Problem 21 - An Introduction to the Mechanics of Solids in SI Units

Question

A long, thin-walled cylinder tank just fits in a rigid cylindrical cavity when there is no pressure in the tank. Estimate the tangential stress in the cylindrical-tank wall 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 tangential or hoop stress in the cylindrical-tank wall when the internal pressure is \( p \), is given by \( \sigma_t = p \cdot r / t \). Without given values for the radius \( r \) and thickness \( t \), the stress cannot be numerically calculated but it's directly proportional to the internal pressure.

Step 1: Understand the concept of hoop stress

Hoop stress, also known as tangential stress or circumferential stress, is a type of stress that occurs in thin-walled pressure vessels. The equation for hoop stress is given by: \[ \sigma_t = p \cdot r / t \], where \( \sigma_t \) is the hoop (tangential) stress, \( p \) is the pressure, \( r \) is the radius of the cylinder, and \( t \) is the thickness of the cylinder wall.

Step 2: Assumptions

Given that the cylinder is thin-walled, an assumption is made that the wall thickness (t) is small relative to the radius (r). Additionally, the material obeys Hooke's law, which implies that the strain in the cylinder wall will be directly proportional to the applied stress, up to the point of the material's elastic limit.

Step 3: Estimate the stress

To solve the problem, simply substitute the given internal pressure (p) into the hoop stress formula. Since the cylinder fits perfectly within the rigid cavity, the wall compresses inward triggering stress due to the pressure. However, no values are provided for the radius and wall thickness, therefore the hoop stress can be expressed directly in terms of the given pressure as: \( \sigma_t = p \cdot r / t \)

Key Concepts

Understanding Thin-Walled Pressure Vessels

Thin-walled pressure vessels are structures that contain fluids or gases under pressure, such as tanks or cylinders. The walls of these vessels are considered thin if their thickness is much smaller than the other dimensions, like the radius or diameter. This thinness allows for specific simplifications in stress analysis.
This is because the stress distribution through a thin wall is relatively uniform, making calculations easier. Engineers often use these assumptions to simplify the analysis using formulas that apply specifically to thin-walled constructions. One common application of this assumption is in determining the hoop stress acting on the walls of a pressure vessel. The thin-walled assumption is valid when the thickness is less than about 1/10th to 1/15th of the radius. If this condition is met, it simplifies the stress calculations greatly and avoids the need for more complex, thick-walled pressure vessel equations.

Hooke's Law and Its Role in Material Behavior

Hooke's law is a principle that describes how solid materials stretch and compress. According to Hooke's law, the amount of deformation or strain experienced by an elastic object is directly proportional to the applied stress, as long as the material's elastic limit is not exceeded. This is expressed in the formula: \( ext{stress} = E imes ext{strain} \),where \( E \) represents the modulus of elasticity of the material.
When applied to thin-walled pressure vessels, it ensures that the deformation of the cylindrical tank wall will be proportionate to the stress imposed by the internal pressure. The relation helps in predicting how much a structure will deform under a certain pressure and thus is critical for ensuring safety and structural integrity.
Knowing that a material behaves according to Hooke's law allows us to predict its deformation under operational conditions, ensuring that the design prevents material failure by maintaining stresses within safe bounds.

Estimating Stress in Thin-Walled Structures

Stress estimation within thin-walled vessels involves calculating the internal stresses generated by the pressure exerted on the walls of the structure. One vital form of stress to consider is hoop stress, which acts tangentially to the circumference of cylindrical vessels. The equation used to determine hoop stress is:\[ \sigma_t = \frac{p \cdot r}{t} \]where \( \sigma_t \) is the hoop stress, \( p \) is the internal pressure, \( r \) is the radius, and \( t \) is the wall thickness.
The estimation helps predict how the structure will sustain the internal pressures and assists in the design and verification process by confirming that stresses remain within acceptable limits. The simplicity of the formula arises from the assumption that the vessel is thin-walled, allowing us to ignore more complex stress distributions present in thicker walls.
When designing or evaluating thin-walled cylindrical tanks, accurately estimating these stresses is key to ensuring functionality without risking the structural integrity of the vessel.