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Steel Designers' Manual - 6th Edition (2003)
The basis of this method revolves around the calculation of a parameter known
as the lowest elastic critical load factor, lcr, for a particular load combination. (N.B.
BS 5950: Part 1 does not give a method for calculating lcr, but refers to reference
[5].)
A detailed treatment of the calculation of lcr is outside the scope of this particular
chapter, and the reader is again referred to reference [5]. Fortunately, the industry
software to carry out the analysis of portal frames is capable of calculating lcr
to BS 5950: Part 1.Accordingly, the task is not as daunting as first appears. However,
it is imperative that the reader understands the background to this particular
methodology.
On determining lcr, the required load factor, lr, is calculated as follows:
lcr � 10 : lr = 1.0
10 > lcr � 4.6 : lr = 0.9lcr/(lcr - 1)
If lcr < 4.6, the amplified moments method cannot be used and a second-order
analysis should be carried out.
The application of lr in the design process is as follows:
• For plastic design, ensure that the plastic collapse factor, lp � lr, and check the
member capacities at this value of lr.
• For elastic design, if lr > 1.0, multiply the ultimate limit state moments and forces
arrived at by elastic analysis by lr, and check the member capacities for these
‘amplified’ forces.
1.4.1.3 Second-order analysis
Design of common structural forms 15
Second-order analysis, briefly referred to above, accounts for additional forces
induced in the frame due to the axial forces acting eccentrically to the assumed
member centroids as the frame deflects under load.
These secondary effects, often referred to as ‘P-Delta’ effects, can be best illustrated
by reference to Fig. 1.9 of a simple cantilever.
As can be seen, the second-order effects comprise an additional moment of PD
due to the movement of the top of the cantilever, D, induced by the horizontal force,
H, in addition to a moment within the member of Pd due to deflection of the
member itself between its end points. (It should be noted that, in certain instances,
second-order effects can be beneficial. Should the force P, above, have been tensile,
the bending moment at the base would have been reduced by PD kNm.)
In the case of a portal frame, there are joint deflections at each eaves and apex,
together with member deflections in each column and rafter. In the case of the
columns and rafters, it is standard practice to divide these members into several subelements
between their start and end nodes to arrive at an accurate representation
of the secondary effects due to member deflections.
As a consequence of these effects, the stiffness of the portal frame is reduced