The design procedures for closed monolithic linings of underground structures of arbitrary cross-section as well as for complex of parallel interaction circular workings of considerable extent subject to internal head and to external pressure of subsurface water are considered in the paper. The above-mentioned problems were solved using the theory of analytic functions of complex variable, the apparatus of conformal mapping, complex rows and the properties of Cauchy intergrals. The algorithms for calculations were obtained and the programmes for electronic computers of EC type were worked out. The design procedures are illustrated by the examples of calculations of the linings of workings of large cross-section. There are shown some dependences of the stresses and forces in the linings on their geometric characteristics and on the relations of the moduli of deformation for the materials of linings and rock mass.


Im Vortrag werden Berechnungsmethodiken fuer geschlossene monolitische Auskleidungen der Untertagebauwerke mit beliebigen Querschnitt sowie fuer einen Komplex paralleler, einander beeinflussender Rundgrubenbaue bedeutender Lange auf Wirkung des Innen- und des Aussendrucks von Grundwasser dargelegt. Die angegebenen Aufgaben sind unter Nutzung der Theorie der analytischen Funktionen der komplexen Veranderlichen, des Apparates der konformen Abbildungen, der Komplexreihen und der Eigenschaften der Koschi-lntegrale gelöst. Es sind Berechnungsalgorithmen erhalten und Programme fuer Rechenmaschinen der Baureihe EC entwickelt worden. Die Methodiken werden an Beispielen der Auskleidungsberechnungen fuer Grubenbaue mit grossen Querschnitt illustriert und einige Abhangigkeiten der Spannungen und Kraftein Auskleidungen von ihren geometrischen Characteristiken und vom Verhaltnis der Verformungswerte der Auskleidungsmaterialen und des Gesteins dargelegt..


Dans le rapport sont exposees les methodes de calcul de la resistance à la poussee interieure et à la pression exterieure des eaux souterraines des revêtements monolithes continus dans des ouvrages souterrains de sections quelconque, ainsi que dans un ensemble d''ouvrages souterrains de section circulaire et de longueur considerable, parallèles et s''affectant les uns aux autres. Les problemès mentionnes sont resolus en appliquant la theorie des fonctions analytiques d''une variable complexe, l''appareil de representations conformes, des series complexes et les proprietes des integrales de Cauchy. On obtenu des algorithmes de calcul et elabore des programmes pour les calculateurs electroniques de la serie EC. Les methodes sont illustrees par des exemples de calcul des revêtements pour des ouvrages de grande section, et par quelques courbes de variation des contraintes et efforts dans les revêtements en fonction des caracteristiques geometriques de ces derniers, et du rapport entre les modules de deformation des revêtements et du rocher.


This paper proposes procedures for designing closed monolithic linings for underground structures of arbitrary cross section, as well as for designing a set of parallel workings, of circular cross section, interacting with one another and of substantial extent, all subject to the action of internal pressure and to the external pressure of subsurface water. The basis of this procedure is the treatment of the lining and the rock massif as a single deformable system. This enables the bearing capacity of the surrounding rock massif to be taken into account to the maximum extent in the design calculations. Formulated and solved for this purpose are the corresponding plane contact problems of elasticity theory for a ring of arbitrary shape (having a single axis of symmetry), reinforcing a hole in a linearly deformable medium having other deformation characteristics, and for a multiply connected piecewise- homogeneous region, i.e. a medium weakened by an. arbitrary number of reinforced circular holes arranged in any manner (in the general case, the holes may be of different radii and have reinforcing rings of various thicknesses, made of various materials). Discussed in the paper is the loading of the linings by internal uniformly distributed pressure. For a series of parallel workings this internal pressure may be different in each working (certain of the workings being empty). Also treated is the case of a jump (discontinuity) in normal stresses at the surface of contact between one or several linings and the massif, this jump being equal to the subsurface water pressure, which varies linearly along the height of the lining.


Considered here is the plane contact problem of elasticity theory concerning the equilibrium of a ring of arbitrary shape (with a single axis of symmetry) in a linearly deformable homogeneous isotropic medium. This ring, whose material is characterized by its modulus of deformation E1 and Poisson''s ratio, functions together with the medium, whose characteristics are to and JO, i.e. at the line of contact L (the basic design layout is given in Fig. 1) the conditions of continuity of the stress and displacement vectors are complied with.

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