Question 13.22: The fixed beam of Fig. 13.30 carries a uniformly distributed...
The fixed beam of Fig. 13.30 carries a uniformly distributed load over part of its span. Determine the values of the fixed-end moments.
Consider a small element δx of the distributed load. We can use the results of Ex. 13.20 to write down the fixed-end moments produced by this elemental load since it may be regarded, in the limit as δx → 0, as a concentrated load. Therefore from Eq. (v) of Ex. 13.20 we have
δM_A = w δx \frac{x(L-x)^2}{L^2}
The total moment at A, M_A , due to all such elemental loads is then
M_A = \int_{a}^{b}{\frac{w}{L^2} } x(L-x)^2 \ dx
which gives
M_A = \frac{w}{L^2} \left[\frac{L^2}{2} (b^2-a^2) – \frac{2}{3}L(b^3-a^3) + \frac{1}{4} (b^4 – a^4) \right] (i)
Similarly
M_B = \frac{wb^3}{L^2} \left(\frac{L}{3} – \frac{b}{4} \right) (ii)
If the load covers the complete span, a = 0, b = L and Eqs (i) and (ii) reduce to
M_A = M_B = \frac{wL^2}{12} (as in Ex. 13.21.)