What is a Valley Curve (Sag Curve)?
A valley curve (also called a sag curve) is a vertical curve with concavity upward (or convexity downward). Unlike a summit curve which forms a hump in the road profile, a valley curve forms a dip or hollow — the road descends into it and then rises out the other side. It occurs wherever two downward gradients meet, or where a downgrade transitions to a flat or upgrade section.
Valley curves present a different design challenge from summit curves: during daytime, there is no sight distance problem because a driver can see the road ahead as it dips and rises. However, at night, with only headlights for illumination, the road ahead in the dip is not visible beyond the headlight beam range — creating a night visibility hazard.
Four Conditions That Create a Valley Curve
| Case | Meeting Gradients | Deviation Angle N |
|---|---|---|
| (a) | −n₁ meets −n₂ (where |n₁| < |n₂|) | N = n₁ + n₂ |
| (b) | −n₁ meets flat (n₂ = 0) | N = n₁ |
| (c) | −n₁ meets +n₂ | N = −(n₁ + n₂) |
| (d) | +n₁ meets +n₂ (where n₁ < n₂) | N = n₂ − n₁ |
Why Valley Curve Uses Cubic Parabola
Unlike summit curves (which use square parabola), IRC recommends the cubic parabola for valley curves. This is because the valley curve needs to be fully transitional — it consists of two equal spiral (transition) curves joined at the lowest point. The cubic parabola satisfies this transitional requirement, introducing curvature gradually from the tangent points. The valley curve is set out using the cubic parabola equation.
Design Criteria for Valley Curves
1. Comfort Criterion (Rate of Change of Centrifugal Acceleration)
At a valley curve, the centrifugal force acts downward (adding to the vehicle’s weight), unlike at a summit where it acts upward. This means the centrifugal force at a valley curve creates additional pressure on the vehicle suspension and passenger discomfort. The comfort criterion limits the rate of change of centrifugal acceleration:
Ls = √(Nv³/C)
Where C = 0.6 m/s³ (IRC), v = speed in m/s, N = deviation angle. This is more restrictive than the C value used for transition curves (0.5–0.8) because passenger discomfort at valley curves is more pronounced.
2. Safety Criterion (Head Light Sight Distance)
At night, visibility is limited to the distance illuminated by headlights. The valley curve must ensure that the headlight beam — projected at the standard beam angle of 1° from a headlight height of 0.75 m — illuminates the road ahead for at least the stopping sight distance (HSD = SSD).
When L > S: L = NS² / (1.5 + 0.035S)
When L < S: L = 2S − (1.5 + 0.035S)/N
The design length is the maximum of the comfort and HSD criteria.
Summit vs Valley Curve — Key Differences
| Property | Summit Curve | Valley Curve |
|---|---|---|
| Shape | Convex upward (hump) | Concave upward (dip) |
| Critical issue | Sight distance (day AND night) | Night visibility + comfort |
| Curve type | Square parabola | Cubic parabola |
| Centrifugal force | Upward (opposing gravity — comfortable) | Downward (adds to weight — uncomfortable) |
| Day visibility | Restricted at peak | No restriction |
| Night visibility | Critical | Very critical (HSD governs) |
| Design criteria | SSD / OSD only | Comfort + HSD (take maximum) |
Key Takeaways
- Valley curve = sag curve = concave upward = dip in road profile
- No daytime sight problem; night visibility (HSD) is the critical issue
- IRC uses cubic parabola (fully transitional) — not square parabola
- Two design criteria: comfort (C = 0.6 m/s³) and HSD safety — take maximum
- Headlight parameters: height h = 0.75 m, beam angle β = 1° (IRC)
- Centrifugal force acts downward at valley — adds to passenger discomfort