## Fourth-Order Ordinary Differential Equations

This chapter presents a finite-element model for fourth-order ordinary differential equations. In solid mechanics, a typical equation is the Euler-Bernoulli beam equation that governs deflection of beams. The principle of virtual displacements is frequently used in solid mechanics to develop finite-element models. Its derivation from the equations of equilibrium follows the same steps as the derivation of the weak form. The principle of minimum total potential energy is usually used for developing finite element models for problems involving elastic deformations. The principle is mathematically equivalent to the weak form if the virtual displacement field is considered to be the test function of the weak form. Interpretation of the weak form of the beam element equation as a statement of virtual displacements illuminates the physical meaning of the weak form and provides clear understanding of the equivalent nodal loads.