Understanding Bending Moment in a Centrally Loaded Beam

Let's consider a simply supported beam of length \(l\) that has a concentrated load \(L\) placed exactly at its center. Simply supported means the beam rests on supports at its ends, allowing rotation but preventing vertical movement.

Calculating Support Reactions

For a simply supported beam with a load \(L\) at the center, the load is symmetrically distributed. Therefore, the reactions at each support are equal.

Each support reaction \(R\) is half of the total load:

\[ R = \frac{L}{2} \]

Determining Bending Moment Along the Beam

The bending moment at any point along the beam is calculated by considering the forces and distances on one side of that point. The bending moment varies along the length of the beam.

For a simply supported beam with a central load, the bending moment is zero at the supports (ends) and maximum at the point where the load is applied, which is the center.

Calculating Maximum Bending Moment

The maximum bending moment occurs at the center of the beam, which is at a distance of \(l/2\) from either support.

Let's calculate the bending moment at the center (at \(x = l/2\)) by considering the left half of the beam. The only force on the left side is the support reaction \(R = L/2\) acting upwards at the left support.

The bending moment \(M\) at the center is the reaction force multiplied by the distance from the support to the center:

\[ M_{\text{max}} = R \times \frac{l}{2} \]

Substitute the value of \(R\):

\[ M_{\text{max}} = \frac{L}{2} \times \frac{l}{2} \]

\[ M_{\text{max}} = \frac{L \times l}{4} \]

The maximum bending moment is \( \frac{L \times l}{4} \). The units are kNm, as \(L\) is in kN and \(l\) is in meters (implicitly, as kNm is the unit). The expression can be written as \(L \times l/4\) kNm.

Comparing with Given Options

Let's look at the provided options:

• Option 1: \(L \times l/4\) kNm
• Option 2: \(L \times l/2\) kNm
• Option 3: \(L/l\) kNm
• Option 4: \(L \times l\) kNm

Our calculated maximum bending moment is \(L \times l/4\) kNm.

Comparing this with the options, we see that Option 1 matches our result.