At

Figure 1-4. Thermal linear gradient across shell wall.

from Ti to To and the growth of the cube is fully restrained:

where Tj = initial temperature, °F T2 = new temperature, °F a = mean coefficient of thermal expansion in./in./°F E = modulus of elasticity, psi v — Poisson's ratio = .3 for steel AT = mean temperature difference, °F

Case 1: If the bar is restricted only in one direction but free to expand in the other direction, the resulting uniaxial stress, a, would be ct= -Ea(T2-Ti)

Case 2: If restraint is in both directions, x and y, then:

Case 3: If restraint is in all three directions, x, y, and z, then erx = oy = crz = —aE AT/1 — 2v

Case 4: If a thermal linear gradient is across the wall of a thin shell (see Figure 1—4), then:

This is a bending stress and not a membrane stress. The hot side is in tension, the cold side in compression. Note that this is independent of vessel diameter or thickness. The stress is due to internal restraint.

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