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Finite layer analysis of surface foundation
Compuk-rs & Sructures
Vol. 45, No. 2. pp. 325-332,
1992
C045-7949/92 $5.00 + 0.00 &?J1992 Pergamon Press Ltd
Printed in Great Britain.
FINITE LAYER ANALYSIS OF SURFACE FOUNDATION S.
and Y. K. CHOW
SWADDIWUDHIPCING
Department of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 05 11 (Received 30 July 1991) Abstract-The behaviour of surface foundations on three-dimensional soil stratum is studied using the finite layer method. The responses resulting from various sources of actions and excitations are considered. Suitable displacement functions are chosen such that the near field compatibility conditions are satisfied and the resulting governing equations are uncoupled termwise. A three-dimensional domain can thus be treated effectively as an equivalent one-dimensional finite element problem making the method suitable for any personal computer. The finite layer solutions are shown to be in good agreement with available
theoretical results.
INTRODUCTION The static rectangular
tion series in the remaining directions. This results in a semi-analytical process involving a collection of a special kind of elements; namely strip, layer and prism elements. Only the finite layer element is discussed in this paper. A finite layer element is a layer of one-dimensional elements across its thickness with two nodal surfaces at the two end plates. A three-dimensional problem is effectively reduced to an equivalent one-dimensional finite element problem. The displacement functions for a typical linear layer element with two nodal surfaces i and i as shown in Fig. 1 are expressed as
and steady-state dynamic response of surface foundations on layered soil media
could, in principle, be studied using a threedimensional finite element model. But, in practice such analyses are prohibitively expensive and are generally not attempted, except for important projects like foundations for offshore gravity platforms and nuclear power stations. In many problems the underlying soil deposit may be considered to be horizontally stratified. In such cases, this feature may be exploited using the finite layer method. With a suitable choice of displacement functions, a threedimensional domain can be effectively treated as an equivalent one-dimensional problem. The finite layer method has previously been used to study the vertical response of rectangular foundations under static loads [ 1,2]. Its two-dimensional equivalent, the finite strip method, has been used to study the static and dynamic response of surface strip foundations [3,4]. In this paper, the behaviour of three-dimensional rectangular foundations on a soil stratum is studied using finite layer method. The responses resulting from various sources of actions and excitations are considered.
m=l
u=
2 5 [Cl-I;bimn+ m=l
w=
n-1
5 i
m-l
T”jmnlvxmvyn(2)
n-l
[Cl -
Owimn
n=l
+
Twjmnlw.xm wym(3)
where t; = z/h; h is the thickness of the layer, U,,, U,, , V,, , Vyn, W,, and WY, are continuous, differentiable functions for u, v and w, respectively and r, , tl , r2, t,, r, and t, are the numbers of functions adopted in the study.
FINITE LAYER METHOD
The standard finite element formulation for the three-dimensional analysis of solids generally employs polynomial displacement functions in all three directions. However, it was pointed out by Cheung [5] that if the material and geometric properties of the domain do not vary significantly in certain directions, the use of standard finite elements is both unnecessary and extravagant. A better proposition is to adopt the finite element procedure only in the direction where changes in geometry or/and properties are to be. considered and use an analytical technique employing continuous, differentiable smooth func-
DISPLACEMENT
FUNCTIONS
For boundary value problems with finite dimensions the chosen displacement functions must satisfy completely at least the geometric boundary conditions at the edges. In foundation problems, the assumed end boundary conditions of the finite model replacing the infinite horizontal extent of the soil are less critical provided that the adopted domains are sufficiently large such that the end conditions do not affect the region of interest significantly. It is more
325
S. SWADDIWUDHIP~NG and Y. K. CHOW
326
V, U,, = cos 7
sin 7
W,, W, = sin y
cos 7
(8) ,
(9)
where A and B are the boundary length of the layer in x and y directions, respectively as indicated in Fig. 1. Orthogonality of the functions can be demonstrated easily. GOVERNING
Rigid base
The governing equations of motion layer problem is expressed as
wlw)~
I
j
+ vw4
for a finite
= {N>L
(10)
where [M] and [K] are the mass and stiffness matrices; {C(r)} and {n(r)} the acceleration and displacement vectors, respectively and {p(t)) the external force vector in the time domain. The explicit forms of the mass and stiffness matrices as well as the consistent force vector based on the adopted displacement fields are given in the Appendix. For harmonic excitation at frequency w
“jm1 ‘jrn aWjrn
LW Fig. 1. Finite layer modelling of soil stratum.
crucial to ensure that the functions are able to produce correctly the response in the near field. If the chosen displacement functions are orthogonal such that the resulting governing equations are uncoupled termwise, it is possible to solve the problem term by term resulting in (i) significantly faster solution times, (ii) substantially reduction in core storage requirement and (iii) a self-terminating algorithm based on a specified degree of required accuracy. The adopted finite layer model, which essentially reduces a threedimensional field to an equivalent one-dimensional finite element problem, enhances further the advantages of the orthogonal property of the displacement functions as the matrices involved in the computation become very narrowly banded. In the present study, the following functions are chosen to span the displacement fields in the x and y directions of the domain: (a)
EQUATIONS
W))
= {P(~))exp(iW
(11)
where {p(o)} is the complex force vector in the frequency domain, the corresponding response is in the form {u(r)} = {ti(w)}exp(ior)
(12)
in which {P(o)} is the complex displacement vector. In view of eqns (11) and (12), eqn (10) becomes
WI - ~2wfl)~~(~)~= {P(o)1
(13)
which is the governing equation in the frequency domain. If the external loading is gradually applied, eqn (10) is independent of time and reduces to static equations of the form
Under vertical and rocking loadings
mu1 = IPI.
(14)
(4) STATIC
(5) w,, WY,= sin 7 (b) Under horizontal
sin y.
(6)
action in the x direction
A
u
(7)
ANALYSIS
Figure 1 shows a uniformly loaded flexible footing of dimension a x b on a layer of soil stratum of depth Zf overlying a rough rigid base. The size of the domain under study is taken as A x B. Though only square footing is considered in this paper, the extension of the approach to cover rectangular footing is obvious and straightforward. Preliminary analyses show that good accuracy is obtained when ten finite layers with finer mesh near the soil surface are adopted. The results presented in
Finite layer analysis of surface.foundation
a2
._____
c
.$:
---
8:
‘0
1.8;60
I
121 0
I 10
20 Number
30 of odd
10 harmmlc
50
60
70
80
terms
Fig. 2. Convergence of vertical deflection for various boundary lengths (H = 20n, a = 6, z = O.la).
Fig. 4. Convergence of vertical deflections at various depths (H = 20a, A = B = 12~).
the paper are based on the above-mentioned modelling. The accuracy of the finite layer method depends largely on (i) the boundary length of the layer elements employed in the analysis and (ii) the number of displacement functions used in the study. The latter in turn depends on the boundary length stated in (i).
gence is slow at the surface but improves markedly at lower level. The accuracy of the finite layer method is demonstrated through the comparison of the finite layer solutions with the analytical results given by Harr [6]. The results as illustrated in Figs 6 and 7 for vertical displacements and normal stresses at various depths, respectively show good agreement.
Flexible footings under umform vertical pressures
Flexible footings under uniform horizontal loads
The influence of the domain size with respect to the loaded area (A/a ratio) on the accuracy of vertical displacements and normal stresses as well as the rate of convergence of the results are illustrated in Figs 2 and 3. The results are improved with a relatively larger domain but the convergence is slower, i.e. more harmonic functions are required for reliable converging solutions. As expected, displacements converge faster than stresses and about 80-odd harmonic functions are required for the convergence of normal stress at the centre of the loaded area on the soil surface when A = B = 12a. To ensure the convergence of vertical displacements for the same domain size requires only about 50 odd harmonic terms. The results presented indicate that the size of the domain should be at least 12 times the size of the loading area. Convergence of deflection and normal stresses at various depth is shown in Figs 4 and 5. The conver-
Figures 8 and 9 show the influence of the boundary length on the accuracy of the horizontal displacements and stresses as well as their rate of convergence. The convergence of the results at various depths was illustrated in Figs 10 and 11. Similar conclusions arrived at earlier for vertical response are applicable for the present case. The only significant difference between the two cases is the rate of convergence of the results. Horizontal responses seem to converge faster than their vertical counterparts and only about 30-40-odd harmonic functions are required to produce values with satisfactory accuracy for A/a = B/b = 12. Comparison of the horizontal stress distribution along the height from the finite layer method with the analytical solutions presented by Milovic and Tournier [7] is illustrated in Fig. 12. Good agreement is observed.
-0 1
se
-0L
Ii-------_-_-_-_ i I\‘; *c___-____
0 fh~mkr
of odd
harmonic
terms
Fig. 3. Convergence of normal stresses at soil surface for various boundary lengths (H/u = 20, (I = b).
in
-.__-----
20
a
LO
Number
of add
harmonic
__----
YI
60
--
m
z-200
I
tzO&
0
terms
Fig. 5. Convergence of normal stresses at various depths (H = 20a, A = B = 12u, (I = b).
S.
328
SWADDIWUDHIWNG
and Y. K.
CHOW
Vertical displacement (mm] 0.0
02
0.1
I
0.0
0.6
I
1.0 -
08
10
12
o/O
zo30-
+
0
1.0 - : 5.0. 0 60-
-
I
Herr I19661 Finite layer
0
Number of odd harmonic 1.0 -
terms
Fig. 9. Convergence of horizontal stresses for various boundary lengths (H = 2a, v = 0.15, z = O.la).
8.0
Fig. 6. Comparison of vertical deflection along the depth. Stress/pressure ratio 02r
ooOO
01
0.6I
06
I
o------
10 2.0 -
$ jo;f 1.0 50
-
Harr (19661 Finite layer
0
60b
Number of odd hormomc terms
Fig. 10. Convergence of horizontal deflections at various depths (H = 2a, v = 0.15, A = B = 12a).
I
::L Fig. 7. Comparison of vertical stress-applied pressure ratio along the depth.
-----A:E:Lo ---A:*:60 -.--b;8=,211
--b:B=16a
10
20
IO
LO
10
Number ol odd hormomc
10
20
30
LO
50
60
m
60
70
80
terms
Fig. Il. Convergence of horizontal stresses at various depths (H = 2a, v = 0.15, A = B = 12a).
Number of odd hormomc terms
Fig. 8. Convergence of horizontal displacements for various boundary lengths (H = 2a, v = 0.15, .r = O.la).
Rigid footings
The approach based on flexibility concept is adopted in the analysis of rigid foundations. The contact area between the footing and the soil stratum is discretized into several subregions and a set of relevant flexibility equations established using unit
load method. The relationship between the interactive loads and the appropriate displacements at the contact region can be obtained by constraining the displacements in the loaded region such that the required deformation pattern of a rigid body motion is satisfied. The method was employed earlier by Cheung and Zienkiewicz [8] to analyse plates on elastic foundation. Details of the procedure were given by Chow [9].
329
Finite layer analysis of surface foundation Hr&.tial-stresslunnifam-shwr ratio 0.1 0.1 0.6 0.B 1.0
Table 1. Comparison of stiffness of rigid footings under vertical load (v = 0.3)
1.2
H /-P
No. of subregions
la
2a
4a
6a
2x2
1.675
4x4
1.825 1.942 1.972
1.342 1.437 1.517 1.493
1.205 1.279 1.295 1.302
1.153 1.220 1.247 1.256
8X8
[91 -
Wilwic & Townier I19711 0
DYNAMIC RESPONSE
finite layer
Fig. 12. Comparison of horizontal stresses at various depths (H = 2a, Y= 0.15).
The stiffnesses of a rigid square footing resting on a soil stratum are defined as k
_
I)-
q,41 WE
v3
w
and
where k is the stiffness, q the pressure, a the width of the footing, w and u the vertical and horizontal displacement components, E and v the Young’s modulus and the Poisson’s ratio of the soil and subscripts tt and h implying vertical and horizontal components, respectively. The stiffnesses of the footings resting on soil strata of various depths under vertical load are given in Table 1. The finite layer solutions obtained from three mesh sizes of loaded regions, 2 x 2, 4 x 4 and 8 x 8 are compared with the values reported by Chow 191.They show good augment and the study indicates that 8 x 8 subregions of loaded area are generally required to produce results with good accuracy. This is similar to the requirement suggested earlier by Chow et al. [3,4] for strip footings. Table 2 shows the stiffness values of the footings resting on a soil stratum of different Poisson’s ratio under horizontal load. The results from finite layer method is compared with those obtained from the formula for a rigid circular footing resting on a soil stratum of depth 2a over a rigid base as suggested by Kausel [lo]. The equivalent area method is adopted in the comparison. ReasonabIe agreement between the two solutions is achieved in all cases except when v is closed to OS. A larger discrepancy observed in the latter is most likely due to the enlarging effect of the boundary influence resulting from the incompressibility condition of the soil when v approaches 0.5. C%S 492-l
In dynamic analysis, the inherent errors of the radiation of stress wave are introduced through the presence of artificially conceived boundaries at the edges. These spurious effects can be rectified by adopting the domain which is sufficiently large such that the reflected stress wave from the ~unda~es are damped out by the soil material damping before reaching the region of interest. The effect of material damping in soil which is usually of hysteretic nature is generally, taken into account by replacing the Young’s modulus of soil, E, by a complex modulus of the form [l I] E* = E(1 + 2ij)
(171
where i = ,/- 1 and j.? the damping ratio. The following dimensionless parameters are introduced: (a) Compliance
functions uG,a Fh = Px F,=-.
(18)
wG,a
(19)
PZ
(b) Impedance functions 1 lu,=-=k,,(k,+iA,c,)(l Fh
+21/Q
(20)
&=$=
+ 2iB).
(21)
L’
k&k, + iA,,c,)(l
In eqns (18X21), u and w are the complex displacement components at the centre of the footing, G, the soil shear modulus, a the width of the foundation,
Table 2. Comparison of stiffness of rigid footings under horizontal load (H = 2~) V
No. of
subregions 2x2 4x4 8x8 DOI
0.0
0.15
0.30
0.49
1.160 1.252 1.307 1.288
1.094 1.180 1.228 1.210
1.073 1.156 1.205 1.165
1.193 1.281 1.341 1.144
330
S. SWADD~WUM~QNG and Y. K.
p, and pz the amplitudes of the disturbances, A,, = oa/2v, the dimensionless frequency, v, = ,/(G,/p) the shear wave velocity of soil, k, the static stiffness of the footing, k the dynamic stiffness coefficient, c the damping coefficient and subscripts h and v denoting horizontal and vertical components, respectively. Earlier studies [3,4] using the finite strip method to study the response of layered soil media indicate that the accuracy of dynamic solutions is more sensitive to the size of the domain than that of static results. This is due largely to an additional effect of radiating stress waves reflecting from the artificially created edge boundaries. These inherent errors are usually rectified through the soil material damping and an appropriate boundary length such that reflected stress waves from boundaries are damped out before reaching the region of interest. For finite layer solutions, the ratio of the boundary length to the width of the footing (A/a) can be estimated from an empirical formula A /a = 300//I.
(22)
It was also observed that the spurious oscillations are less pronounced for vertical vibration and/or at low frequency. The latter is due to the absence of radiating stress waves at frequencies below the fundamental frequency of the system. The studies also pointed out that increasing the size of the domain required more harmonic terms in the Fourier series if convergence of solutions is to be attained. Though the algorithm is self terminating, the number of odd harmonic terms, r and t, required for convergence may be calculated from r = (5/2)(A/a)
+ 10
(23)
f = (5/2)(B/b) + 10.
(24)
and
02
DO
OL
08
10
frequency.
A0
06
Oimtnsiorkrs
12
11
11
IL
i-gz_ _-’
__-- /’
---I02 (ID
00
Ok
06
0‘
Oimtnoionftss
frtqumcy.
10 A0
Fig. 13. Vertical compliance functions for flexible footings with various soil damping ratio (H/o = 2, A/a = 60, Y=
0.4).
CHOW
---
01
p :10x =mx
--p
t
no1 00
’
02
’
01,
’
06
Dimtr6im~
DOI no
’
011
frtqwncy,
’
IO
’
n2
01
06
Oimensiankss
08
12
IL
A0
1
10
fnquency,
12
1
IL
A0
Fig. 14. Horizontal compliance functions of flexible footings with various
material
damping
ratios
(H/a = 2,
A/a = 60, v = 0.4).
The effect of the material damping ratio, /I, on the variations of vertical and horizontal responses of flexible footings are shown in Figs 13 and 14, respectively. As expected, the real parts of both compliance functions decrease when higher values of damping ratio are present. The resonant frequencies of the real parts for both functions are also marginally lower for higher damping ratio. The values are lower comparing to their imaginary counterparts. The same flexibility concept adopted earlier for static analysis is also employed to study dynamic responses of rigid footings. As in static case, the study indicates that discretizing the loaded region into 8 x 8 subregions is sufficient to produce results with acceptable accuracy. Figures 15 and 16 depict respectively the variations of vertical and horizontal compliance functions for flexible and rigid footings. Dynamic responses of flexible foundation are always higher than those of rigid footings. The finite layer solutions are compared with other published results in Figs 17 and 18. The vertical impedance functions of a rigid surface footing resting on a soil stratum of depth H = 2a, /I = 5% and v, = l/3 resulting from the finite layer method and those reported by Chow [9] are depicted in Fig. 17. They show good agreement. The comparison of horizontal impedance functions from finite layer analysis with the values calculated based on an equivalent area of a rigid circular footing presented by Kausel [IO] is illustrated in Fig. 18. The real parts from both approaches agree well while only fair agreement is achieved for the imaginary parts especially when the dimensionless frequency, Ao, is in the range 0.5-0.9, a substantial difference is observed.
Finite layer analysis of surface foundation
i7oL 00
331
’ 02
Dl
0.6
Dimeosmnless DQ
an
a2
01
a6
Oimehonless
08
10
1.2
00
I 1.2
1Q
frequency,
do
..
11
froquey+Ao
IQ 0.e =
06 01 0.2 LIQ
---chsrl1so7~ FititS IqH
___‘_ I
‘,
___-
0.8 IQ 12 Ql ns ~~~io~~sfroquery, A, Fig. 17. Comparison of vertical impedance functions of rigid footings (H = 34 j.i = 5%, v = l/3). 0.0
lI2
01
Oh
Oimmrionloss
Do
IQ
lroqueocy
12
0.D
0.2
IL
( A,
Fig. 15. Vertical compliance functions for flexible and rigid footings (H/a = 2, @= 5%, v = 0.3). CONCLUSIONS
The finite layer method has been adopted to study both static and dynamic responses of surface footings resting on a soil stratum overlying a rough rigid base. Both vertical and horizontal loadings and excitations are considered in the analyses. Appropriate functions satisfying the near field compatibility conditions and producing uncoupling governing equations termwise are chosen to span the displacement field. A threedimensional domain can thus be treated effectively as
one-dimensional finite element an equivalent problem. The method generally occupies small core storage, involves very little data preparation and requires rather short computer time making it suitable for any small personal computer. The accuracy of finite layer solutions depends largely on (i) the choice of domain size such that far field effect is minimal and (ii) the number of harmonic functions spanning the displacement field. The inherent spurious oscillations resulting from the artificially created edge boundaries can be rectified through the proper choice of domain size and the presence of material damping in soils such that the reflected stress waves are damped out before reaching the region of interest. Larger domain is necessary at low soil damping ratio and/or high loading frequency. Results from finite layer analysis agree reasonably well with other theoretical solutions.
10 as
Dimensionless
frequency.
A,
Q~QO--.rL Oimeosiooksr
frequency,
A,
-----limb& Ri$td I,0
Lmmsimlm
12
14
freqlnncy , A,
Fig. 16. Horizontal compliance functions for flexible and rigid footings (H/u = 2, j? = 5%, v = 0.4).
00
0.2
OL Dimensionloss
OK
oe
IO
frcqucncy,
A0
I2
II
Fig. 18. Comparison of horizontal impedance functions of rigid footings (H = 4a, ,!I= 5%, v = l/3).
332
S. SW~DIWUD~~NG and Y. K. REFERENCES
I,, = - 3k,(C,
1. Y. K. Cheung and S. C. Fan, Analysis of pavement and layered foundation by finite layer method. Proceedings 3rd International Conference on Numericaf Methods in Geomechanics. Aachen, pp. 1129-i 13.5(1979).
2. J. R. Booker and J. C. Small, Finite layer analysis of layered pavements subjected to horizontal loading. Proceedings 6th International Conference on Numerical Methods in Geomechanics. Innsbruck, pp. 2109-2113
(1988). 3. Y. K. Chow, S. Swaddiwudhipong and S. A. Lim, Dynamic finite strip analysis of surface foundations. Eart~~e Engng Strucr. Dyn. 16, 457-467 (1988). 4. Y. K. Chow, S. Swadd~wu~ipong and K. F. Phoon, Finite strip analysis of strip footings: horizontal loading. Comput. Geotech. 8, 65-86 (1989). 5. Y. K. Cheung, Finite Strip Method in Structural Analysis. Pergamon Press, Oxford (1976). 6. M. E. Harr, Foundations of Theoretical Soil Mechanics. McGraw-Hill, New York (1966). 7. D. M. Milovic and J. P. Toumier, Stresses and displacements due to a rectangular load on a layer of finite thickness. Soil and Foundations 11, l-28 (1971). 8. Y. K. Cheung and 0. C. Zienkiewicz, Plates and tanks on elastic foundations-an application of the finite element method. ht. J. Solids Srrucrures 1, 451-461 (1965). 9. Y. K. Chow, Vertical defo~ation of rigid foundations of arbitrary shape on layered soil media. Inf. J. Numer. Anal. Me& Geomech. 11, l-15 (1987). 10. E. Kausel, Forced vibrations of circular foundations on layered media, MIT Research Report R74-11, Massachusetts Institute of Technology (1974). 11. F. Molenkamp and I. M. Smith, Hysteretic and viscous material damping. Int. .I. Numer. Anal. Meth. Geomech. 4, 293-311 (1980).
The stiffness matrix of a layer element based on the displacement functions shown in eqns (4)-(6) is expressed as
W”1,= g
12*= 2(k; C, + k:G,)
If the functions of eqns (7)-(9) are adopted to span the displacement field. the stiffness matrix becomes I
0 I 41
+ 6G,
IX = 3k,(C2 - G,) 42
=
4,
I,=(k;C,+k:G,)-6G, Ie2 = -3k,(C,
+ G,)
I,, = 6C, + 2(kS, + k;)G, I43 = -IfJ I53 = -I, &,=
-6C,+(k;+k;;)G,
I44=I,, I54 = I11 I, = -zx, Is, = I22 I65 = -I,2 Ze6= I,, k,=!$!
=p,,,h
E*(l - v,) c1 = (1 + v,)(l - 2vJ VIE* -2vJ
c2=(l+v,)(l E*
where E* = E(1 + Zig), E = soil Young’s modulus and v, = soil Poisson’s ratio. The mass matrix of both sets of displacement fields is as follows:
[Ml,,, = $
symmetric
:
+ G,)
G”=2(1 +v,)3
APPENDIX
&lm = g
CHOW
symmetric 7 2 0 2 0 0 2 1 0 0 2 01002 0 0 10 0 21
in which p is the soil density. The force vectors for vertical and horizontal actions are expressed, respectively as
122 43
-42
0
0
0
42
0
-&,
-153
IA4 0
163 0
4s
--I65
166
where Z,, = (k,$,
+ k;G,) +6G,
where 4, = p,i, 1: = p,Z and
Zz, = 2k,,,k,(C, + G,) I,, = 3k,(C, 14, = (k;C,
I,, = O.SI,,
-G,) + k;G,) - 6G,
px and p, are the horizontal and vertical loads, respectively
acting on the rectangular patch bounded by the lines x = x, , x =,x2, y =y, and y = yz.
Vol. 45, No. 2. pp. 325-332,
1992
C045-7949/92 $5.00 + 0.00 &?J1992 Pergamon Press Ltd
Printed in Great Britain.
FINITE LAYER ANALYSIS OF SURFACE FOUNDATION S.
and Y. K. CHOW
SWADDIWUDHIPCING
Department of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 05 11 (Received 30 July 1991) Abstract-The behaviour of surface foundations on three-dimensional soil stratum is studied using the finite layer method. The responses resulting from various sources of actions and excitations are considered. Suitable displacement functions are chosen such that the near field compatibility conditions are satisfied and the resulting governing equations are uncoupled termwise. A three-dimensional domain can thus be treated effectively as an equivalent one-dimensional finite element problem making the method suitable for any personal computer. The finite layer solutions are shown to be in good agreement with available
theoretical results.
INTRODUCTION The static rectangular
tion series in the remaining directions. This results in a semi-analytical process involving a collection of a special kind of elements; namely strip, layer and prism elements. Only the finite layer element is discussed in this paper. A finite layer element is a layer of one-dimensional elements across its thickness with two nodal surfaces at the two end plates. A three-dimensional problem is effectively reduced to an equivalent one-dimensional finite element problem. The displacement functions for a typical linear layer element with two nodal surfaces i and i as shown in Fig. 1 are expressed as
and steady-state dynamic response of surface foundations on layered soil media
could, in principle, be studied using a threedimensional finite element model. But, in practice such analyses are prohibitively expensive and are generally not attempted, except for important projects like foundations for offshore gravity platforms and nuclear power stations. In many problems the underlying soil deposit may be considered to be horizontally stratified. In such cases, this feature may be exploited using the finite layer method. With a suitable choice of displacement functions, a threedimensional domain can be effectively treated as an equivalent one-dimensional problem. The finite layer method has previously been used to study the vertical response of rectangular foundations under static loads [ 1,2]. Its two-dimensional equivalent, the finite strip method, has been used to study the static and dynamic response of surface strip foundations [3,4]. In this paper, the behaviour of three-dimensional rectangular foundations on a soil stratum is studied using finite layer method. The responses resulting from various sources of actions and excitations are considered.
m=l
u=
2 5 [Cl-I;bimn+ m=l
w=
n-1
5 i
m-l
T”jmnlvxmvyn(2)
n-l
[Cl -
Owimn
n=l
+
Twjmnlw.xm wym(3)
where t; = z/h; h is the thickness of the layer, U,,, U,, , V,, , Vyn, W,, and WY, are continuous, differentiable functions for u, v and w, respectively and r, , tl , r2, t,, r, and t, are the numbers of functions adopted in the study.
FINITE LAYER METHOD
The standard finite element formulation for the three-dimensional analysis of solids generally employs polynomial displacement functions in all three directions. However, it was pointed out by Cheung [5] that if the material and geometric properties of the domain do not vary significantly in certain directions, the use of standard finite elements is both unnecessary and extravagant. A better proposition is to adopt the finite element procedure only in the direction where changes in geometry or/and properties are to be. considered and use an analytical technique employing continuous, differentiable smooth func-
DISPLACEMENT
FUNCTIONS
For boundary value problems with finite dimensions the chosen displacement functions must satisfy completely at least the geometric boundary conditions at the edges. In foundation problems, the assumed end boundary conditions of the finite model replacing the infinite horizontal extent of the soil are less critical provided that the adopted domains are sufficiently large such that the end conditions do not affect the region of interest significantly. It is more
325
S. SWADDIWUDHIP~NG and Y. K. CHOW
326
V, U,, = cos 7
sin 7
W,, W, = sin y
cos 7
(8) ,
(9)
where A and B are the boundary length of the layer in x and y directions, respectively as indicated in Fig. 1. Orthogonality of the functions can be demonstrated easily. GOVERNING
Rigid base
The governing equations of motion layer problem is expressed as
wlw)~
I
j
+ vw4
for a finite
= {N>L
(10)
where [M] and [K] are the mass and stiffness matrices; {C(r)} and {n(r)} the acceleration and displacement vectors, respectively and {p(t)) the external force vector in the time domain. The explicit forms of the mass and stiffness matrices as well as the consistent force vector based on the adopted displacement fields are given in the Appendix. For harmonic excitation at frequency w
“jm1 ‘jrn aWjrn
LW Fig. 1. Finite layer modelling of soil stratum.
crucial to ensure that the functions are able to produce correctly the response in the near field. If the chosen displacement functions are orthogonal such that the resulting governing equations are uncoupled termwise, it is possible to solve the problem term by term resulting in (i) significantly faster solution times, (ii) substantially reduction in core storage requirement and (iii) a self-terminating algorithm based on a specified degree of required accuracy. The adopted finite layer model, which essentially reduces a threedimensional field to an equivalent one-dimensional finite element problem, enhances further the advantages of the orthogonal property of the displacement functions as the matrices involved in the computation become very narrowly banded. In the present study, the following functions are chosen to span the displacement fields in the x and y directions of the domain: (a)
EQUATIONS
W))
= {P(~))exp(iW
(11)
where {p(o)} is the complex force vector in the frequency domain, the corresponding response is in the form {u(r)} = {ti(w)}exp(ior)
(12)
in which {P(o)} is the complex displacement vector. In view of eqns (11) and (12), eqn (10) becomes
WI - ~2wfl)~~(~)~= {P(o)1
(13)
which is the governing equation in the frequency domain. If the external loading is gradually applied, eqn (10) is independent of time and reduces to static equations of the form
Under vertical and rocking loadings
mu1 = IPI.
(14)
(4) STATIC
(5) w,, WY,= sin 7 (b) Under horizontal
sin y.
(6)
action in the x direction
A
u
(7)
ANALYSIS
Figure 1 shows a uniformly loaded flexible footing of dimension a x b on a layer of soil stratum of depth Zf overlying a rough rigid base. The size of the domain under study is taken as A x B. Though only square footing is considered in this paper, the extension of the approach to cover rectangular footing is obvious and straightforward. Preliminary analyses show that good accuracy is obtained when ten finite layers with finer mesh near the soil surface are adopted. The results presented in
Finite layer analysis of surface.foundation
a2
._____
c
.$:
---
8:
‘0
1.8;60
I
121 0
I 10
20 Number
30 of odd
10 harmmlc
50
60
70
80
terms
Fig. 2. Convergence of vertical deflection for various boundary lengths (H = 20n, a = 6, z = O.la).
Fig. 4. Convergence of vertical deflections at various depths (H = 20a, A = B = 12~).
the paper are based on the above-mentioned modelling. The accuracy of the finite layer method depends largely on (i) the boundary length of the layer elements employed in the analysis and (ii) the number of displacement functions used in the study. The latter in turn depends on the boundary length stated in (i).
gence is slow at the surface but improves markedly at lower level. The accuracy of the finite layer method is demonstrated through the comparison of the finite layer solutions with the analytical results given by Harr [6]. The results as illustrated in Figs 6 and 7 for vertical displacements and normal stresses at various depths, respectively show good agreement.
Flexible footings under umform vertical pressures
Flexible footings under uniform horizontal loads
The influence of the domain size with respect to the loaded area (A/a ratio) on the accuracy of vertical displacements and normal stresses as well as the rate of convergence of the results are illustrated in Figs 2 and 3. The results are improved with a relatively larger domain but the convergence is slower, i.e. more harmonic functions are required for reliable converging solutions. As expected, displacements converge faster than stresses and about 80-odd harmonic functions are required for the convergence of normal stress at the centre of the loaded area on the soil surface when A = B = 12a. To ensure the convergence of vertical displacements for the same domain size requires only about 50 odd harmonic terms. The results presented indicate that the size of the domain should be at least 12 times the size of the loading area. Convergence of deflection and normal stresses at various depth is shown in Figs 4 and 5. The conver-
Figures 8 and 9 show the influence of the boundary length on the accuracy of the horizontal displacements and stresses as well as their rate of convergence. The convergence of the results at various depths was illustrated in Figs 10 and 11. Similar conclusions arrived at earlier for vertical response are applicable for the present case. The only significant difference between the two cases is the rate of convergence of the results. Horizontal responses seem to converge faster than their vertical counterparts and only about 30-40-odd harmonic functions are required to produce values with satisfactory accuracy for A/a = B/b = 12. Comparison of the horizontal stress distribution along the height from the finite layer method with the analytical solutions presented by Milovic and Tournier [7] is illustrated in Fig. 12. Good agreement is observed.
-0 1
se
-0L
Ii-------_-_-_-_ i I\‘; *c___-____
0 fh~mkr
of odd
harmonic
terms
Fig. 3. Convergence of normal stresses at soil surface for various boundary lengths (H/u = 20, (I = b).
in
-.__-----
20
a
LO
Number
of add
harmonic
__----
YI
60
--
m
z-200
I
tzO&
0
terms
Fig. 5. Convergence of normal stresses at various depths (H = 20a, A = B = 12u, (I = b).
S.
328
SWADDIWUDHIWNG
and Y. K.
CHOW
Vertical displacement (mm] 0.0
02
0.1
I
0.0
0.6
I
1.0 -
08
10
12
o/O
zo30-
+
0
1.0 - : 5.0. 0 60-
-
I
Herr I19661 Finite layer
0
Number of odd harmonic 1.0 -
terms
Fig. 9. Convergence of horizontal stresses for various boundary lengths (H = 2a, v = 0.15, z = O.la).
8.0
Fig. 6. Comparison of vertical deflection along the depth. Stress/pressure ratio 02r
ooOO
01
0.6I
06
I
o------
10 2.0 -
$ jo;f 1.0 50
-
Harr (19661 Finite layer
0
60b
Number of odd hormomc terms
Fig. 10. Convergence of horizontal deflections at various depths (H = 2a, v = 0.15, A = B = 12a).
I
::L Fig. 7. Comparison of vertical stress-applied pressure ratio along the depth.
-----A:E:Lo ---A:*:60 -.--b;8=,211
--b:B=16a
10
20
IO
LO
10
Number ol odd hormomc
10
20
30
LO
50
60
m
60
70
80
terms
Fig. Il. Convergence of horizontal stresses at various depths (H = 2a, v = 0.15, A = B = 12a).
Number of odd hormomc terms
Fig. 8. Convergence of horizontal displacements for various boundary lengths (H = 2a, v = 0.15, .r = O.la).
Rigid footings
The approach based on flexibility concept is adopted in the analysis of rigid foundations. The contact area between the footing and the soil stratum is discretized into several subregions and a set of relevant flexibility equations established using unit
load method. The relationship between the interactive loads and the appropriate displacements at the contact region can be obtained by constraining the displacements in the loaded region such that the required deformation pattern of a rigid body motion is satisfied. The method was employed earlier by Cheung and Zienkiewicz [8] to analyse plates on elastic foundation. Details of the procedure were given by Chow [9].
329
Finite layer analysis of surface foundation Hr&.tial-stresslunnifam-shwr ratio 0.1 0.1 0.6 0.B 1.0
Table 1. Comparison of stiffness of rigid footings under vertical load (v = 0.3)
1.2
H /-P
No. of subregions
la
2a
4a
6a
2x2
1.675
4x4
1.825 1.942 1.972
1.342 1.437 1.517 1.493
1.205 1.279 1.295 1.302
1.153 1.220 1.247 1.256
8X8
[91 -
Wilwic & Townier I19711 0
DYNAMIC RESPONSE
finite layer
Fig. 12. Comparison of horizontal stresses at various depths (H = 2a, Y= 0.15).
The stiffnesses of a rigid square footing resting on a soil stratum are defined as k
_
I)-
q,41 WE
v3
w
and
where k is the stiffness, q the pressure, a the width of the footing, w and u the vertical and horizontal displacement components, E and v the Young’s modulus and the Poisson’s ratio of the soil and subscripts tt and h implying vertical and horizontal components, respectively. The stiffnesses of the footings resting on soil strata of various depths under vertical load are given in Table 1. The finite layer solutions obtained from three mesh sizes of loaded regions, 2 x 2, 4 x 4 and 8 x 8 are compared with the values reported by Chow 191.They show good augment and the study indicates that 8 x 8 subregions of loaded area are generally required to produce results with good accuracy. This is similar to the requirement suggested earlier by Chow et al. [3,4] for strip footings. Table 2 shows the stiffness values of the footings resting on a soil stratum of different Poisson’s ratio under horizontal load. The results from finite layer method is compared with those obtained from the formula for a rigid circular footing resting on a soil stratum of depth 2a over a rigid base as suggested by Kausel [lo]. The equivalent area method is adopted in the comparison. ReasonabIe agreement between the two solutions is achieved in all cases except when v is closed to OS. A larger discrepancy observed in the latter is most likely due to the enlarging effect of the boundary influence resulting from the incompressibility condition of the soil when v approaches 0.5. C%S 492-l
In dynamic analysis, the inherent errors of the radiation of stress wave are introduced through the presence of artificially conceived boundaries at the edges. These spurious effects can be rectified by adopting the domain which is sufficiently large such that the reflected stress wave from the ~unda~es are damped out by the soil material damping before reaching the region of interest. The effect of material damping in soil which is usually of hysteretic nature is generally, taken into account by replacing the Young’s modulus of soil, E, by a complex modulus of the form [l I] E* = E(1 + 2ij)
(171
where i = ,/- 1 and j.? the damping ratio. The following dimensionless parameters are introduced: (a) Compliance
functions uG,a Fh = Px F,=-.
(18)
wG,a
(19)
PZ
(b) Impedance functions 1 lu,=-=k,,(k,+iA,c,)(l Fh
+21/Q
(20)
&=$=
+ 2iB).
(21)
L’
k&k, + iA,,c,)(l
In eqns (18X21), u and w are the complex displacement components at the centre of the footing, G, the soil shear modulus, a the width of the foundation,
Table 2. Comparison of stiffness of rigid footings under horizontal load (H = 2~) V
No. of
subregions 2x2 4x4 8x8 DOI
0.0
0.15
0.30
0.49
1.160 1.252 1.307 1.288
1.094 1.180 1.228 1.210
1.073 1.156 1.205 1.165
1.193 1.281 1.341 1.144
330
S. SWADD~WUM~QNG and Y. K.
p, and pz the amplitudes of the disturbances, A,, = oa/2v, the dimensionless frequency, v, = ,/(G,/p) the shear wave velocity of soil, k, the static stiffness of the footing, k the dynamic stiffness coefficient, c the damping coefficient and subscripts h and v denoting horizontal and vertical components, respectively. Earlier studies [3,4] using the finite strip method to study the response of layered soil media indicate that the accuracy of dynamic solutions is more sensitive to the size of the domain than that of static results. This is due largely to an additional effect of radiating stress waves reflecting from the artificially created edge boundaries. These inherent errors are usually rectified through the soil material damping and an appropriate boundary length such that reflected stress waves from boundaries are damped out before reaching the region of interest. For finite layer solutions, the ratio of the boundary length to the width of the footing (A/a) can be estimated from an empirical formula A /a = 300//I.
(22)
It was also observed that the spurious oscillations are less pronounced for vertical vibration and/or at low frequency. The latter is due to the absence of radiating stress waves at frequencies below the fundamental frequency of the system. The studies also pointed out that increasing the size of the domain required more harmonic terms in the Fourier series if convergence of solutions is to be attained. Though the algorithm is self terminating, the number of odd harmonic terms, r and t, required for convergence may be calculated from r = (5/2)(A/a)
+ 10
(23)
f = (5/2)(B/b) + 10.
(24)
and
02
DO
OL
08
10
frequency.
A0
06
Oimtnsiorkrs
12
11
11
IL
i-gz_ _-’
__-- /’
---I02 (ID
00
Ok
06
0‘
Oimtnoionftss
frtqumcy.
10 A0
Fig. 13. Vertical compliance functions for flexible footings with various soil damping ratio (H/o = 2, A/a = 60, Y=
0.4).
CHOW
---
01
p :10x =mx
--p
t
no1 00
’
02
’
01,
’
06
Dimtr6im~
DOI no
’
011
frtqwncy,
’
IO
’
n2
01
06
Oimensiankss
08
12
IL
A0
1
10
fnquency,
12
1
IL
A0
Fig. 14. Horizontal compliance functions of flexible footings with various
material
damping
ratios
(H/a = 2,
A/a = 60, v = 0.4).
The effect of the material damping ratio, /I, on the variations of vertical and horizontal responses of flexible footings are shown in Figs 13 and 14, respectively. As expected, the real parts of both compliance functions decrease when higher values of damping ratio are present. The resonant frequencies of the real parts for both functions are also marginally lower for higher damping ratio. The values are lower comparing to their imaginary counterparts. The same flexibility concept adopted earlier for static analysis is also employed to study dynamic responses of rigid footings. As in static case, the study indicates that discretizing the loaded region into 8 x 8 subregions is sufficient to produce results with acceptable accuracy. Figures 15 and 16 depict respectively the variations of vertical and horizontal compliance functions for flexible and rigid footings. Dynamic responses of flexible foundation are always higher than those of rigid footings. The finite layer solutions are compared with other published results in Figs 17 and 18. The vertical impedance functions of a rigid surface footing resting on a soil stratum of depth H = 2a, /I = 5% and v, = l/3 resulting from the finite layer method and those reported by Chow [9] are depicted in Fig. 17. They show good agreement. The comparison of horizontal impedance functions from finite layer analysis with the values calculated based on an equivalent area of a rigid circular footing presented by Kausel [IO] is illustrated in Fig. 18. The real parts from both approaches agree well while only fair agreement is achieved for the imaginary parts especially when the dimensionless frequency, Ao, is in the range 0.5-0.9, a substantial difference is observed.
Finite layer analysis of surface foundation
i7oL 00
331
’ 02
Dl
0.6
Dimeosmnless DQ
an
a2
01
a6
Oimehonless
08
10
1.2
00
I 1.2
1Q
frequency,
do
..
11
froquey+Ao
IQ 0.e =
06 01 0.2 LIQ
---chsrl1so7~ FititS IqH
___‘_ I
‘,
___-
0.8 IQ 12 Ql ns ~~~io~~sfroquery, A, Fig. 17. Comparison of vertical impedance functions of rigid footings (H = 34 j.i = 5%, v = l/3). 0.0
lI2
01
Oh
Oimmrionloss
Do
IQ
lroqueocy
12
0.D
0.2
IL
( A,
Fig. 15. Vertical compliance functions for flexible and rigid footings (H/a = 2, @= 5%, v = 0.3). CONCLUSIONS
The finite layer method has been adopted to study both static and dynamic responses of surface footings resting on a soil stratum overlying a rough rigid base. Both vertical and horizontal loadings and excitations are considered in the analyses. Appropriate functions satisfying the near field compatibility conditions and producing uncoupling governing equations termwise are chosen to span the displacement field. A threedimensional domain can thus be treated effectively as
one-dimensional finite element an equivalent problem. The method generally occupies small core storage, involves very little data preparation and requires rather short computer time making it suitable for any small personal computer. The accuracy of finite layer solutions depends largely on (i) the choice of domain size such that far field effect is minimal and (ii) the number of harmonic functions spanning the displacement field. The inherent spurious oscillations resulting from the artificially created edge boundaries can be rectified through the proper choice of domain size and the presence of material damping in soils such that the reflected stress waves are damped out before reaching the region of interest. Larger domain is necessary at low soil damping ratio and/or high loading frequency. Results from finite layer analysis agree reasonably well with other theoretical solutions.
10 as
Dimensionless
frequency.
A,
Q~QO--.rL Oimeosiooksr
frequency,
A,
-----limb& Ri$td I,0
Lmmsimlm
12
14
freqlnncy , A,
Fig. 16. Horizontal compliance functions for flexible and rigid footings (H/u = 2, j? = 5%, v = 0.4).
00
0.2
OL Dimensionloss
OK
oe
IO
frcqucncy,
A0
I2
II
Fig. 18. Comparison of horizontal impedance functions of rigid footings (H = 4a, ,!I= 5%, v = l/3).
332
S. SW~DIWUD~~NG and Y. K. REFERENCES
I,, = - 3k,(C,
1. Y. K. Cheung and S. C. Fan, Analysis of pavement and layered foundation by finite layer method. Proceedings 3rd International Conference on Numericaf Methods in Geomechanics. Aachen, pp. 1129-i 13.5(1979).
2. J. R. Booker and J. C. Small, Finite layer analysis of layered pavements subjected to horizontal loading. Proceedings 6th International Conference on Numerical Methods in Geomechanics. Innsbruck, pp. 2109-2113
(1988). 3. Y. K. Chow, S. Swaddiwudhipong and S. A. Lim, Dynamic finite strip analysis of surface foundations. Eart~~e Engng Strucr. Dyn. 16, 457-467 (1988). 4. Y. K. Chow, S. Swadd~wu~ipong and K. F. Phoon, Finite strip analysis of strip footings: horizontal loading. Comput. Geotech. 8, 65-86 (1989). 5. Y. K. Cheung, Finite Strip Method in Structural Analysis. Pergamon Press, Oxford (1976). 6. M. E. Harr, Foundations of Theoretical Soil Mechanics. McGraw-Hill, New York (1966). 7. D. M. Milovic and J. P. Toumier, Stresses and displacements due to a rectangular load on a layer of finite thickness. Soil and Foundations 11, l-28 (1971). 8. Y. K. Cheung and 0. C. Zienkiewicz, Plates and tanks on elastic foundations-an application of the finite element method. ht. J. Solids Srrucrures 1, 451-461 (1965). 9. Y. K. Chow, Vertical defo~ation of rigid foundations of arbitrary shape on layered soil media. Inf. J. Numer. Anal. Me& Geomech. 11, l-15 (1987). 10. E. Kausel, Forced vibrations of circular foundations on layered media, MIT Research Report R74-11, Massachusetts Institute of Technology (1974). 11. F. Molenkamp and I. M. Smith, Hysteretic and viscous material damping. Int. .I. Numer. Anal. Meth. Geomech. 4, 293-311 (1980).
The stiffness matrix of a layer element based on the displacement functions shown in eqns (4)-(6) is expressed as
W”1,= g
12*= 2(k; C, + k:G,)
If the functions of eqns (7)-(9) are adopted to span the displacement field. the stiffness matrix becomes I
0 I 41
+ 6G,
IX = 3k,(C2 - G,) 42
=
4,
I,=(k;C,+k:G,)-6G, Ie2 = -3k,(C,
+ G,)
I,, = 6C, + 2(kS, + k;)G, I43 = -IfJ I53 = -I, &,=
-6C,+(k;+k;;)G,
I44=I,, I54 = I11 I, = -zx, Is, = I22 I65 = -I,2 Ze6= I,, k,=!$!
=p,,,h
E*(l - v,) c1 = (1 + v,)(l - 2vJ VIE* -2vJ
c2=(l+v,)(l E*
where E* = E(1 + Zig), E = soil Young’s modulus and v, = soil Poisson’s ratio. The mass matrix of both sets of displacement fields is as follows:
[Ml,,, = $
symmetric
:
+ G,)
G”=2(1 +v,)3
APPENDIX
&lm = g
CHOW
symmetric 7 2 0 2 0 0 2 1 0 0 2 01002 0 0 10 0 21
in which p is the soil density. The force vectors for vertical and horizontal actions are expressed, respectively as
122 43
-42
0
0
0
42
0
-&,
-153
IA4 0
163 0
4s
--I65
166
where Z,, = (k,$,
+ k;G,) +6G,
where 4, = p,i, 1: = p,Z and
Zz, = 2k,,,k,(C, + G,) I,, = 3k,(C, 14, = (k;C,
I,, = O.SI,,
-G,) + k;G,) - 6G,
px and p, are the horizontal and vertical loads, respectively
acting on the rectangular patch bounded by the lines x = x, , x =,x2, y =y, and y = yz.