Human Factors Guidelines for Road Systems: Third Edition (2025)
Chapter: 32 Equations
CHAPTER 32
Equations
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Road User Sampling Rate |
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Road User Information Database for Making Decisions |
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Long Description.Information open parenthesis t close parenthesis equals Information open parenthesis t minus 1 close parenthesis plus changes during delta t |
Where:
t = time
Δt = sampling interval
Sight Distance |
Sight Distance |
= |
Distance traveled while driver perceives, makes |
+ |
Distance traveled while the driver completes an appropriate maneuver (MT) |
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Stopping Sight Distance (SSD) Design Values |
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US Customary: Long Description.SSD equals 1.47 Vt subscript RT end subscript plus 0.039 start fraction V superscript 2 over a end fraction |
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Metric: Long Description.SSD equals 0.278 Vt subscript RT end subscript plus 0.039 start fraction V superscript 2 over a end fraction |
Where:
tRT = perception-reaction time
V = design speed (mi/h) [km/h]
a = deceleration level (ft/s2) [m/s2]
Note: US Customary units in parentheses, metric units in square brackets.
Page number |
Decision Sight Distance (DSD) for Avoidance Maneuvers |
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Stopping |
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US Customary: Long Description.DSD equals 0.147 V subscript t end subscript plus 1.075 start fraction V superscript 2 over a end fraction |
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Metric: 
Long Description.DSD equals 0.278 V subscript t end subscript plus 0.039 start fraction V superscript 2 over a end fraction |
Where:
t = time (s)
rural road: 3.0 s
urban road: 9.1 s
V = design speed (mi/h) [km/h]
a = deceleration level (ft/s2) [m/s2]
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Speed/Path/Direction Change |
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US Customary: Long Description.DSD equals 0.147 V subscript t |
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Metric: 
Long Description.DSD equals 0.278 V subscript t |
t = time, s
rural road: 10.2–11.2 s
suburban road: 12.1–12.9 s
urban road: 14.0–14.5 s
V = design speed (mi/h) [km/h]
Note: US Customary units in parentheses, metric units in square brackets.
Storage Length Calculations |
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Left Turn Capacity |
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Long Description.c equals start fraction V subscript 0 end subscript e superscript negative v subscript 0 end subscript t subscript c end subscript over 3600 end superscript divided by 1 minus e superscript negative v subscript 0 end subscript t subscript f end subscript over 3600 end superscript end fraction |
Where:
c = left-turn capacity (veh/h)
V0 = major-road volume conflicting with the minor movement, assumed to be equal to one-half of the two-way major-road volume (veh/h)
tc = critical gap (s)
tf = follow-up gap (s)
Left Turn Storage Length |
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Long Description.SL equals open curly brace start fraction l n open square bracket P open parenthesis n greater than N close parenthesis close square bracket divided by l n open square bracket v over c close square bracket end fraction close curly brace multiplied by VL |
Page number |
Where:
SL = storage length (ft) [m]
P(n > N) = probability of turn-lane overflow
v = left-turn vehicle volume (veh/h)
c = left-turn capacity (veh/h)
VL = average length per vehicle (ft) [m]
Note: US Customary units in parentheses, metric units in square brackets.
Yellow Timing Interval Time Plus the Red Clearance Interval Time |
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Non-Dilemma Change Period |
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Long Description.CP equals Y plus R |
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Yellow Timing Interval |
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Long Description.Y equals t plus start fraction V over 2a plus 2Gg end fraction |
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Red Clearance Interval |
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Long Description.R equals start fraction W plus L over V end fraction |
Where:
CP = non-dilemma change period (s)
Y = yellow timing interval (s)
R = red clearance interval (s)
t = perception-reaction time (nominally 1 s)
V = approach speed (ft/s) [m/s]
g = percent grade (positive for upgrade, negative for downgrade)
a = deceleration (ft/s2) [m/s2]
G = gravity (32.2 ft/s2) [9.8 m/s2]
W = width of intersection, curb-to-curb (ft) [m]
L = length of vehicle (typical 20 ft) [typical 6 m]
Note: Imperial units in parentheses, metric units in square brackets.
Gate Delay and Gate Interval Times |
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Gate Delay |
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Long Description.open square bracket t plus start fraction v over 2 open parenthesis a plus G times g close parenthesis end fraction plus D over V close square bracket |
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Gate Interval Time |
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Long Description.open square bracket start fraction 1 over v end fraction open parenthesis W subscript ght end subscript plus L close parenthesis close square bracket |
Page number |
Where:
t = driver perception-reaction time (s)
V = approach speed (m/s)
a = deceleration rate on level pavement (m/s2)
G = acceleration resulting from gravity (m/s2)
g = grade of approach lanes, percent/100
D = distance between stop bar and gates (m)
L = length of the vehicle
Wght = distance between entry and exit gates (m)
Distance between Entry and Exit Gates (Wght) |
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For α ≤ 90°: 
Long Description.W subscript ght equals start fraction W subscript t over sin alpha end fraction plus start fraction 2W subscript h over tan alpha end fraction plus start fraction 2W subscript g over sin alpha end fraction |
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For α > 90°: 
Long Description.W subscript ght equals start fraction W subscript t over sin open parenthesis 180 minus alpha close parenthesis end fraction plus start fraction 2W subscript h over tan open parenthesis 180 minus alpha close parenthesis end fraction plus start fraction 2W subscript g over sin open parenthesis 180 minus alpha close parenthesis |
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Where:
α = crossing angle (degrees)
Wght = distance between entry and exit gates (m)
Wt = width of railroad track (m)
Wh = width of approaching lane of the highway (m)
Wg = distance from track edge to gate (m)
CIE Veiling Luminance Model |
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Long Description.start fraction L subscript veil over I subscript glare end fraction equals start fraction 10 over theta superscript 3 end fraction plus open square bracket 5 over theta superscript 2 close square bracket multiplied by open square bracket 1 plus open parenthesis A over 62.5 close parenthesis superscript 4 close square bracket |
Where:
Lveil = veiling luminance
Iglare = luminous intensity of glare source
θ = glare angle
A = driver age
Contrast Ratio (light-emitting CMS) |
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Long Description.Contrast ratio equals Luminance subscript max over Luminance subscript min |
Where:
Luminancemax = luminance emitted by the area or element of greatest intensity (text)
Luminancemin = luminance emitted by the area or element of least intensity (background)
Note: Suggested contrast ratio range = 8–12
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Display Time for Phases of Dynamic Message Signs |
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Display Time for the Entire Message |
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Long Description.T open parenthesis s close parenthesis equals start fraction Legibility Distance open parenthesis feet close parenthesis over Traveling Speed open parenthesis feet per second close parenthesis end fraction |
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Time Needed for Each Phase |
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Long Description.open parenthesis t subscript 1 close parenthesis equals 2x Long Description.open parenthesis t subscript 2 close parenthesis equals 2y |
Where:
t1 = time for phase 1
t2 = time for phase 2
x = number of information units in phase 1
y = number of information units in phase 2
Make sure that the time required for the total message is less than or equal to the time available:
Long Description.T greater than or equal to B plus t subscript 1 end subscript plus t subscript 2 |
Where:
B = blanking time between phases
Luminance Contrast |
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Long Description.start fraction L subscript stripe minus L subscript pavement over L subscript pavement end fraction |
Where:
Lstripe = the luminance of the pavement marking
Lpavement = the luminance of the pavement
Bicycle Crossing Time (initially stopped/standing and rolling) |
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Long Description.BCT subscript standing end subscript equals PRT plus start fraction V over 2a end fraction plus start fraction open parenthesis W plus L close parenthesis over V |
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Long Description.BCT subscript rolling end subscript equals start fraction BD plus W plus L over V end fraction comma BD equals PRT multiplied by V plus start fraction V squared over 2a |
Where:
BCTstanding = bicycle crossing time (s), initially stopped
BCTrolling = bicycle crossing time (s), initially rolling
W = intersection width (ft) [m]
Page number |
L = bicycle length (typical 6 ft) [1.8 m]
V = attained bicycle crossing speed (typical 14.7 ft/s) [4.5 m/s]
BD = braking distance (ft) [m]
PRT = perception-reaction time (typical 1 s)
a = bicycle acceleration (typical 1.5 ft/s2) [0.5 m/s2]
Note: Imperial units in parentheses, metric units in square brackets.
Bicycle Minimum Green Time |
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Long Description.BMG equals BCT subscript standing end subscript minus Y minus R subscript clear |
Where:
BMG = bicycle minimum green time (s)
Y = yellow change interval (s)
BCTstanding = bicycle crossing time (s)
Rclear = all-red interval (s)
All-Red and Extension Time |
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Long Description.BCT subscript rolling end subscript lesser than or equal to T subscript extension end subscript plus Y plus R subscript clear |
Where:
BCTrolling = bicycle crossing time (s)
Textension = extension time (s)
Y = yellow change interval (s)
Rclear = all-red interval (s)
Sight Distance |
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Long Description.d subscript SD equals kVt subscript prt plus kVt subscript man |
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Long Description.d subscript SD equals Vt subscript prt plus Vd subscript man superscript v |
Where:
dSD = required sight distance
V = velocity of the vehicle(s)
tprt = perception-reaction time
tman = maneuver time
dmanV = distance required to execute a maneuver at velocity V
k = a constant to convert the solution to the desired units (feet, meters)
Reading Time (Sign Comprehension) |
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Reading Time (Sign Placement) |
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Long Description.BRT equals 0.5 x plus 1 y |
Page number |
Where:
BRT = base reading time (s)
x = the number of critical words/numbers in the message
y = the number of critical symbols in the message
Reading Time (Complex Signs in High-Speed Conditions) |
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Long Description.Reading Time open parenthesis s close parenthesis equals 0.31 open parenthesis Number of Familiar Words close parenthesis plus 1.94 |
Note: Reading distance is obtained by multiplying the reading time by the travel speed.
Information Presentation Distance (Sign Placement) |
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Long Description.Information Presentation Distance equals Reading Distance plus Decision Distance plus Maneuver Distance |
Legibility Distance (Sign Placement) |
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Long Description.Legibility Distance equals Information Presentation Distance minus Advance Placement |
Minimum Letter Height (Signs) |
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Long Description.Minimum Letter Height open parenthesis in. close parenthesis equals start fraction Legibility Distance open parenthesis feet close parenthesis over Legibility Index open parenthesis feet over in. close parenthesis end fraction |
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Abbreviations and acronyms used without definitions in TRB publications: |
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A4A |
Airlines for America |
AAAE |
American Association of Airport Executives |
AASHO |
American Association of State Highway Officials |
AASHTO |
American Association of State Highway and Transportation Officials |
ACI–NA |
Airports Council International–North America |
ACRP |
Airport Cooperative Research Program |
ADA |
Americans with Disabilities Act |
APTA |
American Public Transportation Association |
ASCE |
American Society of Civil Engineers |
ASME |
American Society of Mechanical Engineers |
ASTM |
American Society for Testing and Materials |
ATA |
American Trucking Associations |
CTAA |
Community Transportation Association of America |
CTBSSP |
Commercial Truck and Bus Safety Synthesis Program |
DHS |
Department of Homeland Security |
DOE |
Department of Energy |
EPA |
Environmental Protection Agency |
FAA |
Federal Aviation Administration |
FAST |
Fixing Americaʼs Surface Transportation Act (2015) |
FHWA |
Federal Highway Administration |
FMCSA |
Federal Motor Carrier Safety Administration |
FRA |
Federal Railroad Administration |
FTA |
Federal Transit Administration |
GHSA |
Governors Highway Safety Association |
HMCRP |
Hazardous Materials Cooperative Research Program |
IEEE |
Institute of Electrical and Electronics Engineers |
ISTEA |
Intermodal Surface Transportation Efficiency Act of 1991 |
ITE |
Institute of Transportation Engineers |
MAP-21 |
Moving Ahead for Progress in the 21st Century Act (2012) |
NASA |
National Aeronautics and Space Administration |
NASAO |
National Association of State Aviation Officials |
NCFRP |
National Cooperative Freight Research Program |
NCHRP |
National Cooperative Highway Research Program |
NHTSA |
National Highway Traffic Safety Administration |
NTSB |
National Transportation Safety Board |
PHMSA |
Pipeline and Hazardous Materials Safety Administration |
RITA |
Research and Innovative Technology Administration |
SAE |
Society of Automotive Engineers |
SAFETEA-LU |
Safe, Accountable, Flexible, Efficient Transportation Equity Act: A Legacy for Users (2005) |
TCRP |
Transit Cooperative Research Program |
TEA-21 |
Transportation Equity Act for the 21st Century (1998) |
TRB |
Transportation Research Board |
TSA |
Transportation Security Administration |
U.S. DOT |
United States Department of Transportation |
Transportation Research Board
500 Fifth Street, NW
Washington, DC 20001
