4.5. Example 2.5

Two straights AC and CB intersect at C at a chainage of 86.22 chains at a deflection angle of 620. They are to be smoothly connected by a simple curve of radius 12 chains. Find the tangent length, length of curve and the chainages of the starting and end points of the curve. Also, find the length of the long chord.

In this example, we're tasked with connecting two straight sections of a road (AC and CB) at a point of intersection (C) with a circular curve. The chainage is a way to express positions along the road in chain lengths, often used in civil engineering. The deflection angle of 620 indicates how much the path will change direction at the connection. We need to calculate several key components, including the tangent length, the length of the curve, and chainages for points related to the curve.

Imagine you have two straight paths in a park that meet at an angle and you want to connect them with a curved footpath, similar to a water slide joining two straight segments. You need to know how long that curved segment will be and how far along each straight path to make the connection. Just like measuring a distance for the slide and ensuring it fits well, engineers need to calculate these values accurately.

Tangent Length T = R tan(Δ/2) = 12 tan(62/2) = 7.21 chains.

The tangent length (T) helps in determining how far from the intersection point (C) the curve starts. Here, we are using the formula T = R tan(Δ/2), where R is the radius of the curve, and Δ is the deflection angle in degrees. Dividing the deflection angle by 2 helps in finding the tangent distance efficiently.

Think of the tangent length as the ramp leading to a water slide. The steeper the angle (the more turns you want) means a longer ramp before you start the slide. In this case, we're finding how long that ramp needs to be before it curves with the path.

Length of the curve (l) = πRΔ/180 = 12 * 62 π/ 180 = 12.985 chains.

To find the actual length of the curve connecting the two straight paths, we use the formula for arc length, which is l = πRΔ/360 for degrees converted into a fraction of a full circle. We multiply the radius by the angle and a constant factor to convert it into a distance along the curve. This gives us how long the road will actually be along the curve.

Consider the length of a cookie cutter shaped like a circle that you are using to cut dough. The total distance around the circle's edge (in this case, how far a car would travel along the road) can be calculated like how you'd measure the dough's cut length using the radius and the angle you used to create the cut.

Chainage at point of curve (T1) = Chainage at the point of intersection – tangent length = 86.22 – 7.21 = 79.01 chains. Chainage at point of tangency (T2) = Chainage at point of curve (T1) + Length of curve = 79.01 + 12.985 = 91.995 chains.

The chainage at the point of curve (T1) is found by subtracting the tangent length from the chainage at the intersection point. This gives the exact position where the curve starts. Then, to find where the curve ends (T2), we add the length of the curve (l) to the starting chainage.

Imagine you're on a road trip and the GPS tells you that you've driven 86 miles to a junction (the intersection). If you know that there's a ramp (tangent) that takes you off the highway for about 7 miles before connecting to a new road (the curve), you subtract that distance to find where the new road starts. When you finally reach the end of the curved road, you add the distance traveled along the curve to know how far you've gone overall.

Length of Long chord (L) = T1 T2 = 2R sin(Δ/2) = 2 * 12 chains sin(62/2) = 12.361 chains.

To find the length of the long chord connecting the start and end of the curve directly, we apply the formula for the chord length. The long chord acts as a straight line across the circle's segment formed by the curve. It's calculated using the radius and the sine of half the deflection angle, providing the direct distance across the circle.

Think of this long chord as if you're stretching a rope directly from one end of a curved path to the other end, skipping the curve itself. Just like measuring a straight distance across a rounded table, you're finding out how far it is without following the curve.

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