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Advanced Civil Engineering Handbook › Chapter 11

Chapter 11 of 14 · 31 formulas · 78 one-liners

Railway Engineering — formulas & one-liners

Broad Gauge track components, ballast, creep, curves and cant, points and crossings, gradients, stations, signalling, track maintenance and tunnels, with Indian Railways values (IRPWM 2020, IRSOD).

11.1 Gauges, rails, sleepers & fastenings

Sleepers per rail length (sleeper density)
M = rail length in metres (BG 13 m, MG 12 m); x = 3 to 7 depending on traffic and speed
Sleepers per km from sleeper spacing
s = centre-to-centre sleeper spacing (m)
s = 0.60 m gives about 1660 sleepers/km, the standard BG density on PSC sleepers
Expansion gap at rail joints
L = rail length (mm); α = 11.5 × 10⁻⁶ per °C for rail steel; Δt = temperature change (°C)
Thermal force in a fully restrained (LWR) rail
E = modulus of rail steel (about 2.1 × 10⁵ MPa); A = rail cross-sectional area; Δt = rail temperature minus stress-free (destressing) temperature
Extra distance travelled by outer wheel on a curve (coning)
G = gauge (centre-to-centre of rails); θ = angle subtended by the curve (radians)
Coning lets the outer wheel run on a larger tread diameter to reduce this slip
  1. Gauges in India: Broad Gauge 1676 mm, Metre Gauge 1000 mm, Narrow Gauges 762 mm and 610 mm; metros mostly use standard gauge 1435 mm.
  2. Project Unigauge (from 1992) aims to convert MG and NG lines to Broad Gauge.
  3. Gauge is the clear distance between the inner (running) faces of the two rails.
  4. Flat-footed rails are standard on Indian Railways; the 60 kg/m (60E1, UIC 60) section is used on BG trunk and high-density routes and 52 kg/m on other BG routes.
  5. 90R rail weighs about 44.6 kg/m; the number in the rail designation is its weight in lb per yard.
  6. Standard rail length is 13 m for BG and 12 m for MG; rails are now also supplied as long welded panels (up to 260 m) to cut field welds.
  7. Most IR rails are of 880 grade (minimum UTS 880 MPa); 1080 grade head-hardened rails are used for heavy-haul routes and turnouts.
  8. Limit of vertical rail-head wear: 13 mm for 60 kg, 8 mm for 52 kg and 5 mm for 90R rails (IRPWM).
  9. Rails are tilted inward at 1 in 20 to match the 1 in 20 coning of wheel treads.
  10. Standard BG wooden sleeper measures 2740 × 250 × 130 mm; BG PSC mono-block sleeper is 2750 mm long.
  11. Prestressed concrete (PSC) mono-block sleepers are now standard; CST-9 cast-iron and steel trough sleepers are older types.
  12. Elastic fastenings on PSC sleepers consist of ERC (elastic rail clip, e.g. Mk-III/Mk-V), GFN liners and a grooved rubber sole plate (GRSP).
  13. Plant welding is done by flash-butt welding; field welding is done by alumino-thermic (thermit) welding.
  14. Rail joints on Indian Railways are usually suspended joints; joints are kept square on straight track and staggered on curves.
  15. Battering (hammering of rail ends), hogging (bending down of rail ends), wheel burns and corrugation are common rail defects.
  16. Eastern and Western Dedicated Freight Corridors are built for a design axle load of 32.5 t.

11.2 Creep, ballast & formation

Minimum depth of ballast below sleeper (45° load dispersion)
d = ballast cushion (mm); S = sleeper spacing (mm); b = width of sleeper (mm)
Ensures pressure bulbs of adjacent sleepers just meet at formation level
  1. Creep is the longitudinal movement of rails along the track; it is greater on falling gradients, in the direction of heavier traffic and on the outer rail of curves.
  2. Theories of creep include the wave theory, percussion theory and drag (or dragging) theory.
  3. Creep is measured at creep posts and should be adjusted when it exceeds 150 mm.
  4. Creep is resisted by creep anchors and elastic fastenings that grip the rail firmly to the sleeper.
  5. Track ballast is hard broken stone of 65 mm nominal size; aggregate abrasion value should not exceed 30% and impact value 20% (IRS GE-1, normal limits).
  6. Ballast transfers and spreads sleeper load to the formation, provides drainage, and gives lateral and longitudinal resistance to the track.
  7. Formation (subgrade) is the prepared earthwork bed on which ballast rests; its top is sloped from the centre towards the sides for drainage.
  8. A blanket layer of coarse granular material is provided over weak or fine-grained formation soils to prevent mud-pumping.
  9. Short welded rail (SWR) on BG is normally 3 rails welded together (39 m), expanding and contracting over its full length.
  10. A long welded rail (LWR) has a central portion that does not move with temperature; its minimum length on BG is 250 m.
  11. The end portions of an LWR that move with temperature are called breathing lengths; switch expansion joints (SEJ) are provided there.
  12. LWR/CWR are destressed (fastened) within a prescribed rail temperature range to avoid buckling in summer and fractures in winter.
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11.3 Curves, cant & transitions (BG, IRPWM 2020)

Degree of curve (30.5 m chord)
D = degree of curve (°); R = radius (m)
Versine of a curve
V = versine; C = chord length; R = radius (same units)
V (mm) = 125C²/R with C and R in metres
Versine on a 20 m chord (curve realignment)
V in mm; R in m
Equilibrium cant
e = cant (mm); G = centre-to-centre of rail heads (1750 mm for BG); V = equilibrium speed (km/h); R (m)
Equilibrium cant on BG
e (mm); V (km/h); R (m)
Cant deficiency
Vmax = maximum permissible speed; Ca = actual cant provided
Cant excess
Vg = booked speed of the slowest (goods) trains
Maximum permissible speed on a transitioned BG curve
V (km/h); Ca, Cd (mm); R (m)
Transition length – rate of change of cant
L (m); Ca (mm); Vm = maximum speed (km/h)
Based on 35 mm/s; adopt the largest of L1, L2, L3
Transition length – rate of change of cant deficiency
Cd = cant deficiency (mm)
Based on 35 mm/s
Transition length – cant gradient
Ca (mm); L3 (m)
Cant gradient 1 in 720; in exceptional cases a cant gradient of 1 in 360 (and 55 mm/s for L1, L2) may be used
Rate of change of cant along a transition
dC/dt (mm/s); Ca (mm); V (km/h); L = transition length (m)
Shift of circular curve
L = transition length; R = radius
Cubic parabola transition
y = offset at distance x from the start of transition; L = transition length
  1. Maximum cant on BG is normally 165 mm; on MG it is 90 mm.
  2. Permissible cant deficiency on BG is 75 mm, increased to 100 mm on Group A and B routes for nominated rolling stock running above 100 km/h.
  3. Permissible cant excess on BG is 75 mm.
  4. On MG, cant deficiency is limited to 50 mm and cant excess to 65 mm.
  5. Cant deficiency arises for trains faster than the equilibrium speed; cant excess arises for trains slower than it.
  6. Cant gradient on a transition should desirably not exceed 1 in 720 (exceptionally 1 in 360).
  7. Indian Railways uses the cubic parabola as the transition curve.
  8. Maximum degree of curve is 10° (R = 175 m) on BG, 16° on MG and 40° on NG.
  9. Group A routes of Indian Railways are those cleared for speeds up to 160 km/h.
  10. Negative cant occurs on the turnout curve when a turnout takes off from a curved main line on the outside of the curve.

11.4 Points & crossings

Crossing number – right-angle (Cole's) method (used on IR)
α = crossing angle; crossing expressed as 1 in N
Crossing number – centre-line method
α = crossing angle
Crossing number – isosceles triangle method
α = crossing angle
Curve lead (Coles' method)
G = gauge; N = crossing number
Switch lead
Ro = radius of outer turnout rail; d = heel divergence
Lead of turnout
CL = curve lead; SL = switch lead
Radius of turnout curve (Coles' method)
G = gauge; N = crossing number
  1. A turnout consists of a pair of points (switches), a crossing, lead rails and check rails.
  2. Points (switches) consist of a pair of tongue rails and stock rails; the tongue rail is thin at its toe and is fixed at the heel.
  3. The crossing (frog) consists of the nose, wing rails and check rails; check rails guide the opposite wheel so the flange passes the gap safely.
  4. The actual nose of crossing (ANC) lies a little behind the theoretical nose of crossing (TNC), where the gauge faces would intersect.
  5. Standard turnouts on IR: 1 in 8.5 for goods lines and sidings, 1 in 12 for passenger running lines and 1 in 16 for higher-speed turnouts.
  6. The larger the crossing number N, the flatter the crossing angle and the higher the permissible speed on the turnout.
  7. A diamond crossing is formed where two tracks cross each other; it has two acute and two obtuse crossings.
  8. A crossover connects two parallel tracks using two turnouts; a scissors crossover provides crossovers in both directions with a diamond in between.
  9. A triangle or turntable is used to reverse the direction of locomotives.

11.5 Gradients & train resistance

Grade compensation on curves (BG)
D = degree of curve
MG 0.03D %, NG 0.02D %; compensated gradient = ruling gradient − GC
Curve resistance (BG)
Rc and W (train weight) in the same units (t); D = degree of curve
MG 0.0003WD; NG 0.0002WD
Gradient resistance
W = train weight; g = gradient (%) (or W/n for a gradient of 1 in n)
Hauling capacity of a locomotive
μ = coefficient of adhesion between wheel and rail; n = number of driving axles; Wa = load per driving axle
  1. Ruling gradient is the steepest gradient to which a section is laid; it fixes the maximum load a locomotive can haul.
  2. Usual ruling gradients on IR: 1 in 150 to 1 in 200 in plains and 1 in 100 to 1 in 150 in hilly areas.
  3. Momentum gradient is steeper than the ruling gradient and can be climbed using the momentum gained on a preceding falling or level stretch.
  4. Pusher (helper) gradient needs an extra banking engine; Bhor Ghat on the Mumbai–Pune line has a gradient of 1 in 37.
  5. Gradients in station yards should not be steeper than 1 in 400; 1 in 1200 is desirable to prevent rolling of stabled vehicles.
  6. Grade compensation is given on curves so that the total resistance on a curve on a ruling gradient does not exceed that on the straight.

11.6 Stations, yards & signalling

  1. Block stations are classified as Class A, B and C; non-block (flag) stations are Class D; special-class stations do not fit these categories.
  2. A block section is the stretch between the last stop signal of one block station and the first stop signal of the next.
  3. Marshalling yards sort and re-form goods trains; they may be flat, gravity or hump yards, the hump yard being the most efficient.
  4. Home signal is the first stop signal of a station; starter controls departure from a line; advanced starter is the last stop signal, beyond which the block section starts.
  5. Warner and distant signals are permissive signals that warn the driver of the aspect of the stop signal ahead.
  6. Interlocking ensures that a signal cannot be taken 'off' unless the points are correctly set and locked and conflicting signals are 'on'.
  7. Under the absolute block system, only one train is allowed in a block section at a time.
  8. In automatic block signalling, track circuits or axle counters detect trains and signals change automatically.
  9. Kavach is the indigenous Automatic Train Protection (ATP) system developed by RDSO.
  10. IRSOD specifies the maximum and minimum fixed dimensions and maximum moving dimensions to keep clearances between rolling stock and structures.
  11. Per IRSOD, height of a high-level BG passenger platform above rail level is 760 mm minimum and 840 mm maximum.
  12. The fouling mark shows the point beyond which a vehicle on one track would infringe the clearance of the adjoining converging track.

11.7 Track maintenance & tunnels

  1. Track geometry parameters monitored are gauge, cross-level, twist, unevenness and alignment.
  2. Track recording cars record track geometry; the oscillation monitoring system (OMS) records vertical and lateral accelerations to judge ride quality.
  3. Ultrasonic flaw detection (USFD) is used to detect internal flaws in rails and welds.
  4. Mechanised maintenance uses tie-tamping machines, ballast cleaning machines (BCM) and dynamic track stabilisers (DTS).
  5. Measured shovel packing (MSP) corrects track levels by adding measured quantities of stone chips under the sleepers.
  6. Through packing is a systematic attention to every sleeper of the track, while picking up slack is attention to isolated low spots.
  7. Horseshoe-shaped tunnels are common in rock for railways and roads; circular sections suit soft ground and high external pressure.
  8. The New Austrian Tunnelling Method (NATM) uses the surrounding rock as part of the support, with shotcrete, rock bolts and continuous monitoring.
  9. Drill-and-blast and tunnel boring machines (TBM) are used in rock; shield tunnelling, needle-beam and fore-poling methods are used in soft ground.
  10. Cut-and-cover construction is used for shallow tunnels such as metro stations and underpasses.
  11. Mucking is the removal of excavated material from the tunnel face.
  12. Alignment is transferred into a tunnel through shafts, for which the Weisbach triangle method is used.
  13. A pilot tunnel (drift) driven ahead of the main tunnel helps in investigation, drainage and ventilation.
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Values follow the current Indian Standards and IRC / Indian Railways documents named in each item. Always check the latest amendment of a code before using a value in design.