The bar of length 2L (see the figure) is loaded along its axes with the load q(x)=q_{0} x / L The left end is fixed. At the right end,the spring

of stiffness k connects the bar to the fixed support. It is fully relaxed when no external load exists. The elastic modulus E is uniform, and the cross-sectional area varies as A(x)=A_{0}(3-x / L) \text { The governing equation of the problem is } \frac{d}{d x}\left(E A(x) \frac{d u(x)}{d x}\right)+q(x)=0 \text {. } (a) (20%) Write the boundary conditions in terms of displacement and its derivative. Which one is the essential and which one is the natural boundary condition? [Hint: Watch the signs!] (b) (40%) Define the one condition that must be satisfied by the otherwise arbitrary test function w(x). Then derive the weak form of the problem. (c) (40%) Discretize the problem using the 2 finite elements (el and e2) and three nodes (seethe figure). The interpolation functions are piecewise linear and are illustrated in the figure. Derive the finite element equations for the two unknowns: \left[\begin{array}{ll} K_{22} & K_{23} \\ K_{32} & K_{33} \end{array}\right]\left\{\begin{array}{l} u_{2} \\ u_{3} \end{array}\right\}=\left\{\begin{array}{l} F_{2} \\ F_{3} \end{array}\right\} \text { Compute the coefficients } K_{22}, K_{23}, K_{33}, F_{2} \text {, and } F_{3} \text { in terms of known quantities } k, q_{0}, A_{0}, E \text { and } L