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Correlating the Unconfined Compressive Strength of Rock with the Compressional Wave Velocity Effective Porosity and Schmidt Hammer Rebound Number Using Artificial Neural Networks

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Abstract

In this research, a series of artificial neural networks for the prediction of the unconfined compressive strength of rock were trained and developed. A data and site independent database were compiled from 367 datasets reported in the literature, using the Schmidt hammer number Rn, compressional wave velocity Vp, and effective porosity ne as input parameters. Different types of Schmidt hammer numbers were consolidated using the artificial neural network developed by the authors, which correlates N with L-type Schmidt hammer numbers with less than ± 20% deviation from the experimental data for 97.27% of the specimens. Of the various soft computing models developed in this study (ANN-LM, ANN-PSO, and ANN-ICA), the highest accuracy was obtained with the ANN-ICA, which predicts the unconfined compressive strength of various rock types and formation methods with less than ± 20% deviation from the experimental data for 86.36%. The closed-form equation of the ANN-ICA model is incorporated into a graphical user interface, which is made available as supplementary material, allowing the verification of the reported results by different researchers.

Highlights

  • The unconfined compressive strength of rock was correlated with the compressional wave velocity, effective porosity and Schmidt hammer rebound.

  • A parametric analysis was performed to obtain a suitable architecture for the artificial neural network.

  • Metaheuristic-optimization techniques were applied to optimize the weights and biases of the artificial neural network model.

  • The prediction accuracy of the hybrid machine learning model developed in this research is higher than that currently reported in the literature.

  • The optimum developed artificial neural network is presented as a closed-form analytical equation, which has also been integrated into a graphical user interface and made available as supplementary material

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Data Availability

The raw/processed data required to reproduce these findings will be made available on request.

Abbreviations

ANN(s):

Artificial neural network(s)

BPNN:

Back propagation neural network

CS:

Compressive strength

HTS:

Hyperbolic tangent sigmoid transfer function

Li:

Linear transfer function

LS:

Log-sigmoid transfer function

MAPE:

Mean absolute percentage error

MSE:

Mean square error

\(n_{{\text{e}}}\) :

Effective porosity

\(N_{{{\text{ip}}}}\) :

Number of input parameters

\(N_{{\text{n}}}\) :

Number of hidden layers

\(N_{{{\text{op}}}}\) :

Number of output parameters

\(N_{{{\text{td}}}}\) :

Number of datasets

NRB:

Normalized radial basis transfer function

PLi:

Positive linear transfer function

R:

Pearson correlation coefficient

RB:

Radial basis transfer function

\(R_{{\text{n}}}\) :

Schmidt hammer rebound number

SM:

Soft max transfer function

SSE:

Sum square error

SSL:

Symmetric saturating linear transfer function

TB:

Triangular basis transfer function

UCS:

Unconfined compressive strength

\(V_{{\text{p}}}\) :

Ultrasonic pulse velocity

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Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering and Mechatronics, PHENIKAA University, Yen Nghia, Ha Dong, Hanoi, 12116, Vietnam

    Tien-Thinh Le

  2. PHENIKAA Research and Technology Institute (PRATI), A&A Green Phoenix Group JSC, No.167 Hoang Ngan, Trung Hoa, Cau Giay, Hanoi, 11313, Vietnam

    Tien-Thinh Le

  3. Computational Mechanics Laboratory, School of Pedagogical and Technological Education, Heraklion, 14121, Athens, Greece

    Athanasia D. Skentou, Anna Mamou & Panagiotis G. Asteris

Authors
  1. Tien-Thinh Le
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  2. Athanasia D. Skentou
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  3. Anna Mamou
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  4. Panagiotis G. Asteris
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Contributions

Conceptualization: PGA; methodology: T-TL; software: PGA and T-TL; formal analysis and investigation: ADS and AM; writing—original draft preparation: ADS, AM, PGA and T-TL; writing—review and editing: ADS, AM, PGA and T-TL; supervision: PGA.

Corresponding author

Correspondence to Panagiotis G. Asteris.

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Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

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Informed consent was obtained from all individual participants included in the study.

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Electronic supplementary material

Below is the link to the electronic supplementary material.

603_2022_2992_MOESM1_ESM.xlsx

The Graphical User Interface (GUI) proposed in Section 6 is appended to this manuscript as a supplementary material (Excel file) (XLSX 293 kb)

Appendix

Appendix

1.1 Appendix A Weight and bias matrices for the closed-form prediction equation

$$\left[ I \right] = \left[ {\begin{array}{*{20}r} \hfill {1.661397} & \hfill { - 0.14178} & \hfill { - 1.35389} \\ \hfill { - 0.64739} & \hfill {2.447476} & \hfill { - 1.9973} \\ \hfill {0.713735} & \hfill {5.003011} & \hfill { - 1.92216} \\ \hfill { - 3.5355} & \hfill {0.036428} & \hfill {0.40145} \\ \hfill {0.66559} & \hfill {2.339007} & \hfill { - 2.44231} \\ \hfill { - 0.0997} & \hfill {0.809266} & \hfill {1.566401} \\ \hfill { - 0.89215} & \hfill { - 0.0328} & \hfill {1.812075} \\ \end{array} } \right]$$
(14)
$$\left[ {b_{i} } \right] = \left[ {\begin{array}{*{20}r} \hfill {3.604827} \\ \hfill { - 2.39426} \\ \hfill { - 0.81611} \\ \hfill { - 3.29987} \\ \hfill {3.862143} \\ \hfill { - 0.89642} \\ \hfill {4.053752} \\ \end{array} } \right],\;\left[ {b_{1} } \right] = \left[ {\begin{array}{*{20}r} \hfill { - 1.36574} \\ \hfill {1.573937} \\ \hfill { - 1.5509} \\ \hfill { - 1.86935} \\ \hfill {0.952506} \\ \hfill {1.470129} \\ \hfill { - 0.04467} \\ \hfill { - 1.15646} \\ \hfill { - 0.36454} \\ \hfill {0.358028} \\ \hfill { - 0.58351} \\ \hfill { - 1.31097} \\ \hfill {1.571746} \\ \hfill { - 3.20568} \\ \end{array} } \right],\;\left[ {b_{2} } \right] = \left[ {\begin{array}{*{20}r} \hfill {1.654426} \\ \hfill { - 1.23268} \\ \hfill {1.066539} \\ \hfill { - 0.65889} \\ \hfill { - 4.25272} \\ \hfill {0.791742} \\ \hfill { - 0.56185} \\ \hfill { - 0.56814} \\ \hfill { - 1.20397} \\ \hfill {1.788574} \\ \hfill {0.915082} \\ \end{array} } \right],\;\left[ {b_{o} } \right] = 0.111311$$
(15)
$$\left[ {L_{1} } \right] = \left[ {\begin{array}{*{20}r} \hfill { - 0.8782} & \hfill { - 0.99962} & \hfill {1.499121} & \hfill { - 2.64494} & \hfill {0.361302} & \hfill {2.095666} & \hfill { - 0.98309} \\ \hfill { - 0.69253} & \hfill {1.304393} & \hfill {2.228084} & \hfill { - 0.08536} & \hfill {1.085533} & \hfill { - 2.19959} & \hfill { - 0.39296} \\ \hfill {1.05315} & \hfill { - 0.68625} & \hfill {1.342234} & \hfill { - 0.20784} & \hfill { - 1.4139} & \hfill {0.145148} & \hfill { - 0.83731} \\ \hfill {0.405814} & \hfill {1.081347} & \hfill {0.343433} & \hfill { - 0.54708} & \hfill { - 0.13617} & \hfill { - 1.547} & \hfill {0.666412} \\ \hfill {0.174296} & \hfill { - 0.97901} & \hfill { - 2.13542} & \hfill {0.242037} & \hfill {0.347969} & \hfill {0.284862} & \hfill {0.707297} \\ \hfill { - 1.20541} & \hfill { - 0.39423} & \hfill {1.153361} & \hfill { - 1.47751} & \hfill { - 0.95343} & \hfill {2.165322} & \hfill { - 0.52944} \\ \hfill {1.398231} & \hfill { - 0.47024} & \hfill {1.066001} & \hfill { - 0.02978} & \hfill {0.443553} & \hfill {0.503076} & \hfill { - 0.30821} \\ \hfill { - 2.06296} & \hfill { - 0.8647} & \hfill {1.27019} & \hfill {2.210488} & \hfill { - 0.19995} & \hfill {0.757884} & \hfill { - 2.3009} \\ \hfill { - 0.22636} & \hfill {0.235273} & \hfill {0.628726} & \hfill { - 0.58286} & \hfill {0.469869} & \hfill { - 1.86198} & \hfill { - 1.76095} \\ \hfill {0.073986} & \hfill {1.415642} & \hfill {0.373092} & \hfill { - 3.31245} & \hfill {0.404094} & \hfill { - 0.29845} & \hfill {0.286848} \\ \hfill { - 2.48805} & \hfill {5.286997} & \hfill {0.038757} & \hfill { - 0.60762} & \hfill {0.032864} & \hfill { - 1.9888} & \hfill { - 2.47023} \\ \hfill { - 0.19311} & \hfill { - 0.84632} & \hfill {1.323591} & \hfill { - 3.94162} & \hfill {1.259942} & \hfill {1.002719} & \hfill {0.560537} \\ \hfill { - 0.19565} & \hfill {0.84086} & \hfill { - 1.41437} & \hfill { - 1.41863} & \hfill {0.80495} & \hfill {0.358905} & \hfill { - 1.72667} \\ \hfill { - 4.40114} & \hfill {3.63015} & \hfill {3.089973} & \hfill {1.425097} & \hfill { - 2.61846} & \hfill {3.559342} & \hfill { - 2.16638} \\ \end{array} } \right]$$
(16)
$$\left[ {L_{2} } \right]^{T} = \left[ {\begin{array}{*{20}r} \hfill { - 0.28694} & \hfill {5.254582} & \hfill {0.076577} & \hfill {0.290431} & \hfill {2.044863} & \hfill { - 0.83704} & \hfill {0.1835} & \hfill { - 0.45579} & \hfill { - 0.21553} & \hfill {0.327125} & \hfill {1.220407} \\ \hfill {1.03021} & \hfill {0.295219} & \hfill {1.352103} & \hfill { - 0.05127} & \hfill { - 3.12788} & \hfill { - 2.60804} & \hfill { - 0.17164} & \hfill { - 1.35779} & \hfill { - 1.58765} & \hfill { - 2.1358} & \hfill { - 0.78562} \\ \hfill {0.337542} & \hfill { - 4.35133} & \hfill {0.837567} & \hfill { - 0.42361} & \hfill {0.637845} & \hfill {2.217968} & \hfill { - 0.28846} & \hfill {0.187174} & \hfill { - 0.21943} & \hfill {0.547103} & \hfill { - 1.1829} \\ \hfill {0.146508} & \hfill {0.486228} & \hfill {0.808219} & \hfill { - 0.89259} & \hfill {1.781647} & \hfill {1.230272} & \hfill { - 1.12393} & \hfill {0.004264} & \hfill {0.097414} & \hfill {1.253491} & \hfill {5.276541} \\ \hfill { - 0.58895} & \hfill {1.460256} & \hfill { - 0.02959} & \hfill { - 1.67226} & \hfill {0.87549} & \hfill {1.057835} & \hfill { - 0.08871} & \hfill {0.091339} & \hfill {0.153623} & \hfill {1.157143} & \hfill { - 3.36874} \\ \hfill { - 0.12108} & \hfill {0.393446} & \hfill { - 0.89965} & \hfill {0.295674} & \hfill {0.084553} & \hfill { - 0.44635} & \hfill { - 0.80773} & \hfill {1.194954} & \hfill { - 1.53646} & \hfill {0.028673} & \hfill { - 2.78681} \\ \hfill { - 0.05375} & \hfill { - 0.11875} & \hfill { - 0.21255} & \hfill { - 1.24414} & \hfill {0.699623} & \hfill { - 1.5526} & \hfill {0.427779} & \hfill {1.340979} & \hfill {0.202754} & \hfill { - 0.33351} & \hfill { - 2.08414} \\ \hfill {0.396514} & \hfill { - 0.36695} & \hfill { - 0.59537} & \hfill {0.835505} & \hfill {0.971163} & \hfill { - 1.99312} & \hfill {1.457704} & \hfill {0.415827} & \hfill { - 0.21418} & \hfill { - 0.04104} & \hfill { - 1.26643} \\ \hfill { - 0.09565} & \hfill { - 0.70107} & \hfill { - 0.3862} & \hfill { - 0.56253} & \hfill { - 0.23784} & \hfill {0.40787} & \hfill {0.603849} & \hfill { - 0.6314} & \hfill {0.848649} & \hfill {0.187479} & \hfill { - 3.58737} \\ \hfill {1.385888} & \hfill { - 2.36476} & \hfill { - 0.28875} & \hfill { - 2.45499} & \hfill {1.647075} & \hfill {0.413473} & \hfill { - 0.47977} & \hfill {0.912148} & \hfill { - 0.7176} & \hfill {0.649352} & \hfill { - 0.36682} \\ \hfill {0.711066} & \hfill { - 0.84289} & \hfill {0.339678} & \hfill { - 0.0194} & \hfill {1.119607} & \hfill { - 2.49373} & \hfill {1.728109} & \hfill { - 0.33904} & \hfill {0.239314} & \hfill { - 1.19006} & \hfill {4.463031} \\ \hfill { - 0.10513} & \hfill {1.344628} & \hfill { - 0.65572} & \hfill { - 0.32968} & \hfill {4.40827} & \hfill { - 0.90321} & \hfill {0.676455} & \hfill {0.654538} & \hfill { - 0.79358} & \hfill {1.892678} & \hfill { - 4.03022} \\ \hfill { - 1.55037} & \hfill {3.668548} & \hfill { - 0.43446} & \hfill { - 0.08346} & \hfill { - 1.19047} & \hfill { - 0.44219} & \hfill {0.462182} & \hfill { - 0.01463} & \hfill { - 0.23651} & \hfill {0.628005} & \hfill { - 3.95906} \\ \hfill { - 0.57508} & \hfill {5.171249} & \hfill {0.267539} & \hfill {0.206091} & \hfill {0.175508} & \hfill { - 0.37796} & \hfill {0.975063} & \hfill { - 0.0881} & \hfill { - 0.5813} & \hfill { - 0.30869} & \hfill {3.272834} \\ \end{array} } \right]$$
(17)
$$\left[ {L_{3} } \right]^{T} = \left[ {\begin{array}{*{20}r} \hfill { - 1.67299} \\ \hfill {0.327106} \\ \hfill {1.503866} \\ \hfill {0.97786} \\ \hfill { - 0.36676} \\ \hfill {0.557426} \\ \hfill { - 0.51132} \\ \hfill {1.381175} \\ \hfill { - 0.9985} \\ \hfill { - 1.86068} \\ \hfill { - 0.46603} \\ \end{array} } \right]$$
(18)

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Le, TT., Skentou, A.D., Mamou, A. et al. Correlating the Unconfined Compressive Strength of Rock with the Compressional Wave Velocity Effective Porosity and Schmidt Hammer Rebound Number Using Artificial Neural Networks. Rock Mech Rock Eng 55, 6805–6840 (2022). https://doi.org/10.1007/s00603-022-02992-8

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  • DOI: https://doi.org/10.1007/s00603-022-02992-8

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