Thermal Stress in a Restrained Member

A free member heated by ΔT would stretch by strain αΔT and feel nothing. Clamp both ends and it cannot move, so the material must carry exactly the stress needed to undo that strain: σ = EαΔT. Notice what is missing — length and area do not appear. A 3 m pipe spool and a 300 m pipeline reach the same thermal stress for the same temperature swing, which is the single most counter-intuitive result in the subject. Steel with E = 200 GPa and α = 12 × 10⁻⁶/°C heated 50 °C develops 200 × 10⁹ × 12 × 10⁻⁶ × 50 = 120 MPa, roughly half of A36's yield, from a temperature change any black pipe sees between a winter install and a summer steam-up.

Railways learned this the hard way: continuously welded rail is laid pre-tensioned at a "stress-free temperature" precisely so that summer heat does not build enough compression to throw the track sideways into a sun kink. The same physics puts expansion loops, bellows and slide guides into every hot-water and steam distribution system, and cracks the anchor lugs of any pipe rigidly clamped at both ends. The trap in the arithmetic is unit pairing: α in 1/°F must go with ΔT in Fahrenheit degrees, α in 1/K with kelvin or Celsius degrees — the two scales differ by 9/5 and mixing them is a 44% error.