Отримано загальний розв’язок задач згину круглих пластин, товщина яких змінюється за експоненціальним законом із застосуванням вироджених гіпергеометричних функцій Куммера. Розв’язано задачу контакту циліндричної оболонки з круговою пластиною змінної товщини в загальному вигляді. Запропоновано методику мінімізації маси пластинчастих елементів конструкцій кругової форми. Розроблена конструкція зони переходу від днища до стінки, міцність якої перевірена методом скінчених елементів у реальному проектуванні
In the bodies of cylindrical apparatuses that operate under pressure, one of the weak elements is a flat bottom whose thickness is increased by 4…5 times in comparison with the wall thickness. This is due to the fact that the bottom is exposed to a more unfavorable bending deformation compared to the wall that "works" on stretching. In order to reduce specific metal consumption for the bottom, we propose the optimization of the shape of a radial cross-section by a rational redistribution of the material: to increase thickness of the bottom in the region of its contact with the wall and to significantly reduce it in the central zone. To describe a variable thickness of the bottom, we applied the Gauss equation with an arbitrary parameter that determines the intensity of change in the thickness in radial direction.
We have obtained a general solution to the differential equation of the problem on bending a bottom at a given law of change in its thickness, which is represented using the hypergeometric Kummer's functions. A technique for concretizing the resulting solution was proposed and implemented, based on the application of conditions of contact between a cylindrical shell and a bottom. The solution derived was used to minimize the mass of the bottom. We have designed a zone of transition from the bottom to the wall whose strength was verified by the method of finite elements under actual conditions.
