RL of B is known, not RL of A. How is the book worked?

Compute falls from the readings and work back from B to get A

Use only the HI method forward from A

Asked 3/31
2061 Baisakh · 10 marks

Level book, RL of B 1380.500

Short answer

Working backward from B (RL 1380.500 m) using the HI method gives RL of A = 1389.730 m. Over the 360 m length AB, a fall of 9.230 m results in a gradient of 1 in 39 (approximately 2.56%).

The following consecutive readings were taken with a level and a 4 m levelling staff on a continuously sloping ground at a common interval of 30 metres.

0.855 (on A), 1.545, 2.335, 3.115, 3.825, 0.455, 1.380, 2.055, 2.855, 3.455, 0.585, 1.015, 1.850, 2.755, 3.845 (on B).

The R.L. of B was 1380.500. Make a level book and apply usual checks. Determine the gradient of line AB.

Assumptions: the staff positions are 30 m apart; the readings 3.825 and 3.455 are foresights on change points, and the next readings (0.455 and 0.585) are backsights on the same points after the instrument is shifted. This gives 13 points, with A to B = 12 x 30 = 360 m. RL of B is known, so the levels are worked back from B.

Point	Chainage	BS	IS	FS	HI	Fall	RL (m)
A	0	0.855			1390.585		1389.730
2	30		1.545			0.690	1389.040
3	60		2.335			0.790	1388.250
4	90		3.115			0.780	1387.470
5	120	0.455		3.825	1387.215	0.710	1386.760
6	150		1.380			0.925	1385.835
7	180		2.055			0.675	1385.160
8	210		2.855			0.800	1384.360
9	240	0.585		3.455	1384.345	0.600	1383.760
10	270		1.015			0.430	1383.330
11	300		1.850			0.835	1382.495
12	330		2.755			0.905	1381.590
B	360			3.845		1.090	1380.500

HI is shown on the row where a new set-up starts. RL = previous RL + rise - fall.

Arithmetic checks

\begin{aligned} \Sigma BS - \Sigma FS &= 1.895 - 11.125 = -9.230 \text{ m}\\ \Sigma \text{Rise} - \Sigma \text{Fall} &= 0.000 - 9.230 = -9.230 \text{ m}\\ \text{Last RL} - \text{First RL} &= 1380.500 - 1389.730 = -9.230 \text{ m} \end{aligned}

All three are equal, so the arithmetic is correct.

Gradient of the line AB

RL of A = 1389.730 m, RL of B = 1380.500 m, difference = 9.230 m (A is higher).

\text{Gradient} = \frac{9.230}{360} = \frac{1}{39.00}

Answer: RL of A = 1389.730 m; the line AB falls from A to B at 1 in 39.0 (2.56 %).

✓ Answer checked by Claude Opus 5.5 (high effort)

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