Vertical Curves for Highways
1993 Edition

Introduction to Vertical Curves per IRC SP 23

Definition of Summit Curves:

• Convex vertical curves linking two grade lines with slopes ( +n_1 ) and ( -n_2 ).
• Typically applied where an uphill meets a downhill or another uphill grade.

Parabolic Curve Equation:

[ y = ax^2 + bx + c ]

Origin set at the tangent point ( A ), with horizontal distance ( x ):

[ a = \frac{2N}{L^2} ]

Where:

• ( N = n_1 + n_2 ) denotes the algebraic sum of grade slopes in decimal form
• ( L ) is the horizontal length of the curve

Radius of Curvature ( R ):

[ R = \frac{L^2}{2N} ]

Determining Length of Summit Curve ( L ):

For overtaking sight distance ( S ) and driver's eye height ( H = 1.2 , m ):

[ L = 96 N S^2 ]

Sight Distance Considerations:

• Utilize overtaking or intermediate sight distances from established tables.
• Driver's eye height standardized at 1.2 m.

Summary Table

Diagrammatic Representation

graph LR
A[Tangent Point A] -- Length L --> C[Tangent Point C]
A -- Grade slope n1 --> D[Grade Intersection]
C

IRC SP 23: Gradient Guidelines for Various Terrain Categories

Definitions:

• Standard Gradient: Preferred maximum slope for economical and safe road design.
• Maximum Gradient: Highest permissible slope considering vehicle capabilities.
• Exceptional Gradient: Steepest slope allowed only for short segments under special conditions.

Recommended Usage:

• Employ standard gradient generally.
• Use maximum gradient under specific considerations.
• Resort to exceptional gradient only sparingly where unavoidable.

flowchart LR
    A[Terrain Classification] --> B{Gradient Category}
    B --> C[Standard Gradient]
    B --> D[Maximum Gradient]
    B --> E[Exceptional Gradient]
    C --> F[Economical, Safe Design]
    D --> G[Vehicle Performance Constraints]
    E --> H[Exceptional Cases Only]

This guide assists in selecting appropriate gradients based on terrain and altitude per IRC SP 23.

Summary of Design Speeds According to IRC SP 23

1. Design Speeds by Road Category and Terrain

Urban Road Design Speeds (Plains):

2. Sight Distance Values Corresponding to Design Speeds

Purpose of Vertical Curves as per IRC SP 23

Vertical curves are integral to highway design, providing a gradual transition between differing slopes to:

• Enhance driver comfort and safety by preventing abrupt grade changes.
• Provide sufficient sight distances for stopping and overtaking maneuvers.
• Facilitate effective drainage through maintained cross slopes.

Essential Specifications & Calculations:

1. Types of Vertical Curves:

   • Crest (Summit) Curve: Convex upward curve located at hill crests.
   • Sag (Valley) Curve: Concave upward curve situated in dips or valleys.

2. Determination of Vertical Curve Length (L): Based on the algebraic difference in gradients (A) and design speed (V). For crest curves ensuring stopping sight distance:

   [ L = \frac{A \times V^2}{46.5} ]

   Where:

   • ( L ) is curve length in meters
   • ( A ) is algebraic difference between gradients (percentage)
   • ( V ) is design speed in km/h

3. Recommended Minimum Curve Lengths for Comfort:

4. Gradient Difference Calculation:

[ A = |g_2 - g_1| ]

Where ( g_1 ) and ( g_2 ) represent the initial and final slope gradients.

Summary:

Vertical curves ensure smooth slope transitions, enhancing safety and driver comfort. Their length depends on speed and gradient change, designed to provide adequate sight distances.

graph LR
A[Initial Grade g1] --> B[Calculate Vertical Curve Length L]
C[Final Grade g2] --> B
B --> D[Smooth Slope Transition]
D --> E[Improved Driving Safety & Comfort]

Summit Curve Essentials According to IRC SP 23

1. Definition and Characteristics:

• Summit curves are convex vertical curves used where an ascending grade meets a descending one.
• Typically modeled as parabolic curves for straightforward calculations and uniform sight distances.

2. Parabolic Curve Formula:

[ y = ax^2 + bx + c ]

With origin at tangent point ( A ), vertical offset ( y' ) from grade line, and horizontal distance ( x ):

[ a = \frac{2N}{L^2} ]

Where:

• ( N = n_1 + n_2 ) is the algebraic sum of the slopes (in decimals)
• ( L ) is the length of the summit curve

3. Radius of Curvature Calculation:

[ R = \frac{L^2}{2N} ]

4. Length Determination of Summit Curve:

Depends on the deviation angle ( N ) and sight distance ( S ) (overtaking or intermediate):

[ L = \frac{96 \times N \times S^2}{H} ]

Where:

• ( S ) is sight distance in meters
• ( H = 1.2 ) m, the driver's eye height

5. Sight Distance Recommendations:

• Use either overtaking or intermediate sight distance values from IRC SP 23's Table 4.
• Ensure minimum curve length accommodates these sight distances.

6. Practical Considerations:

• Transition curves are generally not utilized for summit curves.
• Parabolic shape maintains consistent sight distance.
• Length ( L ) refers to the horizontal projection of the curve.

Summary Table:

Key Aspects of Valley Curves per IRC SP 23

Definition: Valley curves are concave vertical curves connecting descending and ascending grades or two descending slopes.

1. Deviation Angle ( N )

• Calculated as ( N = \theta_1 - \theta_2 ), the algebraic difference between grade angles.

2. Length of Valley Curve ( L )

• Determined primarily by headlight sight distance during night driving.
• Headlight height assumed as 0.75 m with a beam angle of 1° upwards.
• Length must be at least equal to the safe stopping sight distance ( S ).

For cases where ( L > S ):

[ L = 1.5S + 0.055N ]

Where:

• ( L ) is valley curve length in meters
• ( S ) is stopping sight distance in meters
• ( N ) is deviation angle in degrees

3. Valley Curve Length Based on Design Speed and Grade Difference

• ( A ) is the algebraic grade difference (%)

4. Minimum Vertical Curve Lengths (Table 7)

Practical Guidelines for Vertical Curve Design on Highways (IRC SP 23)

Core Concepts:

1. Types of Vertical Curves:

   • Crest (Summit) Curve: Convex upward
   • Sag (Valley) Curve: Concave upward

2. Fundamental Parameters:

   • ( L ): Length of vertical curve (m)
   • ( A ): Algebraic difference of grades (%)
   • ( G_1, G_2 ): Initial and final grade slopes (%)
   • ( S ): Stopping sight distance (m)
   • ( h_1, h_2 ): Heights of driver's eye and object (m)

3. Minimum Curve Length Expressions:

• Typically, ( h_1 = 1.2 , m ) and ( h_2 = 0.15 , m )
• Sight distance ( S ) depends on design speed and standards.

4. Gradient Change Rate:

[ \text{Rate of gradient change} = \frac{A}{L} ]

• Should be comfortable for drivers, usually between 0.03% and 0.05% per meter.

Example Summary Table for Minimum Lengths:

flowchart LR
    A[Input Grades G1 & G2] --> B[Compute Algebraic Difference A]
    B --> C[Determine Sight Distance S]
    C --> D[Calculate Minimum Length L]
    D --> E[Verify Design Criteria]

IRC SP 23: Key Formulas and Sample Calculations for Summit Curves

Fundamentals of Summit Curves

• Crest vertical curves connect an upward slope to a downward slope.
• Preferred shape is parabolic for ease of calculation and maintaining consistent sight distances.

Parabolic Curve Relation

Let:

• ( n_1, n_2 ) be the slopes before and after the summit (decimal form)
• ( L ) be the length of the curve (m)
• ( x ) be horizontal distance from the start (m)
• ( y ) be vertical offset from the tangent grade (m)
• ( N = |n_1 + n_2| ) be the algebraic sum of the slopes

Equation:

[ y = \frac{N}{2L} x^2 ]

Radius of Curvature

[ R = \frac{L^2}{N} ]

Length Calculation for Sight Distance ( S )

• Driver's eye height ( H = 1.2 ) m
• Deviation angle ( N )
• Sight distance ( S ) (safe stopping, intermediate, or overtaking)

When ( L > S ):

[ L = \frac{96 N S^2}{H} ]

Sight Distance Values

• Safe stopping distance based on design speed and driver reaction time
• Intermediate distance for overtaking by slower vehicles
• Overtaking distance for safe passing at higher speeds

Design Procedure

1. Identify slopes ( n_1, n_2 ) and calculate ( N ).
2. Select appropriate sight distance ( S ) from tables.
3. Compute curve length ( L ) using the formula.
4. Use ordinate values for vertical offsets from example datasets.

IRC SP 23: Calculating Summit Curve Length for Safe Stopping Sight Distance

Primary Formula:

[ L = \frac{A S^2}{200 (f + G)} ]

Where:

• ( L ): Length of summit curve (m)
• ( A ): Deflection angle of the curve (degrees)
• ( S ): Stopping sight distance (m)
• ( f ): Coefficient of longitudinal friction (commonly 0.35)
• ( G ): Road grade in decimal form (e.g., 0.02 for 2%)

Minimum Length Values (Typical)

Note: Exact values depend on terrain and design speed.

Summary:

• Stopping sight distance determines the minimum summit curve length to ensure safe vehicle stopping.
• Curve length grows with increasing sight distance and deflection angle.
• Reference minimum lengths to avoid abrupt vertical changes.

flowchart LR
    A[Intersection of Grade Lines] --> B[Summit Curve]
    B --> C[Curve Length L]
    C --> D[Ensures Adequate Stopping Sight Distance]

IRC SP 23: Summit Curve Length Calculation for Intermediate Sight Distance

Formula for Curve Length ( L' ):

[ L' = \frac{N^2}{M} ]

Where:

• ( L' ): Length of summit curve (m)
• ( N ): Deviation angle (m) (ordinate at grade line intersection)
• ( M ): Vertical ordinate from grade line intersection to summit curve (m)

Notes:

• Deviation angle ( N ) represents lateral offset at the intersection.
• Ordinate ( M ) represents vertical offset defining the curve shape.
• Minimum lengths derived from relevant IRC tables ensure adequate intermediate sight distance.

Conceptual Summary Table:

flowchart LR
    A[Grade Line Intersection] --> B[Deviation Angle (N)]
    A --> C[Ordinate (M)]
    B & C --> D[Calculate L' = N^2 / M]
    D --> E[Check Minimum Length]
    E --> F[Design Summit Curve for Intermediate Sight]

Utilize this approach along with minimum length tables for safe intermediate sight distance compliance on summit curves.

Summit Curve Length Calculation for Overtaking Sight Distance (IRC SP 23 - Plate 3)

Key Parameters:

• ( N ): Deviation angle between gradients in radians
• ( S ): Required overtaking sight distance (m)
• ( H = 1.2 ) m: Driver’s eye height

Length Formula When ( L > S ):

[ L = 96 \times N \times S^2 ]

Where:

• ( L ): Length of summit curve (m)
• ( N ): Deviation angle (radians)
• ( S ): Overtaking sight distance (m)

Additional Information:

• Summit curve modeled as a parabolic curve.
• Length ( L ) approximates the horizontal projection.
• Overtaking sight distances are sourced from IRC SP 23’s Table 4.
• Deviation angle measured at the intersection of the two grades.

Summary Table

flowchart LR
    A[Intersecting Grade Lines at Angle N]
    B[Parabolic Summit Curve]
    C[Driver's Eye Height H = 1.2 m]
    D[Required Sight Distance S]
    E[Calculated Curve Length L = 96 * N * S^2]
    A --> B
    B --> C
    B --> D
    C --> E
    D --> E

Use this formula to design summit curves that ensure safe overtaking visibility per IRC SP 23 standards.

Specifications for Valley Curve Lengths per IRC SP 23:

1. Definitions:

• Valley Curve: Vertical curve concave upward, connecting descending and ascending slopes.
• Deviation Angle (N): Algebraic difference between the two grade angles.

2. Length Determination:

• Length must accommodate night-time headlight sight distance.
• Headlight height assumed at 0.75 m and beam angle 1° upward.
• Curve length must be at least equal to the safe stopping sight distance ( S ).

3. Length Formula (when length exceeds sight distance):

[ L = 1.5S + 0.055N ]

Where:

• ( L ): Valley curve length (m)
• ( S ): Stopping sight distance (m)
• ( N ): Deviation angle (degrees)

4. Length Multipliers Based on Speed and Grade Difference ( A ):

Where ( A ) is algebraic difference in grades (%).

5. Minimum Length Requirements (Table 7):