The maximum bending moment of the center of laminated spring of span L due to load W is given by-
WL/4
Calculating Maximum Bending Moment in a Laminated Spring
A laminated spring, often used in vehicle suspension, acts like a simply supported beam when subjected to a load. The question asks for the maximum bending moment at the center of such a spring with a span \(L\) under a load \(W\) applied at its center.
Let's analyze the setup:
- The laminated spring is supported at its ends, similar to a simply supported beam.
- A concentrated load \(W\) is applied exactly at the center of the span.
- The span of the spring is \(L\).
For a simply supported beam with a concentrated load \(W\) at the center, the reactions at the supports are equal due to symmetry, each being \(W/2\).
Consider a section of the beam at a distance \(x\) from one support (say, the left support). The bending moment \(M(x)\) at this section is the sum of the moments of all forces to one side of the section.
For \(0 \le x \le L/2\), the bending moment \(M(x)\) is caused by the reaction force at the left support:
\[ M(x) = R_{left} \times x = \frac{W}{2} \times x \]The bending moment is zero at the supports (\(x=0\), \(x=L\)) and increases linearly from the supports towards the center.
The maximum bending moment occurs at the point where the shear force is zero, which is at the center of the span (\(x = L/2\)) for this loading condition.
Substitute \(x = L/2\) into the bending moment equation:
\[ M_{max} = M\left(\frac{L}{2}\right) = \frac{W}{2} \times \frac{L}{2} = \frac{WL}{4} \]So, the maximum bending moment at the center of the laminated spring (acting as a simply supported beam with central load) is \(\frac{WL}{4}\).
Let's compare this result with the given options:
- Option 1: \(\frac{WL}{4}\)
- Option 2: \(WL\)
- Option 3: \(\frac{WL}{2}\)
- Option 4: \(\frac{WL}{6}\)
Our calculated maximum bending moment matches Option 1.
| Location | Bending Moment Formula | Value at Center (\(x=L/2\)) |
|---|---|---|
| At a distance \(x\) from support (for \(0 \le x \le L/2\)) | \(M(x) = \frac{W}{2} \times x\) | N/A |
| At the center (\(x=L/2\)) | \(M_{max} = M\left(\frac{L}{2}\right)\) | \(\frac{WL}{4}\) |
Therefore, the maximum bending moment at the center of the laminated spring under a central load \(W\) is \(\frac{WL}{4}\).
Revision Table: Laminated Spring Bending Moment
| Concept | Description | Formula/Value |
|---|---|---|
| Laminated Spring Model | Simply supported beam | - |
| Load Applied | Concentrated load \(W\) at center | \(W\) |
| Span | Distance between supports | \(L\) |
| Support Reactions | Equal reactions at each end | \(W/2\) |
| Location of Maximum Bending Moment | At the center of the span | \(x = L/2\) |
| Maximum Bending Moment Value | Calculated at the center | \(\frac{WL}{4}\) |
Additional Information: Beam Bending Concepts
Understanding beam bending is crucial in structural and mechanical engineering. Here are a few key concepts related to this problem:
- Bending Moment: It is the internal moment that resists the bending of a beam caused by external forces. It varies along the length of the beam.
- Simply Supported Beam: A beam supported by a pin support at one end and a roller support at the other. This allows for rotation at both ends but prevents vertical movement.
- Shear Force: The internal force acting perpendicular to the beam's axis, which resists the tendency of one part of the beam to slide past another.
- Relationship between Shear Force and Bending Moment: The bending moment at any point is the integral of the shear force diagram up to that point. The maximum bending moment occurs where the shear force is zero or changes sign.
- Laminated Spring: A spring made of several curved plates (leaves) of varying lengths, stacked together. They are designed to absorb shock and store energy, commonly used in vehicle suspension systems. Although complex internally, for overall bending analysis, it's often approximated as a beam.
Calculating bending moments is essential for designing beams and springs to ensure they can withstand the applied loads without failure.
Important Questions from Bending Moment
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The point of contraflexure is the point at which ___________ changes its sign.
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
