The thickness that decides who fails

A bolt in a tee stub is pulled by the load and pushed by the flange it is holding down. The second is prying, and how much of it there is depends on one dimension in a way that is not gradual.

The figure above plots that dependence for a stub carrying 140 kN per bolt: prying disappears above 31.91 mm of flange, and below 22.75 mm the flange has become a mechanism and the bolt has stopped being the thing that decides. The nine millimetres between those two numbers is where the whole of the design question lives, and a designer who reads the flange thickness off a table has picked one of three behaviours without being told there were three.

Three regimes, not a curve #

The tee stub has three distinct behaviours and the flange thickness picks between them.

Mechanism. A thin flange forms plastic hinges at the web and at the bolt line, becomes a four-hinge mechanism, and collapses. The bolt is not the thing that fails — it may be nowhere near its capacity — and the connection’s strength is the flange’s.

One hinge. A middle flange forms a hinge at the web only. The bolt reaches its capacity while the flange is still holding at the bolt line, and the failure is the bolt’s, with prying present.

No prying. A thick flange does not bend enough for its tip to bear on anything, so the prying force is zero and the bolt carries exactly the load applied to it.

Those are three failure modes with three different capacities and three different amounts of warning, and moving between them is a matter of millimetres.

Go past the peak and the behaviour changes character again, because the flange has stopped forming a hinge at the bolt line and the prying is now a correction rather than the answer.

Prying peaks in the middle, which is the least intuitive feature of the whole subject: neither the thinnest nor the thickest flange produces the largest bolt force.

Which free body produced the number #

The free body is half of the flange, cut on the plane of the web and at the flange tip.

Crossing the web cut is the applied tension $T$ per bolt and the moment of the hinge that forms there. Crossing nothing at the tip is a contact force $Q$ — the prying force — which exists only because the flange has bent enough for its tip to bear on the surface it is bolted to.

Taking moments about the bolt line gives $Q$ directly, and vertical equilibrium then gives the bolt force as $T + Q$. Two features of that free body explain everything above.

$Q$ is a contact force, so it cannot be negative. A flange that does not bend enough to touch has $Q = 0$, and there is no smooth continuation into “negative prying”. That is the discontinuity at 31.91 mm.

And $Q$ acts at the flange tip, at a lever arm $n$ beyond the bolt. So the prying force is levered against the bolt’s own distance $m$ from the web: $Q \approx T\,(\text{something})\,m/n$, and the two distances matter more than the thickness does.

The whole of the subject is a lever with three points on it — the web, the bolt, the tip — and the flange’s thickness decides only whether the lever is stiff enough to work.

Why the thickness appears squared and the lever arm does not #

The two boundaries in the figure come from setting two different things equal, and it is worth doing both by hand, because the exponents explain why the geometry wins.

The mechanism boundary is where the flange’s own yield-line resistance equals the moment the applied load puts into it. A plastic hinge across a strip of flange of width $p$ and thickness $t$ carries $M_p = p t^2 f_y / 4$ — the section is a rectangle, its plastic modulus is $p t^2/4$, and there is nothing else in it. The applied moment at the web is $T m$ minus whatever the prying at the tip gives back. Setting them equal and solving for $t$ puts the thickness under a square root: $t \propto \sqrt{T m / (p f_y)}$.

The no-prying boundary is where the bolt-line hinge has just disappeared, so the flange is a cantilever from the web with the bolt as its only support, and the tip lifts clear. That gives a second expression, again with $t$ squared and $m$ linear.

So in both boundaries the thickness enters squared and the lever arm enters linearly — but they enter on opposite sides. Doubling $m$ demands $\sqrt{2} \approx 1.41$ times the thickness to stay in the same regime, which is a 41 per cent increase in a dimension that costs money and welding, against a 30 mm change in where a hole is drilled.

That asymmetry is the whole reason the practical advice runs the way it does. It is also why the effect is invisible in a spreadsheet that solves for capacity at a fixed geometry: the capacity moves smoothly and nothing announces that the answer came from a different equation than it did on the previous line. The regimes are visible only when the thickness is swept, which is what the curve above is for.

And note which quantity is absent from both boundaries: the bolt. Its diameter, its grade and its preload appear nowhere in deciding which regime the flange is in. The flange decides, and the bolt then either survives what the flange hands it or does not.

Geometry beats thickness #

The two lever arms are the strong variables, and the arithmetic says so plainly.

Read that the other way and it is a saving. Bringing the bolts in from 80 mm to 50 takes 8.45 mm off the flange thickness at which prying stops — 40.36 down to 31.91 — on every stub of this shape, bought by drilling the holes 30 mm closer to the web. That is a plate two standard thicknesses lighter for a change that costs nothing to fabricate.

There is a limit, and it is a spanner. The bolt cannot be closer to the web than a socket will fit, which is typically 1.2 to 1.5 times the bolt diameter plus the root radius — and that clearance, rather than any structural quantity, sets the smallest $m$ available. A connection’s prying behaviour is decided by the width of a tool.

The bolt is a spring, and preload moves the whole picture #

Everything above treats the bolt as a rigid point that simply resists. It is not: a bolt in tension is a spring of stiffness $A_s E / L_b$, where $L_b$ is its grip length plus an allowance for the head and nut, and its extension is what lets the flange lift.

That matters twice.

A longer bolt prys less. Its extension under the same force is greater, so the flange rotates about the bolt line rather than bending in front of it, and the tip is slower to bear. A stub bolted through a thick column flange with a long grip is a different connection from the same plate on a thin one, and nothing in the geometry of the tee says so.

And a preloaded bolt barely prys at all until it decompresses. The preload has already clamped the flange to what it bears on, so the contact at the tip is pre-existing rather than newly created, and the applied load first unclamps the interface before it can add to the bolt. Up to the decompression load the bolt force rises by only the ratio of the bolt’s stiffness to the joint’s — often less than a fifth of what is applied — and past it the behaviour reverts to the arithmetic in this essay.

The near-cancellation is worth understanding. The flange’s yield-line resistance goes as $f_y t^2$, so raising the grade allows a thinner flange for the same mechanism load; raising the applied load demands a thicker one. The two changes are nearly equal and opposite, which is why the boundaries barely move and why a rule of thumb about flange thickness works across grades and load levels better than it has any right to.

A preloaded joint has one more property this essay does not model: it does not slip, which is a different thing to design for entirely and is decided by friction rather than by any of the three regimes.

The regime that is ductile is the weak one #

Choosing between the regimes is a design decision, and it is the one place in connection design where the strongest option is not obviously the right one.

The mechanism regime is ductile. The flange yields, deforms visibly, and the bolt is not the thing that fails. A connection in this regime gives warning and redistributes.

The no-prying regime is strong and brittle. The bolt carries exactly what is applied, so it can be sized exactly — and when it goes, it goes suddenly, because a bolt in tension has almost no deformation capacity.

Codes require ductility at the joints that need it — a seismic connection, a member relied on for robustness, a joint whose rotation capacity a plastic analysis assumes — and they get it by forbidding the strong regime: the flange must be thin enough to hinge before the bolt breaks. That is a deliberate weakening, and it is the same argument as the part that is meant to be weak applied to a plate rather than to a link.

So the design question is not how to remove prying but which of the three regimes the joint is meant to be in, and the answer depends on what the joint is for.

Where the prying force goes #

The prying force is a contact force, which means it has to be resisted by whatever the flange is bearing on — and that surface is somebody else’s member.

For a beam-to-column end plate, the tee’s tip bears on the column flange, which is itself a tee stub of exactly the same kind, spanning between the column’s web and its own free edges. So the prying force in one component is an applied force on another, and 73.94 kN pressing on a slender column flange is not a small load to hand over quietly. That is one of the reasons a joint is assembled as springs rather than checked as a single object: the components load each other.

For a base plate the tip bears on grout, which does not behave like steel at all. Grout crushes, creeps and is sometimes not there — a plate bedded on shims with the grout poured afterwards has no contact at the tip until the grout hardens and shrinks away from the underside of the plate. The connection at the ground is the one place where the prying force can honestly be taken as zero, and the honest reason is that nothing is reliably in the way.

And for a tee welded to the face of a plate, the contact runs through the weld root at the toe of the flange, which is loaded through its thickness. That is the direction a rolled plate is weakest in, and it is the mechanism behind the tearing nobody tests for: a prying force is a through-thickness tension applied at exactly the detail where the inclusions lie.

None of these appear on the tee stub’s own free body, because the free body was cut at the contact and the contact was drawn as an arrow. An arrow is a promise that something on the other side of the cut can take it.

The same component, in a real joint #

The tee stub is not a connection anybody builds; it is a component, and its value is that a real joint is made of several of them.

Each bolt row in that plate is a tee stub — twice over, in fact, since the column flange is one and the end plate is another, and the row’s capacity is the smaller of the two. Making a moment cross a gap is a matter of assembling those components, and prying is what each one contributes.

The plastic distribution in the figure is available only if the rows are ductile enough to redistribute — which brings the regime question back at the scale of the joint. A joint whose rows are all in the no-prying regime cannot use the plastic distribution, and loses the factor of 1.22 that made it worth having.

The same amplification, at a stress range #

Everything so far is a strength argument, and the amplification is worse where strength is not the criterion.

A bolt in a connection carrying live load sees a stress range — the difference between loaded and unloaded — and that range is amplified by the same factor the peak is. A stub whose bolt sees 1.53 times the applied force sees 1.53 times the applied variation, and fatigue life goes as roughly the inverse cube of the range. A factor of 1.53 on stress is a factor of 3.6 on life.

The detail is poor to begin with. A bolt loaded in tension has its stress concentrated at the first engaged thread, where the load transfer between nut and bolt is most abrupt, and the category assigned to that detail is one of the lowest in any fatigue table — around 50 N/mm² at two million cycles for a rolled thread, and lower for a cut one. It is also a detail whose category does not improve with the steel’s grade, which is the general rule that makes fatigue a geometry problem rather than a material one.

So the sensible design for a fatigue-loaded tension connection is the opposite of the sensible design for a ductile one. Fatigue wants the no-prying regime — thick flange, close bolts, no amplification — and preload on top of it, because a preloaded bolt below decompression sees almost none of the applied variation. Ductility wants the mechanism regime.

A joint asked for both has to be told which it is, and that is a decision about what the structure is for rather than a calculation. The one thing that cannot be done is to leave it to whatever flange thickness came off the section table.

What to carry away #

A bolt carries more than it is given, by up to about 1.5. The excess is a contact force at the flange tip, levered against the bolt’s own distance from the web.

There are three regimes and the flange thickness picks between them. Prying is largest in the middle one, and the transitions are millimetres apart.

The lever arms beat the thickness. Moving the bolt line does more than adding plate, and the smallest bolt spacing available is set by a spanner rather than by mechanics.

And the ductile regime is the weaker one. A joint required to deform is required to be in it, which is a deliberate choice to be worse at one thing in order to be better at another.

Where the model stops #

The tee is a two-dimensional strip. A real end plate is a plate with bolts in a pattern, and its yield-line mechanism is two-dimensional — circular, side, corner or group patterns, each with its own effective length.

Contact is assumed at the flange tip. In reality it is distributed over a region whose extent depends on the flange’s own curvature, and taking it as a point force at the tip is conservative for the bolt and unconservative for the flange’s moment.

The bolt is a point force with no stiffness. A bolt in tension is a spring, its extension unloads the prying, and a preloaded bolt has almost no extension until the preload is exceeded — so a preloaded connection prys much less than this arithmetic says until it decompresses.

The two tee stubs are treated separately. The column flange and the end plate deform together, and their combined behaviour is not the smaller of two separate ones — which is the approximation the component method makes and lives with.

And nothing here is a fatigue statement. A bolt in a prying connection sees a stress range amplified by the same 1.53, at a detail category that is poor to begin with.

The ladder from here #

Later rungs on this anchor: the yield-line patterns of a real end plate, and the effective lengths that turn a plate into an equivalent tee. Preloaded connections, where the bolt’s own stiffness delays the prying and the calculation is a contact problem. The column flange as the other tee stub, and how the two combine. Rotation capacity of a bolt row, which is what a plastic moment distribution is spending. Prying in a base plate, where the “flange” is bearing on grout and the contact is not at a tip. And the component method’s assembly, where each row’s stiffness and strength go into a joint’s own moment–rotation curve.

Prying was measured before it was modelled. Early riveted and bolted tension connections failed at loads well below the fastener’s capacity, the discrepancy was attributed to workmanship for years, and the lever mechanism was identified in the 1960s during the work that produced the first bolted-connection design guides. What made it a component rather than a correction was the European component method of the 1990s, which took the tee stub — an object nobody builds — and made it the unit that every moment connection is assembled from.