Traffic engineering

1. Design “Sag Curve”.
Where ;
Y = the lowest point of the curve
C = height of PVC from Datum where x = 0.
From the equation of parabolics, differentiation of ....
𝑌 = 𝑎𝑥!
+ 𝑏𝑥 + 𝑐
!"
!"
= 2𝑎𝑥 + 𝑏  curve gradient
therefore, at PVC, x = 0
!"
!"
= 𝑏 and G1 = b ( in percentage) = initial gradient
Rate of slope changes
From the curve gradient, we do the 2nd differentiation to
get the rate of slope changes.
!"
!"
= 2𝑎𝑥 + 𝑏
PVC PVT
PVI
C
‐G1
+G2
Y = ax2 + bx+ c
cdifferentiation
Parabolics Equation
Y
Datum
T T

2.
!"
!"
= 2𝑎, where the rate of slope changes from geometric design
is
!!!!!
!
Therefore, 2𝑎 =
!!!!!
!
,
a =
!!!!!
!!
Example
A 500-meter equal-tangent sag vertical curve has the PVC at station
100+00 with an elevation of 1000 m. The initial grade is -4% and the final
grade is +2%.
Determine the stationing and elevation of the PVI, the PVT, and the
lowest point on the curve
Solution
Where T(tangent) = 500 m
PVC PVT
PVI
1000 m
‐4% +2%
Y = ax2 + bx+ c
cdifferentiation
Parabolics Equation
100+00
Y
x

3.
a. Determine the stationing at PVI and PVT
We know that, the curve length, L for vertical curve design is ...
L = 2T,
So that, PVI = PVC + tangent
= (100+00) + 500 m
= (100+00) + (5+00) = 105+00 stn.
PVT = PVC + L
= (100+00) + 1000 m
= (100+00) + (10+00) = 110+00 stn
b. Determinethe elevation of PVI and PVT from datum
Heigth of PVI = 1000 −
!!
!"#
𝑥 5𝑠𝑡𝑛 = 980 m
PVC
PVI
T = 500m
‐4%
1000 m
???
Datum

4.
Heigth of PVT = 980 +
!!
!"#
𝑥 5𝑠𝑡𝑛 = 990 m
c. The lowest point on the curve
From the parabolics equation, 𝑌 = 𝑎𝑥!
+ 𝑏𝑥 + 𝑐
Find, a = ?
b = ?
c = Elevation of PVC = 1000 m
b = G1 = ‐4%
= −4
𝑠𝑡𝑛
a =
!!!!!
!!
=
!!(!!)
!(!")!"#
where, L = 2T = 2(500)
1000m = 10 stn
= 6
20𝑠𝑡𝑛 = 0.3/stn
From the equation of gradient,
!"
!"
= 2𝑎𝑥 + 𝑏
When
!"
!"
= 0, 2ax + b = 0
2(0.3/stn)x + (‐4) = 0
PVT
PVI
T = 500m
+2%
980m
???
Datum

5.
2(0.3/stn)x + (‐4) = 0
!.!!
!"#
− 4 = 0
0.6x = 4 stn
x = 6.67 stn = 6.7 stn = 670 m = 6+70 stn
So, the distance of the lowest point of curve is...
Dist point from PVC = PVC + x
= (100+00) + (6+70)
= 106+70 stn
The lowest point of curve,
Y = ax2 + bx + c
= 0.3
𝑠𝑡𝑛 6.7!
𝑠𝑡𝑛 + −4/𝑠𝑡𝑛 6.7𝑠𝑡𝑛 + 1000m
= 986.6 m