Deflections of Beams with Varying Rectangular Cross Section

Abstract

Closed-for solutions are presented for bending beams with linearly and (in the binomial form) parabolically varying depth and for bending beams with linearly varying width. The solutions are achieved by transforming the fourth-order differential equations with function coefficients into fourth-order differential equations with constant coefficients. For each law of variability of the cross section, four load conditions are considered: Concentrated, uniformly distributed, linearly, and parabolically varying distributed. The solutions depend on four integration constants, and they can be applied to any mechanical and kinematical end conditions. Dimensionless graphs for maximum deflection of three special beams under typical load conditions are given as a function of the ratio between the minimum and the maximum heights. For the statically indeterminate beam, dimensionless stress (of the flexural stress component) is also given. Though the solution expressions require a little computational effort (a pocket calculator is sufficient) the dimensionless graphs are useful for analyzing similar problems.