On the assumptions of the generalized plane stress problem and the Filon average

In this article, we prove that the assumption of vanishing normal stress for the generalized plane stress problem can be further weakened, so that the normal stress is harmonic. The stress state under this harmonic assumption can also be decomposed into the plane stress state and the shear stress state. Finally, a necessary and sufficient condition is presented for the existence of the biharmonic Airy stress function for the Filon averaged in-plane stresses, which is shown to be weaker than the assumption of the harmonic normal stress.