Simply Supported Beam & Point of Contraflexure

A point of contraflexure, also known as a point of inflection, is a crucial concept in structural mechanics. It refers to a specific location along a beam where the bending moment is zero and, more importantly, where the bending moment changes its sign from positive to negative or vice versa. This change in sign indicates a change in the curvature of the beam.

Bending Moment in a Simply Supported Beam with UDL

Let's consider a simply supported beam of length \(L\) carrying a uniformly distributed load (UDL) of intensity \(w\) per unit length. In such a beam:

• The supports are at the ends, typically providing vertical reactions but no moment resistance.
• The reactions at each support are equal, calculated as \(\frac{wL}{2}\).
• The bending moment at any section at a distance \(x\) from one end can be calculated using the following formula:

\[ M_x = \frac{wL}{2}x - \frac{wx^2}{2} \]

Let's analyze the bending moment at critical points:

• At the supports (i.e., at \(x = 0\) and \(x = L\)), the bending moment is zero.
• At the center of the beam (i.e., at \(x = \frac{L}{2}\)), the bending moment is maximum and positive:

\[ M_{\text{max}} = \frac{wL}{2}\left(\frac{L}{2}\right) - \frac{w}{2}\left(\frac{L}{2}\right)^2 = \frac{wL^2}{4} - \frac{wL^2}{8} = \frac{wL^2}{8} \]

Throughout the entire span between the two supports, for a simply supported beam subjected only to a UDL, the bending moment is always positive. It starts at zero at one support, increases parabolically to a maximum positive value at the center, and then decreases parabolically back to zero at the other support.

Existence of Point of Contraflexure

For a point of contraflexure to exist, the bending moment must not only be zero but also change its sign (i.e., pass from positive to negative or vice versa) at that point. In the case of a simply supported beam carrying a UDL:

• The bending moment is zero only at the ends (supports) of the beam.
• Between the supports, the bending moment is always positive.
• Since the bending moment does not change its sign anywhere within the span of the beam (it's always positive), there is no point where its curvature reverses.

Therefore, a point of contraflexure does not exist within the span of a simply supported beam subjected solely to a uniformly distributed load. These points typically appear in beams where there are overhanging sections or in continuous beams where load conditions can cause sections to experience both positive and negative bending moments.