**Bending Moment in a Cantilever Beam**

A cantilever beam is a type of structural member that is fixed at one end and free at the other end. It is commonly used in various civil engineering applications such as bridges, buildings, and support structures. When a moment is applied at the free end of a cantilever beam, it induces a bending moment at its support.

**Definition of Bending Moment**

Bending moment is a measure of the internal forces that cause a beam to bend. It is the algebraic sum of the moments about a particular point in a beam due to the external loads and the internal forces. Bending moment is expressed in units of force multiplied by distance (Nm or lb-ft) and represents the amount of bending that occurs at a specific point along the beam.

**Bending Moment Induced by a Moment Load at the Free End**

When a moment load, denoted as P, is applied at the free end of a cantilever beam, it induces a bending moment at its support. The magnitude of the bending moment at the support can be calculated using the following formula:

M = P * L

Where:
M = Bending moment at the support
P = Applied moment load at the free end
L = Span or length of the cantilever beam

The bending moment induced by the moment load is equal to the product of the applied moment load and the span of the beam. This equation assumes that the beam is homogeneous, isotropic, and in the linear elastic range.

**Explanation**

When a moment load is applied at the free end of a cantilever beam, it causes the beam to deform. As a result, internal forces are generated within the beam to resist the applied load and maintain equilibrium. These internal forces, known as bending moments, cause the beam to bend and induce stresses in the cross-section of the beam.

At the support of the cantilever beam, the bending moment induced by the applied moment load is maximum. This is because the support acts as a fixed point, preventing any rotation or movement. The bending moment at the support is equal to the product of the applied moment load and the span of the beam. This can be understood intuitively by considering the beam as a lever, where the moment load at the free end creates a moment arm equal to the span of the beam.

The bending moment induced at the support of a cantilever beam is crucial in determining the structural integrity and design of the beam. It helps engineers analyze the beam's behavior under different loading conditions and ensures that the beam can safely resist the applied loads without excessive deflection or failure.

In conclusion, when a moment load is applied at the free end of a cantilever beam, it induces a bending moment at its support. The bending moment at the support can be calculated using the formula M = P * L, where M is the bending moment, P is the applied moment load, and L is the span of the cantilever beam. Understanding the bending moment induced by the applied load is essential for designing and analyzing the structural behavior of cantilever beams.