Series Element Model

The series element model involves a semirigid connection spring, an inelastic spring and an elastic member all connected in series. The nature of the compound element is indicated in Figure 2, where parameters Rn, Re, Rc, Rp denote the rotational stiffnesses of the compound member end, elastic member end, semirigid connection spring, and the member plasticity spring, respectively. Only end 1 of the member is considered, while end 2 may or may not have the same nature as end 1.

Figure 2. Compound-rotational element at the member end

The case in Figure 2(a) is conventionally used in structural analysis, where a beam-to-column connection at node 1 is assumed as either a pinned connection (Rn = 0) or fixed connection (Rn = Re). This assumption makes analysis and design simple both for hand and computer-based analysis. However, if the effect of actual connections on structural responses must be considered, the model of the semirigid connection shown in Figure 2(b) is introduced in the analysis and design of structures. Another case is also popular in rigid-plastic analysis, where a plastic hinge is abruptly formed rather than gradually degrading from initial yield to full yield states. To improve the accuracy in this latter method, the inelastic model shown in Figure 2(c) is used to model the property of the gradual stiffness degradation due to the presence of plasticity. Finally, if both semirigid connection and plasticity behaviour may occur at the same time, the series-element model shown in Figure 2(d) should be introduced in analysis and design, as in the work of Yau and Chan (1994) where, however, the influences of plasticity and semirigid connections were considered separately. To facilitate structural
analysis accounting for both connection and inelasticity stiffness degradations and their interactions, an integrated compound element is needed and is investigated in detail as follows.

The rotational deformation involving nonlinear connection and inelasticity behaviour indicated in Figure 2(d) is graphically represented in Figure 3(a) using two springs. It can be verified that the two series-connected springs may be substituted by the compound element shown in Figure 3(b) involving a single spring. The compound stiffness R that reflects the combined stiffnesses Rc and Rp can be derived as in the following (Liu 2005).

Provided that a moment M is applied at the joint in Figure 3(a), the relative rotations 0c and 0p are given by,

0c = M / Rc; 0p = M / Rp (1)

Since the total relative rotation 0 between the joint and the elastic member end is the summation of rotations induced by the connection and inelastic springs, from Eqs (1) the rotation 0 can be expressed


accounts for both connection and plasticity stiffness.

2.1 Determining stiffness Rc

It remains to determine the stiffness of the compound element through Eq. (3). The inelastic stiffness Rp of the member is directly available in the literature (Grierson et al 2005). Thus, only the connection stiffness Rc in Eq. (3) need be established herein.

Several semirigid connection models have been proposed (Xu 1994), and of these models the four – parameter power model, originally proposed for modeling the post-elastic stress-strain relation (Richard et al 1975), is commonly used in analysis. Recently, this model has been further confirmed to be effective and accurate for predicting the behaviour of end-plate connections on the basis of experimental data for extended-end-plate and flush-end-plate connections (Kishi et al 2004). Thus, the following four-parameter model,

is employed in this study to simulate the behaviour of semirigid connections. In Eq. (4), 0c denotes the relative rotation of the semirigid connection, and the four parameters Rce, Rcp, M0, and у are the initial rotation stiffness, strain-hardening/softening stiffness, reference moment, and shape parameter of the
connection, respectively. The initial yield moment Mcy and corresponding rotation 9cy determine the elastic stiffness Rce= Mcy/9cy, while the reference moment M0, strain-hardening/softening stiffness Rcp, and rotation capacity 9u determine the ultimate moment capacity to be,

Mu = M0 – QuRcp (5)

where the rotation capacity 0u depends on the connection type and can be determined from the results of existing research (e. g., Bjorhovde et al 1990). The four parameters in Eq. (4) can be found for different types of connections from an existing database of experimental results (Xu 1994).

Differentiating Eq. (4) with respect to rotation 0c determines the tangent stiffness of the connection to be (Richard et al 1975),

where Rce is the elastic rotational stiffness at the initial condition 0c = 0, while Rcp is the strain – hardening/softening stiffness when rotation 0c tends to infinity (for practical steel structures, 0c can reach at most to the limiting rotation capacity of the connection when fracture occurs (Bjorhovde et al 1990).


[1 + (RJC /Mu) ]1/*

It is seen from Eqs (4) and (6) that the four-parameter model reduces to a linear model with Rc = Rce when Rcp tends to Rce, whereas a bilinear model is reached when the shape parameter у approaches to infinity. If Rcp is set to zero (i. e., strain-hardening/softening is ignored), Eq. (4) reduces to the following three-parametric model that was previously suggested by Kishi and Chen (1987),

where the reference moment M0 is replaced by the ultimate moment Mu. Note that the rotation 0c can be explicitly obtained from Eq. (7) as,

0c = M / Rce [1 – (M / Mu У f *

Using an expression for post-elastic rotation previously derived by the authors (Grierson et al 2005, Xu et al 2005), and the connection rotation given by Eq. (8), the total relative rotation 0 of the compound element can be explicitly expressed as,

which represents the moment-rotation relationship of the compound element. The benefit of using the three-parameter model is that the rotation 0c of the connection is directly obtained in the nonlinear analysis for the given moment M; the disadvantage is that the strain-hardening or softening nature of the connection is omitted. In contrast, strain hardening/softening is accounted for in the four-parameter model, but an iterative procedure is needed to find the relative rotation 0c of the connection. Both of the connection models are considered for the verification analysis presented later.