Transcript Surveying-II

Surveying-II
SURVEYING-II
Surveying-II
Horizontal Alignment
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Horizontal Alignment
• Along circular path, vehicle undergoes centripetal
acceleration towards center of curvature (lateral
acceleration).
• Balanced by superelevation and weight of vehicle (friction
between tire and roadway).
• Design based on appropriate relationship between design
speed and curvature and their relationship with side
friction and super elevation.
Vehicle Cornering
Fcp
Rv
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Fcn
Fc

e
W
Ff

Wn
1 ft
Ff
Wp
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• Figure illustrates the forces acting on a vehicle
during cornering. In this figure,  is the angle of
inclination, W is the weight of the vehicle in pounds
with Wn and Wp being the weight normal and
parallel to the roadway surface respectively.
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• Ff is the side frictional force, Fc is the centrifugal force
with Fcp being the centrifugal force acting parallel to the
roadway surface, Fcn is the centrifugal force acting
normal to the roadway surface, and Rv is the radius
defined to the vehicle’s traveled path in ft.
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• Some basic horizontal curve relationships can be derived
by summing forces parallel to the roadway surface.
Wp + Ff = Fcp
• From basic physics this equation can be written as
WV 2
WV 2
W sin   fs[W cos  
sin  ] 
cos 
gRv
gRv
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• Where fs is the coefficient of side friction, g is the
gravitational constant and V is the vehicle speed (in ft per
second).
• Dividing both the sides of the equation by W cos ;
sin 
cos  WV 2 sin 
WV 2 cos 
W
 fs[W

]
cos 
cos 
gRv cos 
gRv cos 
tan   f s
V2

(1  f s tan  )
gRv
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• The term tan is referred to as the super elevation of
the curve and is denoted by ‘e’.
• Super elevation is tilting the roadway to help offset
centripetal forces developed as the vehicle goes
around a curve.
• The term ‘fs’ is conservatively set equal to zero for
practical applications due to small values that ‘fs’
and ‘’ typically assume.
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• With e = tan, equation can be rearranged as
2
V
Rv 
g ( f s  e)
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• In actual design of a horizontal curve, the engineer
must select appropriate values of e and fs.
• Super-elevation value ‘e’ is critical since
• high rates of super-elevation can cause vehicle
steering problems at exits on horizontal curves
• and in cold climates, ice on road ways can reduce
fs and vehicles are forced inwardly off the curve
by gravitational forces.
• Values of ‘e’ and ‘fs’can be obtained from AASHTO
standards.
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Horizontal Curve Fundamentals
• For connecting straight tangent sections of roadway
with a curve, several options are available.
• The most obvious is the simple curve, which is just
a standard curve with a single, constant radius.
• Other options include;
• compound curve, which consists of two or
more simple curves in succession ,
• and spiral curves which are continuously
changing radius curves.
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Basic Geometry
Tangent
Horizontal
Curve
Tangent
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Tangent Vs. Horizontal Curve
• Predicting speeds for tangent and horizontal segments is
different
• May actually be easier to predict speeds on curves than
tangents
• Speeds on curves are restricted to a few well defined
variables (e.g. radius, superelevation)
• Speeds on tangents are not as restricted by design
variables (e.g. driver attitude)
Elements of Horizontal Curves
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PI

E


M
L
PC
PT
R
R




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• Figure shows the basic elements of a simple
horizontal curve. In this figure
• R is the radius (measured to center line of the
road)
• PC is the beginning point of horizontal curve
• T is tangent length
• PI is tangent intersection
•  is the central angle of the curve
• PT is end point of curve
• M is the middle ordinate
• E is the external distance
• L is the length of the curve
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Degree of Curve
• It is the angle subtended by a 100-ft arc along the
horizontal curve.
• Is a measure of the sharpness of curve and is frequently
used instead of the radius in the actual construction of
horizontal curve.
• The degree of curve is directly related to the radius of the
horizontal curve by
5729.6
D
R
PI

E
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

M
L
PC
PT
R
R




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• A geometric and trigonometric

analysis of figure, reveals the T  R tan
2
following relationships




1
E  R
 1
 cos(  )



2


 

M  R1  cos( ) 
2 

100
L
D
Stopping Sight Distance and Horizontal Curve
Design
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SSD
Ms
Sight
Obstruction
Highway
Centerline
Rv
Critical
inside lane
s
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• Adequate stopping sight distance must also be
provided in the design of horizontal curves.
• Sight distance restrictions on horizontal curves
occur when obstructions are present.
• Such obstructions are frequently encountered in
highway design due to the cost of right of way
acquisition and/or cost of moving earthen materials.
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• When such an obstruction exists, the stopping sight
distance is measured along the horizontal curve
from the center of the traveled lane.
• For a specified stopping sight distance, some
distance, Ms, must be visually cleared, so that the
line of sight is such that sufficient stopping sight
distance is available.
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• Equations for computing SSD relationships for horizontal
curves can be derived by first determining the central
angle, s, for an arc equal to the required stopping sight
distance.
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• Assuming that the length of the horizontal curve exceeds the
required SSD, we have
100 s
SSD 
D
Combining the above equation with following
we get;
5729.6
D
R
57.296 SSD
s 
Rv
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• Rv is the radius to the vehicle’s traveled path, which is
also assumed to be the location of the driver’s eye for
sight distance, and is again taken as the radius to the
middle of the innermost lane,
• and s is the angle subtended by an arc equal to SSD in
length.
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• By substituting equation for s in equation of middle
ordinate, we get the following equation for middle
ordinate;

 28.65SSD  
 
Ms  Rv 1  cos

R
v




• Where Ms is the middle ordinate necessary to provide
adequate stopping sight distance. Solving further we get;
Rv 
1  Rv  M s
SSD 
cos 
28.65 
Rv




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Max e
• Controlled by 4 factors:
• Climate conditions (amount of ice and snow)
• Terrain (flat, rolling, mountainous)
• Frequency of slow moving vehicles which influenced
by high superelevation rates
• Highest in common use = 10%, 12% with no ice and
snow on low volume gravel-surfaced roads
• 8% is logical maximum to minimized slipping by
stopped vehicles
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Radius Calculation (Example)
Design radius example: assume a maximum e of 8% and
design speed of 60 mph, what is the minimum
radius?
fmax = 0.12 (from Green Book)
Rmin = _____602__________
15(0.08 + 0.12)
Rmin = 1200 feet
Radius Calculation (Example)
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For emax = 4%?
Rmin = _____602_________
15(0.04 + 0.12)
Rmin = 1,500 feet
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Sight Distance Example
A horizontal curve with R = 800 ft is part of a 2-lane
highway with a posted speed limit of 35 mph. What is the
minimum distance that a large billboard can be placed
from the centerline of the inside lane of the curve without
reducing required SSD? Assume p/r =2.5 and a = 11.2
ft/sec2
SSD = 1.47vt + _________v2____ = 246 ft
30( a___  G)
32.2
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Horizontal Curve Example
• Deflection angle of a 4º curve is 55º25’, PC at station 238 +
44.75. Find length of curve,T, and station of PC.
• D = 4º
•  = 55º25’ = 55.417º
• D = _5729.58_ R = _5729.58_ = 1,432.4 ft
R
4
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Horizontal Curve Example
•
•
•
•
D = 4º
 = 55.417º
R = 1,432.4 ft
L = 2R = 2(1,432.4 ft)(55.417º) = 1385.42ft
360
360
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Horizontal Curve Example
•
•
•
•
D = 4º
 = 55.417º
R = 1,432.4 ft
L = 1385.42 ft
• T = R tan  = 1,432.4 ft tan (55.417) = 752.29 ft
2
2
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Stationing Example
Stationing goes around horizontal curve.
For previous example, what is station of PT?
PC = 238 + 44.75
L = 1385.42 ft = 13 + 85.42
Station at PT = (238 + 44.75) + (13 + 85.42) =
252 + 30.17
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Suggested Steps on Horizontal Design
1. Select tangents, PIs, and general curves make sure
you meet minimum radii
2. Select specific curve radii/spiral and calculate
important points (see lab) using formula or table
(those needed for design, plans, and lab
requirements)
3. Station alignment (as curves are encountered)
4. Determine super and runoff for curves and put in
table (see next lecture for def.)
5. Add information to plans
Geometric Design – Horizontal
Alignment (1)
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• Horizontal curve
– Plan view, profile, staking, stationing
– type of horizontal curves
– Characteristics of simple circular curve
• Stopping sight distance on horizontal
curves
• Spiral curve
+0
24
0
23+00
22+00
21+00
20+00
19+00
18+00
0
17+0
+0
16
0
+0
15
0
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Plan view and profile
plan
700
700
600
500
profile
400
300
200
15+00
16+00
17+00
18+00
19+00
20+00
21+00
22+00
23+00 24+00
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Surveying and Stationing
• Staking: route surveyors define the geometry of a
highway by “staking” out the horizontal and vertical
position of the route and by marking of the crosssection at intervals of 100 ft.
• Station: Start from an origin by stationing 0, regular
stations are established every 100 ft., and numbered
0+00, 12 + 00 (=1200 ft), 20 + 45 (2000 ft + 45) etc.
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Horizontal Curve Types
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Curve Types
1. Simple curves with spirals
2. Broken Back – two curves same direction (avoid)
3. Compound curves:
multiple curves connected
directly together (use with caution) go from large radii
to smaller radii and have R(large) < 1.5 R(small)
4. Reverse curves – two curves, opposite direction
(require separation typically for superelevation
attainment)
1. Simple Curve
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Circular
arc

R

Straight road
sections
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2. Compound Curve
Circular arcs
R1
Straight road
sections
R2
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3. Broken Back Curve
Circular arc
Straight road
sections
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4. Reverse Curve
Circular arcs
Straight road
sections
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5. Spiral
R = Rn
R=
Straight road
section
Angle measurement
90
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60
30
180
(a) degree
0
(b) Radian
1   / 180radians  0.0174532
1radian  (180 /  )  57.2957
radians
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As the subtended arc is proportional to the radius of the
circle, then the radian measure of the angle.
Is the ratio of the length of the subtended arc to the radius of
the circle
radian
measure  R  arc length
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•
 •
•
•
•
•
•
•
•
•
•
•
•
Define horizontal Curve:
Circular Horizontal Curve Definitions
Radius, usually measured to the centerline of the road, in ft.
= Central angle of the curve in degrees
PC = point of curve (the beginning point of the horizontal curve)
PI = point of tangent intersection
PT = Point of tangent (the ending point of the horizontal curve)
T = tangent length in ft.
M = middle ordinate from middle point of cord to middle point of
curve in ft.
E = External distance in ft.
L = length of curve
D = Degree of curvature (the angle subtended by a 100-ft arc* along
the horizontal curve)
C = chord length from PC to PT
PI

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T
E
L
PC

2
90 
M
C

2

2
90 
R
R

2


2
PT

2
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Key measures of the curve
M  R[1  cos(  / 2)]
100(180 /  ) 18000 5729.57
D


R
R
R
or ,
100
D
 57.2957
R

T  R tan
2
1
E  R[(
)  1]
cos(  / 2)
L
180
R

C  2 R sin
2
180 / 
Note

converts from radians to degrees
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Example:
A horizontal curve is designed with a 2000-ft radius, the
curve has a tangent length of 400 ft and the PI is at station
103 + 00, determine the stationing of the PT.
Solution:
T  R tan

2
400  2000 tan

2
  22.62
L

180
R  789.58 ft
PC  (103  00)  (4  00)  99  00
PT  PC  L  (99  00)  (7  89.58)  106  89.58
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Spiral Curve:
Spiral curves are curves with a
continuously changing radii,
they are sometimes used on
high-speed roadways with
sharp horizontal curves and
are sometimes used to
gradually introduce the super
elevation of an upcoming
horizontal curve
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Length of Spiral
o AASHTO also provides recommended spiral length
based on driver behavior rather than a specific
equation.
o Superelevation runoff length is set equal to the
spiral curve length when spirals are used.
o Design Note: For construction purposes, round your
designs to a reasonable values; e.g. round
o Ls = 147 feet, to use Ls = 150 feet.
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Thanks