Simply Supported Beam Bending Moment Diagram Explained

The bending moment diagram (BMD) is a graphical representation that illustrates how the internal bending moment varies along the length of a structural element, such as a beam. For structural engineers, comprehending the BMD is vital for ensuring the safety and efficiency of designs.

Bending Moment Diagram Shape for Central Load

Let us consider a simply supported beam. This type of beam rests on two supports, typically at its ends, which allow for rotation but prevent vertical movement. When such a beam of length \(L\) is subjected to a concentrated or point load \(W\) applied exactly at its center, specific reactions and moment distributions occur.

• Support Reactions: Due to the symmetrical nature of the loading (a central point load on a simply supported beam), the vertical reactions at both supports will be equal. If we denote the supports as A and B, the reactions \(R_A\) and \(R_B\) are: \[R_A = R_B = \frac{W}{2}\]
• Bending Moment Calculation:
  • At the extreme ends of a simply supported beam (i.e., at the supports), the bending moment is always zero. Thus, at \(x = 0\) (left support) and \(x = L\) (right support), the bending moment \(M = 0\).
  • As we move from either support towards the center of the beam, the bending moment increases linearly. For any section at a distance \(x\) from the left support (where \(0 \le x \le L/2\)), the bending moment \(M_x\) can be calculated as: \[M_x = R_A \cdot x = \frac{W}{2} \cdot x\]
  • The maximum bending moment occurs precisely at the center of the beam, where the point load is applied. At \(x = L/2\), the maximum bending moment \(M_{max}\) is: \[M_{max} = \frac{W}{2} \cdot \frac{L}{2} = \frac{WL}{4}\]

Isosceles Triangle Formation

Based on the calculated bending moment values and their distribution along the beam:

• The bending moment starts at zero at the left support.
• It then increases linearly from zero to its maximum value of \(\frac{WL}{4}\) at the exact midpoint of the beam.
• Continuing from the center, the bending moment decreases linearly back to zero at the right support.

This linear variation, starting from zero at both ends and peaking symmetrically at the center, graphically forms a triangular shape. Since the highest point of the diagram (the maximum bending moment) is exactly at the midpoint of the beam's span, the two non-base sides of this triangle are of equal length. A triangle characterized by two sides of equal length and a base is defined as an isosceles triangle.

Therefore, the bending moment diagram for a simply supported beam loaded in its center is an isosceles triangle.