Problem 2.4. Pressure vessel’s stresses

Determine principal stresses in point A of the cylindrical pressure vessel given in the Figure. The thickness of the wall is 3 mm, the radius is 300 mm, working pressure is 2 bars (1 bar = 105 Pa ≈ 1 atm).

Illustrate the stresses with the Mohr circle.
Determine the stresses in a coordinate system rotated 40 degrees
from the horizontal axis.

Stresses can be determined from the pressure vessel formula (Eqs.11-8 and 11-10).


Solve Problem

Solve

Stress, σx [MPa]=

Stress, σy[MPa]=

Stress, τxy[MPa]=

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Steps

Step by step

Step 1. Determine hoop and axial stresses in the wall of the pressure vessel. Use pressure vessel formula.

The formula is derived in Section 11.1.2 (Eqs.11-8 and 11-10).

Check hoop stress

σ1=pRt=2×0.1×3003=20 MPa

Check axial stress

σ2=pR2t=2×0.1×3002×3=10 MPa

Step 2. Find principal directions and illustrate the stresses with the Mohr circle.

Check Mohr circle

In the coordinate system attached to the hoop and axial directions only tensile forces (and no shear) arise in the wall of the pressure vessel. Thus this directions are the principal directions. 

See explanation of the Mohr circle in Section 2.1.1 and in Figure 2.7

Step 3. Transform stresses into the rotated coordinate system.

Check transformed stresses

Eq.(2-9)

σxσyτxy=Tσσ12=cos240sin2402sin40cos40sin240cos2402sin40cos40sin40cos40sin40cos40cos240sin240σ1σ20=0.590.410.980.410.590.980.490.490.1720100σxσyτxy=15.8914.134.92MPa

Step 4. Draw transformed stresses on the Mohr cicle.

Check figure

See explanation of the Mohr circle in Section 2.1.1 and in Figure 2.7

Results

Worked out solution

Hoop and axial stresses in the wall of the pressure vessel can be determined by the pressure vessel formula.

The formula is derived in Section 11.1.2 (Eqs.11-8 and 11-10).

The hoop stress is

σ1=pRt=2×0.1×3003=20 MPa.

The axial stress is

σ2=pR2t=2×0.1×3002×3=10 MPa.

In the coordinate system attached to the hoop and axial directions only tensile forces (and no shear) arise in the wall of the pressure vessel. Thus this directions are the principal directions as it is illustrated by the Mohr circle.

See explanation of the Mohr circle in Section 2.1.1 and in Figure 2.7

 Stress transformation into rotated coordinate system results in

 
Eq.(2-9)

σxσyτxy=Tσσ12=cos240sin2402sin40cos40sin240cos2402sin40cos40sin40cos40sin40cos40cos240sin240σ1σ20=0.590.410.980.410.590.980.490.490.1720100σxσyτxy=15.8914.134.92MPa

Transformed stresses are shown also on the Mohr circle.