1
Geometric Design &
Vertical Alignment
Transportation Engineering - I
Geometric Design
Outline
1. Concepts
2. Vertical Alignment
a.
b.
c.
d.
Fundamentals
Crest Vertical Curves
Sag Vertical Curves
Examples
3. Horizontal Alignment
a. Fundamentals
b. Super elevation
4. Other Non-Testable Stuff
Concepts
• Alignment is a 3D problem broken
down into two 2D problems
– Horizontal Alignment (plan view)
– Vertical Alignment (profile view)
• Stationing
– Along horizontal alignment
– 12+00 = 1,200 ft.
Stationing
Horizontal Alignment
Vertical Alignment
From Perteet Engineering
Vertical
Alignment
Vertical Alignment
• Objective:
– Determine elevation to ensure
• Proper drainage
• Acceptable level of safety
• Primary challenge
– Transition between two grades
– Vertical curves
Sag Vertical Curve
G1
G2
G1
Crest Vertical Curve
G2
Vertical Curve Fundamentals
• Parabolic function
– Constant rate of change of slope
– Implies equal curve tangents
y  ax  bx  c
2
• y is the roadway elevation x stations
(or feet) from the beginning of the
curve
Vertical Curve Fundamentals
Choose Either:
• G1, G2 in decimal form, L in feet
G1
PVC
• G1, G2 in percent, L in stations
PVI
δ
G2
PVT
L/2
L
x
y  ax2  bx  c
Where: G1: Initial roadway grade( initial tangent grade)
G2: Final roadway grade
A: Absolute value of the difference in grades
L: Length of vertical curve measured in a horizontal plane
PVC: Initial point of the vertical curve
PVI: Point of vertical intersection ( intersection of initial and final grades)
PVT:
Final point of the vertical curve
• Vertical curves are almost arranged
such that half of the curve length is
positioned before the PVI and half
after and are referred as equal
tangent vertical curves.
• A circular curve is used to connect
the horizontal straight stretches of
road, a parabolic curve is usually
used to connect gradients in the
profile alignment.
• It provides a constant rate of change of
slope and implies equal curve lengths.
+
+
-
+
Level
-
+
CREST VERTICAL CURVES
-
-
+
-
+
Level
SAG VERTICAL CURVES
+
Vertical Curve
• For a vertical curve, the general form
of the parabolic equation is;
Y = ax2 + bx + c
1
where, ‘y’ is the roadway elevation of
the curve at a point ‘x’ along the
curve from the beginning of the
vertical curve (PVC).
‘C’ is the elevation of the PVC since
x=0 corresponds the PVC
dy
b
dx
Slope of Curve
• To define ‘a’ and ‘b’, first derivative of
equation 1 gives the slope.
dy
 2ax  b
dx
• At PVC, x=0;
dy
 b
dx
2
or
dy
G 
dx
3
G1  b
Where G1 is the initial slope.
• Taking second derivative of
equation1, i.e. rate of change of
slope;
dy 2
 2a
2
dx
4
• The rate of change of slope can also
be written as;
dy2 G2  G1

2
dx
L
5
2a 
G2  G1
L
• Equating equations 4 and 5
G2  G1
2a 
L
6
• or
G 2  G1
a
2L
7
Fundamentals of Vertical
Curves
• For vertical curve design and
construction, offsets which are
vertical distances from initial tangent
to the curve are important for vertical
curve design.
PVI
PVC
PVC
PVT
PVT
PVI
PVC
PVC
PVT
PVI
PVC
PVT
PVT
• A vertical curve also simplifies the
computation of the high and low
points or crest and sag vertical
curves respectively, since high or low
point does not occur at the curve
ends PVC or PVT.
• Let ‘Y’ is the offset at any distance ‘x’
from PVC.
• Ym is the mid curve offset & Yt is the offset
at the end of the vertical curve.
• From an equal tangent parabola, it can be
written as;
A
2
y
x
200 L
where ‘y’ is the offset in feet and
‘A’ is
8
the absolute value of the difference in
grades(G2-G1, in %), ‘L’ is length of
vertical curve in feet and ‘x’ is distance
from the PVC in feet.
Putting the value of x=L in eq. 8
A L 2
ym 
( )
200 L 2
AL
ym 
800
A
yf 
* L2
200 L
AL
yf 
200
• First derivative can be used to
determine the location of the low point,
the alternative to this is to use a k-value
which is defined as
L
k
A
where ‘L’ is in feet and ‘A’ is in %.
• This value ‘k’ can be used directly to
compute the high / low points for
crest/ sag vertical curves by
x=kG1
where ‘x’ is the distance from the
PVC to the high/ low point. ‘k’ can
also be defined as the horizontal
distance in feet required to affect a
1% change in the slope.
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
•
•
•
•
•
Solution : The curve length is stated to be 500 meters. Therefore, the PVT is at
station 105+00 (100+00 + 5+00) and the PVI is in the very middle at 102+50,
since it is an equal tangent curve. For the parabolic formulation, equals the
elevation at the PVC, which is stated as 1000 m. The value of b equals the initial
grade, which in decimal is -0.04. The value of a can then be found as 0.00006.
Basic Equation of the parabola: y = ax2+bx+c
At x= 0 y= C=1000 b= G1= -0.04 , a = G2-G1/2L =(0.02-0.04))/2x500=0.00006
Using the general parabolic formula, the elevation of the PVT can be found:
y = 0.00006x2+ (-0.04x)+c = 0.00006(500)+(-0.04*500)+1000= 995m
Since the PVI is the intersect of the two tangents, the slope of either tangent and
the elevation of the PVC or PVT, depending, can be used as reference. The
elevation of the PVI can then be found as : y = -0.04(250)+1000= 990 m
To find the lowest part of the curve, the first derivative of the parabolic formula
can be found. The lowest point has a slope of zero, and thus the low point
location can be found: dy/dx = 0.00012x-0.04 = 0 x = 0.04/0.00012=
333.333 m
Using the parabolic formula, the elevation can be computed for that location. It
turns out to be at an elevation of 993.33 m, which is the lowest point along the
curve.
Sight Distances
• Sight Distance is a length of road surface which a
particular driver can see with an acceptable level
of clarity. Sight distance plays an important role in
geometric highway design because it establishes an
acceptable design speed, based on a driver's ability
to visually identify and stop for a particular,
unforeseen roadway hazard or pass a slower vehicle
without being in conflict with opposing traffic.
• As velocities on a roadway are increased, the
design must be catered to allowing additional viewing
distances to allow for adequate time to stop. The two
types of sight distance are:
• (1) stopping sight distance and (2) passing sight
distance.
Stopping Sight Distance
At every point on the roadway, the minimum sight distance provided
should be sufficient to enable a vehicle traveling at the design speed
to stop before reaching a stationary object in its path. Stopping sight
distance is the aggregate of two distances:
•brake reaction distance and braking distance.
•Brake reaction time is the interval between the instant that the
driver recognizes the existence of an object or hazard ahead and
the instant that the brakes are actually applied. Extensive studies
have been conducted to ascertain brake reaction time. Minimum
reaction times can be as little as 1.64 seconds: 0.64 for alerted
drivers plus 1 second for the unexpected signal.
•Some drivers may take over 3.5 seconds to respond under similar
circumstances. For approximately 90% of drivers, including older
drivers, a reaction time of 2.5 see is considered adequate. This
value is therefore used in Table on next page
Vehicle Stopping Distance
• Vehicle stopping distance is calculated by
2
the following formula
v1
d
where
V1
f
G
2 g ( f  G)
initial speed of vehicle
friction
percent grade
Distance Traveled During
Perception/ Reaction Time
• It is calculated by the following
formula
dr = V1* tr
where V1 Initial Velocity of vehicle
tr time required to perceive and
react to the need to stop
• Hence formula for the Stopping sight
distance will be;
2
V1
SSD 
 V1t r
2 g ( f  G)
SSD and Crest Vertical
Curve
• In providing the sufficient SSD on a vertical curve,
the length of curve ‘L’ is the critical concern.
• Longer lengths of curve provide more SSD, all
else being equal, but are most costly to construct.
• Shorter curve lengths are relatively inexpensive to
construct but may not provide adequate SSD.
• In developing such an expression, crest and sag
vertical curves are considered separately.
• For the crest vertical curve case, consider the
diagram.
SSD and Crest Vertical Curve
S
H1
PVI
PVT
PVC
H2
L
S = Sight distance (ft)
H1= height of driver’s eye above road-way surface (ft)
L = length of the curve (ft), H2= height of roadway object (ft) ,
A = difference in grade
Lm= Minimum length required for sight distance.
Minimum Length of the
Curve
• For a required sight distance S is
calculated as follows;
• If the sight distance is found to be
less than the curve length (S>L)
200( H1 
Lm  2S 
A
H 2 )2
• for sight distances that are greater
than the curve length (S<L)
AS 2
Lm 
200( H1  H 2 ) 2
• For the sight distance required to
provide adequate SSD, standard
define driver eye height H1 is 3.5 ft
and object height H2 is 0.5 ft. S is
assumed is equal to SSD. We get
SSD > L
SSD < L
1329
Lm  2 SSD 
A
ASSD2
Lm 
1329
• Working with the above equations
can be cumbersome.
• To simplify matters on crest curves
computations, K- values, are used.
L = K*A
where k is the horizontal distance in
feet, required to affect 1 percent
change in slope.
SSD and Sag Vertical Curve
• Sag vertical curve design differs from crest
vertical curve design in the sense that
sight distance is governed by night time
conditions, because in daylight, sight
distance on a sag vertical curve is
unrestricted.
• The critical concern for sag vertical curve
is the headlight sight distance which is a
function of the height of the head light
above the road way, H, and the inclined
upward angle of the head light beam,
relative to the horizontal plane of the car,
β.
• The sag vertical curve sight distance
problem is illustrated in the following
figure.
S
H
β
PVT
PVC
PVI
L
• By using the properties of parabola
for an equal tangent curve, it can be
shown that minimum length of the
curve, Lm for a required sight
distance is ;
• For S>L
200 ( H  S tan  )
Lm  2S 
• For S<L
A
AS 2
Lm 
200( H  S tan  )
• For the sight distance required to
provide adequate SSD, use a head
light height of 2.0 ft and an upward
angle of 1 degree.
• Substituting these design standards
and S = SSD in the above equations;
• For SSD>L
400  3.5SSD
Lm  2 SSD 
• For SSD<L
ASSD2
Lm 
400  3.5SSD
A
• As was the case for crest vertical
curves, K-values can also be
computed for sag vertical curves.
• Caution should be exercised in using
the k-values in this table since the
assumption of G=0 percent is used
for SSD computations.
Engineering vs. Politics
• A current roadway has a design speed of 100 km/hr, a coefficient
of friction of 0.1, and carries drivers with perception-reaction
times of 2.5 seconds. The drivers use cars that allows their eyes
to be 1 meter above the road. Because of ample road kill in the
area, the road has been designed for car cases that are 0.5
meters in height. All curves along that road have been designed
accordingly.
• The local government, seeing the potential of tourism in the area
and the boost to the local economy, wants to increase the speed
limit to 110 km/hr to attract summer drivers. Residents along the
route claim that this is a horrible idea, as a particular curve called
"Dead Man's Hill" would earn its name because of sight distance
problems. "Dead Man's Hill" is a crest curve that is roughly 600
meters in length. It starts with a grade of +1.0% and ends with (1.0)%. There has never been an accident on "Dead Man's Hill"
as of yet, but residents truly believe one will come about in the
near future.
• A local politician who knows little to nothing about
engineering (but thinks he does) states that the 600-meter
length is a long distance and more than sufficient to handle
the transition of eager big-city drivers. Still, the residents
push back, saying that 600 meters is not nearly the distance
required for the speed. The politician begins a lengthy
campaign to "Bring Tourism to Town", saying that the
residents are trying to stop "progress". As an engineer,
determine if these residents are indeed making a valid point
or if they are simply trying to stop progress?
• Using sight distance formulas from other sections, it is found
that 100 km/hr has an SSD of 465 meters and 110 km/hr
has an SSD of 555 meters,
2
V1
SSD 
 V1t r
2 g ( f  G)
•
Given the criteria stated above. Since both 465 meters and
555 meters are less than the 600-meter curve length, the
correct formula to use would be:
AS 2
Lm 
200( H1  H 2 ) 2
• For flash animation please visit the
following site”
• http://street.umn.edu/flash_curve_cre
st.html
Problem for Pondering
• Problem:
• To help prevent future collisions between cars and trains, an
at-grade crossing of a rail road by a country road is being
redesigned so that the county road will pass underneath the
tracks. Currently the vertical alignment of the county road
consists of an equal tangents crest vertical curve joining a
4% upgrade to a 3% downgrade. The existing vertical curve
is 450 feet long, the PVC of this curve is at station
48+24.00, and the elevation of the PVC is 1591.00 feet. The
centerline of the train tracks is at station 51+50.00. Your job
is to find the shortest vertical curve that provides 20 feet of
clearance between the new county road and the train tracks,
and to make a preliminary estimate of the cut at the PVI that
will be needed to construct the new curve.
• Study the solution and resolve the
question by making the sketches etc