Railway Track Infrastructure, AREMA Continuous Welded Rail (CWR) Thermal Buckling
Continuous Welded Rail (CWR) Neutral Temp & Buckling Force
Calculate Rail Temperature Excursion Above Neutral (°C), Constrained Axial Thermal Stress (MPa), Single-Rail Axial Force (kN), and Two-Rail Track Panel Force (kN).
With zero expansion joints over kilometers of track, a 62 °C summer rail builds nearly 100 tonnes of lateral sun-kink buckling compression!
Modern high-speed and heavy-haul **Continuous Welded Rail (`CWR`)** flash-butt and thermite welds `UIC60` / `136RE` steel rails (`Cross-Sectional Area A ≈ 7,670` mm², `Young’s Modulus E = 207` GPa, `Thermal Expansion alpha = 11.5 * 10^{-6}` /°C) into multi-kilometer jointless ribbons anchored to concrete ties so zero longitudinal strain (`epsilon = 0`) can occur! Hydraulic rail tensors stretch cold rail during installation to lock in a deliberate **Rail Neutral Temperature `T_N`** (`35–43 °C`, where axial stress equals ZERO!). Whenever solar heating or Arctic cold shifts Actual Steel Temperature `T_{rail}` away from `T_N` by **`Delta_T = T_{rail} - T_N`** (`°C`), full axial restraint induces **Thermal Stress `sigma_{th} = (E * 1,000) * (alpha * 10^{-6}) * Delta_T`** (`MPa`, `+` compression / `-` tension!), **Single-Rail Axial Force `F_1 = (A * sigma_{th}) / 1,000`** (`kN`), and **Full Track-Panel Force `F_{panel} = 2 * F_1`** (`kN`—resisted in summer solely by crushed-granite crib/shoulder ballast lateral resistance, or risking catastrophic **"sun-kink" lateral track buckling**)!
Verify on a `UIC60` (`A = 7,670` mm²) Mainline Track Anchored at `T_N = 35.0 °C`—Rail_Section_Area `A` = 7,670 mm² (B2), Rail_Modulus `E` = 207 GPa (B3), Expansion_Coefficient `alpha` = 11.5 ppm/°C (B4), Installed_Neutral_Temp `T_N` = 35.0 °C (B5), and Sunlit_Summer_Rail_Temp `T_{rail}` = 62.0 °C (`~42 °C` air + solar gain, B6): Excursion `Delta_T` equals `+27.0 °C` (E2), Compressive Stress hits `+64.27` MPa (E3), Single-Rail Compression reaches `+493.0` kN (E4), and Two-Rail Track Panel Compression hits **`+986.0` kN (`100.5` tonnes-force!)** (E5)!
| AREMA CWR Thermal Force KPI (UIC60 A=7,670 mm², TN=35 °C: Summer +62 °C vs Winter -18 °C) | Formula | Verified Output |
|---|---|---|
| Excursion From Neutral dT (°C) | Actual_Rail_Temp_C - Neutral_Temp_TN_C | +27.0 °C (-53.0 °C in winter freeze) |
| Constrained Axial Stress (MPa) | (E_GPa * 1000) * (alpha * 1e-6) * dT_C | +64.27 MPa (-126.17 MPa tensile pull!) |
| Single-Rail Axial Force F1 (kN) | (Rail_Area_mm2 * Stress_MPa) / 1000 | +493.0 kN (-967.7 kN pull-apart load!) |
| Two-Rail Track Panel Force (kN) | 2 * Single_Rail_Force_kN | +986.0 kN (-1,935.4 kN winter tension) |
Constrained Thermoelastic CWR Sun-Kink Compression & Pull-Apart Tension Formula
Validate `Rail_Area A` in `[4,000, 12,000]` mm², `Modulus E` in `[180, 225]` GPa, `alpha` in `[9.0, 14.0]` ppm/°C, `Neutral_Temp T_N` in `[15, 50] °C`, and `Actual_Rail_Temp` in `[-55, 80] °C`!
=LET(ar_m, B2, em_g, B3, al_p, B4, tn_c, B5, tr_c, B6, dt_c, tr_c - tn_c, sg_m, (em_g * 1000) * (al_p * 0.000001) * dt_c, IF(OR(ar_m < 4000, ar_m > 12000, em_g < 180, em_g > 225, al_p < 9, al_p > 14, tn_c < 15, tn_c > 50, tr_c < -55, tr_c > 80), "Invalid CWR Input", ROUND((2 * ar_m * sg_m) / 1000, 1)))Polar-Vortex -18 °C Winter Rail-Break Pull-Apart Tension Check (Set Rail Temp B6 = -18 °C)
Drop rail temperature during a `-18.0 °C` winter night (`B6 = -18`, `Delta_T = -53.0 °C` in E2): Axial Stress flips to **`-126.17` MPa tension** (E3) and Single-Rail Pull-Apart Force reaches **`-967.7` kN** (`-1,935.4` kN panel tension in E5)—explaining why setting `T_N` *too high* snaps thermite welds open by `50–80` mm in January, triggering track-circuit occupancy red signals!
=IF(ISTEXT(E5), E5, ROUND((B2 * (B3 * 1000) * (B4 * 0.000001) * (B6 - B5)) / 1000, 1))Try the example
Use the original values or change them to test your understanding.