Geometric Stiffness Matrix Method for Second-Order Effects

The conventional matrix stiffness method of analysis (Art. 3.39) may be modified to include directly the influences of second-order effects described in Art. 3.46. When the response of the structure is nonlinear, however, the linear relationship in Eq. (3.145), P = Kdelta, can no longer be used. An alternative is a numerical solution obtained through a sequence of linear steps. In each step, a load increment is applied to the structure and the stiffness and geometry of the frame are modified to reflect its current loaded and deformed state. Hence Eq. (3.145) is modified to the incremental form

In this way, the analysis ensures that the equations of equilibrium are sequentially being formulated for the deformed geometry and that the bending stiffness of all members is modified to account for the presence of axial forces.
Inasmuch as a piecewise linear procedure is used to predict nonlinear behavior, accuracy of the analysis increases as the number of load increments increases. In many cases, however, good approximations of the true behavior may be found with relatively large load increments. The accuracy of the analysis may be confirmed by comparing results with an additional analysis that uses smaller load steps.

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