Estimation of the reliability of a rectangular frame made of the porous metal in terms of its rigidity on the basis of probabilistic approach

This article is aimed to solve one of the problems of mechanics of deformable solids such as the analysis of the influence of material porosity on the structural elements under random loads. The rigidity of the legs and girder of rectangular frame made of the porous material is studied. The porosity distribution law is specified to be rational. In this case, the porosity distribution is opposite to the normal stress distribution law due to the bending-buckling. The elements of rectangular frame are bent during buckling. In the bar cross-section, the stresses are distributed unevenly. Therefore, it is reasonable to distribute porosity in the same manner. The porosity is minimum in the region of maximum stress, and, near the neutral axis of the bend, the maximum porosity is observed. The load is shared symmetrically between the frame legs and is oriented to the axes of the latter. The effect of random load, whose law of variation corresponds to a random stationary process, is considered. It should be noted that the presence of porosity in the material breaks the main postulate of the mechanic of deformable solids that is the hypothesis of continuity. To solve this problem, it is considered that the porous material is continuous, and its mechanical characteristics are consistent with the porosity. In particular, Young 's modulus for the porous material is approximated by power polynomial whose factors are defined by experimental data processing using the least squares method.
A correlation function is used for the random stationary process that is represented as a formula for determination of the frame reliability by rigidity of the construction during the specified operating period. The energy approach is applied to solve the deterministic problem of the theory of stability and to determine the critical force. An asymmetric buckling is used as it is the most probable form. The critical load is defined by condition of the equality of deformation potential energy to the external force energy. Estimation of the frame reliability at the assigned loads and porosity distribution law is demonstrated numerically. For comparison, the frame made of a continuous material is considered. Analysis shows that insignificant decrease in the reliability of the frame porous structure as compared to the continuous structure (about 2%) provides saving in the material of about 26.7%.