A column unrestrained at one end with length L and subjected to horizontal load H and vertical load P (Fig. 3.92a) can be used to illustrate the general concepts of second-order behavior. If E is the modulus of elasticity of the column material and I is the moment of inertia of the column, and the equations of equilibrium are formulated on the undeformed geometry, the first-order deflection at the top of the column is delta1 = HL^3/3EI, and the firstorder moment at the base of the column is M1 = HL (Fig. 3.92b). As the column deforms, however, the applied loads move with the top of the column through a deflection S. In this
case, the actual second-order deflection S = delta 2 not only includes the deflection due to the horizontal load H but also the deflection due to the eccentricity generated with respect to the neutral axis of the column when the vertical load P is displaced (Fig. 3.92c). From equations of equilibrium for the deformed geometry, the second-order base moment is M2 = HL + Pdelta2 (Fig. 3.92d). The additional deflection and moment generated are examples of second-order effects or geometric nonlinearities.
In a more complex structure, the same type of second-order effects can be present. They may be attributed primarily to two factors: the axial force in a member having a significant influence on the bending stiffness of the member and the relative lateral displacement at the ends of members. Where it is essential that these destabilizing effects are incorporated within a limit-state design procedure, general methods are presented in Arts. 3.47 and 3.48.