Soft computing based closed form equations correlating L and N-type Schmidt hammer rebound numbers of rocks

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Abstract

This paper reports the results of soft computing-based models correlating L and N-type Schmidt hammer rebound numbers of rock. A data-independent database was compiled from available measurements reported in the literature, which was used to train and develop back propagating neural networks, genetic programming and least square method models for the prediction of L-type Schmidt hammer rebound numbers. The results show that the highest predictive accuracy was obtained for the neural network model, which predicts the L type Schmidt hammer rebound number, with less than ±20% deviation from the experimental data for 97.27% of the samples. The optimum neural network is presented as a closed form equation and is also incorporated into an Excel-based graphical user interface, which directly calculates the Rn(L) number for any input Rn(N) = 12.40–75.97 and which is made available as supplementary material.

Introduction

The Schmidt hammer rebound number is a hardness index, which is correlated with the unconfined compressive strength of rock. The laboratory testing procedure involves pressing a spring-loaded piston perpendicular to a flat specimen surface and the rebound height of the piston is used as an indication of the rock’s hardness. Although the testing procedure is outlined in detail by ASTM-D5873 [12] and ISRM (2007) [48] standards, an ambiguity as to the recommended hammer type and therefore different applied impact energies, varying water contents of the tested specimen and different data reduction techniques may render measurements made by different researchers not directly comparable. The effect of varying water content is generally eradicated as specimen are generally dried to constant mass prior to testing and any ambiguities associated with different data reduction methodologies are likely to be less significant than those associated with the application of different hammer types and therefore different impact energies. Although a number of first order linear expressions have been proposed in the literature which correlate L with N-type Schmidt hammer numbers, these were validated on a limited number of data (less than 65 data per linear equations) and may therefore have been prone to over-fitting and unrealistic parameter estimation [18], [17], [40], [45] (Del Potro and Hürlimann, 2008). The ability to correlate N- with L-type measurements is a critical step in consolidating the significant amount of different Schmidt hammer rebound data reported in the literature, which will enable the compilation of site independent - unbiased databases which may be used to calibrate advanced data analysis models for the prediction of the unconfined compressive strength of rock [71], [80], [50], [86], [38], [74], [56], [84], [37], [62], [63], [6], [7], [46], [55], [81], [1], [85], [36].

Section snippets

Research Significance

Correlating N- with L-type Schmidt hammer measurements is a critical step in consolidating the significant amount of different hammer rebound data reported in the literature, which use the hardness index as an input parameter to predict the unconfined compressive strength of rock. Although a number of linear expressions have been proposed in the literature which correlate L with N-type Schmidt hammer measurements, these were validated on a limited number of data (less than 65 data per linear

Operating principle

The operating principle of the Schmidt hammer involves pressing a spring-loaded piston perpendicular to a flat specimen surface and the rebound height of the piston is used as an indication of the rock’s hardness. The ratio of the maximum stretch of the spring at rebound x2 to the maximum stretch of the spring when fully loaded x1 is termed the Schmidt hammer rebound number RN = x2/x1 %. Depending on the applied impact energy two Schmidt hammer types are available, the N-type (2.207 Nm impact

Computational predictive models

This section presents the basic principles and constitutive models underpinning the computational predictive methods and techniques used in this research. A brief review of the basic principles of the Least Squares Method (LSM), Artificial Neural Networks (ANNs), with a focus on back-propagation neural networks (BPNNs), and Genetic Programming (GP) techniques will be presented.

Least square methods models

Using the least square error method presented in the previous section the most suitable n-order polynomials were determined, for which the closets fit with the 183 experimental datasets was achieved. To this end the coefficients of first, second and third order polynomial analytical solutions for which the closets fit with the experimental data was achieved was determined. The accuracy of the proposed analytical solutions increased with increasing polynomial order (Table 4). The predictive

Conclusions

The aim of this paper was to consolidate the N and L type Schmidt hammer numbers of rock reported in the literature into a data independent database comprising 183 datasets, which was used to train and develop advanced data analysis models correlating N with L-type measurements. The data analysis included back propagating neural networks (BPNN), genetic programming (GP) and least square method (LSM). The following main conclusions can be drawn:

  • The BPNN1-7-1 model, GP model and the 3d order

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank Dr. Rodrigo del Potro and Prof. Marcel Hürlimann for providing part of the data accompanying the publication entitled “A Comparison of Different Indirect Techniques to Evaluate Volcanic Intact Rock Strength”.

References (86)

  • O. Katz et al.

    Evaluation of mechanical rock properties using a Schmidt Hammer

    Int J Rock Mech Min Sci

    (2000)
  • S. Kahraman

    Evaluation of simple methods for assessing the uniaxial compressive strength of rock

    Int J Rock Mech Min Sci

    (2001)
  • M. Milovančević et al.

    Vibration analyzing in horizontal pumping aggregate by soft computing

    Measurement

    (2018)
  • E. Momeni et al.

    Prediction of uniaxial compressive strength of rock samples using hybrid particle swarm optimization-based artificial neural networks

    Measurement

    (2015)
  • I.T. Ng et al.

    Predictive model for uniaxial compressive strength for Grade III granitic rocks from Macao

    Eng Geol

    (2015)
  • A. Özbek

    Variation of Schmidt hammer values with imbrication direction in clastic sedimentary rocks

    Int J Rock Mech Min Sci

    (2009)
  • D. Petković et al.

    Adaptive neuro-fuzzy approach for wind turbine power coefficient estimation

    Renew Sustain Energy Rev

    (2013)
  • D. Petković et al.

    Wind farm efficiency by adaptive neuro-fuzzy strategy

    Int J Electr Power Energy Syst

    (2016)
  • D. Petković

    Prediction of laser welding quality by computational intelligence approaches

    Optik

    (2017)
  • P. Samui

    Support vector machine applied to settlement of shallow foundations on cohesionless soils

    Comput Geotech

    (2008)
  • P. Samui et al.

    Utilization of a least square support vector machine (LSSVM) for slope stability analysis

    Scientia Iranica

    (2011)
  • F.I. Shalabi et al.

    Estimation of rock engineering properties using hardness tests

    Eng Geol

    (2007)
  • S. Shamshirband et al.

    Support vector regression methodology for wind turbine reaction torque prediction with power-split hydrostatic continuous variable transmission

    Energy

    (2014)
  • R.W. Poole et al.

    Consistency and repeatability of Schmidt hammer rebound data during field testing

    Int J Rock Mech Min Sci Geomech Abstr

    (1980)
  • A. Tuğrul et al.

    Correlation of mineralogical and textural characteristics with engineering properties of selected granitic rocks from Turkey

    Eng Geol

    (1999)
  • E. Yaşar et al.

    Estimation of rock physicomechanical properties using hardness methods

    Eng. Geol.

    (2004)
  • I. Yılmaz et al.

    Correlation of Schmidt hardness with unconfined compressive strength and Young's modulus in gypsum from Sivas (Turkey)

    Eng Geol

    (2002)
  • G. Aggistalis et al.

    Correlating uniaxial compressive strength with Schmidt hardness, point load index, Young’s modulus, and mineralogy of gabbros and basalts (Northern Greece)

    Bulletin of the International Association of Engineering Geology-Bulletin de l'Association Internationale de Géologie de l'Ingénieur

    (1996)
  • M. Apostolopoulou et al.

    Prediction of compressive strength of mortars using artificial neural networks

    1st international conference TMM_CH, transdisciplinary multispectral modelling and cooperation for the preservation of cultural heritage, Athens, Greece, 10-13 October

    (2018)
  • D.J. Armaghani et al.

    Prediction of the uniaxial compressive strength of sandstone using various modeling techniques

    Int J Rock Mech Min Sci

    (2016)
  • D.J. Armaghani et al.

    Application of several non-linear prediction tools for estimating uniaxial compressive strength of granitic rocks and comparison of their performances

    Eng Comput

    (2016)
  • D.J. Armaghani et al.

    Prediction of the strength and elasticity modulus of granite through an expert artificial neural network

    Arabian J Geosci

    (2016)
  • D.J. Armaghani et al.

    A comparative study of ANN and ANFIS models for the prediction of cement-based mortar materials compressive strength

    Neural Comput Appl

    (2020)
  • F. Arıkan et al.

    Characterization of weathered acidic volcanic rocks and a weathering classification based on a rating system

    Bull Eng Geol Environ

    (2007)
  • ASTM. Standard test method for determination of rock hardness by rebound hammer method. Designation D5873;...
  • P.G. Asteris et al.

    Neural network approximation of the masonry failure under biaxial compressive stress

  • P.G. Asteris et al.

    Anisotropic masonry failure criterion using artificial neural networks

    Neural Comput Appl

    (2016)
  • P.G. Asteris et al.

    Self-compacting concrete strength prediction using surrogate models

    Neural Comput Appl

    (2017)
  • P.G. Asteris et al.

    Concrete compressive strength using artificial neural networks

    Neural Comput Appl

    (2019)
  • C. Ayday et al.

    Correlations between L and N-type Schmidt hammer rebound values obtained during field testing

  • A. Azimian

    Application of statistical methods for predicting uniaxial compressive strength of limestone rocks using nondestructive tests

    Acta Geotech

    (2017)
  • N. Barton et al.

    The shear strength of rock joints in theory and practice

    Rock mechanics

    (1977)
  • A. Basu et al.

    Evaluation of rock mechanical behaviors under uniaxial compression with reference to assessed weathering grades

    Rock Mech Rock Eng

    (2009)
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