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.
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Rampa compensada em curva (ferrovia)
A rampa compensada de um trecho ferroviário em curva é i_c = i − 700/R, a partir da rampa (greide) real do trecho i (em ‰, por mil) e do raio da curva R. Quando uma rampa coincide com uma curva, o trem enfrenta duas resistências ao mesmo tempo: a resistência da subida (a gravidade, proporcional à rampa) e a resistência adicional da curva (o atrito extra das rodas, especialmente dos frisos, contra os trilhos ao mudar de direção). Se nada fosse feito, esse trecho em curva-e-rampa exigiria um esforço trator maior que o de uma rampa equivalente em reta, criando um ponto crítico. Para evitar isso, a rampa real na curva é reduzida (compensada), descontando-se um valor equivalente à resistência da curva — comumente estimado como 700/R em ‰ (uma aproximação empírica usual; alguns manuais usam 500/R ou 600/R conforme a bitola e o tipo de via). Assim, a rampa compensada é a rampa equivalente que o trem efetivamente 'sente', já considerando o efeito penalizador da curva. Esse conceito é essencial no projeto geométrico de ferrovias porque garante que o esforço trator exigido seja uniforme ao longo de toda a linha: sem a compensação, uma única curva em rampa criaria um 'gargalo de tração' que limitaria o peso de todos os trens da ferrovia (um trem só é tão pesado quanto o permite seu trecho mais difícil). O projetista, portanto, reduz deliberadamente o greide nos trechos em curva, mantendo a resistência total dentro do valor da rampa máxima de projeto. É um exemplo elegante de como a geometria da via é pensada de forma integrada com a física da tração. Informe a rampa real e o raio da curva.
Related Tools
Railway Minimum Curve Radius
Calculate the minimum railway curve radius for a design speed, R = (B·V²) ÷ (127·(h_max + I_max)), from the gauge B (mm), speed V (km/h), maximum allowable cant h_max (mm) and maximum allowable cant deficiency I_max (mm). The minimum radius is set by combining the two comfort/safety limits available to 'absorb' lateral acceleration at the desired speed: the maximum buildable cant (limited by overturning risk of slow/stopped trains) and the maximum deficiency allowed to passengers. The larger these limits, the smaller the radius for a given speed — but both have normative caps. This is central to railway alignment: it defines how sharp a curve can be without speed reduction. Sharper curves require slowing down, penalizing travel time and line capacity — so high-speed railways need huge radii (kilometers). Enter the gauge, speed, maximum cant and maximum deficiency.
Railway Curve Maximum Speed
Calculate the maximum allowable speed on a railway curve, V = √(127·R·(h_a + I) ÷ B), from the curve radius R (m), the applied cant h_a (mm), the allowable cant deficiency I (mm) and the gauge B (mm). It is the inverse of curve design: given an existing curve (radius and cant) and the permitted deficiency, it finds the maximum speed trains can run safely and comfortably. Speed is limited because above it the cant deficiency would exceed the allowable — passengers would feel excessive lateral force and wheel-rail wear and risk would rise. This is fundamental in railway operation: it defines each section's maximum speeds (line speed profile) and travel time. Raising speed on existing curves needs more cant (limited), more allowed deficiency (tilting trains) or, ultimately, larger-radius regrading — an expensive work. Enter the radius, applied cant, allowable deficiency and gauge.
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.
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