Summary: An assumed-strain finite element technique is presented for linear, elastic small-deformation models. Weighted residual method (reminiscent of the strain – displacement functional) is used to weakly enforce the balance equation with the natural boundary condition and the kinematic equation (the strain – displacement relationship). A priori satisfaction of the kinematic weighted residual serves as a condition from which strain – displacement operators are derived via nodal integration. A variety of element shapes is treated: linear triangles, quadrilaterals, tetrahedra, hexahedra, and quadratic (six-node) triangles and (27-node) hexahedra. The degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials. A straightforward constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Furthermore, the numerical inf – sup test is applied in select problems and several variants of the proposed formulations (linear triangles, quadrilaterals, tetrahedra, hexahedra, and 27-node hexahedra) pass the test. Examples are used to illustrate the performance with respect to sensitivity to shape distortion and the ability to resist volumetric locking.