A compressed air tank is supported by two cradles as shown in the figure. The cradles don’t exert any longitudinal force on the tank and stresses occurred on the tank is only due to pressure of compressed air inside the tank. The cylindrical body of the tank is manufactured from 10 mm steel plate (Material ASTM A204 Steel) by butt welding along a helix which forms an angle of 30° with the transverse plane. The end caps are spherical and have a thickness of 6mm. The outer diameter of the vessel is 0.7 m. For an internal gage pressure of 1.5 MPa, determine:

a) Principal and maximum shear stress on the spherical end caps

b) Longitudinal and tangential stress (principal stresses) on the cylindrical body

c) Material yield criteria for stresses occurred on spherical cap and cylindrical section. Design factor is given as 5.

d) Normal stress perpendicular to the weld and shear stress parallel to the weld.

**Step 1:** Write down input parameters which are defined in sample example
including material properties.

INPUT PROPERTIES SUMMARY | ||

Parameter | Value | Unit |

Vessel outer radius (r_{o}) |
0.35 | m |

Vessel thickness(end caps) (t_{s}) |
6 | mm |

Vessel thickness (cylindrical body) (t_{c}) |
10 | mm |

Vessel inner radius (end caps) (r_{si}) |
0.344 | m |

Vessel inner radius (cylindrical body) (r_{ci}) |
0.340 | m |

Gage pressure (p_{g}) |
1.5 | MPa |

Helix angle (θ) | 30 | deg |

Design factor (n_{d}) |
5 | --- |

Yield Strength (A204 Steel) (Sy) | 275 | MPa |

Elastic modulus(A204 Steel) (E) | 200 | GPa |

Poisson's ratio(A204 Steel) (v) | 0.29 | --- |

Elongation at break(A204 Steel) (ε_{brk}) |
21% | --- |

**Step 2 :** Visit "
Thin Walled Pressure Vessel Stress Calculations" page to calculate
principal and maximum shear stresses on spherical end caps. Calculate principal stresses and maximum shear stress on the
spherical end caps by using the values summarized in step 1 and given below.

INPUT PARAMETERS TO CALCULATE PRINCIPAL AND MAXIMUM SHEAR STRESSES ON THE SPHERICAL END CAPS | ||

Parameter | Value | Unit |

Gage Pressure of Fluid (p_{g}) |
1.5 | MPa |

Vessel Wall Thickness (t) | 6 | mm |

Vessel Inside Radius (r) | 0.344 | m |

For spherical end caps, thin-walled assumption is ok so we can use results. Required results for clause a) are summarized in the following table.

PRINCIPAL AND MAXIMUM SHEAR STRESSES ON THE SPHERICAL CAP | ||

Parameter | Value | Unit |

Principal stress 1 (Tangential direction) (σ_{1}) |
43 | MPa |

Principal stress 2 (Longitudinal direction) (
σ_{2}) |
43 | |

Maximum shear stress (τ_{max}) |
21.5 |

**Step 3:** Calculate principal stresses and maximum shear stress on the cylindrical
body by using the values summarized in step 1 with "Stresses in Thin-Walled Pressure
Vessel" calculator.

INPUT PARAMETERS TO CALCULATE PRINCIPAL AND MAXIMUM SHEAR STRESSES ON THE CYLINDRICAL BODY | ||

Parameter | Value | Unit |

Gage Pressure of Fluid (p_{g}) |
1.5 | MPa |

Vessel Wall Thickness (t) | 10 | mm |

Vessel Inside Radius (r) | 0.340 | m |

For cylindrical body, thin-walled assumption is ok so we can use results. Required results are summarized in the following table. This is the answer of clause b).

PRINCIPAL AND MAX. SHEAR STRESSES ON CYLINDRICAL BODY | ||

Parameter | Value | Unit |

Principal stress 1 (Tangential direction) (σ_{1}) |
51 | MPa |

Principal stress 2 (Longitudinal direction) (σ_{2}) |
25.5 | |

Maximum shear stress (
τ_{max}) |
25.5 |

**Step 4:** Selected material (A204 Steel) is ductile since elongation at
break is greater than 5%. For the evaluation of yield criteria for a ductile
material with plane stress state, we can use "
Yield Criteria for Ductile Materials" page.
Evaluate yield criteria of spherical end cap and cylindrical body with the
values and results summarized in step 1, step 2 and step 3.

INPUT PARAMETERS FOR SPHERICAL END CAPS | ||

Parameter | Value | Unit |

Max. principal stress (σ_{max}) |
43 | MPa |

Min principal stress (σ_{min}) |
43 | |

Yield strength (S_{y}) |
275 | |

Design factor (n_{d}) |
5 |

RESULTS FOR SPHERICAL END CAPS | |||

Parameter | Condition to be met for safe design | Status | |

MSS theory | (σmax) < Sy/n | 43<55 | Ok |

DE theory | (σmax^2 - σmax*σmin + σmin^2)^0.5 < Sy/n | 43<55 | Ok |

INPUT PARAMETERS FOR CYLINDRICAL BODY | ||

Parameter | Value | Unit |

Max. principal stress (σ_{max}) |
51 | MPa |

Min principal stress (σ_{min}) |
25.5 | |

Yield strength (S_{y}) |
275 | |

Design factor (n_{d}) |
5 |

RESULTS FOR CYLINDRICAL BODY | |||

Parameter | Condition to be met for safe design | Status | |

MSS theory | (σmax) < Sy/n | 51<55 | Ok |

DE theory | (σmax^2 - σmax*σmin + σmin^2)^0.5 < Sy/n | 44.2<55 | Ok |

According to results found above, both spherical end caps and cylindrical body satisfy design requirements and no yielding is expected on the material. This is the answer of clause c).

**Step 5:** To be able to find stresses which are perpendicular and shear to the weld on
the cylindrical body, plane stress transformation is needed. Tangential and
longitudinal stresses have been found in step 3. Helix angle is given as 30°
with transverse plane (also tangential stress), so 30° transformation is
required with calculated stresses to solve clause d) of the sample example. Go
to the "Plane Stress and Transformations" page to calculate plane stresses in
different directions. Calculate the transformation shown in the figure with values
calculated in step 3.

INPUT PARAMETERS | ||

Parameter | Value | Unit |

Normal stress (σ_{x}) |
25.5 | MPa |

Normal stress (σ_{y}) |
51 | |

Shear stress (τ_{xy}) |
0 | |

Transformation angle (θ) | 30 | deg |

RESULTS | ||

Parameter | Value | Unit |

Normal stress after transformation (σ_{x}') |
31.9 | MPa |

Normal stress after transformation (σ_{y}') |
44.6 | |

Shear stress after transformation (τ_{xy}') |
11 |

After the plane stress transformation, shear stress is calculated as 11 MPa and perpendicular stress is 31.9 MPa. This is the answer of clause d).

The problem is solved with calculators below.

Calculator | Usage |

Thin Walled Pressure Vessel Stress Calculations | To calculate stresses on spherical end caps and cylindrical body. |

Yield Criteria for Ductile Materials | To evaluate material condition against yielding for ductile material which is under static loading. |

Plane Stress and Transformations | To calculate normal stress perpendicular to the weld and shear stress parallel to the weld. |

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