# Parameter Analysis on Hardening Soil Model of Soft Soil for Foundation Pits Based on Shear Rates in Shenzhen Bay, China

Authors: Yanbin Fu; Siyue He; Sizhan Zhang; Yong Yang
Journal: Advances in Materials Science and Engineering
 (2020)

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## Abstract

The hardening soil (HS) model is the most commonly used constitutive models of soft soil of foundation pits of PLAXIS software in numerical analysis, and its parameters are prerequisite for accurate calculation. In this paper, relevant parameters of the HS model in Shenzhen Bay in China were studied through one-dimensional consolidation tests and triaxial shear tests. Analytical methods of reference secant stiffness and failure ratio of soft soil were systematically studied, the influence of shear rates on reference secant stiffness and failure ratio of soft soil was analyzed, and the relationship between stiffness parameters and compressive modulus of soft soil was established. The results showed that reference secant stiffness and failure ratio of soft soil obtained by different analytical methods were quite different, and the errors of reference secant stiffness and failure ratio of soft soil obtained by stress-strain curves were the smallest and the stability was the best; at the same time, with increase of shear rates, the peak deviator stress and reference secant stiffness of soft soil increased, but failure ratio did not change much. The research results could provide a reference of parameter analysis of soft soil for the HS model in the numerical analysis and similar working conditions of foundation pits.


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## Body

## 1. Introduction

With rapid development of economy, more and more foundation pits adjacent to metro tunnels, metro stations, buildings, and municipal pipelines have been built in soft soil in coastal areas. Excavation of foundation pits has great impact on these adjacent projects and even causes surface collapse. It is difficult to analyze the influence of excavation of foundation pits on surrounding environment in soft soil reasonably by traditional analytical methods and relevant normative methods [1, 2]. The numerical analysis method has become the most effective method for excavation of foundation pits in soft soil because it can take into account the stratification and nature of soil, distribution and nature of support systems, construction process of excavation and support structures, and influence on surrounding environment [7–10].The key point in numerical simulation is to adopt reasonable calculation parameters and constitutive models. The Mohr–Coulomb (MC) model, Duncan–Chang (DC) model, Drucker–Prager (DP) model, modified Cam–Clay (CC) model, cap model, and hardening soil (HS) model are commonly used in numerical simulation of excavation of foundation pits in soft soil [7–8]. The applicable conditions for different constitutive models of cemented clay and soft soil are shown in Table 1. The MC model is usually used in the rapid analysis of numerical simulation. When soft soil has shear-softening effect, yield surface with cap shape is suitable, such as DP model, modified CC model, and cap model [11–13], among which, the HS model can consider hardening characteristics of soft soil and distinguish the difference between loading and unloading, and its stiffness parameters depend on stress history and stress path; its calculation results can give reasonable wall deformation and soil deformation behind wall at the same time, which is suitable for numerical analysis of excavation of foundation pits in soft soil [14–16]. Therefore, the HS model has become one of the most widely used constitutive models in the numerical analysis of excavation of foundation pits in soft soil. However, the HS model has many parameters, and it is difficult to obtain more complete model parameters [17, 18].Table 1
The applicable conditions for different constitutive models of cemented clay and soft soil.
Constitutive models
Analysis type
Soft soil
Cemented clay
OCR > 1
OCR≈1
OCR < 1
Sandy soil
Low compressibility
High compressibility
MC model
Serviceability limit state
Bearing capacity limit state
+
++
+
DC model
Serviceability limit state
+
++
++
+
Bearing capacity limit state
+
++
++
+
DP model
Serviceability limit state
++
+
+
++
++
++
Bearing capacity limit state
++
+
+
++
++
++
Modified CC model
Serviceability limit state
+
+
+
Bearing capacity limit state
++
++
++
Cap model
Serviceability limit state
+
++
++
+
+
+
Bearing capacity limit state
++
++
++
++
++
++
HS model
Serviceability limit state
+++
++
++
+++
+++
+++
Bearing capacity limit state
+++
++
++
+++
+++
+++
OCR: over consolidation ratio; +: generally applicable, the effect is not ideal; ++: applicable, good effect; +++: applicable, very good effect.At present, the main methods to determine parameters of HS model are the back-analysis method based on measured data and the laboratory test method [19, 20]. The back-analysis method can obtain sensitive parameters, and determination of other parameters depends on engineering experience; because of complexity of foundation pits, it is sometimes difficult for measured data to truly reflect actual state of projects [21, 22]. The laboratory test method can directly test the physical and mechanical properties of soft soil, and it is an effective method to obtain the parameters [23, 24]. Up to now, some studies have obtained some parameters of the HS model through the laboratory test, and only a few have obtained relatively complete parameters of the HS model [2, 15, 25, 26]. In this paper, the parameters of the HS model of soft soil in Shenzhen Bay in China were obtained through one-dimensional consolidation tests and triaxial shear tests. Based on theory of nonlinear elastic deformation of soft soil, analytical methods of reference secant stiffness and failure ratio in parameters of the HS model were systematically studied, and influence of shear rates on parameters of HS model was summarized.


## 2. HS Model

The HS model is a constitutive model based on Vermeer’s double hardening soil model proposed by Schanz and Vermeer [27, 28]. It consists of a hyperbolic shear yield surface and an elliptic cap yield surface in the P-Q plane, as shown in Figure 1(a). It model can consider both shear hardening and compression hardening of soft soil and satisfy Mohr–Coulomb failure criterion. It is suitable for simulating different types of soft soil in foundation pits. This model contains 11 parameters; it can be divided into three categories: strength parameter, stiffness parameter, and advanced parameter [29, 30], as shown in Table 2. In this paper, effective cohesion, effective friction angle, reference tangent stiffness, reference secant stiffness, reference unloading/reloading stiffness, failure ratio, and other parameters were studied. These parameters could be obtained from one-dimensional consolidation tests and triaxial shear tests. Stress-strain curves were very important in parameter analysis of the HS model [31–33], and typical stress-strain curves of triaxial shear tests are shown in Figure 1(b).Figure 1
(a) Yield surface in the P-Q plane and (b) stress-strain curves of the HS model.
(a)
(b)Table 2
Main parameters of the HS model [29, 30].
Strength parameter
Stiffness parameter
Advanced parameter
Effective cohesionc′
Reference secant stiffnessE50ref
Reference stiffness stresspref
Effective friction angleφ′
Reference tangent stiffnessEoedref
Unloading/reloading Poisson’s ratiovur
Dilation angleψ
Side pressure coefficientK0
Reference unloading/reloading stiffnessEurref
—
Failure ratioRf
Power exponent of stiffness-stress level correlationm


## 3. Materials and Methods

### 3.1. Materials

To obtain more accurate parameters of the HS model, one-dimensional consolidation tests and triaxial shear tests were carried out of soft soil in Shenzhen Bay in China. The physical parameters of soil samples are shown in Table3.Table 3
Physical parameters of soil samples.
Soil samples
Depth (m)
Water content (%)
Density (g·cm−3)
Specific gravity
Void ratio
T1
2.2∼3.4
48.3
1.67
2.70
1.40
T2
4.0∼5.5
47.5
1.74
2.71
1.31
T3
6.5∼7.8
48.4
1.64
2.69
1.42


### 3.2. Methods

#### 3.2.1. One-Dimensional Consolidation Tests

In this test, the compression characteristics of soft soil in Shenzhen Bay were studied by using the WG lever consolidation equipment. Consolidation tests of T1, T2, and T3 soil samples were carried out, respectively. Height (H) of soil samples was 2 cm, and section area (S) was 30 cm2. During one-dimensional consolidation tests, soil samples were placed in rigid retaining ring, and only vertical deformation occurred under lateral restriction of retaining ring and the ring knife, so as to achieve the purpose of one-dimensional compression.


#### 3.2.2. Triaxial Shear Tests

The test equipment was a fully automatic and high-precision triaxial apparatus from GDS Company, UK. Height (H) of soil samples was 7.6 cm, and diameter (D) was 3.9 cm. In this section, triaxial shear tests at the same shear rate and at different shear rates were carried out, respectively. Sample preparation and installation process in triaxial shear tests is shown in Figure 2. At the same shear rate, through consolidated drainage shear (CD) tests on T1, T2, and T3 soil samples, effective stress intensity index, reference secant stiffness, and failure ratio of soft soil in Shenzhen Bay were determined under reference stiffness stress pref=100kPa. The shear rate was set to V = 0.002 mm/min and confining pressure was 100 kPa; the shear rate could reduce the influence of pore water pressure on strength and prevent excess pore water pressure [34]. At different shear rates, consolidated undrained shear (CU) tests were carried out on the T1 soil sample to study the shear properties of soft soil in Shenzhen Bay. To obtain reference secant stiffness and failure ratio of soft soil at different shear rates, shear rates were set to 0.05 mm/min and 0.15 mm/min and the shear rates could be controlled to ensure that the pore water pressure increases uniformly and conforms to the actual engineering [32].Figure 2
Sample preparation and installation process in triaxial shear tests. (a) Sample preparation. (b) Sample saturator. (c) Sample pasted with filter paper. (d) Sample installation.
(a)
(b)
(c)
(d)


## 3.1. Materials

To obtain more accurate parameters of the HS model, one-dimensional consolidation tests and triaxial shear tests were carried out of soft soil in Shenzhen Bay in China. The physical parameters of soil samples are shown in Table3.Table 3
Physical parameters of soil samples.
Soil samples
Depth (m)
Water content (%)
Density (g·cm−3)
Specific gravity
Void ratio
T1
2.2∼3.4
48.3
1.67
2.70
1.40
T2
4.0∼5.5
47.5
1.74
2.71
1.31
T3
6.5∼7.8
48.4
1.64
2.69
1.42


## 3.2. Methods

### 3.2.1. One-Dimensional Consolidation Tests

In this test, the compression characteristics of soft soil in Shenzhen Bay were studied by using the WG lever consolidation equipment. Consolidation tests of T1, T2, and T3 soil samples were carried out, respectively. Height (H) of soil samples was 2 cm, and section area (S) was 30 cm2. During one-dimensional consolidation tests, soil samples were placed in rigid retaining ring, and only vertical deformation occurred under lateral restriction of retaining ring and the ring knife, so as to achieve the purpose of one-dimensional compression.


### 3.2.2. Triaxial Shear Tests

The test equipment was a fully automatic and high-precision triaxial apparatus from GDS Company, UK. Height (H) of soil samples was 7.6 cm, and diameter (D) was 3.9 cm. In this section, triaxial shear tests at the same shear rate and at different shear rates were carried out, respectively. Sample preparation and installation process in triaxial shear tests is shown in Figure 2. At the same shear rate, through consolidated drainage shear (CD) tests on T1, T2, and T3 soil samples, effective stress intensity index, reference secant stiffness, and failure ratio of soft soil in Shenzhen Bay were determined under reference stiffness stress pref=100kPa. The shear rate was set to V = 0.002 mm/min and confining pressure was 100 kPa; the shear rate could reduce the influence of pore water pressure on strength and prevent excess pore water pressure [34]. At different shear rates, consolidated undrained shear (CU) tests were carried out on the T1 soil sample to study the shear properties of soft soil in Shenzhen Bay. To obtain reference secant stiffness and failure ratio of soft soil at different shear rates, shear rates were set to 0.05 mm/min and 0.15 mm/min and the shear rates could be controlled to ensure that the pore water pressure increases uniformly and conforms to the actual engineering [32].Figure 2
Sample preparation and installation process in triaxial shear tests. (a) Sample preparation. (b) Sample saturator. (c) Sample pasted with filter paper. (d) Sample installation.
(a)
(b)
(c)
(d)


## 3.2.1. One-Dimensional Consolidation Tests

In this test, the compression characteristics of soft soil in Shenzhen Bay were studied by using the WG lever consolidation equipment. Consolidation tests of T1, T2, and T3 soil samples were carried out, respectively. Height (H) of soil samples was 2 cm, and section area (S) was 30 cm2. During one-dimensional consolidation tests, soil samples were placed in rigid retaining ring, and only vertical deformation occurred under lateral restriction of retaining ring and the ring knife, so as to achieve the purpose of one-dimensional compression.


## 3.2.2. Triaxial Shear Tests

The test equipment was a fully automatic and high-precision triaxial apparatus from GDS Company, UK. Height (H) of soil samples was 7.6 cm, and diameter (D) was 3.9 cm. In this section, triaxial shear tests at the same shear rate and at different shear rates were carried out, respectively. Sample preparation and installation process in triaxial shear tests is shown in Figure 2. At the same shear rate, through consolidated drainage shear (CD) tests on T1, T2, and T3 soil samples, effective stress intensity index, reference secant stiffness, and failure ratio of soft soil in Shenzhen Bay were determined under reference stiffness stress pref=100kPa. The shear rate was set to V = 0.002 mm/min and confining pressure was 100 kPa; the shear rate could reduce the influence of pore water pressure on strength and prevent excess pore water pressure [34]. At different shear rates, consolidated undrained shear (CU) tests were carried out on the T1 soil sample to study the shear properties of soft soil in Shenzhen Bay. To obtain reference secant stiffness and failure ratio of soft soil at different shear rates, shear rates were set to 0.05 mm/min and 0.15 mm/min and the shear rates could be controlled to ensure that the pore water pressure increases uniformly and conforms to the actual engineering [32].Figure 2
Sample preparation and installation process in triaxial shear tests. (a) Sample preparation. (b) Sample saturator. (c) Sample pasted with filter paper. (d) Sample installation.
(a)
(b)
(c)
(d)


## 4. Results and Analysis

### 4.1. Analysis of Reference Tangent Stiffness

Results of one-dimensional consolidation tests and fitting curves of soil samples are shown in Figures3 and 4. Figure 3 shows the relationship between axial load (p) and void ratio (e) of soil samples. Figure 4 shows the relationship between axial strain (ε1) of soil samples and axial load (p).Figure 3
Axial load and void ratio curves of one-dimensional consolidation tests (p∼e).Figure 4
Axial strain and axial load curves of one-dimensional consolidation tests (ε1∼p).As can be seen from Figure3, the larger axial load (p) on soil samples is, the smaller void ratio (e) of soil samples is, and the smaller void ratio of soil samples after consolidation and compression is. The compressive modulus (Es,1−2) of soil samples was 2.26 MPa, 2.34 MPa, and 2.16 MPa, respectively, when the axial load was as follows: p1=100kPa to p2=200kPa.As can be seen from Figure4, the axial strain (ε1) of soil samples increased with increase in the axial load (p) on soil samples. After derivation calculation, reference tangent stiffness (Eoedref) of soil samples were 1.99 MPa, 2.09 MPa, and 2.06 MPa, with an average value of 2.05 MPa.


### 4.2. Analysis of Strength Parameter

In this paper, the strength parameter was determined according to the relationship between Mohr–Coulomb strength envelope andKf′ line [2]. The results of CD tests and fitting curves of soil samples are shown in Figure 5:(1)φ′=arcsintanα′,c′=a′cosφ′,where tanα′ is the slope of fitting line and a′ is the intercept.Figure 5
Results of CD tests and fitting curves of soil samples (p∼q).Effective friction angle (φ′) of T1, T2, and T3 soil samples was 21.5°, 21.3°, and 21.3°, respectively, and the average value was 21.4°; effective cohesion (c′) of T1, T2, and T3 soil samples was 7.1 kPa, 6.7 kPa, and 7.3 kPa, respectively, and the average value was 7.0 kPa.


### 4.3. Analysis of Reference Secant Stiffness and Failure Ratio

#### 4.3.1. Reference Secant Stiffness and Failure Ratio at the Same Shear Rate

CD tests were carried out on soil samples. Effective confining pressure was 100 kPa. Three methods were proposed to analyze the reference secant stiffness (E50ref) and failure ratio (Rf).Figure 6
ε
1
∼q curves of CD tests.Figure 7
1
/
ε
1
∼
1
/
q relation curves.(1)
Analyzing by the relation ofε1∼q: according to Kondner et al. [35], ε1∼q curves of soil samples were fitted in the form of hyperbolic equation as follows:
(2)
q
=
σ
1
−
σ
3
=
ε
1
a
+
b
ε
1
.ε
1
∼q curves and fitting curves of soil samples are shown in Figure 6.Because there is no yield point inε1∼q curves of soil samples, combined with guideline of geotechnical test method [34], failure stress (qf) is deviator stress corresponding to strain ε1=15%. Reference secant stiffness (E50ref) was the slope of a straight line connecting the origin and the corresponding point (0.5qf). The failure ratio (Rf) of soil samples could be obtained from the following equation:(3)Rf=qfqa,where qa is the asymptotic value of deviator stress of soil samples, which could be obtained by fitting the hyperbolic equation.(2)
Analyzing by relation of1/ε1∼1/q: according to Kondner [35], the relationship between the axial strain and deviator stress of soil samples was approximately a hyperbola, which could be expressed as follows:(4)ε1=1Eiq1−q/qa,q<qf,where Ei is the initial stiffness, qa is the asymptotic value of deviator stress, and qf is the ultimate deviator stress.By transforming equation (4), the 1/ε1∼1/q correlation can be obtained as follows:(5)1q=1Ei1ε1+1qa,where data 1/ε1 and 1/q could be obtained by transforming experimental data. The correlation between them is shown in Figure 7. The relationship between Ei and E50 is as follows:(6)Ei=2E502−Rf.(3)
Analyzing by relation ofε1∼ε1/q: according to Schanz et al. [22, 27], the yield curves of soil samples of CD tests could be expressed as follows:(7)ε1=qa2E50σ1−σ3qa−σ1−σ3,q<qf.Becauseq=σ1−σ3, equation (7) can be transformed into the following equation:(8)ε1=q2E501−q/qa.Further conversion of equation (8) led the ε1∼ε1/q relationship as follows:(9)ε1q=ε1qa+12E50,q<qf.The ratio of actual deviator stress to ultimate deviator stress was defined as stress level and expressed as(10)S1=qqf.According to the Mohr–Coulomb failure criterion, there were following relations:(11)qf=σ1−σ3f=2σ3sinφ+2cosφ1−sinφ.Through linear fitting ofε1∼ε1/q curves, parameters qa,E50 could be obtained. To avoid excessive error of linear fitting, two points of stress levels S1=70% and S1=95% were selected to fit on the basis of the DC model. ε1∼ε1/q curves are shown in Figure 8.Figure 8
ε
1
∼
ε
1
/
q curves.E
50
ref and Rf were obtained by using above three methods, and results are shown in Figure 9.Figure 9
R
f and E50ref obtained by using the above three methods. (a) Rfcontrast. (b) E50ref contrast.
(a)
(b)From Figure9(a), it can be seen that the failure ratio (Rf) difference obtained by using ε1∼q curves was the smallest and the stability was the best, and Rf difference obtained by using 1/ε1∼1/q curves was the greatest, and difference between the minimum and the maximum was 0.48, reaching 52.7%. For the same soil samples, the lowest difference of Rf was T1 and the biggest difference of Rf was T3. Among them, the largest difference of Rf was obtained by using 1/ε1∼1/q curves, reaching 0.91.From Figure9(b), it can be seen that the difference of reference secant stiffness (E50ref) between ε1∼q curves and ε1∼ε1/q curves was small, 1/ε1∼1/q curves were the biggest, and difference between minimum and maximum was 0.78 MPa, reaching 32.0%. For the same soil samples, the smallest difference of E50ref was T1 and the biggest difference of E50ref was T3. The largest difference of E50ref obtained by 1/ε1∼1/q curves was 2.44 MPa, which was 0.99 MPa larger than the minimum.Based on the above analysis, the following conclusions could be drawn: failure ratio (Rf) and reference secant stiffness (E50ref) obtained by using ε1∼q curves were less error and more stable, which were more suitable for parameter analysis of the HS model. Therefore, to reduce the error, the reference secant stiffness (E50ref) in CU tests at different shear rates was solved by ε1∼q curves, and the stiffness relationship was mainly based on ε1∼q curves.


#### 4.3.2. Reference Secant Modulus at Different Shear Rates

When effective confining pressure of soil samples was 100 kPa, shear rates and stress-strain curves in CU tests are shown in Figure10.Figure 10
ε
1
∼
q curve based on different shear rates in CU tests.From Figure10, it can be seen that peak deviator stress increased to some extent with increase of shear rates in a certain range. Compared with stress-strain curves obtained from CD tests, stress-strain curves obtained from CU tests tended to enter “hardening” stage ahead of time. According to fitting hyperbola, reference secant stiffness (E50ref) and failure ratio (Rf) of the HS model were obtained, as shown in Table 4. It could be seen that, within a certain range of shear rates, with increase of shear rates, shear strength failure value (qf), shear strength progressive value (qa), and reference secant stiffness (E50ref) also increased gradually, and the failure ratio (Rf) changed little with shear rates and was close to 1.Table 4
Analytical results of partial parameters based on different shear rates in CU tests.
Soil sample
Shear rates (mm/min)
q
f (kPa)
0.5
q
f (kPa)
q
a (kPa)
E
50
ref (MPa)
R
f
T1
0.05
59.5
29.8
61.7
5.61
0.96
0.15
107.8
54.0
113.6
7.40
0.95


### 4.4. Analysis of Unloading and Reloading Reference Secant Moduli

Figure11 shows the stress-strain curves of soil samples under loading-unloading-reloading under CD tests of effective confining pressure = 100 kPa.Figure 11
ε
∼
q curves under loading-unloading-reloading under CD tests. (a) T1. (b) T2.
(a)
(b)
(c)It can be seen that the stress-strain relationship has hysteresis loops, and the curve of unloading stage decreased rapidly. When unloading to a deviator stress, the curve of the unloading stage decreased slowly. When the unloading stage completed, the curve rose quickly and then slowly. Reference unloading/reloading stiffness (Eurref) of soil samples was the slope of line connecting the two ends of hysteresis loops. Eurref of the soil samples was 11.76 MPa, 13.03 MPa, and 15.12 MPa, respectively.


## 4.1. Analysis of Reference Tangent Stiffness

Results of one-dimensional consolidation tests and fitting curves of soil samples are shown in Figures3 and 4. Figure 3 shows the relationship between axial load (p) and void ratio (e) of soil samples. Figure 4 shows the relationship between axial strain (ε1) of soil samples and axial load (p).Figure 3
Axial load and void ratio curves of one-dimensional consolidation tests (p∼e).Figure 4
Axial strain and axial load curves of one-dimensional consolidation tests (ε1∼p).As can be seen from Figure3, the larger axial load (p) on soil samples is, the smaller void ratio (e) of soil samples is, and the smaller void ratio of soil samples after consolidation and compression is. The compressive modulus (Es,1−2) of soil samples was 2.26 MPa, 2.34 MPa, and 2.16 MPa, respectively, when the axial load was as follows: p1=100kPa to p2=200kPa.As can be seen from Figure4, the axial strain (ε1) of soil samples increased with increase in the axial load (p) on soil samples. After derivation calculation, reference tangent stiffness (Eoedref) of soil samples were 1.99 MPa, 2.09 MPa, and 2.06 MPa, with an average value of 2.05 MPa.


## 4.2. Analysis of Strength Parameter

In this paper, the strength parameter was determined according to the relationship between Mohr–Coulomb strength envelope andKf′ line [2]. The results of CD tests and fitting curves of soil samples are shown in Figure 5:(1)φ′=arcsintanα′,c′=a′cosφ′,where tanα′ is the slope of fitting line and a′ is the intercept.Figure 5
Results of CD tests and fitting curves of soil samples (p∼q).Effective friction angle (φ′) of T1, T2, and T3 soil samples was 21.5°, 21.3°, and 21.3°, respectively, and the average value was 21.4°; effective cohesion (c′) of T1, T2, and T3 soil samples was 7.1 kPa, 6.7 kPa, and 7.3 kPa, respectively, and the average value was 7.0 kPa.


## 4.3. Analysis of Reference Secant Stiffness and Failure Ratio

### 4.3.1. Reference Secant Stiffness and Failure Ratio at the Same Shear Rate

CD tests were carried out on soil samples. Effective confining pressure was 100 kPa. Three methods were proposed to analyze the reference secant stiffness (E50ref) and failure ratio (Rf).Figure 6
ε
1
∼q curves of CD tests.Figure 7
1
/
ε
1
∼
1
/
q relation curves.(1)
Analyzing by the relation ofε1∼q: according to Kondner et al. [35], ε1∼q curves of soil samples were fitted in the form of hyperbolic equation as follows:
(2)
q
=
σ
1
−
σ
3
=
ε
1
a
+
b
ε
1
.ε
1
∼q curves and fitting curves of soil samples are shown in Figure 6.Because there is no yield point inε1∼q curves of soil samples, combined with guideline of geotechnical test method [34], failure stress (qf) is deviator stress corresponding to strain ε1=15%. Reference secant stiffness (E50ref) was the slope of a straight line connecting the origin and the corresponding point (0.5qf). The failure ratio (Rf) of soil samples could be obtained from the following equation:(3)Rf=qfqa,where qa is the asymptotic value of deviator stress of soil samples, which could be obtained by fitting the hyperbolic equation.(2)
Analyzing by relation of1/ε1∼1/q: according to Kondner [35], the relationship between the axial strain and deviator stress of soil samples was approximately a hyperbola, which could be expressed as follows:(4)ε1=1Eiq1−q/qa,q<qf,where Ei is the initial stiffness, qa is the asymptotic value of deviator stress, and qf is the ultimate deviator stress.By transforming equation (4), the 1/ε1∼1/q correlation can be obtained as follows:(5)1q=1Ei1ε1+1qa,where data 1/ε1 and 1/q could be obtained by transforming experimental data. The correlation between them is shown in Figure 7. The relationship between Ei and E50 is as follows:(6)Ei=2E502−Rf.(3)
Analyzing by relation ofε1∼ε1/q: according to Schanz et al. [22, 27], the yield curves of soil samples of CD tests could be expressed as follows:(7)ε1=qa2E50σ1−σ3qa−σ1−σ3,q<qf.Becauseq=σ1−σ3, equation (7) can be transformed into the following equation:(8)ε1=q2E501−q/qa.Further conversion of equation (8) led the ε1∼ε1/q relationship as follows:(9)ε1q=ε1qa+12E50,q<qf.The ratio of actual deviator stress to ultimate deviator stress was defined as stress level and expressed as(10)S1=qqf.According to the Mohr–Coulomb failure criterion, there were following relations:(11)qf=σ1−σ3f=2σ3sinφ+2cosφ1−sinφ.Through linear fitting ofε1∼ε1/q curves, parameters qa,E50 could be obtained. To avoid excessive error of linear fitting, two points of stress levels S1=70% and S1=95% were selected to fit on the basis of the DC model. ε1∼ε1/q curves are shown in Figure 8.Figure 8
ε
1
∼
ε
1
/
q curves.E
50
ref and Rf were obtained by using above three methods, and results are shown in Figure 9.Figure 9
R
f and E50ref obtained by using the above three methods. (a) Rfcontrast. (b) E50ref contrast.
(a)
(b)From Figure9(a), it can be seen that the failure ratio (Rf) difference obtained by using ε1∼q curves was the smallest and the stability was the best, and Rf difference obtained by using 1/ε1∼1/q curves was the greatest, and difference between the minimum and the maximum was 0.48, reaching 52.7%. For the same soil samples, the lowest difference of Rf was T1 and the biggest difference of Rf was T3. Among them, the largest difference of Rf was obtained by using 1/ε1∼1/q curves, reaching 0.91.From Figure9(b), it can be seen that the difference of reference secant stiffness (E50ref) between ε1∼q curves and ε1∼ε1/q curves was small, 1/ε1∼1/q curves were the biggest, and difference between minimum and maximum was 0.78 MPa, reaching 32.0%. For the same soil samples, the smallest difference of E50ref was T1 and the biggest difference of E50ref was T3. The largest difference of E50ref obtained by 1/ε1∼1/q curves was 2.44 MPa, which was 0.99 MPa larger than the minimum.Based on the above analysis, the following conclusions could be drawn: failure ratio (Rf) and reference secant stiffness (E50ref) obtained by using ε1∼q curves were less error and more stable, which were more suitable for parameter analysis of the HS model. Therefore, to reduce the error, the reference secant stiffness (E50ref) in CU tests at different shear rates was solved by ε1∼q curves, and the stiffness relationship was mainly based on ε1∼q curves.


### 4.3.2. Reference Secant Modulus at Different Shear Rates

When effective confining pressure of soil samples was 100 kPa, shear rates and stress-strain curves in CU tests are shown in Figure10.Figure 10
ε
1
∼
q curve based on different shear rates in CU tests.From Figure10, it can be seen that peak deviator stress increased to some extent with increase of shear rates in a certain range. Compared with stress-strain curves obtained from CD tests, stress-strain curves obtained from CU tests tended to enter “hardening” stage ahead of time. According to fitting hyperbola, reference secant stiffness (E50ref) and failure ratio (Rf) of the HS model were obtained, as shown in Table 4. It could be seen that, within a certain range of shear rates, with increase of shear rates, shear strength failure value (qf), shear strength progressive value (qa), and reference secant stiffness (E50ref) also increased gradually, and the failure ratio (Rf) changed little with shear rates and was close to 1.Table 4
Analytical results of partial parameters based on different shear rates in CU tests.
Soil sample
Shear rates (mm/min)
q
f (kPa)
0.5
q
f (kPa)
q
a (kPa)
E
50
ref (MPa)
R
f
T1
0.05
59.5
29.8
61.7
5.61
0.96
0.15
107.8
54.0
113.6
7.40
0.95


## 4.3.1. Reference Secant Stiffness and Failure Ratio at the Same Shear Rate

CD tests were carried out on soil samples. Effective confining pressure was 100 kPa. Three methods were proposed to analyze the reference secant stiffness (E50ref) and failure ratio (Rf).Figure 6
ε
1
∼q curves of CD tests.Figure 7
1
/
ε
1
∼
1
/
q relation curves.(1)
Analyzing by the relation ofε1∼q: according to Kondner et al. [35], ε1∼q curves of soil samples were fitted in the form of hyperbolic equation as follows:
(2)
q
=
σ
1
−
σ
3
=
ε
1
a
+
b
ε
1
.ε
1
∼q curves and fitting curves of soil samples are shown in Figure 6.Because there is no yield point inε1∼q curves of soil samples, combined with guideline of geotechnical test method [34], failure stress (qf) is deviator stress corresponding to strain ε1=15%. Reference secant stiffness (E50ref) was the slope of a straight line connecting the origin and the corresponding point (0.5qf). The failure ratio (Rf) of soil samples could be obtained from the following equation:(3)Rf=qfqa,where qa is the asymptotic value of deviator stress of soil samples, which could be obtained by fitting the hyperbolic equation.(2)
Analyzing by relation of1/ε1∼1/q: according to Kondner [35], the relationship between the axial strain and deviator stress of soil samples was approximately a hyperbola, which could be expressed as follows:(4)ε1=1Eiq1−q/qa,q<qf,where Ei is the initial stiffness, qa is the asymptotic value of deviator stress, and qf is the ultimate deviator stress.By transforming equation (4), the 1/ε1∼1/q correlation can be obtained as follows:(5)1q=1Ei1ε1+1qa,where data 1/ε1 and 1/q could be obtained by transforming experimental data. The correlation between them is shown in Figure 7. The relationship between Ei and E50 is as follows:(6)Ei=2E502−Rf.(3)
Analyzing by relation ofε1∼ε1/q: according to Schanz et al. [22, 27], the yield curves of soil samples of CD tests could be expressed as follows:(7)ε1=qa2E50σ1−σ3qa−σ1−σ3,q<qf.Becauseq=σ1−σ3, equation (7) can be transformed into the following equation:(8)ε1=q2E501−q/qa.Further conversion of equation (8) led the ε1∼ε1/q relationship as follows:(9)ε1q=ε1qa+12E50,q<qf.The ratio of actual deviator stress to ultimate deviator stress was defined as stress level and expressed as(10)S1=qqf.According to the Mohr–Coulomb failure criterion, there were following relations:(11)qf=σ1−σ3f=2σ3sinφ+2cosφ1−sinφ.Through linear fitting ofε1∼ε1/q curves, parameters qa,E50 could be obtained. To avoid excessive error of linear fitting, two points of stress levels S1=70% and S1=95% were selected to fit on the basis of the DC model. ε1∼ε1/q curves are shown in Figure 8.Figure 8
ε
1
∼
ε
1
/
q curves.E
50
ref and Rf were obtained by using above three methods, and results are shown in Figure 9.Figure 9
R
f and E50ref obtained by using the above three methods. (a) Rfcontrast. (b) E50ref contrast.
(a)
(b)From Figure9(a), it can be seen that the failure ratio (Rf) difference obtained by using ε1∼q curves was the smallest and the stability was the best, and Rf difference obtained by using 1/ε1∼1/q curves was the greatest, and difference between the minimum and the maximum was 0.48, reaching 52.7%. For the same soil samples, the lowest difference of Rf was T1 and the biggest difference of Rf was T3. Among them, the largest difference of Rf was obtained by using 1/ε1∼1/q curves, reaching 0.91.From Figure9(b), it can be seen that the difference of reference secant stiffness (E50ref) between ε1∼q curves and ε1∼ε1/q curves was small, 1/ε1∼1/q curves were the biggest, and difference between minimum and maximum was 0.78 MPa, reaching 32.0%. For the same soil samples, the smallest difference of E50ref was T1 and the biggest difference of E50ref was T3. The largest difference of E50ref obtained by 1/ε1∼1/q curves was 2.44 MPa, which was 0.99 MPa larger than the minimum.Based on the above analysis, the following conclusions could be drawn: failure ratio (Rf) and reference secant stiffness (E50ref) obtained by using ε1∼q curves were less error and more stable, which were more suitable for parameter analysis of the HS model. Therefore, to reduce the error, the reference secant stiffness (E50ref) in CU tests at different shear rates was solved by ε1∼q curves, and the stiffness relationship was mainly based on ε1∼q curves.


## 4.3.2. Reference Secant Modulus at Different Shear Rates

When effective confining pressure of soil samples was 100 kPa, shear rates and stress-strain curves in CU tests are shown in Figure10.Figure 10
ε
1
∼
q curve based on different shear rates in CU tests.From Figure10, it can be seen that peak deviator stress increased to some extent with increase of shear rates in a certain range. Compared with stress-strain curves obtained from CD tests, stress-strain curves obtained from CU tests tended to enter “hardening” stage ahead of time. According to fitting hyperbola, reference secant stiffness (E50ref) and failure ratio (Rf) of the HS model were obtained, as shown in Table 4. It could be seen that, within a certain range of shear rates, with increase of shear rates, shear strength failure value (qf), shear strength progressive value (qa), and reference secant stiffness (E50ref) also increased gradually, and the failure ratio (Rf) changed little with shear rates and was close to 1.Table 4
Analytical results of partial parameters based on different shear rates in CU tests.
Soil sample
Shear rates (mm/min)
q
f (kPa)
0.5
q
f (kPa)
q
a (kPa)
E
50
ref (MPa)
R
f
T1
0.05
59.5
29.8
61.7
5.61
0.96
0.15
107.8
54.0
113.6
7.40
0.95


## 4.4. Analysis of Unloading and Reloading Reference Secant Moduli

Figure11 shows the stress-strain curves of soil samples under loading-unloading-reloading under CD tests of effective confining pressure = 100 kPa.Figure 11
ε
∼
q curves under loading-unloading-reloading under CD tests. (a) T1. (b) T2.
(a)
(b)
(c)It can be seen that the stress-strain relationship has hysteresis loops, and the curve of unloading stage decreased rapidly. When unloading to a deviator stress, the curve of the unloading stage decreased slowly. When the unloading stage completed, the curve rose quickly and then slowly. Reference unloading/reloading stiffness (Eurref) of soil samples was the slope of line connecting the two ends of hysteresis loops. Eurref of the soil samples was 11.76 MPa, 13.03 MPa, and 15.12 MPa, respectively.


## 5. Discussion

Compared with other constitutive models, the HS model can well describe the mechanical behavior of shear hardening of soil samples, but in practical engineering, there were many parameters and it was difficult to determine. By analyzing the relationship between compression modulus (Es1−2) and reference secant stiffness (Eoedref) and reference secant stiffness (E50ref) and reference unloading/reloading stiffness (Eurref) of the one-dimensional consolidation tests and triaxial shear tests, it was more convenient to determine the parameters of the HS model [36–39]. The relationship of stiffness parameters of soil samples in Shenzhen Bay in China is shown in Table 5. In recent years, comparisons of parameters of soil and cemented clay of the HS model obtained by researchers are shown in Table 6 [2, 40–46].Table 5
Relationship of stiffness parameters of soil samples.
Soil sample
E
s
1
−
2 (MPa)
E
oed
ref (MPa)
E
50
ref (MPa)
E
ur
ref (MPa)
Relationship of stiffness parameters
T1
2.26
1.99
1.71
11.76
E
oed
ref
=
0.88
E
s
1
−
2 E50ref=0.76Es1−2
E
ur
ref
=
5.20
E
s
1
−
2
T2
2.34
2.09
1.86
13.03
E
oed
ref
=
0.89
E
s
1
−
2 E50ref=0.79Es1−2
E
ur
ref
=
5.57
E
s
1
−
2
T3
2.16
2.06
2.08
15.12
E
oed
ref
=
0.95
E
s
1
−
2 E50ref=0.96Es1−2
E
ur
ref
=
7.00
E
s
1
−
2Table 6
Comparisons of parameters of soil and cemented clay of the HS model [2, 40–46].
Soil types
E
s
1
−
2 (MPa)
HS model
Relationship of stiffness parameters
E
oed
ref (MPa)
E
50
ref (MPa)
E
ur
ref (MPa)
c
′ (kPa)
φ
′ (°)
R
f
Soft soil (Shanghai)
2.2
1.9
2.0
15.6
3.0
27.3
0.54
E
oed
ref
=
0.9
E
s
1
−
2
E
50
ref
=
0.9
E
oed
ref
E
ur
ref
=
7.8
E
50
ref
=
8.2
E
oed
ref
Gault clay
4.3
14.8
44.5
0.05
33.0
0.96
E
50
ref
=
3.5
E
oed
ref
E
ur
ref
=
3
E
50
ref
=
10.4
E
oed
ref
Silty clay (Taipei)
2.4
6.6
19.7
0.05
29.0
0.95
E
50
ref
=
2.8
E
oed
ref
E
ur
ref
=
3
E
50
ref
=
8.3
E
oed
ref
Clay (Blodgett)
1.6
2.4
10.0
0.05
24.1
0.70
E
50
ref
=
1.5
E
oed
ref
E
ur
ref
=
4.3
E
50
ref
=
6.25
E
oed
ref
Lacustrine clay
37.6
37.6
150.4
30
26
E
50
ref
=
E
oed
ref
E
ur
ref
=
4
E
50
ref
=
4
E
oed
ref
Soft soil (Tianjin)
3.4
4.4
14.0
10
25.2
E
50
ref
=
0.5
∼
1.8
E
oed
ref
E
oed
ref
,
E
50
ref
,
and
E
ur
ref were under reference stress pref=100kPa.From Tables5 and 6, the stiffness parameters of soft soil and cemented clay of the HS model had the following rules: Eoedref≈0.9Es1−2, E50ref=0.7∼1.0Es1−2, and Eurref=3∼8E50ref=4∼11Eoedref, and stiffness parameters in different regions had big differences, so it was better to select parameters of the HS model in the numerical simulation based on field sampling and indoor tests; the relationship of stiffness parameters of soil samples of the HS model in Shenzhen Bay was similar to those obtained in similar studies in Shanghai, Blodgett, and Tianjin, and the value was smaller than that of gault clay and lacustrine clay.


## 6. Conclusion

Based on one-dimensional consolidation tests and triaxial shear tests, the parametersc′,φ′,Eoedref,E50ref,Eurref, and Rf of the HS model of soft soil in Shenzhen Bay in China were determined, which could provide a reference for the numerical analysis of similar foundation pits. The main findings were as follows:(1)
Based on results of CD tests at the same shear rate,E50ref and Rf were discrete by using 1/ε1∼1/q curves and ε1∼ε1/q curves, while ε1∼q curves had smaller error and better stability. It was more suitable for parameter analysis of the HS model E50ref and Rf.(2)
Within a certain range of shear rates in CU tests (≤0.15 mm/min), with increase of shear rates, shear strength failure value (qf) and reference secant modulus (E50ref) increased gradually, while the failure ratio (Rf) changed little with shear rates, which was close to 1.(3)
The stiffness parameter ratio (Eoedref/Es,1−2) of soft soil in Shenzhen Bay was 0.92; the average (E50ref/Es,1−2) was 0.86; and the average (Eurref/Es,1−2) was 5.92. Based on Es,1−2 obtained from one-dimensional consolidation tests, the parameters Eoedref,E50ref,andEurref could be obtained quickly.



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*Source: 7810918-2020-02-18.xml*