Structural Engineering & Statics
where the load goes
Concept

Moment rotation — where it appears

The relation between the moment a connection transfers and the angle it turns through, which classifies it as pinned, rigid or semi-rigid. The boundaries are multiples of the beam's own stiffness, so the same detail is rigid in a short stiff member and semi-rigid in a long slender one.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Moment against rotation, for three real joints. Three connections on one plot, with the classification boundaries for a beam of EI/L = 14000 drawn as rays through the origin. web cleats is pinned, flush end plate is semi-rigid, extended end plate is semi-rigid. The boundaries are multiples of EI/L, so the same joint is rigid on a short stiff beam and semi-rigid on a long slender one.

Neither pinned nor rigid, which is every real connection

Frame analysis offers two options for a joint and reality supplies a continuum between them. Worse, the boundaries are not properties of the connection at all — the same end plate is rigid on a short stiff beam and semi-rigid on a long slender one.

connections · Joint stiffness The end bolts do the work and the middle ones very nearly nothing. A lap of 10 bolts at 75 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 1.16 of their nominal share and the middle ones 0.89. 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 0.861, and the end bolt has to slip 1.16 mm before the rest catch up.

The joint that has to be as good as the member

A splice exists because members come in lengths and structures do not. It has to deliver the same force, at the same stiffness, in the same distribution across the section, through a discontinuity — and each of those three requirements is met by a different feature of the detail, with the third one usually left to look after itself.

connections · Splice The loaded end stops; the unloaded one takes over. The moments at the four floor beam ends of the frame of two 8 m bays and two 4 m storeys, its beam ends joined to the columns by end plates of 3.0 EI/L initial stiffness and 120 kN·m resistance, under λ times 20 kN/m on the left bay and 20 kN at each floor, against λ: on the softening joints (solid) and on joints that keep their initial stiffness (dashed). At λ = 1 the loaded beam's inner end carries 78 kN·m and the unloaded beam's inner end, across the same column, 13. The loaded end reaches its resistance at λ = 2.69; at λ = 3 the four ends carry 89, 120, 81, 88 kN·m, against 97, 235, 39, 68 on stiff joints. The unloaded inner end has gone from 13 to 81, 6.2 times as much for three times the load.

The joint that softens under its own load

A bolted end plate is stiff until its moment passes two thirds of its resistance and then softens until, at its resistance, it is a third as stiff. Put such joints at every beam end of a frame under a pattern load and a push, and the most loaded joint softens first and hands its share across the column to the joint on the other side, which carries six times as much for three times the load. The redistribution grows as the frame approaches its strength. Every joint ends at its own stiffness, so the single secant stiffness an elastic analysis is allowed asks the loaded joint for half as much again as it can carry, and gives the exterior panel three quarters of what it gets.

internal-forces · Corner moment 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.

Joint stiffnessConnectionEnd platePattern loadingRedistributionRotation capacitySwayBolt groupClassificationContinuityContraflexureDuctility

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