In-Depth Notes on Vertical Alignment and Curves Design

Design of the Alignment

• Vertical Alignment:

  • Composed of straight sections known as grades, linked by vertical curves.
  • Key components:
  • Length of Crest Vertical Curves.
  • Length of Sag Vertical Curves.

• Horizontal Alignment:

  • Composed of straight sections (tangents) connected by curves:
  • Simple Curves.
  • Reverse Curves.
  • Transition Curves.

Design of Vertical Alignment

• Key Objectives:
  • Recognize the components of vertical curves.
  • Distinguish between crest and sag vertical curves.
  • Understand relationships characterizing vertical curves.
  • Elements influencing roadway grades selection.

Overview of Vertical Alignment

• Vertical curves enable a smooth transition from one tangent grade to another for vehicular travel.
• Two types of vertical curves:
  • Crest Vertical Curves.
  • Sag Vertical Curves.
• Elements:
  • Grades.
  • Minimum lengths of crest and sag vertical curves.
  • Elevations at the crest and sag points.
  • Design methods for these curves.

Design of Vertical Curves

• Shape: Typically parabolic.
• Length Measurement: Measured along the horizontal alignment.
  • Significant in road planning using horizontal maps.
• Primary Design Criteria:
  • Crest Vertical Curves:
  • Ensure minimum stopping sight distance.
  • Sag Vertical Curves:
  • Ensure minimum stopping sight distance via vehicle headlights, comfortable ride, adequate drainage, and aesthetic appearance.

Properties of Vertical Curves

• Terms:
  • PVI: Point of Vertical Intersection.
  • PVC: Point of Vertical Curve.
  • PVT: Point of Vertical Tangent.
• Calculating algebraic difference:
  $A = G<em>2 - G</em>1$

Stopping Sight Distance (SSD) for Vertical Curves

For Crest Curves

• Case 1: If sight distance (S) > length of the vertical curve (L):

  • Traditional values are $H<em>1=1.2 ext{m}$ and $H</em>2=0.15 ext{m}$; revised values are $H<em>1=1.1 ext{m}$ and $H</em>2=0.6 ext{m}$.

• Case 2: If sight distance (S) < length of vertical curve (L).

Example Problems

Example 15.1: Minimum Length of Crest Vertical Curve

• Given: +3% and -2% grade; design speed = 96 km/h; perception-reaction time = 2.5 s; deceleration = 3.41 m/s².
• SSD Calculation:
  $SSD = 0.278ut + \frac{u^2}{a(254(9.81-G))}$
• Resulting SSD = 180.96 m.
• Minimum Length Calculation:
  $L_{min} = \frac{A}{S^2} = \frac{658}{5 \times 180.96} = 248.83 ext{ m}$

Example 15.2: Maximum Safe Speed on a Crest Vertical Curve

• Given: +4.4% and -4.4% grades; curve length = 82 m.
• Calculate maximum safe speed using SSD equation from Chapter 3.
• Result: Maximum safe speed = 55.4 km/h; recommended posted speed = 48 km/h.

Sag Vertical Curves

• Key Requirements:
  • Minimum SSD through headlights.
  • Comfort in operation.
  • Adequate drainage.
  • Aesthetic considerations.

Minimum Length of Sag Vertical Curves

Criteria Considered

• Headlight Sight Distance Requirement:
  • If S > L: Use $L = 2S$.
• Comfort Criterion:
  • Maintain radial acceleration < 0.3 m/s².
  • Typical length = 75% of the headlight sight distance requirement.
• Appearance Criterion:
  • $L_{min} = 30A$ where A = algebraic difference in grades.
• Drainage Criterion:
  • Must maintain a minimum slope of 0.35% within 15 m of the curve's lowest point.

Example 15.3: Minimum Length of Sag Vertical Curve

• Given: -5% to +2% with design speed = 64 km/h.
• SSD Calculation:
  $SSD = 0.278ut + \frac{3.41}{254(0.35 - -.05)}$
• Length Calculation based on various criteria leads to the minimum length being determined as 210 m based on combined requirements.

K-values and Their Use

• K: Length of vertical curve per percent change in grades.
• Equation: $L = KA$
• Tables provide K-values based on design speeds; assist in calculating curve minimum lengths conveniently by combining grade differences.

Additional Example Problems and Solutions

Example 15.4 and 15.5: Design of Vertical Curves

• Detailed calculations and elevation computations for crest and sag curves using given grades and speeds.
• Emphasized updating elevation points at interpolation intervals of 30 m to ensure accuracy in design.

Factors Affecting Grade Selection

• Steep Grades impact speed, especially in heavy vehicles.
• Recommended maximum grades vary from 5% for 70 mph to 7%-12% for lower speeds.
• Critical in minimizing operational impacts on traffic flow and safety.

Summary of Grade Recommendations

• Range from 2% to 8% typically depending on the type of road and expected vehicle speeds, ensuring overall road functionality and safety.