4.6. Example 2.6

Chainage at the point of intersection = 2056.44 m, Angle of intersection = 1300, Radius of curve (R) = 50 m

The section begins with essential details about the curve that connects two straight road alignments. The chainage represents a specific location on the road measured from a defined starting point. Here, the chainage at the point of intersection is given as 2056.44 meters, and the angle of intersection at this point is 130 degrees. The radius of the curve has also been specified as 50 meters, which indicates how sharply the curve bends. The angle of intersection helps determine how the two straight sections align with each other at the curve.

Imagine you're driving on a highway that makes a turn at a junction. You can think of chainage like the mile markers you see along the side of the road, which tell drivers how far they've traveled. Knowing the angle at which the roads meet helps drivers anticipate how sharply they need to turn their steering wheel as they enter the curve.

Deflection angle (Δ) = 1800 - 1300 = 500 Tangent Length T= R tan Δ/2 = 50 tan 50/2 = 23.32 m Length of the curve (l) = πRΔ/ 180 = 50 * 50 π/ 180 = 43.63 m

To construct the curve accurately, we calculate several key measurements. The deflection angle is calculated by subtracting the intersection angle from 180 degrees, which gives us 50 degrees, signifying how much the curve deviates from a straight line. The tangent length is determined using the formula T = R tan(Δ/2), where R is the radius of the curve. The result tells us how far the road travels along a straight path before entering the curve. Lastly, the length of the curve is computed using the formula for the arc length, which involves multiplying the radius (50 m) by the deflection angle and applying a conversion factor (π/180) to convert to meters.

Think of driving onto a roundabout. The ‘deflection angle’ is like how much you need to steer your car away from driving straight ahead. The ‘tangent length’ is analogous to the distance your car would travel on a straight path before you actually start turning into the roundabout. Finally, the ‘length of the curve’ is the actual distance around the circular path of the roundabout itself.

Chainage at point of curve (T1) = Chainage at the point of intersection – tangent length = 2056.44 – 23.32 = 2033.12 m Chainage at point of tangency (T2) = Chainage at point of curve (T1) + Length of curve = 2033.12 + 43.63 = 2076.75 m

This part involves determining the specific chainage at important points along the curve. First, we find the chainage at the point where the curve begins (T1) by subtracting the tangent length from the initial chainage at the intersection. Then, we calculate the chainage at the endpoint of the curve (T2) by adding the length of the curve to T1. These calculations help in marking the locations on the roadway where drivers will transition into and out of the curve, ensuring clarity and safety.

Consider planning a scenic drive. When you reach a junction, you recognize that you need to start turning your steering wheel (T1) a certain distance before you reach the curve's full extent (T2). Just like how you would measure out how far along the road you should start preparing for the turn, these chainages mark where the curve actually starts and ends, helping you make smooth transitions.

Length of Long chord (L) = T1 T2 = 2R sin(Δ/2) = 2 * 50 sin(50/2) = 42.26 m

The final calculation here is about finding the length of the long chord. The long chord is the straight line that connects the points where the curve begins and ends. It's calculated using the radius and the deflection angle. This relationship explains how 'flat' or straight the curve seems when you look at it as a whole. Using the formula for the long chord, the length is derived, contributing to an accurate representation of how much roadway is actually laid out in a straight line along what was once curved.

Imagine you laid a piece of string from one end of a circular piece of pizza to the other—this string represents the long chord of the curve. Instead of driving on the entire circle, if you drove straight across the pizza, the length of that string would represent how much distance you save versus traveling along the curve. It helps you visualize the direct path compared to the curved one.

Starting from centre of long chord, offset at 5 m interval are calculated for half of the long chord, i.e., 42.26/2 = 21.13m y = √ (R2 – x2) - √ [R2- (L/2)2]

For accurately setting out the curve in the field, offsets are calculated from the long chord at specified intervals. This means we measure out perpendicular distances (offsets) from the straight line (long chord) to the curve at regular intervals (5 meters in this case). The formula described helps to compute these offsets by transforming the circular geometry into measurable distances. This ensures that workers or engineers can replicate the planned curve accurately on the ground.

Picture a construction team laying down the path for a new bike trail that will curve around trees. They lay a straight line (long chord) where the finish line will be and then calculate how far off the straight line they need to go to create the curve at 5-meter intervals. These offsets are essential for laying down the exact path the trail should take.

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