Problem 20 - An Introduction to the Mechanics of Solids in SI Units

Question

A long, thin-walled cylindrical tank of length \(L\) just fits between two rigid end walls when there is no pressure in the tank. Estimate the force exerted on the rigid walls by the tank when the pressure in the tank is \(p\) and the material of which the tank is made follows Hooke's law.

Step-by-Step Solution

Answer

The force exerted by the tank on the rigid walls is given by \(F = p \cdot \pi r^2\), where \(F\) is the Force, \(p\) is the pressure in the tank and \(r\) is the radius of the cylindrical tank.

Step 1: Derive relationship of Pressure and Force

Since pressure is force per unit area, we can express force as the product of pressure and area. In this case, the area is the cross-sectional area of the cylinder, which can be computed using the formula for the area of a circle, \(A = \pi r^2\), where \(r\) is the radius of the cylinder. Therefore, the force exerted by the pressure on the ends of the tank is given by \(F = p \cdot A = p \cdot \pi r^2\).

Step 2: Apply Hooke's Law

Next, Hooke's law, which states that \(F = k \cdot \Delta L\), where \(F\) is the force applied, \(k\) is the spring constant, and \(\Delta L\) is the change in length (in this case, the change in length of the cylinder as a result of the pressure). From step 1, we can equate \(F = p \cdot \pi r^2\) to \(k \cdot \Delta L\), and solve for \(\Delta L\).

Step 3: Compute force exerted on the walls

Finally, the force exerted on the walls by the tank is the change in length multiplied by the spring constant, \(\Delta L \cdot k\). This can be replaced by \(F = p \cdot \pi r^2\) obtained from step 1, as the force exerted on the walls by the tank when the pressure is \(p\). Hence, the force exerted by the tank on the rigid walls is equal to the internal pressure times the cross-sectional area of the tank.

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in the mechanics of solids. It relates to the behavior of materials within the elastic range, meaning they return to their original shape after the force is removed. This principle states that the force required to extend or compress a spring by some distance is proportional to that distance. Mathematically, it is expressed as:\[ F = k \cdot \Delta L \]where:

• \( F \) is the force applied,
• \( k \) is the spring constant, representing the material's stiffness, and
• \( \Delta L \) is the change in length.

In the context of a cylindrical pressure vessel, if the material behaves according to Hooke's Law, it means that under a certain internal pressure, the vessel will expand. However, this expansion will be proportional and directly related to the amount of pressure applied. This ensures predictable behavior useful in engineering calculations.
Understandably, if the material doesn't have elastic properties, Hooke's Law wouldn't apply, and the object might not return to its original form after the pressure is relieved.

Cylindrical Pressure Vessels

Cylindrical pressure vessels are widely used in various industries for storing gases or liquids at a certain pressure. The cylinder structure supports pressure exerted on its walls both circumferentially and longitudinally. Understanding their properties is crucial for safe design and operation.
The main surface area that feels the impact of the internal pressure comprises two parts:

• The ends of the cylinder, where the pressure acts perpendicularly, contributing to axial stress.
• The cylindrical wall, where the pressure acts circumferentially, contributing to hoop stress.

Both stresses are crucial to evaluate when analyzing a vessel's strength. When internal pressure is applied, stress is generated in the material, which creates a force on the ends of the vessel. The pressure multiplied by the cross-sectional area (\( \pi r^2 \)) gives the total force.
To ensure safety, the vessel must be able to withstand this force without yielding or failing. Engineers use equations derived from mechanics of solids to calculate stresses and determine the necessary wall thickness and material type to safely handle the internal pressure.

Stress and Strain Analysis

Stress and strain analysis is an essential part of assessing materials used in constructing pressure vessels. This analysis helps predict how a material will react under different forces.Stress is the internal force experienced by the material per unit area, usually measured in Pascals (Pa). It can be calculated using:\[ \text{Stress} = \frac{F}{A} \]where:

• \( F \) is the force applied, and
• \( A \) is the cross-sectional area.

Strain, on the other hand, is the deformation or displacement experienced by the material, typically a ratio without units. It is calculated as:\[ \text{Strain} = \frac{\Delta L}{L} \]where:

• \( \Delta L \) is the change in length, and
• \( L \) is the original length.

In designing pressure vessels, engineers use stress and strain analysis to ensure that materials operate within their elastic limit as predicted by Hooke's Law. By analyzing these factors, safety margins and design specifications can be determined to prevent material failure.