Application of numerical modeling and genetic programming to estimate rock mass modulus of deformation Ebrahim Ghotbi Ravandi a,⇑ , Reza Rahmannejad b , Amir Ehsan Feili Monfared c , Esmaeil Ghotbi Ravandi d a Mineral Industries Research Center, Shahid Bahonar University of Kerman, Kerman 76175-133, Iran b Mining Engineering Department, Shahid Bahonar University of Kerman, Kerman 76175-133, Iran c Chemical Engineering Department, Shahid Bahonar University of Kerman, Kerman 76175-133, Iran d Civil Engineering Department, Shahid Bahonar University of Kerman, Kerman 76175-133, Iran article info Article history: Received 20 December 2012 Received in revised form 15 January 2013 Accepted 2 February 2013 Available online xxxx Keywords: Modulus of deformation (E m ) Displacement Numerical modeling Genetic programming (GP) Back analysis abstract Estimation of the rock mass modulus of deformation (E m ) is one of the most important design parameters in designing many structures in and on rock. This parameter can be obtained by in situ tests, empirical relations between deformation modulus and rock mass classification, and estimating from laboratory tests results. In this paper, a back analysis calculation is performed to present an equation for estimation of the rock mass modulus of deformation using genetic programming (GP) and numerical modeling. A database of 40,960 datasets, including vertical stress (r z ), horizontal to vertical stresses ratio (k), Pois- son’s ratio (m), radius of circular tunnel (r) and wall displacement of circular tunnel on the horizontal diameter (d) for input parameters and modulus of deformation for output, was established. The selected parameters are easy to determine and rock mass modulus of deformation can be obtained from instru- mentation data of any size circular galleries. The resulting RMSE of 0.86 and correlation coefficient of 97% of the proposed equation demonstrated the capability of the computer program (CP) generated by GP. Ó 2013 Published by Elsevier B.V. on behalf of China University of Mining & Technology. 1. Introduction Rock mass modulus of deformation (E m ) is one of the most important design parameters in a rock engineering practice and is used in designing many structures in and on rock. It refers to the ratio of stress to the corresponding strain during loading of a rock mass, including elastic and inelastic behavior. Several meth- ods are available to estimate rock mass modulus of deformation including in situ tests, empirical relations between deformation modulus and rock mass classification, geophysical (usually seis- mic) methods and estimating from laboratory tests results. Despite their great features, the stated methods have some restrictions in the application. In situ tests are time-consuming and expensive, and require large scale galleries and difficult procedures. In situ tests are usually ignored in small sized projects but since labora- tory tests on small specimens cannot predict the deformability of rock mass, in situ tests which provide direct information on defor- mability are preferred for site investigation studies. Empirical rela- tions are indirect methods that relate the rock mass deformation modulus to index properties such as RQD or to rock mass classifi- cations systems such as RMR and Q. These empirical relations were developed by different authors due to difficulties encountered dur- ing the in situ tests. As shown in Table 1, the most widely known empirical equations were studied by Nicholson et al. [1–5]. Although the empirical equations for the indirect estimation of the deformation modulus of rock mass are simple and cost-effec- tive, the equations include some uncertainties relating to the lim- ited data availability, variability of rock type and the heterogeneous nature of the rock masses [6]. Another method for obtaining rock mass deformation modulus is back analysis. Back analysis is an indirect technique that has been used to determine the mechanical properties of rock masses by using field measure- ments of displacements. Since displacements of rock masses in- duced by excavation can be measured easily and reliably, the displacement based on back analysis techniques have been always a research topic since 1970s [7–12]. Numerical modeling has been used to investigate a variety of problems in geotechnics and geoengineering. If extensive geotech- nical and geological data are available, then comprehensive predic- tions of deformations and stability can be made by numerical stress analyses. If not, the model can still be used to perform parametric studies, providing insight into the possible range of responses of a system, given the likely ranges for the various parameters. This understanding of the key parameters can then help set priorities for site investigation and material testing, which 2095-2686/$ - see front matter Ó 2013 Published by Elsevier B.V. on behalf of China University of Mining & Technology. http://dx.doi.org/10.1016/j.ijmst.2013.08.018 ⇑ Corresponding author. Tel.: +98 9132979928. E-mail address: Ghotbi_Ebrahim@yahoo.com (E. Ghotbi Ravandi). International Journal of Mining Science and Technology xxx (2013) xxx–xxx Contents lists available at ScienceDirect International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst Please cite this article in press as: Ghotbi Ravandi E et al. Application of numerical modeling and genetic programming to estimate rock mass modulus of deformation. Int J Min Sci Technol (2013), http://dx.doi.org/10.1016/j.ijmst.2013.08.018