Three-Dimensional Expansion of the Terzaghi Arching Formula Considering Inclined Sliding Surfaces and Examination of Its Effects
Publication: International Journal of Geomechanics
Volume 17, Issue 7
Abstract
Terzaghi proposed a two-dimensional (2D) formula for arching based on the assumption of a vertical sliding surface induced in the upper part of soil mass attributable to the downward movement of a trapdoor. This study expands the 2D Terzaghi arching formula to a three-dimensional (3D) formula in which inclined sliding surfaces under general 3D excavation conditions are considered. The effects of the formula under various conditions were investigated. The assumption of the original 2D formula that the maximum shear strength is mobilized on a vertical surface is different from reality, because an actual sliding surface caused by downward movement of a trapdoor is an inclined surface. Furthermore, the 2D Terzaghi formula is limited to conditions of plane strain and does not consider 3D excavation conditions. Therefore, to examine the practical effects of these issues and to consider the arching effect in 3D excavation conditions, a formula was developed to expand the Terzaghi arching formula to inclined sliding surfaces under downward movement in 3D conditions. Using the expanded formula, the effects of excavation, ground, and surcharge conditions on vertical stress were examined, and the results were compared with those of the 2D Terzaghi formula. The induced vertical stress was highly affected by the inclination angle and the longitudinal excavation length. The degree of influence depended on the excavation, ground, and surcharge conditions. It is expected that the results from this study will provide a better understanding of various arching phenomena in the future.
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References
Adachi, T., Kimura, M., and Kishida, K. (2003). “Experimental study on the distribution of earth pressure and surface settlement through three-dimensional trapdoor tests.” Tunnelling Underground Space Technol., 18(2–3), 171–183.
Adachi, T., Kimura, M., Kishida, K., Kosaka, K., and Sakayama, Y. (1999). “The mechanical behavior of tunnel interaction through three dimensional trapdoor tests.” J. Geotech. Eng., 285–299 (in Japanese).
Ahmadi, A., and Hosseininia, E. S. (2013). “An experimental investigating the width and height of a stable arch formed in granular materials by using a new developed trapdoor apparatus.” Conf. on Int. Civil, Structural and Environmental Engineering, ACSEE, New York, 20–24.
Bierbaumer, A. (1913). Die dimensionierung des tunnelmauerwerkes, W. Engelmann, Leipzig (in German).
Cain, W. (1916). Earth pressure, retaining walls and bins, John Wiley & Sons, New York.
Caquot, A. (1934). Equilibre des massifs ii frottement interne, GauthierVillard, Paris (in French).
Chau, H. Y., and Bolton, M. D. (2006). “The use of centrifuge tests in the study of arching.” Proc., Physical Modeling in Geotechnics—6th ICPMG’06, Taylor & Francis, London, 1075–1080.
Chen, R. P., Huang, W. Y., and Tseng, C. T. (2011). “Stress redistribution and ground arch development during tunneling.” Tunnelling Underground Space Technol., 26(1), 228–235.
Chevalier, B., Combe, G., and Villard, P. (2012). “Experimental and discrete element modeling studies of the trapdoor problem: Influence of the macro-mechanical frictional parameters.” Acta Geotech., 7(1), 15–39.
Chevalier, B., and Otani, J. (2010). “3-D arching effect in the trap-door problem: A comparison between X-ray CT scanning and DEM analysis.” Proc., GeoFlorida 2010, ASCE, Reston, VA, 570–579.
Costa, Y. D., Zornberg, J. G., Bueno, B. S., and Costa, C. L. (2009). “Failure mechanism in sand over a deep active trapdoor.” J. Geotech. Geoenviron. Eng., 1741–1753.
Engesser, F. (1882). Über den Erddruck gegen innere Stützwände, Vol. 16, Deutsche Bauzeitung, Berlin, 91–93.
Evans, C. H. (1983). “An examination of arching in granular soils.” M.S. thesis, Massachusetts Institute of Technology, Cambridge, MA.
Falaknaz, N., Aubertin, M., and Li, L. (2015). “Numerical investigation of the geomechanical response of adjacent backfilled stopes.” Can. Geotech. J., 52(10), 1507–1525.
Getzler, Z., Gellert, M., and Eitan, R. (1970). “Analysis of arching pressures in ideal elastic soil.” J. Soil Mech. Found. Div., 96(SM4), 1357–1372.
Iglesia, G. R., Einstein, H. H., and Whitman, R. V. (2014). “Investigation of soil arching with centrifuge tests.” J. Geotech. Geoenviron. Eng., 1–13.
Koutsabeloulis, N. C., and Griffiths, D. V. (1989). “Numerical modeling of the trapdoor problem.” Géotechnique, 39(1), 77–89.
Ladanyi, B., and Hoyaux, B. (1969). “A study of the trap door problem in a granular mass.” Can. Geotech. J., 6(1), 1–15.
Li, L., Aubertin, M., and Belem, T. (2005). “Formulation of a three dimensional analytical solution to evaluate stresses in backfilled vertical narrow openings.” Can. Geotech. J., 42(6), 1705–1717.
Li, L., Dube, J., and Aubertin, M. (2013). “An extension of Marston’s solution for the stresses in backfilled trenches with inclined walls.” Can. Geotech. J., 42(6), 1705–1717.
Marston, A. (1930). “The theory of external loads on closed conduits in the light of the latest experiments.” Bulletin 96, Iowa State Univ. Engineering Experiment Station, Ames, IA.
McNulty, J. W. (1965). “An experimental study of arching in sand.” Ph.D. thesis, Univ. of Illinois at Urbana–Champaign, Champaign, IL.
Moradi, G., and Abbasnejad, A. (2013). “The state of the art report on arching effect.” J. Civil Eng. Res., 3(5), 148–161.
Nielson, F. D. (1966). “Soil structure arching analysis of buried flexible structures.” Ph.D. thesis, Univ. of Arizona, Tucson, AZ.
Nunes, M. A., and Meguid, M. A. (2009). “A study on the effects of overlying soil strata on the stresses developing in a tunnel.” Tunnelling Underground Space Technol., 24(6), 716–722.
Ono, K., and Yamada, M. (1993). “Analysis of arching action in granular mass.” Géotechnique, 43(1), 105–120.
Paikowsky, S. G., DiRocco, K. J., and Xi, F. (1993). “Interparticle contact force analysis and measurements using photoelastic techniques.” Proc., 2nd Int. Conf. on Discrete Element Methods, Massachusetts Institute of Technology, Cambridge, MA, 449–461.
Paikowsky, S. G., and Hsienjen, S. T. (2002). “Experimental examination of the arching effect mechanism on the micro level.” Proc., 3rd Int. DEM Conf., ASCE, Reston, VA, 222–228.
Pardo, G. S., and Sáez, E. (2014). “Experimental and numerical study of arching soil effect in coarse sand.” Comput. Geotech., 57, 75–84.
Pirapakaran, K., and Sivakugan, N. (2007). “Arching within hydraulic fill stopes.” Geotech. Geol. Eng., 25(1), 25–35.
Sakaguchi, H., and Ozaki, E. (1992). “Analysis of the formation of arches plugging the flow of granular materials.” Proc., 2nd Int. Conf. on Discrete Element Method, Massachusetts Institute of Technology, Cambridge, MA, 153–163.
Santichaianaint, K. (2002). “Centrifuge modelling and analysis of active trapdoor in sand.” Ph.D. thesis, Univ. of Colorado at Boulder, Boulder, CO.
Sardrekarimi, J., and Abbasnejad, A. R. (2010). “Arching effect in fine sand due to base yielding.” Can. Geotech. J., 47(3), 366–374.
Singh, S., Shukla, S. K., and Sivakugan, N. (2011). “Arching in inclined and vertical mine stopes.” Geotech. Geol. Eng., 29(5), 685–693.
Sivakugan, N., Widisinghe, S., and Wang, V. Z. (2014). “Vertical stress determination within backfilled mine stopes.” Int. J. Geomech., 1–5.
Spangler, M. G., and Handy, R. L. (1982). “Loads on underground conduits.” Soil engineering, Harper and Row, New York, 727–763.
Terzaghi, K. (1943). Theoretical soil mechanics, John Wiley & Sons, New York.
Vardoulakis, I., Graf, B., and Gudehus, G. (1981). “Trap-door problem with dry sand: A statical approach based upon model test kinematics.” Int. J. Numer. Anal. Methods Geomech., 5(1), 57–78.
Völlmy, A. (1937). “Eingebettete rohre.” Thesis/dissertation, Mitt. Inst. Baustatik, Eidgen. Tech. Hochschule, Zurich, Switzerland, Mitt. No. 9 (in German).
Vorster, T. E. B. (2005). “The effects of tunnelling on buried pipes.” Ph.D. thesis, Cambridge Univ., Cambridge, U.K.
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© 2016 American Society of Civil Engineers.
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Received: Feb 1, 2016
Accepted: Sep 21, 2016
Published online: Nov 28, 2016
Discussion open until: Apr 28, 2017
Published in print: Jul 1, 2017
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