Experimental Study on the Axial Deformation Characteristics of Compacted Lanzhou Loess under Traffic Loads

Abstract

It is beneficial to the sustainable development of expressway engineering to reuse excavated soil as roadbed filling material. There are a large number of filling projects using loess as a filling material in Northwest China. In this paper, the loess subgrade of an expressway in Lanzhou is taken as the research object, and a series of experimental studies are conducted using a hollow cylindrical torsion shear system to simulate the formation of a “heart-shaped” stress path and the principal stress rotation (PSR) under long-term traffic loads. The effects of the vertical cyclic dynamic stress ratio, torsion shear stress ratio, initial static shear stress, and intermediate principal stress coefficient on the axial plastic deformation and rebound deformation of compacted loess in Lanzhou were studied. The results show that the vertical cyclic stress ratio (VCSR) has a significant effect on the axial deformation of compacted loess in Lanzhou. When the VCSR is less than 0.6, all the axial strain curves develop stably with the number of cycles. With an increasing VCSR, the axial plastic deformation increases obviously, and the axial rebound deformation also increases. The vertical cyclic dynamic stress of the specimen is constant. Moreover, increasing the torsional shear stress ratio (that is, increasing the amplitude of cyclic shear stress) can greatly increase the development of axial deformation, but it has no effect on the rebound deformation curve. When the initial static shear stress exists in the specimen, the larger the initial static stress ratio (SSR) is, the larger the axial plastic deformation. The axial plastic deformation increases by approximately 33% for every 0.1 increase in the SSR. The rebound deformation of different SSRs fluctuates at the initial stage of cyclic loading, but the final stable rebound deformation is basically the same as that at the initial stage of cyclic loading. The intermediate principal stress coefficient has no effect on the development of axial strain, and the effect on axial rebound deformation is negligible. Finally, the calculation model of the axial plastic strain of Lanzhou compacted loess under traffic loads is obtained. The research results can provide a reference for the durability and settlement prediction in loess engineering.

1. Introduction

The realization of sustainable development of expressway engineering is a complex and systematic subject, which runs through the decision-making, design, construction, operation, and other stages of expressway engineering. The construction of an expressway in the mountainous areas of northwest China is of great significance to the sustainable economic development of the region. However, soil accumulation caused by a large number of excavation works in the construction of the expressway will occupy land resources. To alleviate the contradiction of the increasing shortage of available land resources, the reuse of excavated soil is in line with the concept of sustainable development. Many high-fill projects using loess as filler have appeared in Lanzhou, Yan ‘an and other places in northwest China. For example, some sections of expressways under construction in Lanzhou New District use the surrounding excavated loess, which is used as a roadbed after layered filling and compaction. After compaction, the pores of loess decrease, the particles rearrange and re-cement, and the soil becomes more homogeneous. It has greatly improved in terms of the structure, bearing capacity, physical and mechanical properties, and deformation. The properties of compacted loess are significantly different from those of undisturbed loess [1,2,3,4], which is worth further research. In addition, the compacted loess roadbed of an expressway will be affected by traffic loads for a long time. Traffic loads are a kind of long-term cyclic dynamic load with strong randomness and are closely related to the axle load, vehicle speed, pavement smoothness, and traffic flow of vehicles. It produces vertical dynamic stress and horizontal dynamic stress in the subgrade and forms a complex dynamic stress path in the subgrade soil, which causes plastic cumulative deformation of the subgrade soil, affects the durability of the ex-press way, causes uneven settlement of the subgrade, and affects driving safety and comfort. Research on the axial dynamic deformation characteristics of compacted loess under traffic load is helpful to accurately predict the settlement of the subgrade, reduce the operation and maintenance cost of expressways, enhance the disaster resistance of expressways, so as to prolong the service life of expressways, minimize the impact of expressway construction on the environment, coordinate expressway construction with environmental, economic, and social development, and realize the concept of sustainable development [5,6,7].

The existing research results show that the traffic load will form a “heart-shaped” stress path in the soil, and the phenomenon of principal stress axis rotation (PSR) occurs [8,9,10,11]. In the process of vehicle load moving from far to near and then moving away, the soil element in the subgrade is subjected to the combined action of vertical cyclic stress, horizontal cyclic stress, and shear stress. The amplitude and direction of the shear stress are constantly changing, which makes the principal stress axis rotate continuously. The rotation of the principal stress axis can increase the axial strain, shear strain, and volumetric strain to a certain extent. The settlement deformation during the road operation period is predicted by conventional dynamic triaxial test results, and the cumulative axial deformation is underestimated due to the neglect of the rotation of the principal stress axis [12,13]. Gräbe et al. [14,15] studied the influence of principal stress axis rotations on the permanent deformation and elastic properties of rail transit foundation materials through a series of hollow cylinder tests. This shows that the rotation of the principal stress axis can accelerate the development of permanent strain and reduce the elastic properties of the foundation materials. Inam et al. [16], Gallage et al. [17], and Dareeju et al. [18] evaluated the deformation characteristics of unsaturated sand subgrade materials under moving wheel loads through torsional multiring shear apparatus tests. It was proven that the rotation of the principal stress axis caused by traffic loads will further increase the cyclic plastic deformation accumulation of the subgrade, and an empirical formula for predicting plastic deformation was proposed. The rotation of the principal stress axis under traffic loads also has a non-negligible effect on the permanent deformation of saturated sand. The coupling effect of the confining pressure and vertical cyclic stress ratio with the rotation of the principal stress axis was more obvious [19]. Except for sand, the influence of traffic loads on the deformation of soft soil materials has also been the focus of many scholars. Xiao et al. [20] studied the influence of subway traffic loads on the settlement of normally consolidated soft clay. When the principal stress axis rotates, the excess pore water pressure and cumulative deformation of normally consolidated soft clay increase significantly, and the cumulative deformation increases by 9–23% compared with that without PSR. Shi et al. [21] and Wang et al. [22] took a large number of overconsolidated soft clay roadbeds in the construction of the overload preloading method in offshore engineering as the research background and studied the deformation characteristics of overconsolidated soft clay under traffic load stress paths. The overconsolidation ratio and the initial static stress ratio have obvious influences on the axial deformation and pore water pressure. Lin et al. [23] studied the change in the microstructure of undisturbed soft clay before and after cyclic loading under traffic loads. The sample may be damaged by fewer cycles at higher stress ratios or lower frequencies. Qian et al. [24], Pang et al. [25], and Guo et al. [26] conducted hollow cylindrical tests on saturated soft clay and believed that under long-term traffic loads, the effects of drainage conditions, medium principal stress coefficient, deviator stress, and cyclic stress ratio on the corresponding strain components should be considered. To describe the cyclic degradation and plastic accumulation behavior of soil under the principal stress rotation of traffic loads, Du et al. [27,28] proposed an elastoplastic model to reproduce the cyclic behavior of natural soft clay with the principal stress rotation under traffic loads, and introduced the influence of small strain stiffness and anisotropic elasticity in the model. An in-depth study of principal stress axis rotation caused by traffic loads can establish a constitutive model or empirical formula of soil considering principal stress axis rotation, which can truly reflect the mechanical properties of roadbed soil in the operation and maintenance stage, and provide a scientific basis for the design, construction, operation, and maintenance of expressway engineering.

It can be seen from the previous research results that the principal stress axis rotation caused by traffic loads has an important influence on the plastic cumulative deformation and rebound deformation of soil, and it aggravates or slows down the development of deformation after coupling with the confining pressure, cyclic stress ratio, initial static shear stress, consolidation ratio, drainage conditions, and other factors. At present, research on sand and soft clay is mainly focused on, and there are few studies on the deformation characteristics of loess under traffic loads. In this paper, the compacted loess roadbed of the expressway in Lanzhou is taken as the research background. The heart-shaped stress path and the principal stress rotation formed by the traffic load in the roadbed are simulated by a hollow cylindrical torsional shear test. The deformation characteristics of compacted loess in Lanzhou were studied. The research results are particularly important for improving the service performance and deformation control accuracy of loess roadbeds, controlling operation and maintenance costs, and reducing the impact on the environment.

2. Materials and Methods

2.1. Test Apparatus

The hollow cylindrical torsion shear system is one of the most advanced instruments used in soil dynamics research. The instruments used in the experiment were manufactured by Geotechnical Consulting & Testing Systems (GCTS), LLC in Tempe, Arizona, USA. It uses hollow cylindrical specimens, known as GCTS hollow cylindrical torsion shear systems, as shown in Figure 1. It consists of a pressure chamber, signal conditioning and wireless control system (SCON), computer-aided testing system software (CATS), hydraulic pump, pressure control panel, air pump, etc. The pressure chamber provides an internal and external closed space for the hollow cylindrical sample and is filled with water, and the internal and external pressures are uniformly applied to the sample. The signal conditioning and wireless control system is an integrated microprocessor-based digital servo controller. It can accept pressure, displacement sensors, or other analog input signals, and can also be connected to a computer through wireless communication. The CATS software communicates with the SCON through a communication protocol, controls and saves the information of all sensors, projects, samples, specimens, etc., and displays the test status in real time. The hydraulic pump provides stable pressure and power. It has two states of high pressure and low pressure, which are connected with a pressure control panel, axial loading actuator, and torsion loading actuator, respectively. The pressure control panel mainly controls the external pressure, internal pressure, and back pressure, and connects the pressure chamber through the pipeline. The air pump can not only provide gas pressure but also provide vacuum pressure, which plays an important role in sample saturation.

The GCTS hollow cylinder torsional shear system can be divided into independent axial force (W), torque ($M_T$), internal confining pressure ($P_i$), external confining pressure ($P_o$) and back pressure ($P_b$). The instrument not only has its own regular waveforms, such as sine waves and triangular waves, but can also input custom waveforms to achieve complex dynamic stress paths, such as principal stress axis rotation, through the coupling of various loads. The maximum coupling loading frequency can reach 5 Hz. The heart-shaped stress path formed by traffic loads can be simulated by maintaining the internal and external pressure constant and applying the coupling effect of the axial force and torque. The test data can be automatically collected and saved using the data acquisition system, and the data acquisition time interval can be set as needed.

2.2. Test Soil Material

The soil materials used in the test were all from the site of the expressway under construction in Lanzhou New District. The expressway subgrade adopted loess soil materials abandoned by excavation on-site, which were compacted and vibrated to meet the relevant specifications. In this paper, the same soil samples were taken from the engineering construction site, and the limit water content test, compaction test, and particle analysis test were carried out according to the ASTM standard. The compaction test was carried out using a light compaction test, and the particle analysis test was carried out using a laser particle size analyzer. The basic physical properties of the Lanzhou loess are shown in

Table 1. Physical and mechanical properties of Lanzhou loess.

Soil Types	Liquid Limit/%	Plastic Limit/%	Plasticity Index	$\mathbf{Maximum}\mathbf{Dry}\mathbf{Density},{\mathit{\rho }}_{\mathit{d}\mathit{m}\mathit{a}\mathit{x}}/\mathbf{g}/{\mathbf{cm}}^3$	Optimum Moisture $\mathbf{Content}, {\mathit{\omega }}_{\mathit{o}\mathit{p}}/\%$	Grain Composition/%
						>0.075 mm	0.005~0.075 mm	<0.005 mm
Loess	24	15.9	8.1	1.72	14.2	0.66	90.58	8.76

. According to the Unified Soil Classification System (USCS) classification standard, the Lanzhou loess sample belongs to low liquid limit clay, and the soil code is CL. The height of the hollow cylinder sample used in the test was 200 mm, the outer diameter was 100 mm, and the inner diameter was 60 mm. A dry density of 0.96${\rho }_{dmax}$ (1.65 g/cm³) was used as the control standard for the sample preparation of the remolded soil sample, and the moisture content of the sample was taken as the optimal moisture content of 14.2%. All the test samples were unsaturated, and the specific gravity of the Lanzhou loess was 2.71. The calculated saturation of the samples was 59.8%. First, the soil material was dried, crushed, and sieved by 2 mm, and the moisture content of the soil material was matched to 14.2% by the spray method. The soil was sealed and stored for more than 3 days with a fresh-keeping bag to make the moisture content in the soil more uniform. The drying method was used to further check the moisture content of the soil before sample preparation. Then, a specially designed sampling tool was used to compact the soil sample in eight layers using the compaction method. After the sample preparation was completed, it was further weighed and checked to ensure the accuracy of the moisture content. Finally, for the sample whose water content did not reach 14.2%, the mass of water required for the completed hollow cylindrical sample was calculated. Water was gently sprayed onto the surface of the sample, and the total mass of the sample was checked. The completed hollow cylinder sample was placed in a moisturizing cylinder for more than 3 days. The compacted loess hollow cylinder sample is shown in Figure 2. Because the sample was unsaturated, although the drain valve was opened during the test, no water flowed out. After the end of the dynamic cycle test, two groups of samples were taken from the removed hollow cylinder samples, and the water content after the test was measured by the drying method. The calculated average value was checked with a target water content of 14.2 %. After verification, it was found that the water content of the samples after the test was basically unchanged.

2.3. Stress Loading and Stress Path Analysis

The axial stress ${\sigma }_z$, radial stress ${\sigma }_r$, circumferential stress ${\sigma }_{\theta }$, and shear stress ${\tau }_{z\theta }$ of the soil element of the hollow cylindrical specimen are related to the applied external load as follows:

$${\sigma }_z=\frac{W}{\pi \left(R_o^2-R_i^2\right)}+\frac{P_o{R}_o^2-P_i{R}_i^2}{R_o^2-R_i^2}$$

(1)

$${\sigma }_r=\frac{P_o{R}_o+P_i{R}_i}{R_o+R_i}$$

(2)

$${\sigma }_{\theta }=\frac{P_o{R}_o-P_i{R}_i}{R_o-R_i}$$

(3)

$${\tau }_{z\theta }=\frac{3M_T}{2\pi \left(R_o^3-R_i^3\right)}$$

(4)

In the formula, $R_o$ is the outer radius of the hollow cylindrical specimen, and the value is 50 mm; $R_i$ is the inner radius of the hollow cylinder sample, and the value is 30 mm.

According to the analysis of the stress state of the hollow cylindrical specimen by Hight et al. [29], the axial load W, the external confining pressure $P_{\mathrm{o}}$, the internal confining pressure $P_i$, and the torque $M_T$ can be controlled by the stress parameters of the mean principal stress p, the intermediate principal stress coefficient b, the deviatoric stress ratio η, and the principal stress direction angle α. These stress parameters are expressed as follows:

$$\mathsf{\alpha }=\frac{1}{2}\mathrm{arc}\tan \frac{2{\tau }_{z\theta }}{{\sigma }_z-{\sigma }_{\theta }}$$

(5)

$$\mathrm{b}=\frac{{\sigma }_2-{\sigma }_3}{{\sigma }_1-{\sigma }_3}$$

(6)

$$\mathsf{\eta }=\frac{q}{p}$$

(7)

$$\mathrm{p}=\frac{{\sigma }_1+{\sigma }_2+{\sigma }_3}{3}$$

(8)

$$\mathrm{q}=\frac{1}{\sqrt{2}}\left(\sqrt{{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2}\right)$$

(9)

In the formula, ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$ are the major principal stress, the intermediate principal stress, and the minor principal stress, respectively, and q is the generalized shear stress.

The heart-shaped dynamic stress path formed by traffic loads in soil can be coupled by custom axial stress and shear stress waveforms. The axial stress and torsional shear stress waveform functions are as follows:

$${\sigma }_z^{cyc}=\Delta {\sigma }_z^{max}\left[\frac{1}{2}\cos \left(2\omega t\right)-\cos \left(\omega t\right)+0.5\right]$$

(10)

$${\tau }_z^{cyc}=\Delta {\tau }_z^{max}\left[\sin \left(\omega t\right)-\frac{1}{2}\sin \left(2\omega t\right)\right]$$

(11)

In the formula, ${\sigma }_z^{cyc}$ is the vertical stress amplitude, ${\tau }_z^{cyc}$ is the torsional shear stress amplitude, ω = 2π/T, and T is the loading period, with T = 1 s. The corresponding bidirectional loading frequency of this test is 1 Hz, and t is the loading time.

Figure 3a is the waveform diagram of axial loading, and Figure 3b is the waveform diagram of torsional loading. During the loading process, the internal and external confining pressures are maintained constant, and the ideal heart-shaped stress path formed by coupling in the ${\tau }_{z\theta d}~\left({\sigma }_{zd}-{\sigma }_{\theta d}\right)/2$ deviatoric stress plane is shown in Figure 4a. The actual heart-shaped stress path formed during 10,000 cycles of loading is shown in Figure 4b. The self-defined waveform function can meet the test requirements.

2.4. Test Scheme

The existing research results show that [30] the dynamic stress caused by traffic loads in subgrade soil will decay sharply with depth, and the dynamic stress influence depth of the standard axle load is approximately 3 m. Therefore, a relatively low consolidation stress was selected for the experiment in this paper, and the mean effective principal stress is $p_0^{\prime }=50kPa$. Since there may be initial static shear stress on the soil element under the high-fill subgrade, the initial static deviatoric stress or shear stress superimposed on the bidirectional cyclic dynamic stress will make the soil deformation have some different properties. In particular, the parameter initial static deviatoric stress ratio is defined and is expressed by the static stress ratio (SSR) [31] to measure the influence of the initial shear stress level on the soil deformation characteristics, as shown in Formula (12). In addition, the dynamic stress generated by traffic loads varies at different depths of subgrade soil. To express the influence of the vertical cyclic stress level, the parameter vertical cyclic stress ratio (VCSR) [19] is used, as shown in Formula (13). To study the relationship between the magnitude of bidirectional dynamic stress loading, the torsional shear stress ratio δ is defined, as shown in Formula (14) [19]. In this paper, the effects of the vertical cyclic dynamic stress ratio, initial static stress ratio, torsional shear stress ratio and intermediate principal stress coefficient on the plastic deformation of compacted loess are studied. The specific test scheme is shown in

Table 2. Test plan.

Sample Number	${\mathit{p}}_0^{\prime }/\mathbf{kPa}$	b	${\mathit{q}}_{\mathbf{s}}/\mathbf{kPa}$	${\mathit{\tau }}_{\mathit{z}\mathit{\theta }}^{\mathit{c}\mathit{y}\mathit{c}}/\mathbf{kPa}$	${\mathit{\sigma }}_{\mathit{z}}^{\mathit{c}\mathit{y}\mathit{c}}/\mathbf{kPa}$	SSR	VCSR	δ	Vibration Number N
Ⅰ01	50	0	0	10	20	0	0.2	1/2	10,000
Ⅰ02	50	0	0	15	30	0	0.3	1/2	10,000
Ⅰ03	50	0	0	20	40	0	0.4	1/2	10,000
Ⅰ04	50	0	0	25	50	0	0.5	1/2	10,000
Ⅰ05	50	0	0	30	60	0	0.6	1/2	10,000
Ⅱ01	50	0	0	40	40	0	0.4	1	10,000
Ⅱ02	50	0	0	20	40	0	0.4	1/2	10,000
Ⅱ03	50	0	0	10	40	0	0.4	1/4	10,000
Ⅱ04	50	0	0	0	40	0	0.4	0	10,000
Ⅲ01	50	0	10	20	40	0.1	0.4	1/2	10,000
Ⅲ02	50	0	20	20	40	0.2	0.4	1/2	10,000
Ⅲ03	50	0	30	20	40	0.3	0.4	1/2	10,000
Ⅳ01	50	0.25	20	20	40	0.2	0.4	1/2	10,000
Ⅳ02	50	0.5	20	20	40	0.2	0.4	1/2	10,000
Ⅳ03	50	0.75	20	20	40	0.2	0.4	1/2	10,000
Ⅳ04	50	1	20	20	40	0.2	0.4	1/2	10,000

.

$$\mathrm{SSR}=\frac{q_s}{2p_0^{\prime }}$$

(12)

$$\mathrm{VCSR}=\frac{{\sigma }_z^{cyc}}{2p_0^{\prime }}$$

(13)

$$\mathsf{\delta }=\frac{{\tau }_{z\theta }^{cyc}}{{\sigma }_z^{cyc}}$$

(14)

where $q_s$ is the initial static deviator stress; $p_0^{\prime }$ is the initial mean effective principal stress; ${\sigma }_z^{cyc}$ is the axial cyclic dynamic stress amplitude; and ${\tau }_{z\theta }^{cyc}$ is the cyclic shear stress amplitude.

3. Results and Discussion

3.1. Effect of the Vertical Cyclic Stress Ratio (VCSR)

The axial strain ${\epsilon }_a$ development curves with vertical cyclic stress ratios of 0.2, 0.3, 0.4, 0.5, and 0.6 are shown in Figure 5 when the mean effective principal stress of Lanzhou compacted loess is 50 kPa and the torsional shear stress ratio is 0.5. Figure 6 shows the axial cumulative deformation ${\epsilon }_a^p$ curves of different vertical cyclic stress ratios. It can be seen from the figure that the five vertical cyclic stress ratios selected did not reach the failure stress of the sample. All the axial strain curves were stable with the number of vibrations. The axial strain rate accumulated rapidly from the initial stage and then decreased gradually with the number of cycles. After 1000 cycles, it basically stabilized. As the vertical cyclic stress ratio increased, the axial strain also increased. This was mainly because the pore structure of the sample was further compacted at the initial stage of cyclic loading, and the axial deformation was large. After the pore structure is compacted to a certain extent by cyclic dynamic stress, the cyclic dynamic stress cannot overcome the effect between soil particles, and the axial deformation enters the ‘dynamic stability’ state. Compared with the saturated remolded loess studied by other scholars [32], when the vertical cyclic stress ratio also reached 0.6, the saturated remolded loess sample was destroyed under small vibration times. However, the axial strain of the compacted loess in Lanzhou was still small, and the axial cumulative deformation was approximately 1%. This indicates that the water content has an aggravating effect on the plastic deformation of compacted loess under traffic loads. Compared with the research results of soft clay under traffic loads [33], it was found that soft clay has a large cumulative deformation and failure when the vertical cyclic stress ratio reaches 0.3, indicating that Lanzhou compacted loess is also a good subgrade filling material. The cumulative deformation under a traffic load with a vertical cyclic stress ratio less than 0.6 is small. In addition, when the VCSR was 0.2, the axial strain was small and finally reached approximately 0.1%. When the VCSR increased to 0.3, the axial strain increased greatly to 0.6%, and the growth rate was approximately six times. Therefore, in road engineering, special attention should be given to the increase in axial deformation when the vertical cyclic stress ratio is between 0.2 and 0.3.

Figure 7 shows the axial rebound deformation curves of different vertical cyclic stress ratios. It can be seen from the diagram that the larger the vertical cyclic stress ratio is, the greater the axial rebound deformation ${\epsilon }_a^r$ at the same vibration number. When the VCSR is 0.2 and 0.3, the rebound deformation is almost unchanged with the increase in the number of vibrations. When the VCSR increases to 0.4, 0.5, and 0.6, the axial rebound deformation curve fluctuates in the first ten cycles and then gradually increases with the number of vibrations. After 1000 cycles, the situation tends to be stable, which is a certain increase from the initial state. According to the definition of the resilient modulus proposed by SEED [34] as the ratio of the cyclic dynamic stress to the resilient strain, the resilient modulus of compacted loess in Lanzhou has a certain increase after a large number of cycles; therefore, the axial partial stiffness of loess samples has a certain increase and tends to be stable, which is consistent with Qian et al.’s conclusion that soil exhibits cyclic strengthening at a lower stress level [35].

3.2. The Effect of the Torsional Shear Stress Ratio (δ)

Figure 8 shows the axial deformation development curve of compacted loess in Lanzhou when the mean principal stress is 50 kPa, the initial static stress ratio is 0, the vertical cyclic stress ratio is 0.4, and the torsional shear stress ratios are 0, 1/4, 1/2, and 1. Figure 9 shows the axial cumulative deformation ${\epsilon }_a^p$ curves of different torsional shear stress ratios. As shown in Figure 8, the torsional shear stress ratio had a significant effect on the axial deformation of the compacted loess in Lanzhou. When δ is zero, there is only vertical cyclic dynamic stress, and the axial deformation without cyclic shear stress is the smallest. Maintaining the vertical cyclic dynamic stress unchanged at 40 kPa and gradually increasing the torsional shear stress, the shear stress makes the soil particles more prone to dislocation; therefore, the axial deformation of the soil sample still increases greatly when the vertical cyclic stress ratio remains unchanged. Figure 9 shows that the cumulative deformation increases by approximately 50.9%, 99.2%, and 274.6% when the torsional shear stress ratio is 1/4, 1/2, and 1, respectively, compared with that without dynamic shear stress. From Figure 10, it can be seen that the torsional shear stress ratio has little effect on the axial rebound deformation curve. The rebound deformation of the sample is different at the initial stage of compression–torsion coupling, but after ten cycles of compression– torsion coupling, the original ‘anisotropy’ of the sample is gradually eliminated. The rebound deformation curves of the samples with different torsional shear stress ratios are basically coincident, and they are basically stable at approximately 0.06% after long-term cyclic loading.

3.3. Effect of the Initial Static Stress Ratio (SSR)

Figure 11 shows the axial strain development curve of Lanzhou compacted loess with a mean principal stress of 50 kPa, a VCSR of 0.4, a torsional shear stress ratio of 0.5, and initial static stress ratios of 0, 0.1, 0.2, and 0.3. Figure 12 shows the curve of the axial cumulative deformation ${\epsilon }_a^p$ with the number of cycles under different initial static stress ratios. As shown in the figure, the static deviatoric stress also has a significant effect on the axial cumulative deformation of compacted loess in Lanzhou. The axial cumulative deformation is the smallest when the initial SSR is zero. With an increase in the initial SSR, the axial cumulative deformation increases significantly. For every 0.1 increase in the SSR, the axial cumulative deformation increases by approximately 33%. The final axial cumulative deformation when the SSR is 0.3 is approximately two times that when the SSR is 0, indicating that the initial static deviatoric stress in the soil can increase the axial cumulative deformation after superposition with the dynamic stress. Figure 13 shows that the SSR has little effect on the axial rebound deformation of Lanzhou compacted loess under a long-term traffic load. The rebound deformation fluctuates at the initial stage of cyclic loading. With an increase in the number of vibration cycles, the final stable rebound deformation is basically the same as the initial stage. The final stable rebound deformation of different SSRs is approximately 0.07%.

3.4. Influence of the Intermediate Principal Stress Coefficient (b)

Figure 14 shows the axial strain development curve of Lanzhou compacted loess with different intermediate principal stress coefficients when the mean principal stress is 50 kPa, the initial static stress ratio is 0.4, the vertical cyclic dynamic stress is 40 kPa, and the torsional cyclic shear stress is 20 kPa. It can be seen from the figure that the intermediate principal stress coefficient has no effect on the axial strain development of the Lanzhou compacted loess, and the axial strain development curves of the different intermediate principal stress coefficients almost coincide. From Figure 15, it can be seen that the axial cumulative deformation curves of different intermediate principal stress coefficients are almost coincident. After 10,000 compression–torsion coupling cycles, the axial deformation is basically stable at approximately 0.9%. Figure 16 shows that the intermediate principal stress coefficient has a weak effect on the axial rebound deformation curve and can be basically ignored. In general, with the increase in the vibration times, the axial rebound deformation increases first, then decreases gradually after 10 times, and increases gradually after 50 times. Compared with the initial value, the growth is relatively gentle. Finally, the rebound deformation of different intermediate principal stress coefficients is basically stable at approximately 0.06%.

3.5. Calculation Model of the Axial Cumulative Deformation

Lanzhou compacted loess tended to stabilize at different vertical cyclic stress ratios and torsional shear stress ratios after 1000 cycles, and there was no linear correlation between the vertical cumulative deformation and vibration number after 1000 cycles, as shown in Figure 17. Referring to the study of Wang et al. [36], the empirical formula of the cumulative strain can be established as:

$$\lg {\epsilon }_a^p=\lg {\epsilon }_{a,1000}^p+k\lg \frac{N}{1000}$$

(15)

$$\mathrm{or}{\epsilon }_a^p={\epsilon }_{a,1000}^p{\left(\frac{N}{1000}\right)}^k$$

(16)

In the formula, lg is the abbreviation of Common logarithm with base 10, that is, $\lg {\epsilon }_a^p={\log }_{10}{\epsilon }_a^p$, k is the slope of the relationship curve in the double logarithmic coordinates of $\lg {\epsilon }_a^p~\lg N$, and ${\epsilon }_{a,1000}^p$ is the vertical cumulative plastic deformation of the 1000th cycle.

To simplify the calculation of the model parameters, the total cyclic stress ratio (CSR) proposed by Cai et al. [37] was introduced to consider the influence of the vertical cyclic stress ratio and torsional shear stress ratio:

$$\mathrm{CSR}=\mathrm{VCSR}\sqrt{1+4{\delta }^2}$$

(17)

The Lanzhou loess corresponding relationship between the slope of the fitted line in Figure 17 and the CSR plotted in Figure 18. It can be seen from the figure that with an increase in the total cyclic stress ratio, the slope gradually decreases, indicating that with the increase in the total cyclic stress ratio, the soil has a smaller and more stable vertical cumulative deformation after 1000 cycles. The relationship between the axial plastic deformation and CSR of the Lanzhou loess in the 1000th cycle was established. As shown in Figure 19, the axial plastic deformation and CSR have a linear relationship in the 1000th cycle.

Therefore, the calculation model of the vertical plastic deformation of the Lanzhou loess can be expressed as:

$${\epsilon }_a^p=\left(a\ast CS{R}^b\right)\ast {\left(\frac{N}{N_f}\right)}^{\left(m\ast CSR+n\right)}$$

(18)

where CSR is the total cyclic stress ratio, N is the number of cycles, $N_f=1000$, $\mathrm{a}=0.0431$, $\mathrm{b}=-1.104$, $\mathrm{m}=1.5013$, and $\mathrm{n}=-0.2246$.

To verify the accuracy of the above calculation model, a set of verification tests that were different from the above stress conditions were carried out on Lanzhou loess. The average principal stress was 50 kPa, the vertical cyclic stress ratio VCSR was 0.5, the torsional shear stress ratio δ was 0.2, and the initial SSR was 0. The axial cumulative deformation curve and the model calculation curve obtained by the test are shown in Figure 20. It can be seen from the figure that the final axial deformation error between the two is only 3.1%, indicating that the calculation results of the calculation model can be used as a reference for settlement prediction.

4. Conclusions

In this paper, based on the engineering background of using compacted loess as a roadbed in the Lanzhou expressway in China, the heart-shaped stress path and principal stress rotation formed in soil under traffic loads were simulated by a hollow cylinder test. The effects of the vertical cyclic stress ratio (VCSR), torsional shear stress ratio (δ), initial static stress ratio (SSR), and intermediate principal stress coefficient (b) on the axial deformation were studied. Moreover, a calculation model of the axial cumulative deformation was obtained. The research results can provide a reference for durability research in loess roadbed engineering. The main conclusions are as follows.

(1)

The vertical cyclic stress ratio has a significant effect on the axial deformation of Lanzhou compacted loess, and the axial strain increases with an increasing vertical cyclic stress ratio at the same number of cycles. The strain increases rapidly at the beginning of the cycle, then the strain growth rate slows, and the axial plastic deformation tends to be stable after 1000 cycles. After a large number of cycles, the resilient modulus of the Lanzhou compacted loess has a certain increase, and the axial partial stiffness has a certain increase and tends to stabilize.

(2)

The torsional shear stress ratio also has a significant influence on the axial deformation of Lanzhou compacted loess. Maintaining the vertical cyclic dynamic stress unchanged and increasing the cyclic shear stress, the axial deformation also increases greatly. The torsional shear stress ratio has little effect on the axial rebound deformation curve. After long-term cyclic loading, the axial rebound deformation curves of different torsional shear stress ratios almost coincide and finally stabilize at approximately 0.06%.

(3)

When there is an initial static deviatoric stress or shear stress in the soil, the development of axial cumulative deformation can be aggravated under long-term traffic loads. For every 0.1 increase in the SSR, the axial cumulative deformation increases by approximately 33%. Although the rebound deformation is different and fluctuates at the beginning of the cycle, the final stable rebound deformations of the different SSRs are basically the same.

(4)

Under traffic loads, the intermediate principal stress coefficient has no effect on the axial strain development of Lanzhou compacted loess, and the effect on axial rebound deformation is negligible.

(5)

The introduction of the total cyclic stress ratio (CSR) can consider both VCSR and δ to reduce the parameters of the calculation model of the vertical plastic cumulative deformation of Lanzhou compacted loess. The calculation model has good applicability through experimental verification for the soil preparation in this study.