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Figure 5.33. Illustration of step-wise calculation of second order effects.

a) Horizontal load Ho (without vertical load) gives deformation yo.

b) Vertical load V on deformed structure gives additional deformation yi.

c) Hi is an equivalent horizontal load that would give the same deformation yi.

d) Vertical load V and deformation yi give additional deformation yi.

e) H2 is equivalent horizontal load giving the same deformation yi etc

Figure 5.33. Illustration of step-wise calculation of second order effects.

a) Horizontal load Ho (without vertical load) gives deformation yo.

b) Vertical load V on deformed structure gives additional deformation yi.

c) Hi is an equivalent horizontal load that would give the same deformation yi.

d) Vertical load V and deformation yi give additional deformation yi.

e) H2 is equivalent horizontal load giving the same deformation yi etc

The total equivalent horizontal force is

If ki = Hi/Hi-i is < i, then the sum H will be finite (i.e. the structure is stable). With increasing number of steps, in a linear analysis, the ratio k will sooner or later become constant. In other words, the following terms will form a geometric series.

The simplest alternative is to assume that all terms, including Ho, will form a geometric series. The total equivalent horizontal force is then obtained as

Hn Hn

which is equivalent to expression (H.8).

Note. Expression (7-15), including the definition of k, can also be expressed in terms of y or a relevant M.

It can also be shown that the final value of k is equal to the ratio V/VB, where V is the total vertical load and Vb is the global buckling load. Thus, the method of stepwise calculation can be seen as just a different formulation or derivation of the method based on a magnification factor.

If the distribution of Hi is significantly different from that of Ho, the accuracy can be improved by including one or more steps: