In this work we present a bipotential approach for estimating the bearing capacity of shallow strip footing in the context a non-associated Mohr–Coulomb soil. To this end, the considered problem is formulated in a rigorous mathematical framework coupling the bipotential concept and the limit analysis theory. Analytical quasi-bounds are given based upon a statically and plastically trial stress field and a Prandtl-like collapse mechanism.


Limit analysis [1,2,3] is a powerful method for the direct determination of the collapse load of structures subjected to proportional loadings and operating beyond the elastic limit. The constitutive laws are supposed rigid-perfectly plastic, modelled by a plastic domain and an associated plastic flow rule. Typically, the task is to predict the ultimate load factors using the lower and upper theorems related to static and kinematic approaches respectively.

For a long time, limit analysis has been used to evaluate the bearing capacity of foundations and the slop stability for geomaterials modelled by associated plastic laws. In particular, a problem of practical interest for engineers is the bearing capacity of a semi-infinite soil foundation with the standard law of Mohr–Coulomb, a cohesion c and a friction angle φ. Prandtl–Hill analytical solutions provide the exact limit load [4].

Another development in computing the bearing capacity of foundations is achieved by using the finite element method and the finite difference method [5,6,7]. Note also that numerical limit analysis bounds involving linear/non-linear programming are proposed in the literature [8,9]. A notable advantage of these numerical methods is the study of three-dimensional problems with complex geometries and loadings.

It is noteworthy that the classical limit analysis theorems are restricted to standard materials with associated flow rule (the plastic strain rate is normal to the yielding surface). However, many experimental observations showed that for geomaterials and polymers, the dilatancy angle is lower than the friction one and thus the plasticity is not associated. Decidedly this affects the failure mechanism and the plastic limit loads. The assessment of the closed-form expression of the bearing capacity of a strip footing remains open. Many numerical results are proposed in the literature. Moreover, a simple but widely used formula for computing the limit load has been proposed by Drescher and Detournay [10].

In this paper the bipotential theory and the slip-line method have been used to derive estimates of the ultimate loads of rigid strip resting on a non-standard Mohr–Coulomb half-plane. The punch is subjected to a normal pressure and the weight of substrate is neglected.

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