Distributed Loads and Their Resultants

A distributed load acts continuously over a length or area instead of being applied at one single point. In beam and statics problems, a load distributed along a length is usually described by its intensity \(q\), measured in force per unit length, for example N/m or kN/m.

Distributed loads are commonly used to model effects such as the weight of a beam, pressure acting along a member, or many closely spaced forces that are more conveniently represented as one continuous load.

Types of distributed loads

The load intensity may remain constant or vary from point to point. The most common cases are:

• Uniformly distributed load (UDL): the intensity is constant over the loaded length. Its load diagram is a rectangle.
• Triangular distributed load: the intensity changes linearly from zero to a maximum value, or from a maximum value to zero.
• Trapezoidal distributed load: the intensity changes linearly between two nonzero values.
• General nonuniform load: the intensity varies according to a function \(q(x)\) and may have a curved load diagram.

In equilibrium calculations, it is often convenient to replace a distributed load by a single equivalent concentrated force, called its resultant. The replacement is equivalent when it produces the same total force and the same moment.

Uniform load

Start with the simplest case. If a constant load intensity \(q\) acts over a length \(L\), its resultant is

\[R=qL.\]

The resultant acts at the middle of the loaded length:

\[x_R=\frac{L}{2}.\]

This follows directly from the rectangular load diagram: its area is \(qL\), and its centroid is at the center.

Triangular load

For a load that changes linearly from zero to \(q_{max}\), the load diagram is a triangle. Its resultant equals the area of that triangle:

\[R=\frac{q_{max}L}{2}.\]

The resultant passes through the centroid of the triangle: \(2L/3\) from the zero-intensity end, or \(L/3\) from the maximum-intensity end.

Trapezoidal load

A trapezoidal load is often easiest to understand by splitting it into two simple loads: a uniform rectangular part and a triangular part. Find the resultant of each part, place each force at the centroid of its diagram, and then combine the forces and their moments.

General rule: area and centroid

The previous cases reveal a useful geometric rule: the magnitude of the resultant equals the area under the load-intensity diagram, and its line of action passes through the centroid of that area.

For rectangles, triangles, and combinations of simple shapes, this rule usually lets you solve the problem without integration.

General nonuniform load

When the load varies according to an arbitrary function \(q(x)\), the same area-and-centroid idea is written mathematically using integration. For a load acting from \(x=a\) to \(x=b\),

\[R=\int_a^b q(x)\,dx.\]

The location of the resultant follows from equality of moments:

\[x_R=\frac{\int_a^b xq(x)\,dx}{\int_a^b q(x)\,dx}.\]

Thus, the integral formulas are not a separate rule: they are the general form of the same area-and-centroid method used for rectangular, triangular, and trapezoidal load diagrams.

Sign-changing loads

If \(q(x)\) changes direction, positive and negative parts of the load diagram must be treated algebraically using the chosen sign convention. The expression for \(x_R\) applies when \(R\ne0\). If the net resultant force is zero, the distributed system may instead reduce to a pure couple.