Hyperbolic Model for Tropical Residual Soils

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME
121
HYPERBOLIC CONSTITUTIVE MODEL FOR TROPICAL RESIDUAL
SOILS
Nagendra Prasad.K1
, Sulochana.N2
1
Professor, Dept. of Civil Engineering, SV University, Tirupati, India, (corresponding author)
2
Sulochana.N , Research Scholar, Dept. of Civil Engineering, SV University College of
Engineering, Tirupati & Lecturer in Civil Engineering, Govt. Polytechnic for Women,
Palamaner, Chittoor District, A.P.
ABSTRACT
The stress-strain response of natural soils depends on soil state, stress history and
drainage conditions. Many constitutive models are available for describing the stress-strain
relationship for different soil types. It is desirable to have a comprehensive model, based on
sound principles of continuum mechanics, capable of describing the soil behaviour under any
type of loading. The model parameters involved in such models most often, require elaborate
experimental procedures to evaluate them. There are many instances when a problem posed
to an engineer may not necessarily require such a complex material model. For example, a
simple undrained analysis may be sufficient for the immediate or end of construction (this
will be always critical condition) of structures on clayey soils. Depending on specific field
situation, it may be possible to analyze the problem with much simpler model. Therefore,
there is a need to develop a realistic and simple model whose parameters can be determined
easily with simple procedures. The cardinal aim of the present paper is to develop a simple
constitutive relationship using hyperbolic approach, based on analysis of test results on five
different types of soils. Combination of stress ratio and mean principal stress is identified to
capture the strain softening behaviour of residual soils. The model developed is applied to
predict the stress-strain response for other soils found in literature. The model predictions are
quite comparable and model parameters are easily determinable.
Keywords: Tropical residual soils, hyperbolic model, stress-strain-pore pressure response,
yield stress, confining pressure.
INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
TECHNOLOGY (IJCIET)
ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 4, Issue 3, May - June (2013), pp. 121-133
© IAEME: www.iaeme.com/ijciet.asp
Journal Impact Factor (2013): 5.3277 (Calculated by GISI)
www.jifactor.com
IJCIET
© IAEME

122
1. INTRODUCTION
Soils are very complicated engineering materials, whose constitutive response
depends on many compositional and environmental factors. The availability of high-speed
computers and powerful numerical techniques (such as the finite element method) makes it
possible to incorporate the non-linear behaviour of materials into the analysis of soil systems
and soil structure interaction problems. Some advanced soil models have been proposed for
the non-linear stress-strain behaviour of soils, including the hypo elastic models, the hyper
elastic models and the plasticity models. However, these models require the determination of
many parameters for the investigated soils.
The explicit nature of stress-strain response of tropical residual soil mostly depends
on fabric and nature of cement bonding in addition to the usual factors such as current state,
stress history, stress path and drainage conditions. Despite the availability of quite a good
number of constitutive relations concerning the behaviour of clays, there are still a large
number of problems which have not been satisfactorily tackled. Among these, strain
softening behaviour of tropical residual soils during deformation process is of importance.
Strain softening is an important phenomenon causing concern as regards the design problems
associated with estimation of bearing capacity, stability and deformation.
Most often tropical residual soils are treated as overconsolidated soils because they
also exhibit similar features like strain softening, higher initial stiffness, etc. But a closer
study of the test results of different tropical residual soils found in literature would reveal that
the behaviour of these soils in undrained shear is very much different from that of
uncemented overconsolidated soils. Most important difference is that softening here is
associated with continued positive pore pressures whereas softening in overconsolidated soils
is associated with negative pore pressures. Probably, this is because there is an additional
component of resistance from cementation bonds.
Thus there is a need for development of a realistic and simple model comprising of
easily determinable constitutive parameters which is capable of capturing the most important
aspects of the behaviour. For instance, the hyperbolic elastic models (Duncan and Chang,
1970) are still widely used in the non linear finite element analysis of uncemented soils is one
such example. The reasons for using hyperbolic models are ascribed to its simplicity and well
defined constants associated with the model. It is well known that hyperbolic model was
originally formulated to fit the undrained triaxial test results with only two constants to be
defined. It subsequently grew in strength and came to be applied to realistic boundary value
problems involving both drained and undrained conditions with corresponding modifications
using incremental approach. The inherent capability of the hyperbolic form to capture the
softening behaviour of tropical residual soils has not been attempted in the past (Nagendra
Prasad et al. 1999).
2. BACKGROUND INFORMATION
In a tropical region, residual soil layers can be very thick, sometimes extending to
hundreds of meters before reaching un-weathered rock. Unlike the more familiar transported
sediment soil, the engineering properties and behaviour of tropical residual soils may vary
widely from place to place depending upon the rock of origin and the local climate during
their formation; and hence are more difficult to predict and model mathematically. Despite
their abundance and significance, our knowledge and understanding of these soils is not as

123
extensive as that of transported sediment soil (Huat et al. 2012). However, with respect to
residual soil, both its interaction mechanism and its failure behaviour in soil composites are
not well understood due to limited study (Mofiz et al. 2010).
Tropical soils appear in large regions of the world and have been less studied than
soils from temperate climates, particularly with respect to critical state and limit state
conditions. Most geo-materials are structured in nature and this natural structure affects the
behavior of tropical soils. Structural features affecting soil behavior include soil cementation
and soil fabric.
Sarma et al. (2008) observed that the consolidation properties of soils indicate an
insight on the compressibility behaviour of soils with associated expulsion of water.
However, determination of such properties involves considerable time, cost and rigorous
testing process. Further, natural state of partial saturation and soil-moisture is not simulated
in the standard consolidation procedures. The sampling technique is also not specific for the
Oedometer tests and sampling disturbance influences the results considerably. As such,
modified methodologies of Odometer test for field simulation as well as simple correlations
of the consolidation parameters with fundamental properties are always preferred by
practising engineers.
Karmakar et al. (2004) brought out that the soil undergoes both elastic and plastic
deformation when subjected to loading. The basic requirement for integrated analyses of
movements and failure of a soil mass is a constitutive relationship capable of modelling
stress-strain behaviour of soil up to and beyond failure. Development of such a relationship
generally involves separating the elastic and plastic behaviour. This is achieved using a well-
defined curve known as the yield locus located in a shear stress-normal stress space. If the
stress state of a soil plots inside the yield locus, it is considered to be elastic and undergoes
recoverable deformation. On the other hand, if a particular stress path puts the stress state of
the soil on or outside the yield locus, plastic or irrecoverable deformation of soil occurs.
Elasto-plastic constitutive models help to distinguish between the recoverable and
irrecoverable deformations for understanding the stress strain behaviour of soil during
loading and unloading. In order to develop a simple framework, a mechanistic approach is
needed based on well planned experimental investigation.
3. SCOPE OF THE PAPER
Particularly soils in the Southern Indian Region are residual in nature (those derived
by in-situ weathering of rocks). 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. This is in contrast to the situation while
sedimentary soil deposits are encountered. Quite often cementation in rock would be left
behind due to varied degrees of weathering.
The objective of this paper is to develop a simple practical procedure for representing
the nonlinear, stress dependent, inelastic stress-strain behaviour of tropical residual soil
during undrained shear. Accordingly, the relationship described has been developed in such a
way that values of the required parameters may be derived from the results of the standard
laboratory triaxial tests. The formulations are proposed within the framework of a hyperbolic
relation of stress ratio (q/p) and also of effective mean principal stress (p) with strain. The

124
formulation incorporates stress dependency, nonlinearity, strain softening aspects of the
behaviour quite effectively. It must be pointed out that this is an attempt to circumvent the
difficulty associated with the choice of complicated constitutive models while dealing with
undrained field situations.
4. EXPERIMENTAL INVESTIGATIONS
4.1 Residual Soils Tested
In order to understand the mechanisms involved in shear and compression behaviour
in relation to naturally sedimented soils, a detailed experimental program has been
undertaken on undisturbed soil samples extracted from regional soil deposits in Tirupati and
its surroundings. These soil deposits are residual in nature which has been subjected to a
number of wetting and drying cycles. Owing to increase in construction activity concerning
these soils, there is a need to comprehensively understand the mechanisms involved in shear
and compression behaviour. The investigation considers the laboratory testing on
representative field samples (both undisturbed and remolded) which have been extracted
from the bottom of test pits of depths ranging from 1.8m to 3.5m. It may be seen that these
soils represent wide spectrum of residual soils encountered in practice in the region. The
liquid limit values range from 27-92 and fine fraction ranging from 32-79. The basic soil
properties of the soils considered are shown in table 1.
Table 1: Soil Properties
S.No. Description
Vinayaka
Nagar
Gayathri
Nagar
Renigunta
Muni Reddy
Nagar
Tiruchanur
Depth of Sampling, m
2.70 3.50 2.50 1.80 2.70
1 % Gravel 3.00 1.00 6.60 0.50 7.00
2 %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 Plastic limit (%) 30.10 22.17 45.30 20.75 35.24
6 Plasticity index (%) 11.90 10.83 46.70 6.25 19.76
7 Void ratio (eo) 0.610 0.601 0.620 0.605 0.550
8 Percent < 425µ 68 79 32 77 50
9
Modified liquid
limit, (WL )M %
28.56 26.00 29.44 21.00 27.50
10 eo/eLM 1.080 0.800 0.785 0.750 0.850
11 IS Classification CI CL SC SC SC
12 Field density, kN/m3
19.58 19.85 19.62 19.18 19.54
13
Natural moisture
content, %
16.70 17.73 19.86 16.25 14.92
14
Yield stress σy in
kPa
80 76 67 64 80

125
4.2 Analysis of Test Data
The test results are analysed using the effective mean normal and deviatoric stress
parameters p and q as given by:
3
2 31 σσ +
=p (1)
31 σσ −=q (2)
Where σ1 = axial stresses on a cylindrical sample.
σ3 = radial stresses on a cylindrical sample.
The deviatoric strain is expressed by:
)2(
3
2
31 εεε −=s (3)
31 2εεε +=v (4)
Where ε1 = axial strains
ε3 = radial strains.
For undrained tests, εv = 0 and hence εs = ε1. Axial strains were measured externally and the
deviatoric stresses were calculated from the readings of pressure controller and the current
sample area using conventional area correction.
Figures 1 and 2 shows the stress-strain-pore pressure response of tropical residual
soils for two confining pressures (50kPa and 100kPa) of two soils. From the figures 1 and 2,
it may be observed that the strain softening is associated with positive pore pressure. The
other soils are also following the same trend. It is well known that it is not possible to get a
unique plot of q/po versus εs for tropical residual soils while it is possible in the case of
normally consolidated clays. This may be attributed to the fact that the evolution of
cementation bond resistance and subsequent softening during deformation process is not
proportional to the initial confining pressures, there by being more predominant for low
confining pressures in comparison to the equivalent unbonded response.
Effective stress paths of the two samples are presented in figures 3 and 4 for the
confining pressures tested. The other soils are also following the same trend. These stress
paths indicate that mean effective stress decreases during strain hardening and strain
softening process. The specimens tested under different confining pressures tend to reach the
critical states corresponding to remoulded situation if cementation bonds were not present. It
turns out that critical state is approached only slowly at large strains. The results indicate that
it is the type of soil that determines the critical state parameters and not the initial state or
cementation bonding. The results show that the value of stress ratio (η=q/p) upon reaching
respective peak values remains nearly constant for two confining pressures as indicated in
figures 3 and 4. It may be further observed from figures 1 and 2 that the pore water pressure
continues to be positive even in the softening region indicating that the behaviour is not
similar to that of overconsolidated soils as is frequently reported. Strain softening associated
with positive pore pressures is perhaps peculiar feature concerning the behaviour of tropical
residual soils. This may be ascribed to the additional stress transfer on to the pore pressure as
a consequence of debonding with progressing shearing. This stress transfer seems to occur in
such a way that the value of η remains fairly a constant with distortional strain.

126
Fig.2 Stress-strain-pore pressure response of
Gayathri nagar soil
Fig.4 Effective stress paths of
Gayathri nagar soil
Fig.3 Effective stress paths of
Vinayaka nagar soil
Fig.1 Stress-strain-pore pressure response
of Vinayaka nagar soil

127
An examination of data obtained from experimental results of tropical residual soils
indicates that p versus εs (figures 5 and 6) and η versus εs (figures 7 and 8) relations are
hyperbolic. Other soils are also following the same trend. These observations form the basis
for the formulations proposed in this paper. The two constant hyperbolic relations have been
utilized with advantage to analyze the consolidated undrained triaxial test results. Variation
of stress ratio (q/p) and the mean principal stress with deviatoric strain in terms of hyperbolic
relation takes the form as
)( 22 s
s
Xba ε
ε
η
+
= (5)
Fig.6 Mean principal stress-strain
response of Gayathri Nagar soil
Fig.7 Stress ratio-strain response
of Vinayaka Nagar soil
Fig.8 Stress ratio-strain response
of Gayathri Nagar soil
Fig.5 Mean principal stress-strain
response of Vinayaka Nagar soil

128
)(
)(
33 s
s
o
Xba
pp
ε
ε
+
=− (6)
Where η = q/p
εs = shear strain
po = Initial mean principal stress
Equations 5 and 6 can be transformed into the linear form as presented below in order to be
able to make them suitable for experimental verification.
)( 22 s
s
Xba ε
η
ε
+= (7)
)(
)(
33 s
o
s
Xba
pp
ε
ε
+=
−
(8)
The experimental data of five soils is plotted in the form represented by equations 7
and 8 and are shown in figures 9 and 10. A good straight line can be fitted to the experimental
data between εs/η versus εs and εs/(po-p) versus εs for all the soils of selected confining
pressures. This is a good indication of the applicability of the form proposed to represent the
stress-strain response of tropical residual soils.
Elimination of εs in equations 5 and 6 yields:
)1()(1
)(
2
2
3
3
b
a
ppb
ppa
o
o
−
=
−−
− η
(9)
which describes the undrained stress path of tropical residual soil.
Fig.9 Transformed stress ratio-
strain curves
Fig.10 Transformed mean
principal stress-strain curves

129
For meaningful application of the relations proposed it is necessary to determine the
exact nature of the parameters a2, b2, a3 and b3 in relation to the confining pressure
normalized with yield stress. Figures 11-14 shows the variation of these constants with the
confining pressure normalized with yield stress.
Experimental results indicate that it is convenient to express the parameters a3 and b3
in terms of initial confining pressures normalized with yield stress in the form of a power
function and a2 and b2 in the form of linear relationship with confining pressure normalized
with yield stress on log scale (Equations 10 and 11).
Fig.12 Variation of parameter b2 with
confining pressure normalized with yield
stress
Fig.13 Variation of parameter a3 with
confining pressure normalized with yield stress
Fig.14 Variation of parameter b3 with
confining pressure normalized with yield stress
Fig.11 Variation of parameter a2 with
confining pressure normalized with yield
stress

130
488.0ln326.02 +








=
y
op
a
σ
(10a)
481.0ln143.02 +








=
y
op
b
σ
(10b)
28.1
3 109.0
−








=
y
op
a
σ
(11a)
87.1
3 011.0
−








=
y
op
b
σ
(11b)
The hyperbolic constants a2, b2, a3 and b3 as obtained by the equations mentioned
above are used to compute the stress-strain response. In computing the above parameters, in
addition to the experimental results, data from the literature related to tropical residual soils is
also used. The test data of reddish lateritic soil (Futai et al. 2004) which is sampled at 1m
depth and having yield strength of 100kPa. This soil is tested under different confining
pressures ranging from 25kPa to 400kPa. The experimental data is plotted in the transformed
hyperbolic form with εs / η and εs / (po-p) on y-axis and deviatoric strain on x-axis and are
presented in figures 15 and 16. The computed and the observed plots of q versus εs of soils
tested are presented in figures 17 and 18 respectively. The close agreement between the
computed and experimental results seems to confirm the applicability of the hyperbolic
model for the tropical residual soils in undrained shear. Other soils are also following the
same trend. However, these formulations may not be applicable for very low confining
pressures where the pore pressure response does not follow a hyperbolic variation with strain.
Fig.15 Transformed stress ratio-strain
curves of lateritic soil
Fig.16 Transformed mean principal
stress-strain curves of lateritic soil

131
Fig.17 Experimental and predicted stress-
strain curves of Vinayaka Nagar soil
Fig.18 Experimental and predicted stress-
strain curves of Gayathri Nagar soil
Fig.19 Experimental and predicted stress-strain curves of saprolitic soil

132
4.3 Application to other experimental investigations
It is desirable to consider the proposed mathematical form in relation to other
published literature in order to assess its general applicability. The test data of tropical
residual saprolitic soil (Futai et al. 2004) which is sampled at 5m depth and having yield
strength of 260kPa is examined in this connection. This soil is tested under different
confining pressures ranging from 25kPa to 690kPa. The equations 10 and 11 are used to
predict the stress-strain characteristics. The close agreement between predicted and
experimental values of saprolitic soil (Futai et al. 2004) is once again well demonstrated by
comparative plot shown in figure 19.
5. CONCLUDING REMARKS
Based on the analysis of test results of carefully planned experimental programme, the
following concluding remarks may be made.
1. The strain softening behaviour associated with positive pore water pressures noticed
in the residual soils can be captured using hyperbolic approach with appropriate
modifications.
2. The combination of stress ratio (η=q/p) and mean principal stress (p) is used to
represent the non-linear stress dependent behaviour of residual soils.
3. Four parameters are involved in the proposed hyperbolic model which can be
determined from simple consolidated undrained triaxial tests and one dimensional
compression tests.
4. The model parameters are found to have functional relationship with the yield stress
value (σy) in one dimensional oedometer compression.
5. The model developed has been applied to other soil data and the applicability is
evidenced from the model predictions being in close agreement with observed
behaviour.
REFERENCES
1. Bujang B.K. Huat, David G. Toll, Arun Prasad (2012) - Handbook of Tropical Residual
Soils Engineering- Published 24th May 2012 by CRC Press.
2. Duncan J.M. and Chang C.Y. (1970) - Nonlinear analysis of stress and strain in soils.
Journal of the Soil Mechanics and Foundations Division, ASCE, 1970, 96, No. SM5,
1629-1653.
3. Karmakar1, S., Sharma.J. and Kushwaha.R.L.(2004),” Critical state elasto-plastic
constitutive models for soil failure in tillage – A review”, Canadian biosystems
engineering, volume 46. 2004.
4. M. M. Futai, M. S. S. Almeida, and W. A. Lacerda, (2004) Yield, Strength, and
Critical State Behavior of a Tropical Saturated Soil, Journal Of Geotechnical And
Geoenvironmental Engineering © ASCE / November 2004, 1169 -1179
5. Mofiz M. and Mohammad Nurul Islam M. (2010)- Assess the Stress-Strain and
Interfacial Frictional Behaviour of Nonwoven Geotextile Reinforced Residual Soils-
GeoFlorida 2010: Advances in Analysis, Modeling & Design (GSP 199) © 2010
ASCE.

133
6. Nagendra Prasad, K., Srinivasa Murthy, B.R., Sitharam, T.G., and Vatsala, A. (1999)-
Hyperbolic stress-strain and pore pressure response of sensitive clays – Indian
Geotechnical Journal, 29 (3), 221-241.
7. Sarma, M.D. & D. Sarma,D. (2008) “Prediction of Consolidation Properties of
Partially Saturated Clays” The 12th
International Conference of International
Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6
October, 2008 Goa, India
8. Nagendra Prasad.K, Manohara Reddy.R, Chandra.B and Harsha Vardhan Reddy.M,
“Compression Behaviour of Natural Soils”, International Journal of Civil Engineering
& Technology (IJCIET), Volume 4, Issue 3, 2013, pp. 80 - 91, ISSN Print:
0976 – 6308, ISSN Online: 0976 – 6316.
9. Nagendra Prasad.K, Sivaramulu Naidu.D, Harsha Vardhan Reddy. M and Chandra.B,
“Framework for Assessment of Shear Strength Parameters of Residual Tropical
Soils”, International Journal of Civil Engineering & Technology (IJCIET), Volume 4,
Issue 2, 2013, pp. 189 - 207, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.

Hyperbolic Model for Tropical Residual Soils

Recomendados

Recomendados

Más contenido relacionado

La actualidad más candente

La actualidad más candente (19)

Destacado

Destacado (20)

Similar a Hyperbolic Model for Tropical Residual Soils

Similar a Hyperbolic Model for Tropical Residual Soils (20)

Más de IAEME Publication

Más de IAEME Publication (20)

Último

Último (20)

Hyperbolic Model for Tropical Residual Soils