Vertical Curve  

• Vertical Curve are provided at the intersections of different grades to smoothen the vertical profile.
• The vertical curve used in the highway are of two types
  1. Summit Curve
  2. Valley Curve

Note: This topic is part of the second chapter of Highway Engineering. I suggest reading this topic in the context of the complete chapter: Geometric Design of Highway. If you want to read the entire Highway Engineering, click here: Highway Engineering.

Summit Curve

• Summit curve are vertical curve with convexity upward & concavity downward.
• It can formed by two gradient in following ways.

 Some Important Points of Summit Curve

• Convexity upward & Concavity downward.
• Vertical Point Of Intersection (VPI) always lies above the curve.
• Summit curve are designed only for sight distance criteria.
• Summit curve are made with square parabola shape.
• Ideal shape of summit curve is circular.
• Centrifugal force acts in upwards direction due to which effective weight of vehicle reduces. Hence there is no discomfort in summit curve.

  Derivation of Summit Curve  

1.General Equation of Summit Curve

y’= ax² + bx + c

At point A, (y’=0, x=0)

then we put the value of (y’=0, x=0)

0= 0+0+c

c=0

y’= ax² + bx + 0

y’= ax² + bx

\(\frac{\mathrm{d} y’}{\mathrm{d} x}= 2ax+b\)=Gradient

At point A, x=0, \(\frac{\mathrm{d} y’}{\mathrm{d} x}= +n_1\)

+n₁=2a·0+b

b=+n₁

At point B, x= L_S, \(\frac{\mathrm{d} y’}{\mathrm{d} x}= -n_2\)

-n₂=2a·L_S + n₁

\(a=-[\frac{n_1+n_2}{2L_S}]\).

\(a=-[\frac{N}{2L_S}]\)

General Equation 

\(y’=[-\frac{N}{2L_S}]x^{2}+n_1x\)

2.Position of Summit / Highest / Crest Point of Curve From VPC

we know that at crest point of curve, slope is zero.

\(\frac{\mathrm{d} y’}{\mathrm{d} x}=0\)

2ax+b=0

x=-b/2a

we know that

b=+n₁ & \(a=-[\frac{N}{2L_S}]\).

so \(x=\frac{n_1}{N}L_S\)

3.Apex Equation of Summit Curve

tanθ= n₁ = (y+y’)/x

y= n₁x-y’

\(y’=[-\frac{N}{2L_S}]x^{2}+n_1x\)

\(y= [n_1x-([-\frac{N}{2L_S}]x^{2}+n_1x)\)

\(y= [\frac{N}{2L_S}]x^{2}\)

4.Position pf VPI from VPC

From ΔAVE

tanθ₁=n₁=VE/x

VE= n₁x…….(1)

From ΔVDB

tanθ2=n2=VD/(L_S-x)

VD= n₂L_S – n₂x…….(2)

y’_B = DE = VE- VD……..(3)

\(y’_B=[-\frac{N}{2L_S}]x^{2}+n_1x\) (x= L_S) ……(4)

the value of VE, VD, y’_B from equation 1,2,4 put in Equation 3.

x=L_S/2 

Case-1: When curve is symmetrical (n₁=n₂)

• Vertical line passing through VPI will bisect length of curve into two equal half.
• This line always pass through summit point of curve.
• In this case R.L of starting point & end point will be same.

Case-2: When curve is unsymmetrical (n_1≠n₂)

• Vertical line passing through VPI will bisect length of curve into two equal half.
• This line will never pass through summit point of curve.
• In this case the curve will be tilted & R.L of starting point & end point will be different.

5.Calculating Ordinates of Summit Curve

RL of a point at a distance x

RL_x = (RL of A) + n₁x – y

RL_x = (RL)_A + n₁x – (Nx²/2L )

RL_B = (RL)_A + n₁L – (NL²/2L )

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  Length Of Summit Curve  

Case-1: When L_S> SD (Sight Distance)

H= height of drivers eye

h= height of obstruction

s= sight distance

\(L_S=\frac{N\cdot s^{2}}{2(\sqrt{H}+\sqrt{h})^{2}}\).

As per IRC

for SSD, H= 1.2m, h= 0.15m.

\(L_S=\frac{N\cdot s^{2}}{4.4}\)

for OSD/ISD, H=1.2m, h=1.2m.

\(L_S=\frac{N\cdot s^{2}}{9.6}\)

Case-2: When L_S< SD (Sight Distance)

\(L_S=2s-\frac{2(\sqrt{H}+\sqrt{h})^{2}}{N}\).

As per IRC

for SSD, H= 1.2m, h= 0.15m.

\(L_S=2s-\frac{4.4}{N}\)

for OSD/ISD, H=1.2m, h=1.2m.

\(L_S=2s-\frac{9.6}{N}\)

Note:

For calculation of sight distance in design of summit curve & valley curve, effect of gradient should not be taken.

hence

\(SSD= 0.2Vtr + \frac{V^{2}}{254f}\)

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Valley Curve

• Valley curve are vertical curve with concavity upward & convexity downward.
• It can formed by two gradient in following ways.

Some Important Points of Valley Curve

1.Convexity downward & Concavity upward.

2.Vertical Point Of Intersection (VPI) always lies below the curve.

3.Valley curves are designed taking following consideration into account

1. Comfort to passengers
2. Headlight sight distance
3. Drainage in valley curve

Note:

Maximum Distance visible through Head light of vehicle is called Head Light Sight Distance.

Generally HSD = SSD

As per IRC

• height of head light =h=0.75m
• Beam angle = β=1°

4.Valley Curves is usually made up of two transition curve of equal length without having a circular curve in between.

5.In valley curve, centrifugal force and gravity force both acts downward, hence create discomfort for passengers.

6.Position of lowest point of valley curve from starting of curve.

\(x=L_V(\frac{n_1}{2N})^{\frac{1}{2}}\)

L_V= length of valley Curve

N= Change in Gradient

n₁= 1_st gradient.

  Length Of Valley Curve  

Length of valley curve is designed on the basis of max of following two criteria.

Criteria 1: As per Comfort Condition 

In this criteria the rate of change of centrifugal acceleration is limited to a comfortable zone of about 0.6m/sec³, then the length of transition curve is given by

\(L_T=\frac{v^{3}}{CR}\)……(1)

and \(R=\frac{L_T}{N}\)………(2)

the value of R from equation 2 put in equation 1.

\(L_T=\sqrt{\frac{Nv^{3}}{C}}\)

and L_V = 2L_T

\(L_V=2\sqrt{\frac{Nv^{3}}{C}}\)

As per IRC

c= 0.6m/s³ for cubic parabola

v= change in speed

N = change in gradient

unit change:

\(L_V=0.378\sqrt{Nv^{3}}\)

v in kmph

L_V in m

Criteria 2: As per HSD 

Case-1: When L_V> HSD 

s= HSD of horizontal

β= Beam angle

h= height of headlight

\(L_V=\frac{N\cdot s^{2}}{2h+2\cdot s\cdot tan\beta }\).

As per IRC

for β=1°, h= 0.75m.

\(L_V=\frac{N\cdot s^{2}}{1.5+0.035\cdot s }\).

Case-2: When L_V< HSD 

\(L_V=2s -\frac{2h+2s\cdot tan\beta }{N}\).

As per IRC

for β=1°, h= 0.75m.

\(L_V=2s -\frac{1.5+0.035\cdot s}{N}\).