Types and design of curves on roads  

        As i have already explained in another blog regarding radius of curvature of existing road and its instantaneous point radius calculations in Radius of curvature blog , here i am writing about curves on new road project.
 

1) Circular curves

    a) Simple Curves:- 

                                    The alignment of Road consists of Straight sections called Tangents connected by circular curves; on main-line roads, the tangents and circular curves are connected by Spiral transition curves.

    b) Compound Curves:- 

                                      A compound curve is formed by 2 simple curves that have a common tangent

    c) Broken-back Curves:-

                                    Two simple curves in the same direction connected by a short tangent, known as " Broken-Back" curves, are undesirable and are often revised by compounding with a new curve.

    d) Reversed Curves:- 

                                    A reversed curve is formed by 2 simple curves that have a common Tangent and a common point of Tangency, the curves lie on opposite side of tangent. Reverse curves are ordinarily used to connect parallel lines. In rail tracks, reverse curves are undesirable except crossover tracks.

The Indian Road Congress recommends the use of the Spiral as Transition Curve in the Horizontal alignment of highway due to following reasons

1) The Spiral curve satisfies the requirements of an ideal transition.

2) The geometric property of Spiral is such that the calculations and setting out the curve in the field is simple and easy.

2) Vertical curves

                            a) Vertical curves are used to connect grade lines of the profile of Roads and Railways.

                            b) Parabolic curves has been adopted for this purpose

                            c) The length depends upon the algebraic difference between gradients and on whether curve is at " Summit " or a " Sag ".

                            d) In highways, sight distance across the summit is governing factor, thus the length may be selected as following conditions

                                    i) L > S --> Sight distance is entirely on the curve

                                    ii) L < S --> Sight distance overlaps the curve and extends on the tangents

To design a horizontal curve on a road as per Indian Standards, the following steps can be followed:

Determine the design speed of the road: 

    The design speed is the maximum safe speed at which a vehicle can travel on the road. It is based on factors such as terrain, traffic volume, and road alignment. The design speed for the road should be determined based on the prevailing conditions.

Example:-

Radius of curve (R) = 200 meters

Coefficient of lateral friction (f) = 0.15

Design superelevation rate (e) = 7%

Solution:

Design speed (V) can be calculated using the following formula:

V = √(f × g × R × e)

where:

g is the acceleration due to gravity (9.81 m/s²)

Substituting the given values into the formula, we get:

V = √(0.15 × 9.81 × 200 × 0.07)

V = √19.4766

V = 4.41 m/s (rounded to 4.4 m/s)

To convert the speed to kilometers per hour (km/h), we can multiply the speed in meters per second (m/s) by 3.6:

V = 4.4 × 3.6

V = 15.84 km/h (rounded to 16 km/h)

Therefore, the design speed of the road is 16 km/h for the given radius of curve, coefficient of lateral friction, and design superelevation rate.

Calculate the super-elevation rate: 

    The super-elevation rate is the amount of banking required on a curve to counteract the centrifugal force acting on a vehicle. The super-elevation rate is calculated based on the design speed and the radius of the curve.

Example:

Design speed (V) = 60 km/h

Radius of curve (R) = 300 meters

Coefficient of lateral friction (f) = 0.15

Solution:

The super-elevation rate (e) can be calculated using the following formula:

e = (V² / (g × R)) - f

where:

g is the acceleration due to gravity (9.81 m/s²)

First, we need to convert the design speed from kilometers per hour (km/h) to meters per second (m/s):

V = 60 / 3.6

V = 16.67 m/s

Substituting the given values into the formula, we get:

e = (16.67² / (9.81 × 300)) - 0.15

e = (277.89 / 2943) - 0.15

e = 0.0943 or 9.43% (rounded to two decimal places)

Therefore, the super-elevation rate for a design speed of 60 km/h and a radius of curve of 300 meters is 9.43%.

Determine the radius of the curve: 

    The radius of the curve is the distance between the center of the curve and the point where the road intersects the tangent at the beginning and end of the curve. The radius of the curve should be determined based on the design speed and the super-elevation rate.

Example:

Design speed (V) = 50 km/h

Coefficient of lateral friction (f) = 0.20

Super-elevation rate (e) = 6%

Solution:

The radius of the curve (R) can be calculated using the following formula:

R = (V² / (g × e × f)) + (f / g)

where:

g is the acceleration due to gravity (9.81 m/s²)

First, we need to convert the design speed from kilometers per hour (km/h) to meters per second (m/s):

V = 50 / 3.6

V = 13.89 m/s

Substituting the given values into the formula, we get:

R = (13.89² / (9.81 × 0.06 × 0.20)) + (0.20 / 9.81)

R = (192.86 / 0.11748) + 0.02040

R = 1640.28 meters

Therefore, the radius of the curve for a design speed of 50 km/h, a coefficient of lateral friction of 0.20, and a super-elevation rate of 6% is 1640.28 meters (rounded to two decimal places).

Determine the length of the curve: 

    The length of the curve is the distance between the tangent points at the beginning and end of the curve. The length of the curve should be determined based on the design speed and the radius of the curve.

Example:

Design speed (V) = 70 km/h

Radius of curve (R) = 250 meters

Rate of lateral friction (f) = 0.15

Super-elevation rate (e) = 4%

Solution:

The length of the curve (L) can be calculated using the following formula:

L = (2πR × α) / 360

where:

α is the central angle of the curve in degrees

First, we need to calculate the super-elevation required (e_req) using the following formula:

e_req = (V² / (127R × f)) × 100

Substituting the given values into the formula, we get:

e_req = (70² / (127 × 250 × 0.15)) × 100

e_req = 2.45% (rounded to two decimal places)

Since the given super-elevation rate (e) is less than the required super-elevation rate (e_req), we need to increase the length of the curve to provide the required super-elevation.

The required central angle (α_req) can be calculated using the following formula:

α_req = (180 / π) × (e_req / R)

Substituting the given values into the formula, we get:

α_req = (180 / π) × (0.0245 / 250)

α_req = 0.042 degrees (rounded to three decimal places)

The required central angle is very small, so we can assume that the curve is an arc of a circle. Therefore, the central angle is equal to the subtended angle of the curve.

The subtended angle (α) can be calculated using the following formula:

α = (e / (0.01 × R)) × 180 / π

Substituting the given values into the formula, we get:

α = (4 / (0.01 × 250)) × 180 / π

α = 2.3 degrees (rounded to one decimal place)

The length of the curve can be calculated using the formula mentioned above:

L = (2πR × α) / 360

L = (2π × 250 × 2.3) / 360

L = 40.92 meters (rounded to two decimal places)

Therefore, the length of the curve for a design speed of 70 km/h, a radius of curve of 250 meters, a rate of lateral friction of 0.15, and a super-elevation rate of 4% is 40.92 meters (rounded to two decimal places).

Determine the transition length: 

    The transition length is the length of road required to gradually transition from the straight section to the curved section of the road. The transition length should be determined based on the design speed and the length of the curve.

Example:

Design speed (V) = 80 km/h

Radius of curve (R) = 300 meters

Rate of lateral friction (f) = 0.15

Super-elevation rate (e) = 8%

Solution:

The transition length (L_t) can be calculated using the following formula:

L_t = V² / (127Rt)

where:

Rt is the radius of the transition curve

First, we need to calculate the super-elevation required (e_req) using the following formula:

e_req = (V² / (127R × f)) × 100

Substituting the given values into the formula, we get:

e_req = (80² / (127 × 300 × 0.15)) × 100

e_req = 5.82% (rounded to two decimal places)

Since the given super-elevation rate (e) is greater than the required super-elevation rate (e_req), we need to provide a transition curve to gradually increase the super-elevation from the normal cross slope to the full super-elevation.

The length of the transition curve can be calculated using the following formula:

L_t = (V² / (127 × f × (e - e_n))) - (2/3) × R

where:

e_n is the normal cross slope, which is usually assumed to be 2%

Substituting the given values into the formula, we get:

L_t = (80² / (127 × 0.15 × (0.08 - 0.02))) - (2/3) × 300

L_t = 115.95 meters (rounded to two decimal places)

Therefore, the transition length for a design speed of 80 km/h, a radius of curve of 300 meters, a rate of lateral friction of 0.15, and a super-elevation rate of 8% is 115.95 meters (rounded to two decimal places).

Determine the cross slope: 

    The cross slope is the slope of the road perpendicular to the direction of travel. The cross slope should be designed to provide adequate drainage of the road surface.

Example:

Design speed (V) = 60 km/h

Radius of curve (R) = 150 meters

Super-elevation rate (e) = 6%

Length of transition curve (L_t) = 60 meters

Solution:

The cross slope (S) can be calculated using the following formula:

S = e / 100 × R

First, we need to calculate the radius of the transition curve (Rt) using the following formula:

Rt = L_t² / (24 × R)

Substituting the given values into the formula, we get:

Rt = 60² / (24 × 150)

Rt = 10 meters (rounded to two decimal places)

Next, we need to calculate the maximum allowable cross slope (S_max) using the following formula:

S_max = V² / (127R)

Substituting the given values into the formula, we get:

S_max = 60² / (127 × 150)

S_max = 0.248 (rounded to three decimal places)

Since the calculated cross slope (S) is less than the maximum allowable cross slope (S_max), the design is acceptable.

Substituting the given values into the formula for cross slope, we get:

S = 6 / 100 × 150

S = 9 degrees (rounded to one decimal place)

Therefore, the cross slope for a design speed of 60 km/h, a radius of curve of 150 meters, a super-elevation rate of 6%, and a transition length of 60 meters is 9 degrees (rounded to one decimal place).

Determine the superelevation runoff length: 

    The super-elevation runoff length is the length of road required to transition from the super-elevated section of the road to the normal cross slope. The super-elevation runoff length should be determined based on the design speed, the super-elevation rate, and the cross slope.

Example:

Design speed (V) = 50 km/h

Radius of curve (R) = 200 meters

Super-elevation rate (e) = 6%

Length of transition curve (L_t) = 60 meters

Solution:

The super-elevation runoff length (L_r) can be calculated using the following formula:

L_r = V² / (30g × e)

where:

g is the acceleration due to gravity, which is 9.81 m/s²

Substituting the given values into the formula, we get:

L_r = (50/3.6)² / (30 × 9.81 × 0.06)

L_r = 22.42 meters (rounded to two decimal places)

Therefore, the super-elevation runoff length for a design speed of 50 km/h, a radius of curve of 200 meters, a super-elevation rate of 6%, and a transition length of 60 meters is 22.42 meters (rounded to two decimal places).

Prepare a horizontal alignment plan: 

    Based on the above calculations, prepare a horizontal alignment plan that includes the radius of the curve, the length of the curve, the transition length, the super-elevation rate, the cross slope, and the super-elevation runoff length.

Check for compliance with Indian Standards: 

    Finally, ensure that the design of the horizontal curve complies with the Indian Standards for road design, including standards for sight distance, minimum curve radii, and maximum super-elevation rates.