Module 8 · Lesson 8.4
Temperature, restraint and impact
Why a free member does not care about temperature and a restrained one does.
Why this matters
Temperature change is the load case people forget, because nothing visibly pushes on the structure. It produces no stress at all in a member free to move, and very large stresses in one that is not — and the difference is entirely down to restraint, which is a detailing decision rather than a material property.
By the end of this lesson you should be able to
- Calculate free thermal expansion from α and ΔT
- Calculate the stress in a fully restrained member
- Explain why thermal stress does not depend on the member's length or area
- Estimate the effect of a suddenly applied or dropped load
What you should already know
- Direct stress, strain and Hooke's law (Module 7)
- Axial extension δ = PL/AE (Module 7)
What it calculates: the strain a free member undergoes for a temperature change
- εT
- thermal strain (dimensionless)
- α
- coefficient of linear thermal expansion (per °C)
- ΔT
- temperature change (°C)
This assumes
- The member is free to expand or contract
- α is constant over the range
In plain terms: If the member can move, this strain happens and no stress is produced. Strain without stress is a perfectly ordinary state of affairs, and thermal expansion is the clearest example.
Now suppose the member cannot move. Imagine heating it and letting it expand freely by εT = αΔT, then squashing it back to its original length. The squashing is a mechanical strain of −αΔT, and that is what produces stress:
What it calculates: the stress in a member completely prevented from expanding
- σ
- axial stress (compressive for a rise in temperature) (N/mm²)
- E
- Young's modulus (N/mm²)
- α
- coefficient of thermal expansion (per °C)
- ΔT
- temperature change (°C)
This assumes
- The member is fully restrained — the supports do not move at all
- The material stays elastic
In plain terms: Neither the length nor the cross-sectional area appears. A short stubby bar and a long slender one, fully restrained, reach exactly the same thermal stress.
Predict first
Two fully restrained steel bars of the same material are heated by the same amount. One is 2 m long, the other 20 m. How do their thermal stresses compare?
Worked example
A restrained steel member in a temperature rise
Given
- Steel member, length 30.0 m
- Temperature rise ΔT = 25 °C
- α = 12 × 10⁻⁶ per °C
- E = 205 000 N/mm²
Find
The free expansion, and the stress if expansion is fully prevented.
One more effect belongs with material behaviour: what happens when a load arrives suddenly rather than being eased on.
A load lowered gently onto a member does work equal to the area under the load–deflection line, ½Pδ. A load dropped onto it does work equal to the full weight times the distance fallen. The member has to absorb all of that as strain energy, so it deflects further and is stressed harder than the same load applied slowly.
For a weight falling a height h onto a member whose static deflection under that weight would be δst, the deflection and stress are multiplied by an impact factor:
What it calculates: how much a suddenly applied load magnifies deflection and stress
- n
- impact factor (dimensionless)
- h
- height of fall (mm)
- δst
- static deflection under the same load (mm)
This assumes
- The member stays elastic
- No energy is lost to sound, heat or local damage — so this is an upper bound
- The striking mass does not bounce clear
In plain terms: Set h = 0 — the load released while just touching, rather than dropped — and n = 2. A suddenly applied load doubles the effect of the same load applied gradually, even with no fall at all.
Practice
A steel member 12.0 m long is heated by 30 °C. With α = 12 × 10⁻⁶ per °C, how much does it expand if it is completely free to move, in mm?
Practice
The same member is now held between immovable abutments so it cannot expand at all. With E = 205 000 N/mm², what stress does the 30 °C rise produce, in N/mm²?
Practice
A load is released suddenly onto a member while just touching it, with no fall at all (h = 0). By what factor is the deflection greater than if the same load had been applied gradually?
Summary
- Free thermal strain αΔT produces movement and no stress
- Fully restrained, the same temperature change produces σ = EαΔT
- Thermal stress depends on neither length nor area
- Full restraint is an upper bound; real structures shed some of it
- A suddenly applied load doubles deflection and stress even with no drop
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint