Module 13 · Lesson 13.1

The T-stub, and the three ways it fails

Almost every bolted joint contains a piece of plate that bends to deliver tension into a bolt. One model covers all of them, and its three failure modes are derivable.

Why this matters

An end plate, a column flange, an angle cleat, a base plate — they look like different details and they behave identically. Each is a piece of plate that bends, and each delivers its load into bolts that are trying to hold it flat. Reducing all of them to one model is what makes joint design tractable, and it also explains prying, which otherwise looks like an arbitrary penalty. The three failure modes fall out of statics, and one of them matters far more than its resistance suggests: it is the only one that gives any warning.

By the end of this lesson you should be able to

Recognise the T-stub in several details that look unrelated
Derive its three failure modes
Explain prying from equilibrium
Say which mode to aim for, and what it costs

What you should already know

Bolts in tension and the punching check (Module 12)
Prying as a mechanism (Module 12)
Plastic hinges and the plastic moment of a rectangular section (Module 6)
Work equations for a collapse mechanism — the Structural Analysis Fundamentals course covers this

One model, several details

A T-stub is a length of flange, bolted to something stiff, pulled by a web. Nothing else.

Once you look for it, it is everywhere:

Detail	The flange is…	The web is…
Bolted end plate	the end plate	the beam flange
Column flange in bending	the column flange	the column web
Angle cleat in tension	the outstanding leg	the bolt line to the beam
Column base plate	the base plate	the column flange

Each is a plate bending between a stiff line where the load arrives and a bolt that is trying to hold it down. The lever arm from the bolt to the web face is m, and the distance from the bolt to the free edge is n. Those two dimensions, the plate thickness and the bolt strength decide everything.

The one piece that does not come from statics is the effective length — how much of a real plate acts as an equivalent T-stub, when the yield lines can spread sideways, run round a bolt, or be cut off by an edge. That is a yield-line result, it is code-calibrated, and this course takes it as an input rather than tabulating it.

From first principles

The three T-stub failure modes

We want to show: Derive the resistance of a T-stub in each of its three modes, and see why one of them is the only one that gives warning.

Pull the web of a T-stub upwards. Either the flange is weak and bends into a mechanism while the bolts hold, or the flange is strong and the bolts break, or something in between — the flange bends enough to form one hinge and lever its own edge against the support, which loads the bolts harder than the applied force does. Those three pictures are the three modes, and the arithmetic of each is short.

Try it

The three modes, and where ductility is lost

An equivalent T-stub 200 mm long in S355. All three modes are shown at once; the solid bar is the one that governs. Thicken the flange until the joint stops giving warning.

Flange thickness15 mm

Bolt to web face, m45 mm

Bolt to plate edge, n50 mm

Bolt resistance ΣF_t,Rd282 kN

Plastic moment M_pl: 3.99 kNm
Mode 1, flange yields: 355 kN
Mode 2, flange and bolts: 233 kN
Mode 3, bolts alone: 282 kN
Governing mode: 2
Resistance F_T,Rd: 233 kN
Prying force: 24.8 kN
Force in each bolt: 141.0 kN
Gives warning before failure: yes

Mode 2 governs at 233 kN: a hinge forms at the web face, the flange edge prys, and the bolts break carrying 141 kN against the 116 kN applied to them. The prying force of 25 kN is real and is the reason the bolts are working harder than the applied load suggests.

Things worth trying

Start at 15 mm. Mode 2 governs at 233 kN, and the bolt force comes out at exactly the bolt's own resistance — which mode 2 requires, since it is defined by the bolts reaching their limit.
Thicken the flange one step at a time and watch the governing bar move down the list: mode 1 at 8 and 10 mm, mode 2 from 12 to about 18, mode 3 from 20 mm.
Find the thickness where the verdict turns from DUCTILE to BRITTLE. That is the design decision this lab exists for, and it is not marked on any drawing.
Keep going to 30 mm. The mode 3 bar does not move at all — beyond the boundary, extra flange thickness buys nothing whatsoever, because the bolts are what fail.
Watch the prying force as you thicken the flange. It rises to a peak in mode 2 and then vanishes entirely in mode 3, because prying comes from the flange bending.
Now go back to 15 mm and increase the bolt resistance instead. The mode 3 bar rises and the joint moves back towards flange yielding — stronger bolts make a T-stub MORE ductile, not less.
Increase m from 45 to 90 mm. Mode 1 halves, because it goes with 1/m. The distance from the bolt to the web face is the single most sensitive dimension in the detail.
Reduce n below about 25 mm and watch mode 2 fall away. A bolt close to the plate edge has nothing to pry against, so the flange cannot lever and the bolts are reached sooner.

Worked example

An end plate T-stub, thickness by thickness

Given

Equivalent T-stub of effective length 200 mm in S355
Two M20 grade 8.8 bolts, F_t,Rd = 141.1 kN each, so ΣF_t,Rd = 282 kN
m = 45 mm from the bolt to the web face, n = 50 mm to the plate edge
γ_M0 = 1.0

Find

How the resistance and the failure mode change with the plate thickness.

Assumptions

l_eff is a calibrated yield-line result and is given, not derived
n is capped at 1.25m = 56.25 mm, so the full 50 mm counts here
The column flange the plate bolts to is assumed rigid

Predict first

A T-stub governed by mode 3 at 282 kN needs more resistance. The plate is thickened from 20 mm to 30 mm. What happens?

Practice

An equivalent T-stub flange is 200 mm long and 15 mm thick in S355. What is its plastic moment, in kNm? Take γ_M0 = 1.0.

Practice

With M_pl = 4.0 kNm and m = 45 mm, what is the mode 1 resistance in kN?

Practice

With M_pl = 4.0 kNm, m = 45 mm, n = 50 mm and ΣF_t,Rd = 282 kN, what is the mode 2 resistance in kN?

Practice

In that mode 2 case the prying force is 24.8 kN and each bolt takes half the 233 kN. What total force does each bolt carry, in kN?

Check yourself

Why is mode 1 the mode to aim for, despite giving the lowest resistance?

Summary

A T-stub is a flange bending between a web and a bolt — end plates, column flanges, cleats, base plates
Mode 1 = 4M_pl/m, from a four-hinge work equation. The bolts do not appear
Mode 2 = (2M_pl + nΣF_t,Rd)/(m+n), from equilibrium with prying
Mode 3 = ΣF_t,Rd. The flange does not appear
Prying is a consequence of the flange bending, not a code penalty
In mode 2 the bolt force comes out at exactly F_t,Rd — a free consistency check
15 mm gave 233 kN and warning; 20 mm gave 282 kN and none; 25 mm gave nothing more
l_eff is a calibrated yield-line result and is the one piece not derived here

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint