6.1 Vertical Alignment: Tangent Grades, Crest & Sag Vertical Curves

6.1 Vertical Alignment: Tangent Grades, Crest & Sag Vertical Curves

PTOE Exam Focus: Vertical alignment questions on the PTOE examination test candidates on equal-tangent parabolic curve geometry, rate of vertical curvature ($K = L/A$), high/low point station and elevation derivations, and the governing criteria for crest versus sag curves. Master the sight distance formulas ($S \le L$ vs. $S > L$), nighttime headlight beam geometry, rider comfort limits ($a_z \le 1.0\text{ ft/s}^2$), and minimum longitudinal slope requirements for curbed roadway drainage.

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1. Tangent Grades & Vertical Alignment Controls

Vertical alignment represents the longitudinal profile of a roadway centerline, consisting of constant-slope tangent grades connected by smooth vertical curves. The selection of maximum and minimum tangent grades depends directly on highway functional classification, design speed, and terrain topography (flat, rolling, or mountainous):

• Maximum Tangent Grades: AASHTO establishes maximum allowable grades to prevent excessive speed differentials between heavy commercial vehicles and passenger cars. On high-speed freeways in flat terrain, maximum grades are typically restricted to $3\text{--}4%$, whereas low-speed urban collectors or mountainous arterials may permit grades of $6\text{--}9%$ (and up to $12%$ on low-volume local roads).
• Critical Length of Grade: Defined as the maximum length of an ascending grade that a designated heavy commercial vehicle ($200\text{ lb/hp}$ design truck) can traverse without incurring an intolerable speed reduction (standard threshold is a $10\text{ mph}$ speed drop). When an upgrade exceeds this threshold and design hourly truck volumes exceed warrants, a dedicated climbing lane is justified.
• Minimum Tangent Grades: On uncurbed rural highways with adequate cross-slope, longitudinal grade can theoretically be $0.0%$. However, on curbed urban roadways, AASHTO mandates a minimum longitudinal grade of $0.30%\text{ to }0.50%$ (ideally $\ge 0.50%$) to facilitate gutter flow and prevent water ponding, hydroplaning, and sediment deposition.

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2. Mathematical Geometry of Equal-Tangent Parabolic Curves

Highway vertical curves utilize equal-tangent vertical parabolas because a parabola provides a constant rate of change of grade along its horizontal length, delivering a smooth transition in centrifugal vertical acceleration. The curve is established between three critical control points:

                    PVI (Point of Vertical Intersection)
                           /\
  Initial Tangent         /  \         Exit Tangent
  Grade = g1             /    \        Grade = g2
  (+) Upgrade           /      \       (-) Downgrade
                       / Curve  \
                      /__________\
                    PVC          PVT
  |<----------------- L / 2 ------->|<----------------- L / 2 ------->|
  |<-------------------------------- L ------------------------------>|

• PVC (Point of Vertical Curvature): The point of beginning of the vertical curve.
• PVI (Point of Vertical Intersection): The intersection point of the initial and final tangent lines.
• PVT (Point of Vertical Tangency): The point of ending of the vertical curve.
• $L$: The horizontal length of the vertical curve in feet (measured in the horizontal projection, not along the curve arc). In equal-tangent curves, the PVI is located exactly at the midpoint ($L/2$) between PVC and PVT.

General Parabolic Formulations

The elevation $y$ at any horizontal distance $x$ (in feet) from the PVC ($0 \le x \le L$) is expressed by the quadratic polynomial: $y(x) = y_{\text{PVC}} + g_1 x + \left( \frac{g_2 - g_1}{2L} \right) x^2$ Where:

• $y_{\text{PVC}}$ = roadway elevation at the PVC (ft)
• $g_1$ = initial approach tangent grade expressed as a decimal ($g_1 = G_1 % / 100$)
• $g_2$ = exit tangent grade expressed as a decimal ($g_2 = G_2 % / 100$)
• $x$ = horizontal distance from the PVC (ft)
• $L$ = total horizontal length of the vertical curve (ft)

In terms of percentage grades ($G_1$ and $G_2$): $y(x) = y_{\text{PVC}} + \frac{G_1 x}{100} + \frac{(G_2 - G_1) x^2}{200 L}$

Rate of Vertical Curvature ($K$)

The rate of vertical curvature, denoted as $K$, is the fundamental metric used in AASHTO geometric design: $K = \frac{L}{A}$ Where:

• $L$ = horizontal curve length (ft)
• $A$ = algebraic difference in tangent grades ($A = |G_1 - G_2|$, expressed in percent)

Physical Meaning of $K$: $K$ represents the horizontal distance in feet required to effect a $1%$ change in grade. A larger $K$-value corresponds to a flatter, more gradual vertical curve with greater sight distance.

Tangent Offsets ($e$)

The vertical offset from the initial tangent line to the curve at any distance $x$ from the PVC is: $y_{\text{offset}}(x) = a x^2 = \frac{A}{200 L} x^2$ The maximum vertical offset occurs at the PVI ($x = L/2$) and is termed the external distance ($E$): $E = \frac{A L}{800} = \frac{|G_1 - G_2| L}{800}$

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3. Derivation of High and Low Turning Points

The turning point (the high point on a crest vertical curve or the low point on a sag vertical curve) occurs where the instantaneous longitudinal slope of the roadway is exactly zero ($dy/dx = 0$).

$\frac{dy}{dx} = \frac{G_1}{100} + \frac{(G_2 - G_1) x}{100 L} = 0$ Solving for the distance $x_m$ from the PVC: $x_m = \frac{-G_1 L}{G_2 - G_1} = \frac{|G_1| L}{A} = |G_1| \times K$

Elevation of the Turning Point ($y_m$):

Substituting $x_m$ back into the curve elevation equation yields: $y_m = y_{\text{PVC}} + \frac{G_1 x_m}{200} = y_{\text{PVC}} + \frac{G_1^2 L}{200 A}$

Engineering and Drainage Significance:

• Turning Point Location: If $G_1$ and $G_2$ have identical signs (e.g., $+4%$ to $+1%$), the curve is monotonic and no turning point exists within the curve boundaries ($x_m$ falls outside $[0, L]$).
• Drainage Flat Spot Constraint: At the exact apex or sag vertex ($x = x_m$), the grade is $0.0%$. Within a distance of $50\text{ ft}$ on either side of the vertex ($100\text{ ft}$ total window), the longitudinal grade remains flatter than $0.35%$. On curbed roadway sections, AASHTO recommends that $K$ should not exceed $167\text{ ft/%}$ unless special drainage provisions (e.g., warped cross-slopes, closely spaced curb inlets, or flanking inlets) are engineered to prevent hydroplaning and standing water.

AASHTO Design K-Values for Crest and Sag Vertical Curves (Stopping Sight Distance)

4. Crest Vertical Curves: Stopping & Decision Sight Distance

Crest vertical curves are convex curves ($G_1 > G_2$) where sight distance is restricted by the roadway crest itself. AASHTO establishes curve lengths based on Stopping Sight Distance (SSD), using standardized geometric sight line parameters:

• Driver eye height ($h_1$): $3.5\text{ ft}$ above the pavement.
• Object height ($h_2$): $2.0\text{ ft}$ above the pavement (representing the taillight height of a passenger car or a compact roadway hazard).

                   Eye Line (h1 = 3.5 ft)
                   \                         Object (h2 = 2.0 ft)
                    \     Sight Line          /
                     o======================* 
                    /   _---"""""---
                   / _-"   Crest    "-_ \
                  PVI-------------------PVI

Governing Mathematical Formulations:

1. When Sight Distance is Less than Curve Length ($S \le L$): $L = \frac{A S^2}{100 \left( \sqrt{2 h_1} + \sqrt{2 h_2} \right)^2} = \frac{A S^2}{100 \left( \sqrt{2(3.5)} + \sqrt{2(2.0)} \right)^2} = \frac{A S^2}{2158}$ Expressed in terms of the rate of vertical curvature: $K = \frac{L}{A} = \frac{S^2}{2158}$

2. When Sight Distance Exceeds Curve Length ($S > L$): $L = 2S - \frac{200 \left( \sqrt{h_1} + \sqrt{h_2} \right)^2}{A} = 2S - \frac{2158}{A}$

Passing Sight Distance (PSD) & Decision Sight Distance (DSD):

• Passing Sight Distance: Uses an eye height of $h_1 = 3.5\text{ ft}$ and an opposing vehicle height of $h_2 = 3.5\text{ ft}$. The resulting formula yields $K = S^2 / 2800$. Because passing sight distances are substantial ($> 1000\text{ ft}$), crest curve lengths for PSD require $K$-values $4\text{--}8$ times larger than SSD.
• Decision Sight Distance: Applied at complex interchanges, lane drops, or toll plazas where drivers require additional perception-reaction time ($9.5\text{--}14.5\text{ s}$). $K$-values for DSD Avoidance Maneuvers C, D, and E are significantly larger than basic SSD.

5. Sag Vertical Curves: Governing Design Criteria

Sag vertical curves are concave curves ($G_1 < G_2$). Under daytime conditions, driver sight distance on sag curves is generally unrestricted unless an overhead bridge, sign structure, or tree canopy creates a visual obstruction. Therefore, sag vertical curve lengths are determined by four distinct criteria:

1. Headlight Sight Distance Criterion (Primary Nighttime Control)

At night, the distance a driver can see is constrained by the illumination envelope of the vehicle headlights. AASHTO standard assumptions:

• Headlight mounting height ($h_3$): $2.0\text{ ft}$.
• Upward divergence beam angle ($\theta$): $1.0^\circ$.

For $S \le L$: $L = \frac{A S^2}{200 \left( h_3 + S \tan 1^\circ \right)} = \frac{A S^2}{200 \left( 2.0 + S \times 0.017455 \right)} = \frac{A S^2}{400 + 3.5 S}$ $K_{\text{headlight}} = \frac{S^2}{400 + 3.5 S}$

For $S > L$: $L = 2S - \frac{400 + 3.5 S}{A}$

2. Rider Comfort Criterion (Dynamic Vertical Centrifugal Acceleration)

When a vehicle traverses a concave sag curve at design speed $V$, the downward centrifugal force combines with gravity, increasing the apparent weight on the passenger. AASHTO adopts a maximum acceptable vertical acceleration threshold of $a_z \le 1.0\text{ ft/s}^2$ (approximately $0.03 g$): $a_z = \frac{v^2}{R} = \frac{(1.467 V)^2}{R}$ Approximating radius of curvature as $R \approx \frac{L}{\Delta g} = \frac{100 L}{A}$: $a_z = \frac{2.151 V^2 \times A}{100 L} = \frac{A V^2}{46.5 L}$ Setting $a_z = 1.0\text{ ft/s}^2$ and solving for $L$ and $K$: $L = \frac{A V^2}{46.5} \implies K_{\text{comfort}} = \frac{V^2}{46.5}$

Key Exam Insight: On lit urban expressways or where continuous street lighting illuminates the corridor, the headlight sight distance criterion may be relaxed to the rider comfort criterion, allowing substantially shorter sag curve lengths ($K_{\text{comfort}} \approx 40%\text{ to }50%$ of $K_{\text{headlight}}$).

3. Drainage Criterion

Just like crest curves, sag curves develop a flat region near the low point ($x_m = |G_1| K$). To prevent ponding across travel lanes on curbed cross-sections, $K$ should not exceed $167\text{ ft/%}$.

4. General Appearance Criterion

For aesthetic smoothness, very short vertical curves appear as abrupt "kinks" in the longitudinal profile. AASHTO recommends a minimum curve length of $L_{\min} \ge 3 V$ (where $V$ is in mph) or $L_{\min} \ge 100 A$.

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6. Worked PTOE Calculation Examples

Example 1: Crest Vertical Curve Turning Point & Elevation

Problem: A crest vertical curve connects an approach grade of $G_1 = +3.5%$ to an exit grade of $G_2 = -2.5%$. The Point of Vertical Intersection (PVI) is located at Station $145+00$ at Elevation $850.00\text{ ft}$. The design speed is $60\text{ mph}$ ($SSD = 570\text{ ft}$, $K = 151\text{ ft/%}$).

1. Determine the minimum required curve length $L$ rounded up to the nearest $100\text{ ft}$.
2. Find the station and elevation of the PVC and PVT.
3. Determine the station and elevation of the high point (crest turning point).

Solution:

1. Algebraic difference: $A = |3.5 - (-2.5)| = 6.0%$. Minimum curve length: $L = K \times A = 151 \times 6.0 = 906\text{ ft} \to \text{Round to } L = 1000\text{ ft}$. Actual design $K = 1000 / 6.0 = 166.67\text{ ft/%}$.
2. PVC Station = $\text{PVI} - L/2 = (145+00) - (5+00) = \text{Station } 140+00$. PVC Elevation = $\text{Elev}{\text{PVI}} - G_1 (L/2) = 850.00 - 0.035(500) = 832.50\text{ ft}$. PVT Station = $\text{PVI} + L/2 = (145+00) + (5+00) = \text{Station } 150+00$. PVT Elevation = $\text{Elev}{\text{PVI}} + G_2 (L/2) = 850.00 + (-0.025)(500) = 837.50\text{ ft}$.
3. High point distance from PVC: $x_m = \frac{|G_1| L}{A} = \frac{3.5 \times 1000}{6.0} = 583.33\text{ ft}$. High Point Station = $(140+00) + 583.33 = \text{Station } 145+83.33$. High Point Elevation = $y_{\text{PVC}} + \frac{G_1 x_m}{200} = 832.50 + \frac{3.5 \times 583.33}{200} = 832.50 + 10.21 = 842.71\text{ ft}$.