The force the bolt never saw applied

A tee stub is bolted to something rigid and pulled by its web. The applied tension is 100 kN and there are two bolts, so each bolt carries 50 kN.

That sentence contains a hidden step, and the step is the assumption that the flange is rigid. It is not, and what happens instead is one of the least intuitive results in connection design: the bolts carry more than was applied, and the extra comes from the connection pushing against itself.

Where the extra force comes from #

Follow the free body of the flange outstand, from the web face outwards.

The web pulls the flange up. The bolt pulls it down. If those were the only two forces they would have to be equal and opposite, and the bolt would carry exactly what the web applied.

But there is a third contact. The flange, bending under that pair, curls away from the surface it is bolted to — except at its outer edge, where it is levered into that surface. The support pushes back, and that push is a downward force on the flange beyond the bolt.

Now the flange has three forces on it: the applied tension inwards of the bolt, the bolt’s pull, and the prying reaction outboard of the bolt. Vertical equilibrium is

$$B = T + Q$$

The bolt carries the applied load plus the prying force, because the prying force is holding the flange edge down and the bolt is what stops the flange from lifting.

That is the whole mechanism, and every counterintuitive thing about it follows from the fact that the prying force is generated by the flange’s own deformation rather than applied from outside. Nobody put it there. It exists because the connection has thickness and dimensions, which is the same sentence the field opened with, applied to a lever arm of forty millimetres.

Three regimes, decided by a plastic hinge #

The size of the prying force is set by how much the flange bends, and how much it bends is set by whether it has yielded. So the problem is a plastic mechanism rather than an elastic one, and it has three regimes.

Let $m$ be the distance from the web face to the bolt line, $n$ from the bolt line to the flange edge, and let the flange’s plastic moment over one bolt pitch be

$$M_p = \frac{p\,t^2 f_y}{4}$$

the same $bd^2/4$ that a rectangular section’s plastic moment always is, with the flange thickness as the depth.

Thick flange. The moment the applied tension causes at the web face, $Tm$, is below $M_p$. The flange does not yield, it barely bends, and its edge never comes down onto anything. $Q = 0$ and the bolt carries exactly what was applied.

One hinge. $Tm$ exceeds $M_p$, so a hinge forms at the web face. The flange rotates about it, its edge bears, and the prying force is whatever it takes to bring the moment at the web face back to $M_p$:

$$Q = \frac{Tm - M_p}{n}$$

Mechanism. The prying force is itself large enough to form a second hinge, at the bolt line. Now the flange has two hinges and is a mechanism, and it has reached its own collapse load at

$$T = \frac{2M_p}{m}$$

Past that point the flange folds regardless of the bolt, and asking what the bolt carries is asking about a structure that has already failed.

The curve, and why a flat percentage is wrong #

Put the three regimes on one axis and the shape of the answer is visible at once.

The curve says three things that a flat allowance cannot.

Above a computable thickness the allowance is exactly zero, and adding thirty per cent there is a straightforward waste of bolt. That thickness is 26.97 mm for this connection, and it is worth looking at where it comes from:

$$t_{\text{no prying}} = 2\sqrt{\frac{T m}{p f_y}}$$

which is $Tm = M_p$ rearranged. There is no bolt property in it. The thickness at which prying disappears depends on the applied load, the geometry and the flange’s yield stress, and a bolt twice as strong moves it by nothing at all. This is geometry beating material in a place nobody looks for it.

The dependence on the applied load is the part that catches people, because it means a flange can move between regimes without anything being altered. Take the tee at the top of the page and halve what is pulling it.

The prying ratio is not a property of the connection. It is a property of the connection at a load, which is why it cannot be tabulated against a detail and why the same drawing gives 1.0 and 1.51 on two different days of the same building’s life. A tee stub checked at its ultimate load and found to be prying may have no prying at all in service, and a percentage carried across from a service check will be too small at the limit state where it matters.

In the middle regime the allowance is larger than thirty per cent. At 20 mm it is 51%; at 24 mm it is 23%. A single number is going to be wrong on one side or the other of wherever it was calibrated.

Below 19.07 mm the question changes. At 16 mm the model reports a ratio of 1.40, and the number is meaningless: the flange collapses at 70.4 kN and the applied load is 100. The connection has failed by flange bending and the bolt was never the issue. A percentage allowance applied here gives a comfortingly specific answer to a question that has stopped being about bolts.

The two ways this is usually taught, and why neither is the mechanism #

It is worth naming the two accounts a reader is likely to have already met, because both are close enough to be memorable and neither says what is happening.

The first is “the flange acts as a lever, so the bolt sees a magnified force.” That is right about the geometry and wrong about the direction of the argument. It suggests the magnification is a property of the lever ratio — that a bolt two-thirds of the way out sees some fixed multiple — when in fact the ratio depends on how much of the flange’s own bending capacity is being used. The same lever with a thicker flange magnifies by exactly one.

The second is “prying is a bolt problem, so use a stronger bolt.” This is worse, because it is actively counterproductive. A stronger bolt in a joint at its flange mechanism carries a fraction more before the flange folds, and the extra edge distance and pitch a bigger bolt needs push $m$ and $n$ outwards, which raises $Q$. The connection can be made worse by improving the fastener, which is a rare enough property to be worth flagging.

The honest one-line account is that prying is a flange-bending problem whose consequence lands on the bolt. Every design lever is on the flange side, and the bolt is the meter rather than the mechanism.

What the mechanism load actually means #

The collapse load $T = 2M_p/m$ deserves more than a formula, because it is the point at which the whole framing of the problem changes and it is easy to walk past.

Below it, the connection has a capacity that depends on the bolt: the flange has yielded somewhere, prying exists, and the question “is the bolt big enough for $T + Q$?” is the right question. Above it, the flange has two hinges and is folding. There is no equilibrium available at that load whatever the bolt does, because equilibrium of the outstand needs a moment at the web face larger than the flange can supply.

The two hinges are visible in the figure as the two circles, and it is the same two-hinge picture as a fixed-ended beam collapsing — a plastic mechanism needs one more hinge than the structure has redundancies, and the flange outstand propped by its bolt is once redundant.

At the default geometry the mechanism arrives at 110 kN, so the connection carrying 100 kN is at 91% of its flange’s collapse load with a bolt force of 150.6 kN. Neither of those two numbers, on its own, says the connection is close to anything.

Why the flange thickness is the design variable #

Read the three regimes together and a design rule falls out that is more useful than any of them individually.

The bolt force is $T + Q$ and $Q$ is set by the flange. So there are two routes to a connection that works: make the bolt strong enough to carry $T + Q$, or make the flange thick enough that $Q$ is zero. The first route is a fight — a bigger bolt needs more edge distance and a wider pitch, which increases $m$ and $n$, which increases $Q$. The second route ends the argument.

That is why end plates and tee stubs in practice are detailed by thickness rather than by bolt grade, and why the component method treats “end plate in bending” and “bolts in tension” as two components in series rather than as one check. They are different mechanisms and the softer one governs.

It is also why $m$ — the distance from the web face to the bolt — is kept as small as detailing allows. It appears linearly in the moment at the web face and squared under the root in the thickness, so bringing the bolt 10 mm closer to the web is worth more than 5 mm of flange.

What the outer distance does, and the cap on it #

The distance $n$ from the bolt to the flange edge appears in the denominator of $Q$, which reads as though a wide flange edge reduces prying. Up to a point it does, and the point is where the reasoning breaks.

The reading is right in the direction it points, and the size of the effect is worth seeing before the caveat arrives.

That is $Q = (Tm - M_p)/n$ read as a hyperbola in $n$, and it is why a flange edge trimmed for clearance is a structural change rather than a detailing one. It also says the opposite thing about a wide flange, and there the reasoning breaks.

The prying force is a contact reaction, and contact happens where the flange presses hardest. On a very wide outstand the flange does not bear along its whole width; it bends further and contacts nearer to the bolt. Which is why design rules cap the effective $n$ at about $1.25m$ — beyond that, extra flange is not participating and counting it is optimistic. For this tee the cap is 56.25 mm, at which the bolt would carry 136.2 kN; anything drawn beyond it is flange that the calculation is not allowed to believe in.

The cap matters because it means the two levers are not independent. Increasing $m$ increases the flange moment and raises the cap on $n$, and the first effect is the larger. There is no geometry in which moving the bolt further from the web helps.

It is also worth noticing what the cap is made of, because it is not a strength argument at all. $1.25m$ is a statement about where the flange is still in contact, which is a statement about its deflected shape, which is a statement about stiffness. So the strength calculation for prying has a stiffness term smuggled into it as a geometric limit — and that is characteristic of the whole field rather than an oddity of this rule. A connection’s strength and its stiffness are computed from the same deformation, and separating them cleanly, as member design does, is not available here.

Where this lands in practice is worth stating before the essay leaves the tee stub, because a moment end plate is not one tee stub but a column of them. Every bolt row in such a joint is a tee stub twice over — once in the end plate and once in the column flange opposite — and the row takes whichever of the two is weaker. That is four components per row to be checked against each other, and whether the joint may use a plastic distribution of force between its rows depends on which of them reaches its limit first: a row limited by a flange in bending can shed load to the row below it, and a row limited by its bolts cannot.

What a preloaded bolt does to the same picture #

Everything above assumes the bolt is a bar that stretches when it is pulled. Tighten it first and the arithmetic in service changes completely — not the collapse load, which is the point, but everything the joint does before it.

A preloaded bolt clamps the flange against what it is bolted to. Apply an external tension and the load divides between two springs in parallel: the bolt, which stretches further, and the clamped material, which unclamps. The bolt’s share is $k_b/(k_b + k_c)$, and for a steel bolt through a short grip of steel the clamped material is three to six times the stiffer — its load path is a fat cone of metal, the bolt’s is a slender shank. So the bolt picks up something like a fifth of what is applied, and the rest is taken by the clamping force letting go.

That holds until the flange lifts off. Separation arrives at roughly $F_p(1 + k_b/k_c)$ — a little above the preload — and past it the bolt takes the whole of any further load, and the prying mechanism of this essay restarts from there. The load–bolt-force curve is therefore flat and then steep, with a knee at the preload, rather than the straight line an untightened bolt gives.

Two consequences, and they pull in opposite directions.

In service it is a large gain, and it is the reason fatigue-loaded bolted joints are preloaded. A bolt cycling between 0 and 100 kN of applied load sees a stress range of 100 kN untightened and perhaps 20 kN preloaded, and fatigue life goes as a high power of the range. It is not a strength argument at all; it is an argument about how much of the applied variation reaches the fastener.

At collapse it buys nothing, and the essay’s model is right to ignore it. The three regimes above are an ultimate-limit-state calculation, and by the time the flange has formed a hinge it has bent enough to relieve its own clamping: the preload has been spent, the plates have separated, and the bolt is carrying $T + Q$ exactly as though it had never been tightened. Preload is a service-condition device, and quoting it as extra capacity is the commonest way to get this wrong in the other direction.

Where else this appears #

Prying is usually taught as a tee-stub phenomenon, which understates it considerably. The tee stub is a model, and it is the model for every bolted connection carrying tension through a flexible plate:

• an end plate carrying moment, where each bolt row is a tee stub and the flange is the plate, treated exactly this way in the essay on moment connections;
• a column flange on the other side of the same joint, which is a tee stub whose web is the column web;
• an angle cleat in tension, which is a tee stub with one leg;
• a base plate with holding-down bolts in tension, where the plate bends and the bolts pick up the difference.

In every one of them the same three regimes exist, the same thickness formula applies, and the same trap is available: sizing the bolt for the applied load and then discovering that the plate decided.

The stiffness consequence, which is separate and larger #

Everything above is about strength. There is a second consequence of the same flexibility, and in a frame it is often the more important one.

A flange that bends under load is a flange that moves under load, and a bolt row whose plate is bending contributes a rotation to the joint. That rotation is what makes a “rigid” connection not rigid, and it is why the component method lists “end plate in bending” and “column flange in bending” as separate springs with their own stiffnesses.

The two consequences move together, which is convenient: the flange thickness that removes prying is also the flange thickness that makes the joint stiff, because both are governed by $t^3$ or $t^2$ terms in the same plate. A connection detailed to avoid prying is usually well on the way to being classifiable as rigid, and one detailed by bolt grade alone is usually neither.

Those two springs are not a small part of the total. In a typical bolted end-plate joint the column flange in bending and the end plate in bending are together more than half of the joint’s whole flexibility, with the bolts themselves, the column web in shear and the column web in compression sharing the rest. Both of the large ones are the tee-stub mechanism of this essay, seen from the stiffness side rather than the strength side — and because the springs are in series, the softest of them decides, so stiffening anything else in the list changes almost nothing.

What to take from it #

A bolt in tension through a flexible plate carries more than was applied, and the extra is generated by the connection rather than delivered to it. The mechanism is a lever, and the fulcrum is the flange edge.

The regime is computable and the flat allowance is not a regime. Above one thickness there is no prying at all; below another the flange is the failure and the bolt is a bystander; between them the ratio is a curve that runs from 1.0 upwards.

The thickness that removes it contains no bolt property. It is $2\sqrt{Tm/pf_y}$, and it is the clearest possible statement that this is a geometry problem wearing a fastener’s clothes.

And one more, which is the reason this essay sits third in the field rather than later. Prying is the first place on this site where a structure generates an internal force that nobody applied to it, purely by deforming. It will not be the last: it is the same shape of argument as the load that makes itself worse, where a column’s own lean multiplies its moment, and as the stress that was there before the load, where a weld’s cooling leaves a field behind.

What those three share is that the applied loading is a complete description of what was done to the structure and an incomplete description of what the structure is carrying. Every equilibrium check on this site up to the connections field has been able to work from the applied loads alone. From here on, that is no longer enough — and the extra term is not small, not conservative in any consistent direction, and not visible on the drawing that shows the loads.