2.3. Unsolved Numerical Problems

A 4.56° circular curve is to be designed to fit between two intersecting straights. What is the radius of this curve? (Ans: 1256.49 m)

To find the radius of a curve given its angle subtended (in degrees), you can use the formula:

$R = \frac{C}{\theta}$

where $C$ is the chord length subtended by the angle and $\theta$ is the angle in radians. Converting degrees to radians is essential for accurate calculations. However, in this case, the angle given is directly used in a specific formula for circular curves, and we find the radius to be 1256.49 m based on that input.

Think of the curves in a racetrack where the sharpness of the turn affects speed. A larger radius means a gentler turn, which is similar to how a wider turn in a car allows for smoother navigation than a sharp turn.

It was found, while setting a simple circular curve using offsets from the tangents, that for X =16 m, and Y = 0.28 m. Find the radius of the curve. (Ans: R = 457.28 m)

In this problem, we use the concept of offsets to get the radius of a circular curve. The radius can be determined using the relationship between the offset (Y) and the distance from the curve (X). The formula:

$R = \frac{X^2 + Y^2}{2Y}$

is applied here. Substituting the values into this formula gives us the radius R = 457.28 m.

Imagine you're observing a fountain from a distance. The closer you get, the clearer your view becomes. Similarly, the offsets tell us how far we are from the curve, helping us calculate its radius accurately.

A right hand simple circular curve connects two straights AI and IB where A and B are the tangent points. The azimuth of each straight is 500 43' 12" and 880 22' 14", respectively. The curve passes through point C of coordinates (321.25, 178.1) m. If the coordinates of A are (240.40, 125.90) m, compute the radius of curve and degree of curve. Neglecting the existence of point C above and assuming the radius of curve as 450 m and chainage of I as 2140 m, find the (i) chainage of tangents points A and B, (ii) the external distance E and mid-ordinate M, (iii) the length of various chords, and (iv) the corresponding deflection angles. (Ans: R = 428.87 m, Degree of curvature = 040 00' 28", Chainage of A = 1986.59 m, Chainage of B = 2282.29 m, E = 25.43 m, M = 24.072 m, C = 20 m, C = 13.10 m, C = 2.296 m, Δ = 1 2 1 000 51' 13.4", and Δ = 000 08' 46.2")

This problem involves various steps to compute multiple curve-related parameters starting from given coordinates. First, the radius R is recalculated based on geometry derived from the tangent points A and B and the provided angles. Chainages help find the exact position of tangent points, while parameters like the external distance E and mid-ordinate M denote how far the curve deviates from the tangents. Each resultant angle helps in further mapping out the curve's characteristics.

Think of aligning two garden paths with a circular flower bed between them. Each angle helps determine how far the paths drift away from the shortest route, just like how tangents and curves work in road design.

There are two tangents XI and IY for a railroad circular curve where X and Y are the tangent points having coordinates (240.4E, 125.9N) and (253.8E, 218.65N), respectively, and the coordinates of the mid-point on the curve is (60.13E, 195.89N). Compute the radius of curve, the deflection angle, and the length of the curve. (Ans: Δ = 2700 57' 56", R = 104.48 m, L = 494.11 m)

To solve this problem, we start from the given coordinates of the tangent points to find the deflection angle between the two lines. By understanding the angle created and the linear distances, we can calculate the curve's radius and its length. Using formulas for chord lengths and angles in circular motion, we determine all required parameters systematically.

Like sketching paths on a map where you have to calculate how wide your circle (representing a bend) needs to be based on two points of reference. The tighter the path needs to be, the smaller the radius.

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