Product: Abaqus/Standard
This example illustrates the use of the jointed material model in the context of geotechnical applications. We examine the stability of the excavation of part of a jointed rock mass, leaving a sloped embankment. This problem is chosen mainly as a verification case because it has been studied previously by Barton (1971) and Hoek (1970), who used limit equilibrium methods, and by Zienkiewicz and Pande (1977), who used a finite element model. This example also has been extended to study the slope stability of excavated soil medium with the same geometry, by using the Mohr-Coulomb plasticity model with and without the tension cutoff feature.
Geometry and model
The plane strain model analyzed is shown in Figure 1.1.6–1 together with the excavation geometry and material properties. The rock mass contains two sets of planes of weakness: one vertical set of joints and one set of inclined joints. We begin from a nonzero state of stress. In this problem this consists of a vertical stress that increases linearly with depth to equilibrate the weight of the rock and horizontal stresses caused by tectonic effects: such stress is quite commonly encountered in geotechnical engineering. The active “loading” consists of removal of material to represent the excavation. It is clear that, with a different initial stress state, the response of the system would be different. This illustrates the need of nonlinear analysis in geotechnical applications—the response of a system to external “loading” depends on the state of the system when that loading sequence begins (and, by extension, to the sequence of loading). We can no longer think of superposing load cases, as is done in a linear analysis.
Practical geotechnical excavations involve a sequence of steps, in each of which some part of the material mass is removed. Liners or retaining walls can be inserted during this process. Thus, geotechnical problems require generality in creating and using a finite element model: the model itself, and not just its response, changes with time—parts of the original model disappear, while other components that were not originally present are added. This example is somewhat academic, in that we do not encounter this level of complexity. Instead, following the previous authors’ use of the example, we assume that the entire excavation occurs simultaneously.
Solution controls
The jointed material model includes a joint opening/closing capability. When a joint opens, the material is assumed to have no elastic stiffness with respect to direct strain across the joint system. Because of this, and also as a result of the fact that different combinations of joints may be yielding at any one time, the overall convergence of the solution is expected to be nonmonotonic. In such cases the use of *CONTROLS, ANALYSIS=DISCONTINUOUS is generally recommended to prevent premature termination of the equilibrium iteration process because the solution may appear to be diverging.
JOINTED ROCK SLOPE
As the end of the excavation process is approached, the automatic incrementation algorithm reduces the load increment significantly, indicating the onset of failure of the slope. In such analyses it is useful to specify a minimum time step to avoid unproductive iteration.
For the nonassociated flow case UNSYMM=YES is used on the *STEP option. This is essential for obtaining an acceptable rate of convergence since nonassociated flow plasticity has a nonsymmetric stiffness matrix.
Results and discussion
In this problem we examine the effect of joint cohesion on slope collapse through a sequence of solutions with different values of joint cohesion, with all other parameters kept fixed. Figure 1.1.6–2 shows the variation of horizontal displacements as cohesion is reduced at the crest of the slope (point A in Figure 1.1.6–1) and at a point one-third of the way up the slope (point B in Figure 1.1.6–1). This plot suggests that the slope collapses if the cohesion is less than 24 kPa for the case of associated flow or less than 26 kPa for the case of nonassociated flow. These compare well with the value calculated by Barton (26 kPa) using a planar failure assumption in his limit equilibrium calculations. Barton’s calculations also include “tension cracking” (akin to joint opening with no tension strength) as we do in our calculation. Hoek calculates a cohesion value of 24 kPa for collapse of the slope. Although he also makes the planar failure assumption, he does not include tension cracking. This is, presumably, the reason why his calculated value is lower than Barton’s. Zienkiewicz and Pande assume the joints have a tension strength of one-tenth of the cohesion and calculate the cohesion value necessary for collapse as 23 kPa for associated flow and 25 kPa for nondilatant flow.
Figure 1.1.6–3 shows the deformed configuration after excavation for the nonassociated flow case and clearly illustrates the manner in which the collapse is expected to occur. Figure 1.1.6–4 shows the magnitude of the frictional slip on each joint system for the nonassociated flow case. A few joints open near the crest of the slope.
The study of soil slope stability using the Mohr-Coulomb plasticity model is performed for two cases: one without tension cutoff and one including the tension cutoff feature. The tension cutoff feature limits the stress carrying capacity of soil in tension. It can be seen that the maximum principal stress without tension cutoff (see contour plot in Figure 1.1.6–5) is higher than the limiting maximum principal stress (see Figure 1.1.6–6) with tension cutoff as expected. With tension cutoff, one also observes the appearance of the equivalent plastic strain in tension, PEEQT in the region of maximum principal stress. In this case it is also seen that the equivalent plastic strain, PEEQ on the cohesion failure surface is higher compared to the case without tension cutoff. The contour plots for the equivalent plastic strains are not shown.
Input files
jointrockstabil_nonassoc_30pka.inp Nonassociated flow case problem; cohesion = 30 kPa. jointrockstabil_assoc_25kpa.inp Associated flow case; cohesion = 25 kPa.
mc_slopestabil.inp Slope stability analysis, Mohr-Coulomb plasticity without tension cutoff
1.1.6–2
Abaqus Version 6.11 ID: Printed on:
JOINTED ROCK SLOPE
mctc_slopestabil.inp Slope stability analysis, Mohr-Coulomb plasticity with tension cutoff
References
•
Barton, N., “Progressive Failure of Excavated Rock Slopes,” Stability of Rock Slopes, Proceedings of the 13th Symposium on Rock Mechanics, Illinois, pp. 139–170, 1971.•
Hoek, E., “Estimating the Stability of Excavated Slopes in Open Cast Mines,” Trans. Inst. Min. and Metal., vol. 79, pp. 109–132, 1970.•
Zienkiewicz, O. C., and G. N. Pande, “Time-Dependent Multilaminate Model of Rocks – A Numerical Study of Deformation and Failure of Rock Masses,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 1, pp. 219–247, 1977.JOINTED ROCK SLOPE 90 Joint set 1 52.5 Joint set 2 60 70 m E = 28 GPa ν = 0.2 K0= 1/3 ρ = 2500 kg/m3 Joint sets : βa = 45 da = variable Bulk rock : βb= 45 db= 5600 kPa removed in single stage g = 9.81 m/s2 B A O O O O O
Figure 1.1.6–1 Jointed rock slope problem.
1.1.6–4
Abaqus Version 6.11 ID: Printed on:
JOINTED ROCK SLOPE x o x o x o x o x o x o x o o x o x o x o x o x o x o x 120 100 80 60 40 20 da (kPa) +1 0 -1 -2 -3 -4 H or izon ta l di sp lacemen t (mm ) Associated Nonassociated x o x x Point A Point B
Figure 1.1.6–2 Horizontal displacements with varying cohesion.
1 2
3
DISPLACEMENT MAGNIFICATION FACTOR = 3.000E+03
1 2
3
JOINTED ROCK SLOPE 1 2 3 1 2 3 PEQC1 VALUE -6.03E-07 +2.30E-05 +4.66E-05 +7.02E-05 +9.38E-05 +1.17E-04 +1.41E-04 +1.64E-04
Joint set 1 (vertical joints).
1 2 3 1 2 3 PEQC2 VALUE -3.74E-06 +6.47E-06 +1.66E-05 +2.69E-05 +3.71E-05 +4.73E-05 +5.75E-05 +6.78E-05
Joint set 2 (inclined joints).
Figure 1.1.6–4 Contours of frictional slip magnitudes (nonassociated flow).
1.1.6–6
Abaqus Version 6.11 ID: Printed on:
JOINTED ROCK SLOPE
(Avg: 75%) S, Max. In−Plane Principal
−1.320e+03 −1.208e+03 −1.096e+03 −9.832e+02 −8.709e+02 −7.585e+02 −6.462e+02 −5.338e+02 −4.215e+02 −3.091e+02 −1.968e+02 −8.444e+01 +2.790e+01
Figure 1.1.6–5 Maximum principal stress without tension cutoff.
(Avg: 75%) S, Max. In−Plane Principal
−1.320e+03 −1.209e+03 −1.098e+03 −9.874e+02 −8.765e+02 −7.656e+02 −6.547e+02 −5.438e+02 −4.329e+02 −3.220e+02 −2.111e+02 −1.002e+02 +1.069e+01
NOTCHED BEAM UNDER CYCLIC LOADING
1.1.7 NOTCHED BEAM UNDER CYCLIC LOADING
Product: Abaqus/Standard
This example illustrates the use of the nonlinear isotropic/kinematic hardening material model to simulate the response of a notched beam under cyclic loading. The model has two features to simulate plastic hardening in cyclic loading conditions: the center of the yield surface moves in stress space (kinematic hardening behavior), and the size of the yield surface evolves with inelastic deformation (isotropic hardening behavior). This combination of kinematic and isotropic hardening components is introduced to model the Bauschinger effect and other phenomena such as plastic shakedown, ratchetting, and relaxation of the mean stress.
The component investigated in this example is a notched beam subjected to a cyclic 4-point bending load. The results are compared with the finite element results published by Benallal et al. (1988) and Doghri (1993). No experimental data are available.
Geometry and model
The geometry and mesh are shown in Figure 1.1.7–1. Figure 1.1.7–2 shows the discretization in the vicinity of the notch, which is the region of interest in this analysis. Only one-half of the beam is modeled since the geometry and loading are symmetric with respect to the x= 0 plane. All dimensions are given in millimeters. The beam is 1 mm thick and is modeled with plane strain, second-order, reduced-integration elements (type CPE8R). The mesh is chosen to be similar to the mesh used by Doghri (1993). No mesh convergence studies have been performed.
Material
The material properties reported by Doghri (1993) for a low-carbon (AISI 1010), rolled steel are used in this example.
A Young’s modulus of E= 210 GPa and a Poisson’s ratio of = 0.3 define the elastic response of the material. The initial yield stress is = 200 MPa.
The nonlinear evolution of the center of the yield surface is defined by the equation
where is the backstress, is the size of the yield surface (size of the elastic range), is the equivalent plastic strain, and C = 25.5 GPa and = 81 are the material parameters that define the initial hardening modulus and the rate at which the hardening modulus decreases with increasing plastic strain, respectively. The quantity 257 MPa defines the limiting value of the equivalent backstress ; further hardening is possible only through the change in the size of the yield surface (isotropic hardening).
NOTCHED BEAM UNDER CYCLIC LOADING
where is the size of the yield surface (size of the elastic range), = 2000 MPa is the maximum increase in the elastic range, and b = 0.26 defines the rate at which the maximum size is reached as plastic straining develops.
The material used for this simulation is cold rolled. This work hardened state is represented by specifying an initial equivalent plastic strain = 0.43 (so that = 411 MPa) and an initial backstress tensor
using the *INITIAL CONDITIONS, TYPE=HARDENING option. Loading and boundary conditions
The beam is subjected to a 4-point bending load. Since only half of the beam is modeled, the model contains one concentrated load at a distance of 26 mm from the symmetry plane (see Figure 1.1.7–1). The pivot point is 42 mm from the symmetry plane. The simulation runs 3 1/2 cycles over 7 time units. In each cycle the load is ramped from zero to 675 N and back to zero. An amplitude curve is used to describe the loading and unloading. The increment size is restricted to a maximum of 0.125 to force Abaqus to follow the prescribed loading/unloading pattern closely.
Results and discussion
Figure 1.1.7–3 shows the final deformed shape of the beam after the 3 1/2 cycles of load; the final load on the beam is 675 N.
The deformation is most severe near the root of the notch. The results reported in Figure 1.1.7–4 and Figure 1.1.7–5 are measured in this area (element 166, integration point 3). Figure 1.1.7–4 shows the time evolution of stress versus strain. Several important effects are predicted using this material model. First, the onset of yield occurs at a lower absolute stress level during the first unloading than during the first loading, which is the Bauschinger effect. Second, the stress-strain cycles tend to shift and stabilize so that the mean stress decreases from cycle to cycle, tending toward zero. This behavior is referred to as the relaxation of the mean stress and is most pronounced in uniaxial cyclic tests in which the strain is prescribed between unsymmetric strain values. Third, the yield surface shifts along the strain axis with cycling, whereas the shape of the stress-strain curve tends to remain similar from one cycle to the next. This behavior is known as ratchetting and is most pronounced in uniaxial cyclic tests in which the stress is prescribed between unsymmetric stress values. Finally, the hardening behavior during the first half-cycle is very flat relative to the hardening curves of the other cycles, which is typical of work hardened metals whose initial hardened state is a result of a large monotonic plastic deformation caused by a forming process such as rolling. The low hardening modulus is the result of the initial conditions on backstress, which places the center of the yield surface at a distance of 228 MPa away from the origin of stress space. Since this distance is close to the maximum possible distance (257 MPa), most of the hardening during the first cycle is isotropic.
1.1.7–2
Abaqus Version 6.11 ID: Printed on:
NOTCHED BEAM UNDER CYCLIC LOADING
These phenomena are modeled in this example primarily by the nonlinear evolution of the backstress, since the rate of isotropic hardening is very small. This behavior can be verified by conducting an analysis in which the elastic domain remains fixed throughout the analysis.
Figure 1.1.7–5 shows the evolution of the direct components of the deviatoric part of the backstress tensor. The backstress components evolve most during the first cycle as the Bauschinger effect overcomes the initial hardening configuration. Only the deviatoric components of the backstress are shown so that the results obtained using Abaqus can be compared to those reported by Doghri (1993). Since Abaqus uses an extension of the Ziegler evolution law, a backstress tensor with nonzero pressure is produced, whereas the backstress tensor produced with the law used by Doghri (which is an extension of the linear Prager law) is deviatoric. Since the plasticity model considers only the deviatoric part of the backstress, this difference in law does not affect the other solution variables.
The results shown in Figure 1.1.7–4 and Figure 1.1.7–5 agree well with the results reported by Doghri (1993).
Input files
cyclicnotchedbeam.inp Input data.
cyclicnotchedbeam_mesh.inp Element and node data.
References
•
Benallal, A., R. Billardon, and I. Doghri, “An Integration Algorithm and the Corresponding Consistent Tangent Operator for Fully Coupled Elastoplastic and Damage Equations,” Communications in Applied Numerical Methods, vol. 4, pp. 731–740, 1988.•
Doghri, I., “Fully Implicit Integration and Consistent Tangent Modulus in Elasto-Plasticity,” International Journal for Numerical Methods in Engineering, vol. 36, pp. 3915–3932, 1993.NOTCHED BEAM UNDER CYCLIC LOADING 1 2 3 1 2 3 1 2 3 1 2 3 8 ;; ;; ;; ;; 0.33 2.48 9.85 26 6 10 52 R60 P 10.2 y x z
Figure 1.1.7–1 Undeformed mesh (dimensions in mm).
1 2 3 1 2 3 y x z ; ; ; ; 1.64 2.48 1.64 30o R0.4
Figure 1.1.7–2 Magnified view of the root of the notch.
1.1.7–4
Abaqus Version 6.11 ID: Printed on:
NOTCHED BEAM UNDER CYCLIC LOADING
1 2 3
Figure 1.1.7–3 Deformed mesh at the conclusion of the simulation. Displacement magnification factor is 3.
ABAQUS Doghri
NOTCHED BEAM UNDER CYCLIC LOADING ABAQUS alpha11 alpha22 alpha33 Doghri alpha11 alpha22 alpha33
Figure 1.1.7–5 Evolution of the diagonal components of the deviatoric part of the backstress tensor.
1.1.7–6
Abaqus Version 6.11 ID: Printed on:
UNIAXIAL RATCHETTING UNDER TENSION AND COMPRESSION