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Application of a bounding surface model to Boston blue clay
0
al4s-7949/87 s3m i9S7 Pergamon Joum&
i ~0.00 Ltd.
APPLICATION OF A BOUNDING SURFACE MODEL TO BOSTON BLUE CLAY W. 0. MCCARRONand W. F. CHEN School of Civil Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (Receiwd 23 October1986) Abstract-A fo~uIation for an elastic-plastic Bounding Surface model for soils is presented. The model is calibrated and the predicted and measured response compared for laboratory stress paths and a boundary value problem. The predictions arc reasonable, providing confidence in the model formulation, numeric implementation and numeric solution techniques.Possibleimprovementsto the model are discussed.
1. INTRODUCXION
Recently a class of elastic-plastic material models known as Bounding Surface models have been introduced. These models include subsurface plastic behavior in the material response. Since soils often exhibit such behavior, it is expected this class of model may provide an improved representation of soil behavior. This paper presents a particular Bounding Surface model and investigates predictions for laboratory stress paths and a boundary value problem. 2. FORMULATIONOF A SOUNDING SURFACEMODEL
While the application of the classical theories of plasticity to soils have enjoyed some success, the assumption of a distinct xone of elastic behavior is contrary to the many observations of soil response. Dafalias and Popov [l], and Kreig [2] have introduced the Bounding Surface formulation to provide some mechanism to model the occurrence of continuous plastic behavior within the boundaries of a surface in the stress space. The Bounding Surface model may be viewed as an extension of the sublayer models or the multi-surface models of Mroz [3], Iwan [4] and Prevost [q. However, in the Bounding Surface formulation the elastic-plastic response within the funding surface is continuous. In the following, a Bounding Surface model falling within the Critical State classification is introduced. The formulation is that given by Dafalias and Herrmann[6]. A method for simulating the effects of fluid pore pressure in undrained soils is also presented. 2.1 Mathematical formulation The formulation of the constitutive relationship for the Bounding Surface model closely follows that for the classical plasticity models, requiring the definition of the elastic material response, bounding (loading)
surface, flow rule, and hardening rule. In addition, it is necessary to define the direction and magnitude of plastic deformations occurring within the bounding surface. This is accomplished with the aid of a projection rule and plastic modulus. The purpose of the projection rule is to associate a state of stress within the bounding surface with an image point on the surface. Once this association is made, the direction of the plastic strain increment is assumed to be the same as that for the image point on the biding surface. For instance, if an associated flow rule is in use, the plastic strain increment is in the direction of the normal to the bounding surface at the image point. The plastic modulus controls the magnitude of the plastic strains within the bounding surface. The plastic modulus is dependent on the distance between the state of stress and its image point. For large distances from the bounding surface, the plastic modulus is large, resulting in essentially an elastic response. As the state of stress approaches the bounding surface, the plastic modulus approaches the value de&d by the bounding surface and the classical pi~ti~ty formulation. When the state of stress is on the bounding surface, the formulation reduces to a classical plasticity formulation. Plastic deformations within the bounding surface are assumed to occur when the loading function f is positive. The loading function is defined as
where K, is the plastic modulus, F is the expression for the bounding surface, and E# is the stress at the image point on the bounding surface. The plastic modulus K, is retained in eqn (1) to allow for softening (unstable) behavior when both aF/aii(lda, and Kp are negative. This treatment of softening is a simplltication which may require further investigation.
gg7
Once the forms of the bounding surface, potential surface, and plastic modulus K, are selected, the formulation of the constitutive relation is the same as for the classical plasticity models [7]. Further, the s~rnp~~~~t~ouof the c~~stitut~ve retation to matrix
ant, respectively. 1, and J2 are defined as I, = o,,; J? f (l/2) S$,,’
(6)
where
form [S] also takes the same form. The plastic constitutive relation is
The second surface, controhing the strength and defo~~tio~ characteristics of overconso~jd~tcdsoils, is described by where G is the plastic potential function, C>, is the elastic constitutive relation and the plastic modulus Kp has the form
where R, is the cont~bution to the plastic modulus due to the image stress, N is a correction term dependent on the current state of stress (distance from ~~~~g surface and the void ratio eOis a harping parameter. The e~~~c-p~as~c ~nstitutjve relationship is then
da,,= CC:,,- C$,l dew
(4)
Figure I presents the form of the 3o~~d~ng Surface model to be employed here. The harde~ng surface is described by an ellipse of the form
where 4 is the intersection of the ellipse with the hydrostatic axis and R is a material parameter related to the aspect ratio of the ellipse. I, and J, are the first stress invariant and second dctiaaoric stress invari-
where N is the slope of the Critical State line in the J:” - II plane, and A is a rn~te~al parameter controIling the surface shape. 3 does not have the same d&&ion as the parameter R of the often used Cap model Q, 91. However, under some circumstances relations between the hardening surface parameters for the Cap and Bounding Surface models exist. For example, if the hardening surface of the Cap model intersects a Drucker-Prager type surface with an apex at the origin of the stress space (Fig. 2) &en the forgoing relations exist:
The projection rule used in the present fo~u~a~~o~ is known as the radial projection rule. This rule determines the image point on the bounding surface by passing a line through the origin of the stress space and the current state of stress. The image point on the bounding surface is the point of intersection of the surface and the Enc. Thus, the image point is on the
889
Application of a Bounding Surface model to Boston blue clay
Fig. 2. Cap model.
hardening surface F, if the state of stress is below the projection of the critical state line, and on the limiting surface F2 if the state of stress is above the critical state line. For the two surfaces presented above, and the radial projection rule, Dafalias and Herrmann [6] have presented expressions for determining the image point given the state of stress. The image stress is determined as (10)
1, = ?Wll?
and then J’:” = - 9f,.
(13)
The Rp [eqn (3)] contribution to the plastic modulus is evaluated as [8] R,= -;;. l/ 0
(34)
Equation (14) becomes
where w
r(e) = 1 + (K - l)[l + R (K - 2,xy K[l+x2+K(L2)x~
’
x-xy-[(x-Y-l)2+(*z-llly7’~, K(x’ - 1)
BdN
BrN
01)
For the Critical State formulation [IO], aI0 1 K-A -3@-acb lo 1 -t-e,
I
where
(16)
where y---;
RA N
AT=
!z; 8__$
(121
(17)
W. 0. MCCARRON and
890
and C, and C, are rebound and compression indices, respectively. For the specific surfaces presented here,
W.
F. CHEN
where K, is the bulk modulus of water. In practice 11 is not necessary to use the actual bulk modulus, but merely a value several times larger than that of the soil. 3. CALEBRATION OF THE BOUNDING SURFACE MODEL FOR BOSTON BLUE CLAY
2A +N,
0 > hf. I
The correction term is assumed to have the form
(19) where h and m are material parameter, p. is one atmosphere of pressure in the approp~ate units, 6 is the distance between the current state of stress and the image point, and 6, is a reference distance. For the present use, the elastic response is assumed to be characterized by a constant Poisson’s ratio and a stress dependent bulk modulus K=
3.1 Compartion of predicted and measured laboratory response Detailed info~ation concerning the plane strain behavior of Boston blue clay has been presented by Ladd et al. [12], where results from a program of 23 plane strain tests were given. Active and passive load conditions were considered at overconsolidation
(l-eo)zl, K
The Bounding Surface model is calibrated here for the Boston blue clay. Table I presents the model parameters selected. The values of C,, C, and e have been selected to give the reported ratios of Cc/(1 + eo) = 0.2 and C,/(l + e,,) = 0.02 (Fig. 3). Poisson’s ratio v is taken as 0.1. The parameters A and h were determined by trial and error. The parameter m was taken as 0.2, as suggested by Dafalias and Herrmann f6]. The reference distance do is taken as Z,, as suggested by Dafalias and Herrmann 161. The parameter iii is the dominate term in controlling the response of stress paths intersecting the hardening surface. The response of overconsolidated specimens is influenced most by the parameter A. Large values of A result in a flatter limiting surface. Plastic defo~ations within the bounding surface are controlled by the magnitude of It. Large values of h reduce the amount of plastic deformation.
3
2.2 Stitnuiarion of pore pressures To simulate the undrained loading of saturated soils, it is necessary to formufate the constitutive relation for the combined solid-fluid medium. This may be achieved in a simple manner by superimposing over the soil stiffness, a relationship representing the influence of the fluids [ 1I]. We then have D =D,-i-Df,
(21)
where D is the total stiffness, D, is the soil stiffness and D/is the stiffness contribution of the fluids. Since the fluid component may not resist shear distortions, it has the form of an isotropic elastic material with a zero shear modulus. Incremental total stresses are given by {duq = D{ds], while incremental
-60
-60
(22) -120
effective stresses are {do} = D,{dc f
s s ‘G it ” w
‘1‘
_
---.W-
0
(23)
v L
-14’
0.15
0.20 0.25
CJl+eo
and the pore pressure is u = Kwc,,
rl
(24)
0.01 0.02 0.03 C,ll+&J
Fig. 3. Compression and rebound indexes for Boston blue clay.
891
Application of a Bounding Surface model to Boston blue clay
tests are considered. An active pressure condition refers to a failure state in which the major principal stress direction acts in the in situ vertical direction. Thus, if K, is less than one, u, always acts in the vertical direction and no rotation of the principal stresses occurs during shear. Figures 4 through 6 compare the predicted and measured response for three specimens of Boston blue clay having OCRs of 1, 2 and 4, respectively. Figure 7 presents a comparison of the predicted and measured strength ratio &/ad, as a function of the OCR. The overconsolidated specimens (Figs 5 and 6) show a smooth continuous response predicted by the Bounding Surface model. This is in contrast to the sudden change in slope found in the classical plasticity models. The initial cap (hardening surface) positions for the overconsolidated specimens were obtained by considering the maximum previous vertical stress with
Table 1. Bounding Surface model parameters for Boston blue clay
m = 0.2 %:::&t
2:9:‘,
N = 0.2
A = 0.05
h = 1200 v =O.l
ratios (OCRs) of I,2 and 4. Here the OCR is defined as the ratio of the maximum previous vertical stress to the current vertical stress. The specimens used by Ladd were consolidated from a clay slurry under K,, (one-dimensional) conditions. The plane strain equipment, developed at MIT [13], uses a sample with dimensions 3.5 in. high by 3.5 in. wide by 1.4 in. deep (8.9 by 3.6 cm). The vertical and horizontal stresses may be independently varied. The device allows measurements of the three principal stresses as well as the pore pressure. In the following, only the results of active pressure
- -
Bounding
0.400
surface
model
0.600
0.800
1 .ooo
1.200
Axial strain (%)
ia)
lo, + o&2
Axtal rtrrin 1%) (bl
Fig. 4. WPSA
(C)
test for Boston blue clay, OCR = 1.
(ksf)
892
W. 0.
MCCARRO~~
and W. F. CHEN
--
Bounding
-
Test # PSA-6N.ADDetal.
surface
model 19711
Axial strain (%I
Fig. 5. C&UPSA test for Boston blue clay, OCR = 2. & = 0.5. It should be.noted the actual void ratios for the specimens were not used, but rather the ratios C&l + eO) and C,/(l + eO) were assumed constant. The Bounding Surface model is found to overpredict the undrained strength for ove~on~lidated specimens. The excess pore pressures are ah30 in
error. The errors might be attributed to some characteristic of the material which is accounted for in the material model. The Bounding Surface model predicts an initially decreasing mean stress in the plane of deformation [Fig. 4(c)] for a normally consolidated specimen. This
is in contrast to the actual response. It is believed the difference between the two is due to the assumption of a symmetrically located yield surface abut the hydrostatic axis and the use of an associated flow ride. The use of a properly determined nonassociated flow rule might improve the model response. This would be at the expense of additional computation effort and would require additional material parameters for the model. The present use of an associated Row rule is a matter of convenience, However, the flow rule has a significant influence on the dilation and compaction characteristics of the soil. These characteristics govern the generation of excess pore pressures, The manner in which excess pore pressures develop is shown in Figs 4(b), 5(b) and 6(b). In the case of the normally consolidated specimen [Fig. 4(b)], the excess pore pressure builds up too rapidly, indicating that the model is overpredicting the amount of plastic volumetric compression. Two unsatisfactory aspects of the model behavior for the overconso~idat~ Boston blue clay specimens are apparent in Figs 5 and 6. First, it appears that the bounding surface does not contract rapidly enough to give the correct failure strength (or that the initial position is dependent on mvre than the OCR). Second, as the stress state approaches the limiting surface the material begins to dilate due to the use of an associated flow rule. The dilativn results in a reduction in the magnitude of the excess pore pressure. This behavior is not seen in the measured test response, It appears that improved model predictions for Bvston blue clay might be obtained by using separate hardening and softening rules for normatly consohdated and vverconsvhdated states. The results described here have been obtained by using the same rule [eqn (16)] for both hardening and softening conditions.
The Bounding Surface model has been used to represent the foundation material. During the analysis, the state of stress within each element is monitored. Plastic deformations develop when the stress
c
2
2.K?t?3.cw Axialstrain (%I
(a)
Axisi atram (%I lb!
4. ANALYSIS OF A BOUNDARY VALUE PROBLEM This action presents the analysis techniques and analysis results for an embankment construeted on Boston btue clay. Predictions are compared with the actuaf measured response. The analysis was performed with the finite element program NFAP[M], which allows for consideration of material and geometric nonlinearities. Large displacmeents are included through the use of an updated Lagrangian formuhtion [IS]. For instance, perhaps the position of the cap and limiting surface is dependent on more than the degree of overconsolidation. However, in the present case the influence of large displacements is not significant. In the analysis, the foundation has been modeled with eight nvded isoparametric elements. A four-point (2 x 2) Gauss quadrature is used.
4.ooo 5.ooo
i
1.2 1.0
2tf
0.8
0.6 5.4
- - - - Baundlng surfiacemode!
i
l---=-M “l-fi.-
Elev. +5
~_~_____I-l__~~_~____________________
I
I
.A..3
Elev. +9.5 “__““__~’
PWd
2
~-~~~~~~~--~---~-~~~*----~---~~~~~~~--
i
J
----j
y = 125pcf
i
_____
I
I
I
I
Tabte 2. fnitial cap positions and & values for MIT test
embankment & &
Fig. 10. Relationship between K, and OCR for Baston blue clay. specSed. The ~~~~~~~~ stresses have been determined through the ~rec~~s~~idatio~ stress infarma-
tion in Fig. 9 and the K, relationship in Fig. 10.
mg surface model
-11.00
f.2
-12.11 0.8
-9.65 OS
N.C. 0.5
The embankment construction is modeled by a gravity build-up. The stiffening effects of the embankment and over&kg peat and sand layers are negk&d, and only the resultant overburden pressure is included. The circular cross section of the bounding surface model in the deviatoric plane does not allow for an accurate representation of the material response for aB load paths. In the present ana&& the model has been c&brated for active pressure ~nd~~o~s. Xt is believed the active pressure condition beneath the embankment dominates the foundation response.
896
W. 0.
h’fCCARROii
and W. F. C&EN
mations are compared in Fig. 12. The measured lateral deformations are at a distance of IS0 ft (48.8 mf from the centerline, while the predicted values are for 170ft (51.8 m) from the centerline. The predicted pore pressures (Fig. 13) are comparable to the field measurements in those zones of low excess pore pressures. Beneath the center of the embankment, where pore pressures are highest, the predicted values are too large. This discrepancy is similar to that noted with laborato~ specimens above. Figures 1l-13 represent the end of the construction condition,
-110 4 BSmodaliY=a.1~
0 MsasureU NWf, ISWal
-120 -130 -140 $3
2.a Dtsplacsment
3.0
4.0
Itni
Fig, 12. Lateral deformation for MIT test embankment.
The Bounding Surface model parameters given in Table 1 have been used in the analysis. The upper layer of elements is assumed to behave in a drained manner while the remaining elements are assumed to behave in an undrained manner. The initial cap positions were determined by multiplying the current overburden pressure by the OCR and setting &, = 0.5. The resulting stress state is assumed to be on the cap. The exqnion to this procedure is the uppermost Iayer of soil having the largest OCR, In this case the initial cap position is modified to give the correct undrained strength. Table 2 presents the initial cap positions and assumed & for each layer. The settlement histories below the embankment centerline are compared in Fig. I1 with the measured response, The predicted and measured lateral defor-
The Bounding Surface model offers some improvement of the more traditional Critical State and capped soil plasticity madels. The inclusion of subsurface plastic deformations allows for a more realistic repr~en~tio~ of soil behavior. DafaIias and ~er~ann &I also suggest that the tuning surface concept wifl lead to improved predictions for undrained cyclic loading of soils. Application of the Bounding Surface model to Boston blue clay results in overprediction of the strengths for overco~soiidat~ specimens. The excess pore pressures are also overpredicted. It appears that these discrepancies are due to some aspects of the material response which are not included in the material model. The absent model features may include: nonsymmetry of the actual yield (bounding) surface about the hydros~ti~ axis, a nonass~ated Sow rule, separate maddeningand softening rules, and a more ~omplica~ de~~ition of the stress state at which plastic deformation begins. The generalization of the present bounding surface model to include additional aspects of material response would require the identi~cation of additional model parameters. The use of a nonassociated Aow rule would si~i~~antly increase the computational effort in solving boundary value problems since non-
i i c’
1---------
d’
,----__
_a/
-Predicted
Fig. 13. Mcrnsured and predicted excess
pore pressure (feet of water) at MIT test.
i
Ap~~i~tioo of a Bounding Surface model to Boston blue clay
897
TronsianlEd CyclicLoads(Editi by G. N. Pandcand symmetric stiffness matrices result. A more attractive 0. C. Zicnkicwict). John Wiley, New York (1982). procedure is to modify the model parameters to more 7. W. E. Chen and G. Y. Baladi. Soil Plasticity: Theory correctly represent the material behavior in the stress and Implemenrurion.Elsevitr, Amsterdam (1985). range of interest. 8~W. F. Chcn. Piasticify in R&forced Concrere. The 3ounding Surface model does offer AexibiIity Mc~raw-H~~~ New York (t982). in the choice of the projection ruie. Flrture discussion 9. f. S. Sandier, F. L. DiMaggio and G. Y. Baiadi. Creneratizedcap modef for gevlo8icai materials, 2. Geuon this characteristic of the model is’needed. A simple tech. Engng Div., ASCE 102, 683-699 (1976). modifi~t~an in the implementation of the model 1O H-Y, K# and S, Stun. State of the Art: Data reduction would altow the use of separate harde~ng and softening rules for the model. Although this concept is at
odds with Criticat State fo~~~a~o~s~ greater fIetib%ty would be acquired at minimal effort. Analysis of the MIT test embankment with the Bounding Surface model gives reasonable results when compared to the measured response. However, the excess pore presssures are too great. The overall results provide confidence in the model implemutation, ~p~senta~on of pore fluid and numerical solutian techniques,
REFEREF(CES I. Y. F. Dafatias and E. P. Popov. A modei for nonlinearly hardening materials for complex loading. Acfa Me&. 21, 173-192 (1975).
2. R. D. Krieg. A practical two-surface plasticity theory. J. uppl. Mech. 42, 641-646 (1975). 3. Z. Mraz. On the description of anisotropic hardening. I, Me& Phys. SO@ IS, 163-175 (1967). 4, W. D. Iwan. On a class of models for the yielding behavior af composite systems. f. qpPt. h4~ck, 34, SI2-6t? (1967)‘ 5. J. H. Prevost. ~onstit~tive theory for soil. in Limit ~uil~~ri~~ Plastic@ and ~~erali~ed Stress-Slra~ in Geutech~ic~~~g~ee~~g (Edited by R. N. Yong and H-Y. Ka), pp. 74~%14. ASCE (1981). 6. Y. F. Dafalias and L. R. Herrmann. Bolmding surface Fwiw~atians of soif plasticity. In Soi! Meciionics-
* and appli~tion for ~a~~~~ m~etin8. In Laboratory Shear Strength t$Sail (Edited by R. N, Yang and If F. C. Town~~d), pp. 329-386. ASTM, STP 740 (t981). . D, J. Naylor, 6. N. Pande, 8. Simpson and R. Tabb. Finite &+fememin Gearecfvricol&g&ring. Pineridge Press, Swansea (1981). 12. C. C, Ladd, R. Bovee, L. Edgars and J, Rixner. Consolidated-undrained plane strain shear test on Boston blue clay. Rese&h in Earth Physics,Phase
ReDonNa. fS: U.S.Annv EnaincerWaterwaysExpcrim&ar Static&, Contra; Rtpart No. 3--IOI.i197!j. i 3, R. Bovee a& C. C. Ladd. MIT plane strain de*. Research in Earth Physics Phase Report Na. 12, Department of Civil En~n~~ng Research Report R70-24, MIT, Cambridge, MA (1970). 14. T. Y. Chang. A no&n-r bite clement arUysis program, NFAP, Vols i and 2. University of Akron, Akron, OH (1~~). 15 IC. J. Bathe. I;utire Elemonr ptocodvres in Eagheering A@vsb. Prentice-Hall, Englewood Cliffs, NJ (1982). 16. MIT Performance of an embankment on clay, interstate-95 Department of Civil Enginaring, R69-67, MIT, Cambridge, MA (1969). 17. MIT Soil instrumentation for interstate-95 embankment, Saugus, MA. Department of Civil Enginecriag, R69-10, MIT, &&ridge, MA ff%9), IS. T. W. Lambe. Predictions in soif engineering. Georechniqw 23, 14~ZQ2 (1973). 19, D, ~Appolonia, T. Lambe and H. Pouios. Evaluation of pore pressures beneath an ~~krnent. J. Soif Me&. Fo~dati~ Div., ASCE 91, 881-897 (1971). 211 MIT Proceedings of the Foundation afoul Prediction Saturn, Vols 1and 2. Department of Civil Engineering, R75-32, MIT, Cambridge, MA (1975).
al4s-7949/87 s3m i9S7 Pergamon Joum&
i ~0.00 Ltd.
APPLICATION OF A BOUNDING SURFACE MODEL TO BOSTON BLUE CLAY W. 0. MCCARRONand W. F. CHEN School of Civil Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (Receiwd 23 October1986) Abstract-A fo~uIation for an elastic-plastic Bounding Surface model for soils is presented. The model is calibrated and the predicted and measured response compared for laboratory stress paths and a boundary value problem. The predictions arc reasonable, providing confidence in the model formulation, numeric implementation and numeric solution techniques.Possibleimprovementsto the model are discussed.
1. INTRODUCXION
Recently a class of elastic-plastic material models known as Bounding Surface models have been introduced. These models include subsurface plastic behavior in the material response. Since soils often exhibit such behavior, it is expected this class of model may provide an improved representation of soil behavior. This paper presents a particular Bounding Surface model and investigates predictions for laboratory stress paths and a boundary value problem. 2. FORMULATIONOF A SOUNDING SURFACEMODEL
While the application of the classical theories of plasticity to soils have enjoyed some success, the assumption of a distinct xone of elastic behavior is contrary to the many observations of soil response. Dafalias and Popov [l], and Kreig [2] have introduced the Bounding Surface formulation to provide some mechanism to model the occurrence of continuous plastic behavior within the boundaries of a surface in the stress space. The Bounding Surface model may be viewed as an extension of the sublayer models or the multi-surface models of Mroz [3], Iwan [4] and Prevost [q. However, in the Bounding Surface formulation the elastic-plastic response within the funding surface is continuous. In the following, a Bounding Surface model falling within the Critical State classification is introduced. The formulation is that given by Dafalias and Herrmann[6]. A method for simulating the effects of fluid pore pressure in undrained soils is also presented. 2.1 Mathematical formulation The formulation of the constitutive relationship for the Bounding Surface model closely follows that for the classical plasticity models, requiring the definition of the elastic material response, bounding (loading)
surface, flow rule, and hardening rule. In addition, it is necessary to define the direction and magnitude of plastic deformations occurring within the bounding surface. This is accomplished with the aid of a projection rule and plastic modulus. The purpose of the projection rule is to associate a state of stress within the bounding surface with an image point on the surface. Once this association is made, the direction of the plastic strain increment is assumed to be the same as that for the image point on the biding surface. For instance, if an associated flow rule is in use, the plastic strain increment is in the direction of the normal to the bounding surface at the image point. The plastic modulus controls the magnitude of the plastic strains within the bounding surface. The plastic modulus is dependent on the distance between the state of stress and its image point. For large distances from the bounding surface, the plastic modulus is large, resulting in essentially an elastic response. As the state of stress approaches the bounding surface, the plastic modulus approaches the value de&d by the bounding surface and the classical pi~ti~ty formulation. When the state of stress is on the bounding surface, the formulation reduces to a classical plasticity formulation. Plastic deformations within the bounding surface are assumed to occur when the loading function f is positive. The loading function is defined as
where K, is the plastic modulus, F is the expression for the bounding surface, and E# is the stress at the image point on the bounding surface. The plastic modulus K, is retained in eqn (1) to allow for softening (unstable) behavior when both aF/aii(lda, and Kp are negative. This treatment of softening is a simplltication which may require further investigation.
gg7
Once the forms of the bounding surface, potential surface, and plastic modulus K, are selected, the formulation of the constitutive relation is the same as for the classical plasticity models [7]. Further, the s~rnp~~~~t~ouof the c~~stitut~ve retation to matrix
ant, respectively. 1, and J2 are defined as I, = o,,; J? f (l/2) S$,,’
(6)
where
form [S] also takes the same form. The plastic constitutive relation is
The second surface, controhing the strength and defo~~tio~ characteristics of overconso~jd~tcdsoils, is described by where G is the plastic potential function, C>, is the elastic constitutive relation and the plastic modulus Kp has the form
where R, is the cont~bution to the plastic modulus due to the image stress, N is a correction term dependent on the current state of stress (distance from ~~~~g surface and the void ratio eOis a harping parameter. The e~~~c-p~as~c ~nstitutjve relationship is then
da,,= CC:,,- C$,l dew
(4)
Figure I presents the form of the 3o~~d~ng Surface model to be employed here. The harde~ng surface is described by an ellipse of the form
where 4 is the intersection of the ellipse with the hydrostatic axis and R is a material parameter related to the aspect ratio of the ellipse. I, and J, are the first stress invariant and second dctiaaoric stress invari-
where N is the slope of the Critical State line in the J:” - II plane, and A is a rn~te~al parameter controIling the surface shape. 3 does not have the same d&&ion as the parameter R of the often used Cap model Q, 91. However, under some circumstances relations between the hardening surface parameters for the Cap and Bounding Surface models exist. For example, if the hardening surface of the Cap model intersects a Drucker-Prager type surface with an apex at the origin of the stress space (Fig. 2) &en the forgoing relations exist:
The projection rule used in the present fo~u~a~~o~ is known as the radial projection rule. This rule determines the image point on the bounding surface by passing a line through the origin of the stress space and the current state of stress. The image point on the bounding surface is the point of intersection of the surface and the Enc. Thus, the image point is on the
889
Application of a Bounding Surface model to Boston blue clay
Fig. 2. Cap model.
hardening surface F, if the state of stress is below the projection of the critical state line, and on the limiting surface F2 if the state of stress is above the critical state line. For the two surfaces presented above, and the radial projection rule, Dafalias and Herrmann [6] have presented expressions for determining the image point given the state of stress. The image stress is determined as (10)
1, = ?Wll?
and then J’:” = - 9f,.
(13)
The Rp [eqn (3)] contribution to the plastic modulus is evaluated as [8] R,= -;;. l/ 0
(34)
Equation (14) becomes
where w
r(e) = 1 + (K - l)[l + R (K - 2,xy K[l+x2+K(L2)x~
’
x-xy-[(x-Y-l)2+(*z-llly7’~, K(x’ - 1)
BdN
BrN
01)
For the Critical State formulation [IO], aI0 1 K-A -3@-acb lo 1 -t-e,
I
where
(16)
where y---;
RA N
AT=
!z; 8__$
(121
(17)
W. 0. MCCARRON and
890
and C, and C, are rebound and compression indices, respectively. For the specific surfaces presented here,
W.
F. CHEN
where K, is the bulk modulus of water. In practice 11 is not necessary to use the actual bulk modulus, but merely a value several times larger than that of the soil. 3. CALEBRATION OF THE BOUNDING SURFACE MODEL FOR BOSTON BLUE CLAY
2A +N,
0 > hf. I
The correction term is assumed to have the form
(19) where h and m are material parameter, p. is one atmosphere of pressure in the approp~ate units, 6 is the distance between the current state of stress and the image point, and 6, is a reference distance. For the present use, the elastic response is assumed to be characterized by a constant Poisson’s ratio and a stress dependent bulk modulus K=
3.1 Compartion of predicted and measured laboratory response Detailed info~ation concerning the plane strain behavior of Boston blue clay has been presented by Ladd et al. [12], where results from a program of 23 plane strain tests were given. Active and passive load conditions were considered at overconsolidation
(l-eo)zl, K
The Bounding Surface model is calibrated here for the Boston blue clay. Table I presents the model parameters selected. The values of C,, C, and e have been selected to give the reported ratios of Cc/(1 + eo) = 0.2 and C,/(l + e,,) = 0.02 (Fig. 3). Poisson’s ratio v is taken as 0.1. The parameters A and h were determined by trial and error. The parameter m was taken as 0.2, as suggested by Dafalias and Herrmann f6]. The reference distance do is taken as Z,, as suggested by Dafalias and Herrmann 161. The parameter iii is the dominate term in controlling the response of stress paths intersecting the hardening surface. The response of overconsolidated specimens is influenced most by the parameter A. Large values of A result in a flatter limiting surface. Plastic defo~ations within the bounding surface are controlled by the magnitude of It. Large values of h reduce the amount of plastic deformation.
3
2.2 Stitnuiarion of pore pressures To simulate the undrained loading of saturated soils, it is necessary to formufate the constitutive relation for the combined solid-fluid medium. This may be achieved in a simple manner by superimposing over the soil stiffness, a relationship representing the influence of the fluids [ 1I]. We then have D =D,-i-Df,
(21)
where D is the total stiffness, D, is the soil stiffness and D/is the stiffness contribution of the fluids. Since the fluid component may not resist shear distortions, it has the form of an isotropic elastic material with a zero shear modulus. Incremental total stresses are given by {duq = D{ds], while incremental
-60
-60
(22) -120
effective stresses are {do} = D,{dc f
s s ‘G it ” w
‘1‘
_
---.W-
0
(23)
v L
-14’
0.15
0.20 0.25
CJl+eo
and the pore pressure is u = Kwc,,
rl
(24)
0.01 0.02 0.03 C,ll+&J
Fig. 3. Compression and rebound indexes for Boston blue clay.
891
Application of a Bounding Surface model to Boston blue clay
tests are considered. An active pressure condition refers to a failure state in which the major principal stress direction acts in the in situ vertical direction. Thus, if K, is less than one, u, always acts in the vertical direction and no rotation of the principal stresses occurs during shear. Figures 4 through 6 compare the predicted and measured response for three specimens of Boston blue clay having OCRs of 1, 2 and 4, respectively. Figure 7 presents a comparison of the predicted and measured strength ratio &/ad, as a function of the OCR. The overconsolidated specimens (Figs 5 and 6) show a smooth continuous response predicted by the Bounding Surface model. This is in contrast to the sudden change in slope found in the classical plasticity models. The initial cap (hardening surface) positions for the overconsolidated specimens were obtained by considering the maximum previous vertical stress with
Table 1. Bounding Surface model parameters for Boston blue clay
m = 0.2 %:::&t
2:9:‘,
N = 0.2
A = 0.05
h = 1200 v =O.l
ratios (OCRs) of I,2 and 4. Here the OCR is defined as the ratio of the maximum previous vertical stress to the current vertical stress. The specimens used by Ladd were consolidated from a clay slurry under K,, (one-dimensional) conditions. The plane strain equipment, developed at MIT [13], uses a sample with dimensions 3.5 in. high by 3.5 in. wide by 1.4 in. deep (8.9 by 3.6 cm). The vertical and horizontal stresses may be independently varied. The device allows measurements of the three principal stresses as well as the pore pressure. In the following, only the results of active pressure
- -
Bounding
0.400
surface
model
0.600
0.800
1 .ooo
1.200
Axial strain (%)
ia)
lo, + o&2
Axtal rtrrin 1%) (bl
Fig. 4. WPSA
(C)
test for Boston blue clay, OCR = 1.
(ksf)
892
W. 0.
MCCARRO~~
and W. F. CHEN
--
Bounding
-
Test # PSA-6N.ADDetal.
surface
model 19711
Axial strain (%I
Fig. 5. C&UPSA test for Boston blue clay, OCR = 2. & = 0.5. It should be.noted the actual void ratios for the specimens were not used, but rather the ratios C&l + eO) and C,/(l + eO) were assumed constant. The Bounding Surface model is found to overpredict the undrained strength for ove~on~lidated specimens. The excess pore pressures are ah30 in
error. The errors might be attributed to some characteristic of the material which is accounted for in the material model. The Bounding Surface model predicts an initially decreasing mean stress in the plane of deformation [Fig. 4(c)] for a normally consolidated specimen. This
is in contrast to the actual response. It is believed the difference between the two is due to the assumption of a symmetrically located yield surface abut the hydrostatic axis and the use of an associated flow ride. The use of a properly determined nonassociated flow rule might improve the model response. This would be at the expense of additional computation effort and would require additional material parameters for the model. The present use of an associated Row rule is a matter of convenience, However, the flow rule has a significant influence on the dilation and compaction characteristics of the soil. These characteristics govern the generation of excess pore pressures, The manner in which excess pore pressures develop is shown in Figs 4(b), 5(b) and 6(b). In the case of the normally consolidated specimen [Fig. 4(b)], the excess pore pressure builds up too rapidly, indicating that the model is overpredicting the amount of plastic volumetric compression. Two unsatisfactory aspects of the model behavior for the overconso~idat~ Boston blue clay specimens are apparent in Figs 5 and 6. First, it appears that the bounding surface does not contract rapidly enough to give the correct failure strength (or that the initial position is dependent on mvre than the OCR). Second, as the stress state approaches the limiting surface the material begins to dilate due to the use of an associated flow rule. The dilativn results in a reduction in the magnitude of the excess pore pressure. This behavior is not seen in the measured test response, It appears that improved model predictions for Bvston blue clay might be obtained by using separate hardening and softening rules for normatly consohdated and vverconsvhdated states. The results described here have been obtained by using the same rule [eqn (16)] for both hardening and softening conditions.
The Bounding Surface model has been used to represent the foundation material. During the analysis, the state of stress within each element is monitored. Plastic deformations develop when the stress
c
2
2.K?t?3.cw Axialstrain (%I
(a)
Axisi atram (%I lb!
4. ANALYSIS OF A BOUNDARY VALUE PROBLEM This action presents the analysis techniques and analysis results for an embankment construeted on Boston btue clay. Predictions are compared with the actuaf measured response. The analysis was performed with the finite element program NFAP[M], which allows for consideration of material and geometric nonlinearities. Large displacmeents are included through the use of an updated Lagrangian formuhtion [IS]. For instance, perhaps the position of the cap and limiting surface is dependent on more than the degree of overconsolidation. However, in the present case the influence of large displacements is not significant. In the analysis, the foundation has been modeled with eight nvded isoparametric elements. A four-point (2 x 2) Gauss quadrature is used.
4.ooo 5.ooo
i
1.2 1.0
2tf
0.8
0.6 5.4
- - - - Baundlng surfiacemode!
i
l---=-M “l-fi.-
Elev. +5
~_~_____I-l__~~_~____________________
I
I
.A..3
Elev. +9.5 “__““__~’
PWd
2
~-~~~~~~~--~---~-~~~*----~---~~~~~~~--
i
J
----j
y = 125pcf
i
_____
I
I
I
I
Tabte 2. fnitial cap positions and & values for MIT test
embankment & &
Fig. 10. Relationship between K, and OCR for Baston blue clay. specSed. The ~~~~~~~~ stresses have been determined through the ~rec~~s~~idatio~ stress infarma-
tion in Fig. 9 and the K, relationship in Fig. 10.
mg surface model
-11.00
f.2
-12.11 0.8
-9.65 OS
N.C. 0.5
The embankment construction is modeled by a gravity build-up. The stiffening effects of the embankment and over&kg peat and sand layers are negk&d, and only the resultant overburden pressure is included. The circular cross section of the bounding surface model in the deviatoric plane does not allow for an accurate representation of the material response for aB load paths. In the present ana&& the model has been c&brated for active pressure ~nd~~o~s. Xt is believed the active pressure condition beneath the embankment dominates the foundation response.
896
W. 0.
h’fCCARROii
and W. F. C&EN
mations are compared in Fig. 12. The measured lateral deformations are at a distance of IS0 ft (48.8 mf from the centerline, while the predicted values are for 170ft (51.8 m) from the centerline. The predicted pore pressures (Fig. 13) are comparable to the field measurements in those zones of low excess pore pressures. Beneath the center of the embankment, where pore pressures are highest, the predicted values are too large. This discrepancy is similar to that noted with laborato~ specimens above. Figures 1l-13 represent the end of the construction condition,
-110 4 BSmodaliY=a.1~
0 MsasureU NWf, ISWal
-120 -130 -140 $3
2.a Dtsplacsment
3.0
4.0
Itni
Fig, 12. Lateral deformation for MIT test embankment.
The Bounding Surface model parameters given in Table 1 have been used in the analysis. The upper layer of elements is assumed to behave in a drained manner while the remaining elements are assumed to behave in an undrained manner. The initial cap positions were determined by multiplying the current overburden pressure by the OCR and setting &, = 0.5. The resulting stress state is assumed to be on the cap. The exqnion to this procedure is the uppermost Iayer of soil having the largest OCR, In this case the initial cap position is modified to give the correct undrained strength. Table 2 presents the initial cap positions and assumed & for each layer. The settlement histories below the embankment centerline are compared in Fig. I1 with the measured response, The predicted and measured lateral defor-
The Bounding Surface model offers some improvement of the more traditional Critical State and capped soil plasticity madels. The inclusion of subsurface plastic deformations allows for a more realistic repr~en~tio~ of soil behavior. DafaIias and ~er~ann &I also suggest that the tuning surface concept wifl lead to improved predictions for undrained cyclic loading of soils. Application of the Bounding Surface model to Boston blue clay results in overprediction of the strengths for overco~soiidat~ specimens. The excess pore pressures are also overpredicted. It appears that these discrepancies are due to some aspects of the material response which are not included in the material model. The absent model features may include: nonsymmetry of the actual yield (bounding) surface about the hydros~ti~ axis, a nonass~ated Sow rule, separate maddeningand softening rules, and a more ~omplica~ de~~ition of the stress state at which plastic deformation begins. The generalization of the present bounding surface model to include additional aspects of material response would require the identi~cation of additional model parameters. The use of a nonassociated Aow rule would si~i~~antly increase the computational effort in solving boundary value problems since non-
i i c’
1---------
d’
,----__
_a/
-Predicted
Fig. 13. Mcrnsured and predicted excess
pore pressure (feet of water) at MIT test.
i
Ap~~i~tioo of a Bounding Surface model to Boston blue clay
897
TronsianlEd CyclicLoads(Editi by G. N. Pandcand symmetric stiffness matrices result. A more attractive 0. C. Zicnkicwict). John Wiley, New York (1982). procedure is to modify the model parameters to more 7. W. E. Chen and G. Y. Baladi. Soil Plasticity: Theory correctly represent the material behavior in the stress and Implemenrurion.Elsevitr, Amsterdam (1985). range of interest. 8~W. F. Chcn. Piasticify in R&forced Concrere. The 3ounding Surface model does offer AexibiIity Mc~raw-H~~~ New York (t982). in the choice of the projection ruie. Flrture discussion 9. f. S. Sandier, F. L. DiMaggio and G. Y. Baiadi. Creneratizedcap modef for gevlo8icai materials, 2. Geuon this characteristic of the model is’needed. A simple tech. Engng Div., ASCE 102, 683-699 (1976). modifi~t~an in the implementation of the model 1O H-Y, K# and S, Stun. State of the Art: Data reduction would altow the use of separate harde~ng and softening rules for the model. Although this concept is at
odds with Criticat State fo~~~a~o~s~ greater fIetib%ty would be acquired at minimal effort. Analysis of the MIT test embankment with the Bounding Surface model gives reasonable results when compared to the measured response. However, the excess pore presssures are too great. The overall results provide confidence in the model implemutation, ~p~senta~on of pore fluid and numerical solutian techniques,
REFEREF(CES I. Y. F. Dafatias and E. P. Popov. A modei for nonlinearly hardening materials for complex loading. Acfa Me&. 21, 173-192 (1975).
2. R. D. Krieg. A practical two-surface plasticity theory. J. uppl. Mech. 42, 641-646 (1975). 3. Z. Mraz. On the description of anisotropic hardening. I, Me& Phys. SO@ IS, 163-175 (1967). 4, W. D. Iwan. On a class of models for the yielding behavior af composite systems. f. qpPt. h4~ck, 34, SI2-6t? (1967)‘ 5. J. H. Prevost. ~onstit~tive theory for soil. in Limit ~uil~~ri~~ Plastic@ and ~~erali~ed Stress-Slra~ in Geutech~ic~~~g~ee~~g (Edited by R. N. Yong and H-Y. Ka), pp. 74~%14. ASCE (1981). 6. Y. F. Dafalias and L. R. Herrmann. Bolmding surface Fwiw~atians of soif plasticity. In Soi! Meciionics-
* and appli~tion for ~a~~~~ m~etin8. In Laboratory Shear Strength t$Sail (Edited by R. N, Yang and If F. C. Town~~d), pp. 329-386. ASTM, STP 740 (t981). . D, J. Naylor, 6. N. Pande, 8. Simpson and R. Tabb. Finite &+fememin Gearecfvricol&g&ring. Pineridge Press, Swansea (1981). 12. C. C, Ladd, R. Bovee, L. Edgars and J, Rixner. Consolidated-undrained plane strain shear test on Boston blue clay. Rese&h in Earth Physics,Phase
ReDonNa. fS: U.S.Annv EnaincerWaterwaysExpcrim&ar Static&, Contra; Rtpart No. 3--IOI.i197!j. i 3, R. Bovee a& C. C. Ladd. MIT plane strain de*. Research in Earth Physics Phase Report Na. 12, Department of Civil En~n~~ng Research Report R70-24, MIT, Cambridge, MA (1970). 14. T. Y. Chang. A no&n-r bite clement arUysis program, NFAP, Vols i and 2. University of Akron, Akron, OH (1~~). 15 IC. J. Bathe. I;utire Elemonr ptocodvres in Eagheering A@vsb. Prentice-Hall, Englewood Cliffs, NJ (1982). 16. MIT Performance of an embankment on clay, interstate-95 Department of Civil Enginaring, R69-67, MIT, Cambridge, MA (1969). 17. MIT Soil instrumentation for interstate-95 embankment, Saugus, MA. Department of Civil Enginecriag, R69-10, MIT, &&ridge, MA ff%9), IS. T. W. Lambe. Predictions in soif engineering. Georechniqw 23, 14~ZQ2 (1973). 19, D, ~Appolonia, T. Lambe and H. Pouios. Evaluation of pore pressures beneath an ~~krnent. J. Soif Me&. Fo~dati~ Div., ASCE 91, 881-897 (1971). 211 MIT Proceedings of the Foundation afoul Prediction Saturn, Vols 1and 2. Department of Civil Engineering, R75-32, MIT, Cambridge, MA (1975).