4. Solved Examples

A circular curve has 300 m radius and 60° deflection angle. Determine its degree by (a) arc definition and (b) chord definition of standard length 30 m. Also compute the (i) length of curve, (ii) tangent length, (iii) length of long chord, (iv) mid-ordinate and (v) apex distance.

Solution: R = 300 m, D = 60°

(a) Arc definition: s = 30 m, R = (s /D )(180/ π) a 300 = 30 *180 / (D π) a D = 5.730 Ans. a

(b) Chord definition: R sin (D / 2) = s /2 c 300 sin D / 2 = 30 / 2 c D = 5.732 Ans. c

(i) Length of the curve: L = R Δ (π/180) L = 300 * 60 (Δ/180) = 314.16 m Ans.

(ii) Tangent length: T = R tan Δ/2 T= 300 tan 60/2 = 173.21 m Ans.

(iii) Length of long chord: L = 2 R sin Δ /2 c L = 2 * 300 sin 60/ 2 = 300 m Ans. c

(iv) Mid-ordinate: M = R (1- cos Δ/2) = 300 (1- cos 60/2) = 40.19 m Ans.

(v) Apex distance: E = R (sec Δ/2 – 1) = 300 (sec 60/2) = 46.41 m Ans.

In Example 2.1, we are looking at a circular curve defined by a radius and a deflection angle. The first part of the question involves finding the degree of the curve using both arc and chord definitions, which requires some formulas related to the properties of circles. The arc definition uses the radius and the angle to find the curve's degree, while the chord definition does so through sine calculations.

We then calculate various properties of the curve: the length of the curve is determined using the formula involving the radius and deflection angle in degrees. The tangent length gives us the distance from the point of intersection to the curve. The long chord is the straight line distance connecting the two ends of the circular curve, and the mid-ordinate gives us the height of the curve at its midpoint, while the apex distance provides the distance from the curve's center to the point of tangency on the horizontal line. Each of these is calculated using specific formulas provided in the solution.

Imagine driving around a roundabout. The radius of the roundabout dictates how sharp the turn feels. If you increase the radius, your turn becomes smoother—just like finding the curve’s properties helps engineers design safer roads. By using simple measurements and geometry, we can understand how to manage turns, ensuring vehicles can navigate safely at various speeds.

If the approximate perpendicular offset for the mid-point of the circular curve deflecting through 760 38' is 96.1 m, calculate the radius of the curve.

Solution: Δ = 760 38', Ox = 96.1 m Using Perpendicular offset method: O = R - √[R2 - (x)2] x The distance ‘x’ from T for locating the apex point = R sin Δ/2 = R sin 760 38' / 2 = 0.62R m Now 96.1 = R - √[R2 - (0.62R)2] = 0.215 R R = 96.1/ 0.215 = 446.98 m.

In this example, we are determining the radius of a circular curve based on a known offset at the midpoint of the curve. The perpendicular offset (O) is the vertical distance from the long chord to the highest point on the curve. The method used here involves setting up an equation based on the relationship between the offset and the radius.

We first find the distance 'x' from the tangent point necessary to locate the curve's apex. Then, we plug this 'x' value into the offset formula, solve for the radius (R) using algebraic methods. This calculation gives us the radius that relates to the curve's design.

Think of the curve as the arc of a rainbow. The wider the rainbow, the larger its radius. If you know how far you can step back (the offset) from the midpoint, you can deduce how wide the rainbow actually is. This helps in setting up roads that mimic this arc for safe and smooth driving.

Determine the ordinates of the points on a circular curve having a long chord of 100 m and a versed sine of 5 m. The ordinates are to be measured from the long chord at an interval of 10 m.

Solution: Length of long chord (L) = 100 m, Versed sine (O0) = 5 m, Interval = 10 m The versined or mid-ordinate O0 = R - √[R2 - (L/2)2] 5 = R - √[R2 - (100/2)2] R = 252.5 m Ordinates at intervals of 10 m O = √[R2 - (x)2] – (R-O0) 0

In Example 2.3, we need to find the ordinates of points along a circular curve represented by a long chord and a versed sine. The versed sine helps us understand the height of the arc from the chord at midpoint. First, we calculate the radius (R) from this relationship using the formula provided. Following this, at regular intervals along the chord, we find the ordinates, or the vertical measurements, that indicate how high the curve sits above the long chord. This is critical in ensuring the curve meets certain design specifications, particularly in road construction.

Imagine a bow and arrow; the tighter the bowstring (the chord), the more pronounced the curve of the bow (the arc). Measuring the height of the string from the bow at intervals helps ensure that the bow functions properly. Similarly, engineers need to ensure the highway curve does not rise too sharply, providing a safer experience for all drivers.

Two straight alignments AI and IC of a road intersect at I at a chainage (250 chainage + 15 links), angle of deflection being 1200. Calculate the chainages of the point of commencement and the point of tangency if the radius of the right hand circular curve is 200 m. Assume the length of the chain as 30 m.

Solution: Chainage at the point of intersection = (250 + 15) i.e., 250 chains and 15 links. Length of one chain = 30 m so chainage at the point of intersection = 250 x 30 + 15 x 0.2 = 7503 m Deflection angle (Δ) = 1200, Radius of curve = 200 m. Tangent Length T = R tan Δ/2 = 200 tan 120/2 = 346.41m Length of the curve (l) = πRΔ/ 180 = 200 * 120 π/ 180 = 418.88 m Chainage at point of curve (T1) = Chainage at the point of intersection – tangent length = 7503 – 346.41 = 7156.59 m Chainage at point of tangency (T2) = Chainage at point of curve (T1) + Length of curve = 7156.59 + 418.88 = 7575.47 m.

In Example 2.4, we analyze the road's alignment and calculate necessary points related to a circular curve, starting from the intersection. We have a given deflection angle and radius which guide us in finding the tangent length, the length of the curve, and ultimately the chainage points for both the beginning and end of the curve.

The chainage is a method of measuring distances along a road or railway, so we convert our intersection information into useful distances for construction. This technique allows engineers to precisely locate key points when planning and building roadways.

Think about following a winding path in a park where you need to keep track of where you are. The chainage method acts like a map's coordinate system; it helps you know your exact location relative to the starting point, making sure you don't get lost while navigating the curves just like how a road engineer uses this measure to visualize and plan the route.

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