Calculate expansion, stress, and restraint force for a bar between rigid supports, including an initial gap and a uniform temperature change.
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Worked examples
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Example 1
fully restrained steel bar heated 40 °C
Given
length
2 m
modulus
200 GPa
alpha
value 0.000012per degC
temperature change
40 degC
Assumptions
Rigid (unyielding) supports, linear-elastic material, uniform temperature change along the bar.
Solution steps
Free thermal expansion
Unrestrained, the bar would change length in proportion to α, ΔT and L.
δT = α · ΔT · L
δT = 0.96 mm
= 0.96 mm
Stress from the suppressed strain
The supports suppress the entire expansion — the bar carries the strain as compression.
σ = E·(δ_blocked)/L, = E·α·ΔT when there is no gap
suppressed δ = 0.96 mm; σ = 96 MPa (compression)
= 96 MPa
Results
Free thermal expansion δT
0.00096m
Thermal stress σ (compression)
96MPa
Where this answer was checked
source
σ = E·α·ΔT (NCEES FE Reference Handbook) — no gap, SI