Summary:

A method of analyzing stress concentrations and displacements around cavities of arbitrary shape was developed, wherein the plasticity of the material was assumed to be governed by parameters analogous to angle of internal friction and cohesion. The yield function plots as a cone in principal stress space; the stress-strain relations were obtained by regarding the function as a plastic potential. The displacement method of finite element analysis (with triangular elements and linear displacement functions) was employed in a step by step application of load. At each step, increments of displacement which satisfy equilibrium were determined by a recursive process. Stress concentrations around circular openings in an infinite medium were analyzed for arbitrary values of yield parameters and initial state of stress. Results appear to be fairly realistic and reflect the effects of volumetric expansion accompanying yielding, as implied by the stress-strain relations used.

Resume:

Les auteurs developpent une methode d'analyse de la concentration des charges et deplacements au voisinage de cavites de formes arbitraires, pour lesquelles ils admettent que la plasticite du materiau est gouvernee par des paramètres de même nature que l'angle de frottement interne et la cohesion. La fonction de passage à l'etat plastique a la forme d'un cône dans l'espace des contraintes principales; les relations contraintes- deformations ont ete obtenues en considerant la fonction comme un potentiel plastique. La methode de deplacement de l'analyse d'elements finis (elements triangulaires et fonctions de deplacement lineaires) a ete utilisee avec application discontinue et progressive des charges. A chaque palier, l'accroissement de deplacement satisfaisant à un nouvel etat d'equilibre a ete determine par recurrence. Les concentrations de contraintes autour d'ouvertures circulaires dans un milieu infini ont ete etudiees pour des valeurs arbitraires des paramètres de plasticite et des conditions initiales de contraintes. Les resultats apparaissent bien conformes à la realite des phenomènes et rendent compte des effets d'expansion volumetrique accompagnant la deformation plastique, conformement aux relations contraintes-deformations utilisees.

Zusammenfassung:

Die Verfasser berichten ueber eine von ihnen entwickelte Methode fuer die Analyse von Druckkonzentrationen und Druckverschiebungen urn Hohlungen von willkuerlicher Form herum, wobei sie annehmen, dass die Stoffplastizitat von Parametern beherrscht wird, welche dem Winkel von Reibung und Kohasion analog sind. Die Ergebniss- Funktion zeichnet sich aus als ein Kegel im Hauptdruckraum; die Verhaltnisse zwischen Druck und Spannung wurden erhalten, indem man die Funktion als eine plastische Leistungsfahigkeit betrachtet. Die Methode der Verschiebungen-Analyse durch begrenzte Elemente (mit dreieckigen Elementen und Funktionen von linearen Verschiebungen) wurde angewandt durch schrittweisen Zusatz von Ladung. Der jeweilige Zunahme von Verschiebungen nach eingetretenem Gleichgewicht wurde durch einen ruecklaeufigen Prozess bestimmt und festgesetzt. Die Druckkonzentrationen um kreisfoermige Öffnungen herum in unbegrenzten Milieu wurden analysiert fuer willkuerliche Werte von Stoffplastizitat-Parametern und Anfangsstadium von Druck. Die Ergebnisse scheinen der Wirklichkeit ziemlich zu entsprechen, und geben die volumetrische Ausdehnung wieder, welche das Ergebniss begleitet, so wie es durch die angewandten Druck- und Spannungsverhàltnisse angedeutet wird.

Introduction

One of the common problems in rock mechanics is that of estimating stress concentrations and displacements that occur around cavities formed in a rock mass. Observations in the field indicate that the use of classical solutions from the theory of elasticity can be quite unrealistic, particularly if the rock is weak or extensively jointed. On the other hand, reduced tangential stresses near the walls of the cavity suggest that an elasto-plastic analysis may provide solutions which are in better agreement with actual field measurements. Because of the very considerable amount of computations involved in the numerical solution of the problems, not many published results are available at the present time. The classical papers of ALLEN and SOUTHWELL [1] and of JACOBS [9], in which tension plates with holes and notched plates are considered, represent pioneering efforts on the numerical solution of elasto-plastic problems in accordance with the incremental theory of plasticity. ANG and HARPER [2] considered similar problems with the help of a physical model of the governing finite difference equations. In all of the foregoing investigations, the Mises criterion of yielding [8] was used. Because the use of large electronic computers is now becoming quite common, routine elasto-plastic analysis of practical problems in rock mechanics may soon be feasible. For such a general usage, the numerical procedure should have the capability of handling boundary conditions of practical interest. Also, the yield criterion used must account for internal friction in the rock. The object of the present study was to develop a numerical method for the analysis of two-dimensional (plane strain) problems concerning stress concentrations and displacements in the rock mass around an underground opening. By adopting the finite element method to a step-by-step analysis [3] and employing the generalized Mises criterion [6] which accounts for both internal friction and cohesion, the method can be used for openings of arbitrary shape with any selection of elastic constant, yield parameters, and initial state of stress.

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