The circumferential stress is given by:
σ = (pd)/2t
Understanding Circumferential Stress
Circumferential stress, also known as hoop stress, is a stress that occurs in a cylindrical shell (like a pipe or pressure vessel) when it is subjected to internal pressure. This stress acts tangentially around the circumference of the cylinder.
Imagine cutting the cylinder in half lengthwise. The internal pressure acting on the cut surface is resisted by the stress acting along the walls at the cut edges. This stress is the circumferential stress.
For a thin-walled cylindrical shell, the circumferential stress (σc or σh) due to internal pressure can be calculated using a simplified formula based on equilibrium principles.
Derivation of the Circumferential Stress Formula
Consider a thin cylinder with:
- Internal pressure = \(p\)
- Diameter = \(d\)
- Thickness = \(t\)
- Length = \(L\) (consider a unit length, \(L=1\))
Imagine a longitudinal section cutting through the diameter. The total force exerted by the internal pressure on this cut section over a length \(L\) is:
\(F_{pressure} = p \times (\text{projected area})\)
\(F_{pressure} = p \times (d \times L)\)
This force is resisted by the stress acting on the cylinder wall sections at the cut. The resisting force is due to the circumferential stress (σ) acting on the area of the two cut surfaces:
\(F_{resistance} = \sigma \times (\text{area of wall sections})\)
\(F_{resistance} = \sigma \times (t \times L + t \times L)\)
\(F_{resistance} = \sigma \times (2tL)\)
For equilibrium, the pressure force must be balanced by the resisting force:
\(F_{pressure} = F_{resistance}\)
\(p \times d \times L = \sigma \times 2tL\)
Solving for σ:
\(\sigma = \frac{p \times d \times L}{2tL}\)
The length \(L\) cancels out, leaving the formula for circumferential stress:
\(\sigma = \frac{pd}{2t}\)
Formula and Terms
The formula for circumferential stress (σ) in a thin-walled cylinder subjected to internal pressure is:
\(\sigma = \frac{pd}{2t}\)
Where:
- \(\sigma\) is the circumferential stress (hoop stress)
- \(p\) is the internal pressure
- \(d\) is the internal diameter of the cylinder
- \(t\) is the wall thickness of the cylinder
This formula is widely used for thin cylinders, typically when the ratio of diameter to thickness (\(d/t\)) is greater than or equal to 20.
Comparing with Options
Let's look at the given options and compare them with the derived formula:
- \(\sigma = \frac{pd}{2t}\)
- \(\sigma = \frac{pd}{t}\)
- \(\sigma = \frac{pd}{3t}\)
- \(\sigma = \frac{pd}{4t}\)
Our derived formula \(\sigma = \frac{pd}{2t}\) matches option 1.
Option 2, \(\sigma = \frac{pd}{t}\), represents the longitudinal stress in a thin cylinder, not the circumferential stress. The longitudinal stress is typically half the circumferential stress (\(\sigma_{longitudinal} = \frac{pd}{4t}\) for closed ends, which is derived differently).
Options 3 and 4 do not represent standard formulas for stresses in thin-walled cylinders under internal pressure.
Therefore, the correct formula for circumferential stress is \(\sigma = \frac{pd}{2t}\).
Important Questions from Analysis of Thin Cylinder
The longitudinal stress induced in a thin-walled cylindrical vessel of diameter D, thickness t, under pressure P is
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.
In a thin cylinder, the hoop stress is:
