PSI - Issue 32

N.A. Samodelkina et al. / Procedia Structural Integrity 32 (2021) 173–179 N.A. Samodelkina / Structural Integrity Procedia 00 (2019) 000 – 000

175

3

where max  is the maximum tangential stress; p  is the ultimate tangential stress; n  is the mean normal stress; C is the adhesion factor; and  is the inner friction angle. The numerical implementation was carried out by the finite element method [Fadeev, (1987); Zenkevich, (1974)], with the discretisation of the investigated domain into triangular elements of the first order and Goodman contact elements [Fadeev, (1987)]. The elastoplastic nature of deformation of the undermined rock mass was taken into account by successive approximations, based on the method of variable parameters of elasticity [Malinin, (1974)]. The variable parameters of elasticity, equal to the initial deformation properties of the laminated mass, were taken as the zero approximation. Based on the calculated values of stresses and strains of the zero approximation in each finite element, we calculated intensities of stresses i  and strains i  and checked the condition of plasticity (1). When p    max , we found the values of stress intensity * i  corresponding to p  . Based on the values of * i  and i  , we calculated new elasticity parameters:

*

  *

E 3 1 1 2 3 1 2 2 1     E

i i  

i i

i i     *

*

*

;

,

E







i i    *

E 3 1 1 2  

where  , E are the initial strain modulus and the Poisson ratio, respectively. Thus, during the implementation of the plastic stage of deformation, regardless of the initial deformation properties, parameters * * ,  E will be different for each finite element. Furthermore, the computational process continued according to the scheme of the successive approximations: the stresses and strains were calculated again, their intensities were determined, and the variable parameters of elasticity were refined. The convergence of the iterative process was controlled by fulfilling the condition in each finite element

   p max ,

 

where  is a small number. In the calculations, the contact between the OSR and the CSZ was modelled by Goodman elements (Fig. 2). The nature of the shear deformation of the contact was determined by the piecewise linear approximation (Fig. 2c) of a typical diagram of its loading [Baryakhet al., (1996)]. The ultimate shear resistance ( р  ) was calculated according to the Coulomb criterion (1).

Fig. 2. The Goodman contact element (a) and the dependence of the normal and tangential stresses on the normal (b) and shear c) deformations