Question 13.P.4: A uniform, simply-supported beam of span L carries a uniform......
A uniform, simply-supported beam of span L carries a uniformly distributed lateral load of w per unit length. It is propped on a knife-edge support at a distance a from one end. Estimate the vertical force on the prop.
Step-by-Step
If the beam is unpropped, then, from equation (13.15), the downwards vertical deflection at the position of the prop is
EIv = \frac{wz}{24}[L^3 – 2Lz^2 + z^3] (13.15) \\\\ (v)_{z = a} = \frac{wa}{24EI}(L^3 – 2La^2 + a^3)UR is the reaction on the prop, then under the action of R alone the upwards vertical deflection at the prop is, from equation (13.35),
v_D = \frac{Wa^2(L – a)^2}{3EIL} (13.35) \\\\ (v)_{z = a} = \frac{Ra^2(L – a)^2}{3EIL}If there is no resultant deflection at the prop, we have
\frac{Ra^2(L – a)^2}{3EIL} = \frac{wa}{24EI}(L^3 – 2La^2 + a^3)Thus, the reaction on the prop is
R = \frac{wL}{8}\left[\frac{ 1 – 2\left(\frac{a}{L}\right)^2 + \left(\frac{a}{L}\right)^3}{\frac{a}{L}\left(1 – \frac{a}{L}\right)^2} \right]The propping force is least when the prop is at mid-span ; in this case, a/L = 0.5 and R = 5 wL/8.
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