Analytical Solutions for Three-Dimensional Stability of Coastal Slope

Abstract

The theory of limit analysis is presented for three-dimensional stability of coastal slope. In the frictional soils, the failure surface has the shape of logarithm helicoids, with outline defined by log-spirals. Dissipation rate and gravity power are obtained. By solving the energy balance equation, the expression of stability factor for coastal slope is obtained. The influences of the ratio of width and height and slope angle on the stability are evaluated. Numerical results are presented in the form of graphs. Some examples illustrate the practical use of the results.

1 Introduction

Coastal slope failures are major transport process from the upper slope to the ocean (Hutton and Syvitski 2004; Grilli et al. 2009; L’Heureux et al. 2010). Coastal slope failure can take different forms such as translational or rotational slides (Locat 2001). The different failure forms likely represent the different geotechnical and properties of the failed material, (Locat and Lee 2002; Tinti and Bortolucci 2000; Walters et al. 2006). In the absence of direct observations, scientists have made assumptions about failure institutions. The most common assumption is that a failure process is a cascade or an avalanche process (Densmore et al. 1998; Guzzetti et al. 2002; Malamud et al. 2004; McAdoo and Watts 2004; Lee and Stow 2007).

The stability of coastal slope has been given little attention in the past, even though coastal landslides are common. All coastal slope failures are three-dimensional 3D in practical, but two-dimensional 2D model is usually adopted with very simple. However, when the dimensions of coastal slope are clearly limited by adjacent rock formations or existing structures, a three-dimensional analysis of safety may be more appropriate (Ten Brink et al. 2009; Krastel et al. 2001; Locat et al. 2010).

The upper bound solutions for the failure of coastal slope are presented. The soil is treated as a plastic material, satisfying the coulomb’s yield criterion. The failure surface has the shape of logarithm helicoids, with outline defined by log-spirals. Examples are provided to illustrate the stability factor influenced by the angle of internal friction, the unit weight of the soil, distance from sea level to slope top and slope width.

2 Theoretical Formulation

The application of limit analysis to problems involving stability of earth slopes, first performed by Drucker and Prager (1952), involves determining a lower bound on the collapse load by assuming a stress field which satisfies equilibrium and does not violate the yield criterion at any point. An upper bound is obtained by a velocity field compatible with the flow rule in which the rate of work of the external forces equals or exceeds the rate of internal energy dissipation.

The problem considered here is the safety factor of coastal slope with angle β as shown in Figs. 1 and 2. The upper-bound theorem of limit analysis states that the coastal slope will collapse, for any assumed failure mechanism, the rate of work done by the soil weight exceeds the internal rate of dissipation. Equating external and internal energies for any such mechanism thus gives an upper bound on the safety factor.

Fig. 1

Schematic diagram for coastal slope

Fig. 2

Three-dimensional rotational mechanism

A rotational discontinuity mechanism is shown in Fig. 3, in which the failure surface is assumed to pass through the top and the toe of the coastal slope. Similar shape of this mechanism is considered by Michalowski and Drescher (2009) for evaluating critical height of slope. Soil over the failure surface rotates about the center of rotation O (as yet undefined), while the materials below the failure surface static. Failure surface AC is the velocity discontinuous surface. Assumed mechanism can be specified completely by three variables. For the sake of convenience, we select the angles θ₀, θ_h and $r^{\prime}_{0} /r_{0}$. Equation for the logarithmic spiral surface is given by:

Fig. 3

Schematic diagram of the 3D mechanism

$$ r = r_{0} \,e^{{(\theta - \theta_{0} )\tan \varphi }} ,\;r^{\prime} = r^{\prime}_{0}\, e^{{ - (\theta - \theta_{0} )\tan \varphi }} $$

(1)

The rate of work of the external forces (weight) is:

$$ W_{\gamma } = 2\omega \gamma \left[\int_{{\theta_{0} }}^{{\theta_{B} }} {\int_{0}^{{x_{1}^{*} }} {\int_{a}^{y*} {(r_{m} + y)^{2} } } } \cos \theta \;dydxd\theta + \int_{{\theta_{B} }}^{{\theta_{h} }} {\int_{0}^{{x_{2}^{*} }} {\int_{d}^{y*} {(r_{m} + y)^{2} } } } \cos \theta \;dydxd\theta \right] $$

(2)

where γ is unit weight of the soil, ω is angular velocity of the region ABC.

The total rate of external work due to the seawater pressure is.

$$ W_{\text{w}} = - 2\omega \gamma_{\text{w}} \int_{{\theta_{\text{w}} }}^{{\theta_{h} }} {r_{{x_{2} }}^{2} h^{\prime\prime}\sqrt {R^{2} - d^{2} } \cot (\theta + \beta )} d\theta $$

(3)

Internal dissipation of energy can be more specifically written as sum of integrals over top and surface of the coastal slope

$$ D = D_{AB} + D_{BC} $$

(4)

where the functions D_AB, D_BC are defined as:

$$ \frac{{D_{AB} }}{c\cot \varphi } = - 2\omega r_{0}^{2} \sin^{2} \theta_{0} \int_{{\theta_{0} }}^{{\theta_{B} }} {\frac{\cos \theta }{{\sin^{3} \theta }}} \sqrt {R^{2} - a^{2} } d\theta $$

(5)

$$ \frac{{D_{BC} }}{c\cot \varphi } = - 2\omega r_{0}^{2} e^{{2(\theta_{h} - \theta_{0} )\tan \varphi }} \sin^{2} (\theta_{h} + \beta )\int_{{\theta_{B} }}^{{\theta_{h} }} {\frac{\cos (\theta + \beta )}{{\sin^{3} (\theta + \beta )}}} \sqrt {R^{2} - d^{2} } d\theta $$

(6)

where functions r_m, R, $x_{1}^{*}$, $x_{2}^{*}$, $y_{{}}^{*}$, a, d, θ_B, $r_{{x_{1} }}$, $r_{{x_{2} }}$, $r_{{x_{\text{B}} }}$, $h^{\prime\prime}$, θ_w are defined as:

$$ r_{m} = \frac{{r + r^{\prime}}}{2}, R = \frac{{r - r^{\prime}}}{2}, x_{1}^{*} = \sqrt {R^{2} - a^{2}}, x_{2}^{*} = \sqrt {R^{2} - d^{2}}, y_{{}}^{*} = \sqrt {R^{2} - x^{2}}, $$

$$ a = \frac{{\sin \theta_{0}}}{\sin \theta}r_{0} - r_{m},\,d = \frac{{\sin (\beta + \theta_{h})}}{\sin (\beta + \theta)}r_{0} \,e^{{(\theta_{h} - \theta_{0})\tan \varphi}} - r_{m}, $$

$$ \theta_{B} = \arctan \frac{{\sin \theta_{0} \sin \beta}}{{\sin (\theta_{h} + \beta)e^{{(\theta_{h} - \theta_{0})\tan \varphi}} - \sin \theta_{0} \cos \beta}}, r_{{x_{1}}} = \frac{{\sin (\beta + \theta_{h})}}{{\sin (\beta + \theta_{\text{w}})}}r_{0}\, e^{{(\theta_{h} - \theta_{0})\tan \varphi}}, $$

$$ r_{{x_{2}}} = \frac{{\sin (\beta + \theta_{h})}}{\sin (\beta + \theta)}r_{0}\, e^{{(\theta_{h} - \theta_{0})\tan \varphi}}, r_{{x_{\text{B}}}} = \frac{{\sin (\beta + \theta_{h})}}{{\sin (\beta + \theta_{\text{B}})}}r_{0}\, e^{{(\theta_{h} - \theta_{0})\tan \varphi}}, $$

$$ h^{\prime\prime} = r_{{x_{2}}} \sin \theta - r_{{x_{1}}} \sin \theta_{\text{w}}, \frac{{r_{{x_{\text{B}}}}}}{H - h} = \frac{{\sin (\theta_{\text{w}} + \beta)}}{{\sin \beta \sin (\theta_{\text{w}} - \theta_{\text{B}})}} $$

The factor of overall safety is calculated as

$$ F_{{}} = \frac{{D + W_{\text{w}} }}{{W_{\gamma } }} $$

(7)

To avoid lengthy computations, these simultaneous equations may be solved by a numerical procedure. The minimum $F_{\min }$ is calculated with independent variable parameters θ₀, $\theta_{h}$, $r^{\prime}_{0} /r_{0}$.

3 Example

Calculations for parameters are performed using the mechanisms in Fig. 3. For three-dimensional mechanisms, independent variable in minimising F is θ₀, θ_h, $r^{\prime}_{0} /r_{0}$. Finding the least value of F requires a numerical procedure. In the procedure for finding the minimum of safety factor F, the independent variables, θ₀, θ_h and $r^{\prime}_{0} /r_{0}$ are changed sequentially by a single computational loop. The computational loop is repeated until the minimum is found.

The first example considers a coastal slope with the following parameters, slope depth 20 m, slope width 200 m, soil bulk density $\gamma = 18\,{\text{kN/m}}^{ 3}$, sea water density $\gamma_{\text{w}} = 10.22\,{\text{kN/m}}^{ 3}$, distance from sea level to slope top 5 m, and slope angle 70°. Figure 4 shows plots of F versus the angle of internal friction. The factor of safety increases as $\varphi$ increases.

Fig. 4

Variation of safety factor F with the angle of internal friction

Figure 5 shows plots of F versus the unit weight of the soil with the following parameters, slope depth 20 m, slope width 200 m, the angle of internal friction $\varphi = 15^{ \circ }$, sea water density $\gamma_{\text{w}} = 10.22\,{\text{kN/m}}^{ 3}$, distance from sea level to slope top 5 m, and slope angle $\beta = 70^\circ$. The factor of safety decreases with increasing soil bulk density

Fig. 5

Variation of safety factor F with the unit weight of the soil

Figure 6 shows plots of F versus distance from sea level to slope top with the following parameters, slope depth 20 m, slope width 200 m, the angle of internal friction $\varphi = 20^{ \circ }$, sea water density $\gamma_{\text{w}} = 10.22\,{\text{kN/m}}^{ 3}$, soil bulk density $\gamma = 18\,{\text{kN/m}}^{ 3}$, and slope angle $\beta = 70^\circ$. The factor of safety decreases with increasing $h_{\text{w}}$.

Fig. 6

Variation of safety factor F with distance from sea level to slope top

Figure 7 shows plots of F versus slope width with the following parameters, slope depth 20 m, soil bulk density $\gamma = 18\,{\text{kN/m}}^{ 3}$, the angle of internal friction $\varphi = 15^{ \circ }$, sea water density $\gamma_{\text{w}} = 10.22\,{\text{kN/m}}^{ 3}$, distance from sea level to slope top 5 m, and slope angle $\beta = 70^\circ$. The chart shows that F is affected by slope width. The factor of safety decreases with increasing slope width.

Fig. 7

Variation of safety factor F with slope width

Figure 8 shows plots of F versus slope angle with the following parameters, slope depth 20 m, soil bulk density $\gamma = 18\,{\text{kN/m}}^{ 3}$, the angle of internal friction $\varphi = 15^{ \circ }$, sea water density $\gamma_{\text{w}} = 10.22\,{\text{kN/m}}^{ 3}$, distance from sea level to slope top 5 m, and slope width 200 m. The factor of safety decreases with increasing slope angle.

Fig. 8

Variation of safety factor F with slope angle

4 Conclusions

The upper bound method for the three-dimensional analysis of coastal slope stability is presented in this paper for frictional/cohesive soil. Formulas for the stability analysis are obtained through theoretical derivation on the basis of limit analysis theory. Rotational mechanisms are presented for coastal slope stability. The failure surface has the shape of logarithm helicoids, with outline defined by log-spirals.

Examples are provided to illustrate variations of the stability factor with the angle of internal friction, the unit weight of the soil, distance from sea level to slope top and slope width. The stability factor increases with decreasing the soil bulk density, distance from sea level to slope top and slope width. The safety factor increases as φ increases.