Application of Kantorovich-Vlasov Method for Shaped Plate Bending Problem

Abstract

The flexure function \( W(x,y) \) of a ribbed rectangular plate is determined by the Kantorovich-Vlasov method on the basis of the sum of series \( \sum\nolimits_{i = 1}^{n} {W_{i} (y)X_{i} (x)} \) with functions \( X_{i} (x) \) satisfying specified kinematic boundary conditions, while functions \( W_{i} (y) \) are determined as the solutions of differential equations of the fourth order. The problem is complicated by the absence of the target selection of functions \( X_{i} (x) \) required to attain the desired accuracy. A special beam function \( X_{1} (x) \) defined by the initial parameters method was proposed to be used in the first term of series. This function is defined at a single beam with the boundary conditions equal to those of a ribbed plate with respect to the selected coordinate. This allows one to reasonably increase in the accuracy of the specified variation method when one series term is used. The practical calculations of a real object—shaped sheet as per Russian GOST—conducted using the proposed function are provided. Flexure tables were obtained for the shaped sheets provided in the specified standard. The analysis of resulting flexures shows that the use of the sheet profile with more reinforcement ribs is more cost effective as the sheet flexure decreases faster than the sheet weight increases. For example, if one rib is added to S-15-1000-06 profile sheet, sheet flexure in the initial range of 8–12 ribs decreases by no less than 8% with the sheet weight increase by 5%.