  In this paper, we discuss an adaptive hybrid stress finite element method on
quadrilateral meshes for linear elasticity problems. To deal with hanging nodes
arising in the adaptive mesh refinement, we propose new transition types of
hybrid stress quadrilateral elements with 5 to 7 nodes. In particular, we
derive a priori error estimation for the 5-node transition hybrid stress
element to show that it is free from Poisson-locking, in the sense that the
error bound in the a priori estimate is independent of the Lame constant
$\lambda$. We introduce, for quadrilateral meshes, refinement/coarsening
algorithms, which do not require storing the refinement tree explicitly, and
give an adaptive algorithm. Finally we provide some numerical results.
