Multilayered CORC® wires can carry very high currents in background magnetic fields up to 20 T. Mechanical stresses during operation can result in irreversible degradation in the CORC® wires/cables' performance. Different mechanical loads acting on CORC® cable during production, winding, assembly, and electromagnetic operation are bending, axial and transverse loads. The tape's helical shape around the central core allows tapes to experience only a fraction of the total axial strain applied to the entire cable in the case of tensile loads. The winding angle is the main cabling parameter that influences the tensile strain limit of the CORC® cable. The radial contraction of the tape depends on Poisson's ratio of the central core and winding angle. An analytical model is proposed to estimate the tensile strain in CORC® wires and cables. With optimized cabling parameters, the irreversible strain limit of CORC® cables and wires can be as high as 7%, which is 10 to 12 times higher than the irreversible strain limit of single REBCO tapes. The major mechanical stress in fusion and detector magnets in operation is transverse compressive stress. The transverse stress tolerance of the CORC® cables and wires depends mainly on the gap spacing between layers, uniformity, and thickness of the copper layer in REBCO tape, core material, and surrounding material's hardness. A detailed finite element model is developed to study the effect of both tensile and transverse load in CORC® cables and then compared against experimental data.