Study guide
6.1 Vertical Alignment: Tangent Grades, Crest & Sag Vertical Curves
Professional Traffic Operations Engineer PTOE
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1 Introduction & PTOE Exam Overview
1.1 The PTOE Credential: Certification Overview & Industry Value 1.2 Core Industry Reference Standards in Traffic Operations 1.3 Study Guide Architecture, Blueprint Alignment & Exam-Day Strategy
2 Domain 1: Traffic Flow Theory & Fundamental Concepts
2.1 Macroscopic Traffic Stream Models & Fundamental Diagram (q = k × u) 2.2 Microscopic Traffic Characteristics & Car-Following Models 2.3 Hydrodynamic Shockwave Analysis & Traffic Queuing Models
3 Domain 1: Highway & Interrupted Flow Capacity Analysis (HCM)
3.1 Basic Freeway Segments & Multilane Highways (HCM 7th Edition) 3.2 Freeway Weaving, Merging, and Diverging Segments 3.3 Signalized Intersection Capacity & Delay Modeling (HCM)
4 Domain 1: Managed Lanes, Transit Priority, TSMO & Travel Demand Management
4.1 Unsignalized Intersections (TWSC, AWSC) & Modern Roundabouts Capacity 4.2 Managed Lanes, HOV/HOT Operations, & Ramp Metering 4.3 Transit Priority & Multimodal Signal Operations (TSP) 4.4 Transportation Systems Management and Operations (TSMO) & Regional ITS 4.5 Travel Demand Management (TDM) & Trip Reduction Strategies
5 Domain 2: Geometric Design Controls, Criteria, & Sight Distance
5.1 Functional Classification, Design Controls, & Context-Sensitive Design 5.2 Sight Distance Principles: Stopping, Decision, & Passing (AASHTO) 5.3 Horizontal Alignment, Curve Radius, & Superelevation Design
6 Domain 2: Roadway Cross-Sections & Access Management
6.1 Vertical Alignment: Tangent Grades, Crest & Sag Vertical Curves 6.2 Roadway Cross-Section Elements & AASHTO Clear Zone Concepts 6.3 Access Management Principles & Driveway Spacing Criteria
7 Domain 2: Intersection & Interchange Geometric Design
7.1 Intersection Channelization, Turn Lane Design, & Storage Length 7.2 Interchange Configurations & Operational Comparisons (Diamond, Cloverleaf, SPUI) 7.3 Diverging Diamond Interchanges (DDI / DCMI) Operational Principles 7.4 Modern Roundabout Geometric Design Elements & Fastest Paths 7.5 Multimodal Geometric Design: Pedestrian Facilities, Bikeways, & ADA
8 Domain 3: Traffic Safety Concepts, Human Factors, & Safe System
8.1 The Safe System Approach & Vision Zero Principles 8.2 Human Factors, Driver Expectancy, & Perception-Reaction Time (PRT) 8.3 Crash Collision Typologies & Statistical Rate Formulations (MEV, MVMT)
9 Domain 3: Highway Safety Manual (HSM) Predictive Methods & EB
9.1 Highway Safety Manual (HSM) Predictive Methodology & Part C Models 9.2 Safety Performance Functions (SPFs) & Crash Modification Factors (CMFs) 9.3 Empirical Bayes (EB) Method & Regression-to-the-Mean (RTM) Correction 9.4 Economic Appraisal of Safety Projects (Benefit-Cost Analysis, NPV)
10 Domain 3: Safety Countermeasures & Road Safety Audits (RSA)
10.1 Roadway Departure Countermeasures (Rumble Strips, Safety Edge, Barriers) 10.2 Intersection Safety Countermeasures (Sight Triangles, Left-Turn Calming, Roundabouts) 10.3 Vulnerable Road User (VRU) Safety Countermeasures (PHB/HAWK, RRFB, Curb Extensions) 10.4 Road Safety Audits (RSAs) & Systemic Safety Analysis
11 Domain 4: Traffic Signal Warrants & Isolated Timing (MUTCD/ITE)
11.1 MUTCD Traffic Signal Warrants (Warrants 1 through 9) 11.2 Signal Phasing Principles & NEMA Dual-Ring Operations 11.3 Clearance Intervals (ITE Kinematic Equations) & Pedestrian Timing Parameters
12 Domain 4: Signal Coordination, Networks, & ATSPMs
12.1 Actuated Signal Controller Parameters (Passage Time, Gap Reduction, Max Recall) 12.2 Traffic Signal Coordination: Bandwidth, Offsets, & Progression Models 12.3 Traffic Responsive & Adaptive Traffic Signal Control Systems (ASCS) 12.4 Automated Traffic Signal Performance Measures (ATSPMs) & High-Resolution Data
13 Domain 4: Signing, Markings, Work Zones, Grade Crossings & School Zones (MUTCD)
13.1 MUTCD Regulatory, Warning, & Guide Signs Principles and Placement 13.2 Sign Retroreflectivity Standards & Nighttime Visual Maintenance 13.3 Longitudinal Pavement Markings, Delineators, & Object Markers 13.4 Temporary Traffic Control (TTC) & Work Zone Safety (MUTCD Part 6) 13.5 Highway-Rail Grade Crossings: Warning Devices, Preemption & Storage 13.6 School Zone Traffic Control & Student Walk Route Planning
14 Domain 5: Traffic Studies & Applied Statistical Methods
14.1 Applied Statistical Distributions & Sample Size Determination 14.2 Spot Speed Studies, 85th Percentile Speed, & 10-mph Pace 14.3 Volume Count Methodologies & Expansion Factoring (ADT, AADT, PHF)
15 Domain 5: Traffic Impact Studies & Trip Generation (ITE)
15.1 Travel Time, Delay, & Intersection Queue Studies 15.2 Parking Supply, Duration, & Turnover Analysis 15.3 Traffic Impact Study (TIS) Scoping, Horizon Years, & Mitigation 15.4 ITE Trip Generation Handbook: Average Rates vs. Regression Equations
16 Domain 6: Environmental, Institutional, Ethics & Equity Issues
16.1 Vehicle Emissions, Air Quality Conformity, & Decarbonization 16.2 Traffic Noise Modeling (TNM) & Noise Abatement Criteria (NAC) 16.3 Long-Range Transportation Planning & the NEPA Environmental Review 16.4 Professional Engineering Ethics, Tort Liability, & Risk Management 16.5 Transportation Equity, Access & Multimodal Community Considerations
6.1 Vertical Alignment: Tangent Grades, Crest & Sag Vertical Curves
Key Takeaways Explain This Topic AI
Vertical alignment consists of tangent grades connected by equal-tangent parabolic curves defined by the quadratic equation y = y_PVC + g1*x + ((g2 - g1)/(2L))*x^2, where the rate of vertical curvature K = L/A represents the horizontal distance in feet required for a 1% change in grade.
The turning point (high point on crest curves, low point on sag curves) occurs at a distance x_m = (|g1| * L) / A from the Point of Vertical Curvature (PVC); this location creates a localized flat spot that requires strict drainage consideration (K <= 167 for curbed pavements).
Crest vertical curve lengths are governed primarily by Stopping Sight Distance (SSD) using driver eye height h1 = 3.5 ft and object height h2 = 2.0 ft, yielding K = S^2 / 2158 for S <= L.
Sag vertical curve lengths are governed by four independent engineering criteria: (1) Headlight sight distance (h3 = 2.0 ft, 1.0-degree upward divergence, K = S^2 / (400 + 3.5S)), (2) Rider comfort (vertical centrifugal acceleration a_z <= 1.0 ft/s^2, K = V^2 / 46.5), (3) Drainage, and (4) General appearance.
Last updated: August 2026
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 K = L/A 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 ≤ L S \le L S ≤ L vs. S > L S > L S > L ), nighttime headlight beam geometry, rider comfort limits ( a z ≤ 1.0 ft/s 2 a_z \le 1.0\text{ ft/s}^2 a z ≤ 1.0 ft/s 2 ), and minimum longitudinal slope requirements for curbed roadway drainage.
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 – 4 % 3\text{--}4\% 3 – 4% , whereas low-speed urban collectors or mountainous arterials may permit grades of 6 – 9 % 6\text{--}9\% 6 – 9% (and up to 12 % 12\% 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 lb/hp 200\text{ lb/hp} 200 lb/hp design truck) can traverse without incurring an intolerable speed reduction (standard threshold is a 10 mph 10\text{ mph} 10 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 % 0.0\% 0.0% . However, on curbed urban roadways , AASHTO mandates a minimum longitudinal grade of 0.30 % to 0.50 % 0.30\%\text{ to }0.50\% 0.30% to 0.50% (ideally ≥ 0.50 % \ge 0.50\% ≥ 0.50% ) to facilitate gutter flow and prevent water ponding, hydroplaning, and sediment deposition.
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 L 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 L/2 L /2 ) between PVC and PVT.
General Parabolic Formulations
The elevation y y y at any horizontal distance x x x (in feet) from the PVC ( 0 ≤ x ≤ L 0 \le x \le L 0 ≤ x ≤ L ) is expressed by the quadratic polynomial: y ( x ) = y PVC + g 1 x + ( g 2 − g 1 2 L ) x 2 y(x) = y_{\text{PVC}} + g_1 x + \left( \frac{g_2 - g_1}{2L} \right) x^2 y ( x ) = y PVC + g 1 x + ( 2 L g 2 − g 1 ) x 2
Where:
y PVC y_{\text{PVC}} y PVC = roadway elevation at the PVC (ft)
g 1 g_1 g 1 = initial approach tangent grade expressed as a decimal ( g 1 = G 1 % / 100 g_1 = G_1 \% / 100 g 1 = G 1 %/100 )
g 2 g_2 g 2 = exit tangent grade expressed as a decimal ( g 2 = G 2 % / 100 g_2 = G_2 \% / 100 g 2 = G 2 %/100 )
- x x x = horizontal distance from the PVC (ft)
- L L L = total horizontal length of the vertical curve (ft)
In terms of percentage grades ( G 1 G_1 G 1 and G 2 G_2 G 2 ): y ( x ) = y PVC + G 1 x 100 + ( G 2 − G 1 ) x 2 200 L y(x) = y_{\text{PVC}} + \frac{G_1 x}{100} + \frac{(G_2 - G_1) x^2}{200 L} y ( x ) = y PVC + 100 G 1 x + 200 L ( G 2 − G 1 ) x 2
Rate of Vertical Curvature ( K K K )
The rate of vertical curvature, denoted as K K K , is the fundamental metric used in AASHTO geometric design: K = L A K = \frac{L}{A} K = A L
Where:
L L L = horizontal curve length (ft)
A A A = algebraic difference in tangent grades ( A = ∣ G 1 − G 2 ∣ A = |G_1 - G_2| A = ∣ G 1 − G 2 ∣ , expressed in percent)
Physical Meaning of K K K : K K K represents the horizontal distance in feet required to effect a 1 % 1\% 1% change in grade . A larger K K K -value corresponds to a flatter, more gradual vertical curve with greater sight distance.
Tangent Offsets ( e e e )
The vertical offset from the initial tangent line to the curve at any distance x x x from the PVC is: y offset ( x ) = a x 2 = A 200 L x 2 y_{\text{offset}}(x) = a x^2 = \frac{A}{200 L} x^2 y offset ( x ) = a x 2 = 200 L A x 2
The maximum vertical offset occurs at the PVI ( x = L / 2 x = L/2 x = L /2 ) and is termed the external distance ( E E E ): E = A L 800 = ∣ G 1 − G 2 ∣ L 800 E = \frac{A L}{800} = \frac{|G_1 - G_2| L}{800} E = 800 A L = 800 ∣ G 1 − G 2 ∣ L
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 ( d y / d x = 0 dy/dx = 0 d y / d x = 0 ). d y d x = G 1 100 + ( G 2 − G 1 ) x 100 L = 0 \frac{dy}{dx} = \frac{G_1}{100} + \frac{(G_2 - G_1) x}{100 L} = 0 d x d y = 100 G 1 + 100 L ( G 2 − G 1 ) x = 0
Solving for the distance x m x_m x m from the PVC: x m = − G 1 L G 2 − G 1 = ∣ G 1 ∣ L A = ∣ G 1 ∣ × K x_m = \frac{-G_1 L}{G_2 - G_1} = \frac{|G_1| L}{A} = |G_1| \times K x m = G 2 − G 1 − G 1 L = A ∣ G 1 ∣ L = ∣ G 1 ∣ × K
Elevation of the Turning Point ( y m y_m y m ):
Substituting x m x_m x m back into the curve elevation equation yields: y m = y PVC + G 1 x m 200 = y PVC + G 1 2 L 200 A y_m = y_{\text{PVC}} + \frac{G_1 x_m}{200} = y_{\text{PVC}} + \frac{G_1^2 L}{200 A} y m = y PVC + 200 G 1 x m = y PVC + 200 A G 1 2 L
Engineering and Drainage Significance:
Turning Point Location: If G 1 G_1 G 1 and G 2 G_2 G 2 have identical signs (e.g., + 4 % +4\% + 4% to + 1 % +1\% + 1% ), the curve is monotonic and no turning point exists within the curve boundaries ( x m x_m x m falls outside [ 0 , L ] [0, L] [ 0 , L ] ).
Drainage Flat Spot Constraint: At the exact apex or sag vertex ( x = x m x = x_m x = x m ), the grade is 0.0 % 0.0\% 0.0% . Within a distance of 50 ft 50\text{ ft} 50 ft on either side of the vertex ( 100 ft 100\text{ ft} 100 ft total window), the longitudinal grade remains flatter than 0.35 % 0.35\% 0.35% . On curbed roadway sections, AASHTO recommends that K K K should not exceed 167 ft/% 167\text{ ft/\%} 167 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)
Design Speed (mph) Stopping Sight Distance (ft) Crest Curve K-Value (ft/%) Sag Curve K-Value (Headlight, ft/%) Sag Curve K-Value (Comfort, ft/%)
- 30 200 19 37 19.4
- 35 250 29 49 26.3
- 40 305 44 64 34.4
- 45 360 61 79 43.5
- 50 425 84 96 53.8
- 55 495 114 115 65.1
- 60 570 151 136 77.4
- 65 645 193 157 90.9
- 70 730 247 181 105.4
- 75 820 312 206 121.0
4. Crest Vertical Curves: Stopping & Decision Sight Distance
Crest vertical curves are convex curves ( G 1 > G 2 G_1 > G_2 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 h_1 h 1 ): 3.5 ft 3.5\text{ ft} 3.5 ft above the pavement.
Object height ( h 2 h_2 h 2 ): 2.0 ft 2.0\text{ ft} 2.0 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:
When Sight Distance is Less than Curve Length ( S ≤ L S \le L S ≤ L ): L = A S 2 100 ( 2 h 1 + 2 h 2 ) 2 = A S 2 100 ( 2 ( 3.5 ) + 2 ( 2.0 ) ) 2 = A S 2 2158 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} L = 100 ( 2 h 1 + 2 h 2 ) 2 A S 2 = 100 ( 2 ( 3.5 ) + 2 ( 2.0 ) ) 2 A S 2 = 2158 A S 2
Expressed in terms of the rate of vertical curvature: K = L A = S 2 2158 K = \frac{L}{A} = \frac{S^2}{2158} K = A L = 2158 S 2
When Sight Distance Exceeds Curve Length ( S > L S > L S > L ): L = 2 S − 200 ( h 1 + h 2 ) 2 A = 2 S − 2158 A L = 2S - \frac{200 \left( \sqrt{h_1} + \sqrt{h_2} \right)^2}{A} = 2S - \frac{2158}{A} L = 2 S − A 200 ( h 1 + h 2 ) 2 = 2 S − A 2158
Passing Sight Distance (PSD) & Decision Sight Distance (DSD):
Passing Sight Distance: Uses an eye height of h 1 = 3.5 ft h_1 = 3.5\text{ ft} h 1 = 3.5 ft and an opposing vehicle height of h 2 = 3.5 ft h_2 = 3.5\text{ ft} h 2 = 3.5 ft . The resulting formula yields K = S 2 / 2800 K = S^2 / 2800 K = S 2 /2800 . Because passing sight distances are substantial ( > 1000 ft > 1000\text{ ft} > 1000 ft ), crest curve lengths for PSD require K K K -values 4 – 8 4\text{--}8 4 – 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 – 14.5 s 9.5\text{--}14.5\text{ s} 9.5 – 14.5 s ). K K K -values for DSD Avoidance Maneuvers C, D, and E are significantly larger than basic SSD.
Loading diagram... Crest vs Sag Vertical Curve Geometry and Sight Line Controls
5. Sag Vertical Curves: Governing Design Criteria
Sag vertical curves are concave curves ( G 1 < G 2 G_1 < G_2 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 h_3 h 3 ): 2.0 ft 2.0\text{ ft} 2.0 ft .
Upward divergence beam angle ( θ \theta θ ): 1.0 ∘ 1.0^\circ 1. 0 ∘ .
For S ≤ L S \le L S ≤ L : L = A S 2 200 ( h 3 + S tan 1 ∘ ) = A S 2 200 ( 2.0 + S × 0.017455 ) = A S 2 400 + 3.5 S 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} L = 200 ( h 3 + S tan 1 ∘ ) A S 2 = 200 ( 2.0 + S × 0.017455 ) A S 2 = 400 + 3.5 S A S 2 K headlight = S 2 400 + 3.5 S K_{\text{headlight}} = \frac{S^2}{400 + 3.5 S} K headlight = 400 + 3.5 S S 2
- For S > L S > L S > L : L = 2 S − 400 + 3.5 S A L = 2S - \frac{400 + 3.5 S}{A} L = 2 S − A 400 + 3.5 S
- 2. Rider Comfort Criterion (Dynamic Vertical Centrifugal Acceleration)
When a vehicle traverses a concave sag curve at design speed V V 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 ≤ 1.0 ft/s 2 a_z \le 1.0\text{ ft/s}^2 a z ≤ 1.0 ft/s 2 (approximately 0.03 g 0.03 g 0.03 g ): a z = v 2 R = ( 1.467 V ) 2 R a_z = \frac{v^2}{R} = \frac{(1.467 V)^2}{R} a z = R v 2 = R ( 1.467 V ) 2
Approximating radius of curvature as R ≈ L Δ g = 100 L A R \approx \frac{L}{\Delta g} = \frac{100 L}{A} R ≈ Δ g L = A 100 L : a z = 2.151 V 2 × A 100 L = A V 2 46.5 L a_z = \frac{2.151 V^2 \times A}{100 L} = \frac{A V^2}{46.5 L} a z = 100 L 2.151 V 2 × A = 46.5 L A V 2
Setting a z = 1.0 ft/s 2 a_z = 1.0\text{ ft/s}^2 a z = 1.0 ft/s 2 and solving for L L L and K K K : L = A V 2 46.5 ⟹ K comfort = V 2 46.5 L = \frac{A V^2}{46.5} \implies K_{\text{comfort}} = \frac{V^2}{46.5} L = 46.5 A V 2 ⟹ K comfort = 46.5 V 2
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 comfort ≈ 40 % to 50 % K_{\text{comfort}} \approx 40\%\text{ to }50\% K comfort ≈ 40% to 50% of K headlight K_{\text{headlight}} K headlight ).
3. Drainage Criterion
Just like crest curves, sag curves develop a flat region near the low point ( x m = ∣ G 1 ∣ K x_m = |G_1| K x m = ∣ G 1 ∣ K ). To prevent ponding across travel lanes on curbed cross-sections, K K K should not exceed 167 ft/% 167\text{ ft/\%} 167 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 ≥ 3 V L_{\min} \ge 3 V L m i n ≥ 3 V (where V V V is in mph) or L min ≥ 100 A L_{\min} \ge 100 A L m i n ≥ 100 A .
- 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 % G_1 = +3.5\% G 1 = + 3.5% to an exit grade of G 2 = − 2.5 % G_2 = -2.5\% G 2 = − 2.5% . The Point of Vertical Intersection (PVI) is located at Station 145 + 00 145+00 145 + 00 at Elevation 850.00 ft 850.00\text{ ft} 850.00 ft . The design speed is 60 mph 60\text{ mph} 60 mph ( S S D = 570 ft SSD = 570\text{ ft} SS D = 570 ft , K = 151 ft/% K = 151\text{ ft/\%} K = 151 ft/% ).
Determine the minimum required curve length L L L rounded up to the nearest 100 ft 100\text{ ft} 100 ft .
Find the station and elevation of the PVC and PVT.
Determine the station and elevation of the high point (crest turning point).
Solution:
Algebraic difference: A = ∣ 3.5 − ( − 2.5 ) ∣ = 6.0 % A = |3.5 - (-2.5)| = 6.0\% A = ∣3.5 − ( − 2.5 ) ∣ = 6.0% .
Minimum curve length: L = K × A = 151 × 6.0 = 906 ft → Round to L = 1000 ft L = K \times A = 151 \times 6.0 = 906\text{ ft} \to \text{Round to } L = 1000\text{ ft} L = K × A = 151 × 6.0 = 906 ft → Round to L = 1000 ft .
Actual design K = 1000 / 6.0 = 166.67 ft/% K = 1000 / 6.0 = 166.67\text{ ft/\%} K = 1000/6.0 = 166.67 ft/% .
PVC Station = PVI − L / 2 = ( 145 + 00 ) − ( 5 + 00 ) = Station 140 + 00 \text{PVI} - L/2 = (145+00) - (5+00) = \text{Station } 140+00 PVI − L /2 = ( 145 + 00 ) − ( 5 + 00 ) = Station 140 + 00 .
PVC Elevation = Elev PVI − G 1 ( L / 2 ) = 850.00 − 0.035 ( 500 ) = 832.50 ft \text{Elev}_{\text{PVI}} - G_1 (L/2) = 850.00 - 0.035(500) = 832.50\text{ ft} Elev PVI − G 1 ( L /2 ) = 850.00 − 0.035 ( 500 ) = 832.50 ft .
PVT Station = PVI + L / 2 = ( 145 + 00 ) + ( 5 + 00 ) = Station 150 + 00 \text{PVI} + L/2 = (145+00) + (5+00) = \text{Station } 150+00 PVI + L /2 = ( 145 + 00 ) + ( 5 + 00 ) = Station 150 + 00 .
PVT Elevation = Elev PVI + G 2 ( L / 2 ) = 850.00 + ( − 0.025 ) ( 500 ) = 837.50 ft \text{Elev}_{\text{PVI}} + G_2 (L/2) = 850.00 + (-0.025)(500) = 837.50\text{ ft} Elev PVI + G 2 ( L /2 ) = 850.00 + ( − 0.025 ) ( 500 ) = 837.50 ft .
High point distance from PVC: x m = ∣ G 1 ∣ L A = 3.5 × 1000 6.0 = 583.33 ft x_m = \frac{|G_1| L}{A} = \frac{3.5 \times 1000}{6.0} = 583.33\text{ ft} x m = A ∣ G 1 ∣ L = 6.0 3.5 × 1000 = 583.33 ft .
High Point Station = ( 140 + 00 ) + 583.33 = Station 145 + 83.33 (140+00) + 583.33 = \text{Station } 145+83.33 ( 140 + 00 ) + 583.33 = Station 145 + 83.33 .
High Point Elevation = y PVC + G 1 x m 200 = 832.50 + 3.5 × 583.33 200 = 832.50 + 10.21 = 842.71 ft 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} y PVC + 200 G 1 x m = 832.50 + 200 3.5 × 583.33 = 832.50 + 10.21 = 842.71 ft .