Bending Moment in Beams under UDL

Let's analyze the bending moment at the center of two different types of beams when both are subjected to the same uniformly distributed load (u.d.l) and have the same span.

Consider a beam of span $L$ subjected to a uniformly distributed load of intensity $w$ per unit length.

Simply Supported Beam

For a simply supported beam, the maximum bending moment occurs at the center of the span when it is subjected to a uniformly distributed load over its entire length. The formula for the maximum bending moment ($M_{SS}$) at the center is given by:

$\qquad M_{SS} = \frac{wL^2}{8}$

where:

• $w$ is the intensity of the uniformly distributed load
• $L$ is the span of the beam

Fixed Beam

For a fixed beam (or propped cantilever) with both ends fixed, subjected to a uniformly distributed load over its entire length, the bending moment at the center of the span is also a common point of interest. The bending moment at the center ($M_{Fixed}$) for a fixed beam under UDL is given by:

$\qquad M_{Fixed} = \frac{wL^2}{24}$

Note that for a fixed beam, the maximum bending moment occurs at the supports, which is $-\frac{wL^2}{12}$. However, the question specifically asks for the bending moment at the centre of the fixed beam.

Calculating the Ratio

The question asks for the ratio of the bending moment at the center of a simply supported beam to the bending moment at the center of a fixed beam under the same conditions (same span $L$ and same u.d.l. $w$).

Ratio $= \frac{\text{Bending moment at the center of simply supported beam}}{\text{Bending moment at the center of fixed beam}}$

Ratio $= \frac{M_{SS}}{M_{Fixed}}$

Substitute the formulas for $M_{SS}$ and $M_{Fixed}$:

Ratio $= \frac{\frac{wL^2}{8}}{\frac{wL^2}{24}}$

To simplify the expression, we can multiply the numerator by the reciprocal of the denominator:

Ratio $= \frac{wL^2}{8} \times \frac{24}{wL^2}$

The terms $wL^2$ cancel out:

Ratio $= \frac{1}{8} \times 24$

Ratio $= \frac{24}{8}$

Ratio $= 3$

Thus, the ratio of the bending moment at the center of a simply supported beam to the bending moment at the center of a fixed beam, when both are of the same span and subjected to the same u.d.l., is 3.