To assign a plasticity-factor p (see Eq. 8 for definition) to a partially-yielded section (e. g., a section with force point E in Fig. 2) during a loading process, it is assumed that the ratio M/N for the section remains invariant when M^0 (see line O-Oy-E-Op in Fig. 2. Oy and Op thus represent the first – yield and full-yield ‘times’ for the section, respectively.). Defining ^=(Mp/Ny)(N/M), the reduced first-
yield moment Myr for the section in post-elastic range under combined axial force and bending moment is

Myr = My (1 – Fr Fy) (1 + ^ q)

where q =Z/S is shape factor. The reduced full-yield moment Mpr is found from Eq. 2 as

Mpr Mp + (t, Mpr Mp )a = 1

Note that the assumption of N/M as invariant is not necessarily true. This assumption purely serves the purpose of evaluating the plasticity-factor for a partially yielded hinge.

The rotational stiffness of a hinge-spring degrades from infinity to zero as the moment M at the hinge section increases from Myr to Mpr. This degradation is represented by a moment-plastic rotation curve (see Fig. 3). On this curve, point (0, Myr) corresponds to the first-yield of the hinge, while point (0pu, Mpr) corresponds to the full-yield. This curve is expressed as (Xu et al. 2005)

where Opu is the plastic rotation at which the hinge-spring has zero stiffness; and exponent ^ is a parameter dependent on the shape of the cross-section. Upon differentiating moment M with respect to plastic rotation 0p in Eq. 5, the instantaneous rotational stiffness of the hinge-spring is found as

Equation 6 can be rewritten as

A0e 1

p =——— e =

АЄe +АЄp 1 + 3EI RmL

where: A0e=AM»L/3EI is the incremental end rotation of the elastic beam-column (see Fig.1); AM is a moment increment; and A0p=AM/Rm is the incremental rotation of the spring. For a fully elastic section Rm=ro and p=1, for a fully plastic hinge Rm=0 and p=0, while for a partially plastic hinge 0<p<1.

Upon introducing the plasticity-factor by Eq. 8, the stiffness matrix Ke for the hybrid element in Fig.1 is found as (Xu 2001, Hasan et al. 2002),

Ke = S Cs + G Cg (9)

where: S is the standard stiffness matrix for an elastic frame member; Cs is a correction matrix expressed in terms of plasticity-factors p (Hasan et al. 2002); G is the standard geometric stiffness matrix; and Cg is the corresponding correction matrix formulated as a function ofp (Hasan et al. 2002).

A unique feature of the stiffness matrix Ke is that the stiffness of the springs Rm is not directly included in Eq. 9. Instead, the contribution of the hinge-springs to the element stiffness is incorporated through the plasticity-factors p. Such a treatment for Rm significantly improves the accuracy of the element model. To explain this, the Rm-M curves (see Eq. 7) and the corresponding p-M curves for an element of W200x46 section (CISC 2004) are drawn in Figs. 4 and 5, respectively. From Fig. 4, it is seen that spring stiffness Rm is extremely sensitive to moment M in the early post-elastic range. The stiffness Rm is infinite when M=Myr, then Rm dramatically drops as M begins to exceed Myr. Rm is also very sensitive to slight variations in parameter p. For instance, Rm=166323 kN-m/radian for p=1.8 under M=1.03Myr, which is 2.5 times of that for p=1.5 under the same moment. This high sensitivity of Rm to M and p can be interpreted as a difficulty in modeling Rm numerically since some approximation in the Rm model is generally unavoidable. Therefore, if Rm is directly included in the stiffness matrix, considerable errors may be introduced. On the contrary, the plasticity-factor p has a relatively even degradation rate with the increasing of M, as shown in Fig. 5. Thus, some inaccuracy in Rm model does not cause much error inp value. Therefore plasticity-factor p serves as an ‘error filter’ in the hinge model.

There are two key parameters, p and 0pu, in the hinge-spring model Eq. 7. This study makes full use of these two parameters by employing p to simulate the spread of plasticity through the depth of a cross section and 0pu to mimic the spread of plasticity along the length of an element.

It is obvious that parameter p has a significant impact on the degradation rate of the spring stiffness. For instance, the spring having p=1.8 degrades faster than the spring having p=2.1 (see Fig.5). In fact, this variation in degradation rate reflects the geometric difference among various cross sections. For example, an I-section degrades faster than a rectangular section under bending since an I – section has more materials allocated away from its neutral axis than a rectangular section has. Thus, the determination of p value is dependent on the shape of cross section. As it is shown in the numerical examples, p is taken as 1.8 for wide flange I-sections (and it was calibrated with experimental results). It appears that p=2.1 is reasonable for a rectangular section and p=2.4 for a solid circular section. By determining the p value in such a way one can simulate the spread of plasticity through a cross section.

Before determining parameter 0pu, it is instructive to examine how &pu affects the plasticity-factor of a hinge-spring. The relations betweenp and M for different Bpu values are presented in Fig. 6 for the same element of W200x46 section. Figure 6 clearly illustrates that a larger 0pu value will result in a softer hinge-spring. When Bpu value is very small, the hinge-spring behavior is approaching that of a conventional elastic-plastic-hinge.

To simulate the spread of plasticity along the length of a member, the parameter 0pu must consider the distribution of bending moment along the length of the member (called moment gradient). In the following, the moment-curvature-thrust relations for beam-columns will be reviewed first, and then 0pu value will be computed considering different moment gradients.

Moment-Curvature-Thrust Curves (М-ф-А) of Beam-Columns

The moment-curvature-thrust curves for general beam-columns were given by Chen (1971). A typical presentation of these curves is shown in Fig. 7, where the curvature is normalized by фу (Фу=2єу/^, where d=depth of a section, ey=Fy/E). According to Chen (1971), it needs two functions to represent М-ф relation in the post-elastic range.

In this study, the moment-curvature-thrust curves in Fig. 7 are expressed approximately as a unified form as shown in Fig. 8a. In the elastic range (M < Myc, where Myc is the first-yield moment allowing for axial force), ф=М/Е/ (and §yc=Myc/EI at the first-yield). After M>Myc, М-ф relation becomes nonlinear, and ф=фе +ф^ (ф= total curvature, фe=elastic curvature, and ^^plastic curvature). The section is approximately fully yielded at curvature ф/ u, where ф/ can be expressed as a constant ю1 multiplied by §pc (§pc=Mpc/EI, where Mpc is the full-yield moment allowing for axial force). Note that the determination of moments Myc and Mpc does not involve the assumption of M/N being constant (hence different symbols are used). From Eqs. 1 and 2, we have

Myc = My (1 – Fr Fy – N Ny) (10)

Mpc = Mp (1 -(TV Ny )a Ц My (i-(TV Ny У) (11)

 Fig. 8 Moment-curvature relationships: (a) total curvature (b) plastic curvature

For I-sections bending about strong axis, ю1=1.9. The corresponding moment is found to have an average value of 96% of Mpc for 0<N/Ny <0.6 (Chen 1971). The plastic curvature at the onset of full – yield is approximately equal to §pu=f – ф^^-^ф^^ ф^. For applying M^p relation later in this
study, the shaded area to the left of the curve (see Fig. 8b) can be found to be (ю3хф;,„хЬ1) and the centroid of the shaded area to the M=Mpc line is x =©4^1. From Fig. 8a, we have

ФPU =щ{мрс EI)=(ю2 El);SFy[1 -(n Ny)aJ=(2«2^yq d)|1 -(n Ny)a (12)

where: £y=Fy/E, I/S=d/2, and d=depth of a cross section.