Every solid grows a little when it warms: ΔL = αLΔT. A long bridge needs a gap to grow into, or it pushes on its abutments. Two metals bonded together grow by different amounts, so the strip curls, and a thermostat turns that curl into a switch.
Smallest joint gap that survives the hottest day, against span length, from the closed form αL(Tmax−T0) and from stepping the temperature in 0.1 K steps and looking for the first jam.
The science
Linear expansion. A bar of length L changes length by ΔL = α L ΔT, with α the coefficient of thermal expansion (steel about 12, concrete 10, aluminium 23, invar 1.2, brass 19 ×10-6 per kelvin; typical values). A 100 m steel deck heated by 40 K grows by 48 mm.
Joint gap. The total gap between deck and abutments is G(T) = G0 − αL(T − T0), so the gap closes at ΔT = G0/(αL). Beyond that the deck is blocked and carries the blocked strain as compressive stress σ = E α (ΔT − ΔTclose); fully constrained from the start it would be σ = EαΔT (steel: 2.4 MPa per kelvin). The strength limits shown are illustrative.
Bimetal strip. Two bonded layers of thickness t1, t2, moduli E1, E2 and expansion α1, α2 curl with curvature (Timoshenko, 1925) κ = 6(α1 − α2)ΔT(1+m)² / (h[3(1+m)² + (1+mn)(m² + 1/(mn))]), m = t1/t2, n = E1/E2, h = t1+t2. For equal thickness and modulus this reduces to κ = 3ΔαΔT/(2h). The page also solves the force and moment balance of the two layers directly as a 2×2 linear system and compares. The tip of a strip of length Ls rises by (1 − cos κLs)/κ ≈ κLs²/2.
Simplifications, stated honestly. Uniform temperature, free thermal growth, no friction at the bearings, no creep, no wind or traffic load, and linear elastic behaviour; the buckling of a jammed deck is only hinted at by the lift. The deck is one material and the joints are two equal gaps. Strength limits and the bimetal contact gap are illustrative values. Vertical and sideways motion of the joints and strip is exaggerated on screen.