The solution of the classical fourth-order ordinary differential equation for static beam problems by using the finite difference method is reconsidered, but this time for the derivation of feasibility conditions in cases of validity of parametric linear inequality constraints with respect to the loading/geometry of the beam. To this end, the computer algebra system Reduce has been used, but supplemented by its recent REDLOG (REDuce LOGic) package incorporating the efficient Weispfenning computational quantifier elimination algorithms. A particular problem for a finite beam loaded by a triangular loading has been employed as the vehicle for the illustration of the present approach and the derived feasibility conditions are displayed. The finite element method has also been used (instead of the finite difference method) in the same problem. The present results can also be generalized to problems of beams on an elastic foundation, to two-dimensional problems, to optimization problems, etc.