An elastic-plastic stress increment updating scheme, namely, an adaptive step-size control scheme of a second-order Runge-Kutta method, was proposed. In order to model the contact loading of a rigid cylinder on a short beam specimen in the study of three-point short beam shear tests of composite materials, a simple contact modeling scheme was also developed. A three-dimensional nonlinear finite element computer program, incorporating a plasticity model for fiber-reinforced composites, was developed in the present study. The stress increment updating scheme and the contact modeling algorithm were implemented into the program. Some numerical examples were presented and these numerical techniques were shown to be more efficient than other conventional schemes while generating comparable results. The numerical results for the modeling of a short beam shear test method for unidirectional composite materials were also presented as an application of the computer program.