Quiz 1 Equation Sheet
Transportation Models
See INDUSTRIAL ENGINEERING for optimization models and methods, including queueing theory.
DENSITY k (veh/mi)
CAPACITY
CAPACITY
SPEED v (mph)
SPEED v (mph)
VOLUME q (veh/hr)
>
]–’“_
VOLUME q (veh/hr)
DENSITY k (veh/mi)
Vertical Curves: Sight Distance Related to Curve Length
S ≤ L
Crest Vertical Curve
AS2
L =
General equation:
100( 2h1 + 2h2 )
Standard Criteria:
h 1 = 3.50 ft and h2 = 2.0 ft:
2
Sag Vertical Curve
L =
AS2
400 + 3.5 S
(based on riding comfort)
L =
(based on adequate sight distance
under an overhead structure to see an
object beyond a sag vertical curve)
AS2
h +h
800 C − 1 2
2
(
h1 + h2
)
2
2,158
A
L = 2S −
L =
(
A
L = 2S −
Sag Vertical Curve
Sag Vertical Curve
200
L = 2S −
AS2
2,158
L =
(based on standard headlight
criteria)
S > L
(
400 + 3.5 S
A
)
AV 2
46.5
(
h +h
800
C− 1 2
A
2
L = 2S −
)
)
C = vertical clearance for overhead structure (overpass) located within 200
feet of the midpoint of the curve
Horizontal Curves
Side friction factor (based on superelevation)
Spiral Transition Length
0.01e + f =
Ls =
V2
15 R
3.15V 3
RC
C = rate of increase of lateral acceleration
[use 1 ft/sec3 unless otherwise stated]
Sight Distance (to see around obstruction)
HSO = R
[
1 − cos
(
28.65 S
R
)]
HSO = Horizontal sight line offset
CIVIL ENGINEERING
163
Horizontal Curve Formulas
D = Degree of Curve, Arc ‚
PC = Point of Curve (also called BC)
PT = Point of Tangent (also called EC)
PI = Point of Intersection
I
= Intersection Angle (also called Δ)
Angle Between Two Tangents
L
= Length of Curve, from PC to PT
T
= Tangent Distance
E
= External Distance
R
= Radius
LC = Length of Long Chord
M = Length of Middle Ordinate
c
= Length of Sub-Chord
d
= Angle of Sub-Chord
l
= Curve Length for Sub-Chord
R = 5729.58
D
R=
LC
2 sin _ I 2i
T = R tan _ I 2i =
LC
2 cos _ I 2i
LATITUDES AND DEPARTURES
L = RI r = I 100
D
180
+ Latitude
M = R 81 - cos _ I 2iB
– Departure
+ Departure
R
cos _ I 2i
E+R=
R - M = cos _ I 2i
R
c = 2R sin _ d 2i
l = Rd b r l
180
E = R=
1
- 1G
cos _ I 2i
‚€_””^
D 2
164
CIVIL ENGINEERING
– Latitude
Vertical Curve Formulas
TANGENT
OFFSET
L
PVI
x
BACK
TANGENT
FORWARD
TANGENT
E
y
PVT
g
g
2
PVC
1
YPVC
DATUM
VERTICAL CURVE FORMULAS
NOT TO SCALE
L
g2 = Grade of Forward Tangent
= Length of Curve (horizontal)
PVC = Point of Vertical Curvature
a = Parabola Constant
PVI = Point of Vertical Intersection
y = Tangent Offset
PVT = Point of Vertical Tangency
E = Tangent Offset at PVI
g1
= Grade of Back Tangent
r = Rate of Change of Grade
x
= Horizontal Distance from PVC
to Point on Curve
xm = Horizontal Distance to Min/Max Elevation on Curve = -
g1
g1L
2a = g1 - g2
Tangent Elevation = YPVC + g1x and = YPVI + g2 (x – L/2)
Curve Elevation = YPVC + g1x + ax2 = YPVC + g1x + [(g2 – g1)/(2L)]x2
y = ax 2
a=
g2 - g1
2L
E = ab Ll
2
2
r=
g2 - g1
L
EARTHWORK FORMULAS
Average End Area Formula, V = L(A1 + A2)/2
Prismoidal Formula, V = L (A1 + 4Am + A2)/6,
where Am
L
= area of mid-section, and
= distance between A1 and A2
Pyramid or Cone, V = h (Area of Base)/3
AREA FORMULAS
Area by Coordinates: Area = [XA (YB – YN) + XB (YC – YA) + XC (YD – YB) + ... + XN (YA – YN – 1)] / 2
Trapezoidal Rule: Area = w c
h1 + hn
+ h2 + h3 + h4 + f + hn - 1m
2
Simpson’s 1/3 Rule: Area = = w > h1 + 2 e
n-2
! hk o + 4 e
k = 3, 5, f
n-1
! hk o
k = 2, 4, f
+ hn H 3
w = common interval
n must be odd number of measurements
w = common interval
CIVIL ENGINEERING
165
Vertical Curve Offsets
Y=
A
x2
200 L
A = ∣ G1 – G2∣ *A is in percent form.
K=
L
A
x hl = K × G1
Parabolic Equations
y = ax2 + bx +c
Where y = roadway elevation at
distance x from the PVC.
G − G1
a= 2
; b = G1; c = ELEVPVC
2L
*keep in mind that you must use
either station/% or ft/decimal for
x/Gi.
1. Compatible with ITE
No Land Use Code?
Selection of ITE
Rates/Equations,
or Collection of
Local Data
Collect Local Data
Yes
2. Size within Data
No Extremes?
Yes
3. Number of Data
1 or 2 Points?
3-5
If number of data points
between 3 and 5, analysts
are encouraged to collect
local data, but can
proceed to Step 4.
3-5
6+
Source: ITE Trip Generation
Handbook, 2nd Edition
4. Regression Equation?
A
No Yes
B
16
Collect Local Data
Selection of ITE Rates/
Equations, or Collection of Local
Data (cont.)
5. Standard Deviation
No 110 percent?
A
Yes
No
6. Data Cluster Okay?
Yes
Figure 3.1
Use Weighted
Average Rate
Source: ITE Trip Generation
Handbook, 2nd Edition
1
17
Selection of ITE Rates/
Equations, or Collection
of Local Data (cont.)
Figure 3.1
B
Use Regression
Equation
Yes
7. 20 or More Data
Points?
Source: ITE Trip Generation
Handbook, 2nd Edition
No
8a. R2 0.75? And
Within Cluster?
8B. Std Dev 110%?
And Within Cluster?
If 8A is yes & 8B is yes
Choose Line at Cluster
If 8A is yes & 8B is no
Use Regression Equation
If 8A is no & 8B is yes
Use Weighted Average Rate
If 8A is no & 8B is no
Collect Local Data
18
2