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?
0
This question asks about the ratio of bending moments in a specific type of beam under a particular loading condition. We need to determine the bending moment at two locations: the support and the centre, for a simply supported beam carrying a point load at its centre.
Understanding Simply Supported Beams and Bending Moment
A simply supported beam is a beam that is supported at both ends. These supports typically allow rotation but prevent vertical displacement. When a load is applied to the beam, it causes internal stresses, including bending stresses. The bending moment is a measure of the internal resistance of the beam to bending caused by external forces.
For a simply supported beam, the bending moment at the supports is generally zero, provided there are no external moments applied at the supports and no overhanging sections with loads.
Calculating Bending Moment for a Central Point Load
Let's consider a simply supported beam of length \(L\) with a point load \(W\) applied exactly at its centre. The supports are at the ends, let's call them A and B.
First, we find the support reactions. Due to the symmetry of the beam and the load, the reactions at both supports are equal.
Sum of vertical forces = 0:
\(R_A + R_B - W = 0\)
Due to symmetry, \(R_A = R_B\). Therefore,
\(2R_A = W \implies R_A = W/2\)
And \(R_B = W/2\).
Bending Moment at the Support
Let's calculate the bending moment at support A (left support). The bending moment at any point is the sum of the moments of all forces to one side of that point. Considering forces to the left of support A, there are no forces. Thus, the bending moment at support A is:
\(M_A = 0\)
Similarly, considering forces to the right of support B (right support), there are no forces. Thus, the bending moment at support B is:
\(M_B = 0\)
So, the bending moment at the support is always zero for a simply supported beam with standard loading between the supports.
Bending Moment at the Centre
The centre of the beam is at a distance \(L/2\) from either support. Let's calculate the bending moment at the centre by considering the forces to the left of the centre point.
The forces to the left of the centre are the reaction \(R_A\) acting upwards at A.
Bending moment at the centre (\(M_{centre}\)) = \(R_A \times (L/2)\)
\(M_{centre} = (W/2) \times (L/2)\)
\(M_{centre} = \frac{WL}{4}\)
This is the maximum bending moment for this loading condition.
Calculating the Ratio of Bending Moments
The question asks for the ratio of the bending moment at the support to the bending moment at the centre.
Ratio = \(\frac{\text{Bending Moment at Support}}{\text{Bending Moment at Centre}}\)
Ratio = \(\frac{0}{\frac{WL}{4}}\)
Ratio = 0
Therefore, the ratio of the bending moment at the support to the bending moment at the centre is 0.
Summary of Bending Moments
| Location | Bending Moment Formula | Value for Simply Supported Beam with Central Point Load |
|---|---|---|
| Support (ends) | \(M_{support}\) | \(0\) |
| Centre (mid-span) | \(M_{centre}\) | \(\frac{WL}{4}\) |
Based on these values, the ratio is \(0 / (WL/4) = 0\).
Revision Table: Simply Supported Beam Analysis
| Concept | Description |
|---|---|
| Simply Supported Beam | Beam with pinned or roller supports at ends. |
| Point Load | Load concentrated at a single point. |
| Support Reactions | Forces exerted by supports on the beam, balancing external loads. |
| Bending Moment | Internal resisting moment against bending. Zero at supports for standard simply supported beams. Maximum at the point of maximum deflection or where shear force is zero. |
| Central Point Load | Load applied at the midpoint of the beam length. |
Additional Information: Bending Moment Diagrams
Understanding bending moment diagrams is crucial for analyzing beams. For a simply supported beam with a central point load:
- The bending moment is zero at both supports.
- The bending moment increases linearly from the supports towards the centre.
- The bending moment reaches its maximum value (\(WL/4\)) at the exact centre where the point load is applied.
- The bending moment diagram is a triangle with its base along the beam length and its apex at the centre.
This triangular shape visually confirms that the moment at the ends (supports) is zero, while the moment at the centre is a positive maximum value. The ratio of the moment at the support to the moment at the centre is therefore \(0 / (WL/4)\), which is 0.
Important Questions from Bending Moment
Slope and deflection of a cantilever beam carrying a moment M at the free end is given by:
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-
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
