Variable Loading | Solved Example-2

Problem-2:

A beam is subjected to the variable loading as shown in the figure. Find the reactions at support points.

Solution:

The load intensity (w) is given as,

\(w=a−bx^{2}\)

\(\text{At } x = 0 \implies w = w_A = 2400 \text{ N/m} \implies a = 2400 \quad \text{--- (i)}\)

∴a=2400— (i)

\(\text{At } x = 6\text{ m} \implies w = w_B = 1200 \text{ N/m}\)

\(\implies 1200 = 2400 - b(6)^2\)

\(\implies 36b = 1200\)

\(\implies b = 33.33 \quad \text{--- (ii)}\)

\(w = 2400 - 33.33x^2\)

Resultant load due to variable loading on the beam:

\(W_R = \int_{A}^{B} w \cdot dx = \int_{0}^{6} (2400 - 33.33x^2) \, dx\)

\(W_R = 14160.02 \text{ N}\)

The location of resultant load w.r.t the point ‘A’ is given as,

\(x_R = \frac{\int_{A}^{B} x(w \cdot dx)}{\int_{A}^{B} w \cdot dx} = \frac{\int_{0}^{6} x(2400 - 33.33x^2) \, dx}{14160.02}\)

\(x_R = 2.975 \text{ m}\)