Seismic stability of earth slopes with tension crack

Abstract

Cracks at the crest of slopes frequently occur during earthquakes. Such cracks result from limited tension strength of the soil. A tension cut-off in Mohr-Coulomb shear strength can represent this limited strength. Presented is an extension of variational analysis of slope stability with a tension crack considering seismicity. Both translational and rotational failure mechanisms are included in a pseudo-static analysis of slope stability. Developed is a closed-form to assess the seismic stability of slopes with zero tensile strength. The results indicate that the presence of the tension crack has significant effects on the seismic stability of slopes, i.e., leading to small value of the yield acceleration. Considering soil tension strength in seismic slope analysis may lead to overestimation on the stability, as much as 50% for vertical slopes. Imposing tension crack results in transit of the critical failure mode to a straight line from a log-spiral, except for flat slopes with small soil cohesion. Under seismic conditions, large cohesion may increase the depth of crack, moving it closer to the slope.

Keywords

slope stability / tension / crack / limit equilibrium / seismic effect

Introduction

Often cracks occur at the crest of slopes proceeded by delayed total collapse. Many field investigations [¹,²] indicate that earthquakes produce tension cracks thus destabilizing apparent stable slopes. One reason for the occurrence of such cracks that tension stresses at the crests are temporary exceeding of the tensile strength. Baker [³] noted that the tension strength cut-off may seriously affect the safety of slopes by shortening the length of the slip surface while allowing destabilizing hydrostatic water pressure to build up in the crack. Spencer [⁴] introduced tension crack into limit equilibrium (LE) method and then, using the stress normal to the slip surface, estimated rationally the depth of the crack. To determine the tension crack from the tensile strength, Baker [³] used the variational LE method, obtaining closed-form solutions for the problem. Utilizing the safety map introduced by Baker and Leshchinsky [⁵], Baker and Leshchinsky [⁶] investigated the spatial distribution of safety factors in a vertical purely cohesive cut considering tensile cracks. Based on the upper bound of limit analysis (LA), Utili [⁷] and Michalowski [⁸] investigated the effects of cracks on slope stability. Their analyses are limited to the static slope problem. Results showing the effects of seismicity are scarce although earthquakes are known to trigger the formation of tension cracks in apparently stable slopes.

Currently, seismic stability analyses of slopes include pseudo-static analysis based on limit equilibrium or limit analysis as well as deformation based analysis such as finite element (FE). FE can provide some information on the distribution of the tensile stress thus enabling rational assessment of tension crack; however, consideration of discontinuity [⁹–¹¹] is needed. The pseudo-static analysis considers the earthquake loadings in LE or LA as a pseudostatic force, equal to the soil weight multiplied by a seismic acceleration coefficient. Although this implementation is a gross approximation, such an approach is widely used for seismic design in practice. Nevertheless, most pseudo-static analyses of the slope stability [¹²–¹⁴] neglect the tension cracks.

The purpose of this paper is to extend the variational analysis of slope stability with tension crack by Baker [³], into seismic conditions using pseudo-static approach. Both rotational and translational failure mechanisms (log spiral and plane) are considered to find the critical results. Presented is a procedure for evaluating the effects of tension crack on the seismic slope stability to obtain the closed-form solutions for zero tensile strength. Investigated are the effects of the tension crack on the seismic stability of slopes.

Formulation

The formulation follows the notation introduced by Baker [³], Leshchinsky and San [¹³]. For clarity of presentation, a brief description of the derivation procedure and the relevant formulae are reproduced here. As Baker [³] presented, tension cut-off is introduced into Mohr-Coulomb (MC) failure criterion, as shown in Fig. 1. The tensile strength (t) is defined as an additional independent variable characterizing the material, complying the MC criterion with a tension cut-off, as

(1a)\tau ={\displaystyle \frac{c+\sigma \tan \phi }{F_{\mathrm{s}}}},

(1b)\sigma \ge -t,

where t = shear stress, s = normal stress, t = tensile strength, c = cohesion, f = internal friction angle, F_s = factor of safety. It should be noted that the limiting tensile strength is independent of the safety factor F_s. Applying MC criterion the geometry of the critical slip surface derived from the extremization procedure, may be either a log spiral (rotational failure) or a planar (translational failure). The details of the variational derivation are presented elsewhere [³,¹⁵,¹⁶]. For convenience of presentation of results, the following non-dimensional notation is introduced (see Fig. 2):

X={\displaystyle \frac{x}{H}},Y={\displaystyle \frac{y}{H}},\overline{Y}={\displaystyle \frac{\overline{y}}{H}},D_{\mathrm{c}}={\displaystyle \frac{d_{\mathrm{c}}}{H}},L_{\mathrm{c}}={\displaystyle \frac{l_{\mathrm{c}}}{H}},

(2)N_{\mathrm{m}}={\displaystyle \frac{c}{\gamma H{F}_{\mathrm{s}}}},{\psi }_{\mathrm{m}}={\displaystyle \frac{\tan (\phi )}{F_{\mathrm{s}}}},S={\displaystyle \frac{\sigma }{\gamma H}},T={\displaystyle \frac{t}{\gamma H}},

where y(x) and \bar{y}(x) represent the equation of the slip surface and the slope surface; d_c and l_c are the depth of the tension crack and its horizontal distance on the crest from the slope; g = unit weight.

Figure 2 illustrates the potential slip surfaces of homogeneous slopes. The variational extremization yields the rotational failure mechanism in Cartesian coordinate system as

(3)\{\begin{array}{l}X=X_{\mathrm{c}}+A{e}^{-{\psi }_{\mathrm{m}}\beta }\sin \beta \\ Y=Y_{\mathrm{c}}-A{e}^{-{\psi }_{\mathrm{m}}\beta }\cos \beta \end{array},

where A is a constant of integration. The translational failure mechanism is

(4)Y=X\tan \theta +C,

where C is an integration constant, and q is the inclination of the planar slip surface as shown in Fig. (2b). Introducing the horizontal seismic effects into Euler’s equation, variational extremization conducted by Leshchinsky and San [¹³] obtained the function of the normal stress over the log-spiral slip surface as:

(5a)S({\psi }_{\mathrm{m}}\ne \mathrm{0})={\displaystyle \frac{A}{\mathrm{1}+\mathrm{9}{\psi }_{\mathrm{m}}^{\mathrm{2}}}}[(\mathrm{1}+\mathrm{3}{\psi }_{\mathrm{m}}{k}_{\mathrm{h}})\cos \beta +(\mathrm{3}{\psi }_{\mathrm{m}}-k_{\mathrm{h}})\sin \beta ]{\mathrm{e}}^{-{\psi }_{\mathrm{m}}\beta }-N_{\mathrm{m}}{\displaystyle \frac{\mathrm{1}-{\mathrm{e}}^{\mathrm{2}{\psi }_{\mathrm{m}}\beta }}{{\psi }_{\mathrm{m}}}}+B{\mathrm{e}}^{\mathrm{2}{\psi }_{\mathrm{m}}\beta },

(5b)S({\psi }_{\mathrm{m}}=\mathrm{0})=A(\cos \beta -k_{\mathrm{h}}\sin \beta )+\mathrm{2}{N}_{\mathrm{m}}\beta +B,

where k_h is coefficient of the horizontal seismic acceleration; B is unknown constant of integration. Furthermore, the normal stress along the translational slip surface can be also obtained from the same procedure by Leshchinsky and San [¹³] as:

(6)S={\displaystyle \frac{k_{\mathrm{h}}+\tan (\theta -{\phi }_{\mathrm{m}})}{\tan (\theta -{\phi }_{\mathrm{m}})\tan {\phi }_{\mathrm{m}}-\mathrm{1}}}X+B\mathrm{.}

Based on the variationally derived equations of the slip surface (Eqs. (3) and (4)) and the normal stress distribution (Eqs. (5) and (6)), a closed-form solution can be developed for evaluation of the seismic stability of a given slope with a selected tensile strength. For a seismic slope in a limiting equilibrium state, given i, f_m, k_h, and T, there are still several unknown parameters for rotational/translational failure mechanism, as shown in Table 1. These parameters can be determined from the following relations:

1) Geometrical boundary conditions

(7a)Y(X=X_{\mathrm{1}})=\mathrm{0},

(7b)Y(X=X_{\mathrm{2}})=\mathrm{1}-D_{\mathrm{c}}\mathrm{.}

Under seismic conditions, the size of the sliding mass grows exponentially approaching infinity. This is unrealistic as the foundation soil is inhomogeneous. That is, physical constraints may limit the extent of the slip surface. In this study, the potential slip surface is limited to toe failure, essentially assuming ‘competent’ foundation, which does not allow for very deep failures. An additional relation can then be imposed as:

(7c)X_{\mathrm{1}}=\mathrm{0.}

Combining the equations of the slip surface (Eq. (3) or (4)) and these geometrical boundary conditions, one can obtain the following:

Rotational mechanism,

(8a)A={\displaystyle \frac{\mathrm{1}-D_{\mathrm{c}}}{{\mathrm{e}}^{-{\psi }_{\mathrm{m}}{\beta }_{\mathrm{1}}}\cos {\beta }_{\mathrm{1}}-{\mathrm{e}}^{-{\psi }_{\mathrm{m}}{\beta }_{\mathrm{2}}}\cos {\beta }_{\mathrm{2}}}},

(8b)X_{\mathrm{c}}=-A{\mathrm{e}}^{-{\psi }_{\mathrm{m}}{\beta }_{\mathrm{1}}}\sin {\beta }_{\mathrm{1}},

(8c)Y_{\mathrm{c}}=A{\mathrm{e}}^{-{\psi }_{\mathrm{m}}{\beta }_{\mathrm{1}}}\cos {\beta }_{\mathrm{1}}\mathrm{.}

Translational mechanism,

(9a)C=\mathrm{0},

(9b)X_{\mathrm{2}}=(\mathrm{1}-D_{\mathrm{c}})\mathrm{c}\mathrm{o}\mathrm{t}\theta \mathrm{.}

2) Stress boundary condition

(10)S_{\mathrm{2}}=S(\beta ={\beta }_{\mathrm{2}})\mathrm{o}\mathrm{r}S(X=X_{\mathrm{2}})\mathrm{.}

Substituting this relation into Eq. (5) or (6) and solving for B:

Rotational mechanism,

(11a)$$\begin{gathered} B({\psi }_{\mathrm{m}}\ne \mathrm{0})=\{S_{\mathrm{2}}-{\displaystyle \frac{A}{\mathrm{1}+\mathrm{9}{\psi }_{\mathrm{m}}^{\mathrm{2}}}}[(\mathrm{1}+\mathrm{3}{\psi }_{\mathrm{m}}{k}_{\mathrm{h}})\cos {\beta }_{\mathrm{2}}+(\mathrm{3}{\psi }_{\mathrm{m}}-k_{\mathrm{h}})\sin {\beta }_{\mathrm{2}}]{\mathrm{e}}^{-{\psi }_m{\beta }_{\mathrm{2}}} \\ +N_{\mathrm{m}}{\displaystyle \frac{\mathrm{1}-e^{\mathrm{2}{\psi }_{\mathrm{m}}{\beta }_{\mathrm{2}}}}{{\psi }_{\mathrm{m}}}}\}{\mathrm{e}}^{-\mathrm{2}{\psi }_{\mathrm{m}}{\beta }_{\mathrm{2}}}, \end{gathered}$$

(11b)B({\psi }_{\mathrm{m}}=\mathrm{0})=S_{\mathrm{2}}-A(\cos {\beta }_{\mathrm{2}}-k_{\mathrm{h}}\sin {\beta }_{\mathrm{2}})-\mathrm{2}{N}_{\mathrm{m}}{\beta }_{\mathrm{2}}\mathrm{.}

Translational mechanism,

(12)B=S_{\mathrm{2}}-{\displaystyle \frac{k_{\mathrm{h}}+\tan (\theta -{\phi }_{\mathrm{m}})}{\tan (\theta -{\phi }_{\mathrm{m}})\tan {\phi }_{\mathrm{m}}-\mathrm{1}}}{X}_{\mathrm{2}}\mathrm{.}

3) Limiting equilibrium equations for the sliding body

For rotational mechanism, the equilibrium equations at a limit state for horizontal forces, vertical forces and moments about the origin of the coordinate system can be written as follows:

(13a)\overline{H}={\displaystyle {\int }_{X_{\mathrm{1}}}^{X_{\mathrm{2}}}\left\{[N_{\mathrm{m}}+S{\psi }_{\mathrm{m}}]-S{Y}^{\prime }-k_{\mathrm{h}}\left(\overline{Y}-Y\right)\right\}\mathrm{d}X}=\mathrm{0},

(13b)\overline{V}={\displaystyle {\int }_{X_{\mathrm{1}}}^{X_{\mathrm{2}}}\left\{[N_{\mathrm{m}}+S{\psi }_{\mathrm{m}}]Y^{\prime }-\left(\overline{Y}-Y-S\right)\right\}\mathrm{d}X}=\mathrm{0},

(13c)\overline{M}={\displaystyle {\int }_{X_{\mathrm{1}}}^{X_{\mathrm{2}}}\left[(N_{\mathrm{m}}+S{\psi }_{\mathrm{m}})(Y-X{Y}^{\prime })-S(Y{Y}^{\prime }+X)+\left(\overline{Y}-Y\right)X+k_{\mathrm{h}}\left(\overline{Y}-Y\right){\displaystyle \frac{\overline{Y}+Y}{\mathrm{2}}}\right]\mathrm{d}X}=\mathrm{0},

From the moment Eq. (13c), the stability number N_m can be expressed as:

(14)N_{\mathrm{m}}=-{\displaystyle \frac{m_{\mathrm{1}}}{m_2}},

where

(15a)$$\begin{gathered} m_{\mathrm{1}}={\displaystyle {\int }_{{\beta }_{\mathrm{1}}}^{{\beta }_{\mathrm{2}}}\left[\left(\overline{Y}-Y\right)(X-X_{\mathrm{c}})\right](\cos \beta -{\psi }_{\mathrm{m}}\sin \beta )A{\mathrm{e}}^{-{\psi }_{\mathrm{m}}\beta }\mathrm{d}\beta } \\ +k_{\mathrm{h}}{\displaystyle {\int }_{{\beta }_{\mathrm{1}}}^{{\beta }_{\mathrm{2}}}\left[\left(\overline{Y}-Y\right){\displaystyle \frac{\left(\overline{Y}+Y-\mathrm{2}{Y}_{\mathrm{c}}\right)}{\mathrm{2}}}\right](\cos \beta -{\psi }_{\mathrm{m}}\sin \beta )A{\mathrm{e}}^{-{\psi }_{\mathrm{m}}\beta }\mathrm{d}\beta }, \end{gathered}$$

(15b)m_{\mathrm{2}}={\displaystyle {\int }_{{\beta }_{\mathrm{1}}}^{{\beta }_{\mathrm{2}}}[(Y-Y_{\mathrm{c}})-(X-X_{\mathrm{c}})Y^{\prime }](\cos \beta -{\psi }_{\mathrm{m}}\sin \beta )A{\mathrm{e}}^{-{\psi }_{\mathrm{m}}\beta }\mathrm{d}\beta }\mathrm{.}

For translational mechanism, only the equations of force equilibrium in the horizontal and vertical direction (Eqs. (13a) and (13b)) are involved. Moment equilibrium is not needed to solve the problem (same as Coulomb’s problem); however, moment equilibrium can be solved as well and thus determine the location of the resultant force over the slip surface. From the vertical forces equilibrium Eq. (13b), the stability number N_m can be expressed as:

(16)N_{\mathrm{m}}=-{\displaystyle \frac{v_{\mathrm{1}}}{v_2}},

where

(17a)v_{\mathrm{1}}={\displaystyle {\int }_{X_{\mathrm{1}}}^{X_{\mathrm{2}}}\left[S({\psi }_{\mathrm{m}}{Y}^{\prime }+\mathrm{1})-\left(\overline{Y}-Y\right)\right]\mathrm{d}X},

(17b)v_{\mathrm{2}}={\displaystyle {\int }_{X_{\mathrm{1}}}^{X_{\mathrm{2}}}{Y}^{\prime }\mathrm{d}X}\mathrm{.}

4) Transversality condition

(18){\left(g-Y^{\prime }{\displaystyle \frac{\partial g}{\partial Y^{\prime }}}\right)|}_{X=X_i}=\mathrm{0.}

The transversality condition for the slope stability problem was introduced by Baker and Garber [¹⁵]. It was used by Baker [³] in static stability problem of slopes with tension crack. This boundary condition is adopted also here. The transversality condition can be written for rotational and translational failure mechanism, respectively, as:

(19a)$$\begin{gathered} [S(\beta )({\psi }_{\mathrm{m}}\cos \beta +\sin \beta )+N_{\mathrm{m}}\cos \beta -\left(\overline{Y}-Y\right)\sin \beta \\ {-k_{\mathrm{h}}\left(\overline{Y}-Y\right)\left(\overline{Y}+Y+\mathrm{2}{Y}_{\mathrm{c}}\right)/\mathrm{2}A{\mathrm{e}}^{-{\psi }_{\mathrm{m}}\beta }]|}_{\beta ={\beta }_i}=\mathrm{0}, \end{gathered}$$

(19b){\left\{N_{\mathrm{m}}+S(X)[{\psi }_{\mathrm{m}}+\tan (\theta -{\phi }_{\mathrm{m}})]-k_{\mathrm{h}}\left(\overline{Y}-Y\right)-\tan (\theta -{\phi }_{\mathrm{m}})\left(\overline{Y}-Y\right)\right\}|}_{X=X_i}=\mathrm{0.}

Applying this relation at point E (see Fig. 2) and finding that at this point \left(\bar{Y}-Y\right)=D_c, one gets:

(20a)S_{\mathrm{2}}=\left[D_{\mathrm{c}}\sin {\beta }_{\mathrm{2}}+k_{\mathrm{h}}{D}_{\mathrm{c}}{\displaystyle \frac{\mathrm{1}+\mathrm{3}{Y}_{\mathrm{c}}-A{\mathrm{e}}^{-{\psi }_{\mathrm{m}}{\beta }_{\mathrm{2}}}\cos {\beta }_{\mathrm{2}}}{\mathrm{2}A{\mathrm{e}}^{-{\psi }_{\mathrm{m}}{\beta }_{\mathrm{2}}}}}-N_{\mathrm{m}}\cos {\beta }_{\mathrm{2}}\right]/({\psi }_{\mathrm{m}}\cos {\beta }_{\mathrm{2}}+\sin {\beta }_{\mathrm{2}}),

(20b)S_{\mathrm{2}}=[k_{\mathrm{h}}{D}_{\mathrm{c}}+\tan (\theta -{\phi }_{\mathrm{m}})D_{\mathrm{c}}-N_{\mathrm{m}}]/[{\psi }_{\mathrm{m}}+\tan (\theta -{\phi }_{\mathrm{m}})]\mathrm{.}

5) Criterion for tension crack

(21)\begin{array}{l}\lim S(X,Y)=-T\\ X\to X_{\mathrm{2}}\\ Y\to \mathrm{1}-D_{\mathrm{c}}\end{array}\mathrm{.}

When the tension crack is occurred, the following condition should exist:

(22)S(\beta ={\beta }_{\mathrm{2}}\mathrm{or}X=X_{\mathrm{2}})=-T\mathrm{.}

Substituting Eq. (20) into Eq. (22), one obtains:

(23)\overline{S}=S_{\mathrm{2}}+T=\mathrm{0.}

Combining these equations for rotational or translational failure mechanism, a computation scheme is obtained for the closed-form solutions when i, f_m, k_h, and T are given (see Table 1). As Baker [³], Leshchinsky and San [¹³] demonstrated, the problem is reduced to solving a set of two or three nonlinear simultaneous equations for translational and rotational mechanism, respectively. To find the corresponding two or three roots (q and D_c; b₁, b₂ and D_c), the steepest descent algorithm numerical technique is adopted here.

Results and discussions

In the calculations, tensile strength T = 0 is specified thus limiting the results to the extreme (and practical) value of zero tensile strength material with MC criterion. When the acceleration coefficient k_h = 0, the problem degenerates to the static problem. First considered are the two failure modes, the rotational mechanism (RM) and translational mechanism (TM), under static conditions, to verify for a given problem, which one is the critical mode; i.e., which one is rendering the largest value of stability number (N_m). Using both failure mechanisms, Michalowski [⁸] included the presence of cracks into the kinematic approach of limit analysis and obtained the least upper bounds on the critical height (gH/c), which is the reciprocal value of stability number when F_s = 1.0. Following the same procedure of Michalowski [⁸], the authors calculated the stability numbers using the rotational and translational mechanisms, and then compared them with the variational results, as shown in Table 2. Minor differences in stability numbers between LA method of Michalowski [⁸] and this study are observed. The reason for this can be attributed to different considerations on the formation of the tension crack. The rate of work dissipation along the opening crack is included in the energy balance equation to determine the least upper bound by Michalowski [⁸]. Generally, the variationally derived results are in good agreement with the upper-bound solutions of Michalowski [⁸]. The critical failure mode in variational analysis or limit analysis is rotational mechanism for i = 60°, but it transits to translational mechanism for vertical slope. It should be noted that the log-spiral mechanism could completely degenerate to the planar mechanism for vertical slopes. However, for such a case the pole of the log spiral should approach infinity, a numerically difficult problem to solve accurately.

The assessment of slope stability is usually conducted in terms of the factor of safety. However, in seismic stability it is useful and meaningful to use the yield (or critical) earthquake acceleration. When a slope is subjected to its yield acceleration (k_y), an imminent failure will occur; i.e., F_s = 1.0. For example, it is an important characteristic for predicting seismic permanent displacements of a slope by using the sliding block analysis [¹⁷]. The calculated results represent the state of yield acceleration, considering the soil properties and the slope geometry. Since the factor of safety F_s = 1.0, N_m and f_m will be expressed as N and f.

Figures 3-7 show the critical results derived from the rotational mechanism (RM) and translational mechanism (TM) for slopes (i = 90°, 75°, 60°, 45°, 30°) with tension cut-off of zero. For reference, shown also the results for slopes without tension crack. The yield acceleration is given as a function of N = c/gH for different internal friction angles f = 10°, 20°, 30°, and 40°. It can be seen that regardless of failure mechanism, ignoring tension crack will result in larger value of k_y, i.e. overestimation of the seismic stability of the slope. The effects become more profound as the value of c/gH increases or as f decreases. For flat slopes with small cohesion and friction angle, the rotational mechanism is always the most critical. For steep slopes (i = 75° and 90°), the most critical failure mode approaches the translational mechanism. Ignoring the effects of tension crack, the rotational mechanism is the most critical in most cases. The results implies that with zero tension cut-off in seismic analysis of steep slope stability, using a planar failure mechanism is acceptable because of its relative criticality and easy use.

Ling et al. [¹⁴] adopted the planar failure mechanism to evaluate the seismic stability of slopes without tension crack. The critical inclination of the wedge sliding surface (q_c) was obtained as:

(24){\theta }_{\mathrm{c}}={\displaystyle \frac{i+\phi +\mathrm{a}\tan (k_{\mathrm{h}})}{\mathrm{2}}}\mathrm{.}

The critical angle is a function of the seismic acceleration k_h. Based on the closed-form solutions for the translational mechanism, Fig. 8 gives the critical angles (q_c) for steep slopes with and without tension crack. Ignoring the tension crack, the derived critical angle is the same as that calculated from Eq. (24). For vertical slopes, the tension crack has no effects on the critical angle. However, for i = 75°, the critical angle for slopes with tension crack is smaller than when neglecting the crack. The critical inclination still has a linear relation with the horizontal seismic acceleration.

The presence of a tension crack is an important indicator of potential instability for slopes. That is, it implies the maximum depth of the potential tension crack (i.e., the depth at which the normal stress over the slip surface reaches from negative at the crest to zero). The obtained depth and location of the crack from the variational analysis could be useful in designing measures to arrest such crest; i.e., use of planar reinforcement such as geogrids. Selecting the critical results of the yield acceleration in Figs. 3–7, the corresponding values on the depth of the cracks (D_c) and its horizontal location on the crest behind the slope (L_c) are presented (see Figs. 9–13). As might be expected, the depth of the cracks increases as the yield acceleration increases. For flat slopes, the increase of the cracks has a sudden change because of the transition in the critical failure mechanism, as shown in Figs. 5–7. The results derived from less critical failure mechanism are also shown in Figs. 11-13, as the dash lines. It can be seen that the values of D_c or L_c are different for the rotational mechanism (circular symbol) and translational mechanism (square symbol). Without the seismic effects (k_y = 0), small depth of crack could be occurred in the flat slopes, as shown in Figs. 12a and 13a. However, once seismic excitation occurs, the depth of the crack becomes larger, potentially exceeding half the height of the slope. For steep slopes, the crack depth could develop to a large depth. Under static conditions, the maximum depth is D_c = 0.33 for vertical slopes, which indicates that the maximum depth of tension crack does not exceed one third of the slope height. However, when stable slopes are subjected to earthquakes, the depth of the crack will significantly increase, potentially reaching 0.65H for vertical slopes (i.e., doubling the depth). In addition, the internal friction angle has significant effects on the values of D_c and L_c. As the friction angle increases, the crack will be closer to the face of the slope. For vertical slopes, the cohesion of soil has small effects on the adverse location of the crack, as shown in Fig. 9b.

To further demonstrate the effects of the tension crack on the seismic stability, a comparison is made for two cases (see Table 3). When the tension crack is considered in seismic stability analysis, the yield acceleration k_y will largely decrease, especially for vertical slopes. The effects of the tension crack become more significant as the slope inclination increases and the friction angle decreases. Typically, tension crack will decrease the yield acceleration by 50% for vertical slopes. It implies that, neglecting the tension cracks may significantly overestimate the seismic stability of steep slopes. Figures 14 and 15 show the critical slip surfaces and the corresponding critical tension crack for the two cases above. Also illustrated is the normal stress distribution along the slip surface. It can be seen that as the internal friction angle of soil increases (while some cohesion exists), the tension crack becomes deeper and its adverse location is closer to the face of the slope. The critical slip surface is affected by the tension crack. For vertical slopes, the tension crack results in shallower critical slip surface.

Conclusions

Variational analysis of slope stability with tension crack was extended to seismic conditions. Both rotational and translational failure mechanisms are included in the pseudo-static approach. However, failures are limited to toe failures essentially assuming competent foundation. Based on the extremized results, a closed-form solution is developed for the seismic stability of slopes with cut-off of tensile strength to zero. Such an approach is done rationally within the framework of variational analysis, as the stress normal to the slip surface is part of the solution. That is, setting a tension cut-off to zero modifies the variational normal stress distribution to acts on a surface that starts at the tip of a tension crack whereas at this tip, the normal stress is zero and it is determined mathematically. Since it is part of a closed-form solution, the location of the critical slip surface as well as the adverse location and depth of the tension cracks are interrelated, all parts of the solution of a set of nonlinear equations. Based on the presented results, the following conclusions may be drawn:

1) The presence of the tension crack significantly decreases the seismic stability of a slope, typically by 50% for vertical slopes. Neglecting the tension crack will largely overestimate the seismic stability of slopes. As the soil cohesion increases or friction angle decreases, the overestimation become more profound.

2) Ignoring the tension crack, the critical failure mode is the rotational mechanism in most cases. Considering the tension crack, the critical failure mode transits to a translational planar mechanism, except for flat slopes with small soil cohesion.

3) As the yield acceleration of slopes increases, the tension crack becomes deeper and its adverse location is closer to the face of the slope. Comparing with the maximum depth of crack in static conditions (H/3 for vertical slopes), the maximum depth under seismic conditions can exceed one-half of the slope height.

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