Once the stiffness degradation factor of a compound element is determined, as discussed in the previous sections, the structural analysis is readily conducted. This study focuses planar steel frameworks comprised of beam-column members with compact sections, for which plastic deformation is not precluded by local buckling (AISC 2001). The plastic bending, shearing or axial deformation (ф, у or S) of a member under the action of moment, shear or axial force (M, V or P) is concentrated at a member-end section (Xu et al 2005). Figure 6(a) shows a general member with Young’s modulus E, shear modulus G, member length L, cross-section moment of inertia I, sectional area A, and equivalent shear area As. The parameters Rpj, Tpj and Npj are respectively the post-elastic rotational bending, transverse shearing and normal axial stiffness of the member at the two end sections j =1, 2, while Rcj, Tcj and NcJ are respectively the rotational bending, transverse shearing and normal axial stiffness of the connections at the two end sections. Upon adopting a compound element at each member-end, the simplified model shown in Figure 6(b) is obtained, where the determination of the corresponding parameters is discussed in the following.
The evaluation of connection and member rotational stiffnesses RcJ and RpJ in Figure 6(a), and corresponding stiffness degradation factors rcj and rpj, has been discussed in detail in the previous sections. The member transverse shear and normal axial stiffnesses Tpj and Npj were determined in previous research, where it was shown that the corresponding stiffness degradation factors tpJ and npJ are given by (Grierson et al 2005, Xu et al 2005),
tpj = 1/(1 + 3EI/L%); npj = 1/(1 + EA/LNp) (13a, b)
which map TpJ or NpJ є [0, го] into tpJ or npJ є [0, 1]. Similarly, the transverse and normal stiffness degradation factors for the connection can be expressed as,
tcJ = 1/(1 + 3EI/L%); ncj = 1/(1 + EA/LNcj) (14a, b)
where Tcj and NcJ – are the transverse shear and normal axial stiffnesses of the connection.
In this study, it is assumed that: 1) the transverse shear stiffness TcJ or normal axial stiffness NcJ of a connection is infinite when the materials are in the elastic range, and that the corresponding degradation factor tcJ- or ncJ – in Eqs (14) is unity; and 2) the stiffness Tcj or NcJ- is zero when the materials are in the plastic range, and that the corresponding degradation factor tcJ – or ncJ- is zero. Such idealized perfectly elastic-plastic models are shown in Figure 7.
Figure 7. Idealized shear & axial force-displacement relations for
Upon determining the parameters characterizing the compound element, the general planar compound member shown in Figure 6(b) is generated, where f and d (i =1, 2,…, 6) are local-axis joint forces and deformations, respectively. The parameters rj, tj and nj (J =1, 2) in Figure 6(b) are the so – called bending, shearing and axial stiffness degradation factors of the compound element. The factors rj are calculated through Eq. (12), while tj and nj are similarly found as,
Based on the compound model shown in Figure 6(b), the local stiffness matrix k for each element is derived accounting for the effects of shear deformation and geometrical nonlinearity. The local element stiffness matrices are transformed into the global coordinate system and then assembled as the structure stiffness matrix K. If K is nonsingular at the end of an incremental load step, the corresponding incremental nodal displacements Au are solved for and the incremental member-end forces Af and deformations Ad are found. After each load step i, the total nodal displacements u = EAUi and member-end forces f = ZAf and deformations d = ZAdi accumulated thus far over the load history are found. The initial-yield and full-yield conditions for each member-end section are checked to detect plastic behaviour, and the corresponding bending, shearing and axial stiffness degradation factors are found. The degraded stiffnesses Rc, Tc, Nc are determined based on the moments, shear and axial forces given by the analysis results at the current loading level. The degradation factors (rp, tp, np, rc, tc, nc) are applied to modify the element stiffness matrices k and, hence, the structure stiffness matrix K, before commencing the next load step. The incremental-load analysis procedure continues until either a specified load level F is reached or the structure stiffness matrix K becomes singular at a lower load level as a consequence of failure of part or all of the structure. (If the structure has not failed at load level F, the analysis may be continued beyond that level until failure of the structure does occur.)
The final analysis results include the values of the bending, shearing and axial post-elastic compound stiffness degradation factors r, t and n indicating the extent of plastic and connection deformation in the beam-to-column connection regions. Further computational details are provided through the analysis example presented in the following section.