Interior solution for anisotropic strips by polynomial series

Usually, the solution of two-dimensional problems for long rectangular strips is represented by double series expansions of Airy stress function in powers of axial and transverse coordinates. Following this way, the computation of the coefficients becomes rather involved if the degree of the polynomial load prescribed on the long sides increases. For isotropic strips, in a previous authors paper a different representation of solution is described, where the series expansion of Airy stress function is obtained in terms of polynomials of the transverse coordinate multiplying the functions which represent the top and bottom loading. In this case, the first term of the series represents the classical beam theory, and the next terms are corrections involving higher derivatives of loading function. In the present paper, homogeneous anisotropic strips subject to any given continuous distribution of both normal and shear loads are considered. The interior problem is solved by means of a polynomial expansion for the Airy stress function. The polynomial functions defined in transverse direction are determined recursively and those varying along the strip length are obtained in terms of loading functions. Explicit formulas for displacement components are also given. This exact elasticity solution is finally used to derive the set of shear correction factors and to establish the range of validity of Timoshenko-like beam theories.