Temperature loads

Uniform temperature change and through-depth gradients — what each does to a free and a restrained structure, and the coefficients to use.

Loads & combinations 4 min read

A temperature load produces no force in a structure free to move and large forces in one that is not. That makes it the load type most often misread: a zero moment is the right answer for a simply supported beam heated uniformly, and an engineer expecting a number sees a bug.

Elementarium applies temperature to a member or a surface as a uniform change or a gradient across the depth, and both go into a load case like any other action. The tool is under Loads › Apply › Distributed, in the same drop-down as the line loads.

The two components

Component	Strain it produces	Unrestrained structure	Restrained structure
Uniform change $Δ T$	Axial strain $ε$	Extends or shortens, no force	Axial force $N$
Gradient over depth $h$	Curvature $κ$	Bows, no moment	Moment $M$

For a uniform change:

ε = α Δ T N = - E A α Δ T

For a gradient, with $Δ T_{t o p} - Δ T_{b o tt o m}$ the difference between the faces:

κ = \frac{α ( Δ T _{t o p} - Δ T _{b o tt o m} )}{h} M = - \frac{E I α ( Δ T _{t o p} - Δ T _{b o tt o m} )}{h}

A temperature load is a uniform change plus a gradient through the depth. A free member simply moves and carries no force; restrain it and the same strain becomes axial force or moment. Drawn for heating, with the top face warmer than the bottom.

A member is rarely entirely one or the other. A continuous beam heated on its top face is free to extend, so $Δ T$ gives displacement only, but its curvature is restrained by the interior supports, so the gradient gives moments. The solver works this out from the model; the formulas are for judging whether the result is plausible.

Coefficients of thermal expansion

$α$ is a property of the material. The values EN 1991-1-5 gives:

Material	$α$ , per °C
Structural steel	12 × 10⁻⁶
Concrete	10 × 10⁻⁶
Aluminium	24 × 10⁻⁶
Timber, parallel to grain	5 × 10⁻⁶
Stainless steel	16 × 10⁻⁶

For composite steel–concrete structures EN 1994 permits 10 × 10⁻⁶ for both, so the two do not fight over a value that is uncertain anyway.

Orders of magnitude

Steel, 30 °C rise, a 20 m member fully restrained:

ε σ = 12 \times 1 0^{- 6} \times 30 = 360 μ ε = E ε = 210000 \times 360 \times 1 0^{- 6} = 75.6 MPa

That is a third of the yield strength of S235, from a summer's afternoon, if nothing is allowed to move. It is why expansion joints exist, and why a model whose supports restrain the structure more than the real bearings do reports thermal forces the structure will never see.

The same member free at one end extends by $20000 \times 360 \times 1 0^{- 6} = 7.2 mm$ and carries no force at all.

Modelling decisions

Supports decide everything. A temperature load tests the support idealisation harder than any gravity case. A row of pinned bases on a long building restrains the whole floor plate against expansion; real base plates on slotted holes, or real columns bending, do not.

Springs are the realistic middle. A column that bends to let the floor expand is a lateral spring of stiffness $3 E I / L^{3}$ ; a bearing with a known shear stiffness is a spring of that stiffness. Replacing the pin with the spring turns an unbounded thermal force into a finite one that can be checked.

Gradients on surfaces work the same way as on members: a slab warmer on top than below tries to bow, and the restraint against bowing (its own continuity over supports, or a wall along its edge) is where the moment comes from.

One temperature per load case. Summer and winter are two cases, in an exclusive group, so that the combination generator treats them as alternatives instead of adding a rise to a fall.

Combining

Under EN 1990 thermal actions are variable actions with their own ψ factors ( $ψ_{0} = 0.6$ , $ψ_{1} = 0.5$ , $ψ_{2} = 0$ in the recommended values), and they appear in combinations as a leading or an accompanying action like any other. The generator handles that once the case is in the right group; the values are editable fields on the combination, and the national annex may change them.

Thermal actions usually govern nothing at ULS in a building frame and quite a lot at SLS: movement at joints, cracking in a restrained slab, the force on a bearing. Read them under the serviceability combinations first.

Checking a thermal result

What is not modelled

Fire. The stiffness and strength reductions at elevated temperature are not applied; the material is linear elastic at any temperature you enter.
Transient heat flow. The temperature you enter is the temperature; nothing is computed from ambient conditions or exposure.
Non-linear gradients through the depth. The gradient is linear, which is what EN 1991-1-5's simplified approach uses for buildings.

Load cases for the load types alongside temperature; Supports and releases for the springs that make thermal results finite; Prescribed displacements are the other self-equilibrating action, and settlement behaves in the same way.

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