538 soil dynamics and earthquake engineering - wit press€¦ · soil dynamics and earthquake...

Effects of soil-structure interaction on the

seismic response of cable-supported bridges

FLBetti*, A.M. Abdel-GhafW

"Department of Civil Engineering and Engineering

Mechanics, Columbia University, New York, NY

i> Department of Civil Engineering, University of

Southern California, Los Angeles, CA 90089, USA

ABSTRACT

The dynamic soil-structure interaction effects on the response of long-span

cable-supported bridges subjected to traveling seismic waves are presented

in this study. The foundation system is represented by multiple embedded

cassion foundations and the interaction analysis includes both the inter-

action between the foundations and the surrounding soil and the cross-interaction among adjacent foundations. To illustrate the potential imple-

mentation of the analysis, a numerical example is presented in which the

dynamic response of the Vincent-Thomas suspension bridge (Long Beach,

CA) subjected to the 1987 Whittier earthquake is investigated. Although

both kinematic and inertial effects are included in the general procedure,only the kinematic effects of the soil-structure interaction are considered

in the analysis of the test case.

Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509

538 Soil Dynamics and Earthquake Engineering

INTRODUCTION

Cable-supported bridges can be categorized as classical suspension bridges,

(which effectively cover the center span range of 500 m. to 3000 m.),

and contemporary cable-stayed bridges (which effectively cover the center

span range of 200 m. to 1000 m.) and they are increasing in number,

in popularity and in span length in seismically active areas all over the

world.

In calculating the seismic response of such bridges, the assumption

of uniform ground motion at the supports of these horizontally extended

structures can not be considered valid [4,7,8,9]. In fact, the bridge may

be long with respect to the wavelengths of the ground motion in the fre-

quency range of importance to its earthquake response. In addition, the

soil conditions at the foundations sites can be extremely different, con-

tributing to the spatial variability of the ground motion. Thus, different

portions of the bridge can be subjected to significantly different excita-

tion leading to a complicated dynamic interaction problem between the

three-dimensional ground motion inputs and the bridge superstructure.

Since the dynamic soil-structure interaction effect upon structural

response was recognized as being significant, many studies [1,3,5,6,10-13]

have been made and are now in progress in order to analyse the gen-eral dynamic soil-bridge interaction problem during strong earthquake

motions. For example, cable-stayed bridges are relatively stiff structures

and the interaction with the surrounding soil deeply affects the structural

characteristics. However, most of these studies employ some simplify-

ing assumptions addressing the embedment of the foundations and thecross-interaction between foundations through the soil, the traveling wave

effects, the nonlinearity and three-dimensionality of both the structure and

the surrounding soils, etc., focusing their attention either on the behavior

of specific structural components or on very simplified analyses.In this study, an analytical-numerical formulation, based on the Sub-

structure Approach, is used for the three-dimensional analysis of the

soil-foundation-bridge system. This analysis is intended to be an ini-tial step toward a better understanding of the dynamic performance of

cable-supported bridges under earthquake excitation, considering the spa-

tial variability of the ground motion and the complex phenomena of soil-

bridge interaction. The graphic illustration of this analysis is presented

in Fig. 1 where a three-dimensional model of a cable-supported bridge

is subjected to excitation of incoming seismic waves. The earthquake

excitation is represented by arbitrarily inclined three-dimensional plane


Soil Dynamics and Earthquake Engineering 539

waves, propagating in an homogeneous vis co-elastic half-space. In this

way, the spatial variation of the ground motion due to propagation of the

seismic waves can be taken into consideration and by combining the soil-

foundation system with the bridge superstructure, the dynamic response

of the bridge can be easily performed. For the superstructure, a finite el-

ement analysis is performed, using the procedure developed by Niazy [8],

Nazmy and Abdel-Ghaffar [7], while the soil-foundation system is ana-

lyzed using a Boundary Element formulation of the Substructure Deletion

Method, proposed by Betti and Abdel-Ghaffar [2].

SOIL-STRUCTURE INTERACTION ANALYSIS

The soil-bridge interaction effects are investigated by partitioning the en-

tire soil-foundation-structure system into two subsystems: 1) the super-

structure and 2) the soil-foundation system. First, each sub-system is

analyzed independently and then the individual responses are combined

so as to satisfy the displacement continuity and the equilibrium conditions

at the interface. Based on the assumption that the soil and the material

of the structural members remain elastic during the earthquake response

and that only small amplitude vibrations are of concern, it is convenient,

first, to perform the analysis of the soil-bridge interaction in the frequency

domain and, then, to obtain the time histories of the response, using the

Fourier synthesis technique.

Analysis of the SuperstructureThe bridge superstructure can be analysed by using a discrete Finite El-

ement model. In the frequency domain, the equations of motion of the

three-dimensional vibrations of the bridge ( with N degrees of freedom ),

when subjected to earthquake excitation, can be written as:

(1)

where:w : frequency of vibration (rad/sec),

Ua(w) : total displacements of the nodes of the superstructure,

U&(w) : total displacements of the connecting nodes between the super-

structure and the foundation system,R&(w) : interaction forces that the surrounding soil and the foundations

exert on the bridge superstructure.



The matrices [M], [C], and [K] represent the structural mass, damping

and stiffness matrices, respectively.

To take into account the multiple-support excitation, the nodal dis-

placements of the bridge superstructure Ua(w) can be decomposed into a

quasi- ( pseudo- ) static component of the displacement, U%**(w), which

includes the effects of the different displacements at the supports,

, (2)

and into a vibrational ( dynamic ) component of the displacement,

) (3)

with:[$] : the matrix of the eigenvectors for the fixed base bridge super-

structure,

77(0;) : is the vector of the generalized coordinates

[B] : is the " displacement influence matrix " for the superstructure.

Substituting equations (2) and (3) into equation (1) leads to two sets

of coupled equations:

77(0,) = [L]U»(u;), (4)

), (5)

where [L] and [G] are two matrices that depend on the properties (mass,

stiffness, damping) of the bridge superstructure.

Analysis of the Soil-Foundation System

For the analysis of the soil-foundation system, the assumption of rigid

embedded foundations has been used. Through this assumption, the nodaldisplacements of the points at the interface between the superstructure

and the foundation blocks, U&(u;), can be expressed as functions of the

foundations rigid-body motion vector Uo(w):

Ui(w) = [AjUoH (6)

where [A] is the rigid-body influence matrix.The force vector F*°**(w), which represents the forces and moments

that the soil exerts on the rigid foundations, can be expressed as a functionof the foundation displacement vector Uo(u>) and of the foundation input

motion U*o(w):

U%(w)} (7)



where [#oH] is the frequency-dependent impedance matrix for the mul-

tiple embedded rigid-foundation system.For the determination of the impedance matrix [Ko(w)] for the embed-

ded foundations, an alternative formulation of the Substructure Deletion

Method has been used [2]. The proposed method starts from the consid-

eration that the solution of the radiation problem in case of embedded

foundations can be derived from the solution of a flat, homogeneous half-

space (exterior problem) and from the analysis of the excavated portion

of soil (interior problem). In this approach, a Boundary Element formula-

tion is used for the solution of both the interior and the exterior problem.

For the analysis of the finite inclusion of soil removed during the exca-

vation, fundamental solutions for the infinite space have been used while

half-space Green's functions have been applied in the analysis of the flat

homogeneous visco-elastic half-space. To show the validity of the method,

the vertical response of an embedded foundation to a vertical sinusoidal

concentrated force is presented in Fig. 2 and the results show excellent

agreement with those obtained in previous studies [2].Once the impedance matrix for the embedded foundation system has

been obtained, the foundation input motion can be easily obtained as [2]:

* ' (8)

where:[K] : the impedance matrix for the embedded contact surface be-

tween the foundation and the surrounding soil,

U0(x) : the free-field displacement vector at point x,

TJ(x) : the free-field traction vector at point x .It is through the foundation input motion that different types of impinging

seismic waves are included into the analysis.

Analysis of the Global SystemOnce the two subsystems (superstructure and soil-foundation system)

have been analysed, the entire soil-foundation-bridge superstructure can

be recomposed by applying the equilibrium and compatibility equations

at the interface.The equation of motion of the foundation system can be written in

the form:-w=MoUow) = F""(w) + F""(w) (9)



where [Mo] is the mass matrix of the foundation system, while F****(w) isthe vector of the forces and moments exerted on the foundations by the

superstructure,

F""» = -[Af R»(w) = -[Af[G][A]Co(w) (10).

Substituting equations (7) and (10) into (9) leads to the final form of

the equations of motion for the foundation blocks:

[-w»[Mo] + [Af [G][A] + [tfoH]]UoH = [*o(w)]U*o(w) (11)

which represents a system of linear algebraic equations in the unknownsUo(w). Once the unknown displacement components Uo(w) have been

computed, it will be possible to evaluate the dynamic response in the

frequency domain at any point of the bridge, which can be transformed,

using the Fourier synthesis technique, into the time history of the struc-

tural response.

BRIDGE AND SOIL MODEL

To represent the superstructure, a three-dimensional finite element model

of an existing suspension bridge (Vincent Thomas Bridge, Long Beach,CA) has been used (Fig. 1). The bridge has a central span of 457.2 m. andtwo lateral spans of 152.4 m.. The two towers (97.3 m. high) and the an-

chorage abutments support the central and side girders and are connected

to four embedded foundations, with base dimensions equal to 50 x 40 m.and depth of 30 m. Because of the geometry of the bridge structure, the

cross-interaction between adjacent foundations through the soil has been

included only for the side spans. Two different models of the bridge have

been analyzed. In the first model, the foundations of the bridge have been

removed and no soil-structure interaction effects have been considered. In

the second model, the suspension bridge is connected to the four embed-

ded foundations. To emphasize the effect of the spatial variation of the

ground motion and of the soil-structure interaction, the bridge response,

in terms of the three orthogonal components of the displacements, has

been computed at three different locations: 1) at the mid-point of the

central span, 2) at the top of a tower, and 3) at one-forth of the central

span. Fifty modes of vibration (with natural frequencies varying from0.17 Hz. to 1.5 Hz.) have been selected to represent the dynamic response

of the bridge superstructure.



The soil is represented by a uniform elastic half-space with shear wave

velocity 0 equal to 1000 m./sec.; such a value of ft represents the average

soil conditions in the Los Angeles basin. Moreover, this type of hard soil

conditions reduced the effects of the inertial soil-structure interaction and

allow us to focus our attention on the traveling wave effects and on the

kinematic part of the soil-structure interaction. Three orthogonal com-

ponents of the ground motion recorded at a specific location during the

Whittier earthquake (CA, October 1987) have been used, to represent the

free-field ground motion. The external seismic excitation is represented

by incoming compressional and shear waves, with different angles of inci-

dence in both the vertical and horizontal planes. In this way, the effects

of the different motion at the supports and of the rocking and torsional

components of the foundation motion on the structural response have beenemphasized.

NUMERICAL EXAMPLES

In order to investigate the dynamic response of the bridge including the

effects of the soil-structure interaction and of the phase difference related

to the oblique incidence of earthquake waves, incident P- and SV-waves,

impinging either vertically or inclined (30° - 45°) in the vertical plane,

have been considered in this paper. Moreover, such waves also propagate

either parallel to the longitudinal axis of the bridge or inclined in the

horizontal plane at an angle of 45° with respect to that axis. The results

are shown in Tables 1 and 2, where the maximum displacements for thethree points under consideration are represented.

For the case of vertically incident P-waves, the foundations displace-ments are almost in phase and the effects of the kinematic soil-structure

interaction are negligible. This type of excitation excites mainly in-plane,

symmetric modes of vibration of the superstructure, which are attenuated

by the dead load of the bridge. When the angle of inclination differs fromthe vertical one, the different motion of the supports produces longitudinal

displacements of the bridge deck that were absent in the case of vertical

P-waves (Table 1). Anti-symmetric, vertical modes of vibrations are also

amplified. In addition, rocking components of the foundation motion, as-

sociated with the rotation about the transversal axis of the bridge, increasethe maximum vertical displacement at the midspan point of 23.6%. When

the seismic wave does not propagate parallel to the longitudinal axis of

the bridge, rocking components about the longitudinal axis and torsional

components of the motion of the foundations excite out-of-plane modes



of vibration, generating transversal components of the bridge response. Inthis case, the longitudinal and transversal components show percentual

differences of the order of almost 10% and 6%, respectively, while thevertical component of the deck response presents an increase of almost

22% with respect to the case where no soil-structure interaction effects

are considered.

For incoming SV-waves (Table 2), the effects of the soil-structure

interaction are strongly emphasized. Figs. 3 and 4 present the amplitude

of the transfer functions of the longitudinal and vertical displacements,

respectively, for the midspan point in the case of incident SV-waves. The

case of vertically incident SV-wave is very similar to the previous case of

vertical P-wave. The external excitation excites longitudinal and vertical

in-plane modes of vibration and no significant transversal displacement is

shown.

It is when the direction of propagation of the seismic wave is inclined

with respect to the vertical direction that the effects of the soil-bridge in-

teraction become extremely important. For inclined SV-wave, propagating

parallel to the longitudinal axis of the bridge, the interaction effects pro-

duce an increase of over 250% of the vertical deck response (Table 2) while

longitudinal displacements present differences in the order of 14%. Com-

paring the transfer functions of the vertical displacements at the midspanpoint (Fig. 4), it is possible to point out that the inclination of the seismicwaves produces an increase of almost 500% of the amplitude of the trans-

fer functions correspondent to the first vertical mode. Such increment is

drastically increased when the effects of the embedded foundations areincluded in the analysis. The differences in the vertical response of the

bridge superstructure are related to the rocking components of the foun-

dation motion; these components strongly excite the in-plane, symmetric

and anti-symmetric modes of vibration, as shown in Fig.4. Similar re-

sults are obtained in the case of vertically inclined SV-waves propagating

obliquely with respect to the longitudinal axis of the bridge (7 = 45°). The

ground motion will now have a transversal component which will excite

out-of-plane motion of the superstructure while the in-plane components

of the bridge response will present a slight decrease.

CONCLUSIONS

This study represents an attempt for a better understanding of the effects

of the soil-structure interaction and of the spatial variability of the ground

motion on the response of long-span cable-supported bridges subjected to



earthquake excitation. The results clearly show that the multiple-support

excitation and the soil-structure interaction greatly affect the seismic re-

sponse of the bridge superstructure. Symmetric, anti-symmetric in-plane

modes of bridge vibrations as well as out-of-plane modes are excited, de-

pending on the inclination of the seismic wave, and additional quasi-static

deformations have to be included. Incoming waves, inclined both in the

vertical and in the horizontal plane, generate rocking and torsional com-

ponents of the foundation motions that alter the bridge response. Because

of the importance of the rocking components of the foundation motion, it

is important to consider the embedment of the bridge foundations in theanalysis of the soil-foundation systems.

Both the effects of the multiple-support excitation and the effects of

the soil-structure interaction should be carefully considered in the earth-quake response analysis of such extended structures.

REFERENCES

1) Abdel-Ghaffar, A.M., and Trifunac, M.D.: "Antiplane Dynamic Soil-

Bridge Interaction for Incident Plane SH-Wave", Earthquake Engi-

neering and Structural Dynamics, Vol. 5, 1977, pp. 107-129.

2) Betti, R., and Abdel-Ghaffar, A.M.,: "Seismic Response of Embedded

Foundations Using a BEM Formulation of the Substructure Deletion

Method", submitted for publication to the Journal of Engineering

Mechanics, ASCE.

3) Grouse, C.B., Hushmand, B., and Martin, G.R.: "Dynamic Soil-Structure Interaction of a Single-Span Bridge", Earthquake Engineer-

ing and Structural Dynamics, Vol. 15, 1987, pp. 711-729.

4) Durate, R.T.: "Spatially Variable Ground Motion Models for Earth-

quake Design of Bridges and Other Extended Structures", Proceeding

of the Seventh World Conference of Earthquake Engineering, Turkey,1980, pp. 613-620.

5) Esquivel, J.A., and Sanchez-Sesma, F.J.: "Effects of Canyon Topog-

raphy on Dynamic Soil-Bridge Interaction for Incident Plane SH-

Waves", Proceedings of the Sixth World Conference on EarthquakeEngineering, India, 1976, pp. 153-160.

6) Gupta, S.P., and Kumar, A.: "Dynamic Response of Cable Stayed

Bridge Including Foundation Interaction Effect", Proceedings of the

Ninth World Conference on Earthquake Engineering, Tokyo-Kyoto,Japan, 1988. Vol. 6, pp. 501-506.



7) Nazmy, A.S. and Abdel-Ghaffar, A.M.: "Seismic Response Analysis

of Cable-Stayed Bridges Subjected to Uniform and Multiple-Support

Excitations", Report No. 87-Sm-l, Princeton University, May 1987.

8) Niazy, A.M.: "Seismic Performance Evaluation of Suspension

Bridges", Ph.D. Dissertation, Department of Civil Engineering, Uni-

versity of Southern California, Los Angeles, CA, 1991.9) Rubin, L.I., Abdel-Ghaffar, A.M. and Scanlan, R.H.: "Earthquake

Response of Long-Span Suspension Bridges", Report No. 83-SM-13,

Princeton University, Princeton, NJ, 1983.

10) Takemiya, H., Kadotani, T., Saeki, M. and Mori, A.: "Seismic De-

sign of Cable Stayed Three-Span Continuous Bridge with Emphasis

on Soil-Structure Interaction", Proceedings of the International Con-

ference on Cable-Stayed Bridges, Bangkok, Thailand, 1987.

11) Werner, S.D., Lee, L.C., Wong, H.L. and Trifunac, M.D.: "An Eval-

uation of the Effects of Traveling Seismic Waves on the Three- Di-

mensional Response of Structures", Report No. 7720-4514, Agbabian

Associates, El Segundo, 1977.

12) Yamada, Y., Takemiya, II. and Kawano, K.: "Random Response

Analysis of a Non-Linear Soil-Suspension Bridge Pier", Earthquake

Engineering and Structural Dynamics, Vol. 7, 1979, pp. 31-47.

13) Yamada, Y., Takemiya, H., Kawano, K. and Hirano, A.: "EarthquakeResponse Analysis of High-Elevated Multi-Span Continuous Bridges

on Dynamic Soil-Structure Interaction", Journal of Applied Mechan-

ics and Structural Engineering Division, Proceedings of JSCE, No.

328, December 1982, pp. 1-10.



Maximum Displacement ( m.

P Wave

# = 0° , 7 = 0°

P.# 1

P.#2

P.#3

LongitudinalVertical

TransversalLongitudinal

VerticalTransversalLongitudinal

VerticalTransversal

Bridge ModelWithout SSI

2.458E-060.0289903.510E-065.548E-030.0141321.631E-061.580E-030.0239103.002E-06

With SSI3.186E-060.0288343.528E-065.528E-030.0140671.632E-061.576E-030.0237713.029E-06

A%0.0-0.50.0-0.3-0.50.0-0.2-0.60.0

Maximum Displacement ( m. )

P Wave0 = 45° , 7 = 0°

P.#l

P.#2

P.#3




VerticalTransversal

Bridge ModelWithout SSI0.0188980.0217319.732E-050.0244990.0102002.076E-050.0203890.0212221.708E-04

With SSI0.0172700.0268538.834E-050.0230299.901E-032.022E-050.0195620.0249901.681E-04

A%-8.64-23.60.0-6.0-2.90.0-4.1-17.70.0


P Wave9 = 45° , 7 = 45°

P.#l

P.#2

P.#3




VerticalTransversal

Bridge ModelWithout SSI

0.0135160.0217350.0223120.0185540.0102609.90SE-030.0147490.0198530.019803

With SSI0.0132880.0241760.0209210.0181470.0100059.939E-030.0146220.0209870.018197

A%

-1.7

+10.1-6.6

-2.2-2.4+0.3-0.9+5.4-8.8

Table 1 : Maximum Bridge Displacements: Incident P-wave




SV Wave0 = 0° , 7 = 0°

P.# i

P.#2

P-#3




VerticalTransversal

Bridge ModelWithout SSI0.0602200.0245144.157E-040.0798451.265E-033.S60E-040.0594800.0219004.031 E-04

With SSI0.0600020.0243994.137E-040.0797441.257E-033.836E-040.0592430.0218014.024E-04

A%-0.4-0.010.0-0.1-0.6-0.0-0.4-0.4-0.0


SV WaveB = 30° , 7 = 0°

P.# 1

P.#2

P.#3




VerticalTransversal

Bridge ModelWithout SSI0.0963580.0290837.194E-040.1136341.216E-036.711E-040.1050180.0318266.975E-04

With SSI0.0899180.1031665.361E-040.1117610.0305225.363E-040.0902760.1000006.442E-04

A%-6.7

+254.70.0-1.6

+2410.70.0-14.0+214.20.0


SV Wave0 = 30° , 7 = 45°

P.# 1

P.#2

P.#3




VerticalTransversal

Bridge ModelWithout SSI0.0669340.0211470.0659290.0886553.643E-030.1276120.0739170.0254630.061607

With SSI0.0755810.0647800.0678840.0941190.0302840.1207940.0759320.0641910.054066

A%+8.1+206.3+3.0+6.2+731.3-5.3+2.7+ 152.1-12.2

Table 2 : Maximum Bridge Displacements: Incident SV-wave


Soil Dynamics and Earthquake Engineering

VINCENT-THOMAS SUSPENSION BRIDGE

549

Wave Frojit (Plane)

Fig. 1 : Three Dimensional Bridge Model and Incoming Seismic Waves

Mode # 3Frequency : 0.2023 Hz.Period: 4.9431 sec.

ELEVATION

PLAN

Fig. 2 : Example of Mode of Vibration: Mode # 3



- 0°; Q = o°): No Soil-Structure InteractionAMPLITUDE (m.sec.) 80

60

40

20

0

:1 •

} 0.5 1 1.5 2 2.5 3FREQUENCY (Hz.)

_ QO. 9 = 30°): No Soil-Structure Interaction

AMPLITUDE (m.sec.) 100

80

60

40

20

0_Jt) 0.5 1 1-5 2 2.5 2

FREQUENCY (Hz.)

_= Q°;0 = 30°): Soil-Structure Interaction Included

?V)£

UJr̂

£

z<

100

80

60

40

20

0

•_jI

) 0.5 1 1.5 2 2.5 3FREQUENCY (Hz.)

Fig. 3: Transfer Functions of Longitudinal Displacements: Point 1



(7 = 0°; 9 = 0*): No Soil-Structure Interaction

. TRANSFER FUNCTION OF THE VERTICAL DISPLACEMENT OF THE BRIDGE

a 4

0.5 1 1.5 2FREQUENCY (Hz.)

2.5

~ 15

Z, 10LUQ

t 5

(7 = 0°;# = 30°): No Soil-Structure Interaction

TRANSFER FUNCTION OF THE VERTICAL DISPLACEMENT OF THE BRIDGE

0.5 1 1.5 2FREQUENCY (Hz.)

2.5

LJJQ

3a.Z<

(j = 0°;# = 30°): Soil-Structure Interaction Included

-TRANSFER FUNCTION OF THE VERTICAL DISPLACEMENT OF THE BRIDGE

1 1.5 2FREQUENCY (Hz.)

2.5

Fig. 4: Transfer Functions of Vertical Displacements: Point 1


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