The connection is busiest where the beam is not

Joining two beams into one makes the pair four times as stiff, and what does the joining is a row of studs whose spacing is usually a round number written once on a drawing. The demand those studs are meeting is not a round number and is not uniform.

The interface follows the shear, not the moment #

The reason is the same one that makes horizontal shear the shear nobody draws: the force per unit length at an interface is $VQ/I$, and $V$ is the shear force.

A beam’s shear diagram and its moment diagram are the derivative and the integral of each other, so on a simply supported span under a uniform load they are as far apart as two curves can be. The moment peaks at mid-span and is zero at the supports; the shear is zero at mid-span and peaks at the supports. The connection’s busiest section is the beam’s idlest one.

That has a consequence a designer meets before understanding it: the studs run out at the ends of the beam and there are more of them than the mid-span needs. It also explains why a beam with a point load at mid-span needs a different arrangement from one under a uniform load — the shear is constant either side of a point load, so the demand is constant too, and the uniform spacing that is a compromise in one case is exact in the other.

It also inverts the usual relationship between a member and its detail. In most of this collection the detail follows the analysis: the moment is computed, the section is chosen, the connection is designed for what the section can deliver. Here the connection is designed from a different diagram than the section, at a different location, and the two calculations share only the load case. A designer who checks the beam at mid-span and the studs at mid-span has checked one thing twice.

Which free body produced the number #

Take a length $dx$ of the top plank alone, cut along the interface and across the member at each end.

Crossing the two end cuts is the compressive force in that plank, which is the integral of the bending stress over its own area. Crossing the interface is the shear traction. If the moment changes along the beam then the plank’s compression changes too, and the difference has to arrive through the interface — so

$$q = \frac{dC}{dx} = \frac{V Q}{I}$$

The interface force is the rate of change of the flange force, which is why it depends on the shear rather than on the moment, and why it is zero wherever the moment is stationary.

That free body also says what the connectors are for, which is not obvious from the finished beam. They do not carry any of the applied load. They carry the difference between what the top plank has at one section and what it has at the next, and their whole job is to stop the two faces sliding past each other.

There is a second and less obvious consequence of the same free body. Because the interface force is a rate of change, anything that changes the flange force abruptly demands an abrupt transfer. A point load, a change of section, a curtailed cover plate and the end of the beam are all places where $dC/dx$ is locally very large, and all four are places where a uniform stud spacing has no idea anything is happening.

That is why the codes’ otherwise-uniform rule carries a list of exceptions: additional connectors at concentrated loads, at the ends of cover plates and at free ends. Each of them is a place where the smooth $VQ/I$ picture has a step in it, and a step in a shear flow is a finite force to be transferred over almost no length.

What uniform spacing is really claiming #

Placing studs evenly when the demand varies by a factor of 1.81 is not sloppiness. It is a claim about the connector.

A headed stud in a solid slab reaches its capacity at about a millimetre of slip and holds it to six or more. It is, in other words, ductile in exactly the sense a reinforcing bar is: it can be overloaded and will shed what it cannot carry to its neighbours without losing what it already has.

So the uniform arrangement is a plastic distribution, and it is legitimate for the same reason the variable-angle truss is legitimate: the lower-bound theorem permits any equilibrium state the structure can reach, and a ductile connection can reach the uniform one.

The corollary matters more than the rule. A brittle connector cannot be spaced uniformly. A bolt in a bolted composite beam, a notched connection in timber, an epoxy bond — none of them redistributes, so each has to be placed according to the elastic diagram, which is why those systems have close spacing at the ends and wide spacing in the middle while a studded steel beam does not.

The number that decides how composite it is #

Stud count is the wrong variable, and the right one is a dimensionless group.

$\alpha L$ collects the connector stiffness, the two flexural stiffnesses and the axial stiffness of the layers into one number, and the whole of the partial-interaction behaviour is a function of it. Two beams with the same αL behave identically however they got there — many soft connectors or few stiff ones, a long span or a short one.

The shape of that curve is the practical finding. It is steep at the left and flat past about 15, so a real design sits on the flat part where changes to the connection do very little. That is why halving the studs costs most of nothing and why partial shear connection is a viable design strategy rather than a compromise.

Slip, and why it is the quantity that hides #

The interaction curve is drawn against stiffness, and what the stiffness produces is a movement. It is worth looking at the movement directly, because it is small in a way that misleads.

The 12 m beam in the first figure slips 0.47 mm at its end. That is less than the tolerance on almost anything else in the structure, it is invisible, and it is the entire mechanism by which the beam is 96 per cent composite rather than 100. The difference between a composite beam and two loose ones is half a millimetre of movement at each end, distributed along an interface that never moves more than that anywhere.

Two consequences follow and both are about detection.

Nothing on a finished floor shows the degree of interaction. A beam at 60 per cent connection and one at 100 look identical, deflect within a few per cent of each other under service load, and differ substantially only in their ultimate capacity. A construction error in the studs is therefore invisible on the completed structure and shows up only in a load test.

And a connection that has degraded gives no warning. Corrosion at the interface, a cracked slab, studs welded through paint — each reduces the connection, each moves the beam left along the interaction curve, and the curve’s flatness that made partial connection attractive now works the other way: a large loss of connection produces a small change in deflection, right up to the point where it does not.

Two ends of the same beam #

Putting numbers on both bounds makes the size of the effect concrete.

Those two figures are the argument for building composite floors and the argument for not over-connecting them, in one pair. The whole range from loose to bonded is a factor of four in deflection and two in stress; the range from a generous connection to a mean one is a fifth of that.

What “degree of shear connection” actually means #

Two quantities are called the degree of interaction and they are not the same, which is a persistent source of confusion.

The stiffness measure is the one plotted above: how far the beam’s deflection sits between the loose and the bonded bounds. It is a serviceability quantity, it is what αL predicts, and 96 per cent of it is bought easily.

The strength measure is the ratio of the shear force the connectors can transfer to the force needed to develop the full plastic moment — usually written η. It is an ultimate-limit-state quantity, and it is linear in the number of studs in a way the stiffness measure is not.

The two disagree in a useful direction. A beam at η = 0.6 — six studs where ten would give full connection — is well over 90 per cent composite in stiffness, so it deflects almost as little as a fully connected beam while carrying appreciably less moment. Partial connection buys stiffness cheaply and capacity expensively, which is exactly the right shape for a floor beam, since floor beams are usually governed by deflection.

That is the whole design case for partial shear connection, and it rests on the two measures having different shapes rather than on either being approximate.

The stud in a rib #

There is one detail that undoes a good deal of the above, and it is geometric rather than mechanical.

A stud welded through a profiled metal deck sits in a rib, not in solid concrete. The concrete in front of it is a wedge of the rib rather than a half-space, so it can push out sideways, and the stud’s capacity falls — by 20 to 50 per cent depending on the rib’s proportions and on whether the deck’s ribs run along the beam or across it.

The effect is worse than a strength reduction, because it also removes ductility. A stud whose concrete splits out rather than crushing has a falling load-slip curve, and a connector with a falling curve is not ductile, and a connection that is not ductile cannot be spaced uniformly. One detail of the decking therefore changes both the capacity and the design method, and it is chosen by whoever is procuring the deck.

That is the reason the reduction factors for studs in ribs are among the most heavily tested and most frequently revised numbers in composite design: they are a property of a profile that manufacturers keep changing.

There is one arrangement where the rib effect is reversed, and it is worth knowing because it is the commonest one in practice. When the deck’s ribs run parallel to the beam, the stud sits in a continuous trough of concrete rather than in a discrete pocket, and the reduction is much smaller. When they run transverse, the stud sits in a rib of finite length and the reduction is large — and transverse is the arrangement that makes structural sense for the deck, because a deck spans between beams.

So the geometry that is right for the slab is wrong for the connector, and every composite floor is a compromise between the two. That is not a defect anybody can remove; it is the consequence of using one component for two jobs, which is what composite construction is.

The hogging region, where it comes apart #

Everything above is a sagging region. Over an interior support of a continuous composite beam the slab is in tension.

Concrete in tension cracks, so the slab’s contribution to the section is its reinforcement alone, and the composite section over a support is a steel beam plus a few bars. Meanwhile the shear at a support is the largest it gets, so the shear flow at the interface is the largest it gets.

The interaction collapses exactly where the demand on it peaks. A continuous composite beam is therefore a variable-stiffness member — very stiff in the span, much less so over the support — and its analysis needs the cracked length to be known before the moments can be found, which needs the moments to be known first.

That circularity is resolved by iteration or by a code rule assigning a fixed cracked length, usually 15 per cent of the span either side of the support. It is one of the least satisfying rules in the subject and it is standing in for a genuinely non-linear analysis.

What to carry away #

Design the connection from the shear diagram. The moment diagram says where the section is worked hardest; the shear diagram says where the interface is. On a uniformly loaded span those are opposite ends of the beam.

Uniform spacing is a plastic argument, not a simplification. It is available for ductile connectors and unavailable for brittle ones, and whether a connector is ductile can be decided by the shape of a deck profile rather than by anything about the stud.

And the interaction curve is flat where designs live. Stud count is a poor variable for controlling stiffness and a good one for controlling capacity, because the two measures of “how composite” have different shapes — which is a fact about the arithmetic and is the reason a floor can be under-connected on purpose.

Where the model stops #

The connectors are smeared. Newmark’s solution treats the connection as a continuous elastic medium of stiffness $K$ per unit length. Real studs are discrete, and near a large point load the smearing fails — the stud nearest the load carries a share the continuous model does not predict.

The layers are elastic and uncracked. In steel-and-concrete construction the slab cracks in the hogging region and the steel yields at the ultimate limit state, and neither is in the equation.

Slip is assumed small enough not to change the geometry. At 0.47 mm it is; at the several millimetres a plastic design permits, the lever arm between the two layers’ centroids has changed.

Uplift is ignored. The two layers are assumed to stay in contact. A stud resists separation as well as slip, and codes require it to be able to — a headed stud is headed for that reason and for no other.

The two layers are assumed to have the same curvature. They do, if they stay in contact, and the model uses that to relate the slip to the interface force. A beam whose layers can separate has two curvatures and a different equation.

And nothing here computes a fatigue life. The connection carries a stress range at every passage of a live load, at its worst near the supports, and a stud is a welded detail with a poor category.

Three other places on this site carry a force along an interface rather than across a section, and the arithmetic is the same in all of them. A force that arrives along a length is the general statement; shear across a crack that is already there is the concrete version, where the interface is rough rather than studded; and a section made of two materials is what the interface has to deliver for the composite section to exist at all.

The ladder from here #

Later rungs on this anchor: partial shear connection at the ultimate limit state, where the question is a moment capacity rather than a deflection and the answer is a straight line between two points. The stud’s own load-slip curve measured rather than assumed, and the push-off test that produces it. Continuous composite beams with the cracked hogging region resolved by iteration. Composite columns, where the same slip argument runs vertically and there is no bending to drive it. Timber-concrete floors, where the connection is a notch or a screw, αL is genuinely small, and everything comfortable here stops being true. And the historical case: composite action was present in every riveted plate girder and every filler-joist floor for fifty years before anybody counted it, which made a great many old structures considerably stronger than their own calculations.

Newmark’s partial-interaction solution is from 1951 and is one of the few closed-form results in this subject that has never been improved on. What has changed since is everything on the other side of it — the connector, whose behaviour is measured rather than derived, and whose capacity is now a function of a deck profile rather than of a material.