2.1. Rankine’s method of tangential angles

In Rankine’s method, the curve is set out by the tangential or deflection angles using a theodolite and a tape. The deflection angles are calculated to set out the curve (Figure 2.10).

Rankine’s method is used for setting out curves, particularly in surveying and civil engineering. This method utilizes a theodolite—a precision instrument for measuring angles in the horizontal and vertical planes—and a tape measure to establish the curve of a road or railway track. The essence of the method lies in calculating deflection angles, which are the angles between the tangent line and a chord of the curve. These angles allow surveyors to accurately locate points on the curve.

Imagine you're trying to lay out a curved garden path. Instead of measuring directly in a curved line, you lay out straight sections and calculate the necessary angles to connect those sections smoothly. Similarly, Rankine’s method uses calculated angles to maintain the curvature of the path.

If T and T are the tangent points and AB the first tangent to the curve, D, E, F etc., are the successive points on the curve, ϕ is the deflection angle of the curve, R is the radius of the curve, C1, C2, C3, etc., are length of the chords T1D, DE, EF etc., δ, δ, δ etc. are the tangential angles which each of the chords T1D, DE, EF, etc., makes respectively with the tangents at T1, D, E, etc.

In this section, we define several key terms essential for understanding Rankine’s method. Tangent points (T1, T2) are points where straight lines meet the curve. The radius (R) is the distance from the center of the curve to any point on the curve. Each chord (C1, C2, C3) connects points on the curve. Deflection angles (ϕ) refer to the angles at which these chords deviate from the tangent line, allowing the path to bend smoothly. By transforming these angles into measurements, the surveyor can create an accurate layout of the curve.

Think of driving along a curved road. At every point along the curve, your steering wheel must adjust to steer the car smoothly along the path. The angles you are turning at each point are similar to the deflection angles in Rankine’s method, guiding you to maintain a steady course without veering off.

The chord T D can be taken as equal to arc T D = C₁ ∠BT₁D = δ₁ = ∠TOD = 2δ₁ arcTD = ∠TOD in radians radiusOT₁ C₁ or arcTD = 1 = 2δ₁ radians R₁ C₁ or δ₁ = 1 radians/2R.

This segment explains how to compute deflection angles for each chord. The length of the chord T1D is considered equal to the arc along the curve, C1. The angle ∠BT1D (the angle to the tangent line at T1) corresponds to the deflection, δ₁, which is defined in terms of radians based on the radius of the curve. This mathematical relationship is crucial because it helps in determining how sharply or gently the curve will bend.

Imagine you're playing a game of pool. When you hit a ball, the angle at which the cue ball strikes the object ball determines its path across the table. Just as that angle helps you guide the ball to the pocket, the deflection angle helps engineers determine the path of a road. Both rely on precise calculations to achieve the desired outcome.

A general relationship would be as follows: Δ = δ₁ + δ₂ + δ₃ + ... + δn Δ = Δn−1 + δn.

This relationship illustrates that the total deflection angle (Δ) for any point along the curve can be calculated by summing all previous deflection angles (δ₁, δ₂, etc.). Each new angle added represents a new segment of the curve. In this way, surveyors can account for the cumulative turn of the road as they plot its course, ensuring it follows the desired curvature smoothly.

Consider building a winding path through a garden. Each angle you take at a turn essentially compounds the direction you're moving. By keeping track of each turn’s angle, you can ensure the path flows naturally. Just like this, by adding deflection angles, engineers can ensure the road flows well and is easy to navigate.

If the degree of the curve (D) is known, the deflection angle for a 30 m chord is equal to D/2 degrees, and that for the sub-chord of length C₁, it would be; δ₁ = C₁ D/30.

This chunk explains how to verify or check your calculations of deflection angles. When the degree of the curve (D) is known, it allows for an easier calculation of the corresponding deflection angles for different chord lengths. Understanding this relationship helps ensure that the overall layout stays consistent with the required curvature specifications.

Imagine you're measuring out a circular garden bed, and you have a ruler that indicates how 'curvy' that bed is. By knowing the degree of curvature, you can calculate how sharply to bend the edges of the bed to match your vision. Similarly, knowing the degree of the road curve helps planners design it properly for safety and aesthetics.

1. Locate the tangent points T₁ and T₂, and find out their chainages. From these chainages, calculate the lengths of first and last sub-chords and the total deflection angles for all points on the curve.
2. Set up the theodolite at the first tangent point T₁.
3. Set the initial horizontal circle reading to zero and direct the telescope to the intersection point and bisect it.
4. Set the first deflection angle Δ₁ in theodolite, and direct the telescope along T₁D. Along this line, measure T₁D equal in length to the first sub-chord, thus fixing the first point D on the curve.
5. Now set the second deflection angle Δ₂ in theodolite, and direct the line of sight along T₁E. Hold the zero end of the tape at D and swing the other end until the tape is bisected by the line of sight, thus fixing the second point E on the curve.
6. Continue the process until the end of the curve is reached.
7. The end point thus located must coincide with the previously located point (T₂). If not, the distance between them is the closing error. If it is within the permissible limit, only the last few pegs may be adjusted; otherwise the curve should be set out again.

This step outlines the practical process for using Rankine’s method to lay out a curve. Each step builds upon the previous one, starting from establishing key points (tangent points) and moving through to measuring chords and adjusting angles on the theodolite. This orderly approach ensures accuracy and reliability in the layout of the curve, which is essential for safe and effective road design.

Setting out a new road can be likened to drawing a line through a landscape filled with obstacles. Just as an artist has to carefully navigate around trees and rocks, a surveyor must measure accurately to account for natural features while laying out a road. Following each step in Rankine’s method allows for those navigations to be smooth and precise.

If there is small closing error, all points on the curve are moved sideways by an amount proportional to the square of their distances from the tangent point T₁, but if the error is large, the entire process is repeated.

This final part talks about error adjustment in the curve setting. After laying out the curve, if there are discrepancies (closing errors), surveyors can make slight adjustments. For smaller errors, they can adjust points based on their distance from the tangent point. For larger discrepancies, the whole process may need to be redone to ensure accuracy. This aspect emphasizes the importance of precision in engineering tasks.

Think of a chef preparing a complex dish. If they taste the sauce and find it's slightly off, they may add a pinch of salt to adjust the flavor. However, if the dish is significantly off—like if they accidentally used sugar instead of salt—they might need to start over. Similarly, surveyors fine-tune the curve until it meets the necessary standards of accuracy.

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