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De Leon, Verona Alejandrea F. CE121/C1

RAILROAD AND HIGHWAY CURVES In highway or railroad construction, the curves most generally used presently are circular curves although parabolic and other curves are sometimes used. These types of curves are classified as Simple, Compound, Reversed, Spiral Curves, Symmetrical and Unsymmetrical.

Horizontal Curves The surveyor learns to locate points using angles and distances. In construction surveying, the surveyor must often establish the line of a curve for road layout or some other construction. The surveyor can establish curves of short radius, usually less than one tape length, 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 desired. As the radius and length of curve increases, the tape becomes impractical, and the surveyor must use other methods. Measured angles and straight line distances are usually picked to locate selected points, known as stations, on the circumference of the arc. Types of Horizontal Curves A curve may be simple, compound, reverse, or spiral (figure 3-l). 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.

A. Simple Curve A simple curve is a circular arc, extending from one tangent to the next. The point where the curve leaves the first tangent is called the point of curvature (PC) and the point where the curve joins the second tangent is called the point of tangency (PT). The PC and PT are often called the tangent points. If the tangent be produced, they will meet in a point of intersection called the vertex. The distance from the vertex to the curve is called the external distance (measured towards the center of curvature). While the line joining the

middle of the curve and the middle of the chord line joining the PC and PT is called middle ordinate. Sharpness of the curve is expressed in any of the three ways: 1. Degree of Curve (Arc Basis)- Degree of curve is the angle at the center subtended by an arc of 20 m. is the Metric system or 100 ft. in the English system. This is the method generally used in Highway practice. a. Metric System: b. English System:

D=

1145.916 R

D=

5729.58 R

(5 times the metric system)

2. Degree of Curve (Chord Basis)- Degree of curve is the angle subtended by a chord of 20 meters in Metric system or 100 ft. in English system. a. Metric System:

b. English System:

R=

10 D sin ( ) 2

R=

50 D sin ( ) 2

(5 times the metric

system) 3. Radius = length of radius stated

Elements of a simple curve: Figure 3-2 shows the elements of a simple curve. They are described as follows, and their abbreviations are given in parentheses. Point of Intersection (PI) - The point of intersection marks the point where the back and forward tangents intersect. The surveyor indicates it one of the stations on the preliminary traverse. Intersecting Angle (I) - The intersecting angle is the deflection angle at the PI. The surveyor either computes its value from the preliminary traverse station angles or measures it in the field.

Radius (R) - The radius is the radius of the circle of which the curve is an arc. Point of Curvature (PC) - The point of curvature is the point where the circular curve begins. The back tangent is tangent to the curve at this point. Point of Tangency (PT) - The point of tangency is the end of the curve. The forward tangent is tangent to the curve at this point. Length of Curve (LC) - The length of curve is the distance from the PC

to

the PT measured along the curve. Long Chord (L) - The long chord is the chord from the PC to the PT. Tangent Distance (T) - The tangent distance is the distance along

the

tangents from the PI to the PC or PT. These distances are equal on a simple curve. External Distance (E) - The external distance is the distance from the PI to the midpoint of the curve. The external distance bisects the interior angle at the PI. Central Angle

- The central angle is the angle formed by two radii drawn from the

center of the circle (0) to the PC and PT. The central angle is equal in value to the I angle. Middle Ordinate (M) - The central angle is the angle formed by two radii drawn from the center of the circle (0) to the PC and PT. The central

angle is

equal in

value

to the I

angle.

B. Compound Curve Compound curve consists of two or more consecutive simple curves having different radius, but whose centers lie on the same side of the curve, likewise any two consecutive curves must have a common tangent at their meeting point. When two such curves lie upon opposite sides of the common tangent, the two curves then turns a reversed curve. In a compound curve, the point of the common tangent where the two curves join is called the point of compound curvature (PCC).

Elements of compound curve: PC = point of curvature

I1 = central angle of the first curve

PT = point of tangency

I2 = central angle of the second curve

PI = point of intersection

I = angle of intersection = I1 + I2

PCC = point of compound curve

Lc1 = length of first curve

T1 = length of tangent of the first curve

Lc2 = length of second curve

T2 = length of tangent of the second curve

L1 = length of first chord

V1 = vertex of the first curve

L2 = length of second chord

V2 = vertex of the second curve

L = length of long chord from PC to PT

T1 + T2 = length of common tangent

x and y can be found from triangle V1-V2-

measured from V1 to V2

PI.

θ = 180° – I

L can be found from triangle PC-PCC-PT

C. Reversed Curve A reversed curve is formed by two circular simple curves having a common tangent but lies on opposite sides. The method of laying out a reversed curve is just the same as the deflection angle method of laying out simple curves. At the point where the curve reversed in its direction is called the Point of Reversed Curvature. After this point has been laid out from the PC, the instrument is then transferred to this point PRC. With the transit at PRC and a reading equal to the total deflection angle from the PC to the PRC, the PC is backsighted. If the line of sight is rotated about the vertical axis until horizontal circle reading becomes zero, this line of sight falls on the common tangent. The next simple curve could be laid out on the opposite side of this tangent by deflection angle method. Elements of Reversed Curve

PC = point of curvature PT = point of tangency PRC = point of reversed curvature T1 = length of tangent of the first curve T2 = length of tangent of the second curve V1 = vertex of the first curve V2 = vertex of the second curve I1 = central angle of the first curve I2 = central angle of the second curve Lc1 = length of first curve Lc2 = length of second curve L1 = length of first chord L2 = length of second chord T1 + T2 = length of common tangent measured from V1 to V2

D. Spiral Curve Spirals are used to overcome the abrupt change in curvature and super elevation that occurs between tangent and circular curve. The spiral curve is used to gradually change the curvature and super elevation of the road, thus called transition curve. The spiral is a curve with varying radius used on railroads and some modern highways. It provides a transition from the tangent to a simple curve or between simple curves in a compound curve. Elements of Spiral Curve TS = Tangent to spiral

ST = Spiral to tangent

SC = Spiral to curve

LT = Long tangent

CS = Curve to spiral

ST = Short tangent

R = Radius of simple curve Ts = Spiral tangent distance Tc = Circular curve tangent L = Length of spiral from TS to any

Yc = Distance along tangent from TS to point at right angle to SC Es = External distance of the simple curve

point along the spiral Ls = Length of spiral PI = Point of intersection I = Angle of intersection Ic = Angle of intersection of the simple curve p = Length of throw or the distance from tangent that the circular curve has been offset X = Offset distance (right angle distance) from tangent to any point on the spiral Xc = Offset distance (right angle distance) from tangent to SC Y = Distance along tangent to any point on the spiral

θ = Spiral angle from tangent to any point on the spiral θs = Spiral angle from tangent to SC i = Deflection angle from TS to any point on the spiral, it is proportional to the square of its distance is = Deflection angle from TS to SC D = Degree of spiral curve at any point Dc = Degree of simple curve

Vertical Curves In addition to horizontal curves that go to the right or left, roads also have vertical curves that go up or down. Vertical curves at a crest or the top of a hill arecalled summit curves, or oververticals.

Types of Vertical Curves a. Symmetrical Vertical Curve A symmetrical vertical curve is one in which the horizontal distance from the PVI to the PVC is equal to the horizontal distance from the PW to the PVT. In other words, l1 equals l2. b. Unsymmetrical Vertical Curve An unsymmetrical vertical curve is a curve in which the horizontal distance from the PVI to the PVC l2 is 200 feet g1 is –4% g2 is +6% is different from the horizontal distance between the PVI and the PVT. In other words, l1 does NOT equal l2. Unsymmetrical curves are sometimes described as having unequal tangents and are referred to as dog legs. Figure 11-19 shows an unsymmetrical curve with a horizontal distance of 400 feet on the left and a horizontal distance of 200 feet on the right of the PVI. The gradient of the tangent at the PVC is –4 percent; the gradient of the tangent at the PVT is +6 percent. Note that the curve is in a dip.

Sources: Besavilla, VJ. (1987). Surveying for Civil and Geodetic Licensure Exam. Cebu City: VIB Publisher http://faculty.mu.edu.sa/public/uploads/1334330885.7695hl.&vl.curves4.pdf http://www.mathalino.com/reviewer/surveying

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RAILROAD AND HIGHWAY CURVES In highway or railroad construction, the curves most generally used presently are circular curves although parabolic and other curves are sometimes used. These types of curves are classified as Simple, Compound, Reversed, Spiral Curves, Symmetrical and Unsymmetrical.

Horizontal Curves The surveyor learns to locate points using angles and distances. In construction surveying, the surveyor must often establish the line of a curve for road layout or some other construction. The surveyor can establish curves of short radius, usually less than one tape length, 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 desired. As the radius and length of curve increases, the tape becomes impractical, and the surveyor must use other methods. Measured angles and straight line distances are usually picked to locate selected points, known as stations, on the circumference of the arc. Types of Horizontal Curves A curve may be simple, compound, reverse, or spiral (figure 3-l). 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.

A. Simple Curve A simple curve is a circular arc, extending from one tangent to the next. The point where the curve leaves the first tangent is called the point of curvature (PC) and the point where the curve joins the second tangent is called the point of tangency (PT). The PC and PT are often called the tangent points. If the tangent be produced, they will meet in a point of intersection called the vertex. The distance from the vertex to the curve is called the external distance (measured towards the center of curvature). While the line joining the

middle of the curve and the middle of the chord line joining the PC and PT is called middle ordinate. Sharpness of the curve is expressed in any of the three ways: 1. Degree of Curve (Arc Basis)- Degree of curve is the angle at the center subtended by an arc of 20 m. is the Metric system or 100 ft. in the English system. This is the method generally used in Highway practice. a. Metric System: b. English System:

D=

1145.916 R

D=

5729.58 R

(5 times the metric system)

2. Degree of Curve (Chord Basis)- Degree of curve is the angle subtended by a chord of 20 meters in Metric system or 100 ft. in English system. a. Metric System:

b. English System:

R=

10 D sin ( ) 2

R=

50 D sin ( ) 2

(5 times the metric

system) 3. Radius = length of radius stated

Elements of a simple curve: Figure 3-2 shows the elements of a simple curve. They are described as follows, and their abbreviations are given in parentheses. Point of Intersection (PI) - The point of intersection marks the point where the back and forward tangents intersect. The surveyor indicates it one of the stations on the preliminary traverse. Intersecting Angle (I) - The intersecting angle is the deflection angle at the PI. The surveyor either computes its value from the preliminary traverse station angles or measures it in the field.

Radius (R) - The radius is the radius of the circle of which the curve is an arc. Point of Curvature (PC) - The point of curvature is the point where the circular curve begins. The back tangent is tangent to the curve at this point. Point of Tangency (PT) - The point of tangency is the end of the curve. The forward tangent is tangent to the curve at this point. Length of Curve (LC) - The length of curve is the distance from the PC

to

the PT measured along the curve. Long Chord (L) - The long chord is the chord from the PC to the PT. Tangent Distance (T) - The tangent distance is the distance along

the

tangents from the PI to the PC or PT. These distances are equal on a simple curve. External Distance (E) - The external distance is the distance from the PI to the midpoint of the curve. The external distance bisects the interior angle at the PI. Central Angle

- The central angle is the angle formed by two radii drawn from the

center of the circle (0) to the PC and PT. The central angle is equal in value to the I angle. Middle Ordinate (M) - The central angle is the angle formed by two radii drawn from the center of the circle (0) to the PC and PT. The central

angle is

equal in

value

to the I

angle.

B. Compound Curve Compound curve consists of two or more consecutive simple curves having different radius, but whose centers lie on the same side of the curve, likewise any two consecutive curves must have a common tangent at their meeting point. When two such curves lie upon opposite sides of the common tangent, the two curves then turns a reversed curve. In a compound curve, the point of the common tangent where the two curves join is called the point of compound curvature (PCC).

Elements of compound curve: PC = point of curvature

I1 = central angle of the first curve

PT = point of tangency

I2 = central angle of the second curve

PI = point of intersection

I = angle of intersection = I1 + I2

PCC = point of compound curve

Lc1 = length of first curve

T1 = length of tangent of the first curve

Lc2 = length of second curve

T2 = length of tangent of the second curve

L1 = length of first chord

V1 = vertex of the first curve

L2 = length of second chord

V2 = vertex of the second curve

L = length of long chord from PC to PT

T1 + T2 = length of common tangent

x and y can be found from triangle V1-V2-

measured from V1 to V2

PI.

θ = 180° – I

L can be found from triangle PC-PCC-PT

C. Reversed Curve A reversed curve is formed by two circular simple curves having a common tangent but lies on opposite sides. The method of laying out a reversed curve is just the same as the deflection angle method of laying out simple curves. At the point where the curve reversed in its direction is called the Point of Reversed Curvature. After this point has been laid out from the PC, the instrument is then transferred to this point PRC. With the transit at PRC and a reading equal to the total deflection angle from the PC to the PRC, the PC is backsighted. If the line of sight is rotated about the vertical axis until horizontal circle reading becomes zero, this line of sight falls on the common tangent. The next simple curve could be laid out on the opposite side of this tangent by deflection angle method. Elements of Reversed Curve

PC = point of curvature PT = point of tangency PRC = point of reversed curvature T1 = length of tangent of the first curve T2 = length of tangent of the second curve V1 = vertex of the first curve V2 = vertex of the second curve I1 = central angle of the first curve I2 = central angle of the second curve Lc1 = length of first curve Lc2 = length of second curve L1 = length of first chord L2 = length of second chord T1 + T2 = length of common tangent measured from V1 to V2

D. Spiral Curve Spirals are used to overcome the abrupt change in curvature and super elevation that occurs between tangent and circular curve. The spiral curve is used to gradually change the curvature and super elevation of the road, thus called transition curve. The spiral is a curve with varying radius used on railroads and some modern highways. It provides a transition from the tangent to a simple curve or between simple curves in a compound curve. Elements of Spiral Curve TS = Tangent to spiral

ST = Spiral to tangent

SC = Spiral to curve

LT = Long tangent

CS = Curve to spiral

ST = Short tangent

R = Radius of simple curve Ts = Spiral tangent distance Tc = Circular curve tangent L = Length of spiral from TS to any

Yc = Distance along tangent from TS to point at right angle to SC Es = External distance of the simple curve

point along the spiral Ls = Length of spiral PI = Point of intersection I = Angle of intersection Ic = Angle of intersection of the simple curve p = Length of throw or the distance from tangent that the circular curve has been offset X = Offset distance (right angle distance) from tangent to any point on the spiral Xc = Offset distance (right angle distance) from tangent to SC Y = Distance along tangent to any point on the spiral

θ = Spiral angle from tangent to any point on the spiral θs = Spiral angle from tangent to SC i = Deflection angle from TS to any point on the spiral, it is proportional to the square of its distance is = Deflection angle from TS to SC D = Degree of spiral curve at any point Dc = Degree of simple curve

Vertical Curves In addition to horizontal curves that go to the right or left, roads also have vertical curves that go up or down. Vertical curves at a crest or the top of a hill arecalled summit curves, or oververticals.

Types of Vertical Curves a. Symmetrical Vertical Curve A symmetrical vertical curve is one in which the horizontal distance from the PVI to the PVC is equal to the horizontal distance from the PW to the PVT. In other words, l1 equals l2. b. Unsymmetrical Vertical Curve An unsymmetrical vertical curve is a curve in which the horizontal distance from the PVI to the PVC l2 is 200 feet g1 is –4% g2 is +6% is different from the horizontal distance between the PVI and the PVT. In other words, l1 does NOT equal l2. Unsymmetrical curves are sometimes described as having unequal tangents and are referred to as dog legs. Figure 11-19 shows an unsymmetrical curve with a horizontal distance of 400 feet on the left and a horizontal distance of 200 feet on the right of the PVI. The gradient of the tangent at the PVC is –4 percent; the gradient of the tangent at the PVT is +6 percent. Note that the curve is in a dip.

Sources: Besavilla, VJ. (1987). Surveying for Civil and Geodetic Licensure Exam. Cebu City: VIB Publisher http://faculty.mu.edu.sa/public/uploads/1334330885.7695hl.&vl.curves4.pdf http://www.mathalino.com/reviewer/surveying

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