Abstract
The solution to the problem of the stability of a rectangular orthotropic plate is described by the numerical-analytical method of boundary elements. As is known, the basis of this method is the analytical construction of the fundamental system of solutions and Green’s functions for the differential equation (or their system) for the problem under consideration. To account for certain boundary conditions, or contact conditions between the individual elements of the system, a small system of linear algebraic equations is compiled, which is then solved numerically. It is shown that four combinations of the roots of the characteristic equation corresponding to the differential equation of the problem are possible, which leads to the need to determine sixty-four analytical expressions of fundamental functions. The matrix of fundamental functions, which is the basis of the transcendental stability equation, is very sparse, which significantly improves the stability of numerical operations and ensures high accuracy of the results. An analysis of the numerical results obtained by the author’s method shows very good convergence with the results of finite element analysis. For both variants of the boundary conditions, the discrepancy for the corresponding critical loads is almost the same, and increases slightly with increasing critical load. Moreover, this discrepancy does not exceed one percent. It is noted that under both variants of the boundary conditions, the critical loads calculated by the boundary element method are less than in the finite element calculations. The obtained transcendental stability equation allows to determine critical forces both by the static method and by the dynamic one. From this equation it is possible to obtain a spectrum of critical forces for a fixed number of half-waves in the direction of one of the coordinate axes. The proposed approach allows us to obtain a solution to the stability problem of an orthotropic plate under any homogeneous and inhomogeneous boundary conditions.
Highlights
IntroductionThe development level of production at the present stage is characterized by the widespread introduction of new technologies for the manufacture of high-strength materials with orthotropic (orthogonally anisotropic) properties
The development level of production at the present stage is characterized by the widespread introduction of new technologies for the manufacture of high-strength materials with orthotropic properties.Such materials include fiberglass; composite materials reinforced with sequentially alternating layers of fibers in two mutually perpendicular directions; glued wood plates; sheet rolled metals, in which anisotropy begins to appear upon transition to the plastic stage of work, etc.The widespread use of materials with anisotropic properties has given rise to large-scale studies in the field of mechanics of anisotropic structures and, in the first place, plates.In many industries, designs in the form of plates made of orthotropic materials with three planes of symmetry of elastic properties are widely used
The form of fundamental functions is determined by the relation between r and s, which depends on the fixing conditions of the longitudinal edges of the orthotropic plate
Summary
The development level of production at the present stage is characterized by the widespread introduction of new technologies for the manufacture of high-strength materials with orthotropic (orthogonally anisotropic) properties. Such materials include fiberglass; composite materials reinforced with sequentially alternating layers of fibers in two mutually perpendicular directions; glued wood plates; sheet rolled metals, in which anisotropy begins to appear upon transition to the plastic stage of work, etc. Designs in the form of plates made of orthotropic materials with three planes of symmetry of elastic properties are widely used. In well-known monographs and reference books, only the stability problem of a rectangular plate with a hinged support along the contour is solved [1,2,3,4]
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