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.
1 m
Understanding Stress in Seamless Pipes
When a fluid flows through a pipe under pressure, the pipe wall experiences stress. For a thin-walled cylinder like a pipe carrying internal pressure, two principal stresses are induced: circumferential or hoop stress and longitudinal or axial stress. The hoop stress is typically twice the longitudinal stress and is often the critical stress to consider in pipe design, especially for determining the minimum wall thickness or maximum allowable diameter.
Hoop Stress Calculation for Internal Pressure
The hoop stress ($\sigma_h$) developed in a thin-walled cylinder due to internal pressure ($p$) is given by the formula:
$$ \sigma_h = \frac{pd}{2t} $$
Where:
- \( \sigma_h \) is the hoop stress
- \( p \) is the internal pressure
- \( d \) is the internal diameter of the pipe
- \( t \) is the thickness of the pipe wall
Applying Given Values to Find Pipe Diameter
We are given the following information:
- Internal pressure, \( p = 2 \, \text{N/mm}^2 \)
- Thickness of the pipe wall, \( t = 10 \, \text{mm} \)
- Maximum allowable stress (which is the hoop stress), \( \sigma_h = 100 \, \text{N/mm}^2 \)
We need to calculate the diameter of the pipe, \( d \). We can rearrange the hoop stress formula to solve for \( d \):
$$ d = \frac{2 \sigma_h t}{p} $$
Step-by-Step Calculation
Now, substitute the given values into the rearranged formula:
$$ d = \frac{2 \times (100 \, \text{N/mm}^2) \times (10 \, \text{mm})}{2 \, \text{N/mm}^2} $$
Calculate the numerator:
$$ 2 \times 100 \times 10 = 2000 \, \text{N/mm} $$
Now divide by the pressure:
$$ d = \frac{2000 \, \text{N/mm}}{2 \, \text{N/mm}^2} $$
$$ d = 1000 \, \text{mm} $$
Converting Diameter to Meters
The calculated diameter is in millimeters. The options are given in meters. We need to convert millimeters to meters. Recall that 1 meter = 1000 millimeters.
$$ d = 1000 \, \text{mm} \times \frac{1 \, \text{m}}{1000 \, \text{mm}} $$
$$ d = 1 \, \text{m} $$
Comparing with Options
The calculated diameter of the pipe is 1 meter. Let's compare this with the given options:
| Option | Diameter (m) | Diameter (mm) |
|---|---|---|
| 1 | 0.8 m | 800 mm |
| 2 | 0.9 m | 900 mm |
| 3 | 1.1 m | 1100 mm |
| 4 | 1 m | 1000 mm |
Our calculated diameter of 1 m matches Option 4.
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.
The circumferential stress is given by:
In a thin cylinder, the hoop stress is:
