BEARING CAPACITY AND CRITICAL NORMAL STRESS DISTRIBUTION OF SOILS BY METHOD OF VARIATIONAL CALCULUS

BEARING CAPACITY AND CRITICAL NORMAL STRESS DISTRIBUTION OF SOILS BY METHOD OF VARIATIONAL CALCULUS

ABSTRACT

A mathematical technique  is hereby advanced for investigating the bearing capacity and associated normal stress distribution at failure of soil foundations. The stability equations are obtained using the limit equilibrium (LE) conditions. The additions of vertical, horizontal and rotational equilibria are transformed mathematically with respect to the soil shearing strength, leading to the derivation of the equation of the functional Q, and two integral constraints. Generally, no constitutive law beyond the conlomb’s yield criterion is incorporated  in the formulation. Consequently, no constraints are placed on the character of the criticals except the overall equilibrium of the failing soil section. The critical normal stress distribution, dmin, and consmequently the load, Qmin, determined as a result of the minimization of the functional are the smallest stress and load parameters that can cause failure. In other words, for a soil with strength parameters c, ø, ૪, and footing with geometry B, H, when stress d < dmin (c, ø, ૪, B, H) and load Q <Qmin (C, ø, ૪, B, H) foundation is stable. Otherwise the stability would depend on the constitutive character of the foundation soil. In the mathematical method employed, the stability analysis is transcribed as a minimization problem in the calculus of variations. The result of the analysis shows, among others, that the Meycrhoff and Hansen’s Superposition approaches can be derived using the technique of variational calculus, and consequently the representation of the bearing capacity by the three factors Nq, Nc, and N૪ is appropriately possible. Finally, the classical relation between Nc and Nq is again found by the LE approach and is therefore independent of the constitutive law of the soil medium.