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Research Papers

# Kinematic Limit Analysis of Nonassociated Perfectly Plastic Material by the Bipotential Approach and Finite Element Method

[+] Author and Article Information
Ali Chaaba

Ecole Nationale Supérieure d’Arts et Métiers, Marjane II, Beni M’hamed, B. P. 4024 Meknès, Morocco

Lahbib Bousshine

Ecole Nationale Supérieure d’Electricité et de Mécanique, B. P. 8118 Oasis Casablanca, Morocco

Gery De Saxce

LML, U.S.T.L., F-59655 Villeneuve d’Ascq Cedex, France

J. Appl. Mech 77(3), 031016 (Feb 24, 2010) (11 pages) doi:10.1115/1.4000383 History: Received July 19, 2006; Revised September 15, 2009; Published February 24, 2010; Online February 24, 2010

## Abstract

Limit analysis is one of the most fundamental methods of plasticity. For the nonstandard model, the concept of the bipotential, representing the dissipated plastic power, allowed us to extend limit analysis theorems to the nonassociated flow rules. In this work, the kinematic approach is used to find the limit load and its corresponding collapse mechanism. Because the bipotential contains in its expression the stress field of the limit state, the kinematic approach is coupled with the static one. For this reason, a solution of kinematic problem is obtained in two steps. In the first one, the stress field is assumed to be constant and a velocity field is computed by the use of the kinematic theorem. Then, the second step consists to compute the stress field by means of constitutive relations keeping the velocity field constant and equal to that of the previous step. A regularization method is used to overcome problems related to the nondifferentiability of the dissipation function. A successive approximation algorithm is used to treat the coupling question. A simple compression-traction of a nonassociated rigid perfectly plastic material and an application of punching by finite element method are presented in the end of the paper.

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## Figures

Figure 4

Figure 5

Stresses: compression case

Figure 1

Nonassociated flow rule with Drucker–Prager cone

Figure 2

Elastoplastic evolution (associated and nonassociated cases)

Figure 3

Geometry and boundary conditions

Figure 11

Mesh used: 99 linear triangular finite elements

Figure 12

Bearing capacity factor Nc: mesh of 99 linear T3

Figure 13

Bearing capacity factor Nc: mesh of 99 nonlinear T6

Figure 14

Error norm of stress as function of number of iterations φ=30 deg and θ=10 deg

Figure 15

Bearing capacity factor Nc: Mesh with 99 linear T3

Figure 16

Bearing capacity factor Nc: Mesh with 99 nonlinear T6

Figure 17

Velocity field for φ=30 deg and θ=30 deg (99 T3)

Figure 18

Velocity field for φ=30 deg and θ=10 deg (99 T3)

Figure 6

Figure 7

Stresses: traction case

Figure 8

Limit loads (comparison of result): compression

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Limit loads (comparison of result): traction

Figure 10

Mathematical model: geometry and boundary conditions

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