In surveying, for the purpose of plotting, only horizontal distances are required. If the ground is sloping, the horizontal distances are obtained either directly or indirectly. If the slope of the country is upto 3°, it is generally taken as level ground.
The direct and indirect methods of finding horizontal distances on sloping ground are given below:
1. Direct Method (Stepping Method):
The method consists in measuring the line in short horizontal lengths called steps. Suppose it is required to measure the horizontal distance between the points A and B (Fig. 2.19). For convenience and to obtain better results, chaining is done from top of hill to the toe i.e. downhill.
The follower holds the zero end of the chain at A on the ground and the leader with the other handle of the chain and a ranging rod moves in the forward direction at a convenient distance apart.
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The leader then stops and the follower directs him to be in line with B. The leader then stretches the chain in the horizontal position and transfers the end point of the chain on the ground with the help of a plumb-bob or ordinarily with a ranging rod or by dropping a piece of stone.
The follower then moves and reaches the station occupied by the leader and then the leader moves in the forward direction. The process is repeated for several times until the whole line is chained. The lengths of all the steps are added up so as to obtain the total horizontal distance. In this case, the horizontal distance between AB = Ap + Pq + Qr + Rb.
Note:
The vertical distance between the end of the chain and the point vertically below it on the ground such as pP, qQ etc. should not exceed 1.8 m (man’s height) as otherwise the chain cannot be stretched to the horizontal position i.e. a sag would occur in the chain and the measured length will be more than the actual length of the step. Therefore when the slope of the ground is steep, the chain of smaller length is used. For this purpose, the chain may be broken into different portions say 10, 20, 30 or 50 links.
2. Indirect Method:
The whole length of the line is first of all divided into different sections having approximately the same slope. Horizontal distance is calculated for each section separately and the total horizontal distance is then obtained by summing up all the horizontal distances of different sections.
The horizontal distance for each section may be found out by any of the following indirect methods:
(i) The distance along the slope is measured and then the angle of slope i.e. angle between the sloping ground and the horizontal surface (Fig. 2.20) is found with the help of Abney’s level or precisely with theodolite. Knowing the sloping distance say I and angle of slope say α horizontal distance, D can be calculated by the relation, D = I cos α
(ii) The distance along the slope is measured with chain and the difference in elevation between the first and the end stations is found with the help of any levelling instrument (Fig. 2.21).
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Knowing the sloping distance I and the difference in elevation h, the horizontal distance, can be found out by the relation:
(iii) The correction may be applied in the field at every chain length.
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The correction may be calculated as below (fig. 2.22):
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Let α = the angle of slope of the ground.
AD = AB = 1 Chain = 100 links.
Then AC = 100 sec α links and BC = AC – AB = 100 (Sec α- 1) links.
The amount 100 (sec α – 1) is known as hypotenusal allowance.
The chain is stretched in the direction AB and the arrow is placed in advance of the end B of the chain by an amount equal to (BC), the hypotenusal allowance, such that the horizontal equivalent of the total distance i.e. (chain length + hypotenusal allowance)is equal to one chain length. The next chain length starts from C. The process is continued until the end of the line is reached. The required horizontal distance is the equal to number of chains measured.
Comparison between direct and indirect methods of chaining on sloping ground:
Direct or Stepping Method:
It is more convenient and rapid if the slopes are short, steep and of varying degree. The distances measured are sufficiently correct if the end points of the chain are transferred correctly on to the ground and the sag in the chain due to weight of the chain and insufficient pull is avoided.
Indirect Method:
It is more accurate and rapid when the slopes are long, regular and gentle. The third indirect method in which the correction for hypotenusal allowance is applied to each chain length is usually employed in route surveys.
Correction for Slope:
Horizontal distances can also be determined by applying negative corrections to the distance measured along the slope.
The formulae may be derived as follows:
(i) Correction for slope if the vertical distance between first and the last points is known:
In fig. 2.23 let:
I = the slope distance AB
h = the vertical distance BB1 between two points A and B
D = the horizontal distance AB1
Ch= the correction to the
The correction is to be subtracted from the slope distance in order to obtain its horizontal equivalent.
(ii) Correction for slope if the angle of slope is known:
If the angle of slope of the ground is measured with greater precision such as with a theodolite, then the correction for slope should be determined by the following exact formula.
