Seepage failure by heave in sheeted excavation pits constructed in stratified cohesionless soils

Abstract

In this study, experimental and numerical investigations are performed to clarify the seepage failure by heave in sheeted excavation pits in stratified cohesionless soils in which a relatively permeable soil layer (k_upper) lies above a less permeable soil layer (k_lower) between excavation base and wall tip. It is shown that the evaluation of base stabilities of excavation pits against seepage failure by using Terzaghi and Peck’s approach leads to considerably lower critical potential differences than those obtained from the model tests. On the other hand, a relatively good agreement is achieved between the results of the model tests and the finite element (FE) analyses. Further investigations are performed by using axisymmetric excavation models with various dimensions and ground conditions, and a comparison between the results obtained from Terzaghi and Peck’s approach and finite element analyses is given.

Keywords

seepage failure by heave / cohesionless stratified soil / model test / Terzaghi and Peck’s approach / FE analysis

Introduction

During the construction of an excavation pit in groundwater or open water, the main problem is often dominated by seepage flow into the excavation pit. The pore water pressure developed by the seepage flow may lift the excavation base, which is known as seepage failure by heave. This phenomenon in homogeneous cohesionless soils in which no horizontal stratification exists between the excavation base and the wall tip has been investigated by several researches, and various verification methods have been developed [¹–⁹].

The most commonly used verification method was introduced by Terzaghi and Peck [²]. It is found from the two-dimensional model tests performed on homogeneous cohesionless soils that the failure zone lifted by pore water pressure develops in the form of a rectangular-shaped plane. The height and width of the failure plane correspond to the embedment depth of the wall D and its half D/2, respectively (see Fig. 1(a)).

For the verification of seepage failure by heave, the average pore water pressure at the bottom of the heave zone is equated to the total normal stress at the same level. The factor of safety FS is then calculated by using Eq. (3):

(1)(\Delta {{\displaystyle h}}_{\mathrm{a}\mathrm{v}}+h+D)x{{\displaystyle \gamma }}_{\mathrm{w}}=hx{{\displaystyle \gamma }}_{\mathrm{w}}+Dx{{\displaystyle \gamma }}_{\mathrm{s}\mathrm{a}\mathrm{t}}\mathrm{.}

Substituting ({{\displaystyle \gamma }}_{\mathrm{s}\mathrm{a}\mathrm{t}}={{\displaystyle \gamma }}_{\mathrm{w}}+{\gamma }^{\prime }) into Eq. (1) gives

(2)Dx{\gamma }^{\prime }=\Delta {{\displaystyle h}}_{\mathrm{a}\mathrm{v}}x{{\displaystyle \gamma }}_{\mathrm{w}},

(3)FS=(Dx{\gamma }^{\prime })/(\Delta {{\displaystyle h}}_{\mathrm{a}\mathrm{v}}x{{\displaystyle \gamma }}_{\mathrm{w}}),

where {{\displaystyle \gamma }}_{\mathrm{s}\mathrm{a}\mathrm{t}} and {\gamma }^{\prime } are the saturated and submerged unit weights of the upper soil layer, respectively, {{\displaystyle \gamma }}_{\mathrm{w}} is the unit weight of the water, \Delta {{\displaystyle h}}_{\mathrm{a}\mathrm{v}} is the average hydraulic head at the bottom of the heave zone with reference to the water level on the downstream side, h is the height of the water level on the excavation base, D is the embedment depth of the wall, as shown in Fig. 1(a).

The value of \Delta {{\displaystyle h}}_{\mathrm{a}\mathrm{v}} is obtained from the solution of two-dimensional Laplace’s partial differential equation:

(4){{\displaystyle k}}_{y}x{\displaystyle \frac{{\partial }^{\mathrm{2}}h}{\partial y^{\mathrm{2}}}}+{{\displaystyle k}}_{z}x{\displaystyle \frac{{\partial }^{\mathrm{2}}h}{\partial z^{\mathrm{2}}}}=\mathrm{0},

where k_y and k_z are the permeability coefficients of soil, \partial h/\partial y and \partial h/\partial z are the hydraulic gradients in any point within the soil in horizontal (y) and vertical (z) directions, respectively.

However, the experimental and numerical studies demonstrated that the hydraulic gradients arising on the downstream sides of three-dimensional excavation models may be too larger than those obtained from two-dimensional models. As a result, seepage failures usually occur in the corner areas of polygon-shaped excavation pits, where the hydraulic gradients are relatively high [⁴,⁶,⁸,¹⁰,¹¹]. Thus, the evaluation of base stabilities of excavation pits against seepage failure by using Terzaghi’s rectangular-shaped failure plane is questionable.

Some studies on the verification method for three-dimensional cases can be found in Refs. [⁴,⁶,⁸,¹¹]. Davidenkoff [⁴] stated, with regard to seepage failure by heave, the stability of an infinitesimal soil column adjacent to the walls in the corner area is decisive. The height of the infinitesimal soil column is equal to the embedment depth of the wall D. Based on the results of steady-state numerical flow analyses, Aulbach and Ziegler [⁶] introduced a triangular prism, whose height and width are equal to the embedment depth of the wall D and its half D/2, respectively. Koltuk et al. [⁸] stated based on the results of steady-state numerical flow analyses and model tests that the development of quicksand condition on an infinitesimal soil column adjacent to the walls in the corner areas of polygon-shaped excavation pits can be used to assess their base stabilities against seepage failure. Thereby, uncertainties with respect to the shapes of three-dimensional failure bodies are eliminated. The failure concept of Tanaka et al. [¹¹], which is an extension of Terzaghi’s failure criterion, considers additional frictional forces on the sides of various prismatic failure bodies in the corner areas of excavation pits. The height and width of an examined prismatic failure body are varied until a minimum factor of safety is achieved.

Nowadays, various successful numerical techniques have been developed to study the mechanical behavior of engineering materials. Existing techniques for computational fracture can be classified as either discrete or continuum-based [¹²–¹⁹]:

1) Remeshing procedures with extraneous crack path determination, local displacement and strain enrichment, element overlaps, edges repositioning and edge-based fracture with adaptivity, explicit cracking particle method, peridynamics.

2) Fixed-mesh element erosion, smeared band algorithms, viscous-regularized techniques, gradient and non-local continua.

Many applications from structural and civil engineering require the accurate and stable simulation of complex multiphysics processes. Here, it is essential to determine the effect of the variation of input parameters on the model output [²⁰–²⁷]. However, engineering systems mostly contain correlated input parameters, so the variation of one parameter results in variations of other parameters. Thus, it is essential to understand the relations among the uncertain input parameters. Various successful techniques for uncertainty and sensitivity analyses can be found in Refs. [²⁸–³⁰].

By using coupled numerical analyses, Benmebarek et al. [³¹,³²] identified three different shapes of failure zones (boiling, rectangular prism, triangular prism) depending on the friction angle of soil and the friction angle of soil/wall interface as well as the ratio of the dilation angle to the friction angle of soil. Further investigations on seepage failure by using coupled numerical analyses can be found in Refs. [³³–³⁵].

Although many experimental and numerical studies on the assessment of the base stabilities of excavation pits in homogeneous cohesionless soils against seepage failure by heave exist in the literature, a relatively small number of studies treated this problem in stratified cohesionless soils [²,³,³⁶,³⁷].

With respect to the failure mechanisms in stratified cohesionless soils in which a horizontal stratification exists between the excavation base and the wall tip, it should be distinguished between two basic cases: upper soil layer is more permeable than lower layer k_upper/k_lower>1; lower layer is more permeable than upper layer k_upper/k_lower<1. The reason for this differentiation lies in the fact that the most of hydraulic head loss in stratified soils takes place within the relatively low permeable soil layer. Accordingly, the failure mechanism in stratified soils with k_upper/k_lower<1 does not differ basically from that in the homogeneous case. The only difference is that the height of the failure plane does not reach up to the wall tip, but up to the lower edge of the upper soil layer, as shown in Fig. 1(b). The factor of safety FS is also calculated by using Eq. (3).

To verify the safety against seepage failure by heave in the case of k_upper/k_lower>1, the approach introduced by Terzaghi and Peck [²] for homogeneous cohesionless soils loaded with a surcharge filter is used. This is a special form of the case k_upper/k_lower>1, in which the soil stratification exists only on the downstream side. The height of the heave zone suggested for this case is equal to the embedment depth of the wall D while its width is equal to the half of the wall length embedded in the lower soil layer d/2, as shown in Fig. 1(c) [³⁸].

For the verification of seepage failure in stratified cohesionless soils with k_upper/k_lower>1, the average pore water pressure at the bottom of the heave zone is equated to the total normal stress at the same level. The factor of safety FS is then calculated by using Eq. (7):

(5)(\Delta {{\displaystyle h}}_{\mathrm{a}\mathrm{v}}+h+D+d)x{{\displaystyle \gamma }}_{\mathrm{w}}=hx{{\displaystyle \gamma }}_{\mathrm{w}}+Dx{{\displaystyle \gamma }}_{\mathrm{s}a\mathrm{t},\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}+dx{{\displaystyle \gamma }}_{\mathrm{s}\mathrm{a}\mathrm{t},\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}\mathrm{.}

Substituting ({{\displaystyle \gamma }}_{\mathrm{s}\mathrm{a}\mathrm{t}}={{\displaystyle \gamma }}_{\mathrm{w}}+{\gamma }^{\prime }) in Eq. (5) gives

(6)Dx{{\displaystyle {\gamma }^{\prime }}}_{\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}+dx{{\displaystyle {\gamma }^{\prime }}}_{\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}=\Delta {{\displaystyle h}}_{\mathrm{a}\mathrm{v}}x{{\displaystyle \gamma }}_{\mathrm{w}},

(7)FS=(Dx{{\displaystyle {\gamma }^{\prime }}}_{\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}+dx{{\displaystyle {\gamma }^{\prime }}}_{\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}})/\Delta {{\displaystyle h}}_{\mathrm{a}\mathrm{v}}x{{\displaystyle \gamma }}_{\mathrm{w}},

where {{\displaystyle \gamma }}_{sat,upper} and {{\displaystyle \gamma }}_{\mathrm{s}\mathrm{a}\mathrm{t},\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}} are the saturated unit weights of the upper and lower soil layers, {{\displaystyle {\gamma }^{\prime }}}_{\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}} and {{\displaystyle {\gamma }^{\prime }}}_{\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}} are the submerged unit weights of the upper and lower soil layers respectively, g_w is the unit weight of the water, Dh_av is the average hydraulic head at the bottom of the heave zone with reference to the water level on the downstream side, h is the height of the water level on the excavation base, D and d are the lengths of the wall embedded in the upper and lower soil layers respectively (see Fig. 1(c)).

According to the heave zone shown in Fig. 1(c), the width of the heave zone decreases with the increasing thickness of the upper soil layer. As a result, the location of the average hydraulic head to be considered in the verification gets close to the wall. This means that a relatively high average hydraulic head is considered with the increasing thickness of the upper soil layer. However, it is well known from technical practice that an increase of the thickness of the upper layer affects seepage failure by heave favorably because the pore water pressure developing at the bottom level of the heave zone decreases with the increasing thickness of the upper soil layer. Thus, the heave zone shown in Fig. 1(c) seems to be inconsistent with the heave zone suggested for homogeneous case shown in Fig. 1(a), at least in cases in which the thickness of the upper soil layer D is relatively large in comparison to the length of the wall embedded in the lower soil layer d.

The present paper treats the seepage failure by heave in stratified cohesionless soils in which a relatively permeable soil layer lies above a less permeable layer between excavation base and wall tip. For this purpose, two-dimensional model tests are conducted and their results are compared with those obtained from Terzaghi and Peck’s approach and two-dimensional finite element analyses. Furthermore, by using axisymmetric excavation models with various dimensions and ground conditions, a comparison between the results obtained from Terzaghi and Peck’s approach and finite element analyses is given.

Two-dimensional model tests

Test apparatus and test procedure

To investigate the failure mechanism in two-dimensional seepage flows around a wall embedded in stratified cohesionless soils, a test apparatus was designed taking advantage of symmetry. The test apparatus was made of acrylic glass, and its dimensions were: length × width × height= 530 mm × 200 mm × 680 mm. A partition panel with a thickness of 27 mm and a height of 500 mm was used to model the excavation wall (see Fig. 2). De-aerated water was poured into the test box from the bottom inlet at a slow rate. A filter layer with a thickness of 30 mm and a permeability coefficient of 2.3×10⁻² m/s was placed on the bottom of the test box to ensure a uniform rising of the water during the saturation phase of test soils. For the separation, a nonwoven fabric with a thickness of 5 mm was placed between the filter layer and the lower soil layer.

The pore water pressures developing directly along the embedment length of the partition panel were measured with the help of six standpipes with an inner diameter of 5 mm. Starting from the wall tip, three standpipes (S-4, S-5, S-6) were installed on the downstream side and three standpipes (S-1, S-2, S-3) were installed on the upstream side with a vertical spacing of 25 mm at three levels. The ends integrating into the test box of the standpipes were protected against blockage with the help of wire meshes with a spacing of 0.063 mm. The deformations of the soil surface on the up- and down-stream sides were determined visually with the help of six measuring scales that were placed on both sides of the partition panel.

At the beginning of the test, the water levels on both sides of the partition panel were equal. During the test, the water level at the left side of the test apparatus was lowered stepwise by 10 mm while the water level at the right side was kept constant through a continuous water supply and overflow. After the potential difference between up- and down-stream sides DH reached the value required for the development of seepage failure by heave according to Terzaghi and Peck’s approach, it was lowered stepwise by 5 mm. After each lowering of the water level, any change in the test soils was carefully observed. In case of appearance of first deformations on the soil surface or a remarkable change of water heights in the standpipes, the lowering of the water level was stopped and it was waited as long as, until the system reached a new equilibrium of forces and entered into a new stable state. Accordingly, each test took 2–3 h.

Test materials and their preparation

The natural quartz sands and gravelly sands with a specific gravity of G_s = 2.65 were used in the performed model tests. The grain-size distribution curves of the test soils are given in Fig. 3. In Table 1, uniformity coefficients (C_u), curvature coefficients (C_c), median particle sizes (D₅₀), maximum and minimum void ratios (e_max and e_min), permeability coefficients (k), friction angles (j) and dilation angles (y) of the test soils are listed. The void ratios of the test soils placed in the test box (e) and the corresponding relative densities (D_r) are also given in Table 1. In the conducted model tests, the upper soil layer was varied as Soils No. 2, 3, 4, and 5 while Soil No. 1 with g_sat = 17.9 kN/m³, D_r = 30% and a constant thickness of d = 2.5 cm was used as the lower layer.

The test soils were put into the test box with the aim of a very homogeneous and reproducible relative density. To achieve a loose or medium density, the dry soil was placed in the test box by careful tilting of a hand shovel. On the other hand, a denser test soil was achieved by using raining technique. In this technique, the dry soil was pluviated through a raining device that was moved to and fro to spread the soil uniformly. The raining device consisted of a hopper and a 500 mm long pipe with an inverted cone welded at its bottom. The test soil passed through the pipe with 30 mm internal diameter and dispersed at bottom by the 60° inverted cone. The drop-height of the test soils was held constant with the help of a plumb line. After placement of the test soils into the test box, a horizontal soil surface was carefully produced with the help of a spatula. Subsequently, the test soil was saturated with de-aerated water drop by drop. After the water level in the test box reached a level slightly above the soil surface, the test soil was flowed through with de-aerated water under very low potential differences to remove air bubbles in the soil. Accordingly, the saturation phase of the test soils took approximately 5–7 h.

Before the test soil was placed in the test box, its mass was weighed to determine its dry unit weight g_d. The void ratio of the test soil e was then calculated by using Eq. (8):

(8)e=[({{\displaystyle G}}_{\mathrm{s}}x{{\displaystyle \gamma }}_{\mathrm{w}})/{{\displaystyle \gamma }}_{\mathrm{d}}]-\mathrm{1},

where g_w is the unit weight of water and equal to 9.81 kN/m³.

Assuming that the test soil placed in the test box was fully saturated after the saturation phase, its saturated unit weight g_sat was calculated by using Eq. (9):

(9){{\displaystyle \gamma }}_{\mathrm{s}\mathrm{a}\mathrm{t}}={\displaystyle \frac{[({{\displaystyle G}}_{\mathrm{s}}+e)x{{\displaystyle \gamma }}_{\mathrm{w}}]}{\mathrm{1}+e}}.

(9)

Comparison of the results of the model tests and Terzaghi and Peck’s approach

The test configurations and their results are listed in Table 2. Here, DH_{collapse (exp)} represents the potential difference between up- and down-stream sides that led to total collapse in the performed model tests while DH_{Terzaghi&Peck} represents the critical potential difference calculated according to the approach of Terzaghi and Peck. To confirm the reproducibility, each test was conducted twice. When a difference between the results of the same tests existed, the results of the second tests are given in parentheses in Table 2.

To determine the value of ΔH_{Terzaghi&Peck}, the average hydraulic head Δh_av,failure at the bottom of heave zone suggested by Terzaghi and Peck was first calculated by using Eq. (10), which corresponds to Eq. (7) with FS = 1:

(10)\Delta {{\displaystyle h}}_{\mathrm{a}\mathrm{v},\mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{u}\mathrm{r}\mathrm{e}}=[Dx({{\displaystyle \gamma }}_{\mathrm{s}\mathrm{a}\mathrm{t},\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}-{{\displaystyle \gamma }}_{\mathrm{w}})+dx({{\displaystyle \gamma }}_{\mathrm{s}\mathrm{a}\mathrm{t},\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}-{{\displaystyle \gamma }}_w)]/{{\displaystyle \gamma }}_{\mathrm{w}}\mathrm{.}

Subsequently, ΔH_{Terzaghi&Peck} required for the development of Δh_av,failure was determined by means of steady-state groundwater flow analysis. For example, the value of Δh_av,failure for Test No. 10 was calculated as 7.56 cm by using Eq. (10) and the physical properties of the test soils (Soils No. 1 and 4) given in Table 2. It was ascertained from the performed steady-state numerical analysis that a potential difference of ΔH = 1 cm between the up- and downstream sides of the test box led to an average hydraulic head of Δh_av = 0.33 cm at the bottom of the failure zone shown in Fig. 4 with reference to the water level on the downstream side. Then, the critical potential difference required for the development of seepage failure by heave was calculated as ΔH_{Terzaghi&Peck} = (7.56 cm × 1 cm)/0.33 cm= 22.9 cm.

In all conducted tests, the values of ΔH_{collapse (exp)} were higher than ΔH_{Terzaghi&Peck}. On the other hand, in respect of failure behavior, the same observations were made in all tests. The various stages of seepage failure by heave observed in Test No. 10 are shown in Fig. 5. The potential difference that led to a theoretical failure according to Terzaghi and Peck’s approach (ΔH_{Terzaghi&Peck} = 22.9 cm) caused heaves of about 2 mm on the downstream side (see Fig. 5(b)). Further increasing of the potential difference ΔH led to further heaves on the downstream side. However, when the lowering of the water level was stopped, soil reached a new equilibrium of forces and entered into a new stable state. Settlements on the upstream side were appeared shortly before the total collapse occurred (see Fig. 5(c)). The seepage failure by heave (total collapse) took place as a result of progressive deformations that developed under a constant potential difference of DH_{collapse (exp)} = 40 cm (see Fig. 5(d)).

As can be seen in Table 2, the ratios of DH_{collapse (exp)}/DH_{Terzaghi&Peck} in Tests No. 9, 10, and 12 were relatively high. The main problem in the evaluation of these tests was that Soils No. 4 and 5 were unstable against suffusion, so their permeability coefficients changed with increasing potential difference in the test box. Additionally, with increasing potential difference, contact erosion appeared on the downstream side. However, the steady-state flow analyses for these excavation models were carried out by using the constant permeability coefficients of the test soils that were obtained from one-dimensional upward seepage tests under a hydraulic gradient of 1. This induced a strong difference (up to 30%) between the measured and theoretically determined hydraulic gradients. Thus, the values of DH_{Terzaghi&Peck} given in Table 2 for Tests No. 9, 10, 11, and 12 could not be accurately determined. Here, it should be noted that the maximum deviations of the measured hydraulic gradients from the theoretically determined gradients in Tests No. 1 to 8 were smaller than 10% up to the potential differences that led to seepage failure by heave according to Terzaghi and Peck’s approach.

Simulation of the model tests by means of finite element analyses

The model tests presented above were simulated by means of numerical analyses that were performed using finite element (FE) software PLAXIS 2D- Version 2017.

As a result of seepage flows in an excavation pit, the upstream side of the excavation pit is loaded while its downstream side is unloaded. Accordingly, the soils in the following analyses were modeled by using the Hardening Soil Model, which considers a stress-dependent and nonlinear stiffness that is different for loading, unloading and reloading. The limit state in this model is defined by means of the friction angle, the cohesion and the dilation angle of the soil. This soil model requires six main input parameters: cohesion c, internal friction angle j, dilation angle y, reference secant stiffness {{\displaystyle E}}_{\mathrm{50}}^{\mathrm{r}\mathrm{e}\mathrm{f}} reference unloading/reloading stiffness {{\displaystyle E}}_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}, reference tangent stiffness {{\displaystyle E}}_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}} [³⁹].

In the conducted analyses, the friction and dilation angles of the cohesionless test soils (c = 0) were obtained from the performed direct shear tests and given in Table 1.

Generally, the reference secant stiffness {{\displaystyle E}}_{\mathrm{50}}^{\mathrm{r}\mathrm{e}\mathrm{f}} and the reference unloading/reloading stiffness {{\displaystyle E}}_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}} are obtained from drained triaxial test, and correspond to the stiffness of the soil at 50% of the maximum deviatoric stress and the unloading/reloading stiffness for a cell pressure of 100 kN/m², respectively. On the other hand, the reference tangent stiffness {{\displaystyle E}}_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}} is obtained from oedometer test, and corresponds to the tangent stiffness at a vertical stress of 100 kN/m² [³⁹].

It is well known that soil stiffness has an ignorable effect on the limit load obtained from FE-analyses [⁴⁰].{{\displaystyle E}}_{\mathrm{50}}^{\mathrm{r}\mathrm{e}\mathrm{f}} Then, the values of {{\displaystyle E}}_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}} and {{\displaystyle E}}_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}} were assigned to the upper and lower soil layers according to Eq. (11) given in Ref. [³⁹].

(11){{\displaystyle E}}_{oed}^{\mathrm{r}\mathrm{e}\mathrm{f}}={{\displaystyle E}}_{\mathrm{50}}^{\mathrm{r}\mathrm{e}\mathrm{f}},\mathrm{a}\mathrm{n}\mathrm{d}{{\displaystyle E}}_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=\mathrm{3}x{{\displaystyle E}}_{\mathrm{50}}^{\mathrm{r}\mathrm{e}\mathrm{f}}\mathrm{.}

The dimensions of the numerical model corresponded to the dimensions of the test box. In Fig. 6, the numerical model used for the simulations of the performed model tests with an embedment depth of 7.5 cm is shown. The lateral boundaries were fixed in the horizontal direction while the bottom boundary was fixed in the horizontal and vertical directions. The lateral and bottom boundaries of the model were impermeable. The partition panel was modeled as an impermeable, rigid and fixed plate that was connected to the soil via interface elements. The interface friction angle was taken as \delta =\phi /\mathrm{3}, based on the results of the performed direct shear tests on the test soils and wall material.

The Plaxis 2D program allows for a fully automatic generation of finite element meshes. The generation of the mesh is based on a robust triangulation procedure. The mesh generation process takes into account the soil stratigraphy as well as all structural objects, loads and boundary conditions. The user may select either 15-Node or 6-Node triangular elements as the basic type of element to model soil layers and other volume clusters. It should be noted that a mesh composed of 15-node elements gives a much finer distribution of nodes and thus much more accurate results than a similar mesh composed of an equal number of 6-node elements. Regarding the element distribution, distinction is made between five global levels (very coarse, coarse, medium, fine, very fine) [³⁹].

A sensitivity analysis was performed by using 15-noded plain strain elements to ensure that the used element distribution was sufficiently fine for the accurate determination of the critical potential differences DH_{collapse (num)} that led to the numerical limit state in finite element analyses.

Figure 7 shows the effect of mesh density on DH_{collapse (num)}. Up to fine mesh density, the critical potential difference decreases with the increasing number of element, so a very fine mesh has a negligible effect on the critical potential difference in comparison with fine mesh. On the other hand, the effect of mesh density on the shape of failure zone can be ignored (see Fig. 8).

Figure 8(f) shows the effect of a local refinement with 4058 elements near the wall where the flow gradients are concentrated on the failure zone. With respect to DH_{collapse (num)}, the local refinement gives about 1.5% smaller value than that obtained from very fine mesh. Taking into account a long calculation time appearing in the local refinement, the element distribution in the following analyses was chosen as very fine.

At the initial condition, the water levels on the up- and downstream sides were located on the top of the partition panel. The initial stress field was established by using Eq. (12):

(12){{\displaystyle \sigma }}_{\mathrm{h}}=(\mathrm{1}-\sin \phi )x{{\displaystyle \sigma }}_{\mathrm{v}},

where {{\displaystyle \sigma }}_{\mathrm{v}} and {{\displaystyle \sigma }}_{\mathrm{h}} are vertical and horizontal effective stresses.

In the next state, the water level on the upstream side was kept constant while the water level on the downstream side was lowered stepwise by 0.1 cm. The distribution of the pore water pressure in the flow zone was determined for each lowering of the water level by means of a steady-state numerical analysis, and subsequently, it was used in the following stress-strain analysis. This process was continued until a numerical limit state was achieved.

Table 3 shows the critical potential differences DH_{collapse (num)} that led to a numerical limit state in the performed finite element analyses. In addition, the ratios of DH_{collapse (exp)}/DH_{collapse (num)} and DH_{collapse (exp)}/DH_{Terzaghi&Peck} are given for comparison.

In all tests, the critical potential differences obtained from the model tests were higher than those obtained from the numerical analyses and Terzghi and Peck’s approach (DH_{collapse (exp)}/DH_{collapse (num)}>1 and DH_{collapse (exp)}/DH_{Terzaghi&Peck}>1). Compared to the approach of Terzaghi and Peck, the numerical analyses showed a better agreement with the results of the model tests. The difference between the results of the numerical analyses and the model tests was especially pronounced in Tests No. 9, 10, and 12 in which the upper soil layers were unstable against suffusion. A reason for this considerable difference was that the permeability coefficients of Soils No. 4 and 5 changed with increasing potential difference, which could not be considered in the numerical analyses. Another reason was the relatively small friction angles of test soils used in the numerical analyses. It is well known that the friction angles of cohesionless soils decrease with increasing normal stress level [⁴¹]. In the constructed test box, the effective normal stress at the level of the wall tip was between 0.45 and 0.85 kN/m². However, the friction angles of the test soils were determined by using direct shear tests under a normal stress range of 15 to 30 kN/m². Therefore, it can be said that the friction angles used in the numerical analyses were relatively small, which resulted in the obtaining of the relatively small collapse loads.

Figures 9(a) and 9(b) illustrate the deformed mesh and the corresponding distribution of the incremental displacements for Test No. 10. A significant difference between the shapes of the failure zones obtained from the numerical analysis and Terzaghi and Peck’s approach is clearly visible. In contrast to the rectangular-shaped heave zone suggested by Terzaghi and Peck,, a triangle-shaped heave zone with a larger width was obtained from the numerical analysis. This triangle-shaped heave zone is in good agreement with the failure zone shown in Fig. 5(d).

Further investigations by using axisymmetric excavation models

The conducted model tests represented only a limited number of situations in which the ratio of the thickness of the upper layer to the embedment depth D/(D + d) was varied as 0.5 and 0.67 while the ratio of the permeability coefficient of the upper layer to that of the lower layer k_upper/k_lower was varied as 1.1, 1.3, 2.3, 5.6, 7.8, and 9.3.

For this reason, further FE analyses were carried out by using axisymmetric excavation models taking into account the fact that seepage flows in excavation pits are three-dimensional. In the analyses, a relatively wide and narrow excavation pit model with different ratios of D/(D + d) and k_upper/k_lower were used, and their results were compared with those obtained from Terzaghi and Peck’s approach.

A relatively wide circular-shaped excavation pit

To study seepage failure by heave in a relatively wide circular-shaped excavation pit in stratified cohesionless soils, the axisymmetric numerical model shown in Fig. 10 was used. The dimensions of the numerical model were chosen such that the boundary effect on the numerical results was negligibly small. The lateral boundaries were fixed in the horizontal direction while the bottom boundary was fixed in the horizontal and vertical directions. The lateral and bottom boundaries of the model were impermeable. The wall was modeled as an impermeable, rigid and fixed plate with an ignorable thickness. It was connected to the soil via interface elements with an interface friction angle of \delta =(\mathrm{2}x\phi )/\mathrm{3}. The embedment depth (D + d) was equal to 2 m and the soil model was divided into eight parts with a thickness of 0.5 m along the embedment depth, as illustrated in Fig. 11. The element distribution was chosen as very fine, so it consisted of 4579, 15-noded axisymmetric elements.

The analyses were carried out for eight various thickness of the upper soil layer D = 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5 m and three various ratios of k_upper/k_lower = 2.5, 10, 100. In the analyses, the same friction angle was assigned to the upper and lower soil layers (j_upper = j_lower = 35°). The dilation angles were calculated by using Eq. (13) [³⁹]:

(13)\psi =\phi -\mathrm{30}°\mathrm{.}

The saturated unit weights of the upper and lower soil layers were set to g_sat = 20 kN/m³. The same value of reference secant stiffness was assigned to the upper and lower soil layers ({{\displaystyle E}}_{\mathrm{50}}^{\mathrm{r}\mathrm{e}\mathrm{f}} =30000Kn/m²). The values of {{\displaystyle E}}_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}} and {{\displaystyle E}}_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}} were taken as {{\displaystyle E}}_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}={{\displaystyle E}}_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}} and {{\displaystyle E}}_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=\mathrm{3}x{{\displaystyle E}}_{\mathrm{50}}^{\mathrm{r}\mathrm{e}\mathrm{f}} [³⁹].

At the initial state, the water levels on both sides of the wall were located at the soil surface. In the next state, the water level on the downstream side was kept constant while the water level on the upstream side was raised stepwise by 0.1 cm. This process was continued until a numerical limit state was achieved.

Figure 12 illustrates the effect of the thickness of the upper soil layer D on the critical potential differences depending on the ratio of k_upper/k_lower. A significant increase of DH_{collapse (num)} with increasing D was obtained from the numerical analyses, so an almost linear relationship existed between DH_{collapse (num)} and D (see Fig. 12(a)). On the other hand, in the use of the approach of Terzaghi and Peck, a considerably favorable effect of D on DH_{Terzaghi&Peck} could not be seen. The maximum increase of DH_{Terzaghi&Peck} was obtained for k_upper/k_lower = 100, so the value of DH_{Terzaghi&Peck} increased only about 9% when the thickness of the upper layer was increased from 0 m to 3.5 m. There was even a decrease of DH_{Terzaghi&Peck} for k_upper/k_lower = 2.5, which was not plausible (see Fig. 12(b)). The linear relationship between DH_{collapse (num)} and D obtained from the numerical analyses was also confirmed by the model tests conducted by Marsland [³], as shown in Fig. 12(c). Because of the different dimensions of the used excavation models, a direct comparison of both results could not be made.

In Fig. 13, the ratios of DH_{collapse (num)}/DH_{Terzaghi&Peck} are given depending on the ratios of D/(D + d) and k_upper/k_lower. As can be seen here, the critical potential differences obtained from the numerical analyses are higher than those obtained from the approach of Terzaghi and Peck (DH_{collapse (num)}/DH_{Terzaghi&Peck}>1). The ratio of DH_{collapse (num)}/DH_{Terzaghi&Peck} increases with increasing ratio of D/(D + d) and/or k_upper/k_lower, so it can reach up to 1.50 for k_upper/k_lower = 2.5, 1.75 for k_upper/k_lower = 10, and 1.80 for k_upper/k_lower = 100.

For k_upper/k_lower = 10 and four various ratios of D/(D + d) = 0, 0.125, 0.5, 0.875, a comparison of failure zones obtained from the numerical analyses and the approach of Terzaghi and Peck is given in Fig. 14. With increasing ratio of D/(D+ d), the failure zones that developed in the numerical analyses became larger while the failure zones according to the approach of Terzaghi and Peck became smaller. However, it is well known from technical practice that an increase of the thickness of the relatively high permeable, upper soil layer leads to an increase of critical potential difference. This requires an increase of the width of failure body because the average pore water pressure at the bottom of the heave zone decreases with the increasing thickness of the upper layer whereas the total normal stress at the same level remains nearly constant.

In the analyses presented above, the friction angles of the upper and lower soil layers were equal to j = 35°. To examine the effect of the soil friction angle on the value of DH_{collapse (num)}, further numerical analyses were performed for four various values of j = 25°, 30°, 40°, and 45°. The dilation angle was also calculated by using Eq. (13). In the analyses, two various ratios of D/(D + d) = 0.125, 0.875 and k_upper/k_lower = 2.5, 10 were used. As can be seen in Fig. 15, an almost linear relationship exists between j and DH_{collapse (num)} so the value of DH_{collapse (num)} increases with increasing j. For D/(D + d) = 0.125, the effect of the friction angle on the increase of DH_{collapse (num)} can be ignored. For D/(D + d) = 0.875, however, the value of DH_{collapse (num)} increases about 20% for k_upper/k_lower = 2.5 and about 40% for k_upper/k_lower = 10 when the friction angle of the soil layers is increased from j = 25° to 45°. Thereby, the difference between the results of the model tests and numerical analyses can be explained to a certain extent.

Further numerical analyses were carried out to examine the effect of the ratio of the friction angle of the upper soil layer to that of the lower layer j_upper /j_lower on the value of DH_{collapse (num)}. In the analyses, two various ratios of D/(D + d) = 0.125, 0.875 and k_upper/k_lower = 2.5, 10 were used. The friction angle of the lower soil layer was taken as j_lower = 25° while the friction angle of the upper layer was varied as j_upper = 25°, 30°, 35°, 40°, 45°. The analyses showed that the friction angle of the upper soil layer has an ignorable effect on DH_{collapse (num)}. The maximum increase of DH_{collapse (num)} was obtained for D/(D + d) = 0.875 and k_upper/k_lower = 2.5, so the value of DH_{collapse (num)} increased about 9% when the ratio of j_upper/j_lower was increased from 1 to 1.8 (see Fig. 16). Obviously, the friction angle of the lower soil layer plays a relatively significant role in the value of DH_{collapse (num)}. This is due to the fact that a seepage failure starts from the wall tip where the maximum pore water pressure arises. Therefore, the friction angle of the soil adjacent to the wall tip is decisive for the numerical limit state.

A relatively narrow circular-shaped excavation pit

The same investigations were also conducted for a relatively narrow circular-shaped excavation pit, which is illustrated in Fig. 17. The modeling procedure was same like in the relatively wide excavation pit. The only difference was that the relatively narrow excavation model was divided in four parts along the embedment depth. The analyses were carried out for four various ratios of D/(D + d) = 0, 0.125, 0.5, 0.875 and three various ratios of k_upper/k_lower = 2.5, 10, 100. In the analyses, the same friction angle was used for the upper and lower soil layers (j_upper = j_lower = 35°), and the dilation angle was then calculated by using Eq. (13). The element distribution was chosen as very fine, so it consisted of 2058, 15-noded axisymmetric elements.

A comparison of the results of the numerical analyses with those obtained from Terzaghi and Peck’s approach is shown in Fig. 18 depending on the ratios of D/(D + d) and k_upper/k_lower. The critical potential differences obtained from the numerical analyses are higher than those obtained from the approach of Terzaghi and Peck (DH_{collapse (num)}/DH_{Terzaghi&Peck}>1). The value of DH_{collapse (num)}/DH_{Terzaghi&Peck} can reach up to 1.40 for k_upper/k_lower = 2.5, 1.60 for k_upper/k_lower = 10, and 1.95 for k_upper/k_lower = 100 depending on the ratio of D/(D + d). In contrast to the relatively wide excavation pit, the relationship between DH_{collapse (num)}/DH_{Terzaghi&Peck} and D/(D + d) is not linear. There was even a decrease of DH_{collapse (num)}/DH_{Terzaghi&Peck} for k_upper/k_lower = 2.5 and D/(D + d) = 0.875.

Figure 19 illustrates a comparison of the failure zones obtained from the numerical analyses for k_upper/k_lower = 10 with those according to the approach of Terzaghi and Peck. As expected, all the failure zones obtained from the numerical analyses occurred over the whole excavation width. In the approach of Terzaghi and Peck, however, the failure zone for D/(D + d) = 0.875 was in the form of a narrow rectangular-shaped plane, although the pore water pressure that developed on the downstream side was almost uniform.

The FE analyses showed a negligibly small effect of the friction angle of the lower soil layer on the value of DH_{collapse (num)}. This can also be explained by the almost uniform pore water pressure that developed on the downstream side. As a result, the whole soil body in front of the wall was lifted by the pore water pressure, so no failure surface occurred (see Fig. 20). However, due to the side friction \delta =(\mathrm{2}x\phi )/\mathrm{3}, a considerable effect of j on the value of DH_{collapse (num)} was expected, which was not confirmed by the results of the FE analyses.

Conclusions

In this study, experimental and numerical investigations are performed to clarify the seepage failure by heave in sheeted excavation pits in stratified cohesionless soils in which a relatively permeable soil layer (k_upper) lies above a less permeable soil layer (k_lower) between excavation base and wall tip. The following results can be drawn from the performed investigations:

1) The use of the approach of Terzaghi and Peck for the assessment of base stabilities of excavation pits in cohesionless stratified soils against seepage failure by heave leads to considerably lower critical potential differences than those obtained from the model tests. Potential differences leading to seepage failures according to Terzaghi and Peck’s approach correspond to small heaves on the downstream side, and cause no settlements on the upstream side. On the other hand, a relatively good agreement is achieved between the results of the finite element analyses and model tests.

2) In general, the ratio of the critical potential difference obtained from the numerical analysis DH_{collapse (num)} to that obtained from Terzaghi and Peck’s approach DH_{Terzaghi&Peck} increases with the increasing ratio of the thickness of the upper soil layer to the embedment depth, so the ratio of DH_{collapse (num)}/DH_{Terzaghi&Peck} can reach up to 1.50 for k_upper/k_lower = 2.5, 1.75 for k_upper/k_lower = 10, and 1.95 for k_upper/k_lower = 100. It is obvious that the soil deformations occurring before the development of the total collapse (numerical limit state) are decisive to determine the critical potential difference. However, in practice, the deformations in excavation pits result not only from seepage flows but also from excavation stages, wall deformations etc. Thus, all these factors should be considered to determine the critical potential difference by using finite element analyses.

3) In stratified soils, the ratio of the permeability coefficients of the soil layers to each other has a great effect on the pore water pressure developing in the flow zone. However, the permeability coefficients determined in the laboratory may not always be representative because it is difficult to obtain undisturbed samples from cohesionless soils. On the other hand, it is not faultless to estimate the relative densities of cohesionless soils from field penetration tests. In gap-graded soils, the determination of maximum and minimum densities is also difficult due to segregation appearing during sample preparation. Furthermore, the transport and redistribution of fine grains take place under relatively small hydraulic gradients, which results in the change of the permeability coefficient of soils in course of time. For these reasons, the assessment of base stabilities of excavation pits in stratified cohesionless soils against seepage failure requires special consideration.

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