Module 3 · Lesson 3.2
Joints are never pinned or rigid
Those are the two ends of a scale. Everything real is in between, and a connection built stiffer than it was assumed will attract moment whether or not you asked it to.
Why this matters
A connection detailed as 'simple' does not know it is simple. It has whatever rotational stiffness its geometry gives it, and if that stiffness is appreciable it will attract moment — moment that the analysis said was zero and that the connection was never designed to carry. This is one of the few places in design where an assumption can be wrong in a way the structure itself corrects, unhelpfully.
By the end of this lesson you should be able to
- Classify a joint against the beam it connects
- Compute the end moment a real rotational stiffness attracts
- Explain why the classification boundaries differ for braced and unbraced frames
- Recognise when a joint must be modelled as semi-rigid
Stiff or flexible compared with what?
A joint has a rotational stiffness Sj in kNm per radian. On its own that number means nothing: 20 000 kNm/rad is very stiff for a light purlin cleat and quite flexible for a heavy transfer beam.
The comparison that matters is with the beam the joint connects:
ratio = Sj / (EI/L)
That is the right yardstick because a joint's job is to restrain the beam's end rotation, and how well it does so depends on how hard the beam is pushing. A flexible beam rotates readily and is easily restrained; a stiff short beam is not.
The consequence catches people out: the same physical connection can be rigid on one beam and nearly pinned on another. Nothing about the connection has changed. Halve the span or double the beam's I, and a joint that was comfortably rigid becomes semi-rigid.
The boundaries
- Rigid if the ratio is at least 8 for a braced frame, or 25 for an unbraced one.
- Nominally pinned if it is 0.5 or less.
- Semi-rigid in between — which is most connections, honestly assessed.
These are calibrated judgements about when an idealisation is close enough. Nothing physical happens as a joint crosses one.
The unbraced boundary is three times higher, and the reason is worth understanding: in a sway frame the joint stiffness feeds straight into the lateral stiffness of the whole structure, so an error in it propagates to the frame's drift and to αcr. In a braced frame the bracing provides the lateral stiffness and the joint only affects local moments.
Worked example
What a 'simple' connection actually carries
Given
- Beam with I = 29 400 cm⁴, spanning 8.0 m, carrying a UDL of 30 kN/m
- Flexible end-plate connection, measured rotational stiffness 9 700 kNm/rad
- Braced frame; the connection was designed as nominally pinned
Find
Whether the joint classifies as pinned, and what moment it actually attracts.
Assumptions
- Equal stiffness at both ends, and a symmetric load, so the beam is symmetric
- Elastic behaviour — the joint stiffness is the initial value
Try it
How pinned is this connection, really?
Move the joint stiffness and watch the moment migrate between the connection and midspan. Then change the beam without touching the joint — the classification moves anyway, because a joint is only stiff relative to what it connects.
Frame type
- Beam stiffness EI/L
- 7718 kNm/rad
- Ratio Sj/(EI/L)
- 1.26
- Rigid boundary for this frame
- 8 EI/L
- Classification
- semi-rigid
- ρ = kL/EI
- 1.26
- End moment
- 61.7 kNm
- As a fraction of fixed-end
- 39 %
- Midspan moment
- 178.3 kNm
- Statics check: end + mid
- 240.0 = wL²/8
Sj = 1.3 EI/L — SEMI-RIGID. It is neither, and modelling it as either will misrepresent the frame. Either model the rotational stiffness explicitly, or change the connection detail until it classifies.
Things worth trying
- Set Sj to zero. The end moment vanishes and midspan takes the full wL²/8 — a genuine pin. Now raise it slowly and watch how FAST the moment migrates: the connection does not need to be stiff to attract a great deal of moment.
- Find ρ = 2. The end moment is exactly half the fixed-end value, and ρ = 2 is a fairly flexible connection. This is why a 'simple' connection has to be detailed to be genuinely flexible rather than merely called simple.
- Watch the two dashed reference lines. Whatever the joint does, the end and midspan moments always sum to the pinned value — statics decides the total, and the joint only decides the share.
- Now leave Sj alone and change the BEAM. Halve the span or double I and the classification moves, because a joint is only stiff relative to what it connects. The same physical connection is rigid on one beam and nearly pinned on another.
- Switch to an unbraced frame. The rigid boundary jumps from 8 to 25 EI/L, and a joint that was rigid becomes semi-rigid — because in a sway frame the joint stiffness feeds straight into the frame's lateral stiffness and αcr.
- Note what happens to the BEAM as the joint stiffens: it gets safer. That is exactly why this error hides — every beam check passes while the connection quietly takes moment it was never designed for.
Practice
A beam has I = 20 000 cm⁴ and spans 6.0 m. What is EI/L, in kNm/rad? Take E = 210 000 N/mm².
Practice
A beam with ρ = kL/EI = 2 carries a UDL giving a fixed-end moment of 150 kNm. What end moment develops, in kNm?
Practice
A joint has Sj = 60 000 kNm/rad on a beam with EI/L = 7718 kNm/rad, in an unbraced frame. Does it classify as rigid? Give the ratio Sj/(EI/L).
Check yourself
Why is the rigid-joint boundary higher for an unbraced frame than a braced one?
Summary
- A joint's stiffness is meaningless in isolation — judge it against EI/L of the beam
- The same connection can be rigid on one beam and nearly pinned on another
- Rigid at 8 EI/L braced, 25 EI/L unbraced; pinned at 0.5 EI/L — calibrated judgements
- Mend = (wL²/12) · ρ/(ρ + 2), with ρ = kL/EI — derivable, with the right limits
- At ρ = 2 a 'simple' connection already carries HALF the fixed-end moment
- The beam gets safer and the connection gets overloaded, which is why the error hides
- A simple connection is flexible on purpose; stiffening it is not a conservative act
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