ВИЗНАЧЕННЯ НАПРУЖЕНЬ У ТОВСТИХ ПЛИТАХ ПРИ ЛОКАЛІЗОВАНИХ НАВАНТАЖЕННЯХ

Abstract

The problem of invesigating the stresses in the plates, which are subjected to the action of concentrated forces and locally distributed load is considered. The stresses were determined on the basis of the relations of the three-dimensional theory of elasticity using two methods. The first uses the symbolic Lurie method with the additional application of Vashchenko-Zakharchenko expansion formulas. The solution is constructed in the form of series, which exponentially converge at points distant from the applied forces. However, these series converge slowly in the vicinity of the applied load. In this regard, the solution of the problem is constructed by another method, using the Hankel integral transformation. The approach to find special integrals appearing in problems of the theory of elasticity for plates is proposed. After transformations, the relations for determining the stresses that include the integrals of the functions exponentially attenuating at infinity are obtained. The relations for determining the stresses in the plates on the basis of approximate equations using the Kirchhoff-Lev hypotheses are also given. For this case, the solutions for stresses subjected to the action of concentrated forces and under locally distributed load are given in analytical form. There are cases when stresses found by the elementary formulas, obtained on the basis of Kirchhoff-Lev equations with sufficient accuracy for practice, can be applied in plate bending problems. In particular, it is found that the determined stresses at the boundary opposite to the applied local load turn out to be tensile and at the same time practically accurate if the force is greater than the thickness of the plate. In this case the stresses found under the applied load are determined with larger errors. However, these stresses are compressive and are not essential in the calculation of strength of concrete materials.