15. A bar component in the figure is under the uniformly distributed load q due to gravity. For linear elastic material with Young's modulus E and uniform cross-sectional area A, the govern¬

ing differential equation can be written as

<92M AE~d^ + q = Q where u(x) is the downward displacement. The bar is fixed at the top and free at the bottom.

Using Galerkin method and two equal-length finite element, answer the following questions.

(a) Starting from the above differential equation, derive an integral equation using the Galerkin method.

(b) Write the expression of boundary conditions at x = 0 and x = L. Identify whether they are essential or natural boundary conditions.

(c) Derive the assembled finite element matrix equation, and solve it after applying boundary conditions.