UNIVERSIDAD NACIONAL DE INGENIERÍA FACULTAD DE INGENIERÍA MECÁNICA
ESTRUCTURA CON NUDOS NO ARTICULADOS (MARCOS PLANOS) Informe N° 6
CURSO:
Cálculo por elementos nitos.
“G” SECCIÓN: “G”
FECHA DE ENTREGA: 27/11/2015 !f!el "!#n!s! "!#n!s!$$ %nt&on# %nt&on# 'illi!ms 'illi!ms . ALUMNO: !f!el
CÓDIGO: 201(0217)
2015-II
ÍNDICE. 0
*á+. 1. ,n ,nunci!-o
-el
prolem!.
2. "o-el!-o
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i+!.
. . (. "!tri
-e
ri+i-e
loc!l.
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2
. (
-e
ri+i-e
+lo!l.
. .
5. "!tri
-e
fuer!s.
.. .. 5
6. "!tri
-e
esfueros.
..
7. )i!+r!m!
-e
6
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en
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1.ENUNCIADO DEL PROBLEMA. Par al a vi ga most r ada en l a figur a,det er mi ne l as pendi ent esen 2 y 3 y l a de fle x i ó nv e r t i c a le ne lpunt ome di odel ac ar g adi s t r i bui da.
Dat os: A =10 ¿
2
6
E=30 × 10 psi 4
I =100 ¿
2.MODELADO DE LA VI GA. Sepr oceder áahacerelmodel adocon2el ement osfini t os.
2
3.GRADOSDE LI BERTAD NODALES. 3. 1 Encoordenadasl ocal es. El ement o1:
El ement o2:
(
3. 2 Encoordenadasgl obal es.
3. 3 Coordenadasycosenosdi rectores. ,lemento 1 2
No-os 1 2 2 (
= 0 120 120 10
# 0 0 0 10(.:2(
4.MATRI Z DE TRANSFORMACI ÓN.
l
m
1
0
0.5
0.66
El ement o1:
[ ] [
L1=
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
El ement o2: 0.5
0.866
0
0
0
0
−0.866
0.5
0
0
0
0
0
0
1
0
0
0
0
0
0
0.5
0.866
0
0
0
0
−0.866
0.5
0
0
0
0
0
0
1
L2=
]
5.MATRI ZDE RI GI DEZDE ELEMENTOS.
5. 1
'
k e
=
[
Encoordenadas( x’ ;y’ ) :
EA le 0
0
0
0
12 EI
6 EI
3
le
le
6 EI
4 EI
EA le
−
0
0
2
2
le
le
0
0
12 EI
−
−
6 EI
le
3
le
6 EI
2 EI
2
le
2
le
EA le
−
0
0 −
12 EI
0 6 EI
3
2
le 0
−
EA le 0
6 EI
le
le
0
0
12 EI 3
−
2 EI
2
le 0
le
6 EI 2
le
5
−
6 EI 2
le
4 EI
le
]
Par aele l e me nt ofini t o1:
'
k 1=10
6
[
2.5
0
0
−2.5
0
0
0
0.020833
1.25
0
− 0.020833
1.25
0
1.25
100
0
− 1.25
50
−2.5
0
0
2.5
0
0
0
− 0.020833
−1.25
0
0.020833
−1.25
0
1.25
50
0
− 1.25
100
]
Paraelel ement ofini t o2:
'
'
k 2= k 1=10
6
[
2.5
0
0
−2.5
0
0
0
0.02
1.25
0
−0.02
1.25
0
1.25
100
0
−1.25
50
−2.5
0
0
2.5
0
0
0
−0.02
−1.25
0
0.02
− 1.25
0
1.25
50
0
−1.25
100
5. 2 T
]
Encoor denadas( x;y) : '
k e = Le × k e × Le Paraelel ement ofini t o1:
k 1=10
6
[
2.5
0
0
−2.5
0
0
0
0.020833
1.25
0
− 0.020833
1.25
0
1.25
100
0
− 1.25
50
−2.5
0
0
2.5
0
0
0
− 0.020833
−1.25
0
0.020833
−1.25
0
1.25
50
0
− 1.25
100
]
Paraelel ement ofini t o2:
6
k 2= 10
[
0.64062
1.0735
−1.0825
−0.64062
−1.0735
−1.0825
1.0735
1.8801
0.625
−1.0735
− 1.8801
0.625
−1.0825
0.625
100
1.0825
−0.625
50
−0.64062
−1.0735
1.0825
0.64062
1.0735
1.0825
−1.0735
−1.8801
−0.625
1.0735
1.8801
−0.625
1.0825
−0.625
100
−1.0825
0.625
50
6.MATRI ZDE RI GI DEZ GLOBAL. 6
]
[ k ] =10
6
[
2.5
0
0
−2.5
0
0
0
0
0
0
0.020833
1.25
0
− 0.020833
1.25
0
0
0
0
1.25
100
0
− 1.25
50
0
0
0
−2.5
0
0
3.1406
1.0735
− 1.0825
− 0.64062
− 1.0735
− 1.08
0
− 0.020833
−1.25
1.0735
1.9009
− 0.625
− 1.0735
− 1.8801
0.62
0
1.25
50
−1.0825
− 0.625
200
1.0825
− 0.625
0
0
0
− 0.64062
− 1.0735
1.0825
0.64062
1.0735
1.082
0
0
0
−1.0735
− 1.8801
− 0.625
1.0735
1.8801
− 0.62
0
0
0
−1.0825
0.625
50
1.0825
− 0.625
7.MATRI Z DE DEFORMACI ÓN TOTAL.
[Q]
=
[Q
Q2
1
[ Q ] =[ 0
Q3
Q3
0
Q4
Q5
Q4 Q 5
Q6
Q6
Q7
Q8
0
Q9
0
]
Q9 T
8.MATRIZFUERZA. 8. 1 '
F e =
'
[
Encoordenadas( x’ ;y’ ) :
p l e
0
p l e
2
3
[
F 1= 10 0
[ F ] =[ 0 2
8. 2 T
0
p l e
0
12
−5
0
2
2
− 100
0
0
− p l e
0
0
12
−5
2
]
100
T
]
T
]T
Encoor denadas( x;y) : '
F e = L × F e 3
[
F 1= 10 0
[ F ] =[ 0 2
0
−5
0
− 100
0
0
0
0
−5
100
]
T
]T
7
]
T
50
100
9.ECUACI ÓN DE RI GI DEZ.
[ F ] = [ k ] [ Q ]
[][ 0
2.5
0
0
−2.5
0
0
0
0
−5
0
0.020833
1.25
0
−0.020833
1.25
0
0
−100
0
1.25
100
0
−1.25
50
0
0
−2.5
0
0
3.1406
1.0735
−1.0825
− 0.64062
−1.0735
0
− 0.020833
− 1.25
1.0735
1.9009
− 0.625
− 1.0735
−1.8801
100
0
1.25
50
−1.0825
−0.625
200
1.0825
−0.625
0
0
0
0
−0.64062
−1.0735
1.0825
0.64062
1.0735
0
0
0
0
−1.0735
−1.8801
− 0.625
1.0735
1.8801
0
0
0
0
−1.0825
0.625
50
1.0825
−0.625
0
3
10
6
= 10
−5
[ q ] =[ 2.70695 × 10
539.24711
[ q ] =[ 2.70695 × 10
303.0303
−8
1
−8
2
−8
2.70695 × 10
− 1.96654
− 1.97435
206.00114
303.0303
184.09113
−1.97435
− 1.98616
]
]
T
T
10. ESFUERZOSENCADAELEMENTO FI NI TO DELMODELO. 10. 1 Debi doal aflexi ón.
( )[
σ e M =
Ey L
(
σ 1 M =
2
) (
(
)
30 × 10
6
× 5.12
2
1
( )[( E L
)
]
e
)[
(
( )) (
)( ) (
)
((
6 ξ 539.24711 1 + 3 ξ −1 1 −1.96654 −6 ξ 303.0303
10. 2 Debi doal aflexi ón.
σ eN =
) (
(
6 ξ −q1 m+ q2 l + 3 ξ −1 L e q3 −6 ξ −q 4 m+ q5 l + 3 ξ + 1 Le q6
) (q
− q1 l − q 2 m +
4
l + q5 m )
]
σ 1= 0 σ 2=−22537.0059
) ( 1 ) )+ ( 3 ξ + 1 ) ( 1 ) (−1.974
11.
DIAGRAMA DE FLUJO. INICI>
8eer -!tos -e entr!-! *!r! i?1@ Ano-os
8eer posiciones B$
*!r! i?1@ Aelementos C!lcul! lon+itu- -e c!-! elemento$ csenos -irectores$ m!tri -e rot!cin “8”$ m!tri -e ri+i-e # fuer!s.
C!lcul! -espl!!mientos$ l!s re!cciones en los !po#os # los momentos respectios.
*!r! i?1@ Aelementos
ξ
=
1,1
−
1
2
:
N> ;I 1
Imprime re!cciones$ momentos # esfueros ;i B;NE;,?1DF B;N;,?1D ,sfuero i+u!l ! l! sum!
12.
PROGRAMACI ÓN EN MATLAB.
%ESTRUCTURAS CON NUDOS NO ARTICULADOS: MARCOS format long nd=input(IN!RESE EL NUMERO DE NODOS= "# n$=input(IN!RESE EL NUEMRO DE ELEMENTOS= "# E=input(IN!RESE EL MODULO DE OUN!= "# d=input(IN!RESE EL DIAMETRO= "# p$&=input(IN!RESE EL 'ESO ES'ECIICO(gr)f*+m,-"= "# di&p($===(." (/"===="# t+=input(IN!RESE TA0LA DE CONECTI1IDAD(&olo nodo&"= "# n=23# for i=.:nd di&p(IN!RESE LAS CORDENADAS DEL NODO "#di&p(i"#
10
,sfuero i+u!l ! l! rest!
n(i4."=input(N(5"= "# n(i4/"=input(N("= "# $nd l$=23#lm=23# A=pi*67d,/#I=pi7d,6*86# 9r&=$ro&(-7nd"#f=p$&7;<.$)87A#fp=$ro&(-7nd4."#=$ro&(-7nd4."# >i?=$ro&(-7nd"#L=$ro&(-7nd"#9p=$ro&(-7nd"#l$=23#l=23#m=23# for i=.:n$ l$(i"=&@rt((n(t+(i4/"4.")n(t+(i4."4."",/(n(t+(i4/"4/") n(t+(i4."4/"",/"# l(i"=(n(t+(i4/"4.")n(t+(i4."4.""*l$(i"# m(i"=(n(t+(i4/"4/")n(t+(i4."4/""*l$(i"# p&.=t+(i4."7-)/#p&/=t+(i4."7-).#p&-=t+(i4."7-#p&6=t+(i4/"7-) /#p&B=t+(i4/"7-).#p&8=t+(i4/"7-# L(p&.4p&."=l(i"#L(p&.4p&/"=m(i"#L(p&/4p&."=) m(i"#L(p&/4p&/"=l(i"#L(p&-4p&-"=.# L(p&64p&6"=l(i"#L(p&64p&B"=m(i"#L(p&B4p&6"=) m(i"#L(p&B4p&B"=l(i"#L(p&84p&8"=.# 9p(p&.4p&."=E7A*l$(i"#9p(p&.4p&6"=)E7A*l$(i"# 9p(p&/4p&/"=./7E7I*l$(i",-#9p(p&/4p&-"=87E7I*l$(i",/#9p(p&/4p&B"=) ./7E7I*l$(i",-#9p(p&/4p&8"=87E7I*l$(i",/# 9p(p&-4p&/"=87E7I*l$(i",/#9p(p&-4p&-"=67E7I*l$(i"#9p(p&-4p&B"=) 87E7I*l$(i",/#9p(p&-4p&8"=/7E7I*l$(i"# 9p(p&64p&."=)E7A*l$(i"#9p(p&64p&6"=E7A*l$(i"# 9p(p&B4p&/"=)./7E7I*l$(i",-#9p(p&B4p&-"=) 87E7I*l$(i",/#9p(p&B4p&B"=./7E7I*l$(i",-#9p(p&B4p&8"=)87E7I*l$(i",/# 9p(p&84p&/"=87E7I*l$(i",/#9p(p&84p&-"=/7E7I*l$(i"#9p(p&84p&B"=) 87E7I*l$(i",/#9p(p&84p&8"=67E7I*l$(i"# fp(p&.4."=f7m(i"7l$(i"*/#fp(p&/4."=f7l(i"7l$(i"*/#fp(p&-4."=f7l(i"7l$(i", /*./# fp(p&64."=f7m(i"7l$(i"*/#fp(p&B4."=f7l(i"7l$(i"*/#fp(p&84."=) f7l(i"7l$(i",/*./# =L7fp#>i?=>i?L79p7L# 9p=$ro&(-7nd"#L=$ro&(-7nd"#fp=$ro&(-7nd4."# $nd %CONDICIONES DE RONTERA (.4/4-4.-4.64.B"=23 +=23#>+=23#=$ro&(-7nd4."# (B4."=(B4."/#(4."=(4."B#(4."=(4."6# %in+luimo& la& fu$ra& $Ft$rna& +=(6:./4."#>+=>i?(6:./46:./"# (6:./4."=>+G+# %CALCULO DE REACCIONES R.=>i?(.4.:.B"7)(.4."#R/=>i?(/4.:.B"7)(/4."#M-=>i?(-4.:.B"7)(-4."# R.-=>i?(.-4.:.B"7)(.-4."#R.6=>i?(.64.:.B"7)(.64."#M.B=>i?(.B4.:.B"7) (.B4."# for i=.:n$ p&.=t+(i4."7-)/#p&/=t+(i4."7-).#p&-=t+(i4."7-#p&6=t+(i4/"7-) /#p&B=t+(i4/"7-).#p&8=t+(i4/"7-# ESN=E*l$(i"7()(p&.4."7l(i")(p&/4."7m(i"(p&64."7l(i" (p&B4."7m(i""# EM.=E*l$(i",/7d*/7()87()(p&.4."7m(i"(p&/4."7l(i"") 67l$(i"7(p&-4."87((p&64."7m(i"(p&B4."7l(i"")/7l$(i"7(p&84.""# EM/=E*l$(i",/7d*/7(87()(p&.4."7m(i"(p&/4."7l(i""/7l$(i"7(p&-4.") 87((p&64."7m(i"(p&B4."7l(i""67l$(i"7(p&84.""# if aH&(ESNEM."=aH&(ESNEM/" ES(i"=ESNEM/# $l&$
11
ES(i"=ESNEM.#
$nd $nd di&p(===== RESULTADOS ============== "# di&p(REACCION EN 'UNTO(." 5(N"= "# di&p(R."# di&p(REACCION EN 'UNTO(." (N"= "# di&p(R/"# di&p(MOMENTO EN 'UNTO(."(NFmm" = "# di&p(M-"# di&p(REACCION EN 'UNTO(B" 5(N"= "# di&p(R.-"# di&p(REACCION EN 'UNTO(B" (N"= "# di&p(R.6"# di&p(MOMENTO EN 'UNTO(B"(NFmm" = "# di&p(M.B"# di&p(ESUERJOS(M'a"= "# di&p(ES"# EJECUCION DEL PROGRAMA
IN!RESE EL NUMERO DE NODOS= IN!RESE EL NUEMRO DE ELEMENTOS= / IN!RESE EL MODULO DE OUN!=-7.$8 IN!RESE EL DIAMETRO=B IN!RESE EL 'ESO ES'ECIICO (lH)f*in,-"=
12
ESUERJOS= B
13.
)6<.8/
)-<6B.8
B
)<B86
DI BUJO EN ANSYS.
13. 1 Ene ll a doi z q ui e r dos ebus c a“ s t a t i cs t r uc t ur al ”ys el l e v aall a do de r e c hoc o nc l i ci z q ui e r dopr e s i o nado .
13. 2 Luegosel ecci onarl aopci ón“ Geomet ry”conelcl i cder echoy sel ecci onar“ New Geomet ry” .
1(
13. 3 Elpr ogr amat ee nvi ar áaunave nt anadedi buj odondes el epedi r ál as uni dadesenquesevaat r abaj ar .
13. 4 Enl apart esuperi orenl aopci ón“ Cr eat e”seel i gel aopci ón“ Poi nt ”
1
13. 5 Luegol asespeci ficaci onesdell adoi zqui er doi nf er i oren“ Type”se pondr á“ Const ruct i onPoi nt ” ,en“ Defini t i on”sepondr á“ ManualI nput ” yl uegosecol ocar anl asc oor denadasdeseadaspar al uegodarcl i cen “ Gener at e” ,ser epi t eest odependi endodecuant ospunt ossedesea.
15
13. 6 Enl apart esuperi orsese l ecci onara“ Concept ”yses el ecci onará“ Li nes f r om poi nt s”yseuni r ánl ospunt os.
13. 7 Sesel ecci onaráelpl anoquesedeseat r abaj ar ,enest ecasoseusar ael pl a nox yc o nc l i cde r e c hoye l e g i rl ao pc i ó n“ Lo oka t ”
13. 8 Nosi r emosal apart esuper i orye nl aopci ón“ Conce pt ”el egi mos“ Cr oss Se c t i o n”ye l e g i mo se lt i podes e c c i ó nc o ne lq ueq ue r e mo st r a ba j a r .
16
13. 9 Luegoenl apart ei nf er i ori zqui er dasecol ocar ánl asmedi dasdel a secci ónconquesedeseat r abaj ar .
13. 10Enl aopci óndel ai zqui er dasesel eci ona“ Li neBody”yenCr oss Se c c t i o ns es e l e c c i o nal as e c c i ó nq uee s c o g i mo sant e r i o r me nt eyl e damoscl i cen“ Gener at e” .
17
13. 11Cer r amosl ave nt anaynosv amosa“ Model ”del ave nt anapr i nci palde Wor kbenc h.
13. 12En“ Mesh”sehacee n“ Si z i ng”“ El ementsi z e”elt amañoapr oxi madodel mal l adoparaelanál i si smasexact o.Cl i cder echoen“ Mesh”“ Gener at e Mesh” .
1
13. 13En“ St at i cEst r uct ur al ”secol ocar anl ossoport esyf uer zassegúnel pr obl ema.
13. 14En“ Sol ut i on”secol ocanl osr esul t adosquesedeseanobt ener ” .
1:
13. 15Fi nal ment esedacl i cen“ Sol ve ”
20
21
