A cantilever beam is a fundamental structural element widely used in various engineering applications. It is characterized by having one end rigidly fixed to a support, which prevents both translation and rotation, while the other end remains completely free. This fixed end plays a crucial role in how the beam carries loads and how internal forces, such as shear force and bending moment, are distributed along its length.

The bending moment is an internal force within a beam that causes it to bend. It is a measure of the internal resistance of the beam to the external forces and moments that are trying to cause it to deform. The magnitude of the bending moment varies along the length of the beam, depending on the type and distribution of the applied loads. Understanding the distribution of bending moment is essential for designing beams to prevent failure due to bending stresses.

Cantilever Beam Bending Moment at Fixed End

For a cantilever beam subjected to external loads, the bending moment is consistently at its maximum magnitude at the fixed end. Here's a detailed explanation of why this occurs:

• Accumulation of Load Effects: In a cantilever beam, all the external forces acting along the beam's span, from the free end towards the fixed end, contribute to the bending effect at the fixed support. As you move from the free end, where the bending moment is typically zero (assuming no applied moment at that point), towards the fixed end, the cumulative effect of all the loads creates a progressively larger moment.
• Support Reaction Moment: To maintain static equilibrium under the applied loads, the fixed support at the end of the cantilever beam must generate a reaction moment. This reaction moment directly counteracts the moment caused by all the external loads acting on the beam. Since this support moment must balance the entire bending effect from all loads, its magnitude is the largest along the beam's length.
• Point of Maximum Stress: Because the bending moment is maximum at the fixed end, this location experiences the highest bending stresses. Structural engineers pay particular attention to this section during design to ensure the beam can safely withstand these stresses.

Consider a simple case: a cantilever beam of length $L$ with a concentrated load $P$ applied at its free end. The bending moment ($M$) at any section located at a distance $x$ from the free end is given by the formula $M = -P \cdot x$. At the free end, where $x=0$, the bending moment is $M = -P \cdot 0 = 0$. However, at the fixed end, where $x=L$, the bending moment is $M = -P \cdot L$. The magnitude of this bending moment, $P \cdot L$, is the maximum along the beam. Similarly, for a uniformly distributed load $w$ over the entire length $L$, the maximum bending moment at the fixed end is $\frac{wL^2}{2}$. These examples clearly illustrate that the bending moment reaches its maximum value at the fixed end of a cantilever beam under typical loading conditions.