Explicit determination for exact solutions of elastic rectangular beams

In this paper, a method to directly determine explicit expressions of the exact general solutions for isotropic rectangular beam is provided. If the spanwise variation of the bending moment is smooth, the exact solution of the stress function can be expressed in the form of infinite series, whose each term is the product of the bending moment or its higher derivatives and polynomial involving only the longitudinal coordinates, while the polynomials are independent of the distributed loads. First, the explicit exact expression of the stress function is derived by solving recurrence formulae. Then, the convergence and accuracy of the formulae is estimated by retaining different terms. Finally, formulae of the stress and displacement fields are applied to some classical examples with the cases distributed loads in simple polynomials and sine form, and the results obtained are in perfect agreement with the existing exact theory.