## 1 Introduction  
Beam-to-column joints are frequently simplified to be fully rigid or pinned connections in structural analyses and designs, while real beam-to-column joints are semi-rigid connections between the two extreme cases. Compared with rigid connections, semi-rigid connections will reduce total stiffness of steel frames and make lateral displacement increase, thereby causing a second-order effect [1]. Therefore, the rigidity of a semi-rigid joint has a significant effect on the strength and displacement response of steel frames. At present, approaches for investigating mechanical behaviors of semi-rigid joints fall into three categories: experiment, mathematical model and numerical simulation. For the first two approaches, an amount of mature research has been carried out, and a variety of linear or nonlinear semi-rigid connection models were proposed [2; 3; 4; 5; 6]. Furthermore, Eurccode-3 (EC3) [7] provides the M-\(\emptyset\) curves for some specified beam-to column joints of steel structures. Based on the M-\(\emptyset\) curves of EC3, [PERSON] et al. [8] introduced the half initial secant stiffness approach to simplify the full characteristics of nonlinear connection stiffness curve into a bilinear curve in practical design, and analysis results of the two cases were in good agreement. Thus, numerical simulation methods of semi-rigid joints have increasingly attracted the attentions of numerous researchers in recent years. The finite element method (FEM) is the most commonly used method in the numerical simulation of semi-rigid joints. [PERSON] and [PERSON] [9], [PERSON] and [PERSON] [10] considered structural geometric nonlinearity by stability functions, and analyzed nonlinearity and buckling behaviors of semi-rigid beam-to-column joints in conjunction with the geometric stiffness matrix and updated Lagrangian approach. [PERSON] and [PERSON] [11] presented a beam-column element with springs connected in series for geometric and material nonlinear analysis of steel frames with semi-rigid connections, as well as proposed an efficient approach to trace the equilibrium path of steel frames. [PERSON] and [PERSON] [12] performed second-order analysis of steel frames with semi-rigid connections by a displacement-based finite-element approach. [PERSON] and [PERSON] [13; 14], [PERSON] [15] investigated the dynamic responses of steel frames with semi-rigid joints, where a semi-rigid joint was simulated by a zero-length spring element. [PERSON] et al. [16] and [PERSON] et al. [17] presented a multi-degree of freedom spring system to represent physical models of semi-rigid connections and took semi-rigid connections as new elements. This system transferred axial forces, shearing forces as well as bending moments. All the above results are based on the conventional finite-element method. However, in the static or dynamic analyses especially for strong nonlinear and discontinuous problems, the method has some limits such as difficult modeling, plenty of unknowns, long consuming time and convergence difficulty. In order to shorten consuming time, [PERSON] and [PERSON] [18; 19] proposed a numerical procedure based on the beam-column method, and performed nonlinearly elastic and inelastic time-history analysis of three-dimensional semi-rigid steel frames by using the based-displacement finite element method. Compared to SAP2000(Computers and Structures, Inc., California, CA, USA), the procedure captured the second-order effect by using only one element per member, and reduced computational time. Nevertheless, it failed to model fracture behaviors of steel frames with semi-rigid joints. [PERSON] and [PERSON] [20], [PERSON] and [PERSON] [21] carried out an investigation on dynamic collapse of semi-rigid steel frames using vector formed intrinsic finite-element (VFIFE) method. A zero-length spring element was employed to simulate the semi-rigid joints. However, the spring element is involved in the calculation, which makes that the method has a cumbersome calculation process and long consuming time.