THIN WALLED PRESSURE VESSEL STRESS CALCULATION EXAMPLE

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:

Pressure vessel example

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.

Solution:

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

INPUT PROPERTIES SUMMARY
Parameter Value Unit
Vessel outer radius (ro) 0.35 m
Vessel thickness(end caps) (ts) 6 mm
Vessel thickness (cylindrical body) (tc) 10 mm
Vessel inner radius (end caps) (rsi) 0.344 m
Vessel inner radius (cylindrical body) (rci) 0.340 m
Gage pressure (pg) 1.5 MPa
Helix angle (θ) 30 deg
Design factor (nd) 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) (εbrk) 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 (pg) 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) (σ1) 43 MPa
Principal stress 2 (Longitudinal direction) ( σ2) 43
Maximum shear stress (τmax) 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 (pg) 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) (σ1) 51 MPa
Principal stress 2 (Longitudinal direction) (σ2) 25.5
Maximum shear stress ( τmax) 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 (σmax) 43 MPa
Min principal stress (σmin) 43
Yield strength (Sy) 275
Design factor (nd) 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 (σmax) 51 MPa
Min principal stress (σmin) 25.5
Yield strength (Sy) 275
Design factor (nd) 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 (σx) 25.5 MPa
Normal stress (σy) 51
Shear stress (τxy) 0
Transformation angle (θ) 30 deg
RESULTS
Parameter Value Unit
Normal stress after transformation (σx') 31.9 MPa
Normal stress after transformation (σy') 44.6
Shear stress after transformation (τxy') 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).

Summary

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.