Vertical Curve Calculator
Professional vertical curve design tool for highway and railway engineers. Calculate crest and sag curves with full elevation tables, K values, sight distance compliance, and interactive profile chart — following AASHTO, IRC, AREMA, and BS standards.
Vertical Curve Calculator
PVC · PVI · PVT · K Value · Sight Distance · Elevation Table · Profile Chart
Vertical Curve Results
Calculated| Station | Distance x | Tangent Elev | Offset y | Curve Elev | Grade % | Note |
|---|
How Vertical Curve Calculation Works — Step-by-Step
A vertical curve is a parabolic curve in the vertical plane used to join two road or railway gradients (grades) smoothly. It eliminates the abrupt change in grade at the intersection point and ensures driver comfort, sight distance compliance, and drainage. Every highway and railway design requires vertical curves wherever grades change.
Define the Grades and PVI
The Point of Vertical Intersection (PVI) is where the two tangent grade lines meet. G1 is the incoming grade (positive = ascending) and G2 is the outgoing grade. The algebraic difference A = G2 − G1 determines the sharpness of the curve. A negative A value means a crest curve; positive means a sag curve.
Determine PVC and PVT Stations
For an equal-tangent curve: PVC = PVI − L/2 and PVT = PVI + L/2. For unequal-tangent: PVC = PVI − L1, PVT = PVI + L2. PVC is the Point of Vertical Curvature (start of curve); PVT is the Point of Vertical Tangency (end of curve).
Calculate Elevations Along the Curve
The standard parabolic equation for any point at distance x from PVC: Elevation = PVC_elev + (G1/100)×x + (A/(200×L))×x². The second term is the tangent elevation; the third term is the vertical offset (parabolic correction). At x=0 the elevation equals PVC; at x=L it equals PVT.
Find the High Point (Crest) or Low Point (Sag)
The high or low point occurs where the gradient is zero. Distance from PVC: x_HP = −G1×L / A. Valid only when 0 < x < L. This point is critical for drainage design in sag curves and maximum elevation in crest curves.
Calculate K Value and Check Sight Distance
K = L / |A| — the rate of vertical curvature (metres or feet per percent grade change). A higher K value means a flatter curve with better sight distance. Compare the provided K against the minimum K required by AASHTO tables for the design speed and curve type.
Generate Elevation Table and Profile
Compute elevations at regular intervals (5m, 10m, 20m etc.) along the curve for setting-out in the field. The profile chart visualises the tangent grades and parabolic curve together with all control points labelled.
Crest vs Sag Vertical Curves — Comparison
The two types of vertical curves serve very different design purposes and are checked against different criteria.
| Aspect | Crest Curve (Summit) | Sag Curve (Valley) |
|---|---|---|
| Grade Change A | A is negative (G2 < G1) | A is positive (G2 > G1) |
| Shape | Convex upward (like a hill) | Concave upward (like a valley) |
| Critical Check | Stopping Sight Distance (SSD) | Headlight Sight Distance |
| Secondary Check | Passing Sight Distance (PSD) | Riding comfort (vertical accel.) |
| Drainage | Drains to both sides — good | Water collects — design inlet |
| K Criterion | K ≥ K_SSD from AASHTO Table | K ≥ K_HL from AASHTO Table |
| Min K (100 km/h) | K ≥ 55 (SSD) | K ≥ 205 (PSD) | K ≥ 37 (headlight) |
| High/Low Point | High Point at x = -G1L/A | Low Point at x = -G1L/A |
AASHTO K Value Reference Tables — All Design Speeds
The K value determines the minimum curve length for a given design speed. A larger K means a flatter, longer curve. The values below are from AASHTO Green Book 2018 (Policy on Geometric Design of Highways and Streets).
Crest Curves — Minimum K Values
| Design Speed | SSD (m) | K min (SSD) | PSD (m) | K min (PSD) | Min L = K×|A| |
|---|---|---|---|---|---|
| 60 km/h | 110 m | 11 | 490 m | 46 | 11×|A| |
| 70 km/h | 140 m | 17 | 560 m | 70 | 17×|A| |
| 80 km/h | 170 m | 26 | 640 m | 105 | 26×|A| |
| 90 km/h | 200 m | 39 | 720 m | 148 | 39×|A| |
| 100 km/h | 240 m | 55 | 800 m | 205 | 55×|A| |
| 110 km/h | 280 m | 73 | 880 m | 271 | 73×|A| |
| 120 km/h | 320 m | 100 | 960 m | 354 | 100×|A| |
| 130 km/h | 360 m | 133 | 1040 m | 455 | 133×|A| |
Sag Curves — Minimum K Values (Headlight Criterion)
| Design Speed | SSD (m) | K min (Headlight) | Min L for Comfort |
|---|---|---|---|
| 60 km/h | 110 m | 14 | L ≥ AV²/390 |
| 70 km/h | 140 m | 18 | L ≥ AV²/390 |
| 80 km/h | 170 m | 24 | L ≥ AV²/390 |
| 90 km/h | 200 m | 30 | L ≥ AV²/390 |
| 100 km/h | 240 m | 37 | L ≥ AV²/390 |
| 110 km/h | 280 m | 46 | L ≥ AV²/390 |
| 120 km/h | 320 m | 55 | L ≥ AV²/390 |
| 130 km/h | 360 m | 65 | L ≥ AV²/390 |
Worked Example — Crest Vertical Curve NHA Highway
Problem: Design a crest vertical curve at a National Highway chainage 1+000 where a +3% grade meets a −2% grade. PVI elevation = 100.000m. Design speed = 100 km/h. Check AASHTO compliance.
Expert Design Tips for Vertical Curves
Minimum Length Rule
- Minimum curve length should never be less than 3× the design speed in km/h (e.g. for 100 km/h → minimum L = 300m) for appearance and comfort, regardless of sight distance requirements
- For very small grade changes (A < 0.4%), a vertical curve may not be required — check local authority standards
- Always round the final curve length up to the nearest 10m or 25m for practical setting-out
Drainage Considerations
- At the low point of sag curves, install road drainage inlets to prevent water accumulation on carriageway
- Minimum longitudinal grade on any road section should be ≥ 0.5% for adequate drainage — avoid flat grades near sag curve low points
- For sag curves, AASHTO recommends minimum K = 51 (metric) to ensure adequate drainage flow
Setting Out in Field
- Peg out at 5m or 10m intervals using the elevation table generated by this calculator
- Double-check PVC, PVI, and PVT stations with a total station before earthwork begins
- Always recheck the high/low point elevation — this is the critical control point for cut/fill quantities
