The use of coal gangue as roadbed filler can alleviate environmental problems and reduce environmental pollution. In this paper, the multi-vibration cyclic loading test of coal gangue roadbed filling body considering different confining pressure, compactness and gradation parameters is carried out. The deformation characteristics were studied, the residual deformation model was established, and the influence of different factors on the residual deformation value was discussed. The results show that under the cyclic loading of multiple vibration times, the deformation growth rate of coal gangue filler begins to increase, then slows down and tends to be constant. The actual residual deformation of coal gangue filler is not an accurate logarithmic relationship with the number of earthquakes. The shear and volumetric residual strains of 0 ~ 200 earthquakes have a quadratic function relation with logarithmic earthquake number lg (1 + N). There is a logarithmic relationship between the volume residual strain and the logarithmic number lg (1 + N) in 201 ~ 10000 earthquakes. All the influencing factors have a certain influence on the residual deformation value of the coal gangue specimen. According to the grey correlation degree calculation, the influence degree of the influencing factors of the coal gangue filler is ranked as the number of earthquakes, confining pressure, compaction degree and gradation. In addition, according to the grey correlation analysis, the influence of soil particle tumbling on the residual deformation is greater than the influence of soil particle crushing.
Research Article
Experimental Study on Residual Deformation of Coal Gangue Filler under Multi-vibration Cyclic Loading
https://doi.org/10.21203/rs.3.rs-2375948/v1
This work is licensed under a CC BY 4.0 License
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Coal gangue
Multiple earthquakes
Large triaxial test
Residual deformation model
Grey correlation analysis
Gangue refers to waste rock discharged during coal mine construction, coal mining, etc (Liu and Liu 2010), and excessive accumulation will cause serious environmental problems (Tang et al. 2018; Li and Wang 2019; Zhang and Ling 2020). After the coal gangue is broken, it belongs to coarse-grained soil in particle size composition. At present, the granular medium is mainly used as concrete coarse aggregate and fine aggregate filling material, to ensure the performance of concrete under the premise of improving the recycling rate of coal gangue, and made a lot of research results (Long et al. 2019; Zhou et al. 2019; Yu et al. 2022; Liu et al. 2022). In addition, when it reaches a higher density, it has excellent engineering characteristics such as strong bearing capacity, high structural stability and less prone to excessive deformation (Li et al. 2019), combined with these characteristics, the use of gangue as a subgrade filler can effectively achieve the purpose of recycling. Considering that the subgrade filler will produce residual deformation under the action of circulating load (Sawada et al. 2006; Zhou et al. 2020; Li et al. 2021; Yin et al. 2023), and the residual deformation will produce safety hazards when the residual deformation is too large, the residual deformation of gangue is one of the key points of its research on the dynamic characteristics when used as a subgrade filler.
The study of residual deformation of subgrade filler generally includes the establishment of residual deformation model and the investigation of the influencing factors of residual deformation. Residual deformation models are generally divided into two categories, namely dynamic stress-residual deformation models that describe the relationship between a certain vibration under dynamic stress and residual strain (Taniguchi and Whitman 1983; Wang et al. 2020), and vibration-residual deformation models determined directly from the residual deformation curve (such as the Shen-Zhujiang model) (Zhang et al. 2015; Fu et al. 2016; Yang and Cui 2020). Compared with the dynamic stress-residual deformation model, the vibration-residual deformation model (such as the Shenzhujiang model) can describe the volume residual deformation caused by the shear characteristics of the soil, and can effectively avoid the influence of interpolation error (Sheng and Gu 1996). However, the Shen-Zhujiang model cannot accurately describe the residual deformation law of subgrade filler under some specific influencing factors, For example, When studying the vibration residual deformation characteristics of rockfill materials, Wang et al. (2013) found that the data fitted by Shen Zhujiang residual deformation model had greater discreteness; He et al. (2018) found that the actual lg (1 + N) (vibration) residual deformation curve of pile material showed obvious nonlinearity, especially under high stress conditions.
In view of the research on the influencing factors of residual deformation of soil, scholars at home and abroad have found that the influencing factors are diverse, including the selection of filler materials, the granule gradation of fillers, the compaction of fillers, the frequency of cyclic loads, the cyclic stress ratio, and the confining pressure of the sample soil (Wichtmann and Niemunis 2005; Duku et al. 2008; Indraratna 2015; Wichtmann et al. 2019; An et al. 2021; Liu et al. 2022; Liang et al.2022). Among them, Liu et al. (2022) pointed out that the residual deformation of the soil under the action of cyclic load is mainly due to the rearrangement of the soil particles and the fragmentation of the edges and angles of the soil particles, which are closely related to the compactness of the soil; in the indoor test, the confining pressure and compaction of the specimen affect its compactness, and scholars (Nie et al. 2021) have also found that the confining pressure has an effect on the shear residual deformation and volume residual deformation of coarse-grained soil. At the same time, particle gradation also affects the degree of fragmentation of soil particles (Liu et al. 2022). In order to explore the degree of association between multivariates and residual deformation, the gray correlation method is used in this paper, which is a statistical analysis method used to compare and analyze different factors affecting the development and change of a certain parameter (Zheng et al. 2019; Qin et al. 2022), and then investigate the degree of correlation between different factors, which has been widely used in engineering experiments (Wang et al. 2016; Zhu et al. 2021). In the previous research, the small triaxial and medium-sized triaxial tests were generally used, and the maximum particle size of the gangue filler exceeded 20% of the overall diameter of the specimen, resulting in the test not close to the actual situation, in order to solve this problem, this experiment used a large triaxial test to explore the residual deformation development law of the coal gangue roadbed filler, establish a residual deformation model of the gangue filler, and at the same time, use the gray correlation method to study the residual deformation of the gangue roadbed filler under different confining pressures, compaction and grading.
2.1 Specimen preparation
The experimental filler sample studied in this paper, is taken from Xiangtan Tangjiashan coal mine, the particle size range of the filler particles spans a larger span, the uniformity is poor, the particles are scattered, the large particles have obvious edges and corners, the shape is irregular and the degree of hardness is medium, the small and medium particles are affected by weathering, and the particles are blurrier and the softer the texture as the particle size decreases, according to the standard (JTG D30—2015) (2015) remove the particles of more than 60mm, and the soil sample is shown in Fig. 1.
In this paper, the gradation control equation proposed by Talbot according to fractal theory is used to value the gradation parameters (Wu et al. 2020):
$${d_i}={x^A}=\left( {{l \mathord{\left/ {\vphantom {l {{l_{\hbox{max} }}}}} \right. \kern-0pt} {{l_{\hbox{max} }}}}} \right){}^{A}$$
1
Among this, di is the cumulative percentage content of particles smaller than the mass l of particle size in the specimen; i is the percentage of the total mass of particles with a particle size of less than one size; lmax is the maximum particle size, this test takes 60.0mm, A is the graded parameter. Considering that the inhomogeneity coefficient Cu=d60/d10 > 5 and the curvature coefficient Cc= d302/(d60·d10) = 1 ~ 3 are good graded, the A value is 0.3691 ~ 1.1133, in order to make the specimens of different grades more representative, that is, to fully consider the influence of different coarse particle content on the dynamic characteristics of the specimen, the test in this paper is: 0.63, 0.87, 1.11, and the grading parameters are shown in Fig. 2.
Before the large three-axis test, it is necessary to determine the density of the soil body under the maximum pressure compaction according to the standard (JTG D30—2015), through the surface vibration compaction test, and then determine the quality of the soil sample according to the sample volume, and the surface vibration compaction test is carried out by different graded specimens, and the density under the maximum pressure compaction of the three groups of specimens is shown in Table 1.
| Gradation parameters A | A = 0.63 | A = 0.87 | A = 1.11 |
|---|---|---|---|
| Maximum dry density g/cm3 | 1.938 | 1.913 | 1.827 |
According to the Code for the Design of Highway Subgrades (JTG D30-2015), the test was loaded with 3 to 5 layers of compaction to simulate the filling process of subgrade packing in actual engineering, and 5 layers of compaction were used in this test, with a specimen diameter of 300 mm and a height of 600 mm, as shown in Fig. 3.
2.2 Large-scale triaxial test equipment
This study adopts the DJSZ-150 dynamic and static triaxial test system developed by Chengdu Donghua Excellence Technology Co., Ltd. and the West China Geotechnical Instrument Research Institute of Sichuan University, as shown in Fig. 4, which is mainly designed for sample samples with large particle sizes such as gravel, coarse-grained soil and sandy soil, which can be used to explore the dynamic and static responses of crushed stone, coarse-grained soil and sandy soil. The system applies static load to the specimen according to displacement and load control, applies circulating load according to vibration frequency, waveform, amplitude and vibration order, collects axial displacement and volume deformation data through displacement meter and body change tube, and can collect pore pressure, confining pressure and other data.
2.3 Large-scale triaxial test scheme
The compaction levels used in this test are 90%, 93% and 95%, simulating under-compaction, standard compaction and over-compaction, respectively; Isobaric consolidation is used to consolidate the soil of the specimen; The cyclic load is based on the ratio of the general vehicle speed and length, and the load frequency f is 1Hz, and the load waveform is a semi-sine wave (Hardy and Cebon 1994); The specimen belongs to the coarse-grained soil of the gravel type with good gradation, which belongs to the partial drainage material in the actual project, so it is tested under drainage conditions. The basic test parameters in this article are shown in Table 2.
| Test parameters | The parameter value |
|---|---|
| Specimen size | Φ300×600 |
| Consolidation stress ratio | 1.0 |
| Cyclic load waveforms | Semi-sine wave |
| Circulating load amplitude(kPa) | 200kPa Single-level loading |
| Cyclic load frequency (Hz) | 1.0 |
| Number of cycle loads | 30000 times |
| Drainage conditions | Drain |
Considering the long-term nature of the traffic load action, when simulating the traffic load, the number of cyclic load vibrations is determined to be 30,000 times, and a single stage of loading is adopted. Each specimen was controlled by a confining pressure of 50 kPa, a compaction degree of 93%, and a grading parameter A = 0.63, and the specimen number was AY01; On this basis, the three influencing factors of confining pressure, compaction degree and gradation parameters were changed, and two other sets of influencing factors were fixed for comparative test, among which AP01 and AP02 changed the confining pressure size, AC01 and AC02 changed the compaction size, and AG01 and AG02 changed the particle gradation, as shown in Table 3.
| Specimen number | Containment pressure(kPa) | Compaction(%) | Gradation parameters A | Circulating load form |
|---|---|---|---|---|
| AY01 | 50 | 93 | 0.63 | 1.Single-stage loading; 2.The circulating load amplitude takes 200kPa; 3.Number of load vibrations is 30000 |
| AC01 | 100 | 93 | 0.63 | |
| AC02 | 150 | 93 | 0.63 | |
| AP01 | 50 | 90 | 0.63 | |
| AP02 | 50 | 95 | 0.63 | |
| AG01 | 50 | 93 | 0.87 | |
| AG02 | 50 | 93 | 1.11 |
3.1 Definition and value of residual deformation under cyclic loads
According to the residual deformation test scheme designed in this paper, due to the large amount of residual deformation test data, in order to facilitate comparison and calculation, combined with the characteristics of experimental data, the residual deformation is divided into three stages, and the data points are taken: the first to 1000th vibration is the rapid development stage of deformation, the residual deformation develops rapidly, and 1 point is considered to be extracted every 20 points; The 1000th to 5000th vibration is the deformation slowdown stage, the deformation gradually slows down, considering the extraction of 1 point every 100 points; Within the 5000th to 30000th vibration, the deformation is almost flattened, and it is considered to extract 1 point every 500 points. The residual deformation is an irreparable deformation, so the residual deformation is the irreparable deformation of the specimen under cyclic load, the typical residual strain is shown in Fig. 5, and the OA curve is the relationship curve between the residual strain and the vibratile frequency.
3.2 Residual deformation under different confining pressures
The axial residual strain εd and volume residual strain εvd under different confining pressures were extracted, and the axial, volumetric residual strain and vibration N relationship curves εd- N and εvd- N were plotted, respectively, as shown in Fig. 6:
According to Fig. 6, in the case of fixed circulating load amplitude, circulating load frequency and drainage conditions, etc., and the compaction degree and the coarse grain content of the specimen are fixed, the cumulative strain of the specimen increases with the increase of the number of vibrations in each group of specimens, and the rate of deformation increase is first fast and then slow, the rate of deformation increase is maximized in the first 1000 times, and then gradually decreased, the curve tends to be flat when the vibration is about 3000 ~ 8000 times, and the final curve reaches flat, and the larger the surrounding pressure, the smaller the residual deformation of the specimen.
The reason for this phenomenon is that as the number of vibrations increases, the soil of the specimen is gradually compacted and the stiffness gradually becomes larger, resulting in a larger growth in the initial stage of deformation, and then gradually slows down; And the larger the confining pressure, the larger the compaction of the specimen, resulting in a faster increase in stiffness, and the residual deformation increment and residual deformation are smaller. In actual engineering, the residual deformation of the gangue subgrade filler is larger when buried in the shallow layer, and the larger the burial depth, the smaller the cumulative deformation.
3.3 Residual deformation at different degrees of compaction
Extract the axial residual strain εd and volume residual strain εvd under different degrees of compaction, and draw the axial, volumetric residual strain and vibration N relationship curves εd- N and εvd- N, as shown in Fig. 7.
It can be found from Fig. 7 that the basic development law of the basic εd- N and εvd- N relationship curve of each specimen is also the initial growth is larger, and then gradually slows down, and the higher the compaction, the smaller the residual deformation of the specimen, and the compaction of the gravel subgrade is required to reach 90–95%, when the subgrade filler does not meet this requirement, the axial residual deformation will increase, and the compaction degree is equal to 95%, which will significantly reduce the deformation of the gangue subgrade filler.
From Fig. 6 and Fig. 7, it can be found that the increase in confining pressure and the increase in compaction will reduce the final residual deformation of the specimen, this is because the increase in confining pressure and the increase in compaction have increased the compactness of the specimen, the soil compactness of the specimen is higher, and the stiffness is increased faster, thereby effectively reducing the residual deformation of the specimen.
3.4 Residual deformation under different gradations
The axial residual strain εd and volume residual strain εvd under different gradations were extracted, and the axial, volumetric residual strain and vibration N relationship curves εd- N and εvd- N were plotted, as shown in Fig. 8:
It can be seen from Fig. 8 that the increase in the content of graded coarse particles will make the cumulative deformation of the specimen smaller, but when the coarse particle content reaches a certain value, the impact on the axial residual strain is small. The reason for this phenomenon is attributed to: when there are more fine particles in the specimen, the water permeability inside the specimen will be reduced, thereby increasing the internal pore pressure at the beginning of the specimen loading, resulting in a rapid increase in the axial residual strain, while when there are more coarse particles, the pore water can be discharged in time, reducing the cumulative deformation at the beginning of the loading, thereby reducing the final axial residual deformation.
The reason for this phenomenon is that the fine-grained part of the gangue sample is weakly permeable, and at the beginning of loading, the pore water inside the sample cannot be discharged in time, and the pore pressure of the specimen increases rapidly, resulting in a rapid increase in axial residual strain, and then the pore water is gradually discharged, the pore pressure of the sample tends to be stable, and the residual strain of the specimen tends to be stable.
4.1 Residual deformation model of Shenzhu River and its improved model
The Shenzhu River residual deformation model was originally used to analyze the residual deformation of earth-rock dams and is widely used in China. The model was carried out by Sheng (Long et al. 2019) on the large-scale triaxial test of the stone pile with multi-stage and small number of cycle loads, and it was found that the residual deformation was proposed with the increase of the vibratile frequency in line with the semi-logarithmic law. The model takes into account the effects of confining pressure, pre- vibratile shear stress levels and vibratile frequency, and its expression is:
$${\varepsilon _{{\text{vd}}}}={c_{{\text{vd}}}}\lg \left( {1+N} \right)$$
2
$${\gamma _{\text{d}}}={c_{\text{d}}}\lg \left( {1+N} \right)$$
3
$${c_{{\text{vd}}}}={c_1}{\gamma _e}^{{{c_2}}}\exp \left( { - {c_3}{S_l}^{2}} \right)$$
4
$${c_{\text{d}}}={c_4}{\gamma _e}^{{{c_5}}}{S_l}^{2}$$
5
Among them, εvd is the volume residual strain; γd is shear residual strain; N is the time of the shock; cvd and cd are the slope of the volume and shear residual deformation- vibratile correlation curves, is calculated by the empirical formula of Eq. (4) and Eq. (5); γe is the amplitude of the dynamic strain; Sl is the shear stress ratio or stress level; c1、c2、c3、c4、c5 are model parameters.
Although the residual deformation model of the Shenzhu River can take into account the shearability of the soil and obtain the relationship between the volume residual strain and the vibratile degree, scholars found that the follow-up stress ratio increased, and the semi-logarithmic relationship curve of the residual strain- vibratile frequency showed obvious nonlinearity (He et al. 2018), Therefore, the following formula can be used to improve the Shen Zhujiang model:
$${\varepsilon _{{\text{vd}}}}={c^{\prime}_{{\text{vd}}}}{\lg ^n}\left( {1+N} \right)$$
6
$${\gamma _{\text{d}}}={c^{\prime}_{\text{d}}}{\lg ^n}\left( {1+N} \right)$$
7
where c’vd and c’d are the model parameters of residual volume strain and residual shear strain after the model is improved.
4.2 Residual deformation model of gangue filler
In the past, the research on the residual deformation model of the Shenzhu River and its improved model was mostly a multi-stage, small vibration times of dynamic experiments (within 200 vibrations). Considering the engineering background of gangue as a subgrade filler, this paper performs a large Vibration times of single-stage loading of gangue filler samples. Combined with the experimental results of section 3.2 ~ 3.4, the deformation curve of the gangue filler sample within 10,000 times under different conditions enters the gentle stage, that is, the residual strain value tends to be constant with the increase of the vibratile frequency, so this paper mainly analyzes the deformation model of shear residual deformation and volume residual deformation of 0 ~ 10000 vibrations.
Since the DJSZ-150 dynamic and static triaxial test system can only collect axial displacement, it is necessary to convert the axial residual strain into shear residual strain, according to the study of Shen Zhujiang's axial residual strain and shear residual strain as a linear relationship, according to formula (8) for conversion:
$${\gamma _{\text{d}}}=\left( {1+{v_d}} \right){\varepsilon _d}$$
8
Where, vd is the Poisson ratio.
The number of vibration is logarithmically converted, and the shear residual strain- vibration pattern diagram and the volume residual strain- vibration relationship diagram are drawn separately, and they are re-divided into the residual strain-vibration pattern of 0 ~ 200 vibration times, and the residual strain-vibration correlation diagram of 201 ~ 10000 vibration, as shown in Fig. 9.
According to the data collation of Fig. 9, it can be clearly found that at 0 to 200 vibrations, the shear and volume residual strain show obvious nonlinearity with the increase of logarithmic vibration lg (1 + N), and the smaller the confining pressure, the lower the compaction, and the lower the coarse particle content, the more obvious this nonlinearity is. This has a certain error with the Shenzhu River residual deformation model, but it is highly matched with the improved Shenzhu River residual deformation model such as Gong Siyi. From 201 to 10000 vibrations, the shear residual strain increased with the logarithmic vibration lg(1 + N), which was more in line with the prediction results of the Residual Deformation Model of the Shenzhu River, but the volume residual strain increased with the logarithmic vibration lg(1 + N), which also had a certain nonlinearity and was in line with the logarithmic relationship.
According to the above analysis, the data are fitted so that N*= lg(1 + N), the fitted model is shown in Table 4, and the fitting parameters are shown in Table 5.
| Residual strain | Time of shock | Morph model |
|---|---|---|
| Shear residual deformation | 0 ~ 200 | εd=\({c{\prime }}_{\text{d}1}\)·(N*)2 |
| Shear residual deformation | 201 ~ 10000 | εd=\({c{\prime }}_{\text{d}2}\)·N* |
| Volume residual deformation | 0 ~ 200 | εvd=\({c{\prime }}_{\text{v}\text{d}1}\)·(N*)2 |
| Volume residual deformation | 201 ~ 10000 | εvd=\({c{\prime }}_{\text{v}\text{d}2}\)·ln(N*) |
| Fitting Parameter | \({c{\prime }}_{\text{d}1}\) | \({c{\prime }}_{\text{d}2}\) | \({c{\prime }}_{\text{v}\text{d}1}\) | \({c{\prime }}_{\text{v}\text{d}2}\) | |
|---|---|---|---|---|---|
| AY01 | num | 0.32 | 0.99 | 0.2 | 1.41 |
| R2 | 0.81 | 0.99 | 0.98 | 0.98 | |
| AP01 | num | 0.23 | 0.6 | 0.16 | 1.31 |
| R2 | 0.98 | 0.99 | 0.88 | 0.96 | |
| AP02 | num | 0.13 | 0.38 | 0.13 | 1.12 |
| R2 | 0.91 | 0.99 | 0.9 | 0.96 | |
| AC01 | num | 0.41 | 1.08 | 0.22 | 1.61 |
| R2 | 0.76 | 0.99 | 0.9 | 0.73 | |
| AC02 | num | 0.15 | 0.49 | 0.11 | 1.06 |
| R2 | 0.82 | 0.99 | 0.83 | 0.93 | |
| AG01 | num | 0.15 | 0.54 | 0.08 | 0.76 |
| R2 | 0.83 | 0.99 | 0.84 | 0.92 | |
| AG02 | num | 0.17 | 0.58 | 0.06 | 0.7 |
| R2 | 0.86 | 0.99 | 0.85 | 0.88 | |
According to the fitting parameters and fitting degree of the residual deformation model in Table 5, it can be found that the improved model in Table 4 can be used to describe the residual deformation characteristics of coal gangue filler under large vibration times of cycle loads, which effectively improves the fitting accuracy of the residual deformation model of Shenzhu River.
5.1 Grey Association Analysis
In the large-scale triaxial test in this paper, the influencing factors set are limited, and considering the large sample size, the economic cost and labor cost are high, resulting in a limited number of test samples. The grey correlation analysis method focuses on macro analysis and basic situation between various factors, and does not require complex and rich information. The essence of this method is to analyze the degree of influence of each influencing factor on the affected object, and to find out the main factors and secondary factors from it. In view of the development process of residual deformation of gangue filler under cyclic load, the degree of influence of each influencing factor on the deformation value of gangue filler varies, resulting in a certain impact on the rate of deformation process and the results of residual deformation value. Therefore, the grey correlation analysis of coal gangue dynamic large-scale triaxial test is carried out to analyze the data, and the correlation degree of each factor can be found out, that is, the influence degree on the deformation of gangue filler, so as to put forward guiding opinions for practical engineering.
In terms of mathematical expression, the gray association analysis method adopts the degree of influence of the gray correlation degree characterization factor on the main behavior, and its calculation method is summarized as follows:
There is a system feature number column (main behavior), denoted as Y0, where:
$${Y_0}=\left\{ {\left. {{Y_0}\left( l \right)} \right|l=1,2, \cdots ,n} \right\}$$
9
There is also a series of systematic factors, denoted as Yi, i = 1,2,3, …, m(m is the number of factors in the series of comparisons):
$${Y_i}=\left\{ {\left. {{Y_i}\left( l \right)} \right|l=1,2, \cdots ,n} \right\}$$
10
Then normalize the logarithm according to the formula (3), that is, carry out dimensionless and unified data transformation on the original data:
\({\widehat {Y}_i}\left( l \right)=\frac{{{Y_i}\left( l \right) - {{\overline {Y} }_i}}}{{\Delta {Y_{i\hbox{max} }}}}\) \(i \in \{ 0,1,2,3, \ldots , m\}\) (11)
Wherein, \({\overline {Y} _i}\) and \(\Delta {Y_{i\hbox{max} }}\) are the maximum values of the mean and mutual difference of each series, calculated as follows Eq. (12) and Eq. (13), respectively:
$${\overline {Y} _i}=\frac{1}{l}\sum\limits_{{l=1}}^{n} {{Y_i}\left( l \right)}$$
12
$$\Delta {Y_{i\hbox{max} }}=\hbox{max} \left\{ {\left. {\left| {{Y_i}\left( k \right) - {Y_i}\left( l \right)} \right|} \right|k,l=1,2, \cdots ,n} \right\}$$
13
The correlation coefficient between sequence \({\widehat {Y}_0}\left( l \right)\) and series \({\widehat {Y}_i}\left( l \right)\) is 
$${\xi _i}\left( l \right)=\frac{{{\Delta _{\hbox{min} }}+\rho {\Delta _{\hbox{max} }}}}{{{\Delta _i}l+\rho {\Delta _{\hbox{max} }}}}$$
14
where ξi(l) is the correlation coefficient of Yi to Y0 in l sample;
Δmax is the maximum absolute value of the difference between the total sequence and the corresponding time of the reference sequence at all moments as follows:
$${\Delta _{\hbox{max} }}=\mathop {\hbox{max} }\limits_{i} \mathop {\hbox{max} }\limits_{l} \left| {{{\widehat {Y}}_0}\left( l \right) - {{\widehat {Y}}_i}\left( l \right)} \right|$$
15
Δmin is the minimum absolute value of the difference between the time of all sequences and the reference series at all moments as follows:
$${\Delta _{{\text{min}}}}=\mathop {\hbox{min} }\limits_{i} \mathop {\hbox{min} }\limits_{l} \left| {{{\widehat {Y}}_0}\left( l \right) - {{\widehat {Y}}_i}\left( l \right)} \right|$$
16
Δi(k) is the absolute value of the difference between the k-time and the reference series corresponding to the time is calculated as follows (Luo et al. 2021):
$${\Delta _i}\left( l \right)=\left| {{{\widehat {Y}}_0}\left( l \right) - {{\widehat {Y}}_i}\left( l \right)} \right|$$
17
ρ is the resolution coefficient, usually empirically based on ρ value, take ρ = 0.5 (Rekha et al. 2022).
The correlation coefficient ξi(l) only represents the degree of association between the sample data, because it has the characteristics of dispersion and not easy to compare, therefore, the introduction of the concept of correlation degree, that is, the average of the correlation coefficient, can be most of the correlation coefficients can be pooled into a value for centralized treatment. The correlation degree is calculated as follows:
$${r_i}=\frac{1}{n}\sum\limits_{l} {{\xi _i}} \left( l \right)\begin{array}{*{20}{c}} {}&{} \end{array}i=1,2,3, \cdots {\text{n}}$$
18
The degree of association ri represents the degree to which factor Yi influences the main behavior Y0, for l,j∈(1,2,3, ......n), when Yl>Yk, then the system factor sequence Yl is considered to be superior to Yk, or has a higher correlation with the main behavior sequence.
5.2 Grey correlation analysis of residual strains
In order to consider the degree of correlation between the influencing factors on the residual deformation of the specimen under large vibration times, this paper sets the values of the final axial residual strain εd and the final volume residual strain εvd as the residual deformation values after 30,000 times of dynamic load loading, and set them as reference sequences, Y0 and Z0, respectively, and set the confining pressure σ3, compaction degree η, graded parameters A and vibration N as factor sequences, including confining pressure series X1, compaction degree series X2, cascading parameter series X3, and vibration degree X4. According to the numerical value of the factor sequence, the corresponding reference sequence is selected, and the following results are obtained through the calculation in section 3.1, as shown in Table 6.
| Axial residual deformation Y0 | Volume residual deformation Z0 | |||
|---|---|---|---|---|
| Relevance | Sort | Relevance | Sort | |
| Containment pressure X1 | 0.703 | 2 | 0.719 | 2 |
| Compaction degree X2 | 0.694 | 3 | 0.712 | 3 |
| Gradation parameters X3 | 0.681 | 4 | 0.635 | 4 |
| Shock times X4 | 0.852 | 1 | 0.814 | 1 |
According to the basic concept of gray correlation, it can be seen that the reference sequence representing the residual strain in this paper is positively correlated with the degree of correlation between the coefficient sequences, that is, the greater the degree of correlation, the greater the degree of influence, and vice versa.
From Table 6, it can be seen that the three influencing factors all have a certain degree of correlation with the final residual strain of gangue filler, and whether it is for axial residual strain or volume residual strain, the degree of influence is ranked by vibration X4, confining pressure X1, compaction X2, and gradation X3. Considering that the vibration frequency represents the effect of stress history on the soil, the confining pressure and compaction degree both indicate the compactness of the gangue test piece, and the particle gradation indicates the degree of bite between the soil particles. The compactness of the soil reflects the difficulty of rolling the soil particles, and the degree of bite between the soil particles reflects the difficulty of the soil particles breaking during the vibration process. Therefore, under the multi-vibration cycle load, the difficulty of rolling soil particles can affect the residual deformation of soil more than the difficulty of angular fragmentation of soil particles. Reflected in the actual project, considering that the material quality of gangue is general, the quality fluctuation is large, and the natural grading is difficult to achieve good grading, according to the results of the gray correlation analysis, the degree of compaction should be appropriately increased to make the soil more compact, so as to reduce the residual strain under large vibrations.
In this paper, the residual deformation development law of gangue subgrade filler was explored by using a large-scale triaxial experiment, and the correlation degree between different confining pressure, compaction degree, gradation parameters and vibration and the residual deformation value of gangue subgrade filler was analyzed based on the gray correlation analysis method, and the following conclusions were obtained:
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According to the results of the large-scale triaxial test, the axial residual deformation εd and the volume residual deformation εvd of the gangue subgrade filler are both rapidly growing with the development of the vibratile frequency, and then the growth rate is gradually decreasing, and finally the deformation tends to be constant, and the stiffness of the gangue test piece gradually increases with the historical action of stress, reflecting the compactness of the gangue filler. The residual deformation of the gangue subgrade filler is manifested under different influencing factors: the greater the confining pressure, the smaller the residual deformation of the specimen; The higher the compaction, the smaller the residual deformation of the specimen; The increase in the content of graded coarse grains will make the cumulative deformation of the specimen smaller.
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Under the action of single-stage multi-vibration cyclic load, the actual residual deformation of gangue filler is nonlinear in semi-logarithmic coordinates with the development of vibration, which is inconsistent with the form of the Shenzhu River model, which will lead to a large deviation between the calculation results of the Shenzhu River model and the actual situation. By defining N*= lg(1 + N), within 0 to 200 vibration, (N*)2 is substituted for N* to fit the relationship between shear, volume residual strain and vibratile displacement, and ln(N*) is substituted for N* to fit the relationship between volume residual strain and vibratile frequency within 201 ~ 10000 vibration, and the degree of fitting is higher. This provides an idea for the analysis of residual deformation models of subgrade fillers such as coarse-grained soil and gravel under large cycle loads.
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Because the test in this paper is only for the large-scale triaxial dynamic test of single-stage loading, and the number of test samples is small due to the cost of the test, there are certain limitations on the analysis of the deformation parameters of the improved model, and the subsequent number of influencing factors will be increased, and the number of test samples will be expanded to analyze the deformation parameter value of the residual deformation model of gangue filler under multi-vibration cycle loads.
Different confining pressure, compaction degree and grading parameters all have a greater impact on the residual strain of the specimen, according to the influencing factors and the gray correlation of the final residual deformation value after the gangue filler is shaken by the cyclic load 30,000 times, combined with the two main factors affecting the residual deformation of the soil, namely the overturning of soil particles and the fragmentation of soil particles, it can be found that under a multi-vibration of cycle loads, the difficulty of rolling soil particles can affect the residual deformation of soil more than the degree of difficulty of fracture of soil particles.
This work was supported by the National Natural Science Foundation of China (52208341), the Natural Science Foundation of Hunan Province (2020JJ4019), the Scientific Research Project of the Hunan Provincial Education Department (21C0360).
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No competing interests reported.