The longitudinal stress induced in a thin-walled cylindrical vessel of diameter D, thickness t, under pressure P is
Understanding Longitudinal Stress in Thin Cylinders
Thin-walled cylindrical vessels, like pipes or tanks, when subjected to internal pressure, experience stresses within their walls. There are two primary types of stress: hoop stress (circumferential stress) and longitudinal stress (axial stress).
The question asks specifically about the longitudinal stress induced in a thin-walled cylindrical vessel.
What is Longitudinal Stress?
Longitudinal stress (\(\sigma_L\)) acts along the length of the cylinder. It is caused by the internal pressure pushing against the ends of the cylinder, trying to pull it apart longitudinally.
Derivation of Longitudinal Stress Formula
To derive the formula for longitudinal stress, we consider the forces acting on one end of the cylinder. Imagine cutting the cylinder across its length. The internal pressure acts on the circular area of the end cap, and the longitudinal stress in the cylinder wall resists this force.
Let:
- \(p\) be the internal pressure
- \(d\) be the internal diameter of the cylinder
- \(t\) be the thickness of the cylinder wall
The force exerted by the internal pressure on the end cap is the pressure multiplied by the area of the end cap. This area is a circle with diameter \(d\).
Force due to pressure (\(F_P\)) \( = \) Pressure \( \times \) Area
\(F_P = p \times \frac{{\pi d^2}}{4}\)
This force is resisted by the stress in the cylindrical wall acting around the circumference of the cut section. The area of the metal wall resisting this force is approximately the circumference (\(\pi d\)) multiplied by the wall thickness (\(t\)).
Resisting force due to longitudinal stress (\(F_R\)) \( = \) Longitudinal Stress \( \times \) Resisting Area
\(F_R = \sigma_L \times (\pi d t)\)
Note: For a thin-walled cylinder, the thickness \(t\) is much smaller than the diameter \(d\) (\(d/t \ge 10\) or \(15\)), so we can use the mean diameter or internal diameter for the area calculation without significant error. Here, we use the internal diameter \(d\) for simplicity, which is common in introductory treatments.
For equilibrium, the force due to pressure must be balanced by the resisting force due to longitudinal stress:
\(F_P = F_R\)
\(p \times \frac{{\pi d^2}}{4} = \sigma_L \times (\pi d t)\)
Now, we solve for the longitudinal stress (\(\sigma_L\)):
\(\sigma_L = \frac{{p \times \frac{{\pi d^2}}{4}}}{{\pi d t}}\)
\(\sigma_L = \frac{{p \pi d^2}}{{4 \pi d t}}\)
\(\sigma_L = \frac{{p d}}{{4 t}}\)
Comparing with Options
The derived formula for longitudinal stress is \(\frac{{pd}}{{4t}}\). Let's compare this with the given options:
- \(\frac{{pd}}{t}\)
- \(\frac{{pd}}{{2t}}\)
- \(\frac{{pd}}{{4t}}\)
- \(\frac{{pd}}{{8t}}\)
Our derived formula matches option 3.
Note: Option 2, \(\frac{{pd}}{{2t}}\), represents the hoop stress (or circumferential stress) in a thin-walled cylinder, which is typically twice the longitudinal stress.
Therefore, the longitudinal stress induced in a thin-walled cylindrical vessel of diameter D (or d), thickness t, under pressure P (or p) is \(\frac{{pd}}{{4t}}\).
| Stress Type | Formula | Direction |
|---|---|---|
| Hoop Stress (\(\sigma_H\)) | \(\frac{{pd}}{{2t}}\) | Circumferential (around the cylinder) |
| Longitudinal Stress (\(\sigma_L\)) | \(\frac{{pd}}{{4t}}\) | Axial (along the length) |
Revision Table: Thin Cylinder Stresses
Here is a quick summary of the key stresses in thin-walled cylindrical vessels:
| Parameter | Description | Value |
|---|---|---|
| Internal Pressure | Pressure inside the vessel | \(p\) |
| Internal Diameter | Diameter of the vessel | \(d\) |
| Wall Thickness | Thickness of the vessel wall | \(t\) |
| Hoop Stress (\(\sigma_H\)) | Stress acting tangentially | \(\frac{{pd}}{{2t}}\) |
| Longitudinal Stress (\(\sigma_L\)) | Stress acting axially | \(\frac{{pd}}{{4t}}\) |
Remember that the hoop stress is generally the larger of the two stresses and is the primary consideration for failure in cylindrical pressure vessels.
Additional Information: Assumptions and Applications
The formulas for thin-walled pressure vessels are based on certain assumptions:
- The vessel is thin-walled (typically \(d/t \ge 10\) or \(15\)).
- The pressure is uniformly distributed.
- The material is isotropic and homogeneous.
- The stresses are uniformly distributed across the wall thickness.
- End effects are often neglected (away from the ends or joints).
These formulas are fundamental in the design and analysis of pressure vessels, pipelines, boilers, and other structures containing fluids or gases under pressure.
Understanding both longitudinal and hoop stresses is crucial for determining the required wall thickness for a given pressure and material strength, ensuring the vessel operates safely.
Important Questions from Analysis of Thin Cylinder
If the thin cylindrical shell whose diameter is 'd' is subjected to an internal pressure 'p', then the ratio of longitudinal stress to the hoop stress is-
If the thickness of the wall of the cylindrical vessel is less than ________ of its internal diameter, the cylindrical vessel is known as a thin cylinder.
A seamless pipe is to carry a fluid under a pressure of 2 N/mm2. The thickness of the cylinder is 10 mm. Calculate the diameter of the pipe if the maximum stress allowed is 100 N/mm2.
The circumferential stress is given by:
In a thin cylinder, the hoop stress is:
