Investigation of force factors and stresses at singular points of plate elements in special cranes

Abstract

The work addresses studying the possibilities of a numerical-analytical variant of the boundary elements method (BEM) in determining the internal force factors and stresses at singular points when bending thin isotropic plates. The simplest type of singularity has been investigated, a point of application of external concentrated forces and moments. The importance of a given problem is due to the fact that at these points the internal force factors tend towards infinity and it is not possible to determine the size using elementary methods. At the same time, these singular points are the significant stress concentrators (both tangential and normal), which is why calculating the limits to which the internal forces and moments tend is essential to analyze the strength of plate structures. In order to describe an external load, it is proposed to apply the Dirac delta function of two variables. The models of external loads are presented. A given proposal makes it possible to accurately calculate the limits to which the transverse forces, as well as bending and torsional moments, tend at singular points of thin plates. We simulated plate bending using the variational Kantorovich-Vlasov method, which is fully compatible with the models of external load. The internal force factors at the singular points of plates were determined while solving the boundary value problems, formed based on the algorithm of BEM. The MATLAB environment was used for programming and computation. Results of the calculations are characterized by high accuracy and reliability, in particular the errors in determining the deflections of plates at singular points do not exceed 2.0 % and the errors for bending moments are not above 3.0 %. Recommendations have been given to solving different types of boundary problems on bending the plates with singular points based on the proposed approach. It has been established that an accurate model of the external load in the form of concentrated forces and moments fundamentally enables determining the internal forces and moments at the singular points of thin plates applying an algorithm of the variational Kantorovich-Vlasov method. Up to now, there are no data on the importance of internal forces and moments at the singular points of plates. It is also shown that when calculating the internal forces and moments of plates, it is inappropriate to apply a single term from a series of the Kantorovich-Vlasov method; the errors amount to significant magnitudes of the order of 43‒44 %