Three-dimensional elasticity solution for sandwich beams/wide plates with orthotropic phases: The negative discriminant case

In an earlier paper, Pagano (1969) [Pagano NJ. Exact solutions for composite laminates in cylindrical bending. J Compos. Mater. 1969; 3: 398-411] presented the three-dimensional elasticity solution for orthotropic beams (applicable also to sandwich beams) for the cases of: (1) a phase with positive discriminant of the qudratic characteristic equation, which is formed from the orthotropic material constants and further restricted to positive real roots and (2) an isotropic phase, which results in a zero discriminant. The roots in this case are all real, unequal, and positive (positive discriminant) or all real and equal (isotropic case). This purpose of this article is to present the corresponding solution for the cases of (1) negative discrimnant, in which case the two roots are complex conjugates and (2) positive discriminant but real negative roots. The case of negative discriminant is frequently encountered in sandwich construction, where the orthotropic core is stiffer in the transverse than the in-plane directions. Example problems with realistic materials are solved and compared with the classical and the first-order shear sandwich beam theories.