Structural Steel Design / module 4
Analysis methods, imperfections and second-order effects
Seven analysis methods, four stability effects, and one piece of bookkeeping that decides whether the design is safe, wasteful, or wrong in a way nothing reveals.
What you will understand
- Say what each analysis method can and cannot answer
- Derive the moment magnification factor as a geometric series
- Explain what αcr is and why the threshold is a calibrated judgement
- Convert an initial sway imperfection into equivalent horizontal forces
- Allocate each stability effect to exactly one place in the design
- Recognise both the omission error and the double-counting error
Lessons
Second-order effects, from first principles
A frame leans, the vertical load acts through the lean, the frame leans further. Where that stops — and where it does not — is the whole of stability analysis.
Where every stability effect belongs
Four effects, three possible homes, and two symmetrical errors that no individual calculation will ever reveal.
Module summary
- Element type decides what CAN appear; analysis type decides what is looked for
- An LBA gives αcr and the mode shape, and always overstates the real resistance
- Second-order moments follow a geometric series: MII = MI/(1 − 1/αcr)
- At αcr = 10 the amplification is about 11%, which is where the threshold sits
- An imperfection can be replaced by equivalent horizontal forces with the same overturning
- A second-order analysis on a PERFECT symmetric frame finds nothing to amplify
- Global sway has no home in a member check — omit it and it is simply lost
- Member bow lives in χ, so modelling it AND applying χ counts it twice
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 frame has αcr = 6. What is the moment amplification factor?
Practice
A frame carries 3 200 kN of vertical load and has an elastic critical load of 11 200 kN. What is αcr?
Practice
A level carries 5 000 kN with an initial sway imperfection of 1/300. What equivalent horizontal force applies, in kN?
Practice
A first-order sway moment of 150 kNm is found in a frame with αcr = 4. What is the design moment, in kNm?
Practice
In the geometric series for a frame with αcr = 5, what fraction of the first-order sway does the SECOND increment add?
Practice
A portal frame carrying gravity loads only has αcr = 7. It is analysed ELASTICALLY, where the threshold is 10. What amplification factor applies?