  Assumed stress hybrid methods are known to improve the performance of
standard displacement-based finite elements and are widely used in
computational mechanics. The methods are based on the Hellinger-Reissner
variational principle for the displacement and stress variables. This work
analyzes two existing 4-node hybrid stress quadrilateral elements due to Pian
and Sumihara [Int. J. Numer. Meth. Engng, 1984] and due to Xie and Zhou [Int.
J. Numer. Meth. Engng, 2004], which behave robustly in numerical benchmark
tests. For the finite elements, the isoparametric bilinear interpolation is
used for the displacement approximation, while different piecewise-independent
5-parameter modes are employed for the stress approximation. We show that the
two schemes are free from Poisson-locking, in the sense that the error bound in
the a priori estimate is independent of the relevant Lame constant $\lambda$.
We also establish the equivalence of the methods to two assumed enhanced strain
schemes. Finally, we derive reliable and efficient residual-based a posteriori
error estimators for the stress in $L^{2}$-norm and the displacement in
$H^{1}$-norm, and verify the theoretical results by some numerical experiments.
