For the construction of large-scale underground openings such as tunnels, underground power stations, petroleum storages, etc., analysis of stress and deformation of the rock under excavation is important and indispensable. Nowadays, for the prediction of rock stability or the decrease of the bearing capacity around the opening, the finite element method has been successfully applied. However, most conventional finite element analysis limits itself to be within the range of strain-hardening analysis. The ground rock under excavation is characterized by the fact that it itself is an important member with respect to load bearing capacity as well as the object to be excavated. Namely, strain- softening is favorable for the object to be excavated but unfavorable as a load bearing member.
In the rock material to be excavated practically, the stress-strain relation curve shows the so-called strain-softening behavior if the loading condition exceeds the maximum strength. The analysis such as On fall of rock or expansion of side wall', etc., can be carried out if the strain- Softening characteristics can be dealt with within the stress-strain relationship. The practical repeated compression test indicates that the loading even lower than the maximum strength produces the accumulation of permanent plastic strains and finally causes the fracture of the material if loaded cyclically. For analysis of this Phenomenon, it is necessary to include the function so as to be able to treat the cyclic loading condition. The finite element method presented in this paper can analyze the repeated loading, with consideration given to the strain-softening stress-strain relationship.
Basic idea of elastic plastic constitutive equation is the concept of loading surface which is presented by Green-street and Phillips('), The stress-strain curve isassumed to be given from the relation between equivalent stress and equivalent plastic strain. For the practical function form, exponential function is successfully used to express the strain-softening.
To express constitutive relationship between stress and strain, the concept of closed loading surface introduced by Green-street and Phillips is utilized in this paper (Fig. 1). In this analysis, it is assumed that there is no yield surface, which encloses the elastic region. Namely, plastic strain is assumed to be always produced immediately after the initial loading The loading surface passes through the stress point. The loading surface grows larger as the loading proceeds. A stress rate is assumed to be directed toward the outside of loading surface. In the case of loading reversal, new loading surface with new stress-strain relationship will be introduced. After that, if loading increases once again, then the former loading surface will decrease and new loading surface will be produced. (Fig. 2)
(Figure in full paper)
Where p, fi, Vi are density, body force and displacement, respectively. The finite element method can be applied to equations (11)~(14) following the standard approximation procedure. For the interpolation relation of displacement, linear interpolation equation based on the triangular finite element is used. As the basic procedure for the nonlinear iteration, the stress transfer method(3) is employed.