Variational Solution Of Critical Normal Stress Distribution Of Footing On Slope
Keywords:
Variational Solution, mathematical technique, Shear stress, Failure, Coordinate transformationAbstract
A mathematical technique was advanced for investigating the normal stress distribution failure of soil foundations. The stability equations were obtained using the limit equilibrium (LE) conditions. The additions of vertical, horizontal and rotational equilibria were transformed mathematically with respect to the soil shearing strength, leading to the derivation of the equation of the functional Q, and two integral constraints. In the mathematical method employed, the stability analysis was transcribed as a minimization problem using the calculus of variations. Generally, no constitutive law beyond the Coulomb's yield criterion was 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 consequently 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 using the calculus of variations. Key Word: Cohesion, Internal Friction, Vertical load, Stress distribution, Rupture surface, Shear stress, Failure, Coordinate transformation, Polar coordinates.
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