Abstract
To enhance the construction foundation safety in the loess regions of Northwest China, this study systematically investigated the collapsibility characteristics of loess from a construction site in Lanzhou through laboratory collapse tests under constant pressure. Regression modeling was employed to quantitatively evaluate the influence of key geotechnical parameters—including natural density, dry density, moisture content, void ratio, and compression modulus—on the collapsibility coefficient. Significant relationships were established between the collapsibility coefficient and void ratio, natural moisture content, and compression modulus. Furthermore, the standard method was used to determine the variation of potential collapse around piles with depth, revealing an exponential distribution trend. It is worth emphasizing that this study successfully developed an exponential function model based on easily obtainable parameters (natural density, dry density, moisture content, and compression modulus) for quantitatively predicting the collapsibility coefficient. In contrast to existing models, which often rely on single indicators and lack a multi-parameter collaborative evaluation system—resulting in limited engineering applicability—this study integrated five parameters to construct a comprehensive prediction model, filling the gap in multi-index evaluation systems. This model provides a practical tool for more accurately predicting loess collapse settlement and can be directly applied to risk assessment and mitigation design for building foundations and pile-soil interactions in similar loess environments in Northwest China.
Keywords: Collapsible loess, Immersion compression test, Coefficient of collapsibility, Regression analysis
Subject terms: Engineering, Environmental sciences, Materials science, Natural hazards
Introduction
Due to its special structure, loess widely distributed in Northwest China is prone to collapsibility after infiltration with water. This process will lead to the rapid disintegration of the soil structure, causing the significant attenuation of soil strength and the simultaneous decline of foundation bearing capacity. With the deterioration of mechanical properties, the foundation will produce uneven settlement, which will eventually destroy the overall integrity of the soil, and there are potential safety hazards1–4. At present, there are three kinds of experiments to determine the collapsibility of loess, indoor immersion compression test, field static load test and field test pit immersion test. The indoor test is divided into uniaxial and triaxial immersion compression tests. Under the condition of uniaxial lateral confinement and triaxial test, the soil is immersed in water to obtain the collapsibility coefficient.
Many scholars use factor analysis, least square method, multiple regression analysis, fuzzy mathematics, and neural network methods to establish the relationship between the physical and mechanical indexes of collapsible loess and the collapsibility coefficient5–7 to predict the amount of soil collapsibility and evaluate the collapsibility of the site8–11. Zhang Chaozhe12 systematically investigated the effects of moisture content, density, and overburden pressure on loess compressibility and collapsibility through comprehensive laboratory testing. Their study established quantitative relationships between the collapsibility coefficient, electrical resistivity, and fundamental physical indices. Xu Xijiu13 deeply studied the collapsibility and seepage characteristics of undisturbed loess foundation, and revealed the variation characteristics of surface and deep soil settlement with time, immersion range and collapsibility range. Wang feng14 conducted comprehensive field immersion tests at a deep loess site in Lanzhou New Area, complemented by laboratory testing of undisturbed specimens. Their investigation quantitatively analyzed both moisture migration patterns and the temporal evolution of foundation settlement during saturation. Shi Baodong15 employed correlation analysis to examine the relationship between collapsibility coefficient and key soil parameters (dry density, moisture content, void ratio, degree of saturation, and plasticity index) under both undisturbed and disturbed conditions. The study derived regression equations for evaluating loess collapsibility in these states. Wang Qingman16 studied the variation law of water injection, the migration law of water field, the lower limit depth of self-weight collapse and its determination standard during the test process through the field test pit immersion test. The current research is mostly limited to the case analysis of specific site conditions, and has not yet established a multi-index evaluation system, resulting in doubts about the applicability of existing empirical formulas.
Based on the actual engineering project of Lanzhou port area, this paper takes the undisturbed loess of a loess site in Lanzhou as the research object. Through the test of soil with different natural density, dry density, moisture content, void ratio and compression modulus, the relationship between the collapsibility coefficient and the above various influencing factors is studied, and the multiple regression model of the collapsibility coefficient and void ratio, moisture content and compression modulus is established. According to the indoor test data, the standard method is used to calculate the collapsibility of the site, and the exponential model is proposed according to the collapsibility curve, which provides a certain basis for similar projects in the loess site of Lanzhou area, and lays a foundation for the next analysis and calculation of the mechanical characteristics of piles in collapsible loess foundation.
Basic overview of the test
General situation of loess geology in test site
The construction project of multimodal transport logistics park in Lanzhou International Port Area is located in Xiachuan Village, Xigu District, Lanzhou City. It is about 500 m from the Yellow River in the north, about 30m higher than the Yellow River, and about 3 km from the loess hills in the south. The landform type is the second-class terrace on the south bank of the Yellow River. The site is roughly irregular in north–south direction, and the overall terrain is relatively flat. It is slightly inclined from southwest to northeast, and the slope is uniform. The elevation is 1575–1588m, and the height difference is about 13m. The elevation of the northern end of the site is suddenly reduced to about 1770m in the form of artificial slope. The whole site is distributed with different thickness of II ~ III grade collapsible loess silty soil. In the depth range of this drilling, the exposed strata from top to bottom are 1 layer of miscellaneous fill, 2 layer of loess-like silt, 2–1 breccia, 3 layer of saturated silt, 4 layer of pebble and 5 layer of mudstone.
The groundwater is exposed in the site, which belongs to the Quaternary loose rock pore diving, and the water level is 5.2–22.3m. The aquifer is mainly composed of 4 layers of pebbles and 3 layers of saturated silt. It is mainly recharged by atmospheric precipitation and farmland irrigation water infiltration. From southwest to northeast, it flows in the form of underground undercurrent, and finally discharges in the form of spring water to the lower Yellow River.
Basic properties of loess soil samples
To evaluate loess collapsibility characteristics across the region, 35 test points were established in 11 designated zones. Our sampling strategy is based on the geological design of the site. The stable density gradient of typical loess in the south and middle is compared with the high moisture content in the northeast, which is affected by the water level of the Yellow River, while the northwest elevation is lower. A total of 490 undisturbed soil samples were systematically extracted at 1-m depth intervals for laboratory analysis of physical–mechanical properties and collapsibility behavior. The basic physical and mechanical properties of soil samples are shown in Table 1.
Table 1.
Statistics of main physical and mechanical properties of soil samples.
| Physical and mechanical properties | Range values | Mean value | Standard deviation | Coefficient |
|---|---|---|---|---|
| Natural Moisture Content W (%) | 20.4–7.8 | 14.0 | 2.39 | 0.170 |
| Natural Void Ratio e0 | 1.478–0.697 | 0.968 | 0.11 | 0.110 |
| Plastic Limit WP (%) | 17.65–16.66 | 17.11 | 0.15 | 0.009 |
| Liquid Limit WL (%) | 28.9–24.5 | 26.53 | 0.57 | 0.021 |
| Natural Density ρ (g/cm3) | 1.89–1.14 | 1.57 | 0.11 | 0.071 |
| Compression Coefficient a (MPa⁻1) | 1.02–0.02 | 0.235 | 0.11 | 0.566 |
| Compression Modulus Es (MPa) | 25.8–1.78 | 12.7 | 1.05 | 0.678 |
Test scheme
Based on the comprehensive analysis of previous studies on loess collapsibility17–20, 35 test points in 11 areas were sampled every meter. Laboratory collapsibility testing was conducted via oedometer immersion tests in strict compliance with ASTM D5333. Undisturbed ring-knife samples were vertically mounted within the fixed-ring consolidometer’s retaining ring prior to saturation loading. The filter paper and the permeable plate are placed on the upper and lower sides of the sample in turn, and the cover plate is added on the top surface and aligned with the pressure frame. The consolidation pressure was applied to 50kPa, 100kPa, 150kPa and 200kPa according to the requirements, and the deformation reading was recorded every 1h. Following specimen stabilization, distilled water was introduced into the testing chamber to achieve full saturation. Collapsibility testing proceeded under controlled conditions until immersion-induced deformation stabilized. The sampling and test photos are shown in Fig. 1.
Fig. 1.
Indoor test sampling and test photographs.
Results analysis
The relationship between collapsibility coefficient and natural density, dry density
Figure 2 shows the relationship between soil density and depth in this area. The data of a), b), c) and d) are taken from the south, middle, northeast and northwest of the site. The figure demonstrates a strong positive correlation between sampling depth and both natural density and dry density of the loess specimens, with values increasing consistently with depth. The change trend of natural density with depth is more intense, and the change trend of dry density with depth is slower, showing a linear trend as a whole. The dry density is close to the natural density at the top of the soil layer, and as the depth gradually increases, the difference between the natural density and the dry density is also increasing. The natural density presents a relatively loose pattern, and the fitting degree is low. The fitting degree of dry density is very good, and the change trend of soil samples in the four regions is closer, which indicates that there is a certain uneven state of moisture content of soil samples in the site, but the skeleton structure of soil itself is close, and there is no big difference. Because the northern part of the site is close to the Yellow River, the moisture content is slightly different, and the natural density gradually increases from south to north.
Fig. 2.
Graph of density variation with depth.
Figure 3 is the relationship between soil density and collapsibility coefficient. From the diagram, it can be seen that the four loess soil samples in the site all show similar trends. The collapsibility coefficient exhibits an inverse linear correlation with both natural density and dry density in this region. Minimal density values (natural and dry) correspond to peak collapsibility coefficients, where these density metrics converge. Conversely, maximal density values coincide with minimal collapsibility coefficients, demonstrating the widest divergence between natural and dry density measurements. This trend highlights moisture content’s measurable impact on loess collapsibility. The fitting degree between natural density and collapsibility coefficient is low, and the figure is more loose. The fitting degree between dry density and collapsibility coefficient is better, which is basically consistent with the fitting curve. For the loess in this area, when the natural density is close to 1.8g/cm3, or the dry density is close to 1.5g/cm3, the collapsibility coefficient is less than 0.015, which belongs to the non-collapsible loess. The linear regression analysis of natural density, dry density and collapsibility coefficient was carried out. The linear equation was 
, and the fitting results are shown in Table 2. It can also be seen from Table 2 that the R value between natural density and collapsibility coefficient is small, the fitting degree is poor, and the correlation is low. The R value between dry density and collapsibility coefficient is large, the fitting degree is good, and the correlation is high.
Fig. 3.
Graph of the relationship between collapsibility coefficient and density.
Table 2.
Fitting parameters of collapsibility coefficient with density of soil samples.
| Soil sample | Natural density | Dry density | ||||
|---|---|---|---|---|---|---|
| a | b | R | a | b | R | |
| (a) | 0.20017 | − 0.09975 | − 0.72228 | 0.33888 | − 0.2068 | − 0.65607 |
| (b) | 0.19334 | − 0.09151 | − 0.8076 | 0.40969 | − 0.2646 | − 0.6998 |
| (c) | 0.21983 | − 0.10688 | − 0.8264 | 0.42034 | − 0.26889 | − 0.78933 |
| (d) | 0.27916 | − 0.14137 | − 0.82342 | 0.44595 | − 0.27896 | − 0.87949 |
The relationship between collapsibility coefficient and moisture content
One of the necessary conditions for the occurrence of collapse is immersion. The loess without immersion has high strength and strong bearing capacity, and is a foundation soil with good properties. However, when it is affected by immersion, the structure collapses, the strength decreases, and the collapse occurs, which causes great harm to the building. The natural moisture content, representing the initial water state of undisturbed loess, serves as a critical parameter in collapsibility evaluation. This parameter proves particularly significant given loess’s well-documented hydro-sensitivity. Capillary cohesion hypothesis and water film penetration hypothesis are all hypotheses proposed for the change of loess moisture content.
Figure 4 shows the relationship between moisture content and depth in this area. The natural moisture content of the selected soil samples is between 12 and 21%, and the overall change trend is consistent. With the increase of depth, it gradually increases, showing a positive correlation. Because Lanzhou is located in the inland dry climate, the moisture content of the surface soil is low. The sampling is in the winter without rainfall. The moisture content of the surface soil in this area is basically maintained between 12 ~ 15%, and the moisture content of the deep soil is maintained at 17 ~ 20%. The site is 500m north of the Yellow River, and the groundwater level changes greatly. Affected by the change of groundwater level, above the water level line is unsaturated loess-like silt, below the water level line is saturated silt. Since the main purpose of this test is to analyze the collapsibility of loess-like silt, and the collapsibility of saturated silt below the water level line basically disappears completely, the unsaturated loess-like silt above the water level line is taken as the test soil sample. According to the analysis of groundwater level, the maximum sampling depth of the four test points selected in this test is 14m. The soil of this layer has not reached saturation, and the moisture content has not exceeded 21%.
Fig. 4.
Graph of moisture content variation with depth.
Figure 5 is the relationship between moisture content and collapsibility coefficient. The loess collapsibility coefficient exhibits a negative correlation with natural moisture content, consistent with domestic and international research. Lower moisture content corresponds to higher collapsibility coefficients, while higher moisture content reduces collapsibility. At moisture content levels reaching 18%, the collapsibility coefficient typically falls below 0.015, indicating non-collapsible soil. When the natural moisture content is low, the strength of the cementing material connected between the skeleton particles in the soil is high, and the overall structural connection is stable. It is highly sensitive to water, and the structure will collapse rapidly in case of immersion, which has strong collapsibility. When the natural moisture content is high, the moisture content of the cement is high and the connection strength is small. The sensitivity to water is reduced, the space available for particle immersion deformation is reduced, and the collapsibility becomes weak or even no collapsibility.
Fig. 5.
Graph of the relationship between moisture content and collapsibility coefficient.
The linear regression analysis of moisture content and collapsibility coefficient is carried out. The linear equation is 
, and the results are shown in Table 3. The change trend is similar, except that the moisture content of the soil samples in the northeast is more dispersed, the R value is small, and the correlation is weak. The correlation coefficient R values of soil samples in the other three regions are relatively large and the correlation is significant. Due to its proximity to the Yellow River and lower elevation, the northeastern region exhibits distinct collapsibility behavior. The fitting curve for moisture content displays a gentler slope compared to other areas, corresponding to higher baseline moisture levels and elevated collapsibility coefficients.
Table 3.
Fitting parameters of collapsibility coefficient with moisture content of soil samples.
| Soil sample | a | b | R |
|---|---|---|---|
| (a) | 0.18473 | − 0.00894 | − 0.74718 |
| (b) | 0.19032 | − 0.00911 | − 0.79653 |
| (c) | 0.1547 | − 0.00659 | − 0.69026 |
| (d) | 0.24784 | − 0.01313 | − 0.85858 |
The relationship between collapsibility coefficient and void ratio
The number of pores in the loess determines the strength of collapsibility, and the void ratio is positively correlated with the collapsibility coefficient. Figure 6 is the relationship between void ratio and depth in this area. With the increase of depth, the void ratio decreases gradually, the void ratio of the top surface of the soil layer is the largest, and the void ratio of the deep soil is the smallest. The surface soil has a short formation time, low consolidation degree, loose structure and rich large and medium pores. Because it is on the surface, it is less loaded by the upper soil; coupled with weathering erosion, rainfall infiltration and other reasons; the void ratio is relatively large. The deep soil has a long formation age, a high degree of consolidation, and a tight structure. And there is a certain thickness of the overlying soil, long-term self-weight load by the upper soil; in addition, it is close to the groundwater level and is affected by the change of groundwater level. The void ratio is relatively small. From the results, the void ratio of the four areas is between 0.6 and 1.4, and the change trend is roughly the same, which gradually increases with the increase of depth. The void ratio of the whole area gradually decreases from south to north, and the void ratio in the south is the largest, while the void ratio in the northwest does not exceed 1.2.
Fig. 6.
Graph of void ratio variation with depth.
Figure 7 is the relationship between void ratio and collapsibility coefficient. The figure reveals a consistent positive correlation between void ratio and collapsibility coefficient across all four regions, with higher void ratios corresponding to elevated collapsibility. This is consistent with the known theory. For soil samples with large pores, the pores between the skeleton particles in the soil are larger, and the space that can be compressed is larger when the soil is soaked and collapsible, and the corresponding collapsible coefficient is larger. Most of the soil samples in this area are collapsible, and only a small number of soil samples have collapsibility coefficients below 0.015. When the void ratio is less than 0.7, the soil sample basically does not have collapsibility; when the void ratio is greater than 1.2, the soil sample exhibits strong collapsibility. The collapsibility coefficient in the central part of the region changes slowly with the change of void ratio, while the change trend in the northwest is steep. This shows that although in the same area, the location of soil is different, and the structure and void of loess are also different, and these differences also cause the change of collapsibility coefficient. The linear regression analysis of void ratio and collapsibility coefficient is carried out. The linear equation is 
shown in Table 4. The R value of the fitting curve in the central region is low, and the correlation is poor. The R values of the fitting curves of the other three regions are higher and the correlation is better, which further shows that it is more reasonable to use the linear equation to represent the relationship between the collapsibility coefficient and the void ratio.
Fig. 7.
Graph of the relationship between void ratio and collapsibility coefficient.
Table 4.
Fitting parameters of collapsibility coefficient with void ratio of soil samples.
| Soil sample | a | b | R |
|---|---|---|---|
| (a) | -0.04197 | 0.08649 | 0.7181 |
| (b) | − 0.03562 | 0.6182 | 0.6182 |
| (c) | − 0.04861 | 0.09993 | 0.80377 |
| (d) | − 0.06501 | 0.1238 | 0.75272 |
The relationship between collapsibility coefficient and compression modulus
Compression modulus is an important index related to the compressibility of soil. The compression modulus is the ratio of the vertical stress variation of the soil to the corresponding strain variation, which reflects the ability of the soil to resist vertical deformation. The compression deformation of soil is essentially the rearrangement of the structural space of soil. The soil with large compression modulus has stronger ability to resist external pressure and immersion damage, the soil is denser, the connection between skeleton particles is stronger, the structural strength and cementation strength are larger, and the large pores are less, mostly medium and small pores. The smaller the compression modulus of the soil, the smaller the ability to resist external pressure and immersion damage, the loose soil, the weaker the connection between the skeleton particles, the weaker the structural strength and the cementation strength, and the larger the pores.
In this paper, the data of the 50 ~ 200 kPa stage before the test immersion is selected as the analysis object, and the relationship between the collapsibility coefficient and the compression modulus is drawn (Fig. 8). According to the diagram, the compression modulus value is basically between 2 ~ 14 MPa. As the compression modulus increases, the soil becomes more dense, the collapsibility coefficient decreases, and the collapsibility of the soil sample is weaker. The two show an exponential negative correlation. When the compression modulus of the soil is greater than 12 MPa, the soil sample basically has no collapsibility. The exponential curve regression analysis of collapsibility coefficient and compression coefficient of four kinds of soil samples is carried out. The curve equation is 
shown in Table 5. The correlation coefficient R values of the four soil samples are relatively large, and the data are well fitted with the regression analysis curve, and the correlation is significant.
Fig. 8.
Graph of the relationship between compression modulus and collapsibility coefficient.
Table 5.
Fitting parameters of collapsibility coefficient with compression modulus of soil samples.
| Soil sample | a | b | c | R |
|---|---|---|---|---|
| (a) | − 0.02179 | 0.12131 | − 0.10629 | 0.92957 |
| (b) | − 0.03981 | 0.13473 | − 0.08679 | 0.88850 |
| (c) | − 0.06307 | 0.15555 | − 0.09369 | 0.88309 |
| (d) | − 0.00987 | 0.11813 | − 0.14238 | 0.91756 |
Multiple regression analysis of collapsibility coefficient
It can be seen from the previous article that there is a high correlation between each factor and the collapsibility coefficient in the single factor regression analysis equation between dry density, moisture content, void ratio, compression modulus and collapsibility coefficient. However, the collapsibility of loess is complex and cannot be determined by a single factor. It is influenced by many factors. Therefore, multiple physical properties and mechanical properties can be used to comprehensively evaluate collapsibility.
The multiple regression equation of collapsibility coefficient is constructed to better describe the influence degree of each influencing factor on loess collapsibility. The number of independent variables in the multiple regression model is not the more the better. It should be determined according to the sensitivity of specific factors, and consider the actual engineering application to select indicators that are easier to measure. In the regression model, it is required that there is a correlation between the influencing factors and the collapsibility coefficient, and the factors are independent of each other. In this section, three influencing factors of natural void ratio, natural moisture content and compression modulus are selected. Multiple regression analysis was employed to evaluate the correlation between the collapsibility coefficient and key influencing factors of loess-like silt in Lanzhou Port Area, elucidating its relationship with critical physical–mechanical indices. The calculated collapsibility coefficient was compared with the measured value, and the accuracy and practical value were analyzed to provide reference for practical engineering.
According to the above analysis, it is known that there is a linear relationship between the collapsibility coefficient and the natural void ratio and the natural moisture content, and an exponential relationship between the collapsibility coefficient and the compression modulus, that is 
, the multiple linear regression equation of the collapsibility coefficient is assumed. The analysis results are shown in Table 6.
Table 6.
Fitting parameters of collapsibility coefficient with void ratio, natural moisture content, and compression modulus.
| Soil sample | A0 | A1 | A20 | A3 | A43 | R |
|---|---|---|---|---|---|---|
| (a) | 0.11538 | − 0.01653 | − 0.0055 | 0.03867 | − 0.93109 | 0.76826 |
| (b) | 0.12837 | − 0.02034 | − 0.0062 | 0.04276 | − 1.03413 | 0.86765 |
| (c) | 0.96346 | − 0.01343 | − 0.0038 | 0.03487 | − 0.90954 | 0.83654 |
| (d) | 0.10454 | − 0.01532 | − 0.0049 | 0.03649 | − 0.98432 | 0.79235 |
| Comprehensive | 0.12065 | − 0.01754 | − 0.0057 | 0.04021 | − 0.95943 | 0.80125 |
Analysis of collapsible loess foundation soaking collapsibility
Current experimental methods for determining loess collapsibility comprise three approaches: laboratory oedometer tests, field plate load tests, and field test pit immersion tests21,22. Both field methods (plate load tests and test pit immersion) require saturating foundation soils, thereby yielding more realistic real-time deformation characteristics during collapsibility events. However, the experiment is complex, time-consuming and costly. The indoor compression experiment adopts the compression experiment of undisturbed soil or remolded soil, which is greatly affected by the conditions of on-site sampling or sample preparation, and the test operation requirements are more accurate. But the test is convenient and quick, time-consuming and low cost. Therefore, when calculating and predicting the possible collapsible deformation of related construction projects in loess area, the maximum collapsible amount of the foundation can be calculated by taking the undisturbed loess from the site as the test soil sample, carrying out the indoor collapsible test and measuring the relevant parameters. According to the construction standard of collapsible loess area (GB 50025-2018), the calculated value of collapsibility is obtained by indoor compression test. The formula is:
 |
1 |
In the formula: 
represents the collapsibility of the foundation after water immersion and saturation; 
is the collapsibility coefficient of the i-th soil layer; 
is the thickness of the i-th soil layer (mm); α is the probability coefficient of water immersion for the foundation soil at different depths; β is the correction coefficient considering factors such as the stress on the foundation soil.
Single homogeneous loess foundation
When studying a single homogeneous soil layer, it is assumed that the soil is uniform, and the collapsibility coefficient 
is the same. However, due to differences in depth, the correction coefficient α and the water immersion probability coefficient β vary. Based on Eq. (1), the collapsibility is expressed as a piecewise function.
 |
2 |
The formula for calculating the collapsibility is a piecewise function. 
is the correction coefficient for different depths. When substituting this formula into subsequent calculations of pile stress, it is necessary to perform calculations segment by segment based on the collapsibility, which is computationally intensive and complex. This paper proposes fitting the collapsibility curve calculated by Eq. (2) into an exponential function. In subsequent calculations of pile stress, the collapsibility displacement of the loess around the pile is represented by a single function, simplifying the derivation process.
 |
3 |
Assuming the soil is a single homogeneous layer, the collapsibility curves and their exponential fitting curves are calculated using Eq. (2) for different collapsibility coefficients, as shown in Fig. 9. From the figure, it is evident that the variation of collapsibility in the homogeneous soil layer aligns well with the exponential function. Above 6 m, the collapsibility changes significantly, while below 6 m, the collapsibility deformation is relatively small. As the collapsibility coefficient increases, the collapsibility of the upper soil layer changes more dramatically, while the lower soil layer experiences smaller changes. When the collapsibility coefficient increases from 0.03 to 0.07, the displacement at the top of the soil layer (− 1 m) more than doubles, while the change at the lower soil layer (− 14 m) is minimal23–26.
Fig. 9.
Graph of calculated collapsibility variation in homogeneous soil layers.
Layered loess foundation
Based on the actual engineering geological conditions of the Lanzhou Port Area and the collapsibility coefficients obtained from laboratory tests, the calculated collapsibility values and their exponential fitting curves are derived using Eq. (1), as shown in Fig. 10. The top 1-m layer consists of miscellaneous fill and is non-collapsible. The collapsibility changes most significantly within the 1–6-m range, which corresponds to the upper half of the pile and serves as the primary source of collapsibility. Below 6 m, the deeper soil layers are subjected to the self-weight pressure of the overlying soil, resulting in smaller collapsibility coefficients. Additionally, influenced by the probability of water immersion, the collapsibility changes more gradually. Although the layered conditions and collapsibility coefficients vary across different test areas, leading to slight differences in the calculated collapsibility curves, the overall trend remains consistent and aligns well with the exponential curve fitting.
Fig. 10.
Graph of calculated collapsibility variation in soil layers of Lanzhou port area.
For practical application, the multiple regression model offers a direct method for predicting the collapsibility coefficient. The collapse deformation curve (Fig. 10), which guides pile length design, shows that subsidence is primarily confined to the shallow 0–6 m layer. Consequently, it is essential to design piles such that their tips extend through this collapsible zone into a stable stratum. Furthermore, to address the issue of sinking soil around the pile, designers should consider incorporating a pile body isolation layer or utilizing an expanded pile diameter.
Conclusions
To identify the key factors governing loess collapsibility in the Lanzhou region, this study analyzes the relationship between the collapsibility coefficient and relevant physical–mechanical indices based on loess samples collected from the Xigu Port Area. A model suitable for Lanzhou loess was constructed. If it is used in other areas, the parameters in the model need to be adjusted. The following conclusions were reached:
There is a good negative correlation between the natural density, dry density, moisture content, compression modulus and collapsibility coefficient of loess in this site. The relationship between natural density and collapsibility coefficient:
; the relationship between dry density and collapsibility coefficient : 
; the relationship between moisture content and collapsibility coefficient : 
; the relationship between compression modulus and collapsibility coefficient : 
. When the natural density is greater than 1.8 g/cm3, or the dry density is greater than 1.5 g/cm3, the collapsibility coefficient is less than 0.015, and the collapsibility can be ignored. When the natural moisture content is as high as 18% or the compression modulus is greater than 12 MPa, the soil sample basically has no collapsibility.Correlations between the loess collapsibility coefficient and individual parameters vary significantly, with strong relationships observed specifically with compression modulus, dry density, and moisture content. A multiple regression model relating collapsibility coefficient to void ratio, moisture content, and compression modulus was established as follows:
.The collapsible deformation of loess in this area was calculated using a stratified approach, revealing significant variations in magnitude. Based on the collapsible deformation curve, an exponential function
was selected to model the loess immersion behavior. The model demonstrates a high goodness-of-fit, confirming its suitability for representing loess collapsibility.
Acknowledgements
The corresponding author would like to acknowledge the Major Science and Technology Special Project Plan of Gansu Province (Grant No. 24ZDFA010), the Gansu Province Construction of Scientific and Technological Research Project (Grant No. JK2025-13). The financial supports are gratefully acknowledged.
Author contributions
Conceptualization, Z.L.; Methodology, Z.L. and S.Y.; Validation, S.Y. and X.L.; Investigation, L.H.; Data curation, L.H. and Y.Z.; Resources, S.Y., Y.H. and Y.Z.; Writing—original draft, Z.L.; Writing—review and editing, S.Y., Y.H. and J.M. All authors have read and agreed to the published version of the manuscript.
Funding
This research was supported by the Major Science and Technology Special Project Plan of Gansu Province (Grant No. 24ZDFA010), the Gansu Province Construction of Scientific and Technological Research Project (Grant No. JK2025-13).
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Declarations
Conflict of interest
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.