Understanding Bending Stress Variation in a Cantilever Beam with UDL

This explanation details how bending stress changes along the length of a cantilever beam when it's subjected to a uniform load spread across its entire span.

Defining Cantilever Beam and UDL

• Cantilever Beam: A structural element anchored at one end, while the other end remains free.
• Uniformly Distributed Load (UDL): A load that is applied evenly across the entire length of the beam. It's measured in units like Newtons per meter (N/m) or pounds per foot (lb/ft).

Calculating Bending Moment in the Cantilever

To determine the bending stress variation, we first analyze the bending moment ($M$) along the beam. Let's assume:

• The length of the cantilever beam is $L$.
• The UDL has a magnitude of $w$ per unit length.
• We examine a cross-section at a distance $x$ measured from the free end.

The total load acting on the portion of the beam from the free end up to the section at $x$ is the load intensity multiplied by the length of that portion, which is $w \times x$. This resultant load effectively acts at the midpoint of this segment, i.e., at a distance $x/2$ from the section we are considering.

The bending moment ($M$) at the section $x$ is calculated by multiplying this resultant load by its distance from the section:

$$ M(x) = - (w \cdot x) \times \frac{x}{2} $$

$$ M(x) = -\frac{w x^2}{2} $$

The negative sign indicates the direction of the moment (causing tension in the upper fibers and compression in the lower fibers, known as hogging), typical for a cantilever beam under downward load. Importantly, the magnitude of the bending moment ($|M|$) is proportional to the square of the distance ($x$) from the free end.

• At the free end ($x=0$), the bending moment is zero: $M(0) = -\frac{w (0)^2}{2} = 0$.
• At the fixed end ($x=L$), the bending moment reaches its maximum magnitude: $M(L) = -\frac{w L^2}{2}$.

This relationship shows that the bending moment ($M$) varies quadratically, or parabolically, with the distance $x$ along the cantilever's length.

Relating Bending Moment to Bending Stress

The fundamental relationship between bending stress ($\sigma$), bending moment ($M$), the distance from the neutral axis ($y$), and the second moment of area ($I$) is given by the flexure formula:

$$ \sigma = \frac{M y}{I} $$

For a beam having a uniform cross-section throughout its length:

• The second moment of area ($I$) remains constant.
• The maximum bending stress occurs at the points furthest from the neutral axis (the extreme fibers), where $y$ has its maximum value. Let this maximum distance be denoted by $c$.

Thus, the maximum bending stress at a particular section is:

$$ \sigma_{max} = \frac{M c}{I} $$

Determining the Stress Variation Pattern

Considering the flexure formula $\sigma = \frac{M y}{I}$ for a section along the beam:

• The term $y/I$ is constant for the extreme fibers at any given cross-section, and since the cross-section is uniform, $y$ and $I$ are constant along the length.
• Therefore, the bending stress ($\sigma$) is directly proportional to the bending moment ($M$) at that section.
• $$ \sigma \propto M $$
• We previously found that the bending moment varies parabolically with distance from the free end: $M \propto x^2$.
• Combining these relationships, we get: $$ \sigma \propto x^2 $$

This dependency, where stress is proportional to the square of the distance from the free end ($x$), signifies a parabolic variation of bending stress along the length of the cantilever beam.

The bending stress is zero at the free end (where $x=0$ and $M=0$) and increases parabolically towards the fixed end (where $x=L$ and $M$ is maximum).

Final Conclusion on Stress Variation

For a cantilever beam subjected to a uniformly distributed load over its entire length, the bending stress varies parabolically along its length.