Which of the following is the correct expression used to compute the hypotenusal allowance on a sloping ground, where δ = Angle of sloped ground?

Discussion - MCQs on Chain Surveying - Hypotenusal Allowance

A. Measured length on sloped ground (1-cosδ)²

B. Measured length on sloped ground (1-Secδ)²

C. Measured length on sloped ground (1-cosδ)

D. Measured length on sloped ground (Secδ - 1)

Correct Answer: D. Measured length on sloped ground (Secδ - 1)

📝 Detailed Explanation: Hypotenusal Allowance

When measuring distance on sloped terrain, the length measured along the slope (the hypotenuse) is always greater than the true horizontal distance required for mapping. The hypotenusal allowance is the correction applied to account for this difference. 📐

Essentially, it's the extra distance the surveyor needs to measure along the slope to ensure that the horizontal component of that measurement equals a desired standard length (like one chain length). The formula correctly calculates this extra distance based on the slope angle δ.

Derivation of the Formula

Let's derive the formula using the provided diagram:

Let the slope angle be δ.
Let the standard horizontal distance to be measured be one chain length, L. This is represented by AD.
The actual distance measured along the slope is AC.
From the right-angled triangle ΔADC, we have: cos(δ) = AD / AC.
Therefore, the required slope distance is AC = AD / cos(δ) = L sec(δ).
The hypotenusal allowance is the extra length that must be added, which is the difference between the slope distance and the horizontal distance. This is represented by DC in the diagram, if we consider AB to be the true horizontal length L (so AD=AB).
Hypotenusal Allowance (DC) = AC – AD
Substituting the values: Allowance = (L sec(δ)) – L
Factoring out L gives the final formula: Allowance = L (sec(δ) - 1)

💡 Concept: Methods for Chaining on Sloping Ground

There are two primary methods for accurately measuring distances on sloping ground:

1. Hypotenusal Allowance Method (Indirect Method)

In this method, the chain is stretched along the slope (AC). The follower holds the zero end at point A. The leader then places the arrow not at the end of the chain (B), but further up the slope at point C. The distance BC is the pre-calculated hypotenusal allowance. This ensures that the horizontal distance covered (AB or AD) is exactly one chain length. The process is repeated, and the total horizontal distance is simply the number of full chains measured.

2. Stepping Method (Direct Method)

This method involves measuring the line in short horizontal segments called "steps." Instead of laying the chain on the ground, the follower holds the zero end at the starting point, and the leader raises the chain until it is horizontal, using a plumb bob to transfer the endpoint to the ground. This process is repeated in a step-like fashion down the slope. It is a direct method of finding the horizontal distance but can be tedious and less accurate on steep or uneven slopes.