An efficient iterative method is developed for the static analysis of large deflections of an infinite beam with variable cross-section resting on a nonlinear foundation. A pseudo spring constant is added and explicit matrix operators are introduced to perform differentiation through Green's function. The nonlinearity of the problem is handled with quasilinearization. To compute the solution of the quasilinear differential equation with prescribed accuracy, a new discretization method for solving quasilinear differential equations involving up to the 4th order derivative is used. The discretization method is based on relating discretizations of up to the fourth order derivative of the solution with a discretization of the solution by using a suitable Green function. Numerical experiments show that the error incurred by the discretization can be made small for the two first derivatives and that the method proposed in the paper converges fast and has good accuracy.