A. A. Semenov

Mathematical models and algorithms for studying strength and stability of shell structures

In this paper, we describe severalmathematical models of deformation of reinforced shell structures, including those that account for various properties of the material. For the structures composed of orthotropic and isotropicmaterials, we consider linearly elastic and physically nonlinear problems, as well as the problems of creep. All models are constructed on the basis of the functional of total potential energy of the shell deformation. Geometric nonlinearity and transverse shears are taken into account. Strengthening ribs are introduced both in the discrete way and by the method of structural anisotropy. Three different algorithms of studying the strength and stability of shells are presented each of which is most effective for a specific range of problems.

Analysis of the Strength of Shell Structures, Made from Modern Materials, According to Various Strength Criteria

The paper analyzes the possibility of applying five different strength criteria (maximum stress criterion, Mises-Hill, Pisarenko-Lebedev, Fisher, Goldenblat-Kopnov) to calculating the strength of orthotropic shell structures. We consider shallow shells of double curvature, square in plan, panels of cylindrical and conical shells. A geometrically nonlinear mathematical model of their deformation, taking into account transverse shearing, is used. For calculations, the characteristics of modern orthotropic materials are used, such as fiberglass and CFRP. An increase in the areas of the failure of strength conditions with increasing load is shown.

Stability of cylindrical shell panels of modern materials under dynamic loading

Orthotropic cylindrical shell panels under dynamic loading (the load has a linear time dependency) are examined in this paper. Relationships of a mathematical model of their deformation are presented in view of the geometric nonlinearity, transverse shears and orthotropy of the material. The Kantorovich method is applied to form a system of ordinary differential equations. The derived system is solved by the Rosenbrock method. The stability of several types of orthotropic panels of modern materials (fiberglass, carbon fiber reinforced plastic, etc.) is studied and critical load values are obtained.

Strength of orthotropic cylindrical panels with account for geometric nonlinearity

This paper describes the influence of geometric nonlinearity in the analysis of the strength of orthotropic cylindrical panels. The values of maximum permissible loads in the linear and nonlinear versions of calculations of structures made of unidirectional carbon plastics are given, and the loads at which stability loss occurs are determined. The mathematical model accounts for transverse shifts and geometric nonlinearity and stated as the full potential strain energy functional. The calculations are carried out on the basis of the method of solution continuation with respect to the parameter. The strength is estimated by using the maximum stress criterion.

Rationale of the use of the constructive anisotropy method in the calculation of shallow shells of double curvature, weakened holes

Получена: 4 марта 2016 г. Принята: 27 мая 2016 г. Опубликована: 30 июня 2016 г. Приводится геометрически нелинейная математическая модель деформирования изотропных пологих оболочек двоякой кривизны, ослабленных вырезами. Модель основывается на гипотезах теории оболочек Кирхгофа–Лява и представлена в виде геометрических соотношений, физических соотношений и функционала полной потенциальной энергии. Также приводятся выражения для усилий и моментов. Рассмотрено два способа введения вырезов: дискретно и методом конструктивной анизотропии, который позволяет наиболее точно «размазать» нулевую жесткость вырезов по полю оболочки. Для минимизации функционала применяется метод Ритца, что сводит задачу к решению системы нелинейных алгебраических уравнений, которая решается методом Ньютона. Алгоритм реализован в среде аналитических вычислений Maple 2015. Проводится анализ устойчивости пологих оболочечных конструкций двоякой кривизны, выполненных из стали, при действии на них внешней равномерно распределенной поперечной нагрузки и шарнирно-неподвижном способе закрепления контура оболочки. Расчеты производились при наличии разного числа вырезов, при этом фиксировался коэффициент отношения общей площади вырезов к площади оболочки. Таким образом, при увеличении числа вырезов уменьшался их размер. Распределение вырезов по оболочке делалось двумя разными способами. Для всех исследованных конструкций приводятся значения критических нагрузок потери устойчивости. Проводится сравнение значений, полученных при дискретном введении вырезов и методом конструктивной анизотропии. Для нескольких вариантов конструкций показаны графики «нагрузка – прогиб». Для одного варианта оболочки, ослабленной большим числом вырезов, приводятся поля прогибов до и после потери устойчивости, цветом также показана интенсивность напряжений. На основании полученных данных показано, что при увеличении числа вырезов дискретность их ввода теряется и становится возможным использование специально разработанного метода конструктивной анизотропии. Таким образом, сделано обоснование использования данного метода при расчете устойчивости пологих оболочек, ослабленных большим числом вырезов.

Comprehensive study of the strength and stability of shallow shells made of fiberglass

In this paper we report the results obtained in comprehensive studies of the strength and stability of shallow shells made of fiberglass. A regular loss of strength and stability of shells are quantitatively estimated. It is shown that the loss of strength and stability depends on generalized parameters and the radii of curvature. The mathematical model of shell deformation was used based on the Timoshenko hypotheses accounting for the orthotropy of material, on geometric nonlinearity and on lateral shear. The calculation algorithm is developed on the basis of the Ritz method and the best parameter continuation method. As many as 16 options of shells expressed through dimensionless parameters are considered, each corresponding to a series of similar shells.

Models of Deformation of Stiffened Orthotropic Shells under Dynamic Loading

Received 23.03.2016, received in revised form 02.08.2016, accepted 08.09.2016 Two models of deformation of reinforced orthotropic shells under dynamic loading are considered in this paper. One such model is in the form of equations of motion and another model is in the form of a system of ordinary differential equations. Mathematical models are based on the hypotheses of the Kirchhoff – Love theory of shells. They take into account the geometric nonlinearity, orthotropic material properties and reinforcement elements. All relations of the models are in general form, and they can be used for a wide range of structures (shallow shells of double curvature, cylindrical, conical, spherical and toroidal shells and panels, etc.). An important feature of the proposed model is the ability to introduce stiffeners both discretely and by the method of constructive anisotropy (MCA) in accordance with their shear and torsional rigidity. The second model is derived by applying the Kantorovich method to the functional of the total energy of deformation of a shell. The resulting initial value problem is easier to solve than the system of equations of motion in partial derivatives.