A Comparative Study of UCS Results Obtained from Triaxial Tests Under Multiple Failure State Conditions (Test Type II)

Abstract

In any rock engineering project, the uniaxial compressive strength (UCS) is one of the most relevant parameters to be determined as it is used for a variety of purposes. Traditionally, the UCS of a rock sample is obtained by carrying out a uniaxial test on a rock core. The UCS can also be estimated indirectly by correlating it with different parameters such as the pulse velocity (V_p), Schmidt hammer rebound number (R_n), effective porosity (n_e), total porosity (n_t), dry density (γ_d), point load index (I_s50), shear wave velocity (V_s), Brazilian tensile strength (BTS), slake durability index (SDI), or by using an artificial neural network (ANN). This paper presents a comparative study for an additional approach to determine the UCS, namely by converting triaxial test results to the UCS. This research has been conducted to encourage further utilisation of laboratory test results obtained from triaxial tests. Triaxial tests are normally performed to determine the intact rock strength envelope. By further utilisation of triaxial test data to estimate the UCS, the database of the UCS can be enriched for rock engineering projects. The method can also be used in cases of limited samples or when samples are too challenged for carrying out a USC test. The research showed that the UCS derived by this method is more direct and more accurate than many empirical methods. Furthermore, the test procedure described herein (carrying out the UCS and triaxial tests on the same sample) can be used to evaluate the accuracy of the calculated UCS. It is important to bear in mind that the scope of this study is not to develop a new method or replace the traditional uniaxial compression test with the method presented in this paper. The purpose and intention is to document that the test results obtained from triaxial testing can be utilised to also provide values of UCS from the same samples.

1. Introduction

UCS is one of the most commonly applied rock mechanics parameters in any rock engineering project. Together with other parameters such as Young’s modulus, Poisson’s ratio, density, internal friction angle, and cohesion, UCS provides basic information to evaluate the intact rock strength and behaviour, which again are decisive for tunnel/cavern layout, support needs, etc.

Traditionally, and still commonly used today, the UCS of a rock sample is obtained through a uniaxial compression test with apparatus and procedure, which were described in the ISRM-suggested methods and by the ASTM [1,2,3].

In addition to the traditional method, some authors also developed empirical correlations to obtain the UCS based on different parameters, such as pulse velocity (V_p), Schmidt hammer rebound number (R_n), effective porosity (n_e), total porosity (n_t), dry density (γ_d), point load index (I_s50), shear wave velocity (V_s), Brazilian tensile strength (BTS), and slake durability index (SDI). Some of such empirical correlations for different types of rock are summarised and presented in Altindag [4], Le et al. [5], and Skentou et al. [6]. Examples of different empirical correlations for some rocks are presented in Table 1 and Table 2.

Recently, an artificial neural network (ANN)-based model was used to estimate the UCS based on results from three non-destructive tests [5,6]. The selected three non-destructive tests were pulse velocity (V_p), Schmidt hammer rebound number (R_n), and effective porosity (n_e). In this way, a dataset and site-independent database comprising 274 datasets correlating the R_n, V_p, and n_e with the UCS of granite was used to train and develop the ANN models. Based on the results of the study, two closed-form equations were established to estimate the UCS [6].

Many other empirical relations and the ANN model can also be found in Min Wang and Wen Wan [7] and Min Wang et al. [8].

Table 1. Empirical correlations between R_n, V_p, and n_e and the UCS of granite (modified from [4,5,6]).

#	Empirical Relationships	Method	Reference	Range of Input Parameters				UCS Range (MPa)
				ne (%)	Vp (m/s)	Rn (-)
						L-Type	N-Type	From	To
1	UCS = 78.22ne + 201	RA	[9]	0.14–1.07	–	–	–	109.2	193.3
2	UCS = 35.54Vp − 55	RA	[9]	–	4740–6690	–	–	109.2	193.3
3	UCS = 8.36Rn(L) − 416	RA	[9]	–	–	64–72	–	109.2	193.3
4	UCS = 2.208e0.067Rn(N)	RA	[10]	–	–	–	23.9–73.4	11	259
5	UCS = 1.4459e0.0706Rn(L)	RA	[11]	–	–	20.00–65.76	–	6.32	196.5
6	UCS = 0.9165e0.0669Rn(N)	RA	[11]	–	–	-	23.00–75.97	6.32	196.5
7	UCS = 4.24e0.059Rn(N)	RA	[12]	–	–	–	48.30–61.80	60.8	202.9
8	UCS = 124.28ne−0.56	RA	[13]	0.64–3.72	–	–	–	62.4	197
9	UCS = 0.004Vp1.247	RA	[13]	–	2339–5753	–	–	62.4	197
10	UCS = 0.0407Vp − 36.31	RA	[14]	–	1941–4751	–	–	23.77	161.7
11	UCS = e(2.28lnRn(L)−4.04)	RA	[15]	–	–	15–60	–	10.54	244.8
12	UCS = 0.033Vp − 34.83	RA	[16]	–	1682–4657	–	–	32.51	133.5
13	UCS = 2.38e0.065Rn(L)	RA	[17]	–	–	25.89–67.07	–	17.55	198.2
14	UCS = 0.087Vp − 355.8	RA	[17]	–	5384–6250	–	–	91.48	198.2
15	UCS = 228.2e(−1.98ne)	RA	[17]	0.06–0.4	–	–	–	91.48	198.2
16	UCS = −17.88ln(ne) + 60.22	BHM	[18]	0.52–7.23	–	–	–	20.3	112.9
17	UCS = 8.17e0.0004Vp + 3.93	BHM	[18]	–	1160–5935	–	–	20.3	112.9
18	UCS = 7.03e0.0386Rn(L) + 8.39	BHM	[18]	–	–	16.8–55.7	–	20.3	112.9
19	UCS = 4.15e0.0386Rn(L) + 6.08e0.0004Vp − 8.22	BHM	[18]	–	1160–5935	16.8–55.7	-	20.3	112.9
20	UCS = 6.32e0.0004Vp − 9.6ln(ne) + 20.5	BHM	[18]	0.52–7.23	1160–5935	–	–	20.3	112.9
21	UCS = 3.26e0.0386Rn(L) + 5.84e0.0004Vp − 3.56ln(ne) + 0.53	BHM	[18]	0.52–7.23	1160–5935	16.8–55.7	–	20.3	112.9
22	UCS = 1.910Rn(N) − 10.3	RA	[19]	–	–	–	10–72	8	149
23	UCS = 25.952e0.030Rn(L)	RA	[20]	–	–	18–61	–	39	211.9
24	UCS = 0.005Vp1.141	RA	[20]	–	2823–7943	–	–	39	211.9
25	UCS = 0.004Rn(L)2.5972	RA	[21]	–	–	18.8–60	–	6.12	148
26	UCS = 181.58ln(Vp) − 1443.9	RA	[22]	–	4108–7943	–	–	50.2	211.9
27	UCS = 6.311Rn(L) − 194.92	RA	[22]	–	–	40–61	–	50.2	211.9
28	UCS = 0.0367Vp − 31.18		[23]

Note: RA is regression analysis, and BHM is Bayesian hierarchical modelling.

Table 1. Empirical correlations between R_n, V_p, and n_e and the UCS of granite (modified from [4,5,6]).

#	Empirical Relationships	Method	Reference	Range of Input Parameters				UCS Range (MPa)
				ne (%)	Vp (m/s)	Rn (-)
						L-Type	N-Type	From	To
1	UCS = 78.22ne + 201	RA	[9]	0.14–1.07	–	–	–	109.2	193.3
2	UCS = 35.54Vp − 55	RA	[9]	–	4740–6690	–	–	109.2	193.3
3	UCS = 8.36Rn(L) − 416	RA	[9]	–	–	64–72	–	109.2	193.3
4	UCS = 2.208e0.067Rn(N)	RA	[10]	–	–	–	23.9–73.4	11	259
5	UCS = 1.4459e0.0706Rn(L)	RA	[11]	–	–	20.00–65.76	–	6.32	196.5
6	UCS = 0.9165e0.0669Rn(N)	RA	[11]	–	–	-	23.00–75.97	6.32	196.5
7	UCS = 4.24e0.059Rn(N)	RA	[12]	–	–	–	48.30–61.80	60.8	202.9
8	UCS = 124.28ne−0.56	RA	[13]	0.64–3.72	–	–	–	62.4	197
9	UCS = 0.004Vp1.247	RA	[13]	–	2339–5753	–	–	62.4	197
10	UCS = 0.0407Vp − 36.31	RA	[14]	–	1941–4751	–	–	23.77	161.7
11	UCS = e(2.28lnRn(L)−4.04)	RA	[15]	–	–	15–60	–	10.54	244.8
12	UCS = 0.033Vp − 34.83	RA	[16]	–	1682–4657	–	–	32.51	133.5
13	UCS = 2.38e0.065Rn(L)	RA	[17]	–	–	25.89–67.07	–	17.55	198.2
14	UCS = 0.087Vp − 355.8	RA	[17]	–	5384–6250	–	–	91.48	198.2
15	UCS = 228.2e(−1.98ne)	RA	[17]	0.06–0.4	–	–	–	91.48	198.2
16	UCS = −17.88ln(ne) + 60.22	BHM	[18]	0.52–7.23	–	–	–	20.3	112.9
17	UCS = 8.17e0.0004Vp + 3.93	BHM	[18]	–	1160–5935	–	–	20.3	112.9
18	UCS = 7.03e0.0386Rn(L) + 8.39	BHM	[18]	–	–	16.8–55.7	–	20.3	112.9
19	UCS = 4.15e0.0386Rn(L) + 6.08e0.0004Vp − 8.22	BHM	[18]	–	1160–5935	16.8–55.7	-	20.3	112.9
20	UCS = 6.32e0.0004Vp − 9.6ln(ne) + 20.5	BHM	[18]	0.52–7.23	1160–5935	–	–	20.3	112.9
21	UCS = 3.26e0.0386Rn(L) + 5.84e0.0004Vp − 3.56ln(ne) + 0.53	BHM	[18]	0.52–7.23	1160–5935	16.8–55.7	–	20.3	112.9
22	UCS = 1.910Rn(N) − 10.3	RA	[19]	–	–	–	10–72	8	149
23	UCS = 25.952e0.030Rn(L)	RA	[20]	–	–	18–61	–	39	211.9
24	UCS = 0.005Vp1.141	RA	[20]	–	2823–7943	–	–	39	211.9
25	UCS = 0.004Rn(L)2.5972	RA	[21]	–	–	18.8–60	–	6.12	148
26	UCS = 181.58ln(Vp) − 1443.9	RA	[22]	–	4108–7943	–	–	50.2	211.9
27	UCS = 6.311Rn(L) − 194.92	RA	[22]	–	–	40–61	–	50.2	211.9
28	UCS = 0.0367Vp − 31.18		[23]

Note: RA is regression analysis, and BHM is Bayesian hierarchical modelling.

Table 2. A selection of empirical correlations between V_p and the UCS of a variety of rock types, including sedimentary, volcanic, and metamorphic rock (modified from [4,5]). Many other correlations can be found in [5].

Equations (Reference)	Ref.	Correlation Coefficient (r)	Units and Notations	Rock Type	Number of Data
UCS = ax + b	[24]	0.85	--	Sandstones	--
UCS = 0.0642 Vp − 117.99, (MPa)	[25]	0.90	Vp: m/s	Sandstone, coal, quartz mica schist, phyllite, basalt	43
UCS = 56.71 Vp − 192.93, (MPa)	[26]	very small	--	Cement mortar, sandstone, limestone	75
UCS = aVpb	[27]	0.94	--	A wide range of British rock types	150
UCS = 9.95 Vp1.21, (MPa)	[28]	0.83	Vp: km/s	Marl, limestone, dolomite, sandstone, haematite, serpantine, diabase, tuff	48
Vp = 0.00317 UCS + 2.0195	[29]	0.80		--	--
UCS = 0.78 e0.88Vp	[30]	0.73	Vp: km/s	Volcanic group	--
UCS = 0.78 Vp0.88	[30]	0.73	Vp: km/s	Volcanic group	--
UCS = 0.0407 Vp − 36.31, (N/mm2)	[14]	0.85		--	19
UCS = k ρ Vp2 + A, (kg/cm2)	[31]		ρ: g/cm3, Vp: km/s	--	--
UCS = 0.036 Vp − 31.18, (MPa)	[23]		Vp: m/s	--	--
UCS = 0.1564 Vp − 692.41, (MPa)	[32]	0.90	Vp: m/s	Sandstones	9
UCS = 0.0144 Vp − 24.856, (MPa)	[32]	0.71	Vp: m/s	Sandstones	24
UCS = 7.1912 Vp + 26.258, (MPa)	[33]	0.57	Vp: km/s	Sandstone, gravel stone, limestone, mudstone, shale	8
UCS = 21.677 Vp + 21.427	[34]	0.95	Vp: km/s	Limestone, marble, dolomitic limestone, tuff, basalt	8
UCS = 0.0188 Vp + 0.0648	[35]	0.95	Vp: km/s	Sandstone	--
UCS = 2.304 Vp2.4315	[36]	0.97	Vp: km/s	Diorite, quartzite, sandstone, limestone, marble, granadiorite, basalt, travertine, trachyte, tuff, andesite	19
UCS = 12.746 Vp1.194	[4]	0.79	Vp: km/s	Limestone, sandstone, travertine, marl, dolomite, mudrock-shale, slate, siltstone	97
UCS = −7.155 + 6.194 Vp + 9.774 TS	[4]	0.88	Vp: km/s		43
UCS = −10.029 + 5.734 Vp + 10.876 TS − 2.408 Is	[4]	0.90	Vp: km/s		26
UCS = 9.95Vp1.21	[37]			Diabase, dolomite, haematite, limestone, marl, sandstone, serpentine, tuff
UCS = 0.0028Rn(L)2.584	[38]			Limestone, schist, travertine
UCS = (Vp − 2.0195)/0.0317	[39]			Limestone, Marble, Dolomitic limestone, Dolomite, Gravelled limestone

Table 2. A selection of empirical correlations between V_p and the UCS of a variety of rock types, including sedimentary, volcanic, and metamorphic rock (modified from [4,5]). Many other correlations can be found in [5].

Equations (Reference)	Ref.	Correlation Coefficient (r)	Units and Notations	Rock Type	Number of Data
UCS = ax + b	[24]	0.85	--	Sandstones	--
UCS = 0.0642 Vp − 117.99, (MPa)	[25]	0.90	Vp: m/s	Sandstone, coal, quartz mica schist, phyllite, basalt	43
UCS = 56.71 Vp − 192.93, (MPa)	[26]	very small	--	Cement mortar, sandstone, limestone	75
UCS = aVpb	[27]	0.94	--	A wide range of British rock types	150
UCS = 9.95 Vp1.21, (MPa)	[28]	0.83	Vp: km/s	Marl, limestone, dolomite, sandstone, haematite, serpantine, diabase, tuff	48
Vp = 0.00317 UCS + 2.0195	[29]	0.80		--	--
UCS = 0.78 e0.88Vp	[30]	0.73	Vp: km/s	Volcanic group	--
UCS = 0.78 Vp0.88	[30]	0.73	Vp: km/s	Volcanic group	--
UCS = 0.0407 Vp − 36.31, (N/mm2)	[14]	0.85		--	19
UCS = k ρ Vp2 + A, (kg/cm2)	[31]		ρ: g/cm3, Vp: km/s	--	--
UCS = 0.036 Vp − 31.18, (MPa)	[23]		Vp: m/s	--	--
UCS = 0.1564 Vp − 692.41, (MPa)	[32]	0.90	Vp: m/s	Sandstones	9
UCS = 0.0144 Vp − 24.856, (MPa)	[32]	0.71	Vp: m/s	Sandstones	24
UCS = 7.1912 Vp + 26.258, (MPa)	[33]	0.57	Vp: km/s	Sandstone, gravel stone, limestone, mudstone, shale	8
UCS = 21.677 Vp + 21.427	[34]	0.95	Vp: km/s	Limestone, marble, dolomitic limestone, tuff, basalt	8
UCS = 0.0188 Vp + 0.0648	[35]	0.95	Vp: km/s	Sandstone	--
UCS = 2.304 Vp2.4315	[36]	0.97	Vp: km/s	Diorite, quartzite, sandstone, limestone, marble, granadiorite, basalt, travertine, trachyte, tuff, andesite	19
UCS = 12.746 Vp1.194	[4]	0.79	Vp: km/s	Limestone, sandstone, travertine, marl, dolomite, mudrock-shale, slate, siltstone	97
UCS = −7.155 + 6.194 Vp + 9.774 TS	[4]	0.88	Vp: km/s		43
UCS = −10.029 + 5.734 Vp + 10.876 TS − 2.408 Is	[4]	0.90	Vp: km/s		26
UCS = 9.95Vp1.21	[37]			Diabase, dolomite, haematite, limestone, marl, sandstone, serpentine, tuff
UCS = 0.0028Rn(L)2.584	[38]			Limestone, schist, travertine
UCS = (Vp − 2.0195)/0.0317	[39]			Limestone, Marble, Dolomitic limestone, Dolomite, Gravelled limestone

Moreover, as both the empirical and the ANN methods use indirect correlations or models to estimate the UCS, there could be a need for another option, which is a more direct method to estimate the UCS by utilising results from triaxial tests.

Triaxial tests are required in many rock engineering projects to obtain the peak and residual strength of the rock samples. Engineers typically use the results from each triaxial test to obtain the failure envelope (or peak strength envelopes) for the corresponding samples. The question would then be whether results from triaxial tests can be utilised to reliably estimate the UCS for the same rock samples? If the UCS can be estimated from the results of triaxial tests, it could be an additional valuable output from the triaxial tests, enriching the UCS database for the projects. This paper presents the possibility of estimating UCS from triaxial tests. The approach to be described constitutes a way for comparison and evaluation of the reliability of the method by carrying out uniaxial and triaxial tests on the same sample. The results from this method are also compared with other methods.

It is necessary to emphasise that this publication is not aiming to establish any new theory or method for estimating UCS. The theory presented in this publication is fundamental knowledge presented in many rock and soil mechanics textbooks. From the textbooks, the UCS of a rock sample can be calculated in a straightforward way by setting the confinement to zero in any appropriate failure criterion. Despite this clear theory, the authors of this publication found that data from triaxial tests have not been commonly used for calculating UCS. Thus, this publication aims at encouraging further utilisation of results from triaxial tests to estimate the UCS. With appropriate evaluations, the UCS can then be one of the standard output values from triaxial tests.

2. Brief Introduction to Triaxial Tests

The procedures of triaxial tests are described in [2,3]. According to the ISRM [2], there are three types of triaxial tests—namely Type I, Type II, and Type III—as shown in Figure 1, and their typical results are shown in Figure 2. A very brief description of the different test types is as follows:

• Type I: An individual rock specimen is tested with a single pre-set confinement. It is required to perform at least three tests with three different confinements in order to obtain the peak strength envelope. This means that at least three samples are required to fulfil the test and construct the peak strength envelope. Typical results of one test are presented in Figure 1a and Figure 2a;
• Type II: Three different confinement levels are applied in the same sample. At each confinement level, the sample is tested until it tangents its peak strength at that confinement level, and then the confinement is changed to the next level for the test to continue. With the three different confinement levels in one test, it is possible to construct the peak strength envelope for the sample from only one test. Typical results of one test are presented in Figure 1b and Figure 2b;
• Type III: The test is carried out with the initial confinement until the sample tangents its peak strength. The linear portion of the axial stress versus axial strain curve has a slope of V (V = E is the Young’s modulus of the sample). After the initial step, the confinement and the axial stress are simultaneously increased in such a way that the obtained axial stress versus axial strain curve has the same slope as V. Typical results of one test are presented in Figure 1c and Figure 2c.

A more detailed description of the test types can be found in [2].

According to the ISRM [2], with test Type I (“individual test”), individual points on the failure (peak strength) envelope are obtained from several tests (Figure 1a), while with test Type II (“multiple failure state test” as in Figure 1b) and test Type III (“continuous failure state test” as in Figure 1c), the strength envelope is produced with a single test employing a stepwise or continuous procedure as described above.

From a rock mechanics point of view, the UCS of a rock sample can be estimated based on the results of the triaxial tests as follows:

• Fitting a curve based on the results from triaxial tests. Depending on the behaviour of the rock, the fitting curve can be linear or non-linear;
• Plotting the strength envelope in a graph where the vertical axis represents the major principal stress (normally the axial stress during the triaxial test) and the horizontal axis represents minor principal stress (normally the confinement during the triaxial test);
• Extending the strength envelope until it intersects the vertical axis (at this point, the minor principal stress is zero). This intersection point can be interpreted as the UCS of the sample.

In this paper, test Type II has been selected to be applied because (a) different confinement levels can be applied to the same rock sample so it does not require a large quantity of samples as for test Type I, and (b) it is easier to perform than test Type III when it comes to controlling the confinement that applies to the rock sample.

Rock samples used in this study are solely granite. The reason for choosing granite for the study is that (a) granite is a relatively homogenous rock, (b) a general database of granite is available internationally for comparison, and (c) the granite that is tested (Iddefjord granite) is a reference rock type at our laboratory.

3. The Performed Test Procedure and Apparatus

The test procedure in this research was conducted according to the ISRM’s Suggested Methods for Determining the Strength of Rock Materials in Triaxial Compression [2]. In the following, the procedures related to sample preparation, apparatus, and steps of the test will be described.

3.1. Sample Preparation

For these tests, samples consisting of Iddefjord granite were used. Iddefjord granite was used due to its homogenous appearance, and the NTNU/SINTEF laboratory has a comprehensive UCS database for this specific rock type to facilitate the comparison of the test results. To some extent, one can say that Iddefjord granite constitutes a reference rock type for our laboratory.

The core samples were drilled from a cubical rock block of Iddefjord granite of about 30 cm in each dimension. The rock block was intact, without any visible cracks. The rock material is, in this context, considered relatively homogenous, as shown in Figure 3.

After drilling, the cores were controlled to check the intactness and smoothness of the core wall. The qualified long cores were then divided into shorter sections of approximately 120 mm. The two ends of the short cores were flattened to an accuracy of 0.02 mm and perpendicular to the core axis. The preparation of the cores was performed according to sample preparation recommendations in the ISRM’s Suggested Methods for Determining the Strength of Rock Materials in Triaxial Compression [2]. The sample was prepared and tested as soon as possible without any further drying of saturation. Thus, it can be stated that the samples were normal air-dried samples.

Samples A, B, and C after preparation are shown in Figure 4. A sample with installed sensors and ready for testing is presented in Figure 5.

Prior to the triaxial tests, the samples were measured and tested to identify additional characteristics (as shown in

Table 3. Some additional measurements for the samples.

Sample	Diameter (mm)	Height (mm)	Weight (g)	Density (kg/m3)	Sound Travel Time (ms)	Vp (m/s)	ne (%)	Schmidt Hammer Rn(N)	Converted Rn(L)
A	48.39	124.06	598.95	2625	23.8	5213	0.44	74	54
B	48.35	119.21	576.50	2634	24.3	4906	0.44	63	45
C	48.34	119.20	576.71	2636	24.2	4926	0.44	63	45

) to serve for the UCS calculation using empirical methods in a later stage. The exact size (diameter and height) and weight of the samples were measured with standard equipment in the laboratory. Density was calculated based on the obtained dimension and weight of the samples. The sound travel time was measured with a Proceq Pundit Lab ultrasonic instrument, and the sound velocity (Vp) was calculated based on the height of the sample and the measured sound travel time. The rebound value was measured using a Schmidt hammer N-type on the cubical rock block. It is noted that the Schmidt hammer N-type was not used on the rock cores as the energy impact is strong (stronger than the L-type) and may destroy the rock core. The rebound value L-type was then estimated using a correlation proposed by Aday and Goktan [40], which is Rn(N) = 7.124 +1.249Rn(L). Results of the measurements and non-destructive tests are presented in Table 3.

3.2. Test Apparatus

The triaxial test apparatus used in the experiments was a servo-valve-controlled hydraulic press, GCTS RTR-4000, with capacities of 4000 kN in axial load and 140 MPa in confining pressure. Load was measured with a force transducer located inside the triaxial cell. The machine stiffness in the test setup was 476 kN/mm.

3.3. The Various Steps of the Test

To demonstrate the reliability of this method, the triaxial test and the uniaxial test were carried out with the same rock sample to facilitate a consistent comparison. This required a thoughtful test procedure.

At first, an attempt was made to carry out triaxial test Type II at three loading steps, without reaching the point of failure (point C in Figure 2b). After these steps, the sample was tested without confinement until failure to obtain the UCS. It was found in this test that the sample was significantly disturbed after the three loading steps during the initial triaxial test. Indications of fractures were observed at the surface of the rock core. Due to this disturbance, it was no surprise that the result of the consecutive UCS test was quite low—reaching only about 60% of the expected UCS (taken from NTNU/SINTEF’s UCS database of Iddefjord granite).

A second attempt was executed in a procedure where the triaxial tests were run with several loading steps, starting the procedure with zero confinement. The results obtained by this test procedure complied better with the UCS values that can be obtained from the database of the NTNU/SINTEF’s laboratory (database of the UCS obtained by conventional uniaxial test). The test procedure carried out for testing in this paper follows in principle the triaxial test Type II, described in ISRM’s recommendations [2] with one deviation as the triaxial test procedure applied here starts with zero confinement. After the first step (the zero-confinement step), the triaxial test continued with several confinement levels. In this paper, the selected confinement levels were “Step-0 MPa”, “Step-2 MPa”, “Step-5 MPa”, “Step-10 MPa”, “Step-15 MPa”, and “Step-20 MPa”. The range and levels of confinement were selected based on experience obtained from extensive testing on Iddefjord granite and cover the complete behaviour of the Iddefjord granite (non-linear in the lower confinement and more linear in the higher confinement range).

The following concerns related to the test procedure must be considered:

• In “Step-0 MPa” in this test, the indication of corresponding peak strength is accurately monitored with no or minimum disturbance to the sample. This is achieved by high-frequency readings with clear and continuous plotting of the stress–strain curve, strict control of the speed of loading (based on the axial or radial strain rate), and an experienced operator;
• In principle, three loading steps after “Step-0 MPa” are sufficient but in our tests, five loading steps were used with selected confinement levels. The purpose of the five loading steps and the selected confinement levels is (a) to have more points for constructing the peak strength envelope, and (b) to observe the rock behaviour in a wide range of confinement.

All tests were controlled by an axial strain rate of 240 micro-strains per minute. From our experience with this rock, it is known that the sample is hard and brittle. Thus, the rate of 240 με/min was selected so that laboratorians could have enough time to observe and catch the peak loads in every step. A lower rate can be used but it will cause the test to be time-consuming. The rate of 240 με/min is good for the purposes. A new step was initiated by setting a new set point value of confining pressure while the axial strain rate was held constant at 240 micro-strains per minute. At peak axial stress of the final step (20 MPa confining pressure), the test sample was unloaded with axial stress control at 10 MPa per second, and then, from 20 MPa, the axial stress and confining pressure was reduced at the same rate, 1 MPa per second, until the sample was fully unloaded.

After testing with the first loading step (zero confinement), the peak strength was obtained and this was considered to be the “True-UCS”, which is the expected UCS obtained by a conventional uniaxial test. Results obtained from the consecutive loading steps (the non-zero-confinement steps) were used to construct the strength envelope curve (linear or non-linear) of the rock sample. The strength envelope curve was plotted in the principal stress domain, and the intersection of the strength envelope curve with the vertical axis was considered to be the “Estimated-UCS”. The final step was to compare the “True-UCS” with the “Estimated-UCS” to evaluate the reliability of this method for obtaining UCS.

4. Test Results

The results of the three tests for samples A, B, and C are presented in Figure 6, Figure 7 and Figure 8. The obtained peak strength for each confinement is presented in

Table 4. Peak strength at each confinement level for samples A, B, and C.

Confinement σ3 (MPa)	Peak Strength σ1 (MPa)
	Sample A	Sample B	Sample C
0	197.7	205.8	212.8
2	228.1	241.5	243.7
5	278.2	286.9	285.5
10	338.3	348.8	344.1
15	387.8	395.0	390.2
20	426.3	432.5	430.7

.

As can be seen from the figures, the tests were always starting with zero confinement. This is, as stated above, not standard procedure for a triaxial test but it was carried out particularly in this research work in order to find the “True-UCS”. The obtained “True-UCS” was then used to compare with the “Estimated-UCS” based on the subsequence triaxial steps (with different confinements) of the same sample. Thus, the “True-UCS” and the “Estimated-UCS” can be compared in the most consistent way—comparing the results from exactly the same sample.

With varying confinements, including zero confinement, the radial strain rate was larger than the axial strain rate at the near-peak period of every confinement step. This can be observed in the curves of the radial strain and axial stress versus the axial strain in Figure 6, Figure 7 and Figure 8. The tests in this research were executed with an axial strain rate control only. The development of radial strain, i.e., the acceleration of radial strain, can be used as a guide to the operator as to when critical or peak strength is reached in order to change the test to the next confinement level. An experienced operator is required to be able to evaluate and to make such decisions.

Figure 6, Figure 7 and Figure 8 also indicate that after the zero-confinement step, sample A had undergone about 2300 micro-strains of radial strain and 3600 micro-strains of axial strain. Sample B had undergone about 2300 micro-strains of radial strain and 3500 micro-strains of axial strain. Sample C had undergone about 3200 micro-strains of radial strain and 3600 micro-strains of axial strain.

5. Estimating UCS Using Linear and Parabolic Equations

From a rock mechanics point of view, the UCS of a rock sample can be estimated based on the obtained strength at different confinements. As mentioned earlier, three levels of confinement are sufficient but five levels of confinement were chosen to be used in this research (2, 5, 10, 15, and 20 MPa). The purpose has been to produce more data points for a more accurate drawing of the peak strength envelope, and to have better observation of the rock behaviour.

In this research, the estimation was performed using both linear and non-linear equations (or failure criteria) to fit the five obtained data points for each rock sample, plotting in the principal stresses domain. The linear equation is actually the Mohr–Coulomb failure criterion, and many non-linear equations/failure criteria have been presented and discussed by different authors, for example, Hoek and Brown [41], Singh et al. [42], and Shen et al. [43]. In this publication, based on their simplicity and common use, three failure criteria were selected for calculation: linear (Mohr–Coulomb), parabolic, and Hoek–Brown.

The forms of the linear and parabolic equations are as follows:

$${\sigma }_1=k{\sigma }_3+{\sigma }_{ci}$$

(1)

$${\sigma }_1=a{{\sigma }_3}^2+b{\sigma }_3+c$$

(2)

where

• Equation (1) is actually the Mohr–Coulomb failure criterion written in the form of principal stresses;
• σ₁ and σ₃ are major and minor principal stresses, respectively;
• k, σ_ci,a, b, and c are the fitting parameters. The fitting parameters are defined by using the test results and a fitting procedure;
• k is related to the internal friction angle (ϕ) of the rock with the equation as k = (1 + sin(ϕ))/(1 − sin(ϕ));
• σ_ci is also the UCS of the intact rock.

The first option, linear fitting Equation (1), is actually the Mohr–Coulomb failure criterion. The second option is parabolic Equation (2), and the reason to choose a parabolic equation is due to the vast experience with Iddefjord granite at the NTNU/SINTEF laboratory. Based on a large number of triaxial tests being performed in the laboratory over the years, it can be concluded that the obtained strength envelope for this type of rock is curvilinear. The parabolic function fits better than a linear function to the results. The process of fitting the data points and the estimation of the UCS based on the triaxial test results with linear and non-linear options for each sample are presented in Figure 9, Figure 10 and Figure 11.

Results of the “True-UCS” and the “Estimated-UCS” from the tests are presented in

Table 5. The “Estimated-UCS” compared with the “True-UCS” from the tests.

Sample	“True-UCS” MPa	“Estimated-UCS” (Linear Fitting)		“Estimated-UCS” (Parabolic Fitting)
		MPa	Error to the “True-UCS”	MPa	Error to the “True-UCS”
A	197.7	218.5	10.5%	197.26	−0.2%
B	205.8	233.1	11.7%	210.48	2.2%
C	212.8	231.2	8.0%	215.47	1.3%

.

From the estimation of the UCS values (Figure 9, Figure 10 and Figure 11), it can be seen that the coefficient of determination (R²) for the fitting equations is high for both the linear and non-linear equations. However, the R² for the non-linear fitting equation displays in general a better fit than the linear one for the tested samples. It means that a non-linear equation is a more appropriate fit for the obtained data than a linear equation for the Iddefjord granite used in this research.

Further, the test results showed that the “Estimated-UCS” using the parabolic fitting equation is much closer to the “True-UCS” than using the linear fitting equation. The linear fitting equation yielded results with 8 to 12% error, whilst the parabolic fitting yielded results with less than 2.5% error, as shown in Table 5:

• The “Estimated-UCS” using parabolic fitting for sample A provided almost identical results with the “True-UCS” (197.26 versus 197.74 MPa, respectively, means a −0.2% error);
• For sample B, the “True-UCS” is 205.84 MPa, whilst the “Estimated-UCS” using the parabolic equation is 210.48 MPa (means a 2.2% error);
• For sample C, the “True-UCS” is 212.77 MPa, whilst the “Estimated-UCS” using the parabolic equation is 215.47 MPa (means a 1.3% error).

As mentioned above, there is a difference of 1 to 2% between the “True-UCS” and the parabolic fitting “Estimated-UCS” for samples B and C. This difference, although quite small, earns an explanation in the next paragraph.

During the test and the calculation of UCS for sample B, the log shows that the test at stage zero confinement was probably stopped a bit too early. The data curve of axial stress (Sa) versus axial strain (Ea) in Figure 7 shows that the curve is not significantly curving yet. The test with zero confinement could have continued to increase to a slightly higher stress value to gain a higher “True-UCS”. Due to this early stop, the “True-UCS” obtained at this stage of the test might be slightly lower than the actual UCS of the sample. This can be confirmed by the calculation of UCS using the data points from the subsequence steps (steps with confinements of 2, 5, 10, 15, and 20 MPa). The calculation resulted in an “Estimated-UCS” equal to 210.48 MPa (using the parabolic equation), slightly higher than the obtained “True-UCS” (205.84 MPa). A similar observation and comment can be made for sample C to explain the difference between the “True-UCS” (212.77 MPa) and the “Estimated-UCS” (215.47 MPa) for this sample. Nonetheless, the differences between the “True-UCS” and the “Estimated-UCS” in samples B and C are relatively small (2.2% and 1.3%, respectively).

From the observations in samples B and C, and also the estimation result for sample A, it can be stated that if the first step (the zero-confinement step) was not stopped that early, then the “Estimated-UCS” would be identical to the “True-UCS” as demonstrated with sample A.

These initial tests that have been performed indicate that with an appropriate fitting equation, the UCS of a rock core can be derived from triaxial test Type II with a high degree of confidence. The fitting equation can be selected as a linear or non-linear equation, depending on the behaviour of the rock sample under the test.

6. Estimating UCS Using Hoek–Brown Criterion Equation

The database in our lab showed that the appropriate fitting equation for Iddejord granite would be non-linear equations. Therefore, in addition to the parabolic equation to fit the test data, the Hoek–Brown criterion equation for intact rock can also be used to fit the test data for estimating the corresponding UCS. The Hoek–Brown criterion equation for intact rock is non-linear and is presented as follows [41]:

$${\sigma }_1={\sigma }_3+{\sigma }_{ci}{\left(m_i{\displaystyle \frac{{\sigma }_3}{{\sigma }_{ci}}}+1\right)}^{1/2}$$

(3)

where

• σ₁ and σ₃ are major and minor principal stresses, respectively;
• σ_ci is the uniaxial compressive strength of the intact rock;
• m_i is a material constant for the intact rock.

From the test results for each sample, as shown in Table 4, the five data points from each sample were fitted with the best-fit Hoek–Brown criterion equation. It is noted that the data at zero confinement are not included for this fitting process. The fitting work was conducted using the Rocdata program version 5.007 [44]. Results of fitting the Hoek–Brown criterion for the tested data of samples A, B, and C are presented in Figure 12, Figure 13 and Figure 14. The results of “Estimated-UCS” with this method are 201, 215, and 217 MPa for samples A, B, and C, respectively. Results of the calculation of the “Estimated-UCS” and its error are presented in

Table 6. The “Estimated-UCS” using the Hoek–Brown criterion equation compared to the “True-UCS” from the tests.

Sample	“True-UCS” MPa	“Estimated-UCS” (Hoek–Brown Fitting)
		MPa	Error to the “True-UCS”
A	197.7	201.0	1.7%
B	205.8	215.6	4.8%
C	212.8	217.2	2.1%

.

7. Comments on the Fitting Work

Obtained parameters for all three fitting equations (linear, parabolic, and Hoek–Brown criterion) for samples A, B, and C are presented in

Table 7. The fitting equations and obtained fitting parameters.

Sample	Fitting Equation	Fitting Parameters
		k	σci
A	${\sigma }_1=k{\sigma }_3+{\sigma }_{ci}$	10.89	218.49
B		10.56	231.12
C		10.34	231.24
		a	b	c
A	${\sigma }_1=a{{\sigma }_3}^2+b{\sigma }_3+c$	−0.28	16.93	197.26
B		−0.27	16.44	210.48
C		−0.20	14.82	215.47
		mi	σci
A	${\sigma }_1={\sigma }_3+{\sigma }_{ci}{\left(m_i{\displaystyle \frac{{\sigma }_3}{{\sigma }_{ci}}}+1\right)}^{1/2}$	32	201
B		30	216
C		29	217

. As can be seen from the table, the obtained fitting parameters for the samples are fairly similar—indicating an almost homogenous behaviour between tested samples. Some comments can be made for Table 7:

• In the linear fitting equation, the slope of the equation (parameter k) for samples A, B, and C is 10.89, 10.56, and 10.34, respectively. With these values of k, the corresponding internal friction angle for the rock samples are 56, 56, and 55 degrees, respectively.
• The values of “Estimated-UCS” obtained by the linear fitting equation (parameter σ_ci), by parabolic fitting equation (parameter c), and by the Hoek and Brown criterion equation (parameter σ_ci) are not too far from each other, as presented in Table 7;
• The obtained parameter m_i for samples A, B, and C are 32, 30, and 29, respectively. These values are very much comparable to the recommended m_i-value in such programs as RocLab and RocData [44], which is 32 ± 3 for granite;
• The obtained fitting parameters (k, a, b, and c) are relatively close for all samples A, B, and C. This is because the granite in this study is relatively homogenous. It is important to note that the fitting parameters are sample-dependent. Due to the inhomogeneous nature of rock, different rock types or even different samples from the same rock type have different UCS values. Consequently, they should have their own fitting parameters for each individual sample. Thus, it is not recommended to use the obtained fitting parameters in this study as general parameters for other cases. Instead, the form of equations can be used but the fitting parameters for each sample will be estimated directly from the obtained data of triaxial testing for that particular sample;
• The linear and parabolic fitting equations were obtained by built-in function in Excel. The Hoek–Brown fitting equation was obtained from commercial code RocData as mentioned or the “Goal Seek” in Excel can also be used. The mathematical algorithm behind both situations is the least squares method. Interested readers are recommended to refer to different mathematical books for more information on the method;
• The error between “Estimated-UCS” and “True-UCS” comes from two sources, which are (a) the sample’s behaviour and (b) the selection of the fitting equation. If the sample expresses uniform behaviour at every loading step, then it is easier to find a good fitting equation for the tests and vice versa. The right choice of fitting equation will also contribute significant error to the estimate. As demonstrated above, the linear fitting equation gives a larger error than the non-linear equation for the Iddefjord granite samples performed in this research.

8. Comparison of the Estimated UCS with Other Empirical Methods

The obtained UCS from the triaxial test is also compared with other empirical methods (the 28 methods listed in Table 1) and the earlier-mentioned ANN method. The input data for the calculation using empirical methods are presented in Table 3. Results of the calculations from 28 empirical and ANN methods compared to the results from the triaxial test are presented in Figure 15, Figure 16 and Figure 17.

Based on the calculation results and the comparison, the following comments can be made:

• There is a large variety in the results from empirical methods. Most of the empirical methods underestimate the UCS. Results from most of the methods were less than 75% of the “True-UCS”, and some methods even provided a negative value—meaning tension. These poor results are considered as not acceptable. The reason for such poor results may be that each empirical method may have certain limitations of the database and applicable range;
• It seems that method (1) proposed by Tuğrul and Zarif [9] is able to give reasonable estimates for all three samples;
• Some other correlations in Table 2 were randomly tested and it is interesting to find that the equation proposed by Sharma and Singh [25] gives good estimates for all three samples (within ±15% of deviation), even though the study by Sharma and Singh did not cover granitic rock. Unfortunately, there are only three samples in our study so we are not able to comment more on this method other than that our granite samples also support Sharma and Singh’s proposed equation;
• It is not clear in the empirical and ANN methods if their estimated UCS is corresponding to a 50 mm core or other core diameters. It is assumed in this paper that the empirical and ANN methods estimate UCS with a 50-millimeter core diameter;
• The results from method (1) are relatively good for all three samples. The level of error between the results from this method and the “True-UCS” is approximately 15%, which is an accuracy considered acceptable for empirical methods;
• The error between the “Estimated-UCS” based on triaxial tests and the “True-UCS” is within 15% for all tests in this study. With the linear fitting equation, the error is from 9.5 to 13.4%; with the parabolic fitting equation, the error is from −0.2 to 2.3%; and with the Hoek–Brown criterion equation, the error is from 1.7 to 4.8%. The non-linear fitting equations provided much more accurate results than the linear fitting equation in this particular test dataset.

9. Discussion of the Test and Obtained Results

Even though only three samples have been tested in this research study, still, several interesting observations can be made:

• Estimating UCS with empirical methods resulted in a very large variation. Based on 32 empirical methods, calculations yielded results ranging from “−40” to “+330” MPa (“–” means tension, and “+” means compression) for the three granite samples in this research with an expected UCS of about 200 MPa. One probable cause could be that each empirical method is suitable within the limitations of the database that was used to build such empirical formulae. It seems that method (1) produced reasonable estimates compared to the “True-UCS” for all three samples in this research. The rest of the methods, including the ANN model, failed to give results for all three samples that fit to the “True-UCS”;
• On the other hand, the “Estimated-UCS” values based on the triaxial test were close to the “True-UCS” for all three samples in this research. The error between the “Estimated-UCS” based on triaxial tests and the “True-UCS” is within 15% (when using linear fitting equations) and within 5% (when using non-linear fitting equations). It is obvious that non-linear fitting equations provided results with much smaller error than the linear fitting for the test in this research. This may be due to the persistent behaviour of the Iddefjord granite;
• The reliability of the “Estimated-UCS” based on the results from the triaxial test can be evaluated by studying the behaviour of the tested rock sample and the obtained rock strengths at different confinements. Normally, a better coefficient of determination (R²—closer to 1) indicates a more reliable “Estimated-UCS”;
• The triaxial test performed in this research is triaxial test Type II. The method can also be applied for triaxial test Type III as the strength envelope is built for each individual rock sample. The use of results from triaxial test Type I may reduce the accuracy of the “Estimated-UCS” as test Type I is performed on different rock samples, and the inhomogeneity between the samples will have a negative impact on the accuracy of the calculation process;
• It is noted that the triaxial test and the UCS test might be different in the end effects, frictional constraints, alignment of the sample, failure mechanisms, loading rate, and loading path. However, if the triaxial test and UCS test follow a certain standard or method (ISRM’s suggested method for example), then the “Estimated-UCS” is expected to be comparable to the result from a conventional UCS test. As stated in this study, the NTNU/SINTEF laboratory has a comprehensive UCS database (which was obtained by using conventional UCS tests) for Iddefjord granite. The “Estimated-UCS” values obtained from the triaxial test data were compared to the database and found reasonable;
• It is necessary to mention again that the triaxial test procedure in this research study was not conventional. The first stage of the triaxial tests in this research (with zero confinement) was performed to facilitate the comparison between the “True-UCS” and “Estimated-UCS” on the same sample in order to demonstrate the reliability of the method. In a normal triaxial test Type II, the test will always start with a certain confinement. The zero-confinement step in our test procedure may cause some disturbance to the rock samples before carrying out the subsequent non-zero-confinement steps. This is similar to the limitation of the triaxial test Type II itself—after every testing step, the sample responds with a certain disturbance. To keep this disturbance at a minimum level, it requires an experienced operator to stop and move forward the test at the right moment—not too early (causing underestimation of peak strength) and not too late (causing too much disturbance to the sample leading to underestimation of the rock strength).

In this research, Iddefjord granite was used to demonstrate the method for estimating UCS based on triaxial test Type II but the method can be used for other rock types. From a technical point of view, the limitation of using this method is the same as for performing a triaxial test Type II. The rock type must show tendencies in the stress–strain development so that peak stress values can be determined pre-failure/weakening. Some rock types may not be suitable because of too-brittle behaviour and the subsequent non-zero-confinement steps cannot be carried out. After having a successful triaxial test Type II, depending on the indication of the test results, a linear or non-linear strength envelope can be selected for appropriate estimation of the UCS.

As mentioned earlier, the radial strain rate was larger than the axial strain rate at every near-peak point of the tests. Thus, it is essential that the speed of applying load during these peak periods should be controlled by the radial strain rate rather than the axial strain rate as it is normally performed. The adjustment of loading speed is necessary to avoid too-quick loading that may lead to a sudden failure of the rock samples, especially for brittle rock samples.

10. Conclusions and Concluding Remarks

Three Iddefjord granite samples were tested in this research in order to find possible ways to determine the UCS values from triaxial testing. The UCS of the samples were obtained by twenty-eight empirical methods, one ANN-model method, and finally by a triaxial test method. With the triaxial test method, different fitting equations are used, which include linear, parabolic, and Hoek–Brown fitting equations. Based on the results from the test, the following conclusions can be made:

• UCS estimated with empirical methods resulted in a very large variation. Only two out of twenty-nine empirical methods were able to provide the “Estimated-UCS” close to the “True-UCS” (within 15% of error). Thus, great care should be given when using empirical formulae for estimating the UCS;
• The “Estimated-UCS” based on results from the triaxial test is close to the “True-UCS” for all three samples in this research. The behaviour of the rock sample can be observed directly from the test results. Based on the observed behaviour, a proper fitting function (linear or non-linear) can be selected. The reliability of the calculation can also be evaluated based on the value of the coefficient of determination (R²) of the fitting equations;
• It can be stated that it is a much more reliable way to estimate UCS based on results from the triaxial test Type II than using empirical formulae, as demonstrated herein. With a proper fitting equation used, the error of this method can be less than 2.5%. With such accuracy, UCS can be considered to be an additional relevant outcome of triaxial test Type II;
• The method described herein is not limited to granite as used in this research, rather, it is expected to be a universal tool for all hard rock samples. The limitations in using this method for other rock types are just the same as for performing an ordinary triaxial test Type II. For example, too-brittle rock types are not suitable for triaxial tests as the sample will be broken and completely destroyed at a very early stage of the triaxial test Type II. Otherwise, the estimation of UCS can be calculated as soon as a triaxial test Type II has been performed successfully;
• The approach of determining UCS based on triaxial testing has been demonstrated after successfully applying a triaxial test Type II. This method can be applied for triaxial test Type III. The method might be applied with triaxial test Type I, however, inhomogeneity between rock samples will have a greater impact on the accuracy of the calculation when using results from this test type.

As mentioned earlier, this publication is aiming to encourage using fundamental knowledge in rock and soil mechanics to calculate the UCS by utilising data from triaxial tests. It is hoped that with the level of accuracy demonstrated in this publication, the calculation of UCS from triaxial tests becomes more common, and it becomes one of the standard outputs from triaxial tests.