Numerical study on the effect of roughness of shallow foundations on bearing capacity factor Nγ using lower bound finite element analysis

Due to the complexities associated with the determination of bearing capacity coefficient Nγ, researchers have proposed different values for this coefficient. As the roughness of foundation influences the value of this parameter, it is essential to consider its effect on Nγ. In the present research, the values of bearing capacity factor Nγ, has been determined by lower bound finite element method. The problem has been solved for a two-dimensional domain of the soil beneath a strip foundation. The domain was discretized into a three-nodded linear triangular element. In this method, in contrast to the conventional finite element analyses, the primary variables are three components of two-dimensional stress matrix. Differential equations of stress equilibrium in horizontal and vertical directions are the governing equations of problem. The bearing capacity is defined as the maximum load which can be applied to the foundation while not violating the stress constraints. The constraints include stress discontinuity and yield condition constraints. The stress components at a node common in adjacent elements are not essentially equal. The stress at adjacent element can be discontinuous. The condition of stress discontinuity means that the stress components should be such that the tangential and normal forces at common border lines of adjacent elements are in equilibrium. Moreover, the stress components should be such that they do not violate the Mohr-Coulomb failure criterion. The failure criterion was linearized into definite linear segments. The aforementioned equality and non-equality conditions construct a set of linear equations. The maximum load can be applied to the foundation was determined by linear programming as an optimization technique. The Nγ factor was calculated at two conditions of rough and smooth conditions. For the smooth foundation, an additional constraint was applied. The horizontal at the soil-foundation interface was imposed zero. The values of Nγ were compared at different values of soil friction angles. The values of the factor obtained from the current research were compared with the results of other relevant researchers including the results obtained from limit equilibrium, slip line method, upper bound solutions and lower bound solutions. The results indicate that the roughness of foundation affects the value of Nγ factor and consequently affect the bearing capacity. Therefore, in order to calculate the precise and reliable bearing capacity of a foundation, it is essential to take the effect of foundation roughness into account. The results of comparisons with the relevant studies indicate that the results of different methods do not differ substantially for the friction angles less than 30 degrees. On the other hand, the values of factors obtained from different methods differ substantially from each other for the friction angles higher than 30 degrees. This divergence increases by increasing the friction angle. The factors obtained from upper bound solutions are essentially higher than those obtained from the present research which is a lower bound solution. As the exact solution is higher than what obtained from lower bound solution, it could be stated that the reliability index of the factors obtained from the current study is higher than other methods. It could be concluded that the proposed factors can be regarded as a safe estimate for Nγ factor.