Railway engineering Rangwala

Curvature of Track

Overview of Curvature

  • Curvature in railway tracks is necessary to connect important points and avoid obstructions.

  • Example: World’s longest straight railway line in Australia (497 km).

Objections to Curvature

  1. Speed Restrictions: Trains operate slower on curves; limits on train length and weight.

  2. Maintenance Costs: Curved tracks undergo increased wear and tear, raising maintenance costs.

  3. Smoothness of Travel: Curves can make train operation less smooth.

  4. Safety Hazards: Increased risks of collisions, derailments, and other accidents.

    Design of curvature must consider these objections while ensuring safety and efficiency to reduce these objections at minimum level.

Ideal Locations to Avoid Curves

  • Bridges and Tunnels

  • Deep Cuttings

  • Approaches to Bridges

  • Steep Gradients

  • Stations and Yards

  • Level-Crossings

Designation of Curves

Definition of a Curve

  • Curves are designated by either degree or radius. A simple degree of a curve is defined as the angle subtended by a chord (30 m) at the center.

  • Example: 1° results in a larger radius, while 6° corresponds to a sharper curve.

Radius of Curvature Calculations

  • Mathematical relation between radius and degree of curvature:

    • Specific formula for 1° curve yields a radius of 1718.89 m (rounded to 1719 m).

    • Relationship: R = 1719/θ (where θ is in degrees), useful for practical applications up to 10°.

Types of Curves

  • Simple Circular Curves: Uniform curvature.

  • Compound Curves: Multiple curves at different radii.

  • Reverse Curves: Indicates a change in direction.

  • Vertical Curves: A curve in the vertical plane, used to connect gradients smoothly. These are parabolic curves.

  • Transition Curves: A curve with a gradually changing radius, connecting a straight section to a circular curve.

Transition Curves

Definition

  • A transition curve is a curve that connects a straight track to a circular curve, helping to create a smooth change in direction.

Importance of Transition Curves

  • Provide a gradual change in curvature; beginning with straight track leading to circular curve.

  • Helps in maintaining a constant rate of change of radial acceleration, enhancing passenger comfort.

Requirements for Transition Curves

  1. Must be tangential to the straight track.

  2. Full super-elevation attained at the junction with the circular curve.

  3. Rate of increase of curvature equals the rate of increase in cant/super-elevation.

  4. Tangential join to the circular arc ensuring smoothness.

Forms of Transition Curves

  • Examples include Euler’s Spiral or Froud’s cubic parabola (widely used in India).

  • Froud’s cubic parabola formula: Y = Perpendicular offset of curve at distance X from curve commencement.

Length of Transition Curves

  • Indian Railways uses several formulas for determining the transition curve length:

    1. L = 7.20e

    2. L = 0.073 D x Vmax

    3. L = 0.073 e x Vmax

  • Greatest length from all equations is chosen for construction.

Super-elevation (Cant)

Definition and Purpose

  • Super-elevation involves raising the outer rail on curves to counteract centrifugal force.

  • Purposes include ensuring passenger safety, smooth movement, and reducing wear on tracks.

Calculating Super-elevation

  • Formula for determining super-elevation based on train weight (W), velocity (V), radius (R), and gravitational force (g):

    • P = W * v²/gR, with variations for different gauges and standards.

Factors Affecting Super-elevation

  • Frictional resistance between wheels and rails

  • Coning of wheels (slightly conical wheel profile)

  • Flexibility of the vehicle body (springs and suspension)

  • Weighted average speed (considering various train speeds)

Speed of Trains on Curves

Safe Speed Calculation

  • Safe speeds depend on super-elevation, possible transition curves, gauge, and train weight.

  • Empirical formulas for maximum speeds:

    • For B.G. and M.G.: V = 4.4 √(R - 70)

    • For N.G.: V = 3.6 √(R - 6) (max 50 km/h)

Deficiency in Super-elevation

  • Cant deficiency occurs when actual super-elevation falls short of the equilibrium value.

  • A negative super-elevation is calculated based on permissible cant insufficiencies, especially for branch lines diverging from main tracks.

Example Problems

  1. Transition Curve Length Calculation:

  • Given parameters for a BG curve with various cant and speeds.

  1. Speed on Curved Tracks:

  • Calculating maximum speed based on curvature and cant.

  1. Equilibrium Cant Calculations:

  • Determining values across various gauge types and speeds.

Concluding Remarks

  • Effective design and management of curvature on railway tracks is critical for operational efficiency, safety, and passenger comfort.