Elasticity interior solution for orthotropic strips and the accuracy of beam theories

The interior problem of all orthotropic strip subject to any given continuous distribution of normal and shear loads is solved by means of a polynomial expansion for the Airy stress function. The polynomial functions defined in the transverse direction are determined recursively by solving a Fredholm equation of second kind. Explicit formulas for displacements are given. A sufficient condition for the convergence of the series expansion is derived This solution is used to evaluate the error in Timoshenko and higher-order theories. A new beam theory is finally proposed whose error has the same asymptotic form as second-order theories but approaches zero for strips made of strongly orthotropic material