What this calculator does
Steel rail wants to grow and shrink with temperature. In jointed track it can — that’s what expansion gaps are for. In continuously welded rail (CWR) it can’t, so the temperature change is converted into axial force locked inside the rail. This calculator gives you both numbers: the free expansion a length of unrestrained rail would undergo, and the axial force and stress the same temperature change produces in fully restrained CWR.
The formulas
Free (unrestrained) expansion:
ΔL = α × L × ΔT
Restrained rail develops thermal stress and force instead of movement:
σ = E × α × ΔT P = σ × A
with α = 1.15 × 10⁻⁵ /°C for rail steel, E = 207 GPa, and A the rail cross-sectional area. Note the striking result: the force in restrained rail is independent of length — a 10 m rail and a 10 km rail at the same ΔT carry the same axial force.
Why ΔT is measured from the neutral temperature
CWR is installed (or later adjusted) so that it is stress-free at a deliberately chosen stress-free temperature (also called neutral temperature), typically set toward the upper end of the local rail temperature range. Every degree above neutral puts the rail into compression; every degree below puts it into tension. That is why the input here is rail temperature minus neutral temperature, not ambient air temperature — rail in the sun can run 15–20 °C hotter than the air.
What the numbers mean in practice
In hot weather, compression is the enemy: enough compressive force with weak lateral restraint (disturbed ballast, fresh tamping, poor consolidation) is the recipe for a track buckle. In cold weather, tension governs: pull-aparts at joints and rail breaks at defects. A 60 kg/m rail 35 °C above neutral carries roughly 640 kN of compression — which is exactly why networks impose speed restrictions during extreme heat and control when tamping is allowed relative to hot weather.
Worked example
A 110 m closure rail is welded in at neutral temperature. On a hot afternoon the rail reaches 35 °C above neutral. Free expansion would have been 1.15e-5 × 110 000 × 35 ≈ 44.3 mm — but the rail is restrained, so instead it carries 207 000 × 1.15e-5 × 35 ≈ 83.3 MPa of compressive stress, or about 639 kN on a 60 kg/m section.
FAQ
Does rail actually reach these forces? Near mid-track in well-anchored CWR, yes — restraint is close to full. Near expansion switches or track ends, some breathing occurs and force tapers off over the “breathing length”.
What coefficient of expansion should I use? 1.15 × 10⁻⁵ /°C is the standard figure for rail steel; some administrations use 1.13–1.17 × 10⁻⁵. The difference is a few percent.
Indicative figures for planning and checking. Hot-weather management, neutral temperature policy and buckling assessment must follow your network’s engineering standards.