Problem 712 | Propped beam with initial clearance at the roller support

$\delta_R + \Delta = \delta_w + \frac{1}{2}L\theta$
 

 

See Case 1 and Case 3 of Superposition Method for formulas:
$\dfrac{RL^3}{3EI} + \Delta = \dfrac{w_o(\frac{1}{2}L)^4}{8EI} + \dfrac{L}{2} \left[ \dfrac{w_o(\frac{1}{2}L)^3}{6EI} \right]$

$\dfrac{RL^3}{3EI} + \Delta = \dfrac{w_oL^4}{128EI} + \dfrac{w_oL^4}{96EI}$

$\dfrac{RL^3}{3EI} + \Delta = \dfrac{7w_oL^4}{384EI}$

$\dfrac{RL^3}{3EI} = \dfrac{7w_oL^4}{384EI} - \Delta$

$R = \dfrac{7w_oL}{128} - \dfrac{3EI \Delta}{L^3}$

$R = \dfrac{1}{128L^3}(7w_oL^4 - 384EI \Delta)$           answer