Structural Engineering & Statics
where the load goes
Concept

Connection design — where it appears

The sizing of the region where two members meet, in which none of the assumptions member theory rests on hold. Plane sections and Saint-Venant's principle both exclude the disturbed region, so the analysis is an equilibrium model of a stress field rather than a check on a section.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

The end bolts do the work and the middle ones very nearly nothing. A lap of 8 bolts at 70 mm pitch transferring 800 kN between two plates, with the force each bolt actually carries drawn above it and the flat line a division by the bolt count would have given drawn behind. The end bolts carry 1.09 of their nominal share and the middle ones 0.94. The reason is not in the bolts: at the leading end the first plate is carrying everything and the second nothing, so the two strain at different rates and the slip between them is largest there. In the middle they strain alike, there is almost no slip, and a bolt with no slip across it transfers almost no force. The mean over the worst is 0.918, and the end bolt has to slip 1.36 mm before the rest catch up.

The bolts that do not share

Every bolted connection in this collection has divided a force by a number of bolts. That is right for a short joint and wrong for a long one, and the reason has nothing to do with the bolts — it is that the plates they join are elastic, and stretch by different amounts at different points along the lap.

connections · Long joint Six ways for one dowel to fail, and the capacity is the smallest. Johansen's single-shear mechanisms for a 12 mm dowel through 40 and 40 mm members, each drawn as the shape it is: the dowel straight and the timber crushing, the dowel rotating rigidly, one plastic hinge, then two. The capacity under each is that mechanism's own, and the joint's strength is the smallest — 5.02 kN by mode c, which is the dowel rotates rigidly and both members crush. That is the kinematic theorem of plasticity: every mechanism gives an upper bound and the true collapse is the lowest of them. The crushed timber is shaded, and the circles are plastic hinges in the steel.

The smallest of six failures

Everything else in this collection that fails does so in one way at a time. A dowel through timber does not — the wood can crush while the steel stays straight, or one plastic hinge can form in it, or two — and the capacity is the smallest of the six, which is the kinematic theorem of plasticity applied to a joint rather than to a frame.

connections · Dowel yield The width nobody drew. A gusset plate with a brace bolted to it over 240 mm, and the width the profession has agreed to pretend is carrying the force. Everything else on this site arrives with a cross-section; a gusset does not, because it is a piece of steel with something attached somewhere in the middle of it and there is no geometry that says how much of it is working. The answer is the Whitmore section: assume the force spreads at 30° from the first fastener and take the width it has reached at the last, b_eff = w + 2L·tan30° = 367 mm. That is 4.08 times the width anything is actually attached to, and the rule comes from a 1952 master's thesis. It has since been checked against finite element work and holds to about ten per cent, which is fortunate, because moving the assumed angle by ten degrees moves the answer by 34%. On this plate the check that governs is not the stress the rule was written for: it is the Whitmore section buckles, at 721 kN against 1564.

The width nobody drew

Every other member in this collection arrives with a cross-section. A gusset plate does not — it is a piece of steel with a brace bolted to it somewhere in the middle, and no geometry says how much of it is working. The profession's answer is a thirty-degree spread from a 1952 master's thesis, it invents three quarters of the area being checked, and the check it was written for is not the one that governs.

connections · Gusset Force and capacity round a weld group, which do not vary together. Utilisation round the c shape group under 100 kN at 150 mm, walked from one end of the weld to the other, with the force scaled onto the same axis for comparison. The capacity is 1460 N/mm where the force runs along the weld and 1789 where it runs across it — √1.5 more — so the utilisation is not simply the force in different units. The worst point is at 47 degrees to the weld, where the capacity is 1612 N/mm, and the group carries 182 kN against the 165 the along-the-weld value would allow.

The weld that is stronger where it is pulled

A fillet weld pulled across its axis is √1.5 stronger than the same weld pulled along it. On a group under an eccentric load the direction of the pull varies from point to point, so the capacity does too — and the gain that follows is worth twenty per cent at one eccentricity and nothing at all at another.

connections · Weld strength Where the span moment and the support moment cross. Span moment and support moment against the joint's stiffness, for a 6 m beam under 30 kN/m. They move in opposite directions because they add to a constant — the simple-span 135 kN·m is fixed by statics and the joint only decides how it is split. They cross at 68 kN·m, where the joint is delivering 75 per cent of the fixed-end moment, and neither the crossing nor the fraction depends on the beam, the span or the load: it is the point where f·wL²/12 equals wL²/8 − f·wL²/12, which is f = 0.75 for every beam there has ever been.

The joint that was chosen

A joint's stiffness decides how a beam's moment divides between its span and its supports, and the two add to a constant. So there is a stiffness at which they are equal, the beam is sized by the smaller of two numbers rather than the larger of one, and the design moment is half what a simple connection leaves behind.

connections · Joint classification The joint yields first and the span takes the rest. The end and span moments of a beam of span 9 m and flexural rigidity 61,700 kN·m², with a plastic moment of 522 kN·m, on joints of rotational stiffness 41,133 kN·m/rad, 6.00 EI/L, and resistance 261 kN·m, as a uniform load rises to collapse. While elastic the joints carry 0.75 of the fixed-end moment. The joints reach their resistance first, at 51.6 kN/m, and from there every further increment of load goes to the span, which reaches 522 kN·m at 77.3 kN/m. The collapse load is 8(Mp + Mj)/L² = 77.3 kN/m, between the 51.6 at which the beam would collapse on pins and the 103.1 it would reach on rigid full-strength ends, and it contains the joint's strength and not its stiffness.

The joint that has to keep turning

A beam on partial-strength joints collapses at 8(Mp + Mj)/L² however stiff the joints are. Stiffness decides only which yields first, and a joint that yields first has to go on rotating at full moment until the span catches up. A weak joint on a stiff connection has the most turning to do — more than a rigid full-strength one.

connections · Joint classification Made from one side, the throat misses the load and its root takes the difference. Two plates 20 mm thick joined by a partial-penetration butt weld with 10 mm of throat, pulled apart by a force on their mid-thickness, with the stress across the throat drawn beside each joint as a multiple of the mean. Above, welded from one side: the throat runs from the top face down to 10 mm, so the force passes 5.0 mm below its centroid. The moment adds tension at the root and removes it at the face: the root carries 4.00 times the mean stress and the face −2.00, which is compression. The unfused 10 mm below the root is a notch, and it is on the tension side. Below, the same 10 mm of throat split between the two faces: the force passes through its centroid and every part of it carries the mean stress.

The throat that misses the load

A partial-penetration butt weld's throat faces the load squarely, so a millimetre of it carries more than a fillet's, and in every grade what limits it is a cap on normal stress rather than the directional criterion. Made from one side, the throat sits off the plates' line of action. At half penetration it keeps a quarter of that capacity, and the stress the eccentricity adds lands on the root, which is a crack.

connections · Weld strength A butt throat is strongest a little off square. The resultant a millimetre of throat can carry in S355 against the direction of the load, from straight across the weld (0°) to straight along it (90°). The fillet's falls throughout, from 385 to 314 N/mm². The butt's rises from 441 to 478 at 22.7°, because its cap limits only the normal stress and a little shear costs it nothing, and then falls to 314. So the butt's advantage over the fillet is 1.15 straight across, 1.29 at 22.7°, and none straight along.

The shear that helps a butt weld

A partial-penetration butt throat pulled straight across its weld is capped at 0.9fu, which in S355 makes it 15 per cent stronger than a fillet throat. Add a little shear along the weld and the butt throat gets stronger, not weaker, because the cap limits only the normal stress — until the shear is 42 per cent of the pull, where it is 29 per cent stronger than the fillet and carries its largest resultant, 478 N/mm² against 441 straight across. Along a moment connection's web weld the mix of pull and shear sweeps through that peak, and in the direction that is all shear a single-sided throat is as good as two.

connections · Weld strength Out in compression, back in tension. The axial stress in a bare S355 beam (section factor 160 per metre) whose ends are held by the structure around it with 10 per cent of its own axial stiffness, against its temperature, through a compartment fire with an opening factor of 0.04 m^½ and 200 MJ/m² of fuel: heating (solid) and cooling (dashed), with the yield stress at each temperature dotted either side. Heating, the beam pushes against its ends and reaches 134 N/mm² of compression at 628 °C, where the yield stress has fallen to meet it; it then yields, shortening plastically as it follows the falling yield stress up to its peak of 940 °C. Cooling, the thermal expansion comes back out and the plastic shortening does not: the beam passes through zero and ends at 20 °C in 195 N/mm² of tension.

The tension a fire leaves behind

A steel beam held at its ends by the structure around it is pushed into compression as a fire heats it, and yields, because its strength is falling while its expansion is not. When the fire goes out the expansion comes back and the yield does not. Held at a tenth of its own stiffness, a beam that peaked at 134 N/mm² of compression on the way up ends the fire at 195 N/mm² of tension — more than it was ever pushed, arriving hours after the fire was out, and on connections that were designed to carry shear.

materials · Fire The more it carries, the less it pulls when the fire is out. For a 9 m 457 × 191 × 67 steel beam held at its ends at 10 per cent of its own axial stiffness through the fire: the tension pulling on its ends once it has cooled (solid) and the largest tension while it is still hot (dashed), against the load it carries as a share of its cold plastic moment; dotted, the straight beam's 1,669 kN. At 0.10, 1,017 kN cold and 41 hot; at 0.39, 613 and 293; at 0.54, 477 and 397. From 0.56 the beam can find no sag that carries its load near the fire's peak and runs away.

The beam that pays the fire back in sag

A steel beam held at its ends and kept straight through a fire yields short on the way up and pulls on its connections with its plastic shortening on the way down. A beam carrying a floor does not stay straight. Squeezed by its own expansion, it forms a hinge at about 500 °C and bows out of the way, hangs as a catenary while still hot, and keeps a metre of sag when it cools — length it no longer has to be pulled for. It ends the fire pulling a third as hard as the straight beam, nearly whatever holds it, and the hard pull comes earlier instead: at 800 °C, on connections that are as hot as it is.

materials · Fire A floor's robustness belongs to its joints until they break, and to its slab after. For a floor whose interior column is lost, two 8 m-span beams crossing over it (each collapsing at 400 kN) and a slab 16 m square over the four bays round it: the largest floor load a sudden loss can be held at, as a multiple of the two beams' collapse load, against how far the beams' connections can turn before they fracture, for no mesh and for 142, 252 and 393 mm² a metre. No mesh: 0.94 at 0.03 radians and 1.19 at 0.07; 142 mm²: 0.94 at 0.03 radians and 1.29 at 0.07; 252 mm²: 1.50 at 0.03 radians and 1.81 at 0.07; 393 mm²: 2.26 at 0.03 radians and 2.56 at 0.07.

The slab that catches what the joints let go

When an interior column is lost, the beams that crossed over it sag into cables and the slab over the four bays round it sags with them, into a membrane hanging from the floor that did not lose anything. Whether the floor's robustness belongs to its steelwork or to its concrete turns on one number with no span in it: the slab's mesh only starts to carry its full tension when the beams' ends have turned through about 0.09 radians, more than any ordinary joint can follow. So the membrane arrives after the joints have gone. And then it catches the floor — a metre down, alone — and with an ordinary mesh it holds half as much again as the beams could.

structures · Robustness

Named alongside it

The objects these essays reach for when they reach for this one.

DuctilityBearingDirectional methodEccentricityEnd platePlastic hingeStress combinationWeld strengthCatenaryContinuityDesign momentDetailing

All concepts