Budapest University of Technology and Economics

Faculty of Architecture

APPROXIMATE ANALYSIS OF BUILDINGSTRUCTURES SUBJECTED TO EARTHQUAKES

THESIS SUBMITTED TO THE FACULTY OF ARCHITECTURE OF

BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Gabriella Potzta

Supervisor:

Laszlo P. Kollar

Budapest, March, 2002

1

Acknowledgments

I wish to sincerely thank the 6 years work of my supervisor Prof. Laszlo P. Kollar. Im grateful

for his support, all of his help and the knowledge which I owed to him.

Thanks to the chiefs of the departments for providing my Ph.D studies:

Prof. Gyorgy Farkas, Chief of the Department of Structural Engineering,

Prof. Tamas Matuscsak, Chief of the Department of Mechanics and Structures,

Prof. Marta Kurucz, Chief of the Department of Structural Mechanics.

I thank to Prof. Zsolt Gaspar for the possibility to finish my thesis in the Research Group of

Computational Mechanics.

I would like to thank to all of my professors for their help and advices which helped my research

work: Prof. Karoly Zalka, Prof. Endre Dulacska, Prof. Istvan Hegedus, Prof. Lajos Kollar, Prof.

Pal Rozsa.

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1 Introduction

Analysis of high-rise building structures stiffened by shear walls, trusses, coupled shear walls, and

frames requires time consuming numerical computations. The designer may be well served by

approximate methods, which (i) can be used in the preliminary design when some of the structural

dimensions are not yet known, (ii) can verify the results of the more advanced numerical calculation,

and, last but not least (iii) can shed light on the behavior of the structure which may lead to a

better design.

Our aim is to present an approximate analysis of building structures subjected to earthquakes

which is (i) simple, (ii) robust (i.e. it gives results with acceptable accuracy for structures with

very different characteristics), and (iii) which can also handle the torsional vibration of building

structures.

The building is stiffened by an arbitrary combination of lateral load-resisting subsystems (shear

walls, frames, trusses, coupled shear walls, cores). We consider stories with identical masses,

however the mass at the top floor may be different. The stiffnesses of the structure may vary with

the height. The analysis is based on the continuum method. We developed replacement beams of

building structures, and we solved approximately the spatial vibration problem of the replacement

beam. Simple formulas are given to calculate the periods of vibration and the internal forces of a

building structure subjected to earthquakes.

The utility and accuracy of the method is demonstrated by a numerical examples, in which the

approximate solution is compared to the results of a finite element calculation.

1.1 Continuum method

One of the most widely used approximate calculations is based on the continuum method [14],

[33],[34], [40], when the stiffened building structure is replaced by a (continuous) beam.

The simplest replacement beam is a thin-walled beam, characterized by the bending stiffnesses

(D0yy, D0zz, D0yz), the warping stiffness (D) and the torsional stiffness (Dt). (When only a

plane problem is considered, torsion is excluded, and the only parameter that plays a role is the

bending stiffness D0 = D0yy in the x z symmetry plane.) This model is adequate only for solidand slender shear walls.

When a truss is loaded laterally, it may show, depending on the stiffnesses of the elements,

bending (flexural) deformation, shear deformation, or the mixture of those (Fig.1).

Hence the shear deformation must be included and the replacement continuum is a Timoshenko-

beam [39], characterized by the bending (D0 = D0yy) and the shear stiffnesses (S = Szz) in the

x z plane. The bending and the shear stiffnesses of typical structures are given e.g. in [39], [34],[20] and are listed in Table 1.

A wide frame structure can be modelled by a beam which undergoes shear deformation only

and is characterized by the shear stiffness S = Szz.

3

x

z

Figure 1: Flexural deformation, shear deformation, and mixed deformation

D

D

S

0

l

lD

=

D

S

0

a) b)

Figure 2: Replacement beam of a frame (a), the sandwich beam is equivalent to a Timoshenko-

beam supported by a beam with bending deformation only (b)

When a wide frame is braced by solid walls the replacement beam has two stiffnesses: the

shear stiffness (S) due to the frame and the bending stiffness due to the walls (Dl). This model is

referred to a Csonka-beam because P. Csonka developed it for the analysis of building structures

subjected to wind loads [8].

Neither a thin-walled beam, nor a Timoshenko-beam, nor a Csonka-beam is adequate to

characterize a slender frame, or coupled shear walls. The replacement beam can be obtained by

smearing out the beams of the frame along the height, and thus we arrive at the model shown

in Fig.2, which is a sandwich beam [33].

The stiffnesses of the replacement sandwich beam are also included in Table 1 ([14], [37], [8],

[4], D0 is the global bending stiffness, Dl is the local bending stiffness, and S is the shear stiffness).

We note that the sandwich beam is the generalization of the previous models, thus we can

derive them from the sandwich beam with the proper choice of the stiffnesses. This is illustrated

in the following table:

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Stiffnesses of a sandwich beam Choice of the stiffnesses Resulting beam (stiffnesses)

Dl 0 Timoshenko-beam (D0, S)D0 Csonka beam (Dl, S)

D0, Dl, SD0 ,Dl 0

Beam with shear

deformation only (S)

S Thin walled beam (D0 +Dl)S 0 or D0 0 Thin walled beam (Dl)

It is important to note that a sandwich beam with stiffnesses D0, Dl, and S is equivalent to a

Timoshenko-beam (with stiffnesses D0 and S) which is supported laterally by a beam with bending

stiffness Dl (Fig.2). Hence, if we set the stiffness Dl of a sandwich beam equal to zero we obtain a

Timoshenko-beam with stiffnesses D0 and S.

Continuum models were developed by several authors and it was applied successfully for build-

ing structures subjected to wind loads [8], [34], [37], [40],[46], earthquakes [2], [3], [20], [17], [34],

in the dynamic analysis [25], [28], [29], [33], [40], [43], [42], [47], and in the stability analysis [14],

[24], [28], [29], [34], [40], [41], [44], [45].

However, there are two important problems to be solved:

(i) As we stated before the replacement beam of a single lateral load-resisting subsystem (truss,

frame, shear wall etc.) is given in the literature (see Table 1). When there are several parallel lateral

load-resisting subsystems which are connected horizontally along the height the question arises:

how can they be replaced by only one replacement beam? We find answers only for the following

special cases in the literature: (a) When each lateral load-resisting subsystem is a solid wall

(their shear deformation is neglected) the replacement beam is a beam which undergoes bending

deformation only, and its bending stiffness is the sum of the bending stiffnesses of the individual

walls. (b) When there are frames which can be modeled as beams undergo shear deformation only

and solid walls undergo bending deformation only, the replacement beam is a Csonka-beam, which

has two stiffnesses (with the sandwich notationDl and S, while D0 is infinite), the bending stiffness

is the sum of the bending stiffnesses of the walls, while the shear stiffness is the sum of the shear

stiffnesses of the frames. However, when any of the lateral load-resisting subsystem undergoes both

bending and shear deformation (which is the case of trusses, coupled shear walls, tall frames, and

for wide walls) it can be shown that simple summation (S =Sk, D0 =

D0k, Dl =

Dlk)

may result in a structure which is stiffer by orders of magnitudes than the real structure. We

will show in Section 4.1 how the replacement stiffnesses of the building should be calculated.

(ii) As an example let us consider a structure the cross section of which is shown in Fig.3.

When the structure is subjected to torsion, in the two parallel trusses both shear and bending

deformations occur. The classical (Vlasov) theory of beams does not include the shear deformation

in torsion with warping and, hence, its application may significantly overestimates the torsional

stiffness of the structure. This problem, for arbitrary arrangements of the walls, will be addressed

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StructureReplacement continuum

Siffnesses

wall

b

t = thickness

Timoshenko-beam

D0 = EI I =b3t12

S = AG =AG1.2 A = bt

trussesL

Ad

Ac

d

Ac

h

L

Ad

Acd

h

Timoshenko-beam

D0 =12EAcL

2 D0 =12EAcL

2

S = 2EhL2Ad

d3 S =2Eh

2d3

L2Ad+ L

4Ab

The shear stiffness of trusses

with other type of bracing can

be found in the literature.

frameAci , Ici

Ibi

li

h

ci

0 1 i n

Sandwich beam

Dl =n

i=0EIci

D0 =n

i=0EAcic2i

S =(S1b + S

1c

)1

Sb =n

i=112EIbi

lih,

Sc =n

i=012EIci

h2