Nonlinear Deformation of Flexible Shallow Shells of Complex Shape Made of Materials with Different Resistance to Tension and Compression

IF 0.7 4区 材料科学 Q4 MATERIALS SCIENCE, CHARACTERIZATION & TESTING
O. Z. Galishyn, S. M. Sklepus
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Abstract

A new numerical-and-analytical method is developed for solving geometrically and physically nonlinear problems of bending shallow shells of complex shapes made from materials with different resistance to tension and compression. To linearize the initial nonlinear problem, the method of continuous continuation in the parameter associated with the external load was used. For the variational formulation of the linearized problem, a Lagrange functional was constructed, defined at kinematically possible displacement velocities. To find the main unknowns of the problem of nonlinear bending of a hollow shell (displacements, deformations, stresses), the Cauchy problem for a system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta– Merson method with automatic step selection. The initial conditions are found in the solution to the problem of geometrically linear deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution in the form of a formula, which solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. The problems of nonlinear deformation of a square cylindrical shell and a shell of complex shape with combined fixation conditions are solved. The influence of the direction of external loading, geometric shape, and fixation conditions on the stress-strain state is investigated. It is shown that failure to consider the different behaviors of the material in tension and compression leads to significant errors in calculating the stress-strain state parameters.

Abstract Image

由不同抗拉和抗压材料制成的复杂形状柔性浅壳的非线性变形
为解决由具有不同抗拉和抗压性能的材料制成的复杂形状浅壳弯曲的几何和物理非线性问题,开发了一种新的数值和分析方法。为了使初始非线性问题线性化,采用了与外部载荷相关的参数连续延续法。为了对线性化问题进行变分计算,构建了一个拉格朗日函数,该函数定义于运动学上可能的位移速度。为了找到空心壳体非线性弯曲问题的主要未知数(位移、变形、应力),提出了常微分方程系统的 Cauchy 问题。Cauchy 问题采用自动选择步长的 Runge-Kutta- Merson 方法求解。初始条件可在几何线性变形问题的解中找到。与 Runge-Kutta- Merson 方案相对应的载荷参数固定值的微分方程右边是从拉格朗日函数的变分问题求解中得到的。变分问题采用里兹法结合 R 函数法求解。后者能以公式的形式给出近似解,其解结构完全满足边界条件的全部(一般结构)或部分(部分结构)。解决了具有组合固定条件的方形圆柱形壳体和复杂形状壳体的非线性变形问题。研究了外部加载方向、几何形状和固定条件对应力应变状态的影响。结果表明,不考虑材料在拉伸和压缩时的不同行为会导致应力应变状态参数的计算出现重大误差。
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来源期刊
Strength of Materials
Strength of Materials MATERIALS SCIENCE, CHARACTERIZATION & TESTING-
CiteScore
1.20
自引率
14.30%
发文量
89
审稿时长
6-12 weeks
期刊介绍: Strength of Materials focuses on the strength of materials and structural components subjected to different types of force and thermal loadings, the limiting strength criteria of structures, and the theory of strength of structures. Consideration is given to actual operating conditions, problems of crack resistance and theories of failure, the theory of oscillations of real mechanical systems, and calculations of the stress-strain state of structural components.
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