Rational Design of Nonlinear-Deformable Structurallyheterogeneous Elements of Structures

Abstract

The article deals with the problem of rational design of a nonlinearly deformable heterogeneous composite Timoshenko rod under force impact. The rod has a structure that is symmetrical relative to the force plane, formed by the connection of quasi-homogeneous parts (phases, layers) with different physical properties, taking any geometric shape in space. The structural materials that form the rod have nonlinear elastic properties. To describe the main component of the stress tensor – the normal stress in the longitudinal direction – in each phase, the same type of approximation by entire rational polynomials is taken depending on the deformation. On their basis, compact nonlinear equations were obtained that connect integral forces with generalized deformations of the axial line of the rod. In this system, the rigidity characteristics of higher exponent are figured as coefficients. Nonlinear equilibrium conditions written for the case of large displacements and rotation angles in combination with linear kinematic relations are resolved in the form of the initial parameter method. Based on the strength condition written in the form of a quasi-uniaxial criterion, the designing criterion is formulated for the heterogeneous rod. This criterion is continuous along the longitudinal coordinates and it is discrete along the transverse coordinates. A two-stage algorithm is developed to solve the design problem of the rational design of a nonlinearly deformable layered rod. It makes it possible to identify the geometric functions of the longitudinal profiling of the rod layers presented in a discrete form. Resolution relations were obtained to find the functions of the width and height of the profiled layers. Numerical results are presented to solve the design problem of calculating a compressed-bent I-section rod, in which the flanges and I-beam webs were made of various materials. The presence of geometric restrictions on variable values from below ensured non-degeneracy of the flanges at the pre-support areas. The three characteristic areas were detected in the rod with the implementation of the calculated continuous criterion in the form of two-, one- and zero-point conditions along the transverse coordinate. It is shown that the consideration of the shear stresses in the rod of this flexibility is not relevant.