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ENGINEERING SURVEYS

Circular Curves | Simple CurvesTerminologies used in simple curve PC = Point of curvature. It is the beginning of curve. PT = Point of tangency. It is the end of curve. PI = Point of intersection of the tangents. T = Length of tangent from PC to PI and from PI to PT. It is known as subtangent. R = Radius of simple curve, or simply radius. L = Length of chord from PC to PT. Point Q as shown in the the figure is the midpoint of L. Lc= Length of curve from PC to PT. Point M as shown in the the figure is the midpoint of Lc. E = External distance. It is the nearest distance from PI to the curve. From the above figure, point M is the midpoint of the curve and E is the distance from PI to M. m = Middle ordinate. It is the distance from the midpoint of the curve to the midpoint of the chord. From the figure above, m is the distance MQ. I = Deflection angle (also calledangle of intersectionandcentral angle). It is the angle of intersection of the tangents. The angle subtended by PC and PT at O is also equal to I, where O is the center of the circular curve from the above figure. x = offset distance from tangent to any point in the curve; note that x is perpendicular to T. = offset angle subtended at PC between PI and any point in the curve D = Degree of curve. It is the central angle subtended by a length of curve equal to one station. In English system, one station is equal to 100 ft and in SI, one station is equal to 20 m. Sub chord = chord distance between two adjacent full stations.Sharpness of circular curveThe smaller is the degree of curve, the flatter is the curve and vice versa. The sharpness of simple curve is also determined by radius R. Large radius are flat whereas small radius are sharp.Formulas for Circular CurvesThe formulas we are about to present need not be memorized. All we need is geometry plus names of all elements in simple curve. Note that we are only dealing with circular arc, it is in our great advantage if we deal it at geometry level rather than memorize these formulas. For easy reference, the figure shown in the previous page will be shown again here.

Length of tangent, TLength of tangent (also referred to as subtangent) is the distance from PC to PI. It is the same distance from PI to PT. From the right triangle PI-PT-O,

External distance, EExternal distance is the distance from PI to the midpoint of the curve. From the same right triangle PI-PT-O,

Middle ordinate, mMiddle ordinate is the distance from the midpoint of the curve to the midpoint of the chord. From right triangle O-Q-PT,

Length of long chord, LLength of long chord or simply length of chord is the distance from PC to PT. Again, from right triangle O-Q-PT,

Length of curve, LcLength of curve from PC to PT is the road distance between ends of the simple curve. By ratio and proportion,

An alternate formula for the length of curve is by ratio and proportion with its degree of curve.

SI units: 1 station = 20 m

English system: 1 station = 100 ft

If given the stationing of PC and PT

Degree of curve, DThe degree of curve is the central angle subtended by an arc (arc basis) or chord (chord basis) of one station. It will define the sharpness of the curve. In English system, 1 station is equal to 100 ft and in SI, 1 station is equal to 20 m. It is important to note that 100 ft is equal to 30.48 m not 20 m.Arc BasisIn arc definition, the degree of curve is the central angle angle subtended by one station of circular arc. This definition is used in highways. Using ratio and proportion,

SI units (1 station = 20 m):

English system (1 station = 100 ft):

Chord BasisChord definition is used in railway design. The degree of curve is the central angle subtended by one station length of chord. From the right triangle shaded in green color,

SI units (half station = 10 m):

English system (half station = 50 ft):

Compound CurvesA compound curve consists of two (or more)circular curvesbetween two main tangents joined at point of compound curve (PCC). Curve at PC is designated as 1 (R1, L1, T1, etc) and curve at higher station is designated as 2 (R2, L2, T2, etc).

Elements of compound curve PC = point of curvature PT = point of tangency PI = point of intersection PCC = point of compound curve 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 I = angle of intersection = I1+ I2 Lc1= length of first curve Lc2= length of second curve L1= length of first chord L2= length of second chord L = length of long chord from PC to PT T1+ T2= length of common tangent measured from V1to V2 = 180 I x and y can be found from triangle V1-V2-PI. L can be found from triangle PC-PCC-PTFinding the stationing of PTGiven the stationing of PC

Given the stationing of PI

Reversed CurveReversed curve, though pleasing to the eye, would bring discomfort to motorist running at design speed. The instant change in direction at the PRC brought some safety problems. Despite this fact, reversed curves are being used with great success on park roads, formal paths, waterway channels, and the like.

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 V1to V2Finding the stationing of PTGiven the stationing of PC

Given the stationing of V1

Reversed Curve for Nonparallel Tangents

Reversed Curve for Parallel Tangents

Spiral Curve | Transition CurveSpirals are used to overcome the abrupt change in curvature and superelevation that occurs between tangent and circular curve. The spiral curve is used to gradually change the curvature and superelevation of the road, thus called transition curve.

Elements of Spiral Curve TS = Tangent to spiral SC = Spiral to curve CS = Curve to spiral ST = Spiral to tangent LT = Long tangent ST = Short tangent R = Radius of simple curve Ts= Spiral tangent distance Tc= Circular curve tangent L = Length of spiral from TS to any 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 Yc= Distance along tangent from TS to point at right angle to SC Es= External distance of the simple curve = Spiral angle from tangent to any point on the spiral s= 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 curveFormulas for Spiral CurvesDistance along tangent to any point on the spiral:

At L = Ls, Y = Yc, thus,

Offset distance from tangent to any point on the spiral:

At L = Ls, X = Xc, thus,

Length of throw:

Spiral angle from tangent to any point on the spiral (in radian):

At L = Ls,=s, thus,

Deflection angle from TS to any point on the spiral:

At L = Ls, i = is, thus,

This angle is proportional to the square of its distance

Tangent distance:

Angle of intersection of simple curve:

External distance:

Degree of spiral curve:

Symmetrical Parabolic CurveVertical Parabolic CurveVertical curves are used to provide gradual change between two adjacent vertical grade lines. The curve used to connect the two adjacent grades is parabola. Parabola offers smooth transition because its second derivative is constant. For a downward parabola with vertex at the origin, the standard equation is or .Recall from calculus that the first derivative is the slope of the curve..The value of y' above is linear, thus the grade diagram (slope diagram) for a summit curve is downward and linear as shown in the figure below. The second derivative is obviously constant

which is interpreted as rate of change of slope. This characteristic made the parabola the desirable curve because it offers constant rate of change of slope.Vertical Symmetrical Parabolic CurveIn this section, symmetrical parabolic curve does not necessarily mean the curve is symmetrical at L/2, it simply means that the curve is made up of single vertical parabolic curve. Using two or more parabolic curves placed adjacent to each other is called unsymmetrical parabolic curve. The figure shown below is a vertical summit curve. Note that the same elements holds true for vertical sag curve.Elements of Vertical Curve PC = point of curvature, also known as BVC (beginning of vertical curve) PT = point of tangency, also known as EVC (end of vertical curve) PI = point of intersection of the tangents, also called PVI (point of vertical intersection) L = length of parabolic curve, it is the projection of the curve onto a horizontal surface which corresponds to the plan distance. S1= horizontal distance from PC to the highest (lowest) point of the summit (sag) curve S2= horizontal distance from PT to the highest (lowest) point of the summit (sag) curve h1= vertical distance between PC and the highest (lowest) point of the summit (sag) curve h2= vertical distance between PT and the highest (lowest) point of the summit (sag) curve g1= grade (in percent) of back tangent (tangent through PC) g2= grade (in percent) of forward tangent (tangent through PT) A = change in grade from PC to PT a = vertical distance between PC and PI b = vertical distance between PT and PI H = vertical distance between PI and the curve

Formulas for Symmetrical Parabolic CurveThe figure shown above illustrates the following geometric properties of parabolic curve.Properties of Parabolic Curve and its Grade Diagram1. The length of parabolic curve L is the horizontal distance between PI and PT.2. PI is midway between PC and PT.3. The curve lies midway between PI and the midpoint of the chord from PC to PT.4. The vertical distance between any two points on the curve is equal to area under the grade diagram. The vertical distance c = Area.5. The grade of the curve at a specific point is equal to the offset distance in the grade diagram under that point. The grade at point Q is equal to gQ.Note that the principles and formulas can be applied to both summit and sag curves.rise = run slope

Neglecting the sign of g1and g2

vertical distance = area under the grade diagram

Other formulas