The modulus of deformation of a rock mass is one of the critical parameters in the safe and cost effective design of underground structures. Traditional methods used to determine this property often involve laboratory testing of a group of core samples and "correcting" the laboratory results by using empirical factors related to joint spacing and other conditions. It is well known that core samples are representative of only a limited region in the formation and not necessarily of all the rock surrounding a tunnel. In addition, empirical correction factors have been developed from data having a substantial amount of scatter (Deere et. al., 1967). Other traditional methods such as plate bearing tests and the Goodman jack test provide only localized data and are subject to similar empirical correction factors (Bieniawski, 1978). As a result, traditional methods predict modulus values which are questionable and which err, in most cases, in a non-conservative way. Recently, under DEFENSE NUCLEAR AGENCY (DNA) sponsorship, UTD has developed a methodology for the measurement of the modulus of deformation through monitoring of convergence. This method recognizes that convergence of tunnel walls is the result of the elastic strain, and movement along joints and other anomalies, in thousands of cubic yards of rock surrounding the underground opening. Analytical solutions were derived which relate radial convergence to the elastic modulus (after the theory of elasticity [Timoshenko and Goodier, 1951]) of this large sample of rock, under conditions where induced stresses are not large enough to create a plastic zone around the tunnel. Since the method includes radial convergence which is a composite measurement of elastic strain of the rock, and other strain due to geologic anomalies, the term modulus of deformation is used (Bieniawski, 1978). The calculation of the modulus in this manner is analogous to calculating the modulus of elasticity of a complex composite material from strain and loading conditions. The advantages of using tunnel convergence over other methods is that this method is representative of a substantial "test specimen" complete with fractures, joint fillings, and other anomalies. It is, in fact, the modulus which exists where the rock formation interacts with the tunnel support system.

This paper presents the theoretical basis for the convergence method, and provides equations and charts that can be used to directly calculate the modulus of deformation from field measurements. Case histories are presented and emphasize the differences in magnitude between modulus values obtained from laboratory specimens, empirical methods, and the convergence method described in this paper. In one program UTD utilized the radial convergence measurement technique to obtain the elastic modulus of material at the Nevada Test Site. When the values obtained were used in design equations, they predicted tunnel behavior which agreed very well with experimental observations.


Tunnel convergence is deformation caused by stress redistribution around the periphery of an opening during excavation. Considering the state of stress in an element on the boundary of an opening to be excavated, the state of stress prior to excavation is equal to the free field stress; i.e., the stress state of the element is equal to the undisturbed ground pressure.

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