## 346 General Secondorder Effects

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 A1 = HL3/3EI, 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 8. In this

FIGURE 3.92 (a) Column unrestrained at one end, where horizontal and vertical loads act. (b) First-order maximum bending moment M1 occurs at the base. (c) The column with top displaced by the forces. (d) Second-order maximum moment M2 occurs at the base.

FIGURE 3.92 (a) Column unrestrained at one end, where horizontal and vertical loads act. (b) First-order maximum bending moment M1 occurs at the base. (c) The column with top displaced by the forces. (d) Second-order maximum moment M2 occurs at the base.

case, the actual second-order deflection S = A2 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 + PA2 (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.

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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