2344 Second Order Nonlinear Analysis of Semirigid Framed Structures

The stability of framed structures has long been a concern in the design of such structures. In current design practice, this concern can be addressed by performing the so-called second-order P-D analysis for frames. Unlike the first-order elastic analysis, in which the equilibrium and kinematic relationships of a frame are established with respect to the undeformed geometry of the structure, the second-order elastic analysis is associated with the interaction between forces and deformation of the structure. For that reason, the second-order elastic analysis is referred to as geometrical nonlinear analysis. By inducing additional stresses in the members and by causing the frame members to deform more, the second-order effects (P-D and P-d), have a destabilizing effect on the structure. For rigid frames, the computer-based second-order elastic analysis is often pursued as an iterative procedure and the stiffness matrix of each member i with rigid ends is expressed as

where S,- is the elastic stiffness matrix and is as shown in Equation 23.12 and G, is the geometrical stiffness matrix that accounts for the second-order effects and is expressed as

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