Torsionally Coupled Buildings___Soil Structure Interaction

8/10/2019 Torsionally Coupled Buildings___Soil Structure Interaction

1/15

Turkish J. Eng. Env. Sci.29 (2005) , 143 157.c TUBITAK

An Efficient Seismic Analysis Procedure for Torsionally CoupledMultistory Buildings Including Soil-Structure Interaction

Erkan CELEBI

Sakarya University, Department of Civil Engineering

Sakarya-TURKEY

e-mail: [email protected]

A. Necmettin GUNDUZIstanbul Technical University, Department of Civil Engineering

Istanbul-TURKEY

Received 08.06.2004

Abstract

In this paper, a simplified methodology of analysis for the seismic response of 3-dimensional irregularhigh-rise buildings on a rigid footing resting on the surface of a linear elastic half-space is formulated.An efficient method using modal decomposition and carried out in the frequency domain by using thefast Fourier transform to obtain the structural response of torsionally asymmetric buildings, including soil-structure interaction effects, is presented. Applying this algorithm, full advantage is taken of classical normalmode approximation, and the interaction problem is solved easily and effectively within the framework ofthe Fourier-transformed frequency domain analysis for a fixed-base structure. The matrix formulation ofthis method produces accurate approximation with less computational effort, in spite of using the frequencydependent impedance functions.

Key words: Torsionally coupled multistory buildings, Soil-structure interaction, Frequency domain analy-sis, Impedance functions, Modal decomposition.

Introduction

The comprehensive studies (Chandler and Hutchin-son, 1986; Cruz and Chopra, 1986b; Hejal andChopra, 1989) conducted by a number of researchers

in the past few decades and investigations of the ef-fects of past earthquakes have shown that in build-ings with non-coincident centers of mass and resis-tance, significant coupling may occur between thetranslational and the torsional displacements of thefloor diaphragms even when the earthquake inducesuniform rigid base translations. However, these stud-ies assumed that the asymmetric structure is sup-ported by a rigid foundation. The elastic responseof structures to earthquake ground excitations is in-fluenced by deformability of the foundation medium.Therefore, the rigid foundation assumption repre-

sents an approximation to real conditions. It iswidely recognized that the dynamic response of astructure supported on soft soil may be different fromthe response of a similarly excited, identical struc-ture supported on firm ground. The effects of soil-

structure interaction (SSI) on the dynamic responseof building systems have been the subject of numer-ous investigations in recent years. Most of the pre-vious SSI studies (Chopra and Guiterrez, 1974; Wuand Smith, 1995; Wu, 1997) on buildings, however,were restricted to 2-dimensional (planar) multistoryframes due to the fact that the governing equations ofmotion for structures with many degrees of freedom,as well as the methods of solving these equations,are relatively complicated and involve a considerablecomputational costs.

Various analytical and numerical techniques were

143


2/15

CELEBI, GUNDUZ

also proposed and developed to efficiently simplifySSI analysis, such as transmitting boundaries of dif-ferent kinds, boundary elements, and infinite ele-

ments and their coupling procedures for the mod-elling of unbounded media (Meek and Wolf, 1992;Aydnoglu, 1993b; Wolf and Song, 1996). However,the complicated formulation and intensive computa-tion necessary to obtain the exact solution to thisproblem have so far restricted its common applica-tion to traditional engineering practice. The foun-dation on flexible soil may be idealized in structuraldynamics as a simple spring-dashpot-mass modeland the solution can be performed in the frequencydomain. Since most of the complication in thesolution of the equations of motion results from

the frequency dependence of the dynamic soil stiff-ness, many studies (Ghaffer and Chapel, 1983; Wolf,1997) have been performed to simulate the SSI phe-nomenon by representing the soil with frequency-independent impedance functions by constant pa-rameters. However, this proposed approximationmay not be valid when the soil surface is shal-low relative to the base dimensions of the struc-ture (Tsai et al., 1974). Additional research on SSIproposed by Sivakumaran (1990) and Chandler andSikaroudi (1992) has been dedicated to the appli-cation of modal analysis techniques. Substitutionof structural deformations, in combination with thedynamic SSI force-displacement relationships, intothe governing equations of the whole system re-sults in integro-differential equations for footing dis-placements, which are solved numerically by step bystep integration in time domain. Such a procedure,given by Sivakumaranet al. (1992), requires a largecomputational effort, particularly for 3-dimensionalasymmetric multistory buildings.

In the present paper, an efficient method us-ing modal decomposition and performed in the fre-quency domain by using the fast Fourier transform

algorithm to obtain the dynamic response of torsion-ally asymmetric buildings including the SSI effect ispresented. The effects of the foundation mediumon the structural response are simulated by a se-ries of springs and dashpots representing a theoreti-cal half-space surrounding the base of the structure.The structure investigated herein is presumed to belinear and viscously damped and is supported atthe surface of a homogeneous, isotropic, elastic half-space and is excited at the base. However, to accu-rately represent the elastic half-space, the propertiesof springs and dashpots are required to be depen-

dent on the frequency of excitation. Thus, the gov-erning equations for the structure foundation systemare expressed and solved in the Fourier-transformed

frequency domain. The governing equations are de-veloped considering the motions of each floor and thewhole system. In this method, initially structural de-formations are obtained in terms of foundation dis-placement using a linear combination of vibrationalmodes of the building on a rigid foundation, in com-bination with the dynamic SSI force displacement re-lationships. In this study, the Fourier transforms arecomputed very efficiently by the fast Fourier trans-form algorithm, where they are treated as discretetransforms (Humar and Via, 1993). The numeri-cal approximations of the impedance functions used

in the following study are taken from Veletsos andWei (1971) for lateral and rocking vibration and fromVeletsos and Nair (1974) for torsional vibration. Todemonstrate the proposed method, a detailed para-metric study is made of torsional coupling in a mul-tistory building excited by real earthquake groundmotion (Erzincan, 1992, E-W) for different soil stiff-nesses by using an original program developed andapplied by the authors. For practical purposes thedefinite part of the record time including the peakvalue of acceleration of this input motion is consid-ered. The results represented in frequency responsecurves indicate that this efficient method produces agood approximation to the exact response obtainedby the direct method illustrated in previous studies(Celebi and Gunduz, 2000, 2001). The results pre-sented in this paper show that the seismic responsesof asymmetric buildings including soil structure in-teraction can be significantly different from thosewithout substructure interaction.

Structure-Foundation Model and Analytical

Procedures

Assumptions and equation of motion

The building foundation system considered, asshown in Figure 1, is represented by an N-story3-dimensional superstructure, consisting of shearframes, resting on a rigid square foundation of massmo with a negligible thickness on the surface of alinear homogeneous elastic half-space. The mass ofthis building is considered to be concentrated at eachfloor level, and the floor systems are assumed to berigid rectangular floor decks supported by relativelymassless, axially inextensible columns. The lateralload resisting elements are assumed to be arranged

144


3/15

CELEBI, GUNDUZ

CM

e

Frame B

Frame C

Frame A

d/2

d/2

b

dx

y

gyu&& (t)

hi hi

Floor i Floor i

Pox(t) Poy(t)

To t To t

ugx uox ugy uoy

F

i

x

e

d

r

e

e

r

en

c

e

s

a

x

is

Elevation along x-direction

ox hi

uxi

oy hi

uyi

oxoy

Mox(tMoy(t)

Elevation along y-direction

F

i

xe

d

r

e

e

r

en

c

es

a

x

i

s

Elastic half-space

Structure

Foundation

Flexible

Soil

gxu&& (t)

yx

z z

Footing Displacements

Structural

Deformations

uoy

uyiUiuxi

y

z

x

uoxo

Soil parameters:

G: shear modulus

cs: shear velocity

: Poissons ratio

mo

m1

mi

mN-1

mN

mo

m1

mi

mN-1

mN

Plan

oyox

3-D View

Figure 1. Idealized 3-dimensional asymmetric building foundation system on an elastic half-space.

145


4/15

CELEBI, GUNDUZ

so that the system has no axes of symmetry. Thusthe structure is a 2-way torsionally coupled systemwith 3 degrees of freedom at each floor, namely hori-

zontal displacements uxi, uyi and rotation about thevertical axisui for the ith floor. In addition, due tothe deformability of the foundation, the system has 5more displacement degrees of freedom, namely hori-zontal translations of the foundation uox, uoy, rock-ing rotations of the foundation ox,oyand the twistof the foundation o(Figure 1). The earthquake ex-citation is defined by ugx(t) and ugy(t), the x andy components of the ground acceleration measuredat the surface of the homogeneous half-space in thefar field, and ug(t),gx(t), gy(t) are the rotationalacceleration of the base of the building about the

vertical and horizontal axes, respectively. The equa-tions of motion for a SSI system with 3 N + 5 degreesof freedom can be written in the usual matrix form

[M]{u(t)} + [C]{u(t)} + [K]{u(t)} + [M]({uo1(t)}+

{oh(t)} + {ug1(t)} + {gh(t)}) = {0}

(1a)

where

{u(t)} =

{ux(t)}

{uy(t)}{u(t)}

{uo1(t)} =

uox(t){1}

uoy(t){1}uo(t){1}

{oh(t)} =

ox(t){h}

oy(t){h}

{0}

{ug1(t)} =

ugx(t){1}

ugy(t){1}

ug(t){1}

{gh(t)} =

gx(t){h}

gy(t){h}

{0}

(1b)

In the above expression [M], [C], and [K] repre-sent structural property matrices. Viscous dampingis assumed to be in such a form that the building on arigid foundation admits decomposition into classicalnormal modes (Vaidya et al., 1986). Furthermore,{u(t)} and {h}refer to column vectors of the struc-tural displacements relative to the rigid foundationand the height of floors from the ground level, respec-tively. The acceleration vectors of the rigid founda-tion motion are {uo1(t)} and {oh(t)}, respectively.

In terms of the linear transformation matrix, Eq. (1)can be simplified to

[M]{u(t)} + [C]{u(t)} + [K]{u(t)}+

[M][T]{uot(t)} = {0}(2)

where [T] represents the kinematic transfer matrixfor transmitting total motion of the rigid founda-tion to the structure and the total displacement vec-tor, {uot(t)}, and includes horizontal, rocking andtorsional degrees of freedom of the rigid foundationin addition to the free field motion. The rotationand rocking acceleration components of the free fieldground motion will be disregarded as indicated inEq. (3). That is,

{uot(t)} =

uoxt(t)

uoyt(t)

o(t)

ox(t)

oy(t)

;

[T] =

{1} {0} {0} {h} {0}

{0} {1} {0} {0} {h}{0} {0} {1} {0} {0}

3N5

;

uoxt(t) = uox(t) + ugx(t)

uoyt(t) = uoy(t) + ugy(t)

(3)

In addition, 5 more equations are needed in orderto completely solve the problem. These equationsare developed by considering the equilibrium of thewhole system. They can be further written in openform as

(mo+ {1}T

[mx]{1})uox(t) + {1}T

[mx]{h}ox+

{1}T

[mx]{ux}+Pox(t)=(mo+{1}T

[mx]{1})ugx(t)

(4a)

(mo+ {1}T

[my]{1})uoy(t) + {1}T

[my]{h}oy+

{1}T

[my]{uy}+Poy(t)=(mo+{1}T

[my]{1})ugy(t)

(4b)

146


5/15

CELEBI, GUNDUZ

{1}T[mx]{h}uox(t) + (Itx+ {h}T[mx]{h})ox+

{h}T

[mx]{ux} +Mox(t) = {1}T

[mx]{h}ugx(t)

(4c)

{1}T

[my ]{h}uoy(t) + (Ity+ {h}T

[my ]{h})oy+

{h}T

[my ]{uy} + Moy(t) = {1}T

[my]{h}ugy(t)

(4d)

(mor2o+ {1}

T[m]{1})o(t) + {1}T[m]{u(t)}+

To(t) = 0(4e)

In these equations, {1} is the column vectorwhere each element is unity, and mo and ro are themass and the radius of gyration of the foundation,respectively. Itx, Ity are the total of the mass mo-ment of inertia of the floors and the foundation matwith respect to the x and y axes. {h}is the column

vector of foundation to story heights, [mx], [my] con-sist of floor masses of the structure in terms of thex and y axes, and [m] is the mass polar moment of

inertia about the z axis of the floor mass. Pox(t),Poy(t), Mox(t), Moy(t), To(t) are the horizontal SSIforces, rocking moments and torsional moment, re-spectively. It is noted that they are the dynamicloads imposed by the structure on the foundation.When the foundation medium is flexible these forcescan be related to foundation displacements uox, uoy,ox, oy and o. Dynamic SSI force displacementequations may be specified in the frequency domain.Equation (4) can be expressed more concisely in thefollowing form:

[M1]{u(t)} + [M2]{uot(t)} + {Po(t)} = {0};

{Po(t)}T

= {Pox(t) Poy(t) Mox(t) Moy(t) To(t)}

(5)

where [M1] = [T]T

[M] and [M2] includes compo-nents of superstructure, and these refer to the totalfoundation mass matrix. They are given by

[M1] =

{1}T

[mx] {0}T

{0}T

{0} {1}T[my] {0}T

{0}T

{0}T

{1}T

[m]

{h}T

[mx] {0}T

{0}T

{0}T

{h}T

[my] {0}T

53N

[M2]=

mo+ {1}T

[mx]{1} 0 {1}T

[mx]{h} 0 0

0 mo+ {1}T

[my ]{1} 0 {1}T

[my ]{h} 0

0 0 0 0 m+ {1}T

[m]{1}

{1}T[mx]{h} 0 Itx+ {h}T[mx]{h} 0 0

0 {1}T

[my]{h} 0 Ity+ {h}T

[my]{h} 0

55

(6)

Dynamic soil-structure interaction

The soil is assumed to be an elastic homogeneousisotropic half-space modeled by a massless rigid platesupported by equivalent translational, rotational andtorsional springs and dashpots. Because the foun-

dation stiffness and damping coefficients are depen-dent on excitation frequency, it is most convenientto consider the response of the building foundationsystem to harmonic ground motion. Due to thesefrequencies-based expressions for interaction shearforce, moment, and torque, the interaction problem

147


6/15

CELEBI, GUNDUZ

lends itself readily to formulation in the frequencydomain. Thus, the Fourier transform is applied tothe equations of motion (2) and (5), and the follow-

ing, very simple, results are obtained:

(2[M] + i[C] + [K]){u()} 2[M][T]{uot()}={0};

2[M1]{u()} 2[M2]{uot()}+

S()

{uot()}=

S()

{ug()}

(7)

where {ug()} represents the Fourier transform offree field ground translation. The transformed equa-tions can be rewritten in the matrix notation as fol-lows:

K()

U()

=P()

(8)

The Fourier transformed dynamic stiffness ma-trix and the displacement vector consisting of struc-tural and foundation displacement components take

the following form:

K()

=

2[M]+i[C]+[K] 2[M][T]

2[M1]S()

2[M2]

U()

=

{u()}{uot()}

(9)

where {u()} is the Fourier transform of the {u(t)}.For harmonic excitation of frequency the dynamic

stiffness matrixS()

of the foundation defined

as the ratio of the amplitude of the applied loadP()

to that resulting displacement {uot()} is

written as

S()

=

Khx() 0 Khxrx() 0 00 Khy() 0 Khyry() 0

Khxrx() 0 Krx() 0 00 Khyry() 0 Kry() 00 0 0 0 Kt()

(10)

with Reissners dimensionless frequency parameterao = r/cs (shear-wave velocity cs). The numericalapproximations for the impedance functions used inthis study are taken from Veletsos and Wei (1971)for lateral and rocking vibration and from Veletsosand Nair (1974) for torsional vibration. The founda-tion impedances can be obtained from the solution ofa mixed boundary value problem in elastodynamicsand are generally functions of soil properties, founda-tion size and exciting frequency. The values taken by

the coupling terms are minor, above all for the usualvalues of Poissons ratio of soil between 0.3 and 0.5.For this reason, the terms Khxrx and Khyry are dis-regarded for superficial foundations (Schmid et al.,1988; Siefert, 1996). Assuming exterior terms of thediagonal of the dynamic stiffness matrix, which in-troduces neglible errors for most practical purposes,the impedance functions are described for the terms

of the main diagonal of the matrixS()

as

Kj =Ksj(kj(ao) +iaocj(ao))(1 + 2i) (11)

The internal damping of soil is also taken into con-sideration and is characterized by the damping ratio. In this complex variable notation kj representsthe dimensionless spring coefficient and cj repre-sents the corresponding damping coefficient depend-ing on ao and the Poissons ratio . The real partsof the impedance functions signify force componentsin phase with the displacements and can be termeddynamic stiffness for the foundation. On the otherhand, the imaginary parts are force components in

phase with the velocities and can be interpreted asenergy dissipation by radiation of waves away fromthe foundation into the soil. Therefore, they may betermed foundation damping coefficients.

Static stiffness coefficients may be defined asKSj for each degree of freedom or mode (for in-dices: hx and hy, horizontal displacements along thex and y axes, and for indices rx, ry and t, rotationsalong the x, y and z axes, respectively). It is notedthat Khx= Khy = 8Gr/(2-) is the static horizontalforce necessary to produce a unit horizontal displace-

148


7/15

CELEBI, GUNDUZ

ment of the disk with no restriction on the value ofthe resulting rotation, Krx = Kry = 8Gr

3/3(1-)is the static rocking moment necessary to rotate the

disk through a unit angle with no restriction on thevalue of the resulting horizontal displacement, Kt =16c2sr

3/3 is the static torque necessary to twist thedisk through a unit angle where denotes the massdensity of soil. The SSI relationships given by Eq.(8) were developed for a massless circular rigid disk.In this study, the research has been applied to a rect-angular foundation in an approximate way by useof equivalent values for the radius of the rigid diskro such that the resulting static stiffness coefficientsare the same as those corresponding to the rectan-gular foundation proposed by Thomson and Kobori

(1963).

Modal Analysis

Based on the assumption that the fixed-base struc-ture possesses classical normal modes, the structuralresponse in the complex frequency domain can beexpressed in terms of the mode shapes as

{u()} =

{ux()}

{uy()}

{u()}

= []{q()} (12)

where {q()}is the column vector of the Fouriertransform of the modal displacements and [] is thematrix consisting of normal mode shapes of the fixed-base structure as

{q()} =

q1()

q2()

.

.

q3N()

(3N1)

;

[] =

[xx] [xy] [x]

[yx] [yy] [y]

[x] [y ] [ ]

(3N3N)

(13)

Substituting Eq. (12) intoK()

U()

=

P()

equations of motion of a building system

with a rigid foundation in the frequency domain and

applying the orthogonality conditions, Eq. (8), when

the first row is premultiplied by []T

, introduces

KH()

UH()

=PH()

(14a)

and also note that

KH()

=

[H()]

2[]

2[]T

S()

2[M2]

UH()

=

{q()}

{uot()}

(14b)

where [] represents the modal participation matrixdefined as

[] = []T

[M][T] (15)

A diagonal matrix whose elements of modalstructural transfer functions and typical element cor-responding to the kth mode are written as

[H()] =

H1() 0 0 0

0 H2() 0 0

0 0 . 0

0 0 0 H3N()

3N3N

(16)

Hk() = 2+2ikk+

2k k= 1, 2, 3..............3N

The modal displacement of the structure can then

be written in terms of foundation deformations fromthe first row of Eq. (14a) as given below; the fre-quency variation of the configuration vector can besubsequently computed, by summing up modal re-sponses, as

{q()} =2[H()]1

[]{uot()} (17)

Substituting Eq. (17) in the second row of Eq.(14b), the foundation displacement components canbe obtained from the following equation:

149


8/15

CELEBI, GUNDUZ

{uot()} = D()1

S(){ug()} ugx = 0(18)

in which the total dynamic stiffness matrix,D()

,

is expressed as

D()

=S()

2([M2] +

2[]T

[H()]1

[])

(19)

After Eq. (18) is solved for the foundation dis-placement vector, the modal structural response in

the complex frequency domain can be calculated ac-cording to Eq. (17). Finally, the structural displace-ments of each floor in the frequency domain can becalculated by the mode superposition relations as

{u()} =

{ux()}

{uy()}

{u()}

= []{q()} (20)

The time-domain solution can be obtainedthrough the inverse Fourier transform of Eq. (21)

as follows:

{u(t)} = 1

2

{u()}eitd (21)

With the above computations, a simplified anal-ysis method performed by discrete Fourier trans-form techniques using the fast Fourier transform al-gorithm based on modal decomposition with all vi-bration modes to solve the interaction problem maybe summarized in the following analysis procedure:

Defining the structural and soil properties withan earthquake ground acceleration record,

Step 1. Modal analysis on the superstructure;

a) Solve the eigenvalue problem, determine thenatural frequency and mode shapes.

[M], [K] diag2k, [](k= 1....3N).

b) Evaluate the kinematic transfer matrix fromEq. (3) [T].

c) Set up total foundation mass matrices from Eq.(6) [M1], [M2].

d) Determine the modal participation matrixfrom Eq. (15) [].

Step 2. Frequency domain analysis at each fre-quency ();

a) Fourier transform of ground acceleration,[ug(t)] [ug()].

b) Determine the modal structural transfer func-tions from Eq. (16) [H()].

c) Compute the static soil spring constants KSj (for j = hx, hy, rx, ry and t).

d) Define the foundation impedance functions de-pending on soil properties (,,, and G) fromEq. (11) Kj =Ksj(kj(ao)+iaocj(ao))(1+2i).

e) Set up a total dynamic stiffness matrix from

Eq. (19) D()

.

f) Calculate the translational, rotational and tor-

sional displacements of the foundation fromEq. (18) {uot()}.

g) Compute the modal displacements for eachfloor from Eq. (17) {q()}.

h) Define the structural displacements from Eq.(20) {u()}.

Step 3. Time domain analysis;

a) Determine the displacement response by using

the inverse Fourier transform from Eq. (21) {u(t)}.

b) Calculate the design story shear forces.

c) Design of members.

It is clear that the above design procedure is sim-ilar to that for the analysis of a fixed base structurewith the addition of steps 2c-2f and the alterationof step 2g to include the deformability of the soilmedium.

150


9/15

CELEBI, GUNDUZ

Numerical Analysis

The application of the proposed method of analy-

sis including all modes of vibration in the calcula-tions to evaluate the dynamic SSI effects on the 3-dimensional buildings with eccentricity in only onedirection is considered. The configuration of the soil-structure system is shown in Figure 2. The mainparameters of the dynamical structure foundationmodel and properties of the soil are summarizedin this figure. The monosymmetric 2-story build-ing resting on homogeneous elastic soil through arigid square foundation consists of reinforced con-crete frames joined at each floor level by a rigiddiaphragm. The floors, including the foundation,were assumed to be identically rectangular, with thelength of the building d = 12 m and the width of thebuilding b = 12 m. In this example, the center ofstiffness is assumed to lie at eccentricity e = 1.0 mfrom the center of mass. It has been assumed thatthe floor slab and the foundation mat have the sameeccentricity along the x axis in a direction perpendic-ular to the input motion, and no eccentricity alongthe y axis. The mass mi = 50 t and torsional radiusof gyration ri = 5 m at each floor level (i = 1,2) arethe same. The rigid foundation mat is idealized asa circular plate of radius ro = 5 m and its mass alsotaken to be mo = 50 t. It should be noted that the

radius of the base mass is taken as the radius of acircle having the same area as the plane of the floor.

It is considered that the structure has the sametranslational stiffness in the x and y directions, andthat the SSI translational and rocking stiffnesses arerespectively the same in both horizontal directions.The story stiffnesses of these frames are KyAi, KxCiand KxBi = 26500 kN/m, and the story heights arehi = 3 m, also the same for all floors (i = 1,2). Thedamping ratio of the superstructure in each mode ofvibration is taken as 5%.

The viscoelastic foundation medium is assumed

to have a density of 20 kN/m3

and a Poissons ratioof= 0.33, and the material damping ratio is =0.05. To indicate the significance of SSI effects onstructural response, the shear wave velocity (cs) ofelastic half-space material was selected in a range of300 cs 1500 m/s. The upper limit of 1500 m/sfor the shear wave velocity of the soil may be as-sumed to refer to a very stiff ground condition (fixedbase). The lower limit of 300 m/s for the shear ve-locity is chosen for defining soft soil conditions.

For a parametric study, the key parameters con-

trolling the dynamic structural response includingthe effects of foundation interaction are chosen asthe ratio, m/mo, of the superstructure mass to the

mass of the foundation, the ratio, H/r, of the totalheight of the structure to the radius of the founda-tion base taken here as the same as the floor slabs,and the ratio, e/r, of the eccentricity to the radius ofthe foundation base. The practical range of variablesconsidered in this study is taken as 0.5 m/mo 3 for the mass ratio, 0.05 e/r 0.25 for the ec-centricity ratio, and 0.5 H/r 2 for the heightratio.

The responses of this building foundation systemto the torsional effects were obtained when it wassubjected to the Erzincan, 1992, earthquake (E-W

component, M = 6.8) as shown in Figure 3 with peakacceleration of 0.5 g as the free field ground motionin a direction perpendicular to the direction of theeccentricity.

In Figure 4, the variation of lateral deflection,rocking and twisting (torsional) components of thefoundation base for different shear wave velocitiesof soil are plotted as a function of time and com-pared to the corresponding values for a fixed basestructure. The maximum values of the foundationdisplacements for Erzincan, 1992, excitations are uo= 0.028 m, o = 8.85 103rad and o = 4.25 103rad for a shear velocity of cs= 300 m/s (soft soilcondition) in the case of intermediate high buildingswith moderate eccentricity where m/mo = 1, H/r =1.2, and e/r = 0.2. However, the corresponding val-ues are uo = 0.0025 m, o = 9 104rad and o= 4.2 104rad for relatively rigid based buildings,in which case the structure is considered to be sup-ported by the soil with a shear wave velocity of cs =1500 m/s, as shown in Figure 4. It may be noted thatthe deformability of the foundation soil significantlyinfluences the foundation displacements. Therefore,the SSI effects on the structural response are shownto be more important when the soil becomes softer,

in soil with a lower value of cs.Furthermore, in order to demonstrate the appli-

cation of this modal superposition method to prac-tical problems, a detailed parametric analysis of thesame idealized 3-dimensional model has been solvedwith various controlling parameters (m/mo, H/r, ande/r) to estimate the dynamic behavior of the soilstructure system for 2 ground conditions, for stiffersoil conditions, cs = 1500 m/s, and for softer soilconditions, cs = 500 m/s, as shown in Figures 5-7,respectively.

151


10/15

CELEBI, GUNDUZ

Soil Parameters:

Description Symbol Value

Poissons ratio 0.33Mass density 20 kN/m3

Material damping 0.05

Shear velocity cs 1500, 1000, 700

500, 300 m/s

d

b

H

m2

m1

mo

h1

h2

z

y

x

Structural Parameters:

Description Symbol (i=1,2) Value

Floor mass mj 50 t

Frame

stiffness KyAi, KxCi, KxBi 26.5 MN/m

Foundation tostory height hi 3 m i

Modal dampingratio 0.05

Floor moment of

inertia Ii 120 tm2

Elasticity modulus E 28.5 GPa

Frame B

CM

Frame CFrame A

d

b

0.5d

0.5d

e

x

y

gyu&& (t)

Foundation Parameters:

Description Symbol Value

Mass mo 50 tLength &width b, d 12 m, 12 m

Moment of inertia Io 120 tm2

Ground motion record gyu&& (t)

Erzincan 1992 (E-W component)

Figure 2. Configuration of 2-story building and model parameters for numerical examples.

152


11/15

CELEBI, GUNDUZ

0.4

0.3

0.2

0.10

-0.1

-0.2

-0.3

-0.4

-0.5

0.6

Acceleration,g

Erzincan 1992 E-WM=6.8, max.0.5g

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (s)

Figure 3. Time history acceleration of free-field inputmotion at the surface.

TwistingDisplacement(rad)

0.030.0250.02

0.0150.01

0.0050

-0.005-0.01

-0.015-0.02

-0.025-0.03L

ateralDisplacement(m)

RockingDisplacement(rad)

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (s)

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (s)

cs=1500m/scs=1000m/scs=700m/scs=500m/scs=300m/s

0.08

0.0650.05

0.0350.02

0.005-0.01

-0.025-0.04

-0.055-0.07

-0.085-0.1

0.040.03250.025

0.01750.01

0.0025-0.005

-0.0125-0.02

-0.0275-0.035

-0.0425-0.05

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (s)



Figure 4. Variation of the displacement components ofthe foundation base depending shear velocityin the case of m/mo = 1, H/r = 1.2 and e/r =0.2.

Cs=500 m/s

m/mo=0.5m/mo=1m/mo=2m/mo=3

0 1 2 3 4 5 6 7 8 9 10 11 12Time (s)

0.175

0.1375

0.1

0.0625

0.025-0.0125

-0.05

-0.0875

-0.125

-0.1625

-0.2

x10-3 Cs=1500 m/s8

6.55

3.52

0.5-1

-2.5-4

-5.5-7

-8.5-10

LateralDisplacemen

t(m)

0 1 2 3 4 5 6 7 8 9 10 11 12Time (s)

m/mo=0.5m/mo=1m/mo=2m/mo=3

LateralDispla

cement(m)

Figure 5. Variation of the lateral displacement compo-nents of foundation base depending mass ratiofor shear wave velocity of 1500 m/s and 500m/s in the case of H/r = 1.2 and e/r = 0.2.

Cs=1500 m/s0.02

0.015

0.01

0.005

0

-0.005

-0.01

-0.015

-0.02

-0.025

0.275

0.2

0.125

0.05-0.025

-0.1

-0.175

-0.25

-0.325

-0.4

Cs=500 m/s



0 1 2 3 4 5 6 7 8 9 10 11 12Time (s)

H/r=0.5H/r=1H/r=1.5H/r=2

0 1 2 3 4 5 6 7 8 9 1 0 1 1 12Time (s)

H/r=0.5H/r=1H/r=1.5H/r=2

Figure 6. Variation of the rocking displacement compo-nents of the foundation base depending onheight ratio for a shear wave velocity of 1500m/s and 500 m/s in the case of m/mo = 1 ande/r = 0.2.

153


12/15

CELEBI, GUNDUZ

Cs=500 m/s

e/r=0.10e/r=0.15e/r=0.20e/r=0.25

0 1 2 3 4 5 6 7 8 9 10 11 12Time (s)

0.035

0.0275

0.02

0.0125

0.005-0.0025

-0.01

-0.0175

-0.025

-0.0325

-0.04TorsionalDisplacement(rad)

Cs=1500 m/sx10-3

e/r=0.10e/r=0.15e/r=0.20e/r=0.25

7

5.5

4

2.5

1

-0.5

-2

-3.5

-5

-6.5

-80 1 2 3 4 5 6 7 8 9 10 11 12

Time (s)

TorsionalDisplaceme

nt(rad)

Figure 7. Variation of the torsional displacement com-ponents of foundation base depending on theeccentricity ratio for a shear wave velocity of1500 m/s and 500 m/s in the case of m/mo =1 and H/r = 1.2.

Four different values of height to gyration ra-dius ratio (h/r), eccentricity to gyration radius ratio(e/r), and mass of the structure to mass of the foun-dation ratio (m/mo), depending on the shear wavevelocity of the soil medium, are considered.

It should be noted that an apparent increase oc-curs in the foundation displacement components asthe shear wave velocity of the soil medium decreases.From those time-domain responses, it is observedthat in rigid and soft soil conditions, an obvious in-crease occurs in the horizontal displacement of thefoundation as the mass ratio, m/mo increases, as

shown in Figure 5. For instance, the maximum valueof the foundation lateral displacement is 2.25 103m in the case of m/mo = 1 for the rigid foun-dation medium, whereas the corresponding value is9.35 10 3m in the case of m/mo = 3. In thissituation, the foundation mass is assumed to be rea-sonably small compared to the mass of the super-structure.

Additionally, the lateral displacement of thefoundation based on soil with cs = 500 m/s increasesto almost 200 times the lateral displacement of thefoundation based on soil with a shear velocity of cs=

1500 m/s, for the ratio of m/mo = 3. Furthermore,the rocking rotation of the foundation is affected byany change in the value of the height ratio, H/r, es-

pecially when the soil is softer, as shown in Figure 6.It modifies the rocking rotation of the foundation,decreases for short, squat structures and increasesfor tall, slender structures. The smaller the valueof cs and the greater the value of the height ratio,H/r, the more pronounced the interaction effects be-come on foundation rotation. For instance, for H/r= 2 and cs = 1500 m/s, the peak of the rotationangle is approximately 0.022 rad, whereas for cs =500 m/s this value is 0.25 rad. For squat structures,these rocking displacements decrease by up to 1.5 10 3rad and 0.022 rad for rigid and soft soil condi-

tions, respectively.From Figure 7 it can be seen that the peak values

of torsional displacement of the foundation increasedsignificantly in the case of intermediate high build-ings with a moderate mass ratio for low soil foun-dation stiffness when compared to those obtainedfor rigidly based structures as the eccentricity ra-tio, e/r, increases. The maximum torsional rotationsof the foundation obtained in this analysis are ap-proximately 0.00725 rad and 0.0385 rad for systemshaving a shear wave velocity of soil with cs = 1500m/s and cs = 500 m/s, respectively, when the ec-centricity ratio is taken as e/r = 0.25. Furthermore,it should be noted that when the eccentricity ratioincreased from 0.2 to 0.25, the torsional rotationsalso increased by 2.5 times and 4 times for buildingsfounded on stiff and soft soils, respectively.

Conclusions

In this study a simplified method of analysis is pre-sented using modal decomposition considering allmodes of vibration in the calculation to obtain thestructural responses of torsionally asymmetric build-

ings, including soil-structure interaction effects in thefrequency domain, by using the fast Fourier trans-form. Applying this algorithm, the advantage of clas-sical normal mode approximation is used for almostall vibration modes, and the interaction problem issolved easily and effectively within the framework ofthe Fourier-transformed frequency domain analysisfor a fixed base structure. The matrix formulation ofthis method produces accurate approximation withless computational effort, despite using frequency de-pendent impedance functions.

The results presented in this paper indicate that

154


13/15

CELEBI, GUNDUZ

the earthquake response of a soil-torsionally cou-pled structure interaction can be signicantly differentfrom that calculated with a fixed base model. This

work has been based on the assumptions that thefoundation is supported at the surface of a viscoelas-tic half-space and that the superstructure respondswithin the elastic range. From these time-domain re-sponses, it has been shown that an obvious increaseoccurs in the horizontal and rocking displacementsof the foundation as the predefined mass ratio andheight ratio increase, respectively, for soft soil condi-tions compared to the associated fixed-base system.

This detailed parametric study shows that theeffects of deformability of the foundation mediumcause the dynamic behavior of the structure foun-

dation system to differ, particularly when the shearwave velocity representing the soil stiffness is lowerthan 1000 m/s.

For a medium highrise building with moderateor large eccentricity, increased torsional loadingsmust be taken into account for the combined lateral-torsional response for such structures supported ona moderately and very flexible foundation medium.

Nomenclature

Roman Symbols

ao Reissner parametercj foundation damping coefficients for each

degree of freedom - or the mode jcs velocity of shear wave in the soil[C] viscous damping matrix of structureD()

total dynamic stiffness matrix

e structural (static) eccentricityG shear modulus of soil{h} column vector of heights[H()] modal structural transfer functions

Itx, Ity the total of the mass moment of inertiaof the floors and the foundation mat withrespect to the x and y axes

kj dimensionless spring coefficients for modej

Khx,Khy static translational stiffness of foundationin the x and y directions, respectively

Krx , Kry static rocking moment of foundationabout the x and y axes, respectively

Kt static torque of foundationKj foundation impedance functions for the

mode j

KSj static stiffness for the half-space forthe mode j

[K] stiffness matrix of structure

mi, mo floor and foundation massMox(t),Moy(t)

overturning moments at foundation-soil interface in the x and y directions

[mx], [my] mass matrices consisting of floormasses with respect to the x and yaxes

[m] mass polar moment of inertia, aboutthe z axis of the floor mass

[M] mass matrix of structurePox(t),Poy(t)

horizontal soil-structure interactionforces in the x and y directions, re-spectively

P()

frequency dependent external force

vector{q()} column vector of Fourier transform of

the modal displacementsr, ro radius of floor and foundation (base)

disk, respectivelyS()

dynamic-stiffness matrix of the foun-

dation defined as the ratio of the am-plitude of the applied load

P()

To(t) torsional soil-structure interaction

moment[T] kinematic transfer matrix

uxi, uyi horizontal displacements ofith floor inthe x and y directions, respectively

ui rotation about vertical axisuox, uoy horizontal translations of the founda-

tion with respect to the x and y axesugx(t),ugy(t) free field ground acceleration mea-

sured at the surface{u(t)} column vectors of the structural dis-

placements relative to the rigid foun-dation

{u()} Fourier transform of the{u(t)}{ug(t)} the rotational acceleration of the base

of the building about the vertical axis{uo1(t)} the acceleration vector of the rigid

foundation motion{uot()} Fourier transform of the total dis-

placement vector of the rigid founda-tion in addition to the free field motion

Greek Symbols

internal damping ratio of soil[] normal mode matrix

155


14/15

CELEBI, GUNDUZ

ox, oy rocking rotations of the foundation inthe directions of the x and y axes, re-spectively

gx(t), gy(t) the rotational acceleration of the baseof the building along horizontal axes

[] modal participation matrix

Poissons ratio of soil soil mass densityo twist of the foundation

circular frequency of excitationk natural frequency ofkth mode

References

Aydnoglu, M.N., Consistent Formulation of Di-rect and Substructure Method in Nonlinear Soil-Structure Interaction, Soil Dynamics and Earth-quake Engineering, 12, 403-410, 1993.

Chandler, M. and Hutchinson, G.L., TorsionalCoupling Effects in the Earthquake Response ofAsymmetric Buildings, Engineering Structures, 8,

222-236, 1986.

Chandler, A.M. and Sikaroudi H., Structure-Foundation Interaction in the Earthquake Responseof Torsionally Asymmetric Buildings, Soil Dynam-ics and Earthquake Engineering, 11, 1-16, 1992.

Chopra, A.K. and Guiterrez, J.A., Earthquake Re-sponse Analysis of Multistorey Buildings IncludingFoundation Interaction, Earthquake Engineeringand Structural Dynamics, 3, 65-77, 1974.

Cruz, E.F. and Chopra, A.K., Simplified Proce-dures for Earthquake Analysis of Buildings, Jour-nal of Structural Engineering, 112, 461-480, 1986.

Celebi E. and Gunduz A.N., Efficient Modal Analy-sis of Asymmetric Multi-Storey Buildings IncludingSoil-Structure Interaction, Proc. 4th InternationalColloquium on Computation of Shell and SpatialStructures, Crete, Greece, 5-7 June, 2000.

Celebi E. and Gunduz A.N., Simplified Formula-tions for Soil-Structure Interaction Analysis of Ir-regular High Rise Buildings under Seismic Loading,Proc. 2nd European Conference on ComputationalMechanics, Cracow, Poland, June 26-29, 2001.

Ghaffer-Zadeh, M. and Chapel, F., Frequency In-dependent Impedances of Soil-Structure Systems in

Horizontal and Rocking Modes, Earthquake Engi-neering and Structural Dynamics,11, 523-540, 1983.

Hejal, R. and Chopra, A.K., Earthquake Anal-ysis of a Class of Torsionally-Coupled Buildings,Earthquake Engineering and Structural Dynamics,18, 305-323, 1989.

Hejal, R. and Chopra, A.K., Earthquake Responseof Torsionally-Coupled Frame Buildings, Journalof Structural Engineering, 115, 834-851, 1989.

Humar, J.L. and Via, H., Dynamic Response Anal-ysis in the Frequency Domain, Earthquake Engi-neering and Structural Dynamics, 22, 1-12, 1993.

Liu, S.C. and Fagel, L.W., Earthquake Interactionby Fast Fourier Transform, Journal of EngineeringMechanical Division, ASCE, 97, 1223-1237, 1971.

Meek, J.W. and Wolf J.P., Cone Models for Homo-geneous Soil, Journal of Geotechnical Engineering,ASCE 118, 667-685, 1992.

Richart, F.E., Hall, J.R. and Woods, R.D., Vibra-

tion of Soils and Foundation, Prentice Hall, Inc.,Englewood Cliffs, New Jersey, 1970.

Schmidt, G., Willms, G., Huh, Y. and Gibhard,M., SSI 2D/3D Soil-Structure Interaction, SFB 151,Ruhr Universitat, Bochum, Germany, 1988.

Sieffert, J.G. and Cevaer F., Handbook ofImpedance Functions, Surface Foundations, QuestEditions, Nantes, France, 1991.

Sieffert, J.G., What Ways for the Engineer in Prac-tice, Seismic Design of Foundations, 11th WorldConference on Earthquake Engineering, Acapulco:Elsevier, 1996.

Sivakumaran, K.S., Seismic Analysis of Mono-Symmetric Multistorey Buildings Including Founda-tion Interaction, Journal of Computer and Struc-tures, 36, 99-107, 1990.

Sivakumaran, K.S. and Balendra, T., Seismic Re-sponse of Multistorey Buildings Including Foun-dation Interaction and P- effects, EngineeringStructures, 9, 277-284, 1987.

Sivakumaran, K.S., Lin, M.S. and Karasudhi,P., Seismic Analysis of Asymmetric Building-Foundation Systems, Journal of Computers andStructures, 43, 1091-1103, 1992.

Thomson, W.T. and Kobori, T., Dynamic Compli-

ance of Rectangular Foundations on an Elastic Half-Space, Journal of Applied Mechanics, 30, 579-584,1963.

Tsai, N.C., Niehoff, D., Swatta M. and Hadjian,A.H., The Use of Frequency-Independent Soil-Structure Interaction Parameters, Nuclear Engi-neering and Design, 31, 168-183, 1974.

Vaidya, N.R., Bazan-Zurita, E. and Rizzo P.C., Onthe Validity of Modal Superposition Technique inSoil-Structure Interaction Analysis, Proc. 3rd U.S.National Conf. Earthquake Eng., Charleston, SC, 1,683-692, 1986.

156


15/15

CELEBI, GUNDUZ

Veletsos, A.S. and Wei, Y.T., Lateral and RockingVibrations of Footings, Journal of Soil MechanicsFoundation Division, ASCE, 97 (SM 9),1127-1248,

1971.Veletsos, A.S. and Nair, V.V.D., Torsional Vi-bration of Viscoelastic Foundations, Journal ofGeotechnical Engineering Division ASCE, 100, 225-246, 1974.

Wolf J.P. and Song C., Finite Element Modelling ofUnbounded Media, John Wiley & Sons, West Sus-sex, UK, 1996.

Wolf J.P., Spring-Dashpot-Mass Models for Foun-dation Vibrations, Earthquake Engineering andStructural Dynamics, 26, 931-949, 1997.

Wu, W.H. and Smith, A., Efficient Modal Analy-sis for Structures with Soil-Structure Interaction,Earthquake Engineering and Structural Dynamics,24, 283-299, 1995.

Wu, W.H., Equivalent Fixed-Base Models for Soil-Structure Interaction Systems, Soil Dynamics andEarthquake Engineering, 16, 323-336, 1997.

157

Documents

Torsionally Coupled Buildings___Soil Structure Interaction