Example-3. Title. Description. Cylindrical Hole in an Infinite Mohr-Coulomb Medium

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1 Example-3 Title Cylindrical Hole in an Infinite Mohr-Coulomb Medium Description The problem concerns the determination of stresses and displacements for the case of a cylindrical hole in an infinite elasto-plastic medium subjected to in-situ stresses. The medium is assumed to be linearly elastic, perfectly plastic, with a failure surface defined by the Mohr-Coulomb criterion. This problem tests the Mohr-Coulomb plasticity model, the plane-strain condition and axisymmetric geometry. An isotropic in-situ stress state exists with stresses equal to -30 MPa (tension positive). It is assumed that the problem is symmetric about both the horizontal and vertical axes. The radius of the hole is 1 m and is assumed to be small compared to the length of the cylinder. This permits the use of the plane-strain condition. Cylindrical hole 6 c Pa 30 Fig. 3.1 Cylindrical Hole in an Infinite Mohr-Coulomb Medium

2 Verification Examples Analytical Solution The yield zone radius, R o, is given analytically by a theoretical model based on the solution of Salencon (1969): 1 K 1 p 2 Po q Kp 1 Ro a K p 1 P i q K p 1 where: a = radius of hole; K p 1 sin ; 1 sin q 2ctan 45 2 ; P o = initial in-situ stress magnitude; and P = internal pressure. i The radial stress at the elastic/plastic interface is: 1 re 2Po q K 1 p the stresses in the plastic zone are: q q r r Pi Kp 1 Kp 1 a K p 1 q q r Kp Pi Kp 1 Kp 1 a K p 1 where r = distance to the center of the hole. 2

3 Example-3 The stresses in the elastic zone are: 2 Ro r Po Po re 2 Ro Po Po re r r The displacements in the elastic and plastic regions are given by Salencon (1969). For the elastic region: u r 2Po q Ro Ro Po K p 1 2G r and for the plastic region: r ur 2G q 2 1 Po K p K R R K K K a r 2 Kp1 Kps1 p q o o Pi p ps p 1 KpKps 1 q r 1 Pi Kp K ps Kp 1 a K p 1 where: K ps 1 sin ; 1 sin = dilatation angle; = Poisson s ratio; and G = shear modulus. 3

4 Verification Examples GTS MODEL The problem is first modeled as a two-dimensional plane-strain calculation using quartersymmetry. The boundary conditions applied to the model are shown in Fig The outer boundary is located 10 m (five hole diameters) from the hole center. The model contains 900 rectangular elements oriented in a radial pattern, as indicated in Fig This pattern minimizes the influence of the grid on localization effects. Y P 0 10m P 0 1m X 1m 10m Fig. 3.2 Geometry and mesh for plane-strain model quarter-symmetry The problem is also modeled using axisymmetric geometry. Fig. 3.3 shows the boundary conditions, and the mesh created for this calculation 4

5 Example-3 Y Axis of Symmetry 1m Cylindrical hole 10m P 0 X Fig. 3.3 Geometry and mesh for axisymmetric model At last, the problem is also modeled using 3D solid elements. The same mesh and boundary condition as those of plane-strain model are applied to this model, except that there is a dimension along the thickness direction. All models are subjected to an isotropic compressive stress of -30 MPa. The initial stress state is applied throughout each model first. 5

6 Verification Examples MODEL 1 Analysis Type Unit System Dimension Element Material Boundary Condition Load Case 2D plane-strain material nonlinear analysis Width Height Hole radius m, N 10.0 m 10.0 m 1.0 m 4-node quadrilateral plane-strain element Modulus of elasticity E = MPa Poisson s ratio = Yield criteria Mohr-Coulomb Cohesion c = 3.45 MPa Friction angle Dilatancy angle Left end Bottom end = 30 = 30 Constrain D X Constrain D Y Initial isotropic in-situ compressive pre-stress of -30 MPa. Edge pressure of 30 MPa at right and top ends. 6

7 Example-3 MODEL 2 Analysis Type Unit System 2D axisymmetric material nonlinear analysis m, N Dimension Width Height 10.0 m 10.0 m Element Hole radius 1.0 m 4-node quadrilateral axisymmetric element Modulus of elasticity E = MPa Poisson s ratio = Material Yield criteria Cohesion Friction angle Dilatancy angle Mohr-Coulomb c = 3.45 MPa = 30 = 30 Boundary Condition Top & Bottom end Constrain D Y Load Case Initial isotropic in-situ compressive pre-stress of -30 MPa. Edge pressure of 30 MPa at right end. 7

8 Verification Examples MODEL 3 Analysis Type Unit System 3D solid material nonlinear analysis m, N Dimension Width Height Depth Hole radius 10.0 m 10.0 m 0.2 m 1.0 m Element 8-node hexahedron solid element Modulus of elasticity E = MPa Poisson s ratio = Material Boundary Condition Yield criteria Cohesion Friction angle Dilatancy angle Left end Bottom end Mohr-Coulomb c = 3.45 MPa = 30 = 30 Constrain D X Constrain D Y Load Case Initial isotropic in-situ compressive pre-stress of -30 MPa. Face pressure of 30 MPa at right & top faces. 8

9 Example-3 Results Fig. 3.4, 3.5 and 3.6 show a direct comparison between GTS results and the analytical solution along a radial line for the quarter-symmetry case. Normalized stresses, r Po and Po, are plotted versus normalized radius ra in Fig. 3.4, while normalized displacement ur a is plotted versus normalized radius in Fig Fig. 3.6 shows the yield ratio. For the axisymmetric geometry, Fig. 3.7, 3.8 and 3.9 show a direct comparison between GTS results and the analytical solution. And for the 3D solid geometry, Fig. 3.10, 3.11 and 3.12 show a direct comparison between GTS results and the analytical solution. 9

10 Verification Examples (a) Contour of Stress along X-direction Normalized Stress Analytical(radial) Analytical(tangential) GTS(radial) GTS(tangential) Normalized Radius (b) Graph of tangential and radial stresses Fig. 3.4 Comparison of tangential and radial stresses (model 1) 10

11 Example-3 (a) Contour of displacement magnitude Analytical GTS (model 1) Normalized Displacement Normalized radius (b) Graph of normalized radial displacement Fig. 3.5 Comparison of radial displacement (model 1) 11

12 Verification Examples Fig. 3.6 Plot of equivalent plastic strain (model 1) (a) Contour of stresses along X and Z direction 12

13 Example Normalized Stress Analytical(radial) 0.4 Analytical(tangential) GTS(radial) 0.2 GTS(tangential) Normalized Radius (b) Graph of normalized radial and tangential stresses Fig. 3.7 Comparison of normalized radial and tangential stresses (model 2) (a) Contour of displacement magnitude 13

14 Verification Examples Analytical GTS (model 2) Normalized Displacement Normalized radius (b) Graph of normalized radial displacement Fig. 3.8 Comparison of radial displacement (model 2) Fig. 3.9 Plot of equivalent plastic strain (model 2) 14

15 Example-3 (a) Contour of stress along X-direction Normalized Stress Analytical(radial) 0.4 Analytical(tangential) GTS(radial) 0.2 GTS(tangential) Normalized Radius (b) Graph of normalized radial and tangential stresses Fig Comparison of normalized radial and tangential stresses (model 3) 15

16 Verification Examples (a) Contour of displacement magnitude Analytical GTS (model 3) Normalized Displacement Normalized radius (b) Graph of normalized radial displacement Fig Comparison of normalized radial displacement (model 3) 16

17 Example-3 Fig Plot of equivalent plastic strain (model 3) 17

18 Verification Examples Comparison of Results The yield zone radius values and errors are tabulated below for all three models. As these tables show, the error is less than 4.15%, and this shows that the GTS results are reasonable. Yield zone radius Yield zone Radius Unit: m GTS (mesh dependent) Analytical model 1 model 2 model 3 Value Ratio (%) Value Ratio (%) Value Ratio (%) Reference Salencon, J., Contraction Quasi-Statique D une Cavite a Symetrie Spherique Ou Cylindrique Dans Un Milieu Elastoplastique, Annales Des Ponts Et Chaussees, No. 4, 1969, pp

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