(l6r22 - 5r1r22 + 9r1r2 + 8r1 - 28r2) (l6r^ + 25r12r2 - 96r1r2 - 28r1 + 128r2)

in which rj and r2 are end-fixity factors of the member as defined by Equation 23.7 and L is the length of the member.

Since the correction matrices Ce_,- and that take into account the semirigid connection behavior are in terms of end-fixity factors instead of connection stiffness, the stiffness matrix KfR applies to a beam-column member with any combination of pinned, semirigid, and rigid connections.

The secant-stiffness-based iterative procedure for the second-order analysis of semirigid frames, with nonlinear behavior of connections taken into account, is readily developed through modification of the first-order analysis procedure as described in the following:

Step 1. Input the data, including the parameters for the specified connections from Equation 23.6. Set the iteration index k = 0, assign connection stiffnesses = Re, and set member axial forces N,(k) = 0. Generate elastic stiffness matrix from Equation 23.12 for each member.

Step 2. Set iteration index k = k + 1, update the end-fixity factors r,j using Equation 23.7 for each semirigid member ¿; evaluate member equivalent joint loads; calculate the correction matrices Ce_, and Cg_ by Equations 23.13 and 23.17, respectively. Evaluate geometrical stiffness matrix Gbased on Equation 23.15 and the axial force N"/k_1) from the previous iteration. Generate the member stiffness matrix KSR according to Equation 23.16 for each semirigid member.

Step 3. Assemble the structure stiffness matrix and solve for nodal displacements, member axial forces N;k), and connection moments M,(k).

Step 4. Calculate connection rotations 0,jk) = M,(k)/R,(k) and obtain the corresponding moments M(0,(k)) from Equation 23.6.

Step 5. Check for convergence for specified tolerances e1 and e2 by comparing the connection moments M,(k) obtained through analysis with the moments M(0,f) calculated in Step 23.4 and by also comparing the member axial forces N(k) with the forces N,(k-1) from the previous iteration; if |M(k) - M(0,(k))| < e1 and |Njk) - N(k-1)| < e2, then stop; otherwise, go to Step 6.

Step 6. Update connection secant stiffness Rjk) = M(0,jk))/0,(k) and go back to Step 2.

With the adoption of the end-fixity factor defined in Equation 23.7 and the member stiffness matrix Equation 23.16 accounting for both second-order effects and semirigid connection behavior, it is a straightforward matter to modify linear elastic analysis computer programs for rigid frames to perform second-order analysis of semirigid frames. Figure 23.12 illustrates a program flow chart for the second-order analysis of semirigid frames.

Input structural data and load conditions

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