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This thesis documents the efforts of the writer and his colleagues over the past several years to improve the theoretical foundation of the arbitrary quadrilateral shear panel used in structural analysis codes to model aircraft structures. An equilibrium stress-based element with pure shear resultants on its sides was developed using the principle of complementary virtual work. The internal stress field was derived from a complete polynomial Airy stress function. The element was numerically tested as a pure stress element and a hybrid element to assess the deflection properties for highly distorted planar panels. Linear-stress and quadratic-displacement rods were used, as appropriate, to model the stiffeners required to surround the shear panels. Panel displacements were compared with other well-known shear panels as well as with a finite element model of the shear panel. The pure-stress element, based on a third degree stress polynomial, was finally chosen because it gave displacements in agreement with the other shear panels (but usually on the order of twice the magnitude of the displacement-based finite element model) and panel performance was essentially unchanged with choice of higher-order stress polynomials. Performance of the hybrid version of the panel was spurious and further study is required to understand its behavior.