VARIABLE LOADING ON A BEAM

Variable Loading:

"When the intensity of load (Load per unit length) is varying over the length of the beam, it is known as variable loading."

Intensity of load (w) varies with length (x):     w = f(x)

Diagram Description:

A simply supported beam of span between support A (pinned support) and support B (roller support).

The load intensity varies along the length, starting at \(w_{A}\)at end A and ending at \(w_{B}\)at end B.

At a distance x from end A, a small element of length dx is considered where the load intensity is w.

The total resultant load \(W_{R}\) acts at a distance \(x_{R}\) from point A.

                      \(W_{R}. x=\int_{A}^{B}x.w.dx => x_{R}=\frac{\int_{A}^{B}x.w.dx}{W_{R}} =\frac{\int_{A}^{B}x.w.dx}{\int_{A}^{B}w.dx}\)

Practice Problem 

Find the resultant of variable loading acting on the beam and its location from point A as shown in the figure.

Solution:

Load on the beam of length \(dx = w \times dx\)

Location of resultant load is given as:

\(\therefore \text{Resultant load } (W_R) = \int_{A}^{B} w \cdot dx = \int_{0}^{6} (5x^2) dx \therefore W_R = 5 \left[ \frac{x^3}{3} \right]_0^6 = 360 \text{ N}\)

\(x_R = \frac{\int_{A}^{B} x \cdot w \, dx}{\int_{A}^{B} w \, dx} = \frac{\int_{0}^{6} x \cdot 5x^2 \, dx}{360} = \frac{5}{360} \int_{0}^{6} x^3 \, dx = 4.5 \text{ m}\)

\(\therefore x_R = 4.5 \text{ m}\)