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The decomposed form of the stress state for transversely isotropic beam bending based on elastic theory

In this paper we study a plane stress problem of rectangular beams which are slender and thin elastic bodies with free upper and lower surfaces and in the absence of body forces. Without employing ad hoc stress assumptions, the decomposed form of the stress state for transversely isotropic beam bending is proposed on the basis of the classical elastic theory, and the corresponding decomposition theorem is inextenso presented for the first time. It is shown that the stress state of transversely isotropic beams with free faces can be uniquely decomposed into two parts: the interior state and the Papkovich‐Fadle (P‐F) state. In the proof course of the decomposition theorem, some basic mathematical methods are used only, so the proof is more convenient for being understood. These two states are identical to two equations derived in the refined theory of transversely isotropic beams, so this work practically proves the consistency between the decomposed form and the refined theory of transversely isotropic beams. Moreover, with the separate consideration of the interior state and P‐F state, a relatively simple analytical solution is often practical and desirable, and the numerical computation process is greatly simplified.