Calculation of Active Earth Pressure on a Circular Retaining Wall Based on Energy Method

Abstract

For an axisymmetric circular retaining wall, a calculation method of active earth pressure on the circular retaining wall based on energy method is being proposed. The analysis of the soil behind the wall employs the Coulomb failure mechanism. Assuming the soil to be a completely rigid plastic body, the calculation formula of active earth pressure on the circular retaining wall is being established according to the associated flow law and virtual work principle. The results are indicating that the active earth pressure on the circular retaining wall is lower than that predicted by the Coulomb solution for plane retaining walls. This active earth pressure is increasing as the radius-to-height ratio of the wall increases, eventually aligning with the Coulomb solution for plane walls. Additionally, the inclination angle of the slip surface is being observed to increase with the circumferential stress coefficient, while the active earth pressure is decreasing correspondingly; For non-cohesive soil, when the radius-to-height ratio approaches infinity, the calculation method in this paper is being consistent with Coulomb theory. This method can calculate the active earth pressure on circular retaining wall with cohesive soil behind the wall. As the cohesion increases, the inclination angle of the slip surface rises, and the active earth pressure decreases. The result is being confirmed by model tests. It can provide a reference for the engineering design of circular retaining structures.

1 Introduction

The circular retaining structure, known for its good stress and deformation characteristics, is widely used in various applications. These include retaining walls of circular reservoirs, turns in mountain roads, river bends, and other areas, as well as in practical projects like coal mine shafts. A critical issue in the design of circular retaining walls is the development of a robust and precise methodology for calculating earth pressure. Accurately computing the earth pressure exerted on circular retaining walls necessitates the consideration of three-dimensional spatial effects, a domain where current theoretical research remains notably lacking. The classical Rankine earth pressure theory and Coulomb earth pressure theory have been widely used because of their simple calculation. After Rankine and Coulomb, Paik and Salgado (2003), Lu (2010), Zhu (2018), Deng (2023), and numerous scholars have progressively advanced methods for calculating earth pressure. However, for circular retaining structures, a comprehensive theoretical foundation for earth pressure calculation remains elusive. Consequently, the design of such structures often relies on the conventional calculation methods applicable to plane retaining walls. For example, the plane static earth pressure is used to calculate the earth pressure of the diaphragm wall in the circular foundation pit project of the wagon tipper house of the fourth phase of Qinhuangdao Coal Port Terminal (Liu 1995), and the classical Rankine earth pressure theory is used to calculate the earth pressure of the circular diaphragm wall support of the water pump in Huangjuezhuang Power Plant (Li 1994). However, the existing model tests earth pressure on the circular retaining wall is less than the theoretical value of the plane earth pressure (Tran et al. 2014, Cho et al. 2015, Tan et al. 2018 and Tangjarusritaratorn et al. 2022). Continuing to apply the plane earth pressure theory for calculating the earth pressure on circular retaining walls can lead to conservative designs, resulting in unnecessary costs and resource wastage. Therefore, it’s crucial to explore and develop a method for calculating the active earth pressure on circular retaining walls. Such research not only deepens theoretical understanding but also provides essential technical support for optimizing the design of these structures, thus holding significant theoretical and practical engineering value.

The issue of calculating earth pressure on circular retaining walls is a spatially axially symmetric problem, which has been theoretically explored by various scholars. Derezantzev (1958) using the slip-line method, proposed a theory for the limit earth pressure of granular materials under axial symmetry conditions. According to this theory, the resulting earth pressure distribution is nonlinear and tends to be lower than the values derived using the plane earth pressure calculation method for retaining walls. Xiong et al. (2019, 2020), Xiong and Wang (2020) further improved the earth pressure calculation method of circular retaining wall based on the slip-line method, and achieved some research results. Prater (1977) introduced the hoop stress coefficient to account for the influence of hoop stress. Using the limit equilibrium method, he derived a formula for calculating earth pressure that considers the ratio of the wall’s radius to its height. This approach, grounded in clear mechanical principles, effectively captures the nonlinear distribution of earth pressure along the height of the wall. Kim et al. (2013) assumed that the sliding surface of the soil is Rankine sliding surface and used the lateral earth pressure coefficient, as derived by Paik and Salgado (2003), to calculate the lateral earth pressure on the circular shaft. It was pointed out that the lateral earth pressure on circular retaining walls, estimated by Rankine theory, significant overestimates the earth pressure, a conclusion supported by test results. Numerical modeling is also an approach used to clarify the mechanical behaviors of the surrounding soil and the cylindrical shaft after excavation. Prior to the current study, most numerical simulations of cylindrical shaft problems were conducted using 2D axisymmetric simulations (Meftah et al., 2018). However, recognizing the significance of three-dimensional effects in cylindrical shaft simulations, Chehadeh et al. (2019) and Meftah et al. (2022) employed a 3D finite element analysis to reveal the distribution of earth pressure on a cylindrical shaft.

Current research on the earth pressure affecting circular retaining walls remains limited and not fully developed. Addressing this gap, this paper employs the energy method as its foundational approach. It posits that soil exhibits completely rigid-plastic behavior and follows the associated flow rule at the point of failure. By applying the principle of virtual work, a new formula for calculating the active earth pressure on circular retaining walls is derived and subsequently compared with results from existing model tests.

2 Basic Principle of Energy Method

2.1 Flow Rule

The soil is considered an ideal plastic body, with yield stress at each point in plastic flow state denoted as σ_n and τ. The yield stress is defined as the stress that causes the soil to enter into a plastic flow state. The flow represents the relationship between the yield stress and the plastic strain rate in a plastic body. Mises proposed that the flow rule can be expressed by a plastic potential function f, which satisfies the following equation (GU 2005):

$$ \frac{{\varepsilon_{{\uptau }} }}{{\varepsilon_{{\text{n}}} }} = \frac{{{{\partial f} \mathord{\left/ {\vphantom {{\partial f} {\partial \tau }}} \right. \kern-0pt} {\partial \tau }}}}{{{{\partial f} \mathord{\left/ {\vphantom {{\partial f} {\partial \sigma_{{\text{n}}} }}} \right. \kern-0pt} {\partial \sigma_{{\text{n}}} }}}} $$

(1)

where ε_n is the normal strain rate and ε_τ is the shear strain rate.

For a soil subject to the Mohr-Coulomb strength condition, the yield condition is the strength condition, and the potential function f is:

$$ f = \tau - c - \sigma_{{\text{n}}} \tan \varphi $$

(2)

where c is the cohesion of the soil; φ is the internal friction angle of the soil; τ is the shear stress; σ_n is the normal stress on the shear plane.

When the soil reaches the yield state or the plastic flow state τ = τ_f, the plastic potential function f = 0, and τ_f is the shear strength. This flow rule determined by the yield condition is called the associated flow rule. Substituting the yield Eq. (2) into Eq. (1), we can get:

$$ \frac{{\varepsilon_{{\uptau }} }}{{\varepsilon_{{\text{n}}} }} = - \frac{1}{\tan \varphi } $$

(3)

Equation (3) shows that when the soil is in the state of shear sliding or plastic flow, the angle between the strain velocity vector v on the sliding surface and the sliding surface is φ.

2.2 Principle of Virtual Work

The principle of virtual work states that, for a continuous deformable body, the external virtual work done by a statically admissible stress field on a kinematically admissible displacement field equals the internal virtual work. The principle of virtual power involves replacing the displacement and strain from the principle of virtual work with velocity and strain rate. It is necessary to take into account the energy dissipation on the velocity discontinuity when constructing the dynamic allowable stress field with the velocity discontinuity. If the velocity discontinuity is regarded as a thin layer in which the velocity changes sharply and continuously, the shear strain rate will tend to infinity when the thickness of the thin layer tends to zero, which indicates that the velocity discontinuity must be a slip surface. The internal energy dissipation power on the discontinuity is:

$$ \dot{W} = \int_{S} {\left( {\tau v_{{\text{t}}} - \sigma_{{\text{n}}} v_{{\text{n}}} } \right){\text{d}}S} $$

(4)

where: S is the velocity discontinuity; v_t is the tangential component of the velocity discontinuity; v_n is the normal component of the velocity discontinuity.

Substituting Eq. (2) into Eq. (4), we can get:

$$ \dot{W} = \int_{S} {cv_{{\text{t}}} {\text{d}}S} $$

(5)

Equation (5) is the internal energy dissipation rate on the discontinuity surface.

3 Derivation of the Active Earth Pressure Formula for Circular Retaining Wall

The plan and sectional views of the circular retaining wall are shown in Fig. 1.

Fig. 1.

Plan and section of circular retaining wall

For an axisymmetric circular retaining wall, the Coulomb failure mechanism is used to analyze the soil behind the wall. When the wall moves away from the soil to reach the state of active limit equilibrium, the sliding surface of the soil is a curved surface passing through the heel of the wall, and its generatrix is a straight line, as shown in Fig. 2. A circular retaining wall has a radius of r₀ and a height of H.

Fig. 2.

Analysis model of active earth pressure on circular retaining wall

Fig. 3.

Calculation diagram of energy method

Due to the axial symmetry of the model, any vertical section can be selected for analysis, as illustrated in Fig. 3. It is assumed that the wall is vertical, the wall back is rough, and the displacement mode of the wall is translation. It is assumed that the wall moves centripetally at a velocity of vcos(β − φ), and the relative velocity on the sliding surface is v. δ is the wall-soil-friction angle, φ is the internal friction angle of the soil behind the wall, β is the angle between the generatrix of the sliding surface of the soil behind the wall and the horizontal plane, that is, the inclination angle of the sliding surface of the soil, and G is the gravity of the soil behind the wall.

The gravity and volume of the sliding soil mass are:

$$ G = \gamma V $$

(6)

$$ V = \int_{0}^{H} {\left[ {{\uppi }\left( {r_{0} + \frac{z}{\tan \beta }} \right)^{2} - {\uppi }r_{0}^{2} } \right]} {\text{d}}z = \frac{{{\uppi }H^{3} }}{{3\tan^{2} \beta }} + \frac{{{\uppi }r_{0} H^{2} }}{\tan \beta } $$

(7)

The gravity work power is:

$$ \dot{W}_{G} = \gamma Vv{\text{sin}}\left( {\beta - \phi } \right) = \gamma v{\text{sin}}\left( {\beta - \phi } \right)\left( {\frac{{{\uppi }H^{3} }}{{3\tan^{2} \beta }} + \frac{{{\uppi }r_{0} H^{2} }}{\tan \beta }} \right) $$

(8)

The reaction of the retaining wall to the sliding soil is E_ac, which is equal to the active earth pressure acting on the back of the wall in magnitude and opposite in direction. The work power of the retaining wall on the sliding soil is:

$$ \begin{array}{*{20}l} {\dot{W}_{{E_{{{\text{ac}}}} }} = - 2{\uppi }r_{0} E_{{{\text{ac}}}} \left( {\cos \delta } \right)v\cos \left( {\beta - \phi } \right) - 2{\uppi }r_{0} E_{{{\text{ac}}}} \left( {\sin \delta } \right)v\sin \left( {\beta - \phi } \right)} \hfill \\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - 2{\uppi }r_{0} E_{{{\text{ac}}}} v\cos \left( {\beta - \phi - \delta } \right)} \hfill \\ \end{array} $$

(9)

The cohesive force of cohesive soil is c, and according to the associated flow rule, the internal energy dissipation rate on the velocity discontinuity surface (slip surface AC) is:

$$ \begin{aligned} \dot{W}_{{{\text{AC}}}} & = cv\left( {{\text{cos}}\,\phi } \right)S_{{{\text{AC}}}} = cv\left( {{\text{cos}}\,\phi } \right)\left( {\frac{{2{\uppi }r_{0} H}}{\sin \beta } + \frac{{{\uppi }H^{2} }}{\tan \beta \sin \beta }} \right) \\ \; & = cv\left( {\frac{{2{\uppi }r_{0} H}}{\tan \beta } + \frac{{{\uppi }H^{2} }}{{\tan^{2} \beta }}} \right) \\ \end{aligned} $$

(10)

As the circular retaining wall moves away from the soil, the soil behind the wall will be compressed in the circumferential direction, and the internal energy dissipation power caused by the work done by the circumferential stress is:

$$ \dot{W}_{\theta } = \int\limits_{V} {\sigma_{\theta } \dot{\varepsilon }{\text{d}}V} $$

(11)

Introducing the hoop stress coefficient λ:

$$ \sigma_{\theta } = \lambda \sigma_{1} = \lambda \gamma z $$

(12)

There are others:

$$ {\text{d}}V = 2{\uppi }r{\text{d}}r{\text{d}}z $$

(13)

$$ \dot{\varepsilon } = \frac{{v\cos \left( {\beta - \varphi } \right)}}{r} $$

(14)

Substitute Eqs. (12)–(14) into Eq. (11) to obtain:

$$ \begin{aligned} \dot{W}_{\theta } = & \int\limits_{V} {\sigma_{\theta } \dot{\varepsilon }{\text{d}}V} = \int_{0}^{H} {\int_{{r_{0} }}^{{r_{0} + \frac{H - z}{{\tan \beta }}}} {\lambda \gamma z\frac{{v\cos \left( {\beta - \phi } \right)}}{r}2{\uppi }r{\text{d}}r} {\text{d}}z} \\ = & 2{\uppi }\lambda \gamma v\cos \left( {\beta - \phi } \right)\frac{{H^{3} }}{6\tan \beta } \\ \end{aligned} $$

(15)

According to the principle of virtual work, for sliding soil, the power external by external forces is equal to the power of internal energy dissipation. This can be expressed as:

$$ \dot{W}_{G} + \dot{W}_{{E_{{{\text{ac}}}} }} = \dot{W}_{\theta } + \dot{W}_{{{\text{AC}}}} $$

(16)

Substituting Eqs. (8)–(10) and (15) into Eq. (16) gives:

$$ \begin{array}{*{20}l} {\gamma v{\text{sin}}\left( {\beta - \phi } \right)\left( {\frac{{{\uppi }H^{3} }}{{3\tan^{2} \beta }} + \frac{{{\uppi }r_{0} H^{2} }}{\tan \beta }} \right) - 2{\uppi }r_{0} E_{{{\text{ac}}}} v\cos \left( {\beta - \phi - \delta } \right)} \hfill \\ { = 2{\uppi }\lambda \gamma v\cos \left( {\beta - \phi } \right)\frac{{H^{3} }}{6\tan \beta } + cv\left( {\frac{{2{\uppi }r_{0} H}}{\tan \beta } + \frac{{{\uppi }H^{2} }}{{\tan^{2} \beta }}} \right)} \hfill \\ \end{array} $$

(17)

Eliminate the velocity v and sort it out to get:

$$ \begin{gathered} E_{{{\text{ac}}}} = \frac{{\sin \left( {\beta - \phi } \right)}}{{\cos \left( {\beta - \phi - \delta } \right)}}\left( {\frac{{\gamma H^{3} }}{{6r_{0} \tan^{2} \beta }} + \frac{{\gamma H^{2} }}{2\tan \beta }} \right) \hfill \\ - \frac{{{\text{cos}}\left( {\beta - \phi } \right)}}{{\cos \left( {\beta - \phi - \delta } \right)}}\frac{{\lambda \gamma H^{3} }}{{6r_{0} \tan \beta }} - \frac{{cH\left( {2r_{0} \tan \beta + H} \right)}}{{2r_{0} \tan^{2} \beta \cos \left( {\beta - \phi - \delta } \right)}} \hfill \\ \end{gathered} $$

(18)

In Eq. (18), the inclination angle β of the slip surface is arbitrarily assumed, and only the slip surface with the maximum value of E_ac is the most dangerous slip surface. E_ac is a function of β. According to the condition of ${{{\text{d}}E_{{{\text{ac}}}} } \mathord{\left/ {\vphantom {{{\text{d}}E_{{{\text{ac}}}} } {{\text{d}}\beta }}} \right. \kern-0pt} {{\text{d}}\beta }} = 0$, the angle β at the maximum value of E_ac is obtained, and then the active earth pressure of the circular retaining wall can be obtained by substituting Eq. (18).

Equation (18) is the expression of the resultant force of the active earth pressure of the circular retaining wall. For the determined circular retaining wall, the ratio of the radius to the wall height r₀/H is constant, and the earth pressure intensity at the depth H can be expressed as:

$$ \begin{gathered} \sigma_{{\text{h}}} = \frac{{dE_{{{\text{ac}}}} }}{dh} = \frac{{\sin \left( {\beta - \phi } \right)}}{{\cos \left( {\beta - \phi - \delta } \right)}}\left( {\frac{H}{{6r_{0} \tan^{2} \beta }} + \frac{1}{2\tan \beta }} \right)\gamma h \hfill \\ - \frac{{{\text{cos}}\left( {\beta - \phi } \right)}}{{\cos \left( {\beta - \phi - \delta } \right)}}\frac{\lambda H}{{6r_{0} \tan \beta }}\gamma h - \frac{c}{{2\tan^{2} \beta \cos \left( {\beta - \phi - \delta } \right)}}\left( {2\tan \beta + \frac{H}{{r_{0} }}} \right) \hfill \\ \end{gathered} $$

(19)

It can be seen from Eq. (19) that the earth pressure intensity is distributed in a triangle. Compared with the Coulomb earth pressure for a plane retaining wall, the earth pressure for a circular retaining wall is also related to the radius-height ratio r₀/H and the hoop stress coefficient λ.

4 Model Test Verification

4.1 Model Test 1

Tran and Meguid (2014) conducted an indoor model test out of a shaft. The model parameters are: r₀ = 0.075 m, H = 1 m, γ = 14.7 kN/m³, c = 0, φ = 41°, δ = 0. The results of this method are compared with those from Tran and Meguid (2014). Figure 4 displays the results, which are compared with those derived from Coulomb’s earth pressure theory. T5, T6, and T7 in the figure represent the test results from Tran and Meguid (2014).

The earth pressure values obtained by the method in this study closely match the upper portion of the retaining wall’s measured data from Tran and Meguid (2014). However, a deviation is observed in the lower third of the wall height, where the calculated values differ from the measurements. It due to due to the reduced earth pressure at the base of the wall caused by friction, a factor not reflected in the linear distribution model of this study. Notably, the earth pressure strengths calculated using the classical Coulomb earth pressure theory are substantially greater than those in the findings of Tran and Meguid (2014).

Fig. 4.

Comparison of the proposed method and the experimental data of Tran and Meguid (2014)

4.2 Model Test 2

Cho et al. (2015) conducted a centrifuge model test to investigate the lateral earth pressure on a circular shaft, applying a centrifugal acceleration coefficient of N = 75 g. The vertical shaft of the model, made of aluminum alloy, featured a hollow circular section. It had an embedment height of 0.2 m and an external diameter of 0.08 m. The model is equivalent to a reinforced concrete shaft with a height of 15 m and an outer diameter of 6 m. Model test parameters are: γ = 12.66 kN/m³, c = 0 kPa, φ = 36.95°, Cho et al. (2015) does not give the value of wall-soil friction angle, which is taken in this paper δ = φ/2. The earth pressure values obtained from the model test of Cho et al. (2015) are shown in Fig. 5.

Fig. 5.

Comparison of the proposed method and the experimental data of Cho et al. (2015)

Figure 5 compares the calculation results in this paper with the model test results in Cho et al. (2015) and Coulomb’s earth pressure theory. The results indicate that the calculated values more closely align with the model test of Cho et al. (2015). The earth pressure values derived from Coulomb’s earth pressure theory are higher than those observed in Cho et al. (2015). The enhanced precision of the calculation results of this method, closely aligning with the test earth pressures, can be attributed to the methodological consideration of both the diameter-to-height ratio of the circular retaining wall and the effects of hoop stress.

5 Analysis of Cohesionless Soil Law

5.1 Dip Angle of Slip Surface

For a circular retaining wall with cohesionless backfill, the variation of the slip plane inclination angle β with the ratio of diameter to height r₀/H is shown in Fig. 6. The dip angle of the slip plane calculated by the classical Coulomb earth pressure theory is also shown in the figure.

Fig. 6.

Variation curve of slip surface inclination with the ratio of diameter to height

Figure 6 indicated that the inclination angle of the sliding surface of the soil in the active limit state of the circular retaining wall is greater than that of the Coulomb solution of the plane retaining wall, and the inclination angle of the sliding surface of the soil in the active limit state of the circular retaining wall decreases with the increase of the diameter-height ratio r₀/H, and gradually tends to the Coulomb solution of the plane retaining wall. The greater the internal friction angle, the larger the inclination angle of the sliding plane becomes. The higher the hoop stress coefficient, the larger the inclination angle of the slip surface becomes.

5.2 Active Earth Pressure

For cohesionless soil, when the diameter-height ratio r₀/H of a circular retaining wall tends to infinity, Eq. (18) can be transformed in:

$$ E_{{\text{a}}} = \frac{1}{2}\gamma H^{2} \frac{{\sin \left( {\beta - \varphi } \right)}}{{\tan \beta \cos \left( {\beta - \varphi - \delta } \right)}} $$

(20)

Equation (20) is the active earth pressure expression of plane retaining wall calculated according to the classical Coulomb earth pressure theory. It can be seen that when the ratio of diameter to height r₀/H tends to infinity, the calculation method is consistent with Coulomb’s theory.

For a circular retaining wall, the variation of the active earth pressure with the ratio of diameter to height r₀/H obtained by the energy method is shown in Fig. 7. The Coulomb earth pressure of plane retaining wall is also given in Fig. 7. The ordinate 2E_ac/γH² in Fig. 7 represents the magnitude of the normalized resultant active earth pressure.

Figure 7 shows that the active earth pressure of the circular retaining wall is less than the Coulomb solution of the plane retaining wall, and the active earth pressure of the circular retaining wall increases with the increase of the diameter-height ratio r₀/H, and gradually tends to the Coulomb solution of the plane retaining wall. The greater the value of the circumferential stress coefficient, the lower the active earth pressure of the circular retaining wall.

Fig. 7.

Variation of earth pressure with the ratio of diameter to height

6 Analysis of Clayey Soil Law

The calculation and discussion are based on the cohesionless soil. Equation (18) is derived from the energy method, it can also be used to calculate the active earth pressure of the circular retaining wall with cohesive soil behind the wall.

To investigate the impact of cohesion on the sliding surface and active earth pressure of circular retaining walls, the following parameters have been chosen as reference values for the calculations. Theses parameters include: soil unit weight γ = 18 N/m³, circular retaining wall radius r₀ = 10 m, wall height H = 10 m, soil internal friction angle φ = 30°, wall-soil friction angle δ = 1/2φ, and cohesion c = 10 kPa.

6.1 Dip Angle of Slip Surface

The relationship between the inclination angle of the slip surface and the cohesion is shown in Fig. 8. It can be seen from Fig. 8 that the inclination angle of the sliding surface of the circular retaining wall increases with the increase of the cohesion. A higher internal friction angle corresponds to a greater inclination angle of the sliding plane.

Fig. 8.

Variation of dip angle of slip plane with cohesion

6.2 Active Earth Pressure

The relationship between the active earth pressure and the cohesion of the circular retaining wall is shown in Fig. 9. It can be seen from Fig. 9 that the active earth pressure of circular retaining wall decreases with the increase of cohesion. An increase in the internal friction angle results in a reduction of the active earth pressure.

Fig. 9.

Variation of active earth pressure with cohesion

7 Conclusion

• (1) Utilizing the energy method, this study establishes a calculation method for the active earth pressure on circular retaining walls. It takes into account the internal energy dissipation resulting from hoop stress. The efficacy and applicability of this proposed method have been corroborated by results obtained from model tests.

• (2) The active earth pressure of the circular retaining wall is found to be less than that of plane retaining walls. As the ratio of diameter to height (r0/H) increases, the active earth pressure on circular walls gradually approaches the Coulomb solution for plane walls. Additionally, a higher hoop stress coefficient results in lower active earth pressure for circular walls.

• (3) The proposed method in this paper is applicable for calculating the active earth pressure on circular retaining walls with both cohesionless and cohesive soils. For cohesionless soils, when the ratio of diameter to height r₀/H tends to infinity, the calculation method in this paper is consistent with Coulomb’s theory.

• (4) The earth pressure calculation method obtained in this paper is only a theoretical solution, and there is a certain deviation from the actual value. In future research, we need to consider more influencing factors, such as the uniformity of the soil, the load on the soil surface, the underground water conditions, and the time effect.