Slope and deflection of a cantilever beam carrying a moment M at the free end is given by:
ML/EI and ML2/ 2EI
Understanding Cantilever Beam Behavior
A cantilever beam is a structural element fixed at one end and free at the other. When a load, such as a moment, is applied to the free end, the beam bends, resulting in both slope (rotation) and deflection (displacement) along its length. The amount of slope and deflection depends on the applied load, the beam's length (L), its material properties (represented by the modulus of elasticity, E), and its cross-sectional properties (represented by the moment of inertia, I). The product EI is known as the flexural rigidity of the beam.
Calculating Slope at the Free End of a Cantilever Beam
When a cantilever beam is subjected to a concentrated moment M at its free end, the beam rotates at the free end. This rotation is quantified by the slope. The standard formula for the slope ($\theta$) at the free end of a cantilever beam under a moment M is derived using principles of beam deflection theory, often involving integration of the bending moment equation.
The formula for the slope at the free end is:
\(\theta = \frac{ML}{EI}\)
Where:
- M is the applied moment at the free end.
- L is the length of the cantilever beam.
- E is the modulus of elasticity of the beam material.
- I is the moment of inertia of the beam's cross-section.
The units for slope are typically in radians.
Calculating Deflection at the Free End of a Cantilever Beam
In addition to rotating, the free end of the cantilever beam also displaces downwards (or upwards, depending on the direction of the moment). This displacement is the deflection. The maximum deflection typically occurs at the free end for this loading case. The formula for the deflection (y) at the free end is also derived from beam deflection theory.
The formula for the deflection at the free end is:
\(y = \frac{ML^2}{2EI}\)
Where:
- M is the applied moment at the free end.
- L is the length of the cantilever beam.
- E is the modulus of elasticity of the beam material.
- I is the moment of inertia of the beam's cross-section.
The units for deflection are units of length (e.g., meters, millimeters).
Summary of Slope and Deflection Formulas
For a cantilever beam carrying a moment M at the free end, the slope and deflection at the free end are given respectively as:
- Slope: \(\frac{ML}{EI}\)
- Deflection: \(\frac{ML^2}{2EI}\)
The question asks for the slope and deflection in that specific order.
Matching with Options
Let's examine the given options to find the one that lists the slope (\(\frac{ML}{EI}\)) first and the deflection (\(\frac{ML^2}{2EI}\)) second.
- Option 1: ML2/2EI and ML/EI (Deflection first, then Slope - Incorrect order)
- Option 2: ML/EI and M/EI (Slope formula correct, Deflection formula incorrect)
- Option 3: ML/EI and ML2/ 2EI (Slope formula correct, Deflection formula correct, Correct order)
- Option 4: M/EI and ML/EI (Slope formula incorrect, Deflection formula correct - Deflection listed second)
Based on our standard formulas and the requested order (slope then deflection), Option 3 provides the correct values.
Revision Table: Common Cantilever Beam Formulas
| Loading Condition | Maximum Slope | Location of Max Slope | Maximum Deflection | Location of Max Deflection |
|---|---|---|---|---|
| Concentrated Load P at Free End | \(\frac{PL^2}{2EI}\) | Free End | \(\frac{PL^3}{3EI}\) | Free End |
| Uniformly Distributed Load w over full length | \(\frac{wL^3}{6EI}\) | Free End | \(\frac{wL^4}{8EI}\) | Free End |
| Moment M at Free End | \(\frac{ML}{EI}\) | Free End | \(\frac{ML^2}{2EI}\) | Free End |
Additional Information on Beam Deflection Analysis
Understanding beam deflection is crucial in structural engineering to ensure that structures are not only strong enough to carry loads but also stiff enough to avoid excessive deformation. Excessive deflection can lead to various problems, including damage to finishes, discomfort for occupants, and even failure of attached elements.
Methods used to calculate beam deflection include:
- Double Integration Method: This involves integrating the bending moment equation twice to obtain the deflection equation. Boundary conditions (like zero slope and deflection at the fixed end of a cantilever) are used to solve for integration constants.
- Macaulay's Method: A variation of the double integration method particularly useful for beams with multiple concentrated loads or changes in loading.
- Area-Moment Theorems: Geometrical methods that relate the slope and deflection of a beam to the area under the bending moment diagram (divided by EI).
- Castigliano's Theorem: An energy method that uses strain energy to find deflections.
The flexural rigidity (EI) is a key parameter. A higher EI value means the beam is more resistant to bending and will experience less slope and deflection under the same load. E depends on the material (steel, concrete, wood), and I depends on the shape and dimensions of the cross-section (e.g., rectangular, circular, I-beam).
Important Questions from Bending Moment
Which of the following beams is likely to have the point of contraflexure?
The point of contraflexure is the point at which ___________ changes its sign.
The maximum bending moment of the center of laminated spring of span L due to load W is given by-
If a simply supported beam is loaded with point load W at the centre then what is the ratio of bending moment at the support to the bending moment at the centre?
A uniform beam of span l is rigidly fixed at both supports. It carries a uniformly distributed load w per unit length. The bending moment at mid-span is
