A thin-walled steel cylinder of 1 m diameter is subjected to an internal pressure of 1.5 MPa. Using a safety factor of 1.5 and a yield stress in simple tension of 250 MPa, calculate the wall thickness on the basis of Tresca’s, Von Mises and Rankine’s yield criteria. It may be assumed that the radial stress in the wall is negligible
To calculate the wall thickness of the thin-walled steel cylinder based on Tresca's, Von Mises, and Rankine's yield criteria, we need to follow these steps:
Step 1: Understanding the Given Parameters
- Internal pressure, P = 1.5 MPa
- Safety factor, SF = 1.5
- Yield stress in simple tension, σ_yield = 250 MPa
- Diameter of the cylinder, D = 1 m
- Radial stress, σ_r = negligible
Step 2: Calculating the Hoop Stress (Circumferential Stress)
The hoop stress (σ_h) is the primary stress acting tangentially on the cylinder's wall due to internal pressure. This can be calculated using the following formula:
σ_h = P × D / (2 × t)
Where:
- σ_h = hoop stress
- P = internal pressure
- D = diameter of cylinder
- t = wall thickness
Step 3: Tresca's Yield Criterion
Tresca's yield criterion states that the maximum shear stress should not exceed the yield stress. In this case, as the radial stress is negligible, we only need to consider the hoop stress (σ_h). Therefore:
Tresca's Yield Criterion:
Max. Shear Stress = 0.5 × (σ_1 - σ_2) ≤ σ_yield
In this case, σ_1 = σ_h and σ_2 = 0 (as radial stress is negligible).
0.5 × σ_h ≤ σ_yield
Step 4: Calculating Wall Thickness using Tresca's Yield Criterion
Rearranging the equation from Step 3, we can solve for t (wall thickness):
t = 2 × (σ_h / σ_yield)
Step 5: Von Mises Yield Criterion
Von Mises' yield criterion involves the concept of equivalent stress (σ_eq) which is a measure of combined effects of different normal and shear stresses. The equation for Von Mises' yield criterion is as follows:
Von Mises Yield Criterion:
σ_eq = √(σ_1^2 - σ_1 × σ_2 + σ_2^2) ≤ σ_yield
In this case, we can simplify the equation as σ_1 = σ_h and σ_2 = 0 (as radial stress is negligible):
σ_eq = σ_h ≤ σ_yield
Step 6: Calculating Wall Thickness using Von Mises Yield Criterion
Since σ_h is already provided, we can directly compare it to the yield stress (σ_yield).
Step 7: Rankine's Yield Criterion
Rankine's yield criterion states that the maximum normal stress should not exceed the yield stress. In this case, we need to consider both the hoop stress (σ_h) and the radial stress (σ_r). As the radial stress is negligible, we can ignore it. Therefore:
Rankine's Yield Criterion:
Max. Normal Stress = σ_1 ≤ σ_yield
In this case, σ_1 = σ_h
Step 8: Calculating Wall Thickness using Rankine's Yield Criterion
Since the hoop stress (σ_h) is already provided, we can directly compare it to the yield stress (σ_yield).
By following these steps, you can calculate the wall thickness based on Tresca's, Von Mises', and Rankine's yield criteria for the given thin-walled steel cylinder.