# Experimental Investigation on Uniaxial Compressive Strength of Thin Building Sandstone

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Specimens and Methodology

#### 2.1. Sandstone Specimens

#### 2.2. Test Method

## 3. Results and Discussion

#### 3.1. Failure Mode

**Figure 4.**Failure patterns of sandstone. (

**a**) Group A. (

**b**) Vaneghi et al. [60]. (

**c**) Li et al. [35]. (

**d**) Wasantha et al. [62]. (

**e**) Group B. (

**f**) Group C. (

**g**) Group D. (

**h**) Cylindrical fracture of group B. (

**i**) Cylindrical fracture of group C. (

**j**) Cylindrical fracture of group D. (

**k**) Fakhimi and Hemami [61].

#### 3.2. Stress–Strain Diagram

#### 3.2.1. Experimental Diagram

^{2}. The profiles of each diagram converged, indicating that stable experimental results were obtained. The peak stress and strain were larger for the cylinders of groups B and D than those of group A (Figure 5b,c), whereas the profiles of the diagrams were more scattered than those of group A. For the largest thin cylinder (group D), the strain-softening branch was the least apparent among the four groups of cylinders (Figure 5d). The mean stress–strain diagrams of all the specimens are shown in Figure 5e. Generally, the peak stress and strain increased with decreasing cylinder height (L). This is a characteristic of the size effect of brittle materials [44]. This property was demonstrated in the uniaxial compressive tests of sandstone conducted by Fakhimi and Hemami [61] and will be elaborated in detail in below. The stress–strain diagram for a sandstone cylinder with standard dimensions (group A) is shown in Figure 5f, along with those from the literature. None of these diagrams follow the same trend because of the variable mineral content. The diagram of the standard cylinders in Li et al. [35] followed the same path as the prismatic specimens in Ludovico-Marques et al. [39] until failure at 85.6 N/mm

^{2}. A similar trend was demonstrated in the diagrams of Wasantha et al. [62] and in the present study.

#### 3.2.2. Evolution of Diagram

_{cc}corresponds to the closing of most microcracks in the cylinder under uniaxial loading. The axial stiffness increased nonlinearly. The crack initiation threshold f

_{ci}corresponds to the appearance of new microcracks. The propagation of cracks extended parallel to the applied load. The crack damage threshold f

_{cd}corresponds to unstable crack growth. The correlation between the crack length and stress disappeared, whereas the crack growth velocity increased [65]. Consequently, crack propagation continued unstably until peak stress, f

_{ucs}, and then developed abruptly until complete failure at the post-peak branch of the diagram. The loading stage between the crack initiation and crack damage thresholds is frequently employed to compute the elastic modulus of the material of interest.

#### 3.2.3. Diagram Regression

_{ucs}is the peak compressive stress, ε is the compressive strain, and ε

_{ucs}is the strain corresponding to the peak stress f

_{ucs}.

#### 3.3. Compressive Strength and Porosity

^{2}, indicating the stability of the experimental data. For the cylinders with standard dimensions (group A), the experimental compressive strength was lower than those of the fine- and coarse-grained sandstone [69], which were as large as 158.97 N/mm

^{2}and 131.70 N/mm

^{2}, respectively. The physical uniaxial compressive test is an immediate experimental approach; however, it is expensive in terms of time and labor. An empirical prediction expression with acceptable reliability is necessary, particularly for on-site engineers. Predictors such as porosity, Schmidt hammer rebound number, P wave velocity, and point load strength index are frequently employed in expression regression [70]. Among these, porosity is the simplest parameter and is elaborated herein.

References | Expression | R^{2} | # Samples |
---|---|---|---|

Qi et al., 2022 [16] | f_{ucs} = 110.5 exp(−0.08 n) | 0.72 | 17 sandstone samples |

Farrokhrouz and Asef, 2017 [29] | f_{ucs} = −3.03 n + 107.1 | 0.28 | 299 sandstone samples |

Mishra and Basu, 2013 [38] | f_{ucs} = −55.7 ln(n) + 172.1 | 0.88 | 20 sandstone samples |

Ludovico-Marques et al., 2012 [39] | f_{ucs} = 206.7 exp(−0.129 n) | — | 13 sandstone samples |

Yasar et al., 2010 [74] | f_{ucs} = −2.27 n^{2} + 33.88 n − 16.30 | 0.96 | 11 sandstone samples |

Kılıç and Teymen, 2008 [79] | f_{ucs} = 147.16 exp(−0.0835 n) | 0.93 | 19 rock types, including sandstone |

Sabatakakis et al., 2008 [72] | f_{ucs} = 123.0 exp(−0.12 n) | 0.63 | 95 sandstone samples |

Tugrul, 2004 [78] | f_{ucs} = 195.0 exp(−0.21 n) | 0.79 | 16 different sedimentary rocks, including sandstone |

Chatterjee and Mukhopadhyay, 2002 [80] | f_{ucs} = 64.23 exp(−0.085 n) | 0.92 | 22 samples, including sandstone |

Palchik, 1999 [73] | f_{ucs} = 74.4 exp(−0.04 n) | 0.78 | 16 samples of soft brittle porous sandstone |

Plumb, 1994 [75] | f_{ucs} = 357 (1 − 0.028 n)^{2} | — | 784 sedimentary rocks, mainly sandstone and shale |

#### 3.4. Elastic Modulus and Porosity

^{2}(Figure 11). These results differed from the experimental results of Mousavi et al. [18], Chatterjee et al. [81], and Rice-Birchall et al. [20]. The porosity ratio is supposedly one of the factors that causes variation in the modulus. Empirical expressions were developed to represent the correlations between the elastic modulus and porosity of stone materials (Table 8). These expressions are plotted in Figure 12, along with the experimental results of the current study and those in the literature. The porosity ratios of sandstone in Chatterjee et al. [81] are close to those of the standard cylinder (group A) in the current study. The resulting magnitude was identical when the porosity ratio was the same, that is, 2.48%. In contrast, the porosity ratios obtained by Bedford et al. [82], Mousave et al. [18], and Wong et al. [83] were larger. The elastic modulus of the sandstones reported by Bedford et al. [82] was the smallest with the largest porosity ratio. For the empirical expressions in Table 8, only the expression provided by Salah et al. [84] was developed based on the experimental results for sandstone. The remaining expressions were based on other rock materials. However, the expression of Salah et al. [84] provided the least accurate prediction of the modulus for the sandstone in the current study, whereas it was relatively close to the experimental data reported by Wong et al. [83]. For the cylinders of groups A, B, and C, the expressions suggested by Lashkaripour [85] and Leite and Ferland [86] achieved reasonable predictions compared to the other expressions. However, for the cylinders of group D, the expressions suggested by Armaghani et al. [31] provided a reasonable prediction. Extensive experimental studies are required to obtain a reliable expression for predicting the elastic modulus of sandstone. Note that the expressions of the elastic moduli of the rock materials other than sandstone (e.g., gypsum, artificial rock, and even claystone and mudstone) listed in Table 8 are only for comparison. The expressions for other rocks than the sandstone cannot simply be applied to describe the elastic modulus of the sandstone in this study.

#### 3.5. Compressive Strength and Elastic Modulus

_{ucs}) and elastic modulus (E) is well recognized in the civil engineering community. The ratio of E to UCS is an engineering parameter for the structural design of buildings and underground constructions [39,89]. Based on the experimental data, empirical expressions representing the correlation between UCS and E were developed through direct regression analysis. However, the quantification of the correlation between these two parameters was challenging because of the insufficient experimental results [71]. The suggested expressions (e.g., Somnze et al. [90]) provide an alternative for practicing engineers and researchers in academic and engineering fields. The experimental data and regressed expressions are shown in Figure 13 and Table 9. For the cylinders in the current study, there was no clear trend between E and UCS. For standard cylinders in a group, the corresponding magnitudes of UCS and E were close to the experimental values reported by Heidari et al. [91], Malik and Rashid [92], Cai et al. [52], and Qi et al. [16]. Some of the experimental data in Hawkins and McConnell [93] and Zhang et al. [68] were close to those of groups B and C. Most of the experimental data were enveloped by the expressions provided by Farrokhrouz and Asef [29] and Bradford et al. [94]. The expression suggested by Chatterjee and Mukhopdahyay [80] provided the worst prediction of all the experimental data. The expressions suggested by Sabatakakis et al. [72] and Bell and Lindsay [41] were close to the experimental data of group D. Although the main rock samples in Lacy [95] were not sandstone, the suggested expression generally provided a lower limit of the UCS, whereas the expression of Farrokhrouz and Asef [29] generally provided an upper limit. The magnitude of the UCS generally increases with the elastic modulus, as reported by Mousavi et al. [18], Hawkins and McConnell [93], Heidari et al. [91], and Chatterjee et al. [81]. Most of the experimental data range from 8.0 to 20 GPa in modulus and from 40 to 100 N/mm

^{2}in UCS.

References | Expression | Units | R^{2} | # Sample |
---|---|---|---|---|

Farrokhrouz snd Asef, 2017 [29] | f_{ucs} = 5.49 E^{0.423}/φ^{0.546} | f_{ucs} in MPa, E in GPa, φ is the porosity ratio | 0.8272 | 299 samples of sandstone |

Sabatakakis et al. 2008 [72] | f_{ucs} = E/303 | f_{ucs} and E in MPa | 0.65 | 36 samples of sandstone |

Chatterjee and Mukhopdahyay, 2002 [80] | f_{ucs} = (E − 0.17)/0.73 | f_{ucs} in MPa, E in GPa | 0.93 | 8 samples, including sandstone |

Bell and Lindsay, 1999 [41] | f_{ucs} = (E − 5.6)/0.358 | f_{ucs} in MPa, E in GPa | — | 27 samples of sandstone |

Bradford et al., 1998 [94] | f_{ucs}= 2.28 + 4.1089 E | E in GPa | — | Sandstone sample |

Lacy, 1997 [95] | f_{ucs} = 0.2787 E^{2} + 2.4582 E | f_{ucs} in kpsi and E in Mpsi | 0.84 | 36 samples of weakly consolidated rocks |

^{2}, 1.0 GPa = 1 × 10

^{3}Mpa.

## 4. Size Effect

#### 4.1. Size Effect on Diameter

_{r}is the volume of one element in the specimen, and V is the volume of the specimen. A general expression is provided in logarithmic form as follows:

_{r}ratio can be substituted by the ratio of the corresponding diameters. Hoek and Brown [106] provided an expression representing the correlation of the crack damage stress (f

_{cd}) between a sample with an arbitrary diameter and a sample with a diameter of 50 mm (f

_{c50}), given by

_{c50}) is determined by the rock type. Accordingly, the parameter k was used to indicate the rock types, given by

_{c50}= 25–250 N/mm

^{2}.

_{0}is the maximum aggregate size of the concrete and the maximum grain size of the rock material; d is the characteristic sample size; f

_{t}is the strength of a sample with negligible size, which can be expressed in terms of the intrinsic strength, namely, the strength of the maximum grain (though it is almost impossible to experimentally obtain this parameter); ${\sigma}_{N}$ is the nominal strength of the material; and B and λ are the dimensionless material constants.

_{c}is the strength of a sample with infinite size and can be represented by the intrinsic strength of a large sample with an infinite diameter, and l is a material constant in units of length. In Equation (8), the magnitude of ${\sigma}_{N}$ increases with decreasing d. This trend is demonstrated in Equations (4) and (7).

_{f}is the fractal dimension and σ

_{0}is the strength of a sample with negligible size. If the materials have nonfractal properties, d

_{f}= 1 and σ

_{0}= Bf

_{t}. Equation (9) is then changed to Equation (7). The experimental results of Hawkins [112] and Darlington and Ranjith [113] demonstrated that the size effect model developed by Hoek and Brown [104] did not closely correspond to small specimens. Accordingly, Masoumi et al. [114] proposed a unified size effect law based on uniaxial compression and point-load tests of other sandstone sedimentary rock types in Hawkins [112], which is given by

#### 4.2. Size Effect on Length/Diameter Ratio (L/D)

_{c1}is the UCS of the rock with L/D = 1.0. The f

_{c}value was obtained for cylinders with 1/3 < L/D < 2.0. ASTM C170 [47] suggested a formula to convert the UCS (f

_{n}) of cylinders with non-standard dimensions (L/D < 2.0) into that of the standard, that is, L/D = 2.0, given by

a | b | RSS | R^{2} |
---|---|---|---|

0.835 | 0.361 | 0.142 | 0.936 |

## 5. Conclusions

- (1)
- A columnar vertical fracture was the dominant failure pattern. The stress–strain diagrams of group A converged more than those of the other groups, demonstrating stable mechanical behavior in the standard specimen. The geometry of the diagrams varied among the four groups. The critical strain generally increased with a decrease in the height of the cylinder, whereas the compressive strength exhibited an inverse trend.
- (2)
- The magnitudes of the crack closure stresses of the thin cylinders in groups B, C, and D were identical. A similar trend was observed for the crack initiation stress, crack damage stress, and peak stress. To obtain a representative stress–strain diagram for each group of cylinders, the experimental diagrams were normalized with the peak stress and corresponding critical strain. The normalized stress–strain diagram demonstrated the specific loading behavior of each group of cylinders. To obtain a representative mathematical expression of the diagram, a formula consisting of two parabolas divided by the crack initiation stress was employed for regression.
- (3)
- The correlations between porosity, UCS, and elastic modulus were evaluated based on empirical expressions. The expressions suggested by Lashkaripour [85] and Leite and Ferland [86] provided a reasonably accurate prediction of the UCS of thin cylinders with respect to porosity. However, none of the expressions in the literature achieved a good prediction of the elastic modulus.
- (4)
- The normalized strength was employed to evaluate the size effect on the diameter and L/D ratio of the cylinders. The UCS of group A with standard dimensions was correctly predicted using the expression suggested by Hoek and Brown [106]. However, for thin cylinders, none of the expressions in the literature provided a good prediction. A new expression in terms of L/D was proposed based on the regression analysis of the experimental results.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Progressive failure of brittle material (adapted from [28]).

Mineral | Quartz | Plagioclase | Calcite | Zeolite | Potash Feldspar | Others |
---|---|---|---|---|---|---|

Percentage (%) | 42.4 | 34.9 | 9.2 | 7.3 | 5.2 | 1.0 |

References | Porosity (%) | Density (g/cm^{3}) |
---|---|---|

Current study | 2.38 | 2.46 |

Li et al., 2021 [35] | 5.91 | 2.48 |

Liu et al., 2020 [37] | 20.48 | 1.85 |

Mousavi et al., 2018 [18] | 9.38–20.23 | 1.99–2.88 |

Huang and Xia, 2015 [36] | 17 | 2.15 |

Mishra and Basu, 2013 [38] | 2.89–15.54 | 2.17–2.49 |

Ludovico-Marques et al., 2012 [39] | 3.6–18.6 | 2.18–2.59 |

Shakoor and Barefield, 2009 [40] | 4.12–12.72 | 2.07–2.52 |

Bell and Lindsay, 1999 [41] | 5.6–10.1 | 2.43–2.57 |

O | Si | AL | Ca | K | Fe | Na | Mg |
---|---|---|---|---|---|---|---|

60.1 | 24.2 | 4.6 | 2.8 | 1.2 | 1.2 | 0.9 | 0.9 |

Group # | L (mm) | D (mm) | L/D | Number |
---|---|---|---|---|

A | 100 | 50 | 2.0 | 6 |

B | 25 | 50 | 0.5 | 6 |

C | 30 | 60 | 0.5 | 6 |

D | 75 | 150 | 0.5 | 6 |

Group | a_{1} | b_{1} | a_{2} | b_{2} | c_{2} |
---|---|---|---|---|---|

A | 1.13 | 1.42 | −3.57 | 6.92 | −2.39 |

B | 1.42 | 1.70 | −1.34 | 2.83 | −0.52 |

C | 1.52 | 2.11 | −2.26 | 4.63 | −1.38 |

D | 1.39 | 4.30 | −4.42 | 9.54 | −4.12 |

Group | 1st Portion | 2nd Portion | ||
---|---|---|---|---|

RSS | R^{2} | RSS | R^{2} | |

A | 6.699 | 0.965 | 8.634 | 0.433 |

B | 14.61 | 0.880 | 5.933 | 0.595 |

C | 1.788 | 0.979 | 0.359 | 0.903 |

D | 6.425 | 0.982 | 2.908 | 0.712 |

References | Expression | R^{2} | # Samples |
---|---|---|---|

Salah et al., 2020 [84] | E = 78.926 exp(−0.0852 n) | 0.96 | 49 samples, including sandstone |

Armaghani et al., 2016 [31] | E = 43.899 n^{(−0.556)} | 0.28 | 71 granite samples |

Beiki et al., 2013 [87] | E = exp(−0.10 n + 3.6) | 0.23 | 72 different carbonate rock types |

Beiki et al., 2013 [87] | E = 36.6 (0.91)^{n} | 0.23 | 72 different carbonate rock types |

Yilmaz and Yuksek, 2009 [88] | E = −39.1 ln(n) + 110.31 | 0.83 | 121 samples of gypsum |

Lashkaripour, 2002 [85] | E = 37.9 exp(−0.863 n) | 0.68 | Claystone, clay shale, mudstone, mud shale |

Leite and Ferland, 2001 [86] | E = 10.10 − 0.109 n | 0.74 | Artificial rock |

**Table 10.**Mechanical properties of sandstone (after [116]).

Batch # | Sandstone Type | c (N/mm^{2}) | φ (Degree) | Mean UCS (N/mm^{2}) | L/D Limit |
---|---|---|---|---|---|

1 | Fine | 18.75 | 48.6 | 99.13 | 1.2 |

2 | Fine | 19.47 | 47.1 | 92.87 | 1.1 |

3 | Fine | 31.96 | 27.7 | 101.44 | 0.6 |

4 | Argillaceous | 9.80 | 28.7 | 31.62 | 0.6 |

5 | Medium-coarse | 11.72 | 44.7 | 56.48 | 1.0 |

6 | Fine | 18.11 | 42.6 | 98.08 | 1.0 |

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Huang, B.; Xu, Y.; Zhang, G. Experimental Investigation on Uniaxial Compressive Strength of Thin Building Sandstone. *Buildings* **2022**, *12*, 1945.
https://doi.org/10.3390/buildings12111945

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Huang B, Xu Y, Zhang G. Experimental Investigation on Uniaxial Compressive Strength of Thin Building Sandstone. *Buildings*. 2022; 12(11):1945.
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**Chicago/Turabian Style**

Huang, Baofeng, Yixian Xu, and Guojun Zhang. 2022. "Experimental Investigation on Uniaxial Compressive Strength of Thin Building Sandstone" *Buildings* 12, no. 11: 1945.
https://doi.org/10.3390/buildings12111945