LESSON NOTE ON CHECK ON CLOSED TRAVERSE

CHECK ON CLOSED TRAVERSE

1. Check on angular measurements

(a) The sum of the measured interior angles should be equal to (2N – 4) x 90⁰ where N is the number of sides of the traverse.

(b) The sum of the measured exterior angles should be equal to (2N + 4) x 90⁰.

(c) The algebraic sum of the deflection angles should be equal to 360⁰.

Right-hand deflection is considered positive and left-hand deflection negative.

2. Check on linear measurement

(a) The lines should be measurement once each on two different days (along opposite directions). Both measurements should tally.

(b) Linear measurements  should also be taken by the stadia method. The measurements by chaining and by the stadia method should tally.

LESSON NOTE ON CHECK ON OPEN TRAVERSE

CHECK ON OPEN TRAVERSE

In open traverse, the measurements cannot be checked directly. But some field measurements can be taken to check the accuracy of the work. The methods are discussed below.

1. Taking cut-off lines Cut-off lines are taken between some intermediate stations of the open traverse. Suppose ABCDEF represents an open traverse. Let AD and DG be the cut-off lines. The lengths and magnetic bearings of the cut-off lines are measured accurately. After plotting the traverse, the distances and bearings are noted from the map. These distances and bearings should tally with the actual records from the field

2. Taking an auxiliary point Suppose ABCDEF is an open traverse. A permanent point P is selected on one side of it. The magnetic bearings of this point are taken from the traverse stations A,B,C,D, etc. If the survey is carried out accurately and so is the plotting, all the measured bearings of P when plotted should meet at the point P. The permanent point P is known as the ‘auxiliary point’

LESSON NOTE ON MEHODS OF TRAVERSING

MEHODS OF TRAVERSING

Traverse survey may be conducted by the following methods :

1.     Chain traversing (by chain angle)

2.     Compass traversing (by free needle)

3.     Theodolite traversing (by fast needle) and

4.     Plane table traversing (by plane table)

1.Chain traversing Chain traversing is mainly conducted when it is not possible  to adopt triangulation. In this method, the angles between adjacent sides are fixed by chain angles. The entire survey is conducted by chain and tape only and no angular measurements are taken. When it is not possible to form triangles, as, for example, in a pond, chain traversing is conducted,

The formation of chain angles is

(a) First Method Suppose a chain angle is to be formed to fix the directions of  sides  AB and AD. Tie stations T₁ and T₂ are fixed on lines AB and AD. The distances AT₁, AT₂ and T₁T₂ are measured. Then the angle T1AT2 is said to be the chain angle. So, the chain angle is fixed by the tie line T1T2.

(b) Second Method Sometimes the chain angle is fixed by chord. Suppose the angle between the lines AB and AC is to be fixed. Taking A as the centre and a radius equal to one tape length (15 m), an arc intersecting the lines AB and AC at points P and  Q, respectively, is drawn. The chord PQ is measured and bisected at R.

The angle θ can be calculated from the above equation, and the chain angle  BAC can be determined accordingly.

2. Compass traversing In this method, the fore and back bearings of the traverse legs are measured by prismatic compass and the sides of the traverse by chain or tape. Then the observed bearings are verified and necessary  corrections for local attraction are applied. In this method, closing error may occur when the traverse is plotted. This error is adjusted graphically by using ‘Bowditch’s rule’ (which is described later on).

3. Theodolite traversing In such traversing, the horizontal angles between the traverse legs are measured by theodolite. The lengths of the legs are measured by chain or by employing the stadia method. The magnetic bearing of the starting leg is measured by theodolite. Then the magnetic bearings of the other sides are calculated. The independent coordinates of all the traverse stations are then found out. This method is very accurate.

4. Plane table traversing In this method, a plane table is set at every traverse station in the clockwise or anticlockwise direction, and the circuit is finally closed. During traversing, the sides of the traverse are plotted according to any suitable scale. At the end of the work, any closing error which may occur is adjusted graphically.

LESSON NOTE ON CROSS-STAFF AND OPTICAL SQUARE

CROSS-STAFF AND OPTICAL SQUARE

A. Cross-staff

The cross-staff is a simple instrument for setting out right angles. There are three types of cross-staves.

1. Open
2. French
3. Adjustable

The open cross-staff is commonly used.

Open cross-staff

The open cross-staff consists of four metal arms with vertical slits. The two pairs of arms (AB and BC) are at right angles to each other. The vertical slits are meant for sighting the object and the ranging rods. The crossstaff is mounted on a wooden pole of length 1.5m and diameter 2.5 cm. The pole is fitted with an iron shoe.

For setting out a perpendicular on a chain line, the cross-staff is held vertically at the approximate position. Suppose slits A and B are directed to the ranging rods (R, R₁) fixed at the end stations. Slits C and D are directed to the object (O). Looking through slits A and B, the ranging rods are bisected. At the same time, looking through slits C and D, the object O is also bisected. To bisect the object and the ranging rods simultaneously, the cross staff may be moved forward or backward along the chain line

B. Optical Square

An optical square is also used for setting out right angles. It consist of a small circular metal box of diameter 5 cm and depth 1.25 cm. It has a metal cover which slides round the box to cover the slits. The following are the internal arrangements of the optical square.

1. A horizon glass H is fixed at the bottom of the metal box. The lower half of the glass is unsilvered and the upper half is silvered.

2. A index glass I is also fixed at the bottom of the box which is completely silvered.

3. The angle between the index glass and horizon glass is maintained at 45⁰.

4. The opening ‘e’ is a pinhole for eye E, ‘b’ is a small rectangular hole for ranging rod B, ‘P’ is a large rectangular hole for object P.

5. The line EB is known as horizon sight and IP as index sight.

6. The horizon glass is placed at an angle of 120⁰ with the horizon sight. The index glass is placed at an angle of 105⁰ with the index sight.

7. The ray of light  from P is first reflected from I, then it is further reflected from H, after which it ultimately reaches the eye E

Principle

According to the principle of reflecting surfaces, the angle between the first incident ray and the last reflected ray is twice the angle between the mirrors. In this case, the angle between the mirrors is fixed at 45⁰. So, the angle between the horizon sight and index sight will be 90⁰.

Setting up the perpendicular by optical square

1.     The observer should stand on the chain line and approximately at the position where the perpendicular is to be set up.

2.     The optical square is held by the arm at the eye level. The ranging rod at the forward station B is observed through the unsilvered portion on the lower part of the horizon glass.

3.     Then the observer looks through the upper silvered portion of the horizon glass to see the image of the object P.

4.     Suppose the observer finds that the ranging rod B and the image of object P do not coincide. The he should move forward or backward along the chain line until the ranging rod B and the image of P exactly coincide

5.     At this position the observer marks a point on the ground to locate the foot of the perpendicular.

LESSON NOTE ON PROCEDURE OF PLOTTING

PROCEDURE OF PLOTTING

1. A suitable scale is chosen so that the area can be accommodated in the space available on the map.

2. A margin of about 2 cm from the edge of the sheet is drawn around the sheet.

3. The title block is prepared on the right hand bottom corner.

4. The north line is marked on the right-hand top corner, and should preferably be vertical. When it is not convenient to have a vertical north line, it may be inclined to accommodate the whole area within the map. 
5. A suitable position for the base line is selected on the sheet so that the whole area along with all the objects it contains can be drawn within the space available in the map.

6. The framework is completed with all survey lines, check lines and tie lines. If there is some plotting error which exceeds the permissible limit, the incorrect lines should be resurveyed.

7. Until the framework is completed in proper form, the offsets should not be plotted.

8. The plotting of offsets should be continued according to the sequence maintained in the field book.

9. The main stations, substations, chain line, objects, etc. should be shown as per standard symbols

10. The conventional symbols used in the map should be shown on the right-hand side.

11. The scale of the map is drawn below the heading or in some suitable space. The heading should be written on the top of the map.

12. Unnecessary lines, objects etc. should be erased.

13. The map should not contain any dimensions.

Lesson Note On The importance of geodetic control in the construction process

The importance of geodetic control in the construction process 

The purpose of construction guidance observations Geodetic observations are quite often required during the construction of the engineering structures. The purpose of these observations is either the geometric guidance of the construction or the control of the geometrical quality of the built structures. In both cases surveyors are required to compare the ’as built’ status of the structure with the geometric position and dimensions of the structures on the plans. Construction guidance observations are a part of the construction activities. These observations are carried out continuously during the construction process. The purpose of these observations is to quantify the discrepancies between the built structures and the planned positions and dimensions, thus these discrepancies can be corrected for by applying ’construction/assembly corrections’. Continuous guidance observations are required for example in the case of the construction of reinforced concrete structures using sliding formworks. Let’s imagine that the chimney is being constructed, that has a circular cross‐section.

The purpose of the construction guidance observations is to determine the radius and the center of the cross‐section of the built part. The discrepancies between the ’constructed’ cross‐ sections and the planned cross‐sections provide the geometrical correction for the placement of the sliding formwork. These corrections show not only the correction of the center line, but also the correction of the radius of the structure. Thus both the position of the sliding formwork and the dimension of the formwork can be corrected based on the construction guidance observations. In many cases the purpose of construction guidance observations is to adjust the position and the orientation of structural elements (for example: pillars). These adjustments are done with an iterative procedure. Firstly the structural element is placed approximately to the correct place and it is oriented approximately in the correct direction. Afterwards surveyors measure the exact position and orientation of the element and compute the required corrections to meet the planned position and orientation values. Based on these corrections, construction workers can adjust the placement and the orientation of the structural element. Unfortunately – mainly due to the large size and weight of these structures, this adjustment can not be done in a single step. Thus the new discrepancies in position and orientation must be quantified by the surveyor again, and new correction values are computed. Usually the correction values have a decreasing trend, thus the position and the orientation of the structure should converge to its planned position and orientation. This iterative process is done until the computed correction values are below the given tolerance. Construction control observations are done after the construction process is finished. In this case the purpose of the observations is the determination of the geometric error of the construction. The final approval of the construction usually depends on the results of these construction control observations. In some cases, when the construction process is separated into different individual steps, the results of the construction control observations are taken into consideration during the planning of the next construction step. Such an example can be the construction of the nuclear power plant in Paks, when the small geometric error of the construction of the buildings was taken into consideration in the planning of the technological facilities. In some cases the documentation of geometric construction error helps to determine the cause of malfunctioning structures. It is also important to mention that the positioning observations of the entire structures are usually distinguished from the positioning and dimensional observations of the structural elements. This is because of the fact that – in most cases – a lower accuracy is required for the positioning and the orientation of the entire structure compared to the positioning and the orientation of the structural elements. Imagine that a large hall can be misplaced by several centimeters without any serious consequences, but the misplacement of pillars inside the building by the same amount would lead to the failure of the structure.

LESSON NOTE ON SPHERICAL TRIANGLE

SPHERICAL TRIANGLE

The theodolite measures horizontal angles in the horizontal plane, but when the area becomes large, such as in the case of primary triangulation, the curvature of the earth means that such planes in large triangles called as spherical triangles or geodetic triangles are not parallel at the apices as shown in Fig. 6.8. Accordingly, the three angles of a large triangle do not total 180°, as in the case of plane triangles, but to 180° + £, where £ is known as spherical excess. The spherical excess depends upon the area of the triangle, and it is given by

£=   A_o / R sin1’  seconds

where A = the area of the triangle in sq km, and

R = the mean radius of the earth in km (=6373 km).

The triangular error is given by

£ = Σ  Observed angles – (180° + £)

= A + B + C – (180° – £)

SOME TERMS USED IN LEVELLING OPERATIONS

SOME TERMS USED IN LEVELLING OPERATIONS
1. Bench mark (BM): it is a fixed point on the earth’s surface whose level above ordance
datum is known.
2. Ordnance Datum (OD): it is the mean sea level to which all other levels are related.
3. Back sight (BS): is the first sight taken after the level has been set up. A sight taken to a
point whose height is known or can be calculated.
4. Foresight (FS): The last sight taken. A sight taken to a point whose height is required to
carry on the line of level.
5. Intermediate Sight (IS): it is any other sight taken.
6. Reduced Level (RL): calculated level of a point above or below the datum.
7. Height of Instrument (HI): The height of the line of collimation above the datum.
8. Line of Collimation (LM): It is an imaginary horizontal line drawn between two points.
9. Rise and fall: The difference is height or is level between two is referred to as a rise or fall.
10. Change Point (CP): the point at which both a foresight and then a back sight are taken.
LEVELING INSTRUMENTS
These include:
. A Level e.g. theodolite, transit dumpy level(automatic level),e.t.c.
. A staff
. Devices for angle measurements e.g. graphometer, magnetic compass, prismatic compass,
orientation compass
. Chain or tape
. Pegs, arrows and ranging poles

LESSON NOTE ON INTRODUCTION TO SPHERICAL TRIANGLE

Solution of triangles Solution of triangles, is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
Solving plane triangles Edit  Standard notation for a triangle A general form triangle has six main characteristics (see picture): three linear (side lengths a, b, c) and three angular (α, β, γ). The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three. A triangle can be uniquely determined in this sense when given any of the following:[1][2] Three sides (SSS) Two sides and the included angle (SAS) Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length. A side and the two angles adjacent to it (ASA) A side, the angle opposite to it and an angle adjacent to it (AAS).

Three angles (AAA) on the sphere (but not in the plane). For all cases in the plane, at least one of the side lengths must be specified. If only the angles are given, the side lengths cannot be determined, because any similar triangle is a solution. Main theorems Edit  Overview of particular steps and tools used when solving plane triangles The standard method of solving the problem is to use fundamental relations. Law of cosines {\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos \alpha } {\displaystyle b^{2}=a^{2}+c^{2}-2ac\cos \beta } {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma } Law of sines {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}} Sum of angles {\displaystyle \alpha +\beta +\gamma =180^{\circ }} Law of tangents {\displaystyle {\frac {a-b}{a+b}}={\frac {\mathrm {tan} [{\frac {1}{2}}(\alpha -\beta )]}{\mathrm {tan} [{\frac {1}{2}}(\alpha +\beta )]}}.} There are other (sometimes practically useful) universal relations: the law of cotangents and Mollweide's formula. Notes Edit To find an unknown angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its angle. On the other hand, if the angle is small (or close to 180°), then it is more robust numerically to determine it from its sine than its cosine because the arc-cosine function has a divergent derivative at 1 (or −1). We assume that the relative position of specified characteristics is known. If not, the mirror reflection of the triangle will also be a solution. For example, three side lengths uniquely define either a triangle or its reflection.  Three sides given Three sides given (SSS) Edit Let three side lengths a, b, c be specified. To find the angles α, β, the law of cosines can be used:[3] {\displaystyle \alpha =\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}} {\displaystyle \beta =\arccos {\frac {a^{2}+c^{2}-b^{2}}{2ac}}.} Then angle γ = 180° − α − β. Some sources recommend to find angle β from the law of sines but (as Note 1 above states) there is a risk of confusing an acute angle value with an obtuse one. Another method of calculating the angles from known sides is to apply the law of cotangents.  Two sides and the included angle given Two sides and the included angle given (SAS) Edit Here the lengths of sides a, b and the angle γ between these sides are known. The third side can be determined from the law of cosines:[4] {\displaystyle c={\sqrt {a^{2}+b^{2}-2ab\cos \gamma }}.} Now we use law of cosines to find the second angle: {\displaystyle \alpha =\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}.} Finally, β = 180° − α − γ.  Two sides and a non-included angle given Two sides and non-included angle given (SSA) Edit This case is not solvable in all cases; a solution is guaranteed to be unique only if the side length adjacent to the angle is shorter than the other side length. Assume that two sides b, c and the angle β are known. The equation for the angle γ can be implied from the law of sines:[5] {\displaystyle \sin \gamma ={\frac {c}{b}}\sin \beta .} We denote further D = c / b sin β (the equation's right side). There are four possible cases: If D > 1, no such triangle exists because the side b does not reach line BC. For the same reason a solution does not exist if the angle β ≥ 90° and b ≤ c. If D = 1, a unique solution exists: γ = 90°, i.e., the triangle is right-angled.  Two solutions for the triangle If D < 1 two alternatives are possible. If b < c, the angle γ may be acute: γ = arcsin D or obtuse: γ′ = 180° - γ. The picture on right shows the point C, the side b and the angle γ as the first solution, and the point C′, side b′ and the angle γ′ as the second solution. If b ≥ c then β ≥ γ (the larger side corresponds to a larger angle). Since no triangle can have two obtuse angles, γ is an acute angle and the solution γ = arcsin D is unique. Once γ is obtained, the third angle α = 180° − β − γ. The third side can then be found from the law of sines: {\displaystyle a=b\ {\frac {\sin \alpha }{\sin \beta }}}  One side and two adjacent angles given A side and two adjacent angles given (ASA) Edit The known characteristics are the side c and the angles α, β. The third angle γ = 180° − α − β. Two unknown side can be calculated from the law of sines:[6] {\displaystyle a=c\ {\frac {\sin \alpha }{\sin \gamma }};\quad b=c\ {\frac {\sin \beta }{\sin \gamma }}.} A side, one adjacent angle and the opposite angle given (AAS) Edit The procedure for solving an AAS triangle is same as that for an ASA triangle: First, find the third angle by using the angle sum property of a triangle, then find the other two sides using the law of sines. 

LESSON NOTE ON INTRODUCTION TO SPHERICAL TRIANGLE

Solution of triangles Solution of triangles, is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
Solving plane triangles Edit  Standard notation for a triangle A general form triangle has six main characteristics (see picture): three linear (side lengths a, b, c) and three angular (α, β, γ). The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three. A triangle can be uniquely determined in this sense when given any of the following:[1][2] Three sides (SSS) Two sides and the included angle (SAS) Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length. A side and the two angles adjacent to it (ASA) A side, the angle opposite to it and an angle adjacent to it (AAS).

Three angles (AAA) on the sphere (but not in the plane). For all cases in the plane, at least one of the side lengths must be specified. If only the angles are given, the side lengths cannot be determined, because any similar triangle is a solution. Main theorems Edit  Overview of particular steps and tools used when solving plane triangles The standard method of solving the problem is to use fundamental relations. Law of cosines {\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos \alpha } {\displaystyle b^{2}=a^{2}+c^{2}-2ac\cos \beta } {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma } Law of sines {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}} Sum of angles {\displaystyle \alpha +\beta +\gamma =180^{\circ }} Law of tangents {\displaystyle {\frac {a-b}{a+b}}={\frac {\mathrm {tan} [{\frac {1}{2}}(\alpha -\beta )]}{\mathrm {tan} [{\frac {1}{2}}(\alpha +\beta )]}}.} There are other (sometimes practically useful) universal relations: the law of cotangents and Mollweide's formula. Notes Edit To find an unknown angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its angle. On the other hand, if the angle is small (or close to 180°), then it is more robust numerically to determine it from its sine than its cosine because the arc-cosine function has a divergent derivative at 1 (or −1). We assume that the relative position of specified characteristics is known. If not, the mirror reflection of the triangle will also be a solution. For example, three side lengths uniquely define either a triangle or its reflection.  Three sides given Three sides given (SSS) Edit Let three side lengths a, b, c be specified. To find the angles α, β, the law of cosines can be used:[3] {\displaystyle \alpha =\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}} {\displaystyle \beta =\arccos {\frac {a^{2}+c^{2}-b^{2}}{2ac}}.} Then angle γ = 180° − α − β. Some sources recommend to find angle β from the law of sines but (as Note 1 above states) there is a risk of confusing an acute angle value with an obtuse one. Another method of calculating the angles from known sides is to apply the law of cotangents.  Two sides and the included angle given Two sides and the included angle given (SAS) Edit Here the lengths of sides a, b and the angle γ between these sides are known. The third side can be determined from the law of cosines:[4] {\displaystyle c={\sqrt {a^{2}+b^{2}-2ab\cos \gamma }}.} Now we use law of cosines to find the second angle: {\displaystyle \alpha =\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}.} Finally, β = 180° − α − γ.  Two sides and a non-included angle given Two sides and non-included angle given (SSA) Edit This case is not solvable in all cases; a solution is guaranteed to be unique only if the side length adjacent to the angle is shorter than the other side length. Assume that two sides b, c and the angle β are known. The equation for the angle γ can be implied from the law of sines:[5] {\displaystyle \sin \gamma ={\frac {c}{b}}\sin \beta .} We denote further D = c / b sin β (the equation's right side). There are four possible cases: If D > 1, no such triangle exists because the side b does not reach line BC. For the same reason a solution does not exist if the angle β ≥ 90° and b ≤ c. If D = 1, a unique solution exists: γ = 90°, i.e., the triangle is right-angled.  Two solutions for the triangle If D < 1 two alternatives are possible. If b < c, the angle γ may be acute: γ = arcsin D or obtuse: γ′ = 180° - γ. The picture on right shows the point C, the side b and the angle γ as the first solution, and the point C′, side b′ and the angle γ′ as the second solution. If b ≥ c then β ≥ γ (the larger side corresponds to a larger angle). Since no triangle can have two obtuse angles, γ is an acute angle and the solution γ = arcsin D is unique. Once γ is obtained, the third angle α = 180° − β − γ. The third side can then be found from the law of sines: {\displaystyle a=b\ {\frac {\sin \alpha }{\sin \beta }}}  One side and two adjacent angles given A side and two adjacent angles given (ASA) Edit The known characteristics are the side c and the angles α, β. The third angle γ = 180° − α − β. Two unknown side can be calculated from the law of sines:[6] {\displaystyle a=c\ {\frac {\sin \alpha }{\sin \gamma }};\quad b=c\ {\frac {\sin \beta }{\sin \gamma }}.} A side, one adjacent angle and the opposite angle given (AAS) Edit The procedure for solving an AAS triangle is same as that for an ASA triangle: First, find the third angle by using the angle sum property of a triangle, then find the other two sides using the law of sines.