Structural Engineering & Statics
where the load goes
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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.

Assumes Where the structure meets the ground, and when the bolts start working, The thickness that decides who fails and A joint made of springs in series.

16 min read 7 figures The load must go somewhereWhich failure arrives firstGeometry beats material

Two earlier treatments of a column base answer two different questions and are true of two different plates. The first takes the plate as rigid, puts a moment on it, and follows the pressure block across the base through three regimes until the holding-down bolts start to pull. The second takes the plate as it is — a flexible sheet of steel that can deliver bearing only a limited distance from the column — and finds that most of a large thin plate carries nothing at all. It does that for axial load, and it closes by saying the moment case is still to be worked out on the region actually in bearing.

That case is the one that matters for a base with bolts working, and working it out moves a number nobody expects to move: the force in the bolts.

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.
Fig. 1 A 500 × 400 mm base plate, 20 mm thick, under a 260 mm column carrying 300 kN and 120 kN·m, with its bolts 60 mm from the tension edge. The shaded block is the compression the plate can deliver, within 43 mm of the compressed flange; its resultant is 124 mm from the column’s centre and the bolts carry 264 kN. The dashed block at the plate’s edge is the rigid answer, which needs only 128 kN in the bolts.

Where a rigid plate puts the compression

Take the same base and treat the plate as rigid. The resultant of 300 kN and 120 kN·m sits 400 mm from the column’s centre, well outside the plate’s 250 mm half-length, so no compression-only distribution can balance it and the bolts have to pull.

A base plate, and when the bolts start working. A 500 × 400 mm plate carrying 300 kN and 120 kN·m, so the resultant sits 400 mm from the centre against a kern of 83.33 mm. The plate is in bolts engaged: bearing over 109.68 mm at a peak of 20 N/mm², with the holding-down bolts carrying 138.73 kN. The plate lifts at 25 kN·m and crushes at 67.5 kN·m, and the bolts are not needed until 75 kN·m.
Fig. 2 The same plate and the same actions as a rigid plate. The resultant is 400 mm from the centre, outside the plate, so the base is in its third regime: bearing over 110 mm at the far edge at the full 20 N/mm², with the holding-down bolts carrying 139 kN. The compression is where a rigid plate would put it — against the edge, as far from the bolts as the plate allows.

The pressure block sits at the far edge of the plate because that is the position that balances the moment with the least tension. Pushing the compression as far as possible from the bolts gives the longest lever arm, the longest lever arm gives the smallest pair of forces, and the smallest pair is what a rigid body settles into when the bearing is allowed to go anywhere on its base.

That reasoning has one premise, and it is the whole of the question. The plate has to be able to carry pressure at the edge. A rigid plate can. A 20 mm plate under a 260 mm column cannot, and by a long way.

Where a real plate can put it

A strip of the plate cantilevering outward from the column’s flange can deliver bearing only as far as its own bending capacity reaches, which for a plate of thickness tt on grout of bearing strength fjdf_{jd} is

c=tfy3fjdc = t\sqrt{\frac{f_y}{3f_{jd}}}

For a 20 mm plate at 275 N/mm² on grout at 20 N/mm², that is 43 mm.

The part of a base plate that is delivering anything. A plan of the plate: the 260 × 260 column in the middle, and the shaded region it can reach — its own outline grown by 43 mm, which is t√(f_y/3f_jd) for a 20 mm plate on grout at 20 N/mm². That is 78176 mm² of a 200000 mm² plate, or 39 per cent of it. The corners are outside it and are carrying nothing: a plate cantilevers from the column's perimeter and runs out of bending capacity at c, so making it larger in plan beyond that changes nothing at all, and making it thicker changes everything.
Fig. 3 The plan of the same plate: the column in the middle and the shaded region the plate can reach, its outline grown by 43 mm. Under axial load the whole shaded region bears. Under a moment large enough to engage the bolts, only the part around the compressed flange does — and the plate’s far edge, 120 mm beyond that flange, is outside the region altogether.

Under a moment that engages the bolts, the column’s tension flange is lifting and its part of the effective area is doing nothing. What is left to carry compression is the region round the compressed flange: the flange’s own outline, 260 by 12.5 mm, grown by 43 mm in every direction the plate allows. That is a rectangle 346 mm long and 98 mm wide — a T-stub in compression — and at 20 N/mm² it can deliver 678 kN.

And the resultant of that compression is not at the plate’s edge. It is on the centre line of the flange, 124 mm from the column’s centre, because the region is grown equally inward and outward from the flange. The plate between the flange and its far edge, 120 mm of it, is carrying nothing.

Two forces and one lever arm

With the compression’s position fixed, the base is two forces on a line. The compression CC acts at zcz_c = 124 mm on one side of the column’s centre, the bolt tension TT at ztz_t = 190 mm on the other, and the axial load acts at the centre:

C−T=NC zc+T zt=MC - T = N \qquad C\,z_c + T\,z_t = M

Two equations, two unknowns, no stiffness anywhere. For 300 kN and 120 kN·m:

C=M+Nztzc+zt=564 kNT=M−Nzczc+zt=264 kNC = \frac{M + N z_t}{z_c + z_t} = 564\ \text{kN} \qquad T = \frac{M - N z_c}{z_c + z_t} = 264\ \text{kN}

The lever arm between them is 314 mm. The rigid block’s lever arm was 190 + 223 = 413 mm, and the bolt force that goes with it is 128 kN — or 139 kN with the triangular block the rigid-plate figure draws. The bolts carry about twice as much as either rigid calculation said, for the same column, the same plate and the same actions.

That is not a refinement at the second decimal place. A bolt group sized for 130 kN and carrying 264 is a bolt group at twice its design force, and nothing about the plate looks wrong: it is flat, it is bearing, it has not yielded. The load simply took the only path the plate could offer it.

The bolts start working at half the moment

The shorter lever arm has a second consequence that does not need a capacity calculation at all, and it decides when the holding-down bolts are part of the structure.

A rigid plate needs its bolts only once the resultant of the axial load and the moment leaves the plate, because until then a compression-only pressure block somewhere on the plate can balance it. On this base, under 300 kN, the resultant reaches the plate’s edge at a moment of 300 × 0.25 = 75 kN·m. Below that the rigid calculation has the bolts carrying nothing, however far into its partial-contact regime the plate has gone.

The flexible plate has no pressure block that can wander out to the edge. Its compression can go no further from the column’s centre than the compressed flange’s centre line, 124 mm out, so a compression-only balance exists only while the resultant stays inside that — up to a moment of 300 × 0.124 = 37 kN·m. Past 37 kN·m the bolts are working, at a moment the rigid calculation says is half-way to needing them.

Between the two, the disagreement is not about size but about existence. At 75 kN·m the rigid plate is just reaching the moment at which its bolts engage; the flexible plate’s bolts are already carrying 121 kN. A base designed on the assumption that its bolts are idle under ordinary moments — sized for erection and for uplift cases, as holding-down bolts often are — has bolts doing structural work in exactly the range where nobody checked them.

The same comparison can be put as a distance, which makes it look like a familiar rule. A rigid plate starts to lift when the resultant leaves the middle third of its length, 83 mm from the centre here, and needs its bolts when the resultant reaches the edge at 250 mm. The flexible plate’s equivalent of that second boundary is 124 mm: the region within which it can stand on compression alone is the distance between its flanges, not the length of its plate. The plate beyond the flanges is along for the ride in both regimes.

The pressure block is in equilibrium, and it is not safe

The rigid answer has a property that makes it look trustworthy, and it is worth seeing exactly where the trust breaks.

The pressure block at the edge satisfies every equation of equilibrium: its compression, the bolts’ tension and the column’s load and moment balance exactly. A distribution of forces in equilibrium with the load is what the lower-bound theorem needs to call a load safe — and the rigid-plate treatment of a base itself notes that it is the lower-bound theorem again, which is why it can be trusted to give a safe answer about the concrete.

The theorem asks for one more thing, and the rigid block does not supply it. A lower-bound field has to be in equilibrium and nowhere exceed the strength of any part it passes through. The edge block passes 20 N/mm² of bearing through a strip of 20 mm plate 180 mm from the flange that has to deliver it, and that strip can carry the pressure for only 43 mm. The field is in equilibrium and it violates the plate’s bending capacity, so it is not a lower bound, and the theorem’s guarantee does not attach to it.

The flexible answer is a lower bound. Its compression is inside what the plate can deliver, its bolts inside what they can carry, and its equilibrium is exact. That is why the smaller lever arm and the larger bolt force are the answer to trust: of the two equilibrium states, it is the only one in which no part of the base is being asked for more than it has.

The resistance, both ways

The difference in lever arm becomes a difference in capacity. For a given axial load, the flexible base resists the largest moment that keeps CC within the T-stub’s 678 kN and TT within the bolt row’s 400 kN; the rigid base resists the largest moment that keeps its edge block within the plate and its bolts within the same 400.

A rigid pressure block promises a moment the plate cannot deliver. The moment a 500 × 400 mm, 20 mm base plate under a 260 mm column can resist, against the axial compression it carries, with a bolt row good for 400 kN. The solid line puts the compression where the plate can deliver it, within 43 mm of the compressed flange; the dashed line puts it in a rigid block at the plate's edge. With no axial load they give 126 and 166 kN·m; the flexible plate peaks at 159 kN·m under 271 kN. At the drawn 300 kN the two are 156 and 220 kN·m, a ratio of 1.41, and the applied 120 kN·m is inside both.
Fig. 4 The moment the base can resist against the axial load it carries. The solid line keeps the compression under the flange; the dashed line puts it at the plate’s edge. With no axial load they give 126 and 166 kN·m; the flexible base peaks at 159 kN·m under 271 kN. At 300 kN they are 156 and 220 kN·m, a ratio of 1.41, and the applied 120 kN·m is inside both.

The two curves have the shape every base plate interaction has — resistance rising with a moderate axial load, which helps the bolts, and falling again as a large one uses up the compression — and the rigid curve lies above the flexible one everywhere. At the drawn axial load it promises 41 per cent more moment than the plate can deliver.

At small axial loads the gap is set by the lever arm alone, because the bolts govern both curves and the ratio of the two resistances is the ratio of the two lever arms. At larger axial loads the compression starts to govern the flexible base first, because its compression zone is 98 mm wide rather than the plate’s full 500, and the curves separate further.

Thickness raises the resistance until the bolts decide

The obvious response is to thicken the plate, since thickness is what sets cc.

Thickness raises what the plate can do, until the bolts decide. The moment resistance of a 500 × 400 mm base plate under a 260 mm column at 300 kN, against its thickness, with a bolt row good for 400 kN. The rigid block does not know the thickness exists and gives 220 kN·m throughout. The flexible plate gives 31 kN·m at 8 mm and 164 kN·m at 60 mm, rising until about 25 mm, where the bolts rather than the compression govern and the curve flattens — short of the rigid answer, because the compression never moves out from under the flange.
Fig. 5 The moment resistance at 300 kN against the plate’s thickness. The rigid block does not contain the thickness and gives 220 kN·m throughout. The flexible plate gives 31 kN·m at 8 mm and rises until about 25 mm, where the bolts start to govern; by 60 mm it has reached 164 kN·m and stopped, short of the rigid answer.

Below about 25 mm the compression zone is what limits the base, and thickening the plate enlarges it roughly in proportion. Above that the bolts limit it, and the resistance is the bolt row’s 400 kN times the lever arm plus the axial load’s help — neither of which the plate’s thickness touches.

What does not happen is the one thing the rigid answer needed. A thicker plate enlarges the compression zone, and it enlarges it equally toward the column and away from it, so its resultant stays on the flange’s centre line. Thickness buys more compression in the same place. It does not buy a longer lever arm, and the gap to the rigid answer is a lever-arm gap.

The compression is under the flange, not at the edge of the plate. A 500 × 400 mm base plate, 40 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 86 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.
Fig. 6 The same base with a 40 mm plate. The plate now reaches 86 mm past the flange instead of 43, and the compression zone is twice as wide — but its resultant is still 124 mm from the column’s centre, the lever arm is still 314 mm, and the bolts still carry 264 kN. Doubling the plate did nothing for the force in the bolts.

That is the figure to show anyone who reaches for a thicker plate to reduce a bolt force. At this moment the compression zone was never the problem; the bolts were carrying a force set by where the compression acts, and a thicker plate cannot move that. The plate that would move it outward is one with stiffeners — a gusset from the flange tip to the plate edge, which gives the plate a line of support out where the rigid block wanted the pressure and extends the region within cc of anything stiff, exactly as it does under axial load. A stiffened base is the only kind that can approach the rigid answer, and it approaches it by becoming rigid where it matters.

A longer plate helps the bolts and not the compression

The other obvious response is to lengthen the plate, which gives the moment more room.

A longer plate helps the bolts and not the compression. The moment resistance of a 20 mm base plate under a 260 mm column at 300 kN, against the plate's length, with the bolts always 60 mm from its edge. Lengthening the plate moves the bolts outward in both answers. In the rigid one it moves the compression outward too, so the lever arm grows at both ends and the resistance rises 55.0 kN·m for every 100 mm; with the compression held under the flange only the bolt end moves, and it rises 18.9. From 400 to 900 mm the gap between them grows from 28 to 209 kN·m.
Fig. 7 The moment resistance at 300 kN against the plate’s length, with the bolts always 60 mm from its edge. A longer plate moves the bolts outward in both answers, and in the rigid one it moves the compression outward too: its resistance rises 55.0 kN·m for every 100 mm, the flexible one’s 18.9. Between 400 and 900 mm the gap grows from 28 to 209 kN·m.

Lengthening a plate moves the holding-down bolts further from the column, and that lengthens the tension half of the lever arm in both calculations. In the rigid calculation it also moves the edge block further out, so the compression half grows as well and the resistance climbs at both ends. In the flexible one the compression half is fixed under the flange, so only the bolt end moves.

The two answers agree best on a short plate and worst on a long one, which is the reverse of what intuition says a refinement should do. A long thin plate looks generous, and the rigid calculation rewards its length; the plate itself cannot deliver pressure across that length, and the reward is almost entirely fictional.

What changes on the drawing

Three consequences follow for anyone detailing a moment-resisting base, and each has a clear direction.

The holding-down bolts are sized on the flexible lever arm. Moments about the compressed flange’s centre line, not about the plate’s edge, give the bolt force. On ordinary proportions that is fifty per cent to twice the rigid-block value, and it is the number the bolts actually see.

The bolts are the place to spend length. A longer plate helps only by moving the bolts outward, so if length is going to be added it should carry the bolts with it; a plate lengthened on the compression side is buying nothing.

Stiffeners, not thickness, move the compression. If the lever arm has to grow, the base needs a stiff line out toward the edge on the compression side, and a gusset provides it at a fraction of the steel a plate thick enough to be rigid would need.

And the concrete around the bolts sees the same doubling. A holding-down bolt in tension is only as strong as the cone of concrete it would pull out of the foundation, and that failure is sudden, where a bolt yielding is not. Anchorage design exists to force the brittle failure to be governed by the ductile one: the concrete cone is made stronger than the steel that pulls on it, so the bolt yields first. That arrangement is sized on the bolt force. A cone sized for the 128 kN a rigid block gives and asked for the 264 kN the plate actually delivers has had its margin over the bolt consumed and reversed — and the member that fails first is then the one that gives no warning.

The component method that design codes use for column bases is built on exactly this picture — a joint made of springs and strengths in series, with a compression T-stub under each flange and a tension T-stub at each bolt row — so none of the above is a departure from practice. It is the reason the practice looks the way it does, and the reason a pressure-block calculation carried over from a rigid-plate model gives a bolt force the practice would not.

What the two components leave out

The tension side is taken as the bolts alone. A plate pulled by holding-down bolts is itself a T-stub in bending, and if it is thin it bends, prys and limits the tension long before the bolts do — the prying argument applied to the other half of the same base. Here the bolt row’s 400 kN is taken to govern.

The column’s own compression flange is taken to be strong enough. The flange and a portion of the web deliver the compression into the T-stub, and on a light column under a heavy moment that component can limit the base first.

The bearing strength is a single number. fjdf_{jd} depends on the grout, the concrete under it and how much concrete surrounds the loaded area. A pad of grout under only part of the plate changes the compression zone’s strength and, if it is off-centre, its position.

Shear is carried separately. The base shear goes out through friction under the compression zone, the bolts or a shear key, and a compression zone that is a fifth of the plate’s area supplies correspondingly less friction than the rigid block would have credited.

And the base is treated as a strength, not a spring. How much it rotates under the moment depends on the same lever arm and on the stiffness of every component, which is a separate calculation with its own consequences for the frame above.

Still open: how stiff this base is as a spring

A base plate is not only a set of strengths. It is a rotational spring at the bottom of a column, and a base drawn as a pin is not one for the same reason a base drawn as fixed is not either. The rotational stiffness of a base built from components in series goes as the square of the lever arm between its compression and tension components, divided by the sum of their flexibilities. If the components’ own stiffnesses were unchanged, moving the compression from the plate’s edge to the flange’s centre line would scale that stiffness by (314/413)², a little under six-tenths — a base four-tenths softer than the rigid picture of it. Whether the component flexibilities move enough to change that estimate, and how much of the difference the frame above notices in its drift and its buckling length, is a question about the base as a spring rather than as a strength.

Where this goes next

A base plate, and when the bolts start working. A 500 × 400 mm plate carrying 600 kN and 90 kN·m, so the resultant sits 150 mm from the centre against a kern of 83.33 mm. The plate is in partial contact: bearing over 300 mm at a peak of 10 N/mm², with the holding-down bolts carrying 0 kN. The plate lifts at 50 kN·m and crushes at 120 kN·m, and the bolts are not needed until 150 kN·m. Where the structure meets the ground, and when the bolts start working Push a base plate off its middle third and it lifts off the foundation. The holding-down bolts then carry exactly nothing, and go on carrying nothing until the plate has crushed the concrete underneath it. The part of a base plate that is delivering anything. A plan of the plate: the 260 × 260 column in the middle, and the shaded region it can reach — its own outline grown by 43 mm, which is t√(f_y/3f_jd) for a 20 mm plate on grout at 20 N/mm². That is 78176 mm² of a 200000 mm² plate, or 39 per cent of it. The corners are outside it and are carrying nothing: a plate cantilevers from the column's perimeter and runs out of bending capacity at c, so making it larger in plan beyond that changes nothing at all, and making it thicker changes everything. The plate that is only as big as it is thick A base plate's design is presented as a bearing calculation: an area, a bearing strength, and a check that the pressure fits. The area in that calculation is the plate, and the plate cannot deliver it — a 20 mm plate on a 500 square reaches 43 mm past the column and the corners carry nothing at all. Prying action in a tee stub. A tee stub pulled by its web with 100 kN per bolt. The 20 mm flange is in the one-hinge regime, so the prying force at the flange tip is 50.63 kN and the bolt carries 150.63 kN — 1.51 times what was applied. The flange stops prying entirely at 26.97 mm thick, and collapses on its own at 110 kN. The force the bolt never saw applied Pull a tee stub with a hundred kilonewtons and its bolt carries a hundred and fifty. The extra comes from the flange bending and pressing its own edge against the thing it is bolted to, and no free body of the connection as a point contains it. A base plate, and when the bolts start working. A 550 × 450 mm plate carrying 900 kN and 140 kN·m, so the resultant sits 155.56 mm from the centre against a kern of 91.67 mm. The plate is in partial contact: bearing over 358.33 mm at a peak of 11.16 N/mm², with the holding-down bolts carrying 0 kN. The plate lifts at 82.5 kN·m and crushes at 192.95 kN·m, and the bolts are not needed until 247.5 kN·m. The pinned base that is not pinned A column base drawn as a pin is a plate bearing on grout, and a plate in contact over its whole length resists rotation whether anybody wanted it to or not. The stiffness it delivers depends on the axial load, so the assumption is one a frame can leave and re-enter as its loads change.

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Base plateBearing stressColumnComponent methodConnectionEccentricityEffective areaFoundationHolding-down boltLever armLower-bound theoremPlate bendingT-stub