The transfer matrix method applied to frame-shear wall systems

Compurers & Structures Printed in Great Britain.

Vol. 41,

No. 2, pp. 197-206,

00457949191 s3.00 + 0.00 Pergamoo Press plc

1991

THE TRANSFER MATRIX METHOD APPLIED TO FRAME-SHEAR WALL SYSTEMS S. SYNGELLAKIS

and I. YOUNIZS

Department of Civil Engineering, The University, Southampton

SO9 5NH, U.K.

(Received 7 August 1990) Abstract-A practical method of analysing frameshear wall structures is presented. Its development is based on the transfer matrix technique and structural modelling using beam elements. The basic equations were derived for a general two-dimensional arrangement of frames and shear walls taking into account the shear deformation of all structural elements. Axial deformations are generally neglected but their effect on the structural response of simple frame-shear wall combinations is assessed. An approximate analysis reducing uniform frames to single vertical elements is developed and incorporated into the general formulation. The computer codes implementing both the linear static as well as the free vibration analysis of frame-shear wall systems have been tested for accuracy and efficiency through their extensive application to a wide range of geometries.

NOTATION

A d

E e F G G g I K k L M

cross-sectional area deformation vector elastic modulus wall eccentricity flexibility matrix shear modulus field transfer matrix external joint moment second moment of cross-sectional area stiffness matrix element stiffness member length mass matrix internal moment internal force vector state vector station transfer matrix floor sway displacement rotation Poisson’s ratio shear parameter external horizontal force

m p S U A 6 0 v 4 ICI Indices b beam C column I floor (joint) level i storey (member) level J beam number column or wall number i r reference wall s soil

INTRODUCTION

Frame systems above a certain height are efficiently stiffened against lateral forces by the introduction of shear walls into their design. Combinations of such elements result in interactive stiffness of much higher order than that encountered in conventional frame structures. Consequently, traditional methods of frame analysis need to be modified or new ap-

proaches to be developed for the accurate assessment of the response of tall frame-shear wall systems to static or dynamic loading. The early design stages, usually carried out by a trial and error procedure, focus on finding initial estimates of the structural properties and stress distributions. Thus, although the strength of a building to resist extreme lateral dynamic loads needs ultimately to be checked through a rigorous nonlinear analysis,

linear analyses yielding the response to equivalent static loads as well as the natural frequencies of free vibration of the system are initially more appropriate. Early investigations simplified the analysis by assigning all the lateral loads to shear walls [l]. This approach may have been conservative with regard to wall design but it may also have underestimated the forces in the frames, particularly in their upper part. The most common methods of analysing frame-shear wall systems adopt the replacement of the original structure by an equivalent frame structure so that available frame routines can be applied. In these frame analogies, all columns, beams, and shear walls are modelled as uniform line elements with spans which differ according to the way joint rigidity is accounted for. The wide-column frame analogy has been found the most suitable to shear wall geometries [2] and made applicable with standard frame programs through replacement of the beamrigid arms combination by an equivalent uniform beam [3,4]. The continuous connection approach was developed later as a means of simplifying the problem so that it can be solved quickly by hand calculations. The main disadvantage of the method restricting its practical usefulness is its increasing complexity with a higher number of bays and any additional geometric or material non-uniformity. For this reason, the method has been mainly applied to 197

S. SYNGELLAKIS and

198

coupled shear walls [S]. However, considerable research effort has been reported on its extension to frame-shear wall systems. The technique has been applied in a rigorous way to the special case of a single core wall connected to two exterior columns [S]. More general formulations dealt with any number of rigidly interconnected shear walls or columns [7,8]. Alternative, simpler analyses were developed assuming pin-ended links between vertical elements [9, lo]. Predictions of the linear static and dynamic response of frame-wall systems were also obtained using the finite strip method [ 11, 121.A fair amount of experimentation has been carried out intending to conf%m the validity of the various methods of analysis [2,5]. The potential of analysing frame-wall systems by the transfer matrix method has not been fully explored. This method is applicable to multistorey structures that can he modelled as a non-branching chain of members. A transfer matrix formulation was developed by Oliveto and Santini [ 131for a simplified model of a frame-shear wall combination. The method was more rigourously applied to the free vibration analysis of coupled shear walls 1141.In this paper, transfer matrices are formulated for a general, two-dimensional system of frames, columns and shear walls without imposing severe restrictions on structural and loading symmetry and uniformity. The present analysis is even applicable to space structures with symmet~~al overall plan subjected to a symmetrical load system. The transfer matrix solution of the static problem is subsequently extended to the prediction of the natural frequencies and free vibration modes of frame-shear wall systems. The resulting formulation is not a classical eigenvalue problem, so that its solution is obtained through a searching process involving repeated applications of the transfer matrix procedure. Thus, for multibay, multistorey systems, the size of the problem arrays increases at the expense of accuracy and computational efficiency. Significant reduction in the number of degrees of freedom is achieved by isolating frames with slender elements and treating them as single vertical elements to which a simplified version of the developed analysis applies. NOTATION AND BASIC ASSUMPTIONS

The proposed method of solution assumes the structure discretized into beam elements. Adopting Livesley’s notation [ 151, the displacement and force vectors at any cross-section of such an element are given by

d=

6, S, ii 6

I.Younm

force, and moment of the cross-section, respectively. Neglecting axial deformation allows the dropping of the axial displacement and force components from their respective vectors. The transfer matrix analysis uses the element ~~lib~urn and ~nstitutive equations in the following form PI -t HP, = 0

(1)

PZ= R, (a, - H’d, )t

(2)

where

Ru=

k (I f#)L2

12 -6L

-6L (4+ Cp)L’I

with k = El/L, and E, I, L the Young’s modulus, the second moment of cross-sectional area, and the length of the element, respectively. The superscript t denotes the transpose of a matrix while substripts 1 and 2 indicate the two element ends. Shear deformation is taken into account through the parameter Cp= 12EZ/(G,4,L2) where G is the shear modulus and A, the effective cross-sectional area for shear. Rearranging eqn (2) gives d, = H’d, - FH’p,,

(3)

where F = I&’ is the element flexibiIity matrix. Equations (1) and (2) are easily rearranged into the familiar end-force, end-displacement relationships

(4) where K is the element stiffness matrix. The first of eqns (3) yields an expression for the relative lateral end-displacement A = S,, - S,,

Solving eqn (5) for the shear force gives pvl =(-A@,+Bm,)+(C/L*)A

(6)

with A = 12k/L(2 - (p), 3 = 6/L(2 - 4~)~ and C = 12k/(2 - 4). An alternative expression for A can be obtained in terms of the same set of variables evaluated at end 2

PX P=

pv

,

ii m

where 6,) 6, , 0, px, pv, m are the axial displacement, transverse displacement, rotation, axial force, shear

from which pv2= (A& - Bm,) - (C/L’) A.

(8)

Transfer matrix method and frame-shear wall systems

- Pyl

L

i +l,m

i+lj + 1

i+lj

i+l,l - Pyl

- Pyl

e

199

- Pyl

*

3 I

Joint I

-_

- ~~ w

it

ij

-k

*

ij+ t

0 w

v

-pe

-%2

Element J+ 1

Element J

Etement 1

-ef

I 1

I I

Element m

Fig. 1. Floor level modelled as ‘joint’. Finally, substituting eqn (6) into the second of eqns (3) results in the expression

e*= -

ze,+--1+#

-

6

(2 - 4)k m’ + (2 - rp)L A.

@)

The above relations which are applicable to any beam element facilitate the derivation of the transfer matrices required by the present problem. TRANSFER

MATRIX

FORMULATION

The transfer matrix method is applicable to structures consisting of a non-branching chain of members. An unbraced frame can be visualized as such a structure if its typical ‘member’ is defined as the collection of all columns between any two floors, while its ‘joint’ consists of the connecting medium at any floor level, that is, the effective part of floor slabs

with or without girders. Figures 1 and 2 demonstrate the adopted discretization of the structure and the nosing if its ‘joints’ and ‘members’. These numbers identify as superscripts the corresponding force and deformation components. For the purpose of deriving the necessary relations, a second superscript has been introduced to indicate the column or girder number. Thus, $, and p$ are, respectively, the deformation and internal force vectors at end a (a = 1,2) of a beam element in columni belonging to ‘member’ i. Similarly, df , and d’ are the same vectors at end a of a girder J belonging to ‘joint’ I. According to the above definitions, the displacement vector of ‘member’ i would normally consist of all aii, j=l,..., m, where m is the number of columns. The assumption, however, of neglecting the axial deformation of both columns and beams reduces significantly the number of kinematic variables and hence the size of the problem as it leads to the

Verticalelement j

Level

i+l

Beem J-l Joint I -

Level i -2

-Pvz T ” x2

Fig. 2. Typical structural joint configuration.

S. SYNGELLAKIS and I.YOUNFS

200 conditions Sfa=O; j=l,...,m

61 =ail =...=gim YU

YU

(10) (11)

Ya

n and a = 1,2 so that the deforfor all i=l,..., mation and internal force vectors reduce to

where eiJ is the rigid arm at end tl of beam N accounting for the finite width of the adjacent wall. The above relations make clear why neglecting the axial deformation of the beams and columns leads to conditions (11) and (14). Equations (15) are thus reduced to a single equation 6i+ I Yl

where r denotes a reference vertical element, usually chosen to be the stiffest wall in the system. The transfer matrix solution relies on two matrix equations of the form S;=@Si I

(12)

si,+ ’ = U’ si - S’,

(13)

_ -

hi

(17)

Y2

which gives the first row of the U’ matrix. The subsequent m rows correspond to the m equalities between the first two rotations in eqns (16). The remaining m + 1 equations of the system (13) are obtained by writing the horizontal force equilibrium equation for a typical floor level (Fig. 1) and the moment equilibrium equation for all m joints at that level (Fig. 2)

where s; = {a; (- l)“pb}’ is the state vector according to its standard definition, S’ the external force vector at ‘joint’ Z, and G’, U’, the field and station transfer matrices of the problem, respectively. Field transfer matrix The objective is to express each component of Si only in terms of variables belonging to Sl using and combining the relations (l)-(9) applied to the individual column segments making up ‘member’ i. Only the process of the derivation of matrix G’ will be described herein in order to avoid lengthy and cumbersome expressions. Its first row is obtained from the tirst of eqns (3) applied to the reference column element. The non-zero elements of the m subsequent rows are the coefficients of eqn (9) applied to the m column elements of member i. The condition that L\”= Ls”for allj # r permits A to be always expressed in terms of the variables of the reference column. The (m + 2)th row corresponds to the first of eqns (1) again applied to the reference column segment. Finally, for the last m rows, expressions for all my in terms of elements of S, are obtained from the equilibrium equations (1) in which the shear force is substituted from eqn (6). Station transfer matrix The derivation of the upper half of station transfer matrix u’ relies on the compatibility conditions at joint I. These give the following relations

=

6:.

-

e:Jf3{J

=

0

(14)

_m~+

1-j _

,i’

i

_

+e

ml.J2

1_ m{J

$J-‘~$~-‘-e~p~+g’j=O,

(19)

where $‘is the total external horizontal force on floor I and go the external moment on the junction of floor I and column j. The beam shear forces and moments are expressed in terms of the joint rotations using eqns (4) in conjunction with the kinematic conditions (14) and (16). Substituting the resulting expressions into eqns (19) yields the last m equations of the system (13) _m~+l.j=2~~~-l~~j-l+4(~~~-l+~~~)~~ + 2k;;ei;jfi

+ rng -go,

(20)

where k,2=&[1-~+~(e,+e2+~)]

k,=&[l+$+g(l+;)];

a=1,2.

For the derivation of the (m + 2)th equation of system (13), all terms of the summation in eqn (18), apart from pi: ‘lr, need to be expressed in terms of the components of Si. The first group of terms in eqn (18) depend on $+I according to eqn (6) while the dependence of all components of S:+ ’ , apart from p:: I,‘, on components of Si has been given above. On the other hand, p!2 is expressed in terms of the components of Si according to eqn (8). Performing all

201

Transfer matrix method and frame-shear wall systems the necessary substitutions _yi+lp;:Lr=(cc~+l

where

results in

- cc?@ + y’& + /I’m:

_fli+l(_m;+l,? Setting -j~,[(AU_Ai+l,‘)eB+B”mq _ gi+ ‘.j(-m;+ Ll)] _ *I,

(21) and taking into account yields

where

y&1+$

p=cp,

CT’_~iAi’,

ji=T

cir

the boundary

(R2,F,+ I&)(-pi)

j#r

= r2.

conditions (26)

in which the moments are substituted from eqn (20). Equation (26) can now be solved for the base reactions of the structure from which the state vector Expression (21) is simplified considerably in the case at the base can be formed and the state vectors at any of a uniform structure. Much simpler expressions also result for mi+ *,jalong the side columns and for pi: *sr subsequent level can be readily calculated using eqns (12) and (13). at the top ‘joint’. STATIC RESPONSE

FREE VIBRATIONS

In addition to eqns (12) and (13), the transfer matrix method requires the specification of the boundary conditions at the two ends of the nonbranching structure. At the base, a model for flexible foundation may be introduced in the formulation such that

The mass of the structure is assumed lumped at floor levels while the rotatory inertias of the smaller masses associated with individual joints are neglected. Thus the formulation of the field transfer matrix is not affected by the transition to the dynamic case. In contrast, the station transfer matrix is modified by the introduction of inertia loads at floor levels. In a free vibration analysis, only such forces are acting on the structure. They are given by

d; = -F,p; ,

(22)

where F, is a flexibility matrix depending on the assumed modelling of foundation behaviour. The rigid foundation condition is obtained by setting F, = 0. At the free top end, all internal forces should vanish. Thus, the state vectors at the two ends are given by

that is, they depend only on the unknown base reactions and the top displacement and rotations. With the elements of both the field and station transfer matrices identified and the boundary conditions specified, it is now possible to proceed with the elimination of the state vectors of all intermediate levels from the formulation and the derivation of a single matrix equation in terms of the unknowns at the two ends of the structure. Thus, substituting Si from eqn (12) into (13) gives S+ 1

1 _uQis; -

_sI

B= -& that is, the nodal forces depend on the kinematic variables, they need therefore to be grouped together with the other terms giving the station transfer matrix U, leading to the elimination of the load vectors S’ from the formulation. The transfer matrix procedure in this case would result in an equation of the form (R,,F,+R,)(-pf)=O,

(23)

fori=1,2 ,..., nandZ=l,2 ,..., N,wheren=N is the number of storeys. Repeated application of this recurrent relationship gives the state vector at the top end of the structure as S;+‘=RSt-r,

in accordance with the above assumptions, where a dot above a symbol means differentiation with respect to time and M’ is the lumped mass at level I. For a vibration at a frequency u, the acceleration associated with any degree of freedom d would be given by

(24)

(27)

where now submatrices RX,and R, depend on o. For the system (27) to have a non-zero solution, the determinant of its coefficients must vanish. The values of o for which this condition is satisfied are the natural frequencies of the structure which are determined through a searching process based on the secant method.

S. SYNGELW and I. YOUNES

202

SIMPLIFIED FRAME ANALYSIS

As mentioned earlier, the numerical errors as well as computation time increase to unacceptable levels above a certain structure size. This problem becomes more critical in the free vibration case when a large number of repeated applications of the transfer matrix procedure is required to yield even a limited number of natural frequencies and the associated modes of vibration. Thus the need arises for simplifications of the analysis that would reduce the number of degrees of freedom. This was achieved for simple frames with slender elements and a certain degree of uniformity in both the vertical and the horizontal direction that form part of general frame-shear wall systems. Due to the slenderness of the elements, the shear parameter 4 and the beam rigid arms e, have been dropped from all subsequent equations. The basic ass~ption on which the simplified analysis is based, is that the point of contraflexure lies, approximately, at midspan of columns. This is justified if the beams have a stiffness greater than that of the columns and leads to the approximate relations y my = -G ?-pu,

and

Li & = -yp$

(28)

which, however, should not imply that

Substituting the first of eqn (28) into (5), applied to the column element ‘ij’, and summing over j leads to a relation in which the following approximation is introduced

relation between total shear and average rotation at a particular level. This is achieved by summing eqn (20) over j, and introducing in it the approximation

where m-l k’=

k”

is the total beam stiffness at level I. The above approximation can be justified only if the rotations at a particular level do not vary significantly from their average value. Taking into account eqns (16) and (28) results in -L,+&‘=

-Lip;2+24k’e”’

1 *

(30)

On the other hand, eqn (18) is written

-p;y=p;,+.

(31)

It is now possible to eliminate the rotation from eqn (29) using eqns (30) and (31) applied to level Z - 1. The resulting expression for ti depends on the beam stiffnesses as well as the external load at level Z - 1. An alternative expression for &, depending on the beam stiffnesses and external load at level Z, may also be derived starting from eqn (7) and using eqns (30) and (31) applied to level I. Averaging the right-hand sides of these two equations eliminates the dependance on the external load assuming little variation of structural geometry and inertia properties with height. It also leads to the definition of an average beam flexibility as fi=$(Li,:_‘,

where

c f=l

Li

I

Li::+l)

(32)

p;a=p;m+p$+...+pg is the total shear at end or of ‘member’ i. It may be noted that the above approximation becomes an identity if all columns of level i have the same stiffness. The end result of this summation is thus written &r=Z&

-ffp$

(29)

having defined an average floor rotation e:=;(sk’+@::+...+ebm);

El---l,2

and a collective column flexibility

f:=&j$,;. It is attempted

next to deduce an approximate

so that the final form of the ‘member’ constitutive relation for the reduced system is SC, J 64, - (fr+fb> PiI.

(33)

It should be noted that, for i = 1, the first term within parentheses on the right-hand side of eqn (32) vanishes since the rigidity of the base is infinite. Thus, the beam contribution to flexibility of the first storey is approximately half to that of the other storeys. The elimination of the joint rotations from the formulation reduces significantly the size of the problem. Therefore, provided that a frame satisfying the conditions of uniformity and adequate beam rigidity can be found within a structural system, its modelling according to the analysis of this section is obviously advantageous. The accuracy of the scheme should improve with increasing beam rigidity as it becomes exact in the case of perfectly rigid girders.

Transfer matrix method and frame-shear wall systems NUMERICAL

EXAMPLES

The computer implementation of transfer matrix method generates a numerical error due to the trancation of floating point numbers over successive multiplications. An additional instability arises in this particular application of the method whenever the analysed structure is excessively non-uniform. The deviation of the top stresses from their expected zero values can be a measure of this error which primarily depends on the size of the structure, that is, the number of columns and storeys. Ways of expanding the range of reliability of the method are proposed and their effectiveness was examined by conducting a parametric study using as an example the structure of Fig. 3. A straightforward way of improving the performance of the method is through the use of quad precision variables. However, this considerably increases the running time. For this reason a more economical, iterative procedure was devised. It relies on the assessment of the deviation from nodal equilibrium of the initially computed internal forces and treats any residuals as additional external forces for which the transfer matrix solution is again obtained and superimposed to the previous one. This process is repeated until the error is reduced to an acceptable level. One important limitation of this corrective approach is that the initial deviation must be within 100% of the applied force. The objective of the numerical study was to identify the limit in the size of the chosen structure for which the error remains within an acceptable level. Results with this level fixed at a 5% error in the

203

Table 1. Maximum size, in number of storeys, of structure shown in Fig. 3 for acceptable transfer matrix solution A/B

Standard solution

Iterative scheme

@ad precision

24.5 5.75 2

11 24 26

14 37 40

24 51 55

equilibrium

equations

are given in Table 1. This table

shows that quad precision is certainly the most effective way of dealing with numerical instability but considerable improvement is also achieved by the iterative scheme. The dramatic effect of structural non-unifo~ty on the performance of the method is also very evident from the table. The accuracy of the method was tested through comparison of its predictions with previously obtained experimental and analytical results. The selection of presented examples give an indication of the applicability range of the developed solution, Example 1 The asymmetric perspex model shown in Fig. 4 was tested by Goyal and Sharma [2] for a single concentrated force applied at the top level. The Young’s modulus and Poisson’s ratio of the material were given as 343 MPa and 0.25, respectively. The transfer matrix and experimental results, shown together in Fig. 5, are in excellent agreement in this particular case. Example 2 The araldite model of Fig. 6 was one of two tested by Coull and Subedi [5] for a point load applied at the top. It was analysed by the same authors using the continuous connection model as well as by Cheung and Swaddiwudhipong [ 11J using a finite strip All beams are 15 mm deep

-

-

I

Fig. 3. Model for assessing numerical errors in transfer matrix solution.

All dimensionsare in mm Fig. 4. Test model, example 1 [2].

f.ON

S.SYNGELLAKIS and LYOIJNB

I!

0

x Equivalent frame analogy

/

0.10

I

Lateral displacement, mm

A 0 0 +

Experiment Transfer matrix Finite strip elements Continuous connection

Fig. 5. Comparison of analytical and test results, example 1

a

fln

Lateral displacement, mm

Fig. 7. Comparison of analytical and test results, example 2.

approach. The transfer matrix analysis was carried out with E = 3.17 GN/mr and v = 0.25. Its predictions are shown in Fig. 7 together with the experimental and the other analytical results. Comparison of these curves provides further confirmation of the reliability of the proposed method of solution. Example 3

+

I

-I

1 ,

The free vibration analysis of the three uniform frame-shear wall systems shown in Fig. 8, had been performed by the finite strip and finite element method [ 121. Natural frequencies had been obtained for the following dimensionless numerical data; a = 15, b = 10, shear wall thickness of 0.25, Poisson’s ratio of 0.2 and unit values for Young’s modulus, density, depth and width of beams as well as columns. The results from the previous and the present analyses are listed in Table 2. With regard to the fundamental frequency, their agreement is excellent.

I

15.9

15.9

All dimensions are in mm

Fig. 6. Test model, example 2 [5].

, I

a , a

,a,a,a

I

3 Model 1

I

I

,a,a,a,a I I

Model 2

Fig. 8. Model structures, example 3 [12].

I

Model 3

,

Transfer matrix method and frame-shear wall systems

205

Table 2. Comparison of natural frequencies by various methods-example 3 Natural frequencies (x 103) (rad/s) Method of analysis

Model

Second

Fundamental

1.51

FSM FEM TMM

1.45

1.47

7.34 7.15 7.00

2

FSM FEM TMM

1.34 1.29 1.29

6.31 6.11 5.96

3

FSM FEM TMM

1.24 1.19 1.19

5.65 5.45 5.33

1

FSM = finite strip method, FEM = finite method, and TMM = transfer matrix method.

3

0

Transfer matrix x Equivalent frame analogy

+ Continuous connection

element

0 Lateral displacement, mm

Fig. 10. Comparison of analytical results, example 4.

The smaller values for the second natural frequency resulting from the transfer matrix solution may be attributed to the effect of the shear deformation which had been observed to become more pronounced in higher modes of coupled wall vibrations [ 141. Example 4 The potential of applying the developed analysis to space structures with geometric as well as loading symmetry is demonstrated through an example structure with the plan view shown in Fig. 9. The two-dimensional equivalent of this type of structure consists of half of its vertical elements interconnected through pin-ended rigid beams representing floor slabs which are assumed to transmit no shear forces or moments. The structure had been previously analysed by a simplified method based on the continuous connection technique [9]. It consists of 12 identical storeys, each of 3 m height. The elastic modulus and Poisson’s 2.0m .Y

” .I

1

E

2.5m

3.0m .l

V 1

2.0m

2.5m 1

Y

J

I

I

D

D’

F& c-i

111 A’

&

5’.

2

lb” E

ratio of its material were taken equal to 20GN/m2 and 0.25, respectively. Concerning the connection of columns C and D with the rest of the structure, two possible models have been proposed [9]: (a) vertical interacting forces at points E and G are neglected. This means that columns C and D interact with the rest of the structure only through the slab action; (b) vertical displacement at E and G is zero. This means that columns C and D are part of the frame defined by columns A and A’ connected to them through beams of length twice that of GA and EA. The present transfer matrix solution was obtained under both assumptions above and its predictions were consistent with the continuous connection and frame analogy results. This is demonstrated by Fig. 10 showing the lateral displacement resulting from these three analyses under assumption (a). The static response of the same structure was also obtained by running a version of the program incorporating the simplified frame analysis of the previous section. If assumption (b) regarding the corner beams is adopted, the number of kinematic variables per floor is reduced from 6 to only 2. The approximate solution for the top displacement deviates by only 4% from that obtained using the more rigourous analysis. Free vibration results for this example were also given in the literature [9]. The fundamental frequencies of the structural model under assumptions (a) and (b) described above, were found equal to 3.34 and 4.65 rad/s, respectively. The corresponding values obtained from the present analysis are 3.31 and 4.33 rad/s. Satisfactory agreement was therefore achieved in both cases. DISCUSSION

Fig. 9. Structure plan view, example 4 [9].

Tall buildings with a core wall and only a few slender columns at their perimeter tend to deflect as cantilever beams. In this case, the axial deformation

206

S. SYNGELLAKIS and I. YOUNF.S

of the columns may become important. The present analysis can account for this effect by including the member-end axial displacements as well as the corresponding axial forces in the state vector. The introduction of these additional unknowns does not complicate the formulation. Its more important consequence is the considerable increase in the size of the problem. For this reason, only a special formulation, applicable to simple combinations of a single wall and one or two columns, was developed and computerimplemented. The effect of axial deformation was found to be small in these cases and is expected to diminish further as the structure expands in the lateral direction. It was therefore reasonable to neglect axial deformation from the present general formulation so that it may be more efficiently applied to structures with a large number of storeys and vertical elements. The computer code based on the developed analysis was tested extensively on many solved examples that were found in the literature. In addition to lateral floor displacements and natural frequencies, bending moment and shear force distributions with height were computed and compared with other analytical predictions. The agreement was satisfactory in all cases. However, since all previous analyses were to a certain degree approximate and all experiments involve a degree of uncertainty, the computer program was also verified through comparison of its predictions with those of ANSYS, an advanced finite element software [16]. The model structure of Fig. 6 was also analysed by ANSYS which yielded lateral displacements in full agreement with the transfer matrix solution. The ANSYS runs highlighted the importance of choosing the correct element for modelling the various parts of the structure. Thus, data preparation and the solution itself required much more time than that of the transfer matrix code. Furthermore, the computer space required by ANSYS restricts its use to fairly large computer systems while is only justified when refined analysis is required. The modest storage requirements of the transfer matrix solution make it ideal for use with small work-stations. The computer time and space required by the present two-dimensional analysis were found not excessive in most cases. Thus, the further reduction of the size of the problem by approximating frames as single vertical elements may not be always economically justified. This approximation would be more critical with three-dimensional asymmetric multistorey structures which require substantial reductions in problem size and computer time to be analysed efficiently. It was found convenient however to derive

the simplified analysis in the context of the present, general, two-dimensional transfer matrix formulation. The obvious advantage of the transfer matrix method is that it reduces a problem to a tractable size while, at the same time, offering a certain degree of flexibility over traditional ‘frame’ methods in handling models for local deformation or geometric nonuniformity. The formulation can thus be enhanced by incorporating models for joint flexibility. It may also be extended to deal with more complex geometries, that is, bracing, irregular wall openings, or change of lateral structural size with height. REFERENCES 1. M. Fintel, Ductile shear walls in earthquake resistant multistorey buildings. AC1 JI 71, 296-305 (1974). 2. B. K. Goval and S. P. Sharma. Matrix analvsis of frames wi