The stability of a laminated Voussoir beam: Back analysis of a historic roof collapse using DDA

Abstract

The stability of a horizontally bedded and vertically jointed roof, referred to here as a laminated Voussoir beam, is studied using careful documentation of a historic roof collapse, which occurred in an ancient underground water storage reservoir dated back to ca. 1000 B.C. The roof of the opening failed immediately after the excavation leaving a dome shaped loosened zone, with a span of 7 m and height of 2.5 m, consisting of horizontal beds and vertical joints with mean spacing of 50 and 25 cm, respectively. The ancient engineers erected a massive pillar at the center of the dome in order to passively support the failed roof and the opening remained in service for several hundred years following the failure. Analysis of the roof using iterative Voussoir beam procedure [Beer, G. and Meek, J. L., Design curves for roofs and hanging walls in bedded rock based on Voussoir beam and plate solutions. Trans. Inst. Min. Metall. , 1982, 91 , A18–22.] shows that the roof was more sensitive to failure by shear along the abutments rather than by crushing at hinge zones and that the required friction angle ( φ req. ) for stability would have been 36°. The available friction angle is estimated between 38.6° and 46.4° and, therefore, the result of the iterative solution is considered unconservatively wrong. Results of DDA [Shi, G. -H. and Goodman, R. E., Two-dimensional discontinuous deformation analysis . Int. J. Numer. Anal. Methods Geomech. , 1989, 13 , 359–380; Shi, G. -H., Block System Modeling by Discontinuous Deformation Analysis . Computational Mechanics Publications, Southampton, 1993, pp. 209.] however indicate that φ req. must have been greater than 60°, a shear strength which was not available at the time of construction, thus the immediate failure. Using DDA we further demonstrate that φ req. is related to joint spacing (or block length) in a parabolic function: with increasing joint spacing φ req. decreases to a minimum value and than increases with further increase in joint spacing. This result is attributed to the interaction of two competing forces: one is the stabilizing axial thrust which increases with increasing moment arm in individual blocks (a function of joint spacing or block length), the other is the destabilizing vertical load which increases with increasing weight of overlying blocks.