Transportation engineering

This chapter covers the following topics:

Vertical curves
Horizontal curves
Traffic signal timing
Stopping sight distance
Peak hour factor
Basic freeway segment highway capacity
Traffic flow relationships
Traffic safety equations
Highway pavement design

Vertical curves

A vertical curve is typically modeled with a parabolic equation. The equations below are commonly used to describe the curve shape, the grade change, and key points along the curve.

Equations:

$y = a x^{2}$

$A = ∣ g_{2} - g_{1} ∣ \times 100$

$a = \frac{g _{2} - g _{1}}{2 L}$

$E = a (\frac{L}{2})^{2}$

$r = \frac{g _{2} - g _{1}}{L}$

$K = \frac{L}{A}$

$x_{a} = \frac{B}{2 a} = \frac{g _{1} L}{g _{1} - g _{2}}$

Tangent elevation:

$Y = Y_{P V C} + g_{1} x = Y_{P V T} + g_{2} (x - L /2)$

Curve elevation:

$Y = Y_{P V C} + g_{1} x + a x^{2} = Y_{P V C} + g_{1} x + [\frac{( g _{2} - g _{1} )}{( 2 L )}] x^{2}$

Where:

$P V C$ = point of vertical curvature, or beginning of curve
$P V I$ = point of vertical intersection, or vertex
$P V T$ = point of vertical tangency, or end of curve
$A$ = algebraic difference in grades (percentage)
$a$ = parabola constant
$E$ = tangent offset at PVI
$g_{1}$ = percent grade of back tangent divided by 100
$g_{2}$ = percent grade of forward tangent divided by 100
$h_{1}$ = height of driver’s eyes above the roadway surface ( $f t$ )
$h_{2}$ = height of object above the roadway surface ( $f t$ )
$K$ = rate of vertical curvature
$L$ = length of curve ( $f t$ )
$r$ = rate of change of grade
$S$ = sight distance ( $f t$ )
$x$ = horizontal distance from PVC to point on curve ( $f t$ )
$x_{a}$ = horizontal distance to $min / ma x$ elevation on curve ( $f t$ )
$y$ = tangent offset
$v$ = design speed ( $m p h$ )
$Y$ = elevation ( $f t$ )

Vertical curve: sight distance related to curve length

These equations relate sight distance $S$ and vertical curve length $L$ . The correct form depends on whether the required sight distance fits entirely on the curve.

Case 1: When $S \leq L$

Crest vertical curve (General equation):

$L = \frac{A S ^{2}}{100 ( h _{1} + h _{2} )}$

Standard Criteria (e.g., $h_{1} = 3.5 ft, h_{2} = 2.0 ft$ ):

$L = \frac{A S ^{2}}{2158}$

Sag vertical curve (based on standard headlight criteria):

$L = \frac{A S ^{2}}{100 ( 1 + 3.5 )}$

Standard:

$L = \frac{A S ^{2}}{400}$

Case 2: When $S > L$

Crest vertical curve:

$L = 2 S - \frac{200 ( h _{1} + h _{2} )}{A}$

Standard:

$L = 2 S - \frac{2158}{A}$

Sag vertical curve:

$L = 2 S - \frac{400 ( 1 + 3.5 )}{A}$

Standard:

$L = 2 S - \frac{800}{A}$

Where:

$A$ = vertical difference for the curve
$S$ = sight distance
$h_{1}$ , $h_{2}$ = eye and object heights

Horizontal curves

A horizontal curve is defined by its radius, central angle, and related geometric elements (tangent length, chord length, external distance, and middle ordinate).

Equations:

$R = \frac{5729.58}{D}$

$R = \frac{L C}{2 sin ( d /2 )}$

$T = R tan (d /2) = \frac{L C}{2 cos ( d /2 )}$

$L = R (\frac{d}{180} π)$

$M = R [1 - cos (d /2)]$

$\frac{E + R}{R + M} = cos (d /2)$

$\frac{R - M}{R} = cos (d /2)$

$c = 2 R sin (d /2)$

$l = R d (\frac{π}{180})$

$E = R (\frac{1}{cos ( d /2 )} - 1)$

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 $Δ$ )
$L$ = length of curve, from PC to PT
$L C$ = length of long chord
$M$ = length of middle ordinate
$PC$ = point of curve
$P I$ = point of intersection
$PT$ = point of tangent
$R$ = radius
$S$ = sight distance ( $f t$ )
$T$ = tangent distance
$V$ = design speed ( $m p h$ )

Additional horizontal curve equations:

Side friction factor (based on superelevation):

$0.01 e + f = \frac{V ^{2}}{15 R}$

Spiral transition length:

$L = \frac{3.15 V ^{2}}{RC}$

Where:

$C$ = rate of increase of lateral acceleration

Sight distance (to see around obstruction):

$H SO = R - (R cos (\frac{28.65}{R}))$

Where:

$H SO$ = Horizontal sight line offset

Traffic signal timing

These equations are used to estimate key signal timing intervals (yellow, red clearance, and pedestrian minimum green).

$y = t + \frac{v}{64.4 G}$

$r = \frac{W + l}{v}$

$G_{p} = 3.2 + \frac{L}{S _{p}} + 0.27 N_{p e d}$

Where:

$t$ = driver reaction time ( $sec$ )
$v$ = vehicle approach speed ( $f t / sec$ )
$W$ = width of intersection, curb-to-curb ( $f t$ )
$l$ = length of vehicle ( $f t$ )
$y$ = length of yellow interval to nearest 0.1 sec ( $sec$ )
$r$ = length of red clearance interval to nearest 0.1 sec ( $sec$ )
$G_{p}$ = minimum green time for pedestrians ( $sec$ )
$L$ = crosswalk length ( $f t$ )
$S_{p}$ = pedestrian speed ( $f t / sec$ ), default 3.5 $f t / sec$
$N_{p e d}$ = number of pedestrian in interval
$a$ = deceleration ( $f t / se c^{2}$ )
$\pm G$ = percent grade divided by 100 (uphill grade “+”)

Stopping sight distance

Stopping sight distance combines perception-reaction distance and braking distance. Intersection sight distance is based on the time gap needed to enter or cross the major road.

$SS D = 1.47 V t + \frac{V ^{2}}{30 ( \frac{a}{32.2} \pm G )}$

$I S D = 1.47 V_{maj or} t_{g}$

Where:

$a$ = deceleration ( $f t / se c^{2}$ )
$\pm G$ = percent grade divided by 100 (uphill grade “+”)
$SS D$ = stopping sight distance ( $f t$ )
$I S D$ = intersection sight distance ( $f t$ )
$t$ = driver reaction time ( $sec$ )
$t_{g}$ = time gap for vehicle entering roadway ( $sec$ )
$V$ = design speed ( $m p h$ )
$V_{maj or}$ = design speed of major road ( $m p h$ )

Peak hour factor

Peak hour factor (PHF) compares the hourly volume to the peak 15-minute flow rate within that hour.

$P H F = \frac{Hourly Volume}{4 \times V _{15}}$

Where:

$P H F$ = peak hour factor
$V$ = hourly volume (veh/hr)
$V_{15}$ = peak 15-min. volume (veh/15 min)

Basic freeway segment highway capacity

Parameters for speed-flow curves for basic freeway segments

Parameter	Definition and units	Basic freeway segments
$FFS$	Base segment free-flow speed ( $m p h$ )	Measured or predicted with equation
$FF S_{a d j}$	Adjusted free-flow speed ( $m p h$ )	$FF S_{a d j} = FFS \times S A F$
$S A F$	Speed adjustment factor (decimal)	$S A F = 1.00$ for base conditions
$c$	Base segment capacity ( $p c / h / l n$ )	$c = 2, 200 + 10 (FFS - 50)$ $c \leq 2, 400$ , $55 \leq FFS \leq 75$
$c_{a d j}$	Adjusted segment capacity ( $p c / h / l n$ )	$c_{a d j} = c \times C A F$
$C A F$	Capacity adjustment factor (decimal)	$C A F = 1.00$ for base conditions
$D_{c}$	Density at capacity ( $p c / h / l n$ )	$45$
$BP$	Breakpoint ( $p c / h / l n$ )	$B P_{a d j} = [1, 000 + 40 \times (75 - FF S_{a d j})] \times C A F^{2}$
$a$	Exponent calibration parameter (decimal)	$2.00$

Equations

For $v_{p} \leq BP$

$S = FF S_{a d j}$

For $BP < v_{p} \leq c$

$S = FF S_{a d j} (\frac{c _{a d j}}{D _{c}}) (\frac{v _{p} - BP}{c _{a d j} - BP})^{a}$

Free-flow speed (FFS) equation

This equation predicts $FFS$ from a base value and subtracts adjustments for geometric and operational conditions.

$FFS = BFFS - f_{L W} - f_{R L C} - 3.22 \cdot TR D^{0.84}$

Where

$FFS$ = free flow speed of basic freeway segment ( $m p h$ )
$BFFS$ = base free flow speed of basic freeway segment (default: 75.4 mph)
$f_{L W}$ = adjustment for lane width ( $m p h$ )
$f_{R L C}$ = adjustment for right-side lateral clearance ( $m p h$ )
$TR D$ = total ramp density ( $r am p s / mi$ )

Adjustment to FFS for average lane width for basic freeway and multilane highway segments

Average lane width (ft)	Reduction in FFS, $f_{L W}$ ( $m p h$ )
$\geq 12$	0.0
$\geq 11 - < 12$	1.9
$\geq 10 - < 11$	6.6

Units:

$p c / h / l n$ = passenger cars per hour per lane

Adjustments to FFS for right-side lateral clearance

Adjustments to FFS for right-side lateral clearance, $f_{R L C}$ ( $m p h$ ), for basic freeway segments table can be found from FE Handbook.

Demand flow rate equation

Demand flow rate converts the observed demand volume into an equivalent passenger-car flow rate under base conditions.

$v_{p} = \frac{V}{P H F \times N \times f _{H V}}$

Where:

$v_{p}$ = demand flow rate under equivalent base conditions ( $p c / h / l n$ )
$V$ = demand volume under prevailing conditions ( $v e h / h$ )
$P H F$ = peak-hour factor
$N$ = number of lanes in analysis direction
$f_{H V}$ = adjustment factor for presence of heavy vehicles in traffic stream

Heavy vehicle adjustment factor

$f_{H V} = \frac{1}{1 + P _{T} ( E _{T} - 1 )}$

Where:

$P_{T}$ = proportion of single-unit trucks and tractor-trailers in the traffic stream
$E_{T}$ = passenger-car equivalent (PCE) of single unit truck or tractor-trailer in traffic stream

PCE by type of terrain

Vehicle	Level	Rolling
$E_{T}$	2.0	3.0

Density equation

Density is flow per lane divided by mean speed.

$D = \frac{v _{p}}{S}$

Where:

$D$ = density ( $p c / mi / l n$ )
$v_{p}$ = demand flow rate ( $p c / h / l n$ )
$S$ = mean speed of traffic stream under base conditions ( $m p h$ )

Traffic flow relationships

Greenshields model

The Greenshields model assumes a linear relationship between speed and density, which leads to a parabolic flow-density relationship.

$S = S_{y} (1 - \frac{D}{D _{J}})$

$V = S \cdot D = S_{y} D (1 - \frac{D}{D _{J}})$

$V_{ma x} = \frac{S _{y} D _{J}}{4}$

$D_{o} = \frac{D _{J}}{2}$

Where:

$D$ = density ( $v e h / mi / l n$ )
$S$ = speed ( $m p h$ )
$V$ = flow ( $v e h / h r / l n$ )
$V_{ma x}$ = maximum flow ( $v e h / h r / l n$ )
$D_{o}$ = optimum density ( $v e h / mi / l n$ )
$D_{J}$ = jam density ( $v e h / mi / l n$ )
$S_{y}$ = theoretical speed ( $m p h$ )

Gravity model

The gravity model estimates trips between zones using productions, attractions, and impedance (via friction factors), with optional socioeconomic adjustments.

$T_{ij} = \frac{P _{i} A _{j} F _{ij} K _{ij}}{\sum A _{j} F _{ij} K _{ij}}$

Where:

$T_{ij}$ = number of trips from Zone i to Zone j
$P_{i}$ = trips produced in Zone i
$A_{j}$ = trips attracted to Zone j
$F_{ij}$ = friction factor (inverse of travel time between i and j)
$K_{ij}$ = socioeconomic adjustment factor

Logit Models

Logit models use a utility value for each alternative and convert those utilities into probabilities.

Utility function:

$U_{i} = \sum β_{k} x_{k}^{i}$

Probability (2 modes):

$P (A) = \frac{e ^{U_{A}}}{e ^{U_{A}} + e ^{U_{T}}}$

Probability (n modes):

$P (x) = \frac{e ^{U_{x}}}{\sum e ^{U_{i}}}$

Traffic safety equations

Crash rates at intersections:

$RME V = \frac{A}{A D T \times 365} \times 1, 000, 000$

Where:

$RME V$ = crash rate per million entering vehicles
$A$ = number of crashes
$A D T$ = average daily traffic entering the intersection

Crash rates for roadway segments:

$R V M V = \frac{A}{A D T \times T} \times 1, 000, 000$

Where:

$R V M V$ = crash rate per million vehicle miles
$A$ = number of crashes
$A D T$ = average daily traffic
$T$ = time (days in study period) × length (miles)

Crashes prevented:

$Crashes Prevented = N \times CR (\frac{A D T _{after}}{A D T _{before}})$

Composite reduction factor:

$CR = C R_{1} + (1 - C R_{1}) C R_{2} + (1 - C R_{1}) (1 - C R_{2}) C R_{3} + \dots + (1 - C R_{1}) (1 - C R_{2}) \dots (1 - C R_{m - 1}) C R_{m}$

Highway pavement design

AASHTO structural number equation

The structural number $SN$ is a weighted sum of layer thicknesses, adjusted by drainage coefficients for unbound layers.

$SN = a_{1} D_{1} + a_{2} D_{2} m_{2} + a_{3} D_{3} m_{3} + \dots + a_{m} D_{m} m_{m}$

Where:

$SN$ = structural number for pavement
$a$ = layer coefficient
$D$ = thickness of layer (inches)
$m$ = drainage coefficient

This chapter covers the following topics:

Vertical curves
Horizontal curves
Traffic signal timing
Stopping sight distance
Peak hour factor
Basic freeway segment highway capacity
Traffic flow relationships
Traffic safety equations
Highway pavement design

Vertical curves

A vertical curve is typically modeled with a parabolic equation. The equations below are commonly used to describe the curve shape, the grade change, and key points along the curve.

Equations:

$y = a x^{2}$

$A = ∣ g_{2} - g_{1} ∣ \times 100$

$a = \frac{g _{2} - g _{1}}{2 L}$

$E = a (\frac{L}{2})^{2}$

$r = \frac{g _{2} - g _{1}}{L}$

$K = \frac{L}{A}$

$x_{a} = \frac{B}{2 a} = \frac{g _{1} L}{g _{1} - g _{2}}$

Tangent elevation:

$Y = Y_{P V C} + g_{1} x = Y_{P V T} + g_{2} (x - L /2)$

Curve elevation:

$Y = Y_{P V C} + g_{1} x + a x^{2} = Y_{P V C} + g_{1} x + [\frac{( g _{2} - g _{1} )}{( 2 L )}] x^{2}$

Where:

$P V C$ = point of vertical curvature, or beginning of curve
$P V I$ = point of vertical intersection, or vertex
$P V T$ = point of vertical tangency, or end of curve
$A$ = algebraic difference in grades (percentage)
$a$ = parabola constant
$E$ = tangent offset at PVI
$g_{1}$ = percent grade of back tangent divided by 100
$g_{2}$ = percent grade of forward tangent divided by 100
$h_{1}$ = height of driver’s eyes above the roadway surface ( $f t$ )
$h_{2}$ = height of object above the roadway surface ( $f t$ )
$K$ = rate of vertical curvature
$L$ = length of curve ( $f t$ )
$r$ = rate of change of grade
$S$ = sight distance ( $f t$ )
$x$ = horizontal distance from PVC to point on curve ( $f t$ )
$x_{a}$ = horizontal distance to $min / ma x$ elevation on curve ( $f t$ )
$y$ = tangent offset
$v$ = design speed ( $m p h$ )
$Y$ = elevation ( $f t$ )

Vertical curve: sight distance related to curve length

These equations relate sight distance $S$ and vertical curve length $L$ . The correct form depends on whether the required sight distance fits entirely on the curve.

Case 1: When $S \leq L$

Crest vertical curve (General equation):

$L = \frac{A S ^{2}}{100 ( h _{1} + h _{2} )}$

Standard Criteria (e.g., $h_{1} = 3.5 ft, h_{2} = 2.0 ft$ ):

$L = \frac{A S ^{2}}{2158}$

Sag vertical curve (based on standard headlight criteria):

$L = \frac{A S ^{2}}{100 ( 1 + 3.5 )}$

Standard:

$L = \frac{A S ^{2}}{400}$

Case 2: When $S > L$

Crest vertical curve:

$L = 2 S - \frac{200 ( h _{1} + h _{2} )}{A}$

Standard:

$L = 2 S - \frac{2158}{A}$

Sag vertical curve:

$L = 2 S - \frac{400 ( 1 + 3.5 )}{A}$

Standard:

$L = 2 S - \frac{800}{A}$

Where:

$A$ = vertical difference for the curve
$S$ = sight distance
$h_{1}$ , $h_{2}$ = eye and object heights

Horizontal curves

A horizontal curve is defined by its radius, central angle, and related geometric elements (tangent length, chord length, external distance, and middle ordinate).

Equations:

$R = \frac{5729.58}{D}$

$R = \frac{L C}{2 sin ( d /2 )}$

$T = R tan (d /2) = \frac{L C}{2 cos ( d /2 )}$

$L = R (\frac{d}{180} π)$

$M = R [1 - cos (d /2)]$

$\frac{E + R}{R + M} = cos (d /2)$

$\frac{R - M}{R} = cos (d /2)$

$c = 2 R sin (d /2)$

$l = R d (\frac{π}{180})$

$E = R (\frac{1}{cos ( d /2 )} - 1)$

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 $Δ$ )
$L$ = length of curve, from PC to PT
$L C$ = length of long chord
$M$ = length of middle ordinate
$PC$ = point of curve
$P I$ = point of intersection
$PT$ = point of tangent
$R$ = radius
$S$ = sight distance ( $f t$ )
$T$ = tangent distance
$V$ = design speed ( $m p h$ )

Additional horizontal curve equations:

Side friction factor (based on superelevation):

$0.01 e + f = \frac{V ^{2}}{15 R}$

Spiral transition length:

$L = \frac{3.15 V ^{2}}{RC}$

Where:

$C$ = rate of increase of lateral acceleration

Sight distance (to see around obstruction):

$H SO = R - (R cos (\frac{28.65}{R}))$

Where:

$H SO$ = Horizontal sight line offset

Traffic signal timing

These equations are used to estimate key signal timing intervals (yellow, red clearance, and pedestrian minimum green).

$y = t + \frac{v}{64.4 G}$

$r = \frac{W + l}{v}$

$G_{p} = 3.2 + \frac{L}{S _{p}} + 0.27 N_{p e d}$

Where:

$t$ = driver reaction time ( $sec$ )
$v$ = vehicle approach speed ( $f t / sec$ )
$W$ = width of intersection, curb-to-curb ( $f t$ )
$l$ = length of vehicle ( $f t$ )
$y$ = length of yellow interval to nearest 0.1 sec ( $sec$ )
$r$ = length of red clearance interval to nearest 0.1 sec ( $sec$ )
$G_{p}$ = minimum green time for pedestrians ( $sec$ )
$L$ = crosswalk length ( $f t$ )
$S_{p}$ = pedestrian speed ( $f t / sec$ ), default 3.5 $f t / sec$
$N_{p e d}$ = number of pedestrian in interval
$a$ = deceleration ( $f t / se c^{2}$ )
$\pm G$ = percent grade divided by 100 (uphill grade “+”)

Stopping sight distance

Stopping sight distance combines perception-reaction distance and braking distance. Intersection sight distance is based on the time gap needed to enter or cross the major road.

$SS D = 1.47 V t + \frac{V ^{2}}{30 ( \frac{a}{32.2} \pm G )}$

$I S D = 1.47 V_{maj or} t_{g}$

Where:

$a$ = deceleration ( $f t / se c^{2}$ )
$\pm G$ = percent grade divided by 100 (uphill grade “+”)
$SS D$ = stopping sight distance ( $f t$ )
$I S D$ = intersection sight distance ( $f t$ )
$t$ = driver reaction time ( $sec$ )
$t_{g}$ = time gap for vehicle entering roadway ( $sec$ )
$V$ = design speed ( $m p h$ )
$V_{maj or}$ = design speed of major road ( $m p h$ )

Peak hour factor

Peak hour factor (PHF) compares the hourly volume to the peak 15-minute flow rate within that hour.

$P H F = \frac{Hourly Volume}{4 \times V _{15}}$

Where:

$P H F$ = peak hour factor
$V$ = hourly volume (veh/hr)
$V_{15}$ = peak 15-min. volume (veh/15 min)

Basic freeway segment highway capacity

Parameters for speed-flow curves for basic freeway segments

Parameter	Definition and units	Basic freeway segments
$FFS$	Base segment free-flow speed ( $m p h$ )	Measured or predicted with equation
$FF S_{a d j}$	Adjusted free-flow speed ( $m p h$ )	$FF S_{a d j} = FFS \times S A F$
$S A F$	Speed adjustment factor (decimal)	$S A F = 1.00$ for base conditions
$c$	Base segment capacity ( $p c / h / l n$ )	$c = 2, 200 + 10 (FFS - 50)$ $c \leq 2, 400$ , $55 \leq FFS \leq 75$
$c_{a d j}$	Adjusted segment capacity ( $p c / h / l n$ )	$c_{a d j} = c \times C A F$
$C A F$	Capacity adjustment factor (decimal)	$C A F = 1.00$ for base conditions
$D_{c}$	Density at capacity ( $p c / h / l n$ )	$45$
$BP$	Breakpoint ( $p c / h / l n$ )	$B P_{a d j} = [1, 000 + 40 \times (75 - FF S_{a d j})] \times C A F^{2}$
$a$	Exponent calibration parameter (decimal)	$2.00$

Equations

For $v_{p} \leq BP$

$S = FF S_{a d j}$

For $BP < v_{p} \leq c$

$S = FF S_{a d j} (\frac{c _{a d j}}{D _{c}}) (\frac{v _{p} - BP}{c _{a d j} - BP})^{a}$

Free-flow speed (FFS) equation

This equation predicts $FFS$ from a base value and subtracts adjustments for geometric and operational conditions.

$FFS = BFFS - f_{L W} - f_{R L C} - 3.22 \cdot TR D^{0.84}$

Where

$FFS$ = free flow speed of basic freeway segment ( $m p h$ )
$BFFS$ = base free flow speed of basic freeway segment (default: 75.4 mph)
$f_{L W}$ = adjustment for lane width ( $m p h$ )
$f_{R L C}$ = adjustment for right-side lateral clearance ( $m p h$ )
$TR D$ = total ramp density ( $r am p s / mi$ )

Adjustment to FFS for average lane width for basic freeway and multilane highway segments

Average lane width (ft)	Reduction in FFS, $f_{L W}$ ( $m p h$ )
$\geq 12$	0.0
$\geq 11 - < 12$	1.9
$\geq 10 - < 11$	6.6

Units:

$p c / h / l n$ = passenger cars per hour per lane

Adjustments to FFS for right-side lateral clearance

Adjustments to FFS for right-side lateral clearance, $f_{R L C}$ ( $m p h$ ), for basic freeway segments table can be found from FE Handbook.

Demand flow rate equation

Demand flow rate converts the observed demand volume into an equivalent passenger-car flow rate under base conditions.

$v_{p} = \frac{V}{P H F \times N \times f _{H V}}$

Where:

$v_{p}$ = demand flow rate under equivalent base conditions ( $p c / h / l n$ )
$V$ = demand volume under prevailing conditions ( $v e h / h$ )
$P H F$ = peak-hour factor
$N$ = number of lanes in analysis direction
$f_{H V}$ = adjustment factor for presence of heavy vehicles in traffic stream

Heavy vehicle adjustment factor

$f_{H V} = \frac{1}{1 + P _{T} ( E _{T} - 1 )}$

Where:

$P_{T}$ = proportion of single-unit trucks and tractor-trailers in the traffic stream
$E_{T}$ = passenger-car equivalent (PCE) of single unit truck or tractor-trailer in traffic stream

PCE by type of terrain

Vehicle	Level	Rolling
$E_{T}$	2.0	3.0

Density equation

Density is flow per lane divided by mean speed.

$D = \frac{v _{p}}{S}$

Where:

$D$ = density ( $p c / mi / l n$ )
$v_{p}$ = demand flow rate ( $p c / h / l n$ )
$S$ = mean speed of traffic stream under base conditions ( $m p h$ )

Traffic flow relationships

Greenshields model

The Greenshields model assumes a linear relationship between speed and density, which leads to a parabolic flow-density relationship.

$S = S_{y} (1 - \frac{D}{D _{J}})$

$V = S \cdot D = S_{y} D (1 - \frac{D}{D _{J}})$

$V_{ma x} = \frac{S _{y} D _{J}}{4}$

$D_{o} = \frac{D _{J}}{2}$

Where:

$D$ = density ( $v e h / mi / l n$ )
$S$ = speed ( $m p h$ )
$V$ = flow ( $v e h / h r / l n$ )
$V_{ma x}$ = maximum flow ( $v e h / h r / l n$ )
$D_{o}$ = optimum density ( $v e h / mi / l n$ )
$D_{J}$ = jam density ( $v e h / mi / l n$ )
$S_{y}$ = theoretical speed ( $m p h$ )

Gravity model

The gravity model estimates trips between zones using productions, attractions, and impedance (via friction factors), with optional socioeconomic adjustments.

$T_{ij} = \frac{P _{i} A _{j} F _{ij} K _{ij}}{\sum A _{j} F _{ij} K _{ij}}$

Where:

$T_{ij}$ = number of trips from Zone i to Zone j
$P_{i}$ = trips produced in Zone i
$A_{j}$ = trips attracted to Zone j
$F_{ij}$ = friction factor (inverse of travel time between i and j)
$K_{ij}$ = socioeconomic adjustment factor

Logit Models

Logit models use a utility value for each alternative and convert those utilities into probabilities.

Utility function:

$U_{i} = \sum β_{k} x_{k}^{i}$

Probability (2 modes):

$P (A) = \frac{e ^{U_{A}}}{e ^{U_{A}} + e ^{U_{T}}}$

Probability (n modes):

$P (x) = \frac{e ^{U_{x}}}{\sum e ^{U_{i}}}$

Traffic safety equations

Crash rates at intersections:

$RME V = \frac{A}{A D T \times 365} \times 1, 000, 000$

Where:

$RME V$ = crash rate per million entering vehicles
$A$ = number of crashes
$A D T$ = average daily traffic entering the intersection

Crash rates for roadway segments:

$R V M V = \frac{A}{A D T \times T} \times 1, 000, 000$

Where:

$R V M V$ = crash rate per million vehicle miles
$A$ = number of crashes
$A D T$ = average daily traffic
$T$ = time (days in study period) × length (miles)

Crashes prevented:

$Crashes Prevented = N \times CR (\frac{A D T _{after}}{A D T _{before}})$

Composite reduction factor:

$CR = C R_{1} + (1 - C R_{1}) C R_{2} + (1 - C R_{1}) (1 - C R_{2}) C R_{3} + \dots + (1 - C R_{1}) (1 - C R_{2}) \dots (1 - C R_{m - 1}) C R_{m}$

Highway pavement design

AASHTO structural number equation

The structural number $SN$ is a weighted sum of layer thicknesses, adjusted by drainage coefficients for unbound layers.

$SN = a_{1} D_{1} + a_{2} D_{2} m_{2} + a_{3} D_{3} m_{3} + \dots + a_{m} D_{m} m_{m}$

Where:

$SN$ = structural number for pavement
$a$ = layer coefficient
$D$ = thickness of layer (inches)
$m$ = drainage coefficient