Kinematic limit analysis of the slope encapsulating a laterally loaded pile

Abstract

This study aims to present a theoretical method for the safety factor of a slope encapsulating a laterally loaded pile. The factor can be derived by the object function concerning the external work rate and internal energy dissipation based on kinematic limit analysis, after assuming the log-spiral failure mechanism of the slope. To address the core issues (the critical point depth and lateral force provided by the pile) in the analysis, the modified strain wedge technique and the soil wedge assumption were adopted to evaluate the soil resistance and extra earth pressure around the pile, respectively. Besides, the proposed method was verified by published data and numerical methods, and the slope failure mechanism was revealed by observing two wedge-shaped failure regions around the pile. Furthermore, the variations of normalized safety factors with normalized lateral loads can be empirically fitted by cubic functions, and the normalized safety factor mainly depends on the lateral load and pile location but is not sensitive to the shear strength. The safety factor of the slope encapsulating a laterally loaded pile (FoS₁) can be thus alternatively predicted by scaling the safety factor of the slope without the lateral load (FoS₀) with the corresponding normalized safety factor η.

Appendices

Appendix 1 Stress level

The relationship between the stress level SL and soil strain ε was observed from isotropically consolidated drained and undrained triaxial tests (Ashour et al. 1998). It can be sketched in Fig. 18 and given as Eqs. (37) to (39).

$$\begin{array}{cc}{SL}_{i}=\frac{3.19\varepsilon }{{\left({\varepsilon }_{50}\right)}_{i}}\ \mathrm{exp}\ \left(-3.707{SL}_{i}\right)& \left(\mathrm{Stage\ I}\quad \varepsilon \le {\varepsilon }_{50}\right)\end{array}$$

(37)

$$\begin{array}{cc}{SL}_{i}=\frac{{\lambda }_{i}\varepsilon}{{\left({\varepsilon}_{50}\right)}_{i}}\ \mathrm{exp}\ \left(-3.707{SL}_{i}\right)& \left(\mathrm{Stage\ II}\quad {\varepsilon }_{50}<\varepsilon \le {\varepsilon }_{80}\right)\end{array}$$

(38)

$$\begin{array}{cc}{SL}_{i}=\mathrm{exp}\left[\mathrm{ln}\ 0.2+\frac{100\varepsilon}{m\varepsilon +{q}_{i}}\right]& \left(\mathrm{Stage\ III}\quad \varepsilon >{\varepsilon }_{80}\right)\end{array}$$

(39)

where ε₅₀ denotes the soil strain at 50% stress level; λ_i varies linearly from 3.19 (SL = 0.5) to 2.14 (SL = 0.8); and m = 59.0 and q_i = 95.4ε₅₀.

Fig. 18

Relationship between stress level and soil strain

Appendix 2 Lateral earth pressure coefficient

On the one hand, the lateral stress at point E₀ (Fig. 10), σ_h, was obtained based on the stress equilibrium equation.

$${\sigma }_{\mathrm{h}}={\sigma}_{1}\ {\mathrm{cos}}^{2}\ {\beta}_{0}+{\sigma}_{3}\ {\mathrm{sin}}^{2}\ {\beta}_{0}$$

(40)

Likewise, the lateral stress at point D (Fig. 10), σ_ah, was also obtained as follows:

$${\sigma }_{\mathrm{ah}}={\sigma}_{1}\ {\mathrm{cos}}^{2}\ \psi +{\sigma}_{3}\ {\mathrm{sin}}^{2}\ \psi$$

(41)

Substituting σ₃/σ₁ = 1/N into Eq. (41), the lateral stress σ_ah could be rearranged as:

$${\sigma }_{\mathrm{ah}}=\left({\mathrm{cos}}^{2}\ \psi +\frac{1}{N}\ {\mathrm{sin}}^{2}\ \psi \right){\sigma }_{1}$$

(42)

where N = tan²(π/4 + φ/2).

On the other hand, the vertical stress applied to the differential element E₀JPQ was evaluated, including two components (Fig. 19): (i) vertical stress applied to the quadrangle differential element E₀JG₀Q; (ii) the minor and major principal stresses on the triangular differential element G₀PQ where σ_v is the vertical stress applied on the differential element, consisting of σ_f and σ′_v; the former is parallel to line E₀G₀, and the latter is perpendicular to line E₀G₀, derived as:

$$\frac{{{\sigma }^{\prime}}_{\mathrm{v}}}{{\sigma }_{\mathrm{v}}}=\mathrm{cos}\ \theta$$

(43)

Fig. 19

Stress status on the quadrangle differential element E₀JG₀Q

Substituting that σ_v + σ_ah = σ₁ + σ₃, Eq. (43) could be given as Eq. (44):

$$\frac{{{\sigma }^{\prime}}_{\mathrm{v}}}{{\sigma}_{1}}=\mathrm{cos}\ \theta \left({\mathrm{sin}}^{2}\ \psi +\frac{1}{N}\ {\mathrm{cos}}^{2}\ \psi \right)$$

(44)

For simplification, σ′_v can be replaced by mean stress $\overline{{\sigma }^{^{\prime}}}$ (Fig. 20) where h₀ = cosθ·dz; σ₃ and σ_3v are the minor principal stress on the line G₀Q and its vertical component.

$${\overline{{\sigma }^{\prime}}}_{\mathrm{v}}=\frac{{V}^{\prime}}{S}$$

(45)

where

$${V}^{^{\prime}}={\int }_{{\theta }_{\omega }}^{\pi /2-\omega }\mathrm{d}{V}^{\prime}={\int }_{{\theta }_{\omega }}^{\pi /2-\omega }{\sigma }_{1}R\ \mathrm{sin}\ \psi \left({\mathrm{sin}}^{2}\psi +\frac{1}{N}\ {\mathrm{cos}}^{2}\ \psi \right)\mathrm{d}\psi$$

(46)

$$S=\frac{\mathrm{cos}\ \left({\theta }_{w}+\xi \right)}{\mathrm{cos}\ \left(\theta +\xi \right)}R$$

(47)

Fig. 20

Stress status on the main part of the differential element

Then, Eq. (45) could be rewritten as follows:

$${\overline{{\sigma }^{\prime}}}_{\mathrm{v}}=\frac{\mathrm{cos}\ \left(\theta +\xi \right)}{\mathrm{cos}\ \left({\theta }_{w}+\xi \right)}\ \mathrm{cos}\ {\theta }_{\omega }\left(1-\frac{N-1}{3N}\ {\mathrm{cos}}^{2}\ {\theta }_{\omega }\right){\sigma }_{1}$$

(48)

The average vertical stress ${\overline{\sigma }}_{\mathrm{v}}$ on the differential element could be given as Eq. (49).

$${\tilde{\sigma}}_{\mathrm{v}}=\frac{{\overline{{\sigma}^{\prime}}}_{\mathrm{v}}}{\mathrm{cos}\ \theta }$$

(49)

Substituting Eq. (48) into Eq. (49), it could be finally obtained as:

$${\overline{\sigma }}_{\mathrm{v}}=\frac{\mathrm{cos}\ \left(\theta +\xi \right)}{\mathrm{cos}\ \left({\theta }_{w}+\xi \right)\ \mathrm{cos}\ \theta}\ \mathrm{cos}\ {\theta }_{\omega }\left(1-\frac{N-1}{3N}\ {\mathrm{cos}}^{2}\ {\theta }_{\omega }\right){\sigma }_{1}$$

(50)

Above all, the lateral earth pressure coefficient K_an is given by Eqs. (42) and (50).

$${K}_{\mathrm{an}}=\frac{{\sigma }_{\mathrm{h}}}{{\tilde{\sigma}}_{\mathrm{v}}}=\frac{\mathrm{cos}\ \left({\theta }_{w}+\xi \right)\mathrm{cos}\ \theta }{\mathrm{cos}\ \left(\theta +\xi \right)\ \mathrm{cos}\ {\theta }_{w}}\cdot \frac{3\left(N\ {\mathrm{cos}}^{2}\ {\theta}_{w}+{\mathrm{sin}}^{2}\ {\theta}_{w}\right)}{3N-\left(N-1\right)\ {\mathrm{cos}}^{2}\ {\theta }_{w}}$$

(51)

Appendix 3 Vertical stress

To further investigate the vertical stress within the soil wedge, the triangular element G₀PQ assumed to be in the equilibrium status can be neglected in the analysis of the differential element E₀JPQ. The minor principal stress on the line G₀Q σ₃ and its vertical component σ_3v are derived as:

$${\sigma }_{3}=\frac{{K}_{\mathrm{an}}}{N\ {\mathrm{cos}}^{2}\ {\theta}_{w}+{\mathrm{sin}}^{2}\ {\theta}_{w}}{\overline{\sigma }}_{\mathrm{v}}$$

(52)

$${\sigma }_{3\mathrm{v}}={\sigma}_{3}\ \mathrm{sin}\ \xi \frac{\mathrm{cos}\ \theta }{\mathrm{cos}\ \left(\xi +\theta \right)}$$

(53)

The shear stress on the differential element E₀JG₀Q (Fig. 10) can be regarded as:

$$\tau ={\sigma }_{\mathrm{h}}\ \mathrm{tan}\ \varphi={\overline{\sigma}}_{\mathrm{v}}\ {K}_{\mathrm{an}}\ \mathrm{tan}\ \varphi$$

(54)

The equilibrium equation of the differential element E₀JMQ in the vertical can be derived as Eq. (55), ignoring the vertical stress on the line MG₀.

$$\begin{aligned}DS\mathrm{d}{\overline{\sigma }}_{\mathrm{v}} & +D{K}_{\mathrm{an}}{\overline{\sigma }}_{\mathrm{v}}\ \mathrm{tan}\ \varphi \mathrm{d}z-D{K}_{\mathrm{an}}{\overline{\sigma}}_{\mathrm{v}}\ \mathrm{tan}\ \theta \mathrm{d}z \\ & +D{\sigma}_{3}\frac{\mathrm{sin}\ \xi\ \mathrm{cos}\ \theta}{\mathrm{cos}\ \left(\xi +\theta \right)}\mathrm{d}z=DS{h}_{0}\gamma\end{aligned}$$

(55)

Substituting Eq. (52) into Eq. (55), ${\overline{\sigma }}_{\mathrm{v}}$ is obtained as:

$$\begin{array}{l}{\overline{\sigma }}_{\mathrm{v}}=\frac{\gamma h\ \mathrm{cos}\ \theta }{1-\left({K}_{\mathrm{an}}\mathrm{tan}\ \varphi -{K}_{\mathrm{an}}\mathrm{tan}\ \theta +{m}_{0}\right)\frac{\mathrm{sin}\ {\beta }_{0}}{\mathrm{cos}\ {\theta }_{0}}}\\\qquad\ \times \left[{\left(1-\frac{z}{h}\right)}^{\left({K}_{\mathrm{an}}\mathrm{tan}\ \varphi -{K}_{\mathrm{an}}\mathrm{tan}\ \theta +{m}_{0}\right)\frac{\mathrm{sin}\ {\beta }_{0}}{\mathrm{cos}\ {\theta }_{0}}}-\left(1-\frac{z}{h}\right)\right]\end{array}$$

(56)

Appendix 4 Procedure of the finite element limit analysis

The common procedures of finite element limit analysis are elaborated below (Fig. 21):

• i) Establish the numerical model according to the geometric configuration of the pile-slope system, considering the boundary effect, while the slope and the pile (free-head and fixed-end) are modeled as the Mohr–Coulomb material (associated flow rule) and the linear elastic material, respectively.

• ii) The boundary condition is selected as the standard boundary condition, that is, the lateral displacement is constrained at the left and right boundaries of the model, and the bottom of the model is fixed; the lateral load atop the pile could be applied as the distributed load (Fig. 21a).

• iii) The total element number and initial element number are 10,000 and 1000, respectively; and the adaptive remeshing technique is adopted in the analysis (Fig. 21b).

Fig. 21

Finite element limit analysis: (a) boundary condition and load condition; (b) adaptive remeshing technique