Curves & Gradients
Curves & Gradients – Railway GK for All RRB/RPF Exams
1. Technical Essentials
| Term | Definition / Formula | On-field Value |
|---|---|---|
| Degree of Curve (D) | Angle subtended at centre by a 30.48 m (100 ft) chord | R = 1,749/D (metres) |
| Cant (Superelevation) | Difference in height between outer & inner rail | e = GV²/127R (mm) |
| Equilibrium Speed | Speed at which cant deficiency is zero | Vₑ = √(127 R e/G) km/h |
| Maximum Cant | Broad Gauge (BG): 165 mm MG: 90 mm NG: 65 mm | Permitted 185 mm with CMS |
| Transition Curve | Clothoid (spiral) – rate of change of cant ≤ 55 mm/s (BG) | |
| Grade Compensation | BG: 0.04 % per ° curve, MG: 0.03 %, NG: 0.02 % | |
| Ruling Gradient | BG: 1 in 150 (plain), 1 in 200 (hills) MG: 1 in 200 NG: 1 in 250 | |
| Pusher Gradient | 1 in 37 (BG freight with banker) | |
| Minimum Vertical Curve | Summit: L = 0.4 Δ (m) Valley: L = 0.2 Δ (m) Δ = algebraic difference of grade % |
2. Must-Know Facts & Figures
- Sharpest BG curve in use: 8° (R ≈ 218 m) – Harbour line, Mumbai.
- Indian Railways standard chord length for measuring degree of curve = 30.48 m (100 ft legacy).
- Maximum cant deficiency allowed: BG 75 mm, MG 50 mm, NG 40 mm.
- Transition length (BG) ≥ maximum of (e/0.07) and (0.073 D Vₘₐₓ) metres.
- Minimum radius for 160 km/h (Rajdhani): 4,000 m; for 200 km/h (Gatimaan): 6,000 m.
- Steepest ghat section: Bhor Ghat (Mumbai-Pune) – 1 in 37 (pusher/banker compulsory).
- ROR (Rate of Rise of outer rail) ≤ 55 mm/s for passenger comfort.
- High-speed turnout (1 in 12) curve lead radius: 770 m (BG).
- Formula for shift of transition curve: S = L² / 24R (mm).
- Gradients expressed in: ‰ (per mille) on permanent way drawings; 1‰ = 0.1 %.
- Vertical clearance under over-line structures increased from 4.65 m to 5.30 m (2017) to accommodate double-stack containers.
- CBC (Canted Bearing Curve) introduced 2019 – allows 185 mm cant with CMS (Cast Manganese Steel) liners.
3. Historical Milestones
- 1854: First sharp curve (10°) laid on Bori Bunder–Thane; no superelevation concept.
- 1865: “Macdonald” formula adopted – e = V² / 254R (fps units).
- 1928: Indian Railway Conference Association (IRCA) standardised 100 ft chord.
- 1957: IRCA code limited cant to 150 mm BG; revised to 165 mm 1982.
- 1966: First 1 in 37 pusher gradient on Bhor Ghat electrified—25 kV AC.
- 1998: RDSO issued “Guidelines for High-Speed (160 km/h)” mandating 6,000 m min. radius.
- 2014: Dedicated Freight Corridor (DFC) design: 1 in 200 ruling, 1 in 1000 vertical curves.
- 2020: RDSO approved 200 km/h on existing Rajdhani route – major realignment & 120 mm cant deficiency trial.
4. Current Status & Recent Updates (2022-24)
- Gati Shakti: 4000 km of curves realigned to ≥ 2° (R ≥ 875 m) for 130 km/h freight.
- Kavach (ATP): Automatic permanent speed restriction uploaded for every degree-of-curve database.
- DFCCIL: Western DFC – 1 in 200 ruling, but 1 in 100 compensated for 2° curves; no pusher needed.
- PM-Gati Shakti portal live-streams gradient & curve data to 30 m accuracy.
- 2023 Budget: ₹60,000 crore for traffic facility works – 1,000 curves to be transitioned from circular to clothoid.
- Green Corridors: 1 in 300 ruling gradient targeted to reduce diesel consumption by 8 %.
- New CMS (Curved Modular Sleepers) sanctioned – reduces curve maintenance cycle 3 → 1.5 years.
5. Rapid-Fire MCQs
- The chord length used for defining degree of curve in Indian Railways is
A) 20 m
B) 30 m
C) 30.48 m
D) 100 m
Show Answer
Correct Answer: C
Explanation: In Indian Railways, the degree of curve is defined as the central angle subtended by a chord of 30.48 metres (100 feet).
- For a 4° BG curve, the approximate radius is
A) 437 m
B) 583 m
C) 875 m
D) 1749 m
Show Answer
Correct Answer: A
Explanation: Using the formula $R = \frac{1749}{D}$, for a $4^{\circ}$ curve: $R = \frac{1749}{4} \approx 437 \text{ m}$.
- Maximum permissible cant (BG) with CMS liners is
A) 140 mm
B) 165 mm
C) 185 mm
D) 200 mm
Show Answer
Correct Answer: C
Explanation: For Broad Gauge (BG) tracks equipped with Concrete sleepers and CMS liners, the maximum permissible cant (super-elevation) is 185 mm.
- Grade compensation on BG for a 2° curve is
A) 0.04 %
B) 0.06 %
C) 0.08 %
D) 0.10 %
Show Answer
Correct Answer: C
Explanation: The compensation for BG is 0.04% per degree of curve. For a $2^{\circ}$ curve: $0.04 \times 2 = 0.08%$.
- The steepest ruling gradient for BG in plains is
A) 1 in 100
B) 1 in 150
C) 1 in 200
D) 1 in 400
Show Answer
Correct Answer: B
Explanation: While 1 in 200 is common, 1 in 150 is generally accepted as the steepest ruling gradient for Broad Gauge tracks in plain terrain.
- Which ghat section uses 1 in 37 pusher gradient?
A) Thal Ghat
B) Bhor Ghat
C) Shindawane Ghat
D) Braganza Ghat
Show Answer
Correct Answer: B
Explanation: The Bhor Ghat section on the Central Railway uses a steep 1 in 37 gradient requiring pusher locomotives for heavy trains.
- Minimum vertical curve length for summit when $\Delta = 2 %$ is
A) 0.4 m
B) 0.8 m
C) 1.2 m
D) 2.0 m
Show Answer
Correct Answer: B
Explanation: The length of the vertical curve is calculated as $L = 0.4 \times \Delta$. For $\Delta = 2$: $0.4 \times 2 = 0.8 \text{ m}$.
- Cant deficiency for MG is limited to
A) 40 mm
B) 50 mm
C) 75 mm
D) 100 mm
Show Answer
Correct Answer: B
Explanation: For Meter Gauge (MG) tracks, the maximum permissible cant deficiency is limited to 50 mm.
- The transition curve generally adopted on Indian Railways is
A) Cubic Parabola
B) Lemniscate
C) Clothoid (spiral)
D) Circular
Show Answer
Correct Answer: C
Explanation: The Clothoid (or Euler spiral) is the standard transition curve used to provide a linear change in curvature and cant.
- Equilibrium speed formula uses which constant in denominator?
A) 125
B) 127
C) 130
D) 140
Show Answer
Correct Answer: B
Explanation: The equilibrium cant formula is $e = \frac{GV^2}{127R}$, where 127 is the derived constant for metric units.
- For 160 km/h, minimum curve radius recommended by RDSO is
A) 1750 m
B) 2500 m
C) 4000 m
D) 5000 m
Show Answer
Correct Answer: C
Explanation: For high-speed operations of 160 km/h, RDSO recommends a minimum radius of 4,000 m to ensure safety and comfort.
- Rate of change of cant on BG should not exceed
A) 35 mm/s
B) 45 mm/s
C) 55 mm/s
D) 65 mm/s
Show Answer
Correct Answer: C
Explanation: To avoid passenger discomfort and ensure stability, the rate of change of cant is restricted to 55 mm per second.
- Shift of a transition curve is given by
A) $S = \frac{L}{24R}$
B) $S = \frac{L^2}{24R}$
C) $S = \frac{L^2}{12R}$
D) $S = \frac{LR}{24}$
Show Answer
Correct Answer: B
Explanation: The lateral shift ($S$) required to fit a transition curve between a tangent and a circular curve is $S = \frac{L^2}{24R}$.
- The first electrified ghat gradient was commissioned in the year
A) 1925
B) 1947
C) 1966
D) 1970
Show Answer
Correct Answer: C
Explanation: The electrification of major steep ghat gradients on Indian Railways was successfully commissioned in 1966.
- DFCCIL design ruling gradient is
A) 1 in 150
B) 1 in 200
C) 1 in 400
D) 1 in 1000
Show Answer
Correct Answer: B
Explanation: The Dedicated Freight Corridor Corporation of India (DFCCIL) uses a ruling gradient of 1 in 200 for its heavy-haul tracks.
- The maximum cant deficiency recently trialled for 200 km/h operation is
A) 100 mm
B) 120 mm
C) 150 mm
D) 185 mm
Show Answer
Correct Answer: B
Explanation: Recent trials for increasing sectional speeds to 200 km/h have tested a maximum cant deficiency of 120 mm.
Last Updated: January 2026
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