A multiple precision arithmetic method is applied to derive a numerical solution of a system of ordinary differential equations describing static equilibrium of structural systems. First, it is shown that a numerical approach based upon the widely-used fixed precision arithmetic method defined by IEEE 754 can not give a correct solution because of numerical instabilities appearing in a discrete scheme, even if we apply a quadruple precision floating point number. Then, it is clarified that such numerical instabilities can be effectively removed by introducing the multiple precision arithmetic method, which can give us a correct solution.