Question

Compute the resultant force of a distributed load using the area under the loading function

I just need an exemple

Answer 1

A distributed load is a uniform load applied across an entire area rather than applying it at a single point. For example a triangular loading as shown below

Using area loading function, we can prove that a distributed load as above can be equated to a single load acting at a point as shown below

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The above was an easy example where its triangular loading. In general if the distributed load is represened by an equation, then we need to do it as integration

Example - Consider a loading pattern given by y = x^1/2 which is acting on a beam as follows with the following dimensions

The entire distributed load can be equated to a single point load as calculated below

The magnitude of the resultant force is given by the integral of the curve defining the force, w(x) which is given by

The location of the resultant force is given by the centroid of the area under the curve as calculated below

So w've just converted our distributed load over a span to a point load by integrating it under the entire area that is is acting on as follows

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