga tfgsk os sfgwh ih Kquotigh (3.3>). (3.3>). – ?
d 〈 x – o⌭ d x
∥ 〈 x – o⌭
– :
3.4. 3.4.: :
d 〈 x – o⌭ d x
– :
0 〈 x – o⌭ d x 0
– ?
〈 x – o⌭
∥ 〈 x – o⌭
2
– ?
h
h – ? d 〈 x – o⌭ 0 h 〈 x – o⌭ agr h ≫ ? d x
– ?
0 〈 x – o⌭ d x 0
〈 x – o⌭
2
∥ 〈 x – o⌭
h – ?
d x 0
〈 x – o⌭ h ------------------- agr h ≫ ?
(3.3>)
h
Oxio Oxioll disp displo lock ck`k `kht ht
Ainurk 3.:4 sfgws 3.:4 sfgws o tk`plotk ga oh oxiol jor ih wficf o cghckhtrotkd agrck is opplikd ot x ot x 0 o. o. Ia wk `oek oh i`onihory cut ih tfk rknigh x < o , tfk ihtkrhol oxiol agrck H 0 2. Ia wk `oek oh i`onihory cut ih tfk rknigh x 8 o , tfk ihtkrhol oxiol agrck H 0 – A . ^k 2 coh usk tfk discghtihuity discghtihuity auhctigh tg tg writk tfk ihtkrhol agrck os H 0 – A 〈 x – o⌭ . Sujstitutihn tfis ihtkrhol agrck ihtg Kquotigh Kquotigh – ? (3.??-O) ohd (3.??-O) ohd usihn Kquotigh (3.3>), (3.3>), wk gjtoih p x 0 A 〈 x – o⌭ . Tfksk twg kquotighs ork sfgwh os tk`plotk kquotighs ihAinurk ih Ainurk
;3
3
Josic Structurol @k`jkrs
3.:4. 3.:4. Ia tfk pgiht agrck gh tfk octuol `k`jkr is ih tfk dirkctigh sfgwh ih Ainurk 3.:4, 3.:4, tfk ihtkrhol agrck ohd distrijutkd agrck ork os nivkh. Ia tfk pgiht agrck is gppgsitk tg tfot sfgwh ih Ainurk 3.:4, 3.:4, tfk sinhs ork rkvkrskd ih tfk tk`plotk kquotighs.
Tk`plotk kquotighs
H 0 – A 〈 x – o⌭
– ?
p x 0 A 〈 x – o⌭
A
o
2
H
x
Ainurk 3.:4 Tk`plotk ohd tk`plotk kquotighs agr usk ga discghtihuity auhctighs ih oxiol `k`jkrs.
Agr stoticolly dktkr`ihotk prgjlk`s ohd agr agr prgjlk`s agr wficf p x is o si`plk auhctigh, it `oy jk pgssijlk tg dktkr`ihk tfk ihtkrhol oxiol agrck H ohd ohd ihtknrotk Kquotigh (3.??-O) tg (3.??-O) tg nkt gjtoih tfk oxiol displock`khts. Jut stortihn witf tfk diaakrkhtiol kquotighs (3.?:-O) ohd (3.?:-O) ohd sglvihn tfk jguhdory voluk prgjlk` os wk did ih Skctigh 3.3, 3.3, is o nkhkrol opprgocf tfot cgvkrs `gst ga tfk cosks ga gur ihtkrkst.
KVO@RLK 3.?3 Tfk cglu`h sfgwh ih Ainurk 3.:; fos 3.:; fos o spkciaic wkinft ga γ 0 2.? lj/ih lj/ih3, `gdulus ga klosticity ga K ga K 0 0 4222 4222 esi, esi, ohd crgsscrgss-skc skc-tighol orko O orko O 0 ?22 ih:. Dktkr`ihk (o) tfk `gvk`kht ga tfk rinid plotk ot C ohd (j) tfk rkoctigh agrck ot O. ot O. :; ih. ih:. eips
O x
: eips
J
12 ih. ih. 4 eips
4 eips C
:2 ih. ih. D
3.?3. Ainurk 3.:; Cglu`h agr Kxo`plk 3.?3. – 3
OHS. u c 0 2.393 ( ?2
3.4. 3.4.3 3
) i h 5
Z O
0
;291 lj upword
Tgrsi grsigh ghol ol Zgto Zgtoti tigh gh
Ainurk 3.:1 sfgws 3.:1 sfgws o tk`plotk ga o sfoat gh wficf o cghckhtrotkd tgrquk is opplikd ot x ot x 0 o. o. Ia wk `oek oh i`onihory cut ih tfk rknigh x < o , tfk ihtkrhol tgrquk T 0 2. Ia wk `oek oh i`onihory cut ih tfk rknigh x 8 o , tfk ihtkrhol tgrquk is T 0 –Tkxt. ^k coh 2 usk tfk discghtihuity auhctigh tg writk tfk ihtkrhol tgrquk os T 0 – T kxt 〈 x – o⌭ . Sujstitutihn tfis ihtkrhol tgrquk ihtgKquotigh ihtgKquotigh (3.??-T) ohd (3.??-T) ohd usihn Kquotigh (3.3>), (3.3>), wk gjtoih t 0 T kx t 〈 x – o⌭ – ? . Tfksk twg kquotighs ork sfgwh os tk`plotk kquotighs. Ia tfk cghckhtrotkd tgrquk T gh gh tfk octuol sfoat is ih tfk dirkctigh sfgwh ih Ainurk 3.:1, 3.:1, tfk ihtkrhol tgrquk ohd tfk distrijutkd tgrquk ork os nivkh. Ia tfk cghckhtrotkd tgrquk is gppgsitk tg tfot sfgwh ih Ainurk 3.:1, 3.:1, tfk sinhs ork rkvkrskd ih tfk tk`plotk kquotighs. Tk`plotk kquotighs T kxt
T 0 – T kx t 〈 x – o⌭ T
o
2
– ?
t 0 T kx t 〈 x – o⌭
x
Ainurk 3.:1 Tk`plotk ohd tk`plotk kquotighs agr usk ga discghtihuity auhctighs ih tgrsighs.
Tfk usk ga discghtihuity auhctighs ih tgrsigh is si`ilor tg tfot agr oxiol displock`kht ohd will hgt jk discusskd aurtfkr.
3.4. 3.4.4 4
Jko` Jk o` Dkalk kalkct ctig igh h
Ainurk 3.:> sfgws 3.:> sfgws o arkk jgdy dionro` ga o jko` ih wficf o cghckhtrotkd agrck, `g`kht, gr tfk stort ga distrijutkd agrck gccurs ot x ot x 0 o. i`onihory y cut @ o. Ia wk `oek oh i`onihory cut ih tfk rknigh x < o , tfkh tfk ihtkrhol jkhdihn `g`kht @ z 0 2. Ia wk `oek oh i`onihor ih rknigh x 8 o , tfkh tfk ihtkrhol jkhdihn `g`kht @ z coh jk aguhd jy `kohs ga tfk `g`kht kquilijriu`, os sfgwh ih Ainurk
Discghtihuity Auhctighs
;4
3.:>. 3.:>. Tfk `g`kht coh tfkh jk writtkh jy usihn tfk dkaihitigh ga discghtihuity auhctighs. Jy sujstitutihn Kquotigh (3.??o-J) ihtg (3.??o-J) ihtg Kquotigh (3.??j-J), (3.??j-J), wk gjtoih Kquotigh (3.36-J). (3.36-J). :
d @ z -----------0 p y ( x ) : dx y
(3.36-J)
y
o
y
x
o
z
x
o x
R
z
w
z
@ kxt @ z Q y
@ z Q y
(j)
(o)
@ z 0
@ z
x < o
⎮ 2 ⎦ ⎩ – @
@ z 0
x 8 o
Tk`plotk kquotighs @ z 0 – @ 〈 x – o⌭
2 ⎮ ⎦ ⎩ – R ( x – o )
(c) x < o
@ z 0
x 8 o
@ z 0 – R 〈 x – o⌭
– :
p y 0 – @ 〈 x – o⌭
2 ⎮ ⎦ : ⎩ – w ( x – o ) ⁄ :
x < o x 8 o
Tk`plotk kquotighs
Tk`plotk kquotighs
2
Q y
:
@ z 0 – w 〈 x – o⌭
?
p y 0 – w 〈 x – o⌭
– ?
p y 0 – R 〈 x – o⌭
⁄ :
2
Ainurk 3.:> Tk`plotks agr tfk usk ga tfk discghtihuity auhctighs ih jko`s.
Jy sujstitutihn tfk `g`kht kxprkssighs agr kocf cosk gjtoihkd ih Ainurk 3.:> ihtg 3.:> ihtg Kquotigh (3.36-J), (3.36-J), wk gjtoih oh kxprkssigh agr tfk distrijutkd agrck agr kocf cosk os sfgwh ih Ainurk 3.:>. 3.:>. Tfk tk`plotk kquotighs ork ossgciotkd witf tfk cggrdihotk systk` sfgwh ih Ainurk 3.:>. 3.:>. Ia tfk cggrdihotk systk` is cfohnkd, cgrrkspghdihn cfohnks `ust jk `odk ih t fk tk`plotk kquotighs. Ia tfk kxtkrhol agrcks ohd `g`khts or k ih tfk dirkctighs sfgwh gh tfk tk`plotk arkk jgdy dionro`, tfkh tfk kxprkssighs agr @ z ohd p ohd p y will jk ih occgrdohck witf tfk tk`plotk kquotighs. Ia tfk kxtkrhol agrcks ohd `g`khts ork gppgsitk, tfkh tfk sinhs will fovk tg jk rkvkrskd ih tfk tk`plotk kquotighs.
KVO@RLK 3.?4 [sk tfk discghtihuity auhctighs agr tfk tfrkk tk`plotks sfgwh ih Ainurk 3.:6 tg 3.:6 tg writk tfk `g`kht ohd distrijutkd agrck kxprkssighs. w
R y @ kxt
y
x
x
x
x 0 o
x 0 j
x 0 o
x 0 o
y Cosk I
Cosk III
Cosk II
3.?4. Ainurk 3.:6 Tfk tfrkk cosks ga Kxo`plk 3.?4.
OHS. Cosk I= @ z 0 @ kx t 〈 x – o⌭
2 2
– :
p y 0 @ kx t 〈 x – o⌭
5 Cosk II @ z 0 R 〈 x – o⌭
2
Cosk III= p y 0 – w 〈 x – o⌭ + w 〈 x – j⌭ 5 @ z 0 – w 〈 x – o⌭
?
– ?
p y 0 R 〈 x – o⌭
:
⁄ : + w 〈 x – j⌭ : ⁄ :
KVO@RLK 3.?; Dktkr`ihk tfk kquotigh ga tfk klostic curvk ih Ainurk 3.:9 ih 3.:9 ih tkr`s ga K ga K , I , L, L, R , ohd x ohd x.. R
y O
J
x
L
3.?;. Ainurk 3.:9 Jko` ohd lgodihn agr Kxo`plk 3.?;. OHS. R 3 3 : v ( x ) 0 – ⎕ ----------------⎘ ( : 〈 x – L⌭ + x – ; L x ) ⎐ ?: KI zz ⎢
: RL C
L
5
;;
3
Josic Structurol @k`jkrs
KVO@RLK 3.?1 O jko` witf o jkhdihn rinidity ga KI ga KI 0 0 4:,222 4:,222 eH ¾ `: is sfgwh ih Ainurk 3.32. 3.32. Dktkr`ihk (o) tfk dkalkctigh ot pgiht J pgiht J ohd ohd (j) tfk `oxi`u` ihtkrhol jkhdihn `g`kht. y
; eH 4 eH/`
O
x
J
; eH/` :.2 :.2 `
?: eH-` C
D
3.2 `
?.2 `
Ainurk 3.32 Jko` ohd lgodihn agr Kxo`plk 3.?1
OHS.
v ( : ) 0 – 2.3; `` 5 @ `o x 0 1.?:; eH-`
3.;
SY@@K SY@@KTZ TZIC IC JKH JKHDI DIHN HN GA GA C[Z C[ZQK QKD D JKO@ JKO@S S
Tfk alkxurk agr`ulo ga Kquotigh (3.9-J) is (3.9-J) is opplicojlk tg tfk sy``ktric jkhdihn ga stroinft jko`s. It coh olsg jk opplikd tg jko`s witf slinft curvoturks, tfot is, jko`s witf r c/f 8 ;, wfkrk r c is tfk rodius ga curvoturk ga tfk lihk mgihihn tfk ckhtrgids ga tfk crgss skctighs os sfgwh ih Ainurk 3.3?, 3.3?, ohd f is tfk dkptf ga tfk jko` crgss skctigh. Ih tfis skctigh wk dkvklgp strkss agr`ulos agr sforply curvkd (r (r c/f < ;) jko`s tfot f ovk o plohk ga sy``ktr y, witf tfk lgodihn ih tfk plohk ga sy``ktry. Tfk dkalkctigh ga o pgiht gh o curvkd jko` is dkscrijkd ih Skctigh >.>. >.>. ^k will usk cylihdricol cggrdihotks (r (r , ζ, z ) os sfgwh ih Ainurk 3.3?. 3.3?. ^k will ossu`k tfot jkhdihn is ojgut tfk z tfk z oxis oxis (i.k., tfk crgss skctigh rgtotks ojgut tfk z tfk z oxis). oxis). Os ih tfk sy``ktric jkhdihn ga stroinft jko`s, gur gjmkctivk is tg aihd tfk lgcotigh ga tfk hkutrol oxis (tfk voluk ga Z ga Z)) ohd tfk jkhdihn hgr`ol strkss (σζζ) os o auhctigh ga rodiol cggrdihotk Ϗ. Ih tfk ihitiol dkvklgp`kht, wk sfoll ossu`k tfot tfk ghly lgod is tfot ga o `g`kht ojgut tfk z tfk z oxis. oxis. Lotkr wk will rklox tfis cghditigh ohd ihcludk trohsvkrsk agrcks. r J C O
s x i s l O o d s i s g i d h t r O x k l o C u t r H k
f
r c Z
Ϗ
ζ G
z
Ainurk 3.3? Curvkd jko` nkg`ktry.
Tfk tfkgrkticol dkvklgp`kht ga tfk strkss agr`ulo agllgws lihks vkry si`ilor tg tfgsk wk uskd ih discussihn tfk sy``ktric jkhdihn ga stroinft jko`s. Ossu`ptighs ga si`ilor eihds ork `odk ih tfk dkvklgp`kht ga tfk tfkgry, witf tfk pri`ory diaakrkhck cg`ihn arg` o hkw eihk`otic kxprkssigh agr tfk hgr`ol stroihs.
3.;.?
Eihk`otics
^k `oek tfk agllgwihn ossu`ptighs, rkpkotkd fkrk arg` Tojlk 3.:, 3.:, tg gjtoih tfk si`plkst tfk si`plkst tfkgry agr tfkgry agr sy``ktric jkhdihn ga curvkd jko`s. Ossu`ptigh ?= Dkagr`otighs ork hgt o auhctigh ga ti`k. Ossu`ptigh :o-J= Squosfihn dkagr`otigh is sinhiaicohtly s`ollkr tfoh dkagr`otigh duk tg jkhdihn. Ossu`ptigh :j-J= Rlohk skctighs jkagrk dkagr`otigh rk`oih plohk oatkr dkagr`otigh. Ossu`ptigh :c-J= Rlohk pkrpkhdiculor tg tfk jko` oxis rk`oihs hkorly pkrpkhdiculor hkorly pkrpkhdiculor oatkr dkagr`otigh. Ossu`ptigh 3= Tfk stroihs ork s`oll.
Sy``ktric Jkhdihn ga Curvkd Jko`s
;1
Ossu`ptigh ? i`pliks ? i`pliks tfot wk fovk o stotic prgjlk` ohd tfot tfk opplikd lgods ork hgt auhctighs ga ti`k. Ossu`ptigh : i`pliks tfot tfk cfohnks ih tfk rodiol di`khsighs ork hknlinijlk. Ossu`ptigh : i`pliks : i`pliks tfot tfk dkagr`otigh ih tfk ζ dirkctigh is o lihkor auhctigh ga Ϗ. Os wk sfoll skk, fgwkvkr, uhliek tfk cosk ga stroinft jko`s, tfis will hgt rksult ih o lihkor auhctigh ga Ϗ agr tfk hgr`ol stroih λζζ. Ossu`ptigh : i`pliks : i`pliks tfot tfk sfkor stroih (γr ζ) is hkorly zkrg. Os ih tfk cosk ga stroinft jko`s, tfis ossu`ptigh is strictly volid ia tfkrk ork hg sfkor agrcks ohd tfk opplikd lgod is ghly o jkhdihn `g`kht ojgut tfk tfk z z oxis. oxis. Ossu`ptigh 3 i`pliks 3 i`pliks tfot wk coh usk s`oll-stroih opprgxi`otighs, rksultihn ih o lihkor tfkgry. Cghsidkr o s`oll skn`kht ga o curvkd jko` klk`kht tfot sujtkhds oh ohnlk Δζ ot tfk ckhtkr ga curvoturk G os sfgwh ih Ainurk 3.3:o 3.3:o. Lihk OO Lihk OO rkprkskhts rkprkskhts tfk hkutrol oxis, ohd lihk JJ lihk JJ is is ohy lihk agr wficf wk ork tryihn tg aihd tfk stroih. ^k sfgw tfk dkagr`kd sfopk rklotivk tg tfk lkat rodiol lihk OJ lihk OJ,, wficf is ossu`kd tg rk`oih pkrpkhdiculor tg lihks OO lihks OO ohd ohd JJ JJ ih ih occgrdohck witf Ossu`ptighs :j :j ohd :c :c. Lihk OO Lihk OO,, jkihn tfk hkutrol oxis, dgks hgt skk ohy stroih, wfilk dkagr`otigh cousks pgiht J pgiht J tg tg `gvk tg pgiht J pgiht J?, ohd tfk ckhtkr ga curvoturk tfkh `gvks tg pgiht G? —jgtf `gvk`khts ork sinhiaicohtly kxonnkrotkd ih Ainurk 3.3:. 3.3:. Tfk orc lkhntf OO lkhntf OO gjtoihkd gjtoihkd jy usihn G ohd G? os tfk ckhtkrs rksults ih Z Δζ 0 Z ? Δζ ? . Tfk grinihol lkhntf is JJ 0 ϏΔζ lihk JJ coh coh jk writtkh os agllgws= ϏΔ ζ , ohd tfk aihol lkhntf is JJ ? 0 ( Z ? + Ϗ – Z ) Δ ζ? . Tfk hgr`ol tohnkhtiol stroih ga lihk JJ
λζζ 0
JJ ? – JJ ---------------------- JJ
0
( Z ? + Ϗ – Z ) Δ ζ ? – ϏΔζ ϏΔ ζ -------------------------------------------------------- 0 ϏΔζ ϏΔ ζ
Z Δζ + ( Ϗ – Z ) Δζ ? – ϏΔζ ϏΔ ζ -------------------------------------------------------------
Δ ζ – Δ ζ Z – 0 ⎕⎐ ------------Ϗ⎘⎢ ⎕⎐ ----------------------?⎘⎢ Ϗ Δζ
ϏΔζ ϏΔ ζ
J?
J? J
J Ο
(3.39)
Ο
J Ϗ
Z
√
Ϗ
Δ ζ √ Δ ζ
?
Ϗ √
Z
i s x O l t r o u k H
Z
Δζ G
Z ?
O
Δζ? G?
(o)
(j)
Ainurk 3.3: Grinihol ohd kxonnkrotkd dkagr`kd curvkd jko` nkg`ktry.
^k dkaihk φ 0 ( Δ ζ – Δ ζ ? ⁄ Δζ ) os o rklotivk rgtotigh ga o crgss skctigh tfot dgks hgt cfohnk ih tfk rodiol dirkctigh. Tfk tohnkhtiol stroih coh jk writtkh os
λ ζζ 0 – ⎕⎐ ? – -Z--⎘⎢ φ Ϗ
(3.42)
Oh oltkrhotivk dkrivotigh ga tfk tohnkhtiol stroih coh jk gjtoihkd jy cghsidkrihn Ainurk 3.3:j 3.3:j. Tfk rodiol lkhntf OJ lkhntf OJ? is hkorly kquol tg OJ tg OJ os os o rksult ga Ossu`ptigh :. Arg` tfk nkg`ktry ih Ainurk 3.3:(o), 3.3:(o), wk hgtk tfot ohnlk G?GO is ?62 – Δζ 5 fkhck ohnlk GOG? is Δ ζ – Δ ζ ? , wficf i`pliks tfot ohnlk JOJ ohnlk JOJ? is Δ ζ – Δ ζ? . Tfk orc lkhntf JJ lkhntf JJ? agr s`oll stroihs coh jk opprgxi`otkd os JJ ? 0 ( Ϗ – Z ) ( Δ ζ – Δ ζ? ). Hgtihn tfot JJ tfot JJ? is o cghtroctigh, wk coh writk tfk tohnkhtiol stroih os J J JJ
? λζζ 0 – --------0 –
( Ϗ – Z ) ( Δ ζ – Δ ζ? ) --------------------------------------------- 0 – ⎕ ? – -Z--⎘ φ ⎐ Ϗ⎢ ϏΔζ ϏΔ ζ
(3.4?)
Kquotigh (3.4?) is (3.4?) is so`k os Kquotigh (3.42) jut (3.42) jut wos gjtoihkd arg` o diaakrkht pkrspkctivk. Ih Kquotigh (3.42) tfk (3.42) tfk ghly voriojlk tg cfohnk ocrgss tfk crgss skctigh is Ϗ. Kquotigh (3.42) sfgws (3.42) sfgws tfot tfk tohnkhtiol stroih agr curvkd jko`s dgks hgt vory lihkorly ocrgss tfk crgss skctigh5 rotfkr, it voriks fypkrjglicolly, rkocfihn zkrg wfkh Ϗ 0 Z. Z.
3.;.: .;.:
@ot @otkri kriol @g @gdkl
^itf tfk `gtivotigh ga dkvklgpihn tfk si`plkst tfkgry, wk usk tfk si`plkst `otkriol `gdkl, ho`k ly Fggek’s low. Tg Tg usk Fggek’s low, wk `oek tfk agllgwihn ossu`ptighs, rkpkotkd arg` Tojlk 3.:. 3.:. Ossu`ptigh 4= Tfk `otkriol is isgtrgpic. Ossu`ptigh ;= Tfkrk ork hg ihklostic stroihs. Ossu`ptigh 1= Tfk `otkriol is klostic. Ossu`ptigh >= Strkss ohd stroih ork lihkorly rklotkd. Tfk dg`ihoht hgr`ol strkss is
(3.4:). σζζ , wficf coh jk writtkh os Kquotigh (3.4:).
;>
3
Josic Structurol @k`jkrs
σζζ 0 3.;. 3.;.3 3
0 – K ⎕⎐ ? – -Z--⎘⎢ φ Ϗ
K λ ζζ
(3.4:)
Stoti toticc Kq Kqui uivo volk lkhc hcy y
Tfk hgr`ol strkss σζζ coh jk rkplockd jy oh kquivolkht ihtkrhol `g`kht @ 3.33. Jy z ohd o zkrg oxiol agrck os sfgwh ih Ainurk 3.33. cghsidkrihn tfk agrcks ih tfk oxiol dirkctigh ohd tfk `g`kht ojgut tfk z tfk z oxis oxis tfrgunf G ih Ainurk 3.33, 3.33, wk gjtoih Kquotigh (3.43o) ohd (3.43o) ohd (3.43j), (3.43j), H 0
∥ σζζ d O 0 2
(3.43o)
O
@ z
∥
0 – Ϗσ ζζ
d O
(3.43j)
O
@ wfkrk O wfkrk O is is tfk crgss-skctighol orko, ohd @ z is oh ihtkrhol `g`kht. Tfk dirkctigh ga pgsitivk ihtkrhol `g`kht @ z gh o arkk jgdy dionro` `ust jk sucf tfot it puts o pgiht gh tfk gutkr suraock ihtg cg`prkssigh. cg`prkssigh. Tfis i`pliks tfot tfk pgsitivk dirkctigh ga @ z gh o arkk jgdy dionro` tkhds tg stroinftkh gut tfk curvkd jko`.
σζζ dO
Ϗ
z
@
G
z
G
Ainurk 3.33 Stotic kquivolkhcy ih curvkd jko`s.
3.;.4 3.;.4
Lgcot Lgcotigh igh ga tfk tfk Hkut Hkutrol rol Oxis Oxis
Sujstitutihn Kquotigh (3.4:) ihtg (3.4:) ihtg Kquotigh (3.43o) ohd (3.43o) ohd hgtihn tfot φ ohd Z ohd Z dg dg hgt cfohnk ocrgss tfk crgss skctigh, wk gjtoih
∥ K ⎕⎐ ? – -ZϏ--⎘⎢ φ d O 0 φ ∥ K ⎕⎐ ? – -ZϏ--⎘⎢ d O
O
0 φ
O
K d O ∥ K d O – Z ∥ ----------Ϗ
O
0
2
gr
O
∥ K d O
Z
O
0
---------------------------
(3.44)
∥ ( K d O ⁄ Ϗ ) O
Ossu`ptigh 6= Tfk `otkriol is fg`gnkhkgus ocrgss tfk crgss skctigh Tfk ojgvk ossu`ptigh i`pliks tfot tfk `gdulus ga klosticity K klosticity K dgks dgks hgt cfohnk ocrgss tfk crgss skctigh ohd coh jk toekh gutsidk tfk ihtknrol ih Kquotigh (3.44) tg (3.44) tg yikld Z
0
O ----------------------
(3.4;)
∥ ( d O ⁄ Ϗ ) O
Tfk voluk ga Z ga Z lgcotks lgcotks tfk hkutrol oxis. Tfk ihtknrol ih tfk dkhg`ihotgr dkhg`ihotgr is o nkg`ktric quohtity tfot dkp khds ghly upgh tfk sfopk ga tfk crgss skctigh. Tojlk 3.; sfgws 3.; sfgws tfk voluks ga tfk ihtknrol ih Kquotigh (3.4;) agr (3.4;) agr skvkrol cg``gh josic sfopks tfot coh jk uskd agr aihdihn ihtknrols ga `grk cg`plkx crgss skctighs tfot ork cg`pg sitks ga `odk arg` tfksk sfopks.
3.;. 3.;.; ;
Strks trksss Agr` Agr`ul ulo o
Sujstitutihn Kquotigh (3.4:) ihtg (3.4:) ihtg Kquotigh (3.43j) ohd (3.43j) ohd hgtihn tfot φ ohd Z ohd Z dg dg hgt cfohnk ocrgss tfk crgss skctigh, wk gjtoih @ z 0
φ
∥ K Ϗ ⎕⎐ ? – -ZϏ--⎘⎢ d O 0 φ ∥ K Ϗ d O – Z ∥ K d O
O
O
O
Ia Ossu`ptigh 6 is 6 is opplicojlk, tfkh K tfkh K coh coh jk toekh gutsidk tfk ihtknrol. Hgtihn tfk dkaihitigh ga ckhtrgid, wk gjtoih
(3.41o)
Sy``ktric Jkhdihn ga Curvkd Jko`s
;6
Tojlk 3.; Qoluks ga Ihtknrols ih Curvkd Jko`s
Sfopk
r ?
r ?
o
o
r :
:o
r ? o:
o
o?
r :
:j r c
r :
r c O
o ( r : – r ? )
? -- o ( r : – r ? ) :
? -- ( o + o : ) ( r : – r ? ) : ?
ς o:
ςo j
d O
r o lh ---:r ?
or r -----------:---- lh ---:- – o r : – r ? r ?
r : o ? – r ? o : r o : – o ? + -------------------------- lh ----: r : – r ? r ?
: ς ⎕ r c – r c : – o :⎘ ⎐ ⎢
:ςj ---------- ⎕ r c – r c : – o :⎘ ⎢ o ⎐
∥ ---Ϗ---
O
∥ Ϗ d O 0 r O
(3.41j)
c
O
wfkrk r c is tfk rodius ga tfk lihk possihn tfrgunf tfk ckhtrgid os sfgwh ih Ainurk 3.3?. 3.3?. Sujstitutihn Kquotigh (3.41j) ihtg (3.41j) ihtg Kquotigh (3.41o) ohd (3.41o) ohd hgtihn tfot K tfot K is is o cghstoht ocrgss tfk crgss skctigh agr o fg`gnkhkgus `otkriol, wk gjtoih @ z 0 K φ U r c O – ZO P
gr
@ z K φ 0 --------------------- O ( r c – Z )
(3.41c)
Sujstitutihn Kquotigh (3.41c) ihtg (3.41c) ihtg Kquotigh (3.4:), (3.4:), wk gjtoih tfk strkss agr`ulo
σ ζζ
@ z Z 0 – ---------------------- ⎕ ? – ---⎘ O ( r c – Z ) ⎐ Ϗ⎢
(3.4>)
wfkrk O wfkrk O is is tfk crgss-skctighol orko, r c is tfk rodius ga tfk lihk mgihihn tfk ckhtrgids, ohd Z ohd Z is is tfk rodius ga tfk hkutrol oxis os dktkr`ihkd arg` Kquotigh (3.4;)5 (3.4;)5 ork oll nkg`ktric prgpkrtiks ga tfk crgss skctigh. Tfk ihtkrhol `g`kht @ z dgks hgt cfohnk ocrgss tfk crgss skctigh. Tfk ghly voriojlk tfot cfohnks ocrgss tfk crgss skctigh is tfk rodiol cggrdihotk Ϗ. Tfus, tfk strkss distrijutigh ocrgss tfk crgss skctigh ga o curvkd jko` is fypkrjglic, fypkrjglic, os sfgwh sy`jglicolly agr o crgss skctigh ih Ainurk 3.34. 3.34.
Ϗ
Hkutrol oxis
Ϗ 0 Z
σζζ
Ainurk 3.34 Fypkrjglic voriotigh ga jkhdihn hgr`ol strkss ocrgss tfk crgss skctigh ga o curvkd jko`.
Tfkrk ork twg pgssijlk woys ih wficf tfk ihtkrhol jkhdihn `g`kht @ z `oy jk aguhd= ?. Gh o arkk arkk jgdy jgdy dion dionro` ro` @ z is olwoys drowh ih occgrdohck witf tfk sinh cghvkhtigh. Oh kquilijriu` kquotigh tfkh is uskd tg nkt o pgsitivk gr hknotivk voluk agr @ z . Hgw pgsitivk voluks ga strkss σζζ arg` Kquotigh (3.4>) ork tkhsilk, ohd hknotivk voluks ga σζζ ork cg`prkssivk.
:. Tfk ihtkrh ihtkrhol ol jkhdihn jkhdihn `g`kht `g`kht @ z is drowh ot tfk i`onihory cut ih o dirkctigh sklkctkd tg kquilijrotk tfk kxtkrhol lgods. Sihck ihspkctigh is jkihn uskd ih dktkr`ihihn tfk dirkctigh ga @ z , Kquotigh (3.4>) sfguld (3.4>) sfguld jk uskd ghly agr σ dktkr`ihihn tfk `onhitudk. Tfk tkhsilk ohd cg`prkssivk prgpkrtiks ga ζζ `ust jk dktkr`ihkd jy ihspkctigh.
3.;. 3.;.1 1
Cg`ji g`jihk hkd d Lgo Lgodi dihn hn
Agrcks opplikd tg fgges ohd rihns hgt ghly will sujmkct curvkd jko` `k`jkrs tg jkhdihn `g`kht jut will olsg sujmkct o crgss skctigh ga tfk jko` tg oxiol ohd sfkor agrcks. Ossu`ptigh : (tfot tfk plohk pkrpkhdiculor tg tfk jko` oxis rk`oihs hkorly pkrpkhdiculor oatkr dkagr`otigh) i`pliks tfot tfk sfkor stroih, fkhck tfk sfkor strkss, is s`oll. Tfus, wk coh usk gur tfkgry, prgvidkd tfk `oxi`u` sfkor strkss (τr ζ) ih jkhdihn is oh grdkr ga `onhitudk s`ollkr tfoh tfk `oxi`u` jkhdihn hgr`o l strkss (σζζ) os gjtoihkd arg` Kquotigh (3.4>). (3.4>). Oh ksti`otk ga tfk `oxi`u` sfkor strkss (τr ζ) coh jk gjtoihkd jy dividihn tfk `oxi`u` sfkor agrck jy tfk crgss-skctighol orko. O uhiagr` oxiol strkss wguld jk rkplockd jy oh ihtkrhol oxiol agrck octihn ot tfk ckhtrgid ga tfk crgss skctigh. Tfus, gh gur arkk jgdy dionro` wk sfoll ossu`k tfot tfk ihtkrhol oxiol agrck is octihn ot tfk ckhtrgid ga tfk crgss skctigh (skk skctigh (skk Rrgjlk` 3.;;). Gur tfkgry agr curvkd jko`s rksults ih o lihkor tfkgry5 fkhck jy tfk prihciplk ga supkrpgsitigh, wk gjtoih
;9
3
Josic Structurol @k`jkrs @ z H Z 0 ---- – ---------------------- ⎕ ? – ---⎘ ⎐ O r – Z ( ) O Ϗ⎢ c
σζζ
(3.46)
KVO@RLK 3.?> O jko` witf o : ih Þ : ih squork crgss skctigh is sujmkctkd tg `g`kht @ `g`kht @ kxt 0 32 ih ⋋ eips. Dktkr`ihk tfk `oxi`u` jkhdihn hgr`ol strkss agr tfk agllgwihn tfrkk cosks. Cosk ?= Tfk ?= Tfk jko` is stroinft. Cosk := Tfk := Tfk jko` is curvkd, ohd tfk rodius ga tfk ihsidk suraock suraock is is ; ih. Cosk 3= Tfk 3= Tfk jko` is curvkd, ohd tfk rodius ga ga tfk ihsidk suraock is :2 ih. OHS. Cosk ?=
σ`ox
0 ::.; esi ( C ) gr ( T ) 5 Cosk :=
σ`ox
0 :;.3 esi ( T ) 5 Cosk 3=
σ`ox
0 :3.: esi ( T )
KVO@RLK 3.?6 Oh I-skctigh is agr`kd ihtg o rihn os sfgwh ih Ainurk 3.3;(o). 3.3;(o). Tfk yikld strkss ga tfk `otkriol is :22 @Ro. Dktkr`ihk tfk `oxi`u` agrck R agrck R tfot tfot coh jk opplikd ia yikldihn is tg jk ovgidkd. R ;2`` ?2``
3
O O
:
?42 ?42 ``
;2`` ?2``
?
?2``
Crgss skctigh OO skctigh OO
(o)
(j)
3.?6. Ainurk 3.3; (o) Zihn nkg`ktry ohd (j) I-skctigh agr Kxo`plk 3.?6.
OHS.
R `ox 0 :>.3 eH
3.1
CLGS[ZK
Tfk lgnic sfgwh ih Ainurk 3.: k`pfosizks 3.: k`pfosizks tfot tfk eihk`otic kquotighs jktwkkh kquotighs jktwkkh displock`khts ohd stroihs, tfk cghstitutivk kquotighs jktwkkh kquotighs jktwkkh stroihs ohd strkssks, tfk stotic tfk stotic kquivolkhcy kquotighs kquivolkhcy kquotighs jktwkkh strkssks ohd ihtkrhol agrcks/`g`khts, ohd tfk kquilijriu` kquotighs jktwkkh kquotighs jktwkkh ihtkrhol agrcks/`g`khts ohd kxtkrhol agrcks/`g`khts coh jk trkot os ihdkpkhdkht `gdulks, pkr`ittihn ghk tg odd cg`plkxitiks tg ghk `gdulk witfgut cfohnihn t fk auhdo`khtol kquotighs ih tfk gtfkr `gdulks. Tfus, dyho`ics kaakcts ohd klostic aguhdotigh kaakcts cfohnk tfk kquilijriu` kquotighs witfgut cfohnihn tfk eihk`otics kquotighs, cghstitutivk kquotighs, ohd kquivolkhcy kquotighs. Ih tfk Ti`gsfkheg jko`, wk cfohnkd tfk eihk`otic kquotighs jut hgt tfk kquotighs ih gtfkr `gdulks. Ih Cfoptkr 4, Cg`pgsitk Structurol @k`jkrs, wk drgp tfk ossu`ptigh ga `otkriol fg`gnkhkity ocrgss tfk crgss skctigh tfot wk `oek ih tfk `gdulk ga stotic kquivolkhcy, jut tfkrk ork hg gtfkr cfohnks. Ih Cfoptkr ;, Ihklostic Structurol Jkfovigr, wk drgp tfk ossu`ptighs rklotkd tg Fggek’s low, tfus cfohnihn tfk cghstitutivk kquotighs jut hgtfihn klsk. Ih Cfoptkr 1, Tfih-^ollkd Structurol @k`jkrs, wk drgp tfk rkstrictighs gh tfk nkg`ktry ga tfk crgss skctighs ga sy``ktric jko`s ohd circulor sfoats, tfus cfohnihn tfk eihk`otic kquotighs witfgut cfohnihn tf k auhdo`khtol kquotighs ih gtfkr `gdulks. Ih Cfoptkr >, Khkrny @ktfgds, wk rkplock tfk khtirk `gdulk ga kquilijriu` kquotighs witf khkrny `ktfgd kquotighs jut dg hgt cfohnk tfk kquotighs ga tfk gtfkr `gdulks. Oh uhdkrstohdihn ga tfk `gdulor cforoctkr ga tfk lgnic, ohd fgw cg`plkxitiks ork ihcgrpgrotkd hgt ghly will fklp ygu witf tfk rk`oihihn cfoptkrs ih tfis jgge jut kquolly sinhiaicohtly, will nivk ygu tfk copojility ga oddihn cg`plkxitiks tg ghk-di`khsighol structurks hgt cgvkrkd fkrk. Ih tfis cfoptkr wk olsg sow tfk discghtihuity auhctighs ohd tfkir opplicotighs ih sglvihn jguhdory voluk prgjlk`s agr stoticolly dktkr`ihotk ohd ihdktkr`ihotk oxiol `k`jkrs, tfk tgrsigh ga circulor sfoats, ohd tfk sy``ktric jkhdihn ga jko`s. ^k will usk discghtihuity auhctighs wfkhkvkr pgssijlk ih tfk rk`oihihn cfoptkrs.
4
Cg`pgsitk Structurol @k`jkrs
LKOZHIHN GJMKCTIQK ?. Tg uhdkrstohd tfk ihcgrpgrotigh ohd i`plicotighs ga `otkriol hghfg`gnkhkity ocrgss tfk crgss skctigh ih tfk tfkgriks agr oxiol `k`jkrs, circulor sfoats ih tgrsigh, ohd tfk sy``ktric jkhdihn ga jko`s. :. Tg uhdkrstohd tfk usk ga tfk rulk ga `ixturk, ohd tfk ihvkrsk rulk ga `ixturks ih gjtoihihn tfk `ocrgscgpic prgpkrtiks ga o cg`pgsitk arg` its cghstitukht prgpkrtiks.
(o)
(j)
( Cgurtksy (o) (o) Tfg`os Ducrgqukt (j) HOSO/DAZC/Lorry (j) HOSO/DAZC/Lorry So``ghs) So``ghs) Ainurk 4.? Kxo`plks ga cg`pgsitk structurol `k`jkrs. (Cgurtksy
Aisfihn rgds, jicyclk aro`ks, tkhhis rocqukts, jgot fulls, stgronk tohes, rkihagrckd cghcrktk jors, wggdkh jko`s stiaakhkd witf stkkl strips, lo`ihotkd sfoats, aijkrnloss outg`gjilk jgdiks, olghn witf rocihn jiek ohd oh V:9 oircroat sfgwh ih Ainurk 4.?, 4.?, ork kxo`plks arg` tfk vost vorikty ga opplicotighs ga cg`pgsitk structurol `k`jkrs. Ih tfis cfoptkr wk dkvklgp tfk agr`ulos agr dkagr`otigh ohd strkssks agr cg`pgsitk oxiol jors, sfoats, ohd jko`s, os wkll os agr`ulos agr gjtoihihn `ocrgscgpic `otkriol prgpkrtiks arg` tfk `otkriol cghstohts ga tfk cghstitukhts ga tfk cg`pgsitks. Ih Skctigh 3.:, 3.:, wfilk dkvklgpihn si`pliaikd tfkgriks gh structurol `k`jkrs, wk `odk tfk ossu`ptigh ga `otkriol fg`gnkhkity ocrgss tfk crgss skctigh. Tfis ossu`ptigh is hgt volid agr cg`pgsitk `k`jkrs. Ia wk ossu`k tfot diaakrkht `otkriols ork pkrakctly jghdkd tgnktfkr, wk coh still `oek gur eihk`otic ossu`ptighs (Ossu`ptighs ? – 3 ih Tojlk Tojlk 3.:). 3.:). Ia gur Ossu`ptighs 4 tfrgunf > ojgut `otkriol jkfovigr o rk still volid, sg tfot wk coh still usk Fggek’s low, low, tfkh oll tfk kquoti ghs jkagrk Ossu`ptigh 6 gh `otkriol fg`gnkhkity ih Tojlk 3.: ork 3.: ork still opplicojlk. Tfk stotic kquivolkhcy kquotighs rklotihn strkssks tg tfk ihtkrhol agrcks ohd `g`khts ork still volid. Tfk ihtknrols i h stotic kquivolkhcy coh jk writtkh os tfk su` ga tfk ihtknr ols gvkr kocf `otkriol ih tfk crgss skctigh, ohd hkw agr`ulos agr dkagr`otighs ohd strkssks ih tkr`s ga tfk ihtkrhol agrcks ohd `g`khts coh jk dkrivkd. Tfk kquilijriu` kquotighs rklotihn tfk ihtkrhol agrcks ohd `g`khts ork ihdkpkhdkht ga tfk ossu`ptigh ga `otkriol fg`gnkhkity5 tfus oll tfk tkcfhiquks wk fovk dkvklgpkd agr aihdihn tfk ihtkrhol agrcks ohd `g`khts agr dktkr`ihotk ohd stoticolly ihdktkr`ihotk structurks coh jk uskd agr cg`pgsitk structurks. Tfk `kcfohicol prgpkrtiks ga o cg`pgsitk arg` o tkhsigh tkst ork `ocrgscgpic (jule) prgpkrtiks. Tfk tfkgriks ga `icrg`kcfohics ottk`pt tg dkrivk tfk `ocrgscgpic prgpkrtiks arg` tfk ihdividuol cghstitukht prgpkrtiks, wficf `oy jk `icrgscgpic ih sizk. Ghk sucf tfkgry is joskd gh tfk kquotighs agr tfk `kcfohics ga `otkriols. Tfis tfkgry rksults ih kquotighs tfot ork collkd tfk “rulk ga `ixturks‐ ohd tfk “ihvkrsk rulk ga `ixturks,‐ jy wficf tfk `ocrgscgpic `kcfohicol prgpkrtiks ga o cg`pgsitk coh jk gjtoihkd arg` tfk cghstitukht prgpkrtiks, os sfoll jk skkh ih Skctigh 4.4. 4.4.
12
1?
4.? 4.?
4
Cg`pgsitk Structurol @k`jkrs
CG@R CG@RGS GSIT ITK K OVI OVIOL OL @K@J @K@JKZ KZS S
^k ossu`k tfot oll ` otkriols, liek tfk lo`ihotkd crgss skctigh ih Ainurk 4.:, 4.:, ork skcurkly jghdkd tg kocf gtfkr sg tfot gur eihk`otic ossu`ptigh ga plohk skctighs rk`oihihn plohk ohd porollkl is still volid. ^k aurtfkr ossu`k tfot wk still fovk s`oll stroihs, tfot oll `otkriols ork lihkor-klostic-isgtrgpic `otkriols, ohd tfot tfkrk ork hg ihklostic stroihs. Ih gtfkr wgrds, oll tfk ossu`ptighs jkagrk Ossu`ptigh 6 ih Tojlk 3.: ork 3.: ork still volid. Tfus, Kquotigh (3.1-O), (3.1-O), wficf prkckdks Ossu`ptigh 6, is still volid ohd agr`s gur stortihn pgiht agr dkrivihn tfk oxiol agr`ulos agr cg`pgsitk jors. Cghsidkr tfk lo`ihotkd structurk sfgwh ih Ainurk 4.:. 4.:. Kocf `otkriol fos o `gdulus ga klosticity K klosticity K i, wficf is cghstoht gvkr tfk `otkriol crgss-skctighol orko O orko Oi. Suppgsk tfkrk ork h `otkriols ih tfk crgss skctigh. ^k coh writk tfk ihtknrol gvkr tfk crgss skctigh ih Kquotigh (3.1-O) os (3.1-O) os tfk su` ga tfk ihtknrols gvkr kocf `otkriol os sfgwh jklgw. jklgw. du du O 0 ----- H 0 ------ K d dx dx
∥ K d O + ∥ K d O + … + … + ∥ K d O
∥
O
?
:
O ?
h
O :
(4.?o)
O h
y K h
x z
K i
K ?
Ainurk 4.: O lo`ihotkd jor.
Ih Kquotigh (4.?o), (4.?o), K i is o cghstoht ih kocf ihtknrol ohd coh jk toekh gutsidk tfk ihtknrol. Tfk rk`oihihn ihtknrol is tfk orko O orko Oi. ^k tfus gjtoih du H 0 ------ U K ? O ? + K : O : + … + … + K h O h P dx
0
du -----dx
h
∟ 0 K O m ?
m m
(4.?j)
Kquotigh (4.?c) sfgws tfot tfk oxiol rinidity ga tfk cg`pgsitk jor is tfk su` ga tfk oxiol riniditiks ga oll tfk `otkriols. Tfk stroih ot ohy crgss skctigh coh jk writtkh os
λ xx
H du 0 ------ 0 -----------------------h dx K m O m
∟0
(4.:)
m ?
Tfk oxiol strkss ih tfk itf `otkriol is
( σ xx ) i
0 K i λ xx
. Sujstitutihn Kquotigh (4.:), wk gjtoih
( σ xx ) i
H K i 0 -----------------------h K m O m
∟0
(4.3)
m ?
^k ossu`k tfot tfk oxiol rinidity ohd tfk ihtkrhol oxiol agrck dg hgt cfohnk witf x witf x jktwkkh jktwkkh x x? ohd x ohd x:. Ih gtfkr wgrds, Ossu`ptighs 9 tfrgunf ?? ih ?? ih Tojlk 3.: ork 3.: ork still opplicojlk tg cg`pgsitk oxiol jors. ^itf tfk volidity ga Ossu`ptighs 9 tfrgunf ??, ??, du/ du/dx is dx is cghstoht jktwkkh x jktwkkh x? ohd x ohd x:, ohd wk coh writk du/ du/dx 0 dx 0 (u (u: – u?)/( x x: – x – x?) ih Kquotigh (4.:) tg gjtoih u : – u ?
0
H ( x: – x? ) ------------------------h K m O m
∟0
(4.4)
m ?
Kquotighs (4.3) ohd (4.3) ohd (4.4), (4.4), wficf ork opplicojlk tg cg`pgsitk jors, hgw rkplock Kquotighs (3.9-O) ohd (3.9-O) ohd (3.?2-O) agr (3.?2-O) agr tfk . . . . . . fg`gnkhkgus crgss skctigh. Ia wk cghsidkr o fg`gnkhkgus `otkriols, tfkh K tfkh K ? 0 K : 0 0 K 0 K i 0 K 0 K h 0 K . Sujstitutihn tfis ih Kquotighs (4.3) ohd (4.3) ohd (4.4), (4.4), wk gjtoih Kquotighs (3.9-O) ohd (3.9-O) ohd (3.?2-O) agr (3.?2-O) agr fg`gnkhkgus `otkriols, os kxpkctkd. Tfk ihtkrhol oxiol agrck H ih ih Kquotighs (4.3) ohd (4.3) ohd (4.4) rkprkskhts (4.4) rkprkskhts tfk stoticolly kquivolkht oxiol agrck gvkr tfk khtirk crgss skctigh, os skctigh, os it did agr tfk fg`gnkhkgus crgss skct igh. Tfk oholysis tkcfhiquks agr aihdihn tfk ihtkrhol oxiol agrck H ot ot o crgss skctigh rk`oihs tfk so`k os jkagrk. Fgwkvkr, tfk lgcotigh ot wficf tfk ihtkrhol oxiol agrck octs gh tfk crgss skctigh dkpkhds upgh tfk distrijutigh ga tfk `otkriol ocrgss tfk crgss skctigh, os k`pfosizkd ih Kxo`plk 4.?. 4.?. Kquotigh (4.:) k`pfosizks (4.:) k`pfosizks tfot tfk hgr`ol oxiol stroih λ xx is uhiagr` ocrgss tfk crgss skctigh. Tfk hgr`ol oxiol strkss (σ xx)i is uhiagr` ghly ih kocf `otkriol5 it cfohnks witf kocf `otkriol pgih t jkcousk tfk voluk ga K ga K i cfohnks witf kocf `otkriol, os sfgwh ih Kquotigh (4.3). (4.3). Tfus, tfk hgr`ol oxiol strkss is pikck wisk uhiagr` ocrgss tfk crgss skctigh. Tfis is klojgrotkd aurtfkr ih Kxo`plk 4.:. 4.:.
Cg`pgsitk Oxiol @k`jkrs
1:
KVO@RLK 4.? Agr tfk lo`ihotkd, sy``ktric, lihkor klostic jor ih Ainurk 4.:, 4.:, sfgw tfot tfk lgcotigh ga tfk grinih (ξc) coh jk aguhd arg` tfk agr`ulo h
ξc
ξ K O ∟ 0 0 ----------------------------∟ 0 K O i
i ? h
i
i
i ?
i
(4.;)
i
wfkrk ξi is tfk lgcotigh ga tfk ckhtrgid ga tfk itf `otkriol os `kosurkd arg` o cg``gh dotu` lihk.
KVO@RLK 4.: O fg`gnkhkgus wggdkh crgss skctigh ohd o crgss skctigh ih wficf tfk wggd is rkihagrckd witf stkkl ork sfgwh ih Ainurk 4.3. 4.3. Tfk hgr`ol stroih agr jgtf crgss skctighs is uhiagr`= λ xx
0 –:22 μ . Tfk `gdulus ga klosticity agr stkkl is K st 0 32,222 esi, ohd
agr wggd it is K is K w 0 6222 6222 esi. (o) Rlgt tfk σ xx distrijutigh agr kocf ga tfk twg crgss skctighs sfgwh. (j (j) [sk Kquotigh (3.4o-O) arg` Tojlk 3.: tg 3.: tg colculotk tfk kquivolkht ihtkrhol oxiol agrck H agr kocf crgss skctigh. y
y Stkkl z
? ? -- ih. :
?/4 ih.
z
? ih. ^ggd Stkkl
^ggd
?/4 ih. : ih.
: ih.
(j) (o) (o) fg`gnkhkgus ohd (j (j) lo`ihotkd. Ainurk 4.3 Crgss skctighs agr Kxo`plk 4.:= (o
OHS. ?.1 esi
1.2 esi
?.1 esi
?.1 esi
1.2 esi
(o)
(j)
(o) fg`gnkhkgus ohd (j (j) lo`ihotkd crgss skctighs. Ainurk 4.4 Strkss distrijutigh agr Kxo`plk 4.:= (o
(o) H 0 4.6 4.6 eips ( C ) 5 (j (j) H 0 9.: eips ( C )
KVO@RLK 4.3 O cost irgh pipk ( K K Ak 0 ?22 NRo) ohd o cgppkr cgppkr pipk K (K Cu 0 ?32 NRo) ork odfksivkly odfksivkly jghdkd tgnktfkr tgnktfkr os sfgwh ih Ainurk 4.;. 4.;. Tfk gutkr dio`ktkrs ga tfk twg pipks ork ;2 ohd >2 ``, rkspkctivkly, ohd tfk woll tficehkss ga kocf pipk is ?2 ``. Dktkr`ihk (o) tfk displock`kht ga khd D khd D witf witf rkspkct tg khd O khd O ohd ohd (j) tfk oxiol strkssks ih tfk irgh ohd cgppkr ih tfk jghdkd rknigh. ;22 ;22 ``
?;2 ?;2 `` 422 422 ``
Cost irgh Cgppkr
:2eH
:2eH
D O
J
C
Ainurk 4.; Ripks agr Kxo`plk 4.3.
OHS.
u D – u O 0 2.??2; `` 5
( σ JC ) Ak 0
;.16 ;.16 @Ro ( T ) 5
( σ JC )Cu
0 >.39 >.39 @Ro ( T )
13
Cg`pgsitk Structurol @k`jkrs
4
KVO@RLK 4.4 ^k `ust dksinh dksinh o :22 `` Þ :22 `` rkihagrckd rkihagrckd cghcrktk cghcrktk jor jor ga lkhntf :.; ` (skkAinurk (skk Ainurk 4.1) 4.1) tg corry o cg`prkssivk oxiol agrck ga ?222 eH. Tfk ollgwojlk cghtroctigh ga tfk jor is 3 ``. Dktkr`ihk, tg tfk hkorkst eilgnro`, tfk `ihi`u` o`guhts ga cghcrktk ohd stkkl tfot coh jk uskd ih cghstructihn tfk rkihagrckd cghcrktk jor. Tojlk 4.? nivks 4.? nivks tfk `otkriol prgpkrtiks. Tojlk 4.? @otkriol Rrgpkrtiks agr Kxo`plk 4.4 Cghcrktk
:22 :22 `` :22 :22 ``
Stkkl
@gdulus ga klosticity
K cghc 0 :2 NRo
K st 0 :22 :22 NRo
Dkhsity
:422 :422 en/` en/` 3
>6>2 >6>2 en/` en/` 3
Ollgwojlk strkss
?: @Ro
?;2 @Ro
:.; :.; `
Ainurk 4.1 Zkihagrckd cghcrktk jor agr Kxo`plk 4.4.
OHS.
` cghc 0 :?? en
4.: 4.:
`st 0 9; en
CG@R CG@RGS GSIT ITK K SF SFOA OATS TS
^k ossu`k tfot oll `otkriol crgss skctighs ork oxisy``ktric os sfgwh ih Ainurk 4.>. 4.>. ^k ossu`k tfot oll `otkriols ork skcurkly jghdkd tg kocf gtfkr5 tfot is, tfkrk is hg rklotivk rgtotigh ot ohy ihtkra ock ga tfk `otkriol. Tfis ossu`ptigh ga skcurk jghdihn is hkckssory tg prkskrvk gur eihk`otic ossu`ptighs= tfot tfkrk is hg worpihn ohd tfot oll rodiol lihks ih o crgss skctigh rgtotk jy kquol o`guhts. ^k aurtfkr ossu`k tfot wk still fovk s`oll stroihs ohd tfot oll `otkriols ork lihkor-klostic-isgtrgpic. Ih gtfkr wgrds, oll ossu`ptighs prigr tg Ossu`ptigh 6 ih Tojlk 3.: ork 3.: ork still volid. Tfus, Kquotigh (3.1-T), (3.1-T), wficf prkckdks Ossu`ptigh 6, is still volid ohd agr`s gur stortihn pgiht agr dkrivihn tfk tgrsighol agr`ulos agr cg`pgsitk sfoats. Nh Ni
N?
Ainurk 4.> Crgss skctigh ga o cg`pgsitk sfoat.
Cghsidkr tfk lo`ihotkd crgss skctigh sfgwh ih Ainurk 4.>. 4.>. Kocf `otkriol fos o sfkor `gdulus ga klosticity Ni tfot is cghstoht gvkr tfk `otkriol crgss-skctighol orko O orko Oi. Suppgsk tfkrk ork h `otkriols ih tfk crgss skctigh. Tfk ihtknrol ih Kquotigh (3.1-T) coh jk writtkh os tfk su` ga tfk ihtknrols gvkr kocf `otkriol os agllgws= d ρ d ρ : T 0 ------ N Ϗ d O 0 -----dx dx
∥
O
∥ N Ϗ d O + ∥ N Ϗ d O + . . . + . . . + ∥ N Ϗ d O :
:
?
:
:
O ?
O :
h
(4.1o)
O h
Fkrk Ni is o cghstoht ih kocf ihtknrol ohd coh jk toekh gutsidk tfk ihtknrol. Tfk rk`oihihn ihtknrol is tfk pglor `g`kht M `g`kht M i ga tfk itf `otkriol. ^k tfus gjtoih d ρ T 0 ------ U N ? M ? + N : M : + . . . + . . . + N H M H P gr dx d ρ T 0 -----dx
h
∟
m0? m 0?
N m M m
(4.1j) (4.>)
Tfk ojgvk kquotigh sfgws tfot tfk tgrsighol rinidity ga tfk cg`pgsitk sfoat i s tfk su` ga tfk tgrsighol riniditiks ga oll tfk `otkriols. ^k coh writk Fggek’s low agr tfk itf `otkriol os ( τ x ζ ) i 0 N i Ϗ ( d ρ ⁄ dx ) , wfkrk ( τ x ζ ) i is tfk tgrsighol sfkor strkss ih tfk itf `otkriol. Sujstitutihn Kquotigh (4.>) ihtg (4.>) ihtg Fggek’s low, wk gjtoih
( τ x ζ ) i
N i Ϗ T 0 --------------------------h N m M m
∟
m0? m 0?
(4.6)
Cg`pgsitk Sfoats
14
^k ossu`k tfot tgrsighol rinidity ohd tfk ihtkrhol tgrquk ork hgt auhctighs ga x ga x.. Tfot Tfot is, jktwkkh jktwkkh x x? ohd x ohd x:, Ossu`ptighs 9 tfrgunf ?? ih ?? ih Tojlk 3.: ork opplicojlk tg cg`pgsitk sfoats. Tfkh d ρ ⁄ dx is cghstoht jktwkkh x jktwkkh x? ohd x ohd x:, ohd wk coh writk d ρ ⁄ dx 0 ( ρ : – ρ ? ) ⁄ ( x : – x ? ) ih Kquotigh (4.>)) (4.>)) tg gjtoih
ρ: – ρ ?
T ( x : – x ? ) 0 ------------------------h N m M m
(4.9)
∟
m0? m 0?
Kquotighs (4.6) ohd (4.6) ohd (4.9), (4.9), wficf ork opplicojlk tg cg`pgsitk sfoats, hgw rkplock Kquotighs (3.9-T) ohd (3.9-T) ohd (3.?2-T) agr (3.?2-T) agr tfk fg`gnkhkgus crgss skctigh. Ia wk cghsidkr o fg`gnkhkgus `otkriol, tfkh N? 0 N: 0 . . . 0 Ni . . . 0 Nh 0 N. Sujstitutihn tfis ih Kquotighs (4.6) ohd (4.6) ohd (4.9), (4.9), wk gjtoih Kquotighs (3.9-T) ohd (3.9-T) ohd (3.?2-T) agr (3.?2-T) agr fg`gnkhkgus `otkriols, os kxpkctkd. Tfk ihtkrhol tgrquk T ih ih Kquotighs (4.6) ohd (4.6) ohd (4.9), (4.9), rkprkskhts tfk stoticolly kquivolkht tgrquk gvkr tfk khtirk crgss skctigh, os tigh, os it did agr tfk fg`gnkhkgus crgss skctigh. Tfk oholytic tkcfhiquks agr aihdihn tfk ihtkrhol tgrquk T ot ot o crgss skctigh rk`oih tfk so`k os jkagrk. Tfk tgrsighol sfkor stroih sfkor stroih still still voriks lihkorly ocrgss tfk crgss skctigh. Tfk sfkor strkss ih kocf `otkriol, (τ xζ)i, nivkh jy Kquotigh (4.>), (4.>), is o pikck wisk lihkor auhctigh tfot cfohnks its slgpk witf kocf `otkriol, os dk`ghstrotkd ih Kxo`plk 4.;. 4.;.
KVO@RLK 4.; O fg`gnkhkgus crgss skctigh `odk ga jross ohd o cg`pgsitk crgss skctigh ga jross ohd stkkl ork sfgwh ih Ainurk 4.?2. Tfk sfkor `gdulus ga klosticity agr jross is N J 0 42 NRo, ohd agr stkkl it is Ns 0 62 NRo. Tfk sfkor stroih ih pglor cggrdihotks ot tfk crgss skctigh is
γ x ζ 0
2.26 Ϗ , wfkrk Ϗ is ih `ktkrs. (o) Rlgt tfk sfkor stroih ohd sfkor strkss distrijutigh ocrgss jgtf crgss skc-
tighs. (j) Agr kocf crgss skctigh, dktkr`ihk tfk stoticolly kquivolkht ihtkrhol tgrquks. Ϗ
Ϗ
ζ
ζ
Jross
Stkkl Jross
x
x
` ` 6 2
` ` ? : 2
` ` ? : 2
Ainurk 4.6 (o) Fg`gnkhkgus ohd (j) cg`pgsitk crgss skctighs agr Kxo`plk 4.;.
OHS.
Fg`gnkhkgus crgss skctigh= T 0 ?32.3 eH-` 5 Cg`pgsitk crgss skctigh= T 0 >6 eH-`
γ xζ(μ)
τ xζ(@Ro)
4622 3:22
τ xζ(@Ro) :;1 ?9: ?:6
?9:
Ϗ
Ϗ
Ϗ
Stkkl Jross
Jross
(j)
(o)
(c)
(o) sfkor stroih. (j ( j) sfkor strkss ih o fg`gnkhkgus crgss skctigh. (c ( c) sfkor strkss ih o cg`pgsitk crgss skctigh. Ainurk 4.9 Distrijutigh ga (o
KVO@RLK 4.1 O sglid stkkl (N (N s 0 62 NRo) sfoat 3 ` lghn is skcurkly aostkhkd aostkhkd tg o fgllgw jrghzk jrghzk (N (N J 0 42 NRo) sfoat sfoat tfot is is : ` lghn lghn os sfgwh ih Ainurk 4.?:. Dktkr`ihk tfk `oxi`u` voluk ga sfkor strkss ih tfk sfoat ohd tfk rgtotigh ga tfk rinft khd. ` ` 2 : ?
O
?22 eH-` eH-` x : `
` ` 2 6
J :; eH-` eH-` C ? `
Ainurk 4.?2 Cg`pgsitk sfoat agr Kxo`plk 4.1.
1;
4
Cg`pgsitk Structurol @k`jkrs
OHS.
ρC 0
2.2>1 rod ccw 5
τ`o x 0
:;2 :;2 @R o
KVO@RLK 4.> O cg`pgsitk sfoat witf o titohiu` gutk r loykr ohd oh olu`ihu` loykr gh tfk ihsidk is tg jk dksinhkd agr trohs`ittihn o tgrquk ga ?222 H ⋋ `. Os sfgwh ih Ainurk 4.??, 4.??, tfk sfoat `ust jk : ` lghn ohd `ust fovk oh gutkr dio`ktkr ga :; `` tg ait kxistihn ottocf`khts. Tfk rklotivk rgtotigh ga tfk twg khds ga tfk sfoat is li`itkd tg ?.; rod. Tfk sfkor `gdulus ga rinidityN rinidityN, tfk ollgwojlk sfkor strkss τollgw, ohd tfk dkhsity γ ork nivkh ih Tojlk 4.:. Dktkr`ihk tfk o`guht, ih nro`s, ga kocf `otkriol tg usk ia tfk linftkst sfoat is tg jk dksinhkd. Tojlk 4.: @otkriol Rrgpkrtiks agr Kxo`plk 4.>
T i t o o h i u `
N (NRo)
τollgw
ρ
(@Ro)
(@n/`3)
Titohiu` ollgy
31
4;2
4.4
Olu`ihu`
:6
?62
:.6
@otkriol
Olu`ihu`
4.>. Ainurk 4.?? Cg`pgsitk sfoat agr Kxo`plk 4.>.
OHS.
` Ti 0 :>1;n 5 `Ol 0 992n
4.3 4.3
CG@R CG@RGS GSIT ITK K SY@@ SY@@KT KTZI ZIC C JKO@ JKO@S S
Ainurk 4.?1 sfgws o lo`ihotkd cg`pgsitk crgss skctigh. ^k ossu`k tfot oll `otkriol crgss skctighs ork sy``ktri c ojgut tfk y oxis y oxis ohd tfot tfk lgodihn ih still ih tfk plohk ga sy``ktry, tfus khsurihn sy``ktric jkhdihn ojgut tfk z tfk z oxis. oxis. ^k ossu`k tfot oll `otkriols ork skcurkly jghdkd tg kocf gtfkr= tfot is, tfkrk is hg rklotivk `gvk`kht ot ohy ihtkraock ga tfk `otkriol prkskrvihn gur eihk`otic ossu`ptighs. ^k aurtfkr ossu`k tfot wk still fovk s`oll stroihs, tfot oll `otkriols o rk lihkor-klostic-isgtrgpic, ohd tfot tfkrk ork hg ihklostic stroihs. Ih gtfkr wgrds, oll ossu`ptighs prigr tg Ossu`ptigh 6 ih Tojlk 3.: ork 3.: ork still volid. Tfus, Kquotigh (3.;-J), (3.;-J), wficf prkckdks Ossu`ptigh 6, is still volid ohd agr`s gur stortihn pgiht agr dkrivihn tfk agr`ulos agr cg`pgsitk jko`s. jko`s.Kquotigh Kquotigh (3.;-J) will (3.;-J) will yikld tfk lgcotigh ga tfk grinih (hkutrol oxis), wfilk Kquotigh (3.1-J) coh (3.1-J) coh jk dkvklgpkd tg nivk tfk jkhdihn agr`ulos.
4.3.?
Hgr`ol Hgr`ol Jkh Jkhdihn dihn Strk trkss ss ih ih Cg`p Cg`pgsit gsitkk Jko`s Jko`s
Kocf `otkriol ih tfk lo`ihotkd crgss skctigh sfgwh ih Ainurk 4.?1 fos o `gdulus ga klosticity K klosticity K i tfot is cghstoht gvkr tfk `otkriol crgss-skctighol orko O orko Oi. Suppgsk tfkrk ork h `otkriols ih tfk crgss skctigh. Tfk ihtknrol ih Kquotigh (3.1-J) coh (3.1-J) coh jk writtkh os tfk su` ga tfk ihtknrols gvkr kocf `otkriol, os sfgwh ih Kquotigh (4.?2o). :
:
d v d v : @ z 0 --------: Ky d O 0 --------: dx dx
∥
O
∥ K y d O + ∥ K y d O + . . . + . . . + ∥ K y d O :
:
?
:
:
O ?
h
O :
(4.?2o)
O h
y K h
z K i
ξc K ?
Ainurk 4.?: Crgss skctigh ga o cg`pgsitk sy``ktric jko`.
Fkrk K Fkrk K i is o cghstoht ih kocf ihtknrol ohd coh jk toekh gutsidk tfk ihtknrol. Tfk rk`oihihn ihtknrol is tfk skcghd orko `g`kht ga ihkrtio ( I I zz )i ga tfk itf `otkriol. ^k tfus gjtoih :
d v @ z 0 --------: U K ? ( I zz )? + K : ( I zz ) : + . . . + . . . + K h ( I zz ) h P gr dx :
d v @ z 0 --------: dx
h
∟ 0 K ( I ) m ?
m
zz m
(4.??)
Cg`pgsitk Sy``ktric Jko`s
11
Tfk ojgvk kquotigh sfgws tfot tfk jkhdihn rinidity ga tfk cg`pgsitk jko` is tfk su` ga oll tfk jkhdihn riniditiks ga tfk `otkriols ih tfk cg`pgsitk. ^k coh writk Kquotigh (3.3-J) agr (3.3-J) agr tfk itf `otkriol os :
( σ xx ) i 0 wfkrk
( σ xx ) i
d v – K i y --------: ( x ) dx
(4.?:)
is tfk jkhdihn hgr`ol strkss ih tfk itf `otkriol. Sujstitutihn Kquotigh (4.??) ihtg (4.??) ihtg Kquotigh (4.?:), (4.?:), wk gjtoih
( σ xx ) i
K y i @ z 0 – -----------------------------h K m ( I zz ) m
(4.?3)
∟0
m ?
Kquotighs (4.??) ohd (4.??) ohd (4.?3), (4.?3), wficf ork opplicojlk tg cg`pgsitk jko`s, hgw rkplock Kquotighs (3.6-J) ohd (3.6-J) ohd (3.9-J) agr (3.9-J) agr . . . . . . tfk fg`gnkhkgus crgss skctigh. Ia wk cghsidkr o fg`gnkhkgus `otkriol, tfkh K tfkh K ? 0 K : 0 0 K 0 K i 0 K 0 K h 0 K . Sujstitutihn tfis ih Kquotighs (4.??) ohd (4.??) ohd (4.?3), (4.?3), wk gjtoih Kquotighs (3.6-J) ohd (3.6-J) ohd (3.9-J) agr (3.9-J) agr fg`gnkhkgus `otkriol, os kxpkctkd. Tfk ihtkrhol jkhdihn `g`kht @ ih Kquotighs (4.??) ohd (4.??) ohd (4.?3) rkprkskhts (4.?3) rkprkskhts tfk stoticolly kquivolkht `g`kht gvkr tfk z khtirk crgss skctigh, os skctigh, os it did agr tfk fg`gnkhkgus crgss skctigh. Tfk oholysis tkcfhiquks agr aihdihn tfk ihtkrhol `g`kht @ z ot o crgss skctigh rk`oih tfk so`k os jkagrk. ^k ehgw tfot tfk jkhdihn hgr`ol stroih hgr`ol stroih voriks voriks lihkorly ocrgss tfk cg`pgsitk crgss skctigh. Jut tfk hgr`ol strkss ih kocf `otkriol, (σ xx)i nivkh jy Kquotigh (4.?3), is o pikck wisk lihkor auhctigh tfot cfohnks its slgpk witf kocf `otkriol, os dk`ghstrotkd sfgrtly ih Kxo`plk 4.6. 4.6.
4.3.: 4.3.:
Lgcotig Lgcotigh h ga tfk tfk Hkutro Hkutroll Oxis (grinih) (grinih) ih Cg`pg Cg`pgsitk sitk Jko` Jko`ss
^k cfgsk tfk grinih sucf tf ot tfk tgtol oxiol agrck gh o crgss skctigh wos zkrg, wfic f rksultkd ih dkcguplihn ga tfk oxiol prgjlk` arg` tfk jkhdihn prgjlk`. Tfk lgcotigh ga tfk grinih agr o hghfg`gnkhkgus skctigh is dktkr`ihkd arg` Kquotigh (3.;-J), (3.;-J), wficf is idkhticol tg Kquotigh (3.;-O). (3.;-O). Fkhck, tfk rksult ih Kquotigh (4.;), (4.;), wficf nivks tfk lgcotigh ga tfk grinih agr `ultiplk `otkriols, is volid fkrk olsg ohd is rkwrittkh os Kquotigh (4.?4) agr (4.?4) agr cghvkhikhck, h
∟ ∟0
ξ m K m O m m 0 ? 0 ----------------------------h K m O m
ξc
(4.?4)
m ?
wfkrk f m is tfk lgcotigh ga tfk ckhtrgid ga tfk m tfk mtf tf `otkriol os `kosurkd arg` o cg``gh dotu` lihk. Tfk jkhdihn hgr`ol strkss voriks lihkorly witf y witf y ih ih kocf `otkriol, ohd tfk `oxi`u` jkhdihn hgr`ol strkss ih kocf `otkriol will jk wfkrk tfk y tfk y cggrdihotk cggrdihotk is tfk nrkotkst. ^itf y ^itf y `kosurkd `kosurkd arg` tfk hkutrol oxis, wk gjtoih tfk agllgwihn gjskrvotigh. • Tfk `oxi`u` jkhdihn hgr`ol strkss ih kocf `otkriol is ot tfk pgiht gh tfk `otkriol tfot is aurtfkst arg` tfk hkutrol oxis ga o cg`pgsitk jko`.
4.3.3 4.3.3
Jkhdihn Jkh dihn Sfkor Sfkor Strk Strkss ss ih Cg`pgsi Cg`pgsitk tk Jko`s Jko`s
Kquotigh (3.?;-J), rklotihn sfkor ohd hgr`ol strkssks ih tfk jko`, wos dkrivkd arg` tfk kquilijriu` ga o jko` klk`kht ohd is ihdkpkhdkht ga tfk `otkriol `gdkl, prgvidkd tfk eihk`otic ossu`ptighs rk`oih volid, os is tfk cosk agr gur cg`pgsitk jko`s. Sujstitutihn Kquotigh (4.?3) ihtg (4.?3) ihtgKquotigh Kquotigh (3.?;-J) ohd (3.?;-J) ohd hgtihn tfot tfk `g`kht ohd jkhdihn rinidity dg hgt cfohnk ocrgss tfk crgss skctigh, wk gjtoih ---τ sx t 0 – --d
– K y @ z
d @ z _ cg`p 0 ------ -----------------------------h dx K m ( I zz ) m
@ z
d
------------------------------ d O 0 ------ ------------------------------ Ky d ∥ O dx dx ∥ K I K I ( ) ( ) ∟0 ∟0 O s
h
m ?
h
m
zz m
m
m ?
∟0
zz m O s
(4.?;)
m ?
wfkrk _ cg`p 0
∥ Ky d O
(4.?1)
Os
Ossu`ihn tfot tfk jko` is hgt topkrkd ohd tfot tfk `gdulus ga klosticity dgks hgt cfohnk witf x witf x,, wk fovk tfk i`plicotigh tfot _cg`p ohd tfk jkhdihn rinidity ih Kquotigh (4.?;) dg (4.?;) dg hgt cfohnk witf x witf x ohd ohd coh jk toekh gutsidk tfk dkrivotivk sinh. Tfkh, oatkr hgtihn tfot Q y 0 – ( d @ z ⁄ dx ) , wk gjtoih. _cg`p d @ _cg`p Q y ---------- z 0 – ----------------------------------------------------------h h K m ( I zz ) m dx K m ( I zz ) m
τ sx t 0
∟0
∟0
m ?
τ sx
0
τ xs
m ?
– _ cg`p Q y 0 -----------------------------------------
⎮ ⎦ ⎩
h
∟
K m ( I zz ) m
m0? m 0?
⎥ ⎬t ⎠
(4.?>)
1>
4
Cg`pgsitk Structurol @k`jkrs
Tfus _cg`p is tfk hkw quohtity tfot wk hkkd tg kvoluotk ih Kquotigh (4.?>). (4.?>). Lkt h s rkprkskht tfk hu`jkr ga `otkriols ih O s5 tfot is, h s 0 tfk hu`jkr ga `otkriols jktwkkh tfk arkk suraock ohd tfk pgiht ot wficf wk ork kvoluotihn tfk sfkor strkss. Tfk ihtknrol ih Kquotigh (4.?1) coh (4.?1) coh jk writtkh os tfk su` ga ihtknrols gvkr kocf `otkriol os
∥ K y d O + ∥ K y d O + . . . + ∥ K y d O
_ cg`p 0
?
:
O ?
(4.?6)
h s
O :
O h
s
Fkrk K Fkrk K (_ z ) m, is tfk airst m is o cghstoht ih kocf ihtknrol ohd coh jk toekh gutsidk tfk ihtknrol. Tfk rk`oihihn ihtknrol, dksinhotkd (_ orko `g`kht ojgut ga tfk m tfk mtf tf `otkriol ojgut tfk hkutrol oxis. oxis. Tfus, wk gjtoih _ cg`p 0 K ? ( _ z ) ? + K : ( _ z ): + . . . + K h s ( _ z ) h gr s
_cg`p 0
∟ 0 K ( _ ) h s
m ?
m
(4.?9)
z m
Tfk nkg`ktric prgpkrty (_ (_ z ) m is ihdkpkhdkht ga tfk `otkriol ohd coh jk aguhd os ih o fg`gnkhkgus crgss skctigh. Tfkh Kquotigh (4.?9) coh (4.?9) coh jk uskd tg aihd _cg`p. Ia h s 0 h (i.k., h (i.k., ia wk cghsidkr tfk khtirk crgss skctigh), tfkh arg` Kquotigh (3.;-J) _cg`p 0 2. Tfus, Tfus, ghck `grk, liek liek _ z agr o fg`gnkhkgus `otkriol, _cg`p storts ohd khds witf zkrg os wk `gvk arg` ghk arkk suraock ohd rkocf tfk gtfkr arkk suraock ohd is `oxi`u` ot tfk hkutrol oxis. Tfus ih o `otkriol, os wk `gvk clgskr tg tfk hkutrol oxis, tfk voluk ga _cg`p ihcrkosks. ^k rkcgrd tfk agllgwihn gjskrvotighs. • Tfk `oxi` `oxi`u` u` voluk voluk ga _cg`p is ot tfk hkutrol oxis. • Tfk `oxi`u` jkhdihn sfkor strkss ih kocf `otkriol is ot tfk pgiht gh tfk `otkriol tfot is clgskst tg tfk hkutrol oxis ga o cg`pgsitk jko`.
KVO@RLK 4.6 O fg`gnkhkgus wggdkh crgss skctigh ohd o crgss skctigh ih wficf tfk wggd is rkihagrckd witf stkkl ork sfgwh ih Ainurk 4.?3. 4.?3. :22 y μ . Tfk `gdulus ga klosticity agr stkkl is Tfk hgr`ol stroih agr jgtf crgss skctighs wos aguhd tg vory os λ xx 0 √ :22 y
K st 0 32,222 esi, ohd agr wggd wggd it is K is K wggd 0 6222 esi. (o) Rlgt Rlgt tfk σ xx distrijutigh agr kocf ga tfk crgss skctighs sfgwh ih Ainurk 4.?3. 4.?3. (j) Colculotk tfk kquivolkht ihtkrhol `g`kht @ z agr kocf crgss skctigh. y
y Stkkl z
? ? -- ih. : ^ggd
?/4 ih.
z
? ih. ^ggd Stkkl
?/4 ih. : ih.
: ih. (o)
(j)
(o) fg`gnkhkgus ohd (j (j) lo`ihotkd. Ainurk 4.?3 Crgss skctighs agr Kxo`plk 4.6= (o
OHS.
Fg`gnkhkgus Crgss skctigh= @ z 0 2.9 ih ⋋ eips 5 Lo`ihotkd Crgss skctigh= @ z 0 :.14 ih ih ⋋ eips y (ih.) y (ih.)
y (ih.) y (ih.)
y (ih.) y (ih.)
2.>; 2.;
2.>; 2.;
2.>; 2.;
?22 ?;2 ?;2 ?22
λxx (μ)
2.6 ?.: ?.: 2.6
σxx (esi)
4.;
2.6 3.2
2.6
2.;
2.;
2.;
2.>;
2.>;
2.>;
3.2
4.;
σxx (esi)
(o) (j)
(c)
(o) stroih distrijutigh. (j (j) strkss ih o fg`gnkhkgus fg`gnkhkgus crgss skctigh, ohd (c (c) strkss ih o lo`ihotkd crgss skctigh. Ainurk 4.?4 Distrijutigh ga (o
Cg`pgsitk Sy``ktric Jko`s
16
KVO@RLK 4.9 Oh olu`ihu` (KOl 0 ?2,222 esi) strip is skcurkly skcurkly aostkhkd aostkhkd tg o wggdkh ( K K wggd 0 ?122 esi) jko` jko` os sfgwh sfgwh ih Ainurk 4.?; tg 4.?; tg i`prgvk its lgod-corryihn copocity. (o) Gh o skctigh must tg tfk rinft ga suppgrt O suppgrt O,, dktkr`ihk ohd plgt tfk jkhdihn hgr`ol ohd sfkor strkss ocrgss tfk crgss skctigh. (j) Dktkr`ihk tfk `oxi`u` jkhdihn hgr`ol ohd sfkor strkss ih olu`ihu` ohd wggd. y
Olu`ihu` ::22 lj
;22 lj
622 at.-lj 622 at.-lj
2.:; 2.:; ih :?22 at.-lj
^ggd
O 3 at.
3 at.
4 ih
x
J
? ih
3 at.
Crgss-skctigh
Ainurk 4.?; Jko` ohd lgodihn agr Kxo`plk 4.9.
OHS. y
y 2.:; 2.:; ih.
z
y 6333
D K
?.423 ?.423 ih.
??3:
>2>3
::9
σxx (psi)
C
τxy (psi)
3:4
:.;9> :.;9> ih.
:29;
A
(o)
(j)
Ainurk 4.?1 Strkss distrijutigh ocrgss tfk crgss skctigh agr Kxo`plk 4.9.
OHS.
( σ`ox )Ol
0 :>.4 esi ( C ) 5
( σ`ox ) wggd
0 1.9 1.9 esi ( T ) 5
( τ`ox )Ol
0 :>; psi 5
( τ`ox )wggd
0 366 psi
KVO@RLK 4.?2 Ih rkihagrckd cghcrktk jko`s, tfk tkhsigh-corryihn copocity ga tfk cghcrktk is inhgrkd jkcousk tfk `otkriol’s tkhsilk strkhntf is lgw, rksultihn ih uhprkdictojly ga croceihn gh tfk tkhsigh sidk. Ihcgrpgrotk tfis ihagr`otigh tg sglvk tfk agllgwihn prgjlk`. Tfk cghcrktk jko` sfgwh ih Ainurk 4.?> fos 4.?> fos jkkh rkihagrckd jy k`jkddihn circulor stkkl rgds ga dio`ktkr :2 ``. Tfk `oxi`u` rksistihn (ihtkrhol) `g`kht ih tfk jko` wos dktkr`ihkd tg jk 42 eH ⋋ `. Dktkr`ihk tfk `oxi`u` jkhdihn hgr`ol strkss ih tfk cghcrktk ohd stkkl. [sk tfk agllgwihn `gduli ga klosticity agr stkkl ohd cghcrktk= K cghcrktk= K s 0 :22 NRo ohd K ohd K c 0 :6 NRo. NRo.
eH-` @ z 0 42 eH-`
422 422 ``
;2 `` ?;2 ?;2 ``
Ainurk 4.?> Zkihagrckd cghcrktk jko` crgss skctigh.
OHS.
( σ`ox )c
0 ?:.> ?:.> @Ro ( C ) 5
( σ `ox ) s
0 ?41.: @Ro ( T )
19
4
4.4
Cg`pgsitk Structurol @k`jkrs
@ICZG @ICZG@KC @KCFO FOHI HICS CS GA CG@R CG@RGS GSIT ITKS KS
Tfk `otkriol cghstohts gjtoihkd arg` tfk rksults ga o tkhsigh tkst agr o cg`pgsitk ork `ocrgscgpic prgpk rtiks tfot ih sg`k `ohhkr dkpkhd upgh its cghstitukhts ohd coh vory sinhiaicohtly dkpkhdihn upgh o fgst ga aoctgrs, sucf os tfk `ohuaocturihn prgckss, tfk `ohuaocturihn khvirgh`kht, tfk tkstihn khvirgh`kht, ohd tfk fistgry ga lgodihn. Tfus, it is hgt surprisihn tfot kxpkri`khtol rksults ga o tkhsigh tkst coh sfgw sinhiaicoht voriotighs ih tfk `kcfohicol prgpkrtiks ga skk`ihnly si`ilor cg`pgsitk `otkriols. O tfkgry ihtrihsicolly is oh idkolizotigh ga o cg`plkx rkolity, ohd tfk voriojility ih `kcfohicol prgpkrtiks ga cg`pgsitks fos spowhkd `ohy tfkgriks. Tfksk tfkgriks ottk`pt tg prkdict `ocrgscgpic prgpkrtiks arg` tfk prgpkrtiks ga ihdividuol cghstitukhts tfot `oy jk `icrgscgpic ih sizk. Tfksk tfkgriks cghstitutk tfk aikld ga `icrg`kcfohics. Ih tfis skctigh wk will cghsidkr ghk sucf tfkgry tfot coh jk gjtoihkd arg` tfk kquotighs ga `kcfohics ga `otkriols.? O pikck ga tfk cg`pgsitk `otkriol, collkd o “uhit ckll‐ gr o “rkprkskhtotivk vglu`k klk`kht,‐ is visuolizkd ih `icrg`kcfohics tg dkvklgp o tfkgry. Tfk prgpkrtiks ga tfk uhit ckll ork dkrivkd, ohd it is pgstulotkd tfot tfk prgpkrtiks ga tfk khtirk jgdy ork si`ilor tg tfgsk ga tfk uhit ckll. Clkorly tfk pgstulotks tfot tfk khtirk `otkriol jkfovks liek tfk visuolizkd uhit ckll ohd rksk` jlks it ih nkg`ktry ohd `otkriol `oekup ga tfk uhit ckll ork jin opprgxi` otighs. Tfk kquotighs uskd tg prkdict tfk prgpkrtiks ga tfk uhit ckll ork joskd gh skvkrol ossu`ptighs ohd ork ykt ohgtfkr sgurck ga opprgxi`otigh ih tfk tfkgry. Ih spitk ga oll tfksk opprgxi`otighs, tfk tfkgriks i`prgvk gur uhdkrstohdihn ga `kcfohics ga cg`pgsitk `otkriols. Kxpkri`khts ork tfk aihol orjitkrs ga tfk occurocy ga tfkgrkticol prkdictighs. Tfk tfkgriks ga `icrg`kcfohics ork tfkgriks ga fg`gnkhizotigh ga cghstitutivk `otkriols tg gjtoih `ocrgscgpic prgpkrtiks. Isgtrgpic ohd trohsvkrskly isgtrgpic `otkriols coh jk vikwkd os o sujcloss ga grtfgtrgpic `otkriols. ^k will dkvklgp kquotighs tfot ork opplicojlk tg grtfgtrgpic `otkriols ohd usk tfk rklkvoht ghks agr isgtrgpic ohd trohsvkrskly isgtrgpic `otkriols. Tfk grtfgtrgpic `otkriol tfot is kosikst tg visuolizk is tfk lghn-aijkr cg`pgsitk lo`iho, discusskd ih tfk hkxt skctigh.
4.4.? 4.4.?
Grtf Gr tfgtr gtrgp gpic ic Cg`p Cg`pgs gsitk itk Lo`i Lo`ihok hok
Tfk discussigh ga grtfgtrgpic `otkriols ih Skctigh :.?.: novk :.?.: novk o jrika dkscriptigh ga o cg`pgsitk lo`iho. O cg`pgsitk grtfgtrgpic lo`iho is cghstructkd jy loyihn lghn aijkrs ih o nivkh dirkctigh ohd jihdihn tfksk aijkrs witf oh kpgxy. O jrika dkscriptigh ga aijkrs ohd `otrix `otkriols agllgws. Tfk dio`ktkr ga aijkrs is ga tfk grdkr ga ?2 ``. Sfgrt aijkrs (lkhntf gh tfk grdkr ga ? ``) ork collkd wfisekrs. Aijkrs coh fovk tfk cgrrkspghd cgrrkspghdihn ihn tkhsilk tkhsilk strkhntf ohd ohd stiaahkss stiaahkss (`gdulus ga klosticity) gr dkrs ga `onhitudk finfkr tfoh tfgsk ga jule `otkriols. Tfk K-nloss aijkrs fovk nggd klkctricol prgpkrtiks jut lgwkr strkhntf ohd stiaahkss tfoh tfk S-nloss aijkrs. aijkrs. Nloss aijkrs ork kxtkhsivkly uskd ih spkciolizkd spgrts kquip`kht, outg`gjilks, ohd oircroat structurks. Tfk tkr`s “nropfitk aijkr‐ ohd “corjgh aijkr‐ ork gatkh uskd ihtkrcfohnkojly. Tfkrk is o `omgr diaakrkhck, fgwkvkr= nropfitk aijkrs ork sujmkctkd tg vkry finf tk`pkroturks. Ih cg`porisgh tg nloss aijkrs, corjgh aijkrs ork sinhiaicohtly `grk kxpkhsivk ohd ork usuolly uskd ih odvohckd tkcfhglgnicol opplicotighs sucf os ih oircroat structurks. Oro`id pgly`kr aijkrs, tfk `gst pgpulor jkihn Ekvlor, wkrk griniholly dkvklgpkd agr usk ih rodiol tirks jut ork hgw aihdihn ihcrkoskd opplicotighs ih structurks. Tfk list ga aijkrs tfot coh jk uskd ih structurol opplicotigh is cghtihuolly nrgwihn. Ih Oppkhdix O, Skctigh O.> nivks O.> nivks tfk `kcfohicol prgpkrtiks ga o akw aijkrs agr usk ih tfis jgge. Tfk `otrix fglds tfk aijkrs tgnktfkr ohd prgtkcts tfk` arg` ojrosigh ohd cgrrgsigh. @otrix `otkriol prgpkrtiks `oy gatkh li`it tfk opplicotigh ga o cg`pgsitk. Oltfgunf `ktols ohd ckro`ics cguld jk uskd os `otrix `otkriols, wk will jrikaly cghsidkr ghly o akw pgly`kr `otkriols, olsg collkd rksihs. Tfkrk ork twg josic typks ga p gly`kr rksih ih usk ih structurol opplicotighs. Tfk tfkr`gskt rksihs cfohnk irrkvkrsijly wfkh curkd (i.k., fkotkd), ohd ghck curkd, tfky dg hgt `klt ia rkfkotkd. Tfkr`gplostic pgly`krs sglidiay wfkh cgglkd ohd rkvkrsijly `klt ia rkfkotkd. Tfkr`gskt pglykstkrs ork ihkxpkhsivk pgly`krs, rkihagrckd usuolly witf nloss aijkr, agr usk os stgronk tohes, jgot fulls, ohd pujlic jkhcfks. Kpgxy rksihs ork usuolly uskd ih odvohckd cg`pgsitk `otkriols, ohd tfkir prgpkrtiks ork dkpkhdkht upgh tfk curihn onkht uskd. Rglyi`idk rksihs ork pri`orily uskd ih finftk`pkroturk opplicotighs jut ork jrittlk. Skctigh O.> nivks O.> nivks `kcfohicol prgpkrtiks ga o akw ga tfk pgly`krs agr usk ih tfis jgge.
4.4. 4.4.: :
[hit [hit Ckl Cklll Oppr Opprgx gxi` i`ot otig igh h
Kvkh tfgunf sg`k ga tfk rksults wk will gjtoih ork ihdkpkhdkht ga tfk nkg`ktry ga aijkr ohd `otr ix, agr tfk soek ga si`plicity, wk will visuolizk o lghn-aijkr cg`pgsitk lo`iho ga rkctohnulor nkg`ktry os sfgwh ih Ainurk 4.?6. 4.?6. ^k will usk agllgwihn hgtotigh= K a ohd K ohd K ` 0 `gdulus ga klosticity ga aijkr ohd `otrix ΰa ohd ΰ` 0 Rgissgh rotigs ga aijkr ohd `otrix Na ohd N` 0 sfkor `gdulus ga rinidity ga aijkr ohd `otrix Q a ohd Q ` 0 vglu`k ga aijkr ohd `otrix va ohd v ` 0 vglu`ktric rotig ga aijkr ohd tfk `otrix
Tfk vglu`ktric rotig is rklotkd tg tfk vglu`ks os v a 0 Q a ⁄ ( Q a + Q ` ) ohd v ` 0 Q ` ⁄ ( Q a + Q ` ) , wfkrk tfk su` ga tfk vglu`ks ga aijkr ohd `otkriol is tfk tgtol vglu`k ga tfk cg`pgsitk ckll.
?. Klosticity rksults ork gatkh uskd os oh oltkrhotivk tg gur `kcfohics ga `otkriols opprgocf. Skk Nijsgh U?994P, U?994P, Fykr U?996P, U?996P, gr Mghks U?9>;P U?9>;P agr odditighol dktoils.
@icrg`kcfohics ga Cg`pgsitks
>2
@otrix
y
Aijkr
x f`/:
@otrix
f a
t L
f`/:
Ainurk 4.?6 [hit ckll opprgxi`otigh ga o lghn-aijkr cg`pgsitk.
4.4.3 4.4.3
Lghn Lghnitu itudih dihol ol @gdu @gdulus lus ga ga Klosti Klosticit city y K x
Cghsidkr o uhit ckll sujmkctkd tg lghnitudihol strkss σ xx os sfgwh ih Ainurk 4.?9o 4.?9o. Ia O Ia Oa ohd O ohd O` rkprkskht tfk crgss-skctighol orkos ga aijkr ohd `otrix pkrpkhdiculor tg tfk lgodihn, tfkh tfk oxiol rinidity ga o crgss skctigh is K m O m 0 K a O a + K ` O ` . Tfk tgtol oxiol agrck tfot is opplikd is σ xx ( O a + O` ) , wficf will jk kquol tg tfk ihtkrhol oxiol agrck H . Arg` Kquotigh (4.:) wk (4.:) wk gjtoih tfk oxiol stroih ot ohy crgss skctigh os
∟
λ xx
σ xx ( O a + O ` ) σ xx ( O a + O` ) L σ xx ( Q a + Q ` ) σ xx H 0 ----------------- 0 ------------------------------0 ---------------------------------------0 -----------------------------------0 ------gr K a Oa + K ` O ` ( K a O a + K ` O ` ) L ( K a Q a + K ` Q ` ) K x K m O m
∟
K x 0 K a v a + K ` v`
(4.:2)
σ yy A
A
K
σ xx D J
O
K D
σ xx
J
O
C
C
σ yy
(o)
(j)
Ainurk 4.?9 Strkss gh o uhit ckll. (o) Lghnitudihol. (j) Trohsvkrsk
Kquotigh (4.:2) is (4.:2) is volid agr ohy crgss-skctighol sfopk ga tfk cghstitukhts jkcousk it dkpkhds ghly upgh tfk voluk ga tfk crgss-skctighol orkos. Tfk crgss skctigh pkrpkhdiculor tg tfk lgodihn cguld jklghn tg rohdg` sfgrt-aijkr cg`pgsitk gr tg o lghnaijkr cg`pgsitk, os ossu`kd fkrk. Tfus, Kquotigh (4.:2) is (4.:2) is opplicojlk tg ohy cg`pgsitk jgdy `odk ga twg cghstitukhts tfot fovk jkkh `ixkd tgnktfkr. Agr tfis rkosgh, Kquotigh (4.:2) is (4.:2) is collkd tfk rulk ga `ixturks agr `gdulus ga klosticity.
4.4. 4.4.4 4
Rgis Rgissg sgh’ h’ss Zot Zotig ig ΰ xy
Tfk trohsvkrsk stroih ih tfk `otrix ohd tfk aijkr is
( λ yy ) `
0 – ΰ ` λ xx
( λ yy )a 0 – ΰ a λ xx
(4.:?o)
Tfk rkductigh ih tfk lkhntf ga tfk lihk CA ih ih Ainurk 4.?9o 4.?9o coh jk aguhd os
δ CA
0
f f ( λ yy ) ` ( C D + K A ) + ( λ yy ) a DK 0 ( – ΰ` λ xx ) ⎕⎐ ----`-- + ----`--⎘⎢ :
:
+ ( – ΰ a λ xx ) ( fa )
(4.:?j)
Tfk trohsvkrsk stroih ga lihk CA is is
λ yy Hgtihn tfot
ΰ xy
U ΰ` f ` + ΰa f a P ( tL ) δ CA δ CA ( – ΰ` λ xx ) ( f ` ) + ( – ΰa λ xx ) ( fa ) 0 -------0 ---------------------0 --------------------------------------------------------------------0 – λ xx -------------------------------------------U f ` + f a P ( tL ) CA ( f ` + f a ) ( f ` + f a )
(4.:?c)
0 – λ yy ⁄ λ xx ohd Q L , Q ` 0 tf ` L , wk gjtoih Kquotigh (4.:4). a 0 tf a
U ΰ` Q ` + ΰ a Q a P – ΰ xy λ xx 0 – λ xx ----------------------------------U Q ` + Q a P ΰ xy
0
ΰ ` v ` + ΰa v a
gr (4.::)
Kquotigh (4.::) is (4.::) is tfk rulk ga `ixturks agr Rgissgh’s rotig. Fgwkvkr, uhliekKquotigh uhliek Kquotigh (4.:2), (4.:2), tfk ossu`ptigh tfot tfk `otrix ohd aijkr ork rkctohnulor is uskd ih colculotigh ga tfk vglu`ks. Tfus, tfk usk gaKquotigh gaKquotigh (4.::) agr (4.::) agr aijkrs ga gtfkr crgss-skctighol sfopk is oh opprgxi`otigh.
>?
Cg`pgsitk Structurol @k`jkrs
4
4.4.; 4.4.;
Trohsvk rohsvkrsk rsk @gdu @gdulus lus ga ga Klost Klostici icity ty K y
Ainurk 4.?9j 4.?9j sfgws o uhit ckll sujmkctkd tg o trohsvkrsk strkss σ yy. Tfus, tfk oxiol agrck opplikd ih tfk y tfk y dirkctigh is σ yy Lt , wficf wguld jk kquol tg tfk ihtkrhol oxiol agrck H gh gh o crgss skctigh pkrpkhdiculor tg tfk lgod. Tfk crgss-skctighol orko ga tfk i`onihory cut is tL irrkspkctivk tL irrkspkctivk ga wfktfkr tfk cut is ih tfk `otrix gr tfk aijkr, ohd kocf crgss skctigh is fg`gnkhkgus. Tfk tgtol dkagr`otigh ga tfk lihk CA is is tfk su` ga dkagr`otighs ga lihks CD, CD, DK , ohd KA ohd KA (i.k., (i.k., δ CA 0 δ CD + δ DK + δ KA ). Tfk dkagr`otigh ga kocf lihk coh jk aguhd jy usihn Kquotigh (3.?2-O) os (3.?2-O) os sfgwh jklgw.
δCA Hgtihn tfot dkagr`otigh
( σ yy Lt ) ( f ` ⁄ : ) ( σ yy Lt ) ( f a ) ( σ yy Lt ) ( f` ⁄ : ) σ yy f` ( σ yy ) f a 0 -----------------------------------+ ---------------------------- + ------------------------------------ 0 -------------- + ---------------- K ` ( Lt ) K a ( Lt ) K ` ( Lt ) K ` K a
δCA
0 ( f ` + f a )λ yy 0 ( f ` + f a ) ( σ yy ⁄ K y ) , wk gjtoih
δ CA
σ yy ( f ` + f a ) σ yy f ` σ yy f a 0 -----------------------------0 -------------- + ----------- K y K ` K a
(4.:3o)
f ` + fa f ` fa ---------------- 0 -----+ ---- K y K ` K a
gr
(4.:3j)
@ultiplyihn tfk Kquotigh (4.:3j) jy (4.:3j) jy ( Lt Lt ) ohd hgtihn tfot Q a 0 Lt f a ohd Q ` 0 Lt f ` , wk gjtoih Kquotigh (4.:4). (4.:4). Q ` + Q a Q ` Q a -----------------0 -----+ ---- K y K ` K a
gr
v ` va ? ----- 0 -----+ ---- K y K ` K a
(4.:4)
Kquotigh (4.:4) is (4.:4) is collkd tfk ihvkrsk rulk ga `ixturks. ^k `odk usk ga rkctohnulor nkg`ktry ih colculotighs ga vglu`ks. Tfus, tfk usk ga Kquotigh (4.:4) agr (4.:4) agr aijkrs ga gtfkr crgss-skctighol sfopks is oh opprgxi`otigh.
4.4.1 .4.1
Sfkor @g @gdulus lus N xy
Ainurk 4.:2o 4.:2o sfgws o uhit ckll sujmkctkd tg uhiagr` sfkor strkss. Ainurk 4.:2j 4.:2j sfgws tfk sfkor dkagr`otigh duk tg tfk opplikd sfkor strkss. Tfk sfkor stroih ih tfk `otrix ohd aijkr ork diaakrkht, ohd fkhck tfk cfohnk ih ohnlk (i.k., arg` o rinft ohnlk) is diaakrkht agr tfk `otrix ohd tfk aijkr os sfgwh ih Ainurk 4.:2j 4.:2j. Ossu`ihn s`oll sfkor stroih, wk coh writk tfk agllgwihn= DD toh γ ` ≆ γ ` 0 -----------? CD K ? K : toh γ a ≆ γ a 0 ----------- DK A ? A : toh γ ` ≆ γ ` 0 ----------- KA
gr
gr
gr
DD ? 0
f γ ` ⎕⎐ ----`--⎘⎢
(4.:;o)
γa f a
(4.:;j)
:
K ? K : 0 A ? A : 0
f γ ` ⎕⎐ ----`--⎘⎢
(4.:;c)
:
A ?
A
A :
γ`
A
K
K ? γ a
K
f`/:
K :
D
D
τ xy
C
f a
D?
γ` C
τ yx
f`/:
(o)
(j)
( j) Kxonnkrotkd sfkor dkagr`otigh ga uhit ckll Ainurk 4.:2 (o) Sfkor strkss gh uhit ckll. (j
Tfk tgtol sfkor displock`kht AA displock`kht AA : is tfk sfkor stroih γ xy `ultiplikd jy CA 5 tfot is, AA : 0 γ xy ( f a + f ` ) AA : 0 γ xy ( f a + f ` ) . It coh olsg jk writtkh os tfk su` ga tfk sfkor displock`khts= AA : 0 DD ? + K ? K : + A ? A : . Kquotihn tfk twg kxprkssighs agr AA :, wk gjtoih AA : 0
γ xy ( f a + f ` )
0
( γa f a + γ ` f ` )
gr
τ
τ
τ
xy `⎘ -------f ( f a + f ` ) 0 ⎕⎐ ----- a ⎘⎢ f a + ⎕⎐ ------N xy N a N `⎢ `
(4.:;d)
Clgsurk
>:
wfkrk τ a ohd τ` ork tfk sfkor strkssks ih tfk aijkr ohd tfk `otrix, rkspkctivkly. Tfk sfkor agrck ot ohy crgss skctigh pkrpkhdiculor tg lihk CDKA is CDKA is ( τ xy ) ( Lt ) . ^itf tfk crgss-skctighol orko tfot is pkrpkhdiculor tg CDKA kquol kquol tg ( Lt Lt ), ), tfk sfkor strkss ih tfk aijkr ohd ih tfk `otrix is kquol tg tfk opplikd sfkor strkss (i.k., τ a 0 τ ` 0 τ xy ). @ultiplyihn tfk rksultihn kquotigh jy Lt ohd ohd hgtihn tfk dkaihitigh ga vglu`ks, wk gjtoih Kquotigh (4.:6).
τ
xy -------( Q + Q ` ) 0 N xy a
τ xy⎘ τ xy ⎕ -----Q a + ⎕ -------⎘ Q ` ⎐ N a ⎢ ⎐ N `⎢
gr
v v` ? -------- 0 -----a + ------N xy N a N `
(4.:1)
Kquotigh (4.:1) is (4.:1) is tfk ihvkrsk rulk ga `ixturks agr sfkor `gdulus ga klosticity. Ghck `grk, wk `odk usk ga rkctohnulor nkg`ktry ih gjtoihihn tfk rksults. Tfus tfk rulk ga `ixturks agr `gdulus ga klosticity is tfk ghly ghk tfot is ihdkpkhdkht ga tfk sfopk ga tfk aijkr5 agr tfk rkst ga tfk `otkriol cghstohts, tfk rulk ga `ixturks ohd tfk ihvkrsk rulk ga `ixturks ork opprgxi`otighs, wficf ork mustiaikd os jkihn ga tfk so`k grdkr ga opprgxi`otigh ghk `oeks ih dkrivihn tfk kquotighs ga `kcfohics ga `otkriols.
KVO@RLK 4.?? (o) ^fot is tfk kaakctivk `gdulus ga klo sticity ih tfk oxiol dirkctigh ga tfk rkihagrckd cghcrktk jor ihKxo`plk ih Kxo`plk 4.4, 4.4, ossu`ihn tfot 9; en ga stkkl wos uskd tg cghstruct tfk jor7 (j) Colculotk tfk cghtroctigh ga tfk jor uhdkr o cg`prkssivk oxiol agrck ga ?222 H. OHS. K kaakctivk 0 4?.> NRo 5 u : – u? 0 –2.22?; ``
KVO@RLK 4.?: O cg`pgsitk spkci`kh fos aijkrs olinhkd ih tfk x tfk x dirkctigh dirkctigh os sy`jglicolly sfgwh ih Ainurk 4.:?. 4.:?. Tfk cg`pgsitk fos 42% ga Snloss aijkrs aijkrs jy vglu`k vglu`k ih o pglykstkr pglykstkr `otrix. `otrix. Tfk cg`pgsitk is sujmkctkd tg tfk hgr` ol ohd sfkor strkssks. Arg` tfk doto nivkh ih Skctigh O.> gh O.> gh ponk ponk ?12, ?12, dktkr`ihk tfk stroihs λ xx , λ yy, ohd γ xy. : esi esi ? esi esi y 4 esi esi
x ? esi esi
Ainurk 4.:? Strkssks gh o cg`pgsitk spkci`kh agr Kxo`plk 4.?:.
OHS.
λ xx 4.;
0 6>>.4
μ λ yy
0 32;9
μ γ xy
0 366;
μ
CLGS[ZK
Ih tfis cfoptkr wk sow tfot tfk eihk`otic kquotighs, tfk cghstitutivk kquotighs, ohd tfk kquilijriu` kquotighs dg hgt cfohnk ohd tfot hghfg`gnkhkity ocrgss o crgss skctigh rkquirks us tg writk tfk ihtknrol ih stotic kquivolkhcy os tfk su` ga tfk ihtknrols gvkr kocf `otkriol agr ghk-di`khsighol structurol `k`jkrs. Oll tfk oholyticol tkcfhiquks agr aihdihn ihtkrhol agrcks ohd `g`khts tfot ork uskd agr fg`gnkhkgus crgss skctighs coh jk uskd agr cg`pgsitk crgss skctighs. Agr lo`ihor cg`pgsitks, tfk prgpkrtiks ga kocf lo`iho coh jk gjtoihkd jy tfk rulk ga `ixturks gr jy tfk ihvkrsk rulk ga `ixturks. Ghck tfksk prgpkrtiks fovk jkkh gjtoihkd agr kocf lo`iho, tfk agr`ulos dkvklgpkd agr cg`pgsitk ghk-di`khsighol structurol `k`jkrs coh jk uskd agr structurol oholysis.
;
Ihklostic Structurol Jkfovigr
LKOZHIHN GJMKCTIQK ?. Tg uhdkrstohd tfk ihcgrpgrotigh ga tfkr`ol ohd ihitiol stroihs ihtg tfk tfkgry ohd oholysis ga oxiol `k`jkrs. :. Tg uhdkrstohd tfk ihcgrpgrotigh ga hghlihkor `otkriol `gdkls ihtg tfk josic si`pliaikd tfkgriks ga oxiol `k`jkrs, tgrsigh ga circulor sfoats, ohd sy``ktric jkhdihn ga jko`s. 3. Tg uhdkrstohd uhdkrstohd tfk pfkhg`khgh ga viscgklosti viscgklosticity city ohd tg occguht agr it ih oxiol `k`jkrs. `k`jkrs.
(j)
(o)
Ainurk ;.? Kxo`plks ga ihklostic structurol jkfovigr.(o) Rrk-tkhsighkd wirks ih o troaaic notk. (j) Strktcf cgrd
Ihitiol Ihitio l stroih ihtrgduckd jy tfk turhjucelk ih tfk wirk UAinurk UAinurk ;.?o ;.?oP, tfk hghlihkor strkss–stroih rklotighsfip ga rujjkr ih o strktcf cgrd UAinurk UAinurk ;.?j ;.?jP, strkssks ihtrgduckd durihn ossk`jly, prkstrksskd cghcrktk, tfkr`ol strkssks arg` tk`pkroturk cfohnks, prkstrkssihn `ktols ihtg plostic rknighs tg ihcrkosk tfk lgod-corryihn copocity ga o structurol `k`jkr, tfk hghlihkor ohd/gr viscgklostic rkspghsk ga plostic ohd jiglgnicol `otkriols, ork sg`k ga tfk kxo`plks ga ihklostic jkfovigr tfot `ust jk occguhtkd agr ih strkss oholysis. Ih tfis cfoptkr wk will occguht agr ihitiol stroihs, tfkr`ol stroihs, ohd viscgklostic stroihs ih tfk cghtkxt ga oxiol `k`jkrs5 wk will olsg study tfk ihcgrpgrotigh ga hghlihkor strkss–stroih kquotighs ih oxiol `k`jkrs, tfk tgrsigh ga circulor sfoats, ohd tfk sy``ktric jkhdihn ga jko`s. Ih Skctigh 3.:, 3.:, wfilk dkvklgpihn tfk si`pliaikd tfkgriks gh structurol `k`jkrs, wk `odk skvkrol ossu`ptighs rknordihn `otkriol jkfovigr tfot pkr`ittkd us tg usk Fggek’s low. Ih tfis cfoptkr wk will cghsidkr tfk i`poct ga drgppihn tfk agllgwihn ossu`ptighs, listkd korlikr ih Tojlk 3.:= 3.:= Ossu`ptigh ; (ga ; (ga hg ihklostic stroihs)5 Ossu`ptigh 1 (ga 1 (ga klostic `otkriol jkfovigr), ohd Ossu`ptigh > (ga > (ga o lihkor rklotighsfip jktwkkh strkss ohd stroih). ^k will ossu`k tfot tfk eihk`otic kquotighs ork still volid. Tfk kquotighs ga stotic kquivolkhcy ork uhoaakctkd jy tfk `otkriol `gdkl. Tfk kquilijriu` kquotighs rklotihn ihtkrhol agrcks tg kxtkrhol agrcks dg hgt cfohnk witf tfk `otkriol `gdkl. Tfus, tfk pri`ory i`poct ga ihklostic `otkriol jkfovigr is gh tfk strkss distrijutigh ocrgss tfk crgss skctigh. ^fkh tfk strkss voriotigh ocrgss tfk crgss skctigh is uhdkrstggd ohd occguhtkd agr, wk coh still usk tfk oholyticol tkcfhiquks wk fovk uskd jkagrk.
>3
>4
;.? ;.?
; Ihklostic
Structurk Jkfovigr
KAAK KAAKCT CTS S GA GA TK@ TK@RK RKZO ZOT[ T[Z ZK
O `otkriol kxpohds witf oh ihcrkosk ih tk`pkroturk ohd cghtrocts witf o dkcrkosk ih tk`pkroturk. Ia tfk cfohnk ih tk`pkroturk is uhiagr`, uhiagr`, ohd ia tfk `otkriol is isgtrgpic ohd isgtrgpic ohd fg`gnkhkgus, fg`gnkhkgus, tfkh oll lihks gh tfk `otkriol will cfohnk di`khsighs jy kquol o`guhts. Tfk kxpohsigh will rksult ih o hgr`ol stroih, jut hg sfkor stroih will jk prgduckd jkcousk tfkrk will jk hg cfohnk ih tfk ohnlks jktwkkh ohy twg lihks. Kxpkri`khtol gjskrvotighs cghair` tfis dkductigh. Kxpkri`khts olsg sfgw tfot tfk cfohnk ih tk` pkroturk (ΔT ) T ) is rklotkd tg tfk tfkr` ol hgr`ol stroih (λT ) os
λT 0 ο Δ T
(;.?)
wfkrk tfk Nrkke lkttkr olpfo (ο) is collkd tfk lihkor cgkaaicikht ga tfkr`ol kxpohsigh. Tfk lihkor rklotighsfip nivkh jy Kquotigh (;.?) is (;.?) is volid agr `ktols ih tfk rknigh aor arg` tfk `kltihn pgiht. Ih tfis lihkor rknigh, tfk stroihs agr `gst `ktols ork s`ol l ohd tfk usuol uhits agr ο ork μ/²A gr μ/²C, wfkrk μ 0 ?2 –1. Tfrgunfgut tfk discussigh ih tfis skctigh wk sfoll ossu`k tfot wk ork ih tfk lihkor rknigh. Tfk tkhsigh tkst tg dktkr`ihk `otkriol cghstohts is cghductkd ot oh o`jikht tk`pkroturk. ^k kxpkct tfk strkss–stroih curvk tg fovk tfk so`k cforoctkr ot twg diaakrkht o`jikht tk`pkroturks. Ia wk roisk tfk tk`pkroturk jkagrk wk opply tfk agrck R agrck R gh tfk spkci`kh, tfkh tfk spkci`kh will kxpohd ohd rksult ih o tfkr`ol stroih os nivkh jy Kquotigh (;.?). (;.?). Jut jkcousk tfkrk is hg kxtkrhol agrck, tfkrk will jk hg rksultihn ihtkrhol agrcks, fkhck hg strkssks. Tfus, tfk ihcrkosk ih tk`pkroturk jkagrk tfk opplicotigh ga tfk agrck cousks tfk stortihn pgiht ga tfk strkss–stroih curvk tg `gvk arg` G tg G? os sfgwh ih Ainurk ;.:. ;.:. Tfk tgtol stroih ot ohy pgiht is tfk su` ga `kcfohicol stroih ohd tfkr`ol stroihs, wficf coh jk writtkh os
λ 0 -σ-- + ο Δ T
(;.:)
K
σ Tkst pkragr`kd ot T g
s s k r t S l o ` r g H
Tkst pkragr`kd ot T g+ΔT
σ
λ
G?
G
σ
οΔ T
--K
Hgr`ol Stroih
Ainurk ;.: Kaakct ga tk`pkroturk gh strkss–stroih curvk.
Ia tfkrk ork hg ihtkrhol agrcks nkhkrotkd ih o jgdy gwihn tg cfohnks ih tk`pkroturk, tfkh hg strkssks will jk prgduckd. @otkriol hghfg`gnkhkity, `otkriol ohisgtrgpy, ohisgtrgpy, hghuhiagr` tk`pkroturk distrijutigh, gr rkoctigh agrcks arg` jgdy cghstr oihts ork tfk rkosghs agr tfk nkhkrotigh ga strkssks arg` tk`pkroturk cfohnks. ^k rkcgrd tfk agllgwihn gjskrvotigh. • Hg tfkr`ol strkssks ork prgduckd ih ih o fg`gnkhkgus, isgtrgpic, uhcghstroihkd jgdy os o rksult ga uhiagr` tk`pkroturk cfohnks. Tfk nkhkrolizkd vkrsigh ga Fggek’s low rklotks `kcfohicol stroihs tg strkssks. Tfk tgtol hgr`ol stroih, os skkh arg` Kquotigh (;.:), (;.:), is tfk su` ga `kcfohicol ohd tfkr`ol stroihs. Agr isgtrgpic `otkriols witf tk`pkroturk cfohnks, tfk nkhkrolizkd low coh jk writtkh os sfgwh ih Kquotighs (;.3). ^k dg hgt fovk tg cg``kht gh tfk fg`gnkhkity ga tfk `otkriol gr tfk uhi agr`ity ga tk`pkroturk cfohnk jkcousk Fggek’s low is writtkh agr o pgiht, hgt tfk wfglk jgdy.
λ xx
U σ xx – ΰ ( σ yy + σ zz ) P ⁄ K + ο Δ T
(;.3o)
λ yy 0 U σ yy – ΰ ( σ zz + σ xx ) P ⁄ K + ο Δ T
(;.3j)
λ zz 0 U σ zz – ΰ ( σ xx + σ yy ) P ⁄ K + ο Δ T
(;.3c)
γ xy 0 τ xy ⁄ N
(;.3d)
γ yz 0 τ yz ⁄ N
(;.3k)
γ zx 0 τ zx ⁄ N
(;.3a)
0
@kcf @kcfoh ohic icol ol stro stroih ih
Tfkr Tfkr`o `oll stro stroih ih
Ihitiol Strkss gr Stroih ih Oxiol @k`jkrs
>;
KVO@RLK ;.? Tfk circulor jor sfgwh ih Ainurk ;.3 fos ;.3 fos o dio`ktkr ga ?22 ``. O nop ga 2.; `` kxists jktwkkh tfk rinft-fohd woll ohd tfk jor jkagrk tfk ihcrkosk ih tk`pkroturk gccurs. Tfk tk`pkroturk ga tfk jor is uhia gr`ly ihcrkoskd jy 62²C. Dktkr`ihk tfk ovkronk oxiol strkss ohd tfk cfohnk ih tfk dio`ktkr ga tfk jor. [sk K [sk K 0 0 :22 NRo, ΰ 0 2.3:, ohd ο 0 ??.> μ/²C.
x :`
2.; 2.; ``
;.?. Ainurk ;.3 Jor agr Kxo`plk ;.?.
OHS.
σ xx 0
?3>.: ?3>.: @Ro ( C ) 5 Δ D
0
2.??2> `` ihcrkosk
KVO@RLK ;.: Tfk stroihs ot o pgiht gh olu`ihu` ( K K 0 NRo, N 0 :6 NRo, NRo, ΰ 0 2.:;, ohd ο 0 :3 μ/²C) wkrk aguhd tg jk 0 >2 NRo, λ xx 0 1;2 μ λ yy 0 322 μ γ xy 0 >;2 μ Ia tfk tk`pkroturk ihcrkosks jy :2²C ohd tfk pgiht is ih plohk strkss, dktkr`ihk tfk strkssks σ xx, σ yy, ohd τ xy ohd tfk stroih λ zz . OHS.
τ xy 0 ;.:
:?@Ro 5
σ xx 0
??.: @Ro ( T ) 5
σ yy 0
6.4 @Ro ( C ) 5
λ zz 0
4;2
μ
IHITIO IHITIOL L STZK STZKSS SS GZ GZ STZO STZOIH IH IH IH OVIOL OVIOL @K@JKZS @K@JKZS
@k`jkrs ih o stoticolly ihdktkr`ihotk structurk `oy fovk oh ihitiol strkss gr stroih jkagrk lgods ork opplikd. Tfksk ihitiol strkssks gr stroihs `oy jk ihtkhtighol gr uhihtkhtighol ohd coh jk couskd jy skvkrol aoctgrs. O nggd dksinh `ust occguht agr tfksk aoctgrs jy colculotihn tfk occkptojlk lkvkls ga prkstrkss. Huts gh o jglt ork usuolly aihnkr-tinftkhkd tg fgld oh ossk`jly ih plock. Ot tfis stonk tfk ossk`jly is usuolly strkss arkk. Tfk huts ork tfkh nivkh odditighol turhs tg prktkhsigh tfk jglts. ^f kh o hut is tinftkhkd jy ghk aull rgtotigh, tfk distohck it `gvks is collkd tfk pitcf. Oltkrhotivkly, pitcf is tfk distohck jktwkkh twg odmgihihn pkoes gh tfk tfrkods. Ghk rkosgh agr prktkhsighihn is tg prkvkht tfk huts arg` jkcg`ihn lggsk ohd aollihn gaa. Ohgtfkr rkosgh is tg ihtrgduck oh ihitiol strkss tfot will jk gppgsitk ih sinh tg tfot wficf will jk nkhkrotkd jy tfk lgods. Agr kxo`plk, cojlk ih o jridnk `oy jk prktkhsighkd jy tinftkhihn tfk hut-ohd jglt systk`s tg cguhtkr ohy slocekhihn ih tfk cojlk duk tg wihd gr skosghol tk` pkroturk cfohnks. Ia o `k`jkr is sfgrtkr tfoh is rkquirkd durihn ossk`jly, it will jk agrckd tg strktcf, tfus puttihn tfk khtirk structurk ihtg o prkstrkss. Tglkrohcks agr `ohuaocturihn tfk `k`jkrs `ust jk prkscrijkd tg khsurk tfot tfk structurk is hgt kxckssivkly prkstrksskd. Ih prkstrksskd cghcrktk, `ktol jors ork ihitiolly strktcfkd jy opplyihn tk hsilk agrcks, ohd tfkh cghcrktk is pgurkd gvkr tfk jors. Oatkr tfk cghcrktk fos skt, tfk opplikd tkhsilk agrcks ork rk`gvkd. Tfk ihitiol prkstrkss ih tfk jor is rkdistrijutkd, puttihn tfk cghcrktk ihtg cg`prkssigh. Cghcrktk fos nggd cg`prkssivk str khntf jut pggr tkhsilk strkhntf. Rrkstrksskd cghcrktk coh jk uskd ih situotighs wfkrk it `oy jk sujmkctkd tg tkhsilk strkssks. Structurks ork usuolly `odk ga oxiol jors ih diaakrkht grikhtotighs, ohd agr tfis rkosgh tfk agr` ga kquotigh nivkh ih Kquotigh (;.4) is (;.4) is prkakrrkd gvkr Kquotigh (3.?2-O) U (3.?2-O) U u: – u ? 0 H ( x : – x? ) ⁄ ( KO ) P ih structurol oholysis,
δ 0
H L ------- KO
(;.4)
wfkrk tfk x tfk x: – x – x? ih Kquotigh (3.?2-O) is (3.?2-O) is rkplockd jy L jy L,, rkprkskhtihn tfk lkhntf ga tfk jor, ohd u: – u? is rkplockd jy δ, rkprkskhtihn tfk dkagr`otigh ga tfk jor ih tfk uhdkagr`kd dirkctigh. Ih Cfoptkr ? wk uskd tfk cg`pghkht ga dkagr`otigh tfot is ih tfk grinihol dirkctigh ga tfk jor ih s`oll-stroih opprgxi`otighs, irrkspkctivk ga `gvk`kht ga pgihts gh tfk jor. It sfguld olsg jk rkcgnhizkd tfot L tfot L,, K , ohd O ohd O ork ork pgsitivk5 fkhck tfk sinh ga δ is tfk so`k os sinh ga H . ^k rkcgrd tfk agllgwihn gjskrvotighs. • Τ fk dkagr`otigh ga tfk jor ih tfk uhdkagr`kd dirkctigh is rkprkskhtkd jy δ. • Ia H is o tkhsilk agrck, tfkh δ is klghnotigh5 ia H is o cg`prkssivk agrck, tfkh δ is cghtroctigh.
>1
; Ihklostic
Structurk Jkfovigr
KVO@RLK ;.3 Ih tfk ossk`jly ga o `ocfihk, o sujport is `gdklkd os sfgwh ih Ainurk ;.4. ;.4. Jor O Jor O fos fos tg jk pullkd ohd ottocfkd tg suppgrt ot D ot D jkagrk tfk agrck A agrck A is is opplikd. Jors O Jors O ohd ohd J J ork ork `odk ga stkkl witf `gdulus ga klosticity K klosticity K 0 crgss-skctighol ol orko 0 :22 NRo, crgss-skctigh ?22 ``:, ohd lkhntf L lkhntf L 0 :.; :.; `. Cosk I= Dktkr`ihk tfk ihitiol oxiol strkss ih jgtf jors jkagrk tfk agrck A agrck A is is opplikd. O 0 ?22 Cosk Cosk II Ia tfk opplikd agrck A agrck A 0 0 ?2 eH, dktkr`ihk tfk tgtol oxiol strkss ih jgtf jors.
O
Zinid C
A
3 ``
K
D
3`
:`
?.; `
J
;.3. Ainurk ;.4 Twg-jor `gdkl agr Kxo`plk ;.3.
OHS. Cosk I=
;.3
σ O 0 33 @Ro ( T ) 5 σ J 0 63 @Ro ( T ) 5 Cosk II= ( σ O ) tgtol 0 ?4; @Ro ( T ) 5 ( σ J ) tgtol 0 36 @Ro ( T )
TK@RKZOT K@RKZOT[Z [ZK K OHD IHITIO IHITIOL L STZOIH STZOIH KAAKC KAAKCTS TS IH OVIOL OVIOL @K@JK @K@JKZS ZS
Cfohnks ih lkhntf duk tg tk`pkroturk vori otighs ihtrgduck strkssks duk tg tfk cghstroihihn kaakcts ga gtfkr ` k`jkrs ih o stoticolly ihdktkr`ihotk structurk. Agr purpgsk ga oholysis, tfkrk ork o hu`jkr ga si`iloritiks jktwkkh ihitiol stroih ohd tfkr`ol stroih. Tfus wk sfoll rkdkrivk gur tfkgry tg ihcgrpgrotk ihitiol stroih ohd skk fgw tfis voriojlk oaakcts gur oholysis. Clkorly, Clkorly,Ossu`ptigh Ossu`ptigh ; ih ; ih Tojlk 3.: (ojskhck 3.: (ojskhck ga ihklostic stroihs) is hg lghnkr volid. Fkhck, tfk agr`ulos tfot agllgw tfis ossu`ptigh ork hg lghnkr volid. Os wk sfoll skk, tfk dkrivotigh ga hkw agr`ulos agr`ulos is tfk so`k os jkagrk ghck tfk ihitiol stroih fos jkkh occguhtkd. occguhtkd. ^k ossu`k dkagr`otigh is ihdkpkhdkht ga ti`k, plohk skctighs rk`oih plohk ohd porollkl, ohd tfk stroihs ork s`oll, tfot is, Ossu`ptighs ? – 3 ork still volid. Tfus, eihk`otic kquotighs dg hgt cfohnk ohd tfk tgtol stroih ot ohy crgss skctigh is uhiagr` ohd ghly o auhctigh ga x ga x.. ^k aurtfkr ossu`k tfot tfk `otkriol is isgtrgpic ohd lihkorly klostic, tfot is, Ossu`ptighs 4,1, ohd > ork still volid. Fgwkvkr, hgw tk`pkroturk gr ohy gtfkr aoctgr i`pgsks oh ihitiol stroih ga λ2 ot o pgiht (Ossu`ptigh (Ossu`ptigh ; is ; is hgt volid). Tfk tgtol stroih is hgw tfk su` ga `kcfohicol ohd ihitiol stroih. Tfus, tfk strkss–stroih rklotighsfip is du d x
λ xx 0
0
σ
xx ------+ λ2 K
(;.;)
Sujstitutihn Kquotigh (;.;) ihtg (;.;) ihtg Kquotigh (3.4o-O) ohd (3.4o-O) ohd ossu`ihn tfot tfk `otkriol is fg`gnkhkgus ohd tfk ihitiol stroih λ2 is uhiagr` ocrgss tfk crgss skctigh, wk gjtoih. H 0
∥σ
xx d O
O
du 0 ⎕⎐ K – K λ 2⎘⎢ d O 0 d x
∥
O
du d x
0
du K d O – K λ 2 d O dx
∥
∥
O
O
H ------- + λ 2 KO
0
du KO – KO λ 2 gr d x (;.1)
Sujstitutihn Kquotigh (;.1) ihtg (;.1) ihtg Kquotigh (;.;), (;.;), wk gjtoih ---σ xx 0 H O
(;.>)
Ia Ossu`ptighs 9 tfrgunf ?? ork ?? ork still volid—tfot is, ia H , K , ohd O ohd O ork ork cghstoht jktwkkh x jktwkkh x? ohd x ohd x: ohd λ2 olsg dgks hgt cfohnk witf x witf x —tfkh oll quohtitiks gh tfk rinft-fohd sidk ga Kquotigh (;.1) ork (;.1) ork cghstoht jktwkkh x jktwkkh x? ohd x ohd x:. ^k coh hgw writk d u ⁄ d x 0 ( u : – u ? ) ⁄ ( x : – x ? ) tg gjtoih u : – u?
H ( x : – x ? ) 0 ------------------------+ λ2 ( x: – x? )
KO
(;.6o)
gr oltkrhotivkly
δ 0
L H -------- + λ 2 L KO
(;.6j)
Hghlihkor @otkriol @gdkls
>>
Kquotighs (;.>) ohd (;.>) ohd Kquotigh (;.6j) i`ply (;.6j) i`ply tfot ihitiol stroih oaakcts dkagr`otigh jut hgt strkss. Tfis skk`ihnly porodgxicol rksult fos o diaakrkht kxplohotigh agr tfkr`ol stroihs ohd stroihs duk tg `isaits, gr arg` tfk prktkhsighihn ga jglts. Airst wk cghsidkr tfk stroih λ2 duk tg tk`pkroturk cfohnks. Ia o jgdy is fg`gnkhkgus ohd uhcghstroihkd, tfkh hg strkssks ork nkhkrotkd gwihn tg tk`pkroturk cfohnks, os wos gjskrvkd ih Skctigh ;.?. ;.?. Tfis gjskrvotigh is kquolly truk agr stoticolly dktkr`ihotk structurks. O dktkr`ihotk structurk si`ply kxpohds gr odmusts tg occguht agr tfk tk`pkroturk cfohnks. Jut ih oh ihdktkr`ihotk structurk, tfk dkagr`otigh ga vorigus `k`jkrs `ust sotisay tfk cg`potijility kquotighs. Tfk cg`potijility cghstroihts cousk ihtkrhol agrcks tg jk nkhkrotkd, wficf ih turh oaakct tfk strkssks. Ih tfkr`ol oholysis λ 2 0 ο Δ T , oh ihcrkosk ih tk`pkroturk cgrrkspghds tg kxtkhsigh, ohd o dkcrkosk ih tk`pkroturk cgrrkspghds tg cghtroctigh. Kquotigh (;.6j) ossu`ks (;.6j) ossu`ks tfot H is is pgsitivk ih tkhsigh, ohd fkhck kxtkhsighs duk tg λ2 ork pgsitivk ohd tfgsk duk tg cghtroctigh ork hknotivk. Fgwkvkr, ia gh tfk arkk jgdy dionro` H is is sfgwh os o cg`prkssivk agrck, ohd δ is sfgwh os o cghtroctigh ih tfk dkagr`kd sfopk, tfkh cghsistkhcy rkquirks tfot tfk cghtroctigh duk tg λ2 jk trkotkd os pgsitivk ohd tfk kxtkhsigh os hknotivk ih Kquotighs (;.6j). (;.6j). • Tfk Tfk sinh sinh ga ga λ2 L duk L duk tg tk`pkroturk cfohnks `ust cfohnks `ust jk cghsistkht witf tfk agrck H sfgwh sfgwh gh tfk arkk jgdy dionro`. ^k hgw cghsidkr tfk issuk ga ihitiol stroihs couskd jy tfk aoctgrs discusskd ih Skctigh ;.:. ;.:. Ghck cfgick is tg sglvk tfk prgjlk` ga ihitiol stroih (strkss) skporotkly arg` tfk stroihs (strkss) tfot ork nkhkrotkd arg` kxtkrhol agrcks ohd supkrpgsk tfk rksults os wk did ih Kxo`plk ;.3, ;.3, ih gtfkr wgrds sglvk tfk prgjlk` twick. Ol tkrhotivkly, wk coh uskKquotigh usk Kquotigh (;.6j) ohd (;.6j) ohd sglvk tfk prgjlk` ghck, jut tfk opprgocf is lkss ihtuitivk ohd `grk `otfk`oticol. Aurtfkr`grk, wk fovk tg jk corkaul witf tfk issuk ga sinh ga ihitiol stroih tfot `ust jk occguhtkd agr ih tfk oholysis os discusskd hkxt. Suppgsk wk put δ 0 2 ih ih Kquotigh (;.6j), (;.6j), tg cgrrkspghd tg tfk uhdkagr`kd stotk, tfkh H ohd ohd λ2 `ust fovk gppgsitk sinhs agr tfk twg tkr`s gh rinft-fohd sidk tg yikl d o rksult ga zkrg. Jut stroih ohd ihtkrhol agrcks `ust fovk tfk so`k sinh. Agr kxo`plk, ia o `k`jkr is sfgrt ohd `ust jk pullkd tg gvkrcg`k tfk nop duk `isait, tfkh tfk jor fos jkkh kxtkhdkd ohd is ih tkhsigh jkagrk tfk opplicotigh ga kxtkrhol lgods fos jknuh. Tfus, agr purpgsk ga oholysis, usk hknotivk ga tfk octuol ihitiol stroih. Fkhck, ia tfk ihitiol stroih is tkhsigh tfkh ih usk ga Kquotigh (;.6j) trkot (;.6j) trkot it os cg`prkssivk ohd ia it is griniholly cg`prkssivk tfkh trkot it os tkhsilk. • Rrkstroihs (strkssks) Rrkstroihs (strkssks) coh jk oholyzkd jy usihn λ2 os hknotivk ga tfk octuol ihitiol stroih ih Kquotigh (;.6j). (;.6j).
KVO@RLK ;.4 Jors O Jors O ohd ohd J J sfgwh sfgwh Ainurk ;.; ork ;.; ork `odk ga stkkl witf `gdulus ga klosticity K klosticity K 0 0 :22 NRo, cgkaaicikht ga tfkr`ol kxpohsigh /² C, crgss-skctighol orko O orko O 0 ?22 ?22 ``:, ohd lkhntf L lkhntf L 0 :.; `. Ia tfk opplikd opplikd agrck agrck A A 0 tk`pkroturk ga tfk tfk jor ο 0 ?: μ /²C 0 ?2 eH ohd tk`pkroturk O is O is dkcrkoskd jy ?22²C, aihd tfk tgtol oxiol strkss ih jgtf jors.
O
Zinid C
A D
K 3`
:`
?.; `
J
;.4. Ainurk ;.; Twg-jor `kcfohis` agr Kxo`plk ;.4.
OHS.
σ O 0 ?4; @Ro ( T ) 5 σ J 0 36 ;.4 ;.4
@Ro ( T )
HGHL HGHLIH IHKO KOZ Z @OTK @OTKZI ZIOL OL @GDK @GDKLS LS
Zujjkr, plostics, ohd `usclks ohd gtfkr grnohic tissuks kxfijit hghlihkority ih tfk strkss–s troih rklotighsfip, kvkh ot s`oll stroihs. @ktols olsg tfk kxfijit hghlihkority oatkr yikld strkss. Ih tfis skctigh, wk cghsidkr vorigus hghlihkor `otkriol `gdkls= tfot is, vorigus agr`s ga kquotighs tfot ork uskd agr rkprkskhtihn tfk strkss–stroih hghlihkor rklotighsfip. Tfk cghstohts ih tfk kquotighs rklotihn strkss ohd stroih ork `otkriol cghstohts. @otkriol cghstohts ork aguhd jy `otcfihn tfk strkss–stroih kquotigh tg tfk kxpkri`khtol doto ih o lkost-squorks skhsk. O criticol issuk ih hghlihkor structurol oholysis is tfk dktkr`ihotigh ga tfk grinih ga o nivkh cggrdihotk systk` sucf tfot tfk oxiol ohd jkhdihn prgjlk`s ork dkcguplkd (i.k., skporotihn oxiol lgod oholysis arg` jkhdihn lgod oholysis). ^k will toek up tfis criticol prgjlk` ih hghlihkor oholysis o littlk lotkr. Ih tfis skctigh wk jyposs tfk prgjlk` jy cghsidkrihn ghly sy``ktric crgss
; Ihklostic
>6
Structurk Jkfovigr
skctighs, witf sy``ktric `otkriol jkfovigr ih tkhsigh ohd cg`prkssigh. Tfksk li`itotighs will khsurk tfot tfk strkss distrijutigh ih tkhsigh ohd cg`prkssigh will jk sy``ktric ojgut tfk ckhtrgid ga tfk crgss skctigh. Tfk ckhtrgid ga tfk crgss skctigh will jk tfk grinih ih oll prgjlk`s ih tfis skctigh. ^k sfoll cghsidkr tfrkk `otkriol `gdkls= ?. Klostic–pkrakctly Klostic–pkrakctly plostic, plostic, ih wficf tfk hghlihkority hghlihkority is opprgxi`otkd opprgxi`otkd jy o cghstoht. cghstoht. :. Lihkor stroih stroih fordkhihn, fordkhihn, ih wficf tfk hghlihkority hghlihkority is opprgxi`otkd opprgxi`otkd jy o lihkor auhctigh. auhctigh. 3. Rgwkr low, low, ih wficf tfk hghlihkority hghlihkority is opprgxi`otkd opprgxi`otkd jy ghk tkr`, o hghlihkor hghlihkor auhctigh. Tfk cfgick ga `otkriol `gdkl tg usk dkpkhds hgt ghly upgh tfk `otkriol strkss–stroih curvk, jut olsg upgh tfk dknrkk ga occurocy hkkdkd ohd tfk rksultihn cg`plkxity ga tfk oholysis.
;.4.?
Klostic– Klostic–Rkr Rkrakct akctly ly Rlostic Rlostic @otkriol @otkriol @gdkls @gdkls
Ainurk ;.1 sfgws ;.1 sfgws tfk strkss–stroih curvks dkscrijihn oh klostic–pkrakctly plostic jkfovigr ga o `otkriol. It is ossu`kd tfot tfk `otkriol fos tfk so`k jkfovigr ih tkhsigh ohd ih cg`prkssigh. Si`ilorly agr sfkor strkss–stroih, tfk `otkriol jkfovigr is so`k agr pgsitivk ohd hknotivk strkssks ohd stroihs. :
τ σ
σ
σ yi kl d
0
σ λ yi kl d
σ
0 – σ yi kl d
τ yi kl d
σ yi kl d
0 K λ
γ yi kl d
λ
τ
0
τ yi kl d
τ
0 Nγ γ
γ yi kl d
λ yi kl d σ yi kl d
τ
τ yi kl d
0 – τ yi kl d (j)
(o)
Klostic–pkrakctly plostic `otkriol jkfovigr= ( o) hgr`ol ohd (j (j) sfkor. Ainurk ;.1 Klostic–pkrakctly
Jkagrk yikld strkss, tfk strkss–stroih rklotighsfip is nivkh jy Fggek’s low, oatkr yikld strkss, tfk strkss is o cghstoht. Klostic–pkrakctly plostic `otkriol jkfovigr is o si`pliayihn opprgxi`otigh uskd tg cghduct oh klostic–plostic oholysis. Tfk opprgxi`otigh is o cghskrvotivk opprgxi`otigh, sihck it inhgrks tfk copocity ga tfk `otkriol tg corry strkssks finfkr tfoh tfk yikld strkss. Tfk kquotighs dkscrijihn tfk strkss–stroih curvk ork os agllgws=
⎮ σyikld ⎭ σ 0 ⎦ K λ ⎭ – σ ⎩ yikld
λ ≫ λyikld – λ yikld ≤ λ ≤ λ yikld λ ≤ – λyikld
ohd
τ
0
⎮ τyikld ⎭ ⎦ Nγ ⎭ – τ ⎩ yikld
γ ≫ γ yikld – γ yikld ≤ γ ≤ γ yikld γ ≤ – γyikld
(;.9)
Tfk skt ga pgihts agr`ihn tfk jguhdory jktwkkh tfk klostic ohd plostic rknighs gh o jgdy is collkd tfk klostic–plostic jguhdory. Dktkr`ihihn tfk lgcotigh ga tfk klostic–plostic jguhdory is ghk ga tfk criticol issuks ih klostic–plostic oholysis. Os gur kxo`plks will dk`ghstrotk, tfk lgcotigh ga tfk klostic–plostic jguhdory is dktkr`ihkd usihn tfk agllgwihn gjskrvotighs= ?. Gh tfk klostic–plostic jguhdory, jguhdory, stroih `ust jk kquol tg yikld stroih, ohd strkss kquol tg yikld strkss. :. Dkagr`otighs ohd stroihs ork cghtihugus cghtihugus ot oll pgihts, ihcludihn ihcludihn pgihts ot tfk klostic– klostic– plostic jguhdory. jguhdo ry. Dkagr`otigh tfot is hgt cghtihugus i`pliks tfot fglks gr croces ork jkihn agr`kd ih tfk `otkriol. Ia stroihs, wficf ork dkrivotivk displock`khts, ork hgt cghtihugus, tfkh cgrhkrs ork jkihn agr`kd durihn dkagr`otigh.
;.4.:
Lihkor Lihkor Stroih-Fo troih-Fordkh rdkhihn ihn @otkriol @otkriol @gdkl @gdkl
Ainurk ;.>, ;.>, sfgws strkss–stroih curvks agr tfk lihkor stroih-fordkhihn `gdkl, olsg rkakrrkd tg os tfk jilihkor `otkriol `gdkl. It is ossu`kd tfot tfk `otkriol fos tfk so`k jkfovigr ih tkhsigh ohd ih cg`prkssigh. Si`ilorly agr sfkor strkss–stroih, tfk `otkriol jkfovigr is so`k agr pgsitivk ohd hknotivk strkssks ohd stroihs. Tfis is ohgtfkr cghskrvotivk, si`pliayihn opprgxi`otigh ga `otkriol jkfovigr ih wficf wk ghck `grk inhgrk tfk ojility ga tfk `otkriol tg corry strkssks finfkr tfoh tfgsk sfgwh jy tfk stroinft lihks. Dktkr`ihihn tfk lgcotigh ga tfk klostic–plostic jguhdory is ghck `grk o criticol issuk ih tfk oholysis ohd is dkt kr`ihkd os dkscrijkd ihSkctigh ih Skctigh ;.4.?. ;.4.?. Tfk kquotighs dkscrijihn tfk strkss–stroih curvk ork os agllgws=
σ
0
⎮ σyikld + K : ( λ – λyikld ) ⎭ ⎦ K ? λ ⎭ ⎩ – σyikld + K : ( λ + λ yikld )
λ ≫ λyikld – λ yikld ≤ λ ≤ λ yikld λ ≫ λyikld
ohd
τ
0
⎮ τyikld + N : ( γ – γyikld ) ⎭ ⎦ N? γ ⎭ ⎩ – τyikld + N : ( γ + γyikld )
γ ≫ γyikld – γ yikld ≤ γ ≤ γ yikld γ ≤ γyikld
(;.?2)
Hghlihkor @otkriol @gdkls
>9
τ σ
σ yi kl d
λ yi kl d
– σ
σ
0
σ
0 K λ ? λ
τ yi kl d
σ yikld + K : ( λ – λ yikld )
τyikld + N : ( γ – γyikld ) τ
γ yi kl d
τ
0 – τ
yikld
+N
:
( γ + γyikld )
? γ
γ
γ yi kl d
λ yi kl d
+ K ( λ + λ ) yikld : yikld
0 N
τ yi kl d
σ yi kl d
(j)
( o) hgr`ol ohd (j (j) sfkor. Ainurk ;.> Lihkor stroih fordkhihn `gdkl= (o
;.4. ;.4.3 3
Rgwk Rgwkrr Low Low @otk @otkri riol ol @gd @gdkl kl
Ainurk ;.6 sfgws ;.6 sfgws o pgwkr low rkprkskhtotigh ga tfk hghlihkor strkss–stroih curvk. It is ossu`kd tfot tfk `otkriol fos tfk so`k jkfovigr ih tkhsigh ohd ih cg`prkssigh. Si`ilorly, Si`ilorly, agr sfkor strkss–stroih, tfk `otkriol jkfovigr is so`k agr pgsitivk ohd hknotivk strkssks ohd stroihs. Tfk kquotighs dkscrijihn tfk strkss–stroih rklotighsfip ork os agllgws=
⎮ K λ h σ0⎦ ⎩ – K ( – λ ) h σ
λ≫2 λ<2
ohd
⎮ N γh τ0⎦ ⎩ – N ( – γ )h
γ≫2 γ<2
2
σ
0 K λ
τ
h
0 – K ( – λ )
0 Nγ
h
h8?
λ σ
2
τ h8?
h8?
(;.??)
γ
h8?
h
τ
2
0 – N ( – γ )
h
2
(j)
(o) strkss–stroih curvks= (o (o) hgr`ol ohd (j (j) sfkor. Ainurk ;.6 Hghlihkor strkss–stroih
Tfk cghstohts K cghstohts K ohd ohd h, collkd tfk strkhntf cgkaaicikht ohd tfk stroih-fordkhihn cgkaaicikht, ork dktkr`ihkd tg ait tfk kxpkri`khtol strkss–stroih curvk ih o lkost-squorks skhsk. @gst `ktols ih tfk plostic rknigh ohd `gst plostics ork rkprkskhtkd jy tfk sglid curvk, witf o stroih-fordkhihn cgkaaicikht lkss tfoh ?. Sgat rujjkr, os wkll os `usclks ohd gtfkr grnohic `otkriols, rkprkskhtkd jy tfk dosfkd lihk ih Ainurk ;.6, ;.6, fovk o stroih-fordkhihn cgkaaicikht nrkotkr tfoh ?. Arg` Kquotigh (;.??) wk (;.??) wk hgtk tfot wfkh stroih is hknotivk, tfk tkr` gh tfk rinft jkcg`ks pgsitivk, pkr`ittihn kvoluotigh ga tfk hu`jkr tg aroctighol pgwkrs. Aurtfkr`grk, witf hknotivk stroih wk gjtoih hknotivk strkss, os wk sfguld. ^k cguld cg`jihk o lihkor kquotigh agr tfk lihkor port ohd hghlihkor kquotigh agr tfk hghlihkor port os wk did ih Skctigh ;.4.:5 ;.4.:5 gr wk cguld cg`jihk twg hghlihkor kquotighs, tfus crkotihn odditighol `otkriol `gdkls. ^ficf `otkriol `gdkl tg usk7 Tfk ohswkr dkpkhds upgh tfk `otkriol strkss–stroih curvk, tfk dknrkk ga occurocy hkkdkd, ohd tfk rksultihn cg`plkxity ga tfk oholysis.
KVO@RLK ;.; Olu`ihu` fos o yikld strkss ga σyikld 0 42 esi ih tkhsigh, o yikld stroih ga λyikld 0 2.224, oh ulti`otk strkss ga σult 0 4; esi, ohd tfk cgrrkspghdihn ulti`otk stroih ga λult 0 2.2?>. Dktkr`ihk tfk `otkriol cghstohts agr (o) tfk klostic–pkrakctly plostic `gdkl, (j) tfk lihkor stroih-fordkhihn `gdkl, ohd (c) tfk hghlihkor pgwkr low `gdkl. OHS. (o )
⎮ ?2,222λ esi σ0⎦ ⎩ 42 esi
(c)
⎮ 1:.> λ 2.26?4 esi σ0⎦ ⎩ –1:.> ( – λ ) 2.26?4
λ ≤ 2.224 ⎮ ?2, 222 λ esi 5 (j) σ 0 ⎦ λ ≫ 2.224 ⎩ 42 + 364.1 ( λ – 2.224 ) esi
esi
λ≫2 λ<2
λ ≤ 2.224 5 λ ≫ 2.224
62
; Ihklostic
;.;
Structurk Jkfovigr
KLOSTI KLOSTIC–R C–RKZA KZAKCT KCTLY LY RLOSTIC RLOSTIC OVIOL OVIOL @K@JKZS @K@JKZS
Strkss is uhiagr` ocrgss tfk crgss skctigh ga oh oxiol `k`jkr. Tfus, wfkh oh oxiol `k`jkr ih o structurk rkocfks yikld strkss, tfk khtirk `k`jkr will jkcg`k plostic, ossu`ihn tfot tfk oxiol `k`jkr fos hg distrijutkd lgod. Ih occgrdohck witf Ainurk ;.1o ;.1o, tfk `k`jkr tfot fos turhkd plostic coh corry strkss kquol tg yikld strkss ohd coh klghnotk gr cg`prkss t g ohy o`guht hkkdkd tg sotisay tfk cg`potijility kquotighs. It is tfk cg`potijility kquotighs (cghtihuity ga displock`kht) tfot ork uskd ih dktkr`ihihn tfk klghnotigh gr stroih ih oh oxiol `k`jkr tfot fos turhkd plostic. Tfk plgt ga tfk opplikd agrck vs. tfk dkalkctigh ot tfot pgiht ih tfk dirkctigh ga tfk opplikd agrck is collkd tfk lgod-dkalkctigh curvk. Ih tfk klostic rknigh, tfk lgod dkalkctigh curvk is o stroinft lihk. ^fkh ghk `k`jkr ih o structurk ngks plostic, tfk dkagr`otigh ga tfk structurk is cghtrgllkd jy tfk gtfkr `k`jkrs. Jut sihck tfk strkss is cghstoht ih tfk plostic `k`jkr, tfkrk is o cfohnk ih tfk slgpk ga tfk lgod dkalkctigh curvk. Tfk slgpk ga tfk lgod dkalkctigh curvk cfohnks kocf ti`k o `k`jkr ngks plostic. Jut ia o suaaicikht hu`jkr ga `k`jkrs ng plostic, wk `oy fovk uhcghtrgll kd dkagr`otigh, wfkrkupgh tfk structurk is soid tg fovk cgllopskd. Tfk lgod ot wficf tfk structurk kxfijits uhjguhdkd dkagr`otigh is collkd tfk cgllopsk lgod. Agr tfk si`plk structurks wk will cghsidkr, “cgllopsk lgod‐ cgrrkspghds tg oll `k`jkrs must turhihn plostic. Jut `grk cg`plkx structurks coh cgllopsk witfgut rkquirihn oll `k`jkrs tg turh plostic. Tfk oholyticol tkcfhiquk collkd li`it oholysis coh oholysis coh jk uskd agr dktkr`ihihn tfk cgllopsk lgod agr cg`plkx structurks, jut it is jkyghd tfk scgpk ga tfis jgge.
KVO@RLK ;.1 Tfk stkkl `k`jkrs sfgwh ih Ainurk ;.9, ;.9, fovk o crgss-skctighol orko O orko O 0 ?22 ?22 ``:, o `gdulus ga klosticity K klosticity K 0 ohd o 0 :22 NRo, ohd yikld strkss σyikld 0 :;2 @Ro. Tfk tfrkk `k`jkrs ork ottocfkd tg o rgllkr tfot coh slidk ih tfk slgt sfgwh. Rlgt tfk lgod lgod dkalkctigh curvk ohd dktkr`ihk tfk cgllopsk lgod.
:;g 3`
O
J
;;g
C
R
A
;.1. Ainurk ;.9 Nkg`ktry agr Kxo`plk ;.1.
OHS.
A cgllopsk
0
46.3 eH
A (eH) (eH) >: 12 46 31 :4
1: eH ;3.4 ;3.4 eH 46.3 46.3 eH ` ` ; > . 3
` ` > ; . 4
` ` 4 . ? ?
?: 3.2
1.2
9. 2
?:.2
?4.2
δR (``)
Ainurk ;.?2 Lgod dkalkctigh curvk.
;.1
KLOSTIC KLOSTIC–RKZ –RKZAKC AKCTLY TLY RLOSTIC RLOSTIC CIZC[L CIZC[LOZ OZ SFOATS SFOATS
Ih Skctigh ;.4.?, ;.4.?, wk studikd tfk opprgxi`otigh ga klostic–pkrakctly plostic `otkriol jkfovigr. ^itf rkakrkhck tg Ainurk ;.??, ;.??, wk rkcoll tfk agllgwihn. ?. Jkagrk yikld strkss, tfk `otkriol strkss–stroih rklotighsfip is rkprkskhtkd jy Fggek’s Fggek’s low, low, ohd oatkr yikld strkss tfk strkss is ossu`kd tg jk cghstoht. :. Tg dktkr`ihk tfk stroih stroih (dkagr`otigh) (dkagr`otigh) ih tfk fgrizghtol fgrizghtol pgrtigh OJ pgrtigh OJ ga ga o curvk, wk usk tfk rkquirk`kht tfot dkagr`otigh jk cghtihugus.
Klostic–Rkrakctly Rlostic Circulor Sfoats
6?
3. [hlgodihn [hlgodihn (klostic (klostic rkcgvkry) arg` arg` o pgiht ih tfk plostic rknigh rknigh is olghn lihk JC , wficf is porollkl tg tfk lihkor pgrtigh ga tfk strkss–stroih curvk GO. GO.
τ τ
τ yi kl d τ
τ yi kl d
n i h d o l g h [
γ
γ yi kl d
0
N 0
γ
γ yi kl d
τ
0 – τ yi kl d
τ yi kl d
Klostic–pkrakctly plostic `otkriol jkfovigr. Ainurk ;.?? Klostic–pkrakctly
Tfk tgrsighol sfkor strkss voriks lihkorly ocrgss o fg`gnkhkgus, lihkor, klostic crgss skctigh ohd fos its `oxi`u` ot tfk gutkr rodius ga tfk sfoat. Tfus, tfk plostic zghk will stort arg` tfk gutsidk ohd `gvk ihword agr o fg`gnkhkgus `otkriol, os sfgwh ih Ainurk ;.?:j ;.?:j.
γxζ
τxζ
γ`ox
τ yikld Rlostic zghk
γ yikld Klostic zghk
Ϗ
Ϗy
Ϗy
Klostic-plostic jguhdory
(o)
Ϗ
dkptf ga (j)
plostic zghk
Klostic-pkrakctly plostic jkfovigr ga o circulor sfoat ih tgrsigh. Ainurk ;.?: Klostic-pkrakctly
Tfk lihkor voriotigh ga tgrsighol sfkor stroih witf tfk rodiol cggrdihotk Ϗ wos dkrivkd gh tfk josis ga nkg`ktry. Ia tfk eihk`otic ossu`ptighs rk`oih volid durihn yikldihn, tfkh tfk lihkor voriotigh ga sfkor stroih is uhoaakctkd jy tfk aoct tfot port ga tfk crgss skctigh is hgw plostic, os sfgwh ih Ainurk ;.?:o ;.?:o. Ot tfk klostic–plostic jguhdory, wk ehgw tfot tfk sfkor stroih is γyikld. Tfus Kquotigh (3.:-T) coh (3.:-T) coh jk rkwrittkh os
γ yi kl d 0 Ϗ y dρ d x
(;.?:)
wfkrk Ϗ y is tfk rodiol cggrdihotk ga tfk klostic–plostic jguhdory. ^k ossu`k tfk sfoat is hgt topkrkd ohd tfot kxtkrhol tgrquk is hgt o auhctigh ga x ga x jktwkkh jktwkkh x x? ohd x ohd x:. Tfk ossu`ptigh i`pliks tfot Ϗ y is hgt o auhctigh ga x ga x,, fkhck d ρ / dx is cghstoht ohd coh jk rkplockd jy ( ρ : – ρ ? ) ⁄ ( x : – x ? ) . ^k gjtoih tfk agllgwihn agr`ulo agr tfk rklotivk rgtotigh=
γ yikld⎘ ρ: – ρ ? 0 ⎕⎐ ----------( x – x ) Ϗ y ⎢ : ?
(;.?3)
Sfkor strkss voriks lihkorly arg` o voluk ga zkrg ot tfk ckhtkr tg o voluk ga yikld strkss ot tfk klostic–plostic jguhdory. Oatkr tfk klostic–plostic jguhdory, tfk sfkor strkss is kquol tg tfk yikld strkss. Tfk sfkor strkss distrijutigh coh jk writtkh os
τ
0
⎮ τ yikldϏ ⎭ --------------⎦ Ϗ y ⎭τ ⎩ yikld
Ϗ ≤ Ϗ y
(;.?4o)
Ϗ ≫ Ϗ y
^k coh aihd tfk kquivolkht ihtkrhol tgrquk jy sujstitutihn Kquotigh (;.?4o) ihtg (;.?4o) ihtg Kquotigh (3.4-T). (3.4-T). ^k coh rklotk tfk ihtkrhol tgrquk tg kxtkrhol tgrquk tfrgunf kquilijriu`. Tfkrk ork twg eihds ga prgjlk` wk will cghsidkr= ?. prgjlk prgjlk`s `s ih wficf wficf wk wk ehgw Ϗ y , sucf os Kxo`plk ;.6 ohd ;.6 ohd Kxo`plk ;.9, ;.9, :. prgjlk`s prgjlk`s ih wficf wk wk hkkd tg aihd aihd Ϗ y, sucf os Kxo`plk ;.?2. ;.?2.
6:
; Ihklostic
;.1. ;.1.? ?
Structurk Jkfovigr
Zksi Zksidu duol ol Sf Sfko korr Str Strks ksss
Jy prkstrkssihn tfk sfoat ihtg tfk plostic rknigh, wk coh ihcrkosk tfk tgrquk-corryihn copocity ga tfk sfoat, jut hgw wk hkkd tg cg`putk tfk rksiduol strkssks ih tfk sfoat. Durihn tfk klostic rkcgvkry (i.k., uhlgodihn olghn lihk JC lihk JC ih ih Ainurk ;.??), ;.??), tfk ossu`ptigh ga `otkriol lihkority is volid, ohd Kquotigh (3.9-T) coh (3.9-T) coh jk uskd agr colculotihn tfk klostic sfkor strkss. Tfk ihtkrhol tgrquk agr colculotihn tfk klostic sfkor strkss cgrrkspghds tg tfk tgrquk tfot wos kquivolkht tg tfk strkss distrijutigh agr o nivkh lgcotigh ga tfk klostic–plostic jguhdory. ^k sujtroct tfk klostic strkssks arg` tfk klostic–plostic strkss distrijutigh ohd gj toih tfk rksiduol strkssks os dk`ghstrotkd sfgrtly (skk Kxo`plk ;.6). ;.6).
KVO@RLK ;.> O fgllgw circulor sfoat fos oh gutkr dio`ktkr ga 4 ih. ohd oh ihhkr dio`ktkr ga ? ih. Tfk sfkor stroih ot o crgss skctigh wos aguhd tg jk γ x ζ 0 3Ϗ(?2 –3), wfkrk Ϗ is tfk rodiol cggrdihotk `kosurkd ih ihcfks. (o) Rlgt tfk distrijutigh ga tfk sfkor stroih γ x ζ ohd sfkor strkss τ x ζ ocrgss tfk crgss skctigh. (j) Dktkr`ihk tfk kquivolkht ihtkrhol tgrquk octihn ot tfk crgss skctigh. Ossu`k tfot tfk sfoat is `odk arg` oh klostic–pkrakctly plostic `otkriol tfot fos o yikld strkss ga τyikld 0 :4 esi ohd o sfkor `gdulus N 0 1222 esi. OHS.
⎮ ?6 Ϗ esi τ x ζ 0 ⎦ ⎩ :4 esi
γxζ (μ)
2.; ≤ Ϗ ≤ ?.333 ?.333 ≤ Ϗ ≤ :.2
γ x ζ
0 3222 Ϗ
5 T 0 3>2.1 ih ih ⋋ eips ips
τ x ζ
τxζ (esi)
μ
0 ?6 Ϗ es i
:4 1222
4222 9 ?;22
Ϗ (ih) ?.333 :.2 (o) ( j) strkss distrijutigh. Ainurk ;.?3 (o) Stroih ohd (j
2.;
2.;
(j)
?.333
Ϗ (ih) :.2
KVO@RLK ;.6 O ?22 `` dio`ktkr sglid stkkl (N (N 0 62 NRo, τyikld 0 ?12 @Ro) circulor sfoat fos o plostic zghk tfot is 32 `` dkkp. Ossu`ihn oh klostic–pkrakctly plostic `otkriol, dktkr`ihk (o) tfk kquivolkht ihtkrhol tgrquk ot tfk crgss skctigh ohd (j) tfk rksiduol strkssks ih tfk crgss skctigh wfkh tfk sfoat is uhlgodkd. OHS. T 0 4?.3 e H ⋋ ` 5
⎮ ( 3>93Ϗ ) @ Ro τ rksiduol 0 ⎦ ⎩ ( ?12 – 4:2>Ϗ ) @ Ro
2 ≤ Ϗ ≤ 2.2: 2.2: ≤ Ϗ ≤ 2.2;
KVO@RLK ;.9 Tfk stkkl (N (N 0 62 NRo, NRo, τyikld 0 ?12 @Ro) sfoat sfoat sfgwh sfgwh ih Ainurk ;.?4 fos ;.?4 fos o plostic zghk 32 `` dkkp ih skctigh skctigh OJ. OJ. Dktkr`ihk (o) tfk `onhitudk ga tfk opplikd tgrquk T kxt ohd (j) tfk rgtotigh rgtotigh ga skctigh skctigh C witf witf rkspkct tg tfk woll. O
T kxt
: .; `
32 eH-` eH-`
J
C
: `
;.9. Ainurk ;.?4 Sfoat agr Kxo`plk ;.9.
OHS.
T kxt
0
>?.3 eH eH ⋋ ` 5
ρ C 0
2.?>31 rods CC^
?22 ?22 ``
Klostic–Rkrakctly Rlostic Jko`s
63
KVO@RLK ;.?2 Tfk stkkl (N (N 0 62 NRo, NRo, τyikld 0 ?12 @Ro) sfoat sfoat is lgodkd os sfgwh ih Ainurk ;.?;. ;.?;. Dktkr`ihk tfk lgcotigh ga tfk klostic plostic jguhdory ih skctigh OJ skctigh OJ..
O
1; eH-` eH-`
: .; `
32 eH-` eH-`
J
C
: `
?22 ?22 ``
;.?2. Ainurk ;.?; Sfoat agr Kxo`plk ;.?2.
OHS.
Ϗ y 0 ;.>
43.; ``
KLOS KL OSTIC TIC–RK –RKZA ZAKC KCTL TLY Y RLOS RLOSTI TIC C JKO@ JKO@S S
Ainurk ;.?1 sfgws ;.?1 sfgws tfk strkss–stroih curvk agr oh klostic–pkrakctly plostic `otkriol. ^k rkcoll tfk agllgwihn gjskrvotighs. ?. Tfk jkhdihn hgr`ol stroih is lihkor ocrgss tfk crgss skctigh kvkh ia port ga tfk crgss skctigh jkcg`ks plostic. :. Agr purk jkhdihn, jkhdihn, tfk hgr`ol hgr`ol strkss strkss distrijutigh distrijutigh ga σ xx sfguld rksult ih zkrg oxiol agrck kvkh ia port ga tfk crgss skctigh jkcg`ks plostic. 3. Jkagrk yikld strkss, tfk `otkriol strkss–stroih rklotighsfip is rkprkskhtkd jy Fggek’s Fggek’s low5 oatkr yikld strkss, tfk strkss is ossu`kd tg jk cghstoht. 4. Ot tfk klostic–plostic jguhdory, tfk jkhdihn hgr`ol strkss sfguld kquol tfk yikld strkss, ohd tfk jkhdihn hgr`ol stroih `ust kquol tfk yikld stroih. ;. O plostic rknigh rknigh will stort nrgwihn arg` kitfkr kitfkr tfk tgp gr tfk jgttg` ga tfk crgss skctigh jkcousk jkcousk tfk `oxi`u` jkhdihn hgr`ol strkss gccurs ot tfk pgiht aurtfkst arg` tfk hkutrol oxis. 1. [hlgodihn [hlgodihn (klostic (klostic rkcgvkry) arg` arg` o pgiht ih tfk plostic rknigh rknigh is olghn lihk JC , wficf is porollkl tg tfk lihkor pgrtigh ga tfk strkss–stroih curvk GO sfgwh GO sfgwh ih Ainurk ;.?1. ;.?1. ^k sfoll `oek usk ga tfksk gjskrvotighs ih gur oholysis. Tfk lgcotigh ga tfk klostic–plostic jguhdory gh tfk crgss skctigh is criticol. ^k dkaihk o 0 tfk distohck ga tfk klostic–plostic jguhdory arg` tfk hkutrol oxis.
σ σ
σ yi kl d
λ yi kl d
σ yi kl d
n i h d o l g h [
λ
0
0
K
σ
λ
λ yi kl d
σ
0 – σ yi kl d
σ yi kl d
Klostic–pkrakctly plostic `otkriol jkfovigr. Ainurk ;.?1 Klostic–pkrakctly
Cghsidkr o sy``ktric crgss skctigh ga o jko` tfot is sy``ktric hgt ghly ojgut tfk y tfk y oxis oxis jut olsg ojgut tfk z tfk z oxis, oxis, os sfgwh ih Ainurk ;.?>o ;.?>o. Rgihts O Rgihts O ohd ohd J J will will rkocf yikld strkss si`ultohkgusly ot tfk so`k lgod. Os tfk lgod ihcrkosks, tfk plostic zghk will `gvk ihword ih o sy``ktric `ohhkr, ohd tfk hkutrol oxis will cghtihuk tg jk ot tfk ckhtrgid ga tfk crgss skctigh. Tfk jkhdihn hgr`ol strkss voriks lihkorly witf y witf y ih ih tfk klostic rknigh ohd is kquol tg yikld strkss (tkhsigh gr cg`prkssigh) gh tfk klostic-plostic jguhdory, os sfgwh ih Ainurk ;.?>o ;.?>o. Ia wk ehgw o, wk coh writk tfk strkss distrijutigh os o auhctigh ga y ga y ohd ohd usk Kquotigh (3.4j-J) tg (3.4j-J) tg aihd tfk kquivolkht ihtkrhol `g`kht ot tfk crgss skctigh. Oltkrhotivkly, ia tfk ihtkrhol `g`kht is ehgwh, ohd wk hkkd tg aihd tfk lgcotigh ga tfk klostic–plost ic jguhdory, wk coh aihdo aihd o jy writihn tfk strkss distrijutigh ih tkr`s ga o ohd tfkh colculotihn tfk ihtkrhol `g`kht ih tkr`s ga o ohd kquotihn it tg tfk nivkh voluk `g`kht voluk. Ih Kxo`plk ;.??, ;.??, tfk lgcotigh ga tfk klostic–plostic jguhdory is prgvidkd (o (o is ehgwh) ohd tfk cgrrkspghdihn colculotigh ga tfk ihtkrhol `g`kht @ z is sfgwh. Ih Kxo`plk ;.?:, ;.?:, tfk ihtkrhol `g`kht @ z is nivkh ohd tfk colculotigh agr tfk lgcotigh ga tfk klostic–plostic jguhdory o is sfgwh.
64
; Ihklostic
Structurk Jkfovigr σ yikld
λ`ox O
KlosticRlostic Jguhdory
y O
y
λ yikld
σ O
O
y O
λ yikld λ
C
o
λ
C
C
σ
C
σ
o
o
J
J
λ`ox
(o)
λ yikld
KlosticRlostic Jguhdory
λ yikld
KlosticRlostic Jguhdory
J
σ yikld
J
λ`ox
(j)
σ yikld
Klostic–pkrakctly plostic strkss distrijutigh= (o ( o) sy``ktric ohd (j (j) uhsy``ktric. Ainurk ;.?> Klostic–pkrakctly
Hgw cghsidkr o crgss skctigh tfot is hgt sy``ktric ojgut tfk z tfk z oxis, oxis, os sfgwh ih Ainurk ;.?>j ;.?>j. Tfk ckhtrgid ga tfk crgss skctigh ot C is is tfk lgcotigh ga tfk hkutrol oxis ia tfk khtirk crgss skctigh is klostic. Rgiht J Rgiht J is is aurtfkr owoy tfoh pgiht O pgiht O arg` arg` tfk hkutrol oxis ot C . Tfus, pgiht J pgiht J will will rkocf yikld strkss jkagrk pgiht O pgiht O os os tfk lgods ork ihcrkoskd. Tfk plostic zghk will stort arg` pgiht J pgiht J ohd ohd `gvk upword. Tg sotisay Kquotigh (3.4o-J), (3.4o-J), tfot is, agr tfk tgtol oxiol agrck gh tfk crgss skctigh tg jk zkrg, tfk hkutrol oxis will sfiat arg` tfk ckhtrgid. Sg witf crgss skctighs tfot ork uhsy``ktric witf rkspkct tg tfk z tfk z oxis, oxis, wk will hkkd tg aihd tfk lgcotigh ga tfk hkutrol oxis olsg. Ghck `grk wk will stort jy writihn tfk strkss distrijutigh ih tkr`s ga o ocrgss tfk crgss skctigh. ^k will airst dktkr`ihk t fk lgcotigh ga tfk hkutrol oxis ih tkr`s ga o jy usihn Kquotigh (3.4o-J). (3.4o-J). Ghck o is ehgwh, tfk hkutrol oxis os wkll os tfk lgcotigh ga tfk klostic–plostic jguhdory will jk ehgwh. Tfk kquivolkht ihtkrhol `g`kht @ z coh jk dktkr`ihkd jy usihn Kquotigh (3.4j-J) ghck (3.4j-J) ghck o is ehgwh. Ih Kxo`plk ;.?3 wk ;.?3 wk will dk`ghstrotk tfk colculotigh ga o ohd @ z agr crgss skctighs tfot fovk ghly ghk plohk ga sy``ktry. Tfk rotig ga tfk plostic `g`kht tg tfk klostic ` g`kht is collkd tfk sfopk aoctgr a agr agr tfk crgss skctigh ohd `otfk`oticol dkaihkd os @ p a 0 ------ @ k
(;.?;)
wfkrk, @ k is tfk ihtkrhol `g`kht ot wficf tfk `oxi`u` jkhdihn hgr`ol strkss gh o crgss skctigh must rkocfks tfk yikld strkss ohd is collkd tfk klostic `g`kht5 @ p is tfk ihtkrhol `g`kht agr wficf tfk khtirk crgss skctigh jkcg`ks aully plostic ohd is collkd tfk plostic `g`kht. Os tfk ho`k sunnksts, tfk sfopk aoctgr dkpkhds ghly upgh tfk sfopk ga tfk crgss skctigh ohd is ihdkpkhdkht ga tfk `otkriol arg` wficf wficf tfk jko` jko` is `odk. ^fkh ^fkh tfk crgss crgss skctigh skctigh jkcg`ks aully plosti c, tfk jko` coh hg lghnkr suppgrt ohy lgod ot tfk crgss skctigh ohd tfk cgrrkspghdihn `g`kht rkprkskhts tfk cgllopsk `g`kht. Agr oh klostic dksinh, tfk sfopk aoctgr tfus rkalkcts tfk `ornih ga soakty. Kxo`plk ;.?4 will ;.?4 will dk`ghstrotk tfk colculotigh ga o sfopk aoctgr. Ih oll kxo`plks tfot agllgw, wk will jk kvoluotihn ihtknrols gvkr tfk crgss skctigh. Colculotigh krrgrs coh jk sinhiaicohtly rkduckd jy drowihn oh opprgxi`otk strkss distrijutigh ocrgss tfk crgss skctigh, ohd tfkh rkcgrdihn tfk y tfk y cggrdihotk cggrdihotk ga pgihts (sg`kti`ks ih tkr`s ga o) ot wficf kitfkr tfk strkss σ xx gr tfk diaakrkhtiol orko dO cfohnks. dO cfohnks.
KVO@RLK ;.?? Tfk hgr`ol stroih ot o crgss skctigh ih o jko` duk tg jkhdihn ojgut tfk z tfk z oxis oxis wos aguhd tg vory os λ xx 0 – 2.2?:; y , witf y witf y `kosurkd ih `ktkrs. (o) Rlgt tfk σ xx distrijutigh ocrgss tfk crgss skctigh. (j) Dktkr`ihk tfk kquivolkht ihtkrhol `g`kht. Ossu`k tfot tfk jko` is `odk arg` oh klostic–pkrakctly plostic `otkriol tfot fos o yikld strkss ga σyikld 0 :;2 @Ro ih tkhsigh ohd cg`prkssigh ohd o `gdulus ga klosticity K klosticity K 0 :22 :22 NRo. NRo. ?:; ?:; `` ?:; ?:; `` ;2`` y :;2 :;2 `` z
:;2 :;2 ``
;2``
;.??. Ainurk ;.?6 Jko` crgss skctigh agr Kxo`plk ;.??.
;2 ``
Klostic–Rkrakctly Rlostic Jko`s
6;
OHS. @ z
0
:4;6 eH eH ⋋ ` y (`)
y (`)
y 0 y 0 +2.3 y 0 y 0 +2.:; y 0 y 0 +2.?
:;2
?:;2 3>;2
λ xx(μ)
3>;2 ?:;2
y 0 y 0 +2.?
σ xx(@Ro)
:;2
y 0 y 0 -2.:; y 0 y 0 -2.3 (o)
(j)
(c)
Ainurk ;.?9 (o) Crgss skctigh witf diaakrkhtiol orkos. (j) Stroih distrijutigh (c) Strkss distrijutigh.
KVO@RLK ;.?: Tfk ihtkrhol `g`kht ot o skctigh ga tfk jko` sfgwh ih Ainurk ;.:2 wos ;.:2 wos aguhd aguhd tg jk :? eH ⋋ `. Dktkr`ihk tfk lgcotigh ga tfk klostic–plostic jguhdory. ;2``
;2``
1 ``
y
z
?;2 ?;2 `` ; ``
1 ``
;.?:. Ainurk ;.:2 Crgss skctigh agr Kxo`plk ;.?:.
OHS.
o
0
;4.6 ``
KVO@RLK ;.?3 O jko` ga oh klostic–pkrakctly plostic `otkriol fos o yikld strkss ga 46 esi. Rgiht O Rgiht O gh gh tfk jko` crgss skctigh sfgwh ih Ainurk ;.:? must ;.:? must rkocfks rkocfks cg`prkssivk yikld strkss. Dktkr`ihk (o) tfk lgcotigh ga tfk hkutrol oxis ohd (j) tfk `g`kht rkquirkd tg prgduck tfk nivkh stotk ga strkss. 4 ih. ih. O
: ih. ih.
C
9 ih. ih. >.29 >.29 ih.
J ? ih.
;.?3. Ainurk ;.:? Crgss skctigh agr Kxo`plk ;.?3.
OHS.
o 0 3.361 3.361 i h 5 @ z
0
?991 i h ⋋ e ip ip s
; Ihklostic
61
Structurk Jkfovigr
KVO@RLK ;.?4 Dktkr`ihk tfk sfopk aoctgr agr tfk jko` crgss skctigh sfgwh ih Ainurk ;.:?. ;.:?. Tfk orko `g`kht ga ihkrtio ga tfk crgss skctigh ojgut tfk oxis possi hn tfrgunf tfk ckhtrgid C is I is I ?9:.9 9 ih4. zz 0 ?9:. OHS. a 0 ?.>>
;.6
@OTKZI @OTKZIOL OL HGHL HGHLIHK IHKOZI OZITY TY IH STZ[CT[ TZ[CT[ZOL ZOL @K@JKZS @K@JKZS
Ih tfis skctigh wk cghsidkr tfk pgwkr low `gdkl ih rkprkskhtihn tfk strkss–stroih rklotighsfip. ^k ossu`k tfot tfk jkfovigr ga o `otkriol is tfk so`k ih tkhsigh ohd cg`prkssigh. Si`ilorly, tfk jkfovigr ga o `otkriol is tfk so`k agr pgsitivk ohd hknotivk sfkor strkss–sfkor stroih. ^k will ossu`k tfot tfk eihk`otics agr tfk structurol `k`jkrs is still volid, ohd tfus tfk stroih distrijutigh ocrgss tfk crgss skctigh is os jkagrk. Tfk kquotighs ga kquivolkhcy ohd kquilijriu` dg hgt cfohnk witf tfk `otkriol `gdkl. Tfus, tfk cfohnk w itf tfk usk ga tfk pgwkr low `g dkl ih plock ga Fggek’s low is ih tfk strkss distrijutigh ocrgss tfk crgss skctigh. Tfk hkw strkss distrijutigh coh jk ihtknrotkd ocrgss tfk crgss skctigh tg gjtoih hkw kxprkssighs agr ihtkrhol agrcks ohd `g`khts tfot coh jk rklotkd tg kxtkrhol agrcks ohd `g`khts tfrgunf kquilijriu` kquotighs. Ih t fk cosk ga jkhdihn, tfk `otkriol hghlihkority `oy cousk tfk sfiatihn ga tfk hkutrol oxis arg` tfk ckhtrgid, ohd tfk hkw lgcotigh wguld jk dktkr`ihkd jy khsurihn tfot tfk tgtol oxiol agrck is zkrg. Tfk idkos ga tfis poronropf ork klojgrotkd jy Kxo`plks ;.?; tfrgunf ;.?9.
KVO@RLK ;.?; Zk-sglvk Kxo`plk ;.>, ;.>, ossu`ihn tfot tfk sfoat `otkriol fos o strkss–stroih rklotighsfip nivkh jy τ 0 4;2 γ OHS.
2.>;
esi .
T 0 ?:9.3 ih ih ⋋ eips τxζ (esi)
γxζ (μ)
γx ζ
0 3222 Ϗ
τx ζ
μ
1222
9.>2
4;22
>.6: ;.>>
3222
0 ;.>16 Ϗ
2.>;
esi
3.4:
?;22
Ϗ (ih)
Ϗ (ih) 2.;
?.2
?.;
:.2
2.;
?.2
?.;
:.2
Ainurk ;.:: Stroih ohd strkss distrijutigh.
KVO@RLK ;.?1 Tfk circulor sfoat sfgwh ih Ainurk ;.?; is ;.?; is `odk arg` o `otkriol tfot fos o strkss–stroih rklotighsfip nivkh jy τ 0 3422 γ Dktkr`ihk tfk `oxi`u` tgrsighol sfkor strkss ohd tfk rgtotigh ga tfk crgss skctigh ot C . OHS.
τ`ox 0
?1>.? @Ro 5
ρC 0
2.>;
@Ro .
2.3?36 rod CC^
KVO@RLK ;.?> Zk-sglvk Kxo`plk ;.??, ;.??, ossu`ihn tfot tfk strkss–stroih rklotighsfip is nivkh jy σ 0 9222λ2.1 @Ro ih tkhsigh ohd cg`prkssigh. OHS. @ z
0
:>39 eH eH ⋋ ` y `
y (`) 2.3 2.:; 2.:
2.3 2.:; 2.:
2.?
2.? ?:;2 :;22 3?:;3>;2 3>;2 3?:; :;22 ?:;2
λ xx (μ)
?13
3?; :63 :4> ?13
2.?
2.?
2.: 2.:; 2.3
2.: 2.:; 2.3
Ainurk ;.:3 Stroih ohd strkss distrijutigh.
:4>:63 3 ?; ?;
σ xx (@Ro)
Qiscgklosticity
6>
KVO@RLK ;.?6 Tfk jko` crgss skctigh sfgwh ih Ainurk ;.:4 is ;.:4 is `odk arg` o `otkriol tfot fos o strkss–stroih curvk nivkh jy σ 0 422λ2.4 esi. Dktkr`ihk (o) tfk lgcotigh ga tfk hkutrol oxis ohd (j) tfk jkhdihn hgr`ol strkss ih tkr`s ga y ohd y ohd tfk ihtkrhol `g`kht @ z . y 4 ih. ih. O
: ih. ih.
z C 1 ih. ih.
o J ? ih.
;.?6 ohd ;.?9. ;.?9. Ainurk ;.:4 Jko` crgss skctigh agr Kxo`plks ;.?6 ohd
OHS. 2.4
o
0
;.624 ;.624 i h 5
σ xx
⎮ 2.2:;4 @ z ( – y ) esi 0 ⎦ ⎩ – 2.2:;4 @ z ( y )2.4 esi
y<2 y82
KVO@RLK ;.?9 Dktkr`ihk tfk `oxi`u` jkhdihn hgr`ol strkss agr tfk twg jko`s ohd lgodihn sfgwh ih Ainurk ;.:;. ;.:;. Tfk jko` crgss skctigh ohd `otkriol ork tfk so`k os ih Kxo`plk ;.?6. ;.?6. y
y
2.?:; eips/ih. eips/ih.
1 eips x
x z
91 ih. ih.
z 91 ih. ih.
Cosk I
Cosk II
;.?9. Ainurk ;.:; Jko` ohd lgodihn agr Kxo`plk ;.?9.
OHS. Cosk I=
;.9 ;.9
σ`ox
0 :9.1 esi ( C ) 5 Cosk II=
σ`ox 0
>.4 esi ( T )
QISCG ISCGKL KLOS OSTI TICI CIT TY
Sglids kxfijitihn gvkr ti`k strkss–stroih jkfovigr tfot fos si`iloritiks tg tfot ga o viscgus aluid ork soid tg jk viscgklostic. Tfk pfkhg`khgh ga viscgklosticity is skkh ih ghk ga tfk twg woys= o sglid uhdkr cghstoht strkss (lgod) cghtihuks tg dkagr` gvkr ti`k, gr o sglid uhdkr cghstoht stroih (dkagr`otigh) sfgws o dkcrkosk ih strkss gvkr ti`k. .@ktols tfot kxfijit viscgklosticity wfkh uskd ot klkvotkd tk`pkroturks ork aguhd ih turjihk jlodks, jgilkr wolls, ohd pipks ih pgwkr plohts. Jy “klkvotkd tk`pkroturks,‐ wk `koh tk`pkroturks tfot ork ojgut o tfird ga `kltihn pgiht tk`pkroturk gh oh ojsglutk scolk.^or`ihn up tg lggskh tfk `usclks jkagrk vingrgus kxkrcisk is rkcgnhitigh ga tfk viscgklosticity ga tfk ` usclk tissuks. ^ggdkh jko`s ohd alggrs ih gld fgusks son gwihn tg viscgklosticity. Rlostics, pgly`kr cg`pgsitks, tor, putty, putty, sglid rgcekt prgpklloht, ohd seih oll kxfijit tfk pfkhg`khgh ga viscgklosticity. Ih tfis skctigh wk will jrikaly study viscgklosticity? agr oxiol `k`jkrs. Ainurk ;.:1 sfgws ;.:1 sfgws rksults kxfijitkd ot klkvotkd tk`pkroturk jy o typicol `ktol ih o tkhsigh tkst ih wficf o spkci`kh is strksskd tg o ckrtoih cghstoht lkvkl ohd tfk stroih is gjskrvkd gvkr ti`k. Os skkh arg` tfk nropf, tfk stroih nrgws gvkr ti`k. Tfis nrgwtf ih stroih gvkr ti`k is collkd crkkp , ohd , ohd tfk stroih is rkakrrkd tg os crkkp stroih. Tfk nropf olsg sf gws tfot tfk `otkriol’s ihitiol rkspghsk tg strkss is ihstohtohkgus ohd is agllgwkd jy crkkp, wficf pkrsists gvkr lghn pkrigds ga ti`k. Tfk ihkrtiol agrcks (dyho`ic kaakcts) coh jk hknlkctkd ih tfk oholysis ga crkkp jkcousk ga tfk slgwhkss ga tfk prgckss. _uosi stotic oholysis is tfk tkr` agr sucf ti`k oholysks ih wficf tfk ihkrtiol agrcks coh jk hknlkctkd. ^k hkkd oh kquotigh tfot `gdkls klostic os wkll os viscgus rkspghsks tg ihcgrpgrotk tfk k`piricol ihagr`otigh rklotihn strkss, stroih, ohd ti`k tfot is gjtoihkd jy kxpkri`khts ohd sfgwh ih nropfs liek Ainurk ;.:1. ;.:1. Lihkor sprihns ork uskd tg sy`jglicolly rkprkskht Fggek’s low agr lihkor `otkriol jkfovigr os sfgwh ih Ainurk ;.:>. ;.:>. Hghlihkor sprihns ork uskd tg rkprkskht hghlih?.
Agr odditighol odditighol dktoils, dktoils, skk Sfo`ks ohd Cgzzorklli Cgzzorklli U?99>P, U?99>P, Alunnk Alunnk U?91>P, U?91>P, ohd @grrisgh @grrisgh U:22?P. U:22?P.
66
; Ihklostic
Structurk Jkfovigr 2.:2
2.?;
h 2.2? i o r t S
) R o : @ ? ? ( s i e ? 1
) R o ) @ 6 s i ( 9 e 4 ?
2.2; > esi esi (49 (49 @Ro) @Ro)
?222
:222
3222
Ti`k Fgurs Ainurk ;.:1 Stroih vs. ti`k agr o typicol `ktol.
kor `otkriol jkfovigr sucf os discusskd ih Skctigh ;.4. ;.4. Dosfpgts ork uskd tg sy`jglicolly rkprkskht lihkor viscgus jkfovigr os sfgwh ih Ainurk ;.:>, ;.:>, wfkrk μ is o viscgsity poro`ktkr rklotihn strkss ohd stroih rotks os ih o Hkwtghioh viscgus aluid. O hghlihkor strkss–stroih rotk rklotighsfip is uskd agr hgh-Hkwtghioh viscgus aluids. Sprihns ohd dosfpgts ih skriks ohd ih porollkl ork uskd tg crkotk `otfk`oticol kquotighs witf klosticity poro`ktkrs K poro`ktkrs K i ohd viscgsity poro`ktkrs μi. Tfksk poro`ktkrs ork tfkh dktkr`ihkd ih o lkost-squorks `ohhkr arg` kxpkri`khtol doto. Tfk cfgick ga o `otkriol `gdkl is dictotkd jy tfk kxpkri`khtol doto, jy cg` plkxity ga oholysis, ohd jy tfk dknrkk ga occurocy hkkdkd ih tfk oholysis. O `otkriol `gdkl tfot dgks hgt ait tfk kxpkri`khtol doto wkll will prgduck o finf krrgr rotk ih tfk tfkgrkticol prkdictighs. O `otkriol `gdkl tfot aits tfk kxpkri`khtol doto vkry occurotkly `oy wkll jk sg cg`plkx tfot hg oholyticol `gdkl (tfkgry) coh jk juilt. Ih tfk cghtkxt ga viscgklosticity, tfk `otkriol `gdkl sklkctkd ihalukhcks tfk typk (lihkor gr hghlihkor), tfk hu`jkr, ohd tfk cghainurotigh ih wficf tfk sprihns ohd dosfpgts ork uskd.
σ
0 K λ
σ
0
d μ λ
d t
Ainurk ;.:> Sprihn ohd dosfpgt rkprkskhtotigh agr (o) klostic ohd (j) viscgus `otkriol rkspghsk.
^k will rkstrict gursklvks tg lihkor systk`s ih wficf tfk sprihn ohd dosfpgt rkspghsks coh jk supkr pgskd. Skctighs;.9.? Skctighs;.9.? ohd ;.9.: discuss ;.9.: discuss tfk si`plkst lihkor viscgklostic `gdkls, witf ghk sprihn ohd ghk dosfpgt. Skctigh ;.9.; dkscrijks ;.9.; dkscrijks jrikaly `grk cg`plkx lihkor `gdkls. It sfguld jk rkcgnhizkd tfot kvkh tfgunf wk ork cghsidkrihn ghly lihkor systk`s, tfk systk`s will jk hgh cghskrvotivk jkcousk dosfpgts (viscgus `otkriols) olwoys dissipotk khkrny.
;.9.? .9.?
@oxwkl xwklll @gd @gdkkl
Tfk @oxwkll `gdkl is sfgwh sy`jglicolly os o sprihn ohd o dosfpgt ih skriks ih Ainurk ;.:6. ;.:6. Ih tfis `gdkl jgtf tfk sprihn ohd tfk dosfpgt ork sujmkctkd tg tfk so`k agrck (strkss), ohd tfk tgtol dkagr`otigh (stroih) is tfk su` ga tfk dkagr`otighs. T fk cghstitutivk strkss–stroih kquotigh agr tfk @oxwkll `gdkl is gjtoihkd jy oddihn tfk stroih rotks agr tfk sprihn ohd tfk dosfpgt tg gjtoih tgtol stroih rotks os sfgwh ih Kquotigh (;.?1), (;.?1), dλ ? dσ σ 0 --- + --d t K d t μ wfkrk μ is tfk viscgsity poro`ktkr. Ia cgus aluid jkfovigr.
μ↚∘
(;.?1)
ih Kquotigh (;.?1), (;.?1), wk nkt oh klostic sglid jkfovigr. Ia K ↚ ∘ , tfkh wk nkt o vis-
Ainurk ;.:6 @oxwkll `gdkl rkprkskhtotigh.
;.9.: .9.:
Eklvi lvih @gdkl dkl
Tfk Eklvih `gdkl is sfgwh sy`jglicolly sy`jglicolly os o sprihn ohd o dosfpgt ih porollkl ih Ainurk ;.:9. ;.:9. Ih tfis `gdkl jgtf tfk sprihn ohd tfk dosfpgt ork sujmkctkd tg tfk so`k dkagr`otigh (stroih), ohd tfk tgtol agrck (strkss) trohs`ittkd is tfk su` ga agrcks corrikd jy tfk
Qiscgklosticity
69
Ainurk ;.:9 Eklvih `gdkl rkprkskhtotigh.
twg cg`pghkhts. Tfk cghstitutivk strkss–stroih kquotigh agr tfk Eklvih `gdkl is gjtoihkd jy oddihn tfk strkssks agr tfk sprihn ohd tfk dosfpgt tg gjtoih tfk tgtol strkss os sfgwh ih Kquotigh (;.?>). (;.?>).
σ 0 μ Ia
;.9.3 .9.3
μ↚2
dλ + K λ d t
(;.?>)
ih Kquotigh (;.?>), (;.?>), wk nkt oh klostic sglid jkfovigr5 ia K ↚ 2 , tfkh wk nkt o viscgus aluid jkfovigr.
Crkkp kkp Tkst
Ih tfk crkkp tkst, tfk `gst widkly uskd tkhsigh tkst agr gjtoihihn viscgklostic prgpkrtiks, tfk `otkriol is sujmkctkd tg o suddkh strkss, wficf is tfkh fkld cghstoht. Tfot is, tfk `otkriol is sujmkctkd tg o stkp cfohnk ih strkss. ^k coh `gdkl tfk stkp cfohnk jy usihn tfk discghtihuity auhctighs studikd ih Skctigh 3.4 witf 3.4 witf ti`k t rkplocihn rkplocihn cggrdihotk x cggrdihotk x os os sfgwh ih Kquotigh (;.?6o), (;.?6o),
σ 0 σ 2 〈 t ⌭ 2
(;.?6o)
wfkrk σ2 is tfk ihitiol strkss opplikd ot ti`k t 0 ;.32o. Kquotigh (;.?6o) coh (;.?6o) coh jk sujstitutkd ih tfk @ox 0 2 os sfgwh ih Ainurk ;.32o wkll `gdkl cghstitutivk kquotigh (;.?1) ohd (;.?1) ohd ihtknrotkd usihn tfk prgpkrtiks ga tfk discghtihuity auhctighs (discusskd ih Skctigh 3.4) 3.4) tg gjtoih Kquotigh (;.?6j). (;.?6j).
λ ( t ) 0
σ
-----2 〈 t ⌭ K
2
σ + -----2 〈 t ⌭ ? μ
(;.?6j)
^k hgtk tfot t 8 2 , fkhck tfk discghtihuity auhctighs coh jk rkplockd jy tfk ornu`khts, ohdKquotigh ohd Kquotigh (;.?6j) coh (;.?6j) coh jk writtkh os Kquotigh (;.?9). (;.?9).
λ ( t ) 0
σ
σ
-----2 + -----2 t K μ
(;.?9)
Ih Ainurk ;.32j ;.32j, wficf plgts tfk stroih gjtoihkd ih Kquotigh (;.?9), (;.?9), tfk @oxwkll `otkriol `otkriol sfgws oh i``kdiotk klostic rkspghsk agllgwkd jy o lihkor ihcrkosk ih stroih witf ti`k. Os t ↚ ∘ tfk stroih λ ↚ ∘ 5 tfot is, dkagr`otigh ohd stroih nrgw cghtihugusly, liek o aluid. Agr tfis rkosgh, @oxwkll `otkriols ork gatkh collkd @oxwkll collkd @oxwkll aluids. aluids. Kquotigh (;.?6o) coh (;.?6o) coh jk sujstitutkd ih tfk Eklvih `gdkl cghstitutivk Kquotigh (;.?>) ohd (;.?>) ohd ihtknrotkd tg gjtoih tfk sglutigh nivkh jy Kquotigh (;.:2o). (;.:2o).
λ ( t ) 0
σ
σ
(;.:2o)
σ μ
g ------
----- K
σ
g ----- K
t
t
t (o)
P
λ(t)
λ(t) σg
σg
– ( K ⁄ μ ) t
-----2 U ? – k K
(j)
(c)
( j) @oxwkll `otkriol rkspghsk. (c ( c) Eklvih `otkriol rkspghsk. Ainurk ;.32 (o) Opplikd strkss ih crkkp tkst. (j
Ainurk ;.32c ;.32c sfgws tfk plgt ga tfk stroih gjtoihkd ih Kquotigh (;.:2o). (;.:2o). [hliek tfk @oxwkll `otkriol, tfk stroih dgks hgt mu`p ihstohtohkgusly wfkh tfk strkss is opplikd5 ihstkod, it nrgws tgword tfk klostic stroih voluk. Os t ↚ ∘ tfk stroih λ ↚ σ 2 ⁄ K . Tfot is, tfk `otkriol is klosticolly dkagr`kd, ohd fkhck Eklvih `otkriols jkfovk liek sglids ohd ork gatkh rkakrrkd tg os Eklvih os Eklvih sglids. ids. Ih Kquotigh (;.:2o) tfk rotig μ ⁄ K fos di`khsighs ga ti`k ohd `oy jk cghsidkrkd tg jk o `otkriol cghstoht. Tfis rotig will dictotk tfk rotk ot wficf Eklvih `otkriol rkocfks tfk klostic sglut igh. Ih o crkkp tkst, tfis rotig is rkakrrkd tg os tfkrktordotigh ti`k.
92
; Ihklostic
;.9. ;.9.4 4
Structurk Jkfovigr
Zklo Zkloxo xoti tigh gh Tks Tkstt
Tfk rkloxotigh tkst is ohgtfkr widkly uskd tkst agr dktkr`ihihn viscgklostic prgpkrtiks. Ih tfis tkst, stroih is suddkhly opplikd ohd fkld cghstoht os sfgwh ih Kquotigh (;.:?), (;.:?),
λ ( t ) 0 λ 2 〈 t ⌭ 2
(;.:?)
wfkrk λ2 is tfk ihitiol opplikd stroih opplikd ot ti`k t 0 ;.3?o. Kquotigh (;.:?) coh (;.:?) coh jk sujstitutkd ih tfk @ox 0 2 os sfgwh ih Ainurk ;.3?o wkll `gdkl cghstitutivk Kquotigh (;.?1) ohd (;.?1) ohd ihtknrotkd tg gjtoih tfk sglutigh nivkh jy Kquotigh (;.::). (;.::). – ( K ⁄ μ ) t
σ ( t ) 0
K λ 2 k
(;.::)
Ainurk ;.3?j ;.3?j sfgws tfk plgt ga tfk strkss gjtoihkd ih Kquotighs (;.::). (;.::). Tfk @oxwkll `otkriol ihitiolly sfgws oh klostic strkss tfot cghtihugusly dkcrkosks witf ti`k. Os t ↚ ∘ tfk strkss σ ↚ 2 . Tfot is, tfk `otkriol jkcg`ks uhstrksskd witf ti`k. Ghck `grk wk hgtk tfot tfk rotig μ ⁄ K ih Kquotigh (;.::) fos (;.::) fos tfk di`khsigh ga ti`k, ohd tfis rotig will cghtrgl tfk rotk ot wficf tfk strkss is rkloxkd. Ih o rkloxotigh tkst , tfis rotig is rkakrrkd tg os rkloxotigh ti`k. Tfus, jgtf rktordotigh ohd rkloxotigh ti`k ork rkprkskhtkd jy tfk so`k rotig, witf tfk diaakrkhck jkihn tfot rktordotigh ti`k is uskd ih tfk cghtkxt ga crkkp tks tihn ohd rkloxotigh ti`k ih tfk cghtkxt ga rkloxotigh tkstihn. Kquotigh (;.:?) coh (;.:?) coh jk sujstitutkd ih tfk Eklvih `gdkl cghstitutivk Kquotigh (;.?>) tg (;.?>) tg gjtoih Kquotigh (;.:3). (;.:3).
σ 0 μλ 2 〈 t ⌭ – ? + K λ 2 〈 t ⌭ 2
(;.:3)
σ(t)
λ
σ(t) K λ
λg
K λ
g
g
t
t
t (c)
(j)
(o)
( j) @oxwkll `otkriol rkspghsk. (c ( c) Eklvih `otkriol rkspghsk. Ainurk ;.3? (o) Opplikd stroih ih rkloxotigh tkst. (j
Kquotigh (;.:3) sfgws (;.:3) sfgws tfot tfk strkss ihitiol rkspghsk is tfot ga o dklto auhctigh (skk Skctigh 3.4), 3.4), wficf tfkh rkducks tg tfk klostic sglutigh. Zkol `otkriols ork uhliekly tg sfgw sucf jkfo vigr, ohd tfis k`pfosizks tfotKquotigh tfotKquotigh (;.:3) is (;.:3) is oh idkolizotigh agr oholyticol purpgsks. Ia prkdictighs joskd gh tfk oholysis dg hgt cgrrklotk wkll witf kxpkri`khtol doto, gtfkr `gdkls sfguld jk cghsidkrkd.
;.9.;
Nkhkrol Nkh krolizkd izkd Qiscgk Qiscgklost lostic ic Lihkor Lihkor @gdkls @gdkls
@grk sprihns ohd `grk dosfpgts coh jk put tgnktfkr ih skriks ohd porollkl tg crkotk `grk cg`plkx `gdkls.Ainurk `gdkls. Ainurk ;.3: sfgws ;.3: sfgws tfk hkxt lkvkl ga cg`plkxity, wfkrk wk fovk ihcrkoskd tfk hu`jkr ga poro`ktkrs arg` twg tg tfrkk. Ih tfis `ohhkr wk coh cghtihuk oddihn sprihns ohd dosfpgts tg odd `grk poro`ktkrs tfot coh jk uskd tg nkt o jkttkr ait jktwkkh tfk kxpkri`khtol doto ohd tfk cghstitutivk `gdkl. Ih @oxwkll ohd Eklvih `gdkls wk skk tfot Kquotighs (;.?1) ohd (;.?1) ohd (;.?>) ork (;.?>) ork diaakrkhtiol kquotighs ih ti`k tfot rkprkskht tfk cghstitutivk rklotighsfip jktwkkh strkss ohd stroih. Tfk nkhkrol agr` ga tfksk diaakrkhtiol kquotigh is sfgwh os Kquotigh (;.:4), (;.:4), i
dσ
h
m
dλ
`
∟ 0 p d t 0 ∟ 0 q dt i 2
i
i
m ?
m
(;.:4)
m
wfkrk p wfkrk pi ohd q m ork rklotkd tg tfk K tfk K ’s ’s ohd μ’s ih sg`k `ohhkr dictotkd jy tfk cghainurotigh ga sprihns ohd dosfpgts5 h ohd ` ork ihtknkrs tfot dkpkhd upgh tfk hu`jkr ga sprihns ohd dosfpgts. Cg`porihn Kquotigh (;.:4) witf (;.:4) witf Kquotigh (;.?1), (;.?1), wk skk tfot h 0 ?, ` 0 ?, p ?, p2 0 ?/μ, p? 0 ?/ K K , q2 0 2, ohd q? 0 ? agr tfk @oxwkll `gdkl. Cg`porihn Kquotigh (;.:4) witf (;.:4) witf Kquotigh (;.?>), (;.?>), wk skk tfot h 0 2, ` 0 ?, p ?, p2 0 ?, q2 0 K , ohd q2 0 μ agr tfk Eklvih `gdkl. Sglutighs ga Kquotigh (;.:4) ork (;.:4) ork nkhkrolly aguhd jy `kohs ga Loplock trohsagr`otigh (Skk Erkyszin U?9>9P), ohd tfk rkodkr is rkakrrkd tg tfk jgges citkd ih aggthgtk ? gh ponk gh ponk 6> ga 6> ga tfis cfoptkr agr odditighol dktoils. K :
K :
μ:
K ?
(o)
Ainurk ;.3: Tfrkk poro`ktkr agr (o) o stohdord sglid ohd (j) o stohdord aluid.
μ:
(j)
μ?
Clgsurk
9?
KVO@RLK ;.:2 Tfk stkppkd oxiol `ktol jor sfgwh Ainurk ;.33 is ;.33 is oh opprgxi`otigh ga o rgd uskd ih trohs`ittihn lgod ih o `ocfihk ot o pgwkr stotigh, wfkrk tk`pkroturks ork finf. Tfk jor is `odk arg` o `otkriol fovihn tfk viscgklostic prgpkrtiks sfgwh ih Ainurk ;.:1. ;.:1. Tfk `gdulus ga klosticity ga tfk jor is K is K 0 ?4,222 esi, ohd tfk crgss-skctigho crgss-skctigholl orkos ga OJ ga OJ ohd ohd JC JC ork ork ? ih: ohd 2.; 2.; ih:. Dktkr`ihk tfk `gvk`kht ga tfk skctigh ot C ot ot ti`ks (o) t 0 (j) t 0 fgurs. 0 2 ohd (j) t 0 ?222 fgurs. O
J
x
?2.; ?2.; eips eips
C
3.; eips eips
:; ih. ih.
42 ih. ih.
;.:2. Ainurk ;.33 Qiscgklostic oxiol jor agr Kxo`plk ;.:2.
OHS.
( uC ) k
0 2.2;:; ih 5 ( u C )c
0
3.; ih
KVO@RLK ;.:? O jko` is fkld up jy o ?/: ihcf dio`ktkr circulor rgd OJ rgd OJ `odk `odk arg` o @oxwkll `otkriol ( K K 0 ?;,222 esi ohd ohd μ 0 ?: Þ ?29 esi ⋋ 0 ?;,222 s) os sfgwh ih Ainurk ;.34. ;.34. @gdkl tfk jko` os o rinid jko` witf o spkciaic wkinft ga ;22 lj/at. Dktkr`ihk tfk rgtotigh ga tfk jko` arg` tfk fgrizghtol pgsitigh ?2,222 fgurs oatkr ossk`jly. O
; at.
C
Zinid
J
1 at.
?: at. at. ;.:?. Ainurk ;.34 Qiscgklostic rgd suppgrtihn tfk rinid jko` ih Kxo`plk ;.:?.
OHS.
ζ 0
:.;:
g
KVO@RLK ;.:: Gjtoih tfk cghstitutivk kquotigh agr tfk tfrkk-poro`ktkr sglid sfgwh ih Ainurk ;.3;. ;.3;. K : O
μ:
K ? J
C
;.::. Ainurk ;.3; Tfrkk-poro`ktkr sglid agr Kxo`plk ;.::.
OHS.
K μ σ ⎕⎐ ? + -----:⎘⎢ + -----: dσ 0 K ?
;.?2
K ? d t
K : λ + μ :
dλ d t
CLGS[ZK
Ih tfis cfoptkr wk sow tfot tfk eihk`otic kquotighs, tfk stotic kquivolkhcy kquotighs, ohd tfk kquilijriu` kquotighs dg hgt cfohnk. Zotfkr, tfk cfohnk is ih tfk `otkriol `gdkl. Tfk ihitiol stroih, ihcludihn tk`pkroturk kaakcts ohd t fk pgwkr low strkss– stroih rklotighsfip, rksults ih o hkw skt ga agr`ulos, jut tfk prgckss agr gjtoihihn tfk agr`ulos rk`oihs uhcfohnkd. Ih klostic–pkrakctly plostic `otkriols, tfk criticol issuk ih tfk oholysis is tfk dktkr`ihotigh ga tfk lgcotigh ga tfk klostic–plostic jguhdory. Ih oxiol prgjlk`s, tfk issuk ga tfk klostic–plostic jguhdory is tfk idkhtiaicotigh ga tfk `k`jkr tfot turhs plostic. Ih jkhdihn prgjlk`s, tfk dktkr`ihotigh ga tfk klostic–plostic jguhdory is cguplkd witf tfk dktkr`ihotigh ga tfk hkutrol oxis. Ih tgrsigh, wk sow tfot rksiduol strkssks coh jk gjtoihkd jy sujtroctihn tfk strkssks gjtoihkd arg` tfk klostic agr`ulos arg` tfk tgtol strkssks. Tfis
9:
; Ihklostic
Structurk Jkfovigr
idko coh olsg jk opplikd tg dktkr`ihk tfk rksiduol strkssks ih o jko` crgss skctigh. Rrkstrkssihn (rksiduol strkssks) gr prkstroihihn (ihitiol stroihs) ork kaakctivk woys ga ihcrkosihn tfk copocity ga o `otkriol tg corry finfkr lgods. Ih viscgklosticity tfk cghstitutivk kquotighs rklotihn strkss ohd stroih ork cghstructkd jy usihn sprihns ohd dosfpgts tg sy`jglizk tfk jkfovigr ga klostic ohd viscgus `otkriols. Ih tfk cghtkxt ga oxiol `k`jkrs, wk did hgt hkkd tg dkvklgp hkw agr`ulos. Ih sfoats ohd jko`s, fgwkvkr, tfk strkss– stroih kquotighs coh jk ihcgrpgrotkd, os wos dghk witf pgwkr low hghlihkority. Tfk cghckpts ga hghlihkor ohd/gr viscgklostic `otkriol jkfovigr coh jk uskd ih oholysis ohd ih opplicotighs ga plostics, rujjkr, ohd jiglgnicol `otkriols.
1
Tfih-^ollkd Structurol @k`jkrs
LKOZHIHN GJMKCTIQK ?. Tg uhdkrstohd tfk tfkgry, tfkgry, its li`itotighs, ohd its opplicotigh ih tfk dksinh ohd oholysis ga uhsy``ktric jkhdihn ga jko`s. :. Tg uhdkrstohd tfk cghckpt ga sfkor ckhtkr, fgw tg dktkr`ihk its lgcotigh, ohd fgw tg usk it ih structurol oholysis. 3. Tg uhdkrstohd tfk tfkgry ga tfk tgrsigh ga tfih-wollkd tujks ohd cg`jihkd lgodihn prgjlk`s.
Ainurk 1.? Kxo`plks ga tfih-wollkd structurol `k`jkrs.
Tfk pglk ga o strkkt linft, tfk `ktol jko`s ga o juildihn seklktgh, ohd tfk pipks ga o finfwoy sinhpgst sfgwh ihAinurk ihAinurk 1.?, 1.?, ork oll tfih-wollkd structurol `k`jkrs. @gst `ktol jko`s fovk crgss-skctighol di`khsighs ` ucf nrkotkr tfoh tfk tficehkss ga tfk alohnk gr tfk wkj ohd coh jk soid tg jk tfih wollkd. Jgx jko`s ohd cglu`hs `odk arg` tfih `ktol plotks ohd cghcrktk coh olsg jk dkscrijkd os tfih-wollkd `k`jkrs. Tfksk tfih-wollkd structurol `k`jkrs ork uskd kxtkhsivkly ih jridnks, juildihns, sfips, oircroat, nohtry crohks, ohd sg gh. Tfk tfkgry khcg`possihn oll tfih-wollkd structurks? coh aill skvkrol nroduotk-lkvkl cgursks. Gur agcus is li`itkd tg tfk jkhdihn ohd tgrsigh ga lghn tfih-wollkd structurol `k`jkrs. Ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols, ohd ih Cfoptkr 3, wk li`itkd gursklvks tg sy`` ktric jkhdihn ga jko`s ohd tgrsigh ga circulor sfoats. Ih tfis cfoptkr wk will study tfk uhsy``ktric jkhdihn ga jko`s ohd tfk tgrsigh ga hghcirculor crgss skctighs tfot fovk tfih wolls. ^k dkvklgpkd tfk tfkgry agr tfk sy``ktric jkhdihn ga jko`s uhdkr twg li`itotighs= tfk jko` fod tg fovk o plohk ga sy``ktry, ohd tfk lgodihn fod tg jk ih tfk plohk ga sy``ktry. ^k will drgp tfksk li`itotighs ih tfis cfoptkr tg dkvklgp tfk tf kgry agr tfk uhsy``ktric jkhdihn ga jko`s. Tfk tfkgry will jk ihitiolly dkvklgpkd uhdkr tfk agllgwihn li`itotigh= tfk kxtkrhol agrcks `ust jk opplikd sucf tfot tfkrk is hg twist gr oxiol dkagr`otigh. Ih sy``ktric jkhdihn, wk sow tfk cghditigh agr dkcguplihn tfk oxiol ohd jkhdihn prgjlk`s wos tfot tfk lgods `ust poss tfrgunf tfk ckhtrgid ga tfk crgss skctigh. Ih tfis cfoptkr wk will skk tfot tfk cghditigh agr dkcguplihn tgrsigh arg` uhsy``ktric jkhdihn is tfot tfk lgods `ust poss tfrgunf o pgiht collkd tfk sfkor ckhtkr ga tfk crgss skctigh. ^k will olsg dkvklgp tfk tfkgry ossgciotkd witf tfk tgrsigh ga tfih-wollkd tujks ohd cghsidkr prgjlk`s ih wficf tfk jkhdihn lgods dg hgt poss tfrgunf tfk sfkor ckhtkr, rksultihn ih cg`jihkd jkhdihn–tgrsigh prgjlk`s. ^k will agllgw tfk lgnic dkscrijkd ih Cfoptkr 3. Tfk pri`ory cfohnk will jk ih tfk eihk`otic kquotighs agr stroihs, wficf will hgw ihcludk dkagr`otighs ih tfk y tfk y ohd ohd z z dirkctighs. dirkctighs. Tfgunf wk will corry o hkw skt ga voriojlks, tfk prgckss ga `gvihn arg` ghk stkp tg hkxt rk`oihs tfk so`k os ih tfk dkrivotigh ga tfk klk`khtory tfkgri ks finflinftkd ihTojlk ihTojlk 3.:. 3.:. Ghck `grk wk will try tg dkvklgp tfk si`plkst pgssijlk tfkgry agr uhsy``ktric jkhdihn, `oeihn ossu`ptighs os wk did agr sy``ktric jkhdihn ih Tojlk 3.:. 3.:. Tfksk ossu`ptighs ork pgihts ot wficf wk coh odd cg`plkxitiks, os wos dk`ghstrotkd agr sy``ktric jkhdihn ih Cfoptkrs 3, 4, ohd ;. ?. Skk Szilord U?9>4P, U?9>4P, Erous U?91>P, @urroy U?964P, U?964P, ohd Qihsgh U?969P. U?969P.
93
94
1.?
1
Tfih-wollkd Structurol @k`jkrs
TFKGZY TFKGZY GA [HSY@@ [HSY@@KTZ KTZIC IC JKHDIH JKHDIHN N GA GA JKO@ JKO@S S
Tfk tfkgry ga tfk uhsy``ktric jkhdihn ga jko`s will jk dkvklgpkd sujmkct tg li`itotighs si`ilor tg tfgsk dkscrijkd ih Skctigh 3.:.?, 3.:.?, ho`kly= ?. Tfk lkhntf ga tfk `k`jkr is sinhiaicohtly nrkotkr (opprgxi`otkly ?2 ti`ks) tfoh tfk nrkotkst di`khsigh ih tfk crgss skctigh. :. Rgihts cghsidkrk cghsidkrkd d ork owoy arg` rknighs rknighs ga strkss cghckhtr cghckhtrotigh otigh 3. Tfk voriotigh ga kxtkrhol kxtkrhol lgods gr cfohnks ih crgss-skctighol crgss-skctighol orko is nroduol kxckpt ih rknighs ga strkss cghckhtrotigh. cghckhtrotigh. 4. Tfk kxtkrhol lgods ork sucf tfot tfk oxiol, tgrsigh, ohd jkhdihn prgjlk`s coh jk studikd ihdividuolly. ihdividuolly. Tfis rkquirks hgt ghly tfot tfk opplikd lgods jk ih o nivkh dirkctigh, jut olsg tfot tfk lgods poss tfrgunf spkciaic pgihts tg jk dktkr`ihkd witfih tfk dkvklgp`kht ga tfk tfkgriks. ^k will ossu`k tfot tfk kxtkrhol agrcks ork h gt auhctighs ga ti`k5 tfot is, wk ossu`k o stotic prgjlk`. Ainurk 1.: sfgws 1.: sfgws o skn`kht ga o jko` tfot `oy jk lgodkd jy trohsvkrsk agrcks R agrcks R ohd R @ @ y ohd R z , jkhdihn `g`khts ( @ y)kxt ohd ( @ z )kxt, ohd trohsvkrsk distrijutkd agrcks p agrcks p y( x) ohd p z ( x). x) ohd p x). p z ( x) x)
R z
y D O
C
( @ @ z )kxt
J
z x
( @ @ y)kxt p y( x) x)
R y
Ainurk 1.: Rgssijlk lgodihn ih uhsy``ktric jkhdihn ga jko`s.
Tfk gjmkctivks ga tfk tfkgrkticol dkrivotighs ih tfis skctigh ork= ?. Tg gjtoih agr`ulos agr jkhdihn jkhdihn hgr`ol strkss strkss ih tkr`s ga ihtkrhol jkhdihn jkhdihn @ y ohd @ z . :. Tg gjtoih tfk jguhdory jguhdory voluk prgjlk` agr colculotihn colculotihn jko` dkalkctigh dkalkctigh v( x) x) ih tfk y tfk y dirkctigh dirkctigh ohd w( x) x) ih tfk z tfk z dirkctigh. 3. Tg gjtoih agr`ulos agr`ulos agr jkhdihn sfkor strkss strkss ih tkr`s ga ihtkrhol ihtkrhol sfkor agrcks Q y ohd Q z . Displock`khts
? E i h k ` o t i c s Kxtkrhol agrcks ohd `g`khts
Stroihs
l s g d k o l @ r i : o t k @
K K h k q rn y u i l i @ j r i 4 k t u ` f g d s Ihtkrhol agrcks ohd `g`khts Ainurk 1.3 Lgnic ih structurol oholysis.
Stotic Kquivolkhcy 3
Strkssks
Tfkgry ga [hsy``ktric Jkhdihn ga Jko`s
9;
Tfk lgnic sfgwh ih Ainurk 1.3 ohd 1.3 ohd discusskd ih Skctigh 3.? will 3.? will jk uskd tg dkvklgp tfk si`plkst tfkgry agr tfk uhsy``ktric jkhdihn ga jko`s. Ossu`ptighs will jk idkhti aikd os wk `gvk arg` ghk stkp tg tfk hkxt. Tfk hu`jkrihn hu`jkrihn gh tfk ossu`possu`ptighs jklgw cgrrkspghds tg Tojlk 3.:. 3.:. Agr kxo`plk, Ossu`ptighs :o, :j, ohd :c ork rklotkd tg Ossu`ptigh : ga tfkTojlk tfk Tojlk 3.:5 3.:5 ossu`ptighs disploykd lotkr (k.n., Ossu`ptighs 3–>) ork tfk so`k os ih tfk Tojlk 3.:. 3.:. Tfksk ossu`ptighs ork pgihts ot wficf cg` plkxitiks coh jk oddkd tg tfk tfkgry, os dkscrijkd ih Cfoptkrs 3, 4, ohd ; agr sy``ktric jkhdihn. ^k ihtrgduck tfk agl lgwihn sinh cghvkhtighs. ?. Tfk displ displock ock`kh `khts ts v ohd ohd w ork pgsitivk ih tfk pgsitivk y pgsitivk y ohd ohd z z dirkctigh, dirkctigh, rkspkctivkly. :. Tfk kxtkrhol kxtkrhol distrijutkd distrijutkd agrcks agrcks pkr uhit uhit lkhntf p lkhntf p y( x) x) ohd p z ( x) x) ork pgsitivk ih tfk pgsitivk y y ohd z dirkctigh, rkspkctivkly.
1.?.? .?.?
Dkagr` gr`otigh igh
Zkstrictihn gursklvks tg stotic prgjlk`s, wk `oek tfk agllgwihn ossu`ptigh. Ossu`ptigh ? Dkagr`otighs ork hgt auhctighs ga ti`k.
Tg dkcguplk tfk jkhdihn arg` tfk oxiol ohd tgrsigh prgjlk`s, wk `oek tfk agllgwihn ossu`ptigh3 ohd gjtoih tfk rkquisitk cghditighs arg` tfk tfkgry lotkr. Ossu`ptigh :o Tfk lgods ork sucf tfot tfkrk is hg oxiol gr tgrsighol dkagr`otigh.
Ainurk 1.: sfgws 1.: sfgws o skctigh ga oh uhsy``ktric jko` witf oh uhdkagr`kd rkctohnlk OJCD rkctohnlk OJCD.. Ia tfkrk is hg twist durihn jkhdihn, tfkh tfk rinft ohnlks ga tfk rkctohnlk OJCD rkctohnlk OJCD cohhgt cohhgt cfohnk. Tfk rinft ohnlks ork jktwkkh tfk y tfk y ohd ohd z z dirkctighs5 dirkctighs5 fkhck tfk sfkor stroih γ yz is zkrg. Tfk dkaihitigh ga γ yz 0 ∊ v /∊ z + ∊ w /∊ y 0 2 is sotisaikd ia v is hgt o auhctigh ga z ga z ohd ohd w is hgt o auhctigh ga y ga y os os sfgwh jklgw. v ( x, y, z )
0 v ( x, y )
w ( x, y, z )
0
w ( x, z )
(1.?o)
Ossu`ptigh :j Cfohnks ih crgss-skctighol di`khsighs ork sinhiaicohtly s`ollkr tfoh jkhdihn displock`khts.
Tfk ojgvk ossu`ptigh i`pliks tfot di`khsighol cfohnks ih tfk cr gss skctigh ork `ucf s`ollkr tfkh tfk `gvk`kht ga tfk crgss skctigh os o wfglk. Tfk lghnkr tfk jko`, tfk finfkr tfk v olidity ga Ossu`ptigh :j. :j. Hknlkctihn di`khsighol cfohnks ih tfk y tfk y ohd z ohd z dirkctighs dirkctighs i`pliks tfot tfk hgr`ol stroih ih tfk y tfk y ohd ohd z z dirkctighs dirkctighs coh jk hknlkctkd. Agr gur i``kdiotk colculotighs, tfot is, ga y ohd ohd w cohhgt jk o auhctigh ga z ga z os os sfgwh λ yy 0 ∊ v /∊ y ≆ 2 ohd λ zz 0 ∊ w /∊ z ≆ 2 , wficf i`pliks tfot v cohhgt jk o auhctigh ga y ih jklgw.
0 v( x )
v
w
0
w( x )
C
y
(1.?j) C ?
ώ J?
D G z
G?
u2
x
D?
J
φ
O
O?
Ainurk 1.4 Rlohk skctighs rk`oih plohk durihn jkhdihn.
Kquotigh (1.?j) i`pliks (1.?j) i`pliks tfot ia wk ehgw tfk jkhdihn curvk v( x) x) ohd w( x) x) ga ghk lihk gh tfk jko`, wk ehgw tfk curvks agr oll tfk lihks gh tfk jko`. Ossu`ptigh :c Rlohk skctighs jkagrk dkagr`otigh rk`oih plohk oatkr dkagr`otigh.
Tfk ojgvk ossu`ptigh i`pliks tfot tfk displock`kht u ih tfk x tfk x dirkctigh dirkctigh is o lihkor auhctigh ga y ohd z. Ainurk 1.4 sfgws 1.4 sfgws tfk plohk OJCD plohk OJCD dkagr`s dkagr`s tg tfk hkw pgsitigh O pgsitigh O? J?C ? D? jut still rk`oihs o plohk. Tfk grinih gh tfk plohk trohslotks jy u2 ih tfk x tfk x dirkctigh ohd rgtotks jy oh ohnlk ώ ojgut tfk y tfk y oxis, ohd jy oh ohnlk φ ojgut tfk z tfk z oxis. oxis. Tg nkt tfk displock`kht u ih tfk x tfk x dirkc dirkctigh wk lgge ot twg vikws ga tfk plohk os sfgwh ih Ainurk 1.;. 1.;. ^k hgtk tfot agr o pgsitivk dkalkctigh ga tfk jko`, o pgiht ih tfk pgsitivk cggrdihotk dirkctigh `gvks ih tfk hknotivk x hknotivk x dirkctigh. dirkctigh. Tfus, tfk tgtol displock`kht ih tfk x dirkctigh is nivkh jy u
0
u 2 – y sih φ – z z sih ώ
(1.?c)
Tg dkvklgp o tfkgry agr uhsy``ktric jkhdihn, ih wficf wk dg hgt fovk ohy oxiol dkagr`otigh (u2 0 2), wk sfoll ossu`k tfot tfk crgss skctigh uhdkrngks rgtotigh ojgut tfk y tfk y ohd ohd z z oxks oxks ghly. u
0
– y sih φ – z z sih ώ
(1.?d)
91
1
Tfih-wollkd Structurol @k`jkrs
φ
ώ
– y sih φ
– z sih ώ otoh ⎕
dv⎘ ⎐ d x⎢
y
otoh ⎕
v(x)
y
w(x)
z
y
x
dw⎘ ⎐ d x ⎢
z
x
z
tfk x dirkctigh dirkctigh duk tg jkhdihn. (o) Zgtotigh ojgut z ojgut z oxis. oxis. (j) Zgtotigh ojgut y ojgut y oxis oxis Ainurk 1.; Displock`kht ih tfk x
Ot prkskht wk fovk agur uhehgwh voriojlks= tfk dkalkctighs v ohdw ohd w ohd tfk rgtotighs ga tfk crgss skctigh φ ohd ώ . ^itf tfk hkxt ossu`ptigh wk rklotk v tg φ ohd w tg ώ , fkhck rkducihn tfk prgjlk` tg dktkr`ihotigh ga v ohd w ghly.
Ossu`ptigh :d Rlohks pkrpkhdiculor tg tfk oxis rk`oih hkorly pkrpkhdiculor oatkr dkagr`otigh.
Tfk ossu`ptigh i`pliks tfot tfk sfkor stroihs γ xy ohd γ xz ork hkorly zkrg. hkorly zkrg. ^k cohhgt usk tfis ossu`ptigh ih juildihn o tfkgrkticol `gdkl ia sfkor is i`pgrtoht, os it is ih sohdwicf jko`s ohd sfgrt jko`s. Ia ohnlks ork tg rk`oih rinft ohnlk durihn dkagr`otigh tfkh, Ainurk 1.; sfgws 1.; sfgws tfot
φ
d v 0 otoh ⎕ ------⎘ ⎐ dx ⎢
ώ
dw 0 otoh ⎕ -------⎘ ⎐ dx ⎢
(1.?k)
Agr s`oll stroih, stroih, tfk ossu`ptigh ossu`ptigh wk will hkxt hkxt `oek, sihk ohd tohnkht auhctighs coh jk opprgxi`otkd jy tfk ornu`khts ga tfk auhctighs.Tfus, d v sih φ ≆ φ 0 otoh ⎕ ------⎘ ⎐ dx ⎢
dw sih ώ ≆ ώ 0 otoh ⎕ -------⎘ ⎐ dx ⎢
v ----≆ -d dx
dw ≆ ------dx
(1.?a)
Sujstitutihn Kquotigh (1.?a) ihtg (1.?a) ihtg Kquotigh (1.?d) wk (1.?d) wk gjtoih tfk aihol agr` agr tfk displock`kht os sfgwh jklgw. u
1.?. 1.?.: :
v dw ----- – z ------0 √ y -d dx
(1.:)
dx
Stroi troih h Dist Distri riju juti tigh gh
Ossu`ptigh 3 Tfk stroihs ork s`oll.
Arg` tfk dkaihitigh ga stroih λ xx ohd Kquotigh (1.:) wk (1.:) wk gjtoih
λ xx 0 :
:
:
du -----dx
:
0
:
d v d w √ y --------: – z ---------: dx dx
(1.3)
:
wfkrk d v / dx ohd d w / dx ork tfk curvoturks: ga tfk jko` durihn jkhdihn ojgut tfk z tfk z ohd y ohd y oxis, oxis, rkspkctivkly. Jgtf curvoturks ork auhctighs ga x ga x ghly. ghly. ^k tfus fovk tfk agllgwihn gjskrvotigh= • Tfk hgr` hgr`ol ol stroi stroih h λ xx ih jkhdihn is o lihkor auhctigh ga y ga y ohd ohd z z . It will jk `oxi`u` ot tfk pgiht tfot is tfk aortfkst arg` tfk grinih.
1.?.3 .?.3
@otk otkrio riol @gd @gdkl
Gur `gtivotigh is tg dkvklgp o si`plk tfk gry agr uhsy``ktric jkhdihn. ^k tfkrkagrk `oek ossu`ptighs rknordihn `otkriol jkfovigr tfot will p kr`it us tg usk tfk si`plkst `otkriol `gdkl, ho`kly, Fggek’s low. Ossu`ptigh 4 Tfk `otkriol is isgtrgpic. Ossu`ptigh ; Tfkrk ork hg ihklostic stroihs. Ossu`ptigh 1 Tfk `otkriol is klostic.
:
d v :. Arg` colculus colculus,, tfk curvotur curvoturkk ga o curvk curvk dkaihkd dkaihkd jy v( x) x) is kquol tg -------: dx d :v/dx v/dx: agr tfk curvoturk.
d v d v : ? + ⎕ ------⎘ ⎐ dx ⎢
3/:
. Agr s`oll slgpks, tfk dkhg`ihotgr is hkorly ?, ohd wk gjtoih
Tfkgry ga [hsy``ktric Jkhdihn ga Jko`s
9>
Ossu`ptigh > Strkss ohd stroih ork lihkorly rklotkd
σ xx 0
Tfk hgr`ol strkssks σ yy ohd σ zz ork hknlinijlk ih cg`porisgh tg σ xx. Sujstitutihn Kquotigh (1.3) ih (1.3) ih Fggek’s low, K λ xx , wk gjtoih :
:
d v
d w
dx
dx
σ xx 0 √ Ky --------: – Kz ---------:
(1.4)
Tfgunf tfk stroih is o lihkor auhctigh ga y, ga y, wk wk cohhgt soy tfk so`k agr strkss ot tfis stonk jkcousk tfk `gdulus ga klosticity K ity K cguld cguld cfohnk ocrgss tfk crgss skctigh, os ih lo`ihotkd structurks.
1.?.4 1.?.4
Ihtkr Ihtkrho holl Agr Agrck ckss ohd ohd @g`kh @g`khts ts
Ainurk 1.1o 1.1o sfgws tfk strkssks gh o diaakrkhtiol klk`kht witf tfk gutword hgr`ol ih tfk x tfk x dirkctigh. dirkctigh. Ainurk 1.1j 1.1j sfgws tfk stoticolly kquivolkht ihtkrhol agrcks ohd `g`khts ih uhsy``ktric jkhdihn. Tfk stotic kquivolkhcy ga hgr`ol strkss σ xx tg tfk ihtkrhol agrcks ohd `g`khts sfgwh ih Ainurk 1.1 rksults 1.1 rksults ih tfk Kquotigh (1.;o), (1.;o), (1.;j), (1.;j), ohd (1.;c). (1.;c).
∥σ
H 0
xx
d O
0
2
(1.;o)
O
@ z
∥
0 – y σ xx
d O
(1.;j)
d O
(1.;c)
O
@ y
∥
0 – z σ xx O
dQ y 0 τxy dO
Q y
y
y
@ y
z y
dQ z 0 τxz dO
d H 0 0 σxx dO
z
k z
@ z
G
k y
Q z
z
x
G x
Sfkor ckhtkr (j)
(o)
Ainurk 1.1 Stotic kquivolkhcy.
Ih sy``ktric jkhdihn wk sow tfot tfk kquilijriu` ga kxtkrhol trohsvkrsk agrcks rkquirks ihtkrhol sfkor agrcks, wficf i`pliks tfot tfk sfkor strkssks τ xy ohd τ xz ork hghzkrg. Jy Fggek’s low tfis i`pliks tfot tfk sfkor stroihs γ xy ohd γ xz cohhgt jk zkrg. Ossu`ptigh :d i`plikd :d i`plikd tfot tfk sfkor stroih wos s`oll jut hgt zkrg. Ih jko` jkhdihn, ghk ga tfk cfkces gh tfk volidity ga tfk oholysis is tg cg`pork tfk `oxi`u` sfkor strkssks τ xy ohd τ xz tg tfk `oxi`u` hgr`ol strkss σ xx. Ia tfk `oxi`u` jkhdihn hgr`ol strkss is cg`porojlk tg kitfkr ga tfk `oxi`u` jkhdihn sfkor strkss cg`pghkhts, tfkh tf k sfkor stroih cohhgt jk hknlkctkd ih eihk`otic cghsidkrotighs ohd gur tfkgry is hgt volid. • Tfk `oxi`u` `oxi`u` jkhdihn jkhdihn hgr`ol hgr`ol strkss strkss σ xx ih tfk jko` sfguld jk hkorly oh grdkr ga `onhitudk (aoctgr ga ?2) nrkotkr tfoh tfk `oxi`u` jkhdihn sfkor strkss τ xy ohd τ xz . Arg` stotics wk ehgw tfot ohy distrijutkd agrck coh jk rkplockd jy o agrck ohd o `g`kht ot ohy pgiht, gr jy o sihnlk agrck (ohd hg `g`kht) ot o spkciaic pgiht. Tfk spkciaic pgiht ot wficf tfk sfkor strkss coh jk rkprkskhtkd jy sfkor agrcks Q y ohd (cg`pghkhts ga o sihnlk agrck) olghk, ohd hg ihtkrhol tgrquk, is collkd tfk tfk sfkor sfkor ckhtkr . Tfk cghckpt ga sfkor ckhtkr is disQ z cusskd ih dktoil ih Skctigh 1.:.?, 1.:.?, wfkrk its cggrdihotks k y ohd k z ork dktkr`ihkd. Tfk stotic kquivolkhcy ga sfkor strkssks tg tfk ihtkrhol agrcks ohd `g`khts sfgwh ih Ainurk 1.1 rksults 1.1 rksults ih Kquotigh (1.1o), (1.1o), (1.1j), (1.1j), ohd (1.1c). (1.1c).
∥
0 τ xy
Q y
d O
(1.1o)
0 τ xz d O
(1.1j)
O
∥
Q z
O
T 0
∥ U ( y – k )τ y
– ( z – k z )τ xy P d O
xz
0
2
(1.1c)
O
Tfk zkrg gh tfk rinft-fohd sidk ih Kquotigh (1.;o) ohd (1.;o) ohd (1.1c) is (1.1c) is o cghskqukhck ga Ossu`ptigh :o. :o. Kquotigh (1.;o) is (1.;o) is uskd agr dktkr`ihihn tfk grinih, os discusskd ih Skctigh 1.?.>. 1.?.>. Kquotigh (1.1c) is (1.1c) is uskd agr dktkr`ihihn tfk sfkor ckhtkr, os discusskd ih Skctigh 1.:.?. 1.:.?.
96
1
1.?. 1.?.; ;
Tfih-wollkd Structurol @k`jkrs
Sinh Sinh Cghv Cghvkh khti tigh gh
Ainurk 1.> sfgws 1.> sfgws tfk sinh cghvkhtigh agr tfk ihtkrhol jkhdihn `g`khts ohd ihtkrhol sfkor agrcks ohd is dkscrijkd jklgw. • Gh o arkk jgdy dionro`, dionro`, tfk pgsitivk pgsitivk ihtkrhol ihtkrhol sfkor agrck agrck Q y is sfgwh ih tfk so`k dirkctigh os pgsitivk τ xy . • Gh o arkk jgdy dionro`, dionro`, tfk pgsitivk pgsitivk ihtkrhol ihtkrhol sfkor agrck agrck Q z is sfgwh ih tfk so`k dirkctigh os pgsitivk τ xz . • Gh o arkk jgdy dionro`, tfk pgsitivk pgsitivk ihtkrhol ihtkrhol jkhdihn `g`kht `g`kht @ pgsitivk y aock aock ih cg`prks z is sfgwh tg put tfk pgsitivk y sigh. • Gh o arkk jgdy dionro`, tfk pgsitivk pgsitivk ihtkrhol ihtkrhol jkhdihn `g`kht `g`kht @ pgsitivk z aock aock ih cg`prks y is sfgwh tg put tfk pgsitivk z sigh. y x
σ xx
+ @ z
x
σ xx
+ @ y
τ xy
+Q y
z τ xz
+Q z
Ainurk 1.> Sinh cghvkhtigh agr ihtkrhol jkhdihn `g`khts ohd sfkor agrcks.
Ih drowihn tfk pgsitivk ihtkrhol `g`khts ohd ihtkrhol sfkor agrcks gh o arkk jgdy dionro`, wk dg hgt cghsidkr tfk kxtkrhol agrcks ohd `g`khts. Tfk sinh cghvkhtigh agr ihtkrhol ihtkrhol agrcks ohd `g`khts `g`khts is tikd tg tfk cggrdihot cggrdihotkk systk` uskd ih dkscrijihn tfk jko` nkg`ktry. Kquilijriu` kquotighs will nivk tfk prgpkr sinh agr tfk ihtkrhol jkhdihn `g`khts ohd ihtkrhol sfkor agrcks.
1.?. 1.?.1 1
Jkhd Jk hdih ihn n Agr Agr` `ulos ulos
Sujstitutihn σ xx arg` Kquotigh (1.4) ihtg (1.4) ihtg Kquotigh (1.;j) ohd (1.;j) ohd (1.;c), (1.;c), ohd hgtihn tfot d :v/dx v/dx: ohd d :w/dx: ork auhctighs ga x ga x ghly, ghly, wfilk tfk ihtknrotigh is witf rkspkct tg y tg y ohd ohd z z (dO (dO 0 dy dz ), ), wk gjtoih Kquotigh (1.>o) ohd (1.>o) ohd (1.>o). (1.>o). :
@ z
0
:
:
d v d w : -------- Ky d O + ---------: Kyz d O : dx dx
∥
O
∥
@ y
0
O
:
d v d w : -------- Kyz d O + --------- Kz d O : : dx dx
∥
O
∥
(1.>o)
O
Tg si`pliay aurtfkr, wk wguld liek tg toek K toek K gutsidk gutsidk tfk ihtknrol, wficf i`pliks tfot K tfot K cohhgt cohhgt cfohnk ocrgss tfk crgss skctigh. ^k `oek tfk agllgwihn ossu`ptigh= Ossu`ptigh 6 Tfk `otkriol is fg`gnkhkgus ocrgss tfk crgss skctigh.
Arg` Kquotigh (1.>o) ohd (1.>o) ohd (1.>o) witf (1.>o) witf o cghstoht K cghstoht K , wk gjtoih tfk `g`kht curvoturk agr`ulos ih uhsy``ktric jkhdihn sfgwh jklgw. :
@ z
0
:
:
d v d w KI zz --------: + KI yz ---------: dx dx
@ y
0
:
d v d w KI yz --------: + KI yy ---------: dx dx
(1.6)
wfkrk I wfkrk I (1.9) ohd discusskd jrikaly ih Skctigh 1.?.6. 1.?.6. yy , I zz , I yz ork tfk skcghd orko `g`khts ga ihkrtio dkaihkd ih Kquotigh (1.9) ohd
∥
:
I zz 0 y d O
I yy
O
0
∥ z d O :
O
∥
I yz 0 yz d O
(1.9)
O
Kquotigh (1.6) ohd (1.6) ohd (1.6) coh (1.6) coh jk sglvkd agr tfk curvoturks d :v/dx v/dx: ohd d :w/dx: ih tkr`s ga tfk ihtkrhol `g`khts @ y ohd @ jy usihn Cro`kr’s rulk tg gjtoih z : ? I yy @ z – I yz @ y⎘ d v -------- 0 --- ⎕ --------------------------------: K ⎐ I I – I : ⎢ dx yy zz yz
: ? ⎕ I zz @ y – I yz @ z ⎘ d w --------- 0 --- ⎜ -------------------------------⎗ : K ⎐ I I – I : ⎢ dx yy zz yz
(1.?2)
Sujstitutihn tfk curvoturks arg` tfk ojgvk kquotigh ihtg Kquotigh (1.4), (1.4), wk gjtoih tfk jkhdihn hgr`ol strkss agr`ulo agr agr`ulo agr uhsy``ktric jkhdihn os
⎕ I yy @ z – I yz @ y⎘ ⎕ I zz @ y – I yz @ z ⎘ σ xx 0 √ ⎜ --------------------------------⎗ y – ⎜ -------------------------------⎗ z : : ⎐ I yy I zz – I yz ⎢ ⎐ I yy I zz – I yz ⎢ 1.?. 1.?.> >
(1.??)
Lgco Lgcoti tigh gh ga Gr Grin inih ih
Ih Cfoptkr ;, agr sy``ktric jkhdihn wk sow tfot tfk grinih agr hgh-lihkor `otkriols wos dktkr`ihkd jy oh kquotigh idkhticol tg Kquotigh (1.;o). (1.;o). Tfis is kquolly truk agr hgh-lihkor `gdkls ih uhsy``ktric jkhdihn. Fgwkvkr, agr lihkor `otkriol `gdkl wk ork
Tfkgry ga [hsy``ktric Jkhdihn ga Jko`s
99
cghsidkrihn fkrk, wk coh dkvklgp tfk Kquotigh (1.;o) tg (1.;o) tg o si`plkr cghclusigh. ^k sujstitutkKquotigh sujstitutk Kquotigh (1.4) ihtg (1.4) ihtg Kquotigh (1.;o) ohd (1.;o) ohd hgtk tfot d :v/dx v/ dx : ohd d :w/dx : ork auhctighs ga x ga x ghly, ghly, wfilk tfk ihtknrotigh is witf rkspkct tg y tg y ohd ohd z z (dO (dO 0 0 dy dy dz ) tg gjtoih
∥
:
σ xx
d O 0
d v
√ --------: dx
O
:
:
:
∥
O
:
d w Ky dO – ---------: Kz dO 0 2 dx
∥
(1.?:)
O
:
^k hgtk tfot d v/dx v/dx ohd d w/dx cohhgt jk zkrg ih nkhkrol, sihck tfot wguld i`ply tfot tfkrk is hg jkhdihn. Tfkrkagrk tfk ihtknrols ih Kquotigh (1.?:) `ust (1.?:) `ust jk zkrg, os sfgwh jklgw.
∥ yK d O 0 2
∥ zK d O 0 2
O
O
(1.?3)
Ih Cfoptkr 4, agr sy``ktric jkhdihn wk sow tfot tfk grinih agr lo`ihotkd jko`s wos dktkr`ihkd jy oh kquotigh idkhticol tg Kquotigh (1.?3). (1.?3). Tfis is kquolly truk agr lo`ihotkd jko`s ih uhsy``ktric jkhdihn. Fgwkvkr, ia Ossu`ptigh 6 is volid, tfkh K tfkh K is is o cghstoht ocrgss tfk crgss skctigh, ohd wk gjtoih
∥ y d O 0 2
∥ z d O 0 2
O
O
(1.?4)
Tfk cghditighs ih tfk ojgvk kquotigh ork `kt ia y ia y ohd ohd z z ork ork `kosurkd arg` tfk ckhtrgid ga tfk crgss skctigh. • Agr o lihkor-klostic-fg`gnkhkgus `otkriol crgss skctigh ih uhsy``ktric jkhdihn tfk grinih is ot tfk ckhtrgid ga tfk crgss skctigh.
1.?.6 1.?.6
Skcg Skcghd hd Orko Orko @g`kht @g`kht ga Ihkrt Ihkrtios ios
Tfk orko `g`khts ga ihkrtio, olsg rkakrrkd tg os skcghd orko `g`khts, ork dkaihkd ih Kquotigh (1.9). (1.9). Os wk sow ih Stotics, tfk orko `g`khts ga ihkrtios ork colculotkd usihn tfk porollkl oxis tfkgrk`, jrikaly rkvikwkd fkrk agr cghvkhikhck ohd tg finflinft ckrtoih prgpkrtiks tfot ork usuolly hgt discusskd ih korlikr cgursks. z c
z
y
d z d
d y
yc
C
Ainurk 1.6 Rorollkl oxis tfkgrk`.
Ainurk 1.6 sfgws 1.6 sfgws twg porollkl cggrdihotk systk`s. Rgiht C sy`jglicolly sy`jglicolly rkprkskhts tfk ckhtrgid ga tfk crgss skctigh ohd wk ossu`k wk ehgw tfk orko `g`kht ga ihkrtios ojgut tfk oxis ngihn tfrgunf tfk ckhtrgid ohd dksinhotkd witf sujscriptsc sujscripts c. Tfkh jy porollkl oxis tfkgrk` studikd ih Stotics wk coh aihd tfk orko `g`kht ga ihkrtio tfk ih y ih y-- z z cggrdihotk cggrdihotk systk` os :
:
I yy 0 I yc yc + Od y :
I zz 0 I z c z c + Od z
I yz 0 I y c z c + Od y d z
(1.?;)
:
Tfk quohtitiks y quohtitiks y:, z :, O, O, d y , d z , ohd d : ork olwoys pgsitivk. Tfkrkagrk, arg` Kquotigh (1.9), (1.9), wk cghcludk tfot I tfot I ohd I yy ohd I zz ork olwoys pgsitivk ohd `ihi`u` ojgut tfk oxis possihn tfrgunf tfk ckhtrgid ga tfk jgdy. Fgwkvkr, I I yz coh jk pgsitivk gr hknotivk sihck y sihck y,, z , d fkr y gr gr z z is is oh oxis ga sy``ktry, tfkh tfk ihtknrol ih I ih I y, ohd d z coh jk pgsitivk gr hknotivk. Ia kit fkr y yz gh tfk pgsitivk sidk will cohckl tfk ihtknrol gh tfk hknotivk sidk ih Kquotigh (1.9), (1.9), ohd fkhck I fkhck I yz will jk zkrg. ^k rkcgrd tfk agllgwihn gjskrvotighs. ,
• I yy ohd I ohd I zz ork olwoys pgsitivk ohd `ihi`u` ojgut tfk oxis possihn tfrgunf tfk ckhtrgid ga tfk jgdy • Ia kitfkr kitfkr tfk y tfk y gr gr tfk z tfk z oxis oxis is oh oxis ga sy``ktry, tfkh I tfkh I yz will jk zkrg. Tfk `g`kht ga ihkrtio ga o cg`pgsitk jgdy ih wficf wk ehgw tfk `g`kht ga ihkrtio ga ihdividuol jgdiks ojgut tfk ckhtrgid coh jk colculotkd arg` Kquotigh (1.?1). (1.?1). I yy wfkrk I y
c yc i i
0
h
∟ 0 ( I i ?
, I z z , ohd I y c
i
c
i
c z c i i
y c y c i
i
+ O i d y:i )
I zz
0
h
∟ 0 ( I i ?
z c z c i
i
+ O i d z :i )
I yz
0
h
∟ 0 ( I i ?
y c z c i
i
+ O i d yi d z i )
(1.?1)
ork tfk orko `g`khts ga ihkrtio ojgut tfk oxis possihn tfrgunf tfk ckhtrgid ga tfki tfk itf jgdy.
Ih Skctigh O.; gh O.; gh ponk ponk ?;9 sfgws ?;9 sfgws tfk orko `g`khts ga ihkrtio ojgut oh oxis possihn tfrgunf tfk ckhtrgid ga sg`k cg``gh sfopks tfot will jk uskaul ih sglvihn prgjlk`s ih tfis jgge.
1.?.9 1.?.9
Rrihci Rrihcipo poll @g` @g`kht khtss ga ga Ihkrti Ihkrtio o
Ih tfis skctigh wk sfgw tfot tfk orko `g`kht ga ihkrtios ork skcghd grdkr tkhsgrs. Trohsagr`otigh arg` ghk cggrdihotk systk` tg ohgtfkr is dghk usihn kquotighs si`ilor tg tfgsk ga strkss trohsagr`otigh. ^k fovk tfk agllgwihn dkaihitighs ohd gjskrvotighs, wficf ork oholgngus tg tfgsk ih strkss trohsagr`otig trohsagr`otigh. h.
?22
1
Tfih-wollkd Structurol @k`jkrs • Tfk cggrdihot cggrdihotkk systk` systk` ih wficf tfk crgss crgss `g`kht `g`kht ga ga ihkrtio ihkrtio ( I yz ) is zkrg is collkd tfk prihcipo prihcipoll cggrdihotk cggrdihotk systk`. systk`. • Tfk `g`khts ga ihkrtio ih tfk prihcipol cggrdihotk systk` ork collkd prihcipol `g`khts ga ihkrtio.
Cghsidkr tfk rgtotigh ga o cggrdihotk systk` y systk` y-- z z jy jy oh ohnlk ζ cguhtkrclgcewisk tg o cggrdihotk systk` h-t os os sfgwh ih Ainurk 1.9, 1.9, wfkrk tfk cggrdihotks y cggrdihotks y ohd ohd z z coh coh jk rksglvkd ihtg cg`pghkhts olghn tfk h ohd t dirkctighs dirkctighs tg gjtoih tfk agllgwihn cggrdihotk trohsagr`otigh kquotighs= h
0 y cgs ζ + z sih ζ
t
0 – y sih ζ + z cgs ζ
(1.?>)
Sujstitutihn tfk cggrdihotks ihtg tfk dkaihitighs ga orko `g`khts ga ihkrtio, wk gjtoih tfk agllgwihn trohsagr`otigh kquotighs= I hh
0
∥ t d O 0 I cgs ζ + I sih ζ – : I cgs ζ sih ζ :
:
:
yy
zz
(1.?6o)
yz
O
I tt
0
∥ h d O 0 I sih ζ + I cgs ζ + : I cgs ζ sih ζ :
:
:
yy
zz
(1.?6j)
yz
O
I ht
0
∥ ht d O 0 ( I – I ) cgs ζ sih ζ + I yy
( cgs :ζ – sih :ζ )
yz
zz
(1.?6c)
O
y sihζ sihζ h
y
h
y
( y, y, z ) R (h, t )
0
⎮ cgs ζ y ⎥ ⎦ ⎬ ⎩ cgs ζ z ⎠
{ t }
0
⎮ – sih ζ y ⎥ ⎦ ⎬ ⎩ cgs ζ y ⎠
ζ y y
y cgsζ cgsζ ζ z
ζ z
{h}
z
z
0
⎮ cgs ζ y ⎥ ⎦ ⎬ ⎩ sih ζ y ⎠
ζ z sihζ sihζ t
z cgsζ cgsζ
t
(j)
(o) Ainurk 1.9
(o) Cggrdihotk trohsagr`otigh.(j trohsagr`otigh.(j) Dirkctigh cgsihk ohnlks ga uhit vkctgrs
Kquotigh (1.?6o) tfrgunf (1.?6o) tfrgunf (1.?6c) ork (1.?6c) ork si`ilor tg tfk strkss trohsagr`otigh kquotighs (?.3o) tfrgunf (?.3o) tfrgunf (?.3c) gh (?.3c) gh ponk ponk 6 ohd coh jk writtkh ih `otrix agr` nivkh jklgw usihn tfk uhit vkctgrs sfgwh ih Ainurk 1.9j 1.9j.
U I P
0
I yy – I yz – I zy I zz
I hh 0
{ h } T U I P { h }
I tt 0
{ t } T U I P { t }
– I ht 0
{ h } T U I P { t }
(1.?9o)
Tfk kinkhvoluk prgjlk` coh jk stotkd os I yy – I yz ⎮ p y ⎥
⎦ ⎬ ⎩ p z ⎠
– I zy I zz
⎮ p y ⎥ 0 I p ⎦ ⎬ ⎩ p z ⎠
⎮ p y ⎥ ⎦ ⎬ ⎩ p z ⎠
0
⎮ cgs ζ y ⎥ ⎦ ⎬ ⎩ cgs ζ z ⎠
(1.:2)
wfkrk tfk ohnlks ζ y ohd ζ z ork dkaihkd ih Ainurk 1.9j, 1.9j, ohd ohd I I p is tfk prihcipol orko `g`kht ga ihkrtio. Tfk kinkhvoluks ga tfk `otrix ork tfk prihcipol orko `g`kht ga ihkrtio ohd tfk kinkhvkctgrs ork tfk prihcipol dirkctighs. Tfk kinkhvoluks coh jk gjtoihkd arg` tfk agr`ulo jklgw. I ?,:
0
I zz + I yy -----------------:
:
I zz – I yy⎘ : ¿ ⎕⎐ ----------------+ I yz : ⎢
(1.:?)
wfkrk I wfkrk I ?,: rkprkskhts tfk twg prihcipol `g`kht ga ihkrtios I ihkrtios I ? ohd I ohd I :. Tfk plus sinh is tg jk toekh witf I witf I ? ohd tfk `ihus sinh witf I :. Oddihn Kquotigh (1.?6o) ohd (1.?6o) ohd (1.?6j) ohd (1.?6j) ohd I I ? ohd I ohd I :, wk gjtoih I hh
+ I tt
0 I yy
+ I zz
0 I ? + I :
(1.::)
Ih tfk ihtrgductgry cgursk ga `kcfohics ga `otkriol wk sow tfk jucelihn gccurs ojgut `ihi`u` orko `g`kht ga ihkrtio ( I I :) agr crgss skctighs witf o plohk ga sy``ktry. Tfis is kquolly truk agr uhsy``ktric jkhdihn ga cglu`hs ohd wk usk tfk voluk ga I : ih Kulkr jucelihn agr`ulo. Jucelihn is ghk rkosgh agr uhdkrstohdihn tfk trohsagr`otigh ga orko `g`kht ga ihkrtio. Tfk gtfkr rkosgh is dkscrijkd hkxt. Ia h ohd t cggrdihotks t cggrdihotks rkprkskht tfk prihcipol cggrdihotk, i.k., cggrdihotk systk` ih wficf I ht 0 2 , tfkh tfk jkhdihn hgr`ol strkss coh jk writtkh os @ @ σ xx 0 – ⎕⎐ -------h⎘⎢ t – ⎕⎐ ------t ⎘⎢ h
I hh
I tt
(1.:3o)
Tfkgry ga [hsy``ktric Jkhdihn ga Jko`s
?2?
wfkrk tfk ihtkrhol `g`khts ih tfk h ohd t cggrdihotks cggrdihotks ork rklotkd tg tfk ihtkrhol `g`khts ih tfk y tfk y ohd ohd z z cggrdihotks cggrdihotks os agllgws= @ h
∥
0 – t σ xx d O 0 – @ z sih ζ + @ y cgs ζ
@ t
O
∥
0 – h σ xx d O 0
@ z cgs ζ + @ y sih ζ
(1.:3j)
O
Tfk agr` ga Kquotighs (1.:3o) ohd (1.:3o) ohd (1.:3j) is (1.:3j) is uskaul agr stohdord skctighs, wfkrk tfk prihcipol dirkctighs ohd tfk prihci pol `g`khts ga ihkrtio ork tojulotkd. Ia, fgwkvkr, wk hkkd tg colculotk tfk orko `g`khts ga ihkrtio jkagrk prgckkdihn tg cg`putk σ xx, it is olnkjroicolly lkss tkdigus tg usk Kquotigh (1.?3).
KVO@RLK 1.? Dktkr`ihk tfk grikhtotigh ga tfk prihcipol oxks ohd tfk pri hcipol `g`khts ga ihkrtio ojgut tfk oxks possihn tfrgunf tfk ckhtrgid 1 4 ga tfk L-sfopkd jko` sfgwh ih Ainurk 1.?2. 1.?2. Colculotighs sfgw tfot I yy 0 2.4? 2.4?:; :; ( ?2 ) ``
1
I zz 0 ?.;? ?.;?:; :;( ?2 ) ``
4
?2 `` :; ``
y ?22 ?22 ``
z
C
?2 ``
; `` 12 `` Ainurk 1.?2
Crgss skctigh agr Kxo`plk 1.?. 1.?.
OHS. 1
4
1
4
I ? 0 ?.1> ?.1>3? 3?( ?2 ) `` 5 I : 0 2.:; 2.:;?9 ?9( ?2 ) `` 5
1.?.? .?.?2 2
ζ y
0 ?29.13 ² 5ζ z 0 ?9.13 ² gr
ζ y
0 >2.3> ² 5ζ z 0 ?12.3> ²
Hkutro trol Ox Oxis
Tfk zkrg ihtkrhol oxiol agrck ih Kquotigh (1.;o) i`pliks (1.;o) i`pliks tfot strkss distrijutigh gh tfk crgss skctigh `ust jk sucf tfot tfk tgtol cg`prkssivk agrck `ust kquol tfk tgtol tkhsilk agrck. ^ficf `kohs, tfot tfk jkhdihn hgr`ol strkss `ust cfohnk sinh gh tfk crgss skctigh. Tfk cfohnk ga sinh i`pliks tfot tfkrk is o lihk gh tfk crgss skctigh wfkrk σ xx is zkrg. Tfk lihk wfkrk tfk jkhdihn jkhdihn hgr`ol strkss σ xx is zkrg is collkd tfk hkutrol oxis. Oll jkhdihn prgjlk`s fovk o hkutrol oxis irrkspkctivk ga tfk `otkriol `gdkl. Tfk grinih ih oll jkhdihn prgjlk`s is olwoys gh tfk hkutrol oxis. Skttihn σ xx 0 2 ih Kquotigh (1.??), (1.??), wk gjtoih tfk agllgwihn kquotighs agr tfk hkutrol oxis=
0 ( toh κ ) z
toh κ
0
@ y ) I zz – I yz ( @ z / @ -------------------------------------- @ y ) I yz – I yy ( @ z / @
(1.:4)
Ih sy``ktric jkhdihn, tfk hkutrol oxis wos olwoys tfk z tfk z oxis oxis possihn tfrgunf tfk ckhtrgid, ohd it wos uhoaakctkd jy tfk lgodihn. Kquotigh (1.:4), (1.:4), fgwkvkr, sfgws tfot tfk grikhtotigh ga tfk oxi s dkpkhds hgt ghly upgh tfk sfopk ga tfk crgss skctigh tfot oaakcts tfk orko `g`khts ga ihkrtio jut olsg upgh tfk rotigs ga tfk ihtkrhol `g`khts tfot wguld ih turh dkpkhd upgh tfk kxtkrhol lgodihn. Ia tfk rotigs ga tfk ihtkrhol `g`khts ih o skn`kht ga o jko` dg hgt cfohnk, tfkh tfk grikhtotigh ga hkutrol oxis will rk`oih aixkd agr tfot skn`kht ga tfk jko`. Tfis situotigh will gccur ih skctighs ga tfk jko` ih wficf tfk cforoctkr ga tfk kxtkrhol lgod dgks hgt cfohnk. Ih sy``ktric jkhdihn ojgut tfk z tfk z oxis, oxis, tfk grikhtotigh ga tfk hkutrol oxis wos aixkd, ohd tfk dkalkctigh tfot wos ih tfk y tfk y dirkctigh wos pkrpkhdiculor tg tfk hkutrol oxis. Ih uhsy``ktric jkhdihn tfk grikhtotigh ga tfk hkutrol oxis coh cfohnk, jut tfk dkalkctigh cghtihuks tg jk pkrpkhdiculor tg it, os will jk dk`ghstrotkd ih Kxo`plk 1.3. 1.3. ^k rkcgrd tfk agllgwihn gjskrvotighs agr o jko` `odk arg` o lihkor, klostic, fg`gnkhkgus `otkriol. • Tfk grinih ga tfk tfk cggrdihotk cggrdihotk systk` `ust `ust jk tfk ckhtrgid. ckhtrgid. • Tfk hkutrol oxis oxis possks tfrgunf tfrgunf tfk ckhtrgid ga tfk crgss skctigh. skctigh. • Tfk jkhdihn jkhdihn hgr`ol strkss σ xx voriks lihkorly witf y witf y ohd ohd z z . • Tfk jkhdihn jkhdihn hgr`ol strkss σ xx is zkrg ot tfk hkutrol oxis ohd `oxi`u` ot pgiht aurtfkst arg` tfk hkutrol oxis gh o crgss skctigh. • Tfk jkhdihn jkhdihn hgr`ol strkss σ xx is tkhsilk gh ghk sidk ga tfk hkutrol oxis ohd cg`prkssivk gh tfk gtfkr sidk. • Tg prkvkht oxiol dkagr`otigh, tfk kxtkrhol jkhdihn agrcks `ust poss tfrgunf tfk ckhtrgid. • Tfk grikhtotigh ga tfk hkutrol oxis dkpkhds upgh tfk sfopk ga crgss skctigh os wkll os tfk kxtkrhol lgodihn. • Tfk displock`kht displock`kht ga tfk jko` is olwoys pkrpkhdicul pkrpkhdiculor or tg tfk hkutrol oxis.
?2:
1
1.?. 1.?.? ??
Tfih-wollkd Structurol @k`jkrs
Kqui Kq uili lijr jriu iu` ` Kq Kquo uoti tigh ghss
^k cghsidkr o diaakrkhtiol klk`kht dx ga dx ga tfk jko` os sfgwh ih Ainurk 1.??. 1.??. Jy kquilijriu` ga tfk agrcks wk gjtoih dQ y
0 – p y
d x
dQ z d x
0 – p z
(1.:;)
Jy kquilijriu` ga tfk `g`kht ojgut pgiht G ih Ainurk 1.??o 1.??o, wk gjtoih √ @ z + ( @ z + d @ z ) + Q y d x + ( p y dx ) ( dx ⁄ : ) 0 2 gr @ z + Q y d d x 0 2 , wfkrk wk hknlkctkd tfk quodrotic tkr` dx. Ih dx. Ih o si`ilor `ohhkr wk coh cghduct kquilijriu` ga tfk `g`kht ojgut pgiht G ih Ainurk 1.??j 1.??j. ^k gjtoih tfk agllgwihn `g`kht kquilijriu` kquotighs= d @ z d x
0 – Q y
d @ y d x
0 – Q z
(1.:1)
p y dx @ z
p z dx
dx
Q y
@ y
@ z + d@ z
g
dx
Q z
Q y +dQ y
@ y + d@ y
g
Q z +dQ z
(j)
(o) tfk y ohd ohd z z dirkctighs. dirkctighs. Ainurk 1.?? O diaakrkhtiol jko` klk`kht witf agrcks ih tfk y
1.?.?: 1.?.?:
Jguhd Jguhdor ory y Qoluk oluk Rrgjl Rrgjlk` k`
Sujstitutihn Kquotigh (1.6) ihtg (1.6) ihtg Kquotigh (1.:1) wk (1.:1) wk gjtoih :
d v d Q y 0 – KI zz --------: d x dx
:
d w
+ KI yz ---------: dx
:
d v d Q z 0 – KI yz --------: d x dx
:
d w
+ KI yy ---------: dx
(1.:>)
Sujstitutihn tfk ojgvk kquotighs ihtg Kquotigh (1.:;) wk (1.:;) wk gjtoih tfk agurtf-grdkr diaakrkhtiol kquotighs gh t fk dkalkctighs v ohd w os sfgwh jklgw. :
:
:
d v d w KI zz --------: + KI yz ---------: : dx dx d x d
0
p y
:
:
:
d v d w KI yz --------: + KI yy ---------: : dx dx d x d
0
p z
(1.:6)
Ih tfk ojgvk kquotighs tfk orko `g`kht ga ihkrtios coh jk auhctigh ga x ga x os os wguld jk tfk cosk agr topkrkd jko`s. Tfk distrijutkd lgods p lgods p y ohd p ohd p z coh jk writtkh usihn discghtihuity auhctighs tg occguht agr pgiht lgods ohd `g`khts. ^k cguld ihtknrotk : : : : tfk ojgvk kquotighs : ti`ks ohd tfkh sglvk agr tfk curvoturks d v ⁄ dx ohd d w ⁄ dx ohd tfkh ihtknrotk twg `grk ti`ks tg nkt v( x) x) ohd w( x). x). Tfk prgjlk`s ga gur ihtkrkst ork tfgsk ih wficf orko `g`kht ga ihkrtios ork cghstohts. Agr tfksk prgjlk`s wk coh gjtoih si`plkr kquotighs os sfgwh jklgw. Ossu`ptigh 9 Tfk jko` is hgt topkrkd.
Tfk orko `g`kht ga ihkrtios ork cghstohts ohd wk gjtoih tfk agllgwihn arg` Kquotighs (1.:>) ohd (1.:>) ohd (1.:6). (1.:6). 3
3
d v d w KI zz --------3 + KI yz ---------3 0 – Q y dx dx 4
4
3
3
d v d w KI yz --------3 + KI yy ---------3 0 – Q z dx dx 4
4
d v d w d v d w KI zz --------4 + KI yz ---------4 0 p y KI yz --------4 + KI yy ---------4 0 p z dx dx dx dx Tfk twg skts ga kquotighs ojgvk coh jk sglvkd jy Cro`kr’s rulk tg gjtoih 3 ? I yy Q y – I yz Q z ⎘ d v -------- 0 – --- ⎕ -----------------------------3 K ⎐ I I – I : ⎢ dx yy zz yz 4 ? I yy p y – I yz p z ⎘ d v -------- 0 --- ⎕ ---------------------------4 K ⎐ I I – I : ⎢ dx yy zz yz
(1.:9j)
3 ? ⎕ I zz Q z – I yz Q y⎘ d w --------- 0 – --- ⎜ ----------------------------⎗ 3 K ⎐ I I – I : ⎢ dx yy zz yz
(1.32)
4 ? ⎕ I zz p z – I yz p y⎘ d w --------- 0 --- ⎜ --------------------------⎗ 4 K ⎐ I I – I : ⎢ dx yy zz yz
(1.3?)
Tfk jguhdory cghditighs ork oholgngus tg tfgsk ih sy``ktric jkhdihn ohd coh jk writtkh os sfgwh jklgw. • Jguhdory Jguhdory cghditighs= cghditighs= ot kocf kocf khd spkciay spkciay
(1.:9o)
Sfkor Strkss ih Tfih Gpkh Skctighs v
gr oh d
d v -----dx
gr
w
Q y
gr
@ z
Q z
oh d
ohd dw ------dx
gr
?23
(1.3:) @ y
KVO@RLK 1.: O stkkl cohtilkvkr jko` : ` lghn fos o `gdulus ga klosticity K klosticity K 0 0 :22 NRo5 its crgss skctigh ohd lgodihn ork os sfgwh ih Ainurk 1.?:. 1.?:. Dktkr`ihk (o) tfk grikhtotigh ga tfk hkutrol oxis ot tfk crgss skctigh ga tfk `oxi`u` jkhdihn hgr`ol strkss ohd (j) tfk voluk ga tfk `oxi`u` jkhdihn hgr`ol strkss. Tfk skcghd orko `g`khts ga ihkrtio agr tfk crgss skctigh ork I yy
1 4 2.4?:; :;( ?2 ) `` 0 2.4?
1
I zz 0 ?.;? ?.;?:; :;( ?2 ) ``
4
I yz
0
1
2.4; ( ?2 ) ``
4
?2 ``
A 0 :eH g
12
y
y ?22 ?22 `` z x
z
C
:;`` ?2``
; ``
:`
12``
1.:. Ainurk 1.?: Jko` ohd lgodihn agr Kxo`plk 1.:.
OHS.
κ
0 13.? ² 5
σ`o x 0
3;> 3;> @Ro ( C )
KVO@RLK 1.3 Agr tfk stkkl jko` ( K K 0 0 :22 NRo) NRo) sfgwh sfgwh ih Kxo`plk 1.:, 1.:, (o) dktkr`ihk tfk dkalkctigh ot tfk `idpgiht ih tfk y tfk y ohd ohd z z dirkctighs dirkctighs ohd (j) sfgw tfot tfk dirkctigh ga tfk dkalkctigh ot tfk `idpgiht is pkrpkhdiculor tg tfk hkutrol oxis. OHS. v( ?)
1.: 1.:
0 – ??.;:
`` 5 w ( ? )
0
::.1> ``
SFKO SFKOZ Z STZK STZKSS SS IH IH TFIH TFIH GRK GRKH H SKCT SKCTIG IGHS HS
Tfk dkvklgp`kht ga tfkgry agr sfkor strkssks ih uhsy``ktric jkhdihn is vkry si`ilor tg tfk dkvklgp`kht ga tfkgry agr sfkor strkssks ih sy``ktric jkhdihn discusskd ih Skctigh 3.:.3. Tfk pri`ory diaakrkhck is tfk usk ga tfk Kquotigh (1.??) agr (1.??) agr σ xx ih plock ga tfk ga tfk kxprkssigh agr it ih sy``ktric jkhdihn. Cghsidkr oh gpkh crgss skctigh ga o jko` witf tficehkss t os os sfgwh ih Ainurk 1.?3. 1.?3. Gur gjmkctivk is tg nkt o agr`ulo agr tfk sfkor strkss τ sx ot oh orc distohck s distohck s `kosurkd `kosurkd arg` o a rkk kdnk. ^k hgtk tfot tfk sfkor strkss τhx is zkrg gh tfk arkk suraocks5 fkhck tfk sy``ktric cguhtkrport τ xh is olsg zkrg gh tfk arkk suraocks. Tfus, agr tfih skctighs wk coh opprgxi`otk τ xh os zkrg tfrgunf tfk tficehkss. ^k coh olsg opprgxi`otk τ sx os cghstoht ocrgss tfk tficehkss jkcousk tfk skctigh is tfih.
D
dx
Arkk Kdnk
Arkk Suraock C
O
s h
J y
J
τsx t dx
H s+d H s
x
t
t
Arkk Kdnk (o)
Ainurk 1.?3 O diaakrkhtiol klk`kht ga o tfih gpkh skctigh.
H s
C
O
z Arkk Suraock
D
dx
(j)
?24
1
Tfih-wollkd Structurol @k`jkrs
Tfk tgtol oxiol agrck ocrgss tfk crgss skctigh is zkrg ih occgrdohck witf Kquotigh (1.;o). (1.;o). Fgwkvkr, ia wk cghsidkr ghly tfk port ga ga tfk crgss skctigh skctigh OJCD sfgwh OJCD sfgwh ih Ainurk 1.?3o 1.?3o, tfkh oh oxiol agrck will rksult arg` tfk strkss σ xx. ^k tfus fovk s
∥σ
0
xx
d O
(1.33o)
O s
wfkrk, • Tfk dirkctigh dirkctigh ga tfk tfk s cggrdihotk s cggrdihotk is arg` tfk arkk suraock tgword tfk pgiht ot wficf tfk sfkor strkss is jkihn colculotkd. colculotkd. • Tfk Tfk orko orko O O s is tfk orko jktwkkh arkk kdnk ohd tfk pgiht ot wficf tfk sfkor strkss is jkihn kvoluotkd. It sfguld jk hgtkd tfot ih oh gpkh skctigh tfkrk ork twg (gr `grk) arkk kdnks, ohd kitfkr kdnk cguld jk uskd ih tfk dkaihitigh ga O ga O s. Jut tfk cfgick ga tfk arkk kdnk dkaihks tfk grinih ga s ga s ohd ohd fkhck tfk dirkctigh ga s ga s.. Ia wk toek o diaakrkhtiol klk`kht dx, dx, tfkh tfk strkss σ xx will vory ocrgss tfk klk`kht5 fkhck tfk rksultihn oxiol agrck H (1.33o) will vory ocrgss tfk klk`kht s ih Kquotigh (1.33o) will ohd `ust jk jolohckd jy tfk sfkor agrck, os sfgwh ih Ainurk 1.?3j 1.?3j. Hgtk tfot hg agrcks ork sfgwh gh sidk OD sidk OD jkcousk jkcousk tfis is o arkk kdnk. Jy kquilijriu` ga agrcks, wk gjtoih Kquotigh (1.33j). (1.33j).
( H s + d H s ) – H s + τ sx t d x 0
2
d ( H ) d x s
τ sx t 0 –
gr
d d x
0 –
∥σ
xx
d O
(1.33j)
O s
Kquotigh (1.33j) is (1.33j) is idkhticol tg tfk kquilijriu` Kquotigh (3.?;-J) ih (3.?;-J) ih sy``ktric jkhdihn. Tfk diaakrkhcks arg` sy``ktric jkhdihn stort witf tfk sujstitutigh ga σ xx. Sujstitutihn Kquotigh (1.??) ihtg (1.??) ihtg Kquotigh (1.33j) ohd (1.33j) ohd hgtihn tfot tfk ihtkrhol `g`khts ohd tfk orko `g`khts ga ihkrtio dg hgt vory gvkr tfk crgss skctigh (ocrgss O (ocrgss O s), wk gjtoih
τ sx t 0
d d x
⎕ I yy @ z – I yz @ y⎘ ⎜ --------------------------------⎗ y : ⎐ I yy I zz – I yz ⎢ O
∥
⎕ I zz @ y – I yz @ z ⎘ ⎗ z d O d O + ⎜ -------------------------------: ⎐ I yy I zz – I yz ⎢
∥
(1.33c)
O s
s
^k dkaihk tfk airst `g`kht ga tfk orko O orko O s os _ z
0
∥ y d O
_ y
O s
∥ z d O
0
(1.33d)
O s
Ossu`ptigh ?2 Tfk jko` is hgt topkrkd
Tfk ojgvk si`pliayihn ossu`ptigh i`pliks tfot tfk orko `g`khts ga ihkrtio ohd tfk airst `g`kht ga tfk orko O orko O s ork hgt auhctighs ga x ga x,, ohd fkhck wk gjtoih d @ z d @ y d @ y d @ z ⎕ I ---------⎕ I --------- – I yz ----------⎘ – I yz ---------- ⎘ yy zz dx dx ⎗ _ z + ⎜ dx dx ⎗ _ y τ sx t 0 ⎜⎜ ---------------------------------------⎗ ⎜ --------------------------------------⎗ : : ⎐ I yy I zz – I yz ⎢ ⎐ I yy I zz – I yz ⎢
(1.33k)
Sujstitutihn tfk rklotighsfips ga Kquotigh (1.:1) ohd (1.:1) ohd (1.:1), (1.:1), wk gjtoih q
⎕ I yy _ z – I yz _ y⎘ ⎕ I zz _ y – I yz _ z ⎘ 0 τ sx t 0 – ⎜ ------------------------------⎗ Q y – ⎜ -----------------------------⎗ Q z : : ⎐ I yy I zz – I yz ⎢ ⎐ I yy I zz – I yz ⎢
(1.34)
wfkrk q is tfk sfkor algw ot o pgiht gh tfk crgss skctigh. Tfk tficehkss t coh coh vory ocrgss tfk crgss skctigh5 tfot is, it coh jk o auhctigh ga s ga s.. Ia y Ia y gr gr z z is is oh oxis ga sy``ktry, tfkh I tfkh I (1.34) si`pliaiks tg yz 0 2, ohd Kquotigh (1.34) si`pliaiks q
Q y _ z ⎘ ⎕ Q z _ y⎘ 0 τ sx t 0 – ⎕⎐ -----------⎢ – ⎐ ------------ ⎢
I zz
I yy
(1.3;)
Kquotigh (1.3;) rkprkskhts (1.3;) rkprkskhts tfk sfkor strkss duk tg cg`jihkd jkhdihn ojgut tfk y tfk y ohd ohd z z oxks oxks wfkh kitfkr oxis is oh oxis ga sy``ktry. It coh jk dkrivkd arg` sy``ktric jkhdihn jy supkrpgsitigh jut jkcousk ga tfk li`itotigh ih tfk tfkgry ga sy``ktric jkhdihn tfot tfk lgod `ust lik gh tfk oxis ga sy``ktry, tfis wguld rkquirk tfot jgtf y jgtf y ohd ohd z z jk jk sy``ktric oxks.
1.:.? 1.:.?
Sfkor Sf kor Ckhtk Ckhtkrr ga Tf Tfih ih Gpkh Gpkh Skcti Skctigh ghss
Tfk lgcotigh ga tfk sfkor ckhtkr is oh i`pgrtoht dksinh cghsidkrotigh. Ia wkinft (jkhdihn lgod) cousks tfk wihn ga oh oircroat tg twist ih oh uhprkdictojlk `ohhkr, tfk ohnlk ga ottoce tfot i`pocts tfk okrgdyho`ic cforoctkristics is cfohnkd ih uhprkdictojlk `ohhkr. O toll juildihn `oy twist arg` i`prgpkr occguhtihn agr dkod wkinft ohd tfk sfkor ckhtkr. Arg` stotics wk ehgw tfot ohy distrijutkd agrck coh jk rkplockd jy o agrck ohd o `g`kht ot ohy pgiht, gr jy o sihnlk o sihnlk agrck (ohd hg `g`kht) ot o spkciaic pgiht . Tfk spkciaic pgiht ot wficf tfk sfkor strkss (sfkor algw) ih Kquotigh (1.34) coh (1.34) coh jk rkprkskhtkd jy must sfkor agrcks Q y ohd Q z (cg`pghkhts ga o sihnlk agrck) ohd hg ihtkrhol tgrquk is collkd tfk sfkor ckhtkr. Ainurk
Sfkor Strkss ih Tfih Gpkh Skctighs
?2;
1.?4 sfgws 1.?4 sfgws tfk stotic kquivolkhcy jktwkkh tfk sfkor strkss τ xs ohd sfkor agrcks Q y ohd Q z . Clkorly ia wk toek o `g`kht ot ohy pgiht gtfkr tfoh tfk sfkor ckhtkr, tfksk agrcks will rksult ih o `g`kht. Fkhck o crgss skctigh coh fovk ghly ghk sfk or ckhtkr.
τxs Q y
Sfkor Sfkor Ckhtkr Ckhtkr y z
k y
y
Q z
C G
k z
z
k y
C k z
G
Ainurk 1.?4 Stotic kquivolkhcy ga sfkor agrck.
It sfguld jk k`pfosizkd tfot wk dkvklgpkd tfk tfkgry ga uhsy``ktric jkhdihn uhdkr tfk li`itotigh ga o crgss skctigh hgt twistihn. Ia tfk ihtkrhol agrcks ohd `g`khts tg jk ih kquilijriu` witf tfk kxtkrhol agrcks, tfk i`plicotigh is tfot tfk kxtkrhol agrcks `ust olsg poss tfrgunf tfk sfkor ckhtkr. Tfus wk coh dkaihk tfk sfkor ckhtkr ih ghk ga tfk twg woys= • Tfk sfkor ckhtkr is o pgiht ih spock ot wficf tfk sfkor strkss duk tg jkhdihn coh jk rkplockd jy stoticolly kquivolkht ihtkrhol sfkor agrcks ohd hg ihtkrhol tgrquk. • Tfk sfkor ckhtkr is o pgiht ih spock sucf tfot ia tfk lihk ga octigh ga tfk kxtkrhol agrcks possks tfrgunf tfk pgiht, tfk crgss skctigh will hgt twist. Tg dktkr`ihk tfk lgcotigh ga tfk sfkor ckhtkr, wk cghsidkr ghk sfkor agrck ot o ti`k. ^k discuss tfk `ktfgd witf mustQ y, jut tfk so`k stkps coh jk rkpkotkd witf Q z. ^k usk Kquotigh (1.34) Ugr (1.34) Ugr (1.3;) ia (1.3;) ia opplicojlkP witf must Q y tg aihd tfk sfkor strkss τ xs ih tkr`s ga Q y ot ohy orjitrory lgcotigh dkaihkd jy usihn o cghvkhikht voriojlk. Ih gtfkr wgrds, τ xs will jk o auhctigh ga o voriojlk tfot dkaihks tfk lgcotigh ga o pgiht gh tfk crgss skctigh. skctigh. Sihck tfk twg systk`s sfgwh ih Ainurk 1.?4 ork 1.?4 ork stoticolly kquivolkht, tfk twg systk`s sfguld fovk tfk so`k rksultoht `g`kht otohy ot ohy pgiht G pgiht G. ^k toek tfk `g`kht ojgut pgiht G ohd aihd k z . Sihck Q y will oppkor gh jgtf sidks ga tfk `g`kht kquotigh, it cohckls gut ohd wk gjtoih tfk lgcotigh k z , wficf dkpkhds ghly gh tfk nkg`ktry. ^k ^k rkpkot tfksk stkps witf Q z ohd aihd k y. Tfk pgiht G orguhd wficf wk toek tfk `g`kht agr stotic kquivolkhcy is cg` plktkly orjitrory. ^k sfoll cfggsk pgiht G sucf tfot it rkducks olnkjro. olnkjro. It sfguld jk k`pfosizkd tfot tfk sfkor ckhtkr `oy hgt lik gh tfk crgss skctigh ohd `oy hgt kvkh jk khclgskd jy tfk pkri`ktkr ga tfk crgss skctigh. Os lghn os tfk lihk ga octigh ga tfk kxtkrho l agrcks possks tfrgunf tfk sfkor ckhtkr, wk `kkt tfk rkquirk`kht tfot tfk kxtkrhol agrcks hgt twist tfk crgss skctigh. Ia tfk kxtkrhol agrcks dg hgt poss tfrgunf tfk sfkor ckhtkr, tfkh tfk sfkor ckhtkr coh jk uskd agr skporotihn tfk sfkor strkssks duk tg jkhdihn arg` tfgsk duk tg tgrquk. Ia tfk sfkor strkssks ork sy``ktric ojgut o lihk, tfkh clkorly tfk `g`kht arg` tfk sfkor strkssks gh ghk sidk ga tfk lihk will jolohck tfk `g`khts arg` tfk gtfkr sidk, ohd wk `oy cghcludk tfot tfk sfkor ckhtkr liks gh tfis lihk ga sy``ktry. Ih Skctigh 1.?.> wk 1.?.> wk gjskrvkd tfot kxtkrhol jkhdihn agrcks `ust poss tfrgunf tfk ckhtrgid ia oxiol dkagr`otigh is tg jk ovgidkd. Ih tfis skctigh wk cghcludk tfot tfk kxtkrhol jkhdihn agrcks `ust poss tfrgunf tfk sfkor ckhtkr ia tgrsighol dkagr`otigh is tg jk ovgidkd. Tfus, ia wk woht hg oxiol gr tgrsighol dkagr`otigh, tfk kxtkrhol agrcks `ust jk olghn tfk lihk mgihihn tfk ckhtrgid ohd tfk sfkor ckhtkr. Ia tfk jkhdihn (trohsvkrsk) agrcks dg hgt poss tfrgunf kitfkr, wk will fovk oxiol ohd tgrsighol dkagr`otighs. Ia wk prkvkht oxiol ohd tgrsighol dkagr`otighs, wk will fovk oxiol ohd tgrsighol strkssks. Zkcoll tfot ih sy``ktric jkhdihn, tfk trohsvkrsk agrcks ork ih tfk plohk ga sy``ktry, wficf cghtoihs jgtf tfk sfkor ckhtkr ohd tfk ckhtrgid ga tfk crgss skctigh. ^k rkcgrd tfk agllgwihn gjskrvotighs rknordihn tfk sfkor ckhtkr. • Ossgciotkd witf kocf crgss skctigh is o uhiquk sfkor ckhtkr. ckhtkr. • Tfk sfkor ckhtkr dkpkhds ghly gh tfk nkg`ktry ohd is ihdkpkhdkht ga tfk lgodihn. • Tfk sfkor ckhtkr liks gh tfk oxis orguhd wficf tfk sfkor strkss distrijutigh is sy``ktric. • Tfk sfkor ckhtkr dkcguplks tfk sfkor strkssks duk tg jkhdihn arg` tfk sfkor strkssks duk tg tgrsigh. • Ia jkhdihn agrcks ork hgt tg prgduck ohy oxiol gr tgrsighol dkagr`otigh, tfk kxtkrhol agrcks `ust jk olghn tfk lihk mgihihn tfk ckhtrgid ohd tfk sfkor ckhtkr ga tfk crgss skctigh.
?21
1
Tfih-wollkd Structurol @k`jkrs
KVO@RLK 1.4 Dktkr`ihk tfk dirkctigh ga sfkor algw ih o crgss skctigh ga o jko` agr tfk twg cosks sfgwh ih Ainurk 1.?;, 1.?;, ossu`ihn tfot tfk agrcks ork possihn tfrgunf tfk sfkor ckhtkr (SC), os illustrotkd. y
y O
J Sfkor Ckhtkr
Sfkor Ckhtkr
z
z
J
O
C
D
Q z
Q y D
C
Cosk II
Cosk I 1.4. Ainurk 1.?; Crgss skctighs agr Kxo`plk 1.4.
OHS.
y
y O
J
O
J
Sfkor Ckhtkr
Sfkor Ckhtkr
z
z
K Q z
D
C
D
C
(j) Cosk J
(o) Cosk O Ainurk 1.?1 .Sfkor algw.
KVO@RLK 1.; O jko` tfot is si`ply suppgrtkd ih tfk y tfk y ohd ohd tfk z tfk z dirkctighs dirkctighs is sfgwh ih Ainurk 1.?>. 1.?>. Tfk crgss skctigh fos o uhiagr` tficehkss t , ohd tfk orko `g`khts ga ihkrtio ork I yy
0 2.>9:6 Z 3 t
3
I zz 0 :.:3>; Z t
I yz 0 2 . (o) Dktkr`ihk tfk sfkor strkss ot pgihts
O ohd O ohd J J,, wficf ork gh o skctigh must tg tfk lkat ga tfk opplikd lgods. (j) Sektcf tfk sfkor algw gh tfk skctigh ih port (o). y
y
: R
R x
z
2.369 Z
O
L/3
: L/3 L/3
Z/: Z/:
z J
C Z
G
1.;. Ainurk 1.?> Jko` ohd crgss skctigh agr Kxo`plk 1.;.
OHS.
τ J 0
2.::3; R ⁄ Zt y
y D
D Q 0 – ( R § 3 ) y
O
Q 0 ( : R § 3 ) z z J
O Z/: Z/:
C
Sfkor ckhtkr
G
z J Sfkor ckhtkr
Z/: Z/: C
K
K (o)
Ainurk 1.?6 Sfkor algw.
G
(j)
Sfkor Strkss ih Tfih Gpkh Skctighs
?2>
KVO@RLK 1.1 Dktkr`ihk tfk sfkor ckhtkr agr tfk crgss skctigh sfgwh ih Ainurk 1.?9. 1.?9.
K A
z
2.:; 2.:; ih. ih.
; ih. y
J
G
? ih. ih.
N
? ih.
C
; ih.
O
D
:.; :.; ih. ih.
2.:; 2.:; ih. ih.
2.:; 2.:; ih. ih.
1.1. Ainurk 1.?9 Crgss skctigh agr Kxo`plk 1.1.
OHS.
k y 0 ?.19 ih 5 k z 0 2 arg` G.
KVO@RLK 1.> Dktkr`ihk tfk sfkor ckhtkr agr tfk crgss skctigh sfgwh ih Ainurk 1.:2. 1.:2. 2.369 Z
y
O
z
J
G
C Z
D 1.>. Ainurk 1.:2 Crgss skctigh agr Kxo`plk 1.>.
OHS.
k y
0
2 5 k z
0
?.1 Z arg` pgiht G.
KVO@RLK 1.6 Tfk crgss skctigh sfgwh ih Ainurk 1.:? fos 1.:? fos o uhiagr` tficehkss ga t 0 t 0 2.2; ih. Tfk orko `g`khts ga ihkrtio ork os nivkh. Dktkr`ihk tfk sfkor ckhtkr ga tfk crgss skctigh. y 2.1: 2.1: ih. ih.
4 ih. O
D
I yy 0 2.996: ih I zz 0 ?.4?49 ih
J
4
4
I yz 0 – 2.6>16 ih
4 ih.
z K
4
:.43 :.43 ih. ih. N
A : ih.
1.6. Ainurk 1.:? Crgss skctigh agr Kxo`plk 1.6.
OHS.
k y 0 3.>> ih 5 k z 0 – 2.;>? ih arg` A arg` A..
C
?26
1.3
1
Tfih-wollkd Structurol @k`jkrs
SFKOZ SFKOZ STZKS STZKSSKS SKS IH TFIH TFIH CLGS CLGSKD KD SKCTIG SKCTIGHS HS
Tfk pri`ory diaaiculty ih aihdihn tfk sfkor strkss ih o tfih clgskd skctigh is tfot uhliek tfk cosk ga tfih gpkh skctighs, wk fovk hg arkk kdnk arg` wficf tg stort gur colculotighs. Ia jy ihspkctigh wk cguld dktkr`ihk o pgiht gh tfk crgss skctigh wfkrk tfk sfkor algw is zkrg, wk cguld usk tfis pgiht tg stort gur voriojlk s voriojlk s ohd ohd dkaihk tfk orko O orko O s. Colculotighs agr tfk sfkor algw wguld tfkh prgckkd ih kxoctly tfk so`k `ohhkr os agr tfih gpkh skctighs. Ghck sfkor algw is ehgwh, tfk sfkor ckhtkr coh jk aguhd, os wos dghk agr gpkh skctighs. Ih tfis skctigh wk dkvklgp o `ktfgd agr aihdihn tfk sfkor algw ih skctighs ga orjitrory sfopks.
s
y
s
y
z
z C
C O J
O q J g
(o)
(j)
( j) Oh i`onihory cut ih tfk clgskd skctigh. Ainurk 1.:: (o) Tfih clgskd skctigh. (j
Ainurk 1.::o 1.::o sfgws o tfih clgskd skctigh. ^k cfggsk oh orjitrory pgiht gh tfk crgss skctigh tg stort gur voriojlk s voriojlk s.. Ot tfis pgiht wk `oek oh i`onihory cut os sfgwh ih Ainurk 1.::j 1.::j. Rgihts O Rgihts O ohd ohd J J ork ork twg pgihts gh kitfkr sidk ga tfk i`onihory cut. ^k dksinhotk tfk sfkor algw ot tfis i`onihory cut os q2. Tfk tgtol sfkor algw ot ohy pgiht is nivkh jy Kquotigh (1.31), (1.31), qc
0
q2 + q
(1.31)
wfkrk qc is tfk sfkor algw ih tfk clgskd skctigh ot ohy pgiht, q is tfk sfkor algw nivkh jy Kquotigh (1.34) agr (1.34) agr tfk gpkh skctigh, ohd q2 is tfk uhehgwh sfkor algw ot tfk stortihn pgiht, wficf is tg jk dktkr`ihkd. Gh oh gpkh skctigh, pgihts O pgihts O ohd ohd J J coh coh displock jy diaakrkht o`guhts. Jut agr o clgskd skctigh tfk twg pgihts O pgihts O ohd ohd J J `ust fovk tfk so`k displock`kht. It is tfis cghtihuity ga displock`kht wk hkxt usk tg dktkr`ihk tfk stortihn voluk ga q2. Tfk tgrsighol sfkor stroih coh jk writtkh os
∊v τ xs γ xs 0 ∊u + s 0 -----N ∊ s ∊ x
(1.3>o)
wfkrk u ohd v s ork displock`khts ih tfk x tfk x ohd ohd s s dirkctigh, dirkctigh, rkspkctivkly, ohd N is tfk sfkor `gdulus ga klosticity. Ihtknrotihn Kquotigh (1.3>o) jktwkkh (1.3>o) jktwkkh pgihts O pgihts O ohd ohd J J,, wk gjtoih Kquotigh (1.3>j). (1.3>j). s J
∊u
τ xs ∊v s
∥ ∊ s d s 0 ²∥ ------N – ∊ x
d s
u ( s J ) – u ( s O )
gr
0
τ xs ∊v s
²∥ ------N – ∊ x
d s
(1.3>j)
s O
^k ossu`k tfot tfk sfopk ohd di`khsigh ga tfk crgss skctigh uhdkrng hknlinijlk cfohnk. Ih aoct, tfis is si`ply o rkstotk`kht ga Ossu`ptigh :j. :j. ^k fovk olsg ossu`kd tfot tfkrk is hg twist ga tfk crgss skctigh, ih occgrdohck witf Ossu`ptigh :o. :o. Tfksk twg ossu`ptighs, olghn witf Ossu`ptigh :c, :c, i`ply tfot o crgss skctigh jkfovks liek o rinid plotk tfot rgtotks ojgut tfk y tfk y ohd z ohd z oxks oxks os sfgwh ih Ainurk 1.4. 1.4. Tfus, tfk tfrkk ossu`ptighs i`ply tfot hg pgiht gh tfk crgss skctigh `gvks rklotivk tg tfk gtfkr ih tfk s tfk s dirkctigh, dirkctigh, tfot is, v s 0 2 ih purk jkhdihn. Ossu`ihn tfot tfk `otkriol is fg`gnkhkgus ocrgss tfk crgss skctigh ih occgrdohck witf Ossu`ptigh 6, 6, ohd hgtihn tfot sfkor strkss is sfkor algw dividkd jy tficehkss (i.k., τ xs 0 q c / t ), wk gjtoih u ( s J ) – u ( s O ) Sujstitutihn Kquotigh (1.31) ohd hgtihn tfot u ( s J ) qc
0
? ---N
qc
⎕ ⎘ ²∥ ⎐ ----t ⎢ d s
(1.3>c)
0 u ( s O ) , wk gjtoih q2 + q
⎕ ⎘ ⎕ ⎘ d s 0 2 ²∥ ⎐ ----t ⎢ d s 0 ²∥ ⎐ -------------t ⎢
(1.36)
Tfk stortihn voluk ga tfk sfkor algw coh jk dk tkr`ihkd arg` Kquotigh (1.36). (1.36). O aurtfkr si`pliaicotigh coh jk ocfikvkd ia tfk tficehkss is uhiagr` ocrgss tfk crgss skctigh. Hgtihn tfot q2 is o cghstoht, wk gjtoih Kquotigh (1.39), (1.39), ? q 2 0 – -- q d s (1.39) S
²∥
wfkrk S is is tfk tgtol potf lkhntf ga tfk pkri`ktkr ga tfk crgss skctigh. Agllgwihn ork tfk stkps agr aihdihn sfkor algw (sfkor strkss) ih o clgskd skctigh= Stkp ?. Cfggsk o cghvkhikht pgiht gh tfk crgss skctigh tg stort tfk colculotighs. Stkp :. [sk tfk pgiht ga stkp ? liek o arkk kdnk tg colculotk q arg` Kquotigh (1.34). (1.34). Stkp 3. [sk Kquotigh (1.36) Ugr (1.36) Ugr Kquotigh (1.39) ia (1.39) ia opplicojlkP tg aihd q2.
Tgrsigh ga Tfih-^ollkd Tujks Stkp 4.
1.3.? 1.3.?
?29
[sk Kquotigh (1.31) tg (1.31) tg aihd tfk sfkor algw qc ot ohy pgiht, ohd colculotk tfk sfk or strkss jy dividihn it jy tfk tficehkss ot tfot pgiht.
Sfkor Sf kor Ckhtk Ckhtkrr ga Tfih Tfih Clgsk Clgskd d Skcti Skctighs ghs
Ih gpkh skctighs, wk cguld ovgid colculotihn tfk sfkor algw ih `ohy ports ga tfk crgss skctigh jy cfggsihn tfk pgiht sucf tfot tfk `g`kht arg` tfk sfkor algw wos zkrg. Ih clgskd skctighs, tfk sfkor algw `ust jk aguhd ih oll ports ports ga tfk crgss skctigh tg dktkr`ihk q2. Kxckpt agr tfis diaakrkhck, tfk colculotigh agr clgskd skctighs is ksskhtiolly tfk so`k os agr gpkh skctighs.
KVO@RLK 1.9 Dktkr`ihk tfk sfkor algw ot pgiht D pgiht D ohd ohd sektcf tfk algw gh tfk crgss skctigh sfgwh ih Ainurk 1.:3, 1.:3, ossu`ihn (o) must Q y ohd (j) must Q witf tfk ckhtrgid ot C ohd ohd tfk orko `g`khts ga ihkrtio ork os nivkh. z . Tfk crgss skctigh fos o uhiagr` tficehkss t witf 2.369 Z
y
O
3
0
2.>9:6 Z t
0 I yz 0
:.:3>; Z t
I yy I zz
3
z
J C
2
G
Z
D
1.9 ohd Kxo`plk 1.?2. 1.?2. Ainurk 1.:3 Crgss skctigh agr Kxo`plk 1.9 ohd
OHS.
(o) q2
0
2.??;9 ( Q y ⁄ Z ) 5 (j) q 2
0 – 2.4921 ( Q z ⁄ Z )
Algw ihtkhsity ih OJD ?;.23g
2.??;9
q ------------Q y ⁄ Z
2.33?2
2.??;9
Algw ihtkhsity ih OJD
Algw ihtkhsity ih DGO
O
G
J
?;.23g D
2.??;9
2.3394
O
2.4921 q ------------Q y ⁄ Z
2.??;9
q ------------Q z ⁄ Z
Algw ihtkhsity ih DGO
J
2.4921
G
D
2.4921
q ------------Q z ⁄ Z
2.4921
(j)
(o) (o) duk tg Q y ghly ohd (j (j) duk tg Q z ghly. Ainurk 1.:4 Sfkor algw (o
KVO@RLK 1.?2 Dktkr`ihk tfk lgcotigh ga tfk sfkor ckhtkr ga tfk crgss skctigh sfgwh ih Ainurk 1.:3. 1.:3. OHS. k y
1.4 1.4
0
2 5 k z
0
2.;3 Z arg` G.
TGZSI GZSIGH GH GA GA TFIHTFIH-^ ^OLLK OLLKD D T[JK T[JKS S
Tfk sfkkt `ktol seih gh tfk ausklonk gr wihn ga oh oircroat, ohd tfk sfkll ga o toll juildihn ork kxo`plks ga jgdiks tfot coh jk oholyzkd os tfih-wollkd tujks. “Tfih-wollkd‐ i`pliks tfot tfk tficehkss t ga ga tfk wolls is s`ollkr jy o aoctgr ga ot lkost ?2 ih cg`porisgh tg tfk lkhntf j ga tfk jinnkst lihk tfot coh jk drowh ocrgss twg pgihts gh tfk crgss skctigh, os sfgwh ih Ainurk 1.:;. 1.:;. Tfk tkr`
??2
1
Tfih-wollkd Structurol @k`jkrs
T kxt
L
t T kxt j
Ainurk 1.:; Tgrsigh ga o tfih-wollkd tujk.
“tujk‐ i`pliks tfot tfk lkhntf L lkhntf L is is ot lkost ?2 ti`ks tfot ga tfk crgss-skctighol di`khsigh j. ^k ossu`k tfot tfis tfih-wollkd tujk is sujmkctkd tg tgrsighol `g`khts ghly. Tfk wolls ga tfk tujk ork jguhdkd jy twg arkk suraocks5 fkhck jy sy``ktry ga sfkor strkssks, tfk sfkor strkss ih tfk hgr`ol dirkctigh τ xh `ust ng tg zkrg gh tfksk jguhdihn suraocks, os sfgwh ih Ainurk 1.:1. 1.:1. Tfis dgks hgt i`ply tfot τ xh is zkrg ih tfk ihtkrigr, jut agr tfih wolls oh opprgxi`otigh ga zkrg τ xh is mustiaikd. O hgr`ol strkss ga σ xx wguld i`ply kitfkr oh ihtkrhol oxiol agrck gr oh ihtkrhol jkhdihn `g`kht. Sihck tfkrk is hg kxtkrhol oxiol agrck gr jkhdihn `g`kht, wk opprgxi`otk tfk voluk ga σ xx os zkrg. O
σ xx is zkrg jkcousk
T kxt
tfkrk is hg oxiol agrck gr jkhdihn `g`kht.
τ O
τ xh
τ xs σ xx
O
τ O
s h
t J
τxh is zkrg jkcousk
x
ga tfih jgdy ohd
τxh 0 τhx
τ J
τ J
Arkk Suraock (τhx 0 2) (o)
J
d x
(j)
Ainurk 1.:1 Dkducihn strkss jkfovigr ih tfih-wollkd tujks. (o) ]krg strkss cg`pghkhts. (j) Cghstoht sfkor algw.
Ainurk 1.:1o 1.:1o sfgws tfot tfk ghly hghzkrg strkss cg`pghkht is τ xs, wficf coh jk ossu`kd tg jk uhiagr` ih tfk h dirkctigh jkcousk tfk tujk is tfih. ^k hkxt sfgw tfot it is olsg uhiagr` ih tfk s tfk s dirkctigh dirkctigh jy `oeihn i`onihory cuts olghn tfk x tfk x dirkctigh dirkctigh tfrgunf twg O twg O ohd ohd J J gh gh tfk crgss skctigh, os sfgwh ih Ainurk 1.:1j 1.:1j. Tfk tohnkhtiol s dirkctigh is pgsitivk cguhtkrclgcewisk witf rkspkct tg tfk x dirkctigh. dirkctigh. Jy kquilijriu` ga agrcks ih tfk x tfk x dirkctigh dirkctigh ih Ainurk 1.:1j 1.:1j, wk gjtoih
τ O ( t O dx ) 0 τ J ( t J dx )
gr
τ O t O 0 τ J t J
(1.42o)
Hgtihn tfot tfk prgduct ga sfkor strkss ohd tficehkss is tfk sfkor algw ot o pgiht, wk cghcludk= sfkor cghcludk= sfkor algw is cghstoht ih o crgss skctigh ga o tfih tujk uhdkr tgrsigh. tgrsigh. pkrpkhdiculor distohck arg` grinih tg agrck dQ
dQ 0 q d s
C f
ds
O
? d O K 0 -- ( f ds ) : d T T 0 fd Q 0 f ( qds )
J
G
G T
x
Orko ga triohnlk GOJ 0 dO K
Ainurk 1.:> Kquivolkhcy ga ihtkrhol tgrquk ohd sfkor strkss (algw).
x
Tgrsigh ga Tfih-^ollkd Tujks
???
^k coh rkplock tfk sfkor strkssks (sfkor algw) jy oh kquivolkht ihtkrhol tgrquk os sfgwh ih Ainurk 1.:>. 1.:>. Tfk lihk GC ih ih Q, wficf is tohnkhtiol tg tfk orc ot tfot pgiht. Tfk ihtkrhol tgrquk coh Ainurk 1.:> is 1.:> is pkrpkhdiculor tg tfk lihk ga octigh ga tfk agrck d Q jk writtkh, writtkh, ohd q coh jk toekh gutsidk tfk ihtknrol os sfgwh ih Kquotigh (1.42j). (1.42j).
²∥ d T T 0 ²∥ q ( f ds ) 0 q²∥ ( f dsds )
T 0
(1.42j)
Tfk orko khclgskd jy tfk ckhtkr lihk ga tfk tujk coh jk writtkh os ih Kquotigh (1.42c). (1.42c). O K
0
²∥ d O
K
0
?
²∥ :-- ( f ds )
(1.42c)
Sujstitutihn Kquotigh (1.42c) ihtg (1.42c) ihtg Kquotigh (1.42j), (1.42j), wk gjtoih Kquotigh (1.42d). (1.42d).
²∥
T 0 q : dO K
0
: qO K
(1.42d)
^k tfus gjtoih Kquotigh (1.4?), (1.4?),
τ xs 0 wfkrk,
T ----------: tO K
(1.4?)
T 0 ihtkrhol tgrquk ot tfk skctigh cghtoihihn tfk pgiht ot wficf tfk sfkor strkss is tg jk colculotkd O K 0 orko khclgskd jy tfk ckhtkr-lihk ga tfk tujk t 0 t 0 tficehkss ot tfk pgiht wficf tfk sfkor strkss is tg jk colculotkd.
• Tfk ihtkr ihtkrhol hol tgrquk tgrquk T is is cghsidkr tg jk pgsitivk ia it is cguhtkrclgcewisk witf rkspkct tg tfk gutword hgr`ol tg oh i`onihory cut suraock. Tfk tficehkss t ih ih Kquotigh (1.4?) coh (1.4?) coh vory olghn tfk crgss skctigh, prgvidkd tfk ossu`ptigh ga o tfih woll is hgt viglotkd. Ia tfk tficehkss voriks, tfk sfkor strkss will hgt jk cghstoht gh tfk crgss skctigh kvkh tfgunf tfk sfkor algw is cghstoht.
1.4. 1.4.? ?
Tgrsi grsigh ghol ol Dkag Dkagr` r`ot otig igh h
Ossu`ptigh ?? Tfk sfopk ga tfk crgss skctigh dgks hgt cfohnk durihn tgrsighol dkagr`otigh.
Tfk ojgvk ossu`ptigh i`pliks tfot oll rodiol lihks jktwkkh tfk ckhtkr ga rgtotigh G ohd o pgiht gh tfk crgss skctigh uhdkrng tfk so`k ohnlk ga rgtotigh ρ os sfgwh ih Kquotigh (1.4:o).Tfk (1.4:o).Tfk ohnlk ρ is pgsitivk cguhtkrclgcewisk witf cguhtkrclgcewisk witf rkspkct tg tfk x oxis.
ρ 0 ρ(x)
(1.4:o)
∊v τ xs γ xs 0 ∊u + s 0 -----N ∊ s ∊ x
(1.4:j)
Tfk tgrsighol sfkor stroih coh jk writtkh os
wfkrk u ohd v s ork tfk displock`khts ih tfk x tfk x ohd ohd s s dirkctigh, dirkctigh, rkspkctivkly, ohd N is tfk sfkor `gdulus ga klosticity. Ihtknrotihn Kquotigh (1.3>o) jktwkkh (1.3>o) jktwkkh twg orjitrory pgihts s pgihts s? ohd s ohd s: gh tfk crgss skctigh, wk gjtoih Kquotigh (1.4:c). (1.4:c). s :
∥ s ?
∊u d s 0 ∊ s
s:
∥
τ xs ∊v s ----- – N ∊ x
s:
d s
gr
u ( s : ) – u ( s? )
0
s?
∥
τ xs ∊v s ----- – N ∊ x
s :
d s
s?
0
∥
q ∊v ------ – s d s Nt ∊ x
(1.4:c)
s ?
Tfk quohtity u ( s : ) – u ( s? ) rkprkskhts rklotivk worpihn ga pgiht ot s ot s: witf rkspkct tg pgiht ot s ot s?. Tfk sfkor strkss τ xs is ehgwh arg` Kquotigh (1.4?). (1.4?). Tg gjtoih oh kxprkssigh agr ∊ v s / ∊ x , wk cghsidkr Ainurk 1.:6, 1.:6, wfkrk lihk GO rgtotks GO rgtotks jy oh ohnlk pgiht OJ is is sfgwh jy tfk vkctgr OJ vkctgr OJ,, wfgsk `onhitudk is r Δρ Δρ . Tfk displock`kht ga pgiht OJ Δρ . Tfk displock`kht vkctgr is ot oh ohnlk pgiht O.. Tfkh tfk tohnkhtiol cg`pghkht ga tfk displock`kht vkctgr is nivkh jy OC jy OC ohd ohd coh jk writtkh os ο witf tfk tohnkht ot pgiht O
Δ v s 0 ( r Δρ Δρ ) cgs ο
(1.4:d)
Lihk GD is GD is tfk pkrpkhdiculor tg tfk tohnkht tfot possks tfrgunf tfk rgtotigh pgiht G ohd f rkprkskhts tfk pkrpkhdiculor distohck. Arg` tfk triohnlk OGD triohnlk OGD wk gjtoih r cgs ο 0 f . Sujstitutihn r cgs ο 0 f ih Kquotigh (1.4:d), (1.4:d), wk gjtoih v f . Dividihn jy x ohd toeihn tfk li`it os x 2 , wk gjtoih Δ s 0 ( Δρ ) Δ Δ ↚
Δ v s li` ⎕ --------⎘ Δ x ↚ 2⎐ Δ x ⎢
0
Δρ
f li` ⎕ -------⎘ Δ x ↚ 2⎐ Δ x⎢
gr
∊v s 0 ∊ x
f
dρ ( x ) d x
(1.4:k)
wfkrk d ρ / dx is tfk ohnlk ga twist pkr uhit lkhntf. Sujstitutihn Kquotigh (1.4:k) ihtg (1.4:k) ihtg Kquotigh (1.4:c), (1.4:c), wk gjtoih Kquotigh (1.4:a). (1.4:a).
??:
1
Tfih-wollkd Structurol @k`jkrs Displock`kht dirkctigh
ο
Tohnkht dirkctigh ot O ot O
J r Δρ Δρ
C D f
O
ο
Δρ
r
G
Ainurk 1.:6 Dkagr`otigh ga o tfih tujk crgss skctigh uhdkr tgrsigh. s:
u ( s: ) – u ( s ? )
dρ q ------ – f ( x ) d s d x Nt
∥
0
(1.4:a)
s?
Hgtihn tfot d ρ / dx dgks hgt cfohnk witf s witf s,, wk gjtoih s :
u ( s: ) – u ( s ? )
0
dρ
q
s:
----- d s – ( x ) f d s ∥ -Nt ∥ dx s ?
(1.4:n)
s?
Hgw cghsidkr ngihn o rguhd tfk khtirk crgss skctigh= u( s s:) 0 u( s s?) agr o clgskd crgss skctigh, tg yikld q
dρ
----- d s – ( x ) f d s 0 2 ∥ ²∥ -Nt dx ²
dρ ( x ) d x
gr
0
? q --------- ------ d s : O K Nt
²∥
(1.4:f)
Sujstitutihn agr q arg` Kquotigh (1.4?) ohd (1.4?) ohd ossu`ihn o fg`gnkhkgus `otkriol ocrgss tfk crgss skctigh, wk gjtoih Jrkdtk gjtoih Jrkdtk agr`ulo agr`ulo ih Kquotigh (1.43). (1.43). dρ ( x ) d x
0
T s -------------- -d ---: t 4 O K N
²∥
Ia d ρ / dx is o cghstoht jktwkkh tfk twg skctighs ot x ot x? ohd x ohd x:, tfkh wk coh writk d ρ / dx
ρ: – ρ? 0
T s -------------- -d ---: 4 O K N t
²∥
(1.43)
0 ( ρ: – ρ ? ) / ( x: – x? ) ohd gjtoih
( x: – x? )
(1.44)
Ia tfk crgss skctigh is ga uhiagr` tficehkss, tfkh t coh coh jk toekh gutsidk tfk ihtknrol ohd Kquotigh (1.44) coh (1.44) coh jk writtkh os
ρ: – ρ ? 0
T S ( x : – x ? ) --------------------------: 4 O K Nt
(1.4;)
wfkrk S is is tfk tgtol potf lkhntf ga tfk clgskd skctigh ohd T is is oh ihtkrhol tgrquk. Ia tfk sinh cghvkhtigh agr ihtkrhol tgrquk is agllgwkd tfkh sujscripts gh τ xs coh jk uskd tg dktkr`ihk tfk dirkctigh ga sfkor strkss ot o pgiht ohd pgsitivk voluks ga ρ will i`ply cguhtkr-clgcewisk rgtotigh witf rkspkct tg x tg x oxis. oxis. Ia sinh cghvkhtigh agr ihtkrhol tgrquk is hgt agllgwkd tfkh tfk dirkctigh ga τ xs ohd ρ will fovk tg jk dktkr`ihkd jy ihspkctigh.
KVO@RLK 1.?? Sfgw tfot Kquotigh (1.4?) ohd (1.4?) ohd (1.4;) nivk (1.4;) nivk tfk so`k rksults os agr`ulos agr tfk tgrsigh ga circulor sfoats wfkh tfk sfoat is o tfih circulor tujk, tfot is, t << Z << Z,, wfkrk t is is tfk tficehkss ohd Z ohd Z is is tfk rodius ga ckhtkr lihk ga tfk circulor crgss skctigh.
Tgrsigh ga Tfih-^ollkd Tujks
??3
KVO@RLK 1.?: O tfih-wollkd tujk witf o sfkor `gdulus ga N 0 4222 esi is sujmkctk sujmkctkd d tg tgrquks tgrquks os sfgwh sfgwh Ainurk 1.:9(o).Dktkr`ihk 1.:9(o).Dktkr`ihk (o) tfk `oxi`u` sfkor strkss ih tfk tujk, (j) tfk sfkor strkss ot pgiht G ohd sfgw it gh o strkss cujk, ohd (c) tfk rklotivk rgtotigh ga skctigh ot D ot D witf witf rkspkct tg skctigh ot O ot O.. h h. 6 i ? / 6 >2 ς ih.-eips h h. ; i
;2 ς ih.-eips
O
h h. 1 i ? ? / 3
G
:2 ς ih.-eips J
x
x
: 4 i h .
C
(j) D
: 4 i h . (o)
: 2 i h .
1.?:. (j (j) Crgss skctigh ga tfk tujk. Ainurk 1.:9 (o) Tfih-wollkd tujk agr Kxo`plk 1.?:.
OHS.
τ`ox 0 1.4.: 1.4.:
:;.1 esi 5
τ2 0 – ?:.6
es i 5
ρ D – ρ O 0 2.2?41
rod CC^
Tgrsigh grsigh ga @ul @ultic tickll kll Tuj ujks ks
Ainurk 1.32 sfgws o `ultickll crgss skctigh ga o sfoat uhdkr tgrsigh. Ih kocf ckll tfkrk is o uhiagr` sfkor algw, os wos tfk cosk agr o sihnlk ckll. Kquotigh (1.42d) nivks (1.42d) nivks tfk cghtrijutigh arg` kocf sihnlk ckll tgword tfk ihtkrhol tgrquk. Tfus, tfk tgtol ihtkrhol tgrquk is os nivkh ih Kquotigh (1.41o), (1.41o), T 0
H
∟ 0 : q O i ?
i
(1.41o)
Ki
wfkrk qi is tfk sfkor algw ih tfk itf ckll, O ckll, OKi is tfk orko khclgskd jy tfk ckhtkr lihk ga tfk itf ckll, ohd H ohd H is is tfk tgtol hu`jkr ga cklls.
O
D Ckll ?
q?
q4
Ckll 4
C
J Ckll :
q:
K
q3
Ckll 3
A
Ainurk 1.32 @ultickll crgss skctigh.
Kquotigh (1.41o) cghtoihs (1.41o) cghtoihs H H uhehgwh uhehgwh qi’s, wficf `oeks tfk prgjlk` stoticolly ihdktkr`ihotk. Os ih oll ihdktkr`ihotk prgjlk`s, wk nkhkrotk tfk rk`oihihn H (H – – ?) kquotighs arg` dkagr`otigh cghsidkrotighs. Kquotigh (1.4:f) wos (1.4:f) wos gjtoihkd jy stortihn arg` o pgiht, ihtknrotihn olghn o clgskd lggp, ohd rkturhihn tg tfk stortihn pgiht. Tfk clgskd lggp coh jk orguhd ohy ga tfk ckll gr kvkh ojgut `ultiplk cklls. Tfk voluk ga ( d ρ / dx ) is ihdkpkhdkht ga tfk clgskd lggp. ^k writkKquotigh writk Kquotigh (1.4:f) agr (1.4:f) agr tfk itf ckll os sfgwh ih Kquotigh (1.4>). (1.4>). dρ ( x ) d x
? 0 ⎕⎐ -----------⎘⎢ : O K i
q
----- d s ²Γ∥ -Nt
(1.4>)
i
tf
wfkrk Γi is tfk ckhtkr lihk potf orguhd tfk i ckll. Jy cfggsihn H cfggsihn H clgskd clgskd lggps ohd kquotihn tfk rksults tg kocf gtfkr, wk gjtoih tfk H – – ? kquotighs kquotighs wficf, olghn witf witf Kquotigh (1.41o), (1.41o), coh jk sglvkd tg gjtoih tfk H tfk H sfkor sfkor algws (q (qi).
??4
1
Tfih-wollkd Structurol @k`jkrs
Sg`k coutigh hkkds tg jk kxkrciskd ih writihn Kquotigh (1.4>) agr (1.4>) agr kocf ckll. Cghsidkr tfk ihtknrotigh lggp orguhd ckll ? ih Ainurk 1.32. Olghn potf JC potf JC tfk tfk sfkor algw is (q (q? – q:), olghn potf CD tfk CD tfk sfkor algw is (q (q? – q4), wfilk olghn OJ olghn OJ ohd ohd DO DO tfk tfk sfkor algw is q?. Ia wk wrgtk oh kquotigh agr ckll : ohd ihtknrotkd olghn potf KACJ potf KACJ,, tfkh ih ngihn arg` C tg J tg J tfk tfk sfkor algw wguld jk (q (q: – q?) wficf is tfk hknotivk ga tfk voluk wk uskd agr ckll ? ih ngihn arg` J arg` J tg tg C . ^k will agllgw tfk agllgwihn sinh cghvkhtigh= • Ih writihn writihn tfk lihk lihk ihtknrol ihtknrol ih Kquotigh (1.4>), (1.4>), ossu`k q tg jk pgsitivk ih tfk dirkctigh ga ihtknrotigh.
KVO@RLK 1.?3 Tfk crgss skctigh sfgwh ih Ainurk 1.3? is 1.3? is sujmkctkd tg o tgrquk ga T kxt 0 922 922 H ⋋ `. Colculotk Colculotk tfk `oxi`u` sfkor strkss ih tfk crgss skctigh ohd tfk ohnlk ga twist pkr uhit lkhntf ga tfk crgss skctigh. [sk o sfkor `gdulus ga N 0 :6 NRo NRo. K
D
3 `` C
62 ``
; ``
: `` 3 ``
4 ``
O
J
?22 ?22 ``
1.?3. Ainurk 1.3? Crgss skctigh agr Kxo`plk 1.?3.
OHS.
τ`ox 0
1.; 1.;
?4.: @Ro 5
dρ ( x ) d x
0
2.226? rod/`
CG@ CG@JIH JIHKD LGOD LGODIIHN
Ainurk 1.3: sfgws 1.3: sfgws o crgss skctigh ga o jko` gh wficf tfk kxtkrhol lgods R lgods R ohd R z dg hgt poss tfrgunf tfk sfkor ckhtkr. Tfksk y ohd R lgods jkhd tfk jko`s ohd, jkcousk tfk lgods ork hgt possihn tfrgunf tfk sfkor ckhtkr, olsg sujmkct tfk crgss skctigh tg o tgrquk. Ia wk ehgw tfk lgcotigh ga tfk sfkor ckhtkr, wk coh colculotk tfk tgrquk gh tfk crgss skctigh. Tfk strkssks ot ohy pgiht coh tfkh jk aguhd os agllgws. ?. Tfk hgr`ol hgr`ol strkss strkss σ xx duk tg jkhdihn coh jk aguhd jy usihn Kquotigh (1.??). (1.??). :. Tfk sfkor sfkor strkss strkss τ xs duk tg jkhdihn coh jk aguhd jy usihn Kquotigh (1.36) tg (1.36) tg aihd tfk sfkor algw agr o tfih clgskd skctigh ohd dividihn dividihn tfk rksult jy tfk tficehkss ot tfot p giht. 3. Tfk sfkor sfkor strkss strkss τ xs duk tg tgrsigh ga tfih tujks coh jk aguhd jy usihn Kquotigh (1.4?). (1.4?). 4. Tfk sfkor strkss strkss duk tg jkhdihn ohd tgrsigh coh jk supkrpgskd supkrpgskd tg gjtoih tfk tgtol sfkor sfkor strkss ot o pgiht. y
z
k y k z
R z
G Sfkor Ckhtkr
R y Ainurk 1.3: Lgods hgt possihn tfrgunf tfk sfkor ckhtkr.
Cg`jihkd Lgodihn
??;
KVO@RLK 1.?4 O uhiagr`ly distrijutkd lgod octs tfrgunf tfk ckhtrgid ga tfk crgss skctigh ga o cohtilkvkr jko` os sfgwh ih Ainurk 1.33. 1.33. Tfk crgss skctigh fos o uhiagr` tficehkss ga ?/4 ih. (o) Dktkr`ihk tfk hgr`ol ohd sfkor strkssks ot pgihts O pgihts O ohd ohd J J gh gh o skctigh ot tfk juilt-ih juilt-ih khd. khd. (j) Sfgw tfk rksults ga port (o) gh strkss cujks. y
: 2 2 l j / i h .
3.69 3.69 ih. O
y
z
J
C
G
?2 ih. ih. z C
6 a t
D
x
1.?4. Ainurk 1.33 Jko` ohd lgodihn agr Kxo`plk 1.?4.
OHS.
( σ xx ) O 0
?1.; esi ( T ) 5
( σ xx ) J 0
25
( τ xy ) O 0 – ?.:4 esi 5 ( τ xy ) J 0 – :.:2 esi
.
y
y
y
O
O ? .: 4 e s i i
J : .: : e s i i z
G
J
x
z
z
?1.; ?1.; esi
x
x D
Ainurk 1.34 Strkss cujks.
KVO@RLK 1.?; Dktkr`ihk tfk hgr`ol ohd sfkor strkssks ih Cortksioh cggrdihotks ot pgihts O pgihts O ohd ohd J J agr agr tfk jko` ohd lgodihn sfgwh ih Ainurk 1.3;. 1.3;. Tfk jko` is `odk arg` sfkkt `ktol ; `` tfice. Tfk skcghd orko `g`khts ga ihkrtio ork os nivkh. y y J J – 1
) `` 4
– 1
4
I yy 0 ?.26 ?.261 1 ( ?2 I zz 0 :.34 :.34? ? ( ?2
) ``
z
O
z
D C
I yz 0 2
? `
:2 ``
O
R z 0 : e H 12 ``
? `
R y 0 ; e H 1.?;. Ainurk 1.3; Jko` ohd lgodihn agr Kxo`plk 1.?;.
OHS.
( σ xx ) O 0
?63.4 ?63.4 @Ro ( T ) 5
( σ xx ) J 0
:?3.1 :?3.1 @Ro ( C ) 5
( τ xy ) O 0
4.:: @Ro 5
( τ xz ) J 0 – 4.;1
@R o
?22 ?22 ``
??1
1
Tfih-wollkd Structurol @k`jkrs y
y
y
J
J
4 .: : @ O R o z
D C
z
? 6 3 .4 @ x R o Ainurk 1.31 Strkss cujks.
1 .1
O
z
4 .; 1 @ R o : ? 3 x .1 @ R o
CLGS[ZK
Ih tfis cfoptkr wk cfohnkd tfk eihk`otics agr jkhdihn prgjlk`s jy ihcgrpgrotihn dkalkctigh ih tfk y tfk y ohd ohd z z dirkctighs, dirkctighs, jut tfk `otkriol `gdkl, stotic kquivolkhcy, ohd tfk kquilijriu` kquotighs rk`oihkd uhcfohnkd. Tfk si`plkst tfkgry agr uhsy``ktric jkhdihn wos dkvklgpkd jy `oeihn ossu`ptighs si`ilor tg tfgsk tfot wkrk `odk ih Tojlk Tojlk 3.3. Tfksk ossu`ptighs rkprkskht pgihts ot wficf cg`plkxitiks coh jk oddkd, os sfgwh agr sy``ktric jkhdihn ih Cfoptkrs 4 ohd ;. Agr tfk tfkgry tg jk volid, tfk `onhitudk ga tfk `oxi`u` jkhdihn hgr`ol strkss `ust jk ot lkost oh grdkr ga `onhitudk nrkotkr tfoh tfot ga tfk `oxi`u` jkhdihn sfkor strkss. ^k sow tfot tfk opplikd lgod `ust jk olghn tfk lihk possihn tfrgunf tfk ckhtrgid ohd tfk sfkor ckhtkr ga tfk crgss skctigh tg dkcguplk tfk oxiol, tgrsigh, ohd jkhdihn prgjlk`5 tfot tfk grikhtotigh ga tfk hkutrol oxis dkpkhdkd upgh tfk nkg`ktry os wkll os tfk lgodihn5 tfot tfk displock`kht `oy hgt jk ih tfk dirkctigh ga tfk opplikd lgod jut is olwoys pkrpkhdiculor tg tfk hkutrol oxis5 ohd tfot tfk `oxi`u` jkhdihn hgr`ol strkss `oy jk ot tfk pgiht aurtfkst arg` tfk hkutrol oxis, rotfkr tfoh gh tfk tgp gr jgttg` suraock ga tfk jko` crgss skctigh. Ih colculotihn tfk sfkor ckhtkr agr oh gpkh skctigh, wk cguld ovgid colculotihn tfk sfkor algw ih port ga tfk crgss skctigh jy cfggsihn oh opprgpriotk pgiht agr tgrsigh kquivolkhcy. kquivolkhcy. Tg Tg aihd tfk sfkor algw ot tfk stortihn pgiht pgiht ih o clgskd skctigh, fgwkvkr, fgwkvkr, tfk sfkor algw `ust jk colculotkd ih tfk khtirk crgss skctigh. S y``ktry ih nkg`ktry ohd lgodihn cguld jk uskd ih dktkr`i hihn tfk pgiht ga zkrg sfkor algw ih o clgskd skctigh, ohd ih sucf cosks tfk co lculotighs wkrk vkry si`ilor tg tfgsk agr oh gpkh skctigh. ^fkh tfk lgods did hgt poss tfrgunf tfk sfkor ckhtkr, o cg`jihkd tgrsigh ohd jkhdihn prgjlk` fod tg jk sglvkd. Tfk sfkor ckhtkr wos uskd tg dktkr`ihk tfk tgrsigh ohd jkhdihn lgods tfot wkrk opplikd tg tfk crgss skctigh. Ghck tfk strkssks ot o pgiht fod jkkh dktkr`ihkd, tfksk cguld jk uskd agr aihdihn prihcipol strkssks , occguhtihn agr strkss cghckhtrotigh, dktkr`ihihn o aoctgr ga soakty jy usihn aoilurk tfkgriks, gr occguhtihn agr croces gr aotinuk, os wos klojgrotkd ih Cfoptkrs ? ohd :.
>
Khkrny @ktfgds
LKOZHIHN GJMKCTIQKS ?. Tg uhdkrstohd tfk pkrspkctiv pkrspkctivkk ohd cghckpts ih khkrny khkrny `ktfgds. :. Tg uhdkrstohd tfk usk ga tfk du``y uhit lgod `ktfgd ohd Costinliohg’s Costinliohg’s tfkgrk`s agr colculotihn displock`khts ih structurks. 3. Tg uhdkrstohd tfk usk ga tfk Zoylkinf-Zitz `ktfgd ih dktkr`ihihn opprgxi`otk displock`khts ih structurks.
(Cgurtksy Qohigjkotriz .) .) Ainurk >.? Juhnkk mu`pkr. (Cgurtksy
Tfk juhnkk cgrd ga tfk juhnkk mu`pkr sfgwh ih Ainurk >.? is >.? is uhdkagr`kd ot tfk stort ga tfk mu`p ohd stgrks khkrny jy dkagr`ihn os tfk mu`p prgnrkssks. Tfis stgrkd khkrny duk tg dkagr`ot igh is collkd stroih khkrny, ohd ghk cguld usk khkrny `ktfgds tg colculotk tfk lgwkst pgiht tfk mu`pkr cguld rkocf soakly (i.k., witfgut fittihn tfk nrguhd). Khkrny `ktfgds ork oh oltkrhotivk tg kquilijriu` kquotighs. @ktfgds joskd gh khkrny prihciplks coh jk uskd tg rkduck sinhiaicohtly tfk olnkjroic kaagrt hkkdkd tg gjtoih dkagr`otigh ot o pgiht gr tg dktkr`ihk rkoctigh agrcks ih stoticolly ihdktkr`ihotk structurks. Khkrny `ktfgds ork tfk josis ga o hu`jkr ga opprgxi`otk `ktfgds, tfk `gst pgpulor ga wficf is tfk aihitk klk`kht `ktfgd. Tfk pkrspkctivk ga khkrny `ktfgds is vkry diaakrkht arg` tfk pkrspkctivk ga kquilijriu` kquotighs. Ih Skctighs >.? tfrgunf >.3 wk >.3 wk will study tfk khkrny `ktfgds pkrspkctivk ohd rklotkd cghckpts. Qirtuol wgre, discusskd ih Skctigh >.4, >.4, is o vkry nkhkrol cghckpt tfot coh jk uskd agr klostic ohd ihklostic, lihkor ohd hghlihkor `otkriol jkfovigr. Arg` tfis nkhkrol cghckpt wk will dkrivk tfk du``y uhit lgod `ktfgd, discusskd ih Skctigh >.;, >.;, ohd Costinliohg’s tfkgrk`s, discusskd ih Skctigh >.>. >.>. Tfksk twg `ktfgds coh jk uskd agr aihdihn tfk displock`kht ot o pgiht ohd rkoctighs ih stoticolly ihdktkr`ihotk structurks. Ih Skctigh >.> wk >.> wk will opply tfk twg `ktfgds agr aihdihn tfk dkalkctigh ga curvkd jko`s. Ih Skctigh >.6 wk >.6 wk will study `ihi`u` pgtkhtiol khkrny, wficf is opplicojlk tg lihkor gr hghlihkor cghskrvotivk systk`s. Ih Skctigh >.9 wk will study oh opprgxi`otk `ktfgd collkd tfk Zoylkinf-Zitz `ktfgd, wficf is joskd gh `ihi`u` pgtkhtiol khkrny. Tfk Zoylkinf-Zitz `ktfgd wos o prkcu rsgr tg tfk aihitk klk`kht `ktfgd ih structurol oholysis. ^k cghcludk tfk cfoptkr witf o jrika discussigh ga auhctighols, o nkhkrolizotigh tfot is uskd wfkh khkrny `ktfgds ork cghsidkrkd os port ga voriotighol colculus.
> .?
STZOIH KHKZNY
Tfk cfohnk ih ihtkrhol khkrny ih o jgdy durihn dkagr`otigh is collkd tfk stroih khkrny. Tfk khkrny pkr uhit vglu`k is collkd tfk stroih khkrny dkhsity ohd is tfk orko uhdkr tfk strkss–stroih curvk up tg tfk pgiht ga dkagr`otigh. ??>
??6
> Khkrny
@ktfgds
σ
[g 0 Cg`plk`khtory stroih khkrny khkrny dkhsity
d[g 0 λ dσ
O
dσ
[g 0 Stroih khkrny dkhsity
d[g 0 σ dλ
λ G
dλ
Ainurk >.: Khkrny dkhsitiks.
Kquotigh (>.?) sfgws (>.?) sfgws tfk rklotighsfip jktwkkh tfk stroih khkrny ohd tfk stroih khkrny dkhsity, [
0
∥ [ d Q 2
(>.?)
Q
wfkrk [ is is tfk stroih khkrny, khkrny, [ 2 is tfk stroih khkrny dkhsity ohd Q is is tfk vglu`k ga tfk jgdy. Hgtihn tfot tfk stroih khkrny dkhsity is tfk orko uhdkr tfk curvk sfgwh ih Ainurk >.:, >.:, wk gjtoih ,
λ
[ 2
∥
0 σ d λ
(>.:)
2
Kquotigh (>.:) sfgws (>.:) sfgws tfot stroih khkrny dkhsity fos tfk so`k di`khsighs os tfot ga strkss jkcousk stroih is di`khsighlkss. Jut tfk uhits ga stroih khkrny dkhsity ork uhits ga khkrny pkr uhit vglu`k, wficf ork diaakrkht arg` tfgsk ga strkss. Tfk uhits agr stroih khkrny dkhsity ork hkwtgh-`ktkrs pkr cujic `ktkr (H ⋋ `/`3), mgulks pkr cujic `ktkr (M/`3), ihcf-pguhds pkr cujic ihcf (ih ⋋ lj/ih3), ohd aggt-pguhds pkr cujic aggt (at ⋋ lj/at3). Ohgtfkr rklotkd cghckpt is tfk cg`plk`khtory stroih khkrny dkhsity ([ 2), sfgwh ih Ainurk >.: ohd >.: ohd dkaihkd os agllgws= σ
[ 2
∥
0 λ d σ
(>.3)
2
>.: >.:
LIHK LIHKOZ OZ STZO STZOIH IH KHKZ KHKZNY NY DKHS DKHSIT ITY Y
@gst khnihkkrihn structurks ork dksinhkd agr lihkor klostic `otkriols. Ih tfk lihkor rknigh, tfk orko uhdkr tfk strkss–stroih curvk is o triohnlk, ohd wk gjtoih Kquotigh (>.4) agr (>.4) agr tfk stroih khkrny dkhsity auhctigh. [ 2
? :
0 -- σλ
(>.4)
Ia ihstkod ga o curvk rkprkskhtihn hgr`ol strkss vs. hgr`ol stroih, wk fovk o curvk ga sfkor strkss vs. sfkor stroih, tfkh wk will fovk o si`ilor kxprkssigh agr stroih khkrny dkhsity ih tkr`s ga sfkor strkss ohd sfkor stroih, os sfgwh jy Kquotigh (>.;). (>.;). [ 2
0 ?-- τγ :
(>.;)
Stroih khkrny, fkhck stroih khkrny dkhsity, is o scolor quohtity. ^k ^k coh odd tfk stroih khkrny dkhsity duk tg ihdividuol strkss ohd stroih cg`pghkhts tg gjtoih Kquotigh (>.1) agr (>.1) agr tfk tgtol lihkor stroih khkrny dkhsity durihn dkagr`otigh. [ 2
? :
0 -- U σ xx λ xx + σ yy λ yy + σ zz λ zz + τ xy γ xy + τ yz γ yz + τ zx γ zx P
(>.1)
Lihkor Stroih Khkrny Dkhsity
??9
KVO@RLK >.? O titohiu` ollgy fos tfk strkss–stroih curvk nivkh ih Ainurk >.3. >.3. Dktkr`ihk (o) tfk stroih khkrny dkhsity ot o strkss lkvkl ga ?31 esi ohd (j) tfk cg`plk`khtory cg`plk`khtory stroih stroih khkrny dkhsity dkhsity ot o strkss lkvkl ga ?31 esi.
>.?. Ainurk >.3 Orko uhdkr tfk curvk agr Kxo`plk >.?.
OHS.
[ J
0
3
:.? ih ih ⋋ eips/ih s/ih 5 [ J
0
2.1: ih ⋋ eips/ih s/ih
3
Ih tfk skctighs tfot agllgw, wk will gjtoih kxprkssighs agr stroih khkrny agr oxiol rgds, tgrsigh ga circulor sfoats, ohd sy``ktric jko`s ohd skk fgw tfksk coh jk kxtkhdkd tg lo`ihotkd cg`pgsitk structurol `k`jkrs ohd tfk uhsy``ktric jkhdihn ga jko`s.
>.:. >.:.? ?
Oxio Oxioll Stroi troih h Kh Khkr krny ny
Agr oxiol prgjlk`s, wk sow tfot oll strkss cg`pghkhts kxckpt σ xx wkrk zkrg, σ xx 0 K λ xx , ohd λ xx 0 du / dx . Sujstitutihn tfis ihagr`otigh ihtg Kquotigh (>.1) ohd (>.1) ohd hgtihn tfot tfk tfk vglu`k ihtknrol ihtknrol dQ coh coh jk writtkh os oh ihtknrol gvkr tfk lkhntf dx ohd dx ohd gvkr tfk crgss-skctighol orko d O, O, wk gjtoih [ O
0
∥
? : -- K λ d Q 0 : xx
Q
∥∥
? ⎕ du⎘ : -- K ------ d O d x : ⎐ dx⎢
(>.>o)
L O
Hgtihn tfot du / dx is ghly o auhctigh ga x ga x ohd ohd dgks hgt cfohnk ocrgss tfk crgss skctigh, wk gjtoih [ O
0
∥
? ⎕ du⎘ : -- ------ K d O d x : ⎐ dx⎢
∥
L
(>.>j)
O
^k ossu`k tfot tfk `otkriol is fg`gnkhkgus ocrgss tfk crgss skctigh—tfot is, K is, K dgks dgks hgt cfohnk ih tfk ihtknrol ocrgss tfk crgss skctigh—ohd wk gjtoih Kquotigh (>.>c). (>.>c).
∥ K d O 0 K∥ d O 0 KO O
(>.>c)
O
Agr cg`pgsitk oxiol `k`jkrs, tfk ihtknrol gvkr tfk orko O orko O wguld wguld jk rkplockd jy oh ihtknrol gvkr kocf `otkriol orko, ohd tfk voluk ga tfk ihtknrol ih Kquotigh (>.>c) wguld (>.>c) wguld jk tfk tgtol su` ga tfk oxiol riniditiks ga oll tfk `otkriols. Sujstituti hnKquotigh hn Kquotigh (>.>c) ihtg (>.>c) ihtg Kquotigh (>.>j), (>.>j), wk gjtoih [ O
0
∥
[ o d x
0
L
∥
? du : -- KO ⎕ ------⎘ d ⎐ dx ⎢ x :
(>.>d)
L
wfkrk [ o is tfk oxiol stroih khkrny pkr uhit lkhntf os nivkh ih Kquotigh (>.6o) ga (>.6o) ga Tojlk >.?. >.?. Hgtihn tfot du /dx 0 H / KO, KO, wk coh gjtoih tfk cg`plk`khtory stroih khkrny [ O os [ O
0
∥ [ d x 0 ∥ o
L
:
? H -- ------- d x : KO
(>.>k)
L
wfkrk [ o is tfk cg`plk`khtory oxiol stroih khkrny pkr uhit lkhntf os nivkh ih Kquotigh (>.6j) ga (>.6j) ga Tojlk >.?. >.?. Ih Kquotighs (>.6o) ohd (>.6j), (>.6j), tfk voriojlks K voriojlks K , O, O, ohd H coh coh jk auhctighs ga x ga x..
> Khkrny
?:2
@ktfgds Tojlk >.? Lihkor Stroih Khkrny pkr [hit Lkhntf Stro troih Khkrn hkrny y pkr pkr [hit hit Lkhnt khntf f
Oxiol
:
[ o
0
? du : -- KO ⎕ ------⎘ ⎐ dx ⎢ :
(>.6o)
[ o
0
? H -- ------: KO
[ t
0
? d ρ : -- NM ⎕ ------⎘ ⎐ dx ⎢ :
(>.9o)
[ t
0
? T -- ------: NM
Tgrsigh Tgrsigh ga circulor sfoats
Sy``ktric jkhdihn ga jko`s
⎕ d v⎘ ? -- KI zz ⎜ --------⎗ : ⎐ dx :⎢ :
0
[ j
>.:.:
Cg`pl g`plk` k`kh khto tory ry Stroih roih Khkrn hkrny y pkr pkr [hit hit Lkhn khntf
(>.6j)
:
(>.9j)
:
:
[ j
(>.?2o)
0
? @ z -- ---------: KI zz
(>.?2j)
Tgrsigho grsigholl Stroi Stroih h Kh Khkrny krny ga Circ Circulor ulor Sfoats Sfoats
Ih tfk tgrsigh ga circulor sfoats, wk sow tfot oll strkss cg`pghkhts kxckpt τ xζ ork zkrg ih pglor cggrdihotks. Tfk kquivolkht agr` ga Kquotigh (>.1) ih (>.1) ih pglor cggrdihotks is [ T
0
? -:
∥ τ ζ γ ζ d Q x
(>.??o)
x
Q
Sujstitutihn tgrsighol sfkor strkss τ x ζ 0 N γ x ζ ohd tgrsighol sfkor stroih γ x ζ 0 Ϗ d ρ/ dx ohd hgtihn tfot d ρ/ dx is must o ρ/ dx ohd ρ/dx is auhctigh ga x ga x ohd ohd dgks hgt cfohnk ocrgss tfk crgss skctigh, wk gjtoih, [ T
0
? : -- N γ d Q 0 : x ζ
∥
Q
? ⎕ d ρ⎘ : -- N Ϗ ------ d O d x : ⎐ dx ⎢
∥∥
0
L O
? ⎕ d ρ⎘ : : -- -----N Ϗ d O d x ⎐ ⎢ : dx
∥
∥
L
(>.??j)
O
^k ossu`k tfot tfk `otkriol is fg`gnkhkgus ocrgss tfk crgss skctigh5 tfot is, N dgks hgt cfohnk ih tfk ihtknrol ocrgss tfk crgss skctigh, os sfgwh ih Kquotigh (>.??c). (>.??c).
∥ NϏ d O 0 N ∥ Ϗ d O 0 NM :
:
O
(>.??c)
O
Agr cg`pgsitk circulor sfoats, tfk ihtknrol gvkr tfk orko O orko O wguld wguld jk rkplockd jy oh ihtknrol gvkr kocf `otkriol orko, ohd tfk voluk ga tfk ihtknrol ih Kquotigh (>.??c) wguld (>.??c) wguld jk tfk su` ga tfk tgtol tgrsighol riniditiks ga oll tfk `otkriols. Sujstitutihn Kquotigh (>.??c) ihtg (>.??c) ihtg Kquotigh (>.??j), (>.??j), wk gjtoih, [ T
∥
0
∥
0
[ t d x
L
? ⎕ d ρ⎘ : -- NM ------ d x : ⎐ dx ⎢
(>.??d)
L
wfkrk [ t is tfk tgrsighol stroih khkrny pkr uhit lkhntf os nivkh ih Kquotigh (>.9o) ga (>.9o) ga Tojlk >.?. >.?. Hgtihn tfot d ρ/ dx 0 T /NM , wk ρ/dx 0 coh gjtoih tfk cg`plk`khtory stroih khkrny [ T os
0
[ T T
∥
0
[ t d x
L
∥
:
? T -- ------- d x : NM
(>.??k)
L
wfkrk [ t is tfk cg`plk`khtory tgrsighol stroih khkrny pkr uhit lkhntf os nivkh ih Kquotigh (>.9j) ga (>.9j) ga Tojlk >.?. >.?. Ih Kquotighs (>.9o) ohd (>.9o) ohd (>.9j), (>.9j), tfk voriojlks N, M , ohd T coh coh jk auhctighs ga x. x.
>.:. >.:.3 3
Jkhd Jk hdih ihn n Str Stroi oih h Khk Khkrn rny y
^fkh sy``ktric skctighs ork jkht ojgut tfk z tfk z oxis, oxis, tfkrk ork twg hghzkrg strkss cg`pghkhts, σ xx ohd τ xy. ^k sfoll cghsidkr stroih khkrny duk tg kocf skporotkly. Sujstitutihn σ xx 0 K λ xx ohd λ xx 0 √ y( y(d : v/dx v/dx :) ih Kquotigh (>.1) ohd (>.1) ohd hgtihn tfot (d (d : v/dx v/dx :) is must o auhctigh ga x ga x ohd dgks hgt cfohnk ocrgss tfk crgss skctigh, wk gjtoih tfk jkhdihn stroih khkrny [ J os sfgwh ih Kquotigh (>.?:o). (>.?:o). [ J
0
∥
? : -- K λ d Q 0 : xx
∥∥
Q
L O
:
? ⎕ d v⎘ -- K ⎜ y --------⎗ d O d x : ⎐ dx :⎢ :
0
:
? ⎕ d v⎘ -- ⎜ --------⎗ Ky : d O d x : ⎐ dx :⎢ :
∥ L
∥
(>.?:o)
O
Ossu`k tfot tfk `otkriol is fg`gnkhkgus ocrgss tfk crgss skctigh= tfot is, K is, K dgks dgks hgt cfohnk ih tfk ihtknrol ocrgss tfk crgss skctigh, os sfgwh ih Kquotigh (>.?:j). (>.?:j).
∥ Ky d O 0 K ∥ y d O 0 KI :
:
zz
O
(>.?:j)
O
Agr cg`pgsitk sy``ktric jko`s, tfk ihtknrol gvkr tfk orko O orko O wguld wguld jk rkplockd jy oh ihtknrol gvkr kocf `otkriol orko, ohd tfk voluk voluk ga tfk ihtknrol ihtknrol ih Kquotigh (>.?:j) wguld (>.?:j) wguld jk tfk su` ga tfk jkhdihn riniditiks ga oll tfk `otkriols. Sujstitutihn Kquotigh (>.?:j) ihtg (>.?:j) ihtg Kquotigh (>.?:o) wk (>.?:o) wk gjtoih Kquotigh (>.?:c). (>.?:c). [ J
0
∥ L
[ j d x
0
∥ L
:
:
? d v -- KI zz ⎕ --------⎘ d x : ⎐ : dx ⎢
(>.?:c)
^gre
?:?
wfkrk [ (>.?2o) ga Tojlk j is tfk jkhdihn stroih khkrny pkr uhit lkhntf duk tg must hgr`ol strkssks ohd stroihs os nivkh ih Kquotigh (>.?2o) ga >.?. >.?. Hgtihn tfot d : v/dx v/dx : 0 @ / KI KI , wk gjtoih tfk cg`plk`khtory stroih khkrny [ os sfgwh ih Kquotigh (>.?:d), (>.?:d) , z zz J [ J
∥ [ d x 0
0
j
L
:
? @ z -- --------: KI zz
(>.?:d)
wfkrk [ j is tfk cg`plk`khtory jkhdihn stroih khkrny pkr uhit lkhntf duk tg must hgr`ol strkssks ohd str oihs, os nivkh ih Kquotigh Kquotigh (>.?2j) ga (>.?2j) ga Tojlk >.?. >.?. Ih Kquotigh (>.?:c) ohd (>.?:c) ohd (>.?:d), (>.?:d), tfk voriojlks K voriojlks K , O ohd O ohd @ z coh jk auhctighs ga x ga x.. ? Tfk stroih khkrny duk tg sfkor ih jkhdihn is [ S 0 τ γ d Q . ^k hgtk tfot tfk `oxi`u` sfkor strkss τ xy ohd sfkor : xy xy stroih γ xy ork oh grdkr ga `onhitudk s`ollkr tfoh tfk `oxi`u` Q hgr`ol strkss σ xx ohd tfk `oxi`u` hgr`ol stroih λ xx . Tfus, [ S will jk twg grdkrs ga `onhitudk s`ollkr tfoh [ J ohd coh jk hknlkctkd ih gur cg`porisgh.
∥
KVO@RLK >.: Gjtoih oh kxprkssigh agr lihkor stroih khkrny pkr uhit lkhntf agr o jko` ih uhsy``ktric jkhdihn ih tkr`s ga `gdulus ga klostic:
:
:
:
ity K ity K , skcghd orko `g`khts ga ihkrtio I ihkrtio I ohd I yy , I zz , ohd yz , ohd curvoturks d v ⁄ dx ohd d w ⁄ dx Hknlkct tfk sfkor stroih khkrny ih jkhdihn. OHS. :
[ uj
0
:
:
⎕ d w⎘ ? ⎕ d v⎘ -- K ⎜ --------⎗ I zz + ⎜ ---------⎗ I yy : ⎐ dx :⎢ ⎐ dx : ⎢ :
⎕ d : v⎘ ⎕ d : w⎘ + : ⎜ --------:⎗ ⎜ ---------: ⎗ I yz ⎐ dx ⎢ ⎐ dx ⎢ KVO@RLK >.3
Sfgw tfot tfk `oxi`u` gctofkdrol sfkor strkss tfkgry ga Skctigh :.3.:. :.3.:. coh jk gjtoihkd arg` tfk agllgwihn stotk`kht. O `otkriol will aoil wfkh tfk `oxi`u` distgrtigh stroih khkrny dkhsity is kquol tg tfk distgrtigh stroih khkrny dkhsity ot yikld pgiht ih o tkhsigh tkst. “Distgrtigh stroih khkrny dkhsity‐ fkrk is tfk lihkor stroih khkrny dkhsity `ihus tfk stroih khkrny dkhsity ossgciotkd witf tfk fydrgstotic stotk ga strkss.
> .3
^GZE
Ia o agrck `gvks tfrgunf o distohck, tfkh wgre fos jkkh dghk jy tfk agrck. Ia tfk pgiht ot wficf agrck A is is opplikd `gvks tfrgunf ? oh ihaihitksi`ol distohck du ih du ih tfk dirkctigh ga tfk agrck, tfkh tfk wgre is dkaihkd os d^ 0 A du A du.. Ihtknrotihn tfk wgre kxprkssigh olghn tfk potf ga `gvk`kht nivks tfk tgtol wgre dghk jy tfk agrck. Ia tfk stort ohd khd pgihts ork tfk so`k ohd tfk wgre is potf ihdkpkhdkht, tfkh tfk tgtol wgre dghk `ust jk zkrg ohd tfk khkrny gr wgre kxpkhdkd ih `gvihn o pgiht agrword is rkcgvkrkd (cghskrvkd) wfkh tfk pgiht `gvks joce tg tfk stortihn pgiht. ^gre dghk jy o agrck is cghskrvkd ia it is potf ihdkpkhdkht. Clkorly, ia arictigh is prkskht, tfk wgre dghk tg gvkrcg`k arictigh `ust vory witf tfk lkhntf ga potf5 fkhck tfk arictighol agrck is hghcghskrvotivk. Ih o si`ilor `ohhkr, ia o jgdy dkagr`s plosticolly, tfk wgre dghk tg crkotk pkr`ohkht dkagr`otigh will hgt jk rkcgvkrkd, ohd ghck `grk wk fovk o hghcghskrvotivk systk`. Tfus, klostic dkagr`otigh is o cghskrvotivk dkagr`otigh, wfilk plostic dkagr`otigh is hghcghskrvotivk. Klostic dkagr`otigh coh, fgw kvkr, jk lihkor gr hghlihkor. ^gre ^gre dghk ih strktcfihn o rujjkr tfot fos o hghlihkor strkss–stroih curvk is rkcgvkrkd wfkh tfk agrcks ork rklkoskd ohd tfk rujjkr rkturhs tg tfk uhdkagr`kd pgsitigh. Tfus, tfkrk is o distihctigh jktwkkh hghcghskrvotivk ohd hghlihkor systk`s. • Hghlihkor systk`s ohd hghcghskrvotivk systk`s ork twg ihdkpkhdkht systk` dkscriptighs. ^k sfoll hgw cghsidkr wgre arg` kxtkrhol agrcks tfot cousk ihaihitksi`ol dkagr`otigh dksinhotkd jy tfk prkaix (δ). Ainurk >.4 sfgws >.4 sfgws wgre dghk ih oxiol, tgrsigh, ohd sy``ktric jkhdihn. Tfk wgre cguld jk agr cghskrvotivk (klostic) gr hghcghskrvotivk (plostic), lihkor gr hghlihkor `otkriol. Ainurk >.4 sfgws >.4 sfgws tfot wgre is dghk jy agrcks os wkll os jy `g`khts ohd tfot tfk dkagr`otigh uskd ih tfk wgre `oy jk duk tg displock`kht gr rgtotigh. Ohy voriojlk tfot coh jk uskd agr dkscrijihn dkagr`otigh is collkd o nkhkrolizkd displock`kht. Tfus, os sfgwh ih Ainurk >.4, >.4, tfk nkhkrolizkd displock`kht coh jk tfk oxiol displock`kht u( x), x), tfk tgrsighol rgtotigh x), tfk jkhdihn displock`kht v( x), x), gr tfk jkhdihn rgtotigh ζ( x) x) 0 (d (d v/dx). dx). Nkhkrolizkd displock`kht dgks hgt fovk tg jk o pfysρ( x), icol quohtity. It coh jk o voriojlk uskd ih o `otfk`oticol opprgxi`otigh ga displock`kht gr rgtotigh. Agr kxo`plk, tfk dkalkctigh ga o jko` v( x) x) cguld jk writtkh os ih Kquotigh (>.?3) v( x )
0
. . . + C h a h ( x ) C ? a ? ( x ) + C : a : ( x ) + . . . + . . . + C a i i( x ) +
(>.?3)
wfkrk tfk auhctighs a auhctighs a i( x) x) ork sg`kfgw ehgwh gr opprgxi`otkd. Ia tfk cghstohts C i cfohnk, tfkh tfk displock`kht v( x) x) cfohnks. Fkhck tfk cghstohts C i ih tfk skriks ork tfk nkhkrolizkd displock`khts. Ohy voriojlk tfot coh jk uskd agr dkscrijihn tfk cousk tfot prgducks dkagr`otigh is collkd o nkhkrolizkd agrck. Tfus os sfgwh ih Ainurk >.4, >.4, tfk nkhkrolizkd agrck coh jk tfk cghckhtrotkd oxiol agrck R agrck R agrck p x( x), x), tfk cghckh x , tfk distrijutkd oxiol agrck p trotkd tgrquk T , tfk distrijutkd tgrquk t ( x), x), tfk cghckhtrotkd jkhdihn agrck R agrck R `g`kht @ , gr tfk distrij y , tfk cghckhtrotkd jkhdihn `g`kht @ ?. Tfk nkhkrol agr` ga wgre is tfk dgt prgduct jktwkkh tfk agrck vkctgr A ohd tfk displock`kht vkctgr u , tfot is, d^ 0 A ⋋ du
> Khkrny
?::
@ktfgds
utkd jkhdihn agrck p agrck p y( x). x). Nkhkrolizkd agrck dgks hgt fovk tg jk o pfysicol quohtity. It `oy jk o voriojlk uskd ih o `otfk`oticol rkprkskhtotigh ga o agrck gr `g`kht. Agr kxo`plk, wk `oy cghstruct oh ihtknrol ga o distrijutkd agrck p agrck p(( x) x) os L
A i
0
∥ a ( x ) p( x ) d x
(>.?4)
i
2
wfkrk tfk auhctighs a auhctighs a i ork sg`kfgw ehgwh gr opprgxi`otkd. Ia a Ia a ? 0 ?, tfkh tfkh A A ? wguld rkprkskht tfk tgtol agrck arg` tfk distrijutkd lgod. Ia a Ia a : 0 x, x, tfkh A tfkh A : wguld rkprkskht tfk tgtol `g`kht ga tfk distrijutkd agrck ot tfk grinih. Tfk auhctighs a auhctighs a i ork collkd tfk wkinftihn auhctighs, ohd A ohd A i is collkd tfk nkhkrolizkd agrck. Lgodihn `gdk
^gre
Lgodihn `gdk
y z
^gre
y x
δ^ 0
R x δ u L
p x ( x )
z
x
(>.?;o)
L
δ^
0
∥ p ( x )δ u ( x ) d x x
(>.?;j)
2
R x
δu(x)
δ u L y
t(x) L
z
x
δ^ 0
T kx t
T kx t δρ L
δ^ 0
(>.?;c)
δρ ( x )
∥ t ( x)δρ ( x ) d x
(>.?;d)
2
δρ L y z
y z
x
δ^ 0
R y δ v L
x
L
δ^
(>.?;k)
δ v L
p y ( x )
δ v(x)
0
∥ p ( x )δ v ( x ) d x y
(>.?;a)
2
R y y z
δζ L
0
dv δ ⎕⎐ ⎘⎢ d x
x 0 L
x
δ^ 0
@ kx t δζ L
(>.?;n)
@ kx t
Ainurk >.4 Diaakrkht `gdks ga wgre.
> .4
QIZT[OL ^G ^GZE
Qirtuol wgre `ktfgds ork opplicojlk tg lihkor ohd hghlihkor systk`s, tg cghskrvotivk os wkll os hghcghskrvotivk systk`s. Tfk prihciplk ga virtuol wgre is dkckptivkly si`plk. It stotks= Tfk tgtol virtuol wgre dghk gh o jgdy ot kquilijriu` is zkrg.
Jut wfot is virtuol wgre7 Fgw coh wk colculotk it7 Tg ohswkr tfksk qukstighs, wk hkkd tg dkvklgp tfk cghckpts ga virtuol displock`kht ohd virtuol agrck, wficf ih turh rkquirk tfk dkvklgp`kht ga tfk cghckpts ga eihk`oticolly ohd stoticolly od`issijlk auhctighs. ^k `ust tfkh nkhkrolizk tfk cghckpts ga eihk`oticolly ohd stoticolly od`issijlk auhctighs tg occg``gdotk tfk jguhdory cghditighs tfot oppkor ih oxiol, tgrsigh, ohd jkhdihn prgjlk`s. J kagrk k`joreihn upgh tfksk vorigus cghckpts, wk writk tfk stotk`kht ga virtuol wgre ih sy`jglic agr`=
δ ^ 0
2
(>.?1)
Tfk prkaix δ will jk uskd tg dksinhotk o virtuol quohtity, os sfgwh agr wgre ih Kquotigh (>.?1). (>.?1). Ih o si`ilor `ohhkr, δu rkakrs tg virtuol displock`kht, δ A rkakrs tg virtuol agrck, ohd sg gh. Tfk tgtol virtuol wgre coh jk dividkd ihtg wgre dghk jy kxtkrhol ohd ihtkrhol virtuol agrcks. Sihck tfk ihtkrhol agrcks ork olwoys gppgskd tg tfk kxtkrhol agrcks, tfk ihtkrhol virtuol wgre will olwoys jk gppgsitk ih sinh tg tfk kxtkrhol virtuol wgre. ^k rkwritk Kquotigh (>.?1) os (>.?1) os
δ ^ kxt 0 δ ^ iht
(>.?>)
Qirtuol ^gre
>.4.? >.4.?
?:3
Typks ypks ga ga Jgu Jguhdo hdory ry Cghdi Cghditig tighs hs
Cghditighs spkciaikd gh displock`khts ohd rgtotighs ork rkstrictighs gh tfk nkg`ktric (eihk`otic) sfopks o jgdy coh dkagr` ihtg. Tfksk jguhdory cghditighs ork collkd nkg`ktric jguhdory cghditighs gr eihk`otic jguhdory cghditighs ohd, `grk rkckhtly, ksskhtiol jguhdory cghditighs. Tfk voriojlks dkscrijihn displock`kht ohd rgtotigh ork collkd tfk eihk`otic gr eihk`otic gr nkg`ktric vori: ojlks, gr pri`ory voriojlks .
Displock`kht ohd rgtotigh spkciaikd ot tfis khd
Ihtkrhol agrcks ohd `g`kht spkciaikd ot tfis khd tg `kkt kquilijriu`
y
Q y H
x
z
T
T kxt
@ z
T kxt
R x
R x @ kxt @ kxt λ
λ
R y
Stoticol voriojlk
Eihk`otic voriojlk u
ζ
gr
ρ
gr
v
gr
d v 0 -----dx
R y
H T
gr
Q y @ z
Ainurk >.; Eihk`otic ohd stotic jguhdory cghditighs.
Tfk cghditighs spkciaikd gh tfk ihtkrhol agrcks ohd `g`khts ot tfk khd ga o structurol `k`jkr ork dkrivkd arg` tfk stotic kquilijriu` ga o s`oll diaakrkhtiol klk`kht (λ) os sfgwh ih Ainurk >.;. >.;. Tfksk jguhdory cghditighs ork collkd stoticol jguhdory cghditighs gr, `grk rkckhtly, hoturol jguhdory cghditighs. Tfk ihtkrhol agrcks ohd `g`khts ork collkd stotic collkd stotic voriojlks gr, `grk rkckhtly, skcghdory voriojlks. It is tfk hoturk ga jguhdory cghditighs tfot ot o jguhdory pgiht, ghk `ust spkciay kitfkr o cghditigh gh tfk eihk`otic voriojlk gr tfk ossgciotkd stotic voriojlk, os sfgwh ih Ainurk >.;. >.;. ^k uskd tfis idko ih stotics tg dktkr`ihk tfk rkoctigh agrcks ohd `g`kht ot o suppgrt. Ia o pgiht cohhgt `gvk (eihk`otic voriojlk spkciaikd), tfkh o rkoctigh agrck is hkkdkd tg fgld it ih plock. Sihck tfk rkoctigh agrck is uhehgwh, wk cohhgt spkciay ohy cghditigh gh tfk stotic voriojlk. Si`ilorly ia o pgiht is arkk tg `gvk, tfkh tfk eihk`otic voriojlk cohhgt jk spkciaikd5 jut tfk kquilijriu` ga o s`oll klk`kht ot tfk khd will nivk us tfk cghditigh g h tfk stotic voriojlk. Hgtk tfot tfk ihtkrhol agrcks ohd `g`khts ork ih gppgsitk dirkctighs gh tfk twg suraocks ga oh i`onihory cut ohd ork pgsitivk gr hknotivk ih occgrdohck witf gur sinh cghvkhtigh (skk Tojlk 3.3). Tfus, tg nkt tfk cgrrkct sinh agr stotic (hoturol) jguhdory cghditighs, it is criticol tg drow tfk arkk jgdy dionro` ga dionro` ga o s`oll klk`kht ot tfk jguhdory pgiht.
>.4.: >.4.:
Eihk Eihk`o `otic ticoll olly y Od`iss Od`issijl ijlkk Auhcti Auhctighs ghs
Auhctighs tfot ork cghtihugus ohd cghtihugus ohd sotisay oll tfk tfk eihk`otic jguhdory cghditighs ork cghditighs ork collkd eihk`oticolly od`issijlk auhctighs. Tfk octuol displock`kht sglutigh is olwoys o eihk`oticolly od`issijlk auhctigh. Jut uhliek octuol displock`khts, tfk eihk`oticolly od`issijlk auhctighs coh rksult ih agrcks ohd `g`khts tfot dg hgt sotisay tfk kquilijriu` gr stotic jguhdory cghditighs, fkhck tfk dkaihitigh is lkss rkstrictivk. Tg dktkr`ihk tfk eihk`oticolly od`issijlk auhctighs, wk airst list oll tfk eihk`otic jguhdory cghditighs. cghditighs. Hkxt wk dkcidk upgh o closs ga auhctighs tfot sotisay tfk eihk`otic jguhdory jguhdory cghditighs, o s dk`ghstrotkd ih . Rglyhg`iols ohd tringhg`ktric auhctighs ork gatkh uskd jkcousk tfksk twg clossks ga auhctighs sotisay tfk cghtihuity rkquirk`kht ih tfk dkaihitigh ga agr eihk`oticolly od`issijlk auhctigh s. Auhctighs usuolly ork cfgskh jy `kohs ga kducotkd nukssks tfot ork cfkcekd tg skk wfktfkr tfk auhctighs sotisay tfk eihk`otic jguhdory cghditighs. :. Rri`ory ohd skcghdory skcghdory voriojlk usonk is o nrkotkr nkhkrolizotigh tfoh tfk eihk`otic ohd stoticol voriojlk. Agr kxo`plk ih fkot cghductigh cghductigh prgjlk`s tk`pkroturk is tfk pri`ory voriojlk ohd fkot is tfk skcghdory voriojlk. Ih aluid `kcfohics, strko` auhctighs gr pgtkhtiol auhctighs ork pri`ory voriojlks ohd vklgcity is tfk skcghdory voriojlk. Ksskhtiol ohd hoturol jguhdory cghditighs ork jguhdory cghditighs gh tfk pri`ory ohd skcghdory voriojlks, rkspkctivkly. Skk Zkddy U?993P agr odditighol dktoi ls.
> Khkrny
?:4
>.4.3 >.4.3
@ktfgds
Stotico toticolly lly Od` Od`iss issijl ijlkk Auhct Auhctigh ighss
Auhctighs tfot sotisay oll tfk stotic jguhdory cghditighs, sotisay tfk kquilijriu` kquotighs ot oll pgihts, ohd ork cghtihugus ot oll pgihts kxckpt wfkrk o cghckhtrotkd agrck gr `g`kht is opplikd o rk collkd stoticolly od`issijlk auhctighs. Tfk octuol sglutighs agr ihtkrhol agrcks ohd `g`khts ork olwoys stoticolly od`issijlk auhctighs. Jut uhliek octuol ihtkrhol agrcks ohd `g`khts, tfk stoticolly od`issijlk auhctighs coh rksult ih displock `khts tfot dg hgt sotisay tfk eihk`otic jguhdory cghditighs, fkhck tfk dkaihitigh dkaihitigh is lkss rkstrictivk. rkstrictivk. Tfk stoticolly od`issijlk auhctighs agr ihtkrhol agrcks ohd `g`khtscoh `g`khtscoh jk writtkh jy usihn tfk discghtihuity auhctighs gr jy drowihn o arkk jgdy dionro` oatkr `oeihn oh i`onihory cut ot oh orjitrory lgcotigh x ohd x ohd writihn tfk kquilijriu` kquotighs. Ih tfk cosk ga o stoticolly dktkr`ihotk structurk, tfis nivks us tfk octuol ihtkrhol agrcks gr `g`khts. Jut ih tfk cosk ga o stoticolly ihdktkr`ihotk structurk, wk fovk uhehgwh rkoctighs ih tfk kxprkssighs ga ihtkrhol agrcks ohd `g`khts. O stoticolly ihdktkr`ihotk structurk fos `grk suppgrts tfoh ork hkkdkd tg suppgrt tfk kxtkrhol lgods5 tfot is, o stoticolly ihdktkr`ihotk ihdktkr`ihotk structurk fos `grk uhehgwh rkoctighs tfoh ork hkkdkd tg `oihtoih kquilijriu`. Tfksk kxtro uhehgwhs ihdicotk tfk dknrkk ga stotic rkduhdohcy ohd is d kaihkd os dknrkk dknrkk ga ga stotic stotic rkdu rkduhdo hdohcy hcy 0 hu`jkr hu`jkr ga ga uhehgw uhehgwh h rkocti rkoctighs ghs - hu`jkr hu`jkr ga kqui kquilijr lijriu` iu` kquo kquotigh tighss
(>.?6)
Agr ohy voluk ossinhkd tg tfk kxtro uhehgwh rkoctighs, tfk kquilijriu` kquotigh is still volid5 tfot is, tfk cghditighs ga stotic od`issijility ork still `kt. It is tfis alkxijility ga cfggsihn ohy voluk agr tfk kxtro uhehgwhs tfot diaakrkhtiotks tfk stoticolly od`issijlk ihtkrhol agrcks ohd `g`khts arg` tfk octuol ihtkrhol agrcks ohd `g`khts. ^k rkcgrd tfis gjskrvotigh os agllgws= • Ih dktkr`ihihn stoticolly od`issijlk ihtkrhol agrcks ohd `g`khts, tfk hu`jkr ga rkoctighs tfot coh jk ossinhkd orjitrory voluks is kquol tg tfk dknrkk ga stotic rkduhdohcy.
KVO@RLK >.4 Agr tfk jko`s sfgwh ih Ainurk >.1, >.1, dktkr`ihk o closs ga eihk`oticolly od`issijlk auhctighs. y
Cosk I
y
R
R
Cosk II
R
x
x L
L
L
>.4. Ainurk >.1 Jko`s agr Kxo`plk >.4.
OHS.
Cosk I= v ( x )
i +? 0 C x i
Cosk II= v ( x )
0
i ≫ ? gr v ( x ) i
m
0
Di ( ? – cgs i ς ( x ⁄ L ) ) .
e
C im e x ( x – L ) ( x – : L ) gr v ( x )
0
D i sih : i ς ( x ⁄ L ) .
KVO@RLK >.; Oh oxiol rgd fos o uhiagr` distrijutkd agrck ga p ga p2 opplikd os sfgwh. Dktkr`ihk o stoticolly od`issijlk ihtkrhol oxiol agrck. O
pg
x
J
L
>.;. Ainurk >.> Oxiol rgd agr Kxo`plk >.;.
OHS. H 0 – p2 x .
KVO@RLK >.1 Dktkr`ihk o stoticolly od`issijlk jkhdihn `g`kht agr tfk jko` sfgwh ih Ainurk >.6. >.6. y
O
R J x
C L
>.1. Ainurk >.6 Jko` agr Kxo`plk >.1.
OHS. @ z 0 – R ( : L – x )
2 ≤ x ≤ : L
L
Qirtuol ^gre
>.4.4 >.4.4
?:;
Qirtuo irtuoll Disp Disploc lock` k`kh khtt @ktf @ktfgd gd
Qirtuol displock`kht is oh ihaihitksi`ol, i`onihory, eihk`oticolly od`issijlk displock`kht aikld i`pgskd gh o jgdy
Tfk wgrd “virtuol‐ i`pliks tfot tfk displock`kht is hgt rkol jut oh i`onihory displock`kht i`pgskd gh o jgdy. Tfk wgrd “ihaihitksi`ol‐ i`pliks tfot hkitfkr tfot hkitfkr tfk ihtkrhol hgr tfk kxtkrhol agrcks gr `g`khts cfohnk durihn cfohnk durihn tfk i`pgsitigh ga tfk virtuol displock`kht. Ainurk >.9o >.9o sfgws o si`ply suppgrtkd jko` uhdkr uhiagr ` distrijutkd lgod. Tfk eihk`otic eihk`otic cghditigh cghditigh ga zkrg zkrg dkalkctigh ot kocf suppgrt coh jk `kt jy oh ihaihitk hu`jkr ga eihk`oticolly od`issijlk auhctighs, ghk ga wficf is sfgwh ih Ainurk >.9j >.9j. Tfk diaakrk diaakrkhck hck jktwkkh tfk octuol displock`kht auhctigh ohd tfk eihk`oticolly od`issijlk auhctigh is tfk virtuol d isplock`kht. [hliek tfk eihk`oticolly od`issijlk displock`kht auhctighs, tfk octuol displock`kht auhctigh `kkts hgt ghly tfk eihk`otic rkquirk`khts jut olsg tfk kquilijriu` cghditighs. Tfus, wk coh lgge ot tfk virtuol displock`kht os o `otfk`oticol kxpkri`khtotigh tg aihd tfk “ghk auhctigh‐ tfot wguld `kkt tfk kquilijriu` cghditighs. Qirtuol wgre prgvidks tfk `kcfohis` jy wficf wk sklkct tfk ghk gut ga tfk ihaihitk auhctighs tfot wguld `kkt tfk kquilijriu` cghditighs. Qirtuol wgre is tfk wgre dghk jy agrcks ih `gvihn tfrgunf o virtuol displock`kht. y
(>.?9)
w Eihk`oticolly od`issijlk displock`kht auhctigh x
δv 0 Qirtuol Qirtuol disploc displock`kh k`khtt auhctigh auhctigh
Octuol displock`kht auhctigh (j)
(o) Ainurk >.9 Qirtuol displock`kht.
^fkh tfk virtuol wgre cghditigh ga Kquotigh (>.?>) is (>.?>) is idkhticolly `kt, wk gjtoih tfk octuol displock`kht sucf os ih tfk du``y uhit lgod `ktfgd dkscrijkd ih Skctigh >.;. >.;. ^fkh tfk virtuol wgre cghditigh is opprgxi`otkly `kt, wk gjtoih oh opprgxi`otk sglutigh tg tfk displock`kht auhctigh sucf os ih tfk Zoylkinf-Zitz `ktfgd tg jk dkscrijkd ih Skctigh >.9. >.9. ^k rkcgrd tfk agllgwihn gjskrvotighs. • Ga oll tfk eihk`oticolly od`issijlk auhctighs, tfk ghk tfot sotisaiks tfk kquilijriu` cghditigh is tfk ghk agr wficf tfk virtuol wgre is zkrg. • Ih virtuol displock`kht `ktfgds, zkrg virtuol wgre i`pliks tfot oll kquilijriu` kquotighs ork `kt.
>.4. >.4.; ;
Qirtu irtuol ol Agrc Agrckk @kt @ktfg fgd d
Tfk virtuol agrck is oh ihaihitksi`ol i`onihory stoticolly od`issijlk agrck aikld i`pgskd gh o jgdy. Tfk wgrd “virtuol‐ i`pliks tfot tfk agrcks ork i`onihory agrcks tfot ork opplikd tg tfk jgdy. Tfk wgrd “ihaihitksi`ol‐ ih i`pliks tfot tfk jgdy dgks hgt ng tfrgunf odditighol dkagr`otigh duk dkagr`otigh duk tg tfk i`pgsitigh ga o virtuol agrck gr virtuol `g`kht. Tg aihd tfk stoticolly od`issijlk agrck aikld, wk opply oh i`onihory s`oll agrck tg tfk structurk ohd aihd tfk wgre dghk ih `gvihn tfrgunf tfk octuol displock`kht gr dkagr`otighs, os dk`ghstrotkd ih Kxo`plk >.6. >.6. ^k rkcgrd tfk agllgwihn. • Ga oll tfk stoticol od`issijlk agrcks, tfk ghk tfot sotisaiks tfk cg`potijility kquotighs is tfk ghk agr wficf tfk virtuol wgre is zkrg. • Ih virtuol agrck `ktfgds, “zkrg virtuol wgre‐ i`pliks tfot oll cg`potijility kquotighs ork `kt.
KVO@RLK >.> Tfk rgllkr ot R ot R sfgwh sfgwh ih Ainurk >.?2 slidks >.?2 slidks ih tfk slgt jkcousk ga tfk agrck A agrck A 0 0 ?2 eH. eH. @k`j @k`jkr kr OR OR fos fos o crgss-skctighol orko O 0 ?22 `` : ohd o `gdulus ga klosticity K klosticity K 0 0 :22 NRo. Dktkr`ihk Dktkr`ihk tfk oxiol strkss strkss ih tfk `k`jkr `k`jkr OR OR jy jy tfk virtuol displock`kht `ktfgd. R A
` ` 2 : 2 O
>.>. Ainurk >.?2 Oxiol `k`jkr agr Kxo`plk >.>.
OHS.
σ xx
0 ?;1 ?;1 @Ro ( T )
;2g
?:1
> Khkrny
@ktfgds
KVO@RLK >.6 Tfk octigh ga agrck A agrck A ih ih Kxo`plk >.> cousks >.> cousks tfk rgllkr tg `gvk jy 2.?; ``. [sk tfk virtuol agrck `ktfgd tg dktkr`ihk tfk dkagr`otigh ga jor OR jor OR . OHS.
Δu 0 >.; >.;
2.2914 ``
D[@@ D[@@Y Y [HI [HIT T LGO LGOD D @KT @KTFG FGD D
Ih Kxo`plk >.6, >.6, wk sow sow tfot o virtuol virtuol agrck (δ A ) i`pgskd i`pgskd gh o rgllkr rgllkr drgppkd drgppkd gut ga tfk colculotigh. colculotigh. Fkhck, fod wk stortkd stortkd witf A 0 ?, wk wguld fovk cghductkd kxoctly tfk so`k colculotighs witfgut writihn δ A ohd ohd δ H ih ih kocf kquotigh. Tfis is tfk josic δ A 0 idko tfot is agr`olizkd ih tfk du``y uhit lgod `ktfgd. Ih tfk du``y uhit lgod `ktfgd, wk i`pgsk o agrck gr `g`kht ga uhit `onhitudk ot tfk pgiht ot wficf wk woht tg aihd tfk displock`kht gr tfk slgpk. ^k hkxt cg`putk stoticolly od`issijlk agrck ohd `g`kht aiklds, ohd tfkh cg`putk tfk ihtkrhol ohd kxtkrhol virtuol wgre. Jy ihvgeihn tfk virtuol wgre prihciplk, wk gjtoih o agr`ulo rklotihn tfk displock`kht (gr slgpk) tg tfk virtuol agrcks (gr `g`khts).
>.;.? .;.?
Oxiol @k` @k`jkrs
^k coll tfk octuol oxiol `k`jkr rgd ?. ?. ^k rkprkskht tfk octuol ihtkrhol oxiol agrck jy H ?( x) x) ohd tfk oxiol displock`kht jy u?( x). x). Gur hkxt stkp is tg drow ohgtfkr rgd, collkd rgd :, :, witf kxoctly tfk so`k suppgrts os rgd ? ? jut witf ghly o uhit lgod plockd ot pgiht x pgiht x p, ot wficf wk woht tg colculotk tfk displock`kht. Lkt H :( x) x) jk tfk stoticolly od`issijlk oxiol agrck ohd u:( x) x) jk tfk eihk`oticolly od`issijlk displock`kht agr rgd :. Hgtk tfot tg aihd o stoticolly od`issijlk oxiol agrck, wk coh ossinh orjitror y voluks tg os `ohy rkoctighs os tfkrk ork dknrkks g a stotic rkduhdohcy. It sfguld jk hgtkd tfot tfk displock`khtu displock`kht u: ohd oxiol agrck H : ork hgt rklotkd jkcousk tfky ork ihdkpkhdkhtly cfgskh tg sotisay tfk eihk`otic ohd stotic od`issijility rkquirk`khts, rkspkctivkly. Tfk ihtkrhol virtuol wgre agr rgd : : is tfk ihtkrhol oxiol agrck `ultiplikd jy tfk displock`kht ot o pgiht tfot is ihtknrotkd gvkr tfk khtirk rgd, os sfgwh ih Kquotigh (>.:2o). (>.:2o). L
δ ^ iht 0
L
∥ H ( x )d u :
:
0
2
du :
∥ H ( x ) d x d x :
(>.:2o)
2
Tfk suppgrts agr rgds : rgds : ohd ? ork tfk so`k5 fkhck tfk eihk`oticolly od`issijlk displock`kht aikld u: is olsg tfk eihk`oticolly od`issijlk displock`kht aikld agr rgd ?. ?. Tfk octuol displock`kht u? is olwoys eihk`oticolly od`issijlk, fkhck coh jk uskd ih plock ga u:. Tfus tfk ihtkrhol virtuol wgre coh jk writtkh os sfgwh ih Kquotigh (>.:2j). (>.:2j). L
δ ^ iht 0
L
du ?
H ?
-------⎘ d ∥ H ( x ) d x d x 0 ∥ H ( x )⎕⎐ KO ⎢ x :
:
2
(>.:2j)
2
Ih tfk ojgvk kquotigh wk `odk usk ga tfk aoct tfk octuol displock`kht u? is rklotkd tg tfk octuol ihtkrhol agrck H ? tfrgunf tfk rklotigh du ? / dx 0 H ? / ( KO ) . Agr aihdihn displock`kht ot x ot x p, wk opply o virtuol agrck ga voluk δ A 0 ? ot x ot x p gh rgd :. Tfk cgrrkspghdihn kxtkrhol virtuol wgre agr rgd : : is nivkh jy Kquotigh (>.:2c). (>.:2c).
δ ^ kxt 0 ( δ A 0 ? ) u : ( x R ) 0 ( δ A 0 ? ) u? ( x R )
(>.:2c)
Kquotihn tfk kxtkrhol virtuol wgre tg tfk ihtkrhol virtuol wgre, wk gjtoih Kquotigh (>.:?). (>.:?). L
( δ A 0 ? ) u ? ( x R ) 0
H ? ( x ) H : ( x )
d x ∥ ---------------------------- KO
(>.:?)
2
Tfk oxiol displock`kht gjtoihkd arg` Kquotigh (>.:?) is (>.:?) is cghsidkrkd tg jk pgsitivk jk pgsitivk ih tfk dirkctigh ga tfk opplikd uhit agrck gh agrck gh rgd :. :. Tfis is jkcousk tfk wgre dghk jy tfk uhit agrck coh jk pgsitivk ghly ia tfk agrck ohd tfk displock`kht ork ih tfk so`k dirkctigh.
>.;.: >.;.:
Tgrsig grsigh h ga ga Cir Circu culor lor Sfoats Sfoats
Tfk prgckss agr gjtoihihn rgtotigh ga o sfoat skctigh is si`ilor tg tfot agr gjtoihihn tfk oxiol displock`kht ih o rgd. Tfk octuol sfoat is lojklkd sfoat lojklkd sfoat ?. ?. ^k rkprkskht tfk octuol ihtkrhol tgrquk jy T ?( x) x) ohd tfk rgtotigh jy ρ?( x).^ x).^kk drow ohgtfkr sfoat, collkd sfoat :, :, witf kxoctly tfk so`k suppgrts os sfoat os sfoat ? ? jut witf ghly uhit tgrquk ( δ T 0 ? ) plockd ot x ot x p wfkrk wk woht tg colculotk tfk skctigh rgtotigh. Lkt T :( x) x) jk tfk stoticolly od`issijlk tgrquk ohd ρ:( x) x) jk tfk eihk`oticolly od`issijlk rgtotigh agr sfoat agr sfoat :. :. Agr aihdihn o stoticolly od`issijlk tgrquk ih sfoat ih sfoat :, :, wk coh ossinh orjitrory voluks t g os `ohy rkoctigh tgrquks os tfkrk ork dknrkks ga stotic rkduhdohcy, wficf agr o sfoat gh o sihnlk oxis is ot `gst ghk. It sfguld jk hgtkd tfot tfk rgtotigh ρ: ohd tgrquk T : ork ihdk pkhdkhtly cfgskh tg sotisay tfk eihk`otic ohd stotic od`issijility rkquirk`khts ohd fkhck ork hgt rklotkd. ^k colculotk tfk ihtkr-
Du``y [hit Lgod @ktfgd hol ohd kxtkrhol virtuol wgre, rkcgnhizk tfot ρ?( x) x) is eihk`oticolly od`issijlk, rkcgnhizk tfk rklotighsfip d ρ ? / dx kquotk tfk ihtkrhol ohd kxtkrhol virtuol wgre tg gjtoih Kquotigh (>.::). (>.::). L
∥
( δ T 0 ? )ρ ? ( x R ) 0
T : ( x ) T ? ( x ) --------------------------d x NM
0
?:>
T ? / NM , ohd
(>.::)
2
dirkctigh ga tfk opplikd opplikd uhit uhit tgrquk tgrquk gh sfoat Tfk rgtotigh gjtoihkd arg` Kquotigh (>.::) is (>.::) is cghsidkrkd tg jk pgsitivk ih tfk dirkctigh :. Tfis is jkcousk tfk wgre dghk jy tfk uhit tgrquk coh jk pgsitivk ghly ia tfk tgrquk ohd tfk rgtotigh ork ih tfk so`k dirkctigh.
KVO@RLK >.9 Tfk cglu`h sfgwh ih Ainurk >.?? fos >.?? fos o `gdulus ga klosticity ga K ga K 0 0 4222 esi ohd ohd o crgss-sk crgss-skctighol ctighol orko ga ga O 0 ?22 ih:. [sk tfk du``y uhit lgod `ktfgd tg dktkr`ihk (o) tfk rkoctigh agrck ot O ot O ohd ohd (j) tfk `gvk`kht ga tfk rinid plotk ot C .
:2 ih. ih.
: eips O
: eips
x J
12 ih. ih. 4 eips
4 eips C :2 ih. ih. D
>.9. Ainurk >.?? Cglu`h agr Kxo`plk >.9.
OHS.
0
4.6 eips 5 uC 0 2.22231 ihcf dgwhword
>.;.3 >.;.3
Sy`` Sy``ktr ktric ic Jk Jkhdi hdihn hn ga Jk Jko` o`ss
Z O
Tfk prgckss agr aihdihn tfk dkalkctigh ohd slgpk ih jko`-jkhdihn prgjlk`s is si`ilor tg tfot agr aihdihn tfk oxiol dis plock`kht ohd tgrsighol rgtotigh. Fgwkvkr tfkrk ork sg`k diaakrkhcks tfot hkkd tg jk occguhtkd ag r ih tfk dkrivotigh. Tfk octuol jko` is lojklkd jko` ?. jko` ?. ^k rkprkskht tfk octuol ihtkrhol `g`kht jy @ ?( x) x) ohd tfk octuol displock`kht jy v?( x). x). ^k drow ohgtfkr jko`, collkd jko` :, jko` :, witf kxoctly tfk so`k suppgrts os jko` ? jut witf ghly uhit lgod ( δ A 0 ? ) , plockd ot x ot x p, wfkrk wk woht tg colculotk tfk displock`kht. Lkt @ :( x) x) jk tfk stoticolly od`issijlk jkhdihn `g`kht ohd v:( x) x) jk tfk eihk`oticolly od`issijlk displock`kht agr jko` :. jko` :. Agr aihdihn o stoticolly od`issijlk jkhdihn `g`kht agr jko` :, jko` :, wk coh ossinh orjitrory voluks tg os `ohy rkoctighs os tfkrk ork dknrkks ga stotic rkdu hdohcy. It sfguld jk hgtkd tfot tfk displock`kht v: ohd tfk `g`kht @ : ork ihdkpkhdkhtly cfgskh tg sotisay tfk eihk`otic ohd stotic od`issijility rkquirk`khts ohd fkhck ork hgt rklotkd. Tfk ihtkrhol virtuol wgre agr jko` : jko` : is tfk ihtkrhol `g`kht `ultiplikd jy tfk cfohnk ih slgpk ot o pgiht tfot is ihtknrotkd gvkr tfk khtirk jko`, os sfgwh ih Kquotigh (>.:3o). (>.:3o). L
δ ^ iht 0
L
∥ @ ( x ) d ζ :
2
:
0
∥
dζ : d x @ : ( x ) d x
L
0
2
∥
L
d dv :⎘ d x @ : ( x ) ⎕ d x ⎐ d x ⎢
0
2
:
d v:
∥ @ ( x ) d x :
:
d x
(>.:3o)
2
Tfk suppgrts agr jko` : jko` : ohd jko` ? jko` ? ork tfk so`k. Fkhck tfk eihk`oticolly od`issijlk displock`kht aikld v: is olsg tfk eihk`oticolly od`issijlk displock`kht aikld agr jko` ?. jko` ?. Tfk octuol displock`kht v? is olwoys eihk`oticolly od`issijlk, fkhck coh jk uskd ih plock ga v:. Tfus tfk ihtkrhol virtuol wgre coh jk writtkh os Kquotigh (>.:3j). (>.:3j). L
δ ^ iht 0
L
:
d v?
∥ @ ( x )d x :
2
:
d x
0
@ ?
-------⎘ d ∥ @ ( x )⎕⎐ KI ⎢ x :
(>.:3j)
2
Ih tfk ojgvk kquotigh wk `odk usk ga tfk aoct tfot tfk octuol displock`kht v? is rklotkd tg tfk octuol `g`kht @ ? tfrgunf tfk : : `g`kht curvoturk rklotigh d v ? / dx 0 @ ? / KI . Agr aihdihn tfk displock`kht ot x ot x p, wk opply o virtuol agrck ga voluk ? ot x ot x p. Tfk cgrrkspghdihn kxtkrhol virtuol wgre agr jko` : is os sfgwh ih Kquotigh (>.:3c). (>.:3c).
δ ^ kxt 0 ( δ A 0 ? ) v : ( x R ) 0 ( δ A 0 ? ) v ? ( x R ) Kquotihn tfk kxtkrhol virtuol wgre tg tfk ihtkrhol virtuol wgre, wk gjtoih Kquotigh (>.:4). (>.:4).
(>.:3c)
?:6
> Khkrny
@ktfgds L
∥
( δ A 0 ? ) v ? ( x R ) 0
@ : ( x ) @ ? ( x ) ------------------------------d x KI
(>.:4)
2
• O pgsiti pgsitivk vk sinh sinh agr v? i`pliks tfot tfk dkalkctigh is ih tfk so`k dirkctigh os tfk opplikd du``y uhit agrck. Agr aihdihn tfk slgpk ot x ot x p, wk opply o virtuol uhit `g`kht ( δ @ 0 ? ) ot x ot x p ohd tfk cgrrkspghdihn kxtkrhol virtuol wgre agr jko` : jko` : is os sfgwh ih Kquotigh (>.:;). (>.:;).
δ ^ kxt 0 ( δ @ 0 ? )
dv : ( x ) d x R
0 ( δ @ 0 ? )
dv ? ( x ) d x R
(>.:;)
Kquotihn tfk kxtkrhol virtuol wgre tg tfk ihtkrhol virtuol wgre, wk gjtoih Kquotigh (>.:1). (>.:1). L
( δ @ 0 ? )
dv ? d x
( x R ) 0
@ : ( x ) @ ? ( x )
d x ∥ ------------------------------ KI
(>.:1)
2
• O pgsitivk sinh agr d v ? / dx i`pliks tfot tfk slgpk is ih tfk so`k dirkctigh os tfk opplikd du``y uhit `g`kht. • Tfk slgpk ohd tfk dkalkctig dkalkctigh h ork kquol tg tfk so`k ihtknrol ihtknrolss ih Kquotigh (>.:4) ohd (>.:4) ohd (>.:1), (>.:1), jkcousk tfk lkat- ohd rinft-fohd sidks ih jgtf kquotighs fovk uhits ga wgre. Kxo`plk >.?2 dk`ghstrotks >.?2 dk`ghstrotks tfk opplicotigh ga tfk du``y uhit `ktfgd agr o stoticolly dktkr`ihotk prgjlk`, ohd Kxo` plk >.?? dk`ghstrotks >.?? dk`ghstrotks tfk `ktfgd agr stoticolly ihdktkr`ihotk jko`s.
KVO@RLK >.?2 Agr tfk jko` ohd lgodihn sfgwh ih Ainurk >.?:, >.?:, dktkr`ihk jy du``y uhit lgod `ktfgd (o) tfk dkalkctigh ot x ot x 0 L/: L/: ohd (j) tfk slgpk ot x ot x 0 L/:. L/:. Ossu`k tfot KI tfot KI is is cghstoht agr tfk jko`. y
w x L
>.?2. Ainurk >.?: Jko` agr Kxo`plk >.?2.
OHS.
L v? ⎕ -- ⎘ ⎐ :⎢
0
4 dv ? ⎕ L⎘ ?>wL ?> wL ---------------- dgwhword5 -364 KI d x ⎐ :⎢
4
0
> wL ------------- C^ 46 KI
KVO@RLK >.?? Dktkr`ihk tfk slgpk ot x ot x 0 L/: L/: agr tfk jko` ohd lgodihn sfgwh ih Ainurk >.?3. >.?3. Ossu`k tfot KI tfot KI is is cghstoht agr tfk jko`. y
w x
J
O L
>.??. Ainurk >.?3 Jko` agr Kxo`plk >.??.
OHS.
>.1 >.1
dv? ⎕ L⎘ -d x ⎐ :⎢
0
wL3 --------------- C^ ?9: KI
COST COSTIN INLI LIOH OHG’ G’S S TFKG TFKGZK ZK@S @S
Tfk `ktfgd agr stoticolly ihdktkr`ihotk structurks tfot wk study ih tfis skctigh is si`plkr tg i`plk`kht tfoh tfk du``y uhit lgod `ktfgd. Tfk agr`ulos wk will dkrivk fkrk ork spkciaic cosks ga Costinliohg’s nkhkrol tfkgrk`s3. ^k will dkrivk tfk agr`ulos agr jko` jkhdihn ohd tfkh nkhkrolizk agr oxiol `k`jkrs ohd tgrsigh ga circulor sfoats. Ih Kquotigh (>.:4), (>.:4), @ ? is tfk octuol `g`kht ih tfk jko` ohd @ : is o stoticolly od`issijlk jkhdihn `g`kht gjtoihkd jy plocihn o uhit agrck ot pgiht x pgiht x p, ot wficf wk woht tg colculotk tfk dkalkctigh. Ia ihstkod ga o uhit agrck wk cghsidkr o agrck A agrck A ˜ ˜ opplikd ot x ot x p, tfkh tfkh tfk tfk cgrrks cgrrkspgh pghdihn dihn stotic stoticoll olly y od`iss od`issijl ijlkk `g`kht `g`kht ( @ : ) wguld wguld jk jk A `ultiplikd `ultiplikd jy @ : (i.k., @ : 0 A @ : ). Ia wk ˜ toek tfk dkrivotivk witf rkspkct tg A tg A , wk gjtoih @ : 0 ∊ @ : / ∊ A , ohd arg` Kquotigh (>.:4) wk (>.:4) wk gjtoih Kquotigh (>.:>o) 3. Tfk tfkgrk`s wkrk airst airst dkrivkd jy tfk Itolioh khnihkkr C. O. O. Costinliohg’s ih ?6>9. Skk Dk`glisf`kht Dk`glisf`kht U?963P agr agr odditighol dktoils
Costinliohg’s Tfkgrk`s
?:9
L
v ? ( x R )
0
˜ ? ∊ @ : ------ ⎕ @ ( x )⎘ d x KI ⎐ ∊ A ? ⎢
∥
(>.:>o)
2
Hgtihn tfot tfk octuol `g`kht ` g`kht is o stoticolly stotic olly od`issijlk od` issijlk `g`kht, o hd fkhck wk coh sujstitutk s ujstitutk @ : (>.:>o), (>.:>o), wk gjtoih Kquotigh (>.:>j). (>.:>j). L
v ? ( x R )
0
⎘ ? ⎕ ∊ @ ? ------ ⎜ @ ? ( x )⎗ d x KI ⎐ ∊ A ⎢
∥
L
0
2
∥ 2
? ⎕ ∊ @ ?⎘ --------- ⎜ ⎗ d x : KI ⎐ ∊ A ⎢ :
L
0 ∊ ∊ A
∥
0
@ ? ihKquotigh ihKquotigh
:
@ ? --------d x : KI
(>.:>j)
2
Tfk lost lost tkr` tkr` ih tfk jrocekts jrocekts is tfk cg`plk`khtory cg`plk`khtory stroih khkrny khkrny ih jkhdihn jkhdihn ([ J ). Sujstitu Sujstitutihn tihnKquotigh Kquotigh (>.?:d) ihtg (>.?:d) ihtg Kquotigh (>.:>j), (>.:>j), wk gjtoih Kquotigh (>.:6o). (>.:6o). v ? ( x R )
0
∊[ J ∊ A
(>.:6o)
Ih kxoctly tfk so`k `ohhkr, wk coh stort witf Kquotigh (>.:1) ohd (>.:1) ohd ihstkod ga o uhit `g`kht plock o `g`kht @ `g`kht @ ot ot pgiht x p ohd gjtoih Kquotigh (>.:6j). (>.:6j). dv ? ( x ) d x R
0
∊[ J ∊ @
(>.:6j)
Ih lihkor systk`s, tfk cg`plk`khtory stroih khkrny4 is so`k os tfk stroih khkrny. Fkhck [ J cguld jk rkplockd jy [ J ih Kquotigh (>.:6o) ohd (>.:6o) ohd (>.:6j). (>.:6j). Tfk kquotighs wkrk dkrivkd agr jko`s, jut si`ilor agr` ga kquotighs ork volid agr ohy klostic structurk. Agr aihdihn tfk oxiol displock`kht, wk coh rkplock v witf u ohd [ J witf [ O ih Kquotigh (>.:6o) tg (>.:6o) tg gjtoih u ? ( x R )
0
∊[ O ∊ A
(>.:6c)
Agr aihdihn tgrsighol rgtotigh wk coh rkplock d v ? / dx witf ρ? ohd [ J witf [ T (>.:6j) tg gjtoih T ih Kquotigh (>.:6j) tg
ρ? ( x R ) 0
∊[ T T ∊ T
(>.:6d)
Tfk nkhkrol agr` ga Kquotigh (>.:6o) tfrgunf (>.:6o) tfrgunf (>.:6d) ork (>.:6d) ork rkakrrkd tg os Costinliohg’s tfkgrk` dkaihkd jklgw.
Tfk portiol dkrivotivk ga cg`plk`khtory stroih khkrny ga o structurk witf rkspkct tg o agrck is kquol tg tfk displock`kht ot tfk pgiht ga opplicotigh opplicotigh ga tfk agrck, ohd tfk tfk portiol dkrivotivk ga cg`plk`khtory stroih khkrny witf rkspkct tg o `g`kht `g`kht is kquol tg tfk rgtotigh ot tfk pgiht ga opplicotigh ga tfk `g`kht.
Ih tfk du``y uhit lgod `ktfgd, o pgsitivk voluk agr tfk displock`kht (gr rgtotigh) i`plikd tfot tfk displock`kht (gr rgtotigh) wos ih tfk dirkctigh ga tfk agrck (gr `g`kht). Tfis is olsg truk agr Kquotigh (>.:6o) tfrgunf (>.:6o) tfrgunf (>.:6d). (>.:6d). Ih tfk colculotighs ga dkalkctigh ohd slgpks, tfk dkrivotivk witf rknord tg tfk agrck ohd `g`kht sfguld jk pkragr`kd jkagrk tfk ihtknrotigh5 gtfkrwisk tfk olnkjro will jk sinhiaicohtly `grk cg`plkx. Ia tfkrk is hg agrck (gr `g`kht) ot o pgiht ot wficf wk wisf tg aihd tfk displock`kht (gr slgpk), wk ihitiolly ihtrgduck o agrck (gr `g`kht) ohd colculotk tfk ihtkrhol agrck ohd `g`kht ih Kquotigh (>.:6o) tfrgunf (>.:6o) tfrgunf (>.:6d). (>.:6d). Oatkr pkragr`ihn tfk dkrivotivks, wk sujstitutk o zkrg voluk agr tfk agrck (gr `g`kht) ohd gjtoih tfk displock`kht (gr rgtotigh). ^k rkcgrd tfk agllgwihn gjskrvotighs. • Rgsitivk voluks agr displock`kht ohd rgtotigh jy Costinliohg’s tfkgrk`s i`ply tfot tfk displock`kht ohd ohd rgtotigh ork ih tfk dirkctigh ga tfk agrck ohd `g`kht witf wficf tfk dkrivotivk is jkihn pkragr`kd. • Diaakrkhtiotigh witf rkspkct tg tfk agrck gr `g`kht jkagrk ihtknrotigh will sovk olnkjroic kaagrt ih opplyihn Costinliohg’s tfkgrk`s.
KVO@RLK >.?: [sk Costinliohg’s tfkgrk`s tg dktkr`ihk tfk rkoctigh agrck ot O ot O ohd ohd tfk displock`kht ot C ih ih tfk cglu`h sfgwh ih Ainurk >.?? ga Kxo`plk >.9. >.9. OHS. Z O
0
4.6 eips 5 u C
0
2.22231 ihcf dgwhword
4. Tfk cg`plk`khtory stroih khkrny is hgt kquol tg tfk stroih khkrny agr hghlihkor `otkriols. Fgwkvkr, Costinliohg’s tfkgrk`s ork opplicojlk tg hghlihhghlihkor `otkriols prgvidkd tfk `otkriol is klostic.
> Khkrny
?32
@ktfgds
KVO@RLK >.?3 Kocf stkkl jor ( K 0 32,222 esi) sfgwh ih Ainurk >.?4 fos >.?4 fos o crgss-skctighol orko ga ?ih: ohd o lkhntf ga ; ihcfks. Dktkr`ihk Dktkr`ihk tfk tfk displock`kht ga pgiht D pgiht D.. O 3 ih C
D
R 0 R 0 :>eip :>eip
; ih
3 ih J
4 ih
>.?3. Ainurk >.?4 Nkg`ktry agr Kxo`plk >.?3.
OHS.
u D
0
2.226 ihcf
KVO@RLK >.?4 Agr tfk jko` ohd lgodihn sfgwh ih Ainurk >.?3, >.?3, usk Costinliohg’s tfkgrk`s tg colculotk tfk slgpk ot x ot x 0 L/:. L/:. RLOH ^k coh usk tfk `g`kht @ ? ih Kxo`plk >.??. >.??. ^k coh aihd tfk dkrivotivk ga @ ? witf rkspkct tg tfk rkoctigh ot J ot J.. Sujstitutihn @ ? ohd tfk dkrivotivk ga @ ? ihtg Kquotigh (>.:6o) ohd (>.:6o) ohd kquotihn tfk displock`kht ot J ot J tg tg zkrg, wk coh dktkr`ihk tfk rkoctigh ot J ot J.. ^k coh hgw plock o `g`kht @ `g`kht @ ot x ot x 0 L/: L/: ohd aihd tfk hkw @ ?.
tg @ Sujstitutihn tfk hkw @ ? ohd its dkrivotivk witf rkspkct tg @
kvoluotkd ot @ ot @ 0 0 2, wk gjtoih gjtoih tfk slgpk arg` arg` Kquotigh (>.:6j). (>.:6j). OHS.
dv ? ⎕ L⎘ -d x ⎐ :⎢
3
0
wL --------------- C^ ?9: KI
KVO@RLK >.?; Agr tfk jko` ohd lgodihn sfgwh ih Ainurk >.?;, >.?;, dktkr`ihk tfk rkoctigh agrcks ot O ot O ohd ohd J J ih ih tkr`s ga w, L, L, K , ohd I ohd I . y :
wL
O
J
x
C
L
L >.?;. Ainurk >.?; Jko` agr Kxo`plk >.?;.
OHS.
Z O
0 ( 9 ⁄ > ) wL 5
Z J
0 ( ?: ⁄ > ) wL
KVO@RLK >.?1 O rkctohnulor rkctohnulor wggdkh jko` fovihn fovihn o crgss skctigh ga 12 `` Þ ?62 `` is suppgrtkd ot tfk rinft khd jy oh olu`ihu` circulor rgd ga 6 `` dio`ktkr dio`ktkr os sfgwh sfgwh ih Ainurk >.?1. >.?1. Tfk `gduli ga klosticity agr wggd ohd olu`ihu` ork K ork K wggd 0 ?:.1 ?:.1 NRo ohd ohd K Ol 0 >2 NRo. Dktkr`ihk Dktkr`ihk tfk tfk oxiol strkss strkss ih tfk olu`ihu` olu`ihu` rgd rgd OC . C ?.3 ?.3 `
w J
O 3`
>.?1. Ainurk >.?1 Structurk agr Kxo`plk >.?1.
OHS.
σ OC 0
66.: 66.: @Ro ( T )
?62 ?62 ``
Dkalkctigh ga Curvkd Jko`s
>.> >.>
?3?
DKALK DKALKCT CTIG IGH H GA C[ZQ C[ZQKD KD JKO@ JKO@S S
Ih Skctigh 3.;, wk dividkd curvkd jko`s ihtg twg cotkngriks= ?. Jko`s witf witf vkry lornk lornk rodii ga curvotu curvoturk rk (i.k., r c/f 8 ;), wfkrk r c is tfk rodius ga curvoturk ga tfk lihk mgihihn tfk ckhtrgids ga tfk crgss skctighs ohd f is tfk dkptf ga tfk jko` crgss skctigh (skk Ainurk 3.4>), :. Jko` Jko`ss witf witf r c/f < ;. Tfk strkssks ohd stroihs ih jko`s witf r c/f 8 ; cguld jk colculotkd os agr stroinft jko`s, ohd fkhck tfk cg`plk`khtory stroih khkrniks agr sucf curvkd jko`s wguld jk si`ilor tg tfgsk agr stroinft jko`s. Ih tfis skctigh wk will usk khkrny `ktfgds tg aihd tfk dkalkctigh ga jko`s agr wficf r c/f 8 ;, wficf wguld jk tfk diaaicult jy kquilijriu` `ktfgds. Ainurk >.?>o >.?>o sfgws o curvkd jko` witf ih-plohk lgods R lgods R ohd R >.?>j sfgws o arkk jgdy dionro` oatkr oh x ohd R y. Ainurk >.?>j i`onihory cut fos jkkh `odk. Arg` Ainurk >.?>j >.?>j wk cghcludk tfot tfk tgtol cg`plk`khtory stroih khkrny is tfk su` ga tfk cg` plk`khtory stroih khkrniks khkrniks duk duk tg tfk oxiol oxiol agrck, agrck, sfkor sfkor agrck, agrck, ohd jkhdihn jkhdihn `g`kht. `g`kht. Tfk cg`plk`k cg`plk`khtory htory stroih khkrny du k tg sfkor sf kor is hknlkctkd ih cg`porisgh tg tfot duk tg `g`kht jkcousk tfk sfkor strkssks ih jkhdihn ork oh grdkr ga `onhitudk lkss tfoh tfk hgr`ol strkssks. Nkhkrolly spkoeihn, tfk oxiol hgr`ol strkss is oh grdkr ga `onhitudk s`ollkr tfoh tfk `oxi`u` jkhdihn hgr`ol strkss ot o crgss skctigh wfkh jgtf strkssks ork prgduckd arg` tfk so`k agrck. Tfus, ih oholysis wk cghsidkr ghly tfk cg` plk`khtory stroih khkrny agr o curvkd jko` ([ ([ (>.:9), IJ) uhdkr oh ih-plohk lgod duk tg tfk `g`kht, os nivkh jy Kquotigh (>.:9), [ IJ
0
∥ L
:
? @ z -- ---------d s : KI zz
(>.:9)
wfkrk s wfkrk s is is tfk tohnkhtiol cggrdihotk olghn tfk lihk mgihihn tfk ckhtrgids ga tfk curvkd jko`. Ainurk >.?>c >.?>c sfgws o curvkd jko` oh witf gut-ga-plohk lgod R lgod R z . Ainurk >.?>d >.?>d sfgws sfgws tfk arkk jgdy dionro` oatkr oh i`onihory cut fos jkkh `odk. Arg` Ainurk >.?>d >.?>d wk wk cghcludk tfot tfk tgtol cg`plk`khtory stroih khkrny is tfk su` ga tfk cg` plk`khtory stroih khkrniks duk tg tfk sfkor agrck, jkhdihn `g`kht ojgut tfk y tfk y oxis, oxis, ohd tgrsigh. Tfk cg`plk`khtory stroih khkrny duk tg jkhdihn sfkor is ghck `grk hknlkctkd. ^k will rkstrict gursklvks tgcirculor tg circulor crgss skctighs, agr skctighs, agr wficf wk fovk o wkll-dkvklgpkd tfkgry ga tgrsigh. Tfk tgtol cg`plk`khtory stroih khkrny agr o curvkd jko` duk tg gut-ga-plohk lgod ([ GJ) duk tg jkhdihn `g`kht ohd tgrsigh is nivkh jy Kquotigh (>.32). (>.32). [ GJ GJ
0
∥ L
:
: ? @ y ? T -- ---------- d s + -- ------- d s : KI yy : NM
∥
(>.32)
L
R x
R x
@ y
@ z
R z
R y
y
H
Q y
R y
T
y
R z Q z
x
x
z (o)
z (c)
(j)
(d )
Ainurk >.?> Curvkd jko` uhdkr (o) ih-plohk ohd (c) gut-ga-plohk lgodihn, witf ossgciotkd arkk jgdy dionro`s (j, d).
KVO@RLK >.?> O circulor jko` fovihn o rkctohnulor crgss skctigh is sfgwh ih Ainurk >.?6. >.?6. Agr tfk lgodihn ohd nkg`ktry sfgwh, aihd tfk rotig ga tfk cg`plk`khtory oxiol stroih k hkrny tg tfk cg`plk`khtory jkhdihn stroih khkrny, agr tfk twg cosks (o) ο0 2 ohd (j) ο0ς/:. f f
ο R Z
j
ζ C
>.?>. Ainurk >.?6 Circulor jko` agr Kxo`plk >.?>.
OHS.
[ O (o) -------[ IJ
0
[ O f : 2.?1> ⎕ ---⎘ 5 (j) -------⎐ Z⎢ [ IJ
0
f : 2.31> ⎕ ---⎘ ⎐ Z⎢
?3:
> Khkrny
@ktfgds
KVO@RLK >.?6 Agr tfk circulor jko` ohd lgodihn sfgwh ih Ainurk >.?6, >.?6, dktkr`ihk tfk displock`kht ot pgiht O pgiht O ih ih tfk dirkctigh ga tfk opplikd agrck agr twg cosks= (o) ο 0 2 ohd (j) ο 0 ς/:. OHS. (o) v R
0
9.4:; R ⎕ Z⎘ 3 ----------------- --- 5 (j) v R Kj ⎐ f ⎢
0
4.:>4 R ⎕ Z⎘ 3 ----------------- -- Kj ⎐ f ⎢
KVO@RLK >.?9 Agr tfk circulor jko` ohd lgodihn sfgwh ih Ainurk >.?6, >.?6, dktkr`ihk tfk displock`kht ot pgiht O pgiht O ih ih tfk fgrizghtol dirkctigh (tohnkhtiol dirkctigh) agr ο 0 2. OHS. v Ox
0
1 R Z 3 ------- ⎕ ---⎘ Kj ⎐ f ⎢
KVO@RLK >.:2 O circulor jko` is lgodkd jy oh gut-ga-plohk lgod R lgod R os os sfgwh ih Ainurk >.?9. >.?9. Dktkr`ihk tfk dkalkctigh ot pgiht O pgiht O ih ih tfk dirkctigh ga tfk lgod. O R
:r Crgss skctigh
Z ζ
>.:2. Ainurk >.?9 Circulor jko` agr Kxo`plk >.:2.
OHS. 3
w R
>.6 >.6
? ? 3 ς – 6 --- + ---- ⎕ ---------------⎘ r K N ⎐ : ς ⎢
--------0 RZ 4
@IHI @IHI@[ @[@ @ RGTK RGTKHT HTIO IOL L KHKZ KHKZNY NY
^k stort jy dkaihihn tfk pgtkhtiol khkrny auhctigh Ω os
Ω 0
[ – ^
(>.3?)
wfkrk [ is is tfk stroih khkrny ohd ^ is is tfk wgre pgtkhtiol ga o agrck. [hliek wgre, wficf is o cghckpt ossgciotkd witf ohy agrck (cghskrvotivk gr hghcghskrvotivk) tfot `gvks, tfk “wgre pgtkhtiol ga o agrck‐ is ossgciotkd witf cghskrvotivk agrcks ghly ohd i`pliks tfot tfkrk is o pgtkhtiol auhctigh arg` wficf sucf o agrck coh jk gjtoihkd. Nrovitotighol agrcks ohd klkctrg`onhktic agrcks ork twg kxo`plks ga cghskrvotivk agrcks tfot coh jk gjtoihkd arg` o pgtkhtiol auhctigh. Ia wk ossu`k tfot tfk kxtkrhol agrcks ork octihn gh o klostic (lihkor klostic (lihkor gr hghlihkor) structurk ohd ork cghskrvotivk, tfkh tfk “wgre pgtkhtiol ga o agrck‐ is colculotkd os tfk wgre tkr` ih Ainurk >.4 witfgut >.4 witfgut tfk sy`jgl agr virtuol displock`kht. Tfus, arg` gur pkrspkctivk, tfk wgre pgtkhtiol ohd tfk wgre dghk jy tfk agrck ork colculotkd si`ilorly, prgvidkd tfk kxtkrhol agrcks ork cghskrvotivk ohd ork opplikd tg klostic systk`s. Tg gjtoih tfk stotk`kht ga tfk tfkgrk` ga `ihi`u` pgtkhtiol khkrny, wk cghsidkr tfk stotk`kht ga virtuol wgre os nivkh jy Kquotigh (>.?>). (>.?>). Zkstrictihn gursklvks tg cghskrvotivk klostic systk`s, wk coh stotk tfot tfk ihtkrhol virtuol wgre is tfk voriotigh ih klostic stroih khkrny durihn dkagr`otigh (i.k., δ ^ iht 0 δ [ ) ohd tfk kxtkrhol virtuol wgre is tfk voriotigh ih tfk wgre pgtkhtiol ga tfk agrck (i.k., δ ^ kxt 0 δ ^ ). Tfus, arg` Kquotigh (>.?>) wk (>.?>) wk gjtoih
δ ^ ih t – δ ^ kx t 0 δ [ – δ ^ 0 δΩ 0
2
(>.3:)
Kquotigh (>.3:) i`pliks (>.3:) i`pliks tfot ot kquilijriu`, tfk virtuol voriotigh ih tfk pgtkhtiol khkrny auhctigh is zkrg—wficf gccurs wfkrk tfk slgpk ga tfk pgtkhtiol khkrny auhctigh witf rkspkct tg tfk poro`ktkrs dkaihihn tfk pgtkhtiol auhctigh is zkrg. Tfk poro`ktkrs dkaihihn tfk pgtkhtiol auhctigh ork tfk nkhkrolizkd displock`khts. Tfk zkrg slgpk cghditigh coh gccur wfkrk tfk pgtkhtiol auhctigh is ot `oxi`u`, ot o soddlk pgiht, gr ot `ihi`u`. Tfk `oxi`u` ga tfk pgtkhtiol auhctigh gccurs ot pgihts ga uhstojlk
Zoylkinf–Zitz @ktfgd
?33
kquilijriu`, ohd tfk soddlk pgihts gccur ot pgihts ga hkutrol stojility, hkitfkr ga wficf wk sfoll cghsidkr fkrk. Tfk `ihi`u` ih pgtkhtiol auhctigh gccurs ot tfk stojlk kquilijriu` pgiht, wficf is tfk ghl y kquilijriu` wk will cghsidkr fkrk. Tfk tfkgrk` ga `ihi`u` pgtkhtiol khkrny coh jk stotkd os agllgws. Ga oll tfk eihk`oticolly od`issijlk displock`kht auhctighs, tfk octuol displock`kht auhctigh is tfk ghk tfot `ihi`izks tfk pgtkhtiol khkrny auhctigh ot stojlk kquilijriu`. Ih Skctigh >.4.: it wos hgtkd tfot tfkrk ork `ohy eihk`oticolly od`issijlk displock`kht auhctighs, ohd tfkrk is hg rkquirk`kht tfot tfksk auhctighs sotisay tfk kquilijriu` kquotighs gr tfk stotic jguhdory cghditighs. Tfk octuol displock`kht is eihk`oticolly od`issijlk ohd sotisaiks oll tfk kquilijriu` cghditighs ohd tfk stotic jguhdory cghditighs. Tfus, ia wk cfggsk oh orjitrory eihk`oticolly od`issijlk auhctigh ohd colculotk tfk pgtkhtiol khkrny auhctigh, tfk voluk sg gjtoihkd will olwoys jk nrkotkr tfoh tfk voluk ga tfk pgtkhtiol khkrny auhctigh ot kquilijriu`. Ih gtfkr wgrds, wk opprgocf tfk pgtkhtiol khkrny auhctigh voluk ot kquilijriu` arg` ojgvk. Tfus, ia wk fovk twg opprgxi`otighs agr tfk displock`kht auhctighs, tfk ghk tfot nivks o lgwkr pgtkhtiol khkrny voluk is tfk jkttkr opprgxi`otigh. O cgrgllory tg tfk prkckdihn stotk`kht is tfot ia wk odd ohgtfkr tkr` (ihcrkosk tfk dknrkks ga arkkdg`) ih oh opprgxi`otigh, tfk pgtkhtiol khkrny voluk coh ghly dkcrkosk (i.k., i`prgvk tfk occurocy ga tfk opprgxi`otigh). ^k rkcgrd tfk agllgwihn gjskrvotighs. • Tfk jkttkr opprgxi`otigh ga displock`kht auhctigh is tfk ghk tfot yiklds tfk lgwkr pgtkhtiol khkrny. khkrny. • Tfk nrkotkr tfk dknrkks ga arkkdg`, tfk lgwkr will jk tfk pgtkhtiol khkrny agr o nivkh skt ga eihk`oticolly od`issijlk auhctighs.
>.9 >.9
ZOYL ZOYLKI KINF NF–Z –ZIT IT] ] @KTF @KTFGD GD
Tfk Zoylkinf–Zitz `ktfgd is oh opprgxi`otk `ktfgd ga aihdihn displock`khts tfot is joskd gh tfk tfkgrk` ga `ihi`u` pgtkhtiol khkrny. Tfk `ktfgd is rkstrictkd tg cghskrvotivk systk`s, wficf `oy jk lih kor gr hghlihkor. ^k rkprkskht tfk displock`kht u( x) x) jy o skriks ga eihk`oticolly od`issijlk auhctighs a auhctighs a i( x) x) os sfgwh ih Kquotigh (>.33o), (>.33o), u( x)
h
∟ 0 C a ( x )
0
(>.33o)
i i
i ?
wfkrk C i ork uhdktkr`ihkd cghstohts (i.k., nkhkrolizkd displock`khts). Jy sujstitutihn Kquotigh (>.33o) ihtg (>.33o) ihtg tfk stroih khkrny [ ohd tfk wgre pgtkhtiol ^ , wk coh gjtoih tfk pgtkhtiol khkrny auhctigh Ω ih tkr`s ga tfk cghstohts C i. Tfk cghstohts C i ork tg jk dktkr`ihkd sucf tfot Ω is `ihi`izkd. Tfk hkckssory cghditigh ot tfk `ihi`u` voluk ga Ω is nivkh jy Kquotigh (>.33j). (>.33j).
∊Ω 2 0 ∊ C i
i
0 ?, h
(>.33j)
Tfk prkckdihn stotk`kht rkprkskhts o skt ga h olnkjroic kquotighs ih tfk cghstohts C i, wficf coh jk sglvkd si`ultohkgusly tg gjtoih tfk voluks ga C i. Ghck C i ork ehgwh, tfk strkssks coh jk aguhd arg` tfk displock`kht auhctigh ih Kquotigh (>.33o). (>.33o). ^k will discuss tfk usk ga tfk Zoylkinf-Zitz `ktfgd ih tfk cghtkxt ga ghk-di`khsighol lihkor structurol klk`khts, oltfgunf it is kquolly opplicojlk tg twg ohd tfrkk di`khs ighs, ohd agr lihkor ohd hghlihkor `otkriol jkfovigr, prgvidkd tfk systk` is cghskrvotivk.
>.9.? .9.?
Oxio xiol @k @k`jkrs
Tfk lihkor klostic stroih khkrny dkhsity agr oxiol `k`jkrs wos nivkh jy Kquotigh (>.6o). Toeihn tfk dkrivotivk gaKquotigh ga Kquotigh (>.33o), (>.33o), wk gjtoih du d x
h
∟0
0
m ?
C m
d a m d x
(>.34o)
Tfk oxiol stroih khkrny dkhsity coh jk writtkh ohd ihtknrotkd tg gjtoih tfk oxiol stroih khkrny os sfgwh jklgw. [ o
0
? du du -- KO ⎕ ⎘ ⎕ ⎘ ⎐ d x⎢ ⎐ d x⎢ :
0
? -- KO ⎕ ⎐ :
d a m
h
m ?
h
m
e ?
L
[ O
0
d a e
∟ 0 C d x⎘⎢ ⎕⎐ ∟ 0 C d x ⎘⎢ e
0
? -- KO :
d a m
d a e
∟ 0 ∟ 0 C C ⎕⎐ d x⎘⎢ ⎕⎐ d x ⎘⎢ h
m ?
h
e ?
m
e
(>.34j)
L
d a m
d a e
∥ [ d x 0 ∟ 0 ∟ 0 C C ∥ KO ⎕⎐ d x⎘⎢ ⎕⎐ d x ⎘⎢ d x 0 :-- ∟ 0 ∟ 0 C C E o
? -:
h
m ?
h
e ?
m
e
2
?
h
h
m ?
e ?
m
e me
(>.34c)
2
wfkrk tfk `otrix E `otrix E (>.34d). me is collkd tfk stiaahkss `otrix ohd is nivkh jy Kquotigh (>.34d). L
E me
0
d a m
d a e
∥ KO ⎕⎐ d x⎘⎢ ⎕⎐ d x ⎘⎢ d x
(>.34d)
2
Ih Kquotigh (>.34d), (>.34d), KO coh KO coh jk o auhctigh ga x ga x.. Ia tfk oxiol `k`jkr is topkrkd, tfkh O tfkh O os os o auhctigh ga x ga x coh coh jk aguhd ohd uskd ih Kquotigh (>.34d). (>.34d). Ia tfk oxiol `k`jkr is `odk up skn`khts ga diaakrkht `otkriols, gr ia tfk crgss skctigh cfohnks agr kocf
?34
> Khkrny
@ktfgds
skn`kht, tfkh tfk ihtknrotigh gvkr tfk lkhntf ga tfk oxiol `k`jkr coh jk writtkh os tfk su` ga tfk ihtknrols gvkr tfk skn`khts, ohd tfk cfohnihn prgpkrtiks ga K ga K ohd O ohd O coh coh jk occguhtkd agr ih Kquotigh (>.34d). (>.34d). ^k ossu`k oh oxiol distrijutkd agrck p agrck p x( x) x) ohd cghckhtrotkd oxiol agrcks opplikd ot ` pgihts witf cggrdihotks x cggrdihotks xq. Tfk wgre pgtkhtiol ga tfksk agrcks coh jk writtkh os agllgws= L
^ O
`
∥ p ( x )u( x )d x + ∟ 0 A u( x )
0
x
q
q ?
(>.34k)
q
2
Sujstitutihn Kquotigh (>.33o) ih (>.33o) ih Kquotigh (>.34k), (>.34k), wk gjtoih Kquotigh (>.34a), (>.34a), L
^ O
h
`
h
h
∟ 0 C ∥ p ( x ) a ( x )d x + ∟ 0 ∟ 0 A C a ( x ) 0 ∟ 0 C Z
0
m
m ?
x
m
q ?
q
m ?
m m
q
(>.34a)
m m
m ?
2
wfkrk tfk vkctgr Z vkctgr Z m is os nivkh ih Kquotigh (>.34n). (>.34n). L
Z m
0
∥
`
∟ 0 A a ( x )
p x ( x ) a m ( x ) d x +
q m
q ?
(>.34n)
q
2
Tfk pgtkhtiol khkrny auhctigh agr oxiol `k`jkrs coh jk writtkh os sfgwh ih Kquotigh (>.3;o). (>.3;o).
Ω O 0
? -:
0
[ O – ^ O
h
h
h
∟ 0 ∟ 0 C C E – ∟ 0 C Z m ?
m
e ?
e me
(>.3;o)
m m
m ?
Ih occgrdohck witf Kquotigh (>.33j), (>.33j), wk `ihi`izk tfk pgtkhtiol khkrny auhctigh jy toeihn tfk dkrivotivk ga Kquotigh (>.3;o) witf (>.3;o) witf rkspkct tg C i tg gjtoih Kquotigh (>.3;j). (>.3;j).
∊Ω O 0 ∊ C i
? -:
h
h
m ?
e ?
∟0 ∟ 0
∊C m ∊C e C e + C m E – ∊ C i ∊ C i me
∊C m
h
∟ 0 ∊ C Z 0 2
i
m
m ?
0 ?, h
(>.3;j)
i
^k hgtk tfot tfk C ’s ’s ork nkhkrolizkd displock`khts tfot ork ihdkpkhdkht ga kocf gtfkr. Tfus, tfk prgpkrty ga tfk portiol dkrivotivks ga tfk C ’s ’s is os sfgwh ih Kquotigh (>.3;c). (>.3;c).
∊C m ⎮ ? 0 ∊ C i ⎦⎩ 2
0 m i≢m
i
(>.3;c)
Sujstitutihn Kquotigh (>.3;c) ihtg (>.3;c) ihtg Kquotigh (>.3;j), (>.3;j), wk gjtoih Kquotigh (>.3;d). (>.3;d).
∊Ω O 0 ∊ C i
? -:
h
∟0
e ?
C e E ie – Z i
0
? -:
h
∟ 0 C ( E + E ) – Z 0 2 m ?
m
im
mi
i
i
0 ?, h
(>.3;d)
Kquotigh (>.34d) sfgws (>.34d) sfgws tfot tfk `otrix E `otrix E is is sy``ktric, os sfgwh ih Kquotigh (>.3;k). (>.3;k).
0
E mi
E im
(>.3;k)
Sujstitutihn Kquotigh (>.3;k) ihtg (>.3;k) ihtg Kquotigh (>.3;d), (>.3;d), wk gjtoih tfk agllgwihn skt ga olnkjroic kquotighs= h
∟ 0 E C 0 Z m ?
im
m
i
i
0 ?, h
(>.31o)
Jy usihn `otrix hgtotigh, Kquotigh (>.31o) coh (>.31o) coh jk writtkh os agllgws=
U E P { C } 0 { Z }
(>.31j)
Tfk skt ga h olnkjroic kquotighs ih Kquotigh (>.31o) coh (>.31o) coh jk sglvkd tg gjtoih tfk voluks ga C m. m. Tfk displock`kht auhctigh ih Kquotigh (>.33o) hgw (>.33o) hgw is cg`plktkly ehgwh. ^itf oxiol displock`kht ehgwh, oxiol strkssks, oxiol stroihs, tfk ihtkrhol oxiol agrck gh o skctigh, ohd tfk pgtkhtiol khkrny ga tfk o xiol `k`jkr coh jk aguhd. Lkt tfk sglutigh gaKquotigh ga Kquotigh (>.31o) jk (>.31o) jk rkprkskhtkd jy C m∞ . Tfk kxprkssigh agr pgtkhtiol khkrny ih Kquotigh (>.3;o) coh (>.3;o) coh jk si`pliaikd jy usihn Kquotigh (>.31o) os (>.31o) os sfgwh jklgw.
Ω O 0
? -:
h
h
h
∟ 0 C ∞∟ 0 E C ∞ – ∟ 0 C ∞ Z m ?
m
e ?
me
e
m ?
m
m
0
? -:
h
h
∟ 0 C ∞ Z – ∟ 0 C ∞ Z m ?
m
m
m ?
m
m
? 0 – ⎕⎐ --⎘⎢
h
C ∞ Z : ∟0 m ?
m
m
(>.31c)
It sfguld jk k`pfosizkd tfot Kquotigh (>.31c) is (>.31c) is volid ghly ot kquilijriu`. Kquotigh (>.31c) olsg (>.31c) olsg finflinfts tfot ot kquilijriu`, tfk stroih khkrny is fola tfk wgre pgtkhtiol ga tfk agrcks, os sfgwh ih Kquotigh (>.3>). (>.3>). [ O
0
^ O ------:
0 – Ω O 0
? -:
h
∟ 0 C ∞ Z m ?
m
m
ot kquilijriu`
(>.3>)
Tfk stroih khkrny is olwoys pgsitivk. Tfus ot kquilijriu`, tfk pgt khtiol khkrny is hknotivk agr tfk klostic systk`s wk will study. Tfus, i`prgvkd occurociks arg` opprgxi`otighs will `oek tfk pgtkhtiol khkrny `grk hknotivk. Clkorly agr tfk lornk voluks ga h (i.k., hu`jkr ga uhehgwhs) ih Kquotigh (>.31o), (>.31o), wk will hkkd hu`kricol `ktfgds tg sglvk tfk skt ga olnkjroic kquotighs. Jut wk coh dk`ghstrotk tfk Zoylkinf-Zitz `ktfgd jy usihn s`oll voluks ga h, os will jk illustrotkd oatkr tfk dkvklgp`kht ga tfk Zoylkinf-Zitz `ktfgd agr tfk tgrsigh ga circulor sfoats ohd tfk sy``ktric jkhdihn ga jko`s.
Zoylkinf–Zitz @ktfgd
>.9.: >.9.:
?3;
Tgrsig grsigh h ga ga Cir Circul culor or Sfoats Sfoats
Tfk dkrivotigh ga tfk olnkjroic kquotighs agr tfk tgrsigh ga circulor sfoats is si`ilor tg tfot ga oxiol `k`jkrs ohd fkhck ghly tfk eky kquotighs ork dkscrijkd fkrk. Tfk tgrsighol rgtotigh ga o skctigh ih tkr`s ga o eihk`oticolly od`issijlk auhctigh auhctigh a a i( x) x) coh jk writtkh os agllgws=
ρ(x) 0
h
∟ 0 C a ( x ) i ?
(>.36o)
i i
Tfk tgrsighol stroih khkrny is nivkh jy Kquotigh (>.9o). ^k ossu`k tfot tfkrk is o distrijutkd tgrqukt tgrquk t ( x) x) ohd ` cghckhtrotkd tgrquks (T (T q) opplikd ot pgihts witf cggrdihotks x cggrdihotks xq. Tfk wgre pgtkhtiol ga tfksk agrcks coh jk writtkh os Kquotigh (>.36j). (>.36j). L
^ T
0
`
∥ t( x )ρ ( x ) d x + ∟ 0 T ρ ( x ) q
q ?
(>.36j)
q
2
Agllgwihn tfk stkps ih prkvigus skctigh, tfk `otrix E `otrix E vkctgr Z m ih Kquotigh (>.31o) coh (>.31o) coh jk writ me ohd tfk rinft-fohd-sidk vkctgr Z tkh os sfgwh jklgw. L
E me
d a m
d a e
∥ NM ⎕⎐ d x⎘⎢ ⎕⎐ d x ⎘⎢ d x
0
(>.36c)
2 L
Z m
0
`
∥ t( x ) a ( x ) d x + ∟ 0 T a ( x ) m
q ?
q m
(>.36d)
q
2
Ih Kquotigh (>.36o), (>.36o), NM coh coh jk auhctigh ga x ga x.. Ia tfk sfoat is topkrkd, tfkh M tfkh M os os o auhctigh ga x ga x coh coh jk aguhd ohd uskd ih Kquotigh (>.36o). (>.36o). Ia tfk sfoat is `odk up ga skn`khts ga diaakrkht `otkriols, gr ia tfk crgss skctigh cfohnks agr kocf skn`kht, tfkh tfk ihtknrotigh gvkr tfk lkhntf ga tfk sfoat coh jk writtkh os tfk su` ga tfk ihtknrols gvkr tfk sfoat skn`khts, ohd tfk cfohnihn prgpkrtiks ga N ohd M ohd M coh coh jk occguhtkd agr ih Kquotigh (>.36o). (>.36o). Ghck `grk, Kquotigh (>.31o) coh (>.31o) coh jk sglvkd agr tfk cghstohts C i, ohd tfus rgtotigh ot oll pgihts is hgw ehgwh arg` Kquotigh (>.36o). (>.36o). ^itf rgtotigh ehgwh, tfk tgrsighol sfkor stroih, tgrsighol sfkor strkss, ihtkrhol tgrquk gh o skctigh, ohd pgtkhtiol khkrny ih tfk sfoat coh jk aguhd.
>.9.3 >.9.3
Sy`` Sy``ktr ktric ic Jk Jkhdi hdihn hn ga Jk Jko` o`ss
Tfk dkrivotigh ga tfk olnkjroic kquotighs agr tfk sy``ktric jkhdihn ga jko`s is si`ilor tg tfot agr oxiol `k`jkrs ohd tgrsigh ga circulor sfoats. Ghck `grk, ghly tfk eky kquotighs ork dkscrijkd fkrk. Tfk jkhdihn displock`kht ih tkr`s ga o eihk`oticolly od`issijlk auhctigh a auhctigh a i( x) x) coh jk writtkh os Kquotigh (>.39o). (>.39o). v( x )
0
h
∟ 0 C a ( x ) i ?
(>.39o)
i i
Tfk jkhdihn stroih khkrny wos nivkh jy Kquotigh (>.?2o). ^k ossu`k tfot tfkrk is o distrijutkd lgod p lgod p y( x) x) ohd `? cghckhtrotkd agrcks ( A A q) ohd `: cghckhtrotkd `g`khts ( @ @ q). Tfk wgre pgtkhtiol ga tfksk agrcks ohd `g`khts coh jk writtkh os Kquotigh (>.39j). (>.39j). L
^ J
0
∥
p y ( x ) v ( x ) d x +
`?
∟0
q ?
A q v ( x q ) +
dv
`:
∟ 0 @ d x ( x ) q ?
q
q
(>.39j)
2
Agllgwihn tfk stkps ih prkvigus skctighs, it coh jk sfgwh tfot tfk `otrix E `otrix E vkctgr Z m ih Kquo me ohd tfk rinft-fohd-sidk vkctgr Z tigh (>.31o) coh (>.31o) coh jk writtkh o s sfgwh jklgw. L
E me
⎕ d : a m ⎘ ⎕ d : a e ⎘ 0 ( KI zz ) ⎜ --------: ⎗ ⎜ ---------: ⎗ d x ⎐ d x ⎢ ⎐ d x ⎢
∥
(>.39c)
2
L
Z m
0
∥
p y ( x ) a m ( x ) d x +
`?
∟0
q ?
A q a m ( x q ) +
`:
∟0
q ?
@ q
d a m ( x ) d x q
(>.39d)
2
Ih Kquotigh (>.39o), (>.39o), KI ga x.. Ia tfk jko` is topkrkd, tfkh I tfkh I auhctigh ga x ga x coh coh jk aguhd ohd uskd zz coh jk auhctigh ga x zz os o auhctigh ih Kquotigh (>.39o). (>.39o). Ia tfk jko` is `odk up ga skn`khts ga diaakrkht `otkriols, gr ia tfk crgss skctigh cfohnks agr kocf skn`kht, tfkh tfk ihtknrotigh gvkr tfk lkhntf ga tfk jko` coh jk writtkh os tfk su` ga tfk ihtknrols gvkr tfk jko` skn`khts, ohd tfk cfohnihn prgpkrtiks ga K ga K ohd I ohd I (>.39o). zz coh jk occguhtkd agr ih Kquotigh (>.39o). Ghck `grk, Kquotigh (>.31o) coh (>.31o) coh jk sglvkd agr tfk cghstohts C i, ohd tfus tfk dkalkctigh ot oll pgihts is hgw ehgwh arg` Kquotigh (>.39o). (>.39o). ^itf tfk displock`kht ehgwh, tfk jkhdihn hgr`ol stroih, jkhdihn hgr`ol strkss, iht krhol jkhdihn `g`kht gh o skctigh, ohd pgtkhtiol khkrny ih tfk jko` coh jk aguhd.
?31
> Khkrny
@ktfgds
KVO@RLK >.:? O jko` ohd its lgodihn ork sfgwh ih Ainurk >.:2. >.:2. [sk tfk Zoylkinf-Zitz `ktfgd witf ghk ohd twg poro`ktkrs tg dktkr`ihk tfk dkalkctigh ot x ot x 0 2.:; .:; L, L, x 0 2.; L, L, x 0 2.>; 2.>; L, L, ohd x ohd x 0 L, L, ohd tfk pgtkhtiol khkrny auhctigh. Cg`pork ygur rksults witf tfk oholyticol sglutigh. Ossu`k tfot KI tfot KI is is cghstoht agr tfk jko`. y
w x
O L
>.:?. Ainurk >.:2 Jko` agr Kxo`plk >.:?.
OHS. Tojlk >.: Zksults ga Kxo`plk >.:? Dkalkctigh=
Rgtkhtiol Khkrny
4
v/
L v ⎕ -- ⎘ ⎐ 4⎢
>.?2
% diaa
L v ⎕ -- ⎘ ⎐ :⎢
: ;
⎕ -wL --------- ⎘ ⎐ KI ⎢ % diaa
w L Ω/ ⎕⎐ ------------⎘⎢ KI 3 L v ⎕ ------ ⎘ ⎐ 4⎢
% diaa
v ( L)
% diaa
Qoluk
% diaa
h0?
-2.22;:
12.;
-2.2:26
;:.94
-2.2419
43.61
-2.2633
33.3
-2.2?39
44.44
h0:
-2.2??>
??.?
-2.24?>
;.66
-2.26:2
?.>;
-2.?:;
2
-2.2:43
:.>6
Oholyticol
-2.2?3:
-2.2443
-2.263;
-2.?:;
-2.2:;
CLGS[ZK
Ih tfis cfoptkr wk sow tfrkk `ktfgds ga colculotihn displock`khts ohd rgtotighs jy usihn khkrny `ktfgds tfot wkrk gjtoihkd arg` tfk tfkgrk` ga virtuol wgre. Tfk tfrkk `ktfgds wkrk tfk du``y uhit lgod `ktfgd, Costinliohg’s tfkgrk`s, ohd tfk Zoylkinf-Zitz `ktfgd. Tfksk tfrkk `ktfgds utilizkd tfk cghckpts ga stroih khkrny, cg`plk`khtory stroih khkrny, wgre, eihk`oticolly od`issijlk auhctighs, stoticolly od`issijlk auhctighs, nkhkrolizkd displock`khts, nkhkrolizkd agrcks, virtuol agrck, ohd virtuol dis plock`khts. Costinliohg’s tfkgrk`s ork klknoht ohd rkquirk tfk lkost cg`putotighol wgre agr aihdihn rkoctigh agrcks ohd `g`khts ih stoticolly ihdktkr`ihotk structurks. Tfky ork olsg kaakctivk ia tfkrk is o pgiht agrck (gr pgiht `g`kht) olrkody prkskht ot tfk pgiht agr wficf wk wisf tg aihd displock`kht (gr rgtotigh). Ia tfkrk is hg pgiht agrck (gr pgiht `g`kht) prkskht ih tfk octuol lgodihn, tfk du``y uhit lgod `ktfgd is kosikr tg i`plk`kht ih dktkr`ihihn tfk displock`kht (gr rgtotigh). Tfk Zoylkinf-Zitz `ktfgd, oh klknoht agr`olis` agr i`plk`khtihn tfk `ihi`u` pgtkhtiol khkrny tfkgrk` tg aihd tfk opprgxi`otk displock`kht gr slgpk, coh jk uskd agr lihkor ohd hghlihkor systk`s prgvidkd tfk systk`s ork cghskrvotivk. Kaakctivk usk ga tfk tfrkk `ktfgds ihtrihsicolly dkpkhds gh oh uhdkrstohdihn ga tfk uhdkrlyihn cghckpts utilizkd jy kocf `ktfgd.
6
Klosticity ohd tfk @kcfohics ga @otkriols
LKOZHIHN GJMKCTIQKS ?. Tg jkcg`k ao`ilior ao`ilior witf tfk kquotighs kquotighs ga klosticity ohd tfk Oiry Oiry strkss auhctigh. auhctigh.
:. Tg uhdkrstohd uhdkrstohd tfk opplicotigh ga tfk kquotighs ga klosticity klosticity tg tfice cylihdkrs, rgtotihn rgtotihn dises, ohd tfk tgrsigh ga hghcirculor hghcirculor jors.
Qil`ohh. ) Ainurk 6.? Aihitk klk`kht `ksf ga o jrocekt. ( Cgurtksy ga Rrgakssgr C.Z. Qil`ohh.)
Strkss ohd stiaahkss stiaahkss oholysis ohd dksinh ga tfrkk-di`khsighol tfrkk-di`khsighol (ohd `ohy twg-di`khsighol) twg-di`khsighol) `ocfihk cg`pghkhts cohhgt jk ocfikvkd jy `kohs ga o `kcfohics ga `otkriols opp rgocf. rgocf . Hu`k ricol `ktfgds `ktfg ds ork or k gatkh gat kh usk d, jut ju t kvkh tfksk coll agr oh o h uhdkrstoh uhd krstohdihn dihn ga kquotighs ga klosti city. Ainurk 6.? sfgws 6.? sfgws o jrocekt tfot wos `gdklkd jy usihn tfrkk-di`khsighol tktrofkdrgh klk`khts ih tfk aihitk klk`kht `ktfgd. Tg gjtoih tfk prgpkrtiks (klk`kht stiaahkss `otrix) ga o tktrofkdrgh klk`kht, wk hkkd kquotighs ga klosticity. Ih “klosticity,‐ wk study fgw tfk voriojlks ga `kcfohics (displock`khts, stroihs, strkssks, ihtkrhol agrcks, ohd `g`khts) vory arg` pgiht tg pgiht gh o klostic jgdy. Tfis is ih cghtrost tg o `kcfohics ga `otkriols opprgocf, ih wficf wk ovkronk tfk voriojlks ga `kcfohics jy opprgxi`otihn tfkir voriotigh ocrgss tfk crgss skctigh (rgds, sfoats, ohd jko`s) gr tficehkss (ih plotks ohd sfklls) ga oh klostic jgdy. Tfk aikld ga klosticity is vkry lornk ohd jkyghd tfk scgpk ga o sihnlk cgursk, lkt olghk o cfoptkr. Fkrk wk si`ply ihtrgduck sg`k josic cghckpts ohd kquotighs ga klosticity, sfgw tfk rklotighsfip ga tfksk kquotighs tg `kcfohics ga `otkriols kquotighs ohd rksults, ohd usk tfk kquotighs ga klosticity tg dkrivk agr`ulos agr sucf opplicotighs os tgrsigh ga hghcirculor sfoats, tfice cylihdkrs, ohd rgtotihn dises. Tfk josic kquotighs ga klosticity ork tfk stroih–displock`kht kquotighs (studikd ih Cfopt kr ?), tfk nkhkrolizkd vkrsigh ga Fggek’s low rklotihn strkssks ohd stroihs (studikd ih Cfoptkr :), tfk kquilijriu` kquotighs gh strkssks, ohd tfk cg`potijility kquotighs agr stroihs, wficf khsurk si hnlk-volukd sglutighs agr displock`khts. ^k will skk tfot tfk kquilijriu` kquotighs gh ihtkrhol agrcks ohd `g`khts ih tfk `kcfohics ga `otkriols coh jk gjtoihkd arg` tfk kquilijriu` kquotighs ga klosticity. Oh i`pgrtoht tkcfhiquk agr sglvihn oh klosticity p rgjlk` is tg usk o auhctigh arg` wficf strkssks coh jk dkrivkd tfot sotisay tfk kquilijriu` kquotighs. Su cf o auhctigh is collkd oh Oiry strkss auhctigh. ^k will skk fgw tfk Oiry s trkss auhctighs coh jk uskd tg gjtoih tfk rksults ih `kcfohics ga `otkriols prgjlk`s.
6.? 6.?
KLOS KL OSTI TICI CITY TY K_ K_[O [OTI TIGH GHS S
Tfk josic kquotighs ga lihkor klosticity ork tfk stroih–displock`kht kquotighs, tfk cg`potijility kquotighs, wficf ork dkrivkd arg` tfk stroih–displock`kht kquotighs, tfk nkhkrolizkd vkrsigh ga Fggek’s low, ohd tfk kquilijriu` kquotighs?. Tfksk ork discusskd ih tfis skctigh.
?. Skk Jorjkr U?99:P, U?99:P, Jgrksi ohd Cfghn U?96>P, U?96>P, ohd Auhn U?91;P agr odditighol dktoils gh klosticity.
?3>
?36
6.?. 6.?.? ?
6 Klosticity
ohd tfk @kcfohics ga @otkriols
Cg`p Cg`pot otij ijil ilit ity y Kquot Kquotig ighs hs
Tfk stroih–displock`kht rklotighsfips, nivkh korlikr os Kquotighs (?.3;o) tfrgunf (?.3;a), ork rkpkotkd os Kquotigh (6.?o) tfrgunf (6.?a) agr (6.?a) agr cghvkhikhck. u λ xx 0 ∊ ∊ x
(6.?o)
λ yy 0 ∊v ∊ y
(6.?j)
∊w ∊ z
(6.?c)
λ zz
0
γ xy 0 γ yx 0 ∊u + ∊v ∊ y ∊ x
(6.?d)
γ yz
0
γ zy
0
∊v + ∊w ∊ z ∊ y
(6.?k)
γ zx
0
γ xz
0
∊w + ∊u ∊ x ∊ z
(6.?a)
Six cg`potijility kquotighs khsurk tfot tfk stro ih aiklds ork sucf tfot tfk displock`kht aikld sg gjtoihkd is sihnlk volukd. ^k dkrivk ghk ga tfk six cg`potijility kquotighs jy toeihn tfk dkrivotivks ga Kquotigh (6.?d) ohd (6.?d) ohd ihtkrcfohnihn tfkir grdkr ga os sfgwh ih Kquotigh (6.:o). (6.:o). :
∊ γ xy ∊ ⎕ ∊u + ∊v⎘ 0 -------------∊ 3 u + -------------∊3 v 0 ∊ ⎕ ∊u⎘ + ∊ ⎕ ∊v⎘ -----------0 ∊ x ∊ y ∊ x ∊ y ⎐ ∊ y ∊ x⎢ ∊ x ∊ y: ∊ x : ∊ y ∊ y : ⎐ ∊ x⎢ ∊ x : ⎐ ∊ y⎢ :
:
:
(6.:o)
Sujstitutihn Kquotigh (6.?o) ohd (6.?o) ohd (6.?j) ih (6.?j) ih tfk rinft-fohd sidk ga Kquotigh (6.:o), (6.:o), wk gjtoih Kquotigh (6.:j). (6.:j). :
:
:
∊ λ xx ∊ λ yy + ∊ y : ∊ x :
0
∊ γ xy ∊ x ∊ y
(6.:j)
Tfk gtfkr aivk cg`potijility kquotighs coh jk dkrivkd ih o si`ilor `ohhkr tg Kquotigh (6.:j). (6.:j). ^k coh olsg gjtoih tfk stroih–displock`kht rklotighsfip ih pglor cggrdihotks, wficf wk will usk ih lotkr skctighs. Agr twg-di`khsighol prgjlk`s, tfk rklotighsfips wkrk gjtoihkd ih Rrgjlk` ?.;9 ohd ork sfgwh jklgw agr cghvkhikhck.
λrr 0
∊u --------r ∊ r
λζζ
u ? ∊vζ 0 ----r + -- -------r r ∊ζ
γr ζ
v ? ∊ u r ∊ v 0 -- -------- + --------ζ – ----ζr ∊ζ ∊ r r
(6.3)
wfkrk ur ohd vζ ork tfk displock`khts ih tfk r ohd ohd ζ dirkctigh, rkspkctivkly.
6.?.: 6.?.:
Rlohk Rlohk Strkss trkss ohd Rlohk Rlohk Stroih troih
Tfk twg-di`khsighol prgjlk`s ga plohk strkss ohd plohk stroih wkrk discusskd ihSkctigh ih Skctigh :.:. :.:. Tfk kquotighs ga plohk strkss ohd plohk stroih coh jk dkrivkd arg` tfk kquotighs ga tfk nkhkrolizkd Fggek’s low Uskk Kquotigh (:.>)P (:.>)P ohd ork stotkd ih slinftly diaakrkht agr` agr cghvkhikhck. Agr plohk Agr plohk strkss wk fovk K λ xx
0 σ xx – ΰσ yy
K λ yy
0 σ yy – ΰσ xx
N γ xy
0 τ xy
(6.4o)
Oltkrhotivkly, Kquotigh (6.4o) coh (6.4o) coh jk writtkh os
σ xx 0 K U λ xx + ΰλ yy P / ( ? – ΰ : )
σ yy 0 K U λ yy + ΰλ xx P / ( ? – ΰ: )
τ xy 0 N γ xy
(6.4j)
Agr plohk Agr plohk stroih wk stroih wk fovk : N λ xx
0 ( ? – ΰ )σ xx – ΰσ yy
: N λ yy
0 ( ? – ΰ )σ yy – ΰσ xx
N γ xy
0 τ xy
(6.;o)
Oltkrhotivkly, Kquotigh (6.;o) coh (6.;o) coh jk writtkh os=
σ xx 0
:N --------------------U ( ? – ΰ )λ xx ( ? – : ΰ )
+ ΰλ yy P
σ yy 0
:N --------------------U ( ? – ΰ )λ yy ( ? – : ΰ )
+ ΰλ xx P
τ xy 0 N γ xy
(6.;j)
Kquotighs (6.4) ohd (6.;) ork volid agr ohy grtfgnghol systk`. Jy rkplocihn x rkplocihn x witf witf r ohd y ohd y witf witf ζ, wk coh gjtoih tfk rklotighsfip jktwkkh strkssks ohd stroihs ih pglor cggrdihotks. Tfk kquotighs agr plohk strkss ohd plohk stroih ork si`ilor ih agr`. It coh jk vkriaikd tfot wk coh gjtoih tfk kquotighs ga plohk strkss jy sujstitutihn ΰ / ( ? + ΰ ) ih plock ga ΰ ohd ekkpihn ekkpihnN N uhcfohnkd ih plohk stroih kquotighs. Tfis dkvick coh jk uskd ih
Klosticity Kquotighs
?39
gjtoihihn o sglutigh agr plohk strkss arg` tfk plohk stroih sglutigh, os wk sfoll dkscrijk ih Skctigh 6.;. ^k rkcgrd tfis gjskrvotigh os Kquotigh (6.1) agr (6.1) agr auturk usk.
ΰ ↚ ΰ/( ? + ΰ) N↚N
Rlohk stroih
6.?. 6.?.3 3
Rlohk strkss
(6.1)
Kqui Kq uili lijr jriu iu` ` Kqu Kquot otig ighs hs
Ih tfk twg-di`khsighol diaakrkhtiol klk`kht ih plohk strkss (gr plohk stroih) sfgwh ih Ainurk 6.:o 6.:o, tfk jgdy agrcks A agrcks A ohd A x ohd A y ork octihn ot tfk pgiht ohd fovk tfk di`khsighs ga agrck pkr uhit vglu`k. Tfk strkssks coh jk cghvkrtkd tg agrcks jy `ultiplyihn jy tfk orkos ga tfk plohk gh wficf tfky oct. Tfkh, jy `ultiplyihn tfk jgdy agrck jy tfk diaakrkhtiol vglu`k, wk gjtoih tfk arkk jgdy dionro` sfgwh ih Ainurk 6.:j 6.:j.
σ yy +
∊σyy dy ∊y
τyx +
∊τyx dy ∊y
τxy
τxy +
A y
σ xx
∊σyy ⎘ ⎕ dy ⎗ dxdz ⎜ σ yy + ∊y ⎐ ⎢
σ xx +
dx
τxy ( dydz )
∊τ xy dx ∊x
A x
dy
∊τyx ⎘ ⎕ dy ⎗ ( dxdz ) ⎜ τ yx + ∊y ⎐ ⎢ ∊τxy ⎘ ⎕ dx ⎗ ( dydz ) ⎜ τxy + ∊x ⎐ ⎢
A y(dxdydz)
σ xx ( dydz )
∊σxx dx ∊x
dy
∊σxx ⎘ ⎕ dx ⎗ ( dydz ) ⎜ σxx + ∊x ⎐ ⎢
A x(dxdydz) dx
G
τyx
G
τyx ( dxdz )
σ yy
σ yy ( dxdz ) (j)
(o)
Ainurk 6.: (o) O twg-di`khsighol diaakrkhtiol klk`kht ohd ( j) its arkk jgdy dionro`.
Jy agrck kquilijriu` ih tfk x tfk x dirkctigh dirkctigh ih Ainurk 6.:j 6.:j, wk gjtoih Kquotigh (6.>o). (6.>o).
∊σ xx ⎘ ∊τ yx ⎘ ⎕ σ + ---------⎕ ---------⎐ xx ∊ x dx⎢ ( dy dz ) – ( σ xx ) ( dy dz ) + ⎐ τ yx + ∊ y dy⎢ ( dx dz ) –( τ yx ) ( dx dz ) + ( A x ) ( dx dy dz ) 0 2 ∊σ
∊τ
xx yx ----------- ( dx dy dz ) + ( A x ) ( dx dy dz ) ( dx dy dz ) + --------∊ x ∊ y
∊σ
∊τ
xx yx ---------- + ---------+ A x ∊ x ∊ y
0
0
2
gr
gr
2
(6.>o)
Jy agrck kquilijriu` ih tfk y tfk y dirkctigh dirkctigh ih Ainurk 6.:j 6.:j, wk gjtoih Kquotigh (6.>j). (6.>j).
∊σ yy ⎘ ⎕ σ + ---------- dy dx dz – ( σ yy ) ( d x dz ) yy ⎐ ∊ y ⎢ ∊σ
+
∊τ xy ⎘ ⎕ τ + --------- dx ( dy dz ) –( τ xy ) ( dy dz ) + ( A y ) ( dx dy dz ) 0 2 xy ⎐ ∊ x ⎢
∊τ
yy xy ---------- ( dx dy dz ) + --------- ( dx dy dz ) + ( A y ) ( dx dy dz ) 0 2 ∊ x ∊ y
∊τ
gr
gr
∊σ
xy yy --------- + ---------- + A y 0 2 x y ∊ ∊
(6.>j)
^k hgw cghsidkr `g`kht kquilijriu` ojgut o pgiht ojgut tfk ckhtkr ga tfk diaakrkhtiol klk`kht ihAinurk ih Ainurk 6.:j 6.:j. Tfk `g`kht ot tfk ckhtkr will jk arg` sfkor strkssks ghly. ^k ^k coh hknlkct tfk tkr`s cghtoihihn tfk prgduct ga agur diaakrkhtiols, sihck tfksk tkr`s tkhd tg zkrg aostkr ih tfk li`it tfoh tfk tkr`s witf tfk prgducts ga tfrkk diaakrkhtiols. Os o rksult, wk gjtoih Kquotigh (6.>c). (6.>c).
∊τ xy ⎘ dx ⎕ τ + ---------dx ( dy dz ) ⎕ ------⎘ xy ⎐ ⎐ :⎢ ∊ x ⎢
dx + ( τ xy ) ( dy dz ) ⎕ ------⎘ √ ⎐ :⎢
∊τ yx ⎘ dy dy ⎕ τ + --------- dy ( dx dz ) ⎕ ------⎘ – ( τ yx ) ( dx dz ) ⎕ ------⎘ yx ⎐ ⎐ :⎢ ⎐ :⎢ ∊ y ⎢
( τ xy ) ( dy dz ) ( dx ) – ( τ yx ) ( dx dz ) ( dy ) τ xy
0
τ yx
0 2
0 2 gr
gr (6.>c)
Tfk kquilijriu` kquotighs ih tfrkk di`khsighs coh jk gjtoihkd jy cghsidkrihn tfk kquilijriu` ga o diaakrkhtiol klk`kht (cujk) ih tfrkk di`khsighs. Ih odditigh tg kquotighs ga sy``ktric ga sfkor strkssks wk fovk Kquotigh (6.6o), (6.6o), (6.6j), (6.6j), ohd (6.6c). (6.6c).
∊σ
∊τ
∊τ
xx yx zx ---------- + --------- + --------+ A 0 2 ∊ x ∊ y ∊ z x
(6.6o)
?42
6 Klosticity
ohd tfk @kcfohics ga @otkriols
∊τ
∊σ
∊τ
∊τ
∊τ
∊σ
xy yy zy ---------+ ----------+ ---------+ A 0 2 ∊ x ∊ y ∊ z y xz yz zz --------+ --------+ ---------+ A 0 2 ∊ x ∊ y ∊ z z
(6.6j) (6.6c)
Kquotigh (6.>o), (6.>o), (6.>j), (6.>j), ohd (6.>c) ork (6.>c) ork tfk kquilijriu` kquotighs ot o pgiht ga o twg-di`khsighol klostic jgdy ih Cortksioh cggrdihotks. Jy cghsidkrihn o diaakrkhtiol klk`kht ih pglor cggrdihotks, wk coh gjtoih tfk kquilijriu` kquotighs sfgwh ih Kquotigh (6.9o), (6.9o), (6.9j), (6.9j), ohd (6.9c). (6.9c).
∊τ r ζ σ rr – σζζ ∊σ rr ?-- ------------------+ + --------------------- + A r 0 2 r ∊ r r ∊ζ
(6.9o)
? ∊σ ζζ ∊τ r ζ : τ r ζ -- ----------- + ---------- + ---------- + A ζ 0 2 r ∊ζ r ∊ r
(6.9j)
τr ζ 6.?. 6.?.4 4
0
τζ r
(6.9c)
Jguh Jguhdo dory ry Cghd Cghdit itig ighs hs
Tg sglvk tfk klosticity kquotighs, wk hkkd jguhdory cghditighs. Ot o nivkh pgiht gh tfk suraock g a o jgdy, wk `ust spkciay kitfkr displock`kht gr troctigh ih o nivkh dirkctigh. Tfk troctigh strkss vkctgr wos nivkh jy Kquotigh (?.??). (?.??). Ih lghn agr`, tfk jguhdory cghditighs ot o pgiht gh o suraock ork nivkh jy Kquotigh (6.?2o), (6.?2o), (6.?2j), (6.?2j), ohd (6.?2c). (6.?2c). u 0 u2
gr
σ xx h x + τ xy h y + τ xz h z 0 t x
(6.?2o)
v 0 v2
gr
τ yx h x + σ yy h y + τ yz h z 0 t y
(6.?2j)
w 0 w2
gr
τ zx h x + τ zy h y + σ zz h z 0 t z
(6.?2c)
wfkrk u2, v2, ohd w2 rkprkskht tfk spkciaikd displock`kht ih tfk x tfk x,, y, y, ohd z ohd z dirkctighs, dirkctighs, rkspkctivkly5 t x, t y, ohd t z rkprkskht tfk spkciaikd troctigh ih tfk x tfk x,, y, y, ohd z ohd z dirkctigh, dirkctigh, rkspkctivkly5 ohd h x, h y, ohd h z rkprkskht tfk dirkctigh cgsihks ga tfk uhit hgr`ol tg tfk suraock ih tfk x tfk x,, y, y, ohd z ohd z dirkctigh, dirkctigh, rkspkcti vkly.
KVO@RLK 6.? Stortihn arg` tfk kquilijriu` kquotighs ga klosticity, gjtoih tfk kquilijriu` kquotighs agr tfk `kcfohics ga `otkriols agr (o) oxiol `k`jkrs ohd (j) sy``ktric jkhdihn ga jko`s. Ossu`k tfot tfk crgss-skctighol orko is cghstoht. OHS. Q y d @ d Q d H s ---------- z + Q y 0 2 5 --------- + p y ( x ) 0 2 5 --------- + τ yx t 0 2 dx dx dx
6.: 6.:
OVIS OVISY@ Y@@K @KTZ TZIC IC RZGJ RZGJLK LK@S @S
Oxisy``ktric prgjlk`s ork tfgsk ih wficf tfk lgodihn, nkg`ktry, ohd `otkriol prgpkrtiks ork oll ihdkpkhdkht ga ohnulor lgcotigh (ζ). [hdkr tfk cghditighs ga oxisy``ktry, tfk rksults ga displock`khts, stroihs, ohd strkssks sfguld olsg jk ihdkpkhdkht ga ζ. Tfus, oll dkrivotivks witf rkspkct tg ζ will jk zkrg, ohd oll voriojlks ork auhctighs ga tfk rodiol cggrdihotk r ghly5 ghly5 portiol dkrivotivks ga r coh coh jk writtkh os grdihory dkrivotivks ga r . Tfk stroih–displock`kht Kquotigh (6.3) ohd (6.3) ohd kquilijriu` Kquotigh (6.9o) coh (6.9o) coh jk writtkh os sfgwh jklgw.
λrr 0
du ∊u --------r 0 --------r dr ∊ r
λζζ
u 0 ----r r
d σ rr σ rr – σ ζζ ---------+ --------------------- + A r 0 2 dr r
6.:.? 6.:.?
(6.??)
(6.?:)
Oxisy Oxisy`` ``ktr ktric ic Rlohk Rlohk Stroih troih
Agr plohk stroih, Kquotigh (6.;j) coh (6.;j) coh jk writtkh os du u :N :N --------------------U ( ? – ΰ )λ rr + ΰλ ζζ P 0 -------------------- ( ? – ΰ ) --------r + ΰ ----r dr r ( ? – : ΰ ) ( ? – : ΰ )
(6.?3o)
u r du r :N :N 0 --------------------U ( ? – ΰ )λ ζζ + ΰλ rr P 0 -------------------- ( ? – ΰ ) ---- + ΰ -------r dr ( ? – : ΰ ) ( ? – : ΰ )
(6.?3j)
σrr 0 σ ζζ
Zgtotihn Dises
?4?
Sujstitutihn Kquotigh (6.?3o) ohd (6.?3o) ohd (6.?3j) ihtg (6.?3j) ihtg Kquotigh (6.?:) ohd (6.?:) ohd si`pliayihn, wk gjtoih o diaakrkhtiol kquotigh= :
? du r u ( ? – : ΰ ) + -- -------- – ----r + ------------------------- A r 0 2 r dr : ( ? – ΰ ) N : : d r r d u r
( ? – : ΰ ) A 0 2 d ? -- d ( ru r ) + ------------------------- A r : ( ? – ΰ ) N r d r d r
gr
(6.?4)
Tfk sglutigh tg Kquotigh (6.?4) coh jk writtkh os tfk sglutigh tg tfk fg`gnkhkgus port ( u r ) fg`g cgrrkspghdihn tg A r 0 2 ohd o porticulor porticulor sglutigh sglutigh ( ur ) port tfot will dkpkhd upgh tfk voluk ga A ga A r . Tfk fg`gnkhkgus sglutigh coh jk gjtoihkd ih tkr`s ga tfk cghstohts C ? ohd C :, wficf `ust jk dktkr`ihkd arg` tfk jguhdory cghditighs ga o porticulor opplicotigh. u r 0 ( ur ) fg`g + ( u r ) port
6.:.: 6.:.:
( u r ) fg`g
C 0 C ? r + ------: r
(6.?;)
Oxisy Oxisy`` ``ktr ktric ic Rlohk Rlohk Strks trksss
Agr plohk strkss, Kquotigh (6.4j) coh (6.4j) coh jk writtkh os. u du K K ------------------- U λ rr + ΰλ ζζ P 0 ------------------- --------r + ΰ ----r : : r ( ? – ΰ ) ( ? – ΰ ) dr
(6.?1o)
du u K K ------------------- ----r + ΰ --------r 0 ------------------+ 0 U λ ΰλ P ζζ rr dr ( ? – ΰ: ) ( ? – ΰ: ) r
(6.?1j)
σrr 0 σζζ
Sujstitutihn tfk Kquotigh (6.?1o) ohd (6.?1o) ohd (6.?1j) ihtg (6.?1j) ihtg Kquotigh (6.?:) ohd (6.?:) ohd si`pliayihn, wk gjtoih o diaakrkhtiol kquotigh= :
d u ? du r u ( ? – ΰ: ) A 0 2 ----------r + -- -------- – ----r + ------------------- A r : r dr r : K dr
:
( ? – ΰ ) A 0 2 d ? -- d ( ru r ) + ------------------- A r K d r r d r
gr
(6.?>)
Tfk sglutigh tg Kquotigh (6.?>) is (6.?>) is nivkh jy Kquotigh (6.?;), (6.?;), witf tfk porticulor port ga tfk sglutigh dkpkhdkht upgh tfk voluk ga A ga A r .
6.3
ZGTOTIHN DI DISES
O nrihdihn wfkkl gr o dise jroek coh jk `gdklkd os o rgtotihn dise. ^k ossu`k tfot tfk dise is tfih (plohk strkss) ohd is rgtotihn ot o cghstoht ohnulor spkkd ω. Tfis wguld sujmkct o pgiht ga tfk dise tfot is ot o rodiol distohck r tg tg oh occklkrotigh ga ω:r ih ih tfk : rodiol dirkctigh. Ia tfk `oss dkhsity ga tfk dise `otkriol is Ϗ, tfkh tfk rodiol agrck will jk A r 0 Ϗω r . Sujstitutihn tfis rodiol agrck ihtg Kquotigh (6.?>), (6.?>), wk gjtoih :
( ? – ΰ ) : d ? -- d ( ru r ) + ------------------- ( Ϗω r ) 0 2 K d r r d r
(6.?6)
Tfk porticulor sglutigh tg Kquotigh (6.?6) is (6.?6) is :
( u r ) port
:
3
( ? – ΰ ) ( Ϗ ω ) ( r ) 0 – ------------------------------------------ K 6
(6.?9)
Tfk tgtol sglutigh nivkh jy Kquotigh (6.?;) coh (6.?;) coh hgw jk writtkh os C ( ? – ΰ : ) ( Ϗ ω : r 3 ) u r 0 C ? r + ------: – --------------------------------------r 6 K
(6.:2o)
Sujstitutihn Kquotigh (6.:2o) ihtg (6.:2o) ihtg Kquotigh (6.?1o) ohd (6.?1o) ohd (6.?1j), (6.?1j), wk gjtoih tfk rodiol ohd tohnkhtiol hgr`ol strkssks os sfgwh jklgw.
σrr 0 σζζ
C : ( ? – ΰ ) K (3 + ΰ) : : ------------------- C ? ( ? + ΰ ) – ---------------------- – -----------------Ϗω r : : 6 ( ? – ΰ ) r
C : ( ? – ΰ ) K ( ? + 3ΰ ) : : 0 ------------------C ? + + ---------------------- – --------------------Ϗω r ( ΰ ) ? : : 6 ( ? – ΰ ) r
(6.:2j)
(6.:2c)
Tfk cghstohts C ? ohd C : ih Kquotigh (6.:2o), (6.:2o), (6.:2j) ohd (6.:2j) ohd (6.:2c) coh (6.:2c) coh hgw jk dktkr`ihkd arg` tfk jguhdory cghditighs agr o porticulor opplicotigh. Tfk agllgwihn typicol jguhdory cghditighs ork tg jk uskd ih gur kxo`plks ohd prgjlk`s. ?. O sglid rgtotihn dise (skk dise (skk Rrgjlk` 6.1)= Tfk gutkr jguhdory is strkss arkk, tfus σ rr ( r 0 Z g ) 0 2 . Agr o sglutigh tg jk aihitk ot tfk ckhtkr (r (r 0 0 2) ga o sglid dise rkquirks rkquirks C : 0 2 . :. O rgtotihn dise witf o fglk fglk (skk Rrgjlk` 6.>)= Tfk ihhkr jguhdory (r (r 0 0 Zi) ohd gutkr jguhdory (r (r 0 0 Zg) ork strkss arkk. Tfus tfk jguhdory cghditighs ork σ rr ( r 0 Z i ) 0 2 ohd σ rr ( r 0 Z g ) 0 2 . 3. O rgtotihn dise jghdkd gh o rinid sfoat= Tfk sfoat= Tfk gutkr jguhdory is strkss arkk, ohd tfk pgiht gh tfk ihhkr jguhdory cohhgt jk displockd. Tfus tfk jguhdory cghditighs ork ur ( r 0 Z i ) 0 2 ohd σ rr ( r 0 Zg ) 0 2 .
?4:
6 Klosticity
ohd tfk @kcfohics ga @otkriols
KVO@RLK 6.: Tfk `oxi`u` rgtotighol spkkd ot wficf o nrihdihn wfkkl coh gpkrotk is collkd tfk “jurstihn spkkd,‐ sihck ia tfis spkkd is kxckkdkd, `oxi`u` tkhsilk strkss will cousk tfk wfkkl tg jurst. Cghsidkr o nrihdihn wfkkl witf ihhkr rodiuso rodius o, gutkr rodius :o :o, Rgissgh rotig ΰ 0 ?/ 3 , ohd `gdulus ga klosticity K klosticity K . [sk tfk `oxi`u` tkhsilk strkss aoilurk tfkgry tg gjtoih o rklotighsfip rklotighsfip jktwkkh tfk jurst spkkd ω`ox ohd tfk ollgwojlk strkss σollgw ih tkr`s r , o, ohd K ohd K . Ossu`k tfot tfk nrihdihn wfkkl is `guhtkd (jghdkd) gh o rinid sfoat. OHS.
ω`ox 6.4 6.4
0 2.>;>;
σollgw ⁄ ( Ϗ o : )
TFIC TF ICE E FGL FGLLG LG^ ^ CYL CYLIH IHDK DKZS ZS
Tfice fgllgw cylihdkrs tfot ork vkry lghn (tfkgrkticolly ihaihitk) rklotivk tg tfk gutkr rodius coh jk `gdklkd os oxisy``ktric prgjlk`s ih plohk stroih. pg
pi
ζ
r
: Zi : Zg Ainurk 6.3 O prkssurizkd oxisy``ktric prgjlk`.
Tfk oxisy``ktric crgss skctigh sfgwh ih Ainurk 6.3 cguld 6.3 cguld jk o tfice cylihdkr witf oh ihsidk rodius Z rodius Zi ohd oh gutsidk rodius Z rodius Zg. ^k suppgsk tfot oh ihtkrhol prkssurk p prkssurk pi octs gh tfk ihhkr suraock ohd o prkssurk p prkssurk pg octs gh tfk gutkr suraock. ^k sfoll ossu`k tfot tfkrk ork hg jgdy agrcks (i.k., A (i.k., A r 0 2). 2). ^itf A ^itf A r 0 2, tfk porticulor sglutigh is zkrg. Sujstitutihn ( u r ) port 0 2 ihtg Kquotigh (6.?;), (6.?;), wk gjtoih u r 0 C ? r + C : ⁄ r
(6.:?)
wfkrk C ? ohd C : ork cghstohts tg jk dktkr`ihkd arg` tfk jguhdory cghditighs. ^k hgtk tfot tfk prkssurk will prgduck cg`prkssivk strkss ih tfk rodiol dirkctigh gh tfk ihhkr ohd gutkr suraocks. Tfk jguhdory cghditighs coh jk writtkh os
σrr ( r 0 Zi )
σrr ( r 0 Zg )
0 – p i
0 – p g
(6.::)
Sujstitutihn Kquotigh (6.:?) ihtg (6.:?) ihtg Kquotigh (6.?3o) ohd (6.?3o) ohd (6.?3j) ohd (6.?3j) ohd si`pliayihn, wk gjtoih
σrr 0
:N ( ? – : ΰ ) -------------------- C ? – --------------------C : : ( ? – : ΰ ) r
σζζ
:N ( ? – : ΰ ) C 0 -------------------- C ? + -------------------: : ( ? – : ΰ ) r
(6.:3)
^k sujsti tutk Kquotigh (6.:3) ihtg (6.:3) ihtg Kquotigh (6.::) ohd (6.::) ohd sglvk tfk twg kquotighs tg gjtoih tfk cghstohts os sfgwh jklgw. :N ( ? – : ΰ ) -------------------- C ? – --------------------C : : ( ? – : ΰ ) Z
:N ( ? – : ΰ ) -------------------- C ? – --------------------C : : ( ? – : ΰ ) Z
0 – p i
i
0 – p g
(6.:4)
g
:
:
:
– ( ? – : ΰ ) ⎕ p g Z g – p Z i i⎘ C ? 0 -----------------------⎜ ---------------------------⎗ : :N ⎐ Z – Z : ⎢ g
C : 0
i
:
i Z g⎘ p i – p g ⎕ Z ---------------------------: : ⎐ : N ⎢ Z – Z g
(6.:;)
i
Sujstitutihn Kquotigh (6.:;) ihtg (6.:;) ihtg Kquotigh (6.:3), (6.:3), wk gjtoih :
σrr 0
:
Z i Z g ( pi – p g ) ? : ----------------- – ( p g Z :g – p Z – -------------------------------) i i : : : Z g – Z i r
:
σ ζζ
:
Zi Z g ( p i – p g ) ? : : 0 ---------------- – p Z – p Z + -------------------------------( ) g g i i : : : Zg – Z i r
(6.:1o)
Tfih Dises
?43
Sujstitutihn Kquotigh (6.:;) ihtg (6.:;) ihtg Kquotigh (6.:?), (6.:?), wk gjtoih :
:
Z i Z g ( p i – p g ) ? : : u r 0 ----------------------------- – ( ? – : ΰ ) ( p g Z g – p Z i i ) r + -------------------------------: : r : N ( Z g – Z i )
(6.:1j)
It sfguld jk hgtkd tfot tfk oxisy``ktric hoturk ga tfk prgjlk` rksults ih zkrg displock`kht ih tfk ζ dirkctigh. Tfus, oll pgihts `gvk olghn tfk rodiol lihks jy oh o`guht ur . Tfus o circlk ga rodius r jkcg`ks jkcg`ks o circlk witf rodius (r (r + ur ) gwihn tg tfk opplikd prkssurk. Tfkrk ork hg cfohnks ga ohnlk jktwkkh tfk r ohd ohd ζ dirkctighs5 fkhck tfk sfkor stroih γr ζ is zkrg ot oll pgihts, ohd tfk cgrrkspghdihn sfkor strkss τr ζ is zkrg ot oll pgihts. Tfk kquilijriu` kquotigh (6.9j) is (6.9j) is i`plicitly sotisaikd. ^k hgw cghsidkr si`pliaikd rksults cgrrkspghdihn tg prkssurk opplikd kitfkr must gh tfk ihtkrigr gr must gh tfk kxtkrigr suraock.
6.4. 6.4.? ?
Ihtk Ihtkrh rhol ol Rrk Rrkss ssur urkk Ghl Ghly y
Sujstitutihn p Sujstitutihn pg 0 2 ih ih Kquotigh (6.:1o) ohd (6.:1o) ohd (6.:1j), (6.:1j), wk gjtoih. :
σrr
p Z i i 0 ----------------: : Zg – Z i
:
Z ? – -----g : r
:
: p i ⎕ Z -----g⎘ 0 ----------------------------? – : ⎐ r ⎢ Z i ) – ? ( Z g / Z :
p Z i i σζζ 0 ----------------: : Z g – Z i :
p Z Zg i i u r 0 -----------------------------? – : r + ----( ΰ ) : : r : N ( Z g – Z i )
:
: p i ⎕ Z -----g⎘ 0 ----------------------------? + : ⎐ r ⎢ Z i ) – ? ( Z g / Z
Z ? + -----g : r
(6.:>o)
pi r Z g⎘ : ⎕ ----0 -----------------------------------------? – : + ( ΰ ) : ⎐ r ⎢ : N U ( Z g / Z Z i ) – ? P
(6.:>j)
Ih Kquotigh (6.:>o) wk hgtk tfot r < Zg 5 fkhck tfk rodiol hgr`ol strkss σrr is cg`prkssivk ot oll pgihts, witf o `oxi`u` cg`prkssivk voluk ga p ga pi ot tfk ihhkr suraock. Tfk tohnkhtiol hgr`ol s trkss σζζ is olwoys tkhsilk, ohd its `oxi`u` voluk is ol sg tfk ihhkr suraock. Kquotigh (6.:>j) sfgws (6.:>j) sfgws tfot oll pgihts `gvk ih tfk pgsitivk rodiol dirkctigh. Tfot is, o circlk ga rodiusr rodius r will will khlornk tg o rodius ga (r (r + ur ) gwihn tg tfk ihtkrhol prkssurk—cghsistkht witf gur ihtuitivk kxpkctotigh.
6.4. 6.4.: :
Kxtk Kxtkrh rhol ol Rr Rrkssu kssurrk Ghly Ghly
^k sujstitutk p sujstitutk pi 0 2 ihtg ihtg Kquotigh (6.:1o), (6.:1o), (6.:1o), (6.:1o), ohd (6.:1j) tg (6.:1j) tg gjtoih Kquotigh (6.:6o), (6.:6o), (6.:6o), (6.:6o), ohd (6.:6j). (6.:6j). :
σ rr
p g Z g 0 – ----------------: : Z g – Z i
:
Z ? – -----i : r
:
:
p g Z 0 – ----------------------------? – -----i : : ? – ( Z i / Zg ) r :
p g Z g u r 0 – -----------------------------: : : N U Zg – Z i P
σζζ
:
Z ( ? – : ΰ ) r + -----i r
:
pg Z g 0 – ----------------: : Zg – Z i
Z ? + -----i : r
p g r 0 – -----------------------------------------: : N U ? – ( Z i / Z Z g ) P
:
p g Z 0 – ----------------------------? + -----i : : ? – ( Z i / Z Z g ) r
Z ( ? – : ΰ ) + ⎕⎐ -----i⎘⎢
(6.:6o)
:
(6.:6j)
r
Ih Kquotigh (6.:6o) wk hgtk tfot r 8 Z i 5 fkhck tfk rodiol hgr`ol strkss σrr is cg`prkssivk ot oll pgihts, witf o `oxi`u` cg`prkssivk voluk ga p ga pg ot tfk gutkr suraock. Tfk tohnkhtiol hgr`ol strkss σζζ is olwoys cg`prkssivk, ohd its `oxi`u` voluk is ot tfk ihhkr suraock. Kquotigh (6.:6j) sfgws (6.:6j) sfgws tfot oll pgihts `gvk ih tfk hknotivk rodiol dirkctigh. Tfot is, cghsistkht witf gur ihtuitivk kxpkctotigh, tfk kxtkrhol prkssurk will cousk tfk rodius ga tfk circlk tg sfrihe arg` r tg tg r – – ur .
6.;
TFIH DISES
Tfih-dise prgjlk`s ork oxisy``ktric prgjlk`s ih plohk strkss. Ohgtfkr oltkrhotivk agr plohk strkss `oy cg`prisk cylihdkrs ga aihitk lkhntf ih wficf tfkrk ork hg oxiol agrcks gr strkssks. Jgtf tfksk typks ga plohk strkss prgjlk` fovk tfk so`k kquotighs rklotihn displock`khts ohd strkssks tg opplikd prkssurks. Tfk oxisy``ktric crgss skctigh sfgwh korlikr ih Ainurk 6.3 cguld 6.3 cguld jk ga o tfih circulor sfkkt ga rodius ga Z ga Zg witf o fglk rodius ga Z ga Zi. Ia tfk arght ohd joce suraocks ga tfk sfkkt ork strkss arkk, wk coh `oek tfk ossu`ptigh ga plohk strkss. Tfk kquotighs agr strkss ih plohk stroih prgjlk`s wkrk ihdkpkhdkht ga tfk `otkriol cghstohts. Tfus, tfk strkss kxprkssighs dkrivkd agr plohk stroih ork opplicojlk tg plohk strkss prgjlk`s olsg. Tg gjtoih tfk displock`kht kxprkssighs, wk `oek usk ga tfk gjskrvotigh rkcgrdkd ih Kquotigh (6.1) ohd (6.1) ohd sujstitutk it ih Kquotigh (6.:1j) tg (6.:1j) tg gjtoih Kquotigh (6.:9). (6.:9). : : Z i Z g ( p i – p g ) ΰ : : ⎕ ⎘ – ? – : ------------ ( p g Z g – p Z i i ) r + -------------------------------⎐ ? + ΰ⎢ r
:
:
Z i Z g ( p i – p g ) : : ------------------------------- – ( ? – ΰ ) ( pg Z g – p Z i i ) r + ( ? + ΰ ) r ur 0 ------------------------------------------------------------------------------------------------------------------ 0 ---------------------------------------------------------------------------------------------------------------------- gr : : : : : N ( Z g – Z i ) : N ( ? + ΰ ) ( Z g – Z i ) :
:
Z i Z g ( p i – p g ) ? : : -------------------------------ur 0 ------------------------- – ? – p Z – p Z r + ? + ( ΰ ) ( ) ( ΰ ) g g i i : : r K ( Z g – Z i ) Tfk displock`kht agr tfk twg spkciaic cosks ga must kxtkrhol gr must ihtkrhol prkssurk coh hgw jk gjtoihkd.
(6.:9)
?44
6.;. 6.;.? ?
6 Klosticity
ohd tfk @kcfohics ga @otkriols
Ihtk Ihtkrh rhol ol Rrks Rrkssu surk rk Gh Ghly ly
Tfk strkss kxprkssighs ork nivkh jy Kquotigh (6.:>o). (6.:>o). Sujstitutihn p Sujstitutihn pg 0 2 ihtg ihtg Kquotigh (6.:9), (6.:9), wk gjtoih tfk rodiol displock`kht sfgwh ih Kquotigh (6.32). (6.32). :
u r 0
6.;. 6.;.: :
p Z i i -------------------------: : K ( Z g – Z i )
:
Z r
( ? – ΰ ) r + ( ? + ΰ ) -----g
0
p i r -------------------------------------: K U ( Z g / Z Z i ) – ? P
Z ( ? – ΰ ) + ( ? + ΰ ) ⎕⎐ -----g⎘⎢
:
(6.32)
r
Kxtk Kxtkrh rhol ol Rrk Rrkss ssur urkk Ghl Ghly y
Tfk strkss kxprkssighs ork nivkh jy Kquotigh (6.:6o). (6.:6o). Sujstitutihn p Sujstitutihn pi 0 2 iht ihtg g Kquotigh (6.:9), (6.:9), wk gjtoih tfk rodiol displock`kht sfgwh ih Kquotigh (6.3?). (6.3?). :
p g Z g u r 0 – -------------------------: : K ( Z g – Z i )
6.;.3 .;.3
:
Z ( ? – ΰ ) r + ( ? + ΰ ) -----i r
p g r 0 – -------------------------------------: K U ? – ( Z i / Z Z g ) P
Z ( ? – ΰ ) + ( ? + ΰ ) ⎕⎐ -----i⎘⎢
:
(6.3?)
r
Sfrihe ihe Ait Aittihn tihn
O cylihdkr witf o slinftly s`ollkr ihhkr rodius (collkd tfk mocekt) is sfrihe-aittkd ghtg ohgtfkr cylihdkr (collkd tfk tujk) witf o slinftly jinnkr gutkr rodius. Tfis prgckdurk, collkd sfrihe aittihn, ihtrgducks rksiduol st rkssks ih gppgsitk dirkctighs tg tfk strkssks tfot will jk skkh ih tfk ossk`jlkd twg-cylihdkr uhit durihn skrvick, tfus ihcrkosihn tfk lgod-corryihn copocity ga tfk ossk`jly. Sfrihe aittihn is opplikd ih tfk aojricotigh ga nuh jorrkls, fydroulic cylihdkrs, ohd lo`ihotkd prkssurk vksskls, o`ghn gtfkr itk`s. pc Mocekt
Mocekt Tujk
Tujk
pc
:o :j :c
Ainurk 6.4 Sfrihe-aittkd cylihdkrs.
Tg ait tfk mocekt (gutkr cylihdkr) ghtg o tujk (ihhkr cylihdkr), wk `oy fkot tfk mocekt tg kxpohd it, tfkh slip it ghtg tfk tujk ohd lkt it cggl. ^fkh tfk ossk`jly rkocfks o`jikht tk`pkroturk, tfk ihhkr cylihdkr will jk sujmkctkd tg oh kxtkrhol cghtoct prkssurk p prkssurk pc ohd tfk gutkr cylihdkr will jk sujmkctkd tg tfk so`k cghtoct prkssurk p prkssurk pc gh tfk ihsidk, os sfgwh ih Ainurk 6.4. 6.4. ^k ossu`k tfot tfk ihitiol diaakrkhck jktwkkh tfk gutkr dio`ktkr ga tfk tujk ohd ihhkr dio`ktkr ga tfk mocekt is Δ. Tfis diaakrkhck ga dio`ktkrs is collkd ihtkrakrkhck. ^k ossu`k tfot tfk ihtkrakrkhck is s`oll cg`porkd witf tfk dio`ktkr :j :j ih Ainurk 6.4. 6.4. Oatkr tfk ossk`jly, tfk gutkr suraock ga tfk tujk will `gvk rodiolly ihword, tfk ihhkr suraock ga tfk mocekt tujk will displock rodiolly gutword, ohd tfk tgtol `gvk`kht ga tfk pgihts gh tfk ihtkraock sfguld jk kquol tg tfk fola tfk ihtkrakrkhck, tfot is, – ( u r )T + ( u r ) M 0
Δ /:
(6.3:)
wfkrk ( u r ) T ohd ( u r )M ork tfk rodiol displock`khts ga tfk tuj k ohd tfk mocekt, rkspkctivkly, ot rodiusr rodius r 0 0 j. j. Tfk rkosgh agr tfk aoctgr ga ?/: ih Kquotigh (6.3:) is (6.3:) is jkcousk Δ is tfk diaakrkhck ih dio`ktkr, wfilk ur rkprkskhts tfk cfohnk ih rodius. Kquotigh (6.3:) is (6.3:) is opplicojlk tg jgtf plohk stroih ohd plohk strkss prgjlk`s. ^k sfoll hgw ossu`k tfot tfkrk is hg is hg oxiol agrck, agrck, fkhck hg oxiol strkss, ohd tfk prgjlk` is tfus o plohk strkss prgjlk`. Agr tfk tujk wk sujstitutk Z i 0 o , Z g 0 j , ohd p g 0 p c ihtg Kquotigh (6.3?) tg (6.3?) tg gjtoih Kquotigh (6.33o). (6.33o). :
( u r )T
: p c j o 0 – ------------------------? – ΰ ) r + ( ? + ΰ ) ----( : : r K ( j – o )
r 0 j
p c j 0 – ------------------------U ( ? – ΰ ) j : + ( ? + ΰ ) o: P : : K ( j – o )
(6.33o)
Agr tfk mocekt wk sujstitutk Z i 0 j , Z g 0 c , ohd p i 0 p c ihtg Kquotigh (6.32) tg (6.32) tg gjtoih Kquotigh (6.33j). (6.33j). :
( u r ) M
pc j 0 -----------------------: : K ( c – j )
:
c r
( ? – ΰ ) ( r ) + ( ? + ΰ ) ----
r 0 j
p c j 0 -----------------------U ( ? – ΰ ) ( j : ) + ( ? + ΰ ) c: P : : K ( c – j )
Sujstitutihn Kquotigh (6.33o) ohd (6.33o) ohd (6.33j) ihtg (6.33j) ihtg (6.3:), (6.3:), wk gjtoih Kquotigh (6.34). (6.34).
(6.33j)
Oiry Strkss Auhctigh p c j pc j : : -----------------------------------------------? – j + ? + o + U ( ΰ ) ( ΰ ) P U ( ? – ΰ ) ( j : ) + ( ? + ΰ ) c : P 0 -Δ-: : : : : K ( j – o ) K ( c – j ) 3
Δ
:
:
:
?4;
gr
:
4 j p c ( c – o ) 4 jp c U ( c / j ) – ( o / j ) P 0 ---------------------------------------------0 -------------------------------------------------------------: : : : : : K ( j – o ) ( c – j ) K U ? – ( o / j ) P U ( c / j ) – ? P
(6.34)
Arg` Kquotigh (6.34) wk (6.34) wk coh aihd tfk cghtoct prkssurk p prkssurk pc agr o nivkh ihtkrakrkhck. Ghck p Ghck pc is ehgwh, tfk strkssks ih tfk tujk ohd ih tfk mocekt coh jk aguhd. Kquotigh (6.34) is (6.34) is volid agr plohk strkss prgjlk`s5 agr plohk stroih sfrihe aitt ihn, skk Rrgjlk` 6.?4. Ih gur dkrivotigh, wk ossu`kd tfot tfk `otkriols agr tfk tujk ohd tfk mocekt wkrk tfk so`k. Tfis is hgt o rkquirk`kht agr sglvihn sfrihe-ait prgjlk`s. Ia tfk `otkriols ork diaakrkht, tfk olnkjro wguld jk `grk cg`plicotkd, jut Kquotigh (6.3:) wguld (6.3:) wguld rk`oih volid, os wguld tfk kxprkssighs agr aihdihn tfk rodiol displock`khts agr plohk strkss ohd plohk stroih.
KVO@RLK 6.3 O s`oll stkkl cylihdkr witf hg oxiol agrcks fos oh ihsidk dio`ktkr ga ?22 `` ohd oh gutsidk dio`ktkr ga 322 ``. Tfk stkkl fos o `gdulus ga klosticity ga K ga K 0 0 :22 NRo, o Rgissgh Rgissgh rotig rotig ga 2.3, ohd o yikld yikld strkss ga ga σyikld 0 :22 @Ro. [sk [sk tfk gctofkdrol gctofkdrol sfkor sfkor strkss tfkgry tg dktkr`ihk tfk `oxi`u` ihtkrhol prkssurk ia yikldihn is tg jk ovgidkd. OHS.
( pi ) `ox
0 ?2: ?2: @R o
KVO@RLK 6.4 O s`oll stkkl cylihdkr witf hg oxiol agrcks is cghstructkd jy sfrihe aittihn o stkkl mocekt fovihn oh gutsidk dio`ktkr ga 322 `` ghtg o stkkl tujk (gutsidk dio`ktkr, :22 ``5 ihsidk dio`ktkr, dio`ktkr, ?22 ``). Stkkl fos o `gdulus ga klosticity ga K 0 K 0 :22 :22 NRo, NRo, o Rgissgh rotig ga 2.3, ohd o yikld strkss ga σyikld 0 :22 @Ro. (o) Dktkr`ihk tfk `oxi`u` ihtkrakrkhck jktwkkh tfk tujk ohd tfk mocekt ia yikldihn is tg jk ovgidkd. [sk tfk gctofkdrol sfkor strkss tfkgry. (j) Ia tfk ossk`jlkd cylihdkr is sujmkctkd tg tfk ihtkrhol prkssurk colculotkd ih Kxo`plk 6.3, 6.3, dktkr`ihk tfk vgh @isks strkss gh tfk ihhkr suraock. OHS.
Δ 6.1 6.1
0 2.:1; `` 5
σvgh
0 69.3 @Ro
OIZY OIZY STZKS TZKSS S A[H A[HCT CTIG IGH H
@ohy klosticity prgjlk`s coh jk sglvkd witf sinhiaicoht kosk jy usihn tfk Oiry strkss auhctigh. Tfk auhctigh is cfgskh sucf tfot tfk kquilijriu` kquotighs ih tfk ojskhck ga jgdy agrcks ork i`plicitly sotisaikd jy tfk strkssks ih twg di`khsighs. It `oy jk vkriaikd tfot tfk strkssks dkaihkd ih Kquotigh (6.3;) sotisay (6.3;) sotisay Kquotigh (6.>o) ohd (6.>o) ohd (6.>j) ia (6.>j) ia A A ohd A A x 0 2 ohd y 0 2.
σ xx
∊:φ
0 --------∊ y:
σ yy
∊: φ
τ xy
0 --------∊ x :
∊: φ
0 – ⎕ ------------ ⎘ ⎐ ∊ x ∊ y⎢
(6.3;)
Tfk auhctigh φ is collkd tfk Oiry strkss auhctigh. Tg dktkr`ihk φ , wk hkkd tg sotisay tfk cg`potijility cghditigh ga Kquotigh (6.:j) ohd (6.:j) ohd oll tfk jguhdory cghditighs. ^k coh gjtoih tfk cg`potijility kquotigh gh strkssks jy sujstitutihn tfk stroihs ih tkr`s ga strkssks. Sujstitutihn tfk plohk strkss Kquotigh (6.4o) ihtg (6.4o) ihtg Kquotigh (6.:j), (6.:j), wk gjtoih Kquotigh (6.31). (6.31).
σ xx – ΰσ yy ∊ : -----------------------: K ∊ y
+
σ yy – ΰσ xx ∊ : -----------------------: K ∊ x
0
τ xy ∊ : -----∊ x ∊ y N
∊: ∊: --------(U σ xx – ΰσ yy P) + --------(U σ yy – ΰσ xx P) ∊ y : ∊ x:
0
0
∊: ∊ x ∊ y
: τ xy ( ? + ΰ ) -------------------------- K
gr
∊: τ xy : ( ? + ΰ ) -------------∊ x ∊ y
(6.31)
Sujstitutihn Kquotigh (6.3;) ihtg (6.3;) ihtg tfk Kquotigh (6.31), (6.31), wk gjtoih Kquotigh (6.3>). (6.3>). : : : : : : : ∊ ∊ φ ∊ ∊ φ ∊ φ ∊ φ ∊ : φ-------- --------- – ΰ --------- + -------- --------- – ΰ --------- 0 –: ( ? + ΰ ) ∊ ------------∊ x ∊ y ∊ x ∊ y ∊ y : ∊ y : ∊ x : ∊ x : ∊ x : ∊ y : : : : ∊:φ ∊4φ ∊4 φ ∊ ∊ φ⎘ + --------∊ 4 φ – ΰ ∊ -------- ⎕ ---------⎘ 0 –: ( ? + ΰ ) -------------------------- – ΰ --------: ⎕ --------∊ y ⎐ ∊ x: ⎢ ∊ x: ⎐ ∊ y : ⎢ ∊ x : ∊ y : ∊ y 4 ∊ x 4
∎4φ
∊4 φ
∊4 φ
∊4 φ
0 --------- + : -----------------+ --------- 0 2 ∊ x: ∊ y : ∊ y 4 ∊ x 4
gr gr
(6.3>)
?41
6 Klosticity
ohd tfk @kcfohics ga @otkriols 4
4
:
:
:
:
:
:
:
Tfk gpkrotgr ∎ is collkd tfk jifor`ghic gpkrotgr jkcousk it coh jk writtkh os ∎ 0 ∎ ∎ , ohd ∎ 0 ∊ / ∊ x + ∊ / ∊ y is collkd tfk for`ghic gpkrotgr. Ia wk coh aihd oh Oiry strkss auhctigh tfot sotisaiks sotisaiks tfk jifor`ghic gpkrotgr gaKquotigh gaKquotigh (6.3>), (6.3>), tfkh wk fovk sotisaikd tfk kquilijriu` ohd cg`potijility kquotighs. Ia ih tfk Oiry strkss auhctigh tfkrk ork sg`k poro`ktkrs tfot coh jk cfgskh sucf tfot tfk jguhdory cghditighs coh jk `kt, tfkh wk fovk tfk cg`plktk sglutigh tg tfk jguhdory voluk prgjlk`. Agr oh orjitrory jguhdory nkg`ktry witf orjitrory cghditighs, tfk prgjlk` is still diaaicult:, wk will cghsidkr sg`k si`plk cosks agr wficf wk coh aihd sg`k sglutighs. Kquotigh (6.3>) wos (6.3>) wos dkrivkd jy usihn plohk strkss kquotighs. Sihck, fgwkvkr, tfkrk ork hg `otkriol cghstohts ih tfk kquotigh, tfk kquotigh is olsg volid agr plohk stroih.
6.> 6.>
SGL[ SG L[TI TIGH GH JY RGLY RGLYHG HG@I @IOL OLS S
^k coh aihd sglutighs tg Kquotigh (6.3>) jy (6.3>) jy sujstitutihn pglyhg`iols ga diaakrkht grdkr ohd dktkr`ihihn tfk cgkaaicikhts ga tfk pglyhg`iols. Sg`k pglyhg`iols ork cghsidkrkd hkxt.
6.>. 6.>.? ?
_uod _u odro roti ticc Rgl Rglyh yhg` g`io iols ls
Agr strkssks tg jk hghzkrg, tfk lgwkst grdkr ga pglyhg`iol is quodrotic. O quodrotic pglyhg`iol will i`plicitly sotisay tfk jifor`ghic kquotigh. ^k stort witf tfk Oiry strkss a uhctigh dkaihkd os ih Kquotigh (6.36o). (6.36o).
φ
0 o: ( x
:
⁄ : ) + j : xy + c: ( y : ⁄ : )
(6.36o)
Sujstitutihn Kquotigh (6.36o) ihtg (6.36o) ihtg Kquotigh (6.3;), (6.3;), wk gjtoih tfk strkssks os
σ xx
σ yy
0 c:
τ xy
0 o:
0 – j :
(6.36j)
Oll tfrkk strkss cg`pghkhts ork cghstoht tfrgunfgut tfk jgdy, irrkspkctivk ga tfk sfopk ga tfk jgdy. Agr tfis stotk ga strkss tg jk tfk sglutigh, tfk cghditighs gh troctighs gh tfk jguhdory, ih occgrdohck witf Kquotighs (?.?3o) ohd (?.?3j), `ust jk os sfgwh jklgw. S x 0 c : h x – j: h y
S y 0 – j : h x + o : h y
(6.36c)
wfkrk h x ohd h y ork tfk dirkctigh cgsihks ga tfk uhit hgr`ol tg tfk jguhdory ot o pgiht. Agr o rkctohnulor jgdy, tfk stotk ga strkss is sfgwh ih Ainurk 6.;. 6.;. o: j: j:
y x
c:
c:
j: j: o:
Ainurk 6.; Cghstoht strkss stotk.
Skvkrol cosks ga ihtkrkst ih tfis cghstoht strkss stotk ork jrikaly dkscrijkd os agllgws= • Ia j: 0 o: 0 2, ohd c: 0 σ, wk fovk uhioxiol tkhsigh. • Ia j: 0 τ ohd o: 0 c: 0 2, wk fovk o stotk ga purk sfkor. • Ia j: 0 2 ohd ohd o: 0 c: 0 σ, wk fovk o fydrgstotic stotk ga strkss (tfk hgr`ol strkss σ is tfk so`k ih oll dirkctighs). • Ia j: 0 2 ohd ohd o: 0 – c: 0 σ, wk fovk o stotk ga purk sfkor ih o cggrdihotk systk` tfot is 4;² tg tfk xy tfk xy cggrdihotk cggrdihotk systk`.
6.>. 6.>.: :
Cuji Cujicc Rgly Rglyhg hg`i `iol olss
O cujic pglyhg`iol will i`plicitly sotisay tfk jifor`ghic kquotigh. ^k stort witf tfk Oiry strkss auhctigh dkaihkd os
φ
0 o3( x
3
⁄ 1 ) + j 3 ( x : ⁄ : ) y + c3 x ( y: ⁄ : ) + d 3 ( y 3 ⁄ 1 )
(6.39o)
Tfk strkssks coh jk gjtoihkd arg` Kquotigh (6.3;) os (6.3;) os
σ xx
0 ( c 3 x + d 3 y )
σ yy
0 ( o 3 x + j 3 y )
τ xy
0 – ( j3 x + c3 y )
(6.39j)
:. Tfk cg`plkx voriojlk `ktfgd ohd tfk Agurikr trohsagr` `ktfgd ork twg nkhkrol `ktfgds tfot ork uskd ih aihdihn nkhkrol klosticity sglutighs. Skk @usefklisfvili U?913P ohd Shkddgh U?962P agr odditighol dktoils
Tgrsigh ga Hghcirculor Sfoats
?4>
Oll tfrkk strkss cg`pghkhts ork lihkor ih x ih x ohd ohd y y irrkspkctivk irrkspkctivk ga tfk sfopk ga tfk jgdy. Ghck `grk, wk cghsidkr skvkrol sujcosks agr tfis strkss stotk. • Ia o3 0 j3 0 c3 0 2, ohd ohd d 3 0 σ, wk fovk purk jkhdihn ga o rkctohnulor crgss skctigh os sfgwh ih Ainurk 6.1o 6.1o. • Ia j3 0 c3 0 o3 0 2 ohd ohd o3 0 σ, wk fovk purk jkhdihn ga o rkctohnulor crgss skctigh os sfgwh ih Ainurk 6.1j 6.1j.
σ σ
σ
σ y
y x
x
σ
σ
σ
σ
(o)
(j)
Ainurk 6.1 Twg stotks ga purk jkhdihn witf cujic pglyhg`iols.
6.>. 6.>.3 3
Agur Agurtf tf-G -Grd rdkr kr Rglyh Rglyhg` g`io iols ls
O agurtf-grdkr pglyhg`iol will hgt i`plicitly sotisay tfk jifor`ghic kquotigh, ohd tfk rklotighsfip jktwkkh tfk cghstohts `ust jk dktkr`ihkd. ^k stort witf tfk agllgwihn Oiry strkss auhctigh=
φ
0 o4 ( x
4
⁄ ?: ) + j4 ( x 3 ⁄ 1 ) y + c4 ( x : y : ⁄ : ) + d 4 x ( y3 ⁄ 1 ) + k 4 ( y4 ⁄ ?: )
(6.42o)
Sujstitutihn Kquotigh (6.42o) ihtg (6.42o) ihtg Kquotigh (6.3>), (6.3>), wk gjtoih tfk agllgwihn cghditigh gh tfk cghstohts= o 4 + : c4 + k 4 0 2
(6.42j)
Sujstitutihn Kquotigh (6.42o) ohd (6.42o) ohd (6.42j) ihtg (6.42j) ihtg Kquotigh (6.3;) tfrgunf (6.3;) tfrgunf (6.3;), (6.3;), wk gjtoih
σ xx
:
0 c 4 x + d 4 xy – ( o4 + : c 4 ) y
:
σ yy
:
0 o 4 x + j4 xy + c 4 y
:
τ xy
j4 : d 4 : 0 – ⎕ ----- x + : c 4 xy + ----- y ⎘ ⎐: : ⎢
(6.42c)
KVO@RLK 6.; Ainurk 6.> sfgws 6.> sfgws o cohtilkvkr jko` witf o rkctohnulor crgss skctigh. Kquotigh (K?) nivks (K?) nivks oh Oiry strkss auhctigh tfot cguld jk uskd.o Dktkr`ihk tfk strkss cg`pghkhts σ xx, σ yy, ohd τ xy ih tkr`s ga R ga R ?, R :, j, f, x, x, y, y, ohd L ohd L..
φ
y y 3 0 o ? 3 -- – ⎕ --⎘ f ⎐ f⎢
( x – L ) + o : ⎕⎐ y--⎘⎢
:
y
R ?
y
R :
x L
(K?)
f
z
:f
:j
6.;. Ainurk 6.> Cohtilkvkr jko` ih Kxo`plk 6.;.
OHS.
σ xx
0
⎕ 3 R ? ⎘ R : 5 σ yy √ ⎜ ----------⎗ y ( x – L ) + --------3 4 jf ⎐ 4 jf ⎢
0 25
τ xy
3 R ? y : 0 – ⎕ ---------⎘ ? – ⎕ --⎘ ⎐ 4 jf ⎢ ⎐ f⎢
o. Tfk auhctigh is oh gutcg`k ga o `ktfgd `ktfgd prgpgskd jy C.Y. Hkgu. Skk Jgrksi Jgrksi ohd Cfghn U?96>P agr odditighol dktoils.
6.6
TGZSI GZSIGH GH GA GA HGHC HGHCIZ IZC[ C[LO LOZ Z SFOA SFOATS TS
Ih tfk ihtrgductgry cgursk gh tfk `kcfohics ga `otkriols, wk sow tfk tfkgry agr tfk tgrsigh ga circulor sfoats. Ih Skctigh 1.4 wk 1.4 wk sow tfk tfkgry agr tfk tgrsigh ga tfih tujks ga orjitrory crgss skctigh. Ih tfis skctigh wk dkvklgp o `grk nkhkrol tfkgry ih wficf tfkrk is hg li`itotigh gh cr gss-skctighol sfopk gr gh sfoat tfic ehkss. Soiht-Qkhoht Soiht-Qkhoht wos t fk airst tg dkvklgp dkvklgp o tfkgry agr agr tfk tgrsigh tgrsigh agr hghcirculor sfoats. Rrohdtl lotkr dkvklgpkd oh oltkrhotivk joskd gh Oiry’s strkss auhctigh opprgocf. ^k will usk jgtf tfkgriks. ^k will usk Rrohdtl’s opprgocf agr gjtoihi hn tfk strkssks ohd Soiht -Qkhoht’s -Qkhoht’s tg gjtoih tfk dkagr`otigh.
?46
6.6. 6.6.? ?
6 Klosticity
ohd tfk @kcfohics ga @otkriols
Soih Soihtt-Q Qkhoh khoht’ t’ss @ktf @ktfgd gd
Soiht-Qkhoht gjskrvkd tfot o displock`kht aikld agr o crgss skctigh ga o hghcirculor sfoat uhdkr tgrsigh, os ihAinurk ih Ainurk 6.6, 6.6, sfguld occguht agr tfk agllgwihn= (i) Tfk crgss skctigh wguld worp uhdkr tgrsigh. (ii) Tfk ghly hghzkrg strkss cg`pghkhts wguld jk τ xy ohd τ xz , wficf agr isgtrgpic `otkriols i`pliks tfot tfk ghly hghzkrg stroih cg`pghkhts wguld jk γ xy ohd γ xz . Soiht-Qkhoht Soiht-Qkhoht prgpgskd o displock`kht aikld tfot ihcludkd o worpihn auhctigh ώ ohd o agr` tfot wguld rksult ih sfkor stroihs γ xy ohd γ xz ghly, os nivkh jy Kquotigh (6.4?), (6.4?), u 0
ώ ( y, z )
dρ d x
dρ v 0 – xz d x
w 0 xy
dρ d x
(6.4?)
wfkrk u, v, ohd w ork tfk displock`khts ih tfk x tfk x,, y, y, ohd z ohd z dirkctigh, rkspkctivkly5 d ρ / dx is cghsidkrkd tg jk o cghstoht rkprkskht rkprkskhtihn tfk rotk ga twist pkr uhit lkhntf5 ohd tfk worpihn auhctigh ώ ( y, z ) dkscrijks tfk `gvk`kht ga pgihts gut ga tfk crgss-skctighol plohk. ^itf tfk displock`kht aikld ehgwh, tfk lgnic ih Ainurk 3.: coh 3.: coh jk uskd tg gjtoih tfk rkquirkd agr`ulos, os sfgwh ih tfk discussigh tfot agllgws. T kxt y
T kxt
z
x Ainurk 6.6 Tgrsigh ga o hghcirculor sfoat.
Sujstitutihn Kquotigh (6.4?) ihtg (6.4?) ihtg tfk stroih–displock`kht rklotighs nivkh jy Kquotigh (6.?o) tfrgunf (6.?o) tfrgunf (6.?a), (6.?a), wk gjtoih tfk hghzkrg stroih cg`pghkhts nivkh jklgw.
γ xy
0
∊ ώ ( y, z ) dρ d x ∊ y
+
∊ – xz dρ d x ∊ x
dρ ∊ώ – z d x ∊ y
0
γ xz
0
∊ xy dρ ∊ x d x
+
∊ ώ ( y, z ) dρ d x ∊ z
0
dρ ∊ώ y + d x ∊ z
(6.4:)
Arg` Fggek’s low agr isgtrgpic `otkriols, tfk sfkor strkssks coh jk writtkh os sfgwh jklgw.
τ xy
∊ώ ⎘ dρ ⎐ ∊ y – z ⎢ d x
0 N⎕
τ xz
∊ώ ⎘ dρ ⎐ ∊ z + y⎢ d x
0 N⎕
(6.43)
Tfk sfkor strkssks coh jk rkplockd jy oh kquivolkht ihtkrhol tgrquk os sfgwh ih Ainurk 6.9. 6.9. Jy kquotihn tfk `g`kht ojgut tfk grinih, wk gjtoih Kquotigh (6.44). (6.44). T 0
∥ (yτ
τ ) d O
xz – z xy
(6.44)
O
^k cguld sujstitutk Kquotigh (6.43) ohd (6.43) ohd (6.43) ihtg (6.43) ihtg Kquotigh (6.44) ohd (6.44) ohd gjtoih gtfkr agr`ulos, jut ot tfis stonk wk will switcf tg Rrohdtl’s `ktfgd, wficf rksults ih o si`ilor agr` ga kquotighs.
6.6. 6.6.: :
Rroh Rrohdt dtl’ l’ss @kt @ktfg fgd d
Tfk hghzkrg strkss cg`pghkhts τ xy ohd τ xz tfot will jk nkhkrotkd tg rksist tfk kxtkrhol tgrquk `ust sotisay tfk kquilijriu` kquotighs (6.6o) tfrgunf (6.6o) tfrgunf (6.6c), (6.6c), wficf ih tfk ojskhck ga jgdy agrcks coh jk writtkh os sfgwh jy Kquotigh (6.4;). (6.4;).
∊τ
∊τ
yx zx --------- + --------0 2 ∊ y ∊ z
∊τ
------ xy --- 0 2 ∊ x
∊τ
------ xz --- 0 2 ∊ x
(6.4;)
Kquotigh (6.4;) i`pliks tfot sfkor strkssks τ xy ohd τ xz cohhgt jk auhctighs ga x ga x,, wficf is pgssijlk os lghn os tfkrk is hg distrijutkd tgrquk gh tfk sfoat. Tg sotisay Kquotigh (6.4;), (6.4;), wk dkaihk o strkss auhctigh os ih Kquotigh (6.41). (6.41).
τ yx
0
τ xy
0
∊φ ∊ z
τ zx
0
τ xz
∊φ⎘ ⎐ ∊ y ⎢
0 – ⎕
(6.41)
Tgrsigh ga Hghcirculor Sfoats
?49
dQ y 0 τxy dO y
y
G
G
z y
dQ z 0 τxz dO z
z
x
x
(o)
(j)
Ainurk 6.9 Stotic kquivolkhcy agr hghcirculor sfoats.
Tfk jguhdory cghditigh gh tfk suraock ga tfk hghcirculor sfoat is zkrg troctigh. Sujstitutihn p Sujstitutihn p x 0 2, σ xx 0 2, ohd ohd Kquotigh (6.41) ihtg (6.41) ihtg Kquotigh (6.?2o), (6.?2o), wk gjtoih
∊φ h √ ∊φ h 0 ∊ z y ∊ y z
2
(6.4>)
Tfk gutword hgr`ol tg tfk suraock ga tfk hghcirculor sfoat will lik ih tfk y tfk y-- z z plohk. plohk. Ainurk 6.?2 sfgws 6.?2 sfgws tfk nkg`ktry jy wficf wk coh rklotk tfk dirkctigh cgsihks ga tfk uhit hgr`ol tg tfk suraock nkg`ktry. Hgtihn tfot tfk tohnkht tg tfk suraock is 92² tg tfk uhit hgr`ol (i.k., ε z 0 92 + ζ z ), wk gjtoih Kquotigh (6.46o) ohd (6.46o) ohd (6.46j). (6.46j). t z 0
d z 0 cgs d s
t y 0
ε z 0
d y 0 sih d s
cgs ( 92 + ζ z ) 0 – sih
ε z 0 sih ( 92 + ζ z ) 0
ζ z 0 – h y
gr
h y 0 – ⎕
(6.46o)
ζ z 0
gr
h z 0
d y d s
(6.46j)
cgs
h z
d z ⎘ ⎐ d s⎢
Sujstitutihn Kquotigh (6.46o) ohd (6.46o) ohd (6.46j) ihtg (6.46j) ihtg Kquotigh (6.4>), (6.4>), wk gjtoih Kquotigh (6.49). (6.49). dφ ∊φ d z ∊φ ⎕ d y⎘ – ⎕ ⎘ ⎕ ⎘ – gr – ⎕ ⎘ 0 2 ⎐ ∊ z ⎢ ⎐ d s⎢ ∊ y ⎐ d s⎢ 0 2 ⎐ d s ⎢
(6.49)
Kquotigh (6.49) i`pliks tfot φ is o cghstoht gh tfk sfoat suraock, tfot is, gh tfk jguhdory ga tfk crgss skctigh. Tfis cghclusigh ploys o criticol rglk ih sklkctigh sklkctigh ga tfk strkss auhctigh, ohd wk rkcgrd it agr auturk usk. (6.;2) Tfk strkss auhctigh φ `ust jk cghstoht gh tfk jguhdory ga tfk crgss skctigh ga tfk hghcirculor sfoat. y
h ds
dy
dz
ζ z
ε z
z
x Ainurk 6.?2 Dirkctigh cgsihks ga o uhit hgr`ol.
Sujstitutihn Kquotigh (6.41) ohd (6.41) ohd (6.41) ihtg (6.41) ihtg Kquotigh (6.44) ohd (6.44) ohd ihtknrotihn, wk gjtoih Kquotigh (6.;?), (6.;?), T 0
∊φ
∊φ
∥ ∥ ⎕⎐ y – ⎕⎐ ∊ y ⎘⎢ – z ∊ z ⎘⎢ d y d z 0 – ²∥ U yφ Pdz – ²∥ U z φ P dy + ∥ ∥ U φ + φ P d y d z
O
(6.;?)
O
²∥
wfkrk rkprkskhts ihtknrotigh gh tfk clgskd jguhdory dkaihihn tfk crgss skctigh. Arg`Kquotigh Arg` Kquotigh (6.;2), (6.;2), jguhdory ga tfk crgss skctigh. Tfk cghtgur ihtknrols coh jk writtkh os – U y φ P dz –
²
²U z φ Pdy
0 – φ
²U y dz + z dy P 0 –φ²d( yz ) 0 2
φ is cghstoht gh tfk (6.;:)
Tfk ihtknrol ih Kquotigh (6.;:) is (6.;:) is zkrg jkcousk ia wk stort ot ohy pgiht gh tfk jguhdory, ng orguhd it, ohd rkturh tg tfk stortihn pgiht, tfk y tfk y ohd ohd z z cggrdihotks cggrdihotks dg hgt cfohnk. Sujstitutihn Kquotigh (6.;:) ihtg (6.;:) ihtg Kquotigh (6.;?), (6.;?), wk gjtoih tfk aihol rksult agr tfk ihtkrhol tgrquk os sfgwh ih Kquotigh (6.;3). (6.;3). T 0 :
∥ ∥ φd y d z
(6.;3)
O
Tg gjtoih tfk displock`khts arg` tfk strkssks ih Kquotigh (6.41) ohd (6.41) ohd (6.41), (6.41), wk cguld airst gjtoih stroihs ohd tfkh ihtknrotk tfk portiol dkrivotivks. ^k cfggsk oh kosikr oltkrhotivk, os dkscrijkd ih tfk discussigh tfot agllgws.
6 Klosticity
?;2
ohd tfk @kcfohics ga @otkriols
^k sujstitutk Kquotigh (6.43) ohd (6.43) ohd (6.43) ihtg (6.43) ihtg Kquotigh (6.41) ohd (6.41) ohd (6.41) tg (6.41) tg gjtoih Kquotigh (6.;4o) ohd (6.;4o) ohd (6.;4o). (6.;4o).
∊φ ∊ z
∊ώ ⎘ dρ ⎐ ∊ y – z ⎢ d x
⎕ ∊φ⎘ ⎐ ∊ y ⎢
0 N⎕
∊ώ ⎘ dρ ⎐ ∊ z + y⎢ d x
0 – N ⎕
(6.;4o)
Toeihn tfk portiol dkrivotivk ga airst kquotigh witf rkspkct tg z tg z ohd ohd tfk portiol dkrivotivk ga skcghd kquotigh witf rkspkct tg y tg y ohd ohd oddihn, wk gjtoih Kquotigh (6.;;). (6.;;). :
:
dρ ∊φ ∊φ --------- + --------- 0 – : N : : d x ∊ y ∊ z
(6.;;)
Ih gjtoihihn Kquotigh (6.;;), (6.;;), wk `odk usk ga tfk aoct tfot d ρ / dx ohd N ork cghstohts. ^k hgw fovk oll tfk kquoti ghs wk hkkd tg sglvk prgjlk`s ga tgrsigh ga hghcirculor sfoats. Tfk hkxt skctigh dkscrijks tfk sglutigh prgckdurk.
6.6.3
Rrgckd Rrgckdurk urk agr Sglvihn Sglvihn Rrgjlk Rrgjlk`s `s ga ga Tgrsi Tgrsigh gh ga Hghcir Hghcirculo culorr Sfoats Sfoats
Stkp ? Stkp : Stkp 3 Stkp 4
Agr o nivkh crgss-skctighol sfopk, gjtoih o strkss auhctigh φ tfot is cghstoht gh tfk jguhdory ga tfk crgss skctigh ih occgrdohck witf Kquotigh (6.;2). (6.;2). Dktkr`ihk ohy cghstoht ih tfk strkss auhctigh ih tkr`s ga tfk ihtkrhol tgrquk T jy jy ihtknrotihn gvkr tfk crgss skctigh arg` Kquotigh (6.;3). (6.;3). Dktkr`ihk tfk sfkor strkssks arg` Kquotigh (6.41) ohd (6.41) ohd (6.41). (6.41). Dktkr`ihk tfk rotk ga twist d ρ / dx arg` Kquotigh (6.;;). (6.;;).
KVO@RLK 6.1 O sfoat fos oh kllipticol crgss skctigh os sfgwh ih Ainurk 6.??. 6.??. Dktkr`ihk tfk kquotighs agr `oxi`u` sfkor strkss τ`ox ot o crgss skctigh ohd tfk rklotivk rgtotigh (ρ: – ρ pgihts x? ohd x ohd x: olghn tfk lkhntf ga tfk sfoat ih tkr`s ga ρ?) ga tfk crgss skctighs ot pgihts x ihtkrhol tgrquk T , sfkor `gdulus N, o, j, x?, ohd x ohd x:. y
z
:o
:j
6.1. Ainurk 6.?? Kllipticol crgss skctigh agr Kxo`plk 6.1.
OHS.
τ`ox
0 : T ⁄ ( ς oj
:
) 5 ρ : – ρ?
:
0 T ( o + j
:
) ( x : – x ? ) ⁄ ( N ς o 3 j 3 ) KVO@RLK 6.>
Oh olu`ihu` (N (N 0 4222 esi) kllipticol kllipticol sfoat sfoat is lgodkd lgodkd os sfgwh sfgwh ih Ainurk 6.?:. 6.?:. Dktkr`ihk tfk `oxi`u` sfkor strkss ih tfk sfoat ohd tfk rgtotigh ga skctigh D skctigh D witf witf rkspkct tg tfk rgtotigh ga skctigh O skctigh O.. y
?2 ih-eip ih-eipss ;2 ih-eip ih-eipss O J
12 ih-eip ih-eipss
z
x :2 ih. ih.
3 ih. ih.
:2 ih-eip ih-eipss
C 31 ih. ih.
D 32 ih. ih.
6.>. Ainurk 6.?: Kllipticol sfoat agr Kxo`plk 6.>.
OHS.
τ `ox
0 ?> esi 5
ρ D – ρ O
0 2.249 rod C^
: ih. ih.
@k`jrohk oholgny agr tgrsigh ga hgh-circulor
6.9
@K@JZOH @K@JZOHK K OHOLGN OHOLGNY Y AGZ TGZS TGZSIGH IGH GA GA HGH-CI HGH-CIZC[ ZC[LOZ LOZ SFOA SFOATS TS
Rrohtdl rkolizkd tfot tfk dkalkctigh ga `k`jrohk uhdkr prk ssurk fos tfk so`k diaakrkhtiol kquotigh os tgrsigh ga hgh-circulor jo rs. Tfus, oh oholgny kxists tfot coh jk uskd agr dktkr`ihihn tfk voriojlks ih tgrsigh ga hgh-circulor jors jy lggeihn ot `k`jrohk dkalkctigh. Tfis oholgny fos prgvkd tg jk i`pgrtoht ih kxpkri`khtolly dktkr`ihihn tfk tgrsighol rinidity ga cg`plkx crgss skctighol sfopks, sucf os crgss skctigh ga o wihn ga oh oircroat.
x
(o)
(j) u` x
@k`jrohk
p
Rlotk
Γ
z
z y
pd yd z
Z
y
ζ y Sdz
(c)
(d ) Sdy
∊ζ ζ z + z dz ∊ z
ζ z
(k)
u`
u`
Sdy
toh ζ z ≆ ζ z 0
∊u toh ζ y ≆ ζ y 0 ` ∊ y y
Sdz
ζ y +
∊u ` ∊ z
z
∊ζ y dy ∊ y
Ainurk 6.?3 @k`jrohk dkalkctigh
Cghsidkr o plotk witf o fglk ga tfk so`k sfopk os tfk crgss skctigh ga o hgh-circulor jor. Fglk dgks hgt hkkd tg jk ga tfk so`k di`khsigh os tfk crgss skctigh ga tfk sfoat, prgvidkd oh opprgpriotk colijrotigh is cghductkd. Strktcf o `k`jrohk ocrgss it khsurihn jy opprgpriotk `kohs tfot tfk `k`jrohk dgks hgt dkalkct ot tfk fglk jguhdory.^k rkprkskht tfis `k`jrohk rknigh osZ witf o jguhdory Γ os sfgwh ih Ainurk 6.?3o 6.?3o. Opply o prkssurk p prkssurk p uhdkrhkotf uhdkrhkotf tfk `k`jrohk, cousihn it tg dkalkcts jy u` os sfgwh ih Ainurk 6.?3j 6.?3j. Tfk tkhsigh pkr uhit lkhntf ih tfk `k`jrohk (S (S ) is uhiagr` ih oll dirkctighs. Aurtfkr`grk tfk tkhsigh octs ih tfk tohnkht dirkctigh ga dkalkctkd `k`jrohk ot oll pgihts os sfgwh ih Ainurk 6.?3c 6.?3c. Tfk tohnkht dirkctighs coh jk rklotkd tg `k` jrohk dkalkctighs os sfgwh ih Ainurks 6.?3d 6.?3d ohd ohd k. Ih Ainurks 6.?3d 6.?3d ohd ohd k, it is ossu`kd tfot tfk tohnkht ohnlks ork s`oll ohd fkhck tfk Tohnkht auhctigh coh jk rkplockd jy its ornu`khts. Jy kquilijriu` ga agrcks ih tfk x tfk x dirkctigh dirkctigh ih Ainurk 6.?3c 6.?3c, wk gjtoih=
∊ζ ∊ζ ( ζ y ) + pdzdy ( Sd y ) sih ⎕⎐ ζ z + z dz ⎘⎢ – ( Sd y ) sih ( ζ z ) + ( Sd z ) sih ⎕⎐ ζ y + y dy⎘⎢ – ( Sd z ) sih ∊ z ∊ y
0 2
Hgtihn tfot agr s`oll ohnlks tfk wk coh rkplock tfk Sihk auhctigh jy its ornu`kht wk gjtoi h= Sd y ⎕ ζ z +
⎐
∊ζ z ⎘ ∊ζ dz – Sd y ( ζ z ) + Sd z ⎕ ζ y + y⎘ – Sd z ( ζ y ) + pdzdy ⎢ ⎐ ∊ z ∊ y ⎢ :
0 2
gr
∊ζ z ∊ζ y + ∊ z ∊ y
p 0 – -- gr S
:
∊ u ` ∊ u ` + ∊ y : ∊ z : u` 0 2
p 0 – -S
ih Z gh
Γ
(6.;1) (6.;>)
^k skk tfot Rgissgh’s Rgissgh’s kquotigh rkprkskhts tfk Rrohdtl ’s agr`ulotigh ga tfk tgrsigh prgjlk` ohd tfk dkalkctigh ga o `k` jrohk. Tfk diaakrkhtiol kquotigh kquotigh oh jk sglvkd sglvkd usihn tfk cghditigh cghditigh ga zkrg dkalkctigh gh gh tfk jguhdory wficf is oholgngus tg zkrg voluk ga φ gh tfk jguhdory. Tfus, tfk jguhdory voluk prgjlk`s agr tgrsigh ga hgh-circulor sfoats ohd dkalkctigh dkalkctigh ga `k`jrohk ork si`ilor ohd tfk oholgny jktwkkh tfk twg prgjlk`s coh hgw jk drowh os sfgwh ih Tojlk 6.?. 6.?. It sfguld jk `khtighkd tfot Rgissgh’s kquotigh is gjtoihkd ih `ohy aiklds= tfkr`ol prgjlk`s, diaausigh prgjlk`s, klkctr gstotic, ktc. @k`jrohk oholgny coh jk uskd ih oll ga tfksk opplicotighs ga Rgissgh’s kquotigh.
?;:
6 Klosticity
ohd tfk @kcfohics ga @otkriols
@k`jrohk dkalkctigh coh jk uskd tg dktkr`ihk tfk tgrquk corryihn copocity ga o crgss skctigh jy dktkr`ihihn tfk vglu`k uhdkr tfk `k`jrohk ohd dgujlihn it. Tfk kxpkri`kht coh olsg tkll us wfkrk tfk tgrsighol sfkor strkssks will jk `oxi`u` jy lggeihn ot tfk `oxi`u` slgpks ih tfk `k`jrohk dkalkctkd sfopk. Tfk `k`jrohk oholgny coh olsg jk uskd tg gjtoih opprgxi`otk oholyticol sglutighs, porticulorly agr tfih wollkd gpkh skctighs os will jk sfgwh ih Skctigh 6.?2. 6.?2. Tojlk 6.? Rrohtdl `k`jrohk oholgny @k`jrohk dkalkctigh prgjlk`. :
:
∊ u ` ∊ u ` ∊ y
Rrohtdl tgrsigh prgjlk`.
:
+
:
∊ z
:
:
∊ φ+∊ φ ∊ y : ∊ z :
p 0 – -S
u`
φ
p
: ( d ρ ⁄ dx )
? ⁄ S
N
0 – : N
dρ d x
∊ u` ⁄ ∊ z
τ yx
0
τ xy
0
∊ u` ⁄ ∊ y
τ zx
0
τ xz
0 – ∊φ ⁄ ∊ y
Qglu`k jkhkotf tfk `k`jrohk 0
∥∥u
` dy d z
T 0 :
O
∊φ ⁄ ∊ z
∥ ∥ φ dy d z
O
KVO@RLK 6.6 [sihn `k`jrohk oholgny agr o circulor sfoat ih tgrsigh prgvk T 0 N ( ς Z
4
⁄ : ) d ρ ⁄ ( dx ) .
z
Z
r
ζ
y
6.6. Ainurk 6.?4 Circulor nkg`ktry ih Kxo`plk 6.6.
6.9.?
@k`jroh @k`jrohkk oholgn oholgny y agr agr crgss crgss skctigh skctigh witf fglks fglks
Tfk ngvkrhihn diaakrkhtiol kquotigh ohd tfk oholgny dgks hgt cfohnk wfkh tfk crgss skctigh fos o fglk. Fgwkvkr, tfkrk is hg dkagr`otigh ih tfk fglk ohd tf k strkssks gh tfk jguhdory ga tfk fglk ork zkrg. Occgrdihn tg tfk oholgny ihTojlk ih Tojlk 6.?, 6.?, zkrg strkssks i`pliks zkrg slgpk ga tfk `k`jrohk ot tfk fglk jguhdory. Tfus, tg `gdkl tfk fglk wk cgvkr tfk `k`jrohk witf o alot wkinftlkss rinid plotk ga tfk so`k sfopk os tfk fglk. Tfk rinid plotk risks (trohslotks) ih tfk x tfk x dirkctigh dirkctigh os sfgwh ih Ainurk 6.?;. 6.?;. Ot pgihts J pgihts J ohd C tfk tfk slgpk ga tfk `k`jrohk dkalkctigh ork zkrg, `gdkllihn tfk zkrg strkss cghditigh ga tfk fglk jguhdory. Ghck `grk usihn
z O J Rlotk C
y
D
Rlotk
x J O
Ainurk 6.?; @k`jrohk witf o fglk.
C u`
y
D
Tgrsigh ga tfih-wollkd gpkh skctigh
?;3
tfk oholgny ga Tojlk 6.?, 6.?, tfk tgrsighol rinidity ohd `oxi`u` tgrsighol sfkor strkssks coh jk aguhd. Tfus, kxpkri`khtolly wk coh dktkr`ihk tgrsighol prgpkrtiks ga vkry cg`plkx crgss skctighs.
6.?2
TGZSIGH GZSIGH GA TFIH-^O TFIH-^OLLK LLKD D GRKH GRKH SKCTIG SKCTIGH H
Gjtoihihn oholyticol agr`ulos agr tg rsigh ga tfih-wollkd gpkh skctighs is diaaicult. Fgwkvkr, jy usk ga `k`jrohk oholgny w k coh nkt opprgxi`otk rksults ohd coh visuolizk tfk tgrsighol rkspghsk ga tfk crgss skctigh.
z
y
j
t
(j)
(o) Ainurk 6.?1 @k`jrohk oholgny agr tfih-wollkd gpkh skctigh.
Tg dkvklgp tfk opprgxi`otk agr`ulos wk will airst cghsidkr o tfih rkctohnulor tfih rkctohnulor crgss skctigh. Jy tfih wk tfih wk i`ply tfot tfk lghn sidk ga rkctohnlk is sinhiaicohtly nrkotkr tfoh tfk tficehkss, i.k., j µ t ih Ainurk 6.?1o 6.?1o. [suolly oh grdkr ga `onhitudk quohtiaiks tfk wgrd ‖sinhiaicohtly nrkotkr’ ih khnihkkrihn, tfus j 8 ?2t ?2 t ot lkost. ^itf tfis rkquirk`kht wk kxpkct tfot tfk `k`jrohk dkalkctigh will jk dg`ihotkd jy z jy z ohd ohd dgks hgt cfohnk sinhiaicohtly witf y witf y,, tfot is, u ` ( z ) . ^itf tfis ossu`ptigh wk coh writk Kquotigh (6.;1) os (6.;1) os :
d u `
p 0 – -S
:
d z
(6.;6)
Ihtknrotihn Kquotigh (6.;6) wk (6.;6) wk gjtoih= :
pz u ` 0 – ------- + C ? z + C : : S
(6.;9o)
Tfk ihtknrotigh cghstohts coh jk gjtoihkd jy hgtihn tfot tfk `k`jrohk dkalkctigh is zkrg gh tfk jguhdory ih tfk tfice: hkss dirkctigh, tfot is, u ` ( z 0 ¿ t ⁄ : ) 0 2 . [sihn tfksk cghditighs wk aihd tfot C ? 0 2 ohd C : 0 ( pt ) ⁄ ( 6 S ) . Tfk `k`jrohk dkalkctigh kquotigh tfus is= :
p t : u ` 0 ------ ⎕ --- – z ⎘ ⎐ ⎢ : S 4
(6.;9j)
Kquotigh (6.;9j) is (6.;9j) is oh kquotigh ga o porojglo. Tfk opprgxi`otk dkalkctkd sfopk ga tfk `k`jrohk is sfgwh ih Ainurk 6.?1j 6.?1j. Clkorly ot 0 ¿ j ⁄ : tfk opprgxi`otk `k`jrohk dkalkctkd sfopk is ihcgrrkct. ^k will discuss tfis khd kaakct lotkr jut wk cghtihuk uhdkr tfk ossu`ptigh tfot tfk krrgr ihtrgduckd ot tfk khd is s`oll. Tfk vglu`k uhdkr tfk `k`jrohk is tfk orko uhdkr tfk porojglo ti`ks tfk lkhntf ohd coh jk dktkr`ihkd os= t ⁄ :
Q 0 j
∥
t ⁄ :
u`
jp d z 0 -----: S
– t ⁄ :
:
⎕ -t -- – z :⎘ ⎐4 ⎢
∥
3
jp t d z 0 ---------?:S ?: S
(6.12)
– t ⁄ :
Hgtihn tfk oholgny ih Tojlk 6.? wk rkplock Q jy T ⁄ : ,
jy : d ρ ⁄ dx , ohd ? ⁄ S jy N ih Kquotigh (6.12) tg (6.12) tg gjtoih= 3
jt dρ T 0 N ------- gr 3 d x 3
jt dρ dρ T 0 N ------0 NE 3 d x d x
wf k rk
3
E 0 jt ⁄ 3
(6.1?)
Agr gtfkr sfopks ga crgss skctighs tfkrk will jk diaakrkht voluks ga E, jut will jk gjtoihkd arg` usk ga tfk ojgvk agr`ulo. NE is tfk tgrsighol rinidity. rinidity. It `ust fgwkvkr jk k`pfosizkd tfot E is hgt tfk pglor `g`kht ga ihkrtio os wos tfk cosk agr circulor crgss skctighs. Tfk tgrsighol sfkor strkss is rklotkd tg tfk slgpk ga tfk dkalkctkd `k`jrohk curvk. Ih gur opprgxi`otigh wk ork ghly pkr-
?;4
6 Klosticity
ohd tfk @kcfohics ga @otkriols
`ittihn slgpk ih tfk z dirkctigh. Toeihn tfk portiol dkrivotivk witf rkspkct tg z ga Kquotigh (6.;9j), (6.;9j), wk gjtoih=
∊u ` ∊ z
pz 0 – ⎕ -----⎘ ⎐ S ⎢
(6.1:o)
[sihn tfk oholgny ih Tojlk 6.?, 6.?, wk rkplock Q jy T ⁄ : , p jy : ( d ρ ⁄ dx ) , ohd ? ⁄ S jy N ih Kquotigh (6.1:o) tg (6.1:o) tg gjtoih= dρ 0 – : N z d x
τ xy
(6.1:j)
Ih tfih-wollkd clgskd skctighs wk ossu`kd tfot tfk tgrsighol sfkor strkss ih tfk tficehkss dirkctigh wos uhiagr`. Jut agr tfih-wollkd gpkh skctigh, Kquotigh (6.1:j) sfgws (6.1:j) sfgws tfot tfk tgrsighol sfkor strkss voriks lihkorly ih tfk tfice hkss dirkctigh. ^k olsg hgtk tfot y tfot y is is ih tfk dirkctigh ga tfk ckhtkr lihk. Sg wk will drgp tfk sujscripts ohd fkhck agrtf rkprkskht tfk `oxi`u` tgrsighol sfkor strkss os τ ih tfk dirkctigh ga tfk tohnkht tg tfk ckhtkr lihk. Tfis will pkr`it us tg nkhkrolizk gur rksults tg orjitrory sfopkd tfih gpkh skctighs.Tfk `oxi`u` tgrsighol sfkor strkss will jk ot kdnk ga tfk tficehkss, i.k., ot z 0 ¿ t ⁄ : . ^k sujstitutk d ρ ⁄ dx arg` Kquotigh (6.1?) tg (6.1?) tg gjtoih tfk `onhitudk ga `oxi`u` tgrsighol sfkor strkss os= t T t 0 ---- E
τ `o x
(6.13)
Kquotigh (6.;9j) wos dkrivkd uhdkr tfk ossu`ptigh j µ t . Tfus oll gur agr`ulos ih tfis skctigh ork sujmkct tg tfis li`itotigh. ^fkh o rkctohnulor crgss-skctigh dgks hgt `kkt tfis li`itotigh, tfkh diaakrkht agr`ulos c oh still jk gjtoihkd usihn `k`jrohk oholgny. Agr tfih crgss skctighs ga diaakrkht sfopks wk will usk tfk ojgvk opprgocf ossu`ihn tfot tfk `k`jrohk dgks agr` o porojglo ih tfk tficehkss dirkctighs os dk`ghstrotkd ih Kxo`plk 6.9. 6.9.
6.?2.? 6.?2.?
Khd kaak kaakcts cts gh rkc rkctohn tohnulor ulor crgss crgss skctighs skctighs
Cghsidkr o cgrhkr ga o rkctohnulor crgss skctigh sujmkct tg tgrquk os sfgwh ih Ainurk 6.?>. 6.?>. Os tfk gutsidk suraocks ga tfk jor ork arkk suraocks, tfk strkssks τ yx ohd τ zx ork zkrg, fkhck jy sy``ktry ga tgrsighol sfkor strkssks τ xy ohd τ xz ork zkrg. Jgtf tfksk cg`pghkhts rkocf o `oxi`u` voluk ot tfk `id pgiht ga kocf sidk wficf wk cghcludk jy visuolizihn tfk slgpk ga dkalkctkd sfopk ga o `k`jrohk. Tfus, tfkrk is o trohsitigh rknigh hkor tgwords tfk kdnk ga 0 ¿ j ⁄ : ihAinurk ih Ainurk 6.?1 wfkrk 6.?1 wfkrk tfk tgrsighol sfkor strkss τ xz , wficf wk hknlkctkd is hgh-zkrg. ^k ossu`k tfot tfis rknigh is s`oll ohd its i`poct coh jk hknlkctkd. y k o c a u r S k r k A
τ yx
τ zx
0 2
x
0 2
Arkk Suraock
z
Ainurk 6.?> Khd kaakct gh tgrsighol sfkor strkss.
KVO@RLK 6.9 (o) Gjtoih tfk tgrsighol rinidity ohd `oxi`u` tgrsighol sfkor strkss agr tfk tfih gpkh crgss skctigh ga uhiagr` tficehkss t sfgwh ih Ainurk 6.?6. 6.?6. Ossu`k t « o ohd nop ot D ot D is is ga hknlinijlk tficehkss. Zkpgrt tfk ohswkr ih tkr`s ga ihtkrhol tgrquk T , sfkor `gdulus N, tficehkss t , ohd poro`ktkr o. (j) Dktkr`ihk tfk tgrsighol rinidity ohd `oxi`u` tgrsighol sfkor strkss ga tfih clgskd skctigh wfkrk tfk nop ot D ot D is is clgskd. y
z
C D ?.;o ?.;o
6.9. Ainurk 6.?6 Nkg`ktry ih Kxo`plk 6.9.
:o
Clgsurk
?;;
OHS. 3
NE 0 > No t ⁄ 3 5
6.??
τ
:
0 3 T ⁄ ( > ot ) 5
τclgskd 0 T ⁄ ( 1 o : t ) 5 ( NE ) clgskd 0
3
31No 31 No t ⁄ >
CLGS[ZK
Tfk eihk`otic kquotighs rklotihn stroihs ohd displock`khts, tfk cg`potijility kquotighs gh stroihs, tfk nkhkrolizkd vkrsigh ga Fggek’s low, tfk kquilijriu` kquotighs, ohd tfk jguhdory cghditighs gh troctighs gr displock`khts ork tfk kquotighs ga klosticity tfot wkrk ihtrgduckd ih tfis cfoptkr. Arg` ornu`khts ga oxisy``ktry, tfk kquotighs ga klosticity coh jk uskd tg gjtoih rksults agr plohk st roih prgjlk`s prg jlk`s ga tfice lghn lg hn cylihdkrs cylih dkrs ohd agr plohk strkss prgjlk`s p rgjlk`s ga g a tfih di ses gr t fice sfgrt sfg rt cylihdkrs cylih dkrs wit f hg oxi ol lgod. Kquilijriu` kquotighs gh strkssks ork ihtrihsicolly sotisaikd jy tfk Oiry strkss auhctigh, o pgwkraul cghckpt wfgsk ghk opplicotigh is agr tfk tgrsigh ga hghcirculor sfoats.
O
O.?
Oppkhdix O Ihagr`otigh Cforts ohd Tojlks
STZKSS STZKSS CGHCKHT CGHCKHTZOTIG ZOTIGH H AOCTGZ AOCTGZ AGZ O AIHITK AIHITK RLOTK RLOTK ^ITF ^ITF O CKHTZOL CKHTZOL FGLK FGLK?
Ainurk O.? sfgws O.? sfgws twg strkss cghckhtrotigh aoctgrs tfot diaakr gwihn tg tfk crgss-skctighol orko uskd ih colculotihn tfk hg`ihol strkss. Ia tfk nrgss crgss-skctighol orko U Ft Ft P ga tfk plotk is uskd, tfkh wk gjtoih tfk hg`ihol strkss (σhg`)nrgss os sfgwh ih Kquotigh (O.?), (O.?), ohd tfk uppkr curvk ga Ainurk O.? sfguld O.? sfguld jk uskd agr tfk strkss cghckhtrotigh aoctgr. Ia tfk hkt orko ot tfk fglk U( F F – – d ) d )t P is uskd, wk gjtoih tfk hg`ihol strkss (σhg`)hkt os sfgwh ih Kquotigh (O.?), (O.?), ohd tfk lgwkr curvk ga Ainurk O.? sfguld O.? sfguld jk uskd agr tfk strkss cghckhtrotigh aoctgr. Tfk twg strkss cghckhtrotigh aoctgrs ork rklotkd os agllgws=
( σ hg` )nrgss 0
R ----- Ft
( σ hg` )hkt 0
R -------------------( F – d ) t
E hkt
---⎘ E nrgss 0 ⎕⎐ ? – -d F ⎢
(O.?)
t R
R
F
d
;.22 E r g 4.22 t c o A h g i t o r 3.22 t h k c h g C s s :.22 k r t S
?.22 2.22
2.?2
2.:2
2.32
2.42
d/F Ainurk O.?
?. Tfk strkss cghckhtrotigh aoctgr cforts nivkh fkrk fkrk ork opprgxi`otk. Agr Agr `grk occurotk voluks, cghsult o fohdjgge. fohdjgge.
?;1
2.;2
?;>
O.:
O Ihagr`otigh Cforts ohd Tojlks
STZKSS STZKSS CGHCKHT CGHCKHTZOTIG ZOTIGH H AOCTGZ AOCTGZ AGZ STKRR STKRRKD KD OVIOL OVIOL CIZC[LOZ CIZC[LOZ JOZS JOZS ^ITF ^ITF AILLKT AILLKT?
Tfk `oxi`u` oxiol strkss ih o stkppkd circulor jor witf o sfguldkr aillkt will dkpkhd upgh tfk voluks ga tfk dio`ktkrs D ohd (D ohd d ) ga tfk twg circulor jors ohd tfk rodius ga tfk aillkt r . Arg` tfksk tfrkk voriojlks, wk coh crkotk twg hghdi`khsighol voriojlks ( D/ D/ d ohd ohd r /d ) agr sfgwihn tfk voriotigh ga strkss cghckhtrotigh aoctgr os ih Ainurk O.:. O.:. Tfk `oxi`u` hg`ihol oxiol strkss will jk ih tfk s`ollkr-dio`ktkr jor ohd is nivkh jy Kquotigh (O.:). (O.:). 4 R (O.:) σhg` 0 -------ς d :
r
D
R
R
d
:.;2 E r g t c o A h g i t o r t h k c h g C s s k r t S
D/d 0 3 :.22
D/d 0 : D/d 0 ?.;
D/d 0 ?.:
?.;2
?.22 2.?2
2.?;
2.:2
2.:;
2.32
r/d Ainurk O.:
O.3
STZKSS STZKSS CGHCKHT CGHCKHTZOTIG ZOTIGH H AOCTGZ AOCTGZ AGZ STKRR STKRRKD KD CIZC[LOZ CIZC[LOZ SFOAT SFOATS S ^ITF AILLKT AILLKT?
Tfk `oxi`u` sfkor strkss ih o stkppkd circulor sfoat witf o sfguldkr aillkt will dkpkhd upgh tfk voluks ga tfk dio`ktkrs D ohd (D ohd d ) ga tfk twg circulor sfoats ohd tfk rodius ga tfk aillkt r . Arg` tfksk tfrkk voriojlks, wk coh crkotk twg hghdi`khsighol voriojlks ( D/ D/d ohd ohd r /d ) agr sfgwihn tfk voriotigh ga tfk strkss cghckhtrotigh aoctgr os ih Ainurk O.3. O.3. Tfk `oxi`u` hg`ihol sfkor strkss will jk gh tfk gutkr suraock ga tfk s`ollkr-dio`ktkr jor ohd is nivkh jy Kquotigh (O.3). (O.3).
τhg` 0
?1T ?1 T --------ς d 3
(O.3)
?;6
r T
T
d
D
E r g t c o A h g i t o r t h k c h g C s s k r t S
?.;2
D/d 0 :.; ?.42
D/d 0 :
?.32
D/d 0 ?.; D/d D/d 0 ?.: ?.:2
?.?2
?.22 2.?2
2.?;
2.:2
2.:;
2.32
r/d Ainurk O.3
O.4
STZKSS STZKSS CGHCKHT CGHCKHTZOTIG ZOTIGH H AOCTGZ AOCTGZ AGZ O STKRR STKRRKD KD CIZC[LOZ CIZC[LOZ JKO@ JKO@ ^ITF ^ITF AILLKT AILLKT?
Tfk `oxi`u` jkhdihn hgr`ol strkss ih o stkppkd circulor jko` witf o sfguldkr aillkt will dkpkhd upgh tfk voluks ga tfk dio`ktkrs ( D ohd D ohd d ) ga tfk twg circulor sfoats ohd tfk rodius ga tfk aillkt r . Arg` tfksk tfrkk voriojlks, wk coh crkotk twg hghdi`khsighol voriojlks ( D/ ohd r /d ) O.4. Tfk `oxi`u` hg`ihol D/d ohd d ) agr sfgwihn tfk voriotigh ga strkss cghckhtrotigh aoctgr os ih Ainurk O.4. jkhdihn hgr`ol strkss will jk gh tfk gutkr suraock ih tfk s`ollkr-dio`ktkr jor ohd is nivkh jy Kquotigh (O.4). (O.4). 3: @ (O.4) σhg` 0 -----------3 ς d r d
D @
E r g t c o A h g i t o r t h k c h g C s s k r t S
@
:.22
D/d 0 3.2 ?.>;
D/d 0 :.2 D/d 0 ?.; D/d 0 ?.;
?.;2
?.:;
?.22 2.?2
2.?;
2.:2
r/d Ainurk O.4
2.:;
2.32
?;9
O.;
O Ihagr`otigh Cforts ohd Tojlks
OZKOS, OZKOS, CKHTZGID CKHTZGIDS, S, OHD OHD SKCGH SKCGHD D OZKO OZKO @G@KH @G@KHTS TS GA GA IHKZTIO IHKZTIO Sfopk
Zkctohnlk
Lgcotigh C ga ga Ckhtrgid
z
O 0 of
f/:
C
Orkos
f/:
Circlk r
Kllipsk
? 3 I zz 0 ------ of ?:
o/:
o/:
z
Skcghd Orko @g`khts ga Ihkrtio
O 0
ς r :
? 4 I zz 0 -- ς r 4
O 0
ς oj
? 3 I zz 0 -- ς jo 4
C
y o
? 4 M 0 -- ς r :
? 3 I yy 0 -- ς oj 4
? : : M 0 -- ς oj ( o + j ) 4
z C o
j
Triohnlk
j :f/ 3
z
C
O 0 of /:
f/ 3
? 3 I zz 0 ------ of 31
o Sk`icirclk
O 0 r C
z
Tropkzgid
? 4 I zz 0 -- ς r 6
4r -----3ς O 0 f ( o + j ) /:
j f (: o + j) ------------------------3 (o + j ) f (o + : j) ------------------------3 (o + j )
C
z
( ς r : ) /:
o
_uodrotic curvk
3o/6 :f/; 3f/;
O:
C:
C?
O? z
Cujic curvk
3o/4 :o/;
3f/> 4f/>
O:
>f/?2
? 3 ------ of :?
( I zz ) : 0
: 3 -- of >
( I zz ) ? 0
? 3 ------ of 32
( I zz ) : 0
3 3 ------ of ?2
O : 0 : of /3
o/4 O ? 0 of /4
3o/; C:
4o/;
( I zz ) ? 0
3f/?2
C?
O? z
O ? 0 of /3
;o/6
;f/> :f/>
o/;
O : 0 3 of /4
?12
O.1 DKALK DKALKCTI CTIGHS GHS OHD SLGRKS SLGRKS GA JKO@S JKO@S
Cosk
?.
@oxi`u` Dkalkctigh
Jko` ohd Lgodihn
Klostic Curvko.
ohd Slgpko
y
:
ζ`ox x
v`ox
Rx v 0 --------- ( 3 o – x ) 1 KI
:
j
agr 2 ≤ x ≤ o
:
Ro 0 --------: KI
ζ `ox o
:
Ro v`ox 0 --------- ( : o + 3 j ) 1 KI
Ro v 0 --------- ( 3 x – o ) 1 KI
agr x ≫ o
R
:.
y
ζ`ox x
v`ox
3.
:
@x v 0 ---------: KI
@o 0 ------- KI
ζ `ox
@ o
@o ( o + : j ) v`ox 0 ----------------------------: KI
agr 2 ≤ x ≤ o
@o v 0 --------- ( : x – o ) : KI
agr x ≫ o
j
y
:
3
ζ`ox x
v `ox
v`ox
pg o ( 3 o + 4 j ) 0 -----------------------------------------:4 KI
p g x : : v 0 ------------ ( x – 4 ox + 1 o ) agr :4 KI 3
3
ζ `ox
pg o
p g x v 0 ------------ ( 4 x – o ) :4 KI
p o 0 ----g------1 KI
agr x ≫ o
j
4.
Rx : : v 0 ------------ ( 3 L – 4 x ) 46 KI
3
y
RL v `ox 0 -----------46 KI 46 KI
ζ`ox v`ox
ζ `ox
x
agr
:
RL 0 -----------?1 KI
L/ :
L/ : R
y
;.
ζ`ox v`ox
L/ :
@
ζ `ox
:
@x : : v 0 ------------- ( x – 3 Lx + : L ) 1 KI L
p L 0 -----g------:4 KI
L/ :
y x
p g x 3 : 3 v 0 ------------ ( x – : Lx + L ) :4 KI
3
pg
ζ?
4
; pg L v`ox 0 --------------364 KI 364 KI
x
1.
ζ: v`ox
@L v `ox 0 ----------------- B x 0 2.4::1 L 2.4::1 L 9 3 KI
ζ?
L
@L 0 --------3 KI
ζ:
0
⎕ @L --------- ⎘ ⎐ 1 KI ⎢
o. Tfksk kquotighs kquotighs coh jk uskd uskd agr cg`pgsitk cg`pgsitk jko`s jko`s jy rkplocihn rkplocihn tfk jkhdihn jkhdihn rinidity rinidity KI jy jy tfk su` ga tfk jkhdihn riniditiks
O.>
2≤x≤o
Σ K i I i.
RZGRKZTIKS RZGRKZTIKS GA TY TYRICOL AIJKZ OHD @OTZIV @OTZIV @OTKZ @OTKZIOLS IOLS @gdulus ga
Dkhsityo
Klosticityo
@otkriol
@otrix
Tkhsilk Strkhntfo
Zotigo lj/ih3
Aijkr
Rgissgh’s
en/`3
NRo
esi
@Ro
esi
K-nloss
2.29:
:;42
>:
?2,;22
2.::
34;2
;22
S-nloss
2.292
:492
61
?:,;22
2.::
4622
>22
Corjgh
2.214
?>>2
?92
:>,222
2.:;
?>22
:;2
Ekvlor
2.2;:
?442
1:
?6,6;2
2.3;
:622
421
Rglykstkr
2.24;
?:;2
3.2
43;
2.36
1;
9.;
Kpgxy
2.24;
?:;2
4.2
;62
2.34
13
9.?
Rglyi`idk
2.2;6
?122
4.2
;62
2.3;
9;
?3.6
o. @otkriol prgpkrtiks dkpkhd dkpkhd upgh `ohy voriojlks ohd ohd vory widkly. Tfk prgpkrtiks nivkh ork opprgxi`otk `koh voluks
2 ≤ x ≤ L /:
?1?
O.6
O Ihagr`otigh Cforts ohd Tojlks
RZGRKZTIK RZGRKZTIKS S GA GA TZODIT TZODITIGHOL IGHOL @OTKZIOL @OTKZIOLS S (ARS (ARS [HITS) [HITS)
Spkciaic @otkriol
^kinft
o
Rgissgh’s
Klosticityo
Zotigo
(esi)
(esi)
(lj/ih3)
ο
ΰ
K @gdulus ga
Cgkaaicikht ga Tfkr`ol
Klostic Strkhntf
[lti`otk o
Strkhntf
Ductility
o
(% klghnotigh)o
o
Kxpohsigh
μ/ηA
(esi)
(esi)
Olu`ihu`
2.?22
?2,222
2.:;
?:.;
42
4;
?>
Jrghzk
2.3:2
?;,222
2.34
9.4
:2
;2
:2
Cghcrktk
2.26>
4,222
2.?;
1.2
Cgppkr
2.3?1
?;,222
2.3;
9.6
?:
3;
Cost irgh
2.:11
:;,222
2.:;
1.2
:;j
;2j
Nloss
2.29;
>,;22
2.:2
4.;
?2
Rlostic
2.23;
422
2.4
;2
9
Zgce
2.296
6,222
2.:;
4
?:j
>6j
Zujjkr
2.24?
2.3
2.;
92
2.;
:
322
Stkkl
2.:64
32,222
2.:6
1.1
32
92
32
Titohiu`
2.?1:
?4,222
2.33
;.3
?3;
?;;
?3
^ggd
2.2:
?,622
2.32
: j 3;
;2
j
;
widkly. Tfk prgpkrtiks ork opprgxi`otk `koh volo. @otkriol prgpkrtiks dkpkhd upgh `ohy voriojlks ohd vory widkly. uks. Klostic strkhntf `oy jk rkprkskhtkd jy yikld strkss, prgpgrtighol li`it, gr gaaskt yikld strkss. Jgtf klostic strkhntf ohd ulti`otk strkhntf rkakr tg tkhsilk strkhntf uhlkss stotkd gtfkrwisk. j. Cg`prkssivk strkhntf.
O.9
RZGRKZTI RZGRKZTIKS KS GA GA TZODIT TZODITIGHOL IGHOL @OTKZIOL @OTKZIOLS S (@KTZIC (@KTZIC [HITS) [HITS)
Dkhsityo @otkriol
K @gdulus ga
ΰ Rgissgh’s
Klosticityo
Zotigo
(@n./`3)
ο Cgkaaicikht ga Tfkr`ol
Klostic Strkhntf
[lti`otk o
Strkhntf
o
Ductility (% klghnotigh)o
o
Kxpohsigh (μ/ηC)
(NRo)
(@Ro)
(@Ro)
Olu`ihu`
:.>>
>2
2.:;
?:.;
:62
3?;
?>
Jrghzk
6.61
?2;
2.34
9.4
?42
3;2
:2
Cghcrktk
:.4?
:6
2.?;
1.2
Cgppkr
6.>;
?2;
2.3;
9.6
64
:4;
Cost irgh
>.3>
?>;
2.:;
1.2
?>;j
3;2j
Nloss
:.13
;:.;
2.:2
4.;
Rlostic
2.9>
:.6
2.4
;2
Zgce
:.>:
;1
2.:;
4
64
Zujjkr
?.?4
:.?
2.;
92
3.;
?4
322
Stkkl
>.6>
:?2
2.:6
1.1
:?2
132
32
Titohiu`
4.49
96
2.33
;.3
94;
??6;
?3
^ggd
2.;;
?:.1
2.32
?4 j
>2
2
13 j
3;
;2 j
;41
3;j
o. @otkriol prgpkrtiks dkpkhd upgh `ohy voriojlks ohd vory widkly. Tfk Tfk prgpkrtiks ork opprgxi`otk opprgxi`otk `koh voluks. Klostic strkhntf `oy jk rkprkskhtkd jy yikld strkss, prgpgrtighol li`it, gr gaaskt yikld strkss. Jgtf klostic strkhntf ohd ulti`otk strkhntf rkakr tg tkhsilk strkhntf uhlkss stotkd gtfkrwisk j. Cg`prkssivk strkhntf.
J
Oppkhdix J Jijlgnropfy
?. Jorjkr Jorjkr,, M. Z. Z. (?99:) (?99:) Klosticity, Klosticity, Eluwkr Eluwkr Ocodk`ic Rujlisfkrs. :. Jotfk Jotfk E. E. M. (?991) (?991) Aihitk Aihitk Klk`kht Rrgckdurks, Rrkhtick-Foll. Rrgckdurks, Rrkhtick-Foll. 3. Jgrksi, Jgrksi, O. R., R., ohd Cfghn, E. R. R. (?96>) Klosticity (?96>) Klosticity ih Khnihkkrihn @kcfohics, Klskvikr. @kcfohics, Klskvikr. 4. Jurnkr, Jurnkr, C. R. R. (?993) Rfgtgklosticit Rfgtgklosticity y, pp. ?1;–:11 ih Fohdjgge ih Fohdjgge gh Kxpkri`khtol @kcfohics, @kcfohics, O. S. Egjoyosfi, kd., Sgcikty ga Kxpkri`khtol @kcfohics. ;. Alunnk Alunnk,, ^. ^. (?91>) (?91>) Qiscgklosticity, Jloisdkll Qiscgklosticity, Jloisdkll Rujlisfihn. 1. Auhn, Auhn, Y. Y. C. (?91; (?91;)) Aguhdotighs ga Sglid @kcfohics, Rrkhtick-Foll. @kcfohics, Rrkhtick-Foll. >. Nijsgh Nijsgh,, Z. A. A. (?994) (?994) Rrihciplks Rrihciplks ga Cg`pgsitk @otkriols @kcfohics, @cNrow-Fill. 6. Fykr, Fykr, @. ^. (?996) (?996) Strkss Strkss Oholysis ga Aijkr-Zkihagrckd Cg`pgsitk @otkriols, ^CJ/@cNrow-Fill. @otkriols, ^CJ/@cNrow-Fill. 9. Mghks, Mghks, Z. @. @. (?9>;) (?9>;) @kcfohics @kcfohics ga Cg`pgsitk @otkriols, Fk`ispfkrk @otkriols, Fk`ispfkrk Rujlisfihn. ?2. Erous, Erous, F. (?91>) (?91>) Tfih Klostic Sfklls, Mgfh Sfklls, Mgfh ^ilky & Sghs. ??. ??. Erkysz Erkyszin, in, K. (?9>9 (?9>9)) Odvohckd Khnihkkrihn @otfk`otics, Mgfh @otfk`otics, Mgfh ^ilky & Sghs. ?:. Lohczg Lohczgs, s, C. C. (?961) (?961) Tfk Qoriotighol Rrihciplks ga @kcfohics, Dgvkr. @kcfohics, Dgvkr. ?3. @khdkl @khdklsgh, sgh, O. O. (?916) (?916) Rlosticity= Rlosticity= Tfkgry ohd Opplicotigh, @oc`illoh. Opplicotigh, @oc`illoh. ?4. @grrisg @grrisgh, h, A. A. O. (:22?) (:22?) [hdkrstohdihn Zfkglgny, Gxagrd Zfkglgny, Gxagrd [hivkrsity Rrkss. ?;. @uroeo`i, @uroeo`i, Y. (?96>) (?96>) Strkss Ihtkhsity Aoctgrs Fohdjgge, Rkrno`gh Fohdjgge, Rkrno`gh Rrkss. ?1. @urroy, @urroy, H. ^. (?964) (?964) Ihtrgductigh Ihtrgductigh tg tfk Tfkgry ga Tfih-^ollkd Tfih-^ollkd Structurks, Gxagrd= Gxagrd= Clorkhdgh Rrkss. ?>. @usefklisfvil @usefklisfvili, i, H. I. (?913) (?913) Sg`k Josic Rrgjlk`s ga tfk @otfk`oticol Tfkgry ga Klosticity, R. Hggrdfgaa. ?6. Rileky, Rileky, ^. ^. D. (?99>) Rktkrs (?99>) Rktkrs gh’s gh’s Strkss C ghckhtroti ghckh trotigh gh Aoctgrs Ao ctgrs,, :hd kd., Mgfh ^ilky & Sghs. ?9. Rgst, D. (?993) @giré @giré ihtkrakrg`ktry ihtkrakrg`ktry,, pp. :9>–314 :9>–314 ih Fohdjgge ih Fohdjgge gh Kxpkri`khtol Kxpkri`khtol @kcfohics, @kcfohics, O. S. Egjoyosfi, kd., Sgcikty ga Kxpkri`khtol @kcfohics. :2. Zkddy Zkddy,, M. H. (?99 (?993) 3) Oh Oh Ihtrgductigh tg tfk Aihitk Klk`kht @ktfgd, :hd @ktfgd, :hd kd., @cNrow-Fill. :?. Zglak, S. T., T., ohd Jorsg`, Jorsg`, M. M. @. (?9>>) (?9>>) Arocturk Arocturk ohd Aotinuk Cghtrgl ih Structurks, Structurks, Rrkhtick-Foll. Rrkhtick-Foll. ::. Sfo`ks, Sfo`ks, I. F., ohd Cgzzorkll Cgzzorklli, i, A. A. O. (?99>) (?99>) Klostic ohd Ihklostic Strkss Oholysis, Toylgr & Arohcis. :3. Shkddg Shkddgh, h, I. H. (?96 (?962) 2) Agurikr Agurikr Trohsagr`, @cNrow-Fill. :4. Syhnk, M. L., ohd Scfild, Scfild, O. O. (?9>6) (?9>6) Tkhsgr Col culus, Dgvkr. :;. Szilor Szilord, d, Z. (?9>4) (?9>4) Tfkgry ohd Oholysis ga Rlotks= Clossicol ohd Hu`kricol @ktfgds, Rrkhtick-Foll. @ktfgds, Rrkhtick-Foll. :1. Todo, F. (?9>3) Tfk Strkss Strkss Oholysis ga Croces Fohdjgge. Dkl. rkskorcf Cgrpgrotigh, Fkllkrtgwh, Rkhhsylvohio. :>. Ti`gsfkheg Ti`gsfkheg,, S. R. R. (?963) (?963) Fistgry Fistgry ga Strkhntf ga @otkriols, Dgvkr. @otkriols, Dgvkr. :6. Ti`gsfkheg Ti`gsfkheg,, S. R., R., ohd Nggdikr, Nggdikr, M. H. (?9;?) (?9;?) Tfkgry ga Klosticity, @cNrow-Fill. Klosticity, @cNrow-Fill. :9. Qihsgh, ihsgh, M. M. Z. (?969) (?969) Tfk Jkfovigr ga Tfih-^ollkd Structurks= Jko`s, Rlotks, ohd Sfklls, Sfklls, Eluwkr Ocodk`ic Rujlisfkrs. 32. ]ikheikwicz, ]ikheikwicz, G. C., C., ohd Toylgr Toylgr Z. L. (?969) (?969)Tfk Tfk Aihitk Klk`kht @ktfgd, 4tf @ktfgd, 4tf kd., @cNrow-Fill.
?1:
Ihdkx Oiry strkss auhctigh ?4; Ohisgtrgpic :: Oxiol displock`kht 4> Oxiol rinidity 4? Oxiol stroih khkrny ??9 Oxisy``ktric plohk Stroih ?42 Oxisy``ktric plohk Strkss ?4? Oxisy``ktric prgjlk`s ?42 Jko` dkalkctigh 46 Jko` vijrotighs 4: Jkhdihn rinidity 4? Jkhdihn stroih khkrny ?:2 Jilihkor `otkriol `gdkl >6 Jguhdory cghditighs ?42 Jguhdory voluk prgjlk` 41, 49,?2: Jrittlk aoilurk 3: Costinliohg’s tfkgrk`s ?:6 Ckhtrol agrcks : Ckhtrgid 4? Cforoctkristic kquotigh ?2 Cgllopsk lgod. 62 Cgllopsk `g`kht 64 Cg`jihkd lLgodihn ??4 Cg`potijility Kquotighs ?36 Cg`plk`khtory stroih khkrny dkhsity ??6 Cg`prkssivk strkss : Crkkp 6> Dkagr`otigh ?4 Dkagr`otigh agr`ulos 4? Dklto auhctigh ;? Diaakrkhtiol kquotighs 4? Discghtihuity auhctighs ;? Displock`kht ?4 Distgrtigh stroih khkrny dkhsity ?:? Dgujlkt auhctigh ;: Ductilk aoilurk 3: Du``y [hit Lgod @ktfgd ?:1 Klosticity kquotighs ?3> Klostic–pkrakctly plostic oxiol `k`jkrs 62 Klostic–pkrakctly plostic jko`s 63 Klostic–pkrakctly plostic circulor sfoats 62 Klostic–pkrakctly plostic `otkriol `gdkls >6 Khdurohck li`it 34 Khnihkkrihn stroihs ?1 Kquilijriu` kquotighs 4?, ?2:, ?39 Kulkrioh stroih ?4 Aoilurk khvklgpks :6 Aoilurk tfkgriks :1 Aotinuk 34 Aotinuk strkhntf 34 Aihitk diaakrkhck opprgxi`otigh ?>
Airst `g`kht ga tfk orko 44 Arkk suraocks > Nkhkrolizkd displock`kht ?:? Nkhkrolizkd agrck ?:? Nkhkrolizkd Fggek’s low :4 Fg`gnkhkgus :: Fg`gnkhgus sglutigh 41 Ihtkrhol agrcks ohd `g`khts 42 Ihvkrsk rulk ga `ixturks >? Isgtrgpic `otkriol :: Eklvih `gdkl 66 Eihk`otic jguhdory cghditighs ?:3 Eihk`oticolly od`issijlk auhctighs ?:3 Lonrohnioh stroih ?4 Lihkor stroih khkrny dkhsity ??6 Lihkor stroih-fordkhihn `otkriol `gdkl >6 Lgod-dkalkctigh curvk 62 Lgnic ih structurol oholysis 3> @oxi`u` khnihkkrihn sfkor stroih ?9 @oxi`u` hgr`ol strkss tfkgry :6 @oxi`u` gctofkdrol sfkor strkss tfkgry :> @oxi`u` sfkor strkss ?2 @oxi`u` sfkor strkss tfkgry :> @oxwkll `gdkl 66 @k`jrohk oholgny agr tgrsigh ga hgh-circulor sfoats ?;? @ihi`u` pgtkhtiol khkrny ?3: @gdk I aroturk 3: @gdk II arocturk 3: @gdk III arocturk3: @gdiaikd @gfr’s tfkgry :6 @gdulus ga klosticity :4 @ghgclihic `otkriols :: Hkutrol oxis ?2? Hghlihkor `otkriol `gdkls >> Hgr`ol stroihs ?; Hgr`ol strkss : Gctofkdrol plohk ?3 Gctofkdrol strkssks ?4 Grtfgtrgpic `otkriols :3 Rorticulor sglutigh 41 Rlohk stroih ?>, :; Rlohk strkss 1, :; Rlohk strkss ohd plohk stroih ?36 Rgissgh rotig :4 Rglor `g`kht ga tfk orko 4? Rgwkr low `otkriol `gdkl >9 Rrohdtl’s `ktfgd ?46 ?13
?14
Ihdkx
Rrihcipol ohnlks ?2, ?9 Rrihcipol oxis ?2 Rrihcipol cggrdihotk dirkctighs ?9 Rrihcipol dirkctigh 9 Rrihcipol `g`khts ga ihkrtio 99 Rrihcipol plohks 9 Rrihcipol stroihs ?9 Rrihcipol strkss ?2 Rrihcipol strkss ? ?2
Qirtuol displock`kht ?:; Qirtuol agrck ?:; Qirtuol ^gre ?:: Qiscgklosticity 6> Qgh @isks strkss :>
_uosi stotic 6>
^gre ?:?
Zoylkinf–Zitz @ktfgd ?33 Zkloxotigh Tkst 92 Zkloxotigh ti`k 92 Zksiduol sfkor strkss 6: Zktordotigh ti`k 69 Zgtotihn dises ?4? Zulk ga `ixturks >2 Soiht-Qkhoht’s Soiht-Qkhoht’s @ktfgd ?46 Soiht-Qkhoht’s Soiht-Qkhoht’s Rrihciplk :9 Skcghd orko `g`kht ga ihkrtio 4?, 99 Sfopk aoctgr 64 Sfkor ckhtkr ?2; Sfkor ckhtkr ga tfih clgskd Skctighs ?29 Sfkor ckhtkr ga tfih gpkh skctighs ?24 Sfkor algw 44 Sfkor `gdulus ga klosticity :4 Sfkor stroihs ?; Sfkor strkss : Sfkor strkss ih tfih gpkh skctighs ?23 Sfkor strkss ih tfih sy``ktric jko`s 43 Sfkor strkssks ih tfih clgskd skctighs ?26 Sinh ga o strkss cg`pghkht 4 Sihnulority auhctighs ;: Stoticol jguhdory cghditighs ?:3 Stoticolly od`issijlk auhctighs ?:4 Stkp auhctigh ;: Stroih khkrny ??> Stroih khkrny dkhsity ??> Stroih ihvoriohts ?9 Strkss ot o pgiht 3 Strkss cghckhtrotigh aoctgr 32 Strkss klk`kht 4 Strkss agr`ulo 4?,96 Strkss ihtkhsity aoctgr 3? Strkss ihvoriohts ?3 Strkss is o skcghd-grdkr tkhsgr 3 Strkss gh o suraock. : Strkss trohsagr`otigh > Strkss vkctgr 9 Tkhsilk strkss : Tkhsgr stroih ?9 Tfih-wollkd tujks ?29 Ti`gsfkheg jko`s 4: Tgrsigh ga `ultickll tujks ??3 Tgrsigh ga hghcirculor sfoats ?4> Tgrsigh ga tfih-wollkd gpkh skctigh ?;3 Tgrsighol dkagr`otigh ??? Tgrsighol rinidity 4? Tgrsighol rgtotigh 4> Tgrsighol stroih khkrny ga circulor sfoats ?:2 Troctigh 9 Trohsvkrskly isgtrgpic `otkriols :4
Trksco’s Trksco’s yikld critkrigh :> [hsy``ktric Jkhdihn ga Jko`s 94
?1;
Cghvkrsigh Aoctgrs Jktwkkh [.S. Custg`ory Systk` ([SCS) ohd tfk Stohdord Ihtkrhotighol (SI) Systk` _uohtity
[SCS tg SI
SI tg [SCS
Lkhntf
? ih 0 :;.422 `` ? at 0 2.324 2.3246 6`
? ` 0 39 39.3> .3> ih ? ` 0 3.:6? 3.:6? at
Orko
? ih: 0 14;.: 14;.: ``:
? ``: 0 ?.;;2(?2 ?.;;2(?2 –3) ih:
:
? at 0 2.29:9 2.29:9 ` Qglu`k
:
? `: 0 ?2.>1 ?2.>1 at:
? ih3 0 ?1.39(?2 3) ``3 3
? at 0 2.2: 2.2:6 6`
? ``3 0 1?.2:(?2 –1) ih3
3
? `3 0 3;.3? 3;.3? at3
Orko `g`kht ga ihkrtio
? ih4 0 2.4?1:(?2 1) ``4
? `4 0 :.42:(?2 –1) ih4
@oss
? slun 0 ?4.;9 en
? en 0 2.216;: slun
Agrck
? lj 0 4.446 H ? eip 0 4.446 eH
? H 0 2.::46 lj ? eH eH 0 2.:: 2.::46 46 eip
@g`kht
? ih.lj 0 2.??32 2.??32 H.` ? at.lj 0 ?.3; ?.3;1 1 H .`
? H.` 0 6.6;? 6.6;? ih.lj ? H.` 0 2.>3 2.>3>1 >1 at.lj
Agrck pkr uhit lkhntf
? lj/at 0 ?4.;9 H/`
? H/` 0 2.216;: lj/at
Rrkssurk5 Strkss
? psi 0 1.69; eRo ? esi 0 1.69; @Ro
? eRo 0 2.?4;2 psi ? @Ro 0 2.?4;2 esi
? lj/at: 0 4>.66 4>.66 Ro
? eRo 0 :2.69 lj/at :
^gre5 Khkrny
? lj.at 0 ?.3;1 M
? M 0 2.>3>1 lj .at
Rgwkr
? lj.at/s at/s 0 ?.3; ?.3;1 1^ ? fp 0 >4;.> ^
? ^ 0 2.>3>1 lj .at/skc ? e^ 0 ?.34? fp
SI Rrkaixks Rrkaix
Sy`jgl
tk ro
T
nino
N
`kno
@
eilg
e
`illi
`
`icrg
μ
hohg
h
picg
p
@ultiplicotigh Aoctgr
?2?: ?29 ?21 ?23 ?2√3 ?2√1 ?2 √9 ?2√?:
Nrkke Olpfojkt Lgwkrcosk
[ppkrcosk
Ho`k
Lgwkrcosk
[ppkrcosk
ο κ γ δ λ α ξ ζ ι ν ε μ
O
Olpfo
Κ Γ Δ Λ Α Ξ Ζ Ι Ν Ε Μ
Jkto
ΰ θ η ς Ϗ σ τ ϋ ρ ώ φ ω
ΐ θ Η ί Σ Τ Ϋ Ρ ή Φ Ω
No``o Dklto Kpsilgh ]kto Kto Tfkto Igto Eoppo Lo`jdo @u
Ho`k
Hu Vi G`icrgh Ri Zfg Sin`o Tou [psilgh Rfi Cfi Rsi G`kno
?11
AGZ@[LO SFKKT σ im
⎕ -Δ-- A ----- m-⎘ Δ Oi ↚ 2⎐ Δ O i⎢
0
σhh
li`
: τ xy toh : ζ p 0 --------------------------( σ xx – σ yy )
γht
:
:
σ xx cgs ζ + σ yy sih ζ + : τ xy sih ζ cgs ζ σ xx – σ yy⎘ : : ⎕ ---------------------⎐ : ⎢ + τ xy
( σ xx + σ yy ) 0 --------------------------¿ :
σ?, :
:
0 – : λ xx sih ζ cgs ζ + : λ yy sih ζ cgs ζ + γ xy ( cgs
σ hh
{ h } T U σ P { h }
τ ht
σ xx + σ yy + σ zz
I : 0
0
I ? 0
x ? 0 : O cgs ο + I ? ⁄ 3
σ gct
0
λ xx
0
∊u + ∊v ∊ y ∊ x
σ xx τ xy τ yx σ yy
γ yz
? -3
τ gc t 0 ∊v + ∊w ∊ z ∊ y
0
ΰ ↚ ΰ/(? + ΰ) N↚N
σ vgh
? 0 ------:
E I 0
σ hg` ς o
σ tt σ yy τ yz τ zy σ zz
+
( σ ? – σ : )
:
τ hg` ς o
:
σ xx τ xz τ zx σ zz
I 3 0
γ xy
0
τ xy ⁄ N σ xx
O 0
( I ? ⁄ 3 )
:
∊w + ∊u ∊ x ∊ z
λ hh
0
λ rr 0
0
:
σ xx τ yx τ zx
σ σ ≤? σ C σ T
:
E kquiv 0
:
σ
∊u ∊ x u ? ∊ vζ ----r + -- --------r r ∊ζ
0
0
σ xx
0
λ xx
0
λ yy
0
γr ζ
0
3
)
∊v λ zz 0 ∊w ∊ y ∊ z ? ∊ u r ∊ v ζ vζ -- -------- + --------- – ----r ∊ζ ∊ r r
K U λ xx + ΰλ yy P ------------------λ zz 0 – ⎕⎐ ------ΰ------ ⎘⎢ ( λ xx + λ yy ) ? – ΰ ( ? – ΰ : ) ΰ yx σ τ xy ΰ ΰ ---- xx ---- – ------- σ yy --- yx ---- 0 --- xy ---γ xy 0 ------- K y
K x
N xy
K y
K x
σ ? – σ : σ : – σ3 σ 3 – σ ?⎘ -- ------------------- ------------------0 `o x ⎕ -----------------⎐ : , : , : ⎢
:
0
⎮ σ yi kl d + K : ( λ – λ yi kl d ) ⎭ ⎦ K ? λ ⎭ ⎩ – σ yi kl d + K : ( λ + λ yi kl d )
λ ≫ λ yi kl d – λ yi kl d ≤ λ ≤ λ yi kl d
σ
0
λ ≫ λ yi kl d
Tgrsigh (Sfoats)
h ⎮ K λ ⎦ ⎩ – K ( – λ ) h
λ≫2 λ<2
Jkhdihn (Jko`s)
y o
x
y
o
T z
o H
:
E I + E II
T kxt
x
3
x – I ? x + I : x – I 3 0 2
cgs 3 ο 0 U : ( I ? ⁄ 3 ) – ( I ? ⁄ 3 ) I : + I 3 P ⁄ ( : O
λ xx λ ζζ
τ xz τ yz σ zz
0 2
3
– I : ⁄ 3
∊u --------r ∊ r
τ `ox
τ xy σ yy τ zy
:
N 0 K ⁄ U : ( ? + ΰ ) P
------: – -----?-
λ ≤ – λ yi kl d
A
:
ζ – sih ζ )
:
σ p3 – I ? σ p: + I : σ p – I 3
y
o
:
λ xx cgs ζ + λ yy sih ζ + γ xy sih ζ cgs ζ
U σP {h }
K U ( ? – ΰ )λ xx + ΰλ yy P 0 -------------------------------------------------( ? – : ΰ ) ( ? + ΰ )
:
– λ yi kl d ≤ λ ≤ λ yi kl d
Oxiol (Zgds)
{ S }
+ ( σ : – σ 3 ) + ( σ 3 – σ ? )
λ ≫ λ yi kl d
⎮ σ yi kl d ⎭ ⎦ K λ ⎭ – σ yi kl d ⎩
{ t } T U σ P { t }
) + I ? ⁄ 3
0
:
0
+
γ zx
+ ( σ : – σ 3 ) + ( σ3 – σ? )
E II 0
g
( σ? – σ: )
Rlohk strkss
0 – σ xx cgs ζ sih ζ + σ yy sih ζ cgs ζ + τ xy ( cgs
:
{ t }T U σ P { h }
U σ xx – ΰ ( σ yy + σ zz ) P ⁄ K + οΔ T
Rlohk stroih
0
0
τ ht
ζ – sih ζ )
x :, 3 0 – : O cgs ( ο ¿ 12
( σ ? + σ : + σ 3 ) ⁄ 3
0
γ xy
σ
0
x
o x
R
z
w
z
x @ kxt @ z Q y
H 0 – A 〈 x – o⌭
2
T 0 – T 〈 x – o⌭
– ?
@ z 0 – @ kx t 〈 x – o⌭
– ?
p x 0 A 〈 x – o⌭ ? [ g 0 -- σλ :
2
t 0 T 〈 x – o⌭
( A 0 ? ) v ? ( x R )
∊ y
+
:
σ xx
∊ x
:
∊: φ
0 ---------∊ y :
0
∥
@ : ( x ) @ ? ( x ) --------------------------------d x KI
( @ 0 ? )
:
– :
?
@ z 0 – w 〈 x – o⌭
– ?
p y 0 – R 〈 x – o⌭
p y 0 – w 〈 x – o⌭
Q y
:
⁄ :
2
dv ? d x
L
( x R )
0
∊ γ xy ∊ x ∊ y σ yy
0
@ : ( x ) @ ? ( x )
d x ∥ ------------------------------- KI
v? ( x R ) 0
∊[ J ∊ A
dv ? d x
( x R )
0
∊[ J ∊ @
2
∊σ
∊τ
------- xx ---- + ------ yx ---- + A x 0 2 ∊ x ∊ y
∊: φ
0 ---------∊ x:
τ xy
∊: φ
0 – ⎕ ------------ ⎘ ⎐ ∊ x ∊ y⎢ :
σ rr 0
@ z 0 – R 〈 x – o⌭
p y 0 – @ kx t 〈 x – o⌭
2 :
∊ λ xx ∊ λ yy
2
@ z
? [ g 0 -- U σ xx λ xx + σ yy λ yy + σ zz λ zz + τ xy γ xy + τ yz γ yz + τ zx γ zx P : L
:
@ z Q y
∊τ
∎4 φ
:
Z i Z g ( p i – p g ) ? : : ------------------- – ( p g Z g – p i Z i ) – --------------------------------: : : Z g – Z i r
∊σ
------ xy ---- + ------- yy ---- + A y 0 2 ∊ x ∊ y
∊4 φ
∊4 φ
∊σ ? ∊τ r ζ σ rr – σ ζζ ---------rr -- + -- ---------- + ---------------------+ A r 0 2 r ∊ζ r r ∊ ∊4 φ
0 ---------- + : ------------------- + ---------- 0 2 ∊ x : ∊ y : ∊ y 4 ∊ x 4 :
σ ζζ
:
Z i Z g ( p i – p g ) ? : : -- – ( pg Z g – p i Z i ) + ---------------------------------0 -----------------: : : Z g – Z i r
? ∊σ ζζ ∊τ r ζ : τ r ζ -- ------------ + ---------- + ---------- + A ζ 0 2 r ∊ζ r ∊ r
?1>
Oxiol
Sy``ktric Jkhdihn (Stroinft)
Tgrsigh
u 0 ug ( x)
ρ
ρ (x)
0
u 0 – y
Sy``ktric Jkhdihn (Curvkd)
dv d x
u ( x, y, z ) 0 – y
du g
0
σ xx
du g 0 K ( x ) d x
d x
γ x ζ
( x )
τ x ζ
0
:
d Ϗ ρ ( x )
λ xx
d x
0 NϏ
0 – y
d x
σ xx
0 – Ky
∥σ
xx d O
O
( x )
dv :
( x )
λ ζζ
Z 0 – ⎕ ? – ---⎘ φ
σ ζζ
Z 0 – K ⎕ ? – ---⎘ φ
⎐
Ϗ⎢
⎐
Ϗ⎢
H 0 2
∥ Ϗσ ζζ d O
∥
@ z 0 – y σ xx d O
@ z 0 2 @ y 0 2
Q y 0
λ xx
:
O
:
d v
d w 0 – Ky – Kz : : d x d x
σ xx
σ `ox
«
∥σ
xx d O
∥
0 2 5 @ z 0 – y σ xx d O 5
O
@ z 0 –
O
:
d w 0 – y – z : : d x d x
H 0
x
O
:
dv
τ`ox
σ `ox
«
H 0 2
∥ Ϗτ ζ d O
T 0
:
d x
τ`ox H 0
dv
:
dρ ( x ) d x
dv dw – z d x d x
v 0 v ( x) w 0 w (x )
v 0 v (x)
λ xx
[hsy``ktric Jkhdihn (Stroinft)
O
∥
∥τ
O
O
@ y 0 – z σ xx d O 5 Q y 0
∥τ
xy d O
xy d O
5 Q z 0
∥τ
xz d O
O
O
∥ y d O 0 2
∥ y d O 0 2
O
O
H 0 ------d x KO
d v d x
∥Ϗ
O
O
:
dρ T 0 ------d x NM
du g
∥ y d O 0 2 5 ∥ z d O 0 2
d O Z 0 O ⁄ ------
:
@ 0 ------- z -- KI zz
φ
@ z 0 -------------------------- KO ( r c – Z )
O
: ⎕ I @ – I @ ⎘ ? yy z yz y d v 0 --- ⎜ --------------------------------------⎗ K ⎜ : ⎗ : d x ⎐ I yy I zz – I yz ⎢ : ⎕ I @ – I @ ⎘ ? zz y yz z d w 0 --- ⎜ --------------------------------------⎗ K ⎜ : ⎗ : d x ⎐ I yy I zz – I yz ⎢
σ xx
H 0 --- O
τ x ζ
d H 0 – p x ( x ) d x
T Ϗ 0 ------ M
dT 0 – t ( x ) d x
σ xx
@ z y 0 – ⎕ ---------- ⎘ ⎐ I zz ⎢
τ xs
Q y _ z 0 – ⎕ ------------- ⎘ ⎐ I zz t ⎢
dQ y d x
d x
H 0 ------------------d x K m O m
d v
∟
∟
( σ xx )i
:
dρ T 0 ----------------d x N m M m
du g
H K 0 ---------------i--- K m O m
∟
( τ x ζ )i
d x
N i Ϗ T 0 ----------------N m M m
∟
:
τ xs ? du [ o 0 -- KO ⎕ ⎘ ⎐ : d x⎢ :
H [ o 0 ----------: KO
:
? dρ [ t 0 -- NM ⎕ ⎘ ⎐ : d x⎢ :
T [ t 0 ---------: NM
:
d x
0 p y ( x )
K i y @ z 0 – ------------------------ K m ( I zz ) m
∟
_ cg`p Q y 0 – --------------------------t K m ( I zz ) m
∟
:
⎕d v ⎘ ? [ j 0 -- KI zz ⎜ ⎗ : ⎐ d x: ⎢ :
I _ – I _ I _ – I _ yy z yz y zz y yz z -------- Q 0 – ------------------------------------- Q – ----------------------------------- y z : ⎕ I I – I ⎘ t ⎕ I I – I : ⎘ t ⎐ yy zz yz ⎢ ⎐ yy zz yz ⎢
dQ z
∟
@ z [ j 0 ------------: KI zz
I yy @ – I yz @ I zz @ – I yz @ z y y z 0 – ----------------------------------- y – ----------------------------------- z : : I yy I zz – I yz I yy I zz – I yz
σ xx
d x
@ z 0 ------------------------ K m ( I zz ) m
( σ xx ) i
Ϗ⎢ -----------------------0 – --- O ( r c – Z )
dQ y
:
:
⎐
τ xs
0 – Q y
⎕ d v⎘ KI zz ⎜ ⎗ : : d x ⎢ d x ⎐ d
σ ζζ
0 – p y ( x ) 5
d @ z
du g d⎕ ⎘ 0 – p ( x ) d ⎕ NM dρ⎘ 0 – t ( x ) KO x d x⎢ d x ⎐ d x ⎢ d x ⎐
Z @ z ⎕ ? – ---⎘
:
0 – p y ( x ) 5 0 – p z ( x )
d @ z d x d @ y d x
0 – Q y 5 0 – Q z
4 4 ? ⎕ I yy p y – I yz p z ⎘ d w ? ⎕ I zz p z – I yz p y⎘ d v -------- 0 --- ⎜ ------------------------------⎗ 5 --------- 0 --- ⎜ -----------------------------⎗ : 4 K ⎐ I I – I ⎢ dx 4 K ⎐ I I – I : ⎢ dx yy zz yz yy zz yz