Three Bridges, One Crossing (2D)

One crossing, three load paths

All three bridges span the same distance L and carry the same deck load w (dead load plus the steel of the structure itself, solved by iteration) and the same traffic. What differs is how the load reaches the ground.

Suspension: the parabolic cable

A cable that carries a uniform load per horizontal metre hangs in a parabola of sag f. Taking moments about midspan gives the horizontal pull and the tension at the tower:

H = w L² / (8 f)   Tmax = H √(1 + (4f/L)²)

The tower carries w L / 2 plus the vertical part of the backstay, and the backstay pulls on a gravity anchorage. Real cables hang as a catenary, not a parabola; at f/L = 1/10 the difference in H is about 1.3 % (check 3). Under a partial load the stiffening truss and the cable share the load. With inextensible hangers and the linear (small-deflection) theory the cable takes the extra pull Hq = ∫ M0 y dx / ∫ y² dx and the girder bending moment is M(x) = M0(x) − Hq y(x), where M0 is the simply supported moment and y = 4 f x (L − x) / L².

Arch: the same equation, upside down

A parabolic arch in uniform compression obeys the same H = w L² / (8 f). Here H is a push into the abutments and the rib force at the springing is N = H √(1 + (4f/L)²). For a three-hinged arch the live load bends the rib: Hq = M0(L/2) / f and M(x) = M0(x) − Hq y(x). Because a flat arch (small f) needs a large thrust, steel per metre grows quickly as the rise falls.

Cable-stayed: a fan of short suspension problems

Each stay carries the deck panel between its neighbours, spacing s = L / (2n) on each side of a tower. For two cable planes the tension per stay and the compression it puts into the deck are:

Ti = w s / (2 sin αi)   Ndeck = Σ 2 Ti cos αi   Σ 2 Ti sin αi = w L / 2

The last identity is check 2. The deck is a continuous beam on elastic supports, so between stays it only has local bending; the page uses the fixed-fixed bound q s² / 12 for a uniform load and the exact fixed-fixed formulas for the truck. Steel for the stays is ρ Σ Ti ℓi / σ, plus the tower and only the part of the deck compression that exceeds the girder steel which the deck has anyway for bending (0.2 m²).

Steel per metre of deck

Member mass is m = ρ Σ F ℓ / σallow (force times length over allowable stress, as in the Maxwell and Michell volume arguments). The deck girder itself is the same for all three and sits inside wd; the structure's own weight is added to w and the loop is iterated to convergence. Graph 2 repeats the whole calculation for spans of 150 to 1,400 m. In this simplified model the cable-stayed bridge wins at short and medium spans and the suspension bridge wins beyond roughly 600 to 700 m; the arch is the most expensive in steel because its rib is limited by buckling (a 0.75 factor on its allowable stress), although in practice arches are often built from concrete or masonry which are cheap in compression.

Dynamics: frequency, resonance and flutter

The first vertical mode comes from a beam model: for the suspension bridge the antisymmetric cable mode ω = (2π/L) √((H + EI (2π/L)²) / m) (exact for a shallow inextensible cable, Irvine); for the cable-stayed deck a beam on elastic foundation ω² = (EI (π/L)&sup4; + kstays) / m; for the arch the same with the stiffness from the axial strain of the rib. Vortex shedding at the deck has frequency fs = St U / D and excites the deck near U = f1 D / St. The forced response uses the dynamic amplification 1 / √((1 − r²)² + (2 ζ r)²) with r = fs / f1.

Flutter is estimated with Selberg's formula Uf = 0.44 k b ωθ √((r/b)(1 − (ωh/ωθ)²) μ), with half-width b, radius of gyration r, mass ratio μ = m / (π ρair b²) and shape factor k (1 streamlined box, 0.8 open truss, 0.45 bluff plate girder). The torsion-to-bending frequency ratio ωθ/ωh = (B / 2r) √(1 + boost) is the key: a wide deck hung from two cable planes starts near 1.7, and a closed box raises it. The Tacoma Narrows bridge (1940) had a bluff plate-girder deck, a narrow width and little torsion stiffness.

Numerical checks run in the page

(1) Uniform load: the formula H = wL²/8f is reproduced by the numerical linear theory of the suspension girder (integrals by the trapezoid rule) and by the three-hinge moment balance of the arch, and the girder bending falls to zero. (2) Stay vertical equilibrium, and the lowest string frequency from a finite-difference eigenvalue problem solved by inverse iteration against the exact (π/L) √(H/m). (3) The parabola against a true catenary solved by bisection.

Honest simplifications

The sizing is a conceptual-design model, not a bridge design code: two vertical cable planes, one load case, small-deflection linear statics (no deflection theory for the suspension bridge, no second-order effects, no creep or temperature), stays and hangers sized at one allowable stress, no fatigue, no anchorage or foundation design, rib buckling reduced to one factor, side spans of 0.35 L (suspension) and 0.5 L (cable-stayed) and the arch treated as three-hinged. The frequencies, flutter speed and vortex amplitudes are order-of-magnitude estimates: torsional frequencies and aerodynamic coefficients should come from wind-tunnel tests. Absolute tonnages should not be compared with real bridges; the trends, ratios and the physics of the load paths are the point. The scene shows generic bridges and is not affiliated with any real bridge.

• Load Balanced 500 m crossing and read the table: the arch and suspension girders feel big bending moments, the cable-stayed deck only local panel bending.
• Set the traffic pattern to Left half then Full span: with a full uniform load the suspension girder and the arch rib have zero bending (check 1) while half the span bends them.
• Drag Main span from 300 to 650 m and watch graph 2: where do the steel curves cross?
• Lower the arch rise to 0.1: the thrust, rib force and steel jump, because H = wL²/8f.
• Choose Gale, bluff plate-girder deck and raise the wind to 40 m/s: the suspension deck passes its flutter speed (rolling motion grows), the stiffer systems do not.
• Turn on the load-path flow layer: bright dots show the force travelling deck, hanger or stay, cable or rib, tower or abutment, ground.