J. Inst. Eng. India Ser. A (February–April 2013) 94(1):1–8 DOI 10.1007/s40030-013-0034-y
ORIGINAL CONTRIBUTION
Applicability of Cam-Clay Models for Tropical Residual Soils K. Nagendra Prasad • N. Sulochana U. Venkata Ramana
•
Received: 19 April 2012 / Accepted: 19 August 2013 / Published online: 4 September 2013 The Institution of Engineers (India) 2013
Abstract The development of a critical state framework for saturated soils provides a powerful conceptual model based on the generalized principles of the elastoplastic behavior of frictional materials (Schofield and Wroth, Critical state soil mechanics, 1968]. The model has been modified to meet the requirements of more complex applications (Wheeler, Proceedings of XIV ICSMFE, 1997). An attempt has been made in the present paper to apply conventional Cam-clay models to capture the stress–strain–pore pressure response of tropical residual soils. For this purpose oedometer tests and consolidated undrained triaxial tests were conducted on undisturbed soil specimens and reconstituted soil specimens of five soils collected form the surrounding areas of Tirupati in Andhra Pradesh. Using the Cam-clay models the predictive capabilities of these models are brought out in comparison to the experimental results on tropical residual soils. It has been shown that the Camclay models can only capture the behavior of remolded soils devoid of cementation whereas the Wheeler model can effectively predict the behavior of natural soils
K. Nagendra Prasad (&) N. Sulochana Department of Civil Engineering, College of Engineering, Sri Venkateswara University, Tirupati, Andhra Pradesh, India e-mail:
[email protected] U. Venkata Ramana Irrigation Department, Government of Andhra Pradesh, Tirupati, Andhra Pradesh, India
considered in the present investigation which show strain softening associated with positive pore pressures. Keywords Residual soils Plastic strains Flow rule Hardening rule Yield curve Cam-clay models List of Symbols depv ; deps Plastic volumetric strain and plastic shear strain e e dev ; des Elastic volumetric strain and elastic shear strain depv ; depd Incremental plastic volumetric strain and incremental plastic shear strain k, j, C, Critical state soil mechanics constants. k, L and M slope of normal compression line; j. slope of recompression line; C, specific volume corresponding to a normal stress of 1 kPa on critical state line in lnp-v space; M, stress ratio at critical state; N, specific volume corresponding to a normal stress of 1 kPa on normal consolidation line in lnp-v space p, p0 , q, q0 , Stress components. p ¼ mean principal px0 pm0 and g stress ¼ r1 þr32 þr3 ; 0 0 p00 ¼ effective mean r þr þr principal stress ¼ 1 32 3 ; q ¼ q0 ¼ Shear 0 0 stress ¼ r1 r3 ; px0 ¼ the value of p0 at the intersection of the yield curve with the projection of the critical state line, pm0 ¼ initial size of yield curve, g ¼ qp a, b and l Additional Wheeler’s model constants. a, inclination of the yield surface; b, model parameter which defines the relative effectiveness of plastic shear strains and plastic volumetric strains; l, model parameter controls the state at which k tends towards its target value
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Introduction
Residual Soils Versus Transported Soils
The inherent nature and diversity of geological processes involved in the formation of soil deposits are responsible for a wide variability in the in situ state of the soil. Only a minute fraction of the soil can be sampled and tested because of practical and economic constraints. Most of the conventional mathematical models for describing the soil behavior are based on observed behavior concerning sedimented soils. The soils in the region of present investigation are tropical soils. Tropical soils are found in tropical regions between tropic of Cancer and the tropic of Capricorn enclosed between 23 North and 23 South of the equator. In these regions evaporation is more intense compared to precipitation. The soils are residual having been born and deposited at the place of formation. The characteristic feature of these soils is that the highly weathered soils are found at shallow depths followed by weathered rock, disintegrated rock and hard rock at greater depths. The soils in south Indian region are often residual in nature (those derived by in situ weathering of rock). In residual soils the particles and their arrangement would have evolved progressively as a consequence of physical and chemical weathering. Although the geological study of the formation and structure of in situ residual soils is well advanced, the simple and rapid methods to analyze and assess the engineering properties of these soils have not received the same level of attention. The conventional Cam-clay models and Wheeler model, a logical extension of modified Cam-clay model, were developed for describing the stress–strain behavior for sedimented soils. Accordingly, an attempt has been made to verify the applicability these models for describing the stress–strain response of in situ residual soils. These models help in solving the boundary value problems for appropriate engineering applications.
Residual soils are formed by the in situ physical and chemical weathering of underlying rock, while sedimentary soils are formed by a process of erosion and transportation followed by deposition and consolidation under their own weight. In addition, the latter may undergo further alteration after deposition due to processes such as secondary consolidation, leaching and thixotropic effects [4]. Unloading processes may produce over consolidated clays. Sedimentary soils may also be subjected to the development of inter particle bonds as well as other post deposited effects. As bonds develop with time in residual soils, hardening occurs [3].
Plasticity Models for Soils The development of a critical state framework for saturated soils provides a powerful conceptual model based on the generalized principles of the elastoplastic behavior of frictional materials [12]. The model has been modified with time to meet the requirements of more complex applications [15]. The original Cam-clay model assumes the Roscoe surface to be ‘‘bullet’’—shaped as shown in Fig. 1. However the model predicts larger shear deformations than those observed for small levels of shear stress. In order to overcome this limitation a modified version of the Camclay model was suggested by the researchers replacing the bullet-shaped surface of the Cam-clay model with an elliptic shape and subsequently was extended by the scientists [11] to a model, now known as the modified Camclay model (Fig. 1). The model proposed by Wheeler [15], is an extension of the critical state models, with anisotropy of plastic behavior represented through a rotational component of hardening (Fig. 1). For the sake of simplicity, the model is presented here for the simplified stress space of
General Soil Types Encountered in the Region The development of the ‘classical’ concepts of soil mechanics has been based almost exclusively on the investigation of sedimentary deposits of soil. The natural soils have got components of stiffness represented by cementation bonding and soil skeleton [14]. Sedimented soils exhibit stiffness essentially on account of soil skeleton. As a consequence, the cementation component and its evolution with deformation has to be carefully accounted for. Accordingly, assessment of soil behavior relating to residual soils, which are of frequent occurrence in tropical regions, need greater attention.
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Fig. 1 Yield curves
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the triaxial test, although it has already been extended to general three-dimensional stress space. It has been [13] presented earlier the experimental data from constant water content triaxial tests on a residual soil from Singapore, tested under saturated conditions with measurements of Matric suctions. The functions relating critical state parameters to degree of saturation have been expressed in normalized form by referring them to saturated state. The variations in the observed stress–strain response can be normalized with respect to the state of soil represented by specific volume and normal stress. The researchers have [2] presented a simple constitutive model for structured clays based on an existing constitutive model for reconstituted clays. The scientists [7] present an alternative constitutive model based on classical theory of plasticity and critical state theory for describing the mechanical behavior of soils particularly at unsaturated states. Despite a large number of elastoplastic models, Cam-clay models are used in practice owing to simplicity and easily determinable model parameters. Cam-clay model A number of different theories for the prediction of plastic strains in soils have been developed, mostly by research workers at Cambridge, but the essential characteristics of these theories are the same. The present paper focuses on Cam-clay models owing to their simplicity in terms of determining the model parameters. These are the basis for several more advanced theories which are more complicated. One of the key assumptions of Cam-clay theory is that the flow rule follows normality condition. Thus, if plastic strain increment vector in Fig. 1 is everywhere normal to a yield locus, it is only necessary to specify either the shape of the yield curve or the relationship between deps =depv and the stress state (the flow rule) in order for both the flow rule and the yield curve to be fully specified and elastic shear stains are assumed to be zero. The associated yield curve is given by 0 0 q p þ ln ¼ 1: ð1Þ 0 Mp p0X Elastic volumetric strains are given by deev ¼ j
dp0 : vp0
ð2Þ
The equation of the Cam-clay state boundary surface is obtained as: Mp0 q ¼ ðC þ k j v k ln p0 Þ: kj 0
ð3Þ
The plastic volumetric strains can be computed by Atkinson and Bransby [1]:
depv ¼
ðk j Þ pv
M
q dP þ dq : p
ð4Þ
A second key assumption, which arises from a consideration of the work dissipated during shear, which is the flow rule from which plastic shear strains can be computed. dep desp ¼ h v i M qp
ð5Þ
where p0x is the value of p0 ; at the intersection of the yield curve with the projection of the critical state line. We should also note that the slope of the yield curve is zero at X (Fig. 1), implying that depv =deps is also zero at the critical state. Of course, p0x will be different for different yield curves at the top of different elastic walls, as the response is elastic until the test paths from different initial stresses reach the yield curve. The intersection of critical state line with yield curve which is denoted by px is different for different initial confining pressures leading to a family of yield curves forming the yield surface. The virtue of the Cam-clay theory is that it gives a complete constitutive relationship for soil (Eqs. 2–5) which can describe deformations and pore pressures during drained and undrained loading for a wide variety of stress paths. The soil constants required ðM; k; j; CÞ are few and all can be measured in standard laboratory tests. Modified Cam-Clay model The yield curve for the modified Cam-clay model is given by q2 þ ðp px ÞM 2 p ¼ 0:
ð6Þ
It is assumed that recoverable changes in volume accompany any changes in mean effective stress p0 according to the expression The elastic volumetric strains can be obtained from Eq. 2. It is assumed that recoverable shear strains accompany any changes in deviator stress q according to the expression dees ¼
dq 3G
ð7Þ
With constant shear modulus G. The plastic shear strains are calculated from normality rule: depd 2g p ¼ 2 dev M g2
ð8Þ
It is convenient to make it always pass through the origin of effective stress space, as the sedimented soils devoid of cementation do not show resistance against extension, though this is not essential: it seems reasonable to propose that unless the soil particles are cemented
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together, a soil sample will not be able to support an allround tensile effective stress and that irrecoverable volumetric deformations would develop if an attempt were made to apply such tensile effective stresses.
is a measure of the degree of plastic anisotropy of the soil. For the case of isotropy (a = 0), Eq. (10) reduces to the modified Cam-clay yield curve. For simplicity, an associated flow rule is assumed, and hence:
Wheeler Model
depd 2ðg aÞ ¼ : depv M 2 g2
To make the adoption of anisotropic models for geotechnical design more feasible, an alternative elasto-plastic model for soft clayey soils was proposed by the researchers [15]. The main objective in developing the model was to provide a realistic representation of the influence of plastic anisotropy whilst still keeping the model relatively simple in terms of easily determinable model parameters, so that there would be a realistic chance of widespread application in geotechnical design. This model is an extension of the critical state models, with anisotropy of plastic behavior represented through a rotational component of hardening. The model is applicable to tropical residual soils, where plastic deformations dominate after the peak stress is attained. For simplicity, isotropy of elastic behavior is therefore assumed, and hence the elastic increments of volumetric and deviatoric strains are calculated as deev ¼
jdp0 dq ; deev ¼ 0 vG vp0
The greatest advantage of assuming an associated flow rule is that numerical implementation of the model is far simpler than with a non-associated flow rule. Experimental evidence by the scientists [8, 9] suggests that this assumption is reasonable for many residual soils. The model incorporates two hardening laws. The first one describes changes in size of the yield curve with respect to volumetric deformation and it is similar to that of modified Cam-clay: dp0m ¼
vp0m depv : kj
ð12Þ
The second hardening rule predicts the change of inclination of the yield curve produced by plastic straining, representing the development of anisotropy with plastic strains. It is assumed that plastic volumetric strain attempts to drag the value of a towards an instantaneous target value vm(g) that is dependent on the current value of g, whereas plastic shear strain is simultaneously attempting to drag a towards a different instantaneous target value vd(g) (also dependent on g). ð13Þ da ¼ l ðvm ðgÞ aÞdepv þ bðvd ðgÞ aÞdepd :
ð9Þ
where j is the slope of the swelling line from one dimensional compression test, v is specific volume, G0 is the elastic shear modulus and p0 and q are the mean effective stress and deviatoric stress respectively. The yield curve is sheared ellipse, to account for anisotropy and to satisfy the condition of associated flow rule to make numerical implementation of the model far simpler, as proposed by earlier researches [6, 9], defined by f ¼ ðq ap0 Þ2 ðM 2 a2 Þðp0m p0 Þp0 ¼ 0
ð11Þ
The overall current target value for a will lie between vm(g) and vd(g). Constants l and b control, respectively, the absolute rate at which a heads towards its current target value and the relative effectiveness of plastic shear strains and plastic volumetric strains in determining the current target value. Based on initial yield curve, it has been proposed [10] earlier the following expressions for vm(g) and vd(g):
ð10Þ
where M is the slope of critical state line in q - p0 space and the parameters pm0 and a define the size and the inclination of the yield curve respectively. The parameter a Table 1 Basic soil properties of the soils considered S. No.
Description
Vinayaka Nagar
Gayathri Nagar
Renigunta
Muni Reddy Nagar
Tiruchanur
1. 2.
% Gravel
3.00
1.00
6.60
0.50
7.00
% Sand
46.00
43.00
63.40
57.50
48.00
3.
% Silt ?Clay
51.00
56.00
30.00
42.00
45.00
4.
Liquid limit (%)
42.00
33.00
92.00
27.00
55.00
5. 6.
Plastic limit (%) Plasticity index (%)
30.10 11.90
22.17 10.83
45.30 46.70
20.75 6.25
35.24 19.76
7.
IS classification
CI
CL
SC
SC
SC
8.
Field density (kN/m3)
19.58
19.85
19.62
19.18
19.54
9.
Natural moisture content (%)
16.70
17.73
19.86
16.25
14.92
10.
Depth of sampling (m)
2.70
3.50
2.50
1.80
2.70
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3g 4 g vd ðgÞ ¼ : 3
5
ð14Þ
vv ðgÞ ¼
ð15Þ
In practice the expression for vm(g) in Equation 14 means that plastic volumetric strains attempt to align the yield curve approximately about the current stress point (see [15]). The proposal for vd(g) in Equation 15 corresponds to a significant degree of anisotropy at critical states (a = L/3 at g = L), as suggested by the researchers [10]
Soils Tested The consolidated undrained triaxial tests were conducted on undisturbed soil specimens and reconstituted soil specimens of five soils collected form the surrounding areas of Tirupati in Andhra Pradesh. Table 1 represents the soil properties for soil samples considered in the present investigation. The grain size analysis is carried out using wet sieving. The Normal compression paths of reconstituted soils under oedometer compression are shown in Fig. 2. The model parameters such as N and k can be estimated from the compression paths and the value of j is taken as k/4, as suggested by Atkinson and Bransby [1] for normally consolidated soils. The predictive capabilities of Cam-clay models are brought out in comparison to the experimental results on tropical residual soils (Figs. 3, 4, 5, 6, 7).
Determination of Cam-Clay Model Parameters The critical state parameters viz. N, k, j, C and M can be evaluated based on consolidation and shear test results. The slope of isotropic compression is k and swelling path is j. The specific volumes corresponding to 1 kPa on normal compression line and critical state line in v-lnp0 space 0.9 0.8
Void ratio (e)
0.7 0.6 0.5 0.4
Vinayak Nagar
0.3
Gayathri Nagar Munireddy Nagar
0.2
Tirucahnur
0.1
Renigunta
Fig. 3 Stress–strain–pore pressure response of Vinayaka Nagar soil (predicted and experimental)
indicate N and C respectively. The value of M, is obtained as a ratio of deviatoric stress to mean principal stress at critical state line in q - p0 space. The values of deviatoric stress and mean principal stress at critical state are used to obtain the value of M from one shear test result. Thus the tests needed to evaluate critical state parameters are simple and are performed in routine soil investigations. The Wheeler model involves seven soil constants: five conventional parameters from modified Cam-clay [k, j, L, G0 (or m) and C] and two additional parameters relating to the rotational hardening (b and l). In addition, the initial state of the soil is defined by the stress state and the initial values of the parameters pm0 and a defining the initial size and inclination of the yield curve. The value of pm0 can be obtained after determining the yield points for different test paths and tracing the yield curve. If the initial value of specific volume v is also defined, this replaces the requirement to define a value for the parameter C (the intercept of the critical state line in the v: ln p0 plane corresponding to mean principal stress of 1 kPa). Values of the soil constants k, j, L, C and G0 can be measured in laboratory tests using the data generated from isotropic consolidated undrained or drained tests [15]. This section therefore concentrates on procedures for evaluating the remaining two soil constants (b and l) and the initial values of the parameters pm0 and a.
0 1
10
100
Vertical effective stress (σv ), kPa
Fig. 2 One dimensional compression paths of remolded soils
1000
The inclination of the yield curve, a a will be calculated using the equation
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Fig. 4 Stress–strain–pore pressure response of Gayathri Nagar soil (predicted and experimental) Fig. 6 Stress–strain–pore pressure response of Munireddy Nagar soil (predicted and experimental)
Fig. 5 Stress–strain–pore pressure response of Renigunta soil (predicted and experimental)
a K0 ¼
g2K0 þ 3gK0 M 2 : 3
ð16Þ
The value of aK0 can therefore be estimated using Jaky’s 3ð1k0 Þ simplified formula (K0 = 1-sin /0 ) as gKo ¼ ð1þ2k : 0Þ
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Fig. 7 Stress–strain–pore pressure response of Tiruchanur soil (predicted and experimental)
Initial size of yield curve, pm If the initial inclination of a of the yield curve is estimated using Eq. 16, the value of pm can be determined using
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Table 2 Model parameters obtained from test results Location
k
j
L
a
b
pm
L
M
Vinayaka Nagar
0.0944
0.0236
1.5
0.534
0.857
121.75
60
1.6974
Gayatri Nagar
0.0777
0.0194
1.42
0.788
0.063
90.68
70
1.7639
Renigunta
0.0958
0.0239
1.44
0.736
0.218
121.36
60
1.8639
Munireddy Nagar
0.0711
0.0178
1.47
0.659
0.53
98.81
60
1.9299
Tiruchanur
0.0931
0.0233
1.45
0.659
0.53
98.81
60
1.9134
Eq. 10 when one point on the yield curve is evaluated from single test result. Ideally this, single yield point would be identified by either isotropic or K0 consolidation in a triaxial apparatus. Alternatively, one-dimensional consolidation in an oedometer would be possible, but this would require either measurement or estimation of radial stress, in order to fully define the stress state. Estimation of a point on the yield curve without the performance of any laboratory tests, from the knowledge of the maximum overburden stress applied to the soil deposit, would rarely be satisfactory, because of uncertainties about the depositional history and the possibility of an increase in the yield stress above the maximum pressure previously applied, due to the effects of ageing or inter-particle bonding (Burland 1990). Soil Constant, b The model parameter b defines the relative effectiveness of plastic shear strains and plastic volumetric strains in rotating the yield curve. The soil constant b is given by b¼
3ðM 2 g2K0 3gK0 =4Þ 2ðg2K0 M 2 þ 2gK0 Þ
model predictions. These soil constants are presented in Table 2.
Predictions of Soil Behavior Using Cam-Clay and Wheeler Models The Cam-clay, modified Cam-clay and Wheeler models have been applied to predict the stress–strain behavior of tropical residual soils. The model parameters have been determined from the consolidation and triaxial shear test results. The predictions for the re-molded soil specimens and undisturbed soil specimens are shown in Figs. 3, 4, 5, 6, and 7 for five samples. Figures 3, 4, 5, 6, and 7 also show the stress–strain response of consolidated undrained triaxial tests conducted on remolded soils, reconstituted to the initial density together with stress–strain response of in situ soils.
Conclusion ð17Þ
Soil constant, l The model parameter l controls the rate, at which a tends towards its target value. It is difficult to devise a simple and direct method for experimentally determining the value of l for a given soil. The only solution would appear to be to conduct model simulations with several different values of l and then to compare these simulations with observed behavior in order to select the most appropriate value for l. The type of experimental test required would be one involving significant rotation of the yield curve. Comparisons of observed and predicted behavior could then be made in terms of both the degree of rotation of the yield curve (identified experimentally by unloading and then reloading along a different stress path) and the observed pattern of straining. l is selected based on model simulations with different values and comparing the simulations versus the observed behavior the stress–strain response obtained from the
The Cam-clay models are simple in their formulation and use a few model parameters which can be easily determined by simple experimental procedures. In fact, one isotropic compression test and triaxial shear test would enable determination of soil parameters needed in the model. The Wheeler model which describes kinematic hardening of yield curve is shown to capture the most difficult feature such as strain softening associated with volumetric compression or positive pore pressure changes. The Wheeler model incorporates three additional parameters compared to modified Cam-clay model. With these model parameters the anisotropy of plastic behavior represented through a rotational component of hardening can be predicted. The stress–strain response obtained from the model predictions permit the following observations. •
The remolded stress–strain behavior can be captured by Cam-clay models. The remolded soils devoid of structure show isotropic hardening. This behavior is predicted quite closely by modified Cam-clay model.
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The Cam-clay models show more plastic strains compared to modified Cam-clay model. The Wheeler model predicts the behavior of natural clay. The model predictions clearly indicate that the stress–strain behavior of natural soils comprising of soil skeleton and structure can be predicted by Wheeler model. It may be seen that Wheeler model also predicts the strain softening feature of in situ soils usually noticed in experimental findings. Some of the discrepancies could be ascribed to the degree of anisotropy simulated by rotating hardening model, which may not be truly representing the degree of destructuring of fabric anisotropy during shearing.
6.
7.
8.
9.
10.
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