TRN-09

Greenshields flow

q = k v, v = vf (1 − k/kj). A linear speed–density fundamental diagram.

Reading speed

TrafficGreenshields

Governing equation

q=k v_f\left(1-\dfrac{k}{k_j}\right)

$k \in \left[1, 180\right]$

where

k: Density (veh/km)
v_f: Free speed (km/h)
k_j: Jam density (veh/km)
q: Flow (veh/h)
v: Speed (km/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.

00:0 / 00:08

①

Governing law — red

q=k v_f\left(1-\dfrac{k}{k_j}\right)

$k \in \left[1, 180\right]$

$k = 1$ $q = 89.36$

Along the curve

②

Domain of the variable — blue

k_{\min} = 1,\quad k_{\max} = 180

$k \in \left[1, 180\right]$

$k = 1$ $k = 1$

Along the curve

③

Response curve — green

q=k v_f\left(1-\dfrac{k}{k_j}\right)

$k \in \left[1, 180\right]$

$k = 1$ $q = 89.36$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}q}{\mathrm{d}k}

$k \in \left[1, 180\right]$

$k = 2.126$ $q^{\prime} = 87.27$

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-09 — Greenshields flow) is the form associated with Greenshields. Working symbols: $k$ , $v_f$ , $k_j$ $\rightarrow$ $q$ , $v$ . Capacity is at k = kj/2, qmax = vf kj / 4.

Purpose

Purpose: compute

q

v

from

k

v_f

k_j

in Transport via

q=k v_f\left(1-\dfrac{k}{k_j}\right)

q = k v, v = vf (1 − k/kj). A linear speed–density fundamental diagram. Use it when a real transport question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

k = 30.000\,\mathrm{veh/km}

v_f = 90.000\,\mathrm{km/h}

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

, the governing relation

q=k v_f\left(1-\dfrac{k}{k_j}\right)

yields

q = 2121\,\mathrm{veh/h}

v = 70.7\,\mathrm{km/h}

. A density, a speed, a q–k parabola. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Density k30 veh/kmFree speed v_f90 km/hJam density k_j140 veh/km

Outputs

Flow q2121 veh/h
Speed v70.7 km/h

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A density, a speed, a q–k parabola.

Capacity is at k = kj/2, qmax = vf kj / 4.

Reading speed

Watch on YouTube