Structural Engineering & Statics
where the load goes
Concept

Component method — where it appears

The procedure that treats a joint as an assembly of springs — column web in shear, flange in bending, bolts in tension — each with its own stiffness and strength. The joint's rotational stiffness is their series combination, so the softest component decides it and the strongest contributes almost nothing.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

A joint is springs in series. The five components of an end-plate joint, with each bar the flexibility it contributes. The column flange in bending is 39.62% of the total on its own, and doubling its stiffness raises the joint's by a factor of 1.25 — while doubling the stiffest component buys 1.05. The joint's rotational stiffness is 25227.71 kN·m per radian.

A joint made of springs in series

A connection's stiffness is not a property of the connection. It is the series combination of the flexibilities of everything the force passes through — so the softest component decides, and stiffening any of the others changes almost nothing.

connections · Component method Prying against flange thickness. The ratio of bolt force to applied force, for a tee stub carrying 140 kN per bolt, as the flange thickness varies. Prying disappears above 31.91 mm and the flange has become a mechanism below 22.75 mm, where the shaded region begins and the bolt has stopped being the thing that decides.

The thickness that decides who fails

A bolt in a tee stub carries more than the load applied to it, because the flange bends and levers against its own edge. How much more, and whether the bolt or the flange is the thing that gives way, are both decided by one dimension — and the two regimes it separates fail in completely different ways.

connections · Prying Two ways for the same plate to fold. An end plate 200 mm wide with a bolt 45 mm from the web face and 55 mm from the edge, and the two families of yield line it can collapse along. The circle closes round the bolt and is 283 mm of hinge — a circle round the bolt. The straight pattern runs out to the plate's free edges and is 249 mm — hinges to the plate edges. The plate folds along whichever is cheaper, which here is the fan, and the 200 mm that comes out is the length of the equivalent tee stub — a dimension that is nowhere on the plate and is 100 per cent of its width.

How much of the plate is bending

A tee stub is an object nobody builds, and the whole component method rests on replacing a real end plate with one. The length of the substitute is not a dimension of the plate — it is the length of the cheapest fold the plate can collapse along, and two families of fold compete for it on a criterion with no strength in it at all.

connections · Prying The compression is under the flange, not at the edge of the plate. A 500 × 400 mm base plate, 20 mm thick, under a 260 mm column carrying 300 kN and 120 kN·m, with holding-down bolts 60 mm from the tension edge. A plate this thick can deliver bearing only 43 mm past the compressed flange, so the compression sits under that flange, 124 mm from the column's centre, and the lever arm to the bolts is 314 mm: the bolts carry 264 kN and the flange zone 564 kN. The dashed block is the rigid-plate answer, 54 mm deep at the plate's far edge with its resultant 223 mm out, which needs only 128 kN in the bolts. Both satisfy equilibrium; only one is a plate that can carry the pressure where it is drawn.

The compression that stays under the flange

Put a moment on a base plate and a pressure-block calculation pushes the compression out to the plate's edge, where the lever arm to the holding-down bolts is longest. A plate that is not rigid cannot put it there. The compression stays within a few tens of millimetres of the compressed flange, the lever arm shrinks, and the bolts carry twice what the pressure block said.

connections · Base plate Two collapses for the same plate, and only one of them fits. The same three bolt rows at 90 mm pitch, folding two ways. On the left each row folds on its own pattern — 249 mm of hinge line round each bolt — and the patterns overlap, because 249 mm of fold cannot fit in a 90 mm pitch. On the right the plate does what it can actually do: the hinges run straight from one row to the next, and the whole group folds on 429 mm rather than 746. The arithmetic follows the drawing — 377 kN for the group against 657 for the rows added up.

The rows have to share one fold

Each bolt row of an end plate is checked on its own, and the answers are added up. Two rows ninety millimetres apart cannot each fold the plate on a pattern two hundred and fifty millimetres long, because there is only one plate — so the group folds on one shorter pattern, and the sum was never available.

connections · Prying Two flanges, one bolt, one prying force. An end plate 18 mm thick with its bolt 45 mm from the beam web, bolted to a column flange 14 mm thick with the same bolt 30 mm from the column web; their tips are 35 mm beyond the bolt and bear on each other, so the prying force there is one force acting on both. Checked as two separate tee stubs on rigid bases, the end plate carries 109.3 kN per bolt (mode 2, web hinge and bolt) and the column flange 104.4 (mode 1, both hinges), so the component method gives the joint 104.4. The pair carries 92.3 kN, 12 per cent less, with the prying force at 44.7 kN and hinges in the end plate at its web and the column flange at the bolt line — the end plate's web hinge working with the column flange's bolt-line hinge, a mechanism that exists in neither tee stub on its own.

The hinge that forms in the other flange

An end plate bolted to a column flange is two tee stubs sharing one bolt, and the force at their tips is one force pressing on both. Checked separately, as the component method checks them, the end plate carries 109 kN a bolt and the column flange 104. Together they carry 92 — by a mechanism with one hinge in each flange that neither tee stub has on its own.

connections · Prying 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 Preload makes the base rigid, until the bed lets go. Moment against rotation at the base of a 500 × 400 × 20 mm base plate under a 260 mm column, with two M24 holding-down bolts in each of two rows 60 mm from its ends: with no preload (dashed), 58,609 kN·m per radian, 9.8 times the column's EI/L; with bonded anchors whose free length is 136 mm (dotted), 13.9 EI/L; preloaded to 150 kN a bolt (solid), 45.6 EI/L until the bed under the plate lets go at its tension edge at about 57 kN·m, then softening toward the unpreloaded base. Faint: the slope of EN 1993-1-8's rigid boundary, 30 EI/L.

The base that is rigid until the bed lets go

A column base drawn as fixed is a spring about a tenth as stiff as the rule for "rigid" requires, and most of its softness is the holding-down bolts stretching. Shortening the bolts helps less than it seems: even 60 mm of free length leaves the base below two thirds of the rigid boundary. Preloading them works completely — the base becomes half as stiff again as the boundary asks — but only while the grout under the plate stays in compression. The preload is spent at a definite moment, the same moment the column's own weight would buy for free, and the shorter the bolt the faster creep of the grout takes the preload away.

connections · Base plate The series sum is exact for matched flanges and errs by their mismatch. The pair's stiffness over the series of its two flanges as tee stubs on rigid bases, each with half the bolt's length, against the end plate's lever arm from web to bolt, for end plates 14, 18, 25 mm thick on a column flange 14 mm thick with its bolt 30 mm from its web, tips 35 mm beyond the bolt. Where the end plate matches the column flange — 14 mm thick at 30 mm — the ratio is exactly one: two identical flanges back to back meet on their plane of symmetry, which is the rigid base. A 14 mm end plate is 0.95 of the series with its bolt at 45 mm and 0.85 at 65; an 18 mm end plate is 0.97 of the series with its bolt at 45 mm and 0.86 at 65; a 25 mm end plate is 1.00 of the series with its bolt at 45 mm and 0.93 at 65.

The flanges that lean on each other

A bolted end-plate joint's stiffness is built from springs in series: the end plate as a tee stub on a rigid base, the column flange as another, the bolt shared between them. The rigid base is not a fiction. It is exactly the plane of symmetry between two identical flanges bolted back to back, so for matched flanges the series sum is exact. A real end plate and column flange are not matched, and each tip bears on a tip that moves differently from its own. Solved as a pair, the error is the mismatch: 3% for an 18 mm end plate on a 14 mm column flange with similar lever arms, 12–15% when the end plate's bolt is twice as far from its web.

connections · Prying The pair holds its clamp longer than the stub it is checked by. The tension a bolt at which the clamp lets go, as a share of the 110 kN preload, against the end plate's lever arm from web to bolt, for an end plate 18 mm thick on a column flange 14 mm thick at 30 mm: the pair (solid), the end plate's stub (dashed) and the column flange's (dotted). With the end plate's bolt at 30 mm the pair and the stubs nearly agree. At 45 mm the pair lets go at 88.2 kN and the end plate's stub already at 81.1, the pair holding 9 per cent more; at 60 mm, 82.0 against 71.2, 15 per cent more.

The clamp the pair holds longer

A preloaded bolt clamps an end plate to a column flange, and the joint stays closed until the tension relieves the clamp. The joint diagram says that happens at the preload over one minus the bolt's share — 122 kN for a 110 kN preload — but prying adds to the tension at the bolt line and the clamp lets go at 88. A component model checks each flange as a stub on a rigid base and lets the joint go when its weaker stub does, at 81 kN. The real pair, its tips leaning on a column flange that gives way, pries less than the end plate alone would, and holds its clamp nine per cent longer — fifteen when its lever arms are two to one.

connections · Prying

Named alongside it

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

PryingJoint stiffnessEnd plateLever armT-stubConnectionBase plateBolt tensionCollapse mechanismYield-lineFree bodyMoment connection

All concepts