## M

FIGURE 23.2 Semirigid beam-to-column connections.

The single-angle connection and the weld flange plate with web-angle connection represent very flexible and rather rigid connections, respectively. Also, as shown in Figure 23.1, experimental tests have clearly demonstrated that there is a nonlinear relationship between M and 6 for almost all types of connections over the entire loading range. The nonlinearity is mainly because a connection is an assemblage of several components that interact differently at different levels of applied loads. Material discontinuity of the connection subassemblage, local yielding of some components, local buckling of a plate element, and so on, all contribute to the nonlinearity. One of the primary obstacles to implementing semirigid framing in design practice is that the nonlinear M-6 relationship of semirigid connections complicates the evaluation of the structural response of semirigid frames.

The modeling of the M-6 relationship is a fundamental requirement for any consideration of the interaction of connection and member behavior. The most accurate and reliable knowledge of the behavior of a beam-to-column connection is obtained through experimental testing, but this technique is too expensive for design practice and is usually reserved for research purposes only. With the aim to incorporate connection behavior into structural analysis, connection behavior is generally simulated by means of a mathematical representation of the M-6 relationship. It is difficult to model this nonlinear relationship by rigorous and exact mathematical procedures and the connection behavior modeling in practical design is usually approximate in nature with drastic simplifications. Tests of prototype connections are commonly carried out to obtain actual moment-rotation behavior, which is then modeled approximately by means of different mathematical representation. The degree of the refinement of the mathematical representations of the M-6 relationship depends somewhat on the computational capabilities of available design aids.

23.3.1 Linear Model

The simplest connection model is the single-stiffness linear model that is expressed as follows:

where M is the connection moment and R and 6 are the stiffness and rotation of the connection, respectively.

The connection stiffness R in Equation 23.1 can be the initial connection rotational stiffness, Ri, as shown in Figure 23.3. Alternatively, a secant stiffness of the connection, Rj, which corresponds to a rotation of 60 (= Mu/Ri), can be determined from the initial stiffness Ri, the ultimate moment capacity Mu, and the moment-rotation curve of the connection. R is recommended instead of Ri as a representative connection Connection rotation, d

FIGURE 23.3 Various linear models for semirigid connection.

stiffness value because the initial stiffness of the connection does not represent connection response adequately.

To recognize the degradation of connection stiffness under increased load, the secant stiffness, Rb, determined in accordance with the so-called beam line method can be used in Equation 23.1. As shown in Figure 23.3, the beam line is defined as the straight line connecting two extreme connection conditions for a uniformly loaded beam, the end restrained moment when the connection is fully rigid and the end rotation when the connection is ideally pinned; that is, the rotation is zero for fully rigid end-connections with a resulting end-restrained moment of wL2/12 and the moment is zero when the ends of the beam are pinned with resulting end rotation of wL3/24EI. The connection stiffness Rb is then determined by the intersection of the beam line and moment-rotation curve of the connection. However, it is evident that the beam line does not represent the attainment of yield stress for any condition other than for a simply supported beam and a fixed ended beam, that is, the two end points of the line. To overcome this shortcoming, the bilinear model of the beam line  shown as the dashed line in Figure 23.3 can be used establish a connection stiffness RB, which corresponds to an end rotation midway between pinned and fully fixed.

### 23.3.2 Polynomial Model

While the linear model in Equation 23.1 is simple and easy to use, it does not represent the true behavior of the connection. Bilinear and piecewise linear models have been proposed to provide a closer approximation of connection behavior. With the aim of providing a more accurate representation of nonlinear behavior of semirigid connections, nonlinear empirical models based on curve-fitting experimental results are widely employed to replicate and predict the nonlinear M-6 relationship of various types of beam-to-column connections. Shown in Equation 23.2 is an odd-power polynomial model :

where 6 is the connection rotation and M is the moment acting on the connection. Parameter K is a standardization factor determined by the connection type and geometry and C1, C2, and C3 are curve-fitting constants obtained by using the method of least squares. The standardization factor and curve-fitting constants for various connections are shown in Table 23.1.

### 23.3.3 Three-Parameter Power Model

Chen and Kishi  adopted the power model developed by Richard and Abbott  to represent the moment-rotation characteristics of steel beam-to- column connections. The model is composed of three parameters: initial stiffness Rki, ultimate moment capacity Mu, and a shape parameter n, and can be expressed as in Equation 23.3, taking the shape shown in Figure 23.4.

{1 +[6/6o]n}1/n where 60 is equal to Mu/Rki is the reference plastic rotation. From Figure 23.4, it is recognized that the larger the shape parameter n the steeper the curve, which represents more rigid connections. The shape parameter n can be determined by using the least squares method applied to the differences between the predicted moments and the experimental test moments. The tangent connection stiffness Rkt and connection rotation can be determined from Equation 23.3 as follows:

Rkt " "d6 "(1 + [6/6o]n)(n+1)/n (23:4)

TABLE 23.1 Curve-Fitting Constants and Standardization Constants for Polynomial Model (All Size Parameters are in Centimeters)

Connection types

Curve-fitting and standardization constants

Single web-angle connection

0 0