Combined Loadings - Mechanics of Materials

Combined Loadings - Stress analysis of axial load, torsion, bending, and shear. Cross sections with several internal loads that occur all at once.

Thin-Walled Pressure Tanks

Thin wall -- (r/t) is less than or equal to 10


Cylindrical Tank

This type of tank will be subjected to 2 normal stresses, both in tension. One stress will be in the "hoop" (circumference) direction and the other will be in the longitudinal direction.

• Hoop Stress: σ1 = Pr / t
• Longitudinal Stress: σ2 = Pr / 2t

P = Pressure inside the tank.
r = Inner radius.
t = Thickness of the tanks wall.

As you can see the hoop stress will be twice that of the longitudinal stress.

However, if the ends of the tank were open so the tank would be considered a pipe now, the longitudinal stress would be zero.


Spherical Tank

A spherical tank is very similar to the cylindrical tank except there is no hoop stress.

So therefore, σ1 = σ2 = Pr / 2t

If you were to take a small piece from a cylindrical or spherical tank it would have a biaxial stress. This is just the normal force acting in 2 directions.


State of Stress: The combined loadings on a cross section.

Types of Loadings:

• Normal Force: σ = P/A
• Bending Moment: σ = -My / I -- Flexure Formula
• Thin-walled pressure tank:

σ Hoop = Pr / t
σ longitudinal = Pr / 2t
σsphere = Pr / 2t

• Shear Force: = VQ / It
• Torsional Moment:

For a circular shaft and tube: = Tρ / J
For a closed thin-walled tube: = T / 2tA



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