Determination of the Shear Strength of Unsaturated Loess Samples from Conventional Triaxial Shear Tests Applying Rubber Membrane Correction

Abstract

The shear strength parameters of loess samples are determined from conventional triaxial shear test results and used in the rational design of sustainable geotechnical infrastructures. However, the rubber membrane that is used in the triaxial shear apparatus for applying the all-around pressure to the test specimen has a significant influence on the measured shear strength parameters. In this paper, remolded and undisturbed unsaturated loess samples from northwest China are used in a comprehensive testing program to determine the shear strength from triaxial tests and understand the influence of a rubber membrane. The results show that the measured undrained cohesion from unconsolidated undrained triaxial tests on unsaturated soil specimens with and without a rubber membrane are significantly different. In this study, differences in the shear strength with and without a rubber membrane are assessed from shear strength index values that can be determined from undrained cohesion and the internal friction angle derived from conventional triaxial tests. Experimental results suggest that predominant changes arise mainly in the undrained cohesion values. The change rate of shear strength indices values of undisturbed loess shows a strong correlation with its water content; however, it is weak for remolded loess. The correlation coefficient between error and measured values of all shear strength indices is more than 0.8. Empirical correction relationships for triaxial shear tests with a rubber membrane for three different types of loess were established from the investigations. The simple approach used in this study can be used as a reference to apply corrections to the measured undrained cohesion values of unsaturated loess samples from northwest China.

1. Introduction

The cohesion and the angle of internal friction are the two key parameters required for reliable determination and interpretation of the shear strength of soils. Information about these two parameters is required in the rational design of a sustainable geotechnical infrastructure considering stability aspects [1,2,3]. Triaxial shear testing equipment is widely used for determining the conventional shear strength parameters for saturated soils [4,5]. In a triaxial test, the specimen used for testing is wrapped with a rubber membrane to separate it from water (oil), which facilitates applying boundary stresses and controlling the drainage conditions to determine the drained or undrained shear strength of saturated soils [6]. However, the rubber membrane introduces errors in the measured effective shear strength parameters (i.e., c′ and ϕ′) that are typically determined from consolidation drained (CD) and consolidated undrained test (CU) tests. The rubber membrane can likely become embedded into the pore space of the soil particles with the application of high confining pressure; due to this reason, the measured volume change of the specimen is typically greater than the real value [7]. The measured effective shear strength parameters from the CU tests can be influenced due to the rubber membrane that impacts the volume change behavior and introduces errors in the results (i.e., the effect of membrane compliance) [8]. Both CD and CU tests are performed on saturated soil specimens. Studies from the literature suggest that the rubber membrane contributes to a maximum of 85% volumetric measurement errors [9] and up to a 50% error in the shear strength [10]. Along similar lines, the total shear strength parameters (i.e., c_u and ϕ_u = 0) that are determined from UU tests are also influenced. In addition, the influence of the type and the thickness of the rubber membrane can influence the triaxial test results [11,12].

There are four commonly used methods to address the errors associated with a rubber membrane in soil shear strength testing:

(i)

Change the force conditions: Filling the space between the rubber membrane and the specimen with the liquid rubber, copper, or other materials [12,13,14] to reduce the embedded volume. However, this technique also has an impact on the test results by reducing the actual influence of the axial force conditions of the specimen, triggering new errors;

(ii)

Instrumental compensation method: Water is replenished within the specimen [15,16,17] or between a special dual membrane [18] to compensate for the embedded volume of the rubber membrane. This method compensates for the localized drainage of the specimen due to the embedded rubber membrane and is believed to eliminate 100% of the effect [19];

(iii)

Optimization of measurement systems: Typically, water is continuously injected or withdrawn either by using manual techniques [20] or by using computerized control techniques [15,21,22] to compensate for the effect of membrane compliance. Copper rods of different diameters and the same height are placed in the specimen, and the embedded volume is estimated by the linear relationship between the change of total volume and diameter of copper rods [23]. The copper rod method has been subsequently improved by many scholars [24,25,26]. Various empirical calculations for determining the embedded volume have also been proposed [27,28];

(iv)

Analytical derivation method: Martin et al. [29] proposed a theoretical and computational method based on the generation and development of pore pressure. The analytical expression of the embedded volume has been derived by the elasto-mechanical method used in calibration and applied to correct the measured data. However, this technique is found to not be suitable for well-graded coarse-grained soil samples [6].

The different methods summarized are mainly derived from studies on saturated soils. Studies about the influence of a rubber membrane on the strength of loess deposits from the triaxial shear test are rather limited in the literature, which warrants further investigation. Loess deposits that are widely distributed throughout the world are typically in a state of unsaturated condition. These deposits are extensively found within China, particularly in the Xian, Shanxi, and Henan provinces [30,31,32,33]. The main characteristics of loess, which is regarded as a problematic soil, are wet subsidence and its associated structural problems with the performance of geotechnical infrastructures constructed with them or within them [34,35]. The accuracy of loess parameters is crucial for engineering stability analysis and disaster assessment [36,37] for assuring the sustainability of infrastructures designed in problematic loess deposits. The compressive strength and shear strength are recognized as fundamental mechanical properties by the geotechnical community [38]. According to the Mohr–Coulomb strength theory, which is widely used in civil engineering [39,40,41,42,43,44], the shear strength envelope of saturated samples from unconsolidated undrained tests (UU tests) theoretically should be horizontal (i.e., ϕ_u = 0). However, as discussed, the loess is typically in a state of unsaturated condition. Due to this, UU tests will be influenced due to suction; for this reason, the shear strength envelope may not be horizontal. In this study, UU tests were, however, performed on identical loess samples (remolded or undisturbed) with the same water content, applying different confining pressures quickly. The suction change associated with the application of confining pressure was ignored as samples were sheared quickly. In other words, rigorous interpretation of unsaturated soils in terms of two independent stress state variables, namely, net normal stress and suction, is not extended in this study. However, for quantifying the results, the shear strength index (i.e., the cohesion and the internal friction angle) is determined using the p–q curve (i.e., modified shear strength envelope). As discussed earlier, UU test results with a rubber membrane contribute to errors in the determined shear strength index values. Therefore, in this study, a UU test (σ₃ = 0) (which is a conventional unconfined compression, UC, test) is conducted without a rubber membrane. This test can provide a reliable shear strength index that can be used to correct the UU test with a rubber membrane. The focus of this paper is to (i) quantify the influence of a rubber membrane on the measured shear strength of undisturbed and remolded loess on unsaturated specimens; (ii) identify the influence mechanisms and factors of a rubber membrane on the measured loess shear strength by the UU test; (iii) propose correction relationships considering the influence of a rubber membrane on the shear strength of undisturbed and remolded loess, respectively; and (iv) extend a conventional approach for interpreting the total shear strength of unsaturated soils (i.e., loess deposits of northwest China).

The correction method that is proposed in this study can also be used for other types of soils, extending the approach described in this study. In other words, new relationships have to be developed by performing experimental studies. It is important to note that the proposed correction relationships are valid only for the loess samples used in this study (i.e., from northwest China).

2. Experiments and Methods

2.1. Soil Samples and Physical Properties

All unsaturated loess samples used in this study are collected from man-made high and steep slopes and the palm face of the tunnel projects that are distributed in Lanzhou, Dingxi, and Yanan in northwest China. The specific sampling locations and the related physical parameters are listed in

Table 1. Physical properties of loess samples in different sampling locations.

Sample NO.	Loess Types	Age	Sampling Location	w (%)	ρd (g/cm3)	e	Sr (%)	IP
S1-1-R	Remolded loess	/	Lanzhou	13.00	1.70	0.59	59.67	13.95
S1-2-R				15.00			68.85
S1-3-R				17.00			78.03
S1-4-R				19.00			87.21
S2-1-R			Wangjiagou tunnel	11.00	1.70	0.59	50.49	15.56
S2-2-R				12.00			55.08
S2-3-R				13.00			59.67
S2-4-R				14.00			64.26
S2-5-R				15.00			68.85
S2-6-R				16.00			73.44
S2-7-R				17.00			78.03
S2-8-R				18.00			82.62
S2-9-R				20.00			91.80
S3-1-R			Beiershilipu tunnel	13.00	1.70	0.59	59.67	13.51
S3-2-R				15.00			68.85
S3-3-R				17.00			78.03
S4-UD	Undisturbed loess	Q3	Weijiazui tunnel	25.64	1.55	0.74	93.31	13.12
S5-UD			Wangjiagou tunnel	23.03	1.65	0.64	97.71	15.56
S6-UD
S7-UD		Q2	Xujiachuan tunnel	14.77	1.92	0.41	98.16	17.76
S8-UD
S9-UD			Xinbantashan tunnel	19.64	1.68	0.61	87.34	15.95

Note: S1R: Jiuzhou, Lanzhou; S2R–S8UD: Baoji–Lanzhou tunnels; S9UD: Yanan railway tunnels. S5UD and S7UD are horizontal samples, and S6UD and S8UD are vertical samples. The R in the sample no. means remolded loess, and UD means undisturbed loess.

. To minimize the disturbance, all undisturbed loess samples were processed on-site as cylindrical samples with a diameter of 100 mm and a height of 150 mm. The samples collected from the field were processed in the laboratory into standard specimens with a diameter of 61.8 mm and a height of 125 mm. The remolded samples were prepared by statically compacting them using the hydraulic compression method in the laboratory [45]. The test samples constitute remolded loess (S1R-S3R), Q₃ undisturbed loess (S4UD-S6UD), and Q₂ undisturbed loess (S7UD-S9UD). The water content (w), dry density (ρ_d), void ratio (e), degree of saturation (S_r), and plasticity index (I_P) of all samples are listed in Table 1.

2.2. Experiments

UU tests with and without a rubber membrane on three types of loess samples (S1R–S9UD) were conducted in triaxial shear equipment using the conventional strain-controlled triaxial instrument (TSZ-3) (Figure 1) manufactured by the Nanjing Soil Instrument Factory, which is mainly composed of a pressure chamber, a pressure system, and a pore water pressure system. The maximum load is 30 kN, and the loading rate can be adjusted between 0.002 and 4.5 mm/min, with a relative error ≤ 5%. The confining and axial pressure are digital display and numerically controlled (control error ± 1% FS), with maximum values of 2.0 MPa and 1.0 MPa, respectively. The maximum pore pressure can be measured up to 2.0 MPa, which is also a digital display, and the measurement error is ±1% FS. Rubber membranes with a thickness of 0.2 mm are used for performing the tests and are also supplied by the same manufacturer of this equipment. For each loess sample, four or five identical specimens with the same water content were prepared. On one of the specimens, a UU test (confined pressure is equal to 0) was conducted without a rubber membrane, and the other four (or three) specimens were tested quickly under UU conditions (applying different confined pressures) with a rubber membrane. After the specimens were placed and the confined pressure was set, the axial stress was gradually increased until the specimens failed or deformed (Figure 1b). The modified failure envelope results are summarized (Figure 2). As discussed earlier, the suction change during the test was ignored as the test specimens were sheared quickly. It is important to note that an angle of internal friction has to be measured; unlike saturated soils, the angle of internal friction, φ_u = 0 is not equal to zero.

2.3. Methods

2.3.1. Method 1

The principal stress and shear stress were calculated using Equation (1) from the UU tests performed with a rubber membrane on unsaturated loess specimens. Extending the Mohr–Coulomb strength theory, the modified shear strength envelope (Line K_f) was plotted (Figure 2), and the undrained cohesion c_u and internal friction angle φ_u were calculated by Equation (2) (Method 1). However, the UU tests performed with different confining pressures with a rubber membrane provide c_u and φ_u values, but error exists. The c_u and φ_u are referred to as the measured cohesion and internal friction angle, respectively, in this study.

$$\left\{\begin{array}{l}s=\frac{{\sigma }_1+{\sigma }_3}{2}\\ t=\frac{{\sigma }_1-{\sigma }_3}{2}\end{array}\right.,$$

(1)

$$t=s\sin {\phi }_u+c_u\cos {\phi }_u$$

(2)

where s is the average principal stress, kPa (symbol p is also used in the literature); t (symbol q is also used in the literature) is the shear stress (the symbol q is also used in the literature), kPa; σ₁, σ₃ are the maximum and minimum principal stresses, kPa, respectively; c_u is the undrained cohesion, kPa; and φ_u is the effective internal friction angle, °. The criteria for failure are summarized below.

$$\sin {\phi }_u=\frac{({\sigma }_1-{\sigma }_3)/2}{c_u\cot {\phi }_u+({\sigma }_1+{\sigma }_3)/2},$$

(3)

2.3.2. Method 2

The UU test, with confining pressure equal to 0, as discussed earlier, was conducted without a rubber membrane, which means the measured shear strength of loess was reliable (i.e., without rubber membrane influence). Such a test is a conventional unconfined compression (UC) test. We corrected the shear strength from Method 1 by introducing the UU test (σ₃ = 0), which can reflect the true shear strength. The corrected shear strength envelope (Line K_f0) is the best-fit line of all measured points in the p–q coordinates, which comprises the UU test results with and without a rubber membrane (Method 2). It means the combination of one test without a membrane and some tests with a membrane. Compared with K_f, K_f0 can reflect the true shear strength values. Accordingly, the corrected cohesion c_u0 and internal friction angle φ_u0 can be determined (Equation (2)), and the two parameters are referred to as the corrected cohesion and corrected internal friction angle.

3. Results

3.1. Loess Shear Strength

According to the test data, the corresponding shear strength lines K_f and K_f0 can be obtained by extending Method 1 and 2, respectively (see Section 2.3) using (Equation (2)). Figure 2 summarizes typical results. The shear strength index from the K_f and K_f0 are shown in

Table 2. The shear strength of all loesses from different shear strength lines (K_f and K_f0).

Sample No.	Kf		Kf0		Sample No.	Kf		Kf0
	tanφu	cu	tanφu0	cu0		tanφu	cu	tanφu0	cu0
S1-1-R	0.68	58.32	0.71	50.82	S2-8-R	0.66	35.94	0.69	25.77
S1-2-R	0.69	42.87	0.72	35.53	S2-9-R	0.28	35.2	0.34	22.62
S1-3-R	0.38	31.52	0.41	25.9	S3-1-R	0.52	86.67	0.55	78.56
S1-4-R	0.24	25.08	0.24	24.94	S3-2-R	0.48	56.93	0.53	44.2
S2-1-R	0.68	121.09	0.71	108.68	S3-3-R	0.43	48.51	0.49	34.67
S2-2-R	0.44	121.82	0.47	112.6	S4-UD	0.19	58.85	0.21	49.75
S2-3-R	0.68	73.25	0.7	68.46	S5-UD	0.34	48.42	0.38	37.71
S2-4-R	0.49	61.95	0.54	50.16	S6-UD	0.35	61.72	0.39	50.14
S2-5-R	0.59	59.55	0.66	42.3	S7-UD	0.85	474.97	0.87	458.94
S2-6-R	0.62	62.34	0.64	56.35	S8-UD	0.72	711.72	0.73	695.97
S2-7-R	0.61	48.92	0.67	32.16	S9-UD	0.4	230.82	0.41	218.77

. It can be seen that under the influence of a rubber membrane, the K_f0 are lower than the K_f to different degrees, which relates to the distortion of shear strength from the UU test with a rubber membrane. The measured cohesion c_u obtained by Method 1 for different specimens is significantly higher than the c_u0 obtained from correction by Method 2, and the internal friction angle φ_u is reduced compared with φ_u0 (Figure 2 and Table 2). This phenomenon indicates that the rubber membrane played a damping role during the UU test, resulting in greater shear strength than the actual value.

3.2. Calculation of Shear Strength Error under the Influence of a Rubber Membrane

The measured c_u obtained under the UU test with a rubber membrane is increasing with respect to the corrected c_u0, while φ_u is in contrast, so the following parameters are defined to quantify the error of measured shear strength with a rubber membrane, respectively:

$$\Delta c=c_u-c_{u0},$$

(4)

$$\Delta \phi ={\phi }_{u0}-{\phi }_u,$$

(5)

where Δc is the loss amount of measured cohesion, kPa; and Δφ is the increment amount of the measured internal friction angle.

The change in shear strength can be explained using the parameters introduced below.

$$\eta =\frac{\Delta c}{c_u},$$

(6)

$$\lambda =\frac{\Delta \phi }{{\phi }_u},$$

(7)

where η is the loss ratio of measured cohesion, %; and λ is the increment ratio of the measured internal friction angle, %. In the above equation, the Δc and Δφ are used to characterize the amount of difference, while the η and λ are set to characterize the ratio of difference.

Due to the influence of structure on the undisturbed loess, there are differences in the physical and mechanical properties between the remolded and undisturbed loess. So too are the Q₂ and Q₃ loess due to the different periods and endowment environment of the undisturbed loess. Therefore, the influence degree of the rubber membrane on remolded loess, Q₃ undisturbed loess, and Q₂ undisturbed loess are considered separately using the following index values:

$$\overline{x}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i},$$

(8)

$$s=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}},$$

(9)

$$c_v=\frac{s}{\overline{x}},$$

(10)

where x_i is the measured value of shear strength of each loess sample; n is the total number of samples; $\overline{x}$ is the average shear strength of different soil sample types; s is the corresponding standard deviation; and c_v is the corresponding coefficient of variation.

The errors of different types of loess are calculated according to the above formula (Figure 3 and

Table 3. Comparative analysis of changes in shear strength of different types of loess samples.

Soil Types	Parameters	cu (kPa)	φu (°)	cu0 (kPa)	φu0 (°)	Δc (kPa)	Δφ (°)	η (%)	λ (%)
Remolded loess	Range	31.52–121.82	13.30–34.75	22.62–112.60	15.46–35.73	4.79–17.25	0.64–3.12	6.54–35.74	2.76–19.89
	$\overline{\mathrm{x}}$	61.12	27.57	50.86	29.29	10.26	1.72	19.56	7.21
	s	27.95	6.76	28.22	6.37	3.82	0.78	9.71	5.06
	cv	0.46	0.25	0.55	0.22	0.37	0.46	0.50	0.70
Undisturbed Q3	Range	48.42–61.72	10.58–19.38	37.71–50.14	12.01–21.25	9.10–11.58	1.04–1.87	15.46–22.12	9.30–13.52
	$\overline{\mathrm{x}}$	56.33	16.09	45.87	17.94	10.46	1.85	18.78	12.72
	s	7.00	5.22	7.07	5.15	1.26	0.10	3.33	5.62
	cv	0.12	0.32	0.15	0.29	0.12	0.05	0.18	0.44
Undisturbed Q2	Range	230.82–711.72	21.67–40.28	218.77–695.97	22.10–40.93	12.05–16.03	0.43–0.65	2.21–5.22	1.60–1.92
	$\overline{\mathrm{x}}$	472.50	32.51	457.89	33.06	14.61	0.55	3.60	1.71
	s	240.46	10.24	238.60	9.79	2.22	0.11	1.52	0.18
	cv	0.51	0.31	0.52	0.30	0.15	0.20	0.42	0.11

). The average values of the measured cohesion c_u and internal friction angle φ_u obtained are 61.12 kPa and 27.57°, respectively, for the remolded loess. The corrected c_u0 and φ_u0 are 50.86 kPa and 29.29°, and the Δc, Δφ, η, and λ are 10.26 kPa, 1.72°, 19.56%, and 7.21%, respectively, under the influence of the rubber membrane. In addition, the increment rate of the internal friction angle was higher than 16% when the water content was higher than 19% (Table 1 and Table 3). The shear strength of the specimen is less when it is closer to a fully saturated condition; however, for such scenarios, the test specimen is prone to deformation. In other words, the rubber membrane has a greater impact on the shear strength. A higher standard deviation and coefficient of variation is observed in remolded loess samples (Table 3), indicating the greatly varied shear strength of the remolded loess, which is consistent with Figure 3. It is associated with the mechanical characteristics of the remolded soils that were artificially influenced.

The coefficients of variation of the Q₃ undisturbed loess indices are relatively minimal (Table 3). The average measured cohesion and internal friction angle are 56.33 kPa and 16.09°, and those of corrected parameters are 45.87 kPa and 17.94°, respectively. The Δc, Δφ, η, and λ of Q₃ undisturbed loess are 10.46 kPa, 1.85°, 18.78%, and 12.72%, respectively, while those of Q₂ undisturbed loess are 14.61 kPa, 0.55°, 3.6%, and 1.71%, respectively. Due to the influence structure of Q₂ undisturbed loess, the cohesion value is significantly large compared to Q₃ undisturbed loess and remolded loess (Table 2). Hence, the error amount in the measured shear strength of Q₂ undisturbed loess caused by the rubber membrane is larger, but the error rate is smaller. The increment of cohesion caused by the rubber membrane is between 4.79 and 17.25 kPa (Figure 3 and Table 3), while the loss of internal friction angle is basically within 3°, corresponding to the change rate of 2.21–35.74% and 1.60–19.89%, respectively. These results suggest that the influence of the rubber membrane on the internal friction angle is relatively small in comparison to cohesion.

4. Discussions

4.1. Factors Affecting the Error of Shear Strength Index

4.1.1. Types of Soil Sample

The Q₂ undisturbed loess with the highest shear strength had the larger Δc, as well as the smaller Δφ, η, and λ in comparison to the Q₃ undisturbed and remolded loess, which offered greater resistance to the influence of the rubber membrane. That is to say, the higher the shear strength of the loess, the less it is affected by the rubber membrane. Such a behavior may be attributed to the non-homogeneity and anisotropy of the loess. The cohesion of the horizontal samples (S5, S7) was significantly smaller than that of the vertical samples (S6, S8), whereas the difference in the internal friction angle was relatively small. It is mainly related to the fact that the loess, distributed at different elevations, were subjected to the different influences of self-gravitational stresses.

4.1.2. Water Content

(a)

Relationship to the change amount of shear strength

The shear strength of the undisturbed loess is closely related to the water content naturally. The two types of undisturbed loess, Q₃ and Q₂, basically show a similar change rule that with increased w, the shear strength of them decreases so that the Δc_u and Δφ_u show a downward trend (Figure 4). For remolded loess, the natural structure and shear strength index were damaged after remolding, so the change amount of cohesion, Δc, and internal friction angle, Δφ, is disordered with the changed water content w (Figure 4), indicating that there is no significant correlation between them.

(b)

Relationship to the change rate of shear strength

It can be seen that the loss rate of cohesion, η, increases linearly with an increase in w (Figure 5a). By linearly fitting the η for remolded and undisturbed soils at different w, respectively, Equations (12) and (13) can be determined. The coefficients of determination, R², of the fitting were 0.62 and 0.74, respectively. The linear slopes of the remolded loess were higher than those of the undisturbed loess, and the maximum η of the remolded loess and undisturbed loess was 35.74% and 22.12%, respectively (Table 3). It can be concluded that the η of remolded loess shows larger correlations with w than that of undisturbed loess. It is worth noting that the degree of dispersion of the remolded soil samples is high (Figure 5), which indicates that the cohesion of remolded loess is not only determined by the water content.

$$\eta =2.97w-25.85.$$

(11)

$$\eta =1.62w-21.50.$$

(12)

There is a positive correlation between the w and λ (i.e., the λ increases with the gradual increase in the water content) (Figure 5b). The relationship between them was obtained by fitting Equations (14) and (15), and the R² were 0.58 and 0.95, respectively. It showed a high correlation in the undisturbed loess, while that in the remolded loess was much weaker, which was mainly related to the destroyed structure and shear strength of the remolded loess. The increase rate of λ − w is smaller in the undisturbed loess than in the remolded loess, which is consistent with a. It could be summarized that the change rate of shear strength in remolded loess is influenced by water content to a greater extent, while the fitting relationship between them is better in undisturbed loess.

$$\lambda =1.77\times {10}^{-4}{w}^{3.83}.$$

(13)

$$\lambda =3.78\times {10}^{-6}{w}^{4.66}.$$

(14)

4.1.3. Measured Cohesion c_u

By fitting the measured cohesion, c_u, with the loss rate of cohesion, η, (Figure 6a), it was illustrated that the η showed a decreasing power type with an increasing c_u for all types of loess samples (Equation (16)). R² is 0.80 for these results, suggesting a good fitting effect. Similarly, the relationship between c_u and λ can be obtained (i.e., Equation (17)) with a coefficient of determination, R², value of 0.45. In conclusion, the η of each type of loess shows the same decreasing law in relation to the c_u, which is mainly related to increases in the shear strength of loess with increased cohesion. Therefore, loess is less affected by the rubber membrane in the UU test, which suggests that the higher the shear strength of loess, the lower the change rate in the shear strength (Figure 3). The correlation between the c_u and λ is relatively weak and can be ignored.

$$\eta =497.07c_u^{-0.83}.$$

(15)

$$\lambda =79.71c_u^{-0.63}.$$

(16)

4.1.4. Measured Internal Friction Angle φ_u

By correlating the measured internal friction angle, φ_u, with the increased rate of the internal friction angle, λ, for different types of loess (Figure 7a), the fitted equations for the remolded loess, Q₃ undisturbed loess, and Q₂ undisturbed loess were obtained respectively (Equations (18)–(20)). The R² were 0.71, 0.98, and 0.93, respectively, and the λ of all loess showed a linear decrease with increased φ_u. It was mostly because the shear strength of loess rose as the φ_u increased, increasing resistance to the rubber membrane and therefore lowering the change rate. Similarly, by fitting the relationship between the φ_u and η of all types of loess, the fitting equation and the coefficient of determination were obtained (Figure 7b). Different types of loess showed an obvious difference, and the correlation calculation results are absolutely poor, indicating that the loss rate of cohesion is not very correlated with the measured internal friction angle, whose effect can be ignored.

$$\lambda =-0.63{\phi }_u+24.60.$$

(17)

$$\lambda =-0.47{\phi }_u+18.49.$$

(18)

$$\lambda =-0.02{\phi }_u+2.30.$$

(19)

4.2. Correction for Measured Shear Strength Index

The shear strength of loess as per the discussed procedures is determined by a triaxial shear test; however, there is a certain error between the measured cohesion and internal friction angle and the real value due to the influence of the rubber membrane. To obtain more reliable shear strength of loess for use in engineering projects from triaxial shear test results, it is necessary to correct the measured shear strength. Combining Equations (5) and (7), the correction formula for cohesion can be obtained as below:

$$c_{u0}=c_u-c_u\eta .$$

(20)

Also based on Equations (6) and (8), the correction formula for the internal friction angle can be deduced as follows:

$${\phi }_{u0}={\phi }_u+{\phi }_u\lambda .$$

(21)

In Equations (19) and (20), the measured cohesion, c_u, and internal friction angle, φ_u, as well as the basic physical and mechanical characteristics of the loess, such as water content, w, can be obtained from the triaxial shear tests. However, the loss rate of cohesion, η, and the increment rate of the internal friction angle, λ, are unknown quantities. Hence, it is necessary to establish the relationship between these two indices and the available parameters to realize the final correction. Due to the different properties of remolded and undisturbed loess, the correction methods are discussed separately.

4.2.1. Remolded Loess

For remolded soils, the correlation between water content and loess shear strength is weak due to the structure disruption, which it is difficult to portray in mathematical relationships (Figure 5). Based on the previous analysis, the power function was used to fit the cu–η, which showed a better fitting effect (Figure 6). Hence, Equation (16) was substituted into Equation (21), and the corrected cohesion of the remolded soil was able to be obtained:

$$c_{re}=c_u-497.07c_u^{0.17}.$$

(22)

Similarly, the linear fit of the relationship between the φ_u and λ of the remolded loess was relatively good. For this reason, Equation (18) was substituted into Equation (22) to derive the corrected value of internal friction angle for remolded loess:

$${\phi }_{re}=-0.63{\phi }_u^2+25.60{\phi }_u.$$

(23)

4.2.2. Undisturbed Loess

(a)

Corrections based on measured shear strength indices

Referring to the correction methods for the remolded loess above, empirical equations for correcting the shear strength of Q₃ and Q₂ undisturbed loess can be obtained. Since the c_u–η relationship of all loess generally conforms to the power function (Figure 6), the correction formula for the undisturbed loess is consistent with that for the remolded loess, i.e., Equation (23). By substituting Equations (19) and (20) into Equation (22), respectively, the corrected formula of the internal friction angle for different undisturbed loess can be obtained:

$${\phi }_{\mathrm{Q}3}=-0.47{\phi }_u^2+19.49{\phi }_u.$$

(24)

$${\phi }_{\mathrm{Q}2}=-0.02{\phi }_u^2+3.30{\phi }_u.$$

(25)

(b)

Corrections based on measured water content

The difference between undisturbed and remolded loess is reflected in whether the natural soil structure is maintained or not. It means the shear strength of undisturbed loess is directly related to the water content, so the measured shear strength of undisturbed loess can be corrected by the functional relationship between them that is well-established in this paper. Equations (13) and (15) are substituted into Equations (21) and (22), respectively, and the modified formulas of cohesion and internal friction angle for the undisturbed loess are obtained as follows:

$$c_{\mathrm{un}}=c_u(22.50-0.0162w).$$

(26)

$${\phi }_{\mathrm{un}}={\phi }_u(1+1.81\times {10}^{-15}{w}^{4.66})$$

(27)

In this study, experimental data with rubber membrane were used for samples without rubber membranes for proposing an empirical correction formula. This approach is different from previous studies in which experimental errors caused by rubber membranes were compensated directly in the early test stage; due to this reason, no comparative studies were conducted. It is worth noting that there are also some shortcomings in this study. Due to the limitation of experimental instruments and conditions in the experimental stage, a test with zero confining pressure and a rubber membrane was not carried out. In future, the authors propose carrying out relevant more experimental research studies to further promote the understanding of this problem.

5. Conclusions

In this study, the influence of a rubber membrane on the measured triaxial shear strength was investigated by performing UU tests with and without a rubber membrane of undisturbed and remolded unsaturated loess samples from Lanzhou, Dingxi, and Yanan in northwest China. Based on these investigations, several relationships for corrections were proposed that can be useful for reliable estimation of the mechanical properties of loess from simple conventional tests. The modified method proposed in this paper can also be applied to other types of soils by establishing new empirical relationships. The main conclusions are summarized below:

(1)

The measured undrained cohesion, c_u, from the UU test with a rubber membrane is significantly larger than the corrected value, c_u0, and the differentials are 5–17.5 kPa. The rate of change ranges from 2.21% to 35.74%. Meanwhile, the measured internal friction angle, φ_u, is less than the corrected value, φ_u0, with the amount being 0.43–3.12°. The rate of change ranges from 1.60% to 19.89%, which is significantly less compared to the undrained cohesion;

(2)

The influence degree of the rubber membrane on the measured cohesion, c_u, and internal friction angle, φ_u, of loess in the UU test is inversely proportional to the loess shear strength. The loss rate of cohesion shows a power function decay law with the measured cohesion, while the increased rate of the internal friction angle illustrates a negative linear correlation with the measured internal friction angle. Both the change rate of cohesion and the internal friction angle of undisturbed loess increased linearly with the increase in water content;

(3)

Based on the test data, the relationship between the shear strength indices and influencing factors of types of loess was fitted with the determination coefficient R² basically being more than 0.8. Empirical correction formulae for the measured shear strength indices developed from the UU tests with a rubber membrane of remolded loess, Q₃ undisturbed loess, and Q₂ undisturbed loess were established, respectively. The proposed correction formulae can be used for correcting the mechanical parameters of loess in engineering practice applications and designing sustainable infrastructure in problematic loess deposits of northwest China.

(4)

The proposed correction method in this study can be extended to other soils to develop new relationships. However, it is important to note that the proposed correction relationships are valid only for the mechanical parameters of loess samples from northwest China.