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