# Collapse Potential of Loess under Unloading Effect

Authors: Xin Jin; Tie-Hang Wang; Zai-Kun Zhao; Liang Zhang; Yan-Zhou Hao
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

Collapsible loess is generally characterized by a sudden and substantial decrease in volume that occurs when is applied under constant stress. To evaluate the loess collapse potential, the self-weight collapse and collapse coefficients have been defined by the code for building construction in collapsible loess regions. However, the method in the code does not account for the vertical stress variation. The loess collapse process commonly occurs with stress variation in practice. This paper documents a low-cost, quantitative evaluation scheme using regression analysis to evaluate the loess collapse potential by varying the unloading levels. The results show that the factors that prominently account for loess collapse deformation are the initial pressure, unloading ratio, and collapse completed-ratio. At a constant collapse-completed ratio, the remnant collapse coefficient significantly decreases with the decreasing unloading ratio; at a constant unloading ratio, the remnant collapse coefficient increases with a decreasing collapse-completed ratio. Decreasing unloading and collapse-completed ratios decreased the loess collapse potential with an initial pressure that exceeds the threshold value. Finally, an unloading collapse deformation calculation of loess was prepared to analyze practical project problems of loess based on the unloading collapse test.


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## Body

## 1. Introduction

Loess is an aeolian deposit of yellowish dust mostly formed during the Quaternary period [1–3]. Loess deposits cover 640000 km2, distributed mainly along the middle reaches of the Yellow River Valley, constituting one of the most important parts of the Chinese regional and physical geography and the Loess Plateau. Because of its collapse behavior and other associated geotechnical engineering issues, such as landslides, hydro-consolidation, and wetting-induced collapse, loess belongs to problematic soils and has been the subject of geotechnical research and practice since the 1960s [4–14]. Following the adoption of “the belt and road” strategy, infrastructure construction in the Loess Plateau will develop more vigorously. Therefore, before the design and construction on the loess field, it is important to determine the loess collapse value. The evaluation method for the loess collapse at present mainly concentrates on the constant loading and humidification or dehumidification.At present, there are two ways to study the collapse of loess under constant load: one is the combination of the laboratory test and numerical calculation method; the other is the field immersion test. Rogers et al. [15] through the monitoring results of shear wave velocity and resistivity in the process of loess collapse reflected the collapse deformation characteristics of intact loess under constant pressure. Francisca [9] analyzed data from in situ and laboratory test to establish possible correlations between the high strain elastic properties of loess and results from SPT. Jiang et al. [16, 17] summarized experimental studies on wetting-induced collapsibility in loess using single oedometer, double oedometer, and triaxial wetting tests and presented a distinct element method (DEM) to investigate its macromechanical and micromechanical behavior (compression and collapse behavior) under one-dimensional compression condition. Wang et al. [18] conducted a 40-meter-diameter soaking test in the field with self-weight collapsible loess at a thickness of 36.5 m in order to study the deformation property of large-thickness self-weight collapsible loess. Deng et al. [19] conducted a collapse assessment by combining the laboratory and in situ soaking tests. The reasons for the significant difference in collapse settlement between the in situ soaking tests and the calculated estimates were discussed. Li et al. [20] investigated the functional mechanism of loess collapse for a tunnel structure by conducting a large-field water immersion test on the ground surface above an existing loess tunnel. The subsidence of the loess during the long-term water immersion test was measured and analyzed to reveal the collapsible deformation characteristics of the loess tunnel site. Wang et al. [2] proposed a method for analyzing and evaluating the collapsibility of loess by combining the relationship between the compression and initial structural properties of loess.Because the strength of grain connection of loess is not fixed, it changes with the variety of water content. After the first humidification, the connecting grains between loess grains are weakened properly, and the loess will be rebalanced under the new strength condition after a small amount of collapse occurs. Then, the loess was humidified again, and a little collapse occurred. After many times of wetting, the collapse reaches the maximum value after saturation [21]. In the practical project, the collapse often occurs in the case of different depth loess layers reaching different moisture content, so it is of significance to study the collapse deformation law of loess under the effect of increasing and decreasing moisture. Shao et al. [22–24] based on the triaxial test of Q3 loess, the initial tangent modulus of loess with different water content under the given confining pressure under axial loading is measured, and the damage evolution characteristics of the loess structure during humidification are investigated according to the variety of modulus. Liu et al. [25] proposed the secant modulus algorithm of loess wetting deformation based on a large number of single and double line loess methods, and the collapse deformation or humidification deformation of loess foundation is obtained. Zhang et al. [26] obtained the wetting deformation of loess according to the difference of the e-p curve of the unimmersed loess sample under different initial water content. Xu et al. [27] simulated field conditions involving flood irrigation on a cracked loess plateau surface and studied their effects on the loess deformation. Ng et al. [28] conducted a laboratory test to investigate the effects of the microstructure on the water retention curve and wetting and drying-induced volume change in loess. An et al. [29] studied the settling characteristics of loess through a field immersion test on thick collapsible loess. Weng et al. [30] investigated the mechanisms that influence wetting-induced collapse deformation of loess strata on the lining structure of a subway tunnel. Hou et al. [31] highlighted the dependence of loess collapse deformation on the infiltration processes and estimated the in situ collapse characteristics.At present, the research on loess collapse mainly focuses on the effect of constant load and humidification, which solves a lot of practical engineering problems; however, there is still a lack of research literature and analysis and evaluation methods for loess unloading collapse, which makes the design of loess unloading collapsible engineering lack of scientific basis. The current loess collapse evaluation method does not account for the vertical stress variation. For example, the engineering experience shows that the additional settlement of loess between the piles occurs because of the inundation with construction on bored pile foundations in the self-weight collapsible loess area. Then, the negative friction of the piles [32–34], which develop during the collapse of loess, reacts on the loess and reduces the vertical stress of loess around the piles. The loess collapse often occurs with the unloading process caused by the pile negative friction. Unfortunately, the engineering design and problem analysis are limited by the lack of collapse deformation characteristics’ research of loess between piles.This paper focused on the unloading effect during the loess collapse process and provided an innovative evaluation method for the loess collapse potential of unloading. The loess unloading collapse behavior was analyzed. The loess collapse influence coefficient of unloading was investigated. The research work is significant to supplement and improve the evaluation method of loess collapse and to promote a scientific design method of bored concrete-pile composite foundation in the self-weight collapsible loess site by considering the loess unloading effect.


## 2. Influence Factors of Unloading Collapse

According to the engineering conditions of unloading collapse of the loess site, two factors were considered in the process of loess collapse under the effect of unloading: the completion degree of loess collapse when unloading occurred and the magnitude of the unloading amount during unloading. Therefore, in order to define the process of unloading collapse, the following parameters were defined.The collapse-completed degree of loess was represented by the collapse-completed ratioη, which is calculated using(1)η=S1S×100%,where S1 is the completed collapse amount (mm) of loess under the initial pressure and S is the total collapse amount (mm) of loess under the initial pressure. The initial pressure is unloaded at different degrees when η reaches 20%, 40%, 60%, and 80%.To evaluate the unloading amplitude, the unloading ratioK is calculated using(2)K=p1−Δpp1=prp1,where p1 is the initial pressure (kPa), pr is the remnant pressure after unloading (kPa), and Δp is the vertical pressure released when unloading occurs during loess collapse (kPa).Due to the decrease of vertical pressure during unloading, the collapse deformation of loess should be less than that under constant initial pressurep1. In order to express the value of uncompleted collapse of intact loess under the effect of unloading, the ratio of uncompleted collapse Sr to initial height h0 is defined as the coefficient of collapse reduced by unloading. The remnant collapse characteristic is represented by the remnant collapse coefficient δr, which is calculated using(3)δr=S−S1−S2h0=Srh0,where S is the total collapse deformation of the loess under pressure p1 (mm), S1 is the collapse of loess under pressure p1 before unloading (mm), S2 is the collapse of loess under pressure pr after unloading (mm), Sr is the uncompleted collapse of intact loess after unloading (mm), and h0 is the initial loess sample height (mm).


## 3. Experimental Investigation

### 3.1. Intact Loess in the Test

The tested intact loess was collected from a site in the Tongchuan area, north-western Chinese Loess Plateau. The site investigation and the basic physical properties of soil samples have been carried out in the selection of soil sampling sites. When sampling on-site, it is necessary to ensure the integrity of soil samples without cracks. In order to avoid disturbance to the undisturbed soil sample, the sampling process is all made by manual cutting with soil cutter. After sampling, the soil sample is cut into a soil column with a height of about 30 cm and a diameter of 12 cm. During sampling, the moisture content and density of soil samples shall be measured and recorded. We have avoided tree root, gravel, and other parts and selected the location with relatively uniform soil sample properties for sampling. Since the sampling site is a building foundation pit, the manual cutting method is adopted (the depth of soil sampling is 5.0∼8.5 m). These specimens are yellow-brown in color, porous, and contain a small amount of calcareous nodules. The sampling process is shown in Figure1.Figure 1
Site sampling and laboratory preparation process.
(a)(b)(c)(d)According to the “standard for the soil test method” (GB/T 50123-1999) [35], the physical properties of undisturbed loess are tested. The particle size gradation was measured by the Bettersize 2000 laser particle size analyzer (Figure 2).Figure 2
Bettersize 2000 laser particle size analyzer.Three groups of parallel tests were carried out for each group of samples. If the results meet the reasonable scope stipulated in the standard, the average value is taken as the final index. The basic physical properties of intact loess are shown in Table1.Table 1
Physical properties of intact loess.
Water contentDensityVoid ratioLiquid limitPlastic limitPlastic indexParticles contentω (%)Ρ (g·cm−3)eωL(%)ωp(%)Ip>0.050.05–0.005<0.00513.61.351.2727.317.99.419.3263.5617.12


### 3.2. Collapse Coefficientδs of Loess under Different Initial Pressure p1

According to the “standard for building construction in collapsible loess regions” (GB 50025-2018) [36], the collapse coefficient δs of loess is given as follows:(4)δs=Sh0,where S is the total collapse deformation of the loess under pressure p1 (mm) and h0 is the initial loess sample height (mm).The collapse coefficientδs and initial collapse pressure psh of loess measured by the laboratory test should meet the following requirements:(1)
The quality grade of the soil sample is grade I, which is intact soil sample.(2)
The size of the ring knife:φ = 79.8 mm and h = 20 mm. Before use, the ring knife is cleaned and air dried, and the permeable stone is dried and cooled.(3)
Before loading, the loess sample should be kept in natural water content.(4)
The stability standard of the sample before or after immersion should be that the settlement per hour is not more than 0.01 mm.When determining the collapse coefficient of intact loess, the double-line compression test was adopted. Whenp1 is in the range of 0–200 kPa, the collapse coefficients of intact loess under different p1 actions are measured, as shown in Figure 3 and Table 2.Figure 3
Relationship curves ofδs and (p)1 ((p)1 ≤ 200 kPa).Table 2
δs of loess under different p1 (p1 ≤ 200 kPa).
p1 (kPa)255075100125150175200δs0.00250.00600.01460.02780.04910.06470.09670.0978It can be seen from Figure3 and Table 2, the initial collapse pressure of undisturbed loess is 75 kPa, and the saturated density of overlying soil is calculated according to(5)ρs=ρd1+Sreds,where ρs is the saturated density of loess (g/cm3), ρd is the dry density of soil (g/cm3), Sr is the saturation of soil, Sr = 85%, e is the void ratio of loess, and ds is the relative density of loess particles.Whenρs = 1.67 g/cm3 and p1 = 200 kPa, δs > 0.07, which indicates that the loess in this test has a great collapse potential according to the “standard for building construction in collapsible loess regions” (GB 50025-2018), and the collapse of the loess sample is strong. The collapse coefficient δs under different p1 is shown in Table 3.Table 3
δs of loess under different p1.
p1 (kPa)200250300350400δs0.0980.1180.1420.1610.163


### 3.3. Test Method of Loess Collapse of Unloading

In this paper, the conventional collapsibility test was carried out first, and then, the unloading collapse test was carried out. Under the effect of the initial pressure, the intact loess sample was compressed and stabilized under the natural water content, and then, the loess is soaked in water to let the loess collapse. When the loess completed different collapse amounts, different degrees of vertical unloading were carried out, and the collapse deformation after unloading was measured. (in the unloading collapse test, there is no rebound phenomenon worthy of consideration, and the unloading rebound deformation can be ignored). Second, the intact loess was compressed at a prescribed initial pressure. Then, the loess was subjected to wetting when the compression deformation of loess was stable, and the collapse deformation began to increase. Unloading collapse tests were performed when the loess completed the amount of collapse to some extent, and the unloading collapse deformation of loess was measured. The loess unloading collapse test scheme is shown in Table4.Table 4
Test scheme of unloading collapse tests.
p1 (kPa)η (%)Δp (kPa)20020255075100250402550751003006050100150200350805010015020040050100150200


## 3.1. Intact Loess in the Test

The tested intact loess was collected from a site in the Tongchuan area, north-western Chinese Loess Plateau. The site investigation and the basic physical properties of soil samples have been carried out in the selection of soil sampling sites. When sampling on-site, it is necessary to ensure the integrity of soil samples without cracks. In order to avoid disturbance to the undisturbed soil sample, the sampling process is all made by manual cutting with soil cutter. After sampling, the soil sample is cut into a soil column with a height of about 30 cm and a diameter of 12 cm. During sampling, the moisture content and density of soil samples shall be measured and recorded. We have avoided tree root, gravel, and other parts and selected the location with relatively uniform soil sample properties for sampling. Since the sampling site is a building foundation pit, the manual cutting method is adopted (the depth of soil sampling is 5.0∼8.5 m). These specimens are yellow-brown in color, porous, and contain a small amount of calcareous nodules. The sampling process is shown in Figure1.Figure 1
Site sampling and laboratory preparation process.
(a)(b)(c)(d)According to the “standard for the soil test method” (GB/T 50123-1999) [35], the physical properties of undisturbed loess are tested. The particle size gradation was measured by the Bettersize 2000 laser particle size analyzer (Figure 2).Figure 2
Bettersize 2000 laser particle size analyzer.Three groups of parallel tests were carried out for each group of samples. If the results meet the reasonable scope stipulated in the standard, the average value is taken as the final index. The basic physical properties of intact loess are shown in Table1.Table 1
Physical properties of intact loess.
Water contentDensityVoid ratioLiquid limitPlastic limitPlastic indexParticles contentω (%)Ρ (g·cm−3)eωL(%)ωp(%)Ip>0.050.05–0.005<0.00513.61.351.2727.317.99.419.3263.5617.12


## 3.2. Collapse Coefficientδs of Loess under Different Initial Pressure p1

According to the “standard for building construction in collapsible loess regions” (GB 50025-2018) [36], the collapse coefficient δs of loess is given as follows:(4)δs=Sh0,where S is the total collapse deformation of the loess under pressure p1 (mm) and h0 is the initial loess sample height (mm).The collapse coefficientδs and initial collapse pressure psh of loess measured by the laboratory test should meet the following requirements:(1)
The quality grade of the soil sample is grade I, which is intact soil sample.(2)
The size of the ring knife:φ = 79.8 mm and h = 20 mm. Before use, the ring knife is cleaned and air dried, and the permeable stone is dried and cooled.(3)
Before loading, the loess sample should be kept in natural water content.(4)
The stability standard of the sample before or after immersion should be that the settlement per hour is not more than 0.01 mm.When determining the collapse coefficient of intact loess, the double-line compression test was adopted. Whenp1 is in the range of 0–200 kPa, the collapse coefficients of intact loess under different p1 actions are measured, as shown in Figure 3 and Table 2.Figure 3
Relationship curves ofδs and (p)1 ((p)1 ≤ 200 kPa).Table 2
δs of loess under different p1 (p1 ≤ 200 kPa).
p1 (kPa)255075100125150175200δs0.00250.00600.01460.02780.04910.06470.09670.0978It can be seen from Figure3 and Table 2, the initial collapse pressure of undisturbed loess is 75 kPa, and the saturated density of overlying soil is calculated according to(5)ρs=ρd1+Sreds,where ρs is the saturated density of loess (g/cm3), ρd is the dry density of soil (g/cm3), Sr is the saturation of soil, Sr = 85%, e is the void ratio of loess, and ds is the relative density of loess particles.Whenρs = 1.67 g/cm3 and p1 = 200 kPa, δs > 0.07, which indicates that the loess in this test has a great collapse potential according to the “standard for building construction in collapsible loess regions” (GB 50025-2018), and the collapse of the loess sample is strong. The collapse coefficient δs under different p1 is shown in Table 3.Table 3
δs of loess under different p1.
p1 (kPa)200250300350400δs0.0980.1180.1420.1610.163


## 3.3. Test Method of Loess Collapse of Unloading

In this paper, the conventional collapsibility test was carried out first, and then, the unloading collapse test was carried out. Under the effect of the initial pressure, the intact loess sample was compressed and stabilized under the natural water content, and then, the loess is soaked in water to let the loess collapse. When the loess completed different collapse amounts, different degrees of vertical unloading were carried out, and the collapse deformation after unloading was measured. (in the unloading collapse test, there is no rebound phenomenon worthy of consideration, and the unloading rebound deformation can be ignored). Second, the intact loess was compressed at a prescribed initial pressure. Then, the loess was subjected to wetting when the compression deformation of loess was stable, and the collapse deformation began to increase. Unloading collapse tests were performed when the loess completed the amount of collapse to some extent, and the unloading collapse deformation of loess was measured. The loess unloading collapse test scheme is shown in Table4.Table 4
Test scheme of unloading collapse tests.
p1 (kPa)η (%)Δp (kPa)20020255075100250402550751003006050100150200350805010015020040050100150200


## 4. Characteristic Analysis of Loess Collapse of Unloading

### 4.1. Unloading Collapse Test Results

Consequently, 384 oedometer unloading collapse tests were performed. The outputs of the experimental work are as follows. Figure1 shows that the initial pressure, unloading ratio, and collapse-completed ratio significantly affect the remnant loess collapse. The family of curves of different initial pressures represents the unloading collapse test with different collapse ratios.The collapse potential after unloading decreases with the decreasingK for the same η, whereas the slope of the curves decreases. Thus, with an increasing amplitude of vertical unloading, the collapse deformation of loess gradually decreases. The reason is that the self-weight collapsible loess is generally characterized by sudden and substantial deformations that occur when inundation is applied under constant stress. With the decrease in effective normal stress of the framework particles, the effects of displacement, dislocation, and collapse among the loess particles are weakened. Hence, the collapse deformation of loess decreases.Figure4 also shows that the collapse potential decreases because of an increase in η for the same K. First, the likely reason is the open pores of various diameters and shapes, which are formed by the framework particles that connect with one another by point contact connection because the collapse is processed in parts of dissolving salt crystals and ion exchange among the particles. The connection among the particles can be destroyed. Second, after the loess begins to collapse, its structure is dramatically destroyed, which is closely related to the abundance of large pores in the loess. The presence of large pores decreases with the increasing collapse amount. Thus, the loess deformation will be reduced if collapse occurs again after unloading.Figure 4
Relationship curves ofδr and K. (a) p1 = 200 kPa. (b) p1 = 250 kPa. (c) p1 = 300 kPa. (d) p1 = 350 kPa. (e) p1 = 400 kPa.
(a)(b)(c)(d)(e)


### 4.2. Unloading Effect Coefficienti of Loess Collapse

Based on the results of the unloading collapse test, to investigate the extent of the unloading effect of collapse, the unloading effect coefficienti is defined as follows:(6)i=1−δrδs,where δs is the collapse coefficient of loess under constant initial pressure and δr is the remnant collapse coefficient under the effect of unloading.If the unloading has no effect on the collapse process of loess, the unloading effect coefficienti is 1. The value of i is less than 1, which indicates that unloading has an effect on the collapse process of loess, and the greater the deviation from 1, the greater the influence of unloading on the collapse deformation of loess. According to δs measured by the conventional collapse test of loess and the remnant collapse coefficient δr of different collapse degrees η under the unloading effect, the unloading effect coefficient i is obtained, as shown in Figure 5.Figure 5
Relationship curves ofi and K. (a) p1 = 200 kPa. (b) p1 = 250 kPa. (c) p1 = 300 kPa. (d) p1 = 350 kPa. (e) p1 = 400 kPa.
(a)(b)(c)(d)(e)It can be seen from Figure5 that the unloading effect coefficients under various initial pressures are less than 1, which indicates that unloading makes the collapse of loess smaller, and the influence coefficient is even less than 0.5, and the effect of unloading on collapsible deformation of loess is obvious.Under the same collapse completion ratioη, the effect of unloading on collapsible deformation of loess increases with the decrease of the unloading ratio K, that is, the greater the unloading amount, the greater the effect. Under the same unloading ratio K, with the increase of the collapse completion ratio η, the effect of unloading on collapse deformation gradually weakens, namely, the greater the collapse completion under the initial pressure, the smaller the impact of unloading; on the contrary, the smaller the collapse completion under the initial pressure, the greater the impact of unloading.Whenp1 is 400 kPa, unloading 50 kPa almost has no impact on the collapse process, and i is slightly less than 1. Table 3 also shows that when the initial pressure is greater than 350 kPa, the loess collapse deformation has almost no increase with increasing initial pressure. When the vertical pressure is released from 400 kPa to 350 kPa, the unloading has almost no effect on the collapse process. Thus, there is a threshold value of p1 (350 kPa for the test loess sample). When p1 is greater than the threshold value, neither loading nor unloading affects the collapse deformation of loess.Whenp1 is less than the threshold value, unloading significantly affects the collapse deformation of loess. Therefore, the collapse amount after unloading must be determined when analyzing or solving practical project problems. The results of Figure 4 are used to find a method to determine the coefficient of remnant collapse.To investigate the curve features in Figure4, the nonlinear allometric model is performed using the relationships between the remnant collapse coefficient and the unloading ratio. The relationship is defined as follows:(7)δr=aKb,where the parameters of Equation (7) are obtained based on the regression analysis of unloading collapse tests with different initial pressures. Figure 6 describes the changes of regression parameters a and b with η under different p1.Figure 6
The trend of regression parameters a and b withη under different p1. (a) p1 = 200 kPa. (b) p1 = 250 kPa. (c) p1 = 300 kPa. (d) p1 = 350 kPa. (e) p1 = 400 kPa.
(a)(b)(c)(d)(e)In order to further verify the prediction ability of Equation (7) for residual collapse deformation, under different parameters, the coefficient of determination (R2) under different p1 and η are shown in Table 5.Table 5
The coefficient of determination (R2) of equation (7).
p1 (kPa)η1 (%)204060802000.9620.9580.9650.8872500.9720.9760.9950.9723000.9770.9760.9610.9983500.9940.9990.9580.8264000.9920.9980.9780.983As can be seen from Figure6 and Table 5, the parameter a changes with the change of η, while the parameter b changes little with η under different initial pressures. Therefore, under different p1, take the mean value of b and list the relationship between a and η, as shown in Table 6.Table 6
The relationship betweena, η, and ‾b under different p1.
p1 (kPa)Relationship betweena and ηb¯200a=0.50η−1.03−2.44250a=0.51η−1.06−2.41300a=0.56η−0.98−1.16350a=0.52η−0.92−1.12400a=0.57η−0.95−1.12Furthermore, the expression of the parametera under different initial pressure conditions in Table 6 is simplified, and the expression of a is shown in(8)a=0.53η,b=−2.43,p1≤250kPa,b=−1.13,250p<p1≤400kPa.The loess collapse effect coefficienti of unloading is obtained by substituting Equation (8) into Equation (6). The relation is shown as follows:(9)i=1−0.53ηδsKb.In summary,i needs to obtain only the collapse δs, collapse completion ratio η, and unloading ratio K. Therefore, if i is adopted, rapid analysis of the unloading loess collapse will be realized. Thus, considering the unloading effect, the unloading loess collapse coefficient can be defined as iδs, which can be used to calculate the collapse amount of unloading and provide the basis for loess collapse problems caused by unloading.


## 4.1. Unloading Collapse Test Results

Consequently, 384 oedometer unloading collapse tests were performed. The outputs of the experimental work are as follows. Figure1 shows that the initial pressure, unloading ratio, and collapse-completed ratio significantly affect the remnant loess collapse. The family of curves of different initial pressures represents the unloading collapse test with different collapse ratios.The collapse potential after unloading decreases with the decreasingK for the same η, whereas the slope of the curves decreases. Thus, with an increasing amplitude of vertical unloading, the collapse deformation of loess gradually decreases. The reason is that the self-weight collapsible loess is generally characterized by sudden and substantial deformations that occur when inundation is applied under constant stress. With the decrease in effective normal stress of the framework particles, the effects of displacement, dislocation, and collapse among the loess particles are weakened. Hence, the collapse deformation of loess decreases.Figure4 also shows that the collapse potential decreases because of an increase in η for the same K. First, the likely reason is the open pores of various diameters and shapes, which are formed by the framework particles that connect with one another by point contact connection because the collapse is processed in parts of dissolving salt crystals and ion exchange among the particles. The connection among the particles can be destroyed. Second, after the loess begins to collapse, its structure is dramatically destroyed, which is closely related to the abundance of large pores in the loess. The presence of large pores decreases with the increasing collapse amount. Thus, the loess deformation will be reduced if collapse occurs again after unloading.Figure 4
Relationship curves ofδr and K. (a) p1 = 200 kPa. (b) p1 = 250 kPa. (c) p1 = 300 kPa. (d) p1 = 350 kPa. (e) p1 = 400 kPa.
(a)(b)(c)(d)(e)


## 4.2. Unloading Effect Coefficienti of Loess Collapse

Based on the results of the unloading collapse test, to investigate the extent of the unloading effect of collapse, the unloading effect coefficienti is defined as follows:(6)i=1−δrδs,where δs is the collapse coefficient of loess under constant initial pressure and δr is the remnant collapse coefficient under the effect of unloading.If the unloading has no effect on the collapse process of loess, the unloading effect coefficienti is 1. The value of i is less than 1, which indicates that unloading has an effect on the collapse process of loess, and the greater the deviation from 1, the greater the influence of unloading on the collapse deformation of loess. According to δs measured by the conventional collapse test of loess and the remnant collapse coefficient δr of different collapse degrees η under the unloading effect, the unloading effect coefficient i is obtained, as shown in Figure 5.Figure 5
Relationship curves ofi and K. (a) p1 = 200 kPa. (b) p1 = 250 kPa. (c) p1 = 300 kPa. (d) p1 = 350 kPa. (e) p1 = 400 kPa.
(a)(b)(c)(d)(e)It can be seen from Figure5 that the unloading effect coefficients under various initial pressures are less than 1, which indicates that unloading makes the collapse of loess smaller, and the influence coefficient is even less than 0.5, and the effect of unloading on collapsible deformation of loess is obvious.Under the same collapse completion ratioη, the effect of unloading on collapsible deformation of loess increases with the decrease of the unloading ratio K, that is, the greater the unloading amount, the greater the effect. Under the same unloading ratio K, with the increase of the collapse completion ratio η, the effect of unloading on collapse deformation gradually weakens, namely, the greater the collapse completion under the initial pressure, the smaller the impact of unloading; on the contrary, the smaller the collapse completion under the initial pressure, the greater the impact of unloading.Whenp1 is 400 kPa, unloading 50 kPa almost has no impact on the collapse process, and i is slightly less than 1. Table 3 also shows that when the initial pressure is greater than 350 kPa, the loess collapse deformation has almost no increase with increasing initial pressure. When the vertical pressure is released from 400 kPa to 350 kPa, the unloading has almost no effect on the collapse process. Thus, there is a threshold value of p1 (350 kPa for the test loess sample). When p1 is greater than the threshold value, neither loading nor unloading affects the collapse deformation of loess.Whenp1 is less than the threshold value, unloading significantly affects the collapse deformation of loess. Therefore, the collapse amount after unloading must be determined when analyzing or solving practical project problems. The results of Figure 4 are used to find a method to determine the coefficient of remnant collapse.To investigate the curve features in Figure4, the nonlinear allometric model is performed using the relationships between the remnant collapse coefficient and the unloading ratio. The relationship is defined as follows:(7)δr=aKb,where the parameters of Equation (7) are obtained based on the regression analysis of unloading collapse tests with different initial pressures. Figure 6 describes the changes of regression parameters a and b with η under different p1.Figure 6
The trend of regression parameters a and b withη under different p1. (a) p1 = 200 kPa. (b) p1 = 250 kPa. (c) p1 = 300 kPa. (d) p1 = 350 kPa. (e) p1 = 400 kPa.
(a)(b)(c)(d)(e)In order to further verify the prediction ability of Equation (7) for residual collapse deformation, under different parameters, the coefficient of determination (R2) under different p1 and η are shown in Table 5.Table 5
The coefficient of determination (R2) of equation (7).
p1 (kPa)η1 (%)204060802000.9620.9580.9650.8872500.9720.9760.9950.9723000.9770.9760.9610.9983500.9940.9990.9580.8264000.9920.9980.9780.983As can be seen from Figure6 and Table 5, the parameter a changes with the change of η, while the parameter b changes little with η under different initial pressures. Therefore, under different p1, take the mean value of b and list the relationship between a and η, as shown in Table 6.Table 6
The relationship betweena, η, and ‾b under different p1.
p1 (kPa)Relationship betweena and ηb¯200a=0.50η−1.03−2.44250a=0.51η−1.06−2.41300a=0.56η−0.98−1.16350a=0.52η−0.92−1.12400a=0.57η−0.95−1.12Furthermore, the expression of the parametera under different initial pressure conditions in Table 6 is simplified, and the expression of a is shown in(8)a=0.53η,b=−2.43,p1≤250kPa,b=−1.13,250p<p1≤400kPa.The loess collapse effect coefficienti of unloading is obtained by substituting Equation (8) into Equation (6). The relation is shown as follows:(9)i=1−0.53ηδsKb.In summary,i needs to obtain only the collapse δs, collapse completion ratio η, and unloading ratio K. Therefore, if i is adopted, rapid analysis of the unloading loess collapse will be realized. Thus, considering the unloading effect, the unloading loess collapse coefficient can be defined as iδs, which can be used to calculate the collapse amount of unloading and provide the basis for loess collapse problems caused by unloading.


## 5. Conclusion

(1)
In terms of loess collapse after unloading, the remnant collapse coefficient decreases with the decreasing unloading ratio at constant collapse-completed ratios. The remnant collapse coefficient decreases with the increasing collapse-completed ratio at constant unloading ratios.(2)
There is a threshold value of the initial pressure (the threshold value of the specimen in this paper is 350 kPa). Whenp1 is greater than the threshold value, neither loading nor unloading affects the loess collapse deformation. When p1 is less than the threshold value, the unloading affects the loess collapse deformation. The extent of the loess collapse effect of unloading increases with decreasing K for the same η. The effect of unloading decreases with increasing η for the same K.(3)
The expression ofδr and i can be used to calculate the collapse amount after unloading and provide a practical evaluation method for loess collapse deformation on the effect of unloading.



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*Source: 6617228-2021-01-29.xml*