Yang - 1992 - Axisymmetric Infinite Elements For SSI Analysis PDF
Yang - 1992 - Axisymmetric Infinite Elements For SSI Analysis PDF
Yang - 1992 - Axisymmetric Infinite Elements For SSI Analysis PDF
Engineers are frequently confronted with the problems attenuation and the phase delay in the direction to
in which the domain to be analysed is extended to infinity. The infinite element containing a single wave
infinity, such as the cases of the soil-structure and the component could be successfully applied to problems in
fluid-structure interaction problems. When the which one wave occurs. However, it is not appropriate
dynamic soil-structure interaction problems are for more complex wave propagation problems, in which
analysed, the region near the structure (near field) can several wave components are simultaneously involved.
be modelled using the conventional finite elements. But A dynamic infinite element capable of propagating
simple truncation of the outer area (far field) produces multicomponent waves has been proposed by Medina ~3.
an erroneous response due to the reflected wave energy The shape functions were formulated by using approxi-
from the boundaries. Therefore, to simulate the energy mate expressions for the analytical far field solutions.
dissipation through the boundaries, many different tech- Though acceptible results were obtained using the
niques have been proposed. Most of them utilize energy method, the accuracy of the results was found to deteri-
absorbing boundaries along the interfaces between the orate due to the nonconforming conditions between the
near and far fields ~-5. To a certain extent, those finite and the infinite elements and also between the
methods can serve the purpose, but they are somewhat adjacent infinite elements. Furthermore, the method
restricted by the assumptions made on the special boun- cannot easily be extended to more complex problems
daries, such as the types of wave components to be such as those with layered media, since it is very dif-
transmitted, the homogeneity and the linearity of the soil ficult to obtain the analytical far field solutions, and
media. Recently, the boundary element method (BEM) also to formulate the shape functions based on the
has been widely studied for the analysis of the infinite analytical results.
domain problems. It is based on fundamental solutions This paper presents a dynamic infinite element which
which extend to infinity per se. In spite of its adapt- can deal efficiently with multiple wave components. The
ability to problems which have complex geometries, it shape functions are formulated using more general
requires more study for the treatment of elastoplasticity expressions for the wave components than those in
and nonhomogeneous material behaviours. An alterna- Reference 13. They are in terms of complex exponential
tive approach may be to use infinite elements which can functions of the corresponding wave numbers, and
easily be incorporated with the finite element method. satisfy the Sommerfeld radiation condition. Hence, the
The infinite element concept introduced by Ungless 6 present infinite element may be easily extended to the
and Bettess 7 has been successfully applied to problems problems, for which analytical far field solutions cannot
with unbounded domains, such as those for elasto- easily be computed. The number of wave components
statics 8- ~0, quasistatic load ~t, elastodynamics ~2-14, and included in the shape functions can be increased by
hydrodynamics~5-~8. The shape functions of the dyna- introducing nodeless variables. The compatibility condi-
mic infinite elements were usually developed by employ- tions between the adjacent infinite elements may be
ing a wave function which represents the amplitude preserved by matching the nodeless variables associated
0141-0296/92/060361 - 10
1992 Butterworth- H e i n e m a n n Ltd
wifll the nonzero displacements along the interlace origin. The mappings of the infinite elements t'rom the
betv,.een two neighbouring infinite elements. The com- local coordinates (se, q) to the global coordinates (r, =t
patibility conditions between the finite and the infinite can be defined as
elements can he also kept, except for the radiational For horizontal infinite elements
infinite elements with the surface wave components. The
numerical integrations for constructing the element 3
matrices are carried out efficiently by applying a scheme r : F, MOLj I,-,, -== Lj(r/)=, (1)
based on the G a u s s - L a g u e r r e quadrature. The present / I t=l
method can be verified by comparing the impedance
functions obtained for rigid circular footings on an For radiational infinite elements
elastic half space and also on layered half spaces with
those obtained analytically. 3 3
r = ~_, M(OLiO1)r,, = = y] M(~)LiO1)Z (2)
i=1 t=l
Dynamic infinite elements including nodeless
variables where L/(rl) is a Lagrange polynomial whose value at
Dynamic axisymmetric infinite elements are developed n o d e j is unity, and M ( ( ) is the mapping function for the
for the soil-structure interaction problems, whose infinite direction defined as
domains extend to infinity. Although the element has
only three nodes placed on the interface between the M(()= 1 +~ (0_< c < oo) (3)
near and the far fields as in Figure 1, it can include addi-
tional wave components in its shape functions by The origin of the global coordinate system is placed on
introducing nodeless variables. The concept of the node- the top surface of the underlying half space and on the
less variable is similar to the internal variable in the axis of symmetry as in Figure 1.
classical finite element; for a example, the one which is
added to the 8-node serendipity element to obtain the Displacement shape functions
available shapes in the 9-node Lagrangian element ~. The elastodynamic problems often produce displace-
The additional shape functions corresponding to the ment fields in which several wave components pro-
nodeless variables are constructed by considering the pagate simultaneously. In such problems, a typical
conditions under which the shape functions have zero displacement component in the radial direction may be
values at the nodes as described below. expressed as a combination of several wave components
as tbllows
Geometrical mapping
To analyse a layered half space consisting of several
horizontal layers and a underlying half space as shown
in Figure 1, it is convenient to discretize the far field by l= t 1=1
d__p0 ,/I,0 N
(6)
a,., = F,{Xji U,-
Figure I D e f i n i t i o n o f global and local c o - o r d i n a t e systems I=2
Table 1 Wave functions F ( x ) s for elastic half space and layered half space
Homogeneous
half space e b~ + ik,rl - - e (~ + ik, Ro)~ e-I" + ikoRo)~
Horizontal
Layered layer e (" + ik'R}~ e (~ + i k l R ) ~ -- --
Substituting a~.j into equation (4) gives For an example case employing the shear and the com-
pressional wave components, the shape functions corres-
3 F~(x) ponding to the nodal displacements and nodeless
u(x) = ~a Li (rl) F, (xj) Ui variables are shown in Figure 2 and 3. It can be seen that
j=l the shape functions associated with the nodeless
variables have zero values along the interface between
+ E E LjO;)F,(x) the finite and the infinite elements, while those
/=2 j=l associated with the nodal displacements have unit values
at the corresponding nodes.
FI(xJ) Fi (x))ao (7) Several of the displacement components associated
F 1(xj) with the nodeless variables in equation (7) are no longer
restricted to being zero along the interfaces with 'neigh-
It should be noted that in equation (7) the first term is bouring infinite elements. Hence, to preserve the
represented in terms of the nodal displacements Uj, and displacement continuities along the interfaces, the
the second in terms of the nodeless variables ao(l >_2). nodeless variables inducing nonzero displacements
Hence, the shape functions for the nodal displacements along the interfaces are taken to be equal to those of the
are neighbouring elements. The compatibility condition
between the finite and the infinite elements can be also
F~(x) for j = 1 2, 3 (8) kept, if the wave function, F~(x), has a constant value
~(x) = Li(r/) F, (xj) of the interface. From Table 1, it can easily be seen that
all wave functions, except for the surface wave com-
while those for the nodeless variables are ponents of the radiational elements, have constant values
on the interface. The displacement shape functions in the
circumferential and vertical directions can be constructed
N,,,(x) = Li(T1)IF/(x) - F~(x(xi)
J)~ F~(x)l similarly to the above procedure. Considering three
displacement components in the radial, circumferential
and vertical directions, the total number of the nodal
form=3(l- 1)+j variables for a 3-node infinite element is 9, while the
( j = 1, 2, 3; l = 2 , 3 . . . . . N) (9) number of the nodeless variables becomes 9 ( N - 1).
<Oo Loo
' ~ '
"'Zs
-,.~ ~ ~ ~ . 0 "">s
s.0 .,.0 s.o
0.~ ~ n ~ ~ - - ' i ' ~ _ ~.0 3.0
~.0 ~ 3.0 - 1.9
i. 0.0 & .0 0-0
s b
Figure 2 Example of shape functions corresponding to nodal displacements. (a), real part of N2 (r/, 4); N2 (rl, 4) = L2 b l ) ( F l ( x ) / F 2 (x2)); (b),
real part of N3 (q, ~); N3(~?~) = L3 (r/)(F1 (x)iF1 (x3))
_~.0
5.0 ~.0 ~ ~ .0 5.0
~.0 ~,~
~.0 0.0 ~.0 0.0
a b
Figure 3 Example of shape functions corresponding to nodeless variables. (a) real part of N5(~/, ~); Ns(~, ~) = L2(~)IF2(x) - (F2(x2
F~ (x2})FI(X)]; (b), real part of N6(~/, ~) = L3('q)JF2(x} - (F2(x3)/F1 (x3))F1 (x)]
ei cot kh
~-:: ~ =:_..~_.-.-g.-..o-,~-...-.o- ......
. .e ;~ t ~ .~ , .o
r ~~
~ : L Ilr .a~
~ 0.5
.--?? w
ch
t
-r-.~ d
0.0 I I ! I I I
Z
I~ta I .0 'q~. .....
H = (k h + iaOch)K A AH ~'~
KA = 8Gro/2-v ~'~
U
0.5
._c._c
M = (k m + iaocm)Ktp~ M _~ CL
uE
om 0.0
0 " " ~ ~ ~'. . . . I - cm
K~ = 8Gr30 / 3 ( 1 - v )
Figure 5 Definition o f impedance f u n c t i o n s for a rigid circular -0.5 I I I I i I I
I
I
f o o t i n g on elastic f o u n d a t i o n .0 0.5 1.0 1.5 2.0
R 0 = 2r 0
----A ....
----0----
_._#_.__
Ro/r 0 = 1.5
R o / r 0 = 2.0
R o / r o = 3.0
) Present study
1.5
Applications to elastodynamic p r o b l e m s
"O
c- Circular footing on elastic half space
m
1.5 1.0
/
" Exact ( V e l e t s o s and Wei 21} :/ U
~ 0.8
0
v ,.~.,, ,~..'*.. i ,. O
U 1.0 (.I
tn ~.. .Ir--.~,. ., ,,. ~ 0.6
f o -~ " ~.,i~,,.,.
.,....,./ Q.
E
X
~q
0.5
Present study ".V"
"o
t-
0.4
I
o
" ' + - " Using R a y l e i g h w a v e N
"C. 0.2
L
O
---~-- Using shear w a v e O
--.~-. Using c o m p r e s s i o n a l w a v e -1-
0.0 i i ] 0.0 l i ~
0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0
1.5 1.0
transmitted mainly by the body waves (shear and com- the surface wave components are not included in the
pressional waves). The wave numbers for surface waves shape functions for the underlying half space, which is
are evaluated by considering the continuity condition for modelled by using the radiational elements.
the displacements and stresses between the neighbouring Figure 13 shows, the stiffness and damping coeffi-
layer 23-25. Table 1 shows the wave functions used in cients computed for layered half spaces with various
this example. The same expressions are utilized for the layer depths, which are normalized by the static
wave functions in the radial, circumferential and vertical impedances of the top layer. The results are found to be
directions. As a result of using different wave types for in good agreement with the analytic solutions found by
two different layers, the displacement continuities may Luco 22. The effects of the surface waves may still be
not be preserved on the interface between the layers. significant in the underlying half space, particularly for
Hence, to preserve the displacement continuities, it is the lower exciting frequency range and for the case with
assumed that the wave numbers of the radiational a shallow upper layer. However, reasonable results have
infinite element nearest to the upper horizontal layer are been obtained using the present formulation without
linearly varying from the values of the horizontal layer including surface wave components in the underlying
to those of the radiational infinite element below, as half space region. It may be because, as mentioned in the
shown in Figure 12. The continuity conditions between previous half space example, a combination of two
the finite and the infinite elements are also kept, since shape functions of the body waves incorporating the
I .5 I .0
Present s t u d y
" - + - " Using Rayleigh wave
- - ~ - - Using shear wave 0.8
--~1.-- Using compressional wave
~2
0
0 1.0 O
0.6
oh
C
o
5_
E
-~ 0.4
Oh
0.5 oh
r-
C
"7
"7
o
0
~,
a~
o.2 i -
1.5 1.0
--43--- Using Rayleigh and
shear waves
E
---A--- Using shear and u 0.8
compressional waves
o
0
o
1.0 ,- Using Rayleicjh, shear, and
compressional waves O
u
oh
0.6
c-
2
E
m .~ 0.4
c-
0.5
c
M
o ~ o.2
Exact (Veletsos and Wei 21)
0.0 I t l 0.0 a t l I
0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 .0
Figure 10 Impedance functions of circular footing on elastic half spae for rocking motion (cx = 0.75, Ro/ro = 2.0). (a), using single wave
component; (b), using multiwave components
MeiOJt
l
Horizontal I kR kL
infinite
element ,; t
/ it kR kL .r
Underlying r ,, ^ / f-11 k~ = ( k R + k s ) / 2 + ( k s - k R ) q / 2
Radiational ~ 0~q ,
infinite ~ 1 / ~ = (kL + k p ) / 2 + ( k P k L ) n / 2
element / ~ d.S ~
/ / r ~ , ~'P
1
Z
Underlying
half space
/r^
"~"
2.0 1.0
~
1.5 r H / r 0 = 1.0 ."
=1.0 0.8 30
O
0
%, .. ..0 n* ..o'
u
u
m
m
1.0 O~ 0.6
t-
"~.
E
o
0.5 0.4
t- H/ro=0.5 J ~
C Exact ( Luco221
O
N
o
0.0 ---e--- H / r o = 0 . 5 } t. 0.2
O ---e--- H / r o = 0 . 5 t
---&--- H / r 0 = 1,0 Present s t u d y ---&--- H/ro = l .0 Present s t u d y
---a--- H / r O = 3 . 0 ---a--- H/ro = 3.0
-0.5 0.0
0.0 1.~0 2:0 310 u
q.O 510 .0 0.0 '.o 31o '
4.0 '
5.0 6.0
2.0 1.0
Exact (Luco 22 }
E x a c t ( L u c o 22)
a< ---e--- H / r 0 = 0 . 5 ]
1.5 'o~ 0.8 ---~--- H / r 0 = 1 . 0 j~ Present s t u d y
~ - H / r 0 = O. 5 ---o--- H / r o = 3 . 0
0 0
U
1.0 u 0.6
m C~
u~
C
"5.
E
0.5 .~ 0.4
C r"
U 0.0 8 o.2
O
e -'"~'-" H/ro=0"5 t y L H/ro=O. 5
"--&'-- H / r o = 1 . 0 Present s t u d y
---e--- H / r 0 = 3. 0
-0.5 0.0
, n0 5ou
I ~ ! I 1 I I
oo 21o 3:0 .o 60 0.0 1.0 2.0 3.0 4.0 5.0 6.0
Dimensionless f r e q u e n c y , c0= ~rO/v s Dimensionless f r e q u e n c y , c~0 = ~ r ' 0 / v s
b
Figure 13 Impedance functions for layer half space (c~ = 0.75, Ro/ro = 2.0) (a), horizontal motion; (b), rocking motion
nodeless variables can reasonably simulate the effect of 2 White, W., Valliapan, S. and Lee, I. K. "Unified boundary for finite
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waves. 949 - 964
3 Kausel, E. 'Forced vibrations of circular foundations on layered
media', MIT Research Rep. R?4-11, Soils Publication No. 336,
Conclusions Structures Publication No. 384, MIT, Cambridge, MA, 1974
4 Smith, W. D. "A nonreflecting plane boundary for wave propagation
A dynamic infinite element which is capable of pro- problems', J. Comp. Phys., 1974, 15, 492-503
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integration scheme for computing the element charac- sity of British Columbia, Vancouver. Canada, 1973
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infinite elements can be effectively applied to the com- 8 Bettess, P., "More on infinite elements, Int. J Numer. Meth. Engng,
1980, 15, 1613-1626
plex soil-structure interaction problems, such as those 9 Lynn. P. P. and Hadid, H. A. "Infinite elements with 1/r" type
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10 Curnier. A.. "A static infinite element', hit. J. Numer. Meth. Engng.
1983, 19, 1479-1488
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