The soil-structure interaction at bridge abutments may introduce important changes in the dynamic properties of short to medium span bridges. The paper presents the results obtained, through the use of the Boundary Element Method (B.E.M.) technique in several typical situations, including

semiinfinite and layered media. Both stiffness and damping properties are included

Alarcón Álvarez, Enrique

Gómez Lera, Sagrario

Martínez Cutillas, Antonio

English

The soil-structure interaction at bridge abutments may introduce important changes in the dynamic properties of short to medium span bridges. The paper presents the results obtained, through the use of the Boundary Element Method (B.E.M.) technique in several typical situations, including semiinfinite and layered media. Both stiffness and damping properties are included

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4a 4b 5. 6. AASHTO (1983):&quot;Guide Specifications for seismic design of Highway bridges&quot;

A simple nonlinear model for the investigation of the effects of the gap closure at the abutment joints of short bridges&quot;.

Application of the Boundary Element Method to the analysis of Bridge Abutments&quot;. H.Tajimi (1973):&quot;Dynamic earth pressure on basement wall&quot;.

Boundary Elements approach to the dynamic stiffness of circular foundations&quot;.

Bridge abutments: Formulation of simple model for earthquake response analysis&quot;

Earthquake-induced soil pressures on structures&quot;.

Elastodynamics&quot; Chap 7 in &quot;Progress in Boundary Element Method&quot;.

Dynamic impedances of bridge abutments

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Earthquake Engineering, Tenth World Conference © 1994Balkema, Rotterdam. ISBN 90 5410060 5 
Dynamic impedances of bridge abutments 
E.Alarcon & MS.Gomez-Lera 
Technical University of Madrid, Jose Gutierrez Abascal, Spain 
A. Martinez 
Carlos Fernandez Casado, Oficina Tecnica, Madrid, Spain 
Abstract.- The soil-structure interaction at bridge abutments may introduce important changes in the dynamic properties of short to medium span bridges. 
The paper presents the results obtained, through the use of the Boundary Element Method (B.E.M.) technique in several typical situations, including 
semiinfinite and layered media. Both stiffness and damping properties are included. 
1. INTRODUCTION 0 m wl^$ 
The influence of approaching embankments on the behavior 
of bridges has been recognized since long ago. And this happens both 
at the ultimate limit state and while "in service" conditions. Even 
current codes (ref.l) recommend to simulate the superstructure-
infraestructure interaction by modeling the abutments by suitable 
equivalente springs althrough no specific procedures or figures are 
s uggested. 
That the influence has to be important can be assessed 
qualitatively simply by looking at current designs. Fig. 1, for instance, 
is a recently designed short bridge of the integral abutment type. 
It is easy to see that in order to have a good model it is 
unavoidable to include the infraestructure behavior both in the 
longitudinal and transverse directions. A quantitative estimate of the 
importance of the effect can be obtained using an equivalent single 
degree of freedom system following the ideas described by Wolf. To the 
typical model of reference 2 a spring and dashpot system has been 
added. The model is similar to that of reference 3 altough no gap has 
been included between the deck and the abutment i.e: we are refering to 
integral deck-abutment bridges. The values k, and C. represent the 
stiffness and damping properties of the abutment-embankment system, 
while the soil-structure interaction at the column foundation is 
represented by the couples (kh; Ch) (k#; C$) related to horizontal and 
rotational displacements. Following assumptions similar to those of 
reference 2 it is possible to show that the frequency of the single degree 
of freedom system can be written as 
u 
Ca 
r^VW 
U) 
I _ i — A / W — i D 
a 
F 
•JC 00 
jvao — Jb 00 — —•-j 
f * I 
Figure 2 
ca*2 = t o 2
 + a 
1 
1 
CO 
1 1 
2 (0 2 h 
^hj d-u/v& 
a> 2 
(1) 
10 60 
? 00 1 
DQQOjOeOQT* 
C O . 2 = 
CO h 
k 
m 
2 . * » 
m 
• t * 2 -
k 
m 
k 
mh 2 
Figure 1 while the equivalent damping ratio is 
6847 
C* = F C + F C, + F C h/L 1 
(2a) 
where F, , FM , F, are factors corresponding respectively to the 
damping of the structure, soil damping and abutment damping. The 
definitions are 
1 00 
F. = 1 
1
 + co/ ( 1 1 
0) 2 h d> 
2 ) 
f. ' 1 
,'<-L*-W> 
C- + A 
1 + 0 ) 
CO/ « , * 
1 
1 2 / 1 1 
1 + (0/ (— 
<•>/ " A 2 
) 
2 
^ - « / ( 
1 
* - • - ! - ) 
0) 0) 2 A 6) 
2 
(2b) 
and f, f
 h , fr , ft are the damping ratios associated respectively to the 
structure, the horizontal soil displacement, the rotational soil 
displacement and the top of the abutment displacement. 
In order to simplify the analysis the values proposed in 
reference 1 have been taken to obtain 
k 
h2K 0.8 A
2 
h 
= 0.26 
(3) 
where L is a characteristic length of the column foundation and h is the 
abutment height. 
So depending on the ratio h/L it is possible to plot families 
like those of figure 3. In the horizontal axis the ratio k/kj, between the 
stiffness of the structure and that of the horizontal soil spring has been 
used while the family curves depend on the ratio k./k between the 
abutment stiffness and that the structure. 
In order to normalize the results, the ordinates reflect the 
ratio 
CO •2 0) 
a s 
0) 2 + G> 2 a s 
+ 
1 
( « 
2 * . 1 
"?)( 1 1 
0) h 0) 
2 ) 
(4) 
so when a), is null(that is when there is no deck-abutment connection) 
the lower curve is obtained reflecting the well knowwn effect of the 
reduction in frequency due to foundation flexibility while when,the 
abutment is completely rigid we obtain a unit value. As can be seen the 
variation is substantial and a discretization ignoring the abutment 
influence can be complete unconservative. 
h> 
1 0 0 
0 . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 1 
Figure 3 
The weight of the abutment damping can be understood if we 
compare the factor afecting f. in eq.2 with the factor afecting f and 
the one of fh once the simplifications of eq.3 have been done. Those 
ratios are presented in figure 4 where it is clearly seen that the 
importance of f. is substantial so that correctly evaluating its value is 
a fundamental requisite for a good modelling. 
In a recent paper (ref.4) Wilson and Tan recognized the 
problem and proposed a plane strain trapezoidal model on rigid base 
to simulate the static transverse behavior of the abutment-embankment 
system. They did not provide any clue about how to quantify the 
amount of damping but, when they tried to identify the properties of 
a bridge (Melolan Road Overpan) that suffered a moderate earthquake 
without apparent damage they reported two interesting findings: to 
produce time histories compatible with the observed one it was 
neccesary first to assume a soil shear modulus of about 1/3 of the static 
one and then to include a damping ratio betwen 0.27 & 0.65. Although 
they justified those values on the basis of a nonlinear soil behavior the 
results that are included below show that a linear viscoelastic behavior 
can justify them if the dynamic response is analyzed including a 
substantial part of the soil around the embankment. 
The objective of the paper is then the parametric analysis of 
the longitudinal and transverse dynamic impedances of rigid walls 
acting over layered media representing both the embankment and the 
soil on which it is erected. 
6848 
Fs 
2 . 4 
2 . 2 
2 |-
1 . 0 -
1 . 6 -
1 . 4 h 
1 . 2 
1 
• 6 
.4 f 
. 2 -
<H/l_- 1 ) 
3 0 . 1 . 2 . 3 . 4 . 3 . 6 . 7 . 8 . 9 1 
K/Kh 
1 . 4 
1 . 2 
1 
8 
6 
2 
0 
O Fa C H ^ L _ - 1 ) 
e . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 1 
K/Kh 
Fa 
2 . 4 
2 . 2 
2 
1 . 9 
1 . 6 
1 . 4 
1 . 2 
1 
. 8 
. 6 
. 4 
. 2 
e 
< h / L > 
0 . 1 . 2 . 3 . 4 , 3 . 6 . 7 . 8 . 9 1 
K/Kh 
Fs o 
i 
1 . 4 
1 . 2 
Fa Ch 2 3 
0 . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 9 . 9 1 
K/Kh 
Figure 4a. Ratio of structure to abutment damping factors (Ft/F.) Figure 4b. Ratio of foundation to abutment damping factors (FM/Ft). 
2. NUMERICAL TECHNIQUE AND MODEL 
VALIDATION 
As the main point is the computation of dynamic soil 
impedances the Boundary Element Method (B.E.M.) has been chosen 
as the numerical technique to be used in the parametric analysis. It has 
proved its efficiency in previous similar compromises, see for instance 
references 5 & 6, where some more details can be found. 
The equations of motion are 
where u is the displacement vector, f the body forces, p the material 
density and A , G are the so-called Lam6 parameters. 
Moving to the frequency domain with f = 0 we obtain 
(Pf-C*) u^+C} «,/+«» uf = 0 
(6) 
And it is possible to write an integral equation which on 
smooth boundaries looks as 
(C,2 - C22) « + C22 ii 
c2 = X + 2G 
P 
c2 = G 
P 
dt2 
(5) 
1 
2 
u/(P)+fT/(P,Q) «/«?) = 
dQ 
= JT;t{P,Q) t*(Q) 
dQ 
(7) 
6849 
Irl 
U U 2 w Vxor? 
T * 
1_ 
anpC2 
I
 r .drh I dr . 
an dr r ' a - ; 
- 2 dx dr —r, r,— 
dr «* •'an 
v a* 
2 3r 
r ' •* •* •'an 
C 
• ( - 1 : -2 ) ( 
C 2 
where for tridimensional cases 
dr dr 
2 
* > ' . < 
n J 
(8) 
Figure 5 
Where P and Q are points in the boundary (Fig.5). 60 is the 
boundary of de domain 0 and T"^  ; 
u*j are influence tensors than can be writen as 
a = 2 
T = 
C. 
^ [ K ^ - ^ - K ^ ) } 
nor C. C l c 
X = *,(^ )-(^ )V(i^ ) c 2 c 1 c, 
1 
(9) 
u 12 T»z - O 
^ ^ ^ s V l 
TAJtMl S PROBLEM 
R, t U j . Z ) 
- 3 - 2 - 1 1 
TAJIMI. 
o BEM. 
R 2 ( U J . Z I 
I 
(o) ( b ) 
-L 
y\—i—i—i—i—i—i—i—i—r 
BEM MODEL 
T T 0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
TAJIMI. 
BE M. 
0.5 UJH 
VI 
1.0 1.5 
Figure 6a Figure 6b. Real and imaginary parts of the interface strcsses(above) and 
acting momcnt(bclow). 
6850 
and for three dimensional ones 
a = 4 
-i«r 
T = € 
G c 
r 
+<-
2 c 
"«>r o>2r2 
2 
2
 > * 
<^ r> 
c c 
- ( ^ ) ( ^ - -
c. 2 v « 
r 
- ! < ! ) / • 
1 
) 
C{ ' iur o)2r2 r 
3 C 3C 
X = ( -
2 
-i<i>r 
~GT 
zcor oV 2 r 
-i«r 
c 2 3C 3C 2 
- ( ^ ) ( ^ -c 1 icor o)2r2 
*T+D 
e 
c i 
r (10) 
where r is the distance between P and Q. 
In order to check the accuracy and to test several modelling 
details (mesh size, element size, numerical integration type, etc) (ref.7) 
two classical works, relevant to the topic under discussion, have been 
used; in the first one (ref.8) Tajimi used a wave-propagation technique 
to analyze the response of quarter-space with a rigid comer inducing 
harmonic displacements and rotations. Figure 6 shows that a 
very simple mesh with "constans" boundary elements is able to 
reproduce accurately Tajimi's findings. The only precautions to be taken 
are related to the size element around the corner and at the bottom tip 
of the wall. Also the size of the discretized boundary and of the 
elements is related to the frequency of excitation so that an adaptive 
mesh (ref.6) can be used advantageously which length and size depends 
on the excitation wave length. 
The same happens with figure 7 where the discretization 
corresponds to a problem proposed by Wood (ref.9). Here again both 
the real (stiffness) and imaginary (dissipation) parts of the impedance 
are reproduced without any problems. 
With such a proverful technique it is an easy matter to analyze 
more complicated cases including the halfspace situation or the layered 
one. In figure 8 several meshes are included that are intended lo be use 
both for a longitudinal as well as for a transverse situations. As can be 
seen the discretization includes the interfaces because a piece-wise 
heterogeneous soil is studied. 
3. PARAMETRIC STUDY 
Using the meshes shown in figure 8 several cases have been 
run varying the relative properties of the strata. For the longitudinal 
direction figure 9 show the behavior for a rocking imposed 
displacements. The parametric study includes the Poisson ratio 
variation and the rigid-bottom depth variation. As can be seen several 
oscilations related to the stratum natural frequency can be appreciated. 
The material damping was only 5 %. It is interesting to see that while 
the rotational stiffness is not very much affected by the bedrock depth, 
the damping value reflects the importance of the radiation in the last 
cases.For the transverse situation a Poisson coefficient of 0.3 has been 
selected and a 5 % material damping has been included in ail models. 
The ihear modulus of tho it mum has been varied between 1 and 100 
times that of the embankment and the proportion between the stratum 
depth P and the embankment height H has been varied between 0 and 
3 times. Also a halfspace has been analyzed for completnes. In any 
case the crest embankment width has been taken as twice its height. 
• 
T y y » T x y - 0 
VVVOAVAVAWV VVVvVVVsAVOA 
xy • 0 
ux«u 
Uy»0 
WOOD S PROBLEM 
BEM MODEL. 
L / H • 5.0 
WOOD. 
3.0 
2.0 
MODULUS — v « 0 . 3 
" v " 0.45 
o BEM 
M 1.0 
3 o.o 
\\£ 
-1.0 • 
- 2 . 0 I 
1 
L / H - 5.0 
WOOD. 
3.0 
2.0 
MODULUS v'0.3 
v-0.45 
BEM 
/ 
3 
1.0 
0.0 
Is * 1.0 
- 2 . 0 L 
Figure 7 
6851 
J I T 1 I I 1 I T I T 1 
r™i"- i i T " ~ " i — i — r ~ i — T T 
1—i—r—1—I—i—r-1—i—i 
Kesi'k 
0.8 i 
0.25 C5 0.75 1 1.25 1.5 1.75 
i—i—i—r—i—i—i—i—i—rrrrT—i—i—i—i—i—i—i—i—i 
T 1 1 1—1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
T — I — i 1 r T f-r 
T T — i — i — i — i — r i > 
I—I—T ' T I I — 1 — 1 — I T I 1 ™ r — i — i — i 1-—r • i — i — < 
Figure 8a. Meshes used for longitudinal vibration 
Kesrc 
0.1 
0.25 0.5 
nu-0.00 
n u - 0 3 0 
0.75 1 
wH/cs 
nu-0.10 
nu-0.40 
1.25 1.5 
nu-0.20 
n u - 0 5 0 
1.75 
P 
B 
Gl 
02 
0.2 r 
o 1 -
0.25 0.5 0 7 5 
wH/CS 
1 1.25 1.5 
r 
H 
L 
8 
Czz 
0.25 0.5 
h/H-0 
h/H-10 
0 7 5 1 
wH/CS 
h/H-1 
h/H-20 
h/H-2 
h/H-50 
1.25 
h/H-5 
h/H-100 
1.5 
Figure 8b. Meshes used for transverse vibration Figure 9 
6852 
K 
Gi 
2.S 
1,5 
KVfiANKHEN'T OK fil<J? BAST 
EMBANKMENT * HALFSrT 
0,3 -
o L — i — J . - J . 1 ^ *. J . J L J k 
0.5 1.9 
UJB 
Figure 10. Horizontal stiffness 
The real part (stiffness) of the impedances is plotted in figures 
10, 11 and 12 for the cases of vertical, horizontal and rocking 
displacements. As can be seen when P diminishes or the modulus of the 
stratum increases the curves move towards the rigid base trapezoid case. 
It is interesting to notice the oscillations the horizontal stiffnees, around 
the halspace one. Also noticeable is the reduction of stiffness that is 
produced when the frequncy increases or when a halfspace or stratum 
is added under the trapezoid. 
K_ 
3 1 
I 
i L 
> -
\ 
\ 
'I \ 
\ / 
V 
Figure 11- Horizontal stiffness 
K/Gi 
MV»NKMKNJ "N im:n> h*St 
« -
7 -
1 9 
&5 
<3 
1 3 
I G I / G M 2 ":•£•' 
P-2H.G2 
^ G l 
I S -
? -
1.9 
X 
P"3H,G 2 
^ G, 
\ 
\ 
as 
PHnANKHKNY - MALFSr. ~Z 
J * 1 1 \ 1 _ J I 1 I L J I L j L 
0 3 1.3 
UJB 
K 
Gi-B' 
90 
85 
x i o 2 
Gz/Gt • to 
30 f -
73 
70 
?*<Z 
6!S 
G 2 / G 1 • 2 P - 3 H 
P*H 
60 
33 
30 -
43 
P - 2 H 
EMBANKMENT - KALF*;rA'E 
40 
33 
30 
P - 3 H 
L^ -L L. 25 1 * 1 1 L _ l J _ J \ 1 1 
0 3 1 
J L .J L i J L I 
) 5 
, . !±L8 
Figure 12. Vertical and rocking stiffness 
For the case analyzed in reference 4 it was found that the 
natural adimensional frequency was about 2. For this value and a 
horizontal displacement the halfspace stiffness is 1/3 of that of the 
embankment on a rigid base what agrees with experimental findings. 
On figures 13, 14 and 15 the imaginary parts divided by the 
adimensional frequency are plotted without factoring the internal 
damping effect. 
6853 
Kit C K*l C 
p . 3H 
P - 2 H 
EK&ANKNEKT . HALFSF:.: . 
G2/G>-2] 
EVh;.w.«:KT cm n:;: n*- r 
Figure 13. Vertical disipation 
K u C 
G, 8 1 
Ku C 
2.S 
1.S 
0.5 H 
ENtlANKMtNT > HALFSPACE 
EMBANKMENT ON RIGID BASE 
CI 
P-2H.G 
Gl 
t \ 
^ - P - S H . G 
T7J
 * ifiTrrjjrTvrr 
• . L 
o^ 1.5 
Figure 15. Horizontal disipation 
Finally, figure 16 shows the meshes used for the analysis of 
a threedimensional exemple. 
*•#-
*-i 
Figure 14. Rocking disipation 
It is possible to see the difference between the halfspace and 
the trapezoid on rigid base cases which for the horizontal displacement 
amounts to a 6 fold value. In any case it is advisable to remember that 
2_D damping is very large in comparison with that obtain for a 3_D 
situation. 
»-3 
«-2 
Figure 16 
6854 
0.5 
0.6 
1 1.5 2.5 
m-l 
wH/cs 
m-2 m-3 
1 1.5 2.5 
m-1 
W H / C 3 
m-2 m-3 
Figure 17 
It is important to reduce as much as possible the 
computational effort so that several sizes and distributions have been 
tried. The comparative results can be seen in figure 17 where it can be 
observed that, at least a mesh of the type called 2 will have to be used 
to represent correctly the longitudinal behaviour. 
For the transverse stiffness, figure 18 shows that even with 
the finer mesh it is possible to obtain very accurate results. It is also 
interesting to see that reduction, 
due to the dynamic in stiffness behaviour justifies the 1/3 finding of 
Wilson and also that the radiation damping presents values large 
enough to justify those obtained during the identification parameter 
study of Melolan Road Overpass without having to look for a nonlinear 
behaviour of the soil substructure. 
4. CONCLUSSIONS 
The aim of the paper is to show that it is possible to use 
realistic dynamic models to reproduce the sitffness and damping that 
has been detected in the abutment of real bridges that survived real 
earthquakes without apparent damage. 
On the other hand the numerical technique used, the so-called 
Boundary Element Method, seems to be a proverful tool to analyze 
those kind of dynamic soil-interaction problems. 
REFERENCES 
1. 
2. 
3. 
4a 
4b 
5. 
6. 
7 
8 
9 
AASHTO (1983):"Guide Specifications for seismic design of 
Highway bridges" 
J.P. Wolf (1985): "Dynamic soil-structure interaction". 
Prentice Hall. 
E.A. Maragakis et al (1989): "A simple nonlinear model for 
the investigation of the effects of the gap closure at the 
abutment joints of short bridges". Earth. Eng. Struct. 
Dynamics. Vool. 18; 1163-1178. 
J.C. Wilson & B.S. Tan (1990): "Bridge abutments: 
Formulation of simple model for earthquake response 
analysis" ASCE Jour ENg. Mech. Vol. 116 No. 8. 
Ditto: "Bridge abutments: Asses in g their influence on 
earthquake response of Mcloland Road Overpass" (Same 
reference). 
J. Dominguez& E. Alarc<5n (1981): "Elastodynamics" Chap 
7 in "Progress in Boundary Element Method". Ed. C. 
Brebbia. Pub. Pentech Press. 
E. Alarc6n; J.J. Cano; J. Dominguez (1989): "Boundary 
Elements approach to the dynamic stiffness of circular 
foundations". Inst. Jour. Num. Analyt. Meth. Geomech. Vol 
13; 645-664. 
A. Martinez; E. Alarc6n; M".S. G6mez-Lera; F. Chirino 
(1992): "Application of the Boundary Element Method to the 
analysis of Bridge Abutments". 
H.Tajimi (1973):"Dynamic earth pressure on basement wall". 
World Conference on Earthquake Engineering. Roma. 
J.H. Wood (1973): "Earthquake-induced soil pressures on 
structures". Ph. D. Thesis. Caltech. Pasadena. 
Kzz 
0.6 1 1.6 2.6 
m-1 
wH/cs 
m-2 m-3 
0.5 1 1.5 2.5 
m-1 
wH/cs 
m-2 m-3 
Figure 18 
6855 


Dynamic impedances of bridge abutments

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