## Info

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il y u

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

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

c) Hi is an equivalent horizontal load that would give the same 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: