Trigonometric Levelling and Methods of Contouring

1. Trigonometric Levelling

Introduction

Trigonometric levelling (also called indirect levelling or trigonometrical levelling) is the method of determining the difference in elevation between two points by measuring the horizontal or slope distance between them and the vertical angle subtended at one point to the other. The vertical distance is then computed by trigonometry using the measured distance and angle. It does not require a level line to be carried from point to point as in direct levelling; hence it is rapid and suitable for rough, hilly and inaccessible terrain.

Principle

If the horizontal distance between two stations is $D$ and the vertical angle measured at one station to the other is $\theta$, then for a theodolite whose trunnion axis and the target are at the same height,

$V = D \tan\theta$

The difference in elevation between the two points is then $\Delta h = V$ (plus or minus the correction for curvature and refraction and the height of instrument and signal, as applicable).

Reduction formulae (common cases)

• Base of the object accessible (horizontal distance $D$ measured directly): $V = D \tan\theta$

• Base inaccessible – instrument stations in the same vertical plane (two stations $P_1$ and $P_2$ at a known distance $d$ apart, vertical angles $\theta_1$ and $\theta_2$): $D = \frac{d \tan\theta_2}{\tan\theta_1 - \tan\theta_2}, \qquad V = D \tan\theta_1$

• Correction for curvature and refraction (for long sights): $C = 0.0673\, D^2 \text{ (in metres, } D \text{ in km)}$

Uses

1. Determination of elevation of inaccessible points such as hill tops, towers, spires, church steeples and tall buildings.
2. Fixing reduced levels of stations in hilly terrain where direct levelling is slow and difficult.
3. Establishing vertical control for topographic mapping, contouring and route surveys (roads, canals, transmission lines).
4. Determining heights of clouds, aircraft and other distant objects.
5. Checking and extending benchmarks in mountainous country where fly levelling is impracticable.

2. Methods of Contouring

A contour is an imaginary line on the ground joining points of equal elevation. The following are the methods of contouring:

(a) Direct method The reduced level of a series of points is found by direct levelling (usually by a level and staff) and points of equal elevation are joined to form contours. It is accurate but very slow and costly; used for small areas of high importance such as reservoir sites and dam foundations.

(b) Indirect method The area is divided into a network of points whose elevations are determined, and contours are interpolated between them. It is faster and cheaper. It includes:

1. By cross-sections – Cross-sections are run at regular intervals along a base line; elevations at points on each cross-section are plotted and contours interpolated. Suitable for long narrow strips such as roads, railways and canals.
2. By squares (block levelling) – The area is divided into squares of convenient size (say 5 m to 20 m); spot levels at the corners are taken and contours interpolated. Suitable for small areas, building sites and sports grounds.
3. By radiating lines (radial method) – Radial lines are taken from a central station and spot levels taken along each radial; contours are interpolated. Suitable for hilly areas and for contouring around a hill or a pond.
4. By tacheometry – Horizontal and vertical distances are obtained simultaneously by a tacheometer; spot levels are plotted and contours drawn. Very rapid and suitable for rough and broken ground.
5. By plane table – The plane table is set up and contours are located directly by the intersection or radiation method; no interpolation is needed. Suitable for small-scale mapping.

(c) Indirect method by interpolation After spot levels are obtained, contours are interpolated either arithmetically (by proportional calculation) or graphically (by drawing parallel lines on tracing paper and transferring intersections). Graphical interpolation is quicker and generally preferred.

Choice of method depends on the purpose, scale of map, contour interval, nature of ground and available time and money.

Types of Estimates and Their Uses

An estimate is the approximate calculation of the quantities of materials, labour and cost required to complete a work. The main types are:

1. Preliminary (Approximate) Estimate Prepared at the planning stage without detailed drawings, using rough methods such as unit rate, plinth area, cubical content or service unit method. Used to obtain administrative approval and to decide the feasibility and budget of a project.

2. Detailed Estimate Prepared from complete working drawings, specifications and the schedule of rates. Quantities of each item are measured accurately and priced. Used for tendering, obtaining technical sanction, and for the execution and payment of work.

3. Revised Estimate Prepared when the original sanctioned estimate is exceeded by more than the permissible limit (usually 5%) due to changes in scope, rates or design. Used to obtain fresh technical sanction for the additional expenditure.

4. Supplementary Estimate Prepared when additional work, not included in the original estimate, becomes necessary during execution. Used to sanction extra funds without disturbing the original estimate.

5. Annual Repair (Maintenance) Estimate Prepared every year for the upkeep of an existing structure, based on the previous year's actual expenditure. Used to allot funds for ordinary and special repairs.

6. Quantity Estimate / Bill of Quantities A detailed statement of quantities of all items without rates. Used as the basis for tendering and for checking the detailed estimate.

7. Revised and Supplementary combined is also used when both revision and addition occur together.

Requirements of a Good Estimate

1. Accuracy — quantities and rates must be as close to reality as possible.
2. Completeness — all items of work, including miscellaneous and contingency items, must be covered.
3. Based on standard data — current schedule of rates (DSR), standard specifications and correct drawings.
4. Clear and systematic — proper measurement sheets, abstract forms and units.
5. Adequate contingencies — provision for unforeseen items and price escalation.
6. Reasonable rate analysis — rates must be justified by labour, material and plant output.
7. Comparability — should allow comparison of tenders and control of expenditure.

Basic Components of Rate Analysis

Rate analysis is the determination of the unit rate of an item of work. Its basic components are:

1. Quantity of materials — as per standard norms (e.g., cement, sand, aggregate, bricks, steel), including wastage.
2. Cost of materials — at site, i.e., basic rate + freight + taxes + loading/unloading + storage.
3. Labour — number of skilled and unskilled workers required per unit, with their daily wages, plus mate/ supervisor.
4. Plant and machinery — hire charges or ownership cost (depreciation, repairs, fuel, operator) for mixers, vibrators, hoists, etc.
5. Overheads — general overheads such as office expenses, supervision, insurance, and sometimes water and electricity charges.
6. Contractor's profit — usually 10% of the total cost.
7. Contingencies — usually 3–5% for unforeseen items.

The unit rate is obtained as:

$\text{Rate} = \frac{(\text{Material} + \text{Labour} + \text{Plant}) \times (1 + \%\text{Overhead} + \%\text{Profit} + \%\text{Contingency})}{\text{Quantity of work}}$

The task of rate analysis is to work out the cost of materials, labour and plant per unit of finished work, add overheads and profit, and thus fix a fair and reasonable rate for the item.