Railway Cant Calculator

Equilibrium Cant Formula

Equilibrium cant ($E_{eq}$) is the ideal cant for a given speed where the resultant force is perpendicular to the track. The formula is:

$$E_{eq} = \frac{GV^2}{gR}$$

Where:

• $E_{eq}$ is the Equilibrium Cant (in m)
• $G$ is the gauge of the track (in m, standard gauge is 1.435 m)
• $V$ is the velocity of the train (in m/s)
• $g$ is the acceleration due to gravity ($g \approx 9.81 m/s^2$)
• $R$ is the radius of the curve (in m)

For a standard gauge of 1435mm and velocity in MPH, the simplified formula is:

$$E_{eq} \approx \frac{29.2V^2}{R}$$

Where:

• $E_{eq}$ is the Equilibrium Cant (in mm)
• $V$ is the velocity of the train (in MPH)
• $R$ is the radius of the curve (in m)

Cant Deficiency Formula

Cant Deficiency ($D$) is the difference between the equilibrium cant and the actual cant applied to the track.

$$D = E_{eq} - E_{a}$$

Where:

• $D$ is the Cant Deficiency (in mm)
• $E_{eq}$ is the Equilibrium Cant (in mm)
• $E_a$ is the Applied Cant (in mm)

Transition Length Formula

The minimum transition length ($L$) is determined by the rate of change of cant and cant deficiency. It must be long enough to introduce the full cant and manage the change in lateral acceleration smoothly.

The rate of change of cant is given by:

$$\text{Rate of change} = \frac{\Delta E}{t} = \frac{\Delta E \times V}{L}$$

Rearranging for minimum length, $L$:

$$L = \frac{\Delta E \times V}{\text{Rate of change}}$$

Where:

• $L$ is the minimum transition length (in m)
• $\Delta E$ is the change in cant or cant deficiency (in mm)
• $V$ is the train velocity (in m/s)
• Rate of change is in mm/s

For a velocity in MPH, the simplified formula is:

$$L(m) = \frac{\text{change (mm)} \times V(\text{MPH})}{3333.3 \times \text{rate of change (mm/s)}}$$

Note: The constant 3333.3 is approximately `(1609.34 m/mile) / (3.6 km/h / 1 m/s)`