Beam problem is very important for engineering theoretically and practically. In this thesis we study such kind of nonlinear 4-th order ordiniary differential equations with nonlinear boundary conditions. The well-posedness of these boundary value problems will be discussed. Moreover, we will design different schemes to solve them, through differential equation, integral equation or minimization. Each type can further be discretized by finite difference, finite element or spectral method, etc. In the end we will compare all methods and find the best one.