Mechanical forces play a vital role in the activities of cells and their interaction with biological and nonbiological material. Various experiments have successfully measured forces exerted by the cells when in contact with a substrate, but the intracellular contractile machinery leading to these actions is not entirely understood. Tan , (2003, “Cells Lying on a Bed of Microneedles: An Approach to Isolate Mechanical Force,” Proc. Natl. Acad. Sci. USA, 100 (4), pp. 1484–1489) use a bed of PDMS posts as the substrate for cells and measure the localized mechanical forces exerted by the cell cytoskeleton on the posts. In live cell experiments for this setup, post deflections are measured, and from these results the forces applied by the cell are calculated. From such results, it is desirable to quantify the contractile tensions generated in the force-bearing elements corresponding to the stress fibers within the cell cytoskeleton that generate the loads applied to the posts. The purpose of the present article is to consider the cytoskeleton as a discrete network of force-bearing elements, and present a structural mechanics based methodology to estimate the configuration of the network, and the contractile tension in the corresponding stress fibers. The network of stress fibers is modeled as a structure of truss elements connected among the posts adhered to a single cell. In-plane force equilibrium among the network of stress fibers and the system of posts is utilized to calculate the tension forces in the network elements. A Moore-Penrose pseudo-inverse is used to solve the linear equations obtained from the mechanical equilibrium of the cell-posts system, thereby obtaining a least squares fit of the stress fiber tensions to the post deflections. The predicted network of force-bearing elements provides an approximated distribution of the prominent stress fibers connected among deflected posts, and the tensions in each fibril.