6.2 Crest Vertical Curve Geometry and Sight Distance

6.2 Crest Vertical Curve Geometry and Sight Distance

Geometry of Crest Curves

A crest vertical curve is a convex transition between two tangent grades. These curves are designed using equal-tangent parabolas. In an equal-tangent curve, the horizontal distance from the Point of Vertical Curvature (PVC) to the Point of Vertical Intersection (PVI) is equal to the horizontal distance from the PVI to the Point of Vertical Tangency (PVT). Thus, the total length of the curve ($L$) is divided into two equal halves of length $L/2$.

The primary design parameter is the rate of vertical curvature, $K$, defined as:

$K = \frac{L}{A}$

Where:

• $L$ = Length of vertical curve (feet or meters)
• $A$ = Algebraic difference in grades ($|G_2 - G_1|$, expressed in percent, e.g., $A = 5$ for a change from $+3%$ to $-2%$)

The $K$-value represents the horizontal distance required to effect a $1%$ change in grade. It is a convenient index used by designers to quickly calculate the minimum curve length required for a given design speed: $L_{\min} = K \cdot A$.

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Sight Distance Criteria

The minimum length of a crest vertical curve is dictated by the need to provide adequate Stopping Sight Distance (SSD). When a vehicle traverses a crest, the roadway itself acts as a vertical obstruction blocking the driver's line of sight to an object on the road ahead.

Mathematical Formulation

The relationship between sight distance ($S$), curve length ($L$), and the heights of the driver's eye and the object are modeled mathematically using two cases:

Case 1: Sight Distance is Less Than or Equal to Curve Length ($S \le L$)

In this case, both the driver and the object are within the limits of the vertical curve. The minimum length of the curve is given by:

$L = \frac{A S^2}{100 \left( \sqrt{2 h_1} + \sqrt{2 h_2} \right)^2}$

Where:

• $h_1$ = Driver's eye height above the road surface
• $h_2$ = Object height above the road surface

Using standard AASHTO design criteria for Stopping Sight Distance (SSD):

• Driver's eye height: $h_1 = 3.5\text{ ft}$ ($1.08\text{ m}$)
• Object height (representing a hazard like a stalled car): $h_2 = 2.0\text{ ft}$ ($0.60\text{ m}$)

Substituting these values into the general equation yields:

• US Customary Units: $L = \frac{A S^2}{2158}$
• Metric Units: $L = \frac{A S^2}{658}$

Case 2: Sight Distance is Greater Than Curve Length ($S > L$)

In this case, one or both of the vehicles/objects are located on the tangent grades outside the curve. The equation is:

$L = 2S - \frac{100 \left( \sqrt{2 h_1} + \sqrt{2 h_2} \right)^2}{A}$

Substituting standard AASHTO values yields:

• US Customary Units: $L = 2S - \frac{2158}{A}$
• Metric Units: $L = 2S - \frac{658}{A}$

Exam Tip: When solving exam problems, always assume $S \le L$ first and calculate $L$. If the resulting $L$ is greater than or equal to $S$, your assumption is correct. If the calculated $L$ is less than $S$, you must re-calculate using the $S > L$ equation.

Passing Sight Distance (PSD)

If the highway is a two-lane facility where passing is permitted, the design must provide Passing Sight Distance (PSD). For PSD, the object height ($h_2$) is replaced by the height of an oncoming passenger car, which AASHTO defines as $3.5\text{ ft}$ ($1.08\text{ m}$).

Substituting $h_1 = 3.5\text{ ft}$ and $h_2 = 3.5\text{ ft}$ yields:

• US Customary (PSD): $L = \frac{A S^2}{2800}$ (for $S \le L$) and $L = 2S - \frac{2800}{A}$ (for $S > L$)
• Metric (PSD): $L = \frac{A S^2}{864}$ (for $S \le L$) and $L = 2S - \frac{864}{A}$ (for $S > L$)

Because PSD requires significantly larger sight distances than SSD, designing a crest curve for passing sight distance results in a much longer curve, which is often economically impractical. Passing is typically prohibited over crests unless terrain is exceptionally flat.

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High Point Location and Elevation

Locating the highest point (vertex) of a crest vertical curve is critical for drainage design. To ensure that surface water does not pool on the roadway, catch basins or drainage inlets must be located at or near the high point.

Station of the High Point

For a curve with an equal-tangent parabola, the horizontal distance ($x_m$) from the PVC to the high point is calculated using:

$x_m = \frac{G_1 L}{G_1 - G_2} = \frac{G_1 L}{A}$

Where:

• $G_1$ = Slope of the entering tangent (in percent)
• $L$ = Total length of the vertical curve (in feet or meters)
• $A$ = Algebraic difference in grades ($G_1 - G_2$, in percent)

Note: This equation only applies when the high point lies within the curve limits. If both grades are positive ($G_1 > 0$ and $G_2 > 0$), the highest point is at the PVT ($x_m = L$). If both are negative, the highest point is at the PVC ($x_m = 0$).

Once $x_m$ is determined, the station of the high point is:

$\text{Sta}_{HP} = \text{Sta}_{PVC} + x_m$

Elevation of the High Point

The elevation of the high point ($E_{HP}$) can be calculated by applying the standard parabolic offset equation:

$E_{HP} = E_{PVC} + \left( \frac{G_1}{100} \right) x_m - \frac{A x_m^2}{200 L}$

Alternatively, a simplified shortcut formula can be used:

$E_{HP} = E_{PVC} + \frac{G_1^2 L}{200 A}$

Where $G_1$ and $A$ are expressed in percent (e.g., 3 for $3%$ grade).

Step-by-Step Design Example

Consider a crest curve connecting a $+3%$ grade ($G_1$) to a $-2%$ grade ($G_2$). The PVC station is $20+00$ and its elevation is $100.00\text{ ft}$. The design speed of 60 mph requires a stopping sight distance of 570 feet.

1. Calculate Grade Difference (A): $A = |G_2 - G_1| = |-2 - 3| = 5.0\%$
2. Find Minimum Curve Length (L): Assume $S \le L$ and use US Customary SSD formula: $L = \frac{A S^2}{2158} = \frac{5.0 \cdot 570^2}{2158} = \frac{1,624,500}{2158} \approx 752.8\text{ ft}$ Since $570\text{ ft} \le 752.8\text{ ft}$, our assumption is correct. The design curve length is chosen as $800\text{ ft}$ (rounded up to a standard value).
3. Locate the High Point: $x_m = \frac{G_1 L}{A} = \frac{3.0 \cdot 800}{5.0} = 480\text{ ft}$ $\text{Sta}_{HP} = (20+00) + 480 = 24+80$
4. Calculate High Point Elevation: $E_{HP} = E_{PVC} + \frac{G_1^2 L}{200 A} = 100.00 + \frac{3.0^2 \cdot 800}{200 \cdot 5.0} = 100.00 + \frac{7200}{1000} = 107.20\text{ ft}$