The deck that spans square

A bridge crossing a road at an angle is drawn as a parallelogram. The carriageway runs one way, the abutments run another, and the angle between them — the skew — is set by the alignment of two roads that were laid out for reasons that had nothing to do with structures.

Everything about the deck’s behaviour follows from one sentence, and the sentence is not about skew at all. Load takes the stiffest route to a support. On a rectangular deck the stiffest route is along the span, because that is the shortest way to something that holds it up. On a skew deck the shortest way to something that holds it up is the perpendicular between the two abutments, and that is not the direction the road runs.

The square span, which is shorter than the drawing #

For a deck of carriageway length $L$ at skew angle $\theta$, the perpendicular distance between the abutments is

$$L_{sq} = L\cos\theta.$$

At 30 degrees that is 87 per cent of the drawn span; at 45 degrees, 71 per cent; at 60 degrees, half. Since the stiffness of a plate strip goes as the inverse cube of its span, a strip running square is 1.5 times stiffer than one running along the road at 30 degrees, and four times stiffer at 45. Load goes where the stiffness is, and at high skew there is no contest.

The consequence is a set of numbers that at first look contradictory. Compared with a right deck of the same carriageway length, a 45-degree skew slab has:

• a longitudinal moment of 59 per cent,
• a maximum deflection of 57 per cent,
• and an obtuse-corner reaction 2.42 times the average.

The first two fall because the deck has found a shorter span. The third rises because the load that took the shorter route has to land somewhere, and where it lands is the two corners at which the abutments come closest together.

Zero skew is the check, not the input #

The grillage that produced those numbers was given no instruction about corners. It was given a mesh of longitudinal members along the road, transverse members parallel to the abutments, a uniform load, and a line of supports along each abutment.

Run it at zero skew and the reaction concentration comes out 1.00. That is the check on the model: a right deck loads its abutment uniformly, everybody knows it, and a model that had to be told so would be worth nothing. Run it at 30 degrees and it returns 2.15, at 45 degrees 2.42, and at 60 degrees 2.31 — the concentration turns over near 50 degrees, because by then the deck is spanning so nearly perpendicular that the two obtuse corners have become the whole support.

The acute corner, which stops being a support #

The other end of the same distribution is the one that causes trouble on site.

At 30 degrees the acute corner is carrying 16 per cent of the average reaction per unit length. At 45 degrees it is at six per cent. It is, to any practical purpose, not a support at all: the deck is hanging over it rather than resting on it.

Two things follow. The bearing there is nearly unloaded, which means it can chatter, unseat, or — under an eccentric live load, or a load near the opposite obtuse corner — go into uplift. A bearing designed for a downward reaction and delivered an upward one is a detailing failure rather than a strength one, and it is the commonest defect on skew decks.

And the deck twists. A plate whose support reactions fall away toward one corner is being lifted at that corner relative to the rest, which is a twist, and twist in a plate is carried by torsional moments rather than by bending. The corner region of a skew slab is in almost pure torsion, which puts its principal stresses at 45 degrees to everything the reinforcement is parallel to.

Which free body produced the number #

Take the whole deck as the free body first, because the global statement is the one that constrains everything else.

Vertical equilibrium says the two abutment reactions sum to the applied load, whatever the skew — that is not in question and the grillage reproduces it to twelve figures. Moment equilibrium about the line of one abutment is where the skew enters: the load’s resultant acts at the centroid of the parallelogram, and the lever arm to each abutment line is the perpendicular distance, which is $L\cos\theta$ rather than $L$. So the total on each abutment is unchanged and the arm over which it is developed has shortened.

That is the global half. The distribution along each abutment is not fixed by equilibrium at all — it is a compatibility question, decided by which parts of the deck are stiff enough to attract load. Cut a strip running square between the abutments near an obtuse corner and it is short; cut one near an acute corner and it is long, and it also runs out of deck at both ends. The short strips take the load because they are stiffer, and the short strips land in the obtuse corners.

The corner is a different structure #

Everything above treats the deck as one object with an uneven distribution on it. The corner regions are better read as separate structures, and doing so explains three details that otherwise look like superstition.

Near an obtuse corner the deck is short in the square direction and supported on two sides that meet at an angle greater than a right angle. That is a stiff, heavily loaded, two-way-supported triangle, and it attracts hogging moments across its own diagonal — a negative moment on the top face, running from the corner into the deck, on a slab whose main reinforcement is in the bottom. Top steel in the obtuse corners is not a detailing convention; it is where the moment is.

Near an acute corner the deck is long in the square direction and barely supported, so it is a cantilever hanging off the last stiff strip. It wants to lift, and the corner region is in torsion. Bottom steel across the corner and a hold-down at the bearing are the two things it asks for.

And along the free edges the deck is trying to span longitudinally after all, because there is nothing to the side of it to shed to. The edge is a beam whether or not one was drawn, and an edge stiffening is the usual answer.

The principal moments rotate #

The reinforcement question is the sharpest practical consequence, and it is one about direction rather than about quantity.

In a right deck the principal moment directions are along and across the span, which is where the reinforcement goes, and the two problems coincide. In a skew deck they rotate toward the square direction as the skew increases — not all the way, because the free edges along the carriageway still want to span longitudinally, but far enough that at 45 degrees the largest moment is at a substantial angle to both reinforcement directions.

Steel laid in two orthogonal directions can carry a moment field at any angle to it, but not for free: the yield criterion for an orthogonally reinforced slab requires more total steel to carry a skew moment field than an aligned one, and the excess grows with the angle. So a skew deck reinforced along the road is not wrong, it is inefficient, and the inefficiency is invisible in a calculation that resolves the moments into the reinforcement directions and checks each.

The practical answers are three, and each is used. Reinforce square — bars perpendicular to the abutments — which suits high skew and makes an awkward junction at the free edges. Reinforce along the road, which suits low skew and is easy to detail. Or reinforce along the road and add a heavy corner mesh, which is what most decks do and which is an admission that the corner is a different structure from the rest.

Where the boundary is drawn, and why it moves #

Every code that mentions skew draws a line somewhere between 20 and 30 degrees, below which a skew deck may be analysed as a right deck of the square span and above which it may not.

The numbers above say where the line comes from. At 20 degrees the concentration is 1.81 and the moment ratio 0.91; at 30 degrees they are 2.15 and 0.81. The moment error from treating a 20-degree deck as right is under ten per cent and conservative, because the real deck spans shorter than the model assumes. The reaction error is not conservative and is already 80 per cent at 20 degrees.

So the boundary is really two boundaries that have been collapsed into one. For the moment, treating a skew deck as right is acceptable well past 30 degrees and is conservative. For the reaction, it is unacceptable almost immediately and is unconservative. A single angle in a clause is a compromise between them, and the compromise is safe only because the reaction is usually checked by a bearing schedule that has its own margins rather than by the analysis.

What it does to the abutment #

The abutment sees the mirror image of the deck’s problem, and it sees it as a longitudinal distribution rather than as a point.

A right deck delivers a nearly uniform line load along the bearing shelf, and the abutment is designed as a wall spanning between its own supports under a uniform load. A skew deck delivers a load that is two and a half times the average at one end of the shelf and almost nothing at the other. The wall is then a member under a strongly triangular load, its own bending moment is roughly doubled, and its base pressure distribution follows.

There is a second effect on top of that, and it is the one that makes skew abutments distinctive. The deck’s own thermal movement and its braking forces act along the road; the abutment resists them along its own normal. The two are at the skew angle to each other, so every longitudinal force on the deck has a component sliding along the abutment face, and a skew bridge tends to walk sideways over its life. The remedy is a bearing arrangement that admits it rather than one that pretends the movement is along the road.

The torsion is the mechanism, not a side effect #

It is worth being explicit that the whole of the redistribution above depends on the deck’s torsional stiffness, and that a deck without it behaves completely differently.

A slab has torsional stiffness in abundance — twice its bending stiffness per unit width, which is what lets a plate carry a load by twisting as well as by bending. That is what allows a strip near the acute corner to shed its load sideways to the shorter strips and what produces the corner concentration.

A beam-and-slab deck at the same skew does not do this to nearly the same extent, because its longitudinal stiffness is concentrated in beams that run along the road and its transverse stiffness is a thin slab. The beams cannot span square — they are beams — so the load stays roughly where the tributary rule puts it, and the corner concentration is much weaker.

That is one of the few places in this collection where the choice of deck type changes not the magnitude of an effect but whether the effect exists. A skew slab is a skew problem; a skew beam-and-slab deck is mostly a detailing problem, and knowing which one is being designed decides how much of this page applies.

The measurement that could refuse it #

The model above makes a prediction that a bearing schedule can falsify, and it is worth stating because a claim that cannot be checked against the built structure is weaker than one that can.

Bearings are usually fitted with load cells or at least specified by capacity, and the capacities on a skew deck should be strongly graded along each abutment — a factor of two or more between the obtuse and acute ends at 30 degrees, in the ratios the grillage returns. A schedule with equal capacities along the abutment either belongs to a deck the analysis treated as right, or belongs to one where somebody decided to standardise and absorb the difference. Both happen; the first is a defect and the second is a decision, and the schedule looks identical.

The stronger test is the acute corner. If the analysis says the reaction there is six per cent of the average, the uplift case under an eccentric live load is a short calculation away, and its answer should appear as a hold-down in the detail. A skew deck whose acute corner bearing has no hold-down and no calculation showing it does not need one has a gap in it, and that gap is visible on a drawing without any analysis at all.

What it costs to get right, which is nothing #

Unlike most of the corrections in this collection, the remedy here is free at the right moment and expensive at any other.

The analysis is a grillage, which every bridge office runs as a matter of course, and skewing its mesh costs a line in the input rather than a different program. The reinforcement layout is a decision at scheme stage: square bars, road-direction bars, or road-direction bars with a corner mesh, and all three are ordinary detailing. The bearing schedule is graded rather than uniform, which changes a table.

What is not free is discovering the concentration later. A bearing that is undersized by a factor of two is replaced by jacking the deck, which on a live road is a possession, a temporary support system and a great deal of money. A corner that has cracked because there was no top steel is a repair to the wearing surface as well as to the slab.

So the cost profile is the familiar one for a geometric effect: negligible if the geometry is taken seriously when it is drawn, and dominated by access if it is not. The alignment that produced the skew was fixed before any structural decision was taken, and it is the input that decides most of the behaviour.

Where the model stops #

The grillage is an approximation to a plate. It gets the load path and the reaction distribution well and it does not give a stress field; the corner region, where the moment field is most rotated, is where it is least reliable.

The supports are treated as a line of rigid props. Real bearings are discrete, compressible and sometimes on a flexible bearing shelf, all of which soften the concentration — helpfully, and by an amount that depends on details the deck analysis does not contain.

No live load has been placed. Everything here is uniform loading, which is the case that isolates the geometry. An eccentric vehicle near an acute corner is the case that lifts it, and it is a different calculation with the same machinery.

And the skew is assumed to be the only irregularity. A deck that is skew and curved, or skew with unequal spans, has two effects interacting, and the corner that governs may not be either of the ones this page names.

Where the ladder goes #

Later rungs on this anchor: uplift at the acute corner under eccentric live load, and the hold-down that follows. Reinforcement layouts for skew slabs and the yield criterion for orthogonally reinforced plates under a rotated moment field. Beam-and-slab decks at skew, where most of this does not apply. Skew abutments and the longitudinal force component along the bearing face. Curved and skew decks together. Integral skew abutments, where the thermal ratchet acts at an angle to the deck. And the general form of the argument: a support layout that changes not how much a structure carries but which way it spans.