Load along a Single Axis

A beam of length L subjected to a varying load, which is a combination of parabolic and trapezoidal load distribution along the x-axis.

In the parabolic load, consider a differential element of force dR over the length dx. The resultant load is obtained by integrating the differential element over the length of the loading. It is located at the centroid of the parabolic area, and the centroid's x-coordinate is determined using the moment principle.

The trapezoidal area can be divided into rectangular and triangular areas, with the resultant load acting at their centroids. These centroids are positioned at half the length of the rectangle and one-third of the base length from the vertical side of the triangle.

The resultant load for the trapezoidal area is obtained by adding the individual resultant loads and passes through the centroid of the area, determined by the moment principle.

The total distributed load is the sum of the resultant loads of the trapezoidal and parabolic areas, passing through the centroid of the combined area, determined by the moment principle.