Maximum Bending Moment for Simply Supported Beam

Let's find the maximum bending moment for a simply supported beam carrying a point load at its center.

A simply supported beam is supported at two points and is free to rotate at the supports. A point load is a concentrated load applied at a single point on the beam.

For a simply supported beam of length \(L\) metres with a point load \(W\) kN acting exactly at the center (at a distance \(L/2\) from each support), we can determine the support reactions and the bending moment.

Calculating Support Reactions

Let the supports be at points A and B. Due to the symmetry of the beam and the central load, the reactions at both supports will be equal.

• Total downward load = \(W\) kN
• Total upward reaction = \(R_A + R_B\)
• For equilibrium, \(R_A + R_B = W\)
• Due to symmetry, \(R_A = R_B\)
• Therefore, \(2R_A = W\), which means \(R_A = \frac{W}{2}\) kN
• And \(R_B = \frac{W}{2}\) kN

Calculating Bending Moment

The bending moment at any section of the beam is calculated by considering the forces to one side of the section. For a simply supported beam with a central point load, the maximum bending moment occurs at the point where the shear force is zero, which is under the central point load.

Consider a section at a distance \(x\) from support A, where \(0 \le x \le \frac{L}{2}\). The bending moment \(M(x)\) at this section is due to the reaction at A:

\(M(x) = R_A \times x = \frac{W}{2} \times x\)

The bending moment increases linearly from 0 at support A to its maximum value at the center.

Finding Maximum Bending Moment

The maximum bending moment occurs at the center of the beam, where \(x = \frac{L}{2}\).

Maximum bending moment = \(M\left(\frac{L}{2}\right) = \frac{W}{2} \times \frac{L}{2}\)

Maximum bending moment = \(\frac{W \times L}{4}\)

So, the maximum bending moment is \(\frac{WL}{4}\) kNm.

Comparing this with the given options:

• Option 1: \(W \times \left(\frac{L}{3}\right)\) kNm
• Option 2: \(W \times L\) kNm
• Option 3: \(W \times \left(\frac{L}{2}\right)\) kNm
• Option 4: \(W \times \left(\frac{L}{4}\right)\) kNm

The derived formula matches Option 4.