M.Sc Student Ambarsumian Hachatur Buckling and Vibration of variable Thickness Continuous Plates with Internal Line Hinges and Line Slides Department of Civil and Environmental Engineering Professor Moshe Eisenberger

Abstract

This work gives highly accurate solutions for the buckling loads and for natural frequencies of continuous plates of variable thickness with internal line hinges and line slides, with various combinations of boundary conditions. The line hinges ensure vertical displacement compatibility while discontinuous slope. Case of the line slides ensure slope compatibility but enables discontinuous vertical displacements. The calculation of values of the natural frequencies and critical values of the in-plane forces applied on the ends of the plate core done using the extended Kantorovich method. In the Kantorovich method, the solution for a two dimensional problem is taken in the form . With one of the functions, Y(y) for example, specified ? a priory?, the partial differential equation can be transformed to an ordinary differential equation. The unknown function X(x) is determined by solving an ordinary differential equation for the set of given boundary conditions. As a result the accuracy in the analytical solution in the x direction is usually higher than that in the assumed y direction. After this, the approximate and analytical directions are switched by using the obtained X(x) as the specified function and re-determining Y(y) through another Kantorovich solution process. The iterative procedure is repeated until the result converges to a desired level. In the solution, an exact method for the dynamic natural frequencies and stability analysis of compressed members with variable flexural rigidity is used. It is based on the derivation of the dynamic stiffness matrix for variable cross section members. The terms in the dynamic stiffness matrix are found, using the properties of the shape functions, as follows. The terms in the stiffness matrix are defined as the holding actions at both ends of the strip, due to unit translation or rotation, at each of the four degrees of freedom, one at a time. The stiffness matrix for the strip-element with line hinges or line slides are 3 by 3 and are derived using matrix condensation from the dynamic stiffness matrix of regular strip-element. The calculated shape functions have the special properties, that they are the? ?exact? solution for the differential equations. The natural frequency and buckling loads for variable thickness strip or plate can be found as the frequency or as the in-plane forces, that cause the determinant of the corresponding stiffness matrix to become zero. Several examples are given and compared to published results to demonstrate the accuracy and flexibility of the method.