Every section was somewhere else

Incremental launching builds a bridge deck on the bank, in segments, and pushes it out over the piers. It needs no falsework over whatever is being crossed — a river, a railway, a motorway, a gorge — which is the entire reason it exists.

It also subjects every cross-section of the deck to a history that has nothing to do with the structure that will eventually stand there, and the history is worse than the structure.

The reason is one sentence. In service each section has one bending moment. During the launch, each section passes over every support and through every span.

The envelope of envelopes #

An ordinary continuous bridge has a moment envelope: at each point along the deck, the worst sagging and the worst hogging over all the load arrangements. The envelope is a function of position along the bridge, and the section is varied along it — deeper over the piers, shallower at midspan, more top steel over the supports and more bottom steel in the spans.

A launched bridge’s envelope is a different object. It is a function of position along the deck, and each point of the deck occupies every position along the bridge at some moment of the launch. So the envelope at a given point of the deck is the envelope over all bridge positions, which is very nearly the same at every point.

Every section has to be able to take the worst of everything. The consequence is visible in every launched bridge ever built: a constant-depth box with symmetric top and bottom flanges and prestress in both faces, which is not what a continuous bridge would otherwise be.

That is a real cost. A variable-depth deck of the same span uses substantially less material, and it cannot be launched.

The ordinary object is worth having beside it, because the two are called by the same word and are not the same kind of thing. Here is the envelope a fixed three-span deck of the same 45 m spans has, built by loading each span and not loading it in every combination.

That is already an envelope that is not a structure — no single arrangement produces it, and the deck is designed for a curve nothing draws. A launched deck’s envelope is that object taken one level further out: an envelope over launch positions of an envelope over load arrangements, at every section, and by the time both have been taken the result is nearly flat.

It is worth noticing how unusual that situation is. Load arrangement produces an envelope on an ordinary structure too, and the envelope is a shape: hogging over the supports, sagging in the spans, and the section follows it. Here the envelope is nearly uniform along the deck, which is the one shape a variable section cannot exploit. The structure has been deprived not of capacity but of the variation that a designer would otherwise use.

The cantilever before it lands #

The severest single moment during a launch is easy to identify and easy to compute.

At some point in the advance, the deck’s leading end has left the last pier and has not yet reached the next. It is a cantilever, growing at whatever speed the jacks push, and at the instant before it lands its overhang is a full span. A free cantilever of length $L$ under its own weight $w$ carries

$$M = \frac{wL^2}{2}$$

at its root — which for the 45 m span and 200 kN/m deck drawn is 202,500 kNm. The in-service hogging over a pier for the same deck is 40,500.

A factor of five, for a condition lasting a few hours, on a structure that will spend a hundred years carrying a fifth of it. Designing the deck for that would be absurd, and the alternative is not to reduce the moment but to prevent the cantilever from getting that long.

The nose, which carries nothing #

A launching nose is a light steel extension bolted to the front of the deck, typically a quarter of the deck’s weight per metre and half to three quarters of a span long.

Its purpose is not to carry anything. It is to arrive — to reach the next pier while the deck’s own front end is still well short of it, so that by the time the heavy part is out over the gap it already has a support in front of it.

For the deck drawn, the worst moment anywhere during the launch is 170,156 kNm with no nose and 42,539 kNm with a nose of 0.66 spans — a saving of 76 per cent, and a launch envelope within four per cent of the in-service one.

It is a threshold, not a taper #

The shape of that curve is the finding, and it is not what a designer expects from an optimisation.

At no nose the worst moment is 170,156 kNm. At 0.4 spans it is 140,625 — barely improved. At 0.6 spans it is 44,648, and past that it is flat.

The reason is geometric. The nose helps only if it lands on the next pier before the deck’s own front has advanced far enough to produce the governing moment. A nose that is too short reaches the pier only after that moment has already occurred, so it has no effect on the maximum at all: the deck has already had its worst moment, and the nose arrives afterwards.

There is a length at which the nose starts working and almost nothing between the two states. That is why nose lengths cluster so tightly in practice, around 0.6 to 0.7 of the span, and why the choice is not treated as an optimisation.

Which free body produced the number #

The free body is the deck at one instant of the launch, and it is the sequence of them rather than any single one that matters.

At any position, the deck is a continuous beam on a set of supports whose positions relative to the deck are different at every step: the piers are fixed in the world and the deck slides past them. The supports move, the loading does not, and the moment diagram changes shape continuously.

There are two kinds of state and they alternate. When the deck has $n$ piers under it, it is a well-behaved continuous beam and the moments are close to the in-service ones. When its leading end is between piers it has $n-1$ under it, and the front overhang is a cantilever. The launch is a sequence of continuous-beam states punctuated by cantilever ones, and every large moment belongs to the second kind.

That is also why the launch has to be traced rather than checked at a few positions. The governing state is between two configurations rather than at one of them, and the fineness of the step matters.

Reading it as a moving load, backwards #

There is a way of placing this problem among the ones this collection already contains, and it makes the analysis method obvious.

An influence line answers the question where should the load stand for a fixed structure and a moving load. A launch asks the mirror image: the load is fixed to the structure and the supports move. Change coordinates to the deck and the piers become a set of moving supports; change coordinates to the world and the deck becomes a moving load whose length is the whole bridge.

Neither change of frame makes the problem easier, because an influence line is a property of a structure with fixed supports and this structure does not have any. What the analogy does supply is the right instinct: trace it, do not check it. The governing configuration is not at any obvious position and has to be found by sweeping.

It also says which quantity to sweep for. In an influence-line problem the answer at one section is a maximum over load positions; here the answer at one section is a maximum over launch positions, and the design quantity is the maximum over sections of that — a double envelope, which is why the diagram is nearly flat.

The exchange removes the property that makes an influence line useful in the first place. An influence line is drawn once for a structure and then read for any load position; a launched bridge is a different structure at every step, so there is nothing to draw once.

Which stress, and where #

The moment envelope is the headline and there are three other actions that the launch produces and the finished bridge does not.

Web crippling at the launch bearings. The deck slides over sliding bearings at each pier, so its own web is loaded through a small patch of flange that travels the whole length of the deck. That is a patch load applied everywhere in turn, and it usually decides the web thickness and the stiffener spacing.

Shear reversal. Each section passes through regions of positive and negative shear as the supports move past it, so the shear envelope is also two-sided.

Longitudinal force. The jacks push the deck against friction on every bearing, which accumulates as an axial compression that is largest at the back and zero at the front — a load case with no analogue in service at all.

The three have one property in common and it is the same one the moment envelope has: each of them applies to every section of the deck rather than to a few. A finished bridge’s bearings sit at four or five places and its web is stiffened at those places; a launched deck’s bearing has been at every place, so the stiffening is continuous and the web thickness is the launch’s rather than the service condition’s.

Prestress in both faces #

A prestressed concrete deck launched this way needs prestress top and bottom along its whole length, which doubles the tendon quantity relative to a conventionally built deck.

The usual arrangement is a centroidal prestress — straight tendons at the section’s centroid — applied to every segment as it is cast, providing a uniform compression that keeps both faces in compression under either sign of moment. A second, draped set is added after the launch is complete, to provide the eccentric prestress the finished bridge wants.

That is an unusually explicit example of a structure being designed for two lives. The centroidal prestress is dead weight in service; the draped prestress is absent during the launch. Neither one alone is a design, and the total is more than either.

Where a tendon sits is the load put on backwards and the profile is chosen from the moment diagram it has to cancel. A draped profile suits a beam whose moment has one sign at each section; a centroidal one is the only profile that suits a member which will be hogged and sagged in turn, because it is the only one that is not wrong for either. A launched deck therefore needs both, and carries the first of them for ever.

The temporary pier, which is the other answer #

A nose shortens the effective cantilever by reaching ahead. A temporary pier does the same thing from the other end, by halving the gap.

Put a light trestle at midspan of each span and the deck never cantilevers more than half a span, so the governing moment falls by a factor of four — $wL^2/2$ with $L$ halved. That is a larger reduction than any nose achieves, and it removes the threshold behaviour entirely.

The two are alternatives and the choice is about the obstacle. A temporary pier needs ground under it, which is exactly what a launched bridge usually does not have — if there were somewhere to put a trestle there would probably have been somewhere to put falsework. So temporary piers appear on launched viaducts crossing dry, accessible ground with long spans, and noses appear on everything else.

They are also combined. A long viaduct with one difficult span across a river is often launched with a nose and with temporary piers in the accessible spans, which is a case where the same deck experiences two different launch regimes and the envelope is the worse of them at each section.

Between the cantilever states the deck is an ordinary continuous beam, and those are the states the moment over the support is about. They are well behaved and close to the in-service diagram, and the whole of the difficulty is in the transitions between them — which is the next section’s subject and the one place where a launch differs from a bridge.

What the launch buys #

The comparison is with the alternatives for the same crossing, and it is a comparison of access rather than of material.

Falsework is cheapest where the ground allows it and impossible over water, over a live railway or in a gorge.

Balanced cantilever construction builds outward symmetrically from each pier and needs no ground access either, but it needs a form traveller, a pier that can take an unbalanced moment during construction, and a segment cycle measured in days.

Launching needs a casting yard behind one abutment, a set of sliding bearings, a jacking system, and a deck that can survive its own launch. It is fastest where the spans are equal, the alignment is straight or of constant curvature, and the deck is long enough to amortise the yard.

The equal spans are not a preference. The nose is a fixed length, chosen for one span, and a span longer than the others produces a cantilever the nose cannot reach across — so a viaduct with one long span among short ones needs either a nose sized for the long one, which is oversized everywhere else, or a temporary pier in it. A construction method that is indifferent to the number of spans is very sensitive to their equality, which is the opposite of most.

That last condition is the one that decides it. A launched bridge’s fixed costs are large and its marginal costs are small, so the type appears on long multi-span viaducts and almost never on a single span.

The geometry has to be constant #

A launched deck slides along its own alignment, so the alignment has to be one the deck can slide along.

A straight bridge is trivial. A bridge on a constant-radius curve is also fine, in plan and in elevation, because the deck is cast to that radius and slides along it like a segment of a circle rotating in its own groove. A bridge with a varying radius, a transition curve, or a vertical alignment that changes cannot be launched without the deck fighting its own geometry at every step.

That is a severe planning constraint imposed by a construction method, and it works backwards into the road alignment. The bridge’s construction technique constrains the geometry of the road it carries, which is an unusual direction of influence and is settled long before any structural design begins.

The sign that changes, and what it costs the steel #

The double-sided envelope has a consequence for reinforced and prestressed decks that is easy to state and expensive to satisfy.

A section that will be hogging in service needs top steel; one that will be sagging needs bottom steel. A launched section needs both, everywhere, because it will have been each at some point.

For a reinforced concrete deck that means roughly doubling the flexural reinforcement over the whole length — not quite, because the two envelopes are not equal, but close enough that the saving from curtailment that a conventional deck enjoys is unavailable.

For a steel box it means symmetric flanges. A conventional continuous steel bridge has a heavier bottom flange in the spans and a heavier top flange over the piers, and the transitions are where the flange thickness changes. A launched one is symmetric, so half of every flange is oversized for the service condition at every section.

The launch’s cost is not a temporary works item. It is in the permanent structure, distributed evenly along it, and it is the reason the technique needs a long viaduct to pay for itself.

What the deck is during the launch #

It is worth stating plainly what kind of structure a launched deck is at each moment, because the answer changes and the changes are the design.

Most of the time it is an ordinary continuous beam on three or four supports, and its moments are close to the finished bridge’s. That state is not the problem and it occupies most of the launch.

Close, though, is not the same as bounded by, and it is worth drawing the sequence rather than accepting the sentence. Early in the launch the deck has two spans under it and nothing beyond.

That hogging moment is 50,625 kNm — 25 per cent more than the 40,500 the finished bridge will ever hog, on a deck carrying nothing but itself and standing in one of the states this section called well behaved. The next push adds a pier under it.

Adding a support has reduced the worst moment in the deck, which is the opposite of the direction the launch’s cantilever states move in and is the reason the two kinds of state have to be traced separately. A fourth span does not continue the trend.

So the continuous states run 50,625, 40,500, 43,393 as the piers arrive, and every one of them is a state the section has to survive at every point along the deck. The worst of the well-behaved states is 25 per cent above the service condition, before a single cantilever has been drawn.

Between piers the deck is a continuous beam with a long overhang, and the overhang’s root moment is the governing quantity. That state is brief and it recurs once per span.

And at the very start it is a cantilever off a casting yard, which is why the yard’s trestles are spaced far more closely than the bridge’s piers and why the first push is the one supervised most carefully. It is the most dangerous day in the ordinary sense as well as this one.

Three structures, one deck, and the section is designed for the envelope of all three — with the second of them dominating and the nose existing to shorten it.

Where the model stops #

The deck is a prismatic beam. A real launched deck varies in stiffness at its splices and carries a nose with a different section.

Support settlement was left out of the sequence above, and it is not small. A continuous beam on supports at slightly different heights redistributes moments substantially, and a launched deck slides over bearings whose levels are set by a jacking system rather than by a survey.

A jacking system’s levelling tolerance is therefore a structural quantity rather than a workmanship one, and it is the reason launch bearings are shimmed and surveyed at every push rather than set once.

The launch is quasi-static. In practice it is a stick-slip process, with friction breaking away and the deck moving in surges, which produces dynamic effects the static trace does not have.

Creep and shrinkage are absent. A concrete deck launched over weeks is young at its front and older at its back, and it creeps under the launch’s own stresses — with the strength and stiffness it had on the day rather than the specified ones.

And the trace has a step size. The governing state falls between configurations, so a coarse sweep misses the peak and a fine one costs a solve per step.

Where the ladder goes #

Later rungs on this anchor: nose design as a structure in its own right, including the joint where it meets the deck. Web crippling under a travelling patch load, which usually governs the web. Temporary piers, which shorten the effective span during launch and are the alternative to a longer nose. Launch bearing friction and the jacking force it implies. Settlement sensitivity and the levelling tolerance. Launching curved decks. The casting yard as a structure, and the segment cycle that sets the programme. And the general question this belongs to: what to do when a structure’s worst load case is one that occurs before it exists.