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Fundamentals of Transportation · Wikibooks

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    From Wikibooks, under CC BY-SA 4.0, as the source states it. This page at Wikibooks.

    Transportation networks have analogs with network processes in other systems, such as water networks, structures, and electrical networks. Some of the relationships are outlined below.

    Concept Transportation (traffic) Water (pipe flow / hydraulics) Structures (linear elastic / statics) Electrical (DC circuits)
    Node law
    (conservation / equilibrium)     		Σqin − Σqout = s
    (node source/sink s)     		ΣQin − ΣQout = S
    (node withdrawal/injection S)     		ΣFx = 0,
    ΣFy = 0,
    ΣM = 0     		ΣI = 0
    (Kirchhoff’s Current Law, KCL)
    Element / link law
    (constitutive)     		Kinematics: q = k·v
    Link cost (often nonlinear): t = t(q)     		Head/pressure loss: Δp = Rh·Qn
    (linear case: Q = Ch·Δp)     		Hooke (axial): F = k·δ Ohm: V = R·I
    Also: I = G·V, G = 1/R
    Loop / compatibility law Wardrop user equilibrium (OD pair):
    fr > 0 ⇒ cr = c*
    fr = 0 ⇒ cr ≥ c*     		Loop head balance:
    ΣΔHℓ = 0     		Compatibility (parallel members):
    δ1 = δ2 = …     		ΣV = 0 around any closed loop
    (Kirchhoff’s Voltage Law, KVL)
    “Through” variable
    (adds at node)     		Flow q Discharge Q Member force F Current I
    “Across” variable
    (equal in parallel)     		Travel time / cost t or c Pressure p or head H Displacement δ Voltage V
    Resistance / conductance analog Increasing cost t(q) (generally nonlinear)
    Marginal: dt/dq     		Hydraulic resistance Rh
    Conductance Ch = 1/Rh     		Stiffness k
    Compliance 1/k     		Resistance R
    Conductance G = 1/R

    Notes

    • Hydrostatics (no-flow special case): p = p0 + ρgh (gauge form: p = ρgh).

    Symbols

    Transportation

    • q = flow (vehicles/time)
    • k = density (vehicles/length)
    • v = speed (length/time)
    • t, c = travel time / generalized cost
    • fr = route flow on route r
    • cr = cost on route r
    • c* = minimum cost for the OD pair (under user equilibrium)

    Water (hydraulics)

    • Q = discharge (volume/time)
    • p = pressure
    • H = hydraulic head
    • ρ = fluid density = m/V
    • g = gravitational acceleration
    • h = elevation
    • Rh = hydraulic resistance
    • Ch = hydraulic conductance = 1/Rh
    • n = exponent in loss law (model-dependent)

    Structures

    • F = force
    • M = moment
    • δ = displacement/deflection
    • k = stiffness

    Electrical

    • V = voltage
    • I = current
    • R = resistance
    • G = conductance = 1/R