Issue 30

W. Changfeng et alii, Frattura ed Integrità Strutturale, 30 (2014) 486-494; DOI: 10.3221/IGF-ESIS.30.59 488 Shear Force Displacement K 1 K 2 F cr o Figure 2 : Bilinear element model of the support. Contact and impact elements Presently, analytical methods for contact and impact are mainly: coefficient of reconstitution, Lagrange multiplier and contact finite element method. The contact finite element method is mainly based on Kelvin impact model. Kelvin impact model is a parallel connection of spring and damper. The main parameters include: initial impact stiffness K 1 , yield stiffness K 2 , damping c and initial gap d 0 . The nonlinearity of the restraining devices is generally not considered in existing studies. In this study, considering that the restraining device can be nonlinear, the element and hysteretic model is given in Fig. 3 [8, 9]. 0 1 2 y 0 y 1 2 (a) (b) Figure 3 : Contact element: (a) Mechanical model of the contact element; (b) Hysteresis curve of the contact element. The above model was used to simulate the impact of girder and restraining devices and the impact force is given in Eq. (1) [10, 11]: 1 ( ) ( ) 0 0 0 i j i j i j i j F k u u d c u u u u d F u u d                  (1) In which: k 1 : is the contact and impact stiffness; d : Initial gap; c : damping factor 1 2 1 1 2 2 M M c k M M    , 2 2 ln (ln ) e e      ; e is the hysteretic factor which can be taken as 0.65 for concrete; u i , u j: are respectively the end displacement of the adjacent girder of the contact element; u i , u j: are respectively the end velocity of the adjacent girder of the contact element. Elasto-plastic reinforced concrete model The elasto-plasticity of bridge pier is simulated by Takeda tri-linear model and the skeleton curve of the element is controlled by the moment and curvature at cracking, yield and damage points, as shown in Fig. 4.