Structural Engineering & Statics
where the load goes
Concept

Joint classification — where it appears

The division of connections into nominally pinned, semi-rigid and rigid by their rotational stiffness relative to the beam's. It is a licence rather than a description: a joint in the first or third class may be modelled as a pin or as fully continuous, and one in the middle has to be modelled as it is.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

A force may be moved anywhere, at the price of a couple. A 80 kN force applied 250 mm off the centreline of a body, and the same force applied ON the centreline together with a couple of 20 kNm. The two systems are equivalent: they have the same resultant force and the same moment about every point in space, so no equilibrium equation written about the body can tell them apart. What they are not is the same loading — the stresses inside the body differ, and they differ over a distance of about the body's own depth. The offset is drawn to a scale that keeps the arrow on the body; the number beside it is the real one.

The moment the beam left behind

A beam reaction is drawn arriving on a column's centreline. It arrives on a cleat a hundred millimetres out from the face, and the difference is a couple that goes into the column and has to be shared between the lengths above and below it. Nothing about it appears in a frame model whose members meet at nodes.

internal-forces · Column eccentricity 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 What the frame notices. An 8 m by 4 m portal whose columns stand on bases of every rotational stiffness, in units of the column's EI/L on a logarithmic scale: solid, the portal's sway under a load at the knee as a multiple of the fixed-based portal's; dashed, the columns' effective length factor in sway (fixed bases 1.13, pinned 2.26). Bolts sleeved 300 mm, 5.3 EI/L: 1.45 times the fixed portal's drift, an effective length factor of 1.30 and 75 per cent of its buckling load; as detailed, 9.1 EI/L: 1.28 times the fixed portal's drift, an effective length factor of 1.23 and 83 per cent of its buckling load; rigid by EN 1993-1-8, 30.0 EI/L: 1.09 times the fixed portal's drift, an effective length factor of 1.16 and 94 per cent of its buckling load.

The fixed base the bolts decide

A column base drawn as fixed is a spring, and the component method says what it is made of: the grout squeezed under the flange, the plate bending under the bolt row, and the anchor bolts stretching over their free length. For a 500 mm plate on two M24 bolts the bolts are three fifths of its flexibility, so the base comes out at nine times the column's EI/L — under a third of what EN 1993-1-8 calls rigid — and no thickness of plate can lift it above fifteen. Leave the bolts free in sleeves for erection tolerance and it falls to five. The frame on it drifts 28 per cent more than a fixed-based one and buckles at 83 per cent of the load.

connections · Base plate A bigger top column attracts more moment, and only the joint stops it. The top storey of the edge column of a braced six-storey frame — 4 m storeys, a 254 UC 89 below the splice and a 203 UC 60 above — carrying a 9 m beam at every floor on partial-strength joints of 261 kN·m, with each of seven columns that might be chosen for it: the moment its floor's edge joint delivers at the design load of 60 kN/m (dark) and at the beams' collapse load (light), against what the column can carry at its top end, its plastic moment reduced for the axial force (line). 203×203 UC 46: 148 and 195 against 167; 203×203 UC 60: 171 and 225 against 230; 203×203 UC 71: 189 and 247 against 283; 203×203 UC 86: 205 and 261 against 347; 254×254 UC 73: 218 and 261 against 352; 254×254 UC 89: 232 and 261 against 435; 254×254 UC 107: 243 and 261 against 527. A stiffer column attracts more of the joint's moment until the joint's 261 kN·m caps it; the lightest column that carries what it attracts at collapse is the 203 × 203 UC 60.

The column on the other side of the joint

A partial-strength joint is sold as a fuse: it caps the moment a beam can put into its support at the joint's own resistance. At an interior column that is true. At the edge of a frame the column is not a support that stays still; it is a spring in series with the joint, so the joint delivers less than its resistance in service and almost exactly what a rigid joint would. The column then carries three times the moment the simple-construction rule gave it, and at the top storey, where it stands alone and is lightest, it is weaker than the joint it was meant to be protected by.

connections · Joint classification The first hinge comes latest where the beam's end and its span yield together. The load factor at which the first plastic hinge forms in the two-bay frame of 8 m bays and 4 m storeys, its beams of plastic moment 405 kN·m, under λ times 20 kN/m on the left bay and 20 kN at each floor, against the initial stiffness of its full-strength end plates (1.20 times the beam's plastic moment), softening as EN 1993-1-8 has it and kept linear. With softening, 3.39 at 100 EI/L, 3.78 at 25 and 3.91 at 3; kept linear, 3.32, 3.51 and 3.91. Stiff joints hinge first at the inner end and flexible ones in the span; the first hinge comes latest between, 4.52 at 10.0 EI/L with softening and 4.31 at 5.0 kept linear. Rigid joints give 3.25, and the beam collapses at 5.06 for every joint.

The joint stronger than its beam

A full-strength joint never reaches its own resistance: the beam beside it hinges first. But an end plate softens from two thirds of its resistance, and a joint only a little stronger than its beam is already on the soft part of its curve before the beam yields. The softening sheds moment from the beam's end into its span, so it delays the first hinge — by 8 per cent for a joint at 1.2 times the beam's strength, 18 for one at exactly the beam's, and not at all from 1.5 on. It never moves the collapse. And too much shedding hands the first hinge to the span, so the first hinge comes latest at the stiffness where the end and the span yield together.

internal-forces · Corner moment

Named alongside it

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

Rotational stiffnessContinuityDesign momentEnd plateSemi-rigidBase plateConnection designInteractionJoint stiffnessMoment distributionRedistributionAnchor bolt

All concepts