CL02 - Lecture 5 - Measurement of Angles and Directions.pdf

FUNDAMENTALS OF SURVEYING CESURV30 – ENGR. EARL JAN JUGUETA MEASUREMENT OF ANGLES AND DIRECTIONS  EDJ MEDIANS Direction of line is usually defined b...

FUNDAMENTALS OF SURVEYING CESURV30 – ENGR. EARL JAN JUGUETA MEASUREMENT OF ANGLES AND DIRECTIONS  EDJ MEDIANS Direction of line is usually defined by the horizontal angle it makes with the fixed reference line. This is done with reference to a meridian which lies in a vertical plane passing through a fixed point of reference and through the observer’s position NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ FOUR TYPES OF MERIDIAN True Meridian – this line passes through the geographic north and south poles of the earth and the observer’s position Magnetic Meridian – a fixed line of reference which lies parallel with the magnetic lines of force of the earth Grid Meridian – a fixed line of reference parallel to the central meridian of a system of plane rectangular coordinates Assumed Meridian – this meridian is usually the direction from a survey station to an adjoining station or some well-defined and permanent point NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ UNITS OF ANGULAR MEASUREMENT The Degree – Sexagesimal System. The basic unit which the circle is divided into 360 parts. It is further subdivided to 60 minutes and minutes is divided to 60 seconds. The Grad – Centesimal System. Circle is divided into 400 parts called grads. If is further divided into 100 centesimal minute and centesimal minute is divided into 100 centesimal second. The Mil – Circle is divided into 6400 equal parts called mils. Commonly used in military operations. The Radian – One radian is equal to 180/Π. It is sometimes referred to as the natural unit of angle because there is no arbitrary number in its definition. NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ UNITS OF ANGULAR MEASUREMENT Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ CONVERSION OF ANGLES Sample Problem: Convert the angle 2380 25′ 50′′ into its equivalent in decimal degrees. Solution: Angle = 2380 25′ 50′′ Decimal Equivalent = Deg + Min/60 + Sec/3600 25′ 50′′ = 2380 + + 60 3600 = 238.43060 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ CONVERSION OF ANGLES Sample Problem: Convert 2700 into its equivalent value in grads, mils and radians. Solution: 0 400𝑔 a. Angle in Grads = 270 = 300𝑔 3600 0 6400 𝑚𝑖𝑙𝑠 b. Angle in Mils = 270 = 4800 𝑚𝑖𝑙𝑠 3600 0 2𝜋 𝑅𝑎𝑑 c. Angle in Radians = 270 = 4.7124 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 3600 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DIRECTION OF LINES Defined as the horizontal angle the line makes with an established line of reference. In surveying, directions may be defined by means of: Interior Angles, Deflection Angles, Angles to the right, bearing and azimuths. NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ INTERIOR ANGLES angle between adjacent lines in a closed polygon re-entrant angle – an interior angle that is greater than 180° Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DEFLECTION ANGLES angle between a line and the prolongation of the preceding line Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ANGLES TO THE RIGHT angles that are measured clockwise from the preceding line to the succeeding line Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ BEARING the acute horizontal angle between the reference meridian and the line forward bearing – when the bearing of a line is observed in the direction in which the survey progresses back bearing – when the bearing of the line is observed in an opposite direction Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ BEARING Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ FORWARD AND BACK BEARING Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ AZIMUTH angle between the meridian and the line measured in a clockwise direction from either the north or south branch of the meridian NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ AZIMUTH Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ FORWARD AND BACK AZIMUTH Any line on the earth’s surface has two azimuths – a forward azimuth and a backward azimuth. Depending on which end of the line is considered, these directions differ by 180 degrees from each other since the back azimuth is the exact reverse of the forward azimuth. Rules: 1. If the forward azimuth of the line is greater than 180 deg., subtract 180 deg. to obtain the back azimuth. 2. When the forward azimuth of the line is less than 180 deg., add 180 deg. to determine the back azimuth. NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ FORWARD AND BACK AZIMUTH Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DETERMINING ANGLES FROM BEARING Sample Problem: Compute the angles AOB, COD, EOF and GOH from the following set of lines whose magnetic bearings are given: a. OA, 𝑁390 25′ 𝐸 and OB, 𝑁750 50′ 𝐸 b. OC, 𝑁340 14′ 𝐸 and OD, 𝑁530 22′ 𝑊 c. OE, S150 04′ 𝐸 and OF, 𝑆360 00′ 𝑊 d. OG, 𝑁700 15′ 𝑊 and OH, 𝑆520 05′ 𝑊 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DETERMINING ANGLES FROM BEARING a. Determine Angle AOB Let 𝜃1 = Bearing angle of OA 𝜃2 = Bearing angle of OB 𝛼 = Angle AOB ∝= 𝜃2 − 𝜃1 = 750 50′ - 390 25′ = 𝟑𝟔𝟎 𝟐𝟓′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DETERMINING ANGLES FROM BEARING b. Determine Angle COD Let 𝜃1 = Bearing angle of OC 𝜃2 = Bearing angle of OD 𝛼 = Angle COD ∝= 𝜃1 + 𝜃2 = 340 14′ + 530 22′ = 𝟖𝟕𝟎 𝟑𝟔′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DETERMINING ANGLES FROM BEARING c. Determine Angle EOF Let 𝜃1 = Bearing angle of OE 𝜃2 = Bearing angle of OF 𝛼 = Angle EOF ∝= 𝜃1 + 𝜃2 = 150 04′ + 360 00′ = 𝟓𝟏𝟎 𝟎𝟒′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DETERMINING ANGLES FROM BEARING d. Determine Angle GOH Let 𝜃1 = Bearing angle of OG 𝜃2 = Bearing angle of OH 𝛼 = Angle GOH ∝= 1800 − (𝜃1 + 𝜃2 ) = 1800 − (700 15′ + 520 05′ ) = 𝟓𝟕𝟎 𝟒𝟎′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DETERMINING ANGLES FROM AZIMUTHS Sample Problem: Compute the angles APB, CPD and EPF from the following set of lines whose azimuths are given a. 𝐴𝑍𝐼𝑀𝑛 of line PA = 390 48′ 𝐴𝑍𝐼𝑀𝑛 of line PB = 1150 29′ b. 𝐴𝑍𝐼𝑀𝑠 of line PC = 3200 22′ 𝐴𝑍𝐼𝑀𝑠 of line PD = 620 16′ c. 𝐴𝑍𝐼𝑀𝑛 of line PE = 2190 02′ 𝐴𝑍𝐼𝑀𝑆 of line PF = 1540 16′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DETERMINING ANGLES FROM AZIMUTHS a. Let 𝜆1 = Azimuth from north of line PA 𝜆2 = Azimuth from north of line PB 𝜃 = Angle APB 𝜃 = 𝜆2 − 𝜆1 = 1150 29′ - 390 48′ = 𝟕𝟓𝟎 𝟒𝟏′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DETERMINING ANGLES FROM AZIMUTHS b. Let 𝜆1 = Azimuth from south of line PC 𝜆2 = Azimuth from south of line PD 𝜃 = Angle CPD 𝜃 = 𝜆2 + 3600 − 𝜆1 = 620 16′ + (3600 − 3200 22′ ) = 𝟏𝟎𝟏𝟎 𝟓𝟒′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ DETERMINING ANGLES FROM AZIMUTHS c. Let 𝜆1 = Azimuth from north of line PE 𝜆2 = Azimuth from south of line PF 𝜃 = Angle EPF 𝜃 = 𝜆2 − 𝜆1 − 1800 = 1540 16′ − (2190 02′ − 1800 ) = 𝟏𝟏𝟓𝟎 𝟏𝟒′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ CONVERTING BEARING TO AZIMUTHS Sample Problem: Convert the following bearings to equivalent azimuths a. AB, N250 25′ 𝐸 b. BC, 𝐷𝑢𝑒 𝐸𝑎𝑠𝑡 c. CD, S500 10′ 𝐸 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ CONVERTING BEARING TO AZIMUTHS a. Let ∝= Bearing angle of AB 𝜆1 = Azimuth from south of AB 𝜆2 = Azimuth from north of AB 𝜆1 =1800 00′ + ∝ = 1800 00′ + 250 25′ = 𝟐𝟎𝟓𝟎 𝟐𝟓′ 𝜆2 = ∝ = 𝟐𝟓𝟎 𝟐𝟓′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ CONVERTING BEARING TO AZIMUTHS b. Let ∝= Bearing angle of BC 𝜆1 = Azimuth from south of BC 𝜆2 = Azimuth from north of BC 𝜆1 =1800 00′ + ∝ = 1800 00′ + 900 00′ = 𝟐𝟕𝟎𝟎 𝟎𝟎′ 𝜆2 = ∝ = 𝟗𝟎𝟎 𝟎𝟎′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ CONVERTING BEARING TO AZIMUTHS c. Let ∝= Bearing angle of CD 𝜆1 = Azimuth from south of CD 𝜆2 = Azimuth from north of CD 𝜆1 =3600 00′ − ∝ = 3600 00′ − 500 10′ = 𝟑𝟎𝟗𝟎 𝟓𝟎′ 𝜆2 = 1800 00′ − ∝ = 1800 00′ − 500 10′ = 𝟏𝟐𝟗𝟎 𝟓𝟎′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ CONVERTING AZIMUTH TO BEARING Sample Problem: Convert the following azimuths to equivalent bearings a. 𝐴𝑍𝐼𝑀𝑠 of line AB = 2300 30′ b. 𝐴𝑍𝐼𝑀𝑛 of line BC = 1120 46′ c. 𝐴𝑍𝐼𝑀𝑠 of line CD = 2700 00′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ CONVERTING AZIMUTH TO BEARING a. Let 𝜆 = Azimuth from south of AB ∝ = Bering of AB ∝ = 𝜆 − 1800 00′ = 2300 30′ −1800 00′ = 𝐍𝟓𝟎𝟎 𝟑𝟎′ 𝑬 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ CONVERTING AZIMUTH TO BEARING b. Let 𝜆 = Azimuth from north of BC ∝ = Bering of BC ∝ = 1800 00′ − 𝜆 = 1800 00′ −1220 46′ = S𝟓𝟕𝟎 𝟏𝟒′ 𝑬 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ CONVERTING AZIMUTH TO BEARING c. Let 𝜆 = Azimuth from south of CD ∝ = Bering of CD ∝ = 𝜆 − 1800 00′ = 2700 00′ −1800 00′ = 𝟗𝟎𝟎 𝟎𝟎′ (𝒃𝒆𝒂𝒓𝒊𝒏𝒈 𝒐𝒇 𝑪𝑫 𝒊𝒔 𝑫𝒖𝒆 𝑬𝒂𝒔𝒕) NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MAGNETIC DECLINATION the horizontal angle and direction by which the needle of a compass deflects from the true meridian at any particular locality Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ VARIATIONS IN MAGNETIC DECLINATION Daily Variation Annual Variation also called diurnal variation another form of periodic swing taken by the magnetic meridian with respect to an oscillation of the compass needle the true meridian through a cycle from its mean position over a 24-hour period it usually amounts to only less than 1 minute of arc extreme eastern position of the needle → occurs early in the morning extreme western position of the needle → occurs just about after noon time daily variation is greater in higher latitudes than near the equator NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ VARIATIONS IN MAGNETIC DECLINATION Secular Variation Irregular Variation covers a period of so many years a type of variation uncertain in that its exact cause and character is character and cannot be predicted not thoroughly understood as to amount or occurrence NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MAGNETIC DECLINATION Sample Problem: The magnetic declination in a locality is 20 30′ 𝐸. Determine the true bearing and true azimuths reckoned from north and south of the following lines whose magnetic bearings are given. a. AB, N250 40′ 𝐸 b. AC, S520 12′ 𝐸 c. CD, S500 10′ 𝑊 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MAGNETIC DECLINATION a. Given: 𝑑 = 20 30′ 𝐸 (magnetic declination) ∝ = 250 40′ (bearing) 𝜌 = 𝑑 + ∝ = 20 30′ + 250 40′ = 𝟐𝟖𝟎 𝟏𝟎′ (𝑵𝟐𝟖𝟎 𝟏𝟎′ 𝑬) 𝜋 = 𝜌 = 𝟐𝟖𝟎 𝟏𝟎′ (true azimuth from north) = 1800 + 𝜋 = 1800 + 280 10′ = 𝟐𝟎𝟖𝟎 𝟏𝟎′ (true azimuth from south) NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MAGNETIC DECLINATION b. Given: 𝑑 = 20 30′ 𝐸 (magnetic declination) ∝ = 500 12′ (bearing) 𝜌 = ∝ − 𝑑 = 500 12′ − 20 30′ = 𝟒𝟕𝟎 𝟒𝟐′ (𝑺𝟒𝟕𝟎 𝟒𝟐′ 𝑬) 𝜋 = 1800 − 𝜌 = 1800 − 470 42′ = 𝟏𝟑𝟐𝟎 𝟏𝟖′ (true azimuth from north) = 1800 + 𝜋 = 1800 + 1320 18′ = 𝟑𝟏𝟐𝟎 𝟏𝟖′ (true azimuth from south) NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MAGNETIC DECLINATION c. Given: 𝑑 = 20 30′ 𝐸 (magnetic declination) ∝ = 620 18′ (bearing) 𝜌 = 𝑑 + ∝ = 20 30′ + 620 18′ = 𝟔𝟒𝟎 𝟒𝟖′ (𝑺𝟔𝟒𝟎 𝟒𝟖′ 𝑾) 𝜋 = 1800 + 𝜌 = 1800 + 𝟔𝟒𝟎 𝟒𝟖′ = 𝟐𝟒𝟒𝟎 𝟒𝟖′ (true azimuth from north) 𝜆 = 𝜋 = 𝟔𝟒𝟎 𝟒𝟖′ (true azimuth from south) NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ISOGONIC CHART a chart or map which shows lines connecting points where the magnetic declination of the compass needle is the same at a given time agonic lines – lines connecting parts of the chart with zero magnetic declination oIn areas west of the agonic line, the needle has an easterly declination oIn areas east of the agonic line, the needle has a westerly declination NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ISOGONIC CHART Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ COMPASS SURVEY One of the most basic and widely practiced methods of determining the relative location of points where a high degree of precision is not required. NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ COMPASS SURVEY Traverse Traverse Station a series of lines connecting successive any temporary or permanent point of points whose lengths and directions have reference over which the instrument is set been determined from field up measurements sometimes called angle points because an angle is usually measured at such Traversing stations process of measuring the lengths and Traverse Lines directions of the lines of the traverse for the purpose of locating the position of lines connecting traverse stations and whose lengths and directions are certain points determined NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ TYPES OF COMPASS SURVEY Open Compass Traverse Closed Compass Traverse consists of a series of lines of known consists of a series of lines of known lengths and magnetic bearings which lengths and magnetic bearings which are continuous but do not return to forms a closed the starting point or close upon a point of known position loop, or begin and end at points whose positions have been fixed by other surveys of higher position NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE In a compass traverse there are likely to be discrepancies between the observed forward and back bearings of lines. These may be due to errors of observations or local attractions. Two important steps to perform: oDetermine which among the traverse lines is free from local attraction oPerform the adjustment of successive lines by starting from either end of the selected line The unaffected line is called the best line and is assumed that there are no local attractions on this line NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE AB, Fwd Brg = S410 18′ 𝐸, Back Brg = N410 00′ 𝑊 BC, Fwd Brg = N680 00′ 𝐸, Back Brg = S680 00′ 𝑊 CD, Fwd Brg = N350 00′ 𝐸, Back Brg = S370 00′ 𝑊 DE, Fwd Brg = S430 00′ 𝐸, Back Brg = N420 00′ 𝑊 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE Elementary Surveying, 3rd Edition, Juny La Putt NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE Sample Problem: Given in the tabulation below are the observed forward and back bearings of an open compass traverse. Plot the traverse and adjust the forward and back bearings of each course. Tabulate the answers and show accompanying computations. OBSERVED BEARINGS LINE LENGTH FORWARD BACK AB 400.63 m N250 45′ 𝐸 S250 40′ 𝑊 BC 450.22 m S200 30′ 𝐸 N200 25′ 𝑊 CD 500.89 m S350 30′ 𝑊 N350 30′ 𝐸 DE 640.46 m S750 30′ 𝐸 N750 25′ 𝑊 EF 545.41 m N580 50′ 𝐸 S580 15′ 𝑊 FG 700.05 m N220 05′ 𝐸 S210 55′ 𝑊 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ADJUSTMENT OF AN OPEN COMPASS TRAVERSE Tabulation of adjusted traverse data OBSERVED BEARINGS LINE LENGTH FORWARD BACK AB 400.63 m N250 45′ 𝐸 S250 45′ 𝑊 BC 450.22 m S200 25′ 𝐸 N200 25′ 𝑊 CD 500.89 m S350 30′ 𝑊 N350 30′ 𝐸 DE 640.46 m S750 30′ 𝐸 N750 30′ 𝑊 EF 545.41 m N580 45′ 𝐸 S580 45′ 𝑊 FG 700.05 m N220 35′ 𝐸 S210 35′ 𝑊 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING