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A fully restrained linear-elastic bar develops compressive stress when uniformly heated.
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Formula
σth in Pa; E in Pa; α in K−1; ΔT in K
| Symbol | Meaning | Unit |
|---|---|---|
| Thermally induced axial normal stress | Pa | |
| Young modulus in the temperature range | Pa | |
| Linear expansion coefficient | K−1 | |
| Temperature change | K |
\sigma_{th}=-E\alpha\Delta TCombine zero total axial strain with εtotal=εmechanical+αΔT and Hooke’s law. The minus sign denotes compression for restrained heating; cooling reverses the sign. Assumptions are uniform temperature, constant properties, linear elasticity and rigid axial restraint. Partial restraint, creep and gradients require another model.
Combine zero total axial strain with εtotal=εmechanical+αΔT and Hooke’s law. The minus sign denotes compression for restrained heating; cooling reverses the sign. Assumptions are uniform temperature, constant properties, linear elasticity and rigid axial restraint. Partial restraint, creep and gradients require another model.
A fully restrained linear-elastic bar develops compressive stress when uniformly heated.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Thermal stress in a fully restrained bar” and check its unit and magnitude.
With E=200 GPa, α=12·10−6 K−1 and ΔT=30 K, σth=−72 MPa.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: σth in Pa; E in Pa; α in K−1; ΔT in K.
Formula cluster
A fully restrained linear-elastic bar develops compressive stress when uniformly heated.
With E=200 GPa, α=12·10−6 K−1 and ΔT=30 K, σth=−72 MPa.
Estimate idealized restraint stress in rails, pipes and fixed components.
Here is how to work through a typical Thermal stress in a fully restrained bar (σth = −EαΔT) task step by step:
Task
Work through the numerical example for “Thermal stress in a fully restrained bar” and check its unit and magnitude.
Solution path
With E=200 GPa, α=12·10−6 K−1 and ΔT=30 K, σth=−72 MPa.
Free heating would cause strain αΔT. Rigid supports enforce zero total strain, so the material must develop an equal opposing mechanical strain. Hooke’s law turns that into compressive stress. Partial restraint produces a smaller magnitude. This assumes full restraint, uniform temperature and linear elasticity; the sign denotes compression during heating. Application: Estimate idealized restraint stress in rails, pipes and fixed components. The numerical example With E=200 GPa, α=12·10−6 K−1 and ΔT=30 K, σth=−72 MPa anchors the idea in a concrete measurement case.
For a positive temperature change, the bar tends to lengthen. Preventing that expansion creates compression. Under the usual sign convention compression is negative, hence σth=−EαΔT. Cooling under the same ideal restraint produces tension. This assumes full restraint, uniform temperature and linear elasticity; the sign denotes compression during heating. The corresponding units are σth in Pa; E in Pa; α in K−1; ΔT in K. With E=200 GPa, α=12·10−6 K−1 and ΔT=30 K, σth=−72 MPa. This checks that the reference quantities are compatible.
Not directly. A temperature gradient creates different free strains across a section or along a bar and can also cause bending. The compact equation assumes uniform temperature and purely axial restraint. Real gradients require solving the temperature field, geometry and support conditions together. This assumes full restraint, uniform temperature and linear elasticity; the sign denotes compression during heating. Scope: Combine zero total axial strain with εtotal=εmechanical+αΔT and Hooke’s law. The minus sign denotes compression for restrained heating; cooling reverses the sign. Assumptions are uniform temperature, constant properties, linear elasticity and rigid axial restraint. Partial restraint, creep and gradients require another model. A practical application is Estimate idealized restraint stress in rails, pipes and fixed components.
Write the quantity and unit from the variable list next to every value: σth in Pa; E in Pa; α in K−1; ΔT in K. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example With E=200 GPa, α=12·10−6 K−1 and ΔT=30 K, σth=−72 MPa. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. This assumes full restraint, uniform temperature and linear elasticity; the sign denotes compression during heating.
The background states this starting condition: Combine zero total axial strain with εtotal=εmechanical+αΔT and Hooke’s law. The minus sign denotes compression for restrained heating;. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. This assumes full restraint, uniform temperature and linear elasticity; the sign denotes compression during heating.
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Which formula describes Thermal stress in a fully restrained bar?
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How do you rearrange Thermal stress in a fully restrained bar for Base equation for the target quantity?
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