three-dimensional finite element analysis of shear wall buildings

Three-dimensional ®nite element analysis of shear wallbuildings

N.K. Oztoruna, E. Citipitioglub, N. Akkas c, *aDepartment of Civil Engineering, Gazi University, 06531 Ankara, Turkey

bDepartment of Civil Engineering, Middle East Technical University, Ankara, TurkeycDepartment of Engineering Sciences, Middle East Technical University, Ankara, Turkey

Received 17 October 1995; received in revised form 1 October 1997

Abstract

A three-dimensional ®nite element computer analysis of multistorey building structures, made of pierced shearwalls of open and/or closed cross-sections and ¯at plates, is presented. The computer program developed for thispurpose provides a special and powerful mesh generation subroutine. A graphic program is also developed to

prepare the data interactively by utilizing a screen graphic option. The structure model can be created or modi®edvery easily with the use of the present mesh generation program. The beams or columns can be added or cancelledwith no di�culty at all. The plate ®nite element developed can represent the membrane as well as the bendingbehaviour of the shear wall and the ¯oor components. The program developed is used to obtain solutions to some

realistic structures to determine the bounds of the simplifying assumptions commonly made for the analysis ofmultistorey building structures. The program is also capable of performing analysis by using conventional simpli®edmodels of multistorey structures and of verifying the bounds set for the assumptions. # 1998 Elsevier Science Ltd

and Civil-Comp Ltd. All rights reserved.

1. Introduction

Speci®cation of data is the ®rst contact a user has with

a program. It is likely that some users will have no

knowledge of the intricacies of ®nite element formu-

lation, so data speci®cation should be in a format

natural to the particular problem in hand. Thus ap-

preciable savings in both computation time and man-

hours expended accrue if the input data can be scruti-

nized and any errors detected in some automatic man-

ner before computation begins. For this purpose, error

diagnostic subroutines are generally included in most

of the ®nite element programs [1±4]. For regular

meshes, automatic mesh generating routines save time

and help to reduce errors.

For any general purpose ®nite element analysis the

input data required can be subdivided into three main

classi®cations [1±7]. Firstly, the data required to de®ne

the geometry of the structure and the support con-

ditions must be furnished. The geometry of each indi-

vidual element must be speci®ed by listing in a

systematic way the numbers of the nodal points which

de®ne its outline. Each element is identi®ed by its el-

ement number. With the geometry of the structure

de®ned it is now necessary to specify the boundary

conditions. Secondly, information regarding the ma-

terial properties of the constituent materials must be

prescribed. The ®nal category concerns the loading to

which the structure is subjected [8±10]. Obviously, a

consistent set of units must be employed for all input

data. Provided that all length and force terms are

input in the same respective units, then the resulting

displacements and stresses will be similarly dimen-

sioned.

Since by far the greatest task in any ®nite element

analysis is generally the preparation of the input data

and, in particular, de®nition of the nodal coordinates

Computers and Structures 68 (1998) 41±55

0045-7949/98/$19.00 # 1998 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved.

PII: S0045-7949(98 )00020-0

PERGAMON

* To whom correspondence should be addressed.

and element topology, any savings in e�ort that can bemade in this area will be important. For this purpose

mesh generation programs can be developed. These aregenerally of two types:

1. Where an electronic digitizer is employed to de®neand produce the geometric data.

2. A semi-automatic approach where the structure isdivided into a few large zones and the ®neness of el-

ement subdivision within each is speci®ed. The in-itial data is input in the normal way and thesubdivision proceeds automatically.

After the geometrical input data has been prepared, itis worthwhile to plot this automatically beforeattempting a ®nite element solution. Indeed, a graphi-

cal plot of the mesh o�ers a far better check on thegeometric data than the use of error diagnostic subrou-tines. Since even if no data errors are detected by the

diagnostic subroutines and a ®nite element solution isperformed, it is still possible that the coordinate lo-cation of some nodal points may be incorrect and that

the aspect ratio or distortion of some elements may beunacceptable for an accurate solution. If a sophisti-cated application is envisaged, a preliminary plot of

the mesh can often result in large savings with respectto abortive runs. Graphics programs can also be uti-lized in the processing of the ®nal results [3, 4].Plotting packages have been developed for plotting the

deformed shape of the structure, producing stress con-tours or principal stress vectors, etc. Such plots indi-cate to the engineer the areas where a closer

examination of the stresses is necessary; the computerprintout being employed at this stage. Interactivegraphics systems are already having an impact in this

area also, with programs being developed to allow theengineer to isolate and display critical regions of astructure and to vary the output quantity beingplotted. Ultimately it may be possible to dovetail the

entire operation, with the data being generated and theresults obtained and displayed in one operation, lead-ing eventually to an interactive analysis/design process.

2. Previous studies

Although some approximate methods have been

used to establish solutions for laterally loaded frames,the results obtained are not within acceptable limits.These methods frequently provide unsafe solutions and

should not be used. The Muto method [11] is one ofthe techniques used for lateral load analysis.Everard [12] claims that the portal method solution

was shown to be totally incorrect when compared withthe solutions of computer programs, SAPIV andTAFAP. Rosman [13] presented an approximate

method for the analysis of shear walls with a concen-

trated load applied at the top of the shear wall and auniform load acting along the height of the structure.Parme [14] mentions that, although computer pro-

grams would enable an engineer to determine readilythe interaction between frames and shear walls, therestill remains a need for a rapid manual method of

determining the proportion of load carried by each el-ement. MacLeod [15] investigated di�erent aspects of

shear wall±frame interaction. He also proposed amethod very similar to that given by Rosman [13]. Heindicated that the ®nite element analysis of shear walls

would give the most correct solution. Kristek [16] stu-died a folded plate approach and presented a method

in 1979.Smith and Girgis [17, 18] presented an analysis of

non-planar shear wall assemblies by means of an ana-

logous frame. They also presented another framemodel for the analysis of shear wall systems. Theydeveloped two types of frame models instead of using

wide column analogy for the analysis of shear walls.Lew and Narov [19] provided an approach to analyze

a shear wall as a three-dimensional equivalent frame.Chakrabarti et al. [20] investigated the structural beha-viour of prefabricated shear walls. The results are com-

pared with the results of general purpose computerprograms. Current practice is to utilize the computer

programs [21±23] which consider ¯oors to be in®nitelysti� in their own plane. For structures having long andnarrow rectangular shapes and L or T shapes in plan,

the validity of this assumption should be checked afterthe analysis by comparing the story sway with themaximum relative in-plane de¯ection of the ¯oor dia-

phragm under wind and earthquake loads. Hejal andChopra [24] presented the earthquake response of tor-

sionally coupled buildings for a wide range of the sys-tem parameters. They identi®ed the e�ects of lateral±torsional coupling on building motions, arising from

lack of symmetry in building plan by comparing theresponses with those of corresponding torsionallyuncoupled systems [25]. Dario and Ochoa [26] studied

the seismic behaviour of reinforced-concrete, slendercoupled wall systems and the constitutive elements (i.e.

coupling beams and structural walls). Behr andHenry [27] studied the assumptions of the approximatemethods. Tso [28] clari®ed the de®nitions of eccentri-

city used in two of the approaches to calculate storytorsional moments in the design of torsionally unba-

lanced multistorey structures.Many special purpose computer programs have been

developed for the analysis of building structures [21±

23]. However, most of the programs do not givespecial recognition to the fact that building structuresare of a very special class of structures from the ana-

lytical point of view. Moreover, current design codesdo not specify failure criteria explicitly either and they

N. Oztorun et al. / Computers and Structures 68 (1998) 41±5542

are based on stress resultants (internal forces) actingon the member cross-sections. Consequently, there is

not much use to compute accurate stress distributionsfrom the design point of view to make the code check.All of the special purpose programs are based on

some assumptions in the formulation. In some compu-ter programs such as Super ETABS [23], which analyzethe structure in three dimensions, the slab connecting

the structural elements, walls or columns is taken to bea single element in its own plane. The realistic three-dimensional behaviour of shear walls can not be taken

into account properly. The ACI Committee Report [29]contains a review and an evaluation of various struc-tural systems employed in current building practice,with particular reference to their function in resisting

lateral loads. During the past 30 years, the ®nite el-ement method has become the standard procedure forthe analysis of all types of complex civil engineering

structures. General purpose ®nite element programscan be used for two- or three-dimensional analysis ofcomplex multistorey shear wall building structures [1±

4], but three-dimensional ®nite element analysis of theshear wall buildings by using general purpose ®nite el-ement programs is neither practical nor economical.

Excessive modelling e�ort, time of input preparationand computing cost are not justi®ed.

3. The program, mesh generation and data structure

A three-dimensional ®nite element computer analysisof multistorey building structures made of piercedshear walls of open and/or closed cross-sections and

¯at plates is presented. A computer program, namedTUNAL, based on the ®nite element technique isdeveloped. The program automatically evaluates thestatically equivalent earthquake loads and, when

necessary, modi®es these loads together with theboundary conditions and sectional properties of thestructural components located on the axis of symmetry

by considering the symmetric and/or anti-symmetricconditions. The equivalent horizontal earthquakeloads, calculated in accordance with the Turkish

Earthquake Code [30], corresponding to each storeyare calculated and distributed to the nodal points ofthe ¯oor elements. Storey ¯oor loads are also auto-matically distributed to the nodal points of the ¯oor

slabs in proportion to the ¯oor area surrounding thenodal point. A special rectangular plate ®nite elementwith 6 nodal DOF is used in the program. Most of the

available general purpose ®nite element programs con-sider 5 DOF at the nodal points [1, 2]. The present el-ement is formulated by combining bending and plane

stress cases. The element is fully compatible with space®nite elements and can be used to analyze both shearwalls and ¯oor slabs. The displacement functions of

the plate element are developed using Hermitian in-

terpolation functions. Corners of the element are con-sidered to be rigid. Accordingly, rotations of the edgesconnected to the same nodal point are assumed to be

equal and inplane shear strains, hence the shear stres-ses at the corners of the element are equal to zero. Thee�ect of the rigid corner assumption is to make the el-

ement relatively sti� for representation of the plateshear deformations; however, due to cubic displace-

ment functions used in the derivation, the element isextremely good for representing inplane deformationsalso. The formulation and the sti�ness terms of the

®nite plate element used in TUNAL are given in detailin Refs [31, 32]. Numerical solutions were obtained for

some simple plane stress and plate bending problemswith known analytical solutions to determine the capa-bilities of the plate ®nite element developed. The el-

ement is capable of achieving extremely accurateresults with a relatively small number of elements. Anexample building is shown in Fig. 1 with di�erent ®nite

element models and with a typical deformed shape.This example is used to investigate the e�ect of the

aspect ratio of the element on the results. Number ofelements representing the ¯oors between the axes ofshear walls varies between 8 and 36 and the aspect

ratio of the elements varies betveen 1/6 and 6/1 in theexample problem considered. The di�erences betweenthe results on displacements and stresses of the struc-

tures, having the same geometry and the same bound-ary conditions but modelled by a di�erent number of

elements, is less than 4% for all the cases considered.Three-dimensional computer analysis of building struc-tures using general purpose ®nite element programs is,

in general, neither practical nor economical. Excessivemodelling e�ort, time for input preparation and com-putation cost are not justi®ed. Furthermore, capacities

of commonly available computer programs are limited.For example, SAP90 can, in theory, handle upto

16,000 equations [2]; but in practice, a three-dimen-sional structure with no more than 6000 equationsonly can be analyzed with this worldwide used pro-

gram depending on the number of elements, nodalpoints, loading conditions, band width etc. On the

other hand, TUNAL's capacity is limited simply bythe capacity of the computer available. All variablesare in double precision and, theoretically, a structure

with 750,000 equations can be analyzed. The programhas been used for the analysis of structures withunknowns more than 125,000 and the half-band width

larger than 2500. In addition, component based inputand output options provide a powerful control on the

analysis and save time. The sti�ness matrices of the®nite elements in local and global coordinate systemsare obtained analytically and the sti�ness terms are

de®ned in various subprograms for the elements in thefollowing directions:

N. Oztorun et al. / Computers and Structures 68 (1998) 41±55 43

1. perpendicular to global x-direction;

2. perpendicular to global y-direction;

3. perpendicular to global z-direction;

4. for general orientation of the plate elements.

Matrix multiplication, numerical integration and sti�-

ness transformation are not required. Run time of the

computer program and the round of errors are, thus,

minimized.

Five pre- and post-processing computer programs

have been written. The system macro ¯ow diagram is

presented in Fig. 2. Functions of some of these pre-

and post-processing programs are summarized here for

clarity. Thousands of nodes and elements are required

Fig. 1. An example building with di�erent ®nite element models and typical deformed shapes to investigate the e�ect of aspect

ratio.


to prepare an appropriate model for the ®nite element

analysis. The maximum number of nodes considered in

TUNAL during the present study was 5365. A special

mesh generation program, MESHGEN, has been

developed. This program reads the values of variables

from a data ®le named AXES.DAT and produces ®les

to be used by the ®nite element program TUNAL,

DXF (a program preparing three-dimensional ®les for

the drawing program ACAD) and PLOT (a graphic

program developed to prepare the data interactively by

utilizing a powerful screen graphic option). The input

data of a storey of the building structure with repeti-

tive storey planes having a very large number of DOF

requires no more than a couple of lines of input. The

model can be created or modi®ed easily. The beams or

columns can be added or cancelled easily by typing a

couple of characters. The user will not have to deal

with the nodal points and the elements. It is su�cient

to specify the structural components required such as

moment distribution on any shear wall, or a slab or a

column. The following output ®les are automatically

produced by MESHGEN.

. BB.DAT includes general information such as num-

ber of joints, number of elements, number of joints

subjected to concentrated loads, etc.

. PLATE.DAT includes connectivity array of plate el-

ements together with the material property number

and axis number which is perpendicular to element

face for each element and material properties of

each di�erent material.

. FRAME.DAT includes connectivity array of frame

elements together with the sectional property num-

ber and direction of the element and material and

sectional properties of each di�erent type of sec-tion.

. TRUSS.DAT includes connectivity array of truss el-

ements together with the sectional property numberand material and sectional properties of each di�er-ent type of section.

. SPRING.DAT includes the boundary spring rigid-ities and the joint numbers of the springs where thesprings are attached to consider the elastic behaviour

of the soil optionally.. POINT.DAT includes numbers and coordinates ofthe nodal points.

. LOADS.DAT includes joint numbers and the com-ponents of the loads which are applied to the de®nedjoint, support conditions and external displacementcomponents (optional).

4. Results

Results of a shear wall building structure are pre-

sented. The main objective is to investigate the e�ectof foundation ¯exibility and ¯oor sti�ness on the de-sign parameters. Finite element models of the structureare generated and analyzed using TUNAL program.

A three-dimensional shear wall building structurewhich is planned to be constructed in Turkey is ana-lyzed. Di�erent types of ¯oor plans of the structure

are shown in Figs. 3±5. Only half of the structure ismodelled by utilizing the symmetric behaviour of thestructure about the axis of symmetry. The e�ect of the

¯oor torsion is not considered in this example,although it can be taken into account if necessary.Lateral earthquake loads are applied to half of the

Fig. 2. Macro ¯ow diagram of TUNAL and pre- and post-processors.


structure in the y-direction. Boundary conditions con-

sidering the symmetric behaviour about the y-axis areautomatically generated by the computer program.

Vertical loads are considered to be 12.0 kN/m2 for theevaluation of the equivalent static earthquake loads.

Three di�erent types of ¯oor plans are modelled.The ®rst type of plan geometry is used for the ®rst 16

¯oors including the foundation mat. The ®rst type of¯oor section of the structure is composed of 16 shear

walls in the x-direction and 11 shear walls in the y-

direction as shown in Fig. 3. Then four of the upper¯oors are represented by the second type of plan geo-

metry as shown in Fig. 4. The remaining two ¯oors atthe top are represented by the third type of ¯oor geo-

metry as shown in Fig. 5. The change in plan alongthe elevation can be taken into account very easily.

Fig. 6 shows the ®nite element modelling of the ®rst¯oor. Fig. 7 shows the ®nite element model of the

shear wall PY7 of the second type of plan (i.e. the ®rst

shear wall in the y-direction of storeys 17 and 20).Note that the nodal point and element numbering of

the system is automatically produced by the computerprogram. Both the input and the output can be

obtained on a component basis. This plot option isprovided for theoretical studies and generally is not

required in practice. Deformed shape is plotted inFig. 8, which also shows the change in the plan along

the elevation. Thickness of each shear wall is equal to20 cm. A twenty-two storey structure is modelled by

14 cm thick ¯oor slab plates. The height of the ®rst

storey is equal to 4.11 m and the heights of the remain-ing storeys are equal to 2.79 m. Equivalent lateral

earthquake loads are calculated properly in accordancewith the Turkish Earthquake Code. Lateral loads are

automatically generated by the program and applied iny direction to the nodal points of the slab elements.

Fig. 3. First type of plan view of the example problem.


Two ®nite element models are prepared to see the

e�ect of the boundary conditions.

(a) The foundation of the structure is modelled with

®nite elements representing mat foundation and

linear boundary springs are provided at the nodes

of the ®nite elements which represent the mat foun-

dation to consider the elastic behaviour of the soil.

In-plane motion of the structure at the foundation

level is prevented. E�ect of the soil rigidity is inves-

tigated using this model. Modulus of subgrade

reaction of the soil is considered as 24.0 MN/m3.

This value approximately corresponds to the mod-

ulus of subgrade reaction of Ankara clay. A few

analyses have been performed to investigate the

e�ect of the soil properties, but only one of the nu-

merical results of the analysis is presented in this

study.

(b) Nodal springs and the ®nite elements representing

the mat foundation are not provided in the second

®nite element model. The boundary conditions at

the bottom of the shear walls are modelled to be

completely constrained in each degree of freedom.

Both models are executed ®rst for a structure having

rigid slab plates in the plane of the plate, and then the

same model is executed with slab plates having normal

sti�ness of 14 cm thickness. In total four analyses have

been performed for the structure. The models analyzed

can be summarized as follows:

(i) elastic foundation, ¯exible slab in the plane;

(ii) elastic foundation, rigid slab in the plane;

(iii) rigid foundation, ¯exible slab in the plane;

(iv) rigid foundation, rigid slab in the plane.

Fig. 4. Second type of plan view of the example problem.


All six degrees of freedom for the rigid ¯oor assump-tion are taken into consideration. This way, the resultscan be evaluated at the corresponding degrees of free-

dom. The rigidity of the elements representing thestorey ¯oors is increased by de®ning two thicknessesfor each element. First thickness value is used for theterms corresponding to plane stress and the second

one is for the plate bending part of the equations. Thefollowing are investigated.

4.1. A. In®nitely rigid or ¯exible ¯oor assumption

In earlier works, multistorey building structures wererepresented by models assuming in®nitely rigid or ¯ex-ible ¯oors. In the case of modern high-rise buildings,

¯oors are, in general, neither in®nitely ¯exible norrigid. In this study in®nitely rigid and ¯exible ¯oorassumptions are investigated and the results are com-

pared with those of a slab with appropriate rigidity.Practical bounds are established for the assumed con-ditions.

4.2. B. Shear stresses around shear walls

Localized stresses in the ¯oor diaphragms aroundthe shear walls having open and/or closed cross-sec-

tions may become very critical in the design of ¯oorslabs around the shear walls. A technique whichenables the computation of localized stresses in the

analysis is developed in the study.

4.3. C. Pierced shear walls of open and/or closed cross-section

There are many approximations developed for theanalysis of building structures containing pierced shear

Fig. 5. Third type of plan view of the example problem.


walls. The validity of these assumptions and their

bounds have not yet been thoroughly investigated. The

e�ect of the openings on the sti�ness and on the loca-

lized stresses in shear walls is investigated as a part of

this study.

Bending moment, shear force and axial load dia-

grams at a vertical axis passing through the midpoint

of the width of each shear wall are obtained. It is

observed that the rigid ¯oor assumption of such a

building with complicated geometry and shear wall lo-

cation may have a signi®cant e�ect on the shear distri-

bution along the height of the shear walls.

Consequently, the in-plane rigid ¯oor assumption may

change the shear force distribution along the height

signi®cantly, especially at the lower levels of shear

walls. It is seen that the sign of the shear distribution

along a shear wall may change depending on the rigid-

ity and location of shear wall. Behaviour of the shear

walls is di�erent. Some of the walls have a tendency to

exhibit a cantilever behaviour and some of them show

shear wall behaviour, but behaviour and displacements

of the shear walls are forced to be compatible at the

levels of storey ¯oors by the slab plates. Therefore, a

signi®cant change on the shear distribution on the

walls may occur [31, 32]. A sudden jump in the shear

distribution generally occurs between the bottom of

the wall and the ®rst ¯oor elevation. This e�ect may

continue along a couple of storey elevations on some

shear walls. This change causes tensile and compressive

stresses in the ¯oor slabs. Additional axial tensile

Fig. 6. Finite element model of the ®rst storey ¯oor.


stresses require additional reinforcement and must be

considered in the design. These stresses may exceed theallowable tensile load carrying capacity of the ¯oor

slabs and cause cracking of the slab plate. In this case,

the punching shear reinforcement must be providedduring the design. Additionally it is seen that the in-

ternal stresses may change along the shear walls sud-

denly at the ¯oor levels where the properties of the¯oor plan are changed. Horizontal shear distribution

along the shear wall PY2 in the y-direction is shown inFig. 9. This wall possesses uniform cross sectional

properties along the height of the building. The shear

wall PY6 of the ®rst type of ¯oor continues as PY5 atstoreys 17±20 and as PY6 at storeys 21 and 22. A very

signi®cant jump in shear forces and in in-plane forces

is seen at the elevations corresponding to the change inthe plan geometry. The magnitude, the location and

the sign of the jump in the forces occurring in a shearwall di�er depending on the assumptions on the in-

plane rigidity of the slab and/or on the rigidity of thesoil. For the analysis under consideration, 322 frameelements are used. The total number of elements is

4735, the number of nodal points is 3740, the area inthe plan is 343.664 m2, the number of equations is22440, and the half band width is 1080. The boundary

spring rigidities at the nodal points of the foundationare de®ned to represent the soil rigidity. The programcan automatically evaluate the statically equivalent

earthquake loads and modify these loads together withthe boundary conditions and sectional properties ofthe structural components located on the axis of sym-metry by considering the symmetric and/or anti-sym-

metric conditions. The vertical load applied on thestorey ¯oors is equal to 12.0 kN/m2 and the lateralearthquake load coe�cient is de®ned as 0.13.

Equivalent horizontal earthquake loads for each storeyare calculated in accordance with the TurkishEarthquake Code [30]. Modulus of elasticity and

Poisson's ratio are equal to 210 GPa and 0.17, respect-ively. All of the link beams have 20 cm width and69 cm depth. The horizontal loads correponding to

each storey are calculated by the program and thestorey loads are distributed to the nodal points of the¯oor elements considering the symmetric loading con-ditions. Storey ¯oor loads are automatically distributed

to the nodal points of the ¯oor slabs in proportion tothe ¯oor area surrounding the nodal point. Horizontalearthquake loads are applied in the y-direction for the

example considered. The deformed shapes of the build-ing considered under horizontal earthquake loads arepresented in Fig. 8. The vertical loads are not con-

sidered in this analysis. Maximum and minimumde¯ections and rotations are given in the followingTable 1.

5. Conclusions

A three-dimensional ®nite element technique isdeveloped for the elastic analysis of shear wall building

structures which are constructed using tunnel forms.Shear walls and ¯at plates constitute the vertical andhorizontal load bearing elements in these structures. Ingeneral, structural members are exactly the same in all

of the ¯oors, except that the ®rst ¯oor above the foun-dation may be of di�erent height. Mat (raft) foun-dation is used due to the fact that space between the

shear walls is rather small for the strip foundation. Aspecial purpose ®nite element computer programnamed TUNAL is developed for the analysis.

Rectangular plate ®nite elements having 6 DOF pernode are incorporated. This element is formulatedby combining bending and plane stress cases. The

Fig. 7. Finite element model of the shear wall PY7 of the sec-

ond type of ¯oor.


resulting model is fully compatible with space ®nite el-ements and can be used to analyze shear wall and

¯oor slabs. Rectangular ®nite elements are based oncubic Hermitian displacement functions. Elements

based on these functions represent inplane defor-mations very satisfactorily. Pre- and post-processing

programs which are part of TUNAL allow the utiliz-

ation of minimum amount of input data by eliminatingelement and node numbering and numerical and

graphical display of output on the screen as well as inthe forms of lists or plots. Since the program devel-

oped gives the stress distributions along the structuralcomponents in local and/or global coordinates, re-

inforcements can be calculated easily. Convergence stu-dies are performed on four structures using several

models to check the performance of the rectangular®nite elements. Acceptable results are obtained with

coarse meshes and good convergence observed on themodels tested.

Utilization of general purpose ®nite element pro-

grams such as SAP90 seems to be impractical for theanalysis of such structures due to the large amount of

input required and the limitation of the total DOF.Pseudo-3D analysis programs such as ETABS or gen-

eral purpose programs can be utilized with reasonableinput and computer time by assuming in®nitely rigid

Fig. 8. Deformed shape of the example problem with elastic foundation.


¯oor slabs in their plane. The rigid ¯oor assumption

does not allow the computation of in-plane forces.

Consequently, e�ect of in-plane forces must be neg-

lected in design. One of the main objectives of this

study is to investigate the magnitude of in-plane forces

in the ¯oor slabs. The rigid ¯oor slab is represented by

taking increased thickness for plane stress part of the

rectangular plate element incorporated in TUNAL.

Thus, in order to simulate rigid ¯oor assumption, regu-

lar and increased in-plane sti�nesses are considered for

¯oor slab ®nite elements.

The e�ect of foundation ¯exibility is also investi-

gated. Flexibility of the foundation must be considered

in the design of shear wall structures. The ®nite el-

Fig. 9. Shear distribution along the shear wall PY2.

Table 1

De¯ections and rotations for the building considered

De¯ections (m) Dx Dy Dz

Max. 0.000342 0.07976 0.01256

Min. ÿ0.000788 0.00000 ÿ0.01293Rotations (rad) yx yy yzMax. 0.0032463 0.0018475 0.0002036

Min. ÿ0.0017740 ÿ0.0036477 ÿ0.0002503


ement model of the mat (raft) foundation together

with linear vertical springs at the nodes representing

soil ¯exibility can be generated in the framework of

TUNAL. The e�ect of the change of ¯oor plan in

upper stories is also investigated. The following are the

conclusions:

1. It is customary to assume that ¯oor slabs are in®-

nitely rigid in their plane. The present study

revealed that regular and sti� ¯oor slab assump-

tions resulted in some changes in the distribution of

story shears among the shear walls. However, the

most important conclusion is the realization of sig-

ni®cantly high in-plane forces acting on the ¯oor

slabs at lower and upper stories. These forces are

very critical in the design because they produced

tensile stresses around 3.5 MPa or more, as seen in

Fig. 10. It is not possible to determine these in-

plane forces in the ¯oor slabs with rigid ¯oor

Fig. 10. Inplane stress distribution along the ¯oor slabs in the y direction. (A) For storey 22, axes K12±B12. (B) For storey 1, axes

M15±A15.


assumption. The in-plane forces are not currentlyconsidered in the design of ¯oor slabs.

2. Shear wall building structures constructed by tunnelforms normally contain total shear wall areabetween 4 and 8% of the ¯oor area in each princi-

pal direction. Floor slabs seem to be more criticalthan the shear walls due to large shear wall area.Failure of the ¯oor slabs rather than shear walls is

expected under extreme earthquake conditions.3. As pointed out by earlier research work, foundation

¯exibility is important in the behaviour of shear

wall structures. In this study it is observed that the¯exibility of the foundation has signi®cant e�ect onthe distribution of story shears among shear wallsand the in-plane forces in the ¯oor slabs.

4. Separation of box, U and other types of shear wallsin x- and y-directions into di�erent walls in thepseudo-three-dimensional programs produces sig-

ni®cant errors because longitudinal shear forcealong the shear wall junctions as well warpings intorsion are not taken into consideration.

5. An improved simpli®ed method for the analysisseems to be inappropriate. Three dimensional ®niteelement analysis as presented in this study is the

proper method of solution. The increasing capacityand speed of computers together with appropriatesoftware shall make the ®nite element approach avery convenient design tool.

Acknowledgements

This research is supported by The Scienti®c and

Technological Research Council of Turkey throughproject no. INTAG 515.

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