Load-deformation relations in second-order elasticity

A method for determining overall load-deformation relations for small-strain elastic deformations has been adapted so that second-order elastic effects can be included in certain overall quantities, such as bending stiffness, without explicit determination of the second-order solution. The approach uses Betti's Reciprocal Theorem. The strains are considered small enough that third-order effects can be neglected but there is no restriction on the magnitude of the displacements. The method is applied to obtain the resultant force and moment to second order for cylindrical bars with arbitrary cross section under torsion, extension and torsion, bending by an end load, and twisting and bending to helical form.