Research on the Collapse Coefficient of Collapsible Loess under Unloading

In the geogenetic overburden excavation of underground space projects (such as comprehensive pipe corridors or subways), the foundation stress is in an unloading state. The effect(s) from the unloading on the coefficient of subsidence in the loess area should be considered. In this study, to explore the effect of unloading on the collapsibility of loess, the collapsible loess in the Guanzhong area was considered as the research object. An expression for the unloading collapse coefficient was established based on the unloading stress ratio, unloading collapse ratio, and other parameters. The influence of the unloading on the loess collapse coefficient was studied using an indoor collapsibility test, and the function form and parameters for the expression were determined. As combined with the field test, the accuracy of the expression for the unloading coefficient was verified based on the test value for the specific collapsibility, calculated value for the specific collapsibility, and calculated value for the unloading collapsibility.


Introduction
e contradiction between the rapid expansion of China's urban space demands and limited space resources has become increasingly prominent. As the main carrier of urban infrastructure, underground space has been developed on a large scale, and, accordingly, new underground space projects (such as comprehensive pipe galleries and subways) are continuously being planned and constructed. However, owing to the excavation of the upper soil, the foundation stress is less than the self-weight stress of the original overburden soil; thus, the foundation stress is in an unloading state. In the loess region, if the collapsibility evaluation and subsequent foundation treatment design are conducted according to the "Building Code for Collapsible Loess Region" (GB-50025-2018) [1], the safety coefficient will be excessively high, causing unnecessary economic losses and conflicting with the economic principles. erefore, based on previous studies, this study analyzes and studies loess collapsibility under unloading.
For nearly a half-century, experts and scholars from various countries have conducted qualitative and quantitative research on collapsible loess. In 1966, China issued the first special code for the loess area, that is, "Building Code for Collapsible Loess Area" (BJG-20-66), and gradually introduced the "78 Code," "90 Code," and "04 Code"; the latest "18 Regulations" code provides guarantees of construction safety in the loess area.
Liu et al. [2,3] recognized the special structure of loess earlier and pointed out that the collapsibility of loess is extremely harmful to engineering construction in loess area. Chen and Liu [4] obtained several laws of collapsible deformation of loess through laboratory tests and suggested that the collapsible loess should be treated in different regions. Shao et al. [5] believed that the initial pressure has an important influence on the deformation of collapsible loess. rough the study of loess particles, Assallay [6,7] realized that a certain amount of clay minerals was necessary for collapsibility, but, with an increase in clay content, the loess state would change from collapsibility to noncollapsibility. Shen et al. [8,9] regarded natural loess as a heterogeneous material composed of cemented blocks and weak belts and, based on this analysis, established a binary medium model for loess.
Zhu et al. [10] analyzed the influences of eight soil property indexes (including the pore ratio, dry density, and initial moisture content) on the collapse coefficient and established a regression equation for the loess collapse coefficient and each influencing factor. Liu [11] conducted a detailed analysis of a collapse coefficient of loess and its influencing factors in several areas in northwest China and established a regression relationship between the collapse coefficient and influencing factors.
Wang and Liu [12] derived a collapse coefficient correction formula by setting a correction scale factor and used this factor to correct a collapse coefficient calculated in a laboratory experiment. Hu et al. [13,14] reported that the collapse coefficient of loess increases more significantly at low pressures under special circumstances and showed that the stability time for loess collapsibility was prolonged under various pressures. Chen et al. [15][16][17] believed that the discontinuous distribution of the collapsible loess and complex engineering geological conditions in large thick loess resulted in a high level of collapsibility. Gao et al. [18] established a calculation method for a principal component score for the collapse coefficient by analyzing changes in the structural parameters before and after the collapse.
At present, most research studies on soil collapsibility are conducted under constant pressure and without considering the influence of the upper stress changes on it. However, in practical engineering, the influence of the unloading on the strength characteristics of the geotechnical materials cannot be ignored.
Using laboratory tests, Yang et al. [19,20] found that the unloading rate had a significant impact on the stress-strain relationship and pore pressure variation law of rock-soil mass materials. Li et al. [21] believed that the greater the unloading rate, the greater the damage degree to the expansive soil. Wang et al. [22] found that the axial strain, ultimate strength, and residual strength of the peak values of soft rocks were positively correlated with the unloading stress level and unloading rate.
Owing to the limitations of laboratory tests, field tests can simulate an actual project relatively well, and the test results are relatively reliable. erefore, many scholars have studied the collapsibility of collapsible loess through field experiments.
Huang and Yang [23] summarized a large number of test pit immersion tests and found that the calculated value was lower than the actual value in the areas where the general calculated and measured collapsibility were quite different. In addition, the self-weight collapsibility was strong. Wang et al. [24] conducted an on-site immersion test in conjunction with the Xi'an subway project, studied the characteristics of loess collapsibility deformation at the site, and analyzed the changes in the collapsibility deformation speed with time. Huang et al. [25,26] obtained the characteristics of a slow-fast-slow-stable collapse on the surface and in deep loess through an analysis of immersion in a field test pit. Liu [27] obtained the change characteristics of a cumulative settlement and collapse rate over time through an analysis of field test data.
In terms of these research statuses, the research on collapsible loess is divided into two parts. First, starting from the theoretical aspect, Zhujiang Shen and others established a binary medium model based on the structural properties of loess; Hu Zaiqiang, Fengji Zhu, Zhenghan Chen, Lingxia Gao, and others analyzed the influence of different factors on the characteristics of collapsible loess and established related theoretical research methods. Another part of scholars conducted indoor or on-site tests on collapsible loess under special conditions such as different levels of load conditions, different pH environments, and different moisture contents through a combination of theory and experimentation, through summarizing the laws and theoretical analysis, studying its characteristics.
It can be seen that many scholars have conducted research on loess collapsibility under constant pressure. However, they have failed to consider the impact of the upper pressure changes, and the relevant norms have mostly been derived from collapsibility theory under constant pressure.
us, they cannot be accurately applied to underground space engineering under an unloading state and cannot meet the requirements of the engineering economy.
Jin et al. [28,29] took the unloading collapsibility process of soil between piles in a loess site under water immersion as the background and started the initial research on the unloading collapsibility of loess. e test simulated the collapsibility process of immersion-pressurization-unloading and proposed an expression for the unloading loess collapse coefficient under these working conditions. However, in unloading projects such as subways and integrated pipe corridors, the mode of foundation treatment is usually unloading-immersion or unloading-immersion-pressurization.
Based on the above discussion, analysis, and summary, it remains necessary to study the collapse coefficient of loess under unloading. In this study, the characteristics and deformation mechanism of loess collapsibility under the unloading-immersion mode were studied through laboratory tests. An expression was established for the unloading collapse coefficient, and the accuracy of the calculation formula was verified using field monitoring test data. e research results of this study can provide a theoretical basis for an evaluation method for collapsibility in underground space engineering, provide help for research on the mechanical properties of collapsible loess and disaster prevention and mitigation in northwest China, and provide certain reference value for underground municipal engineering construction in loess areas.

Unloading Collapse Coefficient Expression
is article takes underground space engineering as the background and, based on the literature [29], quantitatively explores the variation law of the unloading collapse 2 Advances in Civil Engineering coefficient under this working condition. e unloading collapse process is described based on the unloading stress ratio, unloading collapse ratio, and unloading collapse attenuation coefficient; based on this information, the unloading collapse coefficient and its coefficient expression are proposed.

Unloading Stress Ratio K.
e unloading stress ratio K is defined to reflect the degree of the pressure reduction in the process of loess unloading: where P x is the upper pressure after unloading, in kPa; P 0 is the initial pressure, in kPa; ΔP is the unloading pressure, in kPa.

Unloading Collapse Ratio η.
By defining the unloading collapse ratio η, the degree of the unloading collapsibility of the loess can be reflected: where S 1 is the unloading collapsibility of the loess, in mm, and S is the loess collapsibility under a constant pressure, in mm.

Unloading Collapsibility Attenuation Coefficient δ r .
Owing to the unloading effect, the collapsibility of the loess will be less than that with a constant pressure. To express the degree of collapsibility, the ratio of the relative collapse reduction S r to the original height h 0 of the soil sample is defined as the unloading collapsibility attenuation coefficient δ r : e unloading collapse process is shown in Figure 1. e BC section represents the total collapse amount S of the undisturbed soil under the initial pressure P 0 , and the BD section represents the process of unloading the load ΔP from the initial pressure. e DE section represents the collapse amount S 1 of the sample under pressure P x after being unloaded and immersed in water.

Unloading Collapse Coefficient δ xs .
e unloading collapse potential is expressed using the unloading collapse parameters of the loess, and the unloading collapse coefficient δ xs is as follows: where δ s is the collapse coefficient under a constant pressure and can be calculated as follows: Substituting equations (2), (3), and (5) into (4), we obtain the expression as follows: It is not difficult to see that δ xs is a reduction of δ s and that η is equivalent to the reduction coefficient. For η, two parameters should be considered: the initial pressure P 0 and the unloading stress ratio K (reflecting the size of the unloading).
A functional relationship is used to represent η as follows: erefore, an expression can be derived as follows: Equation (8) is the expression of the loess unloading collapse coefficient δ xs .
To more intuitively understand the effect of unloading on the collapsibility of the loess, indoor collapsibility tests under constant pressure and unloaded loess were required. e change rule of the collapse coefficient of the collapsible loess under unloading was obtained by comparative analysis.
e dominant factor in f(K, P 0 ) was determined to be the unloading stress ratio K according to the laboratory test data. e functional relationship of η-K was analyzed, and a specific form for the expression of the outlet unloading collapse coefficient was finally fitted.

Laboratory Compression Test of Loess
e soil sample was taken from a construction site in the Tongchuan New District, Shaanxi Province. e terrain was flat, the geomorphic unit was single, and the collapsibility was strong. e basic physical indexes of the soil samples are shown in Table 1. Figure 1: Unloading collapse process.

Loess Collapsibility Test under Constant Pressure.
In this study, the changes in the collapse coefficient and initial pressure of the loess under constant pressure at different buried depths were obtained through laboratory tests, and the collapsibility deformation characteristics of the loess were analyzed. e main test instruments comprised a W-G single rod consolidation apparatus, dial meter (0.01 mm), electrothermal-blast constant temperature drying box, and electronic balance. e test process was divided into two steps: soil sample preparation and loess collapsibility testing under constant pressure. e equipment picture and soil sample are shown in Figure 2.
When preparing the soil samples, a cutting ring with a height of 2 cm and area of 30 cm 2 was selected, and several undisturbed loess cutting ring samples were prepared, with depths of 5 m, 10 m, 15 m, and 20 m. e loading pressure was determined according to the provisions of "Building Standards for Collapsible Loess Area" on the loading pressure of collapsible loess. When the buried depth is less than 10 m, take 200 kPa. When the depth is greater than 10 m, take the saturated self-weight pressure of the overlying soil. In combination with the "Standards for Geotechnical Test Methods" [30], a graded loading method was adopted. e pressure grade was set at 25 kPa and added to the specified pressure. e pressure grades are listed in Table 2.

Loess Subsidence Test under Unloading.
e unloading collapsibility test was mainly divided into three steps: compression consolidation, unloading, and flooding. e specific scheme was as follows.

Test Sampling.
Five pieces were placed at 5 m, 10 m, 15 m, and 20 m, respectively, for a total of 20 pieces, and then the pieces were weighted.

Consolidation Compression.
① Initial pressure setting: according to the "Building Standards in Collapsible Loess Area," the recommended initial pressure value of the soil layer within 10 m was 200 kPa, and the saturated dead weight pressure of the overburden soil under 10 m was taken. According to the geological exploration report calculation, the saturated dead weight pressure P z and initial pressure P 0 values of the overburden soil as borne by the soil at 5 m, 10 m, 15 m, and 20 m are shown in Table 3. ② e pressure: the sample was compressed and consolidated with the W-G single rod type consolidation instrument. e initial pressure was increased step by step to P 0 , and the data were recorded after the deformation stabilized.

Unloading Treatment.
Based on referring to the pressure grade, the unloading quantity △P was classified into 25 kPa, 50 kPa, 75 kPa, and 100 kPa. e upper pressure after unloading is shown in Table 4.

Immersion.
After unloading, the upper pressure was kept unchanged, the sample was immersed in water, and the deformation reading of the sample was observed as stable.

Analysis of Test
Results. e trend of the loess (δ xs -P x ) under unloading was obtained through laboratory tests, as shown in Figure 3.
It can be seen from Figure 3 that the relationship between the loess collapse coefficient and upper pressure after unloading is as follows: ① e change relationship of the loess (δ xs -P x ) under unloading conforms to the trend of constant pressure collapsibility; that is, it occurs in two stages, the compaction stage and unloading collapsibility stage.
In the compaction stage, the structural properties of the loess are not completely destroyed but still have a certain strength, and the structure is relatively compact. In the unloading collapse stage, the upper pressure increases to near the initial pressure of the collapse, and the soil produces collapse and deformation. is develops rapidly until the deformation stabilizes. ② e collapsibility of the loess under unloading is related to the buried depth of the soil. e greater the buried depth, the tighter the connection of the soil particles, the smaller the porosity, and the weaker the collapsibility; the greater the buried depth, the greater the initial pressure of the collapsibility, and the more delayed the process of collapse.
A comparison of the constant pressure, unloading collapse coefficient, and pressure curve is shown in Figure 4.
From a comparative analysis of the constant pressure and unloading collapse coefficient and pressure curve in Figure 4, the following conclusions can be drawn: ① e unloading and constant pressure collapse coefficient of the loess has the same general trend as the pressure curve, showing an increasing curve, but the      Advances in Civil Engineering unloading collapse coefficient is always smaller than the constant pressure collapse coefficient. Under the action of the initial pressure, the soil is compacted and consolidated, and the porosity is reduced. After unloading, as the upper pressure cannot reach the original compaction and stability corresponding to the structural failure conditions, the soil continues to deform after immersion, but the additional deformation is reduced relative to that at the constant pressure condition. Under the corresponding unloading action, the collapse coefficient decreases, but the trends of the two curves are close, and the corresponding pressures at the characteristic points of the curves are similar. ② e collapse coefficient for unloading the loess at a low pressure is small, and the difference from the constant pressure increases with increases in pressure; moreover, the two curves are more separated. However, when the pressure increases to a certain limit value, the coefficient difference reaches a peak value. After that, the slope of the unloading curve increases sharply, the unloading collapse rate is greater than the constant pressure collapse rate, and the two curves gradually become close. e unloading curve slope inflection point is taken as the unloading collapse limit point, and the corresponding pressure is the limit pressure. It can be seen from Figure 4 that the limit pressure is always less than the initial pressure of the collapse. erefore, it can be said that the unloading effect increases the initial pressure of the collapsibility of the loess to a certain extent.
Compared with constant pressure subsidence, unloading can increase the initial subsidence pressure of the loess, and the greater the burial depth, the more evident the improvement. As can be seen from Figure 3, the initial subsidence pressure of the soil at 5 m increased from 109.4 kPa under constant pressure to 111.3 kPa, an increase of 1.5%. e initial subsidence pressure of the soil at 10 m increased from 169.2 kPa under constant pressure to 172.3 kPa, an increase of 1.8%. e initial subsidence pressure of the soil at 15 m increased from 216.8 kPa under constant pressure to 235.5 kPa, an increase of 8.6%. e initial subsidence pressure of the soil at 20 m increased from 331.6 kPa under constant pressure to 362.5 kPa, an increase of 9.3%.

Determination of the Expression for the Unloading Collapse
Coefficient. An orthogonal test was conducted based on the results of the unloading collapsible test. First, the two affecting factors were determined: K and P 0 . Second, the level of each factor was determined: K was taken as 0.8, 0.6, and 0.4, and the P 0 pairs had three test levels: 200, 300, and 400.
In Table 5, L n is the sum of the results of the respective levels of the factors, Ln' is the mean value of the sum of the results of the respective levels of the factors, and R is the range of the factors. e results are shown in Table 5. According to the R range analysis of the results, the range for K (0.41) is much larger than the range for P 0 (0.04), indicating that, within a reasonable range of change, K has a much greater influence on the target result than P 0 . In other words, the unloading stress ratio K has the greatest influence on the η function; K is an important factor, and the initial pressure P 0 is a secondary factor. In summary, the unloading stress ratio K plays a major role in controlling the collapsible deformation of the loess under unloading.
K is also taken as an important factor in the fitting expression f(K, P 0 ), according to the orthogonal test results.
e K-η function is used to fit the unloading test for the soil at 10 m, as shown in Figure 5. It can be seen that the fitting function is approximately an exponential function, so it is expressed by the expression of the exponential function as follows: where a, b, and c are all parameters. A further regression analysis was conducted on all of the unloading collapsibility test data from the laboratory test to determine the fitting curve parameters, as shown in Table 6. e mean values of parameters a, b, and c in Table 6 are substituted into f(K) to determine the expression for the unloading collapse coefficient as follows: To verify the correctness of the unloading collapse coefficient, the results calculated using equation (10) are compared with the test results; the comparison curve is shown in Figure 5. It can be seen that the unloading collapse coefficient expression can better fit the collapse coefficient under different dead weight pressures.
In Figure 6, the calculated value obtained by the expression of the unloading collapse coefficient is basically consistent with the trend of the experimental value curve. e experimental value is generally slightly larger than the calculated value, but the difference does not exceed 0.005. In addition, if the collapse coefficient remains as 0.015, that is, as the limit for judging collapsibility, it can be found that the difference between the calculated initial pressure of the collapsibility and test value will not exceed 5%.

Field Test
According to the survey report and "Building Standards for Collapsible Loess Areas," resampling was conducted at the site, and the original loess samples at 1 m -10 m below the ground (layered height of 1 m) were collected using boreholes. Indoor tests were conducted to obtain the collapse coefficient. e unloading collapse coefficient and total collapsibility were calculated as shown in Tables 7 and 8, and then the self-weight collapse coefficient and self-weight collapsibility were calculated as shown in Tables 9 and 10. e saturation pressure was calculated from the specific 6 Advances in Civil Engineering gravity and pore-solids ratio of the soil; the removal of 2 m of soil was equivalent to unloading at 28 kPa, and its saturation pressure was 35 kPa. In Tables 7-10, the self-weight collapsibility calculated from the unloading collapse coefficient is 120 mm, and the total collapsibility is 468.5 mm. e self-weight collapse coefficient as calculated by the constant pressure test is 314.4 mm, and the total subsidence is 609.5 mm. To further verify the applicability of the expression for the unloading collapse coefficient, a comparative analysis was conducted through field industrial tests.

Test Scheme.
To simulate the unloading conditions of underground space engineering, the foundation pit was excavated at a depth of 2 m. It was determined that the soil layer in the flooded pit was in the unloading state under a saturated deadweight stress. e surface of the foundation pit was taken as the design surface.  Advances in Civil Engineering e area of the immersion test pit was 8100 mm × 7200 mm. To measure the amount of collapsibility and deformation of the formation, 12 stratum subsidence observation punctuation points were set up in the pit, and holes with a diameter of 120 mm were drilled at the punctuation points. e punctuation points were taken at six measuring depths, namely, 1 m, 2 m, 3 m, 5 m, 8 m, and 10 m; each depth was composed of two measuring points as a group, for a total of six groups. A benchmark was made, and a measuring ruler was placed on its surface. e reference point for the observation data was set up on the periphery of the test pit and 1 m away from the edge of the test pit.
To ensure that the soil layer of the site was fully soaked, 32 water injection holes with a diameter of 400 mm and depth of 12 m were arranged in the test pit, and all water injection holes were evenly distributed. e test pit diagram is shown in Figure 7. e duration of immersion was 7 days, and the immersions began at the same time and continued for 26 days. In the process of water immersion, the level was used to observe the benchmark every day, and the settlement amounts of the stratum at 1 m, 2 m, 3 m, 5 m, 8 m, and 10 m in the pit were observed, respectively. e observation accuracy was controlled to be within the specified range. e layout of monitoring points is shown in Figure 8.

Analysis of Test Results.
is test lasted for 26 days, and the entire process was monitored, that is, from the immersion of the test pit to the settlement and deformation stability of the foundation. e variation in the formation settlement over time is shown in Figure 9.
It can be seen from Figure 9 that the formation settlement rate is divided into three stages. In the first stage (0 -5 days), the water has just begun to infiltrate, the soil is not fully soaked, and the settlement deformation is small, accounting for approximately 5% -15% of the total settlement. In the second stage (6 -15 days), the settlement rate of each soil layer in this stage increases sharply, corresponding to the steep drop section in the curve of Figure 9. e settlement amount accounts for approximately 64% -75% of the total settlement amount. In the third stage (16 -26 days), the settlement amount is small, the settlement rate gradually tends to be stable, and the settlement process is basically complete. e settlement amount in this stage accounts for approximately 10%-31% of the total settlement amount. e final observed settlement amount for each soil layer is shown in Table 11. e final settlement of the soil at 10 m is 9.8 mm, and that of soil at 1 m is 83.4 mm. at is, the deeper the soil is buried, the smaller the unloaded subsidence is, and the weaker the collapsibility of the soil is. erefore, it can be inferred that the surface settlement is approximately 85-95 mm, which can be used as the field settlement test value.
In summary, in Table 10, the amount of collapsibility as calculated by the unloading collapse coefficient is 120 mm, and the amount of subsidence as obtained by the constant pressure test coefficient is 314.4 mm. erefore, compared with the surface subsidence value obtained by the field tests (85-95 mm), the amount of subsidence calculated by the unloading coefficient is evidently closer to it. e wet fall at the same time from using the specification of the constant pressure coefficient should be used to calculate the collapsibility of the total quantity and amount for the collapsibility level, that is, to determine the collapse grade, and        should be judged as level III (serious). at from using the wet unloading test of the ground should be judged as level II (medium). e main reason is that there will be a layer of soil subsidence under a constant pressure, but there is no subsidence after unloading. erefore, one less layer of soil will be considered when calculating the subsidence amount, resulting in a large difference in the result. erefore, it can be seen that the criterion has certain limitations in the collapsibility evaluation of loess. When unloading of the loess foundation occurs, the collapsibility evaluation is too conservative, greatly increasing the project budget. However, the total and self-weight collapsibility as calculated by the unloading coefficient are more in line with reality, indicating the necessity and feasibility of this research content.

Conclusions
In this study, collapsible loess under unloading was examined. An unloading collapse coefficient expression was established and then analyzed and verified using field tests.
(1) Based on the unloading stress ratio K, unloading collapsible ratio η, and attenuation coefficient δ r of the unloading collapsibility, an expression is established for the unloading collapse coefficient, and the functional form and specific parameters of the expression are determined from laboratory tests.
(2) rough the laboratory tests, it is found that unloading can increase the initial pressure of the collapsibility to a certain extent, with an increase of 1.5 -9.3%. Moreover, the deeper the soil is buried, the greater the increase is.
(3) e total and deadweight collapsibility as calculated using the unloading coefficient of the collapsibility can more accurately and reliably evaluate the collapsibility of a site under the action of unloading.
Regarding unloading project, the existing codes cannot make an accurate collapsibility evaluation for it. e total collapsibility and self-weight collapsibility calculated by the constant pressure collapsibility coefficient in the specification are used to determine the site's collapsibility level as III (severe), and the site's collapsibility level should be determined as level II (medium) when using the collapsibility under the unloading test. It can be seen that the unloading collapsibility coefficient is more suitable for unloading projects than the constant pressure collapsibility coefficient.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.