Features of calculating tunnel linings for the loads from the weight of rocks by methods of continuum mechanics

Abstract

It is known that calculating structures for their own dead loads in which the gravitational forces are applied to a design model of the completed structure, instantaneous loading (IL) scheme, leads to errors in many cases. In problems of calculating underground structures, for the solution of which models of continuum mechanics are used, this scheme gives divergences so far from the actual picture as to be practically unacceptable.Attention was called to this circumstance in [i] and a general scheme of solution free from the main shortcoming of the IL scheme was proposed. Unfortunately, the fundamental methodological error pointed out in [i] is repeated in many works, theoretical and experimental. The essence of the error is that the IL scheme does not reflect the actual picture of the continuing change in the deformed states of the rock mass and lining: gravitational forces cause initial stresses and strains in the rock undisturbed by tunneling, and only after weakening by a tunnel supported by a lining does the sought deformed state of the mass-lining system occur from the effect of unequalized "removable" stresses applied to the tunnel surface.Let us examine the process of driving an underground tunnel according to [i]. A load P (equal, e.g., to the weight of the overlying rock) is applied using a perfectly stiff crossbar vertical to three bars S M, SC, and S M of the same stiffness (Fig. I). As a result of deformation of the bars the crossbar moves to position I (stage I). The forces in the bars of this case are equal to P/3. Now we remove bar S c and instantaneously replace it by a still undeformed bar* So of the same stiffness and length ~ --AZ. As a result, the unequalized load P/3 with bar S c removed is redistributed over three bars SM, So, and SM, as a consequence of which the crossbar moves to position II (stage If). In this case the force in bar SMwill be equal to (4/9)P and in bar So to (1/9)P. It is obvious that if we simulate the lining by bar SC, stage I will be equivalent to the IL scheme. If we simulate the lining by bar So (actually, this is how it must be done, since it is understood that the lining is being constructed in an already deformed mass), then we will obtain a state of stress corresponding to the proposed [i] scheme for calculating the effect of "removable" stresses. In the given case the IL scheme overestimates the forces in the "lining" (i.e., in bar S C) threefold.For a quantitative evaluation of the indicated error in the IL scheme we investigated a lining supporting a rectangular tunnel (Fig. 2). The calculations were made by the finite element method. As we see from Fig. 2, the maximum forces in the lining calculated according to the scheme in [i] are more than twofold less than according to the IL scheme.Finally, the next problem is an example where the IL scheme cannot yield a solution at all. For example, let it be required to determine the state of stress in the rock mass and a new lining that for some reason replaced the lining installed earlier. It is clear that the stre...