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MODULE 6 (Response of Elastic SDOF system to Earthquake load loading) Dataset · December 2013
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University of Engineering and Technology, Peshawar
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Available Available from: Mohammad Javed Retrieved on: 10 July 2016
University of Engineering & Technology, Peshawar, Pakistan
CE-409: Introduction to Structural Dynamics and Earthquake Engineering
MOD!E " ES#O'SE O1 !I'E& !I'E& E!&S2IC S%D%O%1 S3S2EMS 2O E&25&(E !O&DI'6 #ro$% Dr Dr%% &khtar 'aeem (han ) drakhtarnaeem,n$.uet%edu%.k drakhtarnaeem ,n$.uet%edu%.k
#ro$% Dr Dr%% Mohammad *a+ed m/a+ed,n$.uet%edu%.k
University of Engineering & Technology, Peshawar, Pakistan
CE-409: Introduction to Structural Dynamics and Earthquake Engineering
MOD!E " ES#O'SE O1 !I'E& !I'E& E!&S2IC S%D%O%1 S3S2EMS 2O E&25&(E !O&DI'6 #ro$% Dr Dr%% &khtar 'aeem (han ) drakhtarnaeem,n$.uet%edu%.k drakhtarnaeem ,n$.uet%edu%.k
#ro$% Dr Dr%% Mohammad *a+ed m/a+ed,n$.uet%edu%.k
Earthquake Response of Linear System In this lecture, we will study the earthquake response of linear SDOF systems subjected to earthquake excitations. y definition, linear systems are systems are elastic systems. systems. !hey are also referred to as linearly elastic systems to systems to emphasi"e both properties. $on%linear inelastic system $on%linear elastic system
f s
#inear elastic system
No energy is absorbed by systems
u
f s
&lastic%perfectly plastic system '&lasto plastic system(
Area enclosed by the curve = Energy absorbed by system
u
Effective Earthquake Force *onsider a sin+le story frame with lumped mass. #et the frame at the base displaces by an amount u + due to seismic waes. -s a result lumped mass at the top displaces by an amount ut ,such that
u
t
=
u +u +
/here u+0 1round displacement. ut0!otal displacement at the top end and u 0 Dynamic displacement of lumped mass at the top w.r.t shifted base. )
Effective Earthquake Force !he equation of motion for the frame subjected to the earthquake excitation can be deried by usin+ the usin+ dynamic equilibrium of forces as fI
+ fD + fS
= 2
Effective Earthquake Force Only the relatie motion u between the mass and the base cause structural deformation which produces elastic and dampin+ forces.
!hus for a linear system the inertial force f I is related to the t t acceleration u of the mass by
fI
fD
=
mu 3
=
and f s cu
=
ku
Effective Earthquake Force y substitutin+ the alue of fI , the equation of motion become
t +cu +ku =2 mu + ku = 2 or m' u + + u ( + cu +ku = −mu + 't( u +cu or m *omparin+ with
mu
cu
+
ku
+
p't(
=
p't( = p eff 't( = −mu + 't( !he term on the ri+ht%hand side of the equation may be re+arded as the Effective earthquake force.
Effective Earthquake Force p eff ' t ( mu + 't( =
ase moin+ with u + 't( Effective earthquake force: horizontal ground motion
!hus the +round motion can be replaced by the effective earthquake force 'indicated by the subscript 4 eff 5. Since this force is proportional to the mass, thus, by increasin+ the mass the structural desi+ner increases the effectie earthquake force
Strong motion record
Typical strong motion record
Strong motion record Stron+ earthquakes can +enerally be classified into three +roups
1 !ractically a single shock -cceleration, elocity, and displacement records for one such motion are shown in fi+ure. - motion of this type occurs only at short distances from the epicenter, only on firm +round, and only for shallow earthquakes.
Strong motion record A moderately long" e#tremely irregular motion !he record of the earthquake of &l *entro, *alifornia in 6782, $S component exemplifies this type of motion. It is associated with moderate distances from the focus and occurs only on firm +round. On such +round, almost all the major earthquakes ori+inatin+ alon+ the *ircumpacific elt are of this type.
Strong motion record A long ground motion e#hibiting pronounced prevailing periods of vibration: - portion of the accelero+ram obtained durin+ the earthquake of 6797 in #oma :rieta is shown in fi+ure to illustrate this type. Such motions result from the filterin+ of earthquakes of the precedin+ types throu+h layers of soft soil within the ran+e of linear or almost linear soil behaior and from the successie wae reflections at the interfaces of these layers.
Strong ground motions recorded in various earthquakes
ug
t $igure : %round motions recorded during several earthquakes
Stron+ +round motion $ear source effect
Accelerogram used in these lectures g 1round acceleration, u g 1round elocity, u
1round displacement, u g
$%S component of hori"ontal +round acceleration recoded at &l *entro, *alifornia durin+ the Imperial ;alley earthquake of 6782 68
Equation of motion for SDOF system subjected to EQ excitations
≡ c k mu + cu + ku = −mu + 't( ⇒ u + u + u m m
Since c = ζ c
cr
=
ζ ( =m< ) and n
=
u + =>ω u + < u
⇒
k m
=
u + 't(
=−
<
n
u 't(
=−
Response quantities @esponse is the structural system reaction to a demand comin+ from +round acceleration record
!hus a response quantity may be structural displacement, elocity, acceleration, internal shear, bendin+ moment, axial force etc. t
Sometime, the total acceleration, u o , of the mass would be needed if the structure is supportin+ sensitie equipment and the motion imparted to the equipment is to be determined. 6?
Response quantities One of the important response quantity is total lateral displacement t at the top end of structural system, u o , required to proide enou+h separation between adjacent buildin+s to preent their poundin+ a+ainst each other durin+ an earthquake
!ounding damage" &otel de carlo" 'e#ico city" 1()* earthquake 6A
Solution to equation of motion for SDOF system subjected to EQ excitation = u +=>ω u + 't( n u +< n u =−
!he time ariation of +round displacement, from the +ien time ariation of +round acceleration, can be determined by usin+ any appropriate time steppin+ numerical method. *loser the time interal, more accurate will be solution. !ypically, the time interal is chosen to be 6B622 to 6BC2 of a second, requirin+ 6C22 to )222 ordinates to describe the round motion of aboe +ien &l% *entro , 69 6782, +round acceleration record hain+ a duration of )2 sec.
g Structural disp.,u , due to ground acceleration,u El /entro"1(0" ground acceleration
g , g u go u
= 0.319g
+,-$ system .ith
!
n
= 2.Csec, > = = T n
/orresponding relative displacement at the top end of the +,-$ frame
u" in
=
0.5sec, ζ = 2%
Influence of T n and ζ on Peak displacement, u o , in a liner elastic SDOF system
u +=>ω u +< n
= n
u
!he aboe +ien equation indicates that
u
=−
+
't(
u = f'! ,ζ ( n
!hus any two systems hain+ the same alues of ! n and > will hae the same deformation response u't( een thou+h one system may be more massie than the other or one may be stiffer than the other
Effect of Tn on Deformation response history g , g u go u
=
0.319g
&l *entro +round acceleration
In +eneral, peak alue of displacement at the top end of a SDOF increases with the increase in the time period of the system. @esponse of SDOF systems with different alues of Tn to
=6
Effect of Time Period
22
Effect of ζ on Deformation response history g , g u
go u
= 0.319g
&l *entro +round acceleration
In +eneral, peak alue of displacement at the top end of a SDOF increases with the decrease in the dampin+ ratio of the system 2esponse of +,-$ systems .ith different values of 3 to
=)
Approximate Periods of Vibration (ASCE 7-05)
4here h is the height of building in ft
Approximate Periods of Vibration !hus, structural systems with !n02.Csec, 6 and = sec may be considered as C, 62 and =2 story hei+ht buildin+s, respectiely. - buildin+ with ) story hei+ht can be considered as Eulti DOF system with at least ) DOFs. !o keep the discussion simple at this sta+e, it will be a reasonable assumption to state that ' out of ) natural time periods of the ) story buildin+( we consider only fundamental natural time period '!n02.) sec( to determine the response quantities for the buildin+. #ater on we will discuss how all ) ibration modes 'and the correspondin+ natural time periods( are calculated and are taken into account to find the total response of a buildin+ with DOF 0) ecause the empirical period formula is based on measured response of buildin+s, it should not be used to estimate the period for other types of
Response spectrum concept
- plot of the peak alue of a response quantity as a function of the natural ibration period !n of the system, or a related parameter such as circular frequency
2esponse is the structural system reaction to a demand comin+ from +round acceleration record 'i.e. -ccelero+ram( and when the peak response commodities such as structural system displacement t ( uo ) ( u o ) ( u , elocity and acceleration o ) are plotted a+ainst the structural system natural time period 'or frequencies( will be called spectrum
Response spectrum concept :eak alues of response quantities and shape of response spectrum depends on the accelerogram &ach such plot is for SDOF system hain+ a fixed dampin+ ratio >, and seeral such plots for different alues of > are included to coer the ran+e of dampin+ alues encountered in actual structures. !he deformation response spectrum is a plot of u o a+ainst !n for
u o is the relative velocity response fixed >. - similar plot for t u spectrum, and for o is the total acceleration response spectrum.
Deformation response spectrum
Fi+ure on next slide shows the procedure to determine the deformation response spectrum. !he spectrum is deeloped for &l *entro +round motions, as shown in part 'a( of the fi+ure. !he time ariation of deformation induced by this +round motion in three SDF systems is presented in part 'b( of the fi+ure !he peak alue of deformation D uo, determined for SDF system with different ! n is determined and shown in part 'c( of the Fi+ure
Construction of deformation response spectrum
'a( &l%centro +round acceleration3 'b( Deformation response of three SDF systems with >0= and ! 02.C,6, and = sec3 'c( Deformation response spectrum for >0==7
Pseudo-velocity response spectrum *onsider a quantity 5 for an SDF system with natural frequency
=
;
=
=G !
D
n
!he quantity ; has the unit of elocity and is called relative pseudo- velocity or simply pseudo-velocity. !he prefix pseudo is o u used because ; is not equal to the peak elocity , althou+h it has the correct units.
Pseudo-velocity response spectrum
,
; =D
=G !
n
ζ = = Tn
,
5=,6789Tn
0.5
2.67
33.6
1.0
5.97
37.5
2.0
7.47
23.5
5
ζ = =
Pseudo-acceleration response spectrum , =
ζ = =
A
ζ = =
- =
=
=G D = D !n
Tn
,
A=,6789Tn;7
0.5
2.6 7
1.09g
1.0
5.9 7
0.61g
2.0
7.4 7
0.191g
A caution about Pseudo responses :lease note the followin+ comments re+ardin+ pseudo commodities 6. uo is same as D by definition.
o is not taken as ;, which by definition 0
is not taken as - which by definition0
Displacement Response Spectra for Different Damping values !he hi+her the dampin+, the lower the relatie displacement. -t a period of = sec, for example, +oin+ from "ero to C dampin+ reduces the displacement amplitude by a factor of two. /hile hi+her dampin+ produces further decreases in displacement, there is a diminishin+ return. !he reduction in displacement by +oin+ from C to =2 dampin+ is much less that that for 2 to C dampin+.
,eformation response spectra fr 1(0 El
Pseudo Acceleration Response Spectra for Different Damping Values Dampin+ has a similar effect on pseudo acceleration. $ote, howeer, that the pseudo acceleration at a 'near( "ero period is the same for all dampin+ alues. !his alue is always equal to the peak +round acceleration, 2.)67+, for the +round motion in question. i.e. &l%centro 6782 earthquake
,eformation response spectra fr 1(0 El
Pseudo acceleration (A) Vs peak total acceleration
(u ) t
o
!he term :seudo shall not be conceied by its meanin+ 'i.e. false as defined in &n+lish dictionaries(. In fact it shall be taken as 4an essence similar effect to their releant commodities 5 It can be obsered from below +raph that pseudo acceleration , - , and peak alue of true acceleration, u o t hae almost same alues for systems with Tn 1 sec and 3 1.
≤
It is worth mentionin+ that for elastic system the > seldomly exceed C t as such takin+ - same as u o ne+li+ible effect
Pseudo velocity (V) Vs peak system’s velocity
( u o )
-s shown in below +raph that ; ≈ u o for medium rise buildin+s '2.=H !n H 6 sec( as lon+ as ζ ≤ 2.6 . Similarly ; ≈ 2.9Cu o '2.=H !n H ) sec( for ζ ≤ 2.6
; ;≈ ≈ uu o o
; ≈ 2.9Cu o
Combined D-V-A spectrum !he deformation, pseudo%elocity and pseduo%acceleration spectra are plotted for a wide and practical ran+e of ! n and for a particular alue of > . !he aboe mentioned procedure is repeated for different alues of >. !he results for different alues of > oer a wide ran+e of !n are combined in a sin+le dia+ram, called combined D-V-A diagram, as shown on next slide
Use of D-V-A spectrum @efer to slides =7, )6 and )= for D,; and -, respectiely
$igure: *ombined D%;%- response spectrum for &l *entro +round motion3 3 0 =.
)7
Combined D-V-A spectrum For a +ien earthquake, small ariations in structural frequency 'period( can produce si+nificantly different results 'See ; alue for !n 0 2.C to) sec for &l%centro earthquake(
82
Relation between peak Equivalent static force, f so , and Pseudo acceleration, A
f so = k.u o =
since k =
f so
=
k.u o
= =
⇒
f so
=
=
'< n .m(.u o =
m.'< n .u o (
m.-
Peak Structural Response from the response spectrum
-s already discussed on preious slide, peak alue of the equialent static force fso can be determined as
f so
=
kD
=
m-
!he peak alue of base shear, ;bo, from equilibrium of aboe +ien dia+ram can be written as
;bo
=
f so
=
m.-
Peak Structural Response from the response spectrum
or ;bo
=
w +
.- =
+
.w
/here . is the wei+ht of the structure and g is the +raitational acceleration. /hen written in this form, A9g may be interpreted as the base shear coefficient or lateral force coefficient . It is used in buildin+ codes to represent the coefficient by which the structural wei+ht is multiplied to obtain the base shear
!roblem '>1 !he frame for use in a buildin+ is to be located on slopin+ +round s shown in fi+ure. !he cross sections of the two columns are 62 in. square. Determine the base shears in the two columns at the instant of peak response due to the &l *entro +round motion. -ssume the dampin+ ratio to be C. !he beam is too stiffer than the columns and can be assumed to be ri+id. !otal wei+ht at floor leel 0 62k
+olution
uo is decreasin+ with increase in ! n
+olution contd; *omputin+ the shear force at the base of the short and lon+ columns. u o = D = 2.?A′′ ⇒ - = ( =π B!n ) D =
⇒ - = ( =π B2.)) '2.?A B 6=( =
=
= =8.C6 ftBsec = 2.A?+
>?@
C Damped &lastic Displacement @esponse Spectrum for &l *entro 1round Eotion
/omments: -lthou+h both columns +o throu+h equal deformation, howeer, the stiffer column carries a +reater force than the flexible column. !he lateral force is distributed to the elements in proportion to their relatie stiffnesses. Sometimes this basic principle , if not reco+ni"ed in buildin+ desi+n, lead to unanticipated dama+e of the stiffer elements.
Response Spectrum normalized with peak ground parameters thereby giving the amplification magnitudes for D,V and A e.g., for a system with Tn = 0.5 sec and ζ =0.05. The amplification factors for D,V and A are αD ≈0.2 , αV ≈1.9and αA≈2.3, respectively !seudo
!seudo< acceleration amplification factor
,= ugo for TnB1* sec
u=
- ≈ u +o u= 5ery rigid systems
5ery fle#ible systems
Spectral regions in Response Spectrum -
u +o
≈ const. for !n < !c
or - = const. I u +o 'for !n < !c ( i.e. - directly aries with :1-. !herefore re+ion from !n 0 2 to !c is defined Acceleration sensitive region. Same lo+ic apply in definin+ velocity sensitive and displacement sensitive re+ions
Design Spectrum @esponse spectrum cannot be used for the desi+n of new structures, or the seismic safety ealuation of existin+ structures due to the followin+ reasons @esponse spectrum for a +round motion recorded durin+ the past is inappropriate for future desi+n or ealuation. !he response spectrum is not smooth and ja++ed, specially for li+htly damped structures. !he response spectrum for different +round motions recorded in the past at the same site are not only ja++ed but the peaks and alley are not necessarily at the same periods. !his can be seen from the fi+ure +ien on next slide where the response spectra for +round motions recorded at the same site durin+ past three earthquakes are plotted
$igure: @esponse $igure: @esponse spectra for the $%S component of +round motions recorded at the imperial alley Irri+ation district substation, &l *entro, *alifornia, durin+ earthquakes 87
Design Spectrum Due to the inappropriateness of response spectrum as stated on preious slide, the majority of earthquake desi+n spectra are obtained by aera+in+ a set of response spectra for +round motion recorded at the site the past earthquakes. If nothin+ hae been recoded at the site, the desi+n desi+n spectrum should be based on +round motions recorded at other sites under similar conditions such as ma+nitude of the earthquake, the distance of the site from causatie fault, the fault mechanism, the +eolo+y of the trael path of seismic waes from the source to the site, and the local soil conditions at the site.
Design Spectrum For practical applications, applications, desi+n spectra spectra are presented as smooth cures or strai+ht lines. Smoothin+ is carried , usin+ statistical analysis, out to eliminate the peaks and alleys in the response spectra that are not desirable for desi+n. For this purpose statistical analysis of response spectra is carried out for the ensemble of +round motions. &ach +round motion, for statistical analysis is normali"ed 'scaled up or down( so that all +round motions hae the same peak +round acceleration, say u +o 3other basis for normali"ation can be chosen.
Construction of Design Spectrum @esearchers hae deeloped procedures to construct such desi+n spectra from +round motion parameters. One such procedure is illustrated in +ien fi+ure. !he recommended period alues !a 0 6B)) sec, !b 0 6B9 sec, !e 0 62 sec, and !f 0 )) sec, and the amplification factors J-, J; , and JD for the three spectral re+ions '+ien table on next slide(, were deeloped by the statistical analysis of a lar+er ensemble of +round motions recorded on firm ground 'rock, soft rock, and competent sediments(.
Amplification factors for construction of Design Spectrum
Construction of Design Spectrum (firm soil) /e will now deelop the 98.6 percentile desi+n spectrum for 3=*C
u = 6+ For conenience, a peak +round acceleration +o is selected3 the resultin+ spectrum can be scaled by K to obtain the desi+n spectrum correspondin+ to u +o = K+ u +o u +o +o I = 89 in.BsecB+ and u =? = u +o !he typical alues of u +o , recommended for firm ground, are used. For u +o ratios +ie u +o = 89 inBsec and u +o = )? in
=
6+ , these
Construction of Design Spectrum (firm soil) α =
Lsin+ the alues on preious slide and alues +ien in table ?.7.= 'slide C)( for 98.6 percentile and > 0C, the :seudo% elocity desi+n spectrum can be dawn as shown in Fi+ure ?.7.8
α - =
=.A6
=.) α D
=
=.26
Construction of Design Spectrum (firm soil) Displacement and :seudo%acceleration desi+n spectra can be drawn from pseudo%elocity desi+n spectrum usin+ the relations bein+ already discussed and reproduced here for the conenience
; D=
=
!n = ; =G
;< n
=G = ;. !n
!he :seudo%acceleration and displacement desi+n spectra drawn by usin+ aboe +ien equation are drawn in Fi+ures ?.7.C and ?.7.? on next two slides
Construction of Design Spectrum (firm soil)
Construction of Design Spectrum (firm soil)
Design Spectrum for various values of 3
!seudo< velocity design spectrum for ground motions .ith
6
89 inBsec, and
)? in.
Design Spectrum for various values of 3
:seudo% acceleration desi+n spectrum '98.6 th percentile( drawn on lo+ scale for +round
89 inBsec, and
)? in.
?2
Design Spectrum for various values of 3
$igure: :seudo% acceleration desi+n spectrum '98.6 th percentile( drawn on linear scale for +round motions with u +o 6g , u +o 89 inBsec, and u +o )? in. 3
=
3
=
=
Envelope Design spectrum For some sites a desi+n spectrum is the enelope of two different elastic desi+n spectra as shown below
Nearby fault producing moderate ED
+ite
$ar a.ay fault producing large ED
!roblem '>7 'a( - full water tank is supported on an 92%ft%hi+h cantileer tower. It is ideali"ed as an SDF system with wei+ht w 0 622 kips, lateral stiffness k 0 8 kipsBin., and dampin+ ratio > 0 C. !he tower supportin+ the tank is to be desi+ned for +round motion characteri"ed by the desi+n spectrum of Fi+. ?.7.C scaled to 2.C+ peak +round acceleration. Determine the desi+n alues of lateral deformation and base shear. 'b( !he deformation computed for the system in part 'a( seemed excessie to the structural desi+ner, who decided to stiffen the tower by increasin+ its si"e. Determine the desi+n alues of deformation and base shear for the modified system if its lateral stiffness is 9 kipsBin.3 assume that the dampin+ ratio is still C. *omment on how stiffenin+ the system has affected the desi+n requirements. /hat is the disadanta+e of stiffenin+ the systemM