Second-order analysis

First-order analysis writes equilibrium on the undeformed shape. That is accurate when deflections are small compared with the dimensions, but slender columns and sway frames carrying heavy vertical load behave differently: as the frame sways, the vertical loads act at an offset and create extra moments, which cause more sway. These P-delta effects can raise moments significantly and, in the limit, lead to instability.

Two effects are distinguished. P-big-delta comes from the sway of the whole storey, the vertical load times the storey drift. P-small-delta comes from the curvature of an individual member between its ends. Sway frames are governed mainly by the first; braced members by the second.

A quick estimate uses an amplification factor of one over one minus P over P critical, applied to the first-order sway moments. It shows why a column carrying a fifth of its elastic buckling load already has its moments increased by a quarter. Codes provide such moment magnifiers or require a rigorous second-order analysis when a stability index, comparing the P-delta moment with the first-order storey moment, exceeds a limit.

Software performs second-order analysis by iterating or by using geometric stiffness matrices. Initial imperfections, out-of-plumb columns and member bow, must be included because a perfectly straight model underestimates the effect. Cracking in concrete lowers stiffness and raises second-order moments, so reduced section properties are used.