In structural engineering, the analysis of beams subjected to varying loads is a critical aspect of understanding the behavior and performance of these structural elements. A common scenario involves a beam subjected to a combination of different load distributions.

Consider 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 this case, it is essential to determine the resultant loads, their locations, and the centroid of the combined area to predict the beam's response under these loading conditions.

Firstly, examine the parabolic load distribution. Consider a differential element of force dR acting over a small length dx. To determine the resultant load for the parabolic area, the differential element dR is integrated over the entire length of the load. The location of this resultant load is at the centroid of the parabolic area. The moment principle is applied to find the x-coordinate of the centroid, which states that the first moment of the area about an axis is equal to the product of the area and the distance of its centroid from the axis.

Next, the trapezoidal load distribution is analyzed by dividing it into two rectangular and triangular regions. The resultant loads for these individual regions act at their respective centroids. For the rectangular area, the centroid is positioned at half the length of the rectangle. For the triangular area, the centroid is located one-third of the base length away from the vertical side of the triangle.

By adding the individual resultant loads for the rectangular and triangular areas, the resultant load for the entire trapezoidal area can be determined. The location of the resultant load passes through the centroid of the trapezoidal area, which can also be determined using the moment principle.

Similarly, by adding the resultant loads of both the parabolic and trapezoidal areas, the total resultant load acting on the beam can be determined. The location of the total resultant load passes through the centroid of the combined area, which can be further determined using the moment principle.