Introduction

The point load test has often been reported as an indirect measure of the compressive or tensile strength of rock [1,2,3,4]. It is commonly used in practice due to the ease of testing, simplicity of specimen preparation and possible field application. ISRM [5] suggested that the ratio between uniaxial compressive strength (UCS) and point load index (Is(50)) varies between 20 and 25, although researchers have found various ratio which is depend on rock types.

There are various studies in the literature proposing relationship between Is(50) and UCS on different rock types but not on conglomerates individually: Oztruk and Altinpinar [6], Ferentinou and Fakir [7], Basu et al. [8], Akram and Bakar [9], Quane and Russel [10], Kahraman [11], Fener et al. [12], Basu and Aydin [13], Agustawijaia [14], Cobanoglu and Celik [15], Sabatakakis et al. [16], Kahraman and Gunaydin [17]. Table 1 lists some of the equations correlating the UCS to Is(50). As shown in Table 1, the suggested ratios between UCS and Is indicate a very large range. While some equations conform to y = ax form, the others conform to y = ax + b form. Also, Grasso et al. [18] and Tsiambaos and Sabatakakis [19] derived an equation conforming to y = axb form and Quane and Russel [10] suggested an equation conforming to y = ax2 + bx form for weak rocks.

Table 1 Equations correlating the UCS to the point load index
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Although conglomeratic formations are one of the most common Neogen deposit in Iran, there is little information extant in the literature on the geomechanical characteristics of weak conglomerate rocks. In addition, estimating the geomechanical properties of conglomeratic rocks need high quality samples that can reflect either textural and cement properties. The necessary core samples cannot always be obtained from weak and block-in-matrix conglomeratic rock. For this reason, the predictive models can be employed for the indirect estimation of mechanical parameters. The main objective of this study is to evaluate the correlation between uniaxial and tensile strength with point load index. For this purpose great numbers of specimens which were collected from different formations where major engineering project such as dams, road and tunnel are under construction were tested to evaluate the correlations between the UCS test result, Brazilian test (BTS) result and the corresponding test results of Is(50). The data was analyzed statistically and also by ANN to determine the degree of correlation and the variability of result.

Experimental studies

The conglomeratic rocks in this study are parts of thick synorogenic molasses sequence of upper Pliocene to Pleistocene age, the Bakhtiari and Hezardareh Formations. These conglomerates are composed of the argillaceous-calcareous matrix with different degree of cementation and all three different types of rocks grains (tuff, limestone, marl …) ranging from few millimeters to about a meter. On the basis of thin-section studies carried out in the present study on the samples, conglomerates contained sub-angular quartz, chlorite and to a lesser extent plagioclase, muscovite and biotite as well as varying amounts of opaque minerals, some of which showed evidence of oxidation.

In the present study, 291 NX-sized core specimens for uniaxial compressive strength tests, Brazilian tensile strength test and point load index were prepared. The ends of the specimens were made flattened perpendicular to the axis of specimen. Their sides were smoothed and polished, and specimens were inspected to be free of cracks, fissures, veins and other flaws, which would act as selective planes of weaken and cause an undesirable change of the real properties of the rock. During the testing process premature failures through the surfaces between the grains and the cementation matrix had been seen only in 3 specimens which have been removed from data base. The UCS, PLI and BTS tests were then performed in accordance with the suggestions outlined by ISRM [20]. In addition to this for UCS ISRM [20] also suggests that the diameter of the core specimen should be related to the size of the largest grain in rock specimen by a ratio of at least 10:1. However, it was not possible to satisfy this condition due to the block-in-matrix nature of the conglomerates studied. In addition to the data from 291 samples, the data of 60 specimens for the same material from Aghchai and Khorablu dams were also utilized in this study that makes a total of 351 samples.

The results obtained and their basic test statistics are tabulated in Table 2. The UCS values of the samples ranged between 1.27 and 23 MPa, with an average value of 6.79. While the average value of PLI was 0.5 MPa values varied from 0.0085 to 3.17 MPa. BTS values changed between 0.0038 and 7.8 MPa, with an average value of 0.822 MPa.

Table 2 Basic statistics of the results obtained from tests
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Data processing and analyses

Simple regression models

In order to establish the predictive models among the parameters obtained in this study, simple regression analysis was performed in the first stage of the analysis. All the statistical analysis was performed using SPSS version 17.0. The relation between UCS and BTS with Is(50) were analyzed employing linear, quadratic, power, logarithmic, and exponential functions. Statistically significant and strong correlations were then selected, and regression equations were established among point load index with UCS and BTS (Table 3). Most of obtained relationships were found to be statistically significant according to the Student’s t test at 95% level of confidence. As can see there is a stronger relation between UCS and Is(50). Figures 1 and 2 shows the plot of the UCS and BTS versus point load index. It should be noted that the fitting quadratic line shown in Fig. 1 is physically wrong for high values of PLI.

Table 3 Predictive models for assessing the UCS and BTS
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Fig. 1
figure 1

The relation between Is(50) and UCS

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Fig. 2
figure 2

The relation between Is(50) and BTS

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The coefficient of correlation between the measured and predictive values is a good indicator to check the prediction performance of the model. Figures 3 and 4 shows the relationships between measured and predicted obtained from the models for UCS and BTS, with good correlation coefficient (R2). In this study, the value account for (VAF) and root mean square error (RMSE) indices were also calculated to control the performance of the prediction capacity of predictive models developed in the study, as employed by Cobanglu and Celik [15] and Yilmaz and Yuksek [21].

$${\text{VAF}} = \left[ {1 - \frac{{\text{var} (y - y)}}{{\text{var} (y)}}} \right]$$
(1)
$${\text{RMSE}} = \sqrt {\frac{1}{N}\sum\nolimits_{i = 1}^{N} {(y - y')^{2} } }$$
(2)

where y and y' are the measured and predicted values. The calculated indices are given in Table 4. If the VAF is 100 and RMSE is 0, then the model will be excellent. The obtained values of VAF and RMSE, given in Table 4, indicated high prediction performances.

Fig. 3
figure 3

Cross-correlation of predicted and observed values of UCS for regression model (Quadratic model)

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Fig. 4
figure 4

Cross-correlation of predicted and observed values of BTS for regression model (Power model)

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Table 4 Performance indices (RMSE, VAF) for regression model
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Artificial neural networks (ANN) models

Neural networks may be used as a direct substitute for auto correlation, multivariable regression, linear regression, trigonometric, and other statistical analysis and techniques [22]. Neural networks, with their remarkable ability to derive a general solution from complicated or imprecise data, can be used to extract patterns and detect trends that are too complex to be noticed by either humans or other computer techniques. A trained neural network can be thought of as an ‘‘expert’’ in the category of information it has been given to analyze. This expert can then be used to provide projections given new situations of interest and answer ‘‘what if’’ questions. When a data stream is analyzed using a neural network, it is possible to detect important predictive patterns that are not previously apparent to a non-expert. Thus, the neural network can act as an expert. The particular network can be defined by three fundamental components: transfer function, network architecture, and learning law [23]. It is essential to define these components, to solve the problem satisfactorily.

All data were first normalized and divided into two datasets such as: training (of all data), test (of all data). In this study Neural work profession ∏ software was used in neural network analyses having a three-layer feed-forward back propagation network that consists of an input layer (1 neurons), three hidden layer (9 neurons), and one output layer. The hidden layer has norm tangent hyperbolic transfer function neurons. Cross-correlation between predicted and observed values which shows in Figs. 5 and 6 indicated that the ANN model constructed is highly acceptable for prediction of UCS and BTS. RMSE, VAF and R2 values are tabulated in Table 5.

Fig. 5
figure 5

Cross-correlation of predicted and observed values of UCS for ANN model

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Fig. 6
figure 6

Cross-correlation of predicted and observed values of BTS for ANN model

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Table 5 Performance indices (RMSE, VAF and R2) for ANN model
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Results and discussion

In this paper, use of simple regression and ANN models, for the prediction of UCS and BTS of conglomerate was described and compared.

According to the results of simple regression analyses, there are statistically meaningful relationships between UCS and BTS with point load index. Although, the most studies have suggested a linear relationship between parameters, obtained result indicate there is a quadratic relationship between UCS and Is(50) Eq. 3 and a power relationship between BTS and Is(50) Eq. 4.

$${\text{UCS}} = {\text{ 16 I}}_{{{\text{s}}( 50)}} - 2. 8 7 {\text{ I}}_{{{\text{s}}(50)}}^{2}$$
(3)
$${\text{BTS}} = {\text{I}}{{{\text{s}}( 50)}}^{0. 8 2 6}$$
(4)

In order to predict the UCS and BTS, ANN models having one inputs and one output was applied successfully. The results of the models for prediction of the UCS and BTS showed that the equations obtained from the regression models have high prediction performances and exhibited the more reliable predictions than the ANN model although it had been mentioned ANN models have higher prediction capacity [24], in addition UCS shows a higher correlation with Is(50) than BTS.

The comparison of VAF, RMSE indices and coefficient of correlations (R2) for predicting UCS and BTS can be seen in Fig. 7.

Fig. 7
figure 7

Comparison of the values of RMSE, VAF, R2 for Regression model, ANN models

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Conclusion

The main conclusions arising from the current study, is that the UCS and BTS were found to be correlated with Is(50) through nonlinear relationships. The result obtains from UCS shows higher correlation than BTS for both regression and ANN methods.

This study may be checked further for validation, it is strongly suggested other studies extended to petrography and physical properties of Conglomerates.