Abstract:

Conventional calculations of ultimate bearing capacity of shallow foundations are based on linear Mohr-Coulomb failure criterion. However, experimental data show that the strength envelops of almost all types of rocks are nonlinear over wide range of normal stresses. In this paper, the strength envelope of rock mass is considered to follow Hoek-Brown failure criterion. Hoek-Brown failure criterion introduced into limit analysis theorem. The plastic dissipation power in terms of kinematically admissible velocity fields and a nonlinear optimization formulation is obtained. Then by implementing the acquired formulation into a using nonlinear finite element technique code, the failure mode and the ultimate bearing capacity of a strip footing on a rock mass is calculated. The numerical results are compared with existing limit analysis solutions.

1. INTRODUCTION

Assessment of ultimate bearing capacity of shallow foundations is one of the most common problems in civil engineering. The methods used for assessing the ultimate bearing capacity of shallow foundations mainly fall within one of the four categories:

• the limit equilibrium method,

• the slip-line method,

• the limit analysis method, and

• the numerical methods.

To obtain the maximum load of rock mass under static loading, a step-by-step method based on traditional elastic-plastic analysis is commonly used. However, this incremental approach is often too cumbersome to use in practice because it requires a complete specification of the stress-strain relation and the nonlinear material properties of a rock mass. Therefore, other direct methods such as the slip-line method, limit equilibrium method and limit analysis applied to determine the plastic limit state of a continuous media. 

2. LIMIT ANALYSIS BASED ON A GENERAL YIELD CRITERION

For simplification the stability calculations, with using the plastic theorems of soils, we can ignore some of the equilibrium or compatibility equations. Then, instead of unique answer, we have a bound for ultimate loads. If we ignore the equilibrium equations (admissible stress fields), we have an upper bound for ultimate failure load. So that if this loads affect the structure, it will certainly collapse. It is quite difficult to directly introduce criteria into the plastic limit analysis. A numerical technique based on the nonlinear mathematical programming will be developed to perform kinematic limit analysis for these yield criteria. 

2.1. Mohr-coulomb criterion:

The expression (1) can be regarded as a general yield criterion for frictional materials.

3. KINEMATIC THEOREM OF LIMIT ANALYSIS

An upper bound to the plastic limit load of a structure can be obtained by using the kinematic theorem of limit analysis. The kinematic theorem states: among all kinematically admissible velocities, the real one yields the lowest rate of plastic dissipation power.

4. APPLICATION AND NUMERICAL RESULTS

Optimization of equation 5 is too difficult by means of variation calculus method therefore a numerical method is needed. The finite element method is applied to perform the numerical calculation for the kinematic limit analysis.

4.1. DEFINITION OF PROBLEM

This problem involves finding bearing capacity of a strip footing on a homogenous rock mass.

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