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Finite element applications of a bounding surface plasticity model
Marhl Comput. Modelling, Vol.14, pp.909-914, 1990 Printed inGreatBritain
0895-7177/90 163.00+0.00 Pergamon Press plc
FINITE ELEMENT APPLICATIONS OF A BOUNDING SURFACE PLASTICITY MODEL
Ameir Altaee, Erman Evgin, and Bengt Fellenius
Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, CANADA, KlN 6N5
Abstract. Bardet's bounding surface plasticity model was used to analyze two different boundary value problems: the load-displacement responce of a model-scale footing resting on the sur-face of a sandy silt, and the behaviour of the Leighton Buzzard Sand in the Cambridge Simple Shear Device. Comparisons between experimental observations and the results of the finite element analyses proved the ability of the constitutive model to simulate accurately the soil behaviour in boundary value problems. Key words. bounding surface, finite element, footing, sand, simple shear device. adaptation. In addition, the finite element implementation of the model was verified by comparing the results of finite element calculations for soil behaviour in different laboratory tests with results of calculations made by integrating the constitutive equations numerically following the same stress or strain paths.
INTRODUCTION Much work has been carried out toward the development of constitutive models to describe soil behaviour. The number of publications in this field and the number of speciality conferences and workshops on the subject are rapidly increasing. Constitutive models can be valuable to the analysis of engineering problems if they are implemented in a finite element program.
In finite the present study, the element program which incorporates the bounding surface plasticity model is used to analyze two different boundary value problems. The first application is the analysis of the performance of a model-scale footing resting on the surface of a sandy silt. The second application is the analysis of Leighton Buzzard Sand in the Cambridge Simple Shear Device when subjected to shear under a constant vertical load.
One of the recently developed constitutive models is Bardet's (1986, 1987) bounding surface plasticity model. Bardet (1986; 1987) evaluated the performance of the model with respect to the behaviour of Sacramento River Sand and Fuji River Sand. An independent assessment of its capabilities to simulate the behaviour of a crushed quartz sand was described by Altaee et al. (1988).
BOUNDING SURFACE PLASTICITY In a recent work, Altaee et al. (1989) provided further validations of the model. Predictions were made for the response of a sandy silt in more generalized stress path tests using a true triaxial apparatus (Desai and Siriwardane, 1984). In the same study, Altaee et al. (1989) implemented Bardet's model into a finite element program to facilitate its practical
The boundina surface conceDt in D-afalias plasticity was introduced by and POPOV (1975) for the description of the behaviour of metals subjected to cyclic and monotonic loading. It has been used increasingly in the development of models for the mechanical behaviour of clays, sands, and other engineering materials. 909
910
Proc.
7th Int. Conf
on Mathematical
The bounding surface concept is based observations whereby on experimental the stress-strain curves for monotonic and cyclic loading converge at bounds in the stress-strain space. The plastic modulus is assumed to depend on the proximity of the current state of stress from the bounds. The simulation of both monotonic and hysteretic stress-strain response is carried out by using a continuous function of the distance between the stress state and the bounds.
and Computer
Modeiiing
where 3A is the coordinate of the ellipse summit along the I-axis; lc is the ellipse aspect ratio; M(a) is a parameter which depends on the Lode angle, a, and the slopes of the critical state lines in compression and extension sides of the mean pressure versus deviator stress plane.
MODEL FORMULATION The detailed formulation of the constitutive relation for the bounding surface model can be found in Bardet (1986, 1987) as summarized by Altaee et al. (1988, 1989). A brief account of its formulation is presented in the following. e The elastic strain increment, dE. ,, resulting from a stress incremen=s, do.., is expressed as: iI 1 1 e dEij=98 dijdokk + $uij
J
(1)
B is the bulk modulus, G is the shear modulus and 6.. is the Kronecker 13 delta. P The plastic strain increment, dE.ij' is expressed as:
where,
d,{ = ( f "kl d"kl ) "ij The term between the symbol < > is the loading function, H is the plastic modulus, and n.. is the unit vector normal to the sdunding surface at the image stress point. Plastic strains are assumed to occur when the stress state is on or inside surface and the loading the bounding function is positive. The material is assumed to behave elastically if the negative function loading (unloading) or the loadi:: function is equal to zero (neutral loading). strain plastic The calculation of increments requires an equation for the bounding surface in the stress space, a mapping rule defining the location of the image stress, an evolution rule of plastic surface during the bounding between plastic flow, and a relation point and the image moduli at the current stress state. An ellipsoidal bounding surface in the invariant, I, and the stress first stress second the root of second selected as space is invariant, J, Fig. 1. The ellipse has the shown in following equation:
FIG. 1. Bounding surface, radial and image stress in mapping, I-J invariant space.
The radial mapping rule introduced by Dafalias and Herrmann (1980) is used to define an image point on the bounding which image stress, surface. The to a given stress state corresponds is surface, bounding within the obtained by intersecting the bounding surface with a radial line connecting the projection centre and the actual stress state. The present model uses the coordinate centre of the principal stress space as the projection centre. strain is volumetric plastic The internal only the assumed to be variable controlling the evolution of the bounding surface. The projection of the ellipse summit, which is the point of intersection of the bounding surface with the critical state line, along the mean pressure axis is used as a measure of the bounding surface evolution. The plastic modulus at a stress point, H, is expressed in terms of the plastic and a modulus at the image point, H positive function which depend !2'on the relative distance between the stress corresponding image its and point at the stress. The plastic modulus stress point is given as:
911
Proc. 7th Int. Co@ on Mathematical and Computer Modelling
where 6 is the distance between current stress state and its image point; 6 the maximum possible distance ?@twiin current stress state and its image point; M and M are the slopes of the peak Failure 'line and the critical state line; and ho is a plastic modulus constant.
FINITE ELEMENT APPLICATIONS Analysis of a footinq The finite element program was used to analyze the behaviour of a model-scale footing resting on a sandy silt. The details of the model-scale footing test were reported in publications by Desai et al. (1981), Desai and Siriwardane (1984), and Faruque and Desai (1985). A rigid rectangular box of size 114 x 203 x 876 mm was used as a container. The footing was 76 mm wide, 19 mm thick, and 114 mm long. It was placed at the centre of the box as shown in Fig. 2. Vertical load was applied on the footing in increments. Measurements were taken for vertical displacements corresponding to each load increment.
Bardet (1986, 1987) and Altaee et al. (1988), the material parameters were soil. TWO this for determined conventional triaxial compression tests and one CTClS) and CTClO (Tests isotropic compression test (Test HC) Siriwardane and Desai reported by (1984) were used. The results of these tests are shown in Figs. 3 through 5. The values of the model parameters are shown in Table 1. Figs. 3 and 4 show the comparisons of the calculated and the measured results of Tests CTClO and CTC15, while Fig. 5 shows the comparisons of the calculated and the measured results of Test HC. Confining pressures used in these tests were 69 kPa and 103 kPa, respectively.
- -
Experiments1 Calculated
5 VI
60 -I
2 ;
40.-
7 4
20..
3
EZ’E
s ::
07 -4
-2
0
2
4
6
6
Strain. % n
/L-_L1_o_“_______._---
:
+
/’
FIG. 3. Comparison of stress-strain response of artificial soil in conventional triaxial compression test (CTClO).
FIG. 2. Layout of the test box (modified after Faruque and Desai,1985) (all dimensions are in millimetre) --
Experimental Calculaled -1
Type of soil The soil consisted of 50% of fire clay and 50% of Florida zircon sand. Z_j.rcon has a solid density of 4650 kg/m . It was mixed with 10% of No. 5 SAE mineral oil to reduce the effect of loss of moisture during testing. The soil was classified as sandy silt and referred "Artificial soil" to as (Desai and Siriwardane, 1984)3 Its total density is about 2000 kg/m and its maximum and minimum Sotal densities are 2650 and 1000 kg/m , respectively. Determination of model parameters Nine model parameters are required by the bounding surface plasticity model. outlined in Following the procedures
Strain, %
FIG. 4. Comparison of stress-strain soil in response of artificial conventional triaxial compression test (CTC15).
10
Proc.
912
7th Int. Co&
-
_.
on Mathematical
Er erlmcntal Ca Pculated
and Computer
Modelling
container, vertical horizontal and displacements were not permitted. The vertical load was increased from zero up to 104 kPa in six equal increments. For the purpose of comparison with calculations made by Desai and his coworkers, the finite element mesh used in the present study and the number of load increments were kept identical to that reported in the references.
5
"
10
15
20
Volumetric strain. Z
FIG. 5. Comparison of stress-strain soil in response of artificial isotropic compression test (HC).
A comparison between the observed loaddisplacement relation of the modelscale footing and the results of the finite element analysis obtained in the present study is shown in Fig. 7. The fisure also includes results of the finite element analysis conducted by Desai and Siriwardane (1984) and Faruque and Desai (1985). The comparison shows that the bounding surface plasticity model is able to simulate the behaviour of soil in this boundary value problem.
TABLE 1. Nodelpararneters
for
sandy
K r
= =
0.001 0.472
x
=
0.120
oc
=
35O
=
350
pe vP v
=
35O
=
0.35
P
=
1.90
h”
=
2.00
silt
FIG. 6. Finite element mesh for the strip footing.
Finite element analysis As a first step in the analysis, initial stresses in the soil mass were vertical calculated. The initial stresses were calculated on the bages density (2000 kg/m ). the soil of Horizontal stresses were taken equal to vertical stresses (Ko=l) as reported by Desai et al. (1981). The model-scale footing behaviour was study as a analyzed in the present the plane strain case. Because of symmetry, only one half of the footing was considered. Fig. 6 shows the finite element mesh used in the analysis. At the boundaries of the finite element vertical side of the mesh, the boundary container and the vertical were footing centreline along the assumed smooth (only vertical displacements permitted). At the bottom of the
0
1
2
3
IF4
5
Vertical displacenlenl. ~nlrl
of calculations FIG. 7. Comparisons _. using various moae28 (* after Desai et al., 1981; ** after Faruque and Desai, 1985).
Proc. 7th Int. Conf. on Mathematical and Computer Modelling
Leighton Buzzard Sand Simple Shear Device
in
Cambridge
The Cambridge Simple Shear Device (Roscoe, 1953) accepts 100 mm x 100 mm square base soil sample of 20 mm heigh. Samples in this device are confined between rigid boundaries. Testing soils is ordinarily carried out in two stages: consolidation and shearing. There is a similarity between the loading condition in the simple shear test and loading conditions in some practical problems such as deep foundations and offshore platforms. Testing soils in the simple shear device has been criticized in many studies due to the non-uniformity of stresses and strains developed in the soil sample. Most of the criticism has been based on elastic analysis which exaggerates the non-uniformities of stresses and strains. Carefully conducusing ted experiments simple shear devices have confirmed the existence of non-uniformities in stresses and strains. However, experiments also showed that the middle one-third of soil sample deforms uniformly (Budhu, 1979). In addition, measured shear stresses within the middle one-third of the sample have been shown to be quite close to average shear stress the measured for the full length of the sample. In the present study, a finite element analysis of the behaviour of Leighton Buzzard Sand in the Cambridge Simple Shear Device is carried out for the constant vertical load test of Budhu (1979). The soil is modelled using the bounding surface plasticity model. The material constants of this sand are determined based on the sand properties provided by Budhu and Britto (1987). These constants are shown in Table 2. The metal components of the simple shear device included in the analysis are assumed to behave elastically.
TABLE 2. Model parameters for Leighton Bazzard Sand n r
= =
0.005 0.927
x
=
0.025
=
35O
=
35O
Qc Qe
=
35o
V
=
0.37
P
=
2.20
=
1.00
QP
ho
913
Finite element model The finite element mesh used in the present analysis is shown in Fig. 8. It consists of three different parts; the (ABCD), the soil upper metal platen sample (CDEF), and the lower metal platen (EFGH). The platen surfaces in contact with the soil sample are assumed to be rough. The two side flaps of the device are not included in the their analysis since surfaces in contact with soil are lubricated in tests. Similar boundary conditions were (1987) in used by Budhu and Britto their finite element analysis.
D E
H FIG. 8. Finite element mesh for the simple shear device.
In the finite element analysis, the sample was first consolidated and then horizontal sheared applying by Each increments. displacements in displacement increment of horizontal in full for the nodes was applied located along AD and BC. The nodes located along DE and CF were displaced fractions of the full at different according to increment displacement their location in the mesh. This was done to preserve the effect of the two side flaps on the deformation pattern of the sample. Results of analysis Fig. 9 compares the results of the measured and the analysis finite stress-strain response of the Leighton Buzzard sand. The stress was expressed in terms of shear stress divided by the vertical stress, applied effective whereas the strain is the shear strain calculated by dividing the horizontal displacement by the initial thickness of the soil sample. Two curves for the finite element analysis and another two observation are for the experimental core I' shown. The curves labeled as I8 or measured the correspond to
Proc. 7th Int. ConJ on Mathe pmatical and Computer Modelling
914
calculated results at the core. The other two curves labeled as "average" correspond to the results for the full length of the top boundary. from can be made TWO observations finite First, the present Fig. 9. the confirmed analysis element the that observations experimental difference between the measurements on the sample core and those normally made test along the full top in routine boundary is less than 5%. Second, the agreement between the present finite element analysis and the experimental measurements is an indication of the bounding surface the ability of plasticity model to simulate correctly the behaviour of sand.
0.6,
I
VI
0
2
4
6
6
10
Shear strain,y (%I
FIG. 9. Comparison of normalized shear stress-shear strain response of Leighton Buzzard Sand.
SUMMARY AND CONCLUSIONS Bardet's bounding surface plasticity model was implemented into a finite element program in a previous study. In this paper, the program is used to boundary value analyze two different problems. In the first application, the load-displacement response of a modelscale footing resting on the surface of a sandy silt is analyzed. In the second application, the response of Leighton Buzzard Sand in Cambridge Simple Shear Device is analyzed. In both applications, the results of the finite element calculations agree well with the measurements taken during is, experiments. It therefore, concluded that the bounding surface plasticity model of Bardet describes soils of cohesionless the behaviour value quite well in these boundary problems.
REFERENCES Altaee, A., Evgin, E., and Fellenius, B.H. (1988). Modelling sand behaviour cyclic loading. 41st Canadian for Geotechnical Conference, Waterloo, Kitchener, Ontario. Altaee, A., Evgin, E., and Fellenius, of a (1989). An application B.H. surface plasticity model. bounding Confer42nd Canadian Geotechnical ence Winnipeg, Manitoba. 169-176. Bardet, J.P. (1986). Bounding surface plasticity model for sands. American Civil Engineers, ASCE Society of Mechanics, Journal of Engineering Vo1.112, No. EMll, 1198-1217. surface Bardet, J.P. (1987). Bounding modelling of cyclic sand behaviour. ConstituWorkshop on Preprint, of Fill tive Laws for the Analysis Ottawa, 1-19. Retention Structures, deforBudhu, M. (1979). Simple shear Thesis, sands. Ph. D. mation of University of Cambridge, Cambridge, United Kingdom. and Britto, A. (1987). Budhu, M., Numerical analysis of soils in simple shear devices. Soils and Foundations, Vo1.27, No.2, 31-41. Dafalias, Y.F., and Popov, E.P. (1975). hardening A model of nonlinearly materials for complex loadings. Acta Mech., vol. 21, 173-192. Herrmann, L.R. and Dafalias, Y.F., bounding surface plas(1980). A the model. Proceedings of ticity Symposium on Soils International Under Cyclic and Transient Loading, vo1.1, Kingdom, United Swansea, 335-345. Desai, C.S., Phan, H.V. and Sture, S. and selection Procedure, (1981). application of plasticity models for International Journal for a soil. Numerical and Analytical Methods in Geomechanics, Vol. 5, 295-311. Siriwardane, H.J., and Desai, C.S. for laws Constitutive (1984). engineering .materials. Prentice-Hall Inc., Englewood Cliffs, NJ., 468~~. Faruque, M.O. and Desai, C.S. (1985). Implementation of afogreneralconstitgeological model utive materials. International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 9, 415-436. Roscoe, K.H. (1953). An apparatus for the application of simple shear to soil samples. Proceedings of the 3rd Soil Conference on International Mechanics and Foundation Engineering, VOl.1, 186-191.
0895-7177/90 163.00+0.00 Pergamon Press plc
FINITE ELEMENT APPLICATIONS OF A BOUNDING SURFACE PLASTICITY MODEL
Ameir Altaee, Erman Evgin, and Bengt Fellenius
Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, CANADA, KlN 6N5
Abstract. Bardet's bounding surface plasticity model was used to analyze two different boundary value problems: the load-displacement responce of a model-scale footing resting on the sur-face of a sandy silt, and the behaviour of the Leighton Buzzard Sand in the Cambridge Simple Shear Device. Comparisons between experimental observations and the results of the finite element analyses proved the ability of the constitutive model to simulate accurately the soil behaviour in boundary value problems. Key words. bounding surface, finite element, footing, sand, simple shear device. adaptation. In addition, the finite element implementation of the model was verified by comparing the results of finite element calculations for soil behaviour in different laboratory tests with results of calculations made by integrating the constitutive equations numerically following the same stress or strain paths.
INTRODUCTION Much work has been carried out toward the development of constitutive models to describe soil behaviour. The number of publications in this field and the number of speciality conferences and workshops on the subject are rapidly increasing. Constitutive models can be valuable to the analysis of engineering problems if they are implemented in a finite element program.
In finite the present study, the element program which incorporates the bounding surface plasticity model is used to analyze two different boundary value problems. The first application is the analysis of the performance of a model-scale footing resting on the surface of a sandy silt. The second application is the analysis of Leighton Buzzard Sand in the Cambridge Simple Shear Device when subjected to shear under a constant vertical load.
One of the recently developed constitutive models is Bardet's (1986, 1987) bounding surface plasticity model. Bardet (1986; 1987) evaluated the performance of the model with respect to the behaviour of Sacramento River Sand and Fuji River Sand. An independent assessment of its capabilities to simulate the behaviour of a crushed quartz sand was described by Altaee et al. (1988).
BOUNDING SURFACE PLASTICITY In a recent work, Altaee et al. (1989) provided further validations of the model. Predictions were made for the response of a sandy silt in more generalized stress path tests using a true triaxial apparatus (Desai and Siriwardane, 1984). In the same study, Altaee et al. (1989) implemented Bardet's model into a finite element program to facilitate its practical
The boundina surface conceDt in D-afalias plasticity was introduced by and POPOV (1975) for the description of the behaviour of metals subjected to cyclic and monotonic loading. It has been used increasingly in the development of models for the mechanical behaviour of clays, sands, and other engineering materials. 909
910
Proc.
7th Int. Conf
on Mathematical
The bounding surface concept is based observations whereby on experimental the stress-strain curves for monotonic and cyclic loading converge at bounds in the stress-strain space. The plastic modulus is assumed to depend on the proximity of the current state of stress from the bounds. The simulation of both monotonic and hysteretic stress-strain response is carried out by using a continuous function of the distance between the stress state and the bounds.
and Computer
Modeiiing
where 3A is the coordinate of the ellipse summit along the I-axis; lc is the ellipse aspect ratio; M(a) is a parameter which depends on the Lode angle, a, and the slopes of the critical state lines in compression and extension sides of the mean pressure versus deviator stress plane.
MODEL FORMULATION The detailed formulation of the constitutive relation for the bounding surface model can be found in Bardet (1986, 1987) as summarized by Altaee et al. (1988, 1989). A brief account of its formulation is presented in the following. e The elastic strain increment, dE. ,, resulting from a stress incremen=s, do.., is expressed as: iI 1 1 e dEij=98 dijdokk + $uij
J
(1)
B is the bulk modulus, G is the shear modulus and 6.. is the Kronecker 13 delta. P The plastic strain increment, dE.ij' is expressed as:
where,
d,{ = ( f "kl d"kl ) "ij The term between the symbol < > is the loading function, H is the plastic modulus, and n.. is the unit vector normal to the sdunding surface at the image stress point. Plastic strains are assumed to occur when the stress state is on or inside surface and the loading the bounding function is positive. The material is assumed to behave elastically if the negative function loading (unloading) or the loadi:: function is equal to zero (neutral loading). strain plastic The calculation of increments requires an equation for the bounding surface in the stress space, a mapping rule defining the location of the image stress, an evolution rule of plastic surface during the bounding between plastic flow, and a relation point and the image moduli at the current stress state. An ellipsoidal bounding surface in the invariant, I, and the stress first stress second the root of second selected as space is invariant, J, Fig. 1. The ellipse has the shown in following equation:
FIG. 1. Bounding surface, radial and image stress in mapping, I-J invariant space.
The radial mapping rule introduced by Dafalias and Herrmann (1980) is used to define an image point on the bounding which image stress, surface. The to a given stress state corresponds is surface, bounding within the obtained by intersecting the bounding surface with a radial line connecting the projection centre and the actual stress state. The present model uses the coordinate centre of the principal stress space as the projection centre. strain is volumetric plastic The internal only the assumed to be variable controlling the evolution of the bounding surface. The projection of the ellipse summit, which is the point of intersection of the bounding surface with the critical state line, along the mean pressure axis is used as a measure of the bounding surface evolution. The plastic modulus at a stress point, H, is expressed in terms of the plastic and a modulus at the image point, H positive function which depend !2'on the relative distance between the stress corresponding image its and point at the stress. The plastic modulus stress point is given as:
911
Proc. 7th Int. Co@ on Mathematical and Computer Modelling
where 6 is the distance between current stress state and its image point; 6 the maximum possible distance ?@twiin current stress state and its image point; M and M are the slopes of the peak Failure 'line and the critical state line; and ho is a plastic modulus constant.
FINITE ELEMENT APPLICATIONS Analysis of a footinq The finite element program was used to analyze the behaviour of a model-scale footing resting on a sandy silt. The details of the model-scale footing test were reported in publications by Desai et al. (1981), Desai and Siriwardane (1984), and Faruque and Desai (1985). A rigid rectangular box of size 114 x 203 x 876 mm was used as a container. The footing was 76 mm wide, 19 mm thick, and 114 mm long. It was placed at the centre of the box as shown in Fig. 2. Vertical load was applied on the footing in increments. Measurements were taken for vertical displacements corresponding to each load increment.
Bardet (1986, 1987) and Altaee et al. (1988), the material parameters were soil. TWO this for determined conventional triaxial compression tests and one CTClS) and CTClO (Tests isotropic compression test (Test HC) Siriwardane and Desai reported by (1984) were used. The results of these tests are shown in Figs. 3 through 5. The values of the model parameters are shown in Table 1. Figs. 3 and 4 show the comparisons of the calculated and the measured results of Tests CTClO and CTC15, while Fig. 5 shows the comparisons of the calculated and the measured results of Test HC. Confining pressures used in these tests were 69 kPa and 103 kPa, respectively.
- -
Experiments1 Calculated
5 VI
60 -I
2 ;
40.-
7 4
20..
3
EZ’E
s ::
07 -4
-2
0
2
4
6
6
Strain. % n
/L-_L1_o_“_______._---
:
+
/’
FIG. 3. Comparison of stress-strain response of artificial soil in conventional triaxial compression test (CTClO).
FIG. 2. Layout of the test box (modified after Faruque and Desai,1985) (all dimensions are in millimetre) --
Experimental Calculaled -1
Type of soil The soil consisted of 50% of fire clay and 50% of Florida zircon sand. Z_j.rcon has a solid density of 4650 kg/m . It was mixed with 10% of No. 5 SAE mineral oil to reduce the effect of loss of moisture during testing. The soil was classified as sandy silt and referred "Artificial soil" to as (Desai and Siriwardane, 1984)3 Its total density is about 2000 kg/m and its maximum and minimum Sotal densities are 2650 and 1000 kg/m , respectively. Determination of model parameters Nine model parameters are required by the bounding surface plasticity model. outlined in Following the procedures
Strain, %
FIG. 4. Comparison of stress-strain soil in response of artificial conventional triaxial compression test (CTC15).
10
Proc.
912
7th Int. Co&
-
_.
on Mathematical
Er erlmcntal Ca Pculated
and Computer
Modelling
container, vertical horizontal and displacements were not permitted. The vertical load was increased from zero up to 104 kPa in six equal increments. For the purpose of comparison with calculations made by Desai and his coworkers, the finite element mesh used in the present study and the number of load increments were kept identical to that reported in the references.
5
"
10
15
20
Volumetric strain. Z
FIG. 5. Comparison of stress-strain soil in response of artificial isotropic compression test (HC).
A comparison between the observed loaddisplacement relation of the modelscale footing and the results of the finite element analysis obtained in the present study is shown in Fig. 7. The fisure also includes results of the finite element analysis conducted by Desai and Siriwardane (1984) and Faruque and Desai (1985). The comparison shows that the bounding surface plasticity model is able to simulate the behaviour of soil in this boundary value problem.
TABLE 1. Nodelpararneters
for
sandy
K r
= =
0.001 0.472
x
=
0.120
oc
=
35O
=
350
pe vP v
=
35O
=
0.35
P
=
1.90
h”
=
2.00
silt
FIG. 6. Finite element mesh for the strip footing.
Finite element analysis As a first step in the analysis, initial stresses in the soil mass were vertical calculated. The initial stresses were calculated on the bages density (2000 kg/m ). the soil of Horizontal stresses were taken equal to vertical stresses (Ko=l) as reported by Desai et al. (1981). The model-scale footing behaviour was study as a analyzed in the present the plane strain case. Because of symmetry, only one half of the footing was considered. Fig. 6 shows the finite element mesh used in the analysis. At the boundaries of the finite element vertical side of the mesh, the boundary container and the vertical were footing centreline along the assumed smooth (only vertical displacements permitted). At the bottom of the
0
1
2
3
IF4
5
Vertical displacenlenl. ~nlrl
of calculations FIG. 7. Comparisons _. using various moae28 (* after Desai et al., 1981; ** after Faruque and Desai, 1985).
Proc. 7th Int. Conf. on Mathematical and Computer Modelling
Leighton Buzzard Sand Simple Shear Device
in
Cambridge
The Cambridge Simple Shear Device (Roscoe, 1953) accepts 100 mm x 100 mm square base soil sample of 20 mm heigh. Samples in this device are confined between rigid boundaries. Testing soils is ordinarily carried out in two stages: consolidation and shearing. There is a similarity between the loading condition in the simple shear test and loading conditions in some practical problems such as deep foundations and offshore platforms. Testing soils in the simple shear device has been criticized in many studies due to the non-uniformity of stresses and strains developed in the soil sample. Most of the criticism has been based on elastic analysis which exaggerates the non-uniformities of stresses and strains. Carefully conducusing ted experiments simple shear devices have confirmed the existence of non-uniformities in stresses and strains. However, experiments also showed that the middle one-third of soil sample deforms uniformly (Budhu, 1979). In addition, measured shear stresses within the middle one-third of the sample have been shown to be quite close to average shear stress the measured for the full length of the sample. In the present study, a finite element analysis of the behaviour of Leighton Buzzard Sand in the Cambridge Simple Shear Device is carried out for the constant vertical load test of Budhu (1979). The soil is modelled using the bounding surface plasticity model. The material constants of this sand are determined based on the sand properties provided by Budhu and Britto (1987). These constants are shown in Table 2. The metal components of the simple shear device included in the analysis are assumed to behave elastically.
TABLE 2. Model parameters for Leighton Bazzard Sand n r
= =
0.005 0.927
x
=
0.025
=
35O
=
35O
Qc Qe
=
35o
V
=
0.37
P
=
2.20
=
1.00
QP
ho
913
Finite element model The finite element mesh used in the present analysis is shown in Fig. 8. It consists of three different parts; the (ABCD), the soil upper metal platen sample (CDEF), and the lower metal platen (EFGH). The platen surfaces in contact with the soil sample are assumed to be rough. The two side flaps of the device are not included in the their analysis since surfaces in contact with soil are lubricated in tests. Similar boundary conditions were (1987) in used by Budhu and Britto their finite element analysis.
D E
H FIG. 8. Finite element mesh for the simple shear device.
In the finite element analysis, the sample was first consolidated and then horizontal sheared applying by Each increments. displacements in displacement increment of horizontal in full for the nodes was applied located along AD and BC. The nodes located along DE and CF were displaced fractions of the full at different according to increment displacement their location in the mesh. This was done to preserve the effect of the two side flaps on the deformation pattern of the sample. Results of analysis Fig. 9 compares the results of the measured and the analysis finite stress-strain response of the Leighton Buzzard sand. The stress was expressed in terms of shear stress divided by the vertical stress, applied effective whereas the strain is the shear strain calculated by dividing the horizontal displacement by the initial thickness of the soil sample. Two curves for the finite element analysis and another two observation are for the experimental core I' shown. The curves labeled as I8 or measured the correspond to
Proc. 7th Int. ConJ on Mathe pmatical and Computer Modelling
914
calculated results at the core. The other two curves labeled as "average" correspond to the results for the full length of the top boundary. from can be made TWO observations finite First, the present Fig. 9. the confirmed analysis element the that observations experimental difference between the measurements on the sample core and those normally made test along the full top in routine boundary is less than 5%. Second, the agreement between the present finite element analysis and the experimental measurements is an indication of the bounding surface the ability of plasticity model to simulate correctly the behaviour of sand.
0.6,
I
VI
0
2
4
6
6
10
Shear strain,y (%I
FIG. 9. Comparison of normalized shear stress-shear strain response of Leighton Buzzard Sand.
SUMMARY AND CONCLUSIONS Bardet's bounding surface plasticity model was implemented into a finite element program in a previous study. In this paper, the program is used to boundary value analyze two different problems. In the first application, the load-displacement response of a modelscale footing resting on the surface of a sandy silt is analyzed. In the second application, the response of Leighton Buzzard Sand in Cambridge Simple Shear Device is analyzed. In both applications, the results of the finite element calculations agree well with the measurements taken during is, experiments. It therefore, concluded that the bounding surface plasticity model of Bardet describes soils of cohesionless the behaviour value quite well in these boundary problems.
REFERENCES Altaee, A., Evgin, E., and Fellenius, B.H. (1988). Modelling sand behaviour cyclic loading. 41st Canadian for Geotechnical Conference, Waterloo, Kitchener, Ontario. Altaee, A., Evgin, E., and Fellenius, of a (1989). An application B.H. surface plasticity model. bounding Confer42nd Canadian Geotechnical ence Winnipeg, Manitoba. 169-176. Bardet, J.P. (1986). Bounding surface plasticity model for sands. American Civil Engineers, ASCE Society of Mechanics, Journal of Engineering Vo1.112, No. EMll, 1198-1217. surface Bardet, J.P. (1987). Bounding modelling of cyclic sand behaviour. ConstituWorkshop on Preprint, of Fill tive Laws for the Analysis Ottawa, 1-19. Retention Structures, deforBudhu, M. (1979). Simple shear Thesis, sands. Ph. D. mation of University of Cambridge, Cambridge, United Kingdom. and Britto, A. (1987). Budhu, M., Numerical analysis of soils in simple shear devices. Soils and Foundations, Vo1.27, No.2, 31-41. Dafalias, Y.F., and Popov, E.P. (1975). hardening A model of nonlinearly materials for complex loadings. Acta Mech., vol. 21, 173-192. Herrmann, L.R. and Dafalias, Y.F., bounding surface plas(1980). A the model. Proceedings of ticity Symposium on Soils International Under Cyclic and Transient Loading, vo1.1, Kingdom, United Swansea, 335-345. Desai, C.S., Phan, H.V. and Sture, S. and selection Procedure, (1981). application of plasticity models for International Journal for a soil. Numerical and Analytical Methods in Geomechanics, Vol. 5, 295-311. Siriwardane, H.J., and Desai, C.S. for laws Constitutive (1984). engineering .materials. Prentice-Hall Inc., Englewood Cliffs, NJ., 468~~. Faruque, M.O. and Desai, C.S. (1985). Implementation of afogreneralconstitgeological model utive materials. International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 9, 415-436. Roscoe, K.H. (1953). An apparatus for the application of simple shear to soil samples. Proceedings of the 3rd Soil Conference on International Mechanics and Foundation Engineering, VOl.1, 186-191.