Greenshields Model

Transportation · FE Reference Handbook section

Transportation

12 formulas

10 exam-style examples

~60 min

All Transportation lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Greenshields Model within Transportation. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what greenshields model describes physically and when it applies.
State every one of the 12 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: grades as decimals in curve formulas, percent in the stem.

Lecture

Why this section exists. Greenshields Model is the part of Transportation that lets you connect a vertical or horizontal alignment, or a traffic stream to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as a curve geometry element or a capacity/flow relationship. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. grades as decimals in curve formulas, percent in the stem. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Dense peak-hour traffic queued on an urban arterial at dusk. — Photo 1. Where this shows up in practice: greenshields model.
Wikimedia Commons, CC BY 2.0

Transportation — Greenshields Model: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a vertical or horizontal alignment, or a traffic stream. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 12 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

S	Quantity produced by "S = Sf − D D Vm Sf" — read its definition and unit from the handbook line directly above the equation.
V	Quantity produced by "V = S f D − D D2" — read its definition and unit from the handbook line directly above the equation.
Vm	Quantity produced by "Vm = 4 0 DO Dj" — read its definition and unit from the handbook line directly above the equation.
Do	Quantity produced by "Do = 2 Oversaturated flow" — read its definition and unit from the handbook line directly above the equation.
D	Quantity produced by "D = density (veh/mi)" — read its definition and unit from the handbook line directly above the equation.
Dj	Quantity produced by "Dj = jam density (veh/hr)" — read its definition and unit from the handbook line directly above the equation.
So	Quantity produced by "So = optimum speed (often called critical speed) (mph)" — read its definition and unit from the handbook line directly above the equation.
Sf	Quantity produced by "Sf = theoretical speed selected by the first driver entering a facility (i.e., under zero density and zero" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

Sf Sf
SPEED (mph)
SPEED (mph)
SO SO
0 DO Dj 0 Vm
DENSITY (veh/mi/ln) FLOW (veh/h/ln)
Sf SO
FLOW (veh/h/ln)
D jS f
Dj DENSITY (veh/mi/ln)
AASHTO, A Policy on Geometric Design of Highways and Streets, 6th ed., 2011. Used by permission.
where
flow rate conditions) (mph)

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Greenshields flow, speed and density — Greenshields Model

A freeway segment follows Greenshields with free-flow speed 56 mph and jam density 184 veh/mi/ln. Find the capacity, and the speed and flow at a density of 52 veh/mi/ln.

Given

vf = 56 mph
kj = 184 veh/mi/ln
k = 52 veh/mi/ln

Find

qmax, v(k) and q(k)

Start with the thinking

Greenshields is linear: v = vf(1 − k/kj).
Capacity occurs at k = kj/2, giving qmax = vf·kj/4.

Step-by-step solution

Capacity
$q ma x = v f k j /4 = 56 (184) /4 = 2, 576 v e h / h / ln$
Optimum density
$k_{0} = k j /2 = 92 v e h / mi / ln$
Speed at k
$v = 56 (1 - 52/184) = 40.17 m p h$
Flow
$q = v k = 40.17 (52) = 2, 089 v e h / h / ln$
Utilization
$q / q ma x = 81.1$

Answer: qmax ≈ 2,576 veh/h/ln; at k = 52, v ≈ 40.2 mph and q ≈ 2,089 veh/h/ln

Why the other options are there

qmax = 10,304 veh/h/ln (factor of 4 omitted)
v = 15.8 mph (relation inverted)

Reference: FE Reference Handbook — Transportation → Greenshields Model

Example 2

Greenshields flow, speed and density — Greenshields Model (2)

A freeway segment follows Greenshields with free-flow speed 58 mph and jam density 151 veh/mi/ln. Find the capacity, and the speed and flow at a density of 81 veh/mi/ln.

Given

vf = 58 mph
kj = 151 veh/mi/ln
k = 81 veh/mi/ln

Find

qmax, v(k) and q(k)

Start with the thinking

Greenshields is linear: v = vf(1 − k/kj).
Capacity occurs at k = kj/2, giving qmax = vf·kj/4.

Step-by-step solution

Capacity
$q ma x = v f k j /4 = 58 (151) /4 = 2, 190 v e h / h / ln$
Optimum density
$k_{0} = k j /2 = 76 v e h / mi / ln$
Speed at k
$v = 58 (1 - 81/151) = 26.89 m p h$
Flow
$q = v k = 26.89 (81) = 2, 178 v e h / h / ln$
Utilization
$q / q ma x = 99.5$

Answer: qmax ≈ 2,190 veh/h/ln; at k = 81, v ≈ 26.9 mph and q ≈ 2,178 veh/h/ln

Why the other options are there

qmax = 8,758 veh/h/ln (factor of 4 omitted)
v = 31.1 mph (relation inverted)

Reference: FE Reference Handbook — Transportation → Greenshields Model

Example 3

Greenshields flow, speed and density — Greenshields Model (3)

A freeway segment follows Greenshields with free-flow speed 53 mph and jam density 174 veh/mi/ln. Find the capacity, and the speed and flow at a density of 76 veh/mi/ln.

Given

vf = 53 mph
kj = 174 veh/mi/ln
k = 76 veh/mi/ln

Find

qmax, v(k) and q(k)

Start with the thinking

Greenshields is linear: v = vf(1 − k/kj).
Capacity occurs at k = kj/2, giving qmax = vf·kj/4.

Step-by-step solution

Capacity
$q ma x = v f k j /4 = 53 (174) /4 = 2, 306 v e h / h / ln$
Optimum density
$k_{0} = k j /2 = 87 v e h / mi / ln$
Speed at k
$v = 53 (1 - 76/174) = 29.85 m p h$
Flow
$q = v k = 29.85 (76) = 2, 269 v e h / h / ln$
Utilization
$q / q ma x = 98.4$

Answer: qmax ≈ 2,306 veh/h/ln; at k = 76, v ≈ 29.9 mph and q ≈ 2,269 veh/h/ln

Why the other options are there

qmax = 9,222 veh/h/ln (factor of 4 omitted)
v = 23.1 mph (relation inverted)

Reference: FE Reference Handbook — Transportation → Greenshields Model

Example 4

Greenshields flow, speed and density — Greenshields Model (4)

A freeway segment follows Greenshields with free-flow speed 60 mph and jam density 183 veh/mi/ln. Find the capacity, and the speed and flow at a density of 47 veh/mi/ln.

Given

vf = 60 mph
kj = 183 veh/mi/ln
k = 47 veh/mi/ln

Find

qmax, v(k) and q(k)

Start with the thinking

Greenshields is linear: v = vf(1 − k/kj).
Capacity occurs at k = kj/2, giving qmax = vf·kj/4.

Step-by-step solution

Capacity
$q ma x = v f k j /4 = 60 (183) /4 = 2, 745 v e h / h / ln$
Optimum density
$k_{0} = k j /2 = 92 v e h / mi / ln$
Speed at k
$v = 60 (1 - 47/183) = 44.59 m p h$
Flow
$q = v k = 44.59 (47) = 2, 096 v e h / h / ln$
Utilization
$q / q ma x = 76.3$

Answer: qmax ≈ 2,745 veh/h/ln; at k = 47, v ≈ 44.6 mph and q ≈ 2,096 veh/h/ln

Why the other options are there

qmax = 10,980 veh/h/ln (factor of 4 omitted)
v = 15.4 mph (relation inverted)

Reference: FE Reference Handbook — Transportation → Greenshields Model

Example 5

Greenshields flow, speed and density — Greenshields Model (5)

A freeway segment follows Greenshields with free-flow speed 62 mph and jam density 151 veh/mi/ln. Find the capacity, and the speed and flow at a density of 61 veh/mi/ln.

Given

vf = 62 mph
kj = 151 veh/mi/ln
k = 61 veh/mi/ln

Find

qmax, v(k) and q(k)

Start with the thinking

Greenshields is linear: v = vf(1 − k/kj).
Capacity occurs at k = kj/2, giving qmax = vf·kj/4.

Step-by-step solution

Capacity
$q ma x = v f k j /4 = 62 (151) /4 = 2, 341 v e h / h / ln$
Optimum density
$k_{0} = k j /2 = 76 v e h / mi / ln$
Speed at k
$v = 62 (1 - 61/151) = 36.95 m p h$
Flow
$q = v k = 36.95 (61) = 2, 254 v e h / h / ln$
Utilization
$q / q ma x = 96.3$

Answer: qmax ≈ 2,341 veh/h/ln; at k = 61, v ≈ 37.0 mph and q ≈ 2,254 veh/h/ln

Why the other options are there

qmax = 9,362 veh/h/ln (factor of 4 omitted)
v = 25.0 mph (relation inverted)

Reference: FE Reference Handbook — Transportation → Greenshields Model

Example 6

Greenshields flow, speed and density — Greenshields Model (6)

A freeway segment follows Greenshields with free-flow speed 66 mph and jam density 203 veh/mi/ln. Find the capacity, and the speed and flow at a density of 39 veh/mi/ln.

Given

vf = 66 mph
kj = 203 veh/mi/ln
k = 39 veh/mi/ln

Find

qmax, v(k) and q(k)

Start with the thinking

Greenshields is linear: v = vf(1 − k/kj).
Capacity occurs at k = kj/2, giving qmax = vf·kj/4.

Step-by-step solution

Capacity
$q ma x = v f k j /4 = 66 (203) /4 = 3, 350 v e h / h / ln$
Optimum density
$k_{0} = k j /2 = 101.5 v e h / mi / ln$
Speed at k
$v = 66 (1 - 39/203) = 53.32 m p h$
Flow
$q = v k = 53.32 (39) = 2, 079 v e h / h / ln$
Utilization
$q / q ma x = 62.1$

Answer: qmax ≈ 3,350 veh/h/ln; at k = 39, v ≈ 53.3 mph and q ≈ 2,079 veh/h/ln

Why the other options are there

qmax = 13,398 veh/h/ln (factor of 4 omitted)
v = 12.7 mph (relation inverted)

Reference: FE Reference Handbook — Transportation → Greenshields Model

Example 7

Greenshields flow, speed and density — Greenshields Model (7)

A freeway segment follows Greenshields with free-flow speed 52 mph and jam density 163 veh/mi/ln. Find the capacity, and the speed and flow at a density of 76 veh/mi/ln.

Given

vf = 52 mph
kj = 163 veh/mi/ln
k = 76 veh/mi/ln

Find

qmax, v(k) and q(k)

Start with the thinking

Greenshields is linear: v = vf(1 − k/kj).
Capacity occurs at k = kj/2, giving qmax = vf·kj/4.

Step-by-step solution

Capacity
$q ma x = v f k j /4 = 52 (163) /4 = 2, 119 v e h / h / ln$
Optimum density
$k_{0} = k j /2 = 82 v e h / mi / ln$
Speed at k
$v = 52 (1 - 76/163) = 27.75 m p h$
Flow
$q = v k = 27.75 (76) = 2, 109 v e h / h / ln$
Utilization
$q / q ma x = 99.5$

Answer: qmax ≈ 2,119 veh/h/ln; at k = 76, v ≈ 27.8 mph and q ≈ 2,109 veh/h/ln

Why the other options are there

qmax = 8,476 veh/h/ln (factor of 4 omitted)
v = 24.2 mph (relation inverted)

Reference: FE Reference Handbook — Transportation → Greenshields Model

Example 8

Greenshields flow, speed and density — Greenshields Model (8)

A freeway segment follows Greenshields with free-flow speed 66 mph and jam density 159 veh/mi/ln. Find the capacity, and the speed and flow at a density of 37 veh/mi/ln.

Given

vf = 66 mph
kj = 159 veh/mi/ln
k = 37 veh/mi/ln

Find

qmax, v(k) and q(k)

Start with the thinking

Greenshields is linear: v = vf(1 − k/kj).
Capacity occurs at k = kj/2, giving qmax = vf·kj/4.

Step-by-step solution

Capacity
$q ma x = v f k j /4 = 66 (159) /4 = 2, 624 v e h / h / ln$
Optimum density
$k_{0} = k j /2 = 80 v e h / mi / ln$
Speed at k
$v = 66 (1 - 37/159) = 50.64 m p h$
Flow
$q = v k = 50.64 (37) = 1, 874 v e h / h / ln$
Utilization
$q / q ma x = 71.4$

Answer: qmax ≈ 2,624 veh/h/ln; at k = 37, v ≈ 50.6 mph and q ≈ 1,874 veh/h/ln

Why the other options are there

qmax = 10,494 veh/h/ln (factor of 4 omitted)
v = 15.4 mph (relation inverted)

Reference: FE Reference Handbook — Transportation → Greenshields Model

Example 9

Greenshields flow, speed and density — Greenshields Model (9)

A freeway segment follows Greenshields with free-flow speed 70 mph and jam density 192 veh/mi/ln. Find the capacity, and the speed and flow at a density of 90 veh/mi/ln.

Given

vf = 70 mph
kj = 192 veh/mi/ln
k = 90 veh/mi/ln

Find

qmax, v(k) and q(k)

Start with the thinking

Greenshields is linear: v = vf(1 − k/kj).
Capacity occurs at k = kj/2, giving qmax = vf·kj/4.

Step-by-step solution

Capacity
$q ma x = v f k j /4 = 70 (192) /4 = 3, 360 v e h / h / ln$
Optimum density
$k_{0} = k j /2 = 96 v e h / mi / ln$
Speed at k
$v = 70 (1 - 90/192) = 37.19 m p h$
Flow
$q = v k = 37.19 (90) = 3, 347 v e h / h / ln$
Utilization
$q / q ma x = 99.6$

Answer: qmax ≈ 3,360 veh/h/ln; at k = 90, v ≈ 37.2 mph and q ≈ 3,347 veh/h/ln

Why the other options are there

qmax = 13,440 veh/h/ln (factor of 4 omitted)
v = 32.8 mph (relation inverted)

Reference: FE Reference Handbook — Transportation → Greenshields Model

Example 10

Greenshields flow, speed and density — Greenshields Model (10)

A freeway segment follows Greenshields with free-flow speed 66 mph and jam density 165 veh/mi/ln. Find the capacity, and the speed and flow at a density of 69 veh/mi/ln.

Given

vf = 66 mph
kj = 165 veh/mi/ln
k = 69 veh/mi/ln

Find

qmax, v(k) and q(k)

Start with the thinking

Greenshields is linear: v = vf(1 − k/kj).
Capacity occurs at k = kj/2, giving qmax = vf·kj/4.

Step-by-step solution

Capacity
$q ma x = v f k j /4 = 66 (165) /4 = 2, 723 v e h / h / ln$
Optimum density
$k_{0} = k j /2 = 83 v e h / mi / ln$
Speed at k
$v = 66 (1 - 69/165) = 38.40 m p h$
Flow
$q = v k = 38.40 (69) = 2, 650 v e h / h / ln$
Utilization
$q / q ma x = 97.3$

Answer: qmax ≈ 2,723 veh/h/ln; at k = 69, v ≈ 38.4 mph and q ≈ 2,650 veh/h/ln

Why the other options are there

qmax = 10,890 veh/h/ln (factor of 4 omitted)
v = 27.6 mph (relation inverted)

Reference: FE Reference Handbook — Transportation → Greenshields Model

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a vertical or horizontal alignment, or a traffic stream, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Greenshields Model contains 12 relations; you must be able to find this page in under 15 seconds.
Exam style: a curve geometry element or a capacity/flow relationship.
Unit rule: grades as decimals in curve formulas, percent in the stem.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

grades as decimals in curve formulas, percent in the stem
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.