Route Surveying. Topic Outline

Transcription

1 Route Surveying CE 305 Intro To Geomatics By Darrell R. Dean, Jr., P.S., Ph.D. Topic Outline Horizontal alignment Types of Horizontal Curves Degree of Curve Geometric elements of curve Station ti number computation ti Deflection angle per foot of curve Total deflection angle 1

2 Topic Outline cont. Unit deflection angle Chord length Curve stake out by deflection angles Moving up on the curve Backing in on the curve Tangent offset Radial stake out Horizontal Alignment The geometric pattern of a roadway as represented in a horizontal plane Geometric Elements Tangents - straight line sections Curves 2

3 Horizontal Alignment cont. Tangent Sections Point of Intersection (PI) Each PI is identified by a station number 3

4 Tangent Sections cont. Station Numbers 1 station = 100 ft. accumulative distance along centerline e.g Tangent Sections cont. Deflection Angle at each PI the deflection angle is measured from the prolongation of the previous tangent represented tdby the symbol includes a direction of turning, L or R 4

5 A Horizontal curve is the arc of a circle The radials at the beginning and end of the curve intersect the respective tangents at a 90 angle. Horizontal curves are tangent curves Curves Beginning of curve is called the Point of Curvature - PC The end of the curve is called the Point of Tangency - PT Any point on curve is designated as a POC Curves cont. 5

6 Sta. Nos. increase left to right - sketches Back tangent - name given to tangent before PC Forward tangent - name given to tangent after PT Curves cont. The central angle of the curve equals the deflection angle Curves cont. int. angles ( n 2) 180 int. angles 360 A 90 ( 180 ) A A Central angle Deflection angle 6

7 Types of Horizontal Curves Simple Curve Compound Curve Reverse Curve Broken back Curve Spiral Curve Compound Curve Curve Types cont. 2 or more simple curves, centers on same side of centerline PCC = Point of Compound Curve 7

8 Reverse Curve Curve Types cont. 2 simple curves, centers on opposite sides of centerline PRC = Point of Reverse erse Curve Curve Types cont. Broken Back Curve Two simple curves connected by a relatively short tangent 8

9 Spiral Curve 3 Components Curve Types cont. Spiral Simple Circular Spiral TS = Tan. to Spiral SC = Spiral to Curve CS = Curve to Spiral ST = Spiral to Tan. An index of curve sharpness Arc definition - angle (D a ) subtended by a 100-ft. arc Chord definition - angle (D c ) subtended by a 100-ft. chord Hwy. Curves based on arc definition Degree of Curve 9

10 Degree of Curve cont. Relationship to radius 100 ( ft) 2 R ( ft) Da R D D a a R 2 Da R D ( D a a must be in or decimal degrees) R Degree of Curve cont. Relationship to radius Sin D 2 c 50 R 10

11 Geometric Elements of Curve defl. or delta angle R = radius T = tangent distance L = length of curve LC = long chord E = external Distance M = middle ordinate Geometric Elements cont. Measured or calculated T R tan( / 2) Da R L 100 D 180 R a LC 2 R sin( / 2) 1 E R ( 1) R ( sec( / 2) 1) cos( / 2) M R ( 1 cos( / 2)) 11

12 Geometric Elements cont. Measured or calculated T R tan( / 2) Da R L 100 D 180 R a LC 2 R sin( / 2) E R ( sec( / 2) 1) M R ( 1 cos( / 2)) Computing PC and PT Sta. Nos. Survey and/or Design Elements Required PI Sta. No. R or D a Computed Geometric Elements Required T = tan. distance L = length of curve 12

13 PC and PT Sta. Nos. cont. PC=PI- PI T PT = PC + L These equations are for the first curve on the alignment. Subsequent curves require a different approach Calc. Sta. Nos. - 1st Curve Given: (1) PI at , (2) = , and (3) R = ft. Calculate: PC and PT station numbers for this curve 13

14 Sta. Nos. - 1st Curve cont. T R tan ( / 2 ) T tan( 61 10' 35"/ 2) T ft sta Da 43 03' 02" L 100 D a 100 L 61 10' 35" 43 03' 02" L 14210ft sta.. Sta. Nos. - 1st Curve cont. PI T PC= L PT=

15 Calc. Sta. Nos. for 2nd and Subsequent Curves First: For all curves calculate the geometric elements, i.e., T, L, etc. Calc. Sta. Nos. for 2nd and Subsequent Curves Second: Calculate the tangent distance (S i-1 ) between the previous (PT i -1 )and the (PC i ) of interest 15

16 Subsequent Curve Sta. Nos. cont. Third: Calculate the (PC i ) of interest by adding (S i-1 ) to the previous (PT i -1 ) Subsequent Curve Sta. Nos. cont. Fourth: Calculate the (PT i ) of interest by adding (L i ) to the (PC i ) 16

17 Subsequent Curve Sta. Nos. cont. Data Required for Curve #2 T ft. T T tan(42 49' 34"/ 2) ft. L 42 49' 34" L ft. Subsequent Curve Sta. Nos. cont. Data Required for Curve #2 cont. Calculate distance - S S ( PI PI ) ( T T) i 1 i i 1 i 1 i E.g., g for Curve #2, i=2 S2 1 ( PI2 PI2 1) ( T2 1 T2) S ( PI PI ) ( T T )

18 Subsequent Curve Sta. Nos. cont. Calculation of S 1 for curve # 2 cont. 1 S 1 = ( ) - ( ) = ft. Subsequent Curve Sta. Nos. cont. Data Required for Curve #2 cont. PT 1 = T ft. T L ft ft. S 1 = ft. PC 2 = PT 1 + S 1 = = = PT 2 = PC 2 + L 2 = = =

19 Calculate Sta. No. for End of Project (E.O.P.) The station for the end of the line will change as curves are introduced to connect tangents. Sta. No. for E.O.P. cont. Steps to Calculate E.O.P Calculate l distance (S n )f from the last tpoint of Tangency (PT n ) to the alignment terminus S n = (PI n + 1 -PI n ) - T n n = number of curves PI n + 1 = Sta. no. of terminus after last curve and before curves are introduced PI n = PI of last curve on the alignment T n = tangent dist. for the last curve 19

20 Sta. No. for E.O.P. cont. Steps to Calculate E.O.P. cont. 2. Calculate the E.O.P. station number E.O.P. = PT n + S n n = number of curves PT n = PT of last curve on the alignment Sta. No. for E.O.P. cont. E. g., calculate the E.O.P. for the following case Given: T 2 = 65.08, PT 2 = S 2 = ( ) = = ft. E.O.P. = = =

21 Sta. No. for E.O.P. cont. Deflection angle per foot of curve = deflection angle in minutes per foot of curve Da 2 Da ( units deg. per ft.) D a 03. Da ( units min. per ft.) 21

22 Total Deflection (d t ) Angles Total deflection angles are used to stake out horizontal curves. They are measured at the PC by backsighting the PI, setting zero, and turning the angle toward the curve Total Deflection cont. Total Deflection = d t d ( 03. D l)/ 60 t a Where: l = arc length in ft. l = POC (sta. no.) - PC (sta. no.) D a = degree of curve Units for d t = decimal degrees 22

23 Total Deflection cont. Example Calculation d ( 03. D l)/ 60 t l = l = ft. d t = (0.3) (6.75) /60 = a Unit Deflection Angle (d u ) The unit deflection angle is the angle between the local tangent and the chord for any arc on the curve. d u = ((0.3) D a l)/60 l = POC#2 - POC#1 l = arc length ft. 23

24 Unit Deflection Angle (d u ) cont. Example Calculation d u = ((0.3) D a l)/60 l = arc length (ft.) l = l = ft. d u = ((0.3) )/60 d u = Chord Length For Any Arc Chord length equation 1 2 Chord sin( du ) R Chord 2 R sin( du ) du (. 03 Da l)/ 60 l arclengthft. Chord 2 R sin(0.3 Da l / 60) Note: May substitute d t for d u 24

25 Chord Length cont. Example Calculation l = = ft. = arc length R ft. 645 ' dt / 60 dt 3 57' 38" chord sin( 3 57' 38") chord ft. Curve Stake out by the Deflection Angle Method The Deflection Angle Method is an angledistance intersection method. Angles are measured after backsighting the PI and setting zero Chord distances are measured from the last station staked 25

26 Deflection Angle Method cont. Procedures Office: Calculate total deflections for full and half stations and chord lengths for all arc segments as required Deflection Angle Method cont. Procedures: cont. sample problem total deflection angles D a = R = ft. = L = ft. PC = PT = Station l = arc length (ft.) dt PC = '00" '55" PT =

27 Deflection Angle Method cont. Procedures: cont. sample problem total deflection angles D a = R = ft. = L = ft. PC = PT = Station l = arc length (ft.) PC = PT = dt 0 00'00" 01 32'55" 12 34'42" 23 36'30" 34 38'17" 35 00'00" Deflection Angle Method cont. Procedures: cont. sample problem chord lengths D a = R = ft. = L = ft. PC = PT = Arc (ft.) du chord (ft.) '55"

28 Deflection Angle Method cont. Procedures: cont. sample problem chord lengths D a = R = ft. = L = ft. PC = PT = Arc (ft.) du 01 32'55" 05 30'54" 11 01'48" 00 21'42" chord (ft.) Deflection Angle Method cont. Field Procedures Occupy PI set PC and PT on alignment Occupy PC, Backsight the PI, set zero Layoff the total deflection angle for the first station and set hub at chord distance from PC 28

29 Deflection Angle Method cont. Field Procedures cont. In turn lay off the total deflection angle for each station and set a hub for the station at the intersection of the line of sight and the chord distance from the last station staked. Deflection Angle Method cont. Field Procedures cont. Checks: Measure distance between PTs as set from PI and PC, call it the L.E.C. R.E.C = L.E.C / ( 2T + L ) Set midpoint on curve and measure distance to PI: Compare to E 29

30 Moving up on the Curve Obstructions sometimes prohibits the measuring of total deflection angles to all stations on the curve. A station on the curve may be occupied and the same total deflection angles computed for using at the PC may be used to complete the layout. Moving up on the Curve cont. The last visible station on the curve is staked and then occupied, i.e., you move up on the curve. The first time you move up you backsight the PC. 30

31 Moving up on the Curve cont. Orientation is always such that the circle reading, as you sight along the projection of the chord from the last station occupied through the station you are occupying, is set to equal the total deflection to the last station occupied. In the case of backsighting the PC the setting should be zero as you sight along the projection Moving up on the Curve cont. Stake out of stations proceeds as if the procedure was taking place at the PC. That is, the total deflections calculated to be measured from the PC may now be used at the station occupied on the curve. 31

32 Moving Up on the Curve cont. Backing the Curve in from PT The same total deflection angles calculated for staking stations on the curve from the PC may also be used to stake the same stations from the PT. The PT is occupied and rather than backsighting the PI, the PC is the backsight point. 32

33 Backing the Curve in from PT The PC is backsighted and the circle reading is set to zero. Stations are located with the same angledistance intersection technique as previously described. The total deflection angle to each station is as calculated for layout from the PC. Backing in the Curve cont. Consider Triangle AOB dt d t Therefore, curve stakeout from the PC or PT with the same set of total deflection angles is possible. 33

34 Tangent Offset Method Procedures measure distance (x) along the tangents from PC or PT measure perpendicular offset (y) to curve Requirements total deflection angle chord distance Tangent Offset Method cont. Note for first half of curve arc length calculated from PC for second half of curve arc length calculated from PT 34

35 Tangent Offset Method cont. Offset Equations x chord cos( dt ) y chord sin( d ) d 03. D l/ 60 t l arc length ( ft.) chord 2 R sin( d t ) a t Tangent Offset Method cont. Find x and the offset, y, for this example: x chord cos( dt ) y chord sin( d ) d 03. D l/ 60 t l arc length ( ft.) chord 2 R sin( d t ) a t 35

36 Tangent Offset Method cont. Answers: l = ft. d t = chord = ft. x = ft. y = 6.22 ft. Radial Stake out Calculate coordinates of PC Azimuth from the PC to any station is the azimuth of PC to PI plus the total deflection angle to the station. Remember left deflections are negative Distance from the PC to any station is the chord distance. 36

37 Radial Stake Out cont. With the coordinates of the PC known and the direction and distance to each station known, calculate the station coordinates. Radial stake out from control points can then be achieved. 37

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