Рассмотрен изгиб шарнирно закрепленной балки на упругом основании. Начальный прогиб и неоднородность жесткости основания заданы с точность до малых параметров. Получено условие, определяющее границу области сходимости метода малого параметра. Найдена функция, характеризующая прогиб, с точностью до величин четвертого порядка малости. Проанализирован случай, когда малые параметры являются случайными величинами. The bending of a pivotally fixed beam on an elastic base is considered. The initial deflection and inhomogeneity of the base stiffness are set up to small parameters. A condition is obtained that defines the boundary of the convergence region of the small parameter method. A function describing the deflection is found up to the fourth order of smallness. The case when small parameters are random variables is analyzed.

The paper considers the plane strain of a compressed elastic strip made from incompressible material. The deviation of the cross-section contour from the rectangle, as well as the intensity of the force applied to the upper and the lower edge of the section are known up to small parameters. The authors have carried out a study of the problem solution analyticity with respect to small near zero parameters. With the help of the perturbation method, they have found a solution for a particular case of the function describing the deviations up to the second order of smallness. It has been shown that if an experimental research is carried out and the applied force meets the condition described, the average stress and displacement values will differ from those typical for homogeneous state only by a second order infinitesimal.

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