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
1. The chord length used for defining degree of curve in Indian Railways is
**Answer:** 30.48 m2. For a 4° BG curve, the approximate radius is
**Answer:** 437 m (1749/4)3. Maximum permissible cant (BG) with CMS liners is
**Answer:** 185 mm4. Grade compensation on BG for a 2° curve is
**Answer:** 0.08 % (0.04 % × 2)5. The steepest ruling gradient for BG in plains is
**Answer:** 1 in 1506. Which ghat section uses 1 in 37 pusher gradient?
**Answer:** Bhor Ghat7. Minimum vertical curve length for summit when Δ = 2 % is
**Answer:** 0.8 m (0.4 × 2)8. Cant deficiency for MG is limited to
**Answer:** 50 mm9. The transition curve generally adopted on Indian Railways is
**Answer:** Clothoid (spiral)10. Equilibrium speed formula uses which constant in denominator?
**Answer:** 12711. For 160 km/h, minimum curve radius recommended by RDSO is
**Answer:** 4,000 m12. Rate of change of cant on BG should not exceed
**Answer:** 55 mm/s13. Shift of a transition curve is given by
**Answer:** S = L² / 24R14. The first electrified ghat gradient was commissioned in the year
**Answer:** 196615. DFCCIL design ruling gradient is
**Answer:** 1 in 20016. The maximum cant deficiency recently trialled for 200 km/h operation is
**Answer:** 120 mmRevision Tip: Remember “1749” (R = 1749/D) & “127” (V & e formulae) – they solve 50 % of curve questions!
BETA
