Incremental single-step method, as described in the following, is employed to conduct nonlinear analysis for steel frameworks.
1) Discretize frame members into elements. Initialize plasticity-factors to be pi=p2=1 for all elements.
2) Form element stiffness matrix Ke for each element, which involves using plasticity-factors from previous loading step to compute Cs and Cg (p=1 for the first loading step). Then assemble overall structure stiffness matrix K. If the tangent stiffness matrix K is found to be singular, which indicates the structure collapses, terminate analysis.
3) Solve for incremental nodal displacements by AF=K»Au, where u and F are the vectors of overall nodal displacements and loads respectively. Calculate incremental deformations and forces for each element. Update the total overall nodal displacements and loads, and member internal forces by u=ZAu, F=ZAF, and f=ZAf-, respectively, where fj is the vector of member internal force.
4) Check the yielding status for each hinge-spring at the ends of each element. The combined actions of axial force and bending moment at a hinge-spring are, from Eqs. 1 and 2, denoted as p1=M/My+N/Ny and p2=M/Mp+(N/Ny)a. Then update p for each spring as below:
(i) If p1 < (1- Fr/Fy), the hinge-spring is still fully elastic with p=1.
(ii) If p1>(1-Fr/Fy) and p2<1, the hinge-spring is partially plastic. Calculate Myr and Mpr from Eqs. 3 and 4, respectively. If the hinge-spring is found to be yielding for the first time, calculate its Bpu value using 0pu-K relation. Compute the post-elastic rotational stiffness of the hinge-spring from Eq. 7 and the plasticity-factor p from Eq. 8.
(iii) If p2 >1, the hinge-spring is fully plastic with p=0.
5) Go back to step 2.
The inelastic analysis is illustrated for two structures comprised of steel members of wide flange I-section. All the members are oriented with their webs in the loading plane. The exponent p in Eq. 5 is taken as 1.8. Young’s modulus is E=2×105 MPa.