Train Movement Resistance (Davis)
Calculate a train's specific movement resistance by the Davis equation, R = A + B·V + C·V², from coefficient A (rolling resistance and mechanical friction, speed-independent), B (resistance proportional to speed, from flange friction and oscillations), C (aerodynamic resistance, proportional to speed squared) and the speed V (km/h). The Davis equation, from the 1920s and still standard in railway engineering, describes the total resistance to motion the locomotive must overcome on straight, level track, per unit weight (N/t or kgf/t). At low speed the constant and linear terms (friction) dominate; at high speed the quadratic aerodynamic term dominates, decisive for high-speed trains (hence their careful streamlining). Davis resistance, plus grade (gravity) and curve resistances, sets the required tractive effort, energy consumption and locomotive traction capacity. It is the basis of traction calculation and train performance. Enter coefficients A, B and C and the speed.
Result
—
Train movement resistance (Davis)
The specific running resistance of a train is given by the classic Davis equation, R = A + B·V + C·V², from three coefficients and the speed V. Coefficient A stands for rolling resistance and mechanical friction (bearings, wheel-rail contact), independent of speed; coefficient B, the resistance proportional to speed, tied to flange friction (the wheel flange rubbing against the rail) and to the swaying of the consist; and coefficient C, the aerodynamic resistance, which grows with the square of speed. The Davis equation, formulated by W. J. Davis Jr. in the 1920s and still the standard of railway engineering today, describes the total resistance to forward motion that the locomotive must overcome on straight and level track, expressed per unit of weight (N/t or kgf/t). The behaviour is telling: at low speeds, the constant and linear terms dominate (the mechanical friction); at high speeds, the quadratic aerodynamic term takes over and grows explosively — which is why high-speed trains carry that carefully engineered aerodynamic profile, with long noses and fairings, since above roughly 250 km/h most of the energy goes into pushing the air aside. Davis resistance, added to grade resistance (gravity) and curve resistance (the extra friction on curves), sets the required tractive effort, the energy consumption and the pulling capacity demanded of the locomotive. It underpins every traction and performance calculation for a rail consist — from sizing the locomotive to forecasting journey times and fuel burn. Enter coefficients A, B and C and the speed.
Related Tools
Paper Tensile Index
Compute the paper tensile index, index = tensile strength (N/m) / grammage (g/m²), in N·m/g, normalizing the strength by the grammage to allow comparing papers of different weights. It is one of the most important mechanical properties, linked to fiber strength, inter-fiber bonding and refining. Packaging and sack papers require a high tensile index. Enter the tensile strength and the grammage.
Bolt Tensile Stress
Calculate the tensile stress in a bolt, σ = F_b ÷ A_t, from the total bolt tensile force F_b (N) and the tensile stress area A_t (mm²). It is the basic strength check of a tensioned bolt: the acting stress (force over resisting area) must be below the material strength with a safety margin. The force F_b is the total load the bolt carries — in a preloaded joint, the preload plus the fraction of external load reaching the bolt (F_i + C·P). The resulting stress is compared with the proof strength S_p (the limit up to which the bolt can be loaded without permanent deformation — typically 85-90% of yield) or the ultimate strength, per the criterion. The bolt strength class (marked on the head: 8.8, 10.9, 12.9 metric; or SAE grades 2, 5, 8) sets these allowable stresses — a class 8.8 bolt has a proof strength of 580-600 MPa, a 12.9 reaches ~970 MPa. Verifying σ does not exceed the allowable, considering preload and service load, is essential: overloaded bolts yield (losing preload) or break. With the fatigue and separation checks, it defines the tensioned joint's safety. Enter the total bolt force and the tensile area.
Adhesion Tractive Effort (Locomotive)
Calculate a locomotive's maximum tractive effort limited by adhesion, F = μ·W, from the wheel-rail adhesion coefficient μ (typically 0.25-0.35 dry, less with rain, ice or leaves) and the adhesive weight W (N, the locomotive weight on powered axles). Tractive effort is the force the locomotive applies to pull the train, with two limits: power (engine) and adhesion (wheel-rail friction). At low speed and starting, ADHESION limits — however powerful the engine, if the demanded force exceeds μ·W, the wheels spin, losing traction and wearing wheels and rails. So locomotives concentrate weight on powered axles (adhesive weight) and use anti-slip systems and sand application to boost friction. The steel-on-steel railway contact has very low rolling resistance (the train's great energy advantage) but precisely therefore limited adhesion — the fundamental paradox of rail traction. This defines the maximum train a locomotive can start and pull on a grade. Enter the adhesion coefficient and the adhesive weight.
Curve-Compensated Grade (Railway)
Calculate the compensated grade of a railway section on a curve, i_c = i − 700/R, from the actual section grade i (in ‰, per mille) and the curve radius R (m). When a grade coincides with a curve, the train faces both the climb resistance (gravity) and the extra curve resistance (added wheel-rail friction when changing direction). So the total resistance does not exceed that of the maximum tangent grade, the actual grade on the curve must be reduced (compensated) — subtracting a value equivalent to the curve resistance, commonly estimated as 700/R (in ‰, a usual empirical approximation; some manuals use 500/R or 600/R by gauge). Thus the compensated grade is the equivalent grade the train 'feels' including the curve. This is essential in railway geometric design: it keeps the required tractive effort uniform along the line, preventing a curve-on-grade from creating a critical point (a 'traction bottleneck') that would limit all trains' weight. The designer reduces the grade on curved sections to compensate. Enter the actual grade and the curve radius.
Tensile Strength from Brinell Hardness
Estimate a carbon steel's tensile strength (Rm) from the Brinell hardness, Rm ≈ 3.45·HB, in MPa. There is a remarkably robust empirical correlation between hardness and strength in steels, which lets you estimate strength from a hardness test — fast, cheap and almost non-destructive — instead of a tensile test. Useful in inspection and quality control. Enter the Brinell hardness (HB).
Paper Breaking Length
Compute the breaking length (self-rupture) of paper, L = tensile index / 9.80665, in km — the length of a paper strip that, hung from one end, would break under its own weight. It is an intuitive, classic way to express tensile strength, independent of grammage. Common papers break around 3–8 km; high-strength papers, more. Enter the tensile index (N·m/g).
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.