## 3 Member Discrete Element Modeling for Semi-Rigid Connections  
### Virtual Zero-Length Spring Element  
In this study, a semi-rigid joint is simulated by a multi-degree-of-freedom spring element with two particles but without actual mass and length. The basic concept of the spring element is similar to that of References [13; 18]. The moment-rotation relationship of the zero-length spring element is expressed as a diagonal tangent stiffness matrix in other numerical approaches taking the FEM as a representation. In addition, the stiffness matrix needs to be integrated with structural total stiffness matrix. However, in the MDEM, the zero-length spring element is not directly involved in the calculation, but is employed only to modify the stiffness coefficient of each contact element adjacent to the semi-rigid connection. In addition, then the modified stiffness coefficient of the contact element is directly applied in the next time step. The whole procedure is simple and effective. Furthermore, computational time of the MDEM after considering semi-rigid connections has no increase. This is not only the nature difference between the zero-length spring element of the MDEM and that of other methods, but also the advantage of the MDEM processing semi-rigid connection problem. The MDEM model of a simple frame is displayed in Figure 2. If the joint D (i.e., particle D) is a semi-rigid connection, the processing mode of the semi-rigid connection is given in Figure 3, that is, a zero-length spring element is supplemented at the particle. The spring element contains two particles E and F with same coordinates, and its translational and rotational stiffness matrix can be expressed as  
\[\mathbf{k}_{s}=\left[ \begin{array}{cc}R_{x}&&\\ &R_{y}&\\ &&R_{z}\end{array} \right]\qquad\mathbf{k}_{\theta}=\left[ \begin{array}{cc}R_{ \theta x}&&\\ &&R_{\theta y}&\\ &&R_{\theta z}\end{array} \right]\, \tag{9}\]  
where \(R_{n}\) and \(R_{\theta n}\) are the translational and rotational stiffness vectors of a zero-length spring element with respect to \(n\) axis (\(n=x,y,z\)). In addition, for a two-dimensional problem, the zero-length spring element can degrade to three degrees of freedom which is two translational degrees of freedom and one rotational degree of freedom. The comparison of Figures 1 and 3 shows that the force state of the zero-length spring element is identical with that of the contact element, except that the former is dummy while the length of the latter is related to connected particles. This characteristic makes that the MDEM is able to simulate a semi-rigid connection by directly modifying the contact element stiffness. Only the bending stiffness of a semi-rigid joint is frequently taken into account in steel frames. The relationship between the moment and the relative rotation of a semi-rigid connection can be expressed as follow:  
\[\Delta M=R_{\theta}\Delta\theta_{r}, \tag{10}\]  
where \(R_{\theta}\) is the linear or nonlinear stiffness of a semi-rigid connection (i.e., zero-length spring element stiffness), which is obtained by experiments or mathematic models; \(\Delta\theta_{r}\) denotes the relative relation between particles E and F.  
Taking the semi-rigid connection of this type in Figure 2 as an example, the derivation detail of the rotational stiffness coefficient of the contact element 1 is given. When the joint D is rigid, the rotational stiffness of the members 1 and 2 are \(K_{\theta 1}=E_{1}I_{1}/L_{1}\) and \(K_{\theta 2}=E_{2}I_{2}/L_{2}\), respectively, \(L_{1}\) and \(L_{2}\) are the lengths of the members 1 and 2. When the joint D is semi-rigid, the semi-rigid connection stiffness is  
Figure 3: Zero-length spring element at a semi-rigid connection. Figure 2: MDEM (Member Discrete Element Method) model of a simple steel frame. assumed to be linearly distributed to the contact elements 1 and 2 in accordance with the stiffness of the members 1 and 2.