DesignGyan2324148026Response-Spectrum-Method-Of-Analysis.ppt

Response spectrum analysis

1. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Chapters –5 & 6
Chapter -5
RESPONSE SPECTRUM
METHOD OF ANALYSIS

2. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Introduction
Responsespectrummethodisfavouredby
earthquakeengineeringcommunitybecauseof:
Itprovidesatechniqueforperformingan
equivalentstaticlateralloadanalysis.
Itallowsaclearunderstandingofthe
contributionsofdifferentmodesofvibration.
Itoffersasimplifiedmethodforfindingthe
designforcesforstructuralmembersfor
earthquake.
Itisalsousefulforapproximateevaluation
ofseismicreliabilityofstructures.
1/1

3. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
Theconceptofequivalentlateralforcesforearth-
quakeisauniqueconceptbecauseitconvertsa
dynamicanalysispartlytodynamic&partlyto
staticanalysisforfindingmaximumstresses.
Forseismicdesign,thesemaximumstressesare
ofinterest,notthetimehistoryofstress.
Equivalentlateralforceforanearthquakeis
definedasasetoflateralforcewhichwill
producethesamepeakresponseasthat
obtainedbydynamicanalysisofstructures.
Theequivalenceisrestrictedtoasinglemodeof
vibration.
1/2

4. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
A modal analysis of the structure is carried out
to obtain mode shapes, frequencies & modal
participation factors.
Using the acceleration response spectrum, an
equivalent static load is derived which will
provide the same maximum response as that
obtained in each mode of vibration.
Maximum modal responses are combined to
find total maximum response of the structure.
1/3
Theresponse spectrum method of analysis is
developed using the following steps.

5. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Thefirststepisthedynamicanalysiswhile,the
secondstepisastaticanalysis.
Thefirsttwostepsdonothaveapproximations,
whilethethirdstephassomeapproximations.
Asaresult,responsespectrumanalysisis
calledanapproximateanalysis;butapplications
showthatitprovidesmostlyagoodestimateof
peakresponses.
Methodisdevelopedforsinglepoint,single
componentexcitationforclassicallydamped
linearsystems.However,withadditional
approximationsithasbeenextendedformulti
point-multicomponentexcitations&fornon-
classicallydampedsystems.
Contd…
1/4

6. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Equation of motion for MDOF system under
single point excitation
(5.1)
Using modal transformation, uncoupled sets of
equations take the form
 is the mode shape; ω
iis the natural frequency
is the more participation factor; is the
modal damping ratio.
Development of the methodg
x   Mx Cx Kx MI 2
2 ; 1 (5.2)
i i i i i i i g
z z z x i m       T
i
i T
ii




MI
M
1/5i
 λ
i ξ
i

7. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Response of the system in the ith mode is
(5.3)
Elastic force on the system in the ith mode
(5.4)
As the undamped mode shape satisfies
(5.5)
Eq 5.4 can be written as
(5.6)
The maximum elastic force developed in the ith
mode
(5.7)
Contd…i i i
x=φz si i i i
f =Kx =Kφz i
 2
i i i
Kφ =ω Mφ 2
si i i i
f=ω Mφ z 2
simax i i imax
f =Mφ ω z
1/6

8. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Referringtothedevelopmentofdisplacement
responsespectrum
(5.8)
Using ,Eqn5.7maybewrittenas
(5.9)
Eq5.4canbewrittenas
(5.10)
 istheequivalentstaticloadfortheithmode
ofvibration.
 isthestaticloadwhichproducesstructural
displacementssameasthemaximummodal
displacement.
Contd…
max
,
i
i i d i i
zS   max
i
ia
S
i
s i i e
f M P 11
max max


i
i si e
x K f K P 2
ad
SS P
ei
1/7P
ei

9. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Sincebothresponsespectrum&modeshape
propertiesarerequiredinobtaining,itisknown
asmodalresponsespectrumanalysis.
Itisevidentfromabovethatboththedynamic&
staticanalysesareinvolvedinthemethodof
analysisasmentionedbefore.
Asthecontributionsofresponsesfromdifferent
modesconstitutethetotalresponse,thetotal
maximumresponseisobtainedbycombiningmodal
quantities.
Thiscombinationisdoneinanapproximatemanner
sinceactualdynamicanalysisisnowreplacedby
partlydynamic&partlystaticanalysis.
Contd…P
ei
1/8

10. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Three different types of modal combination rules
are popular
ABSSUM
SRSS
CQC
Contd…
Modal combination rules
ABSSUM stands for absolute sum of maximum
values of responses; If is the response quantity
of interestx max
1
m
i
i
xx


(5.11)
is the absolute maximum value of
response in the ith mode.maxi
x
2/1

11. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
The combination rule gives an upper bound to the
computed values of the total response for two
reasons:
It assumes that modal peak responses occur at
the same time.
It ignores the algebraic sign of the response.
Actual time history analysis shows modal peaks
occur at different times as shown in Fig. 5.1;further
time history of the displacement has peak value at
some other time.
Thus, the combination provides a conservative
estimate of response.
Contd…
2/2

12. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
0 5 10 15 20 25 30
-0.4
-0.2
0
0.2
0.4
Top floor displacement (m)
t=6.15
0 5 10 15 20 25 30
-0.4
-0.2
0
0.2
0.4
Time (sec)
First generalized displacement (m)
t=6.1
(a) Top storey displacement
(b) First generalized displacement
2/3
Fig 5.1

13. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
0 5 10 15 20 25 30
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Time (sec)
Second generalized displacement (m)
t=2.5
(c) Second generalized displacement
Fig 5.1 (contd.)
2/3

14. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
SRSS combination rule denotes square root of sum
of squares of modal responses
For structures with well separated frequencies, it
provides a good estimate of total peak response.
Whenfrequenciesarenotwellseparated,some
errorsareintroducedduetothedegreeof
correlationofmodalresponseswhichisignored.
TheCQCrulecalledcompletequadratic
combinationruletakescareofthiscorrelation.
Contd…2
max
1
(5.12)
m
i
i
xx


2/4

15. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Itisusedforstructureshavingcloselyspaced
frequencies:
Second term is valid for & includes the effect
of degree of correlation.
Due to the second term, the peak response may be
estimated less than that of SRSS.
Various expressions for are available; here
only two are given :
Contd…2
1 1 1
(5.13)
m m m
i ij i j
i i j
x x x x 
  
  ij
2/5i

j

16. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
(Rosenblueth & Elordy) (5.14)
(Der Kiureghian)(5.15)
BothSRSS&CQCrulesforcombiningpeakmodal
responsesarebestderivedbyassuming
earthquakeasastochasticprocess.
If the ground motion is assumed as a stationary
random process, then generalized coordinate in
each mode is also a random process & there
should exist a cross correlation between
generalized coordinates.
Contd… 
 
2
2
2
2
1
14
ij
ij
ij ij


  


  
   
3
2 2
22
2
81
1 4 1
ij ij
ij
ij ij ij
  

   


  
2/6

17. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Becauseofthis,existsbetweentwomodal
peakresponses.
BothCQC&SRSSrulesprovidegoodestimatesof
peakresponseforwidebandearthquakeswith
durationmuchgreaterthantheperiodofstructure.
Becauseoftheunderlyingprincipleofrandom
vibrationinderivingthecombinationrules,the
peakresponsewouldbebettertermedasmean
peakresponse.
Fig5.2showsthevariationofwithfrquency
ratio.rapidlydecreasesasfrequencyratio
increases.
Contd…
2/7i

j i

j i

j

18. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Fig 5.2
Contd…
2/8

19. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Asbothresponsespectrum&PSDFrepresent
frequencycontentsofgroundmotion,arelationship
existsbetweenthetwo.
Thisrelationshipisinvestigatedforthesmoothed
curvesofthetwo.
Here a relationship proposed by Kiureghianis
presented
Contd…0
2.8
( ) 2ln (5.16 b)
2
p







2/9


2
2
0
,24
(5.16 a)
g
ff
x
D
S
p


 

    


 
 

20. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
0 10 20 30 40 50 60
0
0.01
0.02
0.03
0.04
0.05
Frequency (rad/sec)
PSDF of acceleration
(m
2sec
-3/rad)
Unsmoothed PSDF from Eqn 5.16a
Raw PSDF from fourier spectrum
0 10 20 30 40 50 60 70 80 90 100
0
0.005
0.01
0.015
0.02
0.025
Frequency (rad/sec)
PSDFs of acceleration (m
2
sec
-3/rad)
Eqn.5.16a
Fourier spectrum of El Centro
Unsmoothed
5 Point smoothed
Fig5.3
2/10
Example5.1:ComparebetweenPSDFsobtained
fromthesmootheddisplacementRSPandFFTof
Elcentrorecord.

21. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Degreeoffreedomisswaydegreeoffreedom.
Swayd.o.fareobtainedusingcondensation
procedure;duringtheprocess,desiredresponse
quantitiesofinterestaredeterminedandstoredin
anarrayRforunitforceappliedateachsway
d.o.f.
Frequencies&modeshapesaredetermined
usingMmatrix&condensedKmatrix.
For each mode find (Eq. 5.2) & obtain P
ei
(Eq. 5.9)
Application to 2D framesi
 
1
2
1
(5.17)
N
r
ir
r
i N
r
ir
r
W
W








2/11

22. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Obtain ; is the modal peak
response vector.
Use either CQC or SRSS rule to find mean peak
response.
Example 5.2 : Find mean peak values of top dis-
placement, base shear and inter storey drift between
1
st
& 2
nd
floors.
Contd…( 1... )
j ej
R RP j r R
j 23
4
12
3
ω =5.06rad/s; ω =12.56rad/s;
ω =18.64rad/s; ω = .5rad/s
2/12
Solution :   
   
;
;
TT
12
TT
34
φ = -1 -0.871 -0.520 -0.278 φ = -1 -0.210 0.911 0.7 52
φ = -1 0.738 -0.090 -0.347 φ = 1 -0.843 0.268 -0.14 5

23. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Approaches
Disp (m)
Base shear in terms of
mass (m)
Drift (m)
2 modesall modes2 modesall modes2 modesall modes
SRSS
0.9171 0.917
1006.5581006.658 0.221 0.221
CQC
0.9121
0.905 991.172 991.564 0.214 0.214
ABSSUM 0.9621 0.971 1134.5461152.872 0.228 0.223
Time history0.8921 0.893 980.098 983.332 0.197 0.198
Table 5.1
Contd…
2/13

24. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Analysis is performed for ground motion applied to
each principal direction separately.
Following steps are adopted:
Assume the floors as rigid diaphragms & find
the centre of mass of each floor.
DYN d.o.f are 2 translations & a rotation; centers
of mass may not lie in one vertical (Fig 5.4).
Apply unit load to each dyn d.o.f. one at a
time & carry out static analysis to find
condensed K matrix & R matrix as for 2D frames.
Repeat the same steps as described for 2D
frame
Application to 3D tall frames
3/1

25. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
3/2C.G. of mass line

1
CM

2
CM

3
CM
L
L
L
L

g
x

x
(a) C.G. of mass line

1
CM

2
CM

3
CM
L
L
L
L L

g
x

x
(b)
Figure 5.4:

26. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Example 5.3 : Find mean peak values of top floor
displacements , torque at the first floor &
at the base of column A for exercise for problem
3.21. Use digitized values of the response spectrum
of El centro earthquake ( Appendix 5A of the book).
Results are obtained following the steps of
section 5.3.4.
Results are shown in Table 5.2.
Contd…1 2 3
4 5 6
ω =13.516rad/s; ω =15.138rad/s; ω =38.731rad/s;
ω =39.633rad/s ; ω =45.952rad/s; ω =119.187rad/s XY
V and V
3/3
Solution :

27. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Approac
hes
displacement (m)
Torque
(rad) V
x(N) V
y(N)
(1) (2) (3)
SRSS
0.1431 0.00340.0020214547 44081
CQC
0.1325 0.00310.0019207332 43376
Time
history
0.1216 0.00230.0016198977 41205
TABLE 5.2
Contd…
Results obtained by CQC are closer to those of
time history analysis.
3/4

28. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Responsespectrummethodisstrictlyvalidfor
singlepointexcitation.
Forextendingthemethodformultisupport
excitation,someadditionalassumptionsare
required.
Moreover,theextensionrequiresaderivation
throughrandomvibrationanalysis.Therefore,itis
notdescribedhere;butsomefeaturesaregiven
belowforunderstandingtheextensionofthe
methodtomultisupportexcitation.
Itisassumedthatfutureearthquakeis
representedbyanaveragedsmoothresponse
spectrum&aPSDFobtainedfromanensemble
oftimehistories.
RSA for multi support excitation
3/5

29. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
3/6
Lack of correlation between ground motions at
two points is represented by a coherence function.
Peak factors in each mode of vibration and the
peak factor for the total response are assumed to
be the same.
A relationship like Eqn. 5.16 is established
between and PSDF.
Mean peak value of any response quantity r
consists of two parts:d
S

30. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…2
1
2 ; 1.. (5.18)
s
i i i i i ki k
k
z z z u i m  

     (5.19)
ik
ki
ii



T
T
MR
M
3/7
•Pseudostaticresponseduetothe
displacementsofthesupports
•Dynamicresponseofthestructurewith
respecttosupports.
Usingnormalmodetheory,uncoupled
dynamicequationofmotioniswrittenas:

31. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
If the response of the SDOF oscillator to
then
Total response is given by
 are vectors of size m x s (for s=3 &
m=2)
Contd…  
 
  
11
1 1 1
(5.21)
(5.22)
(5.23)
sm
k k i i
ki
s m s
k k i ki ki
k i k
r t a u t z t
r t a u t z
rt



  




  


TT
a u t z t
3/8k ki
u is z 1
(5.20)
s
i ki ki
k
zz

 β
φ and z

32. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Assuming to be random
processes, PSDF of is given by:
Performingintegrationoverthefrequencyrange
ofinterest&consideringmeanpeakaspeak
factormultipliedbystandarddeviation,
expectedpeakresponsemaybewrittenas:
Contd… 
     

T
β 1 11 1 21 1 31 2 12 2 22 2 32
T
11 21 31 12 22 32
φ = β β β β β β ( 5.24a)
z = z z z z z z ( 5.24b)  tr t ,u t and z ()rt (5.25)
rr
S   
T T T T
uu zz uz zu
a S a S a S S a
   
   
3/9

33. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…

 
....
 





1 2 3 S
12
T T T T
uu uz βD βD zz βD βD zu
T
1 p 2 p 3 p S p
T
βD 1 11 11 1 21 21 1 s1 s1 m 11 1m
ij i j j
E max r t = b b+b φ +φ φ + φ b ( 5.26)
b = a u a u a u a u ( 5.27a)
φ = φ β D φ β D φ β D ...φ β D ( 5.27b)
D =Dω ,ξ i=1,..,s ; j=1,..,m ( 5.27c)
 and are the correlation matrices
whose elements are given by:,
uu u z
ll zz
l i j i j
ij
α
u u u u
uu -α
1
=S ω dω ( 5.28)
σσ
3/10

34. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…i kj i k
i kj
α
*
u z j u u
uz -α
1
= h S ω dω ( 5.29)
σσ ki lj k l
ki lj
α
*
z z i j u u
zz -α
1
= hh S ω dω ( 5.30)
σσ 

i k i k g
11
22
u u u u u22
coh i,k1
S = S S coh i,k = S ( 5.31)
ωω  
k l k l g
11
22
u u u u u
S =S S coh k,l =coh k,l S ( 5.33) 

i j i j g
11
22
u u u u u44
coh i,j1
S = S S coh i,j = S ( 5.32)
ωω
3/11

35. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd… ij i j j
D =Dω ,ξ
Forasingletrainofseismicwave,
thatisdisplacementresponsespectrumfora
specifiedξ;correlationmatricescanbeobtained
if isadditionallyprovided; canbe
determinedfrom (Eqn5.6).
If only relative peak displacement is required,third
term of Eqn.5.26is only retained.
Steps for developing the program in MATLAB is
given in the book.coh( i,j )  jj
Dω ,ξ ug
S
Example 5.4Example 3.8 is solved for El centro
earthquake spectrum with time lag of 5s.
3/12

36. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
Solution :The quantities required for calculating the
expected value are given below:12
11 11 11 21 11 31 12 12 12 22 12 32
21 11 21 21 21 31 22 12 22 22 22 32
1 1 1 1 1 1 1 1
; ; ,
0.5 1 0.5 1 1 1 1 3
12.24rad/s ; 24.48rad/s
1111
;
1113
0.0259 0.0259 0.0259 -
T
D
ww

           
           

     
  
     
     




 



TT
φ φ r
a

11 21 31 1
12 22 32 2
12
1 1 1 2
21
0.0015 -0.0015 -0.0015
0.0129 0.0129 0.0129 0.0015 0.0015 0.0015
( 12.24) 0.056m
( 24.48) 0.011m
0
5 10
, 0 ; exp ; exp
22
0
D D D D
D D D D
coh i j




   





    
    

   
  
   
   


3/13

37. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis1 0.873 0.765
0.873 1 0.873
0.765 0.873 1
0.0382 0.0061 0.0027 0.0443 0.0062 0.0029
0.0063 0.0387 0.0063 0.0068 0.0447 0.0068
0.0027 0.0063 0.0387 0.0029 0.0068 0.0447
1 0.0008 0.0001 0.0142
0.0008 1 0













uu
uz
zz
0.0007 0.0001
.0008 0.0007 0.0142 0.0007
0.0001 0.0008 1 0.0001 0.0007 0.0142
0.0142 0.0007 0.0001 1 0.0007 0.0001
0.0007 0.0142 0.0007 0.0007 1 0.0007
0.0001 0.0007 0.0142 0.0001 0.0007 1









Contd…
3/14

38. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Mean peak values determined are:
Contd…12
12
( ) 0.106 ; ( ) 0.099
( ) 0.045 ; ( ) 0.022
tot tot
rel rel
u m u m
u m u m


For perfectly correlated ground motion1 0 0
0 1 0 nullmatrix
0 0 1
1 1 1 0 0 0
1 1 1 0 0 0
1 1 1 0 0 0
0 0 0 1 1 1
0 0 0 1 1 1
0 0 0 1 1 1















uu uz
zz
3/15

39. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
Mean peak values of relative displacement1
2
RSA RHA
u =0.078m ; 0.081m
u =0.039m ; 0.041m
It is seen that’s the results of RHA & RSA match
well.
Another example (example 3.10) is solved for a time
lag of a 2.5 sec.
Solutionisobtainedinthesamewayandresults
aregiveninthebook.Thecalculationsteps
areselfevident.
3/16

40. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Cascaded analysis
Cascaded analysis is popular for seismic analysis
of secondary systems (Fig 5.5).
RSAcannotbedirectlyusedforthetotalsystem
becauseofdegreesoffreedombecome
prohibitivelylarge;entiresystembecomes
nonclasicallydamped.
4/1
Secondary System
x
g
..
..
k
c
m
x
a= x
f+ x
g
......
Secondary system mounted
on a floor of a building frame
SDOF is to be analyzed for
obtaining floor response spectrum
x
f
Fig 5.5

41. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
Inthecascadedanalysistwosystems-primary
andsecondaryareanalyzedseparately;outputof
theprimarybecomestheinputforthesecondary.
In this context, floor response spectrum of the
primary system is a popular concept for
cascaded analysis.
The absolute acceleration of the floor in the figure
is
PseudoaccelerationspectrumofanSDOFis
obtainedfor;thisspectrumisusedforRSAof
secondarysystemsmountedonthefloor.a
x a
x
4/2

42. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
Example 5.6For example 3.18, find the mean peak
displacement of the oscillator for El Centro earthquake.
forsecondarysystem=0.02;forthemain
system=0.05;floordisplacementspectrumshownin
theFig5.6isused
Solution
4/3 
0 5 10152025303540
0
0.5
1
1.5
Frequency (rad/sec)
Displacement (m)
Usingthisspectrum,
peakdisplacementofthe
secondarysystemwith
T=0.811sis0.8635m.
Thetimehistoryanalysis
fortheentiresystem(with
CmatrixforP-Ssystem)is
foundas0.9163m.
Floor displacement response
spectrum (Exmp. 5.6)

43. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Approximate modal RSA
Fornonclassicallydampedsystem,RSAcannot
bedirectlyused.
However,anapproximateRSAcanbeperformed.
Cmatrixfortheentiresystemcanbeobtained
(usingRayleighdampingforindividualsystems
&thencombiningthemwithoutcouplingterms)
 matrixisobtainedconsideringalld.o.f.&
becomesnondiagonal.
Ignoringoffdiagonalterms,anapproximate
modaldampingisderived&isusedforRSA.1
2
0
0
C
C
C



  T
C
4/4

44. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Seismic coefficient method
Seismiccoefficientmethodusesalsoasetof
equivalentlateralloadsforseismicanalysisof
structures&isrecommendedinallseismiccodes
alongwithRSA&RHA.
Forobtainingtheequivalentlateralloads,ituses
someempiricalformulae.Themethodconsistsof
thefollowingsteps:
•Usingtotalweightofthestructure,base
shearisobtainedby
isaperioddependentseismiccoefficient (5.34)
bh
V W C
h
C
4/5

45. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
•Baseshearisdistributedasasetoflateral
forcesalongtheheightas
bearsaresemblancewiththatforthe
fundamentalmode.
•Staticanalysisofthestructureiscarriedout
withtheforce .
Differentcodesprovidedifferentrecommendations
forthevalues/expressionsfor . ( ) (5.35)
i b i
F V f h (i = 1,2...... n)
i
F ( )
i
fh h
C & ( )
i
fh
4/6

46. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
4/7
Distribution of lateral forces can be written as j j j1
j j j1
j j1
jb
j j1
jj
jb
jj
k
jj
jb k
jj
S
a1
F=ρ×W ×φ× ( 5.36)
1j j j1 g
F W ×φ
= ( 5.37)
∑F ΣW ×φ
W×φ
F = V × ( 5.38)
ΣW×φ
W ×h
F = V ( 5.39)
ΣW×h
W ×h
F = V ( 5.40)
ΣW×h

47. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
4/8
Computation of base shear is based on first mode.
Following basis for the formula can be put forward.
 
i
i
i
i
ae
bi
bb
a e
i
a1
b
S
a
i
V=ΣF =( ΣW×φ× )×λ ( 5.41)
b ji j ji i gi
S
V = W ( 5.42)
g
V≤Σ V ( 5.43)
S
≤Σ W i= 1to n ( 5.44)
g
S
V = ×W ( 5.45)
g

48. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Seismic code provisions
Allcountrieshavetheirownseismiccodes.
Forseismicanalysis,codesprescribeallthree
methodsi.e.RSA,RHA&seismiccoefficient
method.
Codes specify the following important factors for
seismic analysis:
•Approximate calculation of time period for
seismic coefficient method.
• plot.
•Effect of soil condition on h
CVs T a
&
h
SA
or C
gg
5/1

49. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
•SeismicityoftheregionbyspecifyingPGA.
•Reductionfactorforobtainingdesignforces
toincludeductilityinthedesign.
•Importancefactorforstructure.
Provisions of a few codes regarding the first three
are given here for comparison. The codes include:
•IBC –2000
•NBCC –1995
•EURO CODE –1995
•NZS 4203 –1992
•IS 1893 –2002
5/2

50. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
IBC –2000
• for class B site,
•for the same site, is given byh
C A
g 0.4 7.5 0 0.08s
1.0 0.08 0.4s (5.47)
0.4
0.4s
nn
n
n
n
TT
A
T
g
T
T


  


  

 
 1
1
1
1.0 0.4s
(5.46)0.4
0.4s
h
T
C
T
T






5/3

51. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
T may be computed by
 can have any reasonable distribution.
Distribution of lateral forces over the height
is given byi
F 1
(5.49)
k
jj
ib N
k
jj
j
Wh
FV
Wh


 2
1
1
1
2 (5.48)
N
ii
i
N
ii
i
Wu
T
g Fu











5/4

52. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
Distribution of lateral force for nine story frame is
shown in Fig5.8 by seismic coefficient method .  
1 1 1 1
k={ 1;0.5 T +1.5 ; 2 for T≤0.5s ;0.5 ≤T ≤2.5s; T ≥2.5s ( 5.50)
0 2 4
1
2
3
4
5
6
7
8
9
Storey force
Storey
T=2sec
T=1sec
T=0.4sec
W
2
W
W
W
W
W
W
W
W
9@3m
Fig5.8
5/5

53. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
NBCC –1995
• is given by
•For U=0.4 ; I=F=1, variations of with T
are given in Fig 5.9. h
C e
he
CU
C = ; C =USIF ( 5.51a);( 5.51b)
R A
S&
g
0 0.5 1 1.5
1
1.5
2
2.5
3
3.5
4
4.5
Time period (sec)
Seismic response factor S
Fig5.9
5/6

54. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
•For PGV = 0.4ms
-1
, is given by
•T may be obtained by
•S and Vs T are compared in Fig 5.10 for
v = 0.4ms
-1
, I = F = 1; (acceleration and
velocity related zone) 





1
N 2
2
ii
1
1 N
ii
1
Fu
T =2π ( 5.53)
g Fu A
g 




n
n
n
1.2 0.03 ≤T ≤0.427s
A
= ( 5.52)0.512
T >0.427sg
T A/g hv
z =z
5/7

55. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time period (sec)
A/g
S
S or A/g
Fig5.10
5/8

56. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
•Distribution of lateral forces is given by




1
t 1 b 1
b1
0T ≤0.7 s
F = 0.07T V 0.7<T <3.6s ( 5.55)
0.25V T≥3.6 s  

ii
i b t N
ii
i=1
Wh
F = V -F ( 5.54)
Wh
5/9

57. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…






1c
1
e -
3
c
1c
1
A
0≤T ≤T
g
C = ( 5.57)
TA
T≥T
gT
5/10
EURO CODE 8 –1995
•Base shear coefficient is given by
• is given by
•Pseudo acceleration in normalized form is given
by Eqn 5.58 in which values of T
b,T
c,T
d s
C e
C e
s
C
C = (5.56)
q b c d
T T T
hard 0.1 0.4 3.0
med 0.15 0.6 3.0
soft 0.2 0.8 3.0
(A is multiplied by 0.9)
are

58. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
5/11
•Pseudo acceleration in normalized form
,
is given by 











0
n
nb
b
b n c
c
c n dg
n
cd
nd2
n
T
1+1.5 0 ≤T ≤T
T
2.5 T ≤T ≤T
A
= ( 5.58)T
2.5 T ≤T ≤Tu
T
TT
2.5 T ≥T
T

59. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Rayleigh's method may be used for calculating T.
Distribution of lateral force is
Variation of are shown in
Fig 5.11. 

i i1
ib N
i i1
i=1
ii
ib N
ii
i=1
Wφ
F = V ( 5.59)
Wφ
Wh
F = V ( 5.60)
Wh / & /
e go go
c u A u
5/12

60. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
Time period (sec)
A/u
g0
C
e/u
g0
C
e
/u
g0
or
A
/u
g0
..
..
..
..
..
Fig 5.11
5/13

61. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
NEW ZEALAND CODE ( NZ 4203: 1992)
•Seismic coefficient & design response curves
are the same.
•For serviceability limit,
is a limit factor.
•For acceleration spectrum, is replaced by T. 
 
b 1 s 1
b s 1
C T =C T ,1 RzL T ≥0.45 ( 5.61a)
=C 0.4,1 RzL T ≤0.45 ( 5.61b) s
L 1
T
6/1

62. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
•Lateral load is multiplied by 0.92.
•Fig5.12 shows the plot of
•Distribution of forces is the same as Eq.5.60
•Time period may be calculated by using
Rayleigh’s method.
•Categories 1,2,3 denote soft, medium and hard.
•R in Eq 5.61 is risk factor; Z is the zone factor;
is the limit state factor.1
b
c vs T for s
l
6/2

63. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.2
0.4
0.6
0.8
1
1.2
Time period (sec)
Category 1
Category 2
Category 3
C
b
Fig5.12
6/3

64. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
IS CODE (1893-2002)
• are the same; they are
given by:a
e
S
C vs T & vs T
g
•Timeperiodiscalculatedbyempirical
formulaanddistributionofforceisgivenby:
2
jj
jb N
2
jj
j=1
Wh
F = V ( 5.65)
Wh 






a
1+15T 0≤T ≤0.1s
S
= 2.5 0.1 ≤T ≤0.4s for hard soil ( 5.62)
g
1
0.4≤T ≤4.0s
T
6/4

65. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…













a
a
1+15T 0≤T ≤0.1s
S
= 2.5 0.1 ≤T ≤0.55s for m edium soil ( 5.63)
g
1.36
0.55≤T ≤4.0s
T
1+15T 0≤T ≤0.1s
S
= 2.5 0.1 ≤T ≤0.67s for softsoil ( 5.64)
g
1.67
0.67≤T ≤4.0s
T
6/5
For the three types of soil S
a/g are shown in Fig
5.13
Sesmic zone coefficients decide about the PGA
values.

66. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
6/6
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
Time period (sec)
Hard Soil
Medium Soil
Soft Soil
Spectral acceleration coefficient (S
a
/g)
Variations of (Sa/g) with time period T
Fig 5.13
Contd…

67. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
Example 5.7:Seven storey frame shown in Fig 5.14
is analyzed with
For mass: 25% for the top three & rest 50% of live
load are considered.1 2 3
T =0.753s ; T =0.229s ; T =0.111s
R = 3; PGA = 0.4g ; for NBCC,PGA≈0.65g
Solution:
First period of the structure falls in the falling
region of the response spectrum curve.
In this region, spectral ordinates are different
for different codes. -3 7 -2
-1
Concrete density =24kNm ; E = 2.5×10 kNm
Live load =1.4kNm
6/7

68. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
6/8
A Seven storey-building frame for analysis
Fig 5.14
5m 5m 5m
7@3m
All beams:-23cm 50cm
Columns(1,2,3):-55cm 55cm
Columns(4-7):-:-45cm 45cm
Contd…

69. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
Table 5.3:Comparison of results obtained by different codes
Codes
Base shear (KN)
1st Storey Displacement
(mm)
Top Storey Displacement
(mm)
SRSS CQC SRSS CQC SRSS CQC
3 all 3 all 3 all 3 all 3 all 3 all
IBC 33.5133.6633.5233.680.740.740.740.7410.6410.6410.6410.64
NBCC 35.4635.6635.4635.680.780.780.780.7811.3511.3511.3511.35
NZ
4203
37.1837.2637.237.290.830.830.830.8312.0012.0012.0012.00
Euro 848.3448.4148.3548.421.091.091.091.0915.9415.9415.9415.94
Indian44.1944.2844.2144.290.990.990.990.9914.4514.4514.4514.45
6/9

70. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Contd…
0 2 4 6 8 10 12 14 16
1
2
3
4
5
6
7
Displacement (mm)
Number of storey
IBC
NBCC
NZ 4203
Euro 8
Indian
Comparison of displacements obtained by different codes
Fig 5.15
6/10

71. T.K. Datta
Department Of Civil Engineering, IIT Delhi
Response Spectrum Method Of Analysis
Lec-1/74