For the finite element-based solution of problems involving contact in physical systems, a discretised description of surfaces is considered for their physical interactions. The decision of the scheme of discretization and element topology used depends upon the geometries of the bodies and the problem in consideration. Multiple strategies have been developed for the enforcement of contact constraints, namely, Node-to-Node (NTN), Node-to-Segment (NTS), and Segment-to-Segment (STS) formulations [1]. The NTN method has the limitation of being restricted in use for conformally discretized surfaces. While the widely popular NTS method [2] overcomes this limitation, the choice of slave and master surface introduces a biasing in the formulation. Also, it fails the contact patch test and changing node-segment contact pairs during sliding cause jumps in the nodal forces. Although the STS method [3] overcomes these issues, it too suffers from biasing with the choice of master-slave surface, and the dual pass strategy designed to overcome this biasing adds to the complexity of the algorithm. By employing a middle surface for unbiased contact traction resolution, the STS method has been utilized to solve problems with different discretization of first and second-order elements. Here, the interpenetration between contacting surfaces is penalized in the form of contact tractions, which act as nodal forces on the facets of contacting elements in a manner that maintains the force-moment equivalency and the equilibrium of contact tractions. Using regularized Mohr-Coulomb frictional law, an unbiased tangential traction calculation method is also developed for this STS method. It considers the pairs of contacting segments for any relative tangential movement and calculates the frictional forces based on the elastoplastic analogy. These implementations are tested with several benchmarks for quasi-static and dynamic cases of contact between solids, including self-contact in an explicit time integration framework. [1] Wriggers, P. (2006). Computational contact mechanics (Vol. 2, p. 25). T. A. Laursen (Ed.). Berlin: Springer. [2] Zavarise, G., & De Lorenzis, L. (2009). A modified node‐to‐segment algorithm passing the contact patch test. International journal for numerical methods in engineering, 79(4), 379-416. [3] Puso, M. A., & Laursen, T. A. (2004). A mortar segment-to-segment contact method for large deformation solid mechanics. Computer methods in applied mechanics and eng