Symmetrical Parabolic Curve

Symmetrical Parabolic Curve Vertical Parabolic Curve Vertical 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 Curve In 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

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h1 = vertical distance between PC and the highest (lowest) point of the summit (sag) curve

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h2 = vertical distance between PT and the highest (lowest) point of the summit (sag) curve

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g1 = grade (in percent) of back tangent (tangent through PC) g2 = grade (in percent) of forward tangent (tangent through PT) 

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A = change in grade from PC to PT

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a = vertical distance between PC and PI

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b = vertical distance between PT and PI H = vertical distance between PI and the curve

Formulas for Symmetrical Parabolic Curve The figure shown above illustrates the following geometric properties of parabolic curve. Properties of Parabolic Curve and its Grade Diagram 1.

The length of parabolic curve L is the horizontal distance between PI and PT. 2.

3. 4.

5.

PI is midway between PC and PT.

The curve lies midway between PI and the midpoint of the chord from PC to PT.

The vertical distance between any two points on the curve is equal to area under the grade diagram. The vertical distance c = Area. 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 g1 and g2

vertical distance = area under the grade diagram

Other formulas

Problem 01 | Stationing and elevation of crossdrainage pipe Problem 01 A grade of -4.2% grade intersects a grade of +3.0% at Station 11 + 488.00 of elevations 20.80 meters. These two center gradelines are to be connected by a 260 meter vertical parabolic curve. 1.

At what station is the cross-drainage pipes be situated? A.

Sta 11 + 493.42 km

B.

Sta 11 + 509.67 km

C.

Sta 11 + 515.14 km

D.

Sta 10 + 600.00 km

2.

If the overall outside dimensions of the reinforced concrete pipe to be installed is 95 cm, and the top of the culvert is 30 cm below the subgrade, what will be the invert elevation at the center? A.

25.76 m

B.

22.15 m

C.

27.32 m

D.

21.83 m

Solution HideClick here to show or hide the solution

From the grade diagram:

The cross-drainage pipe should be at the lowest point of the curve. Stationing of the lowest point indicated as point A in the figure:

Answer for Part 1: [ B ] Vertical distance between PC and PI:

Vertical distance between PC and the lowest point A:

Elevation of the lowest point A:

Answer for Part 2: [ D ]

Problem 02 | Elevation of the first quarter point of parabolic sag curve Board Exam Problem A descending grade of 6% and an ascending grade of 2% intersect at Sta 12 + 200 km whose elevation is at 14.375 m. The two grades are to be connected by a parabolic curve, 160 m long. Find the elevation of the first quarter point on the curve. A.

16.082 m

B.

15.575 m

C.

17.175 m

D.

13.936 m

Solution HideClick here to show or hide the solution

From the grade diagram:

Horizontal distance from the lowest point to point Q:

Grade at point Q by ratio and proportion of triangles:

Elevation of PC:

Difference in elevation between PC and Q:

Elevation of the first quarter point Q:

[ C ] answer

Problem 03 | Elevation of the summit of vertical parabolic curve Board Problem A grade line AB having a slope of +5% intersect another grade line BC having a slope of –3% at B. The elevations of points A, B and C are 95 m, 100 m and 97 m respectively. Determine the elevation of the summit of the 100 m parabolic vertical curve to connect the grade lines. A.

98.32 m

B.

99.06 m

C.

97.32 m

D.

96.86 m

Solution HideClick here to show or hide the solution Horizontal distance between A and B = (100 - 95)/0.05 = 100 m Horizontal distance between B and C = (100 - 97)/0.03 = 100 m

The figure above place the parabolic curve at the middle-half:

Distance from PC to the summit:

Elevation of the summit:

[ B ] answer