Sample Problem : 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:
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 
Symbol 
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 : Go to "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.
For spherical end caps, thinwalled assumption is ok so we can use results.
Required results are summarized in the following table. This
is the answer of clause a) of the sample example.
PRINCIPAL AND MAX. SHEAR STRESSES ON SPHERICAL CAP 
Parameter 
Symbol 
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 ThinWalled Pressure
Vessel" calculator.
For cylindrical body, thinwalled assumption is ok so we can use results.
Required results are summarized in the following table. This
is the answer of clause a) of the sample example.
PRINCIPAL AND MAX. SHEAR STRESSES ON CYLINDRICAL BODY 
Parameter 
Symbol 
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.
Evaluation Of Spherical End Caps 

Evaluation Of Cylindrical Body 

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) of the sample example.
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.
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) of the sample
example.
Summary
The problem is fully solved with calculators which are summarized as
follows.