Upper bound solution to seismic active earth pressure of submerged backfill subjected to partial drainage

Abstract

In waterfront geotechnical engineering, seismic and drainage conditions must be considered in the design of retaining structures. This paper proposes a general analytical method to evaluate the seismic active earth pressure on a retaining wall with backfill subjected to partial steady seepage flow under seismic conditions. The method comprises the following steps: i) determination of the total head, ii) upper bound solution of seismic active earth thrust, and iii) deduction for the earth pressure distribution. The determination of total head h(x,z) relies on the Fourier series expansions, and the expressions of the seismic active earth thrust and pressure are derived by using the upper bound theorem. Parametric studies reveal that insufficient drainage and earthquakes are crucial factors that cause unfavorable earth pressure. The numerical results confirm the validity of the total head distribution. Comparisons indicate that the proposed method is consistent with other relevant existing methods in terms of predicting seismic active earth pressure. The method can be applied to the seismic design of waterfront retaining walls.

Graphical abstract

Keywords

seismic active earth pressure / partial seepage flow / pore pressure / anisotropy / upper bound theorem

1 Introduction

Determination of earth pressure under seismic conditions is essential in the design of retaining walls because damages caused by earthquakes on retaining structures can lead to catastrophic failure [1–9]. The seismic pressure that acts on retaining walls is usually assessed by using a pseudo-static approach presented by Okabe [10] and Mononobe and Matsuo [11] and described by Ebeling and Morrison [12]. Thereafter, pseudo-static procedures have also been developed to estimate the seismic earth pressure for c−φ backfills by Iskander et al. [13] and for inhomogeneous backfills by Brandenberg et al. [9], Ebeling and Morrison [12]. In recent decades, a pseudo-dynamic approach that considers time-dependent seismic forces has been proposed by Choudhury and Nimbalkar [1,14], Ghosh [15], and Kolathayar and Ghosh [4] to estimate seismic earth pressure.

In addition to seismic loads, water seepage is a primary factor that affects the lateral earth pressure in many geotechnical engineering problems [12,16–18]. The most common method of analysis is deduction of the pore pressure distribution in seepage fields to determine the effects of seepage on the retaining structures [19]. Seepage field is often classified into the following three categories according to the drainage system layout [20]: hydrostatic field, vertical seepage and lateral seepage fields. The pore pressure distribution in the hydrostatic field is directly solved by using the hydrostatic pressure, in which the pore pressure is proportional to the depth and gravitational field strength. The flow lines of a vertical seepage are along the vertical direction, and the pore pressure is directly proportional to the depth and hydraulic gradient i [17,21]. Lateral seepage is caused by a drainage system along the retaining wall. Existing research provides an expression for the pore water pressure in a lateral seepage flow based on the Fourier expansion technique [22–24]. In addition to the analytical solutions, analysis based on the finite element method (FEM) [25] or boundary element method (BEM) [26,27] was employed in the derivation of the earth pressure.

We must note that in the case of lateral seepage, the pore pressure exerted on a retaining wall is assumed to be zero. However, this is an over-idealised assumption. In practice, the drainage pipelines are arranged at regular intervals in the longitudinal direction, and blockage in the drainage pipeline can frequently occur. Moreover, in addition to imperfect design and pipeline failure, the discharge of some drainage systems must be compromised to protect the groundwater environment. The occurrence of such situations is called partial drainage in our study and it implies that the pore pressure that acts on a retaining wall cannot be completely neglected.

A method to determine the seismic active earth pressure that acts on a retaining wall under the dual influence of seismic load and partial seepage flow is investigated in this study. An outstanding advantage of this method is its ability to find a relationship between the pore pressure distribution and accessible total flow Q by embedding an intermediate quantity, that being exit height d of the equivalent phreatic surface, into the Fourier series expansion equation. Comparisons show that the pore pressure under a partial seepage flow field is roughly identical to the numerical results obtained using the finite difference method (FDM). Furthermore, the upper bound theorem is employed to obtain the analytical solution of the active earth pressure based on the balance between the work rates done by the external forces and by the dissipation of internal energy. A complementary discussion on the different calculations of excess pore pressure is also presented in this study.

2 Total head and pore pressure

2.1 Total head h(x,z) subjected to complete seepage flow

The retaining wall and soil particles carry a load determined by the pore pressure or hydraulic head when water flows through the relevant areas. Figure 1 shows a vertical wall supporting a backfill subjected to seepage flow and the total head is expressed by Eq. (1), according to Barros [22] and Hu and Yang [23]. The bottom boundary is logically considered impermeable in situations with extremely weak vertical seepage. In this study, this type of seepage is called a complete seepage because the drainage system completely discharges the water that flows to the structure-soil surface. The porepressure exerted on the retaining wall is assumed to be zero, i.e., u(x = 0, z = 0).

(1)h(x,\mathit{\text{z}})=H\left(1-\sum _{m=0}^{\mathrm{\infty }}\frac{2}{M^2}{\text{e}}^{-\frac{M\xi x}{H}}\cos \frac{M\mathit{\text{z}}}{H}\right),

where ξ is the anisotropy index of the seepage coefficient as defined in Eq. (2); k_x and k_z represent the horizontal and vertical permeability coefficients, respectively; H represents the height from the water level to the impervious layer. Coefficient M is expressed by Eq. (3).

(2)\xi =\sqrt{k_{\mathit{\text{z}}}/k_{\mathit{\text{x}}}},

(3)M=\frac{(2m+1)\text{π}}{2}.

2.2 Total head h(x,z) subjected to partial seepage flow

Let us consider two particular cases.

Case 1: The drainage system works with full efficiency, as presented in Section 2.1. The left boundary conditions can be expressed as u(0,z) = 0 or h(0,z) = z;

Case 2: When the drainage system does not work, the retaining wall bears the full pore pressure. The left boundary condition is expressed as u(0,z) = γ_w(H − z) or h(0,z) = H [20].

When a partial seepage flow occurs, the water is implied to be discharged from the structural plane with a controlled or moderate flow rate Q, and the Q value varies from that in the completely drained case (Case 1) to that in the completely undrained case (Case 2). Thus, the pore pressure exerted on the retaining structure surface should satisfy 0 < u < γ_w (H − z).

Generally, determining the actual pore pressure distribution in a partial seepage flow is very difficult. Therefore, we performed the following processes. Because pore pressure u(0,0) at the wall toe is greater than zero, an equivalent phreatic surface determined by the hydrostatic gradient was developed using the impervious layer as the reference level, whose exit point height is d (see Fig. 2(a)). Under the influence of the gravity gradient, the left boundary condition was divided into two phases, i.e., one in 0 < z < d (with nonzero pore pressure) and another in d < z < h (with zero pore pressure), as expressed in Eq. (4) and presented in Fig. 2(b).

(4)

Note that the above-mentioned boundary conditions differ from those proposed by Santos and Barros [27], who indicated that only a part of the fill was affected by seepage. However, the drainage system completely discharged the water; thus, no pore pressure was exerted on the retaining structure. In contrast, our study focuses on the results induced by inherently inadequate drainage of a drainage system.

The right boundary at an infinite distance from the wall is permeable with a continuous supply of water to guarantee a constant total head H as follows:

(5)\underset{x\to +\mathrm{\infty }}{lim}h(x,\mathit{\text{z}})=H.

The bottom surface is an impervious layer, i.e.,

(6){\frac{\mathrm{\partial }h(x,\mathit{\text{z}})}{\mathrm{\partial }\mathit{\text{z}}}}_{\mathit{\text{z}}=0}=0.

The area in d < z < h can be regarded as the total seepage field with a head difference of H − d, whereas the area in 0 < z < d is considered a hydrostatic field, as presented in Fig. 2(b). Correspondingly, the total head h(x,z) is also satisfied.

i) The total head in d < z < h can be obtained using h − d, H − d, and z − d of the partial seepage model to replace h, H and z in Eq. (1), respectively. The result is expressed in the form of a Fourier serial expansion as follows:

(7)h(x,\mathit{\text{z}})=\left(H-d\right)\left(1-\sum _{m=0}^{\mathrm{\infty }}\frac{2}{M^2}{\text{e}}^{-\frac{M\xi x}{\left(H-d\right)}}\cos \frac{M\left(\mathit{\text{z}}-d\right)}{\left(H-d\right)}\right)+d.

ii) The total head is fixed in the vertical direction at the height of 0 − d, and its value is determined by the continuous condition at z = d in Eq. (7), as expressed in Eq. (8).

(8)h(x,\mathit{\text{z}})=\left(H-d\right)\left(1-\sum _{m=0}^{\mathrm{\infty }}\frac{2}{M^2}{\text{e}}^{-\frac{M\xi x}{\left(H-d\right)}}\right)+d.

Pore pressure u(x,\mathit{\text{z}}) at any point inside the soil mass is given by Eq. (9), which serves as an essential precondition for the calculation of the pore water resultant force in subsequent sections.

(9)u(x,\mathit{\text{z}})={\gamma }_{\text{w}}\left[h(x,\mathit{\text{z}})-\mathit{\text{z}}\right].

To determine d, we associate it with the total flow Q. According to Darcy’s law, the horizontal component of the proposed seepage velocity along the soil-structure interface can be expressed as Eq. (10).

(10){v_x(x,\mathit{\text{z}})}_{x=0}=-k_x{\frac{\mathrm{\partial }h(x,\mathit{\text{z}})}{\mathrm{\partial }x}}_{x=0}=2k_x\xi \sum _{m=0}^{\mathrm{\infty }}\frac{1}{M}\cos \frac{M\left(\mathit{\text{z}}-d\right)}{\left(H-d\right)}.

Hence, the total water flow Q can be obtained by the following integration

(11)Q={\int }_d^H{v_x(x,\mathit{\text{z}})}_{x=0}\text{d}\mathit{\text{z}}=\frac{8\mathrm{G}{k}_x\xi \left(H-d\right)}{{\text{π}}^2},

where G = 0.915966···, which is known as Catalan’s constant [28]. Equation (11) indicates that the total flow Q almost linearly varies with the horizontal component of the permeability coefficient k_x, anisotropic index ξ and effective total head difference H − d. Because Q is easily obtained, designers can deduce height d by using Eq. (11).

It is noteworthy that the derivation of the seepage velocity and water flow can only be applied to a laminar flow condition. When the flow pattern becomes turbulent, the nonlinear relationship between the velocity and hydraulic gradient should be replaced to calculate the total water flow.

2.3 FDM validation

A finite-difference analysis was performed to compare the proposed hydraulic boundary condition with the actual drainage boundary (left boundary) and to verify the accuracy of the proposed analytic solution for pore pressure in Eqs. (7) and (8) by using the commercial FLAC software. A vertical rigid retaining wall with a unit height rested against a fully saturated backfill and was supplied with a controlled drainage system along with the soil-structure interface, as presented in Fig. 3. The backfill parameters were set as k_x = 1 × 10⁻⁵ and ξ = 0.8. Water infiltrated from the right boundary at a distance of quintuple units and was partially drained from the wall-soil interface. The left hydraulic boundary condition was applied using the apply discharge command in FLAC3D to simulate the pore pressure surplus at the wall toe in the real situation. The equivalent height of the phreatic exit point was set to d = 0, 0.4 and 0.8 m, and the corresponding water discharge was obtained by Eq. (11). The pore pressure distributions along the surface at angles of θ = 30°, 45°, and 60° were investigated. To ensure the precision of the analytical calculation under the influence of the expansion series, the truncation error in Eqs. (7) and (8) was considered to converge to a negligible value when m > 100. Figure 3(b) plots the calculation results of grid point pore pressure when d = 0.4. The range in simulated results for u > 0 at the left hydraulic boundary seemed wider than the equivalent measures in Fig. 2, indicating that the hydraulic gradient near the wall was slightly less than that determined by gravity. Despite the above difference given by contours, Fig. 4 indicates that the FDM results are roughly consistent with those of the proposed method, with a relative error of no more than 10%. It implies that the simplified treatment is still feasible when calculating the pore pressure on the failure surface.

2.4 Discussion on the distribution of total head

Figures 5(a) and 5(b) present the effects of d and ξ on the distribution of the total head in two dimensions with x = 0 to 2.0 and z = 0 to 1.0, respectively. We can observe that the parameters d and ξ have different effects on the distribution of the total head. The d value necessarily depends on the boundary condition, which is associated with the drainage efficiency along with the wall-soil interface. A highly efficient drainage system undoubtedly causes the total head to drop and the caving-in range to expand. In particular, the distribution under a complete drainage condition intersects at the origin. Conversely, ξ affects the curvature of the total head, and an increasing anisotropy in the backfill in terms of permeability (a smaller ξ value) can exaggerate the range of dewatering.

3 Upper bound analysis of the seismic active earth pressure

3.1 Upper bound theorem that accounts for pore water pressure

The upper bound theorem [29–31] has been widely used in civil engineering. The load calculated by equating the rate of internal energy dissipation to the rates induced by external forces is assumed to be not less than the actual external load for any given strain rate field {\epsilon }_{ij} and kinematically admissible velocity field v_i. In the interest of stability of the saturated slopes with variable water levels, Michalowski [32,33] incorporated the work rate of the pore pressure into the kinematically admissible velocity field for limit analysis as

(12)\underset{V}{\int }{\sigma }_{ij}{\epsilon }_{ij}\text{d}V⩾\underset{S}{\int }{T}_i{v}_i\text{d}S+\underset{V}{\int }{F}_i{v}_i\text{d}V+\underset{V}{\int }u{\epsilon }_{ii}\text{d}V+\underset{S}{\int }u{n}_i{v}_i\text{d}S,

where {\sigma }_{ij} is the stress field relative to the associated flow rule, T_i is the surcharge load applied on boundary S, F is the body force on area V, u is the pore water pressure and n_i is the normal unit vector toward the outside of the boundary of the admissible velocity field. The third and fourth terms at the right side of Eq. (12) represent the work done by the pore pressure on the bulk strain and boundary, respectively. When the soil is regarded as an ideally rigid-plastic material, bulk strain {\epsilon }_{ii}=0; thus, \underset{V}{\int }u{\epsilon }_{ii}\text{d}V=0. The internal energy dissipation and work done by the pore pressure only occur at the velocity discontinuity surface S.

3.2 Active failure mechanism

Two assumptions are made to satisfy the prerequisites of the upper bound theorem.

1) The backfill soil is considered as a perfectly rigid-plastic material satisfying the Mohr−Coulomb failure criterion. Thus, the discontinuous velocityv_i and the velocity discontinuity surface always intersect with an angle of φ [34], where φ represents the internal friction angle of the soil.

2) The failure surface is considered to be planar.

The case analyzed in this study is presented in Fig. 6, where a vertical retaining wall rests against a backfill subjected to a partial seepage flow that considers the effects of seismic accelerations.

The loads that induce the external work rates on the wedge comprise seismic active earth resultant force E_a, internal seismic forces Q_h and Q_v, gravity force F_g, pore water resultant force F_u1 and an excess pore water resultant force F_u2 induced by an earthquake. Note that the direction of Q_h must be the same as that of wall movement since seismic forces are considered as unfavorable factors; meanwhile, the direction of Q_v is synchronised with the P-wave transmission. The internal energy dissipation is related to the possible cohesion of the backfill if it is reinforced using cement slurry or other cementitious materials.

Comparing the present failure mechanism with the three existing models in literature (Choudhury et al. [1,2], Wang et al. [16] (Passive) and Ghosh and Sharma [35]), we can find that the main differences are:

i) The existing models rarely consider partial seepage conditions. The seismic model proposed by Wang et al. [16] counted in pore pressure but referred only to the completely drained and passive state.

ii) Almost all published models employ the limit equilibrium method. Treatment of the reaction forces imposed on the wedge block is required in existing models, whereas it was ignored in our study since its work rate is zero.

iii) In the present work, the seismic forces were considered as pseudo-static forces, which are identical to that of Wang et al. [16]. The advantage of discarding the pseudo-dynamic method [1,2,35] is its convenience for quick analysis and applications by engineers.

3.3 Determination of seismic active earth thrust

The determination of pore pressure presented in Section 2 (Eqs. (7)–(9)) allows a very easy assessment of the resultant pore force. According to Refs. [22] and [26], resultant pore water force F_u1 in the soil can be directly obtained by integrating pore water pressure u along any assumed slip surface.

(13)\begin{array}{rl}{F}_{\text{u1}}& ={\int }_0^{d}u(a\mathit{\text{z}},\mathit{\text{z}})\csc \theta \text{d}\mathit{\text{z}}+{\int }_d^{H}u(a\mathit{\text{z}},\mathit{\text{z}})\csc \theta \text{d}\mathit{\text{z}}\\ & =\frac{1}{2}{\gamma }_{\text{w}}{H}^2{U}_1\csc \theta ,\end{array}

with

(14)\begin{array}{rl}{U}_1=& 1-\frac{{\left(H-d\right)}^2}{H^2}\sum _{m=0}^{\mathrm{\infty }}\{\frac{4{\mathrm{e}}^{-\frac{Ma\xi H}{H-d}}}{M^3\left[{\left(a\xi \right)}^2+1\right]}\\ & \left(\sin M-a\xi \cos M+a\xi {\mathrm{e}}^{Ma\xi }\right)+\frac{4}{M^{3}a\xi }\left(1-{\mathrm{e}}^{-\frac{Ma\xi d}{H-d}}\right)\},\end{array}

where a is the cotangent of inclination angle θ.

Under seismic conditions, excess pore water pressure may be induced inside the backfill, which can be obtained by using the pseudo-static method given by Refs. [16,36,37] in the plane strain state.

(15)\begin{array}{rl}\mathrm{\Delta }u=& {\gamma }_{\text{s}}[\beta \frac{\left(k_{\text{h}}+k_{\text{v}}\right)(1+\nu )}{3}\\ & +\sqrt{2}\alpha \sqrt{{\left(k_{\text{h}}+k_{\text{v}}\right)}^2\left({\nu }^2-\nu +1\right)-3k_{\text{h}}{k}_{\text{v}}}],\end{array}

where {\gamma }_{\text{s}} is the unit weight of the saturated soil; \nu is the Poisson’s ratio of the soil mass, and \nu \approx 0.5 for saturated soil; α and β are the coefficients of the excess pore water pressure, and \beta =1.0 for saturated soil [38]; k_{\text{h}} and k_{\text{v}} are the seismic acceleration coefficients in the horizontal and vertical directions, respectively, and the relationship k_{\text{h}}⩾k_{\text{v}} is provided by previous researchers [11,39].

The total excess pore water pressure induced by earthquakes, i.e., F_{\text{u2}}, which results from excess pore water pressure \mathrm{\Delta }u expressed in Eq. (15) and acts along the planar failure surface, is given by

(16)F_{\text{u2}}={\int }_0^H\mathrm{\Delta }u\csc \theta \text{d}\mathit{\text{z}}={\gamma }_{\text{s}}H{U}_2\csc \theta ,

where

(17)U_2=\beta \frac{\left(k_{\text{h}}+k_{\text{v}}\right)(1+\nu )}{3}+\sqrt{2}\alpha \sqrt{{\left(k_{\text{h}}+k_{\text{v}}\right)}^2\left({\nu }^2-\nu +1\right)-3k_{\text{h}}{k}_{\text{v}}}.

The total horizontal and vertical seismic inertia forces Q_{\text{h}} and Q_{\text{v}} acting on the soil wedge can be expressed as Eqs. (18) and (19), respectively:

(18)Q_{\text{h}}=k_{\text{h}}{F}_{\text{g}},

(19)Q_{\text{v}}=k_{\text{v}}{F}_{\text{g}}.

The work rates done by external forces, i.e., gravity, seismic inertia forces, active earth thrust and pore pressure, can be calculated as follows:

(20)W_{\mathrm{G}}=\frac{1}{2}{\gamma }_{\text{s}}{H}^2\frac{1}{\tan \theta }\cdot V_0\sin (\theta -\phi ),

(21)\begin{array}{rl}{W}_{\text{Q}}=& -k_{\text{v}}\cdot \frac{1}{2}{\gamma }_{\text{s}}{H}^2\frac{1}{\tan \theta }\cdot V_0\sin (\theta -\phi )\\ & +k_{\text{h}}\cdot \frac{1}{2}{\gamma }_{\text{s}}{H}^2\frac{1}{\tan \theta }\cdot V_0\cos (\theta -\phi ),\end{array}

(22)W_{\text{Ea}}=-E_{\text{a}}{V}_0\cos (\theta -\phi -\delta ),

(23)W_{\text{u}}=\left(F_{\text{u1}}+F_{\text{u2}}\right)\cdot V_0\sin \phi .

The internal energy dissipation generated by the cohesion of surfaces OA and OB is derived as:

(24)\begin{array}{rl}N=& \frac{H}{\sin \theta }c\cdot V_0\cos \phi +c\frac{\tan \delta }{\tan \theta }H\cdot V_0\sin (\theta -\phi )\\ =& cH{V}_0\left(\frac{\cos \phi }{\sin \theta }+\frac{\tan \delta \sin (\theta -\phi )}{\tan \theta }\right).\end{array}

According to the energy conservation law, the internal energy dissipation balances the work rate done by the external forces at the limit state,

(25)W_{\text{G}}+W_{\text{Q}}+W_{\text{Ea}}+W_{\text{u}}=N.

The formula for calculating the active earth thrust is derived as follows:

(26)E_{\text{a}}=\frac{1}{2}{\gamma }_{\text{s}}{H}^2{K}_{\text{as}},

with

(27)K_{\text{as}}=f_{\text{s}}-k_{\text{v}}{f}_{\text{Qv}}+k_{\text{h}}{f}_{\text{Qh}}+\frac{{\gamma }_{\text{w}}}{{\gamma }_{\text{s}}}{f}_{\text{w}}-\frac{2c}{{\gamma }_{\text{s}}H}{f}_{\text{c}},

where f_{\text{s}}, f_{\text{Qv}}, f_{\text{Qh}}, f_{\text{w}}, and f_{\text{c}} are dimensionless coefficients related to the saturated weight, seismic acceleration components, seepage, and cohesion, respectively, as expressed in Eqs. (28)–(31).

(28)f_{\text{s}}=f_{\text{Qv}}=\frac{\sin (\theta -\phi )}{\tan \theta \cos (\theta -\phi -\delta )},

(29)f_{\text{Qh}}=\frac{\cos (\theta -\phi )}{\tan \theta \cos (\theta -\phi -\delta )},

(30)f_{\text{w}}=\left(U_1+\frac{{\gamma }_{\text{s}}}{{\gamma }_{\text{w}}}\frac{2}{H}{U}_2\right)\frac{\csc \theta \sin \phi }{\cos (\theta -\phi -\delta )},

(31)f_{\text{c}}=\frac{\cos \phi \tan \theta +\sin \theta \tan \delta \sin (\theta -\phi )}{\sin \theta \tan \theta \cos (\theta -\phi -\delta )}.

Note that E_a is a single-variable function of θ. When θ varies from 0° to 90°, the maximum value of E_a can be obtained by using a tabulation tool or Matlab codes.

4 Comparisons and validation

4.1 Complete drainage case

Wang et al. [16] obtained the seismic passive earth pressure coefficients when a drainage system was established along with the wall-soil interface, by using the limit equilibrium method. The expression for total head h(x,z) expressed in Eq. (1) with ξ = 1.0 was adopted.

On the basis of the same method described in a published work [16], the formula for calculating the resultant active earth pressure can be derived as follows:

(32)

with

(33)\begin{array}{rl}{K}_{\text{a1}}=& \frac{1}{\cos (\theta -\delta )+\sin (\theta -\delta )\tan \phi }\\ & [\cot \theta (\sin \theta -\cos \theta \tan \phi )\left(1-k_{\text{v}}\right)\\ & +k_{\text{h}}\cot \theta \left(\cos \theta +\sin \theta \tan \phi \right)+g{U^{\prime }}_1\csc \theta \tan \phi ]\end{array},

(34)K_{\text{a2}}=\frac{U_2\csc \theta \tan \phi }{\cos (\theta -\delta )+\sin (\theta -\delta )\tan \phi },

where g={\gamma }_{\text{w}}/{\gamma }_{\text{s}}. U_1^{\prime } can be expressed as Eq. (35), which can also be considered as a special case of the proposed method with d = 0 and ξ = 1.0.

(35)U_1^{\prime }=1-\sum _{m=0}^{\mathrm{\infty }}\frac{4{\sin }^2\theta {\text{e}}^{-\cot \theta M}}{M^3}\left(\cot \theta {\text{e}}^{\cot \theta M}-\cot \theta \cos M+\sin M\right).

4.2 Undrained case

Under the undrained condition, the coefficient of seismic active earth pressure can be obtained by setting d = H.

(36)\begin{array}{rl}{K}_{\text{as}}=& \left(1-k_{\text{v}}\right)\frac{\sin (\theta -\phi )}{\tan \theta \cos (\theta -\phi -\delta )}+k_{\text{h}}\frac{\cos (\theta -\phi )}{\tan \theta \cos (\theta -\phi -\delta )}\\ & +\left(\frac{{\gamma }_{\text{w}}}{{\gamma }_{\text{s}}}+\frac{2}{H}{U}_2\right)\frac{\csc \theta \sin \phi }{\cos (\theta -\phi -\delta )}\\ & -\frac{2c}{{\gamma }_{\text{s}}H}\frac{\cos \phi \tan \theta +\sin \theta \tan \delta \sin (\theta -\phi )}{\sin \theta \tan \theta \cos (\theta -\phi -\delta )}\\ =& f_{\text{s}}-k_{\text{v}}{f}_{\text{Qv}}+k_{\text{h}}{f}_{\text{Qh}}+\frac{{\gamma }_{\text{w}}}{{\gamma }_{\text{s}}}{f}_{\text{w}}-\frac{2c}{{\gamma }_{\text{s}}H}{f}_{\text{c}}.\end{array}

4.3 Comparisons

For the purpose of demonstrating the validity of the proposed method, it is necessary to analyze the variation in the seismic active earth pressure with the drainage efficiency. The default values of the parameters are considered as follows for comparative analysis: H = 5, γ_s = 20 kN/m³, c = 0, ξ = 1, and γ_w = 10 kN/m³. The parameters related to the calculation of the excess pore pressure are β = 1.0, α = 0.75, and ν = 0.5 according to Refs. [14,40]. The calculated results obtained using the method used by Wang et al. [16] are listed in Table 1 for comparison.

Table 1 indicates that as the working efficiency of the drainage system decreases from maximum to minimum, the seismic active earth thrust increases by approximately one-third to two-thirds. We can clearly observe that the effects of the drainage efficiency on the active earth pressure are significant and that the seepage should be considered as an indispensable factor. We must also note that the relationship between the seismic active earth thrust and parameter d is nonlinear; this can be attributed to the nonlinear relationship between the pore pressure and the drainage efficiency. In this study, the results obtained when d = 0 are identical to those obtained by Wang et al. [16], whereas the results are unfavorable in magnitude when compared with those in the latter approach when the variation in d is considered.

4.4 Comparisons with other analytical methods

The selected parameters are consistent with those described in the previous section. Theoretically, the difference of the results calculated by all methods regardless of the water is supposed to be negligible. The comparisons of the results obtained using other analytical methods [1,3,16,35,41] are listed in Table 2. However, we need to emphasize that only the isotropic seepage case with k_h = 0.1 and k_v = 0 is provided. The results listed in Table 2 represent the classic cases of previous analysis involving numerical solutions using FEM [3], pseudo-dynamic method [1,35,41] and pseudo-static method [16]. In contrast, we note that under the special case when f_w = 0, the results in our study are essentially consistent with those of published literature despite the different methods used.

However, an unfavorable K_as value can be observed once the drainage conditions are considered. It indicates that the influence of seepage should be considered to quantitatively ensure the stability of the retaining wall. In the general analytical procedures proposed by Wang et al., only the isotropic case of permeability of soils and the complete drainage system was presented, which could be regarded as a particular solution in our study with d = 0 and ξ = 1.

5 Parametric analysis

The effects of drainage efficiency, anisotropy of permeability, seismic acceleration and wall-soil friction on the seismic active earth thrust are considered in Eq. (27). For more general cases, a set of parameter presuppositions are provided as follows unless special statements are specified: φ = 30°, δ/φ = 1/3, γ_s = 20 kN/m³, H = 5 m, d = 2 m, c = 10 kPa, ξ = 0.8, k_h = 0.1, k_v = 0.5k_h, γ_w = 10 kN/m³, β = 1.0, α = 0.75, and ν = 0.5. The number of iterations of m in Eq. (14) was set to 100, and the results were obtained by an extremum solver in which the MATLAB software was used.

Since the plain failure surface was assumed, the critical inclination angle and coefficient of seismic active earth thrust relative to the different parameters were investigated in detail.

5.1 Effects of drainage efficiency

Drainage efficiency d/H was the core feature considered in this method. Additionally, anisotropic index ξ was also taken into account, and a consistent tendency where the horizontal permeability coefficient was greater than the vertical permeability coefficient is recognized with v_z < v_x (ξ < 1.0) [ 42,43].

Figure 7 implies that with the reduction in the drainage efficiency (increasing d/H), the values of θ_cr and K_as exhibit a nonlinear trend with d/H. The θ_cr value initially decreases but subsequently increases when d/H changes from zero to a unit, which is less apparent when ξ increases. Nevertheless, the K_as value nonlinearly increases regardless of the anisotropy. The magnitude of θ_cr changes in the ranges of 2°−5° for ξ = 0.8 and 3.5°−7.1° for ξ = 0.4. However, the K_as values increase by 0.18−0.31 for ξ = 0.8 and 0.21−0.35 for ξ = 0.4. This result implies that the change in the rupture angle and active earth pressure with respect to the drainage efficiency change is relatively more significant when the anisotropy is more remarkable. We must point out that anisotropy does not affect the values of θ_cr and K_as under the undrained condition, which is consistent with reality under a completely hydrostatic condition.

5.2 Effects of seismic acceleration coefficients k_h and k_v

Figure 8 presents the comparisons of θ_cr and K_as with different seismic acceleration components for k_h = 0, 0.1 and 0.2 and k_v/k_h = 0, 0.5 and 1. Figure 8 indicates that the horizontal component of the seismic acceleration reduces θ_cr and increases K_as. For example, under the same premise of k_v = 0, the values of θ_cr corresponding to k_h = 0.1 and 0.2 decrease by 6.4°−7.5° and by 15.2°−20.9°, respectively, compared with those under static conditions. Simultaneously, the K_as values increase by 0.11−0.12 and 0.26−0.29, respectively. Meanwhile, the vertical component k_v plays a distinct role when compared with k_h because it reduces the K_as values to a certain extent. The magnitude of θ_cr decreases by 1.5° (k_h = 0.1) and 8.5° (k_h = 0.2) when k_v increases from zero to k_h (see Fig. 8(a)). Simultaneously, the value of K_as also decreases (see Fig. 8(b)). The increasing acceleration of the lateral seismic forces is a critical factor that causes the variation in the rupture angle and seismic active earth pressure.

5.3 Effects of wall-soil friction

Figure 9 presents the variations in θ_cr and K_as with the internal friction angle of the soils for different ratios of δ/φ. Figure 9(a) implies that the value of θ_cr slightly decreases with an increase in the wall-soil friction and the d/H values. θ_cr generally increases with the increase in φ, but an exception occurs when d = 0.4 H and δ/φ = 1, which is untypical because when φ exceeds 40°, a slight fall in the rupture angle and a moderate increase in K_as are observed. Figure 9(b) shows an unchangeable trend in which K_as under partial drainage is more disadvantageous than under full drainage, which is applicable to various soil types. In full drainage cases, the influence of the wall-soil friction on K_as is relatively unremarkable, although some difference can still be observed. An inadequate drainage condition aggravates the discrepancy of the earth pressure coefficient because of the variation in the wall-soil friction. When the wall-soil friction is at a relatively low level (δ/φ = 0 and 0.5), K_as does not considerably change for the same soil type (φ) due to changing δ/φ. However, when the wall-soil friction approaches the internal friction angle of the soils, the previously mentioned decrease in θ_cr leads to an apparent increase in K_as when φ exceeds 40°.

6 Solution to the distribution of seismic active earth pressure

The distribution of seismic active earth pressure cannot be directly obtained by the following differential equation (Eq. (37)) because of the nonlinearity of the pore pressure versus depth h′.

(37)p_{\text{a}}=\frac{\text{d}{E}_{\text{a}}}{\text{d}{h}^{\prime }}.

The soil inside the failure wedge is considered to be in an ideal plastic-equilibrium state [22,23] to obtain the earth pressure distribution. The seismic active earth pressure coefficient is necessary for reformulation along partial depth h′ of the wall face (see Fig. 10).

Two cases are worthy of discussion according to the variation in depth h′.

Case 1: When h^{\prime }⩽H-d, according to Refs. [22,23], proportional coefficient b is introduced, which is defined as

(38)b=\frac{h^{\prime }}{H-d}.

The resultant force of the pore pressure on the wedge, which considers the proportional coefficient, is derived as

(39)\begin{array}{rl}{F}_{\text{uh}}& ={\int }_{(H-d)(1-b)}^{H-d}u(a\left[\mathit{\text{z}}-(1-b)(H-d)\right],\mathit{\text{z}})\csc \theta \text{d}\mathit{\text{z}}\\ & =\frac{1}{2}{\gamma }_{\text{w}}{h}^{\prime 2}{U}_{11}(h^{\prime })\csc \theta ,\end{array}

where

(40)\begin{array}{rl}{U}_{11}(h^{\prime })=& 1-\sum _{m=0}^{\mathrm{\infty }}\frac{4{\text{e}}^{-Mab\xi }}{\left[{(a\xi )}^2+1\right]M^3{b}^2}\{\sin M\\ & -a\xi \cos M-{\text{e}}^{Mab\xi }\left[\sin (1-b)-a\xi \cos M(1-b)\right]\}.\end{array}

Thus, the resultant force of the excess pore pressure is integrated as follows:

(41)F_{\text{u2}}={\int }_0^{h^{\prime }}\mathrm{\Delta }u\csc \theta \mathrm{d}\mathit{\text{z}}={\gamma }_{\text{s}}{h}^{\prime }{U}_2\csc \theta .

According to the upper bound theorem, the coefficient of the partial active earth pressure can be re-derived as

(42)E_{\text{a}}=\frac{1}{2}{\gamma }_{\text{s}}{h}^{\prime 2}{K}_{\text{as}},

(43)

with

(44)f_{\text{w1}}=\left\{U_{11}(h^{\prime })+\frac{{\gamma }_{\text{s}}}{{\gamma }_{\text{w}}}\frac{2}{h^{\prime }}{U}_2\right\}\cdot \frac{\csc \theta \sin \phi }{\cos (\theta -\phi -\delta )}.

Case 2: When H − d < h′ < H, the distribution of the total head is determined by using the new conditions of seepage flow, which can be regarded as a new combination of the changed seepage and hydrostatic water field by modifying the boundary conditions d and H as follows:

At the left boundary:

(45)d^{\prime }=h^{\prime }-(H-d).

At the right boundary:

(46)H^{\prime }=h^{\prime }.

Therefore, the resultant force of the pore pressure in the new coordinate system can be integrated as follows:

(47)\begin{array}{rl}{F}_{\text{uh}}& ={\int }_0^{d^{\prime }}u(a\mathit{\text{z}},\mathit{\text{z}})\csc \theta \text{d}\mathit{\text{z}}+{\int }_{d^{\prime }}^{H^{\prime }}u(a\mathit{\text{z}},\mathit{\text{z}})\csc \theta \text{d}\mathit{\text{z}}\\ & =\frac{1}{2}{\gamma }_{\text{w}}{h}^{\prime 2}{U}_{12}(h^{\prime })\csc \theta ,\end{array}

where

(48)\begin{array}{rl}{U}_{12}(h^{\prime })=& 1-\frac{{\left(H-d\right)}^2}{{h^{\prime }}^2}\sum _{m=0}^{\mathrm{\infty }}\{\frac{4{\mathrm{e}}^{-\frac{M\xi h^{\prime }\cot \theta }{H-d}}}{M^3\left[{\left(\xi \cot \theta \right)}^2+1\right]}\cdot \\ & \left(\sin M-\xi \cot \theta \cos M+\xi \cot \theta {\mathrm{e}}^{M\cot \theta \xi }\right)\\ & +\frac{4}{M^3\xi \cot \theta }\left[1-{\mathrm{e}}^{-\frac{M\xi (h^{\prime }-H+d)\cot \theta }{H-d}}\right]\}.\end{array}

The coefficient of the partial active earth pressure can be re-derived as

(49)

with

(50)f_{\text{w2}}=\left[U_{12}(h^{\prime })+\frac{{\gamma }_{\text{s}}}{{\gamma }_{\text{w}}}\frac{2}{h^{\prime }}{U}_2\right]\frac{\csc \theta \sin \phi }{\cos (\theta -\phi -\delta )}.

In both cases, the coefficient of earth pressure can be expressed in the following unified form:

(51)

where

(52)f_{\text{w}}^{\ast }=\left[U_1^{\ast }(h^{\prime })+\frac{{\gamma }_{\text{s}}}{{\gamma }_{\text{w}}}\frac{2}{h^{\prime }}{U}_2\right]\frac{\csc \theta \sin \phi }{\cos (\theta -\phi -\delta )}.

The distribution of the earth pressure can be obtained by the following differential process.

(53)\begin{array}{rl}{p}_{\text{a}}=& \frac{\mathrm{\partial }{E}_{\text{a}}}{\mathrm{\partial }{h}^{\prime }}={\gamma }_{\text{s}}{h}^{\prime }{f}_{\text{s}}-{\gamma }_{\text{s}}{h}^{\prime }{k}_{\text{v}}{f}_{\text{Qv}}+{\gamma }_{\text{s}}{h}^{\prime }{k}_{\text{h}}{f}_{\text{Qh}}\\ & +\left[{\gamma }_{\text{w}}{h}^{\prime }{U}_1^{\ast }(h^{\prime })+\frac{1}{2}{\gamma }_{\text{w}}{h^{\prime }}^2\frac{\mathrm{\partial }\left(U_1^{\ast }(h^{\prime })\right)}{\mathrm{\partial }{h}^{\prime }}+{\gamma }_{\text{s}}{U}_2\right]\\ & \frac{\csc \theta \sin \phi }{\cos (\theta -\phi -\delta )}-c{f}_{\text{c}}.\end{array}

We should note that when p_a > 0, it is guaranteed to prevent the emergence of a tension zone along with the wall-soil interface; otherwise, p_a = 0.

Figure 11 presents the distribution of the seismic active earth pressure, in which the parameters are the same as those presented in Section 5, except that the cohesion of the backfill is initialised to zero. Generally, improvement in the drainage efficiency leads to a preferential reduction in the earth pressure in the upper part of the retaining wall along the wall back but an unapparent reduction near the bottom, which leads to a nonlinear distribution trend. When d/H = 0.8 and 1.0, the earth pressure distributions are almost identical, which implies that the earth pressure in the undrained cases can be used to replace that under low seepage condition when d/H > 0.8. When d/H < 0.8, the earth pressure distribution can be regarded as a composition of two sections, i.e., the slope of the curves (versus depth h′) when h′ < 0.5( H − d) is smaller, which indicates that some soil failure wedges within this depth are less affected by the pore pressure; thus, it does not influence the seepage. Conversely, when the depth exceeds 0.5(H − d), the seepage effect results in a much larger earth pressure value.

7 Discussion on the excess pore pressure

The calculation of excess pore water pressure is a controversial issue, which has been investigated in previous studies [12,16–18]. The expression of Wang et al. [36] was adopted in our study. Comparing the calculation of excess pore pressure with relevant literature (two methods of Ebeling and Morrison [12]) can help demonstrate the validity of the proposed method and distinguish the different serviceability of the various methods.

7.1 Method I

Excess pore water pressure \mathrm{\Delta }u is believed to be proportional to the initial vertical effective stress {\sigma }_{\mathrm{v}}^{\prime } as follows:

(54)\mathrm{\Delta }u=r_{\text{u}}{\sigma }_{\text{v}}^{\prime },

where r_{\mathrm{u}} is the excess pore water pressure ratio. The initial effective vertical stress can be obtained as

(55){\sigma }_{\text{v}}^{\prime }={\gamma }_{\text{s}}(H-\mathit{\text{z}})-u(x,\mathit{\text{z}}).

Thus, the earth pressure coefficient can be expressed as

(56)K_{\text{as,E1}}=f_{\text{s}}-k_{\text{v}}{f}_{\text{Qv}}+k_{\text{h}}{f}_{\text{Qh}}+\frac{{\gamma }_{\text{w}}}{{\gamma }_{\text{s}}}{f}_{\text{w,E1}}-\frac{2c}{{\gamma }_{\text{s}}H}{f}_{\text{c}},

with

(57)f_{\text{w,E1}}=\left[\left(1-r_{\text{u}}\right)U_1+\frac{{\gamma }_{\text{s}}}{{\gamma }_{\text{w}}}{r}_{\text{u}}\right]\frac{\csc \theta \sin \phi }{\cos (\theta -\phi -\delta )}.

In Method I given by Ebeling and Morrison, partial drainage leads to a change in the effective vertical stress and a change in excess pore water pressure. However, proportional coefficient r_{\text{u}} is not related to the seismic acceleration coefficient and is considered to be ambiguous.

7.2 Method II

Excess pore water pressure \mathrm{\Delta }u is determined as a function of horizontal seismic component k_h and position z as follows:

(58)\mathrm{\Delta }u=\frac{7}{8}{k}_{\text{h}}{\gamma }_{\text{w}}\sqrt{(H-\mathit{\text{z}})H}.

The earth pressure coefficient is expressed as

(59)K_{\text{as,E2}}=f_{\text{s}}-k_{\text{v}}{f}_{\text{Qv}}+k_{\text{h}}{f}_{\text{Qh}}+\frac{{\gamma }_{\text{w}}}{{\gamma }_{\text{s}}}{f}_{\text{w,E2}}-\frac{2c}{{\gamma }_{\text{s}}H}{f}_{\text{c}},

with

(60)f_{\text{w,E2}}=\left(U_1+\frac{7}{6}{k}_{\text{h}}\right)\cdot \frac{\csc \theta \sin \phi }{\cos (\theta -\phi -\delta )}.

The horizontal component of the seismic acceleration is considered in Method II by Ebeling and Morrison [12], and the distribution of excess pore water pressure against the depth is parabolic. Compared with the method of Wang et al. [36], the latter considers the influence of the vertical component of the seismic acceleration. From the above-mentioned analysis, we can notice that neither Ebeling-Morrison Method II nor the method of Wang et al. [36] considers the influence of partial drainage conditions on the excess pore water pressure; it leads to different results in the assessment of earth pressure.

7.3 Comparison

Figure 12 presents the variation in the earth pressure coefficient with a drainage rate of d/H obtained using different methods. The results of K_as in the present study reasonably agree with those obtained using the Ebeling-Morrison methods [12]. In addition, K_as increases gradually when the drainage rate decreases despite some subtle differences. Because the method proposed by Wang et al. is similar to that of Ebeling and Morrison (Method II) [12], which ignore the vertical seismic acceleration, the difference of K_as is relatively fixed in both methods. For example, when k_h = 0.1 and 0.2, the differences are approximately 0.007 and 0.017, respectively, and they are generally neglected. In contrast, the variation corresponding to Method II is relatively unremarkable.

8 Conclusions

In this study, a general analytical method to evaluate the seismic active earth pressure on a retaining wall that rests against a horizontal permeable backfill subjected to a partial steady seepage flow under seismic conditions is proposed. The proposed approach is based on the upper bound theorem in which the anisotropic partial seepage flow, seismic acceleration and friction of the soil-wall interface are considered. The following conclusions are drawn from this study.

1) This study analyses the influence of a partial seepage flow on pore pressure distribution according to equivalent phreatic exit height. A finite-difference analysis is performed to reveal the validity of the proposed analytical expression by comparing the distribution of the pore pressure.

2) According to the limit analysis, the formula for calculating the seismic active earth thrust that acts on a vertical retaining wall under a partial seepage flow is derived. The derived formula based on a complete drainage condition devolves into the case presented by Wang et al.

3) The seismic active earth pressure is re-derived because of the nonlinear distribution of the pore pressure in accordance with depth. A given example reveals that the distribution of earth pressure calculated under the partial drainage case exhibits a nonlinear trend with the depth.

4) The proposed method adopts a more reasonable and accurate formula to consider the pore pressure and excess pore pressure induced by a partial seepage flow and an earthquake. This method can provide some theoretical guidance for the seismic design of waterfront retaining walls.

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