Civil Engineering
Transportation
Queueing models are found in the Industrial Engineering section.
Traffic Signal Timing
v
y = t + 2a ! 64.4 G
W+l
r= v
L
Gp = 3.2 + S + 0.27Nped
p
where
t = driver reaction time (sec)
v = vehicle approach speed (ft/sec)
W = width of intersection, curb-to-curb (ft)
l = length of vehicle (ft)
y = length of yellow interval to nearest 0.1 sec (sec)
r = length of red clearance interval to nearest 0.1 sec (sec)
Gp = minimum green time for pedestrians (sec)
L = crosswalk length (ft)
Sp = pedestrian speed (ft/sec), default 3.5 ft/sec
Nped = number of pedestrian in interval
a = deceleration rate (ft/sec2)
± G = percent grade divided by 100 (uphill grade "+")
Stopping Sight Distance
V2
SSD = 1.47Vt +
30 dc 32.2 m ! G n
a
ISD = 1.47 Vmajor tg
where
a = deceleration rate (ft/sec2)
± G = percent grade divided by 100 (uphill grade "+")
SSD = stopping sight distance (ft)
ISD = intersection sight distance (ft)
t = driver reaction time (sec)
tg = time gap for vehicle entering roadway (sec)
V = design speed (mph)
Vmajor = design speed of major road (mph)
Peak Hour Factor
Hourly Volume V
= =
PHF Hourly Flow Rate 4 * V15
where
PHF = peak hour factor
V = hourly volume (veh/hr)
V15 = peak 15-min. volume (veh/15 min)
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Civil Engineering
Vertical Curves
L
PVI
y = ax 2 x A
y E
A = g2 − g1 PVT g
g −g PVC
2
a = 22L 1 FOR
TAN WARD
g GEN
T
L 2
E = ac 2 m
1
CK NT
BA NGE YPVC
TA
g −g
r = 2L 1
L DATUM
K = A
g g1L VERTICAL CURVE FORMULAS
xm =− 2a1 = g −
1 g2
NOT TO SCALE
Compiled from AASHTO, A Policy on Geometric Design of Highways and Streets, 6th ed., 2011.
Tangent elevation = YPVC + g1x = YPVI + g2 (x – L/2)
Curve elevation = YPVC + g1x + ax2 = YPVC + g1x + [(g2 – g1)/(2L)]x2
where
PVC= point of vertical curvature, or beginning of curve
PVI = point of vertical intersection, or vertex
PVT = point of vertical tangency, or end of curve
A = algebraic difference in grades
a = parabola constant
E = tangent offset at PVI
g1 = grade of back tangent
g2 = grade of forward tangent
h1 = height of driver's eyes above the roadway surface (ft)
h2 = height of object above the roadway surface (ft)
K = rate of vertical curvature
L = length of curve
r = rate of change of grade
S = sight distance (ft)
x = horizontal distance from PVC to point on curve
xm = horizontal distance to min/max elevation on curve
y = tangent offset
V = design speed (mph)
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Civil Engineering
Vertical Curves: Sight Distance Related to Curve Length
S ≤ L S > L
Crest Vertical Curve 2
AS
( )
2
L = 200 h1 + h2
General equation: 100( 2h1 + 2h2 ) 2 L = 2S −
A
Standard Criteria:
AS2 2,158
h 1 = 3.50 ft and h2 = 2.0 ft: L = L = 2S −
2,158 A
Sag Vertical Curve
(based on standard headlight
criteria)
L =
AS2
400 + 3.5 S
L = 2S − ( 400 + 3.5 S
A )
Sag Vertical Curve AV 2
L =
(based on riding comfort) 46.5
Sag Vertical Curve AS2
(based on adequate sight distance
under an overhead structure to see an
L =
( h +h
800 C − 1 2 )
L = 2S −
800
A
C− 1 2
2
(
h +h
)
2
object beyond a sag vertical curve)
C = vertical clearance for overhead structure (overpass) located within 200
feet of the midpoint of the curve
Compiled from AASHTO, A Policy on Geometric Design of Highways and Streets, 6th ed., 2011.
Horizontal Curves
PI
I
E
T
T
100.00
M
PC c LC PT
I/2 I/2
D
R d
R
NOT TO SCALE
R = 5729.58
D
R= LC
2 sin _ I 2i
T = R tan _ I 2i = LC
2 cos _ I 2i
L = RI r = I 100
180 D
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Civil Engineering
M = R 81 - cos _ I 2iB
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
where
c = length of sub-chord
d = angle of sub-chord
D = degree of curve, arc definition
e = superelevation (%)
E = external distance
f = side friction factor
I = intersection angle (also called ∆); angle between two tangents
l = curve length for sub-chord
L = length of curve, from PC to PT
LC = length of long chord
M = length of middle ordinate
PC = point of curve (also called BC)
PI = point of intersection
PT = point of tangent (also called EC)
R = radius
S = sight distance (ft)
T = tangent distance
V = design speed (mph)
Horizontal Curves
V2
0.01e + f =
Side friction factor (based on superelevation) 15 R
3.15V 3
Ls =
Spiral Transition Length 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
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Civil Engineering
Basic Freeway Segment Highway Capacity
Speed Flow Relationship for Basic Freeway Segments
FFS Capacity Breakpoint > Breakpoint ≤ Capacity
(mph) (pc/h/ln) (pc/h/ln) (mph)
75 2,400 1,000 = 75 – 0.00001105 (vp – 1,000)2
70 2,400 1,200 = 70 – 0.00001157 (vp – 1,200)2
65 2,350 1,400 = 65 – 0.00001416 (vp – 1,400)2
60 2,300 1,600 = 60 – 0.00001814 (vp – 1,600)2
55 2,250 1,800 = 55 – 0.00002469 (vp – 1,800)2
* All equations are based on Exhibit 12-6 and Equation 12-1 from the HCM 6th Edition assuming
all calibration factors (CAF and SAF) set to 1.0
where
pc/h/ln = passenger cars per hour per lane
Level of Service (LOS) Density (pc/mi/ln)
A ≤11
B >11 – 18
C >18 – 26
D >26 – 35
E >35 – 45
Demand exceeds capacity
F >45
FFS = BFFS – fLW – fRLC – 3.22 TRD0.84
where
FFS = free flow speed of basic freeway segment (mph)
BFFS = base free flow speed of basic freeway segment (mph); default is 75.4 mph
fLW = adjustment for lane width (mph)
fRLC = adjustment for right-side lateral clearance (mph)
TRD = total ramp density (ramps/mi)
Average Lane Width (ft) Reduction in FFS, fLW (mph)
≥12 0.0
≥11 – 12 1.9
≥10 – 11 6.6
Right-Side
Lanes in One Direction
Lateral
Clearance (ft) 2 3 4 ≥5
≥6 0.0 0.0 0.0 0.0
5 0.6 0.4 0.2 0.1
4 1.2 0.8 0.4 0.2
3 1.8 1.2 0.6 0.3
2 2.4 1.6 0.8 0.4
1 3.0 2.0 1.0 0.5
0 3.6 2.4 1.2 0.6
HCM: Highway Capacity Manual, 6th ed., A Guide for Multimodal Mobility Analysis, Transportation
Research Board of the National Academics, Washington, DC, 2016.
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Civil Engineering
V
vp = PHF # N # f
HV
where
vp = demand flowrate under equivalent base conditions (pc/h/ln)
V = demand volume under prevailing conditions (veh/h)
PHF = peak-hour factor
N = number of lanes in analysis direction
fHV = adjustment factor for presence of heavy vehicles in traffic stream, calculated with
1
fHV =
1 + PT _ ET − 1 i
where
fHV = heavy-vehicle adjustment factor
PT = proportion of single unit trucks and tractor trailers in traffic stream
ET = passenger-car equivalent (PCE) of single unit truck or tractor trailer in traffic stream
PCE by Type of Terrain
Vehicle Level Rolling
ET 2.0 3.0
vp
D= S
where
D = density(pc/mi/ln)
vp = demand flow rate (pc/h/ln)
S = mean speed of traffic stream under base conditions (mph)
Traffic Flow Relationships
Greenshields Model
Sf Sf
SPEED (mph)
SPEED (mph)
DO
SO SO
Dj
0 DO Dj 0 Vm
DENSITY (veh/mi/ln) FLOW (veh/h/ln)
Sf SO
S = Sf − D D Vm Sf
FLOW (veh/h/ln)
j
Sf 2
V = Sf D − D D
j
D jS f
Vm = 4 0 DO Dj
Dj DENSITY (veh/mi/ln)
Do = 2 Oversaturated flow
AASHTO, A Policy on Geometric Design of Highways and Streets, 6th ed., 2011. Used by permission.
304
Civil Engineering
where
D = density (veh/mi)
S = speed (mph)
V = flow (veh/hr)
Vm = maximum flow (veh/hr)
Do = optimum density (sometimes called critical density)
Dj = jam density (veh/hr)
So = optimum speed (often called critical speed) (mph)
Sf = theoretical speed selected by the first driver entering a facility (i.e., under zero density and zero
flow rate conditions) (mph)
Gravity Model
A jFijKij
Tij = Pi > /
A jFijKij H
j
where
Tij = number of trips that are produced in Zone i and attracted to Zone j
Pi = total number of trips produced in Zone i
Aj = number of trips attracted to Zone j
Fij = friction factor that is an inverse function of travel time between Zones i and j
Kij = socioeconomic adjustment factor for travel between Zones i and j
Logit Models
n
Ux = / aiXi
i=1
where
Ux = utility of Mode x
n = number of attributes
Xi = attribute value (time, cost, and so forth)
ai = coefficient value for attributes i (negative, since the values are disutilities)
If two modes, auto (A) and transit (T), are being considered, the probability of selecting the auto Mode A can be written as
eUA
P_ Ai =
e + eUT
UA
If n modes of travel are being considered, the probability of selecting Mode x can be written as:
eUx
P_ x i = n
/ eUx
x=1
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Civil Engineering
Traffic Safety Equations
Crash Rates at Intersections
A # 1, 000, 000
RMEV = V
where
RMEV = crash rate per million entering vehicles
A = number of crashes, total or by type occurring in a single year at the location
V = ADT × 365
ADT = average daily traffic entering intersection
Crash Rates for Roadway Segments
A # 1, 000, 000
RMVM = VMT
where
RMVM = crash rate per million vehicle miles
A = number of crashes, total or by type at the study location, during a given period
VMT = vehicle miles of travel during the given period;
= ADT × (number of days in study period) × (length of road)
ADT = average daily traffic on the roadway segment
Crash Reduction
_ ADT after improvement i
Crashes prevented = N # CR
_ ADT before improvement i
where
N = expected number of crashes if countermeasure is not implemented and if the traffic volume remains the same
CR = CR1 + (1 - CR1)CR2 + (1 - CR1)(1 - CR2)CR3 +. . . + (1 - CR1). . . (1 - CRm -1) CRm
overall crash reduction factor for multiple mutually exclusive improvements at a single site
CRi = crash reduction factor for a specific countermeasure i
m = number of countermeasures at the site
Garber, Nicholas J., and Lester A. Hoel, Traffic and Highway Engineering, 4th ed., Cengage Learning, 2009.
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