The shear whole spans cannot find

The train that is worse than its heaviest axle searched every position of a group of axles on a simple span and found that the worst arrangement never puts the heaviest one at the peak. It ended by naming what the same method would do on a continuous beam, where the negative regions of an influence line turn pattern loading into a search over subsets, and the shear envelope as the thing that disagrees with the moment envelope about everything.

Those turn out to be one question. A building’s continuous beam is not designed by walking a load across it. It is designed by loading its spans in a short list of patterns — every span, alternate spans, two adjacent spans and alternate ones beyond — and taking the worst of each quantity over the list. The envelope that results is not itself an equilibrium state, which is well known and harmless. What is less well known is that the list itself is chosen for the bending moment, and that for the shear inside a span it does not contain the worst arrangement at all.

A beam of three spans and a section in the middle one #

The beam here is three equal spans of 8 m, pinned at every support, with a dead load of 20 kN/m everywhere and an imposed load of 15 kN/m that may stand anywhere. Its influence lines are found literally: a unit load is placed at each of 160 points a span, the three-moment equation gives the two inner support moments for that position, and the moment and shear at the section being watched follow from statics of the span it is in. The ordinates are exact to the arithmetic, and a uniform load’s effect is the area under the line wherever the load stands.

The question asked of each section is the one an influence line answers: where must the imposed load stand to make this quantity as large as it can be? The answer is everywhere the ordinate has the sign being sought, and nowhere else. The question the design method asks is narrower: of the arrangements that load whole spans or leave them empty, which is worst? The two agree wherever the influence line keeps one sign over each span, because then every span is either worth loading or not, and a whole-span pattern can do exactly what the line asks. They disagree wherever the line changes sign inside a span.

The line that changes sign at the cut #

The section in the first figure is 3.2 m into the second span, two fifths of the way along it. Its moment influence line keeps one sign over every span: positive throughout the span it is in, negative throughout both neighbours. That is what the patterns are built on. To make this section’s sagging moment largest, load the second span and leave the first and third empty, and no partial arrangement can improve on it.

The shear line does something the moment line never does. It runs negative from the second span’s left support up to the section, jumps by exactly one as the unit load passes the cut, and runs positive from the section to the right support. A load just to the left of the section pushes that section’s shear one way; a load just to the right pushes it the other. In the first span the line is positive, and in the third it is negative, so to make the shear at this section as large as possible the load should cover the whole of the first span, the part of the second span to the right of the section, and nothing else.

No pattern of whole spans can do that. A whole-span pattern either loads the second span — both sides of the section, so that the two lobes largely cancel — or leaves it empty and loses the positive lobe altogether. The jump at the section is the reason, and it is not a property of this beam: every shear influence line jumps by one at its own section, on any beam, because the load passing the section moves from one side of the free body to the other. The moment line is continuous at the section and only has a kink there, which is why it can keep one sign across a whole span.

One span, where the answer is old #

On a simple span the consequence is long known, and the worst place to stand drew it: a uniform load over part of a span produces more shear at a section than the same load over all of it. The largest imposed shear at a distance $x$ from the left support comes from loading from the section to the far support, and it is

$$V_{\max}(x) = \frac{q\,(L - x)^2}{2L},$$

a parabola from $qL/2$ at the support to $qL/8$ at mid-span. The full-span load gives a straight line through zero at mid-span.

Hand design for a simple span has an answer for this that is both simple and safe. It draws a straight line from the support shear to $qL/8$ at the middle, which is the chord of the parabola and so lies above it everywhere: at a quarter span the line gives 37.5 kN where the beam needs 33.8. The line is tangent to the truth at its far end and conservative everywhere else. It is also, as the next section shows, the wrong thing to carry to a continuous beam, because the number at its far end belongs to the simple span.

Three spans, and the envelope with a waist #

The second figure in the beam’s own terms is the full shear envelope, dead and imposed together, with the imposed load placed exactly and then restricted to whole spans.

At every support the two envelopes coincide, because a section at a support has nothing of its own span to one side of it: the shear line there keeps one sign over each span, like a moment line, and whole spans are the exact arrangement. The largest shear in the beam, 170 kN just to the right of the second inner support, is found correctly by the patterns, and so is every support value. A designer who sizes shear reinforcement at the supports and nowhere else has lost nothing.

Away from the supports the envelopes separate, and they separate most in the middle of each span, where the exact envelope has a waist and the pattern envelope pinches nearly shut. At the middle of the second span the dead load contributes no shear at all — the span is symmetric and so is its dead load — and the whole of the design shear is imposed. Placed exactly, the imposed load produces 23.8 kN either way. Restricted to whole spans, the best it can do is 10.0 kN, by loading the first span alone and letting the difference of the two support moments drive a shear through the empty middle span. The patterns find 42% of the shear.

The simple span’s rule, borrowed #

The obvious repair is to fall back on the simple span’s line: from the support value to $qL/8$ at the middle. The line was safe there. On an inner span it is not.

Near the support the rule and the exact envelope agree, as they did on the simple span. By a third of the way along the exact curve has risen above the line, and at the middle it stands at 158% of the rule’s $qL/8$. Past the middle it stays well clear of the line, flattening towards the 10 kN that loading the first span alone produces at the far support, where the rule has fallen to nothing.

The reason the rule fails is that $qL/8$ is a fact about a span with no neighbours. The imposed load on the right half of the second span produces the simple span’s share at mid-span, less a small correction from the support moments that half-load induces. Loading the first span as well adds a shear of its own: the first span’s load hogs the first inner support more than the second, and the difference between the two support moments, divided by the span, is a shear that runs uniformly through the whole second span. That second term has no counterpart on a simple span, and it adds two thirds again to the first.

How much the patterns find #

The size of the gap depends on how many spans there are and where in the span the section is.

On a simple span the patterns have only one choice, the whole span or nothing, and past the middle they find none of the positive shear at all. On a continuous beam there is always a neighbouring span whose load drives some shear through the middle, so the share never falls to nothing, but it falls to between 42 and 58 per cent at mid-span and recovers only towards the supports. The worst position is just past the middle for two spans, at the middle for three and five. Within a fifth of a span of the left support, every continuous beam’s patterns find more than 95 per cent.

The figure is of shares, and a share can be large where the shear is small. The absolute shortfall is what matters to the reinforcement, and that is set by the next figure: how much of the design shear at mid-span is imposed in the first place.

Dead load hides it #

A large dead load makes every quantity in the beam less sensitive to where the imposed load stands, because more of each effect is fixed. At mid-span of a symmetric inner span the dead load contributes no shear, so there it hides nothing; at the quarter point it contributes most of the shear and hides most of the gap.

At the quarter point the gap is a few per cent at any mix of loads, 3 kN of 83 for this beam. At mid-span it grows in proportion to the imposed share: with dead load alone there is no shear at the middle at all, and with the whole load imposed the patterns find 23.3 kN of the 55.4 the beam can see. A beam that carries mostly its own weight is hardly affected. A beam that carries mostly imposed load — a lightweight floor under heavy storage, a crane beam, a footbridge — has a mid-span shear the patterns understate by more than half.

That is the regime in which the understatement can reach a design. Shear reinforcement in a concrete beam is laid out from the envelope; the stirrups are closed up near the supports and opened to a minimum spacing in the middle third, where the envelope says the concrete alone will do. A minimum is usually generous, and the 23.8 kN here would be carried by the concrete of any reasonable beam with room to spare. In a deep, heavily loaded transfer beam carrying mostly imposed load, or in a steel plate girder whose web is thinned or whose stiffeners are spaced out towards mid-span, the middle third is where a design that trusts the pattern envelope has the least margin it believes it has.

The moment, which the patterns nearly always find #

The patterns are not wrong about everything, and the last figure is what they were built for.

Over most of each span the two moment envelopes are indistinguishable. The moment influence line for a section in the middle part of a span keeps one sign over every span, so whole spans are exact. Near an inner support the line changes sign inside its own span — a load on the far part of the span near the support hogs the section, a load on the near part sags it — and the patterns begin to miss. A tenth of the way along the second span, the worst hogging moment is 137.7 kN·m against the patterns’ 131.2, 5% more. That is the zone over which top reinforcement is curtailed, and the moment over the support has more to say about why that region is where a design’s envelope matters most. The zone is narrow, about a fifth of a span, which is why the moment patterns have survived as a design method for a century.

So the two quantities do not disagree about everything. They disagree about where the influence line changes sign. For the moment that happens only near the supports; for the shear it happens at every section, because the shear line always jumps at its own cut.

The mid-span shear by hand #

The 23.8 kN can be checked with two standard coefficients. For three equal spans with the first span alone loaded, the first inner support moment is $-qL^2/15$ and the second is $+qL^2/60$. Their difference over the span is a shear through the second span of

$$\frac{qL^2/60 + qL^2/15}{L} = \frac{qL}{12} = 0.0833\,qL = 10.0 \text{ kN},$$

which is exactly the pattern envelope’s value at mid-span, because loading the first span is the only whole-span pattern that drives positive shear through the middle of the second.

To that the exact placement adds the right half of the second span. On a simple span that half would give $qL/8 = 0.125\,qL$. On the continuous span its load also hogs both inner supports, slightly unequally, and the difference takes 0.0104 qL away, leaving 0.1146 qL. The sum is

$$0.0833\,qL + 0.1146\,qL = 0.198\,qL = 0.198 \times 15 \times 8 = 23.8 \text{ kN},$$

which is the figure’s value. The left half of the second span, which whole-span loading cannot help including, contributes −0.1146 qL and is what cancels the right half when the whole span is loaded.

What a designer does with it #

Bridge design has always loaded partial lengths, because influence lines are where bridge design starts and a traffic lane is free to stop anywhere. Building codes allow the span patterns because a floor’s imposed load is usually small beside its dead load and because the patterns are exact where the reinforcement is decided. The defensible practice for a beam carrying a large share of imposed load is to take the support shears from the patterns, which are exact there, and to draw the shear in the middle of an inner span to about $0.2\,qL$ rather than to $qL/8$ — or to load partial spans, which any stiffness program will do if asked.

An influence line is a deflected shape gives the quickest way to see which kind of section is being designed. Cut the beam at the section, release the quantity, and move it by a unit: for a moment the released beam kinks and stays continuous, and its deflected shape keeps one sign over each span unless the cut is close to a support; for a shear the released beam steps, and the step puts opposite signs on the two sides of the cut whatever the span. A designer who sketches that step has seen why no whole-span pattern can find the shear inside a span.

What the three-moment beam leaves out #

The beam is prismatic, of equal spans, on unyielding knife-edge supports, and analysed elastically. Unequal spans change every number but not the argument: the shear line still jumps at its own section, so the patterns still miss the shear inside every span, by an amount that grows as a short span sits beside long ones. A concrete beam that cracks redistributes moment between the supports and the spans and changes the support moments’ difference, which is the very term that drives shear through an unloaded span. The imposed load is uniform and may stand anywhere; a floor whose imposed load comes in bays, or a beam carrying point loads from secondary beams, has a coarser set of arrangements and lies somewhere between the two envelopes. And the shear envelope, like the moment envelope, is a contour no single loading produces: the 23.8 kN at mid-span and the 170 kN at the support come from different arrangements, and nothing here adds them.

Still open: the slab whose load can stand anywhere in two directions #

Everything here is one-dimensional: a load either stands on a span or does not. A two-way slab is loaded over a plane, its influence surfaces have lobes in both directions, and the patterns a code allows are chequerboards of whole panels. A chequerboard is the two-dimensional version of a whole-span pattern and has the same blind spot: it can load a panel or leave it, but not the part of a panel on one side of a section. Whether a slab’s shear near a column — the punching shear that governs flat slabs — is underestimated by chequerboards in the way a beam’s mid-span shear is, or whether the column’s own surface makes every arrangement close enough, is the question an influence surface would answer, and the surface is where the argument goes next.