Chapter 3. Description of Numerical Programme and Model Properties
3.3 Material Models for Numerical Simulation
3.3.1 Backfill soil
Description of Numerical Programme and Model Properties 52
used in the shaking table tests, had maximum dry weight of 17.66 kN/m3 at the densest state and minimum dry unit weight of 14.03 kN/m3 at the loosest state. The angle of internal friction obtained from triaxial tests on sand at relative density of 63%
was reported as 43° (Krishna 2008).
Constitutive Model for Backfill Soil
In numerical models of GRS walls, the backfill soil is simulated in layer by layer as illustrated in physical model tests (Krishna and Latha 2007, 2009 and Latha and Krishna 2008). As the model is simulated in layers, confining pressure on each zone changes with change in height of fill and hence the modulus. Hence the modulus during construction is modelled as stress dependent deformation modulus (Et), that change as the height of the soil changes. Hence, the backfill soil is modeled as elasto- plastic Mohr Coulomb material coded with hyperbolic soil modulus. The hyperbolic stress dependent soil modulus is expressed as Eq. 3.4 (Duncan et. al. 1980).
( )
n
a a n f
t K P P
c
E R
+
−
− −
= 3
2
3 3
1 .
sin cos
. 2
) )(
sin 1
1 ( σ
φ σ φ
σ σ
φ 3.4
where Kn is the modulus number; n is the modulus exponent; c is the cohesion; σ1 and σ3 are the major and minor effective confining stress respectively; ϕ is the angle of internal friction; Rf is the failure ratio; pa is atmospheric pressure. Krishna (2008) determined the hyperbolic soil parameters as mentioned by Duncan et al. (1980), from laboratory triaxial test data. The hyperbolic model parameters Kn, n and average Rf obtained by Krishna (2008) for backfill sand at 63% relative density were 831, 0.678 and 0.93, respectively.
As FLAC3D do not have built in hyperbolic model, the model is incorporated by FISH subroutines (Itasca 2008). The principal stresses (σ1 and σ3) are extracted by FISH
variables from each element of the model after every 10 steps and stress dependent modulus (Et) is calculated. The modulus is then updated using zone variables in FISH programme.
Numerical Triaxial Tests
Numerical triaxial tests are performed to verify the implementation of hyperbolic soil model parameters in backfill soil. A cylindrical grid of diameter 38 mm and length 76 mm is considered for simulation of numerical triaxial test. The numerical model considered for triaxial test sample is shown in Fig. 3.2
Fig. 3.2 Numerical model of triaxial test sample (dia 38 mm & height 76 mm) The soil is simulated as elasto-plastic Mohr Coulomb material coded with hyperbolic soil modulus proposed by Duncan et al.(1980). The angle of internal friction and other hyperbolic model parameters as mentioned in Krishna (2008) are considered in numerical simulation. The all around confining pressures of 10 kPa, 50 kPa, 100 kPa and 150 kPa are applied to the model. The results obtained are compared with physical triaxial test results (Krishna 2008) and are shown in Fig. 3.3.
Comparison of the results between numerical and physical triaxial tests, at different
Description of Numerical Programme and Model Properties 54
confining pressures, indicates that the deviator stresses in numerical simulations are reasonably comparable to that of physical tests. This validates the hyperbolic model parameters obtained from physical model tests and also its proper implementation in numerical modeling.
Fig. 3.3 Comparison of results between numerical and physical triaxial tests at different confining pressure
Fig. 3.4 shows variation of stress ratio with respect to the octahedral shear strain (γoct) obtained from numerical triaxial tests at different confining pressures. The variation of stress ratio with respect to octahedral shear strain is divided into three zones. They are: (I) Elastic zone - the stress and strain variation is linear and and elastic (II) Non-linear zone – stress and strain interdependency is non-linear and (III) Plastic zone - the stress increase is not significant compared to the strain increment indicating plastic deformation. From the figure it is seen that the elastic zone is extended upto octahedral shear strain of 0.3%, non-linear zone lies within 0.3% - 2.0% strain levels (γoct) and plastic zone exist beyond 2.0% strain (γoct).
0.00 0.03 0.06 0.09 0.12
0 100 200 300 400 500 600 700
σ3=10 kPa σ3=50 kPa σ3=100 kPa
Deviator stress, kPa
Axial strain
Numerical
Physical(Krishna 2008) σ3=150 kPa
Fig. 3.4 Behavior of soil in numerical triaxial compression Constitutive Model for Soil during Cyclic Loading
In the numerical model, subjected to seismic excitation, use of constant shear modulus during cyclic loading is not appropriate due to cyclic/hysteresis nature of soil.
Massing model (Masing 1926) is the simple cyclic model which can incorporate cyclic as well as hysteric behavior of soil. In Massing rule, a skeleton curve is used to express hysteresis loop in unloading and reloading process. A proper hysteresis loop can be simulated efficiently with skeleton curve in Massing rule. The shear behavior of granular soils under cyclic loading is modeled using non-linear and hysteretic constitutive relation according to Massing rule (Masing 1926). Cai and Bathurst (1995); Fakharian and Attar (2007) and Liu et al. (2011) were also adopted the Masing rule for two dimensional numerical simulations of GRS Walls. Schematic representation of shear stress and shear strain relation during unloading and reloading cycles, according to Masing rule, is shown in Fig. 3.5. Here, octahedral shear stress τoct and octahedral shear strain γoct are used to represent stress and strain conditions in
0 1 2 3 4 5 6 7
0 2 4 6
Stress ratio (σ 1/σ 3)
Octahedral shear strain (γoct), % Confining pressure
50 kPa 100 kPa 150 kPa
I
II
III
Zone I - Elastic zone Zone II - Non-linear zone Zone III- Plastic zone
Description of Numerical Programme and Model Properties 56
three dimensional states. The octahedral stress and strain states are calculated from six stresses and strains acting on each element. The octahedral stress invariants, namely octahedral normal stress, σoct and octahedral shear stress, τoct, are stress parameters independent of the choice of reference axis and adopted to analyze the stress variation (Chen and Mizuno, 1990).
!"= #+ %+ & 3.5
' !"= (#− %&%+ #%− &%+ #− &%* % 3.6
where ,% and are the principal stresses on an element. Similarly the octahedral strain invariants are strain parameters independent of the choice of reference axis (Chen and Mizuno, 1990) and are expressed as
+ !" =%(#,--− ,.. &% + /,..− ,001%+ #,00− ,--&%+ 6#,-.%+ ,.0%+ ,0-%&* % 3.7 Where εxx, εyy, εzz, εxy, εyz and εzx are strain parameters acting on each element in three dimensional state.
Fig. 3.5 Non-linear hysteretic stress-strain model of granular soil: (a) stress-strain cap;
(b) unload-reload cycle (modified after Cai and Bathurst 1995)
In the Masing model, the shear modulus is determined on the basis of stress and strain states that may vary during cyclic loading condition. The tangent shear modulus during the first cycle is expressed as
" = 345
67 34589:; _=>?@ABC_=>?ADE 3.8
where FG- is the initial shear modulus, 'FG-_ !" is the maximum octahedral shear stress which is related to shear parameters of soil through cohesion H and internal angle of friction φ, and +I_ !" is the octahedral shear strain. The initial shear modulus FG- is extracted from each soil element by FISH variables before the application of dynamic excitation. The tangent modulus during unloading/reloading cycle is
" = 345
67 345%8345_=>?@|∆B=>?|DE 3.9
where ∆+ !" represents the difference in octahedral shear strain during unloading/reloading cycle. In unloading case as it equals to #+I_ !"− +L_ !"& and in reloading case it is #+I_ !"− +M_ !"&. +I_ !" is the octahedral shear strain at present state and +L_ !" and +M_ !" are octahedral shear strains at starting points of unloading and reloading, respectively, for that cycle.
The tangent bulk modulus N" is expressed in the following form:
N" = O× QG× 7R3
4@S 3.10
where O is the bulk modulus constant and T is the bulk modulus exponent.