Numerical Computation of Buckling Loads by Finite Differences

After introducing the concept of central differences of a function to approximate its derivatives by simple numerical expressions, the writer shows how a problem involving a linear differential equation of the homogeneous type, with homogeneous boundary conditions, may be reduced to a problem involving the solution of a system of simultaneous linear algebraic equations. In this manner solutions of buckling problems may be obtained by purely numerical computations, using a procedure of successive approximations, which is greatly enhanced by a simple method of extrapolation. The errors involved in the procedure are discussed and complete tables are given to be used in connection with extrapolation. The procedure is then applied to various cases of buckling of beams, plates, and shells to demonstrate the technique of its application and to solve a few problems not appearing in the literature.