Route Surveying
Description
www.passpe.com Surveying for California Civil PE License © Dr. Shahin A.Mansour, PE Chapter 10- Horizontal, Spiral and Vertical Curves 225 Chapter 10 Horizontal, Spiral and Vertical Curves Topics to be covered - Types of Horizontal Curves - Deflection Angles, Chord and Offset Calculations - Compound and Reverse Curves - Spiral Curves - Vertical Curves - Geometric Properties of Vertical Curves - High and Low Points on Vertical Curves - Asymmetrical Vertical Curves 16 Sample Problems with Detailed Solutions 10 Supplemental Practice Problems with Detailed Solutions www.passpe.com Surveying for California Civil PE License © Dr. Shahin A. Mansour, PE Chapter 10- Horizontal, Spiral and Vertical Curves 226 Chapter 10- Horizontal, Spiral and Vertical Curves 10-1 INTRODUCTION Horizontal curves may be simple, compound, reverse, or spiral. Compound and reverse curves are treated as a combination of two or more simple curves, whereas the spiral curve is based on a varying radius. Curves of short radius (usually less than one tape length) can be established by holding one end of the tape at the center of the circle and swinging the tape in an arc, marking as many points as may be desired. As the radius and length of curve increases, the tape becomes impractical and the surveyor must use other methods. The common method is to measure angles and straight-line sight distances by which selected points, known as stations, may be located on the circumference of the arc. a) Simple Circular (b) Compound (c) Reverse (d) Spiral Figure 10.1 Types of Horizontal Curves 10-2 TYPES OF HORIZONTAL CURVES Table 10-1 Types of Horizontal Curves Simple Circular Compound Reverse Spiral The simple curve is an arc of a circle. The radius of the circle determines the sharpness or flatness of the curve. The larger the radius, the flatter the curve. This type of curve is the most often used. Frequently the terrain will necessitate the use of a compound curve. This curve normally consists of two simple curves joined together, but curving in the same direction. A reverse curve consists of two simple curves joined together, but curving in opposite directions. For safety reasons, this curve is seldom used in highway construction as it would tend to send an automobile off the road. The spiral is a curve which has a varying radius. It is used on railroads and some modern highways. Its purpose is to provide a transition from the tangent to a simple curve or between simple curves in a compound curve www.passpe.com Surveying for California Civil PE License © Dr. Shahin A. Mansour, PE Chapter 10- Horizontal, Spiral and Vertical Curves 227 10-3 TERMINOLOGY OF HORIZONTAL CURVES Following are the main elements of a simple curve; see Fig.10.2 1. Point of intersection: the point of intersection (PI) is the point where the back and forward tangents intersect. 2. The radius(R): the radius of the circle of which the curve is an arc. 3. The point of curvature: the point of curvature (PC) is the point where the circular curve begins. The back tangent is tangent to the curve at this point. 4. The point of tangency: the point of tangency (PT) is the end of the curve. The forward tangent is tangent to the curve at this point. Note: The terms BC (Beginning of Curve) and EC (End of Curve) are referred to by some agencies as PC (point of curvature) and PT (point of tangency), and by others as TC (tangent to curve) and CT (curve to tangent). 5. The length of curve (L): the distance from the PC to the PT measured along the curve. Figure 10.2 Terminology of Horizontal Curve 6. The tangent distance(T): the distance along the tangents from the PI to the PC or PT. These distances are equal on a simple curve. 7. The central angle (A): the angle formed by two radii drawn from the center of the circle ( O) to the PC or PT. The central angle is equal in value to the intersecting angle (A = I). www.passpe.com Surveying for California Civil PE License © Dr. Shahin A. Mansour, PE 8. Long chord: The long chord (LC or C) is the chord from the PC to the PT. 9. External distance: The external distance (E) is the distance from the PI to the midpoint of the curve. The external distance bisects the interior angle at the PI. 10. Middle ordinate: The middle ordinate (M) is the distance from the midpoint of the curve to the midpoint of the long chord. The extension of the middle ordinate bisects the central angle. 11. Degree of curve: The degree of curve (D) defines the "sharpness" or "flatness" of the curve. There are two common definitions for degree of curve , as follows: Table 10-2 Chord and Arc Definitions for Horizontal Curves Chord Definition Arc Definition The chord definition states that the degree of a curve is the angle formed by two radii drawn from the center of the circle to the ends of a chord 100 ft long. The chord definition is used primarily for civilian railroad construction and is used by the military for both roads and railroads. The arc definition states that the degree of a curve is the angle formed by two radii drawn from the center of the circle to the ends of an arc 100 ft long. This definition is used primarily for highways and streets. Notice that the larger the degree of curve, the "sharper" the curve and the shorter the radius R ft D Sin 50 ) 2 ( = ( 10-1) R R ft D o o 58 . 5729 2 ) 100 )( 360 ( = = t (10-2) The sharpness of a curve is determined by the choice of the radius (R); large radius curves are relatively flat, whereas small radius curves are relatively sharp. Chapter 10- Horizontal, Spiral and Vertical Curves 228 www.passpe.com Surveying for California Civil PE License © Dr. Shahin A.Mansour, PE Chapter 10- Horizontal, Spiral and Vertical Curve 229 10-4 GEOMETRY OF HORIZONTAL CIRCULAR CURVES 2 ) ( A = = Tan R T EC to PI OR PI to BC Tangent (10-3) ) 2 ( 2 2 2 ) ( A = A = = Cos T Sin R C EC to B to BC Chord Long (10-4) | . | \ | A = A = A = D ft radians R R L EC to A to BC e i Curve the Along EC to BC Length Curve ) 100 ( ) ( 360 2 : ) . . ( t (10-5) 2 4 2 ) 2 1 ( ) ( A = A = A ÷ = = Cos E Tan C Cos R M B to A Ordinate Middle (10-6) 4 2 4 ) 1 2 ( 1 ) 2 ( 1 ) ( . A A = A = ÷ A = ÷ A = = Tan Tan R Tan T Sec R Cos R E A to PI Dist External (10-7) Notes: 1- ) ( 2 . 2 1 E R R Cos e i E R R + = A + = A ÷ Cos 2- versed sine (vers) → vers (∆/2) = 1 − Cos (∆/2) 3- external secant (exsec) → exsec (∆/2)= Sec (∆/2)− 1 4- A common mistake is to determine the station of the “EC” by adding the “T” distance to the “PI”. Although the “EC” is physically a distance of “T” from the “PI”, the stationing (chainage) must reflect the fact that the centerline no longer goes through the “PI”. The centerline now takes the shorter distance “L” from the “BC” to the “EC”. www.passpe.com Surveying for California Civil PE License © Dr. Shahin A.Mansour, PE Chapter 10- Horizontal, Spiral and Vertical Curve 230 Sample Problem 10-1: Horizontal Curve Chord, Middle Ordinate & External Distance Given: , R= 1000 ft , 8 3 16 ' = A o PI Sta. @ 6 + 26.57 Find: BC and EC stations, length of chord (C), middle ordinate (M), and external distance (E) Solution: 2 tan A = R T = 1000 tan 8.3167 o = 146.18 ft | . | \ | A = A = A = D ft radians R R L ) 100 ( ) ( 360 .) (deg 2t = 360 6333 . 16 1000 2 × × t = 290.31ft PI at 6 + 26.57 –T 1 + 46.18 BC = 4 + 80.39 : + L 2 + 90.31 EC = 7 + 70.70 : ) 2 ( 2 2 2 A = A = Cos T Sin R C = 2 × 1000 × Sin 8.3167 o = 289.29 ft : 2 2 1 ) 2 1 ( A = A ÷ = Tan C Cos R M = 1000( 1 – Cos 8.3167 o ) = 10.52 ft : 4 2 ) 1 ) 2 ( 1 ( A A = ÷ A = Tan Tan R Cos R E = 1000 (Sec 8.3167 o – 1) = 10.63 ft : www.passpe.com Surveying for California Civil PE License © Dr. Shahin A.Mansour, PE Chapter 10- Horizontal, Spiral and Vertical Curve 231 Sample Problem 10-2: Using Degree of Horizontal Curves Given: 11 ○ , degree of curve D = 6 o , PI Sta. @ 14 + 87.33 = A 5 3 1 2 ' ' ' Find: BC and EC stations Solution: R R ft D o o 58 . 5729 2 ) 100 ( ) 360 ( = = t (10-2) ¬ ft D R 93 . 954 58 . 5729 = = 2 A = Tan R T = 954.93 Tan 5.6799 o = 94.98 ft | . | \ | A = A = A = D ft radians R R L ) 100 ( ) ( 360 .) (deg 2t = 6 3598 . 11 100 × = 189.33 ft PI at 14+ 87.33 –T 00 + 94.98 BC = 13 + 92.35 : + L 01 + 89.33 EC = 15 + 81.68 : Note: A common mistake is to determine the station of the EC by adding the T distance to the PI station. www.passpe.com Surveying for California Civil PE License © Dr. Shahin A.Mansour, PE Chapter 10- Horizontal, Spiral and Vertical Curve 232 10-5 DEFELCTION ANGLES, CENTRAL ANGLE, & CHORD CALCUATIONS The deflection angle is defined as the angle between the tangent and a chord. The following two rules apply for the deflection angles for circular curves: Rule 1: The deflection angle between a tangent and a chord is half the central angle subtended by the arc i.e. the angle between the tangent “BC-PI” and the chord “PC-A” is ½ the central angle “BC-O-A” i.e. α & 2α Rule 2: The angle between two chords is ½ the central angle subtended by the arc between the two chords i.e. the angle “A-BC-B” is ½ the central angle “A- O-B” i.e. β & 2β | . | \ | A | . | \ | = 2 L length arc angle deflection (10-8) | . | \ | A | . | \ | = 2 L length arc angle deflection (10-9) o Sin R A to BC Length Chord 2 ) ( = (10-10) R A to BC length arc t o 2 180 ) ( 0 × = (10-11) L A to BC length arc ) ( 2 = A o (10-12) Figure 10.3 Deflection and Central Angles Realtionship Abbreviations: BC = Beginning of curve PC = Point of curvature TC = Tangent to curve EC = End of curve PT = Point of tangency CT = Curve to tangent www.passpe.com Surveying for California Civil PE License © Dr. Shahin A.Mansour, PE Chapter 10- Horizontal, Spiral and Vertical Curve 233 10-13 GEOMETRIC PROPERTIES OF THE PARABOLA 1. The difference in elevation between the BVC and a point on the g 1 grade line at a distance x units (feet or meters) is g 1 x (g 1 is expressed as a decimal). Figure 10.14 Geometric of a Parabola 2. The tangent offset between the grade line and the curve is given by ax 2 , where x is the horizontal distance from the BVC (PVC); that is, tangent offsets are proportional to the squares of the horizontal distances. 3. The elevation of the curve at distance x from the BVC is given by: c bx ax y + + = 2 (general equation for a parabola) (10-24) 2 2 1 rx x g y y BVC x + + = (10-25) L g g r 1 2 ÷ = (10-26) Where: x = the distance from BVC to a point on the curve r = rate of grade change per station 4. The grade lines (g 1 and g 2 ) intersect midway between the BVC and the EVC ; that is, BVC to PVI = ½ L = PVI to EVC. This is only true for symmetrical vertical curves. 5. The curve lies midway between the PVI and the midpoint of the chord; that is, A‒B = B ‒ PVI = d o which can be calculated as follows: Either: d o = ½ (difference in elevation of PVI and mid-chord elevation) = ½ (elevation of BVC + elevation of EVC) OR: www.passpe.com Surveying for California Civil PE License © Dr. Shahin A.Mansour, PE Chapter 10- Horizontal, Spiral and Vertical Curve 234 d o = 8 2 1 L g g ÷ (10-27) 6. The slope S, in percentage, of the tangent to the curve at any point on the curve is given by the following formula: L g g x g S ) ( 2 1 1 ÷ ÷ = (10-28) Figure 10.15 Crest and Sag Vertical Curves Terminology 7. The distance D in feet from Vertex to P ʹ is given as: D = ) ( ) ( 100 2 1 g g Y Y P H ÷ ÷ ' (10-29) 8- The distance between the curve and the grade line (tangent) “d” is given as” L g g x rx offset d 200 ) ( 2 1 2 2 2 ÷ = = = (L curve length in feet) (10-30) 10-14 HIGH AND LOW POINTS ON VERTICAL CURVES The locations of the curves high and low points are important for drainage considerations; for example, on curbed streets catch basins must be installed precisely at the drainage low point. From equation (10-25), the slope ) ( dx dy is equaled to zero and solving for X: (10-31) 0 1 = + rX g www.passpe.com Surveying for California Civil PE License © Dr. Shahin A.Mansour, PE Figure 10.16 Low Point on a Sag Vertical Curve 2 1 1 1 2 1 1 g g L g g g L g r g X ÷ = ÷ ÷ = ÷ = (10-32) Where X is the distance from BVC to the low or high points. It should be noted that the distance X in the above two equations is different from distance x in equations 10-24 & 10-25. Sample Problem 10-13: Low point on a vertical curve Given: L = 300 ft, g 1 = ‒ 3.2%, g 2 = + 1.8%, PVI at 30 + 30, and elevation = 485.92 Find: Location of the low point and its elevation. Solution: ft Sta g g L g g g L g r g X 00 . 192 . 92 . 1 ) 2 . 3 ( ) 8 . 1 ( ) 3 )( 2 . 3 ( 2 1 1 1 2 1 1 = = ÷ ÷ + ÷ ÷ = ÷ = ÷ ÷ = ÷ = This means that the low point is located at a distance of 192.00 ft from BVC i.e. at Station = [(30 + 30.00) − (1+ 50.00)] + (1 + 92.00) = 30 + 72.00 Remember: All distances used to located a low or a high point or used to determine an elevation of a point on a vertical curve are measured from BVC. 2 2 1 rx x g y y BVC x + + = | | 00 . 72 30 @ 65 . 487 ) 2 92 . 1 )( 00 . 3 ) 2 . 3 ( 8 . 1 ( ) 92 . 1 )( 2 . 3 ( ) 2 . 3 )( 5 . 1 ( 92 . 485 2 + = ÷ ÷ + ÷ + + = Sta ft Chapter 10- Horizontal, Spiral and Vertical Curve 235 known as stations. the flatter the curve. The larger the radius. Spiral The spiral is a curve which has a varying radius. Reverse A reverse curve consists of two simple curves joined together. Compound and reverse curves are treated as a combination of two or more simple curves. PE Chapter 10. As the radius and length of curve increases. a) Simple Circular (b) Compound (c) Reverse (d) Spiral Figure 10. The radius of the circle determines the sharpness or flatness of the curve.www. Spiral and Vertical Curves 10-1 INTRODUCTION Horizontal curves may be simple.Horizontal. may be located on the circumference of the arc. the tape becomes impractical and the surveyor must use other methods. Its purpose is to provide a transition from the tangent to a simple curve or between simple curves in a compound curve Chapter 10.1 Types of Horizontal Curves 10-2 TYPES OF HORIZONTAL CURVES Table 10-1 Types of Horizontal Curves Simple Circular The simple curve is an arc of a circle. It is used on railroads and some modern highways.Horizontal.passpe. Shahin A. but curving in opposite directions. Mansour. compound. For safety reasons. This type of curve is the most often used. but curving in the same direction. This curve normally consists of two simple curves joined together. 226 Compound Frequently the terrain will necessitate the use of a compound curve. Curves of short radius (usually less than one tape length) can be established by holding one end of the tape at the center of the circle and swinging the tape in an arc. Spiral and Vertical Curves . marking as many points as may be desired. whereas the spiral curve is based on a varying radius.com Surveying for California Civil PE License © Dr. or spiral. The common method is to measure angles and straight-line sight distances by which selected points. reverse. this curve is seldom used in highway construction as it would tend to send an automobile off the road. Point of intersection: the point of intersection (PI) is the point where the back and forward tangents intersect. 4. Mansour. The tangent distance(T): the distance along the tangents from the PI to the PC or PT. The point of tangency: the point of tangency (PT) is the end of the curve. The back tangent is tangent to the curve at this point. PE 10-3 TERMINOLOGY OF HORIZONTAL CURVES Following are the main elements of a simple curve. These distances are equal on a simple curve. The point of curvature: the point of curvature (PC) is the point where the circular curve begins.10.2 1. 5. The radius(R): the radius of the circle of which the curve is an arc. Shahin A. 2. Note: The terms BC (Beginning of Curve) and EC (End of Curve) are referred to by some agencies as PC (point of curvature) and PT (point of tangency).2 Terminology of Horizontal Curve 6. The length of curve (L): the distance from the PC to the PT measured along the curve. 3. see Fig. Spiral and Vertical Curves 227 . The central angle is equal in value to the intersecting angle ( = I).www. The forward tangent is tangent to the curve at this point.passpe.Horizontal. 7. Figure 10. and by others as TC (tangent to curve) and CT (curve to tangent). Chapter 10. The central angle ( ): the angle formed by two radii drawn from the center of the circle ( O ) to the PC or PT.com Surveying for California Civil PE License © Dr. www.Horizontal.com Surveying for California Civil PE License © Dr.58 o 2R R D 50 ft Sin( ) R 2 ( 10-1) (10-2) The sharpness of a curve is determined by the choice of the radius (R). The chord definition is used primarily for civilian railroad construction and is used by the military for both roads and railroads. 9. This definition is used primarily for highways and streets. The external distance bisects the interior angle at the PI. Degree of curve: The degree of curve (D) defines the "sharpness" or "flatness" of the curve. Middle ordinate: The middle ordinate (M) is the distance from the midpoint of the curve to the midpoint of the long chord. whereas small radius curves are relatively sharp. Notice that the larger the degree of curve. 10. Long chord: The long chord (LC or C) is the chord from the PC to the PT. 228 Chapter 10. Spiral and Vertical Curves . Shahin A. Arc Definition The arc definition states that the degree of a curve is the angle formed by two radii drawn from the center of the circle to the ends of an arc 100 ft long.passpe. large radius curves are relatively flat. Mansour. The extension of the middle ordinate bisects the central angle. PE 8. as follows: Table 10-2 Chord and Arc Definitions for Horizontal Curves Chord Definition The chord definition states that the degree of a curve is the angle formed by two radii drawn from the center of the circle to the ends of a chord 100 ft long. 11. External distance: The external distance (E) is the distance from the PI to the midpoint of the curve. the "sharper" the curve and the shorter the radius D (360 o )(100 ft ) 5729. There are two common definitions for degree of curve . e 2 Cos 1 ( 1. The centerline now takes the shorter distance “L” from the “BC” to the “EC”.e. ( PI to A) E R 1 R ( Sec 1) 2 Cos ( 2) T Tan R Tan Tan 4 2 4 Notes: R R ) i. BC to A to EC ) : L 2 R R ( radians ) (100 ft ) 360 D (10-5) Middle Ordinate (A to B) M R (1 Cos C ) Tan E Cos 2 2 4 2 (10-6) 1 External Dist . the stationing (chainage) must reflect the fact that the centerline no longer goes through the “PI”.Horizontal. Spiral and Vertical Curve 229 .www.Cos 2 R E RE 2. PE 10-4 GEOMETRY OF HORIZONTAL CIRCULAR CURVES Tangent ( BC to PI OR PI to EC ) T R Tan Long Chord (BC to B to EC ) C 2 R Sin 2 T Cos ( 2) 2 2 (10-3) (10-4) Curve Length ( BC to EC Along the Curve i.A common mistake is to determine the station of the “EC” by adding the “T” distance to the “PI”. Chapter 10.Mansour.external secant (exsec) → exsec (∆/2)= Sec (∆/2)− 1 4. Although the “EC” is physically a distance of “T” from the “PI”.versed sine (vers) → vers (∆/2) = 1 − Cos (∆/2) (10-7) 3.com Surveying for California Civil PE License © Dr. Shahin A.passpe. 3167o – 1) = 10.18 ft 2 L 2 R (deg .3167o = 289. @ 6 + 26.31ft 360 PI at –T BC = +L EC = 6 + 26.com Surveying for California Civil PE License © Dr.passpe. and external distance (E) = 1000 tan 8. length of chord (C).3167o = 146.39 2 + 90.57 Solution: T R tan Find: BC and EC stations.31 7 + 70. R= 1000 ft . middle ordinate (M). PE Sample Problem 10-1: Horizontal Curve Chord.) R (radians) (100 ft ) 360 D 16.57 1 + 46.63 ft 2 4 Cos ( 2) 230 Chapter 10.www.Horizontal.18 4 + 80. Shahin A.29 ft 2 1 M R (1 Cos ) C Tan = 1000( 1 – Cos 8.6333 = 2 1000 = 290.Mansour.3167o) = 10. PI Sta.70 C 2 R Sin 2 T Cos ( 2) = 2 × 1000 × Sin 8. Middle Ordinate & External Distance Given: 16 o 38 .52 ft 2 2 2 1 E R( 1) R Tan Tan = 1000 (Sec 8. Spiral and Vertical Curve . Chapter 10. PE Sample Problem 10-2: Using Degree of Horizontal Curves Given: 11○ 21 35 .Mansour.93 Tan 5.33 ft 6 PI at –T BC = +L EC = 14+ 87. degree of curve D = 6o.93 ft R D T R Tan (10-2) = 954.98 13 + 92.Horizontal.35 01 + 89.com Surveying for California Civil PE License © Dr.33 00 + 94.www. Find: BC and EC stations PI Sta.) R (radians) (100 ft ) 360 D 100 11.33 15 + 81. Shahin A.3598 = 189.58 954.6799o = 94. @ 14 + 87. Spiral and Vertical Curve 231 .58o D R 2 R 5729.passpe.33 Solution: (360 o ) (100 ft ) 5729.98 ft 2 L 2 R = (deg .68 Note: A common mistake is to determine the station of the EC by adding the T distance to the PI station. Shahin A. & CHORD CALCUATIONS The deflection angle is defined as the angle between the tangent and a chord.e. the angle between the tangent “BC-PI” and the chord “PC-A” is ½ the central angle “BC-O-A” i.Horizontal.e.passpe.com Surveying for California Civil PE License © Dr. CENTRAL ANGLE. PE 10-5 DEFELCTION ANGLES. α & 2α Rule 2: The angle between two chords is ½ the central angle subtended by the arc between the two chords i.Mansour.www. Spiral and Vertical Curve .e. β & 2β arc length deflection angle L 2 arc length deflection angle L 2 Chord Length ( BC to A) 2 R Sin arc length ( BC to A) 180 0 2 R 2 arc length ( BC to A) L (10-8) (10-9) (10-10) (10-11) (10-12) Abbreviations: BC = Beginning of curve PC = Point of curvature TC = Tangent to curve EC = End of curve PT = Point of tangency CT = Curve to tangent Figure 10.e. The following two rules apply for the deflection angles for circular curves: Rule 1: The deflection angle between a tangent and a chord is half the central angle subtended by the arc i. the angle “A-BC-B” is ½ the central angle “AO-B” i.3 Deflection and Central Angles Realtionship 232 Chapter 10. Shahin A.Horizontal. The elevation of the curve at distance x from the BVC is given by: y ax 2 bx c (general equation for a parabola) y x y BVC r (10-24) (10-25) g 2 g1 (10-26) L Where: x = the distance from BVC to a point on the curve r = rate of grade change per station 4.www.Mansour.com Surveying for California Civil PE License © Dr. that is.14 Geometric of a Parabola 2. The tangent offset between the grade line and the curve is given by ax2. PE 10-13 GEOMETRIC PROPERTIES OF THE PARABOLA 1.passpe. Figure 10. that is. The grade lines (g 1 and g 2 ) intersect midway between the BVC and the EVC . The curve lies midway between the PVI and the midpoint of the chord. BVC to PVI = ½ L = PVI to EVC. that is. A‒B = B ‒ PVI = do which can be calculated as follows: Either: do = ½ (difference in elevation of PVI and mid-chord elevation) = ½ (elevation of BVC + elevation of EVC) OR: Chapter 10. where x is the horizontal distance from the BVC (PVC). 3. 5. tangent offsets are proportional to the squares of the horizontal distances. Spiral and Vertical Curve 233 rx 2 g1x 2 . This is only true for symmetrical vertical curves. The difference in elevation between the BVC and a point on the g 1 grade line at a distance x units (feet or meters) is g 1 x (g 1 is expressed as a decimal). the slope (dy dx) is equaled to zero and solving for X: g1 rX 0 (10-31) 234 Chapter 10. on curbed streets catch basins must be installed precisely at the drainage low point. PE (10-27) 8 6. Shahin A.passpe.The distance between the curve and the grade line (tangent) “d” is given as” (10-29) rx 2 x 2 ( g 2 g1 ) d offset (L curve length in feet) 2 200 L 10-14 HIGH AND LOW POINTS ON VERTICAL CURVES (10-30) The locations of the curves high and low points are important for drainage considerations.Horizontal. Spiral and Vertical Curve . in percentage. for example. The distance D in feet from Vertex to Pʹ is given as: 100(YH YP ) D= ( g1 g 2 ) 8.15 Crest and Sag Vertical Curves Terminology 7. of the tangent to the curve at any point on the curve is given by the following formula: x( g1 g 2 ) S g1 (10-28) L do = g1 g 2 L Figure 10. The slope S.Mansour.com Surveying for California Civil PE License © Dr. From equation (10-25).www. 2) (3. Sample Problem 10-13: Low point on a vertical curve Given: L = 300 ft.2) This means that the low point is located at a distance of 192. g1 = ‒ 3.00 3.00 ft r g 2 g1 g1 g 2 ( 1.00) = 30 + 72.2) 1.00) − (1+ 50.65 ft @ Sta 30 72. 192.8 (3. and elevation = 485.2%.Mansour. Spiral and Vertical Curve 235 .e. PVI at 30 + 30.92) ( )( ) 487.00)] + (1 + 92. Shahin A.00 X Remember: All distances used to located a low or a high point or used to determine an elevation of a point on a vertical curve are measured from BVC.00 ft from BVC i. g1 g1 L g1 L ( 3.2)(1.www. g2 = + 1.92 Solution: Find: Location of the low point and its elevation.passpe. at Station = [(30 + 30.8%.92 (1.16 Low Point on a Sag Vertical Curve X g1 g1 L g1 L r g 2 g1 g1 g 2 (10-32) Where X is the distance from BVC to the low or high points.2)(3) 1. PE Figure 10.Horizontal.00 2 Chapter 10. It should be noted that the distance X in the above two equations is different from distance x in equations 10-24 & 10-25.5)(3.92 2 485. rx 2 y x y BVC g 1 x 2 1.com Surveying for California Civil PE License © Dr.8) (3.92 Sta.
Comments
Copyright © 2024 UPDOCS Inc.