Understanding Stress from Sudden vs. Gradual Loads

When a load is applied to a material, it causes stress and deformation. The amount of stress developed depends not only on the magnitude of the load but also on how it is applied. There is a significant difference between applying a load gradually and applying it suddenly.

Stress Due to Gradually Applied Load

A gradually applied load starts from zero and increases slowly to its maximum value (let's call it P). As the load increases, the deformation also increases proportionally up to the elastic limit. The stress developed is directly proportional to the load.

Let:

• P be the maximum load applied.
• A be the cross-sectional area of the material.

The stress ($\sigma_{gradual}$) developed due to a gradually applied load is given by:

\(\sigma_{gradual} = \frac{P}{A}\)

Under a gradually applied load, the work done by the load is equal to the strain energy stored in the material. Since the load increases linearly from 0 to P, the work done is the average load multiplied by the total deformation ($\delta L_{gradual}$).

Work done by gradual load = \(\frac{1}{2} P \times \delta L_{gradual}\)

Strain energy stored = \(\frac{1}{2} \sigma_{gradual} \times \epsilon_{gradual} \times V\)

Using \(\sigma = E\epsilon\) and \(V = A \times L\):

Strain energy = \(\frac{1}{2} \frac{\sigma_{gradual}^2}{E} \times (A \times L)\)

Stress Due to Suddenly Applied Load

A suddenly applied load is one that is applied instantaneously and remains constant throughout the deformation process. In this case, the load P is applied at its full value from the beginning.

When the load P is applied suddenly, it causes a deformation ($\delta L_{sudden}$). The work done by this sudden load is simply the load P multiplied by the deformation $\delta L_{sudden}$ because the load P is acting at its full value over the entire deformation.

Work done by sudden load = \(P \times \delta L_{sudden}\)

This work done is converted into strain energy stored in the material. The strain energy stored under sudden loading is given by:

Strain energy stored = \(\frac{1}{2} \sigma_{sudden} \times \epsilon_{sudden} \times V\)

Using \(\sigma = E\epsilon\) and \(V = A \times L\):

Strain energy = \(\frac{1}{2} \frac{\sigma_{sudden}^2}{E} \times (A \times L)\)

Equating work done and strain energy stored for the suddenly applied load:

\(P \times \delta L_{sudden} = \frac{1}{2} \frac{\sigma_{sudden}^2}{E} \times (A \times L)\)

We also know that \(\delta L_{sudden} = \epsilon_{sudden} \times L = \frac{\sigma_{sudden}}{E} \times L\).

Substitute \(\delta L_{sudden}\) in the equation:

\(P \times \left(\frac{\sigma_{sudden}}{E} \times L\right) = \frac{1}{2} \frac{\sigma_{sudden}^2}{E} \times (A \times L)\)

Now, we can simplify this equation. Cancel out the common terms \(\frac{L}{E}\) from both sides:

\(P \times \sigma_{sudden} = \frac{1}{2} \sigma_{sudden}^2 \times A\)

Rearrange the terms to solve for \(\sigma_{sudden}\):

\(\sigma_{sudden} = \frac{2 P \times \sigma_{sudden}}{\sigma_{sudden} \times A}\)

Assuming \(\sigma_{sudden} \neq 0\), we can cancel \(\sigma_{sudden}\) from both sides:

\(\sigma_{sudden} = \frac{2P}{A}\)

Comparison of Stress Values

We found that:

• Stress due to gradually applied load: \(\sigma_{gradual} = \frac{P}{A}\)
• Stress due to suddenly applied load: \(\sigma_{sudden} = \frac{2P}{A}\)

Now, let's compare the two stresses by finding their ratio:

Ratio = \(\frac{\sigma_{sudden}}{\sigma_{gradual}} = \frac{\frac{2P}{A}}{\frac{P}{A}}\)

Ratio = \(\frac{2P}{A} \times \frac{A}{P}\)

Ratio = 2

This means that the stress developed when a load is applied suddenly is 2 times the stress developed when the same load is applied gradually.

Conclusion on Load Application and Stress

The stress developed due to application of a load suddenly is 2 times that due to same load being applied gradually. This significant difference highlights the importance of considering the nature of load application (static vs. dynamic) in structural design to prevent failure.

Revision Table: Load Application Summary

Additional Information on Stress and Strain Energy

This concept is fundamental in understanding impact loads or dynamic loads. While a suddenly applied load is a simplified model of dynamic loading, it provides a crucial insight into the effect of load application speed.

• Strain Energy: It is the energy stored in a material due to deformation caused by applied stress. It is often represented as the area under the stress-strain curve.
• Impact Load: This is a load applied with some initial velocity or from a certain height, causing deformation and stress. The stress developed due to impact is generally even higher than that due to a suddenly applied load and depends on the impact velocity or height of fall.
• Elastic Limit: The analysis above assumes that the material remains within its elastic limit, meaning it returns to its original shape after the load is removed. If the stress exceeds the elastic limit, plastic deformation occurs.
• Dynamic Loading: This involves loads that change over time. Suddenly applied load is a specific type of dynamic loading where the load is applied very quickly.

Understanding the difference between static (gradual) and dynamic (sudden or impact) loading is essential for safe and efficient structural design, especially in applications where loads can be applied rapidly.