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Explicit and exact three-dimensional solutions are derived for the equilibrium problem of plate-like bodies with rectangular or circular cross-sections, made of several layers of transversely isotropic materials, simply supported on the lateral boundary, and loaded on the end faces. The component-wise representation of equilibrium displacement fields has the form of a series, whose typical term is the product of a function of the in-plane coordinates and a function of the transverse coordinate. Examples are presented to demonstrate, by comparison with finite-element solutions, the accuracy achieved when solution series are truncated after few terms