Curve
Introduction.
Curves are regular bends provided in the line of communication like roads, railways etc, and also in canals to bring about the gradual change of direction.
They are used in the vertical plane at all changes of grade to avoid the abrupt change of grade of the apex.
Where the fieldwork is composed of straight lines, setting out is comparatively straight forward. With roads, railways and pipelines, two straights will normally be connected by a curve where there is a change of direction.
Curves to a surveyor includes types of
1. Circular Curves.
2. Transition Curve and
3. Vertical Curve
1. Simple curve
Curves to a surveyor includes types of
1. Circular Curves.
2. Transition Curve and
3. Vertical Curve
Elements of simple curve.
The point at which two straights meet is known as the intersection point, and is signified by I or IP.
The two straights deviate by the angle delta (∆), which is referred as the deviation angle, the deflection angle or the intersection angle.
Let PC = point of curvature = start of the curve = TP1 (back tangent point)
PT = point of tangency = finish of the curve = TP2 (forward tangent point)
IP = point of intersection of the two tangents
∆ = deflection angle of the two tangents
R = radius of the circular arc
R = radius of the circular arc
O = centre of the circle of the curve
As the radius is tangent to the curve, the angle at PC (TP1) = 90°
Similarly the angle at PT (TP2) = 90° Deflection angle = ∆
Internal angle at IP = 180° - ∆
Sum of the internal angles of a 4 sided figure = 360°
Internal angle at O = 360 –(90 + 90 + 180 – ∆) = ∆ (the deflection angle)
Arc Lengths and Circular Measure of Angles
In circular measure of angles, the magnitude of an angle is expressed in radians.
An angle expressed in radians is defined as the ratio of the arc subtended by that angle at the centre of a circle, to the radius of that circle.
Angle at the centre in Radians= Arc Length Radius When the arc length is equal to the radius, the ratio = 1/1=1
therefore the angle at the centre contains 1 radian.
Therefore 1radian/360°= R/2πR=arc distance/circumference of circle.
1 radian =360°/2π
= 180°/π
= 57.2957795°
= 57° 17'44.806"
Crown secant =R(sec∆/2-1)
Mid - ordinate = R(1-cos∆/2)
Arc = R∆/2
Long chord = 2Rsin∆/2
therefore the angle at the centre contains 1 radian.
Therefore 1radian/360°= R/2πR=arc distance/circumference of circle.
1 radian =360°/2π
= 180°/π
= 57.2957795°
= 57° 17'44.806"
Calculations with Curves
Tangent distance (TD)= R.Tan ∆/2Crown secant =R(sec∆/2-1)
Mid - ordinate = R(1-cos∆/2)
Arc = R∆/2
Long chord = 2Rsin∆/2
Chainage of Tangents points
If the chainage of the intersection point is known, the chainage of the tangent points may be calculated.
If the chainage of IP is 2742.60 then the chainage at TP1 and TP2 is as follows:
IP-TP1=TD= R. Tan∆/2
=500tan60°
=866. 03m
Chainage at TP1 =2742.60-866. 03
=1876. 57m
Chainage at TP2 is not change at IP+TD but the chainage TP1+length around the curve.
Arc =R∆
=500*2. 0944
=1047. 20m
Chainage at TP2= 1876. 57+1047. 0944
=2923. 77m
2. Transition curve
Transition curve is a curve in plan which is provided to change the horizontal alignment from straight curve gradually mean the radius of the curve varies between infinity to R or R to infinity.
Requirement of transition curve
• Tangential to straight
• Meet circular curve tangentially
• At origin curvature should zero.
• Curvature should same at junction of circular curve.
• Rate of increase of curvature = rate increase of super elevation
• Length of transition curve=full seper elevation attained.
Purpose of transition curve
• Curvature is increase gradually. •Medium for gradual introduction of superelevation
Provide Extra widening gradually widening
Advantages
. Increase comfort to passenger on curve
• Reduce overturning
• Allow higher speed
• Less wear on gear, tyre on curve
Types of transition curve
1. Cubic parabola -For railway railway
2. Spiral or Clothoid‐Ideal transition‐ Radius α Distance
3. LemniscatesUsed Used for road
3. Vertical Curve
When an vertical curve applied?
1. At an intersection of two slopes on a highway or a roadway2To provide a safe and comfort ride for vehicles on a roadway.
g1 is the slope of lower station of grade line
g2 is the slope of higher station of grade line
BVC is the beginning of vertical curve.
EVC is the ending of vertical curve.
PVI is the point if intersection of the two adjacent grade line
Vertical curve terminology
The algebric change in slope direction is "A" where
A=g2- g1
The geometric curve used in vertical alignment design the vertical axis parabola.
The parabola has the desirable characterstitics
1. A constant rate of slope which contribute to smooth alignment transition.
2. Ease of computation of vertical offsets which permit easily computed curve elevation.
General equation of parabola
y = ax2+bx+c = 0
The slope of this curve at any point is given first derivatives.
dy/dx =2ax+b
The rye of change of slope is given by second derivatives .
d2y/dx2 =2a
2a is a constant
Also 2a is written as A/L.
Length of vertical curve = Total change of grade / Rate of change of grade.
=g2- g1/r
1. The difference in elevation between BVC and a point on g1 grade line at a distance x ie g1x
2. The tngent offset between the grade line and curve is given by ax2.
3. The elevation of curve at distance x from BVC is given by BVC+g1x-ax2.
4. Offsets from two grade lines are symmetrical with respect to PVI.
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