Calculation of underground hydraulic structures by the finite-element method in a nonlinear formulation

Abstract

624.131.522 ~"k~o main types of nonlinearity are important in calculations of underground structures: physical (nonlinearity of the stress--strain relation) and morphological (nonlinearity of the structure of the calculation schemes).The first nonlinearity is a consequence of the known manifestation of the physical and mechanical properties of the material: The rock mass is practically always deformed nonlinearly when driving workings owing to the development of decompression.The more specific type, the second nonlinearity, is determined by the conSiderable effect on the formation of the stress field in the finished structure of technological construction factors and especially the stepwise opening of the section of the workings and sequence of driving them. The latter necessitates examining structurally nonlinear mathematical models, i.e., those models which are transformed during calculation, reflecting the most characteristic stages of development of the structure during its construction.We will examine certain problems related to the construction and realization of geomechanicalmodels of underground structures incorporating the two indicated types of nonlinearity. Construction of Discrete Models of Structures.The development of a discrete model is the most essential stage of the calculation process when using the finite-element method (FEM). Unfortunately, in the majority of cases the given procedure is presently difficult to formalize [i]. Its realization requires from the calculator, apart from a good understanding of the statics of the structure, skill in constructing approximating meshes. Discretization of nonlinear models has a number of characteristic features.When performing this procedure the approximRting mesh is condensed at places of presumed oscillation of the sought functions in order to provide the necessary accuracy, condensing being accomplished in proportion to the increase of their gradient. In the elastic model oscillation is expressed as pronounced concentrations near sources of disturbances of the field of the function --concentrators --and therefore condensation of the approximating mesh is usually performed according to the law of geometric progression (Fig. la). In the nonlinear model the character of oscillation is somewhat more complex --the site of the extremum is generally not known beforehand and the size of the zone of disturbance of the field of the function can be estimated only approximately.Therefore, when solving nonlinear problems it is better to perform discretization of the models within the indicated zones by uniform partition into finite elements of the minimum necessary size (Fig. ib). Thus the character of the approximating meshes in the elastic and nonelastic models is substantially different. .... Translated from Gidr0tekhnicheskoe Stroitel 'stvo, No. 12, pp. 13-i9, December, 1983.