Standard displacement-based finite elements show a locking behavior in the modeling of nearly incompressible materials. Similar phenomenon is observed in volume conserving elasto-plasticity. This limitation of the displacement based elements arises because modeling incompressible material behavior adds kinematic constraints to an element, i.e., the volume at the integral points is required to remain constant. The elements that are not able to resolve these constraints suffer from volumetric locking which causes their response to be too stiff and may lead to overestimation of collapse load when applied to geomechanics. In this paper, stabilized finite elements for mixed displacement-pressure formulation that are based on multiscale variational method are developed. The new formulation allows equal low-order interpolations for both displacement and pressure fields and is suitable for application in real engineering applications. The performance of the elements is evaluated by numerical examples, which involve patch test and representative numerical examples. It is shown that the volumetric locking for the case of near incompressibility or for the isochoric plastic flow is successfully removed.