4
Understanding Beam Bending Moments
This problem asks us to determine the ratio of maximum bending moments for two fundamental types of beams: a cantilever beam and a simply supported beam. Both beams are specified to have the same length and are subjected to the same uniformly distributed load (UDL). To solve this, we need to recall the standard formulas for the maximum bending moment in each beam configuration under a uniformly distributed load.
Cantilever Beam Bending Moment Calculation
A cantilever beam is characterized by having one end fixed and the other end free. When a uniformly distributed load, denoted as 'w' (which represents load per unit length), acts over its entire length 'L', the maximum bending moment occurs at the fixed support. The formula for the maximum bending moment for a cantilever beam under a uniformly distributed load is:
\[ M_{\text{max, cantilever}} = \frac{wL^2}{2} \]Where:
- \(M_{\text{max, cantilever}}\) is the maximum bending moment experienced by the cantilever beam.
- \(w\) is the magnitude of the uniformly distributed load.
- \(L\) is the total length of the beam.
Simply Supported Beam Bending Moment Calculation
A simply supported beam is supported at both ends, typically by a pin support at one end and a roller support at the other. These supports allow for rotation but prevent vertical displacement. When this type of beam is subjected to a uniformly distributed load 'w' over its entire length 'L', the maximum bending moment occurs exactly at the mid-span of the beam. The formula for the maximum bending moment in a simply supported beam under a uniformly distributed load is:
\[ M_{\text{max, simply supported}} = \frac{wL^2}{8} \]Where:
- \(M_{\text{max, simply supported}}\) is the maximum bending moment experienced by the simply supported beam.
- \(w\) is the magnitude of the uniformly distributed load.
- \(L\) is the total length of the beam.
Ratio of Maximum Bending Moments
Now, to find the ratio of their maximum bending moments, we need to divide the maximum bending moment of the cantilever beam by that of the simply supported beam:
\[ \text{Ratio} = \frac{M_{\text{max, cantilever}}}{M_{\text{max, simply supported}}} \]Substituting the formulas we identified:
\[ \text{Ratio} = \frac{\frac{wL^2}{2}}{\frac{wL^2}{8}} \]To simplify this complex fraction, we can multiply the numerator by the reciprocal of the denominator:
\[ \text{Ratio} = \frac{wL^2}{2} \times \frac{8}{wL^2} \]We can observe that the term \(wL^2\) is present in both the numerator and the denominator, allowing us to cancel it out:
\[ \text{Ratio} = \frac{8}{2} \] \[ \text{Ratio} = 4 \]Thus, the ratio of the maximum bending moment of the cantilever beam to that of the simply supported beam, under identical uniformly distributed load and length conditions, is 4.
Key Formulas and Ratio Summary
Here is a summary of the maximum bending moments for the beam types discussed and their ratio:
| Beam Type | Loading Condition | Maximum Bending Moment (\(M_{max}\)) |
|---|---|---|
| Cantilever Beam | Uniformly Distributed Load (w) over length (L) | \( \frac{wL^2}{2} \) |
| Simply Supported Beam | Uniformly Distributed Load (w) over length (L) | \( \frac{wL^2}{8} \) |
This calculation clearly demonstrates that a cantilever beam experiences a maximum bending moment four times greater than a simply supported beam when both are subjected to the same uniformly distributed load over the same length. This difference is critical in structural design and analysis.
Important Questions from Bending Moment
For a simply supported beam carrying distributed load w per unit length, consider the following relations between shear force F and bending moment M.
A. \(\rm w=-\frac{dF}{dx}\)
B. \(\rm w=-\frac{d^2M}{dx^2}\)
C. \(\rm F=\frac{dM}{dx}\)
Which of the above statements is/are correct ?
A simply supported beam of 4 m span carries a uniformly distributed load all over the span of 20 kN. The maximum bending moment is
When a beam is subjected to a bending moment, the strain in a layer is _____ the distance from the neutral axis.
For a cantilever beam of length 2 m, under load of 1 kN/m, the maximum bending moment is
The bending moment diagram for simply supported beam loaded in its center is
