Cantilever Beam Bending Moment Explained

A cantilever beam is a structural element that is fixed at one end and free at the other. When a load is applied to a cantilever beam, it creates internal forces and moments within the beam. The bending moment is one such internal force that causes the beam to bend or deflect. The maximum bending moment in a cantilever beam subjected to a uniform distributed load (UDL) always occurs at the fixed support.

Understanding the Problem Scenario

We are given the following parameters for the cantilever beam:

• Length (L): 2 meters (m)
• Uniform Distributed Load (w): 1 kiloNewton per meter (kN/m)

Our objective is to determine the maximum bending moment that the beam experiences under this specific loading condition.

Maximum Bending Moment Formula

For a cantilever beam subjected to a uniform distributed load (UDL), denoted as 'w', over its entire length 'L', the maximum bending moment (\(M_{max}\)) occurs at the fixed end and is calculated using the following formula:

\[M_{max} = \frac{wL^2}{2}\]

Bending Moment Calculation Steps

Let's substitute the given values into the formula to find the maximum bending moment:

1. Identify the given values:
   • Load, \(w = 1 \text{ kN/m}\)
   • Length, \(L = 2 \text{ m}\)
2. Substitute the values into the formula:

   \[M_{max} = \frac{(1 \text{ kN/m}) \times (2 \text{ m})^2}{2}\]

3. Calculate the square of the length:

   \[(2 \text{ m})^2 = 4 \text{ m}^2\]

4. Perform the multiplication in the numerator:

   \[1 \text{ kN/m} \times 4 \text{ m}^2 = 4 \text{ kN-m}\]

5. Divide by 2 to find the maximum bending moment:

   \[M_{max} = \frac{4 \text{ kN-m}}{2}\]

   \[M_{max} = 2 \text{ kN-m}\]

Final Result of Bending Moment

The calculated maximum bending moment for the given cantilever beam under the specified load is 2 kN-m.

This result indicates the maximum internal rotational force acting within the beam due to the applied load, which is critical for structural design and analysis.