Nonlinear dynamic analysis of damped structures using the transfer matrix technique

Nonlinear dynamic analysis of damped structures using the transfer matrix technique I. A. Akintilo iaa associates (Consulting Engineers) 280 Sylvan Road, Crystal Palace, London SE19 2SB, UK (Received October 1990; revised April 1991)

A conceptually simple and computationally efficient numerical model, based on the transfer matrix formulation is proposed for the prediction of nonlinear dynamic responses of a coupled shear wall structure incorporating damping. With the transfer matrix method, damping is introduced into the formulation through the field and station transfer matrices using the constitutive law. The effect of damping on the responses of a vibrating beam is incorporated into the station transfer matrix while those ot the walls are incorporated into the field transfer matrix. The nonlinear behaviour is approximated as a sequence of successively changing linear systems over a short time interval. The transfer matrix formulation adopts established models of material behaviour in dealing with the inelastic beam element deformation and load reversals. The dynamic solution assumes constant properties within short time increments, and relies on the kinematic relationships of the Wilson 0 method.

Keywords: damped structure, dynamic analysis

Coupled shear walls in high-rise buildings can resist lateral forces arising from winds, blast loading and strong ground motion much more effectively than isolated shear walls. If two shear walls are coupled by means of a series of floor beams of appropriate stiffness and dimensions, a substantial overturning moment can be resisted by composite bending action in a way which is not possible if the shear walls act independently of each other. A well-designed pair of coupled shear walls can also resist extreme dynamic bending if yielding and inelastic deformation is confined to ductile connecting beams. This can be achieved by designing the walls to remain elastic throughout the time-history of the external loading sequence and designing the connecting beams to dissipate kinetic energy through their inelastic response. Many authors have investigated the structural behaviour of coupled shear walls by finite element and frame analogy methods. Mahin and Bertero I employed a wide column frame analogy to study the elastic and nonlinear static and dynamic responses of 18-storey coupled shear wall models to severe earthquake motions. Emphasis was given to the effect of the strength and stiffness of the coupling beams on dynamic behaviour. Keshavarzian and Schnobrich 2 presented a method for the nonlinear analysis of reinforced concrete coupled shear wall structures subjected to both static and 0141-0296/92/030180-08 © 1992 B u t t e r w o r t h - H e i n e m a n n Ltd

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Eng. Struct. 1992, Vol. 14, No 3

dynamic loading. The procedure used a member-bymember modelling technique and has been tested by comparing the computed and measured responses 3'4. Although maximum amplitudes did not agree, elongations of the fundamental period were predicted with reasonable accuracy. Saatcioglu et al. 5,6 idealizing each structural member as a line element studied the dynamic response of a 20-storey coupled shear wall structure. A free vibration analysis of plane - coupled shear walls based on the continuum approach, has been developed by Tso and Chan 7. With the assumption that the connecting beams were extensible, they derived a pair of coupled six-order differential equations, in terms of the lateral displacements of the walls. Dynamic behaviour of stiffened shear walls on flexible foundations using this method, has also be presented by Coull and Tailb 8. Amiya et al 9, analysed a coupled shear wall using the continuum approach by the exact numerical solution of the governing differential equations. Hsu ~0 employed a finite element formulation to trace the response of a three-storey wall, under the simulated El Centro earthquake of May 1940. The results of the analysis were compared with the experimental data, to substantitate the usefulness of the material and the analytical models. Agrawa111, Suidan and Schnobrich 12 , and Yuzugullu and Schnobrich 13 , have extended the use of the plane stress finite element

Nonlinear dynamic analysis of damped structures: I. A. Akintilo method for the nonlinear analysis of reinforced concrete structures in relation to earthquake loads. Cheung and Swaddiwudhipong ~4, using the finite strip method have analysed shear wall-frame structures, by dividing the height of the building into bands of finite strips. The application of the method in analysing coupled shear wall structures subjected to both static and dynamic loading is well established ~5-~8. Unlike the finite element method or the frame analogy, an efficient transfer matrix solution can be easily programmed on mini computers as it mainly requires the manipulation of 6 x 6 matrices. The potential of the theory to predict the linear and nonlinear responses of a coupled shear wall under static and dynamic loading has been firmly established 19-23. The main objective of the work presented in this paper is to test the potential, efficiency and accuracy afforded by the proposed model in predicting dynamic responses of a damped coupled shear wall structure under earthquake loads. A numerical example is given to confirm the potential of this proposed model.

Nomenclature The following symbols are used in this paper de, da, d

the relative joint displacement, velocity and acceleration vector, respectively dj = db the depth of joint at beam-wall junction E modulus of elasticity Eo damped Young's modulus c; eccentricity in wall i G undamped field transfer matrix G° damped field transfer matrix L clear distance between walls Mr yield moment M; diagonal mass matrix P~a force vector at end/3 of member a P~ force vector at end /$ (/$ = 1, 2) of connecting beam S~ state vector at end/$ of memt~er a SA resultant vector of external applied forces U undamped station matrix U° damped station matrix w; width of wall i, i = 1, 2 ~7 frequency of predominant mode Ado, Ad6, Ado incremental relative displacement, velocity and acceleration vectors, respectively AT time interval

Basic assumptions The development of the transfer matrix formulation yielding the incremental responses of the structure over successive short time intervals is based on the following assumptions (i) the structure is constrained to deform in its plane (ii) geometric nonlinearity, that is P - A effects, is neglected

(iii) the mass of the structure is lumped at floor levels and the mass centres coincide with the geometric centres of the joints (iv) the stiffness characteristics of members remain unchanged over each interval (v) Giberson's 24 one-component model for elastoplastic beam deformation with a simple nondegrading hysteresis loop 25 is adopted Theory The transfer matrix method is based on the relation between the state vector at one (usually the left end) of any segment and that of the other end, through the transfer matrix of the segment itself. A 'state vector' is made up of displacements, rotations, internal forces and moments at a particular end of a member. The right end state vector of any segment can in turn be related to the state vector at the left end of the subsequent segment through the transfer matrix of the connecting joint and the resultant external force vector at that level. Therefore, if the state vector at the base of any continuous structural system is known, the state vector at the ends of every other subsequent segment can be determined by the imposition of boundary conditions at the end of that system. With the transfer matrix technique, damping is introduced into the formulation by modifying the transfer matrices of members and those of the joints. A 'member' is meant to be the collection of wall segments between any two floors while a 'joint' is composed of the panels joining the wall segments at each floor level together with the connecting beam and the structurally interacting part of the slab. In this paper, damping is introduced into the formulation through the field and station transfer matrices using the constitutive law 26-29 to represent hysteretic damping (1)

r = k e + c~

where r is defined as the stress resultant vector and e is the associated deformation vector expressed as

r--

I I Iexl q

,

e =

ey

m

The damping matrix c is given as (la)

c=E°k

E where E, E o are the Young's modulus and the damped Young's modulus of the adopted material, respectively. If • = ~ e ~'~, then i~ = ioJ~ e i~t = io~e

(lb)

Substituting (la) and (lb) into (1) gives E~ r = ke + -" E

i~ke

= k(1 - i ~ ) e

Eng. Struct. 1992, Vol. 14, No 3

(2)

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Nonlinear dynamic analysis of damped structures: I. A. Akintilo

where

Field transfer matrix of damped members Eo E

7=--CO

It is assumed that for the vibration at each particular mode Eo E

7. COn

The extension of equation (5) to a coupled shear wall structure entails finding the relationship between nodal displacement vectors at the two ends of every member, floor by floor. Substituting equations (5) and (6) into equation (4) gives HITdIa + Hfdzo + 7 . (Hrd,~ + H/d2o) = k - l r COn

Noting that ( H f ) - I = HIT, substituting equation (7) in the above equation and re-arranging gives

and therefore COn

7. = 7 f COl

(3)

dea = H1THITdla -- Hirk-IHzP1a + 7~ (HlrHtrdl~ + H1rHfd2~)

where 77. is the current damping ratio for the current mode of vibration (CO.). and 7I is the damping ratio corresponding to the fundamental mode (COl). Substituting the damped Young's modulus expressions into the constitutive relations and rearranging gives (4)

e+7" ~=k_lr COn

According to Livesley 3°, the element deformation vector is given as

(.Opt

= Nlldla + 7~ (Nlldl~ - d2a) - NI2Pla

(8)

COn

Arrays di. and Pi. represent the deformation and stress resultants at end, i of wall segment 'a', respectively. A dot above a symbol denotes differentiation with respect to time. By setting (7/00. = X, in equation (8) and writing the final result in a more compact incremental form gives Ad2a = NllAdla - Nl2 " AP1~ + hNllAdl. - XAd2a

(5)

(9)

where for straight beam elements with the origin at the middle of the span, the equilibrium matrices H1 and//2 are defined as

It has also been shown that the stress resultant at both ends of any wall segment can be written as 21'22

e = Hr~d, + Hfd2

AP2. = -N22" APIo

1 0 HI =

_

The velocity vectors at ends 1 and 2 of member 'a' in equation (9) are not known. In order to solve the problem, the Newmark equations with linear acceleration approximation 3t'32 are adopted

1 0

0

0 L 2

=

0

I

1

01

01

0

-L 2

mdia ~--- - d i a A T + 3 m d i a _ 3dia

00 1

2

1

(10)

where, i = 1, 2. Introducing equation (10) into equation (9) and rearranging gives the final field transfer matrix equation

L is the length of the beam element, dl and d2 are the nodal displacement vectors. Equation (5) is obtained from the premise that the virtual work done by the nodal loads Pi must balance the virtual work done by the member stress resultant r. Since the equilibrium matrices H~ and /-/2 do not vary with time, the corresponding derivative of the element deformation vector with respect to time can be written as = H1Tdl + Hrd2

At

,,+a.N,, \ AP2"

0

N,2 N2zA \ - A P , af

(6)

Finally, the member stress resultant vector r is related to each nodal load P~ by Pi=Hir

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(7)

+ ---2

+ 3a

(ll)

Nonlinear dynamic analysis of damped structures: I. A. Akintilo MvT M

such that equation (11) becomes

2

AS2a = GDAS?a -- Sa

where

aS~a=

(

Ad~,

"~

\ ( _ 1)~p~o/, ~ = l, 2

In the present case, the state vector Saa has 12 components consisting of two six-dimensional vectors of deformation and those of internal forces. The damped field transfer matrix GO is 12 x 12 while vector SQ contains all the remaining 12 x 1 arrays as defined in equation (11) taking into consideration the effects of both velocity and acceleration of vibrating wall segments at a particular time of the integration process.

Figure 1

Nondegrading hysteresis loop with bilinear primary

F o r m u l a t i o n o f a d a m p e d station transfer matrix

curve

where the terms N U are defined in Reference 22 and Ot =

The derivation of the damped station transfer matrix begins with the equilibrium conditions applied to each of the two panels (Figure 2) which are written in matrix form as

(1 + 3~'~ -t

-StP,b(t)

- LtPE.(t)

- TlPb(t)

= Mzdc(t)

+ Ug(t)

at/

(12)

Alternatively the state vectors Sa° may be introduced

All coefficient matrices appearing in equation (12)have

[ q

__

#-~--__

I ~_____

71.i} ,

1.

d,

M2(CiG + Og)

M1 (d'G + Ug)

-0 l

u,,

2

r

j-

2

Wall 2

Wall Figure 2

1

Typical joint configuration wttn internal and intertial forces

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Nonlinear dynamic analysis of damped structures: I. A. Akintilo the form

01 oI M2

0

s,=[s

'

0

L2 '

T,=

For a damped beam element, the force-vector in an incremental form is given as

0j o]

f13)

A p b = K b A d b + C~'Ad b

S2

/'2

where

Combining the equation of motion (12) with equation (13) and rearranging gives AP, o(t) = $7 'LtAP2a (t)StlTtKBRtAd2a (t)

-

+ SF'MI(Ada(t) + AUg(t)) Me =

I mi 0 0 1 0 mi 0 0

0

Ji

m~ denotes the masses lumped at joints and j~ represents their moments of inertia

s. =

1 0

0

0

1

0

_(_ei) i d~ 1 2

-01 1

o

0

KB is the stiffness matrix of the connecting beam at a particular level with respect to the current state of deformation and yield condition as described,in, References 22 and 24. P~8 is the force vector at end /3 (/3 = 1, 2) of the connecting beam. P~j is the force vector at the end /3 of the wall /, member j. Ug is the input ground motion vector, dc represents the acceleration vector of the centre of mass of the joint relative to the base. Relationships between the various displacement vectors describing the motion and deformation of the joint are derived by applying the virtual work principle. It then follows that

d~b= Qidi2,,

Li = I 10 01 00 1 0 _dj 1 2

r, =

(14)

(15) where

!1

In general, ' i ' = 1, 2 and w~ and WE represents the width of left and right hand wall segments, respectively.

Qi =

siT(LT)

Ri = T~r(Lir) -i

and

-1

Since the matrix Ri does not vary with time, it is possible to differentiate the latter from the compatibility conditions and express it in an incremental form as Ad b = RIAd:,

(16)

Substituting equation (10) into (16) and then the resulting expression into the equation of compatibility (15) and motion (14) and rearranging gives

The eccentricity in wall i at floor level n, dj is defined as the vertical dimension of the rigid plate element connecting the beams at each floor level to the successive wall segments and should satisfy O<4
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Eng. Struct. 1992, Vol. 14, No 3

as?b=

Io, ° ]Asa vo

s

(17)

L,

Or, more compactly as a s " b = VDASfo - S £

where the superscript D implies a damped system, and U° is the damped station transfer matrix. Other parameters are defined as follows

3

v? = v, + ~ S;'r, CBR, V,=

[ V" + V? I12,

V,2 v22 + v*

] J

Nonlinear dynamic analysis of damped structures: I. A. Akintflo where

while at the free end

Vii = S i - I K B R j

(/./T)-l)

V , ..~- S i - , ( ~

( S°a = SA +

2

o o

so that equation (17) can be re-written in the form

)

Si_~TtCBRt[(At/2)~2 ~ + 3d2a]

( Sa =

i,j=l,

SLIM,

[(6/T)dc(t) + 3d~(t)] - At)g(t) - g

CB = hKB is the damping matrix for the connecting beam. An equivalent undamped response for a coupled shear wall is obtained by setting k = 0.

Modified transfer matrix solution for damped system The procedure for evaluating the state vector at the base for a damped system is slightly different from those developed in References 1 9 - 2 2 due to the new form of equation (11) which can now be rewritten as A S f a = GaDAS?a -- S a

(18)

Similarly, equation (17) can be expressed as aS?~

= U ~oA S 2 ao - - S D

(19)

Now substituting equation (18) into equation (19) gives

-V S'a D - Sa = U)DG~DASt.

(20)

Following the standard transfer matrix approach, the incremental state vector immediately after the joint at the nth floor then becomes AsD+t = (U~G D . . . UDG~ASD~

D D- -- SN

-UNGn SN-I

--( :12)

)

(21)

where

gN= u °s? + sN n = a, b, c . . . . , n, for wall segments a, b, c . . . . , n, respectively N =A, B, C . . . . , N, of intermediate joints at floor levels A, B, C . . . . . N, respectively The state vector at the base is then obtained by applying the boundary conditions as explained in Reference 22. The boundary condition at the base describes its fixity through a flexibility matrix f satisfying.

The second matrix equation of the above system can be solved for -APla from which the state vector at the base and any subsequent level can be readily calculated.

Numerical implementation The developed analysis is implemented through a FORTRAN program written on a VAX computer which allows for unequal pier widths and any type of geometric or material nonuniformity with height. The lower and upper yield moments in the absence of any permanent deformation, plus or minus any effect from P - M interaction, are determined from the constitutive laws for steel and concrete. All states of yield as defined by Giberson 24 are generated by the program. Thus, whenever yield occurs, the hysteresis branch followed by the solution depends on the sign of the bending moment increment. If the latter has the same sign as the current yield moment, inelastic deformation continues. If it has the opposite sign, the deformation becomes elastic with the two new yield moments determined according to the diagram of Figure 1. A small-scale 10-storey structural model of single-bay coupled shear walls is chosen to demonstrate the effectiveness of the proposed analysis in evaluating dynamic responses of damped structures. The material and crosssectional properties of the model are summarized in Table 1. The same model has been dynamically Table 1

-f~Pla

Propertiesof adopted model

No of floors Reinforcement ratio in walls (%) Reinforcement ratio in beams (%) Steel yield stress Width of walls Beam depth Thickness Length of beams Floor height Additional mass per floor

10 1 1.65 496 N m m - 2 178 mm 38.1 mm 25.4 mm 102 mm 229 mm 227.5 kg

Wall cross-sectional properties Yield moment (levels 1 - 5 ) Yield moment (levels 6 - 1 0 ) Act (levels 1 - 5 ) Acr (levels 6 - 1 0 ) /or (levels 1 - 5 ) Icr (levels 6 - 10)

4 1 8 4 . 8 N-m 2 1 1 8 . 3 N-m 2 . 1 8 8 x 10 -3 1.421 x 10 -3 1.022 x 10 -5 5.599 x 10 -e

m2 m2 m4 m4

Connecting beams at all levels Mr Ac~

~vtla =

(22)

Ic~

180.8 N-m 8.72 x 10 - s m 2 1.374 x 10 -e m 4

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Nonlinear dynamic analysis of damped structures: I. A. Akintilo

tested 34 and numerically analysed 2'5"6 for different objectives. The base acceleration records adopted are similar to those recorded in the University of Illinois earthquake simulator. The use of the Wilson 0 numerical integration technique was adopted in this paper because the method automatically negates the condition of stability regarding time step and the least period of vibration. The convergence and stability of the numerical integration scheme was tested by checking dynamic responses for various values of the Wilson coefficient 0. Various values of 0 were tried among which 0 = 1.55 was finally selected based on a comparison between computed and measured results. In the present study, a time increment of At = 0.004 s is adopted. All the nonsignificant motions at the beginning of the record are eliminated.

Results and conclusions A reasonable amount of damping should be introduced into a structure from the point of view of resistance to

24

24

16

16

E

O E-

E

8

E

"O

earthquake loads. The capacity of a structure to dissipate vibrational energy and its ability to counteract the generation of large structural amplitudes during a quasiresonance condition depends largely on this parameter. For this reason various values of X and c~ were considered. The computer responses were similar for very small values of X, so that only the responses for X--0.0002 are given in Figure 4. By comparing the responses of Figures 3 and 4 it can be seen that the computed top displacements as well as the base moment are reduced by more than 5 % when damping is introduced, despite the chosen low value for the damping ratio. A higher value could have been effective on the amplitudes away from the strong motion, but this would require a larger combination of X with a larger value of time increment which would consequently distort the orginal accelerogram. This was not tested since a direct comparison of the computed results and those experimentally observed will not be possible. Earlier work by Syngellakis and Akintilo 2~ revealed that the undamped responses predicted by the transfer

0

8 0

8 0~

-8

r~ O F-

-16

-8 -16

[

-24

0.0

i

I

I

I

I

I

I

[

I

I

-24

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

i t i i [ i i i i i i 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Time (s)

a

Time (s)

a

15

15 10

10 Z" I E

I

E Z

5

Z

0 E O E

E E

5 0

O

-5

-5

{33

0a

-10 -15

-10

[ I J I J ] I I I J I i 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

-15

I

I

I

I

I

I

I

I

i

I

Time (s)

Time (s)

b

b

Figure 3 Transfer matrix predictions of undamped response

Figure 4 Transfer matrix predictions of damped response

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Eng. Struct.

1992,

Vol. 14, No 3

I

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Nonlinear dynamic analysis of damped structures: I. A. Akintilo m a t r i x m e t h o d and t h o s e r e c o r d e d e x p e r i m e n t a l l y a g r e e d v e r y well in pattern, intensity and f r e q u e n c y . A l s o , the p r e d i c t e d r e s p o n s e s h o w s a certain d e l a y in r e a c h i n g its m a x i m u m v a l u e w h i l e the t w o m a x i m a in the n e g a t i v e d i r e c t i o n d i f f e r m o r e significantly than those in the p o s i t i v e d i r e c t i o n . P a r a m e t r i c investigation o f the a d o p t e d structural m o d e l r e v e a l e d that the p r o p o s e d a n a l y t i c a l m o d e l b e c a m e u n s t a b l e for a c o m b i n a t i o n o f l a r g e v a l u e s o f X and s m a l l e r values o f At. This can b e attributed to the resulting l o w e r v a l u e s o f ot w h i c h c o n s e q u e n t l y lead to a d i v e r g e n t solution. T h e solution d i v e r g e s w h e n the value o f o~ is less than 0 . 7 5 . A s a rule o f t h u m b t h e r e f o r e , the values o f X and At should b e c h o s e n such that the resulting v a l u e o f ct lies b e t w e e n 0.75 and 1.0. T h e e x a m p l e g i v e n s h o w s that the d e v e l o p e d f o r m u l a tion can a c c u r a t e l y p r e d i c t the n o n l i n e a r r e s p o n s e s , d a m p e d o r u n d a m p e d , o r a c o u p l e d shear wall structure u n d e r s i m u l a t e d o r real earthquakes. T h e potential o f the p r o p o s e d m o d e l to a c c o m m o d a t e variations o f g e o m e t r y , as well as the m a t e r i a l p r o p e r t i e s , has also b e e n d e m o n s t r a t e d . It should t h e r e f o r e p r o v e a useful a n a l y t i c a l tool w h e n e m b a r k i n g on the p r o d u c t i o n o f c o n c e p t u a l as well as final d e s i g n calculations for p r o j e c t s i n v o l v i n g the use o f shear wall structures, transm i s s i o n t o w e r s , and all o t h e r c o n t i n u o u s systems. N o a t t e m p t was m a d e to i n c o r p o r a t e g e o m e t r i c n o n l i n e a r i t y and h y s t e r e s i s m o d e l s with d e g r a d i n g stiffness in the f o r m u l a t i o n . T h e r e f o r e , further r e s e a r c h w o r k should b e c a r r i e d out in this area.

Acknowledgments T h e a u t h o r wishes to thank D r S. S y n g e l l a k i s o f S o u t h a m p t o n U n i v e r s i t y , U K , for his c o n s t r u c t i v e c r i t i c i s m s a n d P r o f M . A . S o z e n o f the U n i v e r s i t y o f Illinois, U r b a n a , U . S . A . , for p r o v i d i n g the e a r t h q u a k e r e c o r d s u s e d in the o r i g i n a l e x p e r i m e n t a l studies. T h a n k s a r e also due to M r A d e b i s i A d e r e m i for his support.

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