2.3. Simple Circular Curves

In order to compute various parameters of a simple circular curve for its layout, it is necessary to understand various parts of the curve, as shown in Figure 2.3, and explained below.

Tangents or straights: The two straight lines AB and BC, which are connected by the curve, are called the tangents or straights to the curve. The lines AB and BC are tangents to the curve. Line AB is called the first tangent or the rear tangent, and line BC is called the second tangent or the forwarded tangent.

Intersection point: The points of intersection of the two straights (B) is called the intersection or vertex point.

Tangent length: The distance between the two tangent point of intersection to the tangent point (BT and BT ) is called the tangent length.

Right handed curve: When the curve deflects to the right side of the progress of survey, it is termed as right handed curve.

Left handed curve: When the curve deflects to the left side of the progress of survey, it is termed as left handed curve.

Tangent points: The points (T and T ) at which the curve touches the tangents are called the tangent points. The beginning of the curve at T is called the tangent curve point and the end of the curve at T is called the curve tangent point.

Long chord: The line joining the two tangent points (T and T ) is known as the long chord.

Summit or apex: The mid-point (F) of the arc (T FT ) is called summit or apex of the curve.

Length of the curve: The arc T FT is called the length of the curve.

Angle of intersection: The angle between the tangent lines AB and BC (∠ABC) is called the angle of intersection (I).

Deflection angle: It is the angle (φ) by which the forward tangent deflects from the rear tangent (180°-I) of the curve.

Apex distance: The distance from the point of intersection to the apex of the curve BF is called the apex distance.

This chunk discusses the various parts that make up a simple circular curve, which is crucial for understanding how to compute its layout parameters. There are key components like the tangents, which are the straight lines before and after the curve, usually referred to as the rear tangent and the forwarded tangent. The intersection point is where these two tangents meet. The tangent length measures how far these tangents are, while the curve can either deflect to the right (right-handed curve) or to the left (left-handed curve). Tangent points mark where the curve meets the tangents, and the long chord is a straight line connecting these two points. The apex or summit is the highest point of the curve. The angle of intersection is a critical geometric feature that helps in understanding the curve's orientation, and the deflection angle informs how much the curve bends. Finally, the apex distance gives us a measure from the intersection point to the curve’s summit.

Imagine a winding road where you are driving. The straight sections of the road are like the tangents, the point where your path changes direction is the intersection point, and how much the road bends to the left or right represents the left-handed or right-handed curves. The apex of the curve is where you might feel the highest point of the road in that bend. Understanding these parts helps in ensuring a smooth driving experience.

A simple circular curve may be designated either by its radius or by the angle subtended at the centre by a chord of a particular length. In India, a simple circular curve is designated by the angle (in degrees) subtended at the centre by a chord of 30 m (100 ft.) length. This angle is called the degree of curve (D). The relation between the radius and the degree of the curve may be established as follows (Refer Figure 2.4). If R is the radius of the curve in meters, D is the degree of the curve, MN is the chord of 30 m length, and P is the mid-point of the chord, then:

• In ΔOMP, OM = R
• MP = MN = 15 m.
• ∠MOP= D/2.

Thus, applying the sine rule leads to the formulas for calculating the radius in terms of the degree of curve.

This chunk explains how simple circular curves are named or designated based on either their radius or the angle that a chord of a specific length subtends at the center. In India, this specific chord is 30 meters long. The angle formed at the curve's center by this chord is known as the degree of curve (D). This designation is practical because it simplifies communication about curves, especially in planning and engineering contexts. Mathematical relationships derived from trigonometry allow calculations of the radius based on the degree of curve. This means that we can easily determine how sharp a curve is just by knowing the degree of curve, which helps in both design and navigation.

Think of a racetrack where curves are pivotal for the cars' speed and handling. The sharper the curve, the tighter the turn. If you know the degree of the curve, like a 30-meter chord marking a turn, you can predict how fast a car can safely navigate that turn without losing control. Knowing this helps racers and constructors design an optimal racetrack.

From Figure 2.3, I + φ = 180°. ∠T1OT2 = 180° - I = φ (central angle = deflection angle).

• Tangent length = BT = BT = OT tan(φ/2).
• Length of the long chord = 2TE = 2 × OT sin(φ/2).
• Length of the curve = Length of the arc T1FT2 = Rφ (in radians) = (πRφ)/180.
• Apex distance = BF = BO – OF = R sec(φ/2) – R.
• Versine of the curve = EF = FO – OE = R – R cos(φ/2) = R versine (φ/2).

This chunk covers the mathematical aspects of the simple circular curve by introducing specific formulas. It states how the sum of the angle of intersection (I) and the deflection angle (φ) equals 180 degrees, which forms the basis for further calculations. It provides equations for key measurements, including the tangent length, long chord length, the actual length of the curve, apex distance, and versine. Understanding these elements is essential for engineers to precisely design curves that ensure safety and effectiveness in transportation. Each formula links different metrics, allowing for comprehensive analysis and planning.

Picture a large water slide at a park. The measurements and angles are crucial to ensure that the ride is fun yet safe. Just like engineers use these formulas to design curves for roads, they would need to calculate the proper angles and distances to ensure that water flows smoothly without causing riders to slip out or not have enough speed to complete the curve.

A curve may be set out either by (a) linear methods, where chain, tape or EDM is used, or (b) angular methods, where a theodolite with or without a tape or total station is used. Before starting to set out a curve by any method, the exact positions of the tangent points between which the curve would lie, is to be determined. The steps to be used for setting up the curve between two points A and B (Figure 2.3) are as follows:

1. After fixing the directions of the straights, extend them to intersect at point B.
2. Set up a theodolite at the intersection point B and measure the angle of intersection I, using repetition method of angle measurement. Determine the deflection angle φ as (180°-I).
3. Calculate the tangent length as length = R tan(φ/2).
4. From the intersection point B, measure the tangent length BT backward along the rear tangent BA to locate T1.
5. Similarly, locate the position of T2 by measuring the same distance from B along the forward tangent BC.
6. The chainages of tangent points T1 and T2 are determined. The chainage is the distance of point T1 and T2 with respect to a reference point on the road with known chainage.
7. The pegs are then fixed at equal intervals (normally 30 m) on the curve. This distance should actually be measured along the arc, but in practice, it is measured along the chord, as the difference is negligible.

This chunk outlines the methods used to set out horizontal curves, which is integral for actual field applications. The two primary methods are linear and angular, using different instruments to achieve accuracy. The process begins with determining the tangent points and following systematic steps to ensure precision. This involves measuring the angle of intersection and calculating necessary lengths, such as the tangent length, ensuring that the curve connects to the straight paths correctly. By adhering to these steps, surveyors and engineers can effectively plan out the location and design of curves in transportation projects.

Think about laying out a new bike path in a park. First, you’d need to figure out where the path should connect to existing walkways. By measuring angles and distances accurately, you can ensure that the path flows smoothly, just as engineers carefully lay out curves on roads or railroad tracks to make them safe and navigable.

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