A cantilever beam with a span length of L m carries a uniform moment of intensity ‘M’ N-m/m. Which of the following statements is correct?

Explanation:

The cantilever beam with a span length of L meters carrying a uniform moment of intensity 'M' N-m/m throughout its length is an interesting case to analyze in structural mechanics. To understand the correct statement regarding the shear force distribution along the length of the cantilever beam, we need to delve into the fundamental concepts of beam theory and moment distribution.

Understanding Shear Force and Moment Distribution:

In beam theory, the shear force and bending moment are two critical parameters that describe the internal forces within a beam subjected to external loads. The shear force at any section of the beam is the sum of all vertical forces acting on the beam to the left or right of the section. The bending moment at any section is the sum of moments about that section caused by external loads.

For a cantilever beam subjected to a uniform moment, the moment distribution is constant along the length of the beam. A uniform moment means that at every point along the beam's length, there is a moment of intensity 'M' N-m/m acting.

Shear Force Calculation:

To determine the shear force distribution in the beam, we need to consider the equilibrium conditions of the beam. In a cantilever beam subjected to a uniform moment, the moment is constant along the entire length. Since the moment is uniformly distributed, there are no point loads or distributed loads causing a change in the shear force along the beam.

Key Point: In the absence of vertical loads, the shear force in the beam remains zero throughout its length.

This can be further clarified by considering the differential relationships between shear force and bending moment in beam theory. The shear force (V) and bending moment (M) are related by the following differential equation:

dM/dx = V

Where:

• dM/dx is the rate of change of bending moment with respect to the length of the beam.
• V is the shear force at a particular section.

Since the bending moment 'M' is constant along the length of the beam (uniform moment), its rate of change dM/dx is zero. Therefore, the shear force V must be zero throughout the length of the beam.

Conclusion:

Based on the above analysis, we can conclude that the shear force throughout the length of the cantilever beam carrying a uniform moment of intensity 'M' N-m/m is zero.

The correct answer is option 3.