TRN-01

Greenshields speed–density

Linear v = vf (1 − k/kj); q has a parabolic maximum.

Reading speed

Traffic flowGreenshields 1935HCM

Governing equation

v=v_f\left(1-\dfrac{k}{k_j}\right),\quad q=kv,\quad q_{\max}=\dfrac{v_f k_j}{4}

$k \in \left[1, 200\right]$

where

v_f: Free speed (km/h)
k_j: Jam density (veh/km)
k: Density (veh/km)
v: Speed (km/h)
q: Flow (veh/h)
q_{\max}: Capacity (veh/h)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

v=v_f\left(1-\dfrac{k}{k_j}\right),\quad q=kv,\quad q_{\max}=\dfrac{v_f k_j}{4}

$k \in \left[1, 200\right]$

$k = 1$ $v = 99.29$

Along the curve

②

Domain of the variable — blue

k_{\min} = 1,\quad k_{\max} = 200

$k \in \left[1, 200\right]$

$k = 1$ $k = 1$

Along the curve

③

Response curve — green

v=v_f\left(1-\dfrac{k}{k_j}\right),\quad q=kv,\quad q_{\max}=\dfrac{v_f k_j}{4}

$k \in \left[1, 200\right]$

$k = 1$ $v = 99.29$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}v}{\mathrm{d}k}

$k \in \left[1, 200\right]$

$k = 2.252$ $v^{\prime} = 96.78$

Along the curve

Lecture brief

Historical brief

Highway and traffic theory crystallized with Greenshields’ 1935 speed–density line, then AASHTO stopping sight and equivalent axles. The lab is that operational arithmetic of flow, braking and pavement load. This sheet (TRN-01 — Greenshields speed–density) is the form associated with Greenshields 1935 · HCM. Working symbols: $v_f$ , $k_j$ , $k$ $\rightarrow$ $v$ , $q$ , $q_{\max}$ . Greenshields postulated a linear drop of speed with density. Flow q = k v then peaks at k = kj/2 with qmax = vf kj / 4.

Purpose

Purpose: compute

v

q

q_{\max}

from

v_f

k_j

k

in Transport via

v=v_f\left(1-\dfrac{k}{k_j}\right),\quad q=kv,\quad q_{\max}=\dfrac{v_f k_j}{4}

Linear v = vf (1 − k/kj); q has a parabolic maximum. Use it when a real transport question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

v_f = 100.000\,\mathrm{km/h}

k_j = 140.000\,\mathrm{veh/km}

k = 40.000\,\mathrm{veh/km}

, the governing relation

v=v_f\left(1-\dfrac{k}{k_j}\right),\quad q=kv,\quad q_{\max}=\dfrac{v_f k_j}{4}

yields

v = 71.429\,\mathrm{km/h}

q = 2857.143\,\mathrm{veh/h}

q_{\max} = 3500.000\,\mathrm{veh/h}

. Single homogeneous lane, no ramps. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Free speed v_f100 km/hJam density k_j140 veh/kmDensity k40 veh/km

Outputs

Speed v71.429 km/h
Flow q2857.143 veh/h
Capacity q_{\max}3500.000 veh/h

Reading speed

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Narration of this film

Single homogeneous lane, no ramps.

Greenshields postulated a linear drop of speed with density. Flow q = k v then peaks at k = kj/2 with qmax = vf kj / 4.

Reading speed

Watch on YouTube