🛣️ Highway & Railway Engineering

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.

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Vertical Curve Calculator

PVC · PVI · PVT · K Value · Sight Distance · Elevation Table · Profile Chart

⚙️ Settings

Unit System

Design Standard

🔄 Curve Type

Vertical Curve Type

⛰️ Crest Curve 🏔️ Sag Curve 🔄 Auto Detect

Tangent Type

Equal Tangent Unequal Tangent

📐 Curve Parameters

PVI Station / Chainage (m) Station at point of vertical intersection

PVI Elevation (m) Known elevation at PVI

Initial Grade G1 (%) Positive = ascending, Negative = descending

Final Grade G2 (%) Positive = ascending, Negative = descending

Total Curve Length L (m) Total length of vertical curve from PVC to PVT

🚗 Design Speed & Sight Distance Parameters

Design Speed (km/h) Highway design speed

Road / Track Classification

Eye Height h₁ (m) AASHTO: 1.08m | IRC: 1.20m | Driver eye height

Object Height h₂ (m) AASHTO: 0.60m | IRC: 0.15m | Object on road

Headlight Height h₃ (m) AASHTO: 0.60m | For sag curve headlight criterion

Upward Beam Angle β (°) AASHTO default: 1.0° upward divergence

📋 Elevation Table Interval

📊 Vertical Curve Results

✅ Calculated

📈 Vertical Curve Profile

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.

CORE VERTICAL CURVE FORMULAS: Algebraic Grade Diff: A = G2 − G1 (both in %) K Value: K = L / |A| (m/% or ft/%) PVC Station: PVC = PVI − L/2 (equal tangent) PVT Station: PVT = PVI + L/2 PVC Elevation: PVC_elev = PVI_elev − (G1/100) × L/2 PVT Elevation: PVT_elev = PVI_elev + (G2/100) × L/2 Curve Elevation at x: E(x) = PVC_elev + (G1/100)×x + (A/(200×L))×x² Vertical Offset: y(x) = (A/(200×L)) × x² Grade at x: g(x) = G1 + (A/L)×x (% per unit length) High/Low Point: x_HL = −G1×L/A from PVC High/Low Elevation: E(x_HL) using curve elevation formula

⛰️ 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

For Pakistan National Highway Authority (NHA) design, AASHTO standards apply for motorways (M-roads) and national highways. For Provincial roads, IRC standards are commonly referenced. For Pakistan Railways, AREMA standards govern all railway track design.

📊 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

💡 For Pakistan's motorways (M-1, M-2 etc.) at 120 km/h design speed: minimum K for crest curve SSD = 100. This means for a 5% grade change, minimum curve length = 100 × 5 = 500 metres. For NHA highways at 100 km/h: minimum K = 55.

📝 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.

Given: PVI Station = 1+000.00 m PVI Elevation = 100.000 m G1 = +3.0% G2 = -2.0% L = 200 m Design Speed = 100 km/h Step 1 — Algebraic Difference: A = G2 - G1 = -2.0 - (+3.0) = -5.0% (negative → Crest Curve) |A| = 5.0% Step 2 — K Value Provided: K = L / |A| = 200 / 5.0 = 40.0 m/% Step 3 — Check AASHTO (100 km/h Crest, SSD): K required = 55 → K provided = 40 → FAIL (insufficient!) Min L required = 55 × 5.0 = 275 m Increase L to 275 m minimum for compliance (Using L = 275 m for subsequent calculations) Step 4 — Stations (L = 275 m): PVC Station = 1000 - 275/2 = 862.500 m = 0+862.50 PVT Station = 1000 + 275/2 = 1137.500 m = 1+137.50 Step 5 — Elevations: PVC Elevation = 100.000 - (3.0/100)×137.5 = 95.875 m PVT Elevation = 100.000 + (-2.0/100)×137.5 = 97.250 m Step 6 — High Point: x_HP = -G1 × L / A = -3.0 × 275 / -5.0 = 165.0 m from PVC HP Station = 862.5 + 165.0 = 1+027.50 HP Elevation = 95.875 + (3.0/100)×165 - (5.0/(200×275))×165² = 98.350 m Step 7 — Elevation at Chainage 1+000 (x = 137.5 m): E = 95.875 + (3.0/100)×137.5 + (-5.0/(200×275))×137.5² E = 95.875 + 4.125 - 1.719 = 98.281 m ANSWER: K = 40 < 55 → INADEQUATE. Use L = 275 m minimum. PVC = 0+862.50 at 95.875 m | PVT = 1+137.50 at 97.250 m High Point = 1+027.50 at 98.350 m

💡 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

⚠️ For all structural highway and railway designs, final vertical curve calculations must be verified and signed by a licensed Civil/Transport Engineer. This calculator is for preliminary design and checking purposes.

❓ Frequently Asked Questions

A crest curve (summit curve) is convex upward — it connects an ascending grade to a descending grade (A is negative). It limits forward visibility so it is designed for stopping sight distance. A sag curve (valley curve) is concave upward — it connects a descending grade to an ascending grade (A is positive). Night visibility is limited by headlight range, and comfort is checked against vertical acceleration. Both use the same parabolic equations but have different minimum K values from AASHTO tables.

K value (Rate of Vertical Curvature) = L ÷ |A|, where L is curve length in metres and A is the algebraic grade difference in percent. K represents the length of curve per 1% change in grade. A higher K value means a flatter, more gradual curve. AASHTO prescribes minimum K values for each design speed — e.g. for a 100 km/h crest curve, minimum K = 55. If the provided K is less than the required K, the curve must be lengthened.

The high point (crest curve) or low point (sag curve) is where the gradient = 0. Distance from PVC: x_HL = −G1 × L / A. The point only exists on the curve if 0 < x_HL < L. If x_HL falls outside this range, the high/low point is at one of the tangent ends, not on the curve. Once x_HL is known, substitute it into the curve elevation formula to find the high/low point elevation. This point is critical for drainage design on sag curves.

Pakistan National Highway Authority (NHA) follows AASHTO standards for motorways (M-1, M-2, M-3, M-4 etc.) and national highways. Provincial road departments reference both AASHTO and IRC (Indian Roads Congress) standards. Pakistan Railways follows AREMA (American Railway Engineering and Maintenance-of-Way Association) standards for track geometry. For Gulf-based projects (Saudi Arabia, UAE, Qatar), AASHTO or BS 5400 standards are used depending on the client authority.

Equal tangent (symmetric) vertical curve: the PVI is at the midpoint, so L1 = L2 = L/2. This is the standard type used for most highway design. Unequal tangent (asymmetric) vertical curve: L1 ≠ L2, typically used when geometric constraints (existing structures, drainage points, bridge approaches) prevent the use of a symmetric curve. The parabolic equation is different for each half. This calculator supports both types.

For crest curves, when S < L: L = AS²/[100(√2h₁ + √2h₂)²]. This comes from the geometry of two sight lines from driver eye height h₁ to object height h₂ being tangent to the parabolic curve surface. AASHTO uses h₁ = 1.08m (driver eye height) and h₂ = 0.60m (tail light of vehicle ahead). Substituting these: L = AS²/864. The simplified K relationship: L = K×A, so K = S²/864 when S < L. AASHTO pre-calculates these K values for all standard design speeds in their tables.

Vertical Curve Calculator