_Applied_  
_sciences_  
Chapter 1 Member Discrete Element Method for Static and Dynamic Responses Analysis of Steel Frames with Semi-Rigid Joints  
_[PERSON] 1,*_ and [PERSON] 2  
State Key Laboratory for Geomechanics & Deep Underground Engineering,  
China University of Mining and Technology, Xuzhou 221116, China  
Key Laboratory of Concrete and Pre-Stressed Concrete Structures of the Ministry of Education,  
Southeast University, Nanjing 210018, China; [EMAIL_ADDRESS]  
Correspondence: [EMAIL_ADDRESS]; Tel.: +86-516-8359-0609  
Academic Editor: [PERSON]  
Received: 27 May 2017; Accepted: 7 July 2017; Published: 11 July 2017  
**Featured Application: The MDEM can naturally deal with strong nonlinearity and discontinuity. A unified computational framework is applied for static and dynamic analyses. The method is an effective tool to investigate complex behaviors of steel frames with semi-rigid connections.**  
**Abstract:** In this paper, a simple and effective numerical approach is presented on the basis of the Member Discrete Element Method (MDEM) to investigate static and dynamic responses of steel frames with semi-rigid joints. In the MDEM, structures are discretized into a set of finite rigid particles. The motion equation of each particle is solved by the central difference method and two adjacent arbitrarily particles are connected by the contact constitutive model. The above characteristics means that the MDEM is able to naturally handle structural geometric nonlinearity and fracture. Meanwhile, the computational framework of static analysis is consistent with that of dynamic analysis, except the determination of damping. A virtual spring element with two particles but without actual mass and length is used to simulate the mechanical behaviors of semi-rigid joints. The spring element is not directly involved in the calculation, but is employed only to modify the stiffness coefficients of contact elements at the semi-rigid connections. Based on the above-mentioned concept, the modified formula of the contact element stiffness with consideration of semi-rigid connections is deduced. The Richard-Abbort four-parameter model and independent hardening model are further introduced accordingly to accurately capture the nonlinearity and hysteresis performance of semi-rigid connections. Finally, the numerical approach proposed is verified by complex behaviors of steel frames with semi-rigid connections such as geometric nonlinearity, snap-through buckling, dynamic responses and fracture. The comparison of static and dynamic responses obtained using the modified MDEM and those of the published studies illustrates that the modified MDEM can simulate the mechanical behaviors of semi-rigid connections simply and directly, and can accurately effectively capture the linear and nonlinear behaviors of semi-rigid connections under static and dynamic loading. Some conclusions, as expected, are drawn that structural bearing capacity under static loading will be overestimated if semi-rigid connections are ignored; when the frequency of dynamic load applied is close to structural fundamental frequency, hysteresis damping of nonlinear semi-rigid connections can cause energy dissipation compared to rigid and linear semi-rigid connections, thus avoiding the occurrence of resonance. Additionally, fracture analysis also indicates that semi-rigid steel frames possess more anti-collapse capacity than that with rigid steel frames.  
**Keywords:** steel frames with semi-rigid joints; Member Discrete Element Method; nonlinear semi-rigid connection; geometric nonlinearity; snap-through buckling; dynamic response; fracture