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where E is elastic modulus and L, A, and I are the length, cross-sectional area, and moment of inertia of the member, respectively. The end-fixity factors r1 and r2 in Equation 23.13 are defined by Equation 23.7. Knowing the semirigid beam-column member stiffness matrix from Equation 23.11 for specified values of end-fixity factors reflecting connection stiffness, the analysis of frames with semirigid connections can then be carried out directly using the conventional displacement method.

To take into account the nonlinear behavior of semirigid connections, an iterative procedure is applied to obtain the solution. In each iteration, the member stiffness matrix ksr is modified using the correction matrix Ce-i with updated end-fixity factors r. In addition, when there are member loads on semirigid members, the equivalent joint loads that facilitate matrix analysis of the structure are updated since member end-reactions are also functions of the end-fixity factors. The secant-stiffness-based iterative procedure for the first-order analysis of semirigid frameworks accounting for the nonlinear behavior of connections defined by Equation 23.6 is described in the following [where subscript j (j = 1, 2) denotes the two ends of semirigid member i]:

Step 1. Input data, including the nonlinear parameters and initial stiffness Re defining the specified connections in the connection database. Set the iteration index k = 0 and assign connection stiffness R,(k) = Re.

For each beam-column element i, update the end-fixity factors r,j through Equation 23.7 and compute the corresponding equivalent joint loads. Generate the correction matrix Ce-i and member stiffness matrix KSR through Equations 23.13 and 23.11, respectively. Assemble the structure stiffness matrix and solve for the nodal displacements, member forces, and connection moments

Step 2.

Step 3.

Step 4. Calculate connection rotations = M(k)/R,jk) M(0,(k)) from Equation 23.6.

and obtain the corresponding moments

Step 5. Check convergence by comparing the connection moments Mj ) obtained through analysis with the moments M(6,(k)) calculated in Step 4; if |M,(k) — M(6(k)) | < e (a predefined tolerance), then stop; otherwise, go to Step 6. Update connectk return to Step 2.

Step 6. Update connection secant stiffnesses R(k) = M(0,(k))/0,(k)

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