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Numerical evaluation of bearing capacity of a foundation in strain softening soil
Computers and Geotechnics 13 (1992) 187-198
TECHNICAL NOTE
NUMERICAL EVALUATION OF BEARING CAPACITY OF A FOUNDATION IN STRAIN SOFTENING SOIL S. Pietruszczak and X.Niu
Depamnent of CivilEngineeringand EngineeringMechanics McMoaterUniversi~ Hamilton, Ont. CanadaLSS 4L7
ABSTRACT
The problem involving soil mass supported by a retaining wall and subjected to the load exerted by a rigid footing is considereeL The collapse load of the system is determined using progtr~ivety more advanced plas~city formulations. It is demonstrated that the estimates based on elastosPeo~e~gplasticitydiffer very substantially from those obtained by a d m i t ~ local~ unstable strain response.
INTRODUCTION In engineering practice, the problems of bearing capacity are usually solved within the framework of stability analysis. The material is idealized as being rigid- perfectly plastic and the collapse load is estimated using a limit equilibrium approach or, more rigorously, by employing lower/upper bound theorems. It is reasonable to expect that, the more advanced the material description is, the more accurate the solution to a boundary value problem becomes. In this context, the solution corresponding to an elastic- perfectly plastic model can be regarded as being more reliable. The latter solution requires static stress-deformation analysis, which nowadays does not present major difficulties. A further improvement in material idealization comes from the use of concepts admitting strain hardening. In this case, the displacement field can be predicted more accurately, yet the value of collapse load may not be very sensitive to the details of the deformation history. A problem arises however, when the material exhibits failure modes associated with formation of shear zones triggering an unstable response, i.e. strain softening. This will, in general, have a profound effect not only on the local deformation field, but also on the collapse load. 187 Computers and Geotechnics 0266-352X/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain
188
The objective of this note is to address a practical geotechnical problem and to investigate the sensitivity of the solution to various material idealizations. In particular, the problem similar to that of ref.[1] is considered, involving the stability of soft mass supported by a cantilever retaining wall. The problem is solved by employing progressively more advanced descriptions ranging from perfect plasticity to unstable, strain softening response. In the next section, the mathematical formulation of the problem is outlined followed by the details of numerical analyses.
FORMULATION OF THE PROBLEM
The simplest idealization, in the context of plasticity, is that of an elastic- perfectly plastic material. For soils, the conditions at failure are usually defined by Mohr-Coulomb criterion, which for plane strain deformation considered here, takes the well-known form F = [(ot - a~) 2 + 4 (~. y)2]I12_ (o t + o~)sin@ - 2ccos~ = 0
(I)
where c is the cohesion and ¢ represents the angle of internal friction.In eq.(1) the stress components are referred to a chosen global coordinate system ~'. A simple extension of (1) is the concept of deviatoric hardening [2].The yield surface is expressed in the functional form similar to that of eq.(1), i.e.
or =
[(a£ - a~) 2 + 4 ""(r~")21112'*~" q (O~ + O~ + a ) = 0
(2)
where tt = ~o + (qf
-
qo)A~eP;
,p = 1 [(exP_ eyp)2 + (y~)211/2
2
(3)
Here ~P ffi Sd~ p represents accumulated plastic distortions, whereas ~1o, qf, A and a are material constants. For 11< qo the response is assumed to be elastic whereas for eP-~, ~-~1f. It is quite apparent that by choosing qf =sino and a=2c coto one has f-*F for q-~f, i.e. the conditions at failure remain described by eq.(1). The framework usually involves a nonassociated flow rule and the plastic potential function is selected as
Q ffi[(o _0~)2+4(I.y)211/2 + qc(°~+°~ ÷a)In (a~+oy+a) = 0 O"o
(4)
189 Here ~c = sin~ where @ (defined in the same subspace as O) specifies the orientation of zero*dilatancy line, and o o is determined by the current state of stress. The range of applicability of the above framework extends to granular materials, in a wide range of initial degrees of compaction, as well as to overconsolidated clays. In eq.(3) 11 is a monotonically increasing variable, which ensures that the material characteristics remain stable. For certain materials however (e.g. dense sand), the failure is associated with localization of deformation in a narrow zone and the material characteristics, as measured in a typical experiment, become unstable. Such an instability may have a profound effect on the collapse load of geotechnical structures, as demonstrated later in this paper. In the last decade several approaches for modelling of the localized deformation have emerged, including non-local theories [3,4], concept of gradient limiters [5], introduction of finite elements with embedded localization zones [6], etc. The approach followed here is based on the averaging of material properties in the neighbourhood of the localized zone. Such a description has been originally suggested in ref.[7] and later formulated in more rigorous terms in ref.[8]. The latter formulation has been applied in ref.[9] in the context of finite element analysis. In what follows, the main aspects of this approach are reviewed. The inception of a shear band may be considered as a bifurcation problem. As shown in ref.[10], for the deviatoric hardening model the necessary condition for the off-set of localization reduces to a path-independent criterion F B = {(or + o~ + a)11f - [(o i- 0"~,) 2 + 4 (T~/)2]I/2}2 - 4rlfllc2A E (oi • o~ + a) = 0 -~-~-~
(5) Eq.(5) defines the 'bifurcation surface' which is located in the close proximity of the failure surface F=0, eq.(D. If the stress state satisfies eq.(5) two potential shear bands may form at the orientation
0
=
+ 1 A ~-111_ -~.w~ ~,,c-
11)
(6)
relative to the direction of the minor principal stress. Once the shear band forms, the material becomes inhomogeneous and eqs.(2)-(4) are no longer valid in the entire region. In order to define the material response, consider a domain, of cross-sectional area A, adjacent to the shear band. Choose the local coordinate system, x, in such a way that x-axis remains parallel to the shear band and denote by
190
6 = {~x, ¢~y,¢~z,txy}T;
t = {ix, l}y,~xy}T
the volume averages of stress/strain rates in this domain. The latter are defined by the averaging rule 6 = /~16(1) + ~2 6(2)
(6a)
t ffi /~1 t(1) + /~2}(2)
(6b)
where the indices 1 and 2 correspond to the intact material and the shear band, respectively. The parameters/~ represent the volume fractions, defined as _
~=1
It.
-3,
It
~2 ~-~
(7)
where I and t are the length and the thickness of the shear band, respectively. The thickness of the shear band (approx. 10 times the mean aggregate size) may be considered as negligible compared to the dimension of the homogenization domain. Consequently, the following kinematic and static constraints may be specified ~x = ~(xl)= ~(2), by= 0~1)= ~2);
~.xy= ~.~)= ~.(x~)
(8)
It should be noted that in view of t-,O, ~1--,1, so that 6 (1)* 6, in which case the static constraints in (8), as applied to averages, are rigorous. The deformation field within the shear band may be conveniently expressed in terms of velocity discontinuities ~ = {gy,gx }T. Thus, the decomposition (6b), combined with the kinematic constraint in eq.(8), reduces to [6]t = [6]t (1)+ ~t~
(9a)
where
,
-- ~ ,
[6] =
[01o11 o o
Finally, the stress-strain relations for the constituent materials may be assumed in the following general form 6 (1) = [D] ~(1);
where
6 (2) = [K]~
(10)
[Dl,Oo ]
191
I 1"
Dr3
[D]=
•
.
;
[D41 D42
[K]T = [K12
K42J
3
The problem formulated above, through eqs.(6a),(8),(9) and (10) is mathematically determinate. The solution may be expressed in terms of structural matrices relating the strain rates in the constituents to those in the homogenized material [8] [6] I~(1)
[S] I~ ;
=
I~(1) = [St] • ;
~ = [S2] t
(10)
where [S] = ([/] * ~[B]) -1 ([4]
[Sl] =
1 Sll
0 0 ] S12 S13 ;
+ lIB][8])
(11)
1 ([6] -
[s])
[S2] =
J
S21 S~ Sz3 D21 - Ct
K21 ~ C1
D41 D42 [4]=--~-2--~2
D2~l - --~'1
0
C1
_K42 ,
.¢2J
K22÷ ,D
[B] = K41+/zD42
0
C2
C t = D22. K2I//~;
C 2 = D43. K42//~
(12)
The constitutive matrix [D*] for the homogenized region can be derived directly from the decomposition (6a) ¢~ * [D °] ~;
[D'] = /z 1 [D] [S1] + /~2[K] [S2] = [D] [S1]
(13)
and may be subsequently transformed to the chosen global coordinate system. In the context of the deviatoric hardening model, the matrix [D] can be specified based on eqs.(2)-(4) by following the standard plasticity procedure. The definition of [K], eq.(10),
192
requires identification of material characteristics of the shear band. These characteristics can be investigated in a direct shear apparatus or, perhaps more appropriately, in a shear ring device [11]. In general, the locally unstable response, i.e. ~T b(2)< 0,would trigger the strain softening response in the homogenized domain. Ideally, the shear band properties should be derived from micromechanical considerations, as a simple alternative however, a phenomenological approach may be adopted. The numerical simulations presented in the next section have been completed following the framework outlined in ref.[9]. The response in the elastic range has been defined as 0(2)= KN~y ;
'i'g) = Ks~rx;
0~2)=0~2)= 1 - 2 ~ ' ~
2)
(14)
where K N and K s are the elastic normal and shear stiffnesses, respectively. The yield function has been selected in the form similar to Coulomb criterion f=
r~)-m(o~2)+c0) = 0;
m = mr - (mr - m o ) exp [(-B2g¢ (gyP+ go) ]
(15) Here Co, B and go are material constants, m o is defined by the stress state at the inception of localization, m r is the residual value of m and g ; represents the accumulated tangential component of the irreversible part of the velocity discontinuity vector. The latter is derived from the non-associated flow rule ~. [6] 00(2)~¢3Q;
Q = 'r(2)xy- me O(2)y = corlst.
(16)
where m c =const. The relations (14)-(16) are sufficient to define the constitutive matrix [K], eq.(10), by following the usual plasticity procedure. The implementation of the constitutive relation (13) in the finite element code requires appropriate estimates for the dimension/~ defined in eq.(9b). In ref.[9] a simple procedure has been suggested based on partitioning of the element area according to the actual numerical integration scheme. IfA is the partitioned area associated with a given sampling point, then the approximation h = g A may be employed to define the characteristic dimension of this area. In this case,/~ = (h cosl3) "1 where Ii defines the minimum inclination of the shear band with respect to the global coordinate system chosen. This procedure is the
193
simplest one and can easily be implemented in the context of any type of discretization. It should be noted that the formulation is invariant with respect to the thickness of the shear band, which has formally been eliminated from the considerations.
NUMERICAL RESULTS The geometry of the problem considered in this section is shown in Fig.1. The soil mass, supported by a preeast concrete retaining wall, is subjected to the load exerted by a rigid footing. The wall has the thickness of 0.25m and is assumed to be rough (i.e. no slippage at the interface). The concrete is treated as linearly elastic, with E = 25,000 MPa ; v = 0.2 The soil is considered to be an overconsolidated clay and the following set of material parameters is chosen (after ref.[9]) Elastic constants: Failure parameters: Deviatoric hardening parameters: Shear band material:
E = 30,000 Ida, ¢ = 30 °,
v = 0.3 c = 14.4 I d a (i.e. a=50 I d a )
~c = 0.4, K N = 30,000 MN/m, B = 1 0 0 m "1,
~o = 0.1, A = 0.0002 K s = 20,000 MN/m, go = 0.0001m
m r = 0.7m o,
mc=
0.5m o,
c o = 0.5a
The discretization of the system is shown in Fig.1. The finite element mesh consisting of 65 9-noded Lagrangian elements with 2x2 Gauss quadrature has been adopted. The problem has been solved as displacement-controlled by increasing the footing settlement, s, in an incremental manner. The initial stress method has been employed and the influence of selfweight of soil has been, for simplicity, neglected. The results of the analysis are shown in figures 2 and 3. Fig.2 presents the loaddisplacement characteristics for both the footing and the retaining wall. The collapse load as predicted by the elastic- perfectly plastic formulation is max q = 245 Ida. The employment of the deviatoric hardening model results in only about 10% reduction, the solution admitting the softening mode however, is significantly different and yields max q ~ 160 I d a (i.e. about 35% reduction). In the latter case, the characteristics become
194
unstable after reaching the maximum load intensity. Fig.3 shows the deformed mesh together with the maps of vertical and horizontal displacements. The displacement field, as predicted by the deviatoric hardening/softening model, corresponds to the settlement of s=O.O4m.
It should be noted that, for a sufficiently fine discretization, the solution incorporating strain softening, is virtually insensitive to the mesh density. This aspect has been investigated separately in ref.[9]. Also, since the averaging procedure is invariant with respect to the orientation of the element, the results will not be significantly affected by the mesh alignment.
12 m
I "'
I
6m
I
3m
t
15m
I
q
u
b
i
3m
g 6m
o
rough
Fig.1
Geometry of the problem and the finite element discretization
195 300.
•
i
•
i
•
l
1
Z
200.
v
o I00.
O.
i
I
O.
I
I
50.
i
I
100.
Settlement
i
200.
150.
s (mm)
300.
r
C
~
~ 200.
0
100.
i
0.~
O.
~-~ o-o ~t-.
Fig.2
20.
i
40. Displacement
i
i
60. 80. u (mm)
00.
Elastic- perfectly plastic model Deviatoric hardening model Deviatoric hardening/softening model
Load-displacement characteristics corresponding to different material descriptions
196
L
,0403 m
.0352
(b)
.0302
,0252
.0201
.0151
.D101
.00503
-,000000405
(c)
-,00488
Fig.3
Displacement field at the settlement of s=0.04 m (a) deformed mesh; intensities of vertical Co) and horizontal (c) displacements
197
FINAL REMARKS
The results of the numerical analysis presented here demonstrate the influence of the material idealization on the prediction of the beating capacity. It is quite evident that a simple elastic- perfectly plastic formulation is not adequate in the case of materials displaying locally unstable response, as it overestimates the value of the collapse load quite significantly. Strain softening phenomenon, associated with formation of a shear band, should not be considered strictly as a material property. In such a case, the material parameters can not be uniquely defined as the average mechanical response is clearly affected by the geometry of the sample. In the present approach, the solution corresponding to the softening mode has an objective physical meaning and remains insensitive to the details of discretization. The extension of the outlined homogenization procedure to three-dimensional situations is provided in ref.[9].
REFERENCES
Bakker KJ. and Vermeer P.A., Finite element analyses of sheetpile walls, in: 'Numerical Models in Geomeehanics', Eds. G.N.Pande & W.F.Van Impe, M Jackson & Son Publ., (1986) 409-416. 2.
Pietruszczak S. and Stolle D.F.E., Modelling of sand behaviour under earthquake excitation, lnt.Journ.Nnm.Anal.Meth.Geomeeh., 11 (1987) 221-240.
3,
Bazant Z.P. and Pijaudier-Cabot G., Non-local continuum damage, localization instability and convergence, Journ.Appl.Mech., 55 (1988) 287-293.
4.
Bazant Z.P. and Feng-Bao Lin, Non-local lntJourn.Num.Meth.Eng., 26 (1988) 1805-1823.
5.
Triantafyllidis N. and Aifantis E.C., A gradient approach to localization of deformation,I: Hyperelastic materials, Journal of Elasticity, 16 (1986) 225-237.
.
Belytschko T., Fish J. and Englemann B.E., A finite element with embedded localization zones, Comp.Meth. in Appl.Meeh.&Eng., 70 (1988) 59-89.
.
Pietruszczak S. and Mroz Z., Finite element analysis of deformation of strain softening materials, lntJonrn.Num.Meth.Eng., 17 (1981) 327-334.
yield
limit
degradation,
198 .
Pietruszczak S., On mechanics of jointed media: masonry and related problems, in: "Computer Methods and Advances in Geomechanies", Eds. G.Beer, J.R.Booker & J.P.Carter, Balkema Pub1., (1991) 407-413.
9.
Pietruszczak S. and Niu X., On the description of localized deformation, Comp.Meth. in AppI.Meeh.&Eng., submitted (Oct. 1991).
10.
Pietruszczak S. and Stolle D.F.E., Deformation of strain softening materials, part II, Computers & Geotechnics, 4 (1987) 109-123.
11.
Yoshimi Y. and Kishida T., A ring torsion apparatus for evaluation of friction between soil and metal surfaces, Geotechnical Testing Joum., 4_(1981) 145-152.
Received 3 April 1992; revised version received 1 July 1992; accepted 1 July 1992
TECHNICAL NOTE
NUMERICAL EVALUATION OF BEARING CAPACITY OF A FOUNDATION IN STRAIN SOFTENING SOIL S. Pietruszczak and X.Niu
Depamnent of CivilEngineeringand EngineeringMechanics McMoaterUniversi~ Hamilton, Ont. CanadaLSS 4L7
ABSTRACT
The problem involving soil mass supported by a retaining wall and subjected to the load exerted by a rigid footing is considereeL The collapse load of the system is determined using progtr~ivety more advanced plas~city formulations. It is demonstrated that the estimates based on elastosPeo~e~gplasticitydiffer very substantially from those obtained by a d m i t ~ local~ unstable strain response.
INTRODUCTION In engineering practice, the problems of bearing capacity are usually solved within the framework of stability analysis. The material is idealized as being rigid- perfectly plastic and the collapse load is estimated using a limit equilibrium approach or, more rigorously, by employing lower/upper bound theorems. It is reasonable to expect that, the more advanced the material description is, the more accurate the solution to a boundary value problem becomes. In this context, the solution corresponding to an elastic- perfectly plastic model can be regarded as being more reliable. The latter solution requires static stress-deformation analysis, which nowadays does not present major difficulties. A further improvement in material idealization comes from the use of concepts admitting strain hardening. In this case, the displacement field can be predicted more accurately, yet the value of collapse load may not be very sensitive to the details of the deformation history. A problem arises however, when the material exhibits failure modes associated with formation of shear zones triggering an unstable response, i.e. strain softening. This will, in general, have a profound effect not only on the local deformation field, but also on the collapse load. 187 Computers and Geotechnics 0266-352X/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain
188
The objective of this note is to address a practical geotechnical problem and to investigate the sensitivity of the solution to various material idealizations. In particular, the problem similar to that of ref.[1] is considered, involving the stability of soft mass supported by a cantilever retaining wall. The problem is solved by employing progressively more advanced descriptions ranging from perfect plasticity to unstable, strain softening response. In the next section, the mathematical formulation of the problem is outlined followed by the details of numerical analyses.
FORMULATION OF THE PROBLEM
The simplest idealization, in the context of plasticity, is that of an elastic- perfectly plastic material. For soils, the conditions at failure are usually defined by Mohr-Coulomb criterion, which for plane strain deformation considered here, takes the well-known form F = [(ot - a~) 2 + 4 (~. y)2]I12_ (o t + o~)sin@ - 2ccos~ = 0
(I)
where c is the cohesion and ¢ represents the angle of internal friction.In eq.(1) the stress components are referred to a chosen global coordinate system ~'. A simple extension of (1) is the concept of deviatoric hardening [2].The yield surface is expressed in the functional form similar to that of eq.(1), i.e.
or =
[(a£ - a~) 2 + 4 ""(r~")21112'*~" q (O~ + O~ + a ) = 0
(2)
where tt = ~o + (qf
-
qo)A~eP;
,p = 1 [(exP_ eyp)2 + (y~)211/2
2
(3)
Here ~P ffi Sd~ p represents accumulated plastic distortions, whereas ~1o, qf, A and a are material constants. For 11< qo the response is assumed to be elastic whereas for eP-~, ~-~1f. It is quite apparent that by choosing qf =sino and a=2c coto one has f-*F for q-~f, i.e. the conditions at failure remain described by eq.(1). The framework usually involves a nonassociated flow rule and the plastic potential function is selected as
Q ffi[(o _0~)2+4(I.y)211/2 + qc(°~+°~ ÷a)In (a~+oy+a) = 0 O"o
(4)
189 Here ~c = sin~ where @ (defined in the same subspace as O) specifies the orientation of zero*dilatancy line, and o o is determined by the current state of stress. The range of applicability of the above framework extends to granular materials, in a wide range of initial degrees of compaction, as well as to overconsolidated clays. In eq.(3) 11 is a monotonically increasing variable, which ensures that the material characteristics remain stable. For certain materials however (e.g. dense sand), the failure is associated with localization of deformation in a narrow zone and the material characteristics, as measured in a typical experiment, become unstable. Such an instability may have a profound effect on the collapse load of geotechnical structures, as demonstrated later in this paper. In the last decade several approaches for modelling of the localized deformation have emerged, including non-local theories [3,4], concept of gradient limiters [5], introduction of finite elements with embedded localization zones [6], etc. The approach followed here is based on the averaging of material properties in the neighbourhood of the localized zone. Such a description has been originally suggested in ref.[7] and later formulated in more rigorous terms in ref.[8]. The latter formulation has been applied in ref.[9] in the context of finite element analysis. In what follows, the main aspects of this approach are reviewed. The inception of a shear band may be considered as a bifurcation problem. As shown in ref.[10], for the deviatoric hardening model the necessary condition for the off-set of localization reduces to a path-independent criterion F B = {(or + o~ + a)11f - [(o i- 0"~,) 2 + 4 (T~/)2]I/2}2 - 4rlfllc2A E (oi • o~ + a) = 0 -~-~-~
(5) Eq.(5) defines the 'bifurcation surface' which is located in the close proximity of the failure surface F=0, eq.(D. If the stress state satisfies eq.(5) two potential shear bands may form at the orientation
0
=
+ 1 A ~-111_ -~.w~ ~,,c-
11)
(6)
relative to the direction of the minor principal stress. Once the shear band forms, the material becomes inhomogeneous and eqs.(2)-(4) are no longer valid in the entire region. In order to define the material response, consider a domain, of cross-sectional area A, adjacent to the shear band. Choose the local coordinate system, x, in such a way that x-axis remains parallel to the shear band and denote by
190
6 = {~x, ¢~y,¢~z,txy}T;
t = {ix, l}y,~xy}T
the volume averages of stress/strain rates in this domain. The latter are defined by the averaging rule 6 = /~16(1) + ~2 6(2)
(6a)
t ffi /~1 t(1) + /~2}(2)
(6b)
where the indices 1 and 2 correspond to the intact material and the shear band, respectively. The parameters/~ represent the volume fractions, defined as _
~=1
It.
-3,
It
~2 ~-~
(7)
where I and t are the length and the thickness of the shear band, respectively. The thickness of the shear band (approx. 10 times the mean aggregate size) may be considered as negligible compared to the dimension of the homogenization domain. Consequently, the following kinematic and static constraints may be specified ~x = ~(xl)= ~(2), by= 0~1)= ~2);
~.xy= ~.~)= ~.(x~)
(8)
It should be noted that in view of t-,O, ~1--,1, so that 6 (1)* 6, in which case the static constraints in (8), as applied to averages, are rigorous. The deformation field within the shear band may be conveniently expressed in terms of velocity discontinuities ~ = {gy,gx }T. Thus, the decomposition (6b), combined with the kinematic constraint in eq.(8), reduces to [6]t = [6]t (1)+ ~t~
(9a)
where
,
-- ~ ,
[6] =
[01o11 o o
Finally, the stress-strain relations for the constituent materials may be assumed in the following general form 6 (1) = [D] ~(1);
where
6 (2) = [K]~
(10)
[Dl,Oo ]
191
I 1"
Dr3
[D]=
•
.
;
[D41 D42
[K]T = [K12
K42J
3
The problem formulated above, through eqs.(6a),(8),(9) and (10) is mathematically determinate. The solution may be expressed in terms of structural matrices relating the strain rates in the constituents to those in the homogenized material [8] [6] I~(1)
[S] I~ ;
=
I~(1) = [St] • ;
~ = [S2] t
(10)
where [S] = ([/] * ~[B]) -1 ([4]
[Sl] =
1 Sll
0 0 ] S12 S13 ;
+ lIB][8])
(11)
1 ([6] -
[s])
[S2] =
J
S21 S~ Sz3 D21 - Ct
K21 ~ C1
D41 D42 [4]=--~-2--~2
D2~l - --~'1
0
C1
_K42 ,
.¢2J
K22÷ ,D
[B] = K41+/zD42
0
C2
C t = D22. K2I//~;
C 2 = D43. K42//~
(12)
The constitutive matrix [D*] for the homogenized region can be derived directly from the decomposition (6a) ¢~ * [D °] ~;
[D'] = /z 1 [D] [S1] + /~2[K] [S2] = [D] [S1]
(13)
and may be subsequently transformed to the chosen global coordinate system. In the context of the deviatoric hardening model, the matrix [D] can be specified based on eqs.(2)-(4) by following the standard plasticity procedure. The definition of [K], eq.(10),
192
requires identification of material characteristics of the shear band. These characteristics can be investigated in a direct shear apparatus or, perhaps more appropriately, in a shear ring device [11]. In general, the locally unstable response, i.e. ~T b(2)< 0,would trigger the strain softening response in the homogenized domain. Ideally, the shear band properties should be derived from micromechanical considerations, as a simple alternative however, a phenomenological approach may be adopted. The numerical simulations presented in the next section have been completed following the framework outlined in ref.[9]. The response in the elastic range has been defined as 0(2)= KN~y ;
'i'g) = Ks~rx;
0~2)=0~2)= 1 - 2 ~ ' ~
2)
(14)
where K N and K s are the elastic normal and shear stiffnesses, respectively. The yield function has been selected in the form similar to Coulomb criterion f=
r~)-m(o~2)+c0) = 0;
m = mr - (mr - m o ) exp [(-B2g¢ (gyP+ go) ]
(15) Here Co, B and go are material constants, m o is defined by the stress state at the inception of localization, m r is the residual value of m and g ; represents the accumulated tangential component of the irreversible part of the velocity discontinuity vector. The latter is derived from the non-associated flow rule ~. [6] 00(2)~¢3Q;
Q = 'r(2)xy- me O(2)y = corlst.
(16)
where m c =const. The relations (14)-(16) are sufficient to define the constitutive matrix [K], eq.(10), by following the usual plasticity procedure. The implementation of the constitutive relation (13) in the finite element code requires appropriate estimates for the dimension/~ defined in eq.(9b). In ref.[9] a simple procedure has been suggested based on partitioning of the element area according to the actual numerical integration scheme. IfA is the partitioned area associated with a given sampling point, then the approximation h = g A may be employed to define the characteristic dimension of this area. In this case,/~ = (h cosl3) "1 where Ii defines the minimum inclination of the shear band with respect to the global coordinate system chosen. This procedure is the
193
simplest one and can easily be implemented in the context of any type of discretization. It should be noted that the formulation is invariant with respect to the thickness of the shear band, which has formally been eliminated from the considerations.
NUMERICAL RESULTS The geometry of the problem considered in this section is shown in Fig.1. The soil mass, supported by a preeast concrete retaining wall, is subjected to the load exerted by a rigid footing. The wall has the thickness of 0.25m and is assumed to be rough (i.e. no slippage at the interface). The concrete is treated as linearly elastic, with E = 25,000 MPa ; v = 0.2 The soil is considered to be an overconsolidated clay and the following set of material parameters is chosen (after ref.[9]) Elastic constants: Failure parameters: Deviatoric hardening parameters: Shear band material:
E = 30,000 Ida, ¢ = 30 °,
v = 0.3 c = 14.4 I d a (i.e. a=50 I d a )
~c = 0.4, K N = 30,000 MN/m, B = 1 0 0 m "1,
~o = 0.1, A = 0.0002 K s = 20,000 MN/m, go = 0.0001m
m r = 0.7m o,
mc=
0.5m o,
c o = 0.5a
The discretization of the system is shown in Fig.1. The finite element mesh consisting of 65 9-noded Lagrangian elements with 2x2 Gauss quadrature has been adopted. The problem has been solved as displacement-controlled by increasing the footing settlement, s, in an incremental manner. The initial stress method has been employed and the influence of selfweight of soil has been, for simplicity, neglected. The results of the analysis are shown in figures 2 and 3. Fig.2 presents the loaddisplacement characteristics for both the footing and the retaining wall. The collapse load as predicted by the elastic- perfectly plastic formulation is max q = 245 Ida. The employment of the deviatoric hardening model results in only about 10% reduction, the solution admitting the softening mode however, is significantly different and yields max q ~ 160 I d a (i.e. about 35% reduction). In the latter case, the characteristics become
194
unstable after reaching the maximum load intensity. Fig.3 shows the deformed mesh together with the maps of vertical and horizontal displacements. The displacement field, as predicted by the deviatoric hardening/softening model, corresponds to the settlement of s=O.O4m.
It should be noted that, for a sufficiently fine discretization, the solution incorporating strain softening, is virtually insensitive to the mesh density. This aspect has been investigated separately in ref.[9]. Also, since the averaging procedure is invariant with respect to the orientation of the element, the results will not be significantly affected by the mesh alignment.
12 m
I "'
I
6m
I
3m
t
15m
I
q
u
b
i
3m
g 6m
o
rough
Fig.1
Geometry of the problem and the finite element discretization
195 300.
•
i
•
i
•
l
1
Z
200.
v
o I00.
O.
i
I
O.
I
I
50.
i
I
100.
Settlement
i
200.
150.
s (mm)
300.
r
C
~
~ 200.
0
100.
i
0.~
O.
~-~ o-o ~t-.
Fig.2
20.
i
40. Displacement
i
i
60. 80. u (mm)
00.
Elastic- perfectly plastic model Deviatoric hardening model Deviatoric hardening/softening model
Load-displacement characteristics corresponding to different material descriptions
196
L
,0403 m
.0352
(b)
.0302
,0252
.0201
.0151
.D101
.00503
-,000000405
(c)
-,00488
Fig.3
Displacement field at the settlement of s=0.04 m (a) deformed mesh; intensities of vertical Co) and horizontal (c) displacements
197
FINAL REMARKS
The results of the numerical analysis presented here demonstrate the influence of the material idealization on the prediction of the beating capacity. It is quite evident that a simple elastic- perfectly plastic formulation is not adequate in the case of materials displaying locally unstable response, as it overestimates the value of the collapse load quite significantly. Strain softening phenomenon, associated with formation of a shear band, should not be considered strictly as a material property. In such a case, the material parameters can not be uniquely defined as the average mechanical response is clearly affected by the geometry of the sample. In the present approach, the solution corresponding to the softening mode has an objective physical meaning and remains insensitive to the details of discretization. The extension of the outlined homogenization procedure to three-dimensional situations is provided in ref.[9].
REFERENCES
Bakker KJ. and Vermeer P.A., Finite element analyses of sheetpile walls, in: 'Numerical Models in Geomeehanics', Eds. G.N.Pande & W.F.Van Impe, M Jackson & Son Publ., (1986) 409-416. 2.
Pietruszczak S. and Stolle D.F.E., Modelling of sand behaviour under earthquake excitation, lnt.Journ.Nnm.Anal.Meth.Geomeeh., 11 (1987) 221-240.
3,
Bazant Z.P. and Pijaudier-Cabot G., Non-local continuum damage, localization instability and convergence, Journ.Appl.Mech., 55 (1988) 287-293.
4.
Bazant Z.P. and Feng-Bao Lin, Non-local lntJourn.Num.Meth.Eng., 26 (1988) 1805-1823.
5.
Triantafyllidis N. and Aifantis E.C., A gradient approach to localization of deformation,I: Hyperelastic materials, Journal of Elasticity, 16 (1986) 225-237.
.
Belytschko T., Fish J. and Englemann B.E., A finite element with embedded localization zones, Comp.Meth. in Appl.Meeh.&Eng., 70 (1988) 59-89.
.
Pietruszczak S. and Mroz Z., Finite element analysis of deformation of strain softening materials, lntJonrn.Num.Meth.Eng., 17 (1981) 327-334.
yield
limit
degradation,
198 .
Pietruszczak S., On mechanics of jointed media: masonry and related problems, in: "Computer Methods and Advances in Geomechanies", Eds. G.Beer, J.R.Booker & J.P.Carter, Balkema Pub1., (1991) 407-413.
9.
Pietruszczak S. and Niu X., On the description of localized deformation, Comp.Meth. in AppI.Meeh.&Eng., submitted (Oct. 1991).
10.
Pietruszczak S. and Stolle D.F.E., Deformation of strain softening materials, part II, Computers & Geotechnics, 4 (1987) 109-123.
11.
Yoshimi Y. and Kishida T., A ring torsion apparatus for evaluation of friction between soil and metal surfaces, Geotechnical Testing Joum., 4_(1981) 145-152.
Received 3 April 1992; revised version received 1 July 1992; accepted 1 July 1992