Refined orthotropic beam models based on Castigliano's theorem and an approximate solution of the compatibility equation. (November 2021) - Schoeftner, J.; Gahleitner, J.
Abstract: This paper presents an extension of Castigliano's second theorem for orthotropic beams. The notion "extension" in this context means to find more accurate results not only for the vertical deflection, but also for the horizontal deflection and the rotation angle. For this purpose the Airy stress function is calculated for an orthotropic rectangular strip based on the iterative approximation method from Boley-Tolins when distributed loads or shear tractions are assumed as surface loads. Hence all stress components can be derived from the stress function and may be inserted into the complementary strain energy. Computing the partial derivative of the complementary strain energy with respect to a scalar dummy parameter, the weighted displacement field components over the cross section are obtained. Finally the analytical results from the extended Castigliano theorem (ECT) are verified by two-dimensional (2D) analytical results for an orthotropic cantilever and a clamped-hinged beam and the outcome is also compared to Bernoulli–Euler and to Timoshenko results. The accuracy of the derived results is higher than that derived by the Timoshenko theory because the normal stress in thickness direction is properly taken into account. Moreover it is demonstrated that this lower order normal stress which is caused by distributed loads yields a mean horizontal elongation or shrinkage that is not considered by the Timoshenko beam theory. Hence, highly accurate formulations for laminated composite beams, involving sandwich-structures or multi-layered beams with imperfect bonding, may be derived from the present orthotropic beam model in the future.