Phase2 FAQs > Theory
Below are answers to Theory FAQs for Phase2. Click on the question to read the answer.
What assumptions are being made in a two-dimensional Phase2 analysis?
Phase2 uses a plane strain analysis where two principal in-situ stresses are in the plane of the excavation and the third principal stress is out of plane. This is rarely the case so you have to make some assumptions and decompose the true 3D-stress tensor into 3 orthogonal stresses which are aligned with the 2D model of your excavation (2 in-plane and 1 out-of-plane).
You are also assuming that the cross-section of the excavation is constant and the excavation is of infinite length out of plane. Therefore 3-dimensional end effects are not accounted for in the Plane Strain analysis.
A Plane Strain analysis assumes that there are no shear stresses or strains in the out-of-plane direction.
What does isotropic, homogeneous mean?
Uniform material with one Young's modulus and one Poisson's ratio. The properties are not dependent on directionality or location within the material.
What is the "dilation" material strength parameter?
Dilation is a measure of how much volume increase occurs when the material is sheared. For a Mohr-Coulomb material, dilation is an angle that generally varies between zero (non-associative flow rule) and the friction angle (associative flow rule). For Hoek-Brown materials, dilation is defined using a dimensionless parameter that generally varies between zero and m. Low dilation angles/parameters (i.e. zero) are generally associated with soft rocks while high dilation angles/parameters (i.e. phi or m) are associated with hard brittle rock masses. A good starting estimate is to use 0.333*m or 0.333*phi for soft rocks and 0.666*m or 0.666*phi for hard rocks.
In the case of a very weak rock in which elastic-perfectly plastic behaviour is assumed, it is also generally assumed that there is no volume change. In other words, the dilation angle is zero for this case.
If you wish to use a strain softening model with dilation then the dilation angle should be smaller than the friction angle (for Mohr Coulomb) or m (for Hoek-Brown) - otherwise the analysis will be numerically unstable. What proportion of the friction angle or m should be used is very much a matter of debate and there are no clear guidelines. I generally use something between 1/3 and 1/2 but I have to admit that this is a pure guess. I try to check the sensitivity of the factor but it is sometimes hard to tell whether it makes any significant difference. This is because the process of fracture propagation is very complex and dilation is only one of the factors involved. It probably depends upon the brittleness or ductility of the rock mass as well as the extent of the failure propagation.
How does Phase2 take into account the longitudinal (out-of-plane) spacing of bolts?
The cross-sectional area of a bolt is divided by the out-of-plane spacing to get the effect of the bolts per unit out-of-plane dimension of the model.
Note: when the final bolt results are reported in the Phase2 Interpret program, the bolt results (e.g. axial force) apply to an individual bolt (i.e. the out-of-plane bolt spacing is again used to determine the force and stress per individual bolt).
It is the maximum capacity at failure. The load bearing capacity of the bolt never exceeds this value.
After a bolt fails (force in the bolt exceeds its peak capacity) the load bearing capacity of the bolt is dropped to the residual capacity values. If the residual capacity is less than the peak, the excess load is transferred back into the rock.
A joint is composed of two faces that are attached to each other by normal and shear springs at the nodes. Shear and Normal displacement at the node is composed of both elastic (spring) and plastic (slip) components. What you are seeing is most likely the elastic spring displacements. Think of a joint with infilling of some material, the joint can actually shear elastically with the displacements being a function of the shear modulus of the infilling material. In short, to completely lock the joint, set a very high value of spring stiffness. Also, the slip in the joint occurs when the forces in the shear springs exceed the slip criterion that you set. This results in plastic slip displacement. The combination of both the elastic and plastic gives you the total shear displacement in a joint.
Yes, the program does plasticity so changes in the strength of materials can have a dramatic effect on both the stresses and displacement.
If stresses are entirely within the strength envelopes defined by the cohesion and friction angle (i.e. no yielding is occurring, displacements are elastic), then changing the cohesion and friction angle will have no effect on stresses or displacements (unless the stress exceeds the strength envelope of a material).
The stress conventions used by Phase2 can be found in the following document:
Note: the bending moment convention for liners can actually be specified by the user with the Reverse Liner Orientation option.
The finite element method requires that there be a change in load state, thus putting the model out of equilibrium in order for there to be a change in stresses or displacements. It is inherent in the finite element formulation that if a model is in equilibrium, changing the modulus of a material will not result in any deformation or change in stresses.
First read the preceding question. In order to get deformation, just reducing the modulus is not enough. You also need a change in loading. To accomplish this you can try two methods. First, you could replace the material inside the excavation with a series of distributed loads that get relaxed over a number of stages. Second, each softened material that you are using in each stage should have the Initial Element Loading option set to None. This will have the effect of resetting the internal stresses in each element at the beginning of the stage. This puts the model out of equilibrium and will correctly simulate the behavior you are seeking.
Also see the following links:
Phase2 Developer's Tip:
3D tunnel simulation using the Material Softening method in Phase2
Phase2 Tutorial:
Where can I find more information on the plasticity models used in Phase2?
The following document describes the equations used for the failure surfaces (Mohr-Coulomb, Hoek-Brown, Drucker-Prager, Cam-Clay) and plastic potential flow surfaces in Phase2 :
Here are some references that detail how we implement the plasticity algorithm:
W.F. Chen, Plasticity in Reinforced Concrete, (Chapters 5 & 6), McGraw-Hill Book Company, 1982.
Owen, D. R. J. and Hinton, E., Finite Elements in Plasticity - Theory and Practice, Pineridge Press, Swansea, 1986, 594 pp.
Where can I find more information on the bolt models used in Phase2?
The following document gives a description of the implementation of bolts in Phase2 :
Where can I find more information on the Plain Strand Cable bolt model used in Phase2?
The following document gives a list of references for the model:
See the following document:
The joint formulation in Phase2 is one with normal and shear springs that allow the joint nodes to move relative to each other thus creating normal and shear displacement between once coincident joint nodes. The joints can also behave plastically (they can have shear strength parameters) allowing for slip between the two sides of the crack. References for the finite-element formulation are:
R.E. Goodman et al., "A Model for the Mechanics of Jointed Rock", Journal of the Soil Mechanics and Foundations Division, ASCE, SM3, May 1968.
J. Ghaboussi et al., "Finite Element for Rock Joints and Interfaces", Journal of the Soil Mechanics and Foundations Division, ASCE, SM10, Oct 1973.
C. Desai et al., "Thin Layer Elements for Interfaces and Joints", International Journal for Numerical and Analytical Methods in Geomechanics, vol. 8, 19-43, 1984.
The concept of initial joint deformation comes from a boundary element book by Crouch and Starfield "Boundary Element Methods in Solid Mechanics" page 217 which we incorporated originally in our Examine2D program. The joint springs can either have an initial force in them or not. If the initial force is such that the joint is in equilibrium with the surrounding solid elements then it does not deform. This is the case of no initial joint deformation. If the joint has no initial force in the springs then the joint will deform due to the initial stresses in the adjoining elements. This is the case of initial joint deformation.
The reason the displacements are upward is rebound. The initial stress due to the gravity loading for a particular element is higher than the body force due to the elements above it. This can be because 1) the unit weight you're using for gravity loading is higher than the material unit weight used for body forces and 2) the ground surface elevation is at the top of the slope. If the initial stress is higher than the weight of the material above a certain element it will expand. This is why the surface moves up and is a common phenomenon when modeling surface excavations. Generally, what people do is use the first stage to get the initial condition correct then look at relative displacements between stages. You can do this through the Data->Stage Settings option in Interpret.
Also see:
Phase2 Developer's Tip:
Setting up the Initial Stress State for Surface Models
Also see the Gravity Field Stress topic. The Use Actual Ground Surface option can now be used to obtain a much better estimate of the initial stress field under a non-horizontal ground surface. Previously it was only possible to define a single datum elevation from which to measure the vertical stress.
How can I convert between Mohr-Coulomb and Drucker-Prager strength parameters for use in Phase2?
The following document gives the equations:
How is liner plasticity (yielding) implemented in Phase2?
Phase2 uses the layered beam approach described in Owen and Hinton (1986). See the following document for more information:
If the liner strain after a particular stage is less than the strain at locking, then the liner will have zero axial stiffness in the NEXT stage. Thus in the NEXT stage the liner will have zero axial force in it (moments are not affected). If the liner strain is greater than or equal to the strain at locking after a stage, then the axial stiffness is set to that defined by the user (EA/L) for the NEXT stage and the liner will generally have axial forces in it after the NEXT stage. So for the sliding gap liner to work correctly, you have to have multiple stages with the liner installed and in general relax the boundary stresses (using tractions or core material softening) over a number of stages.
A sliding gap liner with Strain at Locking = 0, is NOT equivalent to a regular liner (i.e. no sliding gap), because for a sliding gap liner, the axial stiffness is always set to zero in the installation stage regardless of the strain at locking value. A sliding gap liner can only develop axial load AFTER the installation stage.
For more information see Tutorial 20 (Liner with Sliding Gap) and the Liner Type: Beam topic.