538 soil dynamics and earthquake engineering - wit press€¦ · soil dynamics and earthquake...
Post on 17-Jun-2018
225 Views
Preview:
TRANSCRIPT
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
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
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.
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
540 Soil Dynamics and Earthquake Engineering
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)
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
Soil Dynamics and Earthquake Engineering 541
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)
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
542 Soil Dynamics and Earthquake Engineering
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.
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
Soil Dynamics and Earthquake Engineering 543
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
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
544 Soil Dynamics and Earthquake Engineering
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
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
Soil Dynamics and Earthquake Engineering 545
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.
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
546 Soil Dynamics and Earthquake Engineering
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.
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
Soil Dynamics and Earthquake Engineering 547
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
LongitudinalVertical
TransversalLongitudinal
VerticalTransversalLongitudinal
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
Maximum Displacement ( m.
P Wave9 = 45° , 7 = 45°
P.#l
P.#2
P.#3
LongitudinalVertical
TransversalLongitudinal
VerticalTransversalLongitudinal
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
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
548 Soil Dynamics and Earthquake Engineering
Maximum Displacement ( m. )
SV Wave0 = 0° , 7 = 0°
P.# i
P.#2
P-#3
LongitudinalVertical
TransversalLongitudinal
VerticalTransversalLongitudinal
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
Maximum Displacement ( m.
SV WaveB = 30° , 7 = 0°
P.# 1
P.#2
P.#3
LongitudinalVertical
TransversalLongitudinal
VerticalTransversalLongitudinal
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
Maximum Displacement ( m. )
SV Wave0 = 30° , 7 = 45°
P.# 1
P.#2
P.#3
LongitudinalVertical
TransversalLongitudinal
VerticalTransversalLongitudinal
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
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
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
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
550 Soil Dynamics and Earthquake Engineering
- 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
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
Soil Dynamics and Earthquake Engineering 551
(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
Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509
top related