Experimental and theoretical investigations have both clearly demonstrated that the bearing capacity of foundations reduces substantially during earthquakes. Unfortunately, a theoretical solution for the three-dimensional bearing capacity in seismic conditions is unavailable and very limited information is available to predict the three-dimensional behaviour of foundations during an earthquake.

The new concept of the discrete element method (DEM) which falls within the framework of the limit equilibrium methodology, in a two dimensional state, was presented by Chang for a bearing capacity of foundations [1]. By developing the concept proposed by Chang [1], a three dimensional formulation of the discrete element method is presented by the authors [2]. In this paper, for the first time, an effort is made to obtain pseudo-static bearing capacity of rectangular shallow foundations. Also, several tables and graphs are provided to demonstrate the applicability of this method. The computations in this research are carried out by means of a DEM program developed, named BCAP (bearing capacity analysis program in three dimensions), written in MATLAB.

To determine the three dimensional bearing capacity of a rectangular shallow foundations by the DEM, a rigid but moving slip body resting on its base is assumed to define the failure mechanism under the footing. The soil mass enclosed in three dimensional a space with assumed failure surfaces is considered as several discrete blocks connected with Winkler springs. Each group of Winkler springs consists of three sets of springs at different orthogonal directions. One set of springs is located in the direction, normal to the contact surface, to simulate the normal stiffness and the two other sets are placed within the contact surface, perpendicular to each other, to simulate the shear resistance on all interfaces. Therefore, compared to the two-dimensional model, there is one set of shear spring added to the contact surface between the two adjacent blocks.

The behaviour of the normal and tangent springs is assumed to be elasto-plastic. The normal springs do not yield in compression but in tension. Based on the Mohr-Coulomb failure criteria, the shear springs yield when the shear strength is reached. The initial values of stiffness in the normal and shear directions between blocks, can be estimated by taking values analogous to the values of Young's modulus (E) and the shear modulus (G), respectively.

In this model, for discretizing the assumed failure surface, pentahedron wedges are used. The geometry of the failure surface is a function of the footing width (B) and length (L), internal friction angle of the underneath soil () and the six independent angles. These six angles are not predefined and are found by iteration to obtain the minimum ultimate bearing capacity. As a result of the existence of six independent angles, a very extensive number of failure surfaces are examined to determine the minimum bearing capacity. Therefore, with a more complex failure surface geometry, the accuracy of the solution is improved in comparison with simpler failure surfaces geometry commonly considered in classical limit equilibrium or limit analysis methods.

Pseudo-static dynamic analysis provides an easy way to estimate the bearing resistance of foundations for any imposed earthquake acceleration. In such a methodology the earthquake body forces are incorporated in to gravity forces. The seismic loading can be applied to the soil mass, soil surcharge and foundation loading, simultaneously.

As no theoretical solutions for 3D bearing capacity under seismic conditions are available in the literature, the comparisons are made with the other methods only for the two-dimensional state. The values obtained by the DEM are comparatively smaller than other methods. Moreover, the relationship between the seismic 2D bearing capacity coefficients with the earthquake horizontal acceleration can almost be expressed by a linear function.

The values of three-dimensional bearing capacity coefficients in static conditions for various internal friction angles and foundation aspect ratios, obtained by the DEM are presented. Also, the values of seismic 3D bearing capacity coefficients related to soil weight, surcharge and cohesion are obtained.

The present method is theoretically more rigorous than the classical limit equilibrium analyses; it offers greater ability to solve bearing capacity problems for complex geometry and loading conditions.

1

Chang, C.S., "Discrete element method for bearing capacity analysis". Comput. Geotech., 1991, 12, 273-288. doi:10.1016/0266-352X(91)90026-C

2

Mirghasemi A.A., Majidi A.R., "Three dimensional bearing capacity analysis of shallow foundations by discrete element method", Int. Conf. on Geotech. Engng, Beyrouth, Lebanon, 2004, 481-486.

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