(Solved): (5) A simply-supported beam has been subjected to an applied moment at midspan. The M(x) equations ...

(5) A simply-supported beam has been subjected to an applied moment at midspan. The $\mathbf{M}(\mathbf{x})$ equations have been supplied. Sketch $\mathbf{v}(\mathbf{x})$ and write the $\mathbf{v}(\mathbf{x})$ equations for $0?{\mathbf{x}}_1?\mathbf{L}/2$ and for $\mathbf{L}/2?{\mathbf{x}}_2?\mathbf{L}$. Hint: Two of the four constraint equations are related to slope and deflection compatibility at $\mathbf{x}=L/2$. Use anti-symmetry to your advantage. Ans. $v_1=\frac{M_0}{24EI}\left[\frac{4x_1^3}{L}?x_{1}L\right];v_2=\frac{M_0}{24EI}\left[\frac{4x_2^3}{L}?12x_2^2+11L{x}_2?3L^2\right]$