## 2343 First Order Nonlinear Analysis of Semirigid Framed Structures

Computer-based analysis of semirigid framed structures can be generally achieved by modifying existing programs of rigid frame analysis in one of two ways. The first method introduces additional ''connection elements'' that model the beam-to-column connections, while the second method modifies the member stiffness matrix to account for the connection flexibility. The application of the first approach is limited to only computer-based analysis and design and is inconvenient for practical use because the beam-column members are separated from the attached end-connections. Therefore, the second approach is considered to be more general as the analysis and design of such structures can also be facilitated manually. Most research and engineering applications have adopted the second approach as the means to incorporate semirigid connection behavior into frame analysis and design.

The implementation of the concept of end-fixity factor into frame analysis is straightforward. For an existing rigid frame analysis computer program, only minor modifications to the member stiffness matrix and the evaluation of member end-reactions due to applied member loads are required. The elastic stiffness matrix of a member i with two semirigid end-connections having stiffness moduli R1 and R2, as shown in Figure 23.6, can be represented by the stiffness matrix for the member taken to have rigid end-connections modified by a correction matrix [21]; that is

where K?R is the stiffness matrix of member i with semirigid end-connections taken into account, Si is the stiffness matrix of the member taken to have rigid ends, and Ce-i is the required correction matrix. For a planar beam-column element with six degrees of freedom, the matrices Si and Ce-i have the following form:

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