SUMMARY, CONCLUSIONS AND FUTURE SCOPE
CHAPTER 3 NUMERICAL MODELLING
3.4. MATERIAL MODELS AND MATERIAL PROPERTIES
In Rslope2d, once the random field is transformed into the desired lognormal field, it is then mapped onto the finite element mesh, which is established according to the user-defined slope geometry. A typical finite element mesh for a 1:1 slope with a height, H = 10 m, is shown in Fig.
3.5. Each element within the slope geometry is 1 m by 1 m in size and it is assigned a random variable of the particular soil property (i.e. c or φ). The computer model uses 8-noded quadrilateral elements with reduced integration for the generation of gravity loads, stiffness matrix generation and stress redistribution phases of the algorithm (Chok, 2009).
Fig. 3.5 Typical finite element mesh used in Rslope2d RFEM model
3.4.1. Material Models in Slope/W Analysis
The commonly available methods used to evaluate FoS in Slope/W are based on limit equilibrium formulations except for the one which uses finite element computed stresses. Many different material models are available in Slope/W that can be used to calculate the FoS of slopes. These material models include ‘Mohr Coulomb model’, ‘Undrained model’, ‘High Strength model’,
‘Bedrock or Impenetrable model’, ‘Bilinear strength model’, ‘Anisotropic Strength model’,
‘Spatial Mohr Coulomb model’, ‘Anisotropic Function model’ and ‘Model for Normal/Shear Function interactions’. In the present study, the Mohr Coulomb material model and Impenetrable/Bedrock model has been used for slope material and foundation material respectively, to compute the stability of the slope.
3.4.2 Material Models in Sigma/W Analysis
Sigma/W includes six inbuilt constitutive models for soils as well as an option for inducing the user-defined constitutive model. For each of these models, the response will be different depending on the type of stress characteristics assigned to the model, namely whether it is assigned with total stress, effective stress with no pressure change, or effective stress with pore-water pressure change.
The in-built models available are (a) Linear elastic model (b) Anisotropic elastic model (c) Hyperbolic model (d) Elastic Plastic model (e) Cam Clay model, and (f) Modified Cam Clay model. For the present study, the elastic-plastic model is utilized. The Elastic-Plastic model in Sigma/W describes an elastic-perfectly plastic relationship. The elastic-perfectly plastic Mohr- Coulomb (M-C) model necessitates 5 input parameters, specifically the strength parameters (cohesion c, angle of internal friction φ and dilatancy angle ψ) and the stiffness parameters (Elastic modulus E and Poisson’s ratio ν). A typical stress-strain curve for this model is shown in Fig. 3.6.
The generated stress is directly proportional to the corresponding strain, until the yield point is reached. Beyond the yield point, the stress-strain curve is perfectly horizontal, indicating substantial increase in strain without change in stress. In Sigma/W, soil plasticity is formulated using the theory of incremental plasticity (Hill, 1950). Sigma/W uses the Mohr-Coulomb yield criterion as the yield function for the Elastic-Plastic model. The following equation provides a common form of the Mohr-Coulomb criterion expressed in terms of principal stresses.
2 1
2 sin cos sin sin cos
3 3 3 3
J I
F = J + − + − −c (3.19)
The Mohr-Coulomb criterion can also be written in terms of the stress invariants I1, I2 and θ. The yield function, F, can then be written as follows (Chen and Zhang, 1991).
2 1
2 sin cos sin sin cos
3 3 3 3
J I
F = J + − + − −c
(3.20)
where,
( ) (
2)
2( )
2 22
1
6 x y y z z x xy
J = − + − + − + = the second deviatoric stress invariant,
1 3
3/ 2 2
1 3 3
3cos 2 J
= − J = the load angle,
2 3
d d d d
x y z z xy
J = − = the third deviatoric stress invariant,
1 x y z
I = + + = the first deviatoric stress invariant,
φ is the angle of internal friction, and c is the cohesion of the soil.
The deviatoric stress idin the ith-direction can be defined as:
1
3
d
i i
= −I (3.21)
where, I = x, y or z.
When the angle of internal friction, φ, is equal to zero, the Mohr-Coulomb yield criterion becomes the Tresca criterion (Smith and Griffiths, 1988):
2 sin
F = J +3−c
(3.22)
The plastic potential function, G, used in Sigma/W has the same form as the yield function, F (i.e., G = F) except the internal friction angle, φ, is replaced by the dilation angle, ψ. Thus, the potential function is given by:
2 1
2 sin cos sin sin cos
3 3 3 3
J I
F = J + − + − −c
(3.23)
The derivatives of the yield function in terms of the stress invariants are computed using the chain rule of differentiation.
1 2
1 2
dI dJ
dF dF dF dF d
d dI d dJ d d d
= + + (3.24)
Derivatives of the Mohr-Coulomb yield function, with respect to the stress invariants, can be written as follows:
1
sin 3 dF
dI
= −
2 2
1 1
sin sin cos
3 3
2 3
dF
dJ J
= + + + (3.25)
2
2 cos sin sin
3 3 3
J
dF J
d
= + + +
The derivatives of the stress invariants with respect to the stresses are:
1 1110
dI d =
2 dx dy zd2 xy
dJ
d
=
3 3 2
2 2
3 3
2 sin 3 2
dJ J dJ
d
d J d J d
= −
(3.26)
3 2 2 2 2
3 3 3
d d d d d d
y z x z x y xy
dJ J J J
d
= + + + −
Similarly, the derivatives of the potential function can be obtained by substituting ψ for φ in Eqn.
3.25.
Fig. 3.6 Elastic-perfectly plastic constitutive relationship (adopted from Sigma/W Manual, 2018)
3.4.3. Material Models in Quake/W Analysis
Quake/W provides three different material models namely (a) linear elastic model, (b) equivalent linear model and (c) nonlinear model. In the present study, nonlinear material model is utilized which can be defined using unit weight (γ), Poisson’s ratio (ν), shear strength parameters (c and φ), damping ratio (ξ) as a constant or a function, pore-water pressure function, recoverable modulus function, maximum shear modulus (Gmax) as a constant or a function, steady state strength and collapse surface angle (for liquefied zones).
3.4.4. Material Models in Rslope2d
The FE based algorithm for slope stability analysis in Rslope2d utilises an elastic-perfectly plastic stress-strain law with a Mohr-Coulomb failure criterion for plain strain conditions. The computer model (Rslope2d) used in present study uses cohesion (c), friction angle (φ), dilation angle (ψ), Young’s Modulus (Es), Poisson’s ratio (υ), and unit weight (γ) of soil as input parameters to conduct the elastoplastic FE slope stability analysis.