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Description
The main objective of this work is to present an idea of the Finite Difference Method (FDM) application for calculations of variable cross-section beams. The differential equation has an equivalent difference equation in FDM [3,4] where unfortunately the flexibility matrix is not symmetric because the bending EJ stiffness is not more constant along the beam [1]. In order to avoid these difficulties, an idea of the solution is developed. Instead of solving fourth-order equations, the calculations are performed in two steps using the FDM approximation of two second-order differential equations. In the first step the matrix equation is solved, equivalent to the first equation describing the dependence between the bending moments and given load. In the next step, after determining the values of bending moments, the second matrix equation is solved. In this step the nodal displacement vector as the solution of the matrix equation equivalent to second-order differential equation is created. The vector on the right-hand side of equations contains the nodal bending moments obtained in the previous step. In both steps each row of the narrow band matrix contains only three number coefficients of the central second-order difference operator and numbers corresponding to boundary conditions, respectively. In order to obtain eigenfrequencies of beams the standard method as the solution of the matrix eigenvalue problem is used. In the stability problem of beams the equation elements containing the normal force N are considered as nodal forces. The most important advantage of the presented idea is that the variable bending stiffness EJ of beams are on the right-hand side of the second order equations with constant coefficients. A significant number of numerical calculations have shown effectiveness and high accuracy of the elaborated method.
Literature
[1] Timoshenko S.P., Goodier J.N. Theory of elasticity, Mc Graw Hill, New York, 1934
[2] Timoshenko S.P., Gere J.M., Theory of elastic stability, Mc Graw Hill, New York, 1961
[3] Levy H., Lessman F., Równania różnicowe skończone, Warszawa, PWN 1966 (in Polish)
[4] Pawlak Z., Rakowski J., Fundamental solutions for regular discrete slabs, Zeitschrift für Angewandte Mathematik und Mechanik, 77, 1997, 261-262
