2. Angular methods of setting out curves

In Angular methods, curves are set out using the deflection angles or tangential angles, calculated with the help of a theodolite and tape. The two main methods utilized in this approach are:

1. Rankine’s method of tangential angles
2. Two theodolites method

In this section, we introduce how curves can be established using angles instead of the traditional linear methods. Angular methods are utilized because they provide precision and ease in areas where measurements might be complicated by obstacles or uncooperative terrain. Rankine's technique focuses on calculating deflection angles to set out curves, while the two theodolite method uses two instruments to enhance accuracy.

Imagine trying to draw a perfect circle on a large field using just a string as a compass. If there are rocks or bushes, it becomes challenging to maintain the string's tautness. Similarly, using angles to define curves helps bypass these obstacles, allowing surveyors to work more efficiently and accurately.

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.

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. The chord T D can be taken as equal to arc T D = C and its relation with deflection angles can be expressed mathematically.

Rankine's method allows surveyors to use precise angles to define curves. By calculating the deflection angles based on the tangent points and chords of the curve, surveyors can accurately plot the curve's path. The key steps involve taking measurements using the theodolite, calculating angles for various segments of the curve, and ensuring the resulting path matches the intended curve shape.

Think of it like directing traffic: if you can see where the cars need to turn (or curve), you can use your hands to signal the exact angle they need to follow. Similarly, using deflection angles helps in guiding the construction of roads or tracks accurately around obstacles.

The deflection angle for a 30 m chord is equal to D/2 degrees, and that for the sub-chord of length C can be calculated as follows:

• For the first chord: $\delta = 1718.9\frac{C_1}{R}$ minutes
• General relationship: $\delta_n = 1718.9\frac{C_n}{R}$

To set out a curve accurately, it is essential to calculate the deflection angles for each segment of the curve. These angles determine how much the curve deviates from a straight line at each chord length. The formula indicates that as the chord length increases, the angle changes, allowing for gradual transitions instead of sharp turns.

Imagine a winding river where the bends (like the deflection angles) vary in sharpness. Just as a mapmaker would note how sharp each turn is to guide a boat safely, surveyors calculate these angles to design roads that are safe and comfortable for vehicles.

The procedure for setting out a curve involves:

1. Locate the tangent points T1 and T2, and find their chainages.
2. Calculate the lengths of first and last sub-chords and the total deflection angles for all points on the curve.
3. Set up the theodolite at the first tangent point.

Setting out a curve is a step-by-step process that begins with identifying the tangent points of the intended curve. These points serve as reference locations from which measurements are taken. After calculating the required dimensions and angles, the surveyor uses the theodolite to accurately mark each point along the curve, ensuring a smooth transition without sharp turns.

Visualize prepping a garden path: you first mark where the path needs to start and end, then measure out sections to determine how the path will curve around existing plants. Just as you’d want a gentle bend in the walkway to avoid tripping, surveyors aim for smooth curves that provide safety and comfort.

The Two Theodolite Method is useful in the absence of distance measurement by tape and when the ground is not favorable for accurate distance measurement. In this method, two theodolites are used to set the curve.

By employing two theodolites, surveyors can achieve a high level of accuracy in setting out curves, especially in complex terrains. Each theodolite can be adjusted independently to measure angles toward different curve points, facilitating precise plotting of the curve without needing to measure distances directly.

Consider a two-person team trying to measure the height of a tree: one person holds a measuring tape while the other uses a protractor to find the right angle for the measurement. Similarly, using two theodolites enables surveyors to gather detailed information about curves, vital for maintaining accuracy across challenging environments.

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