Document Type : Research Paper

**Authors**

Department of Civil Engineering, University of Shahrood, Shahrood, Iran

**Abstract**

One of the conventional methods for temporary support of tunnels is to use steel sets with shotcrete. The nature of a temporary support system demands a quick installation of its structures. As a result, the spacing between steel sets is not a fixed amount and it can be considered as a random variable. Hence, in the reliability analysis of these types of structures, the selection of an appropriate probability distribution function of spacing of steel sets is essential. In the present paper, the distances between steel sets are collected from an under-construction tunnel and the collected data is used to suggest a proper Probability Distribution Function (PDF) for the spacing of steel sets. The tunnel has two different excavation sections. In this regard, different distribution functions were investigated and three common tests of goodness of fit were used for evaluation of each function for each excavation section. Results from all three methods indicate that the Wakeby distribution function can be suggested as the proper PDF for spacing between the steel sets. It is also noted that, although the probability distribution function for two different tunnel sections is the same, the parameters of PDF for the individual sections are different from each other.

**Keywords**

[1] Madani, H. (2005). Tunneling (Volume 4): Design and construction of support systems. ISBN: 964-463-135-8.

[2] Biron, C., Ariogulu, E., & Lucas, J. R. (1983). Design of supports in mines, Wiley, ISBN-10: 0471867268. ISBN-13: 978-463-135-8. 0471867265.

[3] Hjálmarsson, E.H. (2011). Tunnel Support, Use of lattice girders in sedimentary rock, MSc Thesis, University of Iceland.

[4] Carranza-Torres, C., & Diederichs, M. (2009). Mechanical analysis of circular liners with particular reference to composite supports. For example, liners consisting of shotcrete and steel sets. Tunnelling and Underground Space Technology, 24, 506–532. DOI: 10.1016/j.tust. 2009.02.001

[5] Basaligheh, F. (2003). The advantages of using lattice girder with shotcrete instead of steel sets for temporary support of tunnels (in Persian). Proceedings of the Sixth International Conference on Civil Engineering, 5, 265-270.

[6] Basaligheh, F. (2006). Evaluation of two methods for the temporary support of the tunnels from economic point of view (in Persian). Journal of Science and technology, 12, 11-17.

[7] Basaligheh, F., & Keyhani, A. (2013). The advantages of using the method of “equivalent section” in design of shotcrete and steel sets for temporary support of tunnels (in Persian). Proceedings of Seventh National Congress on Civil Engineering. Shahid Nikbakht Faculty of Engineering. Zahedan-Iran.

[8] Basaligheh, F., & Keyhani, A. (2013). New approaches for modelling of two common methods of temporary support for tunnels. Proceedings of Tenth Iranian Tunnelling Conference.

[2] Biron, C., Ariogulu, E., & Lucas, J. R. (1983). Design of supports in mines, Wiley, ISBN-10: 0471867268. ISBN-13: 978-463-135-8. 0471867265.

[3] Hjálmarsson, E.H. (2011). Tunnel Support, Use of lattice girders in sedimentary rock, MSc Thesis, University of Iceland.

[4] Carranza-Torres, C., & Diederichs, M. (2009). Mechanical analysis of circular liners with particular reference to composite supports. For example, liners consisting of shotcrete and steel sets. Tunnelling and Underground Space Technology, 24, 506–532. DOI: 10.1016/j.tust. 2009.02.001

[5] Basaligheh, F. (2003). The advantages of using lattice girder with shotcrete instead of steel sets for temporary support of tunnels (in Persian). Proceedings of the Sixth International Conference on Civil Engineering, 5, 265-270.

[6] Basaligheh, F. (2006). Evaluation of two methods for the temporary support of the tunnels from economic point of view (in Persian). Journal of Science and technology, 12, 11-17.

[7] Basaligheh, F., & Keyhani, A. (2013). The advantages of using the method of “equivalent section” in design of shotcrete and steel sets for temporary support of tunnels (in Persian). Proceedings of Seventh National Congress on Civil Engineering. Shahid Nikbakht Faculty of Engineering. Zahedan-Iran.

[8] Basaligheh, F., & Keyhani, A. (2013). New approaches for modelling of two common methods of temporary support for tunnels. Proceedings of Tenth Iranian Tunnelling Conference.

[9] Wong, L.N.Y., Fang, Q., & Zhang, D. (2013). Mechanical analysis of circular tunnels supported by steel sets embedded in primary linings. Tunnelling and Underground Space Technology, 37, 80–88. DOI: 10.1016/j.tust.2013.03.011

[10] Schwingenschloegl, R., & Lehmann, C. (2009). Swelling rock behaviour in a tunnel: NATM-support vs. Q-support – A comparison. Tunnelling and Underground Space Technology, 24, 356–362. DOI: 10.1016/j.tust.2008.08.007

[11] Nowak, A. a. (2000). Reliability of structures. McGraw Hill, International Edition. ISBN: 0-07-116354-9.

[12] Rezaee-Pajand, H. (2002). Application of statistics and probability for water reservoirs. Sokhan-Gostar Press. ISBN: 964-6932-35-5.

[13] Sanchidrian, J, O. A. (2014). Size distribution functions for rock fragments. International Journal of Rock Mechanics & Mining Sciences, 71, 381-394. DOI:10.1016/j.ijrmms. 2014.08.007.

[14] Milford, R. (1987). Annual maximum wind speeds from parent distribution functions. Journal of wind Engineering and Industrial Aerodynamics, 25, pp. 163-178.

[15] Ghamishon, R., & Malekian,A. (2011). Determination of the most suitable statistical distribution functions for regional floods (Case study of southwest Kerman Province). Proceedings of Sixth National Watershed Sciences and Engineering Conference and forth national congress on soil erosion and sediment, Tarbiat-Modares University.

[16] Meftah-Halghi, M., Zangale, M. E., & Aghili, R. (2012). A Comparison of the most suitable statistical distribution functions for maximum daily flow rate and rainfall (Case study: Hydrometry station of Gonbad Kavos)., Fifth National Watershed Sciences and Engineering Conference , Iran. University of Agricultural Sciences of Gorgan. COI: WATERSHED05_224.http://www.civilica.com/Paper-WATERSHED05-WATERSHED05_224.html

[17] Cochran, W.G. (1952). The χ^2 test of goodness of fit. Institute of Mathematical Statistics, Vol 23, No. 3, pp. 315-345. http://www.jstor.org/stable/2236678

[18] Anderson, T, A. (1952). Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Ann. Math. Statist, 23, pp. 193-212.

[19] Darbandi, S., Mahmudi, S., & Ebrahimi, S., & Shoeybi-Nobarian, M. R. (2012). Introduction and application of Anderson-Darling in river engineering of East Azerbaijan Province. Fifth National Conference on Watersheds and management of soil and water resources.

[10] Schwingenschloegl, R., & Lehmann, C. (2009). Swelling rock behaviour in a tunnel: NATM-support vs. Q-support – A comparison. Tunnelling and Underground Space Technology, 24, 356–362. DOI: 10.1016/j.tust.2008.08.007

[11] Nowak, A. a. (2000). Reliability of structures. McGraw Hill, International Edition. ISBN: 0-07-116354-9.

[12] Rezaee-Pajand, H. (2002). Application of statistics and probability for water reservoirs. Sokhan-Gostar Press. ISBN: 964-6932-35-5.

[13] Sanchidrian, J, O. A. (2014). Size distribution functions for rock fragments. International Journal of Rock Mechanics & Mining Sciences, 71, 381-394. DOI:10.1016/j.ijrmms. 2014.08.007.

[14] Milford, R. (1987). Annual maximum wind speeds from parent distribution functions. Journal of wind Engineering and Industrial Aerodynamics, 25, pp. 163-178.

[15] Ghamishon, R., & Malekian,A. (2011). Determination of the most suitable statistical distribution functions for regional floods (Case study of southwest Kerman Province). Proceedings of Sixth National Watershed Sciences and Engineering Conference and forth national congress on soil erosion and sediment, Tarbiat-Modares University.

[16] Meftah-Halghi, M., Zangale, M. E., & Aghili, R. (2012). A Comparison of the most suitable statistical distribution functions for maximum daily flow rate and rainfall (Case study: Hydrometry station of Gonbad Kavos)., Fifth National Watershed Sciences and Engineering Conference , Iran. University of Agricultural Sciences of Gorgan. COI: WATERSHED05_224.http://www.civilica.com/Paper-WATERSHED05-WATERSHED05_224.html

[17] Cochran, W.G. (1952). The χ^2 test of goodness of fit. Institute of Mathematical Statistics, Vol 23, No. 3, pp. 315-345. http://www.jstor.org/stable/2236678

[18] Anderson, T, A. (1952). Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Ann. Math. Statist, 23, pp. 193-212.

[19] Darbandi, S., Mahmudi, S., & Ebrahimi, S., & Shoeybi-Nobarian, M. R. (2012). Introduction and application of Anderson-Darling in river engineering of East Azerbaijan Province. Fifth National Conference on Watersheds and management of soil and water resources.

Summer and Autumn 2015

Pages 187-203