Cross the hangers and the bending goes

A tied arch is two curved-and-straight chords with the thrust running between them: the arch pushes outward and the tie holds it in, so nothing horizontal reaches the ground. Between them hang the hangers, carrying the deck up to the arch.

Under a load covering the whole span it works exactly as advertised. The arch is in compression, the tie is in tension at the same magnitude, and the frame analysis reproduces the closed-form thrust $wL^2/8f$ to two parts in ten thousand.

Take the load off half the span and the picture changes completely, and the reason has nothing to do with arches.

Vertical hangers make a Vierendeel #

Two chords joined by members perpendicular to them is not a truss. It is a Vierendeel frame, and the defining property of a Vierendeel is that it has no way of carrying shear axially.

In a triangulated truss, a shear across the depth is resolved into a tension in one diagonal and a compression in another. There are no diagonals here. So a shear between the arch and the tie has to be carried by bending in the chords, exactly as it would be in a frame with no bracing.

A uniform load produces no shear between the chords, because the arch’s shape is funicular for it — which is why the whole-span case looks so well behaved. A partial load does produce shear, and the chords bend.

For the arch drawn — 100 m span, 17 m rise, load on half of it — that comes to 3,316 kNm in the tie and 6,234 kNm in the arch, on members whose axial forces are 4,400 kN. Those are the moments a plate girder of that span would carry, in a structure whose whole justification is that it carries load axially.

Inclining them makes a truss #

Give the hangers a slope and they can carry a component along the chords’ direction. That is all that is needed: a member at an angle between two chords resolves a shear into axial forces, which is the definition of truss action.

The same load on the same arch with the hangers crossing at 60 degrees gives 686 kNm in the tie and 831 kNm in the arch — factors of 4.8 and 7.5.

The chords have stopped being beams. They are now the chords of a truss whose web is the hangers, and the moments left in them are the small residuals of a system that is nearly pin-jointed.

That is a change of structural system rather than an optimisation. The geometry of the chords is identical, the number of hangers is identical, the load is identical; what changed is whether a shear can find an axial path.

Which free body produced the number #

Cut vertically through the whole structure at a section within the loaded half, and draw everything crossing the cut.

The arch crossing it carries an axial force and, if it is bending, a moment and a shear. The tie carries the same. And however many hangers the cut passes through carry their own axial forces, which have vertical components.

For vertical hangers, the cut passes through at most one hanger, and its force is vertical — so it contributes to the vertical equilibrium of the free body and nothing to the horizontal. The shear across the section has to be carried by the two chords’ own shears, which means bending.

For crossing hangers, the cut passes through several, some inclined one way and some the other. Their vertical components sum to the shear across the section, and their horizontal components add to the chord forces. Nothing has to bend.

The whole difference is how many hangers a vertical cut crosses, which is a statement about the arrangement and not about the members. That is why Per Tveit’s arrangement is called a network: the criterion is that every section is crossed by enough hangers to resolve its shear.

The hanger that can only pull #

The price is that a hanger has no compression capacity. A slender rod or a locked-coil cable goes slack the moment its computed force turns negative, and a slack hanger has taken its triangle out of the truss.

Under the half-span load drawn, five of the sixteen inclined hangers come out in compression. The vertical arrangement has none — a vertical hanger under a downward load is always in tension, which is the one thing it is good at.

That is the trade, stated plainly. Vertical hangers never go slack and never carry shear. Inclined hangers carry the shear and go slack under the load cases that produce the most shear.

The Vierendeel comparison is exact, not an analogy #

It is worth checking that the frame really is a Vierendeel rather than merely resembling one, because the whole argument rests on it.

A Vierendeel truss carries shear by bending in its chords and its verticals together, in double curvature, with points of contraflexure near the middle of each member. That is exactly what the vertical-hanger arch does under a partial load: the tie bends between hanger points, the arch bends between hanger points, and the hangers — if they were stiff enough to matter — would bend too.

They are not stiff enough to matter, which is the one difference and it makes things worse rather than better. A Vierendeel’s verticals are substantial members that take their share of the bending; a hanger is a rod or a cable with essentially no flexural stiffness, so it contributes nothing and the two chords carry the whole of the shear between them.

A tied arch with vertical hangers is a Vierendeel frame whose verticals have been removed from the bending system, which is the worst arrangement of the two. That is why the moments come out as large as they do.

Designing for slackness rather than against it #

The response is not to prevent hangers going slack, which is impossible for a structure with a moving load. It is to have enough of them that losing some does not matter.

That is why network arches have so many. A conventional tied arch has ten or twelve hangers; a network arch has thirty or forty, at a steep slope, crossing each other three or four times. With that many, a load arrangement that slackens a third of them still leaves a triangulated system at every section.

Two design rules follow, and both are unusual.

The load arrangement matters more than the load. The envelope over arrangements is what governs, and the governing case for hanger slackness is not the one that maximises any force — it is the one that most unbalances the structure. That is an unfamiliar search: most envelopes are searched for a maximum of some quantity, and this one is searched for a pattern — the arrangement that removes the most web from the truss, which is not the arrangement that produces the largest anything.

The hangers are pre-tensioned. Installing them with a locked-in tension shifts the whole force range upward, so a case that would have slackened a hanger merely reduces its tension. That is a self-stress state added on purpose, and it costs nothing but the tensioning.

The shallow arch that becomes possible #

The consequence that decides whether the type is used is not about moments at all. It is about the rise.

A conventional tied arch needs a substantial rise — a fifth of the span or more — because it is relying on arch action to keep the chord moments manageable, and arch action needs depth. A network arch’s chord moments are small regardless, so the rise can be reduced.

The arch drawn is at a rise-to-span of 0.17. Network arches are built at 0.15 and below, which is much shallower than a conventional tied arch of the same span and is the whole visual signature of the type.

That matters for three reasons: less steel in the arch, less material and wind area above the deck, and — for a railway or road crossing — a lower structure over the navigation or the clearance envelope. The efficiency shows up as a slenderer arch rather than as a lighter one, and that is the form the saving usually takes.

The same question asked of a stay system #

The arrangement question here has a close relative one bridge type away, and setting them together shows what is general and what is not.

A cable-stayed bridge’s stays also run at an angle between two chords — the pylon and the deck — and their arrangement is also a design variable with fan, harp and semi-fan alternatives. But a stay’s inclination is doing something different: it is supporting the deck vertically at intervals, and its horizontal component is a compression fed into the deck rather than a shear resolved between two chords.

The distinguishing question is whether a vertical cut crosses more than one of them, and in which directions. A stay system’s cut crosses several stays all leaning the same way; a network arch’s crosses several leaning both ways. Only the second can resolve a shear, and that is why a stay system’s deck still carries substantial local bending while a network arch’s does not.

Inclination alone is not truss action; opposing inclinations are. That distinction is worth carrying to any two-chord system with a web: what matters is whether the web can produce a force couple across a section, and a set of parallel members cannot.

What the chords are actually designed for #

With the bending gone, the chords’ design changes character in a way worth stating.

The tie becomes an almost pure tension member. It is usually the deck itself, or a steel box within it, and its design is governed by the thrust, by fatigue at its splices, and by the fact that it is the one element whose failure is not survivable — a tied arch with a failed tie is a mechanism.

The arch becomes an almost pure compression member, and its design is governed by buckling rather than by strength. Two modes matter: in-plane buckling, restrained by the hangers, and out-of-plane buckling, restrained by whatever bracing runs between two arch ribs — or, on a single-rib arch, by nothing at all.

That is the type’s real design problem, and it is one the network arrangement makes worse rather than better. A slenderer arch at a lower rise carrying the same thrust is a more critical compression member, and the moments that used to be there were at least accompanied by a section large enough to carry them.

Which arrangement of crossings #

The hangers’ arrangement is a design variable with more freedom in it than most, and there is no closed-form answer.

The variables are the number of hangers, their slope, whether the slope is constant or varies along the span, and whether they are attached at equal intervals on the arch, on the tie, or on neither. Tveit’s own designs use a constant slope over most of the span with a variation near the ends, where a constant slope would put the attachments outside the structure.

What the arrangement has to achieve is stated more easily than computed: every vertical section crossed by hangers of both inclinations under every load arrangement, and no hanger so shallow that its axial force becomes large. A steeper hanger is more efficient at carrying vertical load and worse at carrying shear; a shallower one is the reverse.

It is an optimisation over a discrete arrangement against an envelope of load cases, which is why it is done by search rather than by formula and why the published arrangements differ so much between designers.

What it costs to build #

The arrangement has a construction cost that the mechanics does not show, and it is the main reason the type is rarer than its efficiency suggests.

Every hanger crossing is a detail. Thirty or forty hangers crossing three or four times each is a hundred or more intersections, and at every one of them two hangers pass within millimetres of each other. They must not touch, must not clash under any load case, and must not chafe.

The connections are at awkward angles. A vertical hanger meets the arch at a nearly constant angle along the whole span; an inclined one meets it at an angle that varies, so every connection is a one-off.

Erection is harder. A network arch is a mechanism until enough of its hangers are tensioned, so the sequence has to be planned as a structural problem rather than as a logistical one. Conventional tied arches are often erected with the arch complete and the hangers added afterwards; that does not work when the hangers are the web.

Against those, the steel saving is substantial — published comparisons put a network arch at half to two thirds the steel of a conventional tied arch of the same span. The material is cheaper and the fabrication is dearer, and which of the two dominates decides where the type is built.

The load case that does not exist for a beam #

There is a load arrangement question here with no counterpart in ordinary bridge design, and it is worth naming because it changes what “worst case” means.

For a beam, the worst arrangement is the one that maximises a force somewhere — and every envelope in bridge design is built by searching for maxima. For a network arch, the arrangement that matters is the one that slackens the most hangers, and that is not the same search.

A load on half the span slackens five hangers. A load on a quarter, or on two disconnected quarters, or on alternate panels, slackens different sets — and the governing case is whichever leaves the fewest working crossings at some section, which may produce quite modest forces everywhere.

The design case is a topology rather than a magnitude. A structure whose web can disappear has to be checked for the arrangement that removes the most of it, and that arrangement has to be found by enumeration because no influence line points at it.

Where the model stops #

The analysis is linear and elastic. Hanger slackness is a non-linearity: a structure with slack hangers is a different structure, and computing the forces linearly and then noting which came out negative is an approximation rather than a solution.

The deck’s own stiffness is left out. A real tie is a deck with transverse stiffness of its own, which distributes a partial load before it reaches the hangers.

Out-of-plane behaviour is absent. The arch’s buckling out of the plane is the type’s governing check and this two-dimensional model cannot see it.

The thrust is the same in both and the tie is not checked here. Everything on this page is about the bending; the tie’s tension, its splices and its fatigue are unchanged by the hanger arrangement and are what actually governs a tied arch’s deck.

And the hangers are treated as straight bars. Real hangers sag, vibrate, and are cables with their own dynamics — and hanger vibration under wind and rain is a live problem on several built examples.

Where the ladder goes #

Later rungs on this anchor: non-linear analysis with hangers that carry no compression, which is how these are really designed. Hanger arrangement as a search problem, and the published families. Out-of-plane arch buckling and the bracing that prevents it. Hanger vibration, fatigue at the hanger connections, and the replacement details that follow. The tie as a deck, and the fatigue of a member that is simultaneously a tension chord and a bridge deck. Tveit’s own comparisons of steel weight against conventional arrangements. And the general principle underneath: that whether a two-chord structure is a truss or a frame is decided by the angle of its web members and by nothing else.