Module 13 · Lesson 13.2

The component method: weakest, softest, and why they differ

Components in series. The weakest sets the resistance, every one sets the stiffness, and improving the wrong one is the most common wasted effort in joint design.

Why this matters

Module 3 showed that a joint's stiffness changes where the moment goes, and that a 'simple' connection carrying half its fixed-end moment is a real and common problem. This lesson is how that stiffness is actually obtained — and it turns out to rest on one piece of algebra already met in Module 10. Once a joint is treated as components in series, both of the questions it raises answer themselves, and they answer differently.

By the end of this lesson you should be able to

List the components of a bolted end-plate joint
Say why the weakest sets resistance and every component sets stiffness
Identify the softest component and say what improving it does
Calculate M_j,Rd and S_j,ini, and classify the joint
Say when the initial stiffness is the wrong number to have used

What you should already know

The T-stub and its modes (previous lesson)
Joint classification by stiffness (Module 3)
Engesser's series flexibility for built-up members (Module 10)

A joint is a chain

Follow the load through a bolted end-plate connection between a beam and a column. It passes through, in order:

1.the beam flange in compression
2.the column web in transverse compression
3.the column web panel in shear
4.the column web in transverse tension
5.the column flange in bending — a T-stub
6.the end plate in bending — another T-stub
7.the bolts in tension
8.the beam web in tension

Every one of them carries the whole force. They are in series.

Two consequences follow immediately, and they are different consequences:

The weakest component sets the resistance. Everything else's spare capacity is irrelevant, because the force has to get through the weakest one.

Every component sets the stiffness, because the deformations add. Flexibilities in series add: 1/k_eq = Σ 1/k_i, so the SOFTEST component dominates.

That is the same algebra as Engesser's built-up member in Module 10 — a bending flexibility and a shear flexibility adding, so the answer is below both and dominated by the smaller. It is worth recognising the pattern rather than learning it twice.

Take care

The weakest and the softest are usually different components

For the joint in the worked example:

Component	Resistance	k_i	Share of flexibility
Column web panel in shear	520 kN	4.2	39%
Column web in compression	610 kN	6.8	24%
Column flange in bending	380 kN	9.1	18%
End plate in bending	450 kN	12.4	13%
Bolts in tension	564 kN	31.0	5%

The weakest is the column flange at 380 kN. The softest is the column web panel, supplying 39% of the flexibility while being 37% stronger than the component that governs.

So:

Strengthening the column flange raises the resistance to 450 kN and leaves the stiffness exactly unchanged.
Stiffening the web panel raises k_eq by 45% and leaves the resistance exactly unchanged.

A joint that is both too weak and too flexible needs two changes, in two different places. Getting this wrong is the most common wasted effort in joint design — and the calculation will not tell you unless you ask it both questions.

Try it

Weakest, softest, and what each change buys

Five components in series. The bars are each component's share of the joint's total flexibility; the highlighted one is whichever sets the resistance. Watch the two move independently.

column web panel in shear — resistance520 kN

column web in compression — resistance610 kN

column flange in bending — resistance380 kN

end plate in bending — resistance450 kN

bolts in tension — resistance564 kN

column web panel in shear — stiffness4.2

column web in compression — stiffness6.8

column flange in bending — stiffness9.1

end plate in bending — stiffness12.4

bolts in tension — stiffness31.0

Lever arm z400 mm

Resistance F_Rd: 380 kN
Weakest component: column flange in bending
Softest component: column web panel in shear
Same component?: NO
Equivalent stiffness k_eq: 1.645 mm
Smallest individual k_i: 4.2 mm
Lever arm z: 400 mm
Initial stiffness S_j,ini: 55268 kNm/rad
As a multiple of EI/L: 6.3
Joint classification: SEMI-RIGID
If the softest were rigid: 90847 kNm/rad
If the joint were 20% deeper: 79586 kNm/rad

The WEAKEST component is column flange in bending at 380 kN, and the SOFTEST is column web panel in shear, contributing 39% of the joint's flexibility. They are different components, which is the usual case and the practical point: strengthening column flange in bending would not stiffen the joint, and stiffening column web panel in shear would not strengthen it.

Things worth trying

Start at the defaults. The weakest component is the column flange at 380 kN; the softest is the web panel, supplying 39% of the flexibility. They are different components, and the verdict says so.
Raise the column flange's RESISTANCE to 700 kN. The resistance rises to 450 kN and k_eq does not move at all — not by a rounding amount, but exactly not at all.
Now raise the web panel's STIFFNESS from 4.2 to 20. k_eq rises by about 45% and the resistance does not move. Two changes, two effects, no overlap.
Compare k_eq with the smallest individual k_i. It is always below it, because flexibilities add — a free sanity check on any component-method calculation.
Watch the classification as you stiffen the web panel. The joint crosses from semi-rigid to rigid on one component's k, which is a large modelling consequence for a single number.
Now leave every component alone and increase the lever arm instead. z goes from 400 to 480 — 20% — and the stiffness rises 44%, because z is the only term that is squared.
Make the bolts far stiffer, from 31 to 40. Almost nothing happens: they supply 5% of the flexibility, so they can contribute at most 5% of any improvement. Effort belongs where the flexibility is.
Try making every component equally strong. The 'same component?' readout can then turn to yes — and the verdict tells you that is the convenient case and the less common one.

Worked example

A bolted end-plate joint, resistance and stiffness

Given

Bolted end-plate connection to a column flange, three bolt rows
Component resistances and stiffness coefficients as tabulated above
Bolt row lever arms about the compression centre: 460, 370 and 190 mm
Rows 1 and 2 develop 282 kN each; row 3 is limited to 200 kN
Beam 457 × 191 × 74 UB spanning 8.0 m in a braced frame
Overall lever arm z = 400 mm

Find

The joint's moment resistance and stiffness, and how to classify it.

Assumptions

Component stiffness coefficients k_i are calibrated and are given, not derived
The rigid and pinned boundaries are calibrated and unverified here
Row resistances are taken as given; obtaining each is the T-stub work of the last lesson

The initial stiffness is not the stiffness

A joint is not linear. It is stiff up to about two thirds of its moment resistance, and then softens markedly as the components start to yield in turn.

For the joint above, with M_j,Rd = 272 kNm and S_j,ini = 55 268 kNm/rad:

M_Ed / M_j,Rd	Region	Actual stiffness	Rotation
0.50	elastic	55 268	2.5 mrad
0.67	boundary	55 193	3.3 mrad
0.80	softening	33 782	6.4 mrad
0.90	softening	24 579	10.0 mrad
1.00	at resistance	18 494	14.7 mrad

The moment rose by a factor of two between the first and last rows; the rotation rose by a factor of six.

An analysis that used S_j,ini on a joint working at 90% of its resistance has overestimated that joint's stiffness by a factor of 2.25.

When does that matter? Whenever the answer depends on how load is shared — a semi-continuous frame where the joint stiffness decides the split between span and support moments, or an unbraced frame whose α_cr depends on joint stiffness. In those cases the joint's working stiffness is the number the analysis needs, not its initial one.

Try it

When the initial stiffness is the wrong number

The joint from the worked example: M_j,Rd = 272 kNm, S_j,ini = 55 268 kNm/rad. The straight line is what an analysis using the initial stiffness assumes; the curve is what the joint does.

Applied moment160 kNm

Applied moment M_Ed: 160 kNm
Utilisation: 59 %
Region: elastic
Initial stiffness S_j,ini: 55268 kNm/rad
Stiffness at this moment: 55268 kNm/rad
Softening factor μ: 1.00
Rotation: 2.89 mrad
Rotation if it stayed elastic: 2.89 mrad

At 59% of its resistance the joint is still on its initial stiffness. S_j,ini is the right number to have used in the analysis.

Things worth trying

Below about 180 kNm the joint sits on the straight line: S_j,ini is the right number and the two curves are the same.
Cross the dashed two-thirds line and watch them separate. Nothing physical happens at that point — it is where a calibrated softening expression takes over — but the divergence beyond it is real.
At 245 kNm, 90% of the resistance, the joint is 2.25 times less stiff than an analysis using S_j,ini assumed.
Compare the rotation with the 'if it stayed elastic' figure as you go up. The moment roughly doubles between 50% and 100% of the resistance; the rotation rises about six-fold.
This matters wherever the answer depends on how load is shared — a semi-continuous frame where the joint decides the split between span and support moments, or an unbraced frame whose α_cr depends on joint stiffness.
It matters much less in a braced frame with nominally pinned joints, where nothing is being asked of the joint's stiffness in the first place. Knowing which situation you are in is the point.

Practice

Five components have k values of 4.2, 6.8, 9.1, 12.4 and 31.0 mm. What is k_eq?

Practice

With k_eq = 1.645 mm and a lever arm z = 400 mm, what is S_j,ini in kNm/rad? Take E = 210 000 N/mm².

Practice

Three bolt rows carry 282, 282 and 200 kN at lever arms of 460, 370 and 190 mm. What is M_j,Rd in kNm?

Practice

A joint has S_j,ini = 55 268 kNm/rad and the beam it connects has EI/L = 8796 kNm/rad. What is S_j as a multiple of EI/L?

Check yourself

A joint is 20% too flexible. Its weakest component is the column flange in bending; its softest is the column web panel in shear. What should be changed?

Summary

A joint is components in series, so the force passes through every one
The WEAKEST sets the resistance — 380 kN here, the column flange in bending
1/k_eq = Σ1/k_i, so the SOFTEST sets the stiffness — the web panel, at 39%
They are different components, so a weak AND flexible joint needs two changes
Strengthening the weakest left the stiffness exactly unchanged, and vice versa
M_j,Rd = 272 kNm; the top bolt row supplied 48% of it and the bottom 14%
S_j,ini goes with z², so 20% more depth was worth 44% more stiffness
Above two thirds of M_j,Rd a joint softens: at 90% it was 2.25 times less stiff

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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