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Simplified subgrade model for three-dimensional soil-foundation interaction analysis
Soil DynamicsandEarthquakeEngineering15 (1996) 419-429 PII:
ELSEVIER
S0267-7261
(96)00025-5
Copyright © 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 0267-7261/96/$15.00
Simplified subgrade model for three-dimensional soil-foundation interaction analysis Toyoaki N o g a m i Department of Civil and Environmental Engineering, University of Cincinnati, Cincinnati, OH 45221, USA
(Received 17 November 1995; accepted 14 May 1996) A simple mechanical model is presented for the three-dimensional dynamic soil-structure interaction analysis of surface foundations. The model is made of one-dimensional vertical beams with distributed mass and horizontal springs which interconnect the two adjacent beams. Its parameters are rather uniquely related with the soil properties alone and thus are minimally dependent on the loading condition and the foundation conditions like geometry, flexibility and size. Formulations are provided to determine the model parameters from the soil properties. Solving the governing equations of this model, expressions for the subgrade behavior in response to the applied load and soil-foundation interaction are developed in analytical forms for various cases. The dynamic and static response of three-dimensional surface foundations are computed by these expressions. It is verified that the model is well capable of reproducing the threedimensional soil-structure interaction behavior. Copyright © 1996 Elsevier Science Limited
Key words: soil-structure interaction, surface foundation.
INTRODUCTION
rational model is available for the soil at the base of a horizontal structure which rests on the ground surface. This is because the deficiency of the Winkler model is more critical to the latter case than the former case 4 and it does not appear to be possible for developing its rational form. In order to improve this deficiency, a shear coupling mechanism is introduced between the two adjacent vertical springs in the Winkler model for the static condition. 5 Adopting this concept, the author has proposed a model mode of vertical one-dimensional beams with distributed mass and horizontal springs which interconnect the two adjacent beams. 4'6'7 This model is capable o f producing the dynamic effects and the characteristics of a continuous medium. Its parameters are found to be rather uniquely related to the soil properties alone and thus are minimally dependent on the loading condition and the flexibility and size of the foundation. However, it is limited to the two-dimensional condition. This paper studies a threedimensional model in a form similar to that of the previous two-dimensional model, and presents the expressions for the subgrade behavior in response to the directly applied load and soil-structure interaction force in the three-dimensional condition. The foundation behavior which induces the vertical soil-structure interaction is considered herein.
A subgrade medium is treated by various ways for the dynamic response analysis o f foundations. These treatments include the finite element and boundary element methods, analytical solutions developed by solving the wave equations and mechanical models. The finite element method can accommodate various complex features of a subgrade but requires significant effort in computation. Analytical solutions are generally limited to very simple conditions and are often not applicable directly to the real conditions, particularly when the nonlinear behavior and complex boundary conditions are involved. The boundary element method is somewhat inbetween the finite element method and analytical solution. The computation effort associated with this method is far less than that for the finite element method but is not capable o f handling the nonlinear behavior. On the other hand, simplified mechanical models generally result in simple analytical formulations to express the behavior and yet can accommodate the nonlinear behavior in a relatively simple manner. A Winkler model is the simplest form o f a mechanical subgrade mode. A rational dynamic Winkler model has been developed for the soil at the side o f the circular cylindrical embedded structure. 1-3 However, no such 419
T. Nogami
420 SIMPLIFIED SUBGRADE MODEL
A continuous subgrade medium subjected to a vertical load is modeled as a system of one-dimensional vertical elastic beams and horizontal springs uniformly distributed along the beams: Fig. 1 shows schematically a cross-section view of the model. The springs interconnect the two adjacent beams to produce the coupling between the beams. The beam contains uniformly distributed mass to produce the dynamic effects and deforms only in the axial direction. In this mechanism, the vertical normal force is generated by the axial deformation of the beam, while the vertical shear force is by the deformation of the spring which is caused by the differential vertical deformation between the two adjacent beams. This model without springs and mass corresponds to the classical Winkler model. When a beam of a cross-section, Ax x my is considered and the stiffness of horizontal springs in the x and y directions are linarly proportional to 1~rex and 1/my, respectively, the forces at any given location of the beam are expressed in the Cartesian coordinates as
P=
mxAy
(la)
direction; and n = parameter related with spring stiffness. Then, assuming a uniform displacement within a small segment of the thickness A z in a beam, the equilibrium condition of the forces acting on this segment results in
OP
-(mrx +
+ &-mmxAymz =O
(2)
where fb = acceleration; and ATj = difference between the values Tj acting on the two vertical j faces of the beam. When A x and my are infinitely small, the substitution of eqn (1) into eqn (2) leads to the equation of motion of the subgrade model as
nV2w +
02w
k~-z2 - mfb = 0
(3)
and also eqn (1) leads to the expressions for the vertical normal stress and vertical shear stresses in the model as, respectively
P=
k Ow Oz
{txty I = --n
(4a)
{owI
141
Ow
7y
where p = vertical normal stress; tj = vertical shear stress acting on the j plane; and
Ty} =
(~yyAWy)A x
(lb)
where P = vertical normal force induced in the beam; Tj (with j = x or y ) = Vertical shear force acting on the vertical j face (i.e. the face normal to the j direction) of the beam; w = vertical displacement of the beam; mwj = difference between the displacements w of the two beams which stand side by side in the j direction; k = axial stiffness per unit cross-section area of the beam; n/mj = stiffness of the springs oriented in the j direction, per unit width of the beam face normal to t h e j
Surface Foundation
Vertical Beam with Distributed Mass
V2
02
Oq2
= OX2 Jr Oy2
(5)
The beam-spring system is divided into ! horizontal layers and the bottom of the system is fixed. It is assumed that the displacement of the beam within the ith layer is expressed in the form of
W(x,y,z)i=
{q)(Z) T Wa(X,
} {Wb(X,~))}i
(6)
where {~b(z)} is expressed so as to be w(x, y, O) = wa(x, y) and w(x, y, hi) = Wb(X,y) in which hi = thickness of the ith layer. Substituting eqn (6) into eqn (3) and applying Galarkin's procedure for the weighted residual method, eqn (3) is rewritten as
I~=1[ li'(mifbTdzlVZ(W:}i+[ Ii'~ki(a"Tdz]
{w:} i
hi Wa -[Io(amifbrdz]{#b}i={O0}
(7)
Integration of the second term by parts results in
Uniformly Distributed Springs Fig. l. Schematical view of cross-section of ground model considered.
~ __[r/]iV2 Wa Wa I {}iq_[k]i{ Wa}iq_[m]i{} { }Pa i=1 Wb Wb Wb i= Pb i
(8) where
(pa,Pb) T =
vector containing normal tractions
3-D soil-foundation interaction analysis acting on the upper and lower ends of the ith layer; and
in eqn (12). The solution of this equation for r t> R is
/i
[n]i =
I
(bni(aT dz
[k]i =
(b'kiO'I dz
[m]i =
4~rni~r dz
{w(r)} = E CiKo(Air){di} i=l (9)
Then, introducing the compatibility condition and equilibrium condition at the layer-layer interface, eqn (8) leads to the following equation of motions for the entire layered system:
- [U]V2{w(x,y)} + [K]{w(x,y)} + [M]{#(x,y)} = {p(x,y)}
where K0 = zero order modified Bessel function of second kind; C / = unknown constant; {C} = vector containing unknown constants Cl '~ C/; [A(r)] = I × I matrix containing Ko(/~ir){di} at the ith column; Ai and {di} = ith eigenvalue and eigenvector, respectively, which satisfy eqn (12) with {p} = {0}. Then, according to eqn (4b), the integration of the vertical shear stress along the circle of a radius R is
(10a)
Ida(R)] ------ 2 7 r R [ / ] [ dr J{C}
(10b)
where {w} and {#} = vectors of size I, which contain respectively the displacements and accelerations along the top ends of the layers {p} = vector of size I, which contains the external distributed normal forces along the top ends of the layers; [N], [K] and [M] = I x I resultant matrices after superimposing In], [k] and [m], respectively; and
[Ka] = [K] - 2 [M]
(I 1)
It is noted that the vectors {w} and {p} are amplitudes in the frequency domain expression. When a structure (foundation) occupies a portion of the ground surface, the ground response under the structure is described by eqn (10) with {p} which contains the soil-structure interaction force at the first location and zeros at the others, and that outside this area is by eqn (10) with {p} = {0}.
A P P L I C A T I O N S IN THE FREQUENCY-DOMAIN
Ground response to prescribed force A uniform circular line load is applied along the circle of a radius R. For this foundation geometry, it is convenient to rewrite eqn (10) in the cylindrical coordinates such that [N]VZ{w(r, 0)} + [Kd]{w(r, 0)} = {p(r, 0)}
(12)
where r and 0 = coordinates in the radial direction and in the direction normal to it, respectively; and V2
/92 10 1 02 = ~ + -~ r Orr r 2 002
(14)
= [A(r)]{c}
or, in the frequency domain,
-[N]~72{w(x,y)} + [Kd]{W(x,y)} = {p(x,y)}
421
(13)
The origin of the coordinates is taken at the center of the circular line load and this load is treated as an applied shear. Then, w is independent of 0 and {p} is equal to {0}
(15)
where IdA (R)/dr] = matrix containing -AiKt(AiR) {di} at the ith column, with Kt = first order modified Bessel function of second kind. When we take a limit R ~ 0, the above considered circular line load becomes a point load and hence 27rR{t(R)} is equal to the intensity of this point load {Q}. Also, the value RAiKt(Ai R) becomes one when we take a limit R --* 0. Therefore, this limit condition leads eqn (15) to 27r[N][d]{C} = {Q}
(16)
where [d] = I x I matrix containing {di} at the ith column. Substituting eqn (16) into eqn (14), the ground response to a point load is expressed as: {w(r) } = ~ [A (r)] [d ]-l [N]-' {Q}
(17)
An arbitrarily distributed vertical load is applied to the horizontal plane and the origin of the horizontal global coordinates is set at any location. A small area, ds, is considered at the location ~ on the horizontal plane within the loaded area. Then, integrating the response to this load at the location x over the loaded area S, the ground response at x to any arbitrarily distributed load is expressed as {w(x)} =
el,
[A(Ix-
(18)
where x, ( and S are expressed in terms of the global coordinates to evaluate the integration analytically or numerically.
Response of circular rigid massless foundation A circular rigid massless foundation of a radius R rests on the ground surface and is assumed to be subjected to a vertical or rotational excitation. The origin of the cylindrical coordinates is set at the center of the foundation. The ground is divided into the area under
T. Nogami
422
the foundation (r ~< R) and the area outside it (r 1> R). The governing equation for the ground response in the latter area is written in the cylindrical coordinates by eqn (12) with {p} = {0}. The solutions of this equation for {w} are A°(r)]{C}
(191
[al(r)]{C}cosO
{w(r, 0)} =
where the upper and lower expressions are those for the vertical response and rotational response, respectively; [A:(r)] = I × I matrix containing Kj(Air){di} at the ith column, w i t h j = 0 or 1; Ai and {di} = the ith eigenvalue and eigenvector, respectively, which satisfy eqn (12) with {p} = {0}. Then, the vertical shear stress on the 0 plane is
{t(r,O)} = _[N]~Ow(r,O) } { [N][A~o(r)]{C} l. -~r =-[N][A~(r)]{C}cosO
(20) where [Aj] = [dAJdr] = I × I matrix which contains a vector -AiKt(Air){di} for j - - 0 or -Ai(Ko(Air) + Kl (Air)/Ai){di} for j = 1 at the ith column. Eliminating {C} from eqn (20) by using eqn (19) results in the following relation at r = R: {t°(0)) = -[k°]{w°(0)}
(21)
where {w°(0)} = {w(R, 0)}; {t°(0)} = {t(R, 0)}; and [k0] :_ ~ [N][A~(R)][A0(R)] -1
[
(22)
[g] [A;(R)] [AI (R)]-I
The governing equation for the ground response in area r ~ R is eqn (12) with {0} which contains the soilstructure interaction force at the first location and zeros at the others. This equation is split into the following two equations:
-- NAAX72wA(r, O) + KAAWA(r , O) -- NABV2WB(r, O) + KAnwn(r, O) = pa(r, O) + Knnwn(r, 0) = 0
(23b)
where wA and pa = numbers located at the first location in {w} and {p}, respectively; and
{w} =
{::}
NAn] Unn J ' '
{P}=
[Kaa Kan] [Kd] = [ KnA Ken J ' {PA}
0
(24)
The vertical shear stress on the r plane is also split into
tA (r, O) = -N,4A
wA(r, O) = { rWcos00
(26)
where the upper and lower expressions are those for the vertical and rotational excitations, respectively; W and O are vertical and rotational displacement amplitudes of the rigid foundation, respectively. Assuming that wA is known, the solution of eqn (23b) for wB is obtained as
wn(r , O) = WBh(r, O) -- K~IKBAWA(r, O)
(27)
were WA is given in eqn (26) and WBh = solution of the homogeneous equation of eqn (23b), which is
Ao(rlC
WBb(r,O)=
(28/
A l(r) cos OC
where C = vector containing unknown constants C1 ~-" C/_I;Ay = ( 1 - 1) x ( 1 - 1) matrices containing lj(Air){di} at the ith column, with j = 0 or 1, in which Ai and {di} are respectively the ith eigenvalue and eigenvector defined from a homogeneous equation of eqn (23b);/j = j t h order modifed Bessel function of first kind, withj = 0 or 1. Substituting WBin eqn (27) into eqn (25b), the vertical shear stress tn is
tB(r, O) = -- NBBWBh(r, O) - [NB
- NBBIC;
KBA]w'
(r, O)
(29)
where
wtA(r,O)
OwA(r,O) _ : 0 Or
[ cos00
(30a)
(23a)
- NnaVZwA(r, O) + KnAWA(r, O) -- NnnVZwn(r, O)
INAA [U] = [ NnA
where tA and ts---number and vector, respectively, located in {t} such as (tA t~) r. The foundation imposes the following constraint in the ground displacement at soil-foundation contact:
OWn (r, O) OWB(r, O) Or NAB Or
W'Bh(r, O) = OWnh(r, O) = (~,4{)(r)C (30b) Or [. A~(r) cos OC in which Aj = dAj/dr = (I - 1) × (I - 1) matrix which contains AiIl(Air){di} for j = 0 or Ai(Io(Air)+ ll(A:)/Ai){di} f o r j = 1 at the ith column. The continuity at r = R between the two areas, r ~< R and r I> R, requires w° = WB(R < 0) and t° = tn(R, 0), where vectors w° and t ° are located in {w°} and {t °} defined in eqn (21), respectively. Using eqns (26), (28) and (30), wn in eqn (27) and tn in eqn (29) are expressed in terms of W, O and C. These expressions with r = R are substituted into eqn (21) to solve for C. In this manner, unknown C is determined as
EwW
(25a) C=
tn(r, O) = -Nna OWA_~r(r,O) Nnn Own(r'-~rO)
(31)
EoO
(25b) Thus, using eqns (26) and (28) with the above obtained
3-D soil-foundation interaction analysis C, eqn (27) can be rewritten as
B) = ~ (A°(r)Ew - K;~KB,4) W t (A| (r)Eo - Kyl Knaro) cos 0 0
Wn(r,
(32)
where
Ew = -[NnBA~(R) + k°BAo(R)] -1 o - i KBA ] - kBBK~
x
(33a)
423
in which By(r) = f~R Aj_l(r)r2dr = ( I - 1) x ( I - 1) matrix containing (RY/Ai)Iy(AiR){di} at the ith column, with j = 1 or 2; and /j = first kind modified Bessel function o f j t h order. The stiffness of the layered subgrade for the vibration of a circular rigid foundation is therefore St + $2. This stiffness is coupled with the structure model for the soilstructure interaction analysis for a structure on a rigid circular foundation.
Response of rectangular massless rigid foundation
x [[k°x - knnKBn 0 - IKn~]R
+ [NB~ - UnnKffslKnx]]
(33b)
with
k°s]
(33c)
The foundation is subjected to the reaction normal stress over the base area, the vertical shear force at the edge and the applied load. Therefore, the equilibrium condition of these forces acting on the massless foundation requires that
Q - 4 J~/2 ta(R,O) - 4 I~/2 I~ pa(r,O)rdrdO = O
A rectangular massless rigid foundation is assumed to rest on ground surface. Adopting the frequently used approach, a rectangular shape is replaced with an equivalent ellipse. For this foundation shape, it is convenient to rewrite eqn (10) in the elliptical coordinate system such that [N]VR{w(~, 7)} + J[Kd]{W((, ~7)} = J{P(~, ~/)}
(37)
where ( and ~ = elliptical coordinates on the horizontal plane with their origin at the center of the foundation; and v 2
o2
o2
(38a)
(34a)
~
M-4 - 4
/2
J = a] - b2 (cosh 2( - cos 2~/)
(RcosO)(ta(R,O)RdO)
f?I2
(r cos 0) (Px (r, O)rdrdO) = 0
(34b)
where Q and M = amplitudes of applied vertical force and moment, respectively. The expressions of normal stress Pa and vertical shear force ta are given respectively in eqns (23a) and (25a). Substituting these expressions into eqn (34) and expressing wa and wn with eqns (26)-(28) and (31), the following force-displacement relations are obtained:
(38b)
with a I and bl (al > bl) = focal distances of ellipse. The foundation is assumed to occupy the area ( ~< (0. Then, {p((, r/)} in eqn (37) contains the interaction pressure at the first location and zeros at the other locations for the area ( ~< ¢0, and all zeros for the area (/> (0- Equation (37) for each of these two areas is solved to obtain the expressions for the foundation response, in a similar manner as that presented for the circular rigid foundation.
(S 1 + 52)W = Q
(35a)
Response of flexible foundation
( S 1 Jr- 5 2 ) 0 = M
(35b)
A flexible foundation of size, B x L, is assumed to rest on the ground surface. The foundation structure is treated as a bending plate with distributed mass. Assuming that the origin of the coordinates is set at the center of the foundation, the equation of motion of this structure is written in the frequency domain as
where SI and $2 are the resultants of the first and second integrations in eqn (34), respectively, which are respectively
SI
-m-
0 0 -1 0 27rR(kanAo(R)Ew - k~BKBB Kna - kAa) 7r 0 7r 3 0 -! ko
--
kAB&(R)Eo +
(lCAB/ iBKBA -- AA)
(36a)
(39)
(36b)
where p(x, y) = interaction force at the soil-foundation contact; Q(x,y)=external vertical force applied at location (x,y) on the foundation; mf = mass per unit area of the foundation; D = flexural rigidity of the foundation; and V 4 -- V2~72. Since the foundation is located at the ground surface, p in eqn (39) is equal to
{ 2rr(KAB -- N A n N ; 1 KBB)B1 (R)Ew
$2 =
+ R2(KAA -- KABK; KBA) --Tr(KAB - NAnN;sIKoB)B2(R)Eo 71" 4
DV4w(x, y) = w2mff~(x, y) = -p(x, y) + Q(x, y)
T. Nogami
424
p at the first location of {p} in eqn (10). Therefore, substituting eqn (39) into eqn (10), the governing equation for the soil-foundation system in the area under the foundation (Ix[ ~< B and l yl ~< L) is written as [D]V4{w(x, y)} - [N]VZ{w(x, y)} + [K~]{w(x, y)} = {p(x,y)}
(40)
and that outside this area (Ix] t> B and l Yl >t L) is
-[N]V2{w(x,y)} + [Kd]{w(x,y)} = {0}
directly from field loading tests. However, they may also be determined from the soil parameters. In order to find the relations between the elastic model parameters and elastic material constants, two one-dimensional conditions as shown in Fig. 2 are considered for an infinite continous medium and an infinite model: these loading conditions produce Ow/Ox = Ow/Oy = 0 for the first condition and Ow/Ox = 0 for the second condition in the model. In the first condition, the stresses in the vertical direction induced in the continuous medium are
(41) az=(A+2G)~zz
where {p(x, y) } = vector of size I which contains Q(x, y) at the first locations and zeros at the other locations; and [K~] = [K] - wZ[[M] + [My]]
(42)
with [My] = I × I diagonal matrix containing my at the location (1, 1) and zeros at the other locations. The detailed procedure to solve the differential equations for this foundation is explained in Ref. 4 for the twodimensional problem. Following the similar procedure, eqns (40) and (41) are solved to obtain the expressions for the behavior of a foundation coupled with a subgrade medium.
MODEL PARAMETERS The elastic model parameters k and n can be determined
Model
Fig. 2. Infinite medium and model subjected to infinite loads.
~-y =
{00}
(43)
where A and G = Lam6's constants in which the latter is the shear modulus, while those induced in the model are given by eqn (4) with Ow/Ox = Ow/Oy = 0. These two expressions directly indicate that k = A + 2G. With modification, the parameter k is therefore assumed to be k = Fk(A + 2G)
(44)
where Fk is the factor to be determined later. In the second one-dimensional condition, the stresses in the vertical direction induced in the continuous medium are
~z=0
and
rx
Ty
=G
/0w/ OW
(45)
while those induced in the model are given by eqn (4) with Ow/Oz = 0. These two expressions directly indicate that n = G. With modification, the parameter n is therefore assumed to be
n = FnG
Infinite Medium or
and
(46)
where Fn is the factor to be defined later. The response to the point load is the most fundamental and the response to any load, including soilfoundation interaction pressure, can be considered to be the resultant of integration of this response. As indicated in [A(r)] in eqns (14) and (17), the displacement shape, w(r)/w(O), is directly governed by the eigenvalues and eigenvectors, defined from the homogeneous equation of eqn (12), and thus by [N] -1 [Kd]. In view of eqn (9), this implies that the elastic model parameters affect the displacement shape through the kin ratio rather than k or n alone. Therefore, the displacement shape computed by the model can be adjusted by Fk/F,, ratio. Figure 3 shows the shapes of the ground surface displacement for a uniform circular load on a homogeneous ground, which are computed with Fk/Fn = 0.5 and 1. It indicates that Fk = Fn produces the computed results very close to those computed by the continuum solution. Then, assuming Fk = F , ( = F), the values F are backcalculated to match the surface displacement computed by using the present model with that computed by the continuum solution. The displacements at the center of the circular load are used for this backcalculation. The obtained
3-D soil-foundation interaction analysis
[
1,2
~
values shown in Fig. 4 indicate that F varies very little with the variation of the H/R ratio ( H = thickness of soil medium) but varies significantly with the variation of the Poisson's ration t, for ~,/> 0.4. Accordingly, the factor F is assumed to be dependent on the Poisson's ratio only and defined from Fig. 4 as
~ u t i o n r"~Fn = 1
~
- - Present =olullon r-~=. = 0.5 t.0
I"
-'~
H/r° = 4
[8]
I" .~o=~o[8]
o.a
\~
425
v = 0.3
F = 0-73 + 1.4%,
0.8
-
16-10t, 2 + 67"4%)
93.96v 4
-
0
O
(47)
_u O. oll
t~
0.4
The soil material damping is introduced by multiplying the factor 1 + i2D (in which D = material damping parameter and i2 = - 1 ) to the elastic soil constants or to the elastic model parameters.
0.2
0000
'
01,
'
0,'
,it
'
,i,
'
210
214
Distance r/R
C O M P U T E D RESULTS AND VERIFICATIONS
Fig. 3. Displacement shape of the ground surface subjected to uniform circular load.
In order to verify the above defined model and examine
2.O 1.0
v LL ,..."
g
0.8
0.2 0.3 0 0.4
~6
0 U
o
v
1.2
0 0.2 0.3 0.4
r-
o "0
1.6
0.8 0
D. a
0.4
[
0.4 0.2 | 2
0.0 !
'
0.0
-'
'
~'=
i 4
I 6
~'s ' 2~o '
I 8
I . 10
i
Depth H/R
Depth H/R 2.0
1.0
LL
0.8
Presenl solullon
: .%',',:,
1.6
. . _ . - - - - - - ~ ' - ~•.=~ . : - .~ ~ ."~'._,--'~'..~~, . ~ ' t " = ~ ' -
•
O. ® ~
.~
0,6
H~=2
1.2
/, /)]
[]0]
, ~
v=0.4
8 e_e_
~:
04
o.e
--
H/R-4
0 U
----H~=6
m 0.4
O.(l 0.00
*
I 0.10
,
I 0.20
,
Poisson's
I 0.30
Ratio,
*
I 0.40
,
| 0.50
v
Fig. 4. Variation of factor F with soil depth and Poisson's ratio computed at the center of uniform circular load applied.
0.000
'
012
014
00'
'
013
,i0
Distance r/R
Fig. 5. Vertical displacement and contact pressure for rigid
circular foundation subjected to vertical load.
426
T. Nogami
its performance, the ground responses to the directly applied surface load and soil-structure interaction force are compared for various conditions. The study is made for the static condition first and then for the dynamic condition. The elastic model parameters are defined from the elastic soil constants according to eqns (44), (46) and (47). Figures 5 and 6 show the results computed for a static vertical load and moment applied to a rigid circular foundation, respectively. Although the loading conditions are different from those used to define the model parameters, the present approach produces computed results very close to those computed by the continuum solutions. Next, the present approach is applied for the inhomogeneous ground, in which the Young's modulus of the ground continuously increases linearly with depth. Figures 7 and 8 show the computed vertical displacements for this inhomogeneous ground: the former and latter are respectively the vertical displacements of the ground surface, subjected to a uniform vertical
1.2
=
1.0 0.8
t@ 0
0.6
0.4
O.
0.2
HI : ! ! i ! izi "l /' E" m p~E' 1
o.o
,
0.0
,
0.4
,
,
0.8
,
I
1
,
,
1.2
~
,
,
1.6
Distancor/R
Fig. 7. Vertical displacement of inhomogeneous ground subjected to vertical uniform circular load.
P
2.4
v O 4
®
E(0)
0.4
+ .e.o.= (gj ,, H~., [,!
2.0
,\
0.3
23 ~
,
2.0
=_ o
T I ~[s t E(B) H|L/B = 1 ~-
1.6
1.2
[
ev
®
0.8
0
E3
1
0.4
•
0
i 2
i 4
i 6
i
! 8
=
/ 1o
0.0
oo'
oi, ' o','
Depth WR
10 CL
OO
P
2.4
I ~ FOR=0.5 H/R=1 FIAq=2 4
v = 0.4
oi. ' ,Io
Young'sModulusE(0yE(B)
12
"6
&'
T
0~"
2.0
~=
1.6
i.-JE--i E(0)
I~ P~t=o~Uon
"~. J ,zB 4~'~L' I+
,.,
® U
/.~.~.~.~.
O. Ja a
11.8
-
0.4 00.0
0.2
0.4
0.6
0.8
| 1.0
Distance rlR Fig. 6. Rotation and vertica] contact pressure for rigid circular foundation subjected to moment.
0.0
0.0
=
I
I
I
0.2
0.4
0.6
i
e
|
O.8
1.0
Young'sModulusE(0)/E(B)
Fig. 8. Vertical displacement of rigid rectangular foundation on inhomogeneous ground subject to vertical load.
3-D soil-foundation interaction analysis a.
0.8
~
a.
i°i i r
0.0o.o
'
Prmmntsolu~on [1
~" ' o.,
0.4
°,0o
o
0.2
427
018
Go
v = 0.25 D = .05 112
,I.
'
0.1
,]o
0.0
o.O
D
0 05
I
I
0.4
=
0.8
! 1.2
=
I t.6
=
! 2.0
ao
8 8 i
0.8 0.6
• []
m
o~ ®
f,~
0.2
0.0 0.0
"
' 0.8
0.4
"
* 1.2
"
* 1.8
"
' 2.0
0.0 0.0
Go b.
o o£
0.8
ao
1.2
1 .S
2.0
b°
3.O
(3o
/
2.5 H/R
4)
i 0.4
=6
1.2
It
2
2.O
0
4) iv,
1,5
HIR = 4
v = 1/3 1.0
0r0 0
O.5
Go o
2t-
1.5
O
o
2 i
0.0 0.0
0.2
0 4
0 6
0.8
0 0.0
1.0
I 0.2
Distance r/R
0.2 I n¢
0.1
E
-o.o
0tO L)
=
I o.s
I
I o.s
I
I
J
~.o
Distance r/R
(3o
0.6
c= Sx
_E
0
-o.1
#-
I o.4
Q
o.4
O.2 Go 0
0.0
-O.2 2
-0.3
-O.4 o.0
C i
i
!
0.2
0.4
0.6
I
0.8
i
1.0
Distance r/R
Fig. 9. (a) Dynamic vertical stiffness of rigid circular foundation subjected to vertical excitation. (b) Dynamic vertical contact pressure under rigid circular foundation subjected to vertical excitation.
8
1.2 0.0
0.2
0.4
0.6
0.8
1.o
Distance r/R
Fig. 10. (a) Dynamic rotational stiffness of rigid circular foundation subjected to moment excitation. (b) Dynamic vertical contact pressure under rigid circular foundation subjected to moment excitation.
T. Nogami
428
circular load, and the vertical displacements of a rigid rectangular foundation, subjected to a vertical load. In the computation, rectangular shape is replaced with an equivalent ellipse and the ground is treated as a layered medium in which an individual layer is homogeneous. Even for inhomogeneous ground, the present approach can predict the behavior very well. The vertical and rotational dynamic responses are computed for a rigid circular massless foundation on a homogeneous ground. The computed stiffnesses are shown in Figs 9 and 10, respectively, together with the contact pressures: a0 in the figure is nondimensional frequency defined as a0 = ~zR/v~ in which Vs is the shear wave velocity of the ground. The computed vertical dynamic stiffnesses of this foundation on an inhomogeneous ground are also shown in Fig. 11: it is noted that the square foundation was used in the continuum solution but, as it is well known, its vertical response is nearly identical to that of a circular foundation when the areas of the two foundations are identical. All these computed results indicate that the present approach has excellent capability for predicting the dynamic behavior.
a.
.d.... ,,v A 1.8 ~ 1.2
1
1
o m.,.
0.8
P.,
I
®
r"
~
0.4
0.0
.
Primer.
•
solution
[11]
,
.
t
0.4
.
0.8
,
.
1.2
i
•
1.6 ao
,
.
2.0
'
'
"
2.4
"
I
2.8
3.2
P
6
~-
1.6
i ~ E(O) TH/R= 4 E(B)[.-.~LB
iv A 1.2
Hlv
l\
i/3
=
o
_E 0
t'-
0.8
~
mm=am-=m ~
0.4
.
0.0
0.4
i . = . i . . . t . 1 i i 0.8 1.2 1.6 2.0 2.4 2.8 3.2 ao
A simplified subgrade model is presented for the threedimensional soil-structure interaction analysis. This model enables us to develop closed form formulations for the subgrade response which requires much less computation effort compared to that associated with the finite element method. In addition, it can even handle inhomogeneous layered profiles in a straight forward manner. The parameters of the model are two elastic parameters and mass density. They are rather uniquely related with the soil properties alone and thus are minimally dependent on the loading conditions and the foundation conditions such as geometry, size and flexibility. Therefore, once these parameters are determined for any given condition in the foundation and loading, they can be used for other conditions regardless of the static or dynamic condition. When the soil properties are known, the model parameters also can be defined conveniently from these properties by using the formulations presented. The computed results show that the present model has excellent capabilities for predicting the subgrade behavior for both the static and dynamic conditions.
b. ~-"
1.6
1.2
o n?
0.8
rv)
I
~
0.4
Present
•
i
0.4
0.0
solution
[111
I
0.0
018
=
i
I
1.2
=
1.6
I
2,0
ao
P
(~
1.6
o
1.z
(~
0.8
¢~)
0.4
_E
0.0 0.0
,,, r - ~ E(O)
--
-
'
0.4
~
f
i
0.8
1.2 a0
I
1.6
i
?
2.0
CONCLUSIONS
|
Fig. 11. (a) Dynamic vertical stiffness of rigid circular foundation on inhomogeneous ground subjected to vertical excitation (H/B = 1). (b) Dynamic vertical stiffness of rigid circular foundation on inhomogeneous ground subjected to vertical excitation (H/B = 4).
3-D soil-foundation interaction analysis
REFERENCES 1. Baranov, V. A. On the calculation of an embedded foundation. Voprosy dinamiki prochnosti (Riga), 1967, 14, 195-209. 2. Nogami, T. & Novak, M. Coefficient of soil reaction to pile vibration. J. Geotech. Engng., ASCE, 1980, 106(GTS), 565-70. 3. Novak, M. & Beredugo, Y. The effect of embedment on footing vibration. Proc. 1st Canadian Conf. on Earthq. Engng, University of British Columbia, Vancouver, 1971, pp. 111-25. 4. Nogami, T. & Leung, M. B. Simplified mechanical subgrade model for dynamic response analysis of shallow foundations. Int. J. Earthq. Engng Struct. Dyn., 1990, 19, 1041-55. 5. Pasternak, P. L. On a New Method of Analysis of an Elastic
Foundation by Means of Two Foundation Constants. Gosudarstvennogo Izatelstvo Literaturi po Stroitelstvu I Architekutre, Moscow, 1954.
429
6. Nogami, T. Numerical simulation model for dynamic response of foundation medium. J. Engng. Mech., ASCE, 1987, 113(EM12), 1918-32. 7. Nogami, T. & Lam, Y. C. Two-parameter layer model for analysis of slab on elestic foundation. J. Engng. Mech., ASCE, 1987, l13(EMg), 1279-91. 8. Row, R. K. & Brooker, J. R. The behavior of footings restong on nonhomogeneous soil mass with a crust. Part 2. Circular footings. Canadian Geotech. J., 1981, 18, 265-79. 9. Chow, Y. K. Vertical deformation of rigid foundation on arbitary shape on layered media. Int. J. Numer. Anal. Meth. Geomech., 1987, 11, 1-5. 10. Poulos, H. G. & Davis, E. H., Elastic solutions for soil and rock mechanics. Zoiley, 1974. 11. Chow, Y. K. Vertical vibration of three dimensional rigid foundation on layered media. Earthq. Engng. Struct. Dyn., 1987, 15, 585-94.
ELSEVIER
S0267-7261
(96)00025-5
Copyright © 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 0267-7261/96/$15.00
Simplified subgrade model for three-dimensional soil-foundation interaction analysis Toyoaki N o g a m i Department of Civil and Environmental Engineering, University of Cincinnati, Cincinnati, OH 45221, USA
(Received 17 November 1995; accepted 14 May 1996) A simple mechanical model is presented for the three-dimensional dynamic soil-structure interaction analysis of surface foundations. The model is made of one-dimensional vertical beams with distributed mass and horizontal springs which interconnect the two adjacent beams. Its parameters are rather uniquely related with the soil properties alone and thus are minimally dependent on the loading condition and the foundation conditions like geometry, flexibility and size. Formulations are provided to determine the model parameters from the soil properties. Solving the governing equations of this model, expressions for the subgrade behavior in response to the applied load and soil-foundation interaction are developed in analytical forms for various cases. The dynamic and static response of three-dimensional surface foundations are computed by these expressions. It is verified that the model is well capable of reproducing the threedimensional soil-structure interaction behavior. Copyright © 1996 Elsevier Science Limited
Key words: soil-structure interaction, surface foundation.
INTRODUCTION
rational model is available for the soil at the base of a horizontal structure which rests on the ground surface. This is because the deficiency of the Winkler model is more critical to the latter case than the former case 4 and it does not appear to be possible for developing its rational form. In order to improve this deficiency, a shear coupling mechanism is introduced between the two adjacent vertical springs in the Winkler model for the static condition. 5 Adopting this concept, the author has proposed a model mode of vertical one-dimensional beams with distributed mass and horizontal springs which interconnect the two adjacent beams. 4'6'7 This model is capable o f producing the dynamic effects and the characteristics of a continuous medium. Its parameters are found to be rather uniquely related to the soil properties alone and thus are minimally dependent on the loading condition and the flexibility and size of the foundation. However, it is limited to the two-dimensional condition. This paper studies a threedimensional model in a form similar to that of the previous two-dimensional model, and presents the expressions for the subgrade behavior in response to the directly applied load and soil-structure interaction force in the three-dimensional condition. The foundation behavior which induces the vertical soil-structure interaction is considered herein.
A subgrade medium is treated by various ways for the dynamic response analysis o f foundations. These treatments include the finite element and boundary element methods, analytical solutions developed by solving the wave equations and mechanical models. The finite element method can accommodate various complex features of a subgrade but requires significant effort in computation. Analytical solutions are generally limited to very simple conditions and are often not applicable directly to the real conditions, particularly when the nonlinear behavior and complex boundary conditions are involved. The boundary element method is somewhat inbetween the finite element method and analytical solution. The computation effort associated with this method is far less than that for the finite element method but is not capable o f handling the nonlinear behavior. On the other hand, simplified mechanical models generally result in simple analytical formulations to express the behavior and yet can accommodate the nonlinear behavior in a relatively simple manner. A Winkler model is the simplest form o f a mechanical subgrade mode. A rational dynamic Winkler model has been developed for the soil at the side o f the circular cylindrical embedded structure. 1-3 However, no such 419
T. Nogami
420 SIMPLIFIED SUBGRADE MODEL
A continuous subgrade medium subjected to a vertical load is modeled as a system of one-dimensional vertical elastic beams and horizontal springs uniformly distributed along the beams: Fig. 1 shows schematically a cross-section view of the model. The springs interconnect the two adjacent beams to produce the coupling between the beams. The beam contains uniformly distributed mass to produce the dynamic effects and deforms only in the axial direction. In this mechanism, the vertical normal force is generated by the axial deformation of the beam, while the vertical shear force is by the deformation of the spring which is caused by the differential vertical deformation between the two adjacent beams. This model without springs and mass corresponds to the classical Winkler model. When a beam of a cross-section, Ax x my is considered and the stiffness of horizontal springs in the x and y directions are linarly proportional to 1~rex and 1/my, respectively, the forces at any given location of the beam are expressed in the Cartesian coordinates as
P=
mxAy
(la)
direction; and n = parameter related with spring stiffness. Then, assuming a uniform displacement within a small segment of the thickness A z in a beam, the equilibrium condition of the forces acting on this segment results in
OP
-(mrx +
+ &-mmxAymz =O
(2)
where fb = acceleration; and ATj = difference between the values Tj acting on the two vertical j faces of the beam. When A x and my are infinitely small, the substitution of eqn (1) into eqn (2) leads to the equation of motion of the subgrade model as
nV2w +
02w
k~-z2 - mfb = 0
(3)
and also eqn (1) leads to the expressions for the vertical normal stress and vertical shear stresses in the model as, respectively
P=
k Ow Oz
{txty I = --n
(4a)
{owI
141
Ow
7y
where p = vertical normal stress; tj = vertical shear stress acting on the j plane; and
Ty} =
(~yyAWy)A x
(lb)
where P = vertical normal force induced in the beam; Tj (with j = x or y ) = Vertical shear force acting on the vertical j face (i.e. the face normal to the j direction) of the beam; w = vertical displacement of the beam; mwj = difference between the displacements w of the two beams which stand side by side in the j direction; k = axial stiffness per unit cross-section area of the beam; n/mj = stiffness of the springs oriented in the j direction, per unit width of the beam face normal to t h e j
Surface Foundation
Vertical Beam with Distributed Mass
V2
02
Oq2
= OX2 Jr Oy2
(5)
The beam-spring system is divided into ! horizontal layers and the bottom of the system is fixed. It is assumed that the displacement of the beam within the ith layer is expressed in the form of
W(x,y,z)i=
{q)(Z) T Wa(X,
} {Wb(X,~))}i
(6)
where {~b(z)} is expressed so as to be w(x, y, O) = wa(x, y) and w(x, y, hi) = Wb(X,y) in which hi = thickness of the ith layer. Substituting eqn (6) into eqn (3) and applying Galarkin's procedure for the weighted residual method, eqn (3) is rewritten as
I~=1[ li'(mifbTdzlVZ(W:}i+[ Ii'~ki(a"Tdz]
{w:} i
hi Wa -[Io(amifbrdz]{#b}i={O0}
(7)
Integration of the second term by parts results in
Uniformly Distributed Springs Fig. l. Schematical view of cross-section of ground model considered.
~ __[r/]iV2 Wa Wa I {}iq_[k]i{ Wa}iq_[m]i{} { }Pa i=1 Wb Wb Wb i= Pb i
(8) where
(pa,Pb) T =
vector containing normal tractions
3-D soil-foundation interaction analysis acting on the upper and lower ends of the ith layer; and
in eqn (12). The solution of this equation for r t> R is
/i
[n]i =
I
(bni(aT dz
[k]i =
(b'kiO'I dz
[m]i =
4~rni~r dz
{w(r)} = E CiKo(Air){di} i=l (9)
Then, introducing the compatibility condition and equilibrium condition at the layer-layer interface, eqn (8) leads to the following equation of motions for the entire layered system:
- [U]V2{w(x,y)} + [K]{w(x,y)} + [M]{#(x,y)} = {p(x,y)}
where K0 = zero order modified Bessel function of second kind; C / = unknown constant; {C} = vector containing unknown constants Cl '~ C/; [A(r)] = I × I matrix containing Ko(/~ir){di} at the ith column; Ai and {di} = ith eigenvalue and eigenvector, respectively, which satisfy eqn (12) with {p} = {0}. Then, according to eqn (4b), the integration of the vertical shear stress along the circle of a radius R is
(10a)
Ida(R)] ------ 2 7 r R [ / ] [ dr J{C}
(10b)
where {w} and {#} = vectors of size I, which contain respectively the displacements and accelerations along the top ends of the layers {p} = vector of size I, which contains the external distributed normal forces along the top ends of the layers; [N], [K] and [M] = I x I resultant matrices after superimposing In], [k] and [m], respectively; and
[Ka] = [K] - 2 [M]
(I 1)
It is noted that the vectors {w} and {p} are amplitudes in the frequency domain expression. When a structure (foundation) occupies a portion of the ground surface, the ground response under the structure is described by eqn (10) with {p} which contains the soil-structure interaction force at the first location and zeros at the others, and that outside this area is by eqn (10) with {p} = {0}.
A P P L I C A T I O N S IN THE FREQUENCY-DOMAIN
Ground response to prescribed force A uniform circular line load is applied along the circle of a radius R. For this foundation geometry, it is convenient to rewrite eqn (10) in the cylindrical coordinates such that [N]VZ{w(r, 0)} + [Kd]{w(r, 0)} = {p(r, 0)}
(12)
where r and 0 = coordinates in the radial direction and in the direction normal to it, respectively; and V2
/92 10 1 02 = ~ + -~ r Orr r 2 002
(14)
= [A(r)]{c}
or, in the frequency domain,
-[N]~72{w(x,y)} + [Kd]{W(x,y)} = {p(x,y)}
421
(13)
The origin of the coordinates is taken at the center of the circular line load and this load is treated as an applied shear. Then, w is independent of 0 and {p} is equal to {0}
(15)
where IdA (R)/dr] = matrix containing -AiKt(AiR) {di} at the ith column, with Kt = first order modified Bessel function of second kind. When we take a limit R ~ 0, the above considered circular line load becomes a point load and hence 27rR{t(R)} is equal to the intensity of this point load {Q}. Also, the value RAiKt(Ai R) becomes one when we take a limit R --* 0. Therefore, this limit condition leads eqn (15) to 27r[N][d]{C} = {Q}
(16)
where [d] = I x I matrix containing {di} at the ith column. Substituting eqn (16) into eqn (14), the ground response to a point load is expressed as: {w(r) } = ~ [A (r)] [d ]-l [N]-' {Q}
(17)
An arbitrarily distributed vertical load is applied to the horizontal plane and the origin of the horizontal global coordinates is set at any location. A small area, ds, is considered at the location ~ on the horizontal plane within the loaded area. Then, integrating the response to this load at the location x over the loaded area S, the ground response at x to any arbitrarily distributed load is expressed as {w(x)} =
el,
[A(Ix-
(18)
where x, ( and S are expressed in terms of the global coordinates to evaluate the integration analytically or numerically.
Response of circular rigid massless foundation A circular rigid massless foundation of a radius R rests on the ground surface and is assumed to be subjected to a vertical or rotational excitation. The origin of the cylindrical coordinates is set at the center of the foundation. The ground is divided into the area under
T. Nogami
422
the foundation (r ~< R) and the area outside it (r 1> R). The governing equation for the ground response in the latter area is written in the cylindrical coordinates by eqn (12) with {p} = {0}. The solutions of this equation for {w} are A°(r)]{C}
(191
[al(r)]{C}cosO
{w(r, 0)} =
where the upper and lower expressions are those for the vertical response and rotational response, respectively; [A:(r)] = I × I matrix containing Kj(Air){di} at the ith column, w i t h j = 0 or 1; Ai and {di} = the ith eigenvalue and eigenvector, respectively, which satisfy eqn (12) with {p} = {0}. Then, the vertical shear stress on the 0 plane is
{t(r,O)} = _[N]~Ow(r,O) } { [N][A~o(r)]{C} l. -~r =-[N][A~(r)]{C}cosO
(20) where [Aj] = [dAJdr] = I × I matrix which contains a vector -AiKt(Air){di} for j - - 0 or -Ai(Ko(Air) + Kl (Air)/Ai){di} for j = 1 at the ith column. Eliminating {C} from eqn (20) by using eqn (19) results in the following relation at r = R: {t°(0)) = -[k°]{w°(0)}
(21)
where {w°(0)} = {w(R, 0)}; {t°(0)} = {t(R, 0)}; and [k0] :_ ~ [N][A~(R)][A0(R)] -1
[
(22)
[g] [A;(R)] [AI (R)]-I
The governing equation for the ground response in area r ~ R is eqn (12) with {0} which contains the soilstructure interaction force at the first location and zeros at the others. This equation is split into the following two equations:
-- NAAX72wA(r, O) + KAAWA(r , O) -- NABV2WB(r, O) + KAnwn(r, O) = pa(r, O) + Knnwn(r, 0) = 0
(23b)
where wA and pa = numbers located at the first location in {w} and {p}, respectively; and
{w} =
{::}
NAn] Unn J ' '
{P}=
[Kaa Kan] [Kd] = [ KnA Ken J ' {PA}
0
(24)
The vertical shear stress on the r plane is also split into
tA (r, O) = -N,4A
wA(r, O) = { rWcos00
(26)
where the upper and lower expressions are those for the vertical and rotational excitations, respectively; W and O are vertical and rotational displacement amplitudes of the rigid foundation, respectively. Assuming that wA is known, the solution of eqn (23b) for wB is obtained as
wn(r , O) = WBh(r, O) -- K~IKBAWA(r, O)
(27)
were WA is given in eqn (26) and WBh = solution of the homogeneous equation of eqn (23b), which is
Ao(rlC
WBb(r,O)=
(28/
A l(r) cos OC
where C = vector containing unknown constants C1 ~-" C/_I;Ay = ( 1 - 1) x ( 1 - 1) matrices containing lj(Air){di} at the ith column, with j = 0 or 1, in which Ai and {di} are respectively the ith eigenvalue and eigenvector defined from a homogeneous equation of eqn (23b);/j = j t h order modifed Bessel function of first kind, withj = 0 or 1. Substituting WBin eqn (27) into eqn (25b), the vertical shear stress tn is
tB(r, O) = -- NBBWBh(r, O) - [NB
- NBBIC;
KBA]w'
(r, O)
(29)
where
wtA(r,O)
OwA(r,O) _ : 0 Or
[ cos00
(30a)
(23a)
- NnaVZwA(r, O) + KnAWA(r, O) -- NnnVZwn(r, O)
INAA [U] = [ NnA
where tA and ts---number and vector, respectively, located in {t} such as (tA t~) r. The foundation imposes the following constraint in the ground displacement at soil-foundation contact:
OWn (r, O) OWB(r, O) Or NAB Or
W'Bh(r, O) = OWnh(r, O) = (~,4{)(r)C (30b) Or [. A~(r) cos OC in which Aj = dAj/dr = (I - 1) × (I - 1) matrix which contains AiIl(Air){di} for j = 0 or Ai(Io(Air)+ ll(A:)/Ai){di} f o r j = 1 at the ith column. The continuity at r = R between the two areas, r ~< R and r I> R, requires w° = WB(R < 0) and t° = tn(R, 0), where vectors w° and t ° are located in {w°} and {t °} defined in eqn (21), respectively. Using eqns (26), (28) and (30), wn in eqn (27) and tn in eqn (29) are expressed in terms of W, O and C. These expressions with r = R are substituted into eqn (21) to solve for C. In this manner, unknown C is determined as
EwW
(25a) C=
tn(r, O) = -Nna OWA_~r(r,O) Nnn Own(r'-~rO)
(31)
EoO
(25b) Thus, using eqns (26) and (28) with the above obtained
3-D soil-foundation interaction analysis C, eqn (27) can be rewritten as
B) = ~ (A°(r)Ew - K;~KB,4) W t (A| (r)Eo - Kyl Knaro) cos 0 0
Wn(r,
(32)
where
Ew = -[NnBA~(R) + k°BAo(R)] -1 o - i KBA ] - kBBK~
x
(33a)
423
in which By(r) = f~R Aj_l(r)r2dr = ( I - 1) x ( I - 1) matrix containing (RY/Ai)Iy(AiR){di} at the ith column, with j = 1 or 2; and /j = first kind modified Bessel function o f j t h order. The stiffness of the layered subgrade for the vibration of a circular rigid foundation is therefore St + $2. This stiffness is coupled with the structure model for the soilstructure interaction analysis for a structure on a rigid circular foundation.
Response of rectangular massless rigid foundation
x [[k°x - knnKBn 0 - IKn~]R
+ [NB~ - UnnKffslKnx]]
(33b)
with
k°s]
(33c)
The foundation is subjected to the reaction normal stress over the base area, the vertical shear force at the edge and the applied load. Therefore, the equilibrium condition of these forces acting on the massless foundation requires that
Q - 4 J~/2 ta(R,O) - 4 I~/2 I~ pa(r,O)rdrdO = O
A rectangular massless rigid foundation is assumed to rest on ground surface. Adopting the frequently used approach, a rectangular shape is replaced with an equivalent ellipse. For this foundation shape, it is convenient to rewrite eqn (10) in the elliptical coordinate system such that [N]VR{w(~, 7)} + J[Kd]{W((, ~7)} = J{P(~, ~/)}
(37)
where ( and ~ = elliptical coordinates on the horizontal plane with their origin at the center of the foundation; and v 2
o2
o2
(38a)
(34a)
~
M-4 - 4
/2
J = a] - b2 (cosh 2( - cos 2~/)
(RcosO)(ta(R,O)RdO)
f?I2
(r cos 0) (Px (r, O)rdrdO) = 0
(34b)
where Q and M = amplitudes of applied vertical force and moment, respectively. The expressions of normal stress Pa and vertical shear force ta are given respectively in eqns (23a) and (25a). Substituting these expressions into eqn (34) and expressing wa and wn with eqns (26)-(28) and (31), the following force-displacement relations are obtained:
(38b)
with a I and bl (al > bl) = focal distances of ellipse. The foundation is assumed to occupy the area ( ~< (0. Then, {p((, r/)} in eqn (37) contains the interaction pressure at the first location and zeros at the other locations for the area ( ~< ¢0, and all zeros for the area (/> (0- Equation (37) for each of these two areas is solved to obtain the expressions for the foundation response, in a similar manner as that presented for the circular rigid foundation.
(S 1 + 52)W = Q
(35a)
Response of flexible foundation
( S 1 Jr- 5 2 ) 0 = M
(35b)
A flexible foundation of size, B x L, is assumed to rest on the ground surface. The foundation structure is treated as a bending plate with distributed mass. Assuming that the origin of the coordinates is set at the center of the foundation, the equation of motion of this structure is written in the frequency domain as
where SI and $2 are the resultants of the first and second integrations in eqn (34), respectively, which are respectively
SI
-m-
0 0 -1 0 27rR(kanAo(R)Ew - k~BKBB Kna - kAa) 7r 0 7r 3 0 -! ko
--
kAB&(R)Eo +
(lCAB/ iBKBA -- AA)
(36a)
(39)
(36b)
where p(x, y) = interaction force at the soil-foundation contact; Q(x,y)=external vertical force applied at location (x,y) on the foundation; mf = mass per unit area of the foundation; D = flexural rigidity of the foundation; and V 4 -- V2~72. Since the foundation is located at the ground surface, p in eqn (39) is equal to
{ 2rr(KAB -- N A n N ; 1 KBB)B1 (R)Ew
$2 =
+ R2(KAA -- KABK; KBA) --Tr(KAB - NAnN;sIKoB)B2(R)Eo 71" 4
DV4w(x, y) = w2mff~(x, y) = -p(x, y) + Q(x, y)
T. Nogami
424
p at the first location of {p} in eqn (10). Therefore, substituting eqn (39) into eqn (10), the governing equation for the soil-foundation system in the area under the foundation (Ix[ ~< B and l yl ~< L) is written as [D]V4{w(x, y)} - [N]VZ{w(x, y)} + [K~]{w(x, y)} = {p(x,y)}
(40)
and that outside this area (Ix] t> B and l Yl >t L) is
-[N]V2{w(x,y)} + [Kd]{w(x,y)} = {0}
directly from field loading tests. However, they may also be determined from the soil parameters. In order to find the relations between the elastic model parameters and elastic material constants, two one-dimensional conditions as shown in Fig. 2 are considered for an infinite continous medium and an infinite model: these loading conditions produce Ow/Ox = Ow/Oy = 0 for the first condition and Ow/Ox = 0 for the second condition in the model. In the first condition, the stresses in the vertical direction induced in the continuous medium are
(41) az=(A+2G)~zz
where {p(x, y) } = vector of size I which contains Q(x, y) at the first locations and zeros at the other locations; and [K~] = [K] - wZ[[M] + [My]]
(42)
with [My] = I × I diagonal matrix containing my at the location (1, 1) and zeros at the other locations. The detailed procedure to solve the differential equations for this foundation is explained in Ref. 4 for the twodimensional problem. Following the similar procedure, eqns (40) and (41) are solved to obtain the expressions for the behavior of a foundation coupled with a subgrade medium.
MODEL PARAMETERS The elastic model parameters k and n can be determined
Model
Fig. 2. Infinite medium and model subjected to infinite loads.
~-y =
{00}
(43)
where A and G = Lam6's constants in which the latter is the shear modulus, while those induced in the model are given by eqn (4) with Ow/Ox = Ow/Oy = 0. These two expressions directly indicate that k = A + 2G. With modification, the parameter k is therefore assumed to be k = Fk(A + 2G)
(44)
where Fk is the factor to be determined later. In the second one-dimensional condition, the stresses in the vertical direction induced in the continuous medium are
~z=0
and
rx
Ty
=G
/0w/ OW
(45)
while those induced in the model are given by eqn (4) with Ow/Oz = 0. These two expressions directly indicate that n = G. With modification, the parameter n is therefore assumed to be
n = FnG
Infinite Medium or
and
(46)
where Fn is the factor to be defined later. The response to the point load is the most fundamental and the response to any load, including soilfoundation interaction pressure, can be considered to be the resultant of integration of this response. As indicated in [A(r)] in eqns (14) and (17), the displacement shape, w(r)/w(O), is directly governed by the eigenvalues and eigenvectors, defined from the homogeneous equation of eqn (12), and thus by [N] -1 [Kd]. In view of eqn (9), this implies that the elastic model parameters affect the displacement shape through the kin ratio rather than k or n alone. Therefore, the displacement shape computed by the model can be adjusted by Fk/F,, ratio. Figure 3 shows the shapes of the ground surface displacement for a uniform circular load on a homogeneous ground, which are computed with Fk/Fn = 0.5 and 1. It indicates that Fk = Fn produces the computed results very close to those computed by the continuum solution. Then, assuming Fk = F , ( = F), the values F are backcalculated to match the surface displacement computed by using the present model with that computed by the continuum solution. The displacements at the center of the circular load are used for this backcalculation. The obtained
3-D soil-foundation interaction analysis
[
1,2
~
values shown in Fig. 4 indicate that F varies very little with the variation of the H/R ratio ( H = thickness of soil medium) but varies significantly with the variation of the Poisson's ration t, for ~,/> 0.4. Accordingly, the factor F is assumed to be dependent on the Poisson's ratio only and defined from Fig. 4 as
~ u t i o n r"~Fn = 1
~
- - Present =olullon r-~=. = 0.5 t.0
I"
-'~
H/r° = 4
[8]
I" .~o=~o[8]
o.a
\~
425
v = 0.3
F = 0-73 + 1.4%,
0.8
-
16-10t, 2 + 67"4%)
93.96v 4
-
0
O
(47)
_u O. oll
t~
0.4
The soil material damping is introduced by multiplying the factor 1 + i2D (in which D = material damping parameter and i2 = - 1 ) to the elastic soil constants or to the elastic model parameters.
0.2
0000
'
01,
'
0,'
,it
'
,i,
'
210
214
Distance r/R
C O M P U T E D RESULTS AND VERIFICATIONS
Fig. 3. Displacement shape of the ground surface subjected to uniform circular load.
In order to verify the above defined model and examine
2.O 1.0
v LL ,..."
g
0.8
0.2 0.3 0 0.4
~6
0 U
o
v
1.2
0 0.2 0.3 0.4
r-
o "0
1.6
0.8 0
D. a
0.4
[
0.4 0.2 | 2
0.0 !
'
0.0
-'
'
~'=
i 4
I 6
~'s ' 2~o '
I 8
I . 10
i
Depth H/R
Depth H/R 2.0
1.0
LL
0.8
Presenl solullon
: .%',',:,
1.6
. . _ . - - - - - - ~ ' - ~•.=~ . : - .~ ~ ."~'._,--'~'..~~, . ~ ' t " = ~ ' -
•
O. ® ~
.~
0,6
H~=2
1.2
/, /)]
[]0]
, ~
v=0.4
8 e_e_
~:
04
o.e
--
H/R-4
0 U
----H~=6
m 0.4
O.(l 0.00
*
I 0.10
,
I 0.20
,
Poisson's
I 0.30
Ratio,
*
I 0.40
,
| 0.50
v
Fig. 4. Variation of factor F with soil depth and Poisson's ratio computed at the center of uniform circular load applied.
0.000
'
012
014
00'
'
013
,i0
Distance r/R
Fig. 5. Vertical displacement and contact pressure for rigid
circular foundation subjected to vertical load.
426
T. Nogami
its performance, the ground responses to the directly applied surface load and soil-structure interaction force are compared for various conditions. The study is made for the static condition first and then for the dynamic condition. The elastic model parameters are defined from the elastic soil constants according to eqns (44), (46) and (47). Figures 5 and 6 show the results computed for a static vertical load and moment applied to a rigid circular foundation, respectively. Although the loading conditions are different from those used to define the model parameters, the present approach produces computed results very close to those computed by the continuum solutions. Next, the present approach is applied for the inhomogeneous ground, in which the Young's modulus of the ground continuously increases linearly with depth. Figures 7 and 8 show the computed vertical displacements for this inhomogeneous ground: the former and latter are respectively the vertical displacements of the ground surface, subjected to a uniform vertical
1.2
=
1.0 0.8
t@ 0
0.6
0.4
O.
0.2
HI : ! ! i ! izi "l /' E" m p~E' 1
o.o
,
0.0
,
0.4
,
,
0.8
,
I
1
,
,
1.2
~
,
,
1.6
Distancor/R
Fig. 7. Vertical displacement of inhomogeneous ground subjected to vertical uniform circular load.
P
2.4
v O 4
®
E(0)
0.4
+ .e.o.= (gj ,, H~., [,!
2.0
,\
0.3
23 ~
,
2.0
=_ o
T I ~[s t E(B) H|L/B = 1 ~-
1.6
1.2
[
ev
®
0.8
0
E3
1
0.4
•
0
i 2
i 4
i 6
i
! 8
=
/ 1o
0.0
oo'
oi, ' o','
Depth WR
10 CL
OO
P
2.4
I ~ FOR=0.5 H/R=1 FIAq=2 4
v = 0.4
oi. ' ,Io
Young'sModulusE(0yE(B)
12
"6
&'
T
0~"
2.0
~=
1.6
i.-JE--i E(0)
I~ P~t=o~Uon
"~. J ,zB 4~'~L' I+
,.,
® U
/.~.~.~.~.
O. Ja a
11.8
-
0.4 00.0
0.2
0.4
0.6
0.8
| 1.0
Distance rlR Fig. 6. Rotation and vertica] contact pressure for rigid circular foundation subjected to moment.
0.0
0.0
=
I
I
I
0.2
0.4
0.6
i
e
|
O.8
1.0
Young'sModulusE(0)/E(B)
Fig. 8. Vertical displacement of rigid rectangular foundation on inhomogeneous ground subject to vertical load.
3-D soil-foundation interaction analysis a.
0.8
~
a.
i°i i r
0.0o.o
'
Prmmntsolu~on [1
~" ' o.,
0.4
°,0o
o
0.2
427
018
Go
v = 0.25 D = .05 112
,I.
'
0.1
,]o
0.0
o.O
D
0 05
I
I
0.4
=
0.8
! 1.2
=
I t.6
=
! 2.0
ao
8 8 i
0.8 0.6
• []
m
o~ ®
f,~
0.2
0.0 0.0
"
' 0.8
0.4
"
* 1.2
"
* 1.8
"
' 2.0
0.0 0.0
Go b.
o o£
0.8
ao
1.2
1 .S
2.0
b°
3.O
(3o
/
2.5 H/R
4)
i 0.4
=6
1.2
It
2
2.O
0
4) iv,
1,5
HIR = 4
v = 1/3 1.0
0r0 0
O.5
Go o
2t-
1.5
O
o
2 i
0.0 0.0
0.2
0 4
0 6
0.8
0 0.0
1.0
I 0.2
Distance r/R
0.2 I n¢
0.1
E
-o.o
0tO L)
=
I o.s
I
I o.s
I
I
J
~.o
Distance r/R
(3o
0.6
c= Sx
_E
0
-o.1
#-
I o.4
Q
o.4
O.2 Go 0
0.0
-O.2 2
-0.3
-O.4 o.0
C i
i
!
0.2
0.4
0.6
I
0.8
i
1.0
Distance r/R
Fig. 9. (a) Dynamic vertical stiffness of rigid circular foundation subjected to vertical excitation. (b) Dynamic vertical contact pressure under rigid circular foundation subjected to vertical excitation.
8
1.2 0.0
0.2
0.4
0.6
0.8
1.o
Distance r/R
Fig. 10. (a) Dynamic rotational stiffness of rigid circular foundation subjected to moment excitation. (b) Dynamic vertical contact pressure under rigid circular foundation subjected to moment excitation.
T. Nogami
428
circular load, and the vertical displacements of a rigid rectangular foundation, subjected to a vertical load. In the computation, rectangular shape is replaced with an equivalent ellipse and the ground is treated as a layered medium in which an individual layer is homogeneous. Even for inhomogeneous ground, the present approach can predict the behavior very well. The vertical and rotational dynamic responses are computed for a rigid circular massless foundation on a homogeneous ground. The computed stiffnesses are shown in Figs 9 and 10, respectively, together with the contact pressures: a0 in the figure is nondimensional frequency defined as a0 = ~zR/v~ in which Vs is the shear wave velocity of the ground. The computed vertical dynamic stiffnesses of this foundation on an inhomogeneous ground are also shown in Fig. 11: it is noted that the square foundation was used in the continuum solution but, as it is well known, its vertical response is nearly identical to that of a circular foundation when the areas of the two foundations are identical. All these computed results indicate that the present approach has excellent capability for predicting the dynamic behavior.
a.
.d.... ,,v A 1.8 ~ 1.2
1
1
o m.,.
0.8
P.,
I
®
r"
~
0.4
0.0
.
Primer.
•
solution
[11]
,
.
t
0.4
.
0.8
,
.
1.2
i
•
1.6 ao
,
.
2.0
'
'
"
2.4
"
I
2.8
3.2
P
6
~-
1.6
i ~ E(O) TH/R= 4 E(B)[.-.~LB
iv A 1.2
Hlv
l\
i/3
=
o
_E 0
t'-
0.8
~
mm=am-=m ~
0.4
.
0.0
0.4
i . = . i . . . t . 1 i i 0.8 1.2 1.6 2.0 2.4 2.8 3.2 ao
A simplified subgrade model is presented for the threedimensional soil-structure interaction analysis. This model enables us to develop closed form formulations for the subgrade response which requires much less computation effort compared to that associated with the finite element method. In addition, it can even handle inhomogeneous layered profiles in a straight forward manner. The parameters of the model are two elastic parameters and mass density. They are rather uniquely related with the soil properties alone and thus are minimally dependent on the loading conditions and the foundation conditions such as geometry, size and flexibility. Therefore, once these parameters are determined for any given condition in the foundation and loading, they can be used for other conditions regardless of the static or dynamic condition. When the soil properties are known, the model parameters also can be defined conveniently from these properties by using the formulations presented. The computed results show that the present model has excellent capabilities for predicting the subgrade behavior for both the static and dynamic conditions.
b. ~-"
1.6
1.2
o n?
0.8
rv)
I
~
0.4
Present
•
i
0.4
0.0
solution
[111
I
0.0
018
=
i
I
1.2
=
1.6
I
2,0
ao
P
(~
1.6
o
1.z
(~
0.8
¢~)
0.4
_E
0.0 0.0
,,, r - ~ E(O)
--
-
'
0.4
~
f
i
0.8
1.2 a0
I
1.6
i
?
2.0
CONCLUSIONS
|
Fig. 11. (a) Dynamic vertical stiffness of rigid circular foundation on inhomogeneous ground subjected to vertical excitation (H/B = 1). (b) Dynamic vertical stiffness of rigid circular foundation on inhomogeneous ground subjected to vertical excitation (H/B = 4).
3-D soil-foundation interaction analysis
REFERENCES 1. Baranov, V. A. On the calculation of an embedded foundation. Voprosy dinamiki prochnosti (Riga), 1967, 14, 195-209. 2. Nogami, T. & Novak, M. Coefficient of soil reaction to pile vibration. J. Geotech. Engng., ASCE, 1980, 106(GTS), 565-70. 3. Novak, M. & Beredugo, Y. The effect of embedment on footing vibration. Proc. 1st Canadian Conf. on Earthq. Engng, University of British Columbia, Vancouver, 1971, pp. 111-25. 4. Nogami, T. & Leung, M. B. Simplified mechanical subgrade model for dynamic response analysis of shallow foundations. Int. J. Earthq. Engng Struct. Dyn., 1990, 19, 1041-55. 5. Pasternak, P. L. On a New Method of Analysis of an Elastic
Foundation by Means of Two Foundation Constants. Gosudarstvennogo Izatelstvo Literaturi po Stroitelstvu I Architekutre, Moscow, 1954.
429
6. Nogami, T. Numerical simulation model for dynamic response of foundation medium. J. Engng. Mech., ASCE, 1987, 113(EM12), 1918-32. 7. Nogami, T. & Lam, Y. C. Two-parameter layer model for analysis of slab on elestic foundation. J. Engng. Mech., ASCE, 1987, l13(EMg), 1279-91. 8. Row, R. K. & Brooker, J. R. The behavior of footings restong on nonhomogeneous soil mass with a crust. Part 2. Circular footings. Canadian Geotech. J., 1981, 18, 265-79. 9. Chow, Y. K. Vertical deformation of rigid foundation on arbitary shape on layered media. Int. J. Numer. Anal. Meth. Geomech., 1987, 11, 1-5. 10. Poulos, H. G. & Davis, E. H., Elastic solutions for soil and rock mechanics. Zoiley, 1974. 11. Chow, Y. K. Vertical vibration of three dimensional rigid foundation on layered media. Earthq. Engng. Struct. Dyn., 1987, 15, 585-94.