Structural Steel Design / module 9
Beam-columns and frame stability
Almost every real column carries moment as well as axial force, and the two do not simply add. The axial force bends the member further, and the same effect scales up to whole frames — where it becomes a question about stiffness, not strength.
What you will understand
- Explain why a member under N and M is worse than the sum of the two checks
- Derive the amplification factor 1/(1 − N/Ncr) and say what it does not depend on
- Distinguish cross-section interaction, member amplification and the code's interaction factors
- Calculate αcr for a storey from its drift, and say what it means physically
- Apply the amplified sway method to the right moments and not to the wrong ones
- Say what to change when a frame is too flexible, and by how much
Lessons
Why N and M do not simply add
The axial force does not just eat into the moment capacity. It also increases the moment — and that second effect is the one people miss.
Frame stability, and which moments get amplified
The same effect at the scale of a whole frame. It turns into a question about lateral stiffness — and about which moments the amplifier is allowed to touch.
Module summary
- A moment bows the member; the axial force then acts on that bow and adds more moment
- M = M₀/(1 − N/Ncr) — derived, and it depends on stiffness, not strength
- At N/Ncr = 0.5 the moment is DOUBLED; the effect is strongly non-linear
- The straight-line check N/Nb,Rd + M/Mb,Rd ≤ 1 misses this entirely
- A 12 m column here read 0.87 by the straight line and 1.05 once amplified
- C₁ for beams has a counterpart Cm here — both are calibrated, and both reward a favourable moment shape
- αcr = (H/V)(h/δ) is a stiffness ratio and can be read straight off a drift
- Only the SWAY moments are amplified — amplifying everything is wasteful, amplifying nothing is not safe
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Practice
A member has NEd = 600 kN and Ncr = 2400 kN in the plane of bending. What is the amplification factor?
Practice
A first-order moment of 80 kNm sits on that member, with Cm = 1.0. What design moment does the amplification give, in kNm?
Practice
A storey 3.5 m high carries VEd = 3000 kN and sways 14 mm under a horizontal force of 120 kN. What is αcr?
Practice
A storey has αcr = 5. By what factor are its sway moments amplified?
Practice
At what value of N/Ncr does the amplification factor reach 1.25?
Practice
A storey has αcr = 5 and drifts 40 mm. What drift is needed for αcr = 10, in mm?
Check yourself
Which quantity drives the second-order amplification of a beam-column's moment?
Check yourself
A frame is analysed with a genuine second-order analysis that includes the sway imperfections. The sway moments are then amplified by 1/(1 − 1/αcr). What is wrong?