Approximate seismic analysis of inelastic asymmetric structures

" A p p r o x i m a t e se i smlc

analysis of inelastic a s y m m e t r i c structures A. W. Sadek and M. E. Sobaih Department of Structural Engmeermg, Cairo Universtty, Cairo, Egypt

H. S. Esmail Department of Ctvil Engineering, Zagaztg University, Shoubra, Egypt (Recetved October 1988, rewsed July 1990) This paper examines the approximate method of the single element model for analysing inelastic asymmetric structures. It simplifies the analysis by treating only a single resisting element having dynamic as well as yielding properties equivalent to the original structure with many resisting elements. The range of validity of the single element model for earthquake response calculations is defined in the paper. The study considers inelastic asymmetric one-storey structural models whose properties are varied to simulate a wide range of typical structural buildings. These properties include lateral period, torsional-to-lateral frequencies ratio, magnitude of structural eccentricity, number of resisting elements in the original structure, and yield strength of the structural model. The prime concern of the paper is to assess the accuracy of the displacement ductility demand estimated using the approximate method. For long period structures the single element model generally gives conservative estimates of displacement ductility demand. For such a range a better estimate using a modified single element model is proposed in which the single element is located at the plastic centre of strength instead of the centre of stiffness. The accuracy of the approximate method is found to considerably improve as the number of resisting elements in the original structure increases. For highly inelastic stiff structures, no approximate method can reasonably estimate their ductility demand values.

Keywords: earthquake engineering, inelastic behaviour, asymmetric structures

An elastic asymmetric structure with many resisting elements under earthquake excitations can be exactly modelled by an equivalent model with a single element located at the centre of stiffness and having the same overall dynamic properties as the original structure These propertms are lateral period T~, torsional-tolateral frequencies ratio fl, and fractional damping ~. For approximate earthquake analysis of inelastic asymmetric structures, a conceptually similar treatment has been proposed by Kan and Chopra I. The approximate method simplifies the analysis through replacing the original structure with many resisting elements by an equivalent single element model whose yielding properties are synthesized to approximate the original pro0141-0296/92/01049-14 © 1992 But~erworth-Hememann Ltd

pertms. The use of such an equivalent single element model in the time history analysis considerably reduces the computation needed. Nevertheless, its application has been held back because the model was insufficiently tested by Kan and Chopra t especially against stiff asymmetric structures which have been reported 2 to be vulnerable to the combined effect of asymmetry and inelasticity. The present paper aims to identify the range of vahdity of this approximate method and to further substantmte the approximation of the analysis by proposing a modified single element model. The earthquake inelastic response of asymmetric onestorey structural models whose dynamic properties are varied to simulate a wide range of typical structural

Eng. Struct. 1992, Vol. 14, No 1 49

Approximate

seismic

analysis

of inelastlc

asymetrlc

buildings are considered These properties mclude lateral period, torstonal-to-lateral frequencres ratto, magnitude of structural eccentrrcrty, number of resisting elements m the original structure, and yteld strength of the structural model. The prime concern of the paper 1s to assess the accuracy of the displacement ducttlity demand estrmated using the approximate method To avoid results being dependent on a specific ground motion, two different earthquake records are used.

structures:

A. W. Sadek

et al

Plastic region ~, Elastic region

Original structural model Let the original structure be taken as a one-storey structure consrsting of a rigid deck of mass m, square in plan with side length D as shown in Figure 1 The rigrd deck IS supported on four massless mextensrble columns on rigid foundations at the extremities of a square of dimension a. A one-way eccentricity e is introduced by shtftmg the centre of stiffness CS away from the centre of mass CM of the rigid deck along the y-ax.ts. The shift 1s achieved by assigning larger stiffness values to columns 1 and 2 than those assigned to columns 3 and 4. The earthquake ground motion t&(t) is assumed to act horizontally along the x-axis and thus induces coupled lateral and torsional motions, qr and qs respectively

“JV FIgwe

2

Propertles

of resisting

elements

z

Y

-X

a a Y t

r’ T

-o-

cs

e

-X

D

CM

+-

b Ffgure

50

7

Orlglnal

b

structure.

Eng. Struct.

1992,

(a) Isometric;

Vol.

(b) plan

14, No 1

hgure

3

Equwalent

single-element

model:

(a) isometric,

(b) plan

Approximate seismic analysis of inelastic asymetric structures: A. W. Sadek et al.

v. v~p

Equations of motion The incremental form of equations of motion of the above model is given by L,mit yield curve

0

[ 0 0 ] ~ A q+~2 ~ m [ j ( r A # o ) Aq~ = - m I~ + [K,] ~(rAqo)

O~o](.rAdlo) 1

In,t,al y,eld curve

(1) _

where r is the polar radius of gyration of the deck with reference to CM, c0~ and w0 are the uncoupled lateral and torsional frequencies of the structure, respectively, ~"is the fraction of critical damping, and dots in the equations represent differentiation with respect to the time variable t. The overall stiffness matrix [Kt] is assembled from the updated columns stiffness matrices [S], as follows

[d],~[S],[d],

[g,] = ~

-1

Ftgure 4 In,t,al and hmit y,eld curves for 4-element model (elf = 0 4)

(2)

½ V~p

l

For the ith column with position coordinates (x,, y,) with respect to CM, the position matrix [d], is given by

[d], =

[10 -Y"rl x,/r d

Each column is assumed to be elasto-plastic and the yield criterion under the combined action of biaxial shear v~ and v,,. is described by a circular yield curve (Figure 2) and is given by

(a = (vJv,p) 2 + (v,v/v,p)2 = 1

0]

k,

(5)

where k, is the elastic column stiffness. The column is said to be plastic when the tip of the force vector hits the yield curve and the column stiffness matrix is given by

IS],

=

[s]~

-

[s]~

C = 0.5. and TRp/r Vxp = 1 665

~

I -2

S/ \

"\-1 '\

= O. and TRp/r Vxp = 1

C

1 /X ~

2

r.V.p

I//¢ ln,t,al y , e l d c u r v e

(4)

where v,p is the plastic shear capacity of the column along each of the x and y directions. The column is said to be elastic when the tip of the force vector lies inside the yield curve and the matrix [S], is given by

[ s ] , = [s]," =

1

(3)

Yield and stiffness properties of columns

TR TRp

-1

Lirn,t yield curve

Figure 5 Y,eld curves for 4-element and single element models (e/r = 0 4)

Normalized equations of motion Equation (1) can be put into the following nondimensional form

Aao)

t.aUo)

( Auo)

(6)

(8)

(7)

where u~ and uo are the lateral and torsional deformations normalized by the respective yield deformation, fl is the torsional to lateral frequency ratio, and [/¢,] = [K,]/K~. R is the strength reduction factor defined 2 as the ratio of the maximum induced earthquake force mS*~ to the system yield strength Vp.

where

[S] e = kl/(a¢/avj~)

[ (a¢/Ovl~)2

2 + (a~/avjr) 2)

(aC)/avl,)(aO/Ov ~y) ]

x I-(#Ol#vl~)(OOlOVly) (O~l#vl,) z

Eng. Struct. 1992, Vol. 14, No 1

51

Approximate

s e i s m t c a n a l y s t s o f i n e l a s t i c a s y m e t r t c s t r u c t u r e s : A . W. S a d e k e t al. _1_

Present study

r 10

[

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, [,

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those of the original structure W~th regard to the equivalence in the post-elastic range, it is assumed ~ that the total shear force V~-torque TR yield curve of the single element should lie between the initial and limit yield curves of the original structure. The imtial yield curve is defined by combinations of shear force V~ and torque TR at the centre of stiffness for which yielding of any resisting element in the system is initiated The hmlt yield curve is defined by force combinations at which all resisting elements are fully plastic. The initial and hmlt yield curves of the four-element model in Figure 1 are shown m Figure 4 for the eccentricity ratio e/r = 0.4 Yield curves are plotted in the normalized force coordinate system V~/V~p and TR/Tnp where V~p is the total yield shear of the system and TRe is the pure torque at CS causing yield. The yield curve ~b~ of the single element is then approximated by the following expression ~

b

b

I 0.6

x

0.5

%

/ •,,

\

/

I

~, = (vJv, p) 2 + C(K/V,p)(TR/TRp)

\1

\ i

¢

+ (TRITRp) ~ = 1

I

/ 4.Elem.

SEM

---

a=v~

e/r = 0.4 R=5 0.2

I

I

I

i I I

0.1

0.2

0.3

0.5

, I, 1

, 1

r,,I 1.4

2

3

T. Ftgure 6

Companson of maximum response for single- and 4-element models

1

T

I

I

(9)

The yield curve ~b, is mapped onto the alternative force space V,/V~p - TR/rV,p as shown in Figure 5 This is to reflect the relative strength of the original structure in both shear and torque. Selecting an appropriate value for c wdl control the curvature of the yield curve and the value of the ratio TRe/rVxp controls the point of intercept of ~, with the torque axis. Two eqmvalent yield curves are plotted in Figure 5 The first is for torque to shear raUo of 1.665 which is the same as that of the original structure together with c = 0.5. The second curve is for torque to shear ratio of 1.3 and c = 0.0 The reduction of the torque capacity in the second curve Is necessary to bring the eqmvalent yield curve between the initial and limit yield curves.

Sttffness properties The tangential stiffness matrix [K,] of the single element model is given by [K,] = [ K ] " -

1

[K]p

(10)

where [K] ~ is the elastic stiffness matrix given by

.a_~4

Uom

[Kle = ~02r 1

L-e/r

-e/r] f2z J

(11)

and [K] p IS the plastic stiffness matrix whose value depends on the state of the single element as follows Elastic [K] e = [0] 0

I 0.5

t 1

I 15

I 2

Lateral per=od, T~ (s)

(12)

Plastic [K] p = (1/K,(O¢,IOI,'O +

KoR(ae~,IaTR))

Ftgure 7 Normahzed elast=c smooth acceleration spectrum ¢

Equivalent single element model In the eqmvalent model the single element is located at the elastic centre o f stiffness CS as shown in Figure 3. In the elastic range, equivalence is satisfied by assuming dynamic properties, 1 e. T~, fl, and ~, to be the same as

52

Eng. Struct.

1992,

Vol.

14, No 1

[ K2,(Od~./OV.)z K.Ko.(a~,/OV~)(a~,/aT.)

K.KoR(O(~./OV.)(Otb./OTR) ] K~.(a~,/aTR) 2 J

where KoR is the torsional stiffness of the structure evaluated at CS and is given by

KoR = Ko - eEK~

(13)

Approximate seismic analysis of inelastic asymetric structures: A. W. Sadek et al. Taft

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Approximate seismic analysis of Inelastic asymetnc structures: A. W. Sadek et al.

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Approximate seismic analysis of inelastic asymetric structures: A. W. Sadek et al.

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Eng. Struct.

1992,

Vol.

14, No 1

55

A p p r o x i m a t e s e i s m i c analysis o f inelastic a s y m e t r i c s t r u c t u r e s : A. W. Sadek et al.

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Approximate seismic analysis of inelastic asymetric structures: A. W. Sadek et al.

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flexible and stiff structures uncoupled lateral period T, = 0.1 s to 2.2 s strength reduction factor R = 3,5; represent the cases of moderate and severe excitations respectwely. In terms of structural behav]our, such values represent the cases of structures being moderately or severely excited into the inelastic range ground motion two earthquake records are used as the input ground motion, the N-S component of the 1940 EI-Centro earthquake and the $69E component of the 1952 Taft earthquake. The smooth elastic spectrum proposed by Newmark and Hall 4 and shown in Figure 7 is used for normalizing earthquake ground records.

01

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CM

013

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Figure 12 16-element model. System e/D = 0 2, R = 5, a/D = 0 , 7 7 1 6

propertnes

9

=

1 O,

Validity of the single-element model As evidence of the success of the proposed single element model, Kan and Chopra I compared its estimates of maximum torsional and lateral deformations with those obtained from an accurate analyszs using the original structure model. Structures with 9 = ~/3 and whose lateral period ranged from 0.5 s to 2.5 s were used and the comparison is reproduced in Figure 6 (other system parameters are e/r = 0.4, the N-S component of the 1940 El Centro earthquake record). For such a range of periods good agreement was reported. However, it should be noted that the choice of system parameters in Kan and Chopra ~ is rather limited and in favour of the proposed model. Not only did they select torsionally stiff structures, as ~s evident from the large value of 9, but they also hmited the comparnson to structures of long or intermediate lateral periods. The more critical case of the torsionally-flexible laterally-stiff structures, which is reported 2'3 to be very susceptible to inelastic torsional deformations, was not examined. By extending the comparison in Figure 6 down to T, = 0.1 s, a sizable difference is evident between the approximate and accurate estimates of torsional as well as lateral deformations. Th~s discussion indicates that although the single element modelling is a very useful approximate method it has not been properly examined and its range of validity in terms o f system parameters has not been ~dentified. A systematic parametric study ns performed m the next section to establish the range of validity of the method.

Parametric study The following parameters are considered in the present systematic comparison of the approximate and accurate methods • •

eccentricxty ratio e/D = 0.2; represents the case of large eccentnclty uncoupled torsional to lateral frequencies ratio 9 = 0.8, 1, and ,,/3; represent the cases of torsionally

Figures 8 to 10 show the variation of the maximum torsional deformation Uo,,, maximum lateral deformation at CM, ux,,, and the ductihty demand on the most stressed column ~x4 as functions of the lateral period for original and single element models. Values of 9 = 0.8, 1, and ~ and R = 5 are compared. From the figures the following three observations can be made. First from comparing the broken and solid lines in Figure 8 it ns clear that for short period structures ( T , < 0 . 3 s) the maximum rotational deformation responses Uo,, are considerably lower in the single element model than those accurately evaluated using the original structural model. This observation holds true for all the cases of 9 considered. For long period structures, the single element model tends to overestimate the rotational response particularly for torsionally flexible cases (9 = 0.8 and 1). Good agreement is only hmited to the cases of torsionally stiff (9 = x/3) laterally flexible structures (T, > 0.5). Secondly Figure 9 showing Uxm values depicts a similar trend as mentioned above but to a lesser extent and m an unsystematic manner Thirdly the comparison of the ductility demand curves in Figure 10 strongly confirms the first observation. For very surf structures (T, = 0.1 s) the accurate value of ductdity demand may be twice as much as the approximate value. Hence ~t can be said that for such a range accurate analysis is lndlspensible. Effect of level of excaation, R The above discussion is based on R = 5. It is useful, however, to examine the performance of the single element model in approximating moderately excited structures (R = 3). Figure 11 compares the ductdity ratio #x4/l~s, defined as the ratio of the peak ductility demand accurately evaluated using the original model to that evaluated using the equivalent single element model, for R = 3 and 5. It can be seen that ratios assocmted w~th R = 3 (sohd line) are generally closer to unity than the rauos of R = 5 (broken line). The improvement Is pronounced m the short period range (T < 0.5). One can conclude that the accuracy of the approximate model improves for moderately excited inelastic structures.

Effect of number of resistmg elements Discussion so far has been hmited to structures with only four columns. In actual buildings, however, the

Eng. Struct. 1992, Vol. 14, No 1

57

Approxtmate seismic analysm of tnelastic asymetrlc structures: A. W. Sadek et al.

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58

Eng. Struct. 1992, Vol. 14, No 1

i

01

05

I J I iJl

1.0

I

20

r. ), 16-element and ( - - - ) single-element models (e/D = 0.2, R = 5)

Approximate seismic analysis of inelastic asymetric structures: A. W. Sadek et al.

04

03

02

I Maximum possible eccentNcity (+ve side) -__~

~. }s~:7

Ta ft

I

I

D

\ I nltial eccentr tclty

0.1 e/D

0.0

t

I

i

i

I

2

,

6

I

8

10

Time (s) -0.1

9,=1 Tx = 0.1 s R=5 e/D = 0.2

-0.2

-0.3 -0.4 Figure 14

0.4

eccentricity (-ve s~de) I

I

I

Variation of actual eccentric)ty with time for 4-element model

Taft

~_-_~( . . . .

I

Maximum. possible

~-

~ ......

_~___

~--¢-D

0.3

0.2 Initial eccentriciw

0.1

e/D 0.0

2

4

6

8

I

T

10

Time (s) -0.1

D.=I T,=01s R=5 e/D = 0.2

-0.2

-0.3 -- Maximum possible J;;!:il L eccentricity (-ve side)

___-~ -0

4

Figure 15

T

I

Var)atmon of actual eecentrmcit¥ with time for 16-element model

number of resisting elements are typically large. To examine the effect of a number of resisting elements in the original structure on the predictions of the companion single element model, a sixteen-element model (see Figure 12) is used. Figure 13 shows the accurate and approximate estimates of the ductility demand on the most stressed column, /z~6 of such a model. Other system parameters are fl = 0.8, 1, and x ~ , e/D = 0.2, and R = 5. A noticeable improvement in agreement between

the estimates of the original structural model and the companion single element model as compared to the previous case with a lesser number of resisting elements can be observed. It can be seen that the accuracy of the single element model increases as the number of resisting elements in the original structure increases. An attempt is made below to clarify this observation. In the single element model, eccentricity is assumed to remain invariant having a value equal to the initial

Eng. S t r u c t . 1 9 9 2 , V o l . 14, N o 1

59

Approxtmate seismic analysis of inelastic asymetric structures: A. W. Sadek et al.

L

cs 4

cP(IF I es

D

--

ep X

CM

~

CM

X

r

m

D

a

b

Figure 16

Single and modufled

single-element

(a) SEM, (b) p r o p o s e d

models

I

[

1

I es = 0_2 D

2

ELC

ELC

ELC

I

SEM

modified

_

ep=O2D

2

_

I

I

I e,=02D ep=01

_

_

I

I

_

D

I e, = 0 . 2 D

_

ep=OOD

_

=~ 1 5 _'~

:=L

=L 1

1

::L

0.5

I

I

l

I

I

r

L

I

I

I

I

I

05

1

15

2

05

1

15

2

05

1

15

2

I

I

I

I

I

05

1

15

2

T,

T,

[

Taft

Taft

Taft r

I

2~

I

I

I

15

=L

1

_

2

2

..~

_

O5

t

~

L

05

1

15

! 2

h

I

J

I

05

1

15

2

7",

T,

7",

a Ftgure

17

C o m p a r i s o n of ratuo ~/.~4 for single a n d m o d i f i e d

single e l e m e n t

eccentricity e. O n the other h a n d . m the original structure, the location of the effective centre of rigidity changes due to ymlding of resisting elements. Consequently the effectwe eccentricity, defined as the distance from the instantaneous centre of stiffness to CM, will vary from its elastic value. The time variation of the effectwe eccentricity in the 4- and 16-element models are shown in Fzgures 14 and 15, respectively. The maximum possible eccentrlciues on positive and negatwe sides are also shown in Figures 14 and 15. At

60

Eng.

Struct.

1992,

Vol.

14,

No

1

models

(a) R = 5, El = 0 8

the beginning of the excitation, the models are in the elastic range and the effective eccentricity is equal to the elastic value. The onset of yielding induces variations in the value of the actual eccentricity. In the 4-element model, the effective eccentricity varies considerably in magnitude as well as in direcuon. However, in the 16-element model the effective eccentricity also varies but to a lesser extent. Therefore it can be said that the assumption of the fixed locatnon of the single element at the initial CS will be more justified as the number of

Approximate seismic analysis of inelastic asymetric structures: A. W. Sadek et al. ELC

2k

I

ELC

1

I

I e= = 0 . 2 D ep=O 2D

2f

_

I

I

ELC

I

e~ = 0.2 I D

I _

e, =0.1D _

I e~ = 0 . 2 D ep O O D - - _

b",

1.5 =L

I

2 =~ 1

=L 1

1

0.5 0.5

2 _

I

1

1.5

2

I

I

I

I

I

[

I

I

0.5

1

1.5

2

05

1

1.5

2

,

,

T~

T~

Taft

Tab

I

I

I

1-1

I

2fl

I

T~ Taft

'

I

I

_/

.~ 1.5 ~r

=L

_J

2

1

It

-7 ;,,

_

0.5

I

[

I

I

0.5

1

1.5

2

I

I

I

I

I

I

[

I

I

0.5

1

1.5

2

05

1

15

2

rx

T.

ELC

ELC

T~

b ELC

~" 2~----~

I

I-2 _

e, = 0 - 2 D

I

[

I

I

05

1

15

2

/

e, = 0 . 2 0 ep=0.10

I

I

I

1

1.5

2

Tx

-

I

I

I

I

1

I

I

[

[

0.5

1

1.5

2

I

I

Tx

Taft

I

I

e, = 0 . 2 0 ep = 0 " 0 0

T~

Taft

I

I

J-2 _

' ' I

I 05

I

Taft

1

I

[

I

2 ,~ 1.5 =L

%

1

~L

=L -

js~

- ~

~~

_

-

0.5

I

I

I

[

[

I

I

[

I

I

I

I

0.5

1

1.5

2

0.5

1

1.5

2

0.5

1

15

2

Tx

T.

Tx

¢

Figure 17 Compar=son of ratio #x4//~sfor single and modified single element models ( - - - ) modified SEM

resisting elements increases and consequently the use of the approximate method becomes more appropriate. Modified

single

element

model

An alternative definition of eccentricity has been introduced 3 and is based on the strength distribution of

(b) R = 5, fl = 1, (c) R = 5, ft = x/3, (

) SEM,

resisting elements rather than on their stiffness distribution. The strength eccentricity is defined as the offset of the centre of yield strength CP from the centre of mass CM. The centre of strength, (P can be found by taking the first moment of the yield strength values of resisting elements about CM. The ordinates of (P, xp, yp are then given by xp = r.v,yx,/F,v,y, yp = r.v=y,/r.v,~.

Eng. Struct. 1992, Vol. 14, No 1

61

Approximate seismic analysis of inelastic asymetrlc structures: A. W. Sadek et al. In the hght of the findings ~ that the inelastic torsional deformations correlate well with the strength eccentricity ep, a modified single element model is proposed here. The modification IS basically that the single element is to be located at the centre of strength CP as shown in F i g u r e 16. The proposed model assumes the eccentricity to be fixed but equal to the system strength eccentricity on the basis of its influential effect on inelastic torsional responses. The modified single element model is assessed and compared with the single element model as follows. The ratio of ductihty demand is accurately evaluated to that obtained using each of the two approximate models is shown for three structural systems The first structure has equal values of stiffness and strength eccentricities e, = ep = 0.2 D In this case both models are identical and that will give identical results. The other two structures have ep values different from es --- 0 2 D namely e~, = O. 1 D and 0 On this account the two approximate models will give different estimates. F i g u r e ! 7 shows the variation of ductdlty ratio/~,4//z, with period T for the single and modified single element models. Compared with models with [2 = 0.8, 1 and x/3 one can observe that, excluding the short period range, the modified model gives better estimates as indicated by the ductility ratios being closer to unity. For the short period range, both models generally fall tO predict the ductility responses

Conclusions The accuracy of calculating the inelastic earthquake response of simple asymmetric structures via the single element approximation has been examined, and the following conclusions can be drawn.

62

Eng. Struct. 1992, Vol. 14, No 1

Laterally-stiff torsionally-flexible structures with low yield strength are vulnerable to the combined effect of asymmetry and inelasticity Their ductlhty demand responses may very well be exceptionally large. For such structures, the single element approximation is found to be inaccurate and hence an accurate analysis is essential Laterally flexible structures can be reasonably approximated using the single element model in time history analyses. The approximate method tends to overestimate the ductility demand responses A modified single element model is proposed, by locating the single element at the inelastic centre of strength introduced by Sadek 3 rather than at the elastic centre of stiffness. The strength centre can be easily found by taking the first moment of yield strength values of resisting elements about an arbitrary point The proposed modified model is found to be superior to the earlier single element model I as it provides better estimates of ductility responses of inelastic structures with long or intermediate vibrational periods. The accuracy of the estimates provided by the single element approximation improves considerably as the number of resisting elements in the original structure increases.

References 1 Kan.C L and Chopra, A K 'Simplemodelfor earthquakeresponse studies of torsionallycoupledbuildings', J Eng Mech Dlv , ASCE, 1981, 107, (EMS) 935-951 2 Bozorgma,Y and Tso, W K 'Inelasticearthquakeresponseof asymmetric structures', J Struc Dtv, ASCE, 1986, 112, (2) 383-399 3 Sadek, A_ W_ 'Seismic response to melasac structures subjected to bldlrecUonal excitations', PhD Thesis, McMaster Umvers=ty, Harmlton, Ontario, Canada, 1985 4 Newmark, N M and Hall, W J 'Earthquakespectra and design', EarthquakeEngineeringResearchInstitute, Berkeley,Callforma, 1982