The bolt group has no neutral axis

A bracket’s load misses its bolt group’s centroid in the plane of the bolts, and the group answers with a torsion about that centroid. The centroid is not an assumption there; it is where the group’s own geometry puts the centre of rotation, because every bolt in the group resists the torsion the same way and the sum of their moments about the centroid is what has to balance.

Turn the moment through ninety degrees and none of that survives.

Why a bolt group is not a section #

The arithmetic that suggests itself is the section formula: treat each bolt as an area, find the centroid, compute $\sum y^2$, and take $T = My/\sum y^2$. It is the same expression the in-plane calculation used and it is on the same page of every handbook.

It assumes something a bolt cannot do. A section’s fibres carry tension on one side of the neutral axis and compression on the other, and the axis sits at the centroid because that is where the two balance. A bolt carries tension and nothing else. Pushing a bracket against a column flange is not the bolts’ work; it is the plate’s, along whatever part of itself is still in contact.

So the free body has two different kinds of force on it. Above the axis, discrete bolts in tension. Below it, a distributed bearing pressure between two plates, over a width that has nothing to do with the bolt spacing. The two are in equilibrium and there is no reason for the dividing line between them to be at the centroid of the bolts — the bolts are only half the section.

The centroid is where a calculation puts the axis, not where the connection puts it, and the whole of this essay is the distance between those two.

Three defensible axes #

At the group’s centroid. Every bolt is an area, half of them pull and half of them push, and $\sum y^2$ is taken over all of them. This is the section formula applied without asking what pushes, and it is what an analysis program reports if the bolts are modelled as a set of springs and the plate is not modelled at all.

At the plate’s compression edge. The plate is taken to bear on a line along its bottom edge, and the whole group rotates about that line, so every bolt is in tension. This is the assumption behind most bracket design guidance, and it is the cheapest one to apply because it needs no iteration: $T = Md/\sum d^2$ with $d$ measured from the edge.

Wherever the bearing actually is. Give the plate a width to bear over, give the bolts their tensile areas, and require both force and moment to balance with a single strain distribution through the lot. That is a transformed section with a compression block, and the axis comes out of it rather than going into it. Here it sits 40.2 mm above the edge, which is neither of the other two answers.

The third is the one that is a computation rather than a stipulation, and it lands much nearer the second than the first.

The factor of two, and which way it runs #

An axis further down means a longer lever and a smaller force. That is the direction the sweep runs and it is worth dwelling on, because the intuition usually runs the other way.

The moment is carried by a couple: a tension resultant somewhere in the upper part of the connection, and a compression resultant somewhere in the lower. The size of the couple’s forces is the moment divided by the distance between them. Moving the compression down — from the bottom bolts at the centroid model, to a bearing block at the plate’s edge — pushes the two resultants further apart and reduces both forces.

It also spreads the tension. At the centroid the middle row does nothing and the top row does everything; at the edge all three rows pull, in proportion to their distance from the bottom, and the top row’s share falls to a fifth of the total rather than a half.

So the two effects compound, and the top row goes from 50.0 kN to 21.4 — a factor of 2.33. The centroid assumption is conservative on the critical bolt, and it is conservative because it makes both mistakes in the same direction: it puts the compression too high and it declines to use the middle row.

That is the correction this essay owes its own foundation. The essay below states, in one paragraph, that a real bracket bearing on its bottom edge “shortens the lever arm and raises the tension in the top row”. It lengthens it and lowers the tension, and the reason the error is easy to make is that moving a neutral axis down is usually bad news — in a cracked concrete section it is, because there the compression zone is shrinking. Here nothing is shrinking; the compression has simply moved to a place bolts were never going to provide it from.

Where the models agree, and it is not the bolt #

The total tension is nearly the same under all three: 100.0 kN at the centroid, 85.7 at the edge, 87.8 solved. Within fifteen per cent.

That is worth knowing because it says which downstream check the choice affects and which it does not. The force pulling the bracket off the column is essentially model-independent, so the column flange’s own bending — which is a tee stub with its own three modes — sees roughly the same demand whichever axis is assumed. What is model-dependent is how that total is shared between the rows, which is what decides the bolt size.

Adding rows helps a little and does not resolve anything, because the disagreement is about where the compression is and adding bolts does not move it.

Where the choice turns into a bolt #

That is the reason the argument is worth the trouble. The shear term is fixed by the in-plane calculation and is the same under every assumption. The tension term is not, and it takes the utilisation from two-thirds to four-fifths depending on a sentence in a calculation.

A spread of 0.15 of a unit utilisation is a bolt grade, or two extra bolts, or a thicker bracket. And it is decided by something no drawing records: whether the calculation that sized these bolts assumed the plate bears, and where.

The solved axis, once, by hand #

The third model is the only one with an iteration in it, and the iteration is short enough to do on paper, which is worth showing because it is the reason the model is skipped.

Let $x$ be the depth of the axis above the compression edge. Above it the bolts are bars of area $A_b$ at heights $d_i$, each carrying a force proportional to $d_i - x$. Below it the plate bears over a width $b$ with a stress rising linearly from nothing at the axis to a maximum at the edge, so the compression is $\tfrac12 b x^2$ times the same constant of proportionality.

Force balance, with the constant cancelling:

$$n\sum_{d_i>x}(d_i - x) = \tfrac12\,b\,x^{2}$$

with $n = A_b$ times the number of bolts in a row. That is one equation in one unknown, monotone in $x$, and a bisection converges in half a dozen steps. Here $n = 2 \times 245 = 490$ mm², the rows are at 75, 150 and 225 mm, and $b = 200$: the left-hand side at $x = 40$ is $490 \times (35 + 110 + 185) = 161{,}700$ and the right is $100 \times 1600 = 160{,}000$. The root is 40.2 mm.

Then the second moment about that axis fixes the scale. $I = b x^3/3 + n\sum(d_i - x)^2$, which is $8.7\times10^{5} + 5.1\times10^{7}$ — the bearing block contributes under two per cent of it — and the row forces are $M(d_i - x)/I$ per bolt column.

Two things fall out of that arithmetic and both are useful without repeating it.

The bearing block’s own stiffness is nearly irrelevant. It supplies 1.7 per cent of the second moment, so doubling the width it bears over moves the axis a little and the bolt forces hardly at all. That is why the plate’s-edge model is such a good approximation: it is the limit of this one as the bearing gets stiff, and the bearing is already stiff.

And the axis sits close to the compression edge whenever the bolts are slender, which they are — a bolt is a 20 mm bar with a grip of 30 mm, and a plate bearing over 200 mm of width is enormously stiffer per millimetre of depth. The centroid model is what one gets by making the bolts the whole section, and the bolts are the flexible half.

Why the wrong model is the one in the books #

The centroid model persists for a reason that is worth naming, because it is not laziness.

It is the only one of the three that is a single formula covering both eccentricities at once. A bracket has a moment in the plane of the bolts and a moment about an axis in that plane, and if both are treated as a section about the group’s centroid then the two are one calculation — $M_x y/I_x + M_y x/I_y$ with a polar term, which is exactly the biaxial section formula and exactly what a pile group uses. That unity is worth a great deal to whoever is doing it by hand.

It is also the model a finite-element analysis produces by default, because a bolt is naturally modelled as a spring and contact between plates is not naturally modelled at all. A model with no contact in it has no choice but to put the axis at the springs’ centroid, and it will report compression in the lower bolts without complaint.

What it costs is a top-row force that is twice what the connection will see, which on a bracket governed by that bolt is two extra bolts or a grade. That is an ordinary price for a simple rule and it is worth paying knowingly rather than by accident — and the accident is the common case, because nothing in the output says which axis was used.

Which free body produced the number #

The free body is the bracket, cut through the bolts and through the contact between the two plates, and its usefulness here is entirely in what it contains.

It contains the bolt tensions, which are on the cut through the bolts. It contains the bearing pressure, which is on the cut through the contact. Both are internal to the connection and external to this body, and the equations of equilibrium relate them — but only after somebody has said what shape the bearing pressure has and over what width it acts.

Equilibrium has two equations and the problem has more unknowns than that. Vertical force and moment give two conditions; the unknowns are the tension in each row and the depth and intensity of the bearing block. Closing the gap needs a compatibility statement — that the plate stays plane as it rotates — and the compatibility statement is where all three models actually differ.

The centroid model’s compatibility is the bolts stay in a straight line through their own centroid, which is a statement about the bolts alone. The edge model’s is the plate rotates about its edge, which is a statement about the plate alone. Only the third writes one strain distribution through both, and that is why it is the one with an answer rather than an assumption in it.

What the picture cannot show #

The plate does not bend in any of these figures. Every model here assumes the bracket plate is rigid enough to deliver a linear strain distribution, and a plate that is not rigid does something else entirely: it bends between the bolts, lifts off between them, and develops the prying force that makes a bolt carry more than the load applied to it. That is a different line of argument and it governs whenever the plate is thin, which on brackets is often. The three models here are three answers to the rigid-plate question, and the thin-plate question is not one of them.

There is no preload. A preloaded bolt in a tension connection does not see the applied tension until the plates separate, so its force history is nothing like any of these lines — it is nearly flat and then steep. Nothing here knows about it.

The bolts are all the same size and all in the same condition. A row that has been tightened harder, or a hole that is 2 mm oversize on one side, changes which bolt bears first and therefore where the strain distribution starts from. None of the three models has a place to put that, and it is the ordinary state of a bolted connection rather than an exception.

And the bearing width is stipulated. The 200 mm the third model bears over is a width somebody chose: the bracket’s width, or the column flange’s, or whatever part of it the analyst believed was in contact. The solved axis depends on it, and the two stipulated models avoid needing it by being stipulations.

The assumption underneath all three #

Every model on this page assumes that the two plates are in contact over the compression zone and separated over the tension zone, with a clean line between them.

They are not. A bolted joint is in contact in patches around each bolt, wherever the two surfaces happen to touch, and the transition from contact to separation is gradual and depends on the plates’ flatness, the bolt preload, and the paint. The neutral axis is a line drawn through a region, and the three models above are three positions for a line that does not exist.

The patchiness has a direction, too. Two plates pulled apart at the top and pressed together at the bottom are in their best contact exactly where the model needs bearing, so the assumption improves as the moment grows — which is the opposite of how most idealisations behave and is part of why this one has never caused trouble.

That is the honest reason the disagreement has survived: none of the three is right, all three are in equilibrium with the applied moment, and the one that is used is the one somebody’s guidance recommends. The lower-bound theorem covers the choice, provided the connection is ductile enough to redistribute toward whichever state was assumed — and a bolt in tension, unlike a bolt in shear, has very little ductility to redistribute with.

Still open: the bolt that was tightened first #

Everything above treats a bolt as a bar that is stretched by the load applied to it. A preloaded bolt is stretched before any load arrives, and while the plates stay in contact an external tension is shared between the bolt’s stiffness and the joint’s — mostly the joint’s, because a short thick clamped region is much stiffer than a long thin bolt. The bolt’s force therefore barely moves until the plates separate, and then it moves steeply.

That changes the fatigue problem completely and changes the strength problem hardly at all, which is the interesting asymmetry in it. The next essay on prying takes it up, because the same effect is what delays the prying force in a tee stub.

Beyond that, the group that is a line rather than a set of points, where the same three terms become integrals and a weld’s neutral axis really is its own centroid — for the one reason this essay has been about, which is that a weld has no separation to argue about.