Structural Engineering & Statics
where the load goes
Stability

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.

Assumes Held, and not held, Where the structure meets the ground, and when the bolts start working and The ends decide the length that matters.

15 min read 7 figures One support too manyWhich failure arrives firstThe load must go somewhere

Whether a storey is held or not held is a binary classification applied to something continuous, and the same is true one level down, at the bottom of every column in the building.

A base is drawn as a triangle or as a pair of hatched lines, and the analysis takes it as a pin or as a fixity. What is actually there is a steel plate bearing on a layer of grout, held down by two or four bolts, and its behaviour is neither.

A base plate, and when the bolts start working. A 550 × 450 mm plate carrying 900 kN and 40 kN·m, so the resultant sits 44.44 mm from the centre against a kern of 91.67 mm. The plate is in full contact: bearing over 550 mm at a peak of 5.4 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.
Fig. 1 A 550 × 450 mm plate carrying 900 kN with 40 kN·m of moment. The resultant sits 44 mm from the centre against a kern of 92 mm, so the plate is in full contact: bearing over its whole 550 mm at a peak of 5.4 N/mm², a quarter of what the grout can take, with the holding-down bolts carrying nothing at all.

Full contact is a stiff base

A plate in contact over its whole length is a bearing surface, and a bearing surface resists rotation.

Rotate the plate by θ\theta and the pressure distribution tilts: it rises at one end and falls at the other, and the resulting couple opposes the rotation. Nothing has lifted, nothing is in tension, and the resistance comes from the same contact that was carrying the axial load.

The stiffness that produces is not small. For a rigid plate on an elastic grout of subgrade modulus ksk_s, the rotational stiffness is ksBL3/12k_s B L^3/12 — the second moment of the contact area times the bed stiffness — and for ordinary plate sizes on ordinary grout it is comparable with the column’s own EI/LEI/L.

That is the whole of the essay’s finding. A base with no tension capacity, no moment connection and no design intent whatever supplies rotational stiffness of the same order as the member it supports, purely because it is a rectangle pressed against something.

It is worth being clear about what supplies the resistance, because the intuition that a pin cannot carry moment is correct and is about the wrong object.

A pin is a connection with one rotational freedom released, and a released freedom carries no moment by definition. That is a statement about a mechanism, and it is exactly true of a pin.

A base plate has no released freedom anywhere. It is a solid plate on a solid bed, and calling it a pin is a statement about intent rather than about kinematics. The moment it can carry before anything changes is limited by the contact, and the contact is generous: a plate under a substantial axial load can carry a substantial moment before any part of it stops touching.

So the drawing’s triangle is shorthand for “not designed to carry moment”, and the structure reads it as “carries whatever moment its stiffness attracts”. The two are different sentences, and the frame believes the second.

The threshold where it stops

The contact is full only while the resultant stays inside the kern.

e=MN≤L6⟹M≤NL6e = \frac{M}{N} \le \frac{L}{6} \quad\Longrightarrow\quad M \le \frac{NL}{6}

For the plate above that is 82.5 kN·m. Past it, one end lifts, the contact shortens, and the stiffness falls away fast.

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.
Fig. 2 The same plate at 140 kN·m. The resultant is 156 mm out against a kern of 92, so the plate is in partial contact: bearing over 358 mm of its 550 at a peak of 11.2 N/mm². The bolts still carry nothing — they are not needed until 247.5 kN·m — and the base has become considerably softer without any of its parts having done anything.
A base plate, and when the bolts start working. A 550 × 450 mm plate carrying 900 kN and 260 kN·m, so the resultant sits 288.89 mm from the centre against a kern of 91.67 mm. The plate is in bolts engaged: bearing over 220.99 mm at a peak of 22 N/mm², with the holding-down bolts carrying 193.92 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.
Fig. 3 And at 260 kN·m the resultant is 289 mm out, outside the plate itself. The bearing has shortened to 221 mm at the grout’s full 22 N/mm² and the bolts have finally engaged, at 194 kN. This is the only one of the three states in which the detail is doing what its drawing suggests.

Three states, one detail, and the transitions between them are decided by a ratio of moment to axial load rather than by anything on the drawing.

Which free body produced the number

The free body is the plate with the column cut just above it, and the grout replaced by whatever pressure it is delivering.

Crossing the cut are the column’s axial force NN and moment MM. Crossing the underside is a compressive pressure over whatever length is in contact, plus a tension in the bolts if there is one.

Equilibrium gives two equations — vertical forces and moments — and the unknowns are the contact length and the pressure. The bolt force is not one of them until the contact has shortened enough that the pressure block alone cannot balance the moment, which is why it stays at zero over most of the useful range.

The kern condition falls straight out of that free body. A trapezoidal pressure block over the full length has its resultant within L/6L/6 of the centre by construction, so a resultant further out is one no full-contact block can produce.

That is the same free body and the same argument as a body that tips before it slides, applied to a plate rather than to a building — and the kern is the same kern.

The stiffness depends on the load, so the assumption does

The threshold contains NN, which means the base’s classification changes with the load case.

A base plate, and when the bolts start working. A 550 × 450 mm plate carrying 300 kN and 140 kN·m, so the resultant sits 466.67 mm from the centre against a kern of 91.67 mm. The plate is in bolts engaged: bearing over 90.49 mm at a peak of 22 N/mm², with the holding-down bolts carrying 147.94 kN. The plate lifts at 27.5 kN·m and crushes at 76.44 kN·m, and the bolts are not needed until 82.5 kN·m.
Fig. 4 The same plate and the same 140 kN·m, with the axial load reduced from 900 kN to 300. The lift-off threshold has fallen from 82.5 kN·m to 27.5, the bearing has shortened to 90 mm, and the bolts are carrying 148 kN. Under a third of the load the base has gone from partial contact to fully engaged bolts, with the moment unchanged.

A base that is stiff under gravity is flexible under uplift, and uplift arrives in exactly the combination — wind, or seismic overturning — where the sway is largest and the stiffness is most wanted.

That is an unpleasant coincidence and it is not an accident. The load case that unloads a column is the one that pushes the frame sideways, so the base’s stiffness falls at precisely the moment the frame is asking for it.

What it does to the effective length

A frame’s stability depends on its restraints, and a base is one of them.

The alignment chart, computed rather than looked up. The effective length factor against G = (EI/L) of the column ÷ (EI/L) of the beams, for a storey held against sway and for one free to sway. Every point on both curves is the lowest eigenvalue of the assembled frame, swept over 30 beam stiffnesses — not a nomogram, and nothing here is read off a chart. The non-sway curve runs from k = 0.510 at G = 0.02, where the beams are stiff enough to be built-in, to k = 0.992 at G = 50, where they are soft enough to be pins: the whole of it lies between a half and one. The sway curve starts at k = 1.007 and has no upper bound at all, reaching 6.48 at the same G — so the braced frame carries 42.6 times the load of the unbraced one at its worst point on this sweep.
Fig. 5 The effective length factor against the ratio of column to beam stiffness, for a storey held against sway and for one free to sway, every point an eigenvalue of the assembled frame rather than a reading from a nomogram. The non-sway curve runs from 0.510 to 0.992 — the whole of it between a half and one — while the sway curve starts at 1.007 and has no upper bound, reaching 6.48.

The chart’s GG at a base is a ratio of the column’s stiffness to whatever is restraining it, and a base is conventionally entered as G=10G = 10 for “pinned” and G=1G = 1 for “fixed” — two numbers standing in for a continuum.

Neither is what the plate delivers. A base in full contact is somewhere between; one with its bolts engaged is nearer the pin; and the same base is at different points on the curve in different load cases.

One restraint, and several times the load. The same portal — the same columns, the same beam, the same steel — buckling with its head held against sway and with its head free to sway. The braced frame's critical load is 12.37 EI/L² and the swaying one's is 2.96 EI/L², a factor of 4.18, and the effective length factor that comes out of each eigenvalue is 0.893 against 1.826. Both are eigenvalues of the assembled frame at a beam-to-column stiffness ratio of G = 3.00; the buckled shapes are the mode vectors themselves, drawn at 18 per cent of the storey height so that the movement can be seen.
Fig. 6 The same portal buckling with its head held and with its head free: 12.37 EI/L² against 2.96, a factor of 4.18, with effective length factors of 0.893 and 1.826 from the eigenvalues themselves. The restraint is the whole of the difference.

The ends decide the length, and a base plate is one of the two ends. A frame whose base stiffness is a third of what the analysis assumed has a longer effective length, a lower critical load and a larger amplifier — and the amplifier is what multiplies the sway, so an error at the base propagates through the whole storey.

How much stiffness, in numbers

It is worth converting the argument into a comparison, because the whole question is whether the base’s stiffness is of the same order as the column’s or a small fraction of it.

Take the 550 × 450 mm plate on grout with a bed modulus of about 100 N/mm³ — a stiff bedding on a concrete pad. Its rotational stiffness while in full contact is ksBL3/12k_s B L^3/12, which is 100×450×5503/12=6.2×1011100 \times 450 \times 550^3/12 = 6.2\times10^{11} N·mm per radian, or 620,000 kN·m/rad.

The column it supports — a 254 × 254 universal column, 4 m long, with I=1.1×108I = 1.1\times10^8 mm⁴ — has EI/L=210,000×1.1×108/4000=5.8×109EI/L = 210{,}000 \times 1.1\times10^8 / 4000 = 5.8\times10^9 N·mm per radian, or 5,800 kN·m/rad.

The base is a hundred times stiffer than the column. On any classification that ratio is rigid, and a frame analysed with a pin at the bottom of that column is analysing a different structure.

Now soften the assumptions. Halve the bed modulus for imperfect grout: still 50 times. Let the plate lift over a third of its length: the stiffness falls with the cube of the contact length, so a third off takes it to 30 per cent — still 15 times. Allow the plate to bend: perhaps half again.

The conclusion survives every reasonable erosion. Full contact makes a base plate rigid by any code’s definition, and the assumption of a pin is not a small approximation; it is the wrong end of the range.

The two checks want opposite assumptions

There is a structural reason the question is never settled, and it is worth stating plainly.

The column wants the base to be flexible. A stiffer base attracts moment into the column’s bottom, so assuming a pin gives the column a smaller design moment. That is why “pinned” is the conservative assumption for the member.

The frame wants the base to be stiff. A stiffer base reduces the sway, lowers the effective length and reduces the second-order amplification. Assuming a pin is conservative here too.

And the foundation wants the base to be pinned, and it is not. A base assumed pinned delivers no moment to the pad, so the pad is sized for an axial load. A real base in full contact delivers whatever moment its stiffness attracts, which the pad has not been checked for — and this is the only one of the three where the conservative assumption is the wrong one.

The usual resolution is to design the column and the frame on a pinned base and the foundation on a nominal moment, which is two analyses of two structures and is the honest way of admitting that the detail is neither.

What the codes actually say

Design codes handle this with a small set of nominal stiffnesses rather than with a calculation.

A base is classified as rigid if its rotational stiffness exceeds a multiple of the column’s EI/LEI/L — the multiple depending on whether the frame is braced — and as pinned if it falls below a much smaller one. Between them it is semi-rigid, and the analysis is supposed to use the actual value.

In practice a nominal figure of about 10 to 20 per cent of the column’s own stiffness is assigned to a “pinned” base and the frame is analysed with it. That is not a fudge: it is closer to the truth than either extreme, and it is the same treatment a beam-to-column joint gets, where the classification is also a band applied to a continuum.

The residual difficulty is that the number depends on the grout, the plate thickness, the bedding and the axial load, none of which is known when the frame is analysed.

The base during construction, which is a third structure

There is one condition in which the base is genuinely a pin, and it is the condition nobody analyses.

A column is erected, plumbed, and its holding-down bolts are left slack while the grout is placed and cures. During that period the plate is sitting on levelling nuts or shims — two or four point contacts a few tens of millimetres across — and its rotational stiffness is very nearly nothing.

That is the most dangerous day, and it inverts everything above. The frame’s bracing may not be complete, its beams may not be connected, and the one restraint the analysis assumed nothing from is genuinely absent rather than merely uncertain.

The design response is a temporary works calculation with a genuinely pinned base, which is a different structure from the finished one and is checked by different people. What makes it awkward is that the permanent design’s conservatism does not help here: the finished frame was checked with a pin at the base and is safe; the frame during erection has a pin at the base and no beams, and the two omissions are not the same one.

The plate’s own flexibility

Everything above treats the plate as rigid, and it is not.

A base plate, and when the bolts start working. A 800 × 450 mm plate carrying 900 kN and 140 kN·m, so the resultant sits 155.56 mm from the centre against a kern of 133.33 mm. The plate is in partial contact: bearing over 733.33 mm at a peak of 5.45 N/mm², with the holding-down bolts carrying 0 kN. The plate lifts at 120 kN·m and crushes at 305.45 kN·m, and the bolts are not needed until 360 kN·m.
Fig. 7 The same load and moment on a plate 800 mm long rather than 550. The kern has grown to 133 mm so the resultant at 156 mm is only just outside it, the contact runs over 733 mm and the peak pressure has fallen from 11.2 to 5.5 N/mm². The larger plate is stiffer, stronger and further from lift-off — and it is only stiffer if it is thick enough to deliver the pressure at its ends.

A thin plate bends between the column flanges and the pressure it delivers at its extremities is less than the rigid analysis says. That reduces both the strength and the stiffness, and it is why base plate thickness is checked as a cantilever off the column’s own outline rather than being taken as whatever fits.

Making a base plate longer without making it thicker buys a good deal less than the arithmetic suggests, which is the standard failure of the rigid-plate assumption and the reason the effective bearing area is limited to a strip round the column profile rather than being the whole plate.

Why the assumption survives anyway

A simplification this wrong ought to have been abandoned, and it has not been, for three reasons worth separating.

It is conservative for the two checks that get done. The column’s moment and the frame’s sway are both over-estimated by a pinned base. Nobody is punished for conservatism, and the check that is unconservative — the foundation’s — is made by somebody else, often on a different drawing.

The stiffness is genuinely uncertain. The bed modulus of grout is not a specified property, the packing is not inspected, and the plate’s contact depends on how flat the pad was trowelled. A number that could be anywhere in a factor of five is a poor input to an analysis that would then report its output to three figures.

And the base does not stay put. The stiffness falls with the axial load, so a single value is wrong in most load cases even if it is right in one. A frame analysis with one base stiffness is analysing one load case correctly and the rest approximately.

The honest position is the semi-rigid one: use a nominal stiffness, accept that it is a band rather than a number, and check the foundation for the moment that stiffness attracts. The failure mode of the pinned assumption is not a collapsed column; it is a cracked pad, and that is the check the simplification quietly deletes.

What to carry away

A base plate in full contact is a stiff rotational restraint, of the same order as the column’s own stiffness, supplied by a detail nobody designed for it.

The classification is a load case, not a drawing. Lift-off begins at NL/6NL/6, so the same base is stiff under gravity and flexible under uplift.

The bolts engage last. They are not in the load path at all until the contact has shortened past the point where the pressure block can balance the moment — 247 kN·m on the plate here, three times the lift-off moment.

And the column, the frame and the foundation want different assumptions. Two of the three are conservative with a pin and the third is not.

Where the model stops

The plate is rigid and the grout is a bed of springs. Neither is true, and the second is worse than the first — grout is brittle, is often incompletely packed, and delivers a pressure distribution nobody has measured.

Nothing here computes the rotational stiffness. The figures give contact lengths and pressures; converting those to a stiffness needs a bed modulus, which is the least reliable number in the calculation.

The bolts are assumed to be tight and unstressed at the start. A preloaded holding-down bolt changes the lift-off threshold entirely, and a loose one delivers nothing until the plate has moved.

Shear is left out. The base also carries a horizontal force, by friction under the plate, by shear in the bolts, or by a shear key — and friction depends on the axial load, so it falls away in the same load case the stiffness does.

The pad below is assumed rigid. It is not, and a base plate on a flexible pad on compressible ground has three stiffnesses in series — plate, pad and soil — of which the last is usually the smallest and is the one no steelwork calculation contains.

And the whole of it is first-order. A base that has softened has raised the frame’s amplifier, which raises the sway, which raises the base moment, which softens it further. The three quantities are coupled and the calculation above treats the moment as given.

A base condition is a stiffness rather than a name, which is the same observation two other essays make about the other end of a member. A connection is neither pinned nor rigid puts a number on the joint at the top, and the ends decide the length is what either of them does to the buckling calculation.

The ladder from here

Later rungs on this anchor: the sway index computed properly, and where the braced-unbraced threshold actually sits. The notional horizontal force, and its derivation from erection tolerance rather than from wind. Amplified sway methods against direct second-order analysis, and the range over which they agree. Multi-storey critical modes, and why a storey-by-storey check can miss them. The bracing system’s stiffness assembled in series from diagonals, gussets, diaphragms and foundations. And the interaction between sway stability and plastic hinge formation, where the frame’s stiffness falls during the very event that is loading it.

The nominally pinned base is one of the oldest simplifications in frame design and it survives because it is nearly always safe for the member. What it is not safe for is the pad underneath it, and a survey of cracked column pads in industrial buildings would find a good proportion of them under bases that the drawings called pins.

Where this goes next

One restraint, and several times the load. The same portal — the same columns, the same beam, the same steel — buckling with its head held against sway and with its head free to sway. The braced frame's critical load is 16.46 EI/L² and the swaying one's is 5.69 EI/L², a factor of 2.89, and the effective length factor that comes out of each eigenvalue is 0.774 against 1.317. Both are eigenvalues of the assembled frame at a beam-to-column stiffness ratio of G = 1.00; the buckled shapes are the mode vectors themselves, drawn at 18 per cent of the storey height so that the movement can be seen. Held, and not held One horizontal restraint at the head of a storey, carrying no vertical load whatever, moves the critical load of the columns beneath it by a factor of 2.89. Effective length is a property of the frame, not of the member. The ends decide the length that matters. Four columns of identical height and section, buckling under four sets of end conditions. The effective length factor is the fraction of the column that behaves like a pin-ended one, and the buckling load goes as its inverse square. The ends decide the length that matters Four columns of identical height and section, buckling at loads sixteen times apart. Nothing differs but what is holding the two ends. 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. The load being amplified is not the load doing the amplifying. The sway amplifier 1/(1 − ΣP/P_cr) against the storey's total gravity load, as columns are added that carry load and provide no lateral stiffness. The critical load of the storey is fixed at 8203 kN by the bracing that exists, and every leaning column moves the structure along the axis without changing it. At the storey drawn the amplifier is 1.57, so the second-order sway moment is 57% on top of the first-order one — and none of that 70% of the load which is causing it appears in any stability calculation done column by column. The storey stays stable across the whole of this axis. That is the practical reason a gravity-only column is drawn on the frame model rather than designed on its own: it is not being checked, it is being counted. Counted, not checked A column with pinned ends and no bracing cannot buckle on its own, so nothing about it fails a stability check. It still carries load, and load with no stiffness attached lowers the buckling load of everything around it — which is why a gravity-only column is put on the frame model and never designed by itself.

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Base plateBearing pressureBolt tensionEccentricityEffective lengthFoundationFree bodyJoint stiffnessKernRotational stiffnessSecond-orderSway stability