Stability analysis of vegetated slopes under steady transpiration state considering tensile strength cut-off

Abstract


Introduction
Soil slopes are commonly unsaturated due to natural factors like precipitation and transpiration (Song and Tan 2020).Vegetation plays a significant role in natural slope stability (Stokes et al. 2009;Fan and Lai 2013;Das et al. 2018;Zhu and Zhang 2019).Many natural slopes are covered with vegetation, which can change soil suction and shear strength affect the slope stability (Greenwood et al. 2004;Rees and Ali 2012;Ni et al. 2018;Wu et al. 2022) Slope stability analysis is a classical issues of geotechnical engineering (Botero et al. 2020;Khorsandiardebili and Ghazavi 2021).There are three methods to analyze slope stability: limit equilibrium methods (LEMs) (Hammouri et al. 2008;Zhou and Cheng 2013), limit analysis methods (LAMs) (Li et al. 2021;Khoshzaban et al. 2021), and numerical simulation methods (NSMs) (Dyson and Tolooiyan 2019; Meng et al. 2019).LEMs are based on the static equilibrium condition of the slip body and can be divided into slice and global analysis methods (Sun et al. 2016;Shinoda and Miyata 2019).LAMs include upper and lower bound limit analysis.The upper bound limit analysis method is frequently used for the homogeneous slopes stability analysis due to its simple calculation process and rigorous theoretical framework (Michalowski 1995(Michalowski , 2008(Michalowski , 2016;;Kumar and Samui 2006;Yang and Xu 2017;Yang and Deng 2019).NSMs contain finite element, finite difference, and discrete element methods, and have distinct advantages in selecting constitutive models.Some advanced numerical methods have been developed to solve complex engineering problems (Ashford et al. 2001;Gao et al. 2016;Dai et al. 2021;Liu et al. 2021;Guo and Zhao 2021;An et al. 2022).
Rainfall and transpiration are important factors that influence soil suction or water content (Ng et al. 2017).Compared with rainfall, relatively few studies have considered the effect of vegetation transpiration on slope stability.The strength criterion considering root systems is the basis for analyzing the vegetated slopes stability.At present, the most common method is to add a cohesion term reflecting root reinforcement.The cohesion can be obtained by the Wu and Waldron model (WW model) (Wu 1976;Waldron 1977) or the fiber bundle model (FBM) (Pollen and Simon 2005).
Due to the simplicity and fewer input parameters, the WW model has become the most common model for investigating vegetated slope stability (Ng 2017;Tan et al. 2019;Zhu and Zhang 2019;Su et al. 2021).These studies show that vegetation roots have hydrological and mechanical effects on unsaturated soil slopes (Tosi 2007;Chirico et al. 2013;Cislaghi et al. 2017;Masi et al. 2021).The hydrologic effect is to reduce soil water content and increase soil suction by transpiration (Simon and Collison, 2002;Feng et al. 2020).The mechanical effect is to increase soil strength by root reinforcement (Feng et al. 2020).Both hydrological and mechanical effects are related to the root architecture.Some analytical solutions of slope pore-water pressure considering root architecture have been derived (Yuan and Lu 2005;Ng et al. 2015), which are generally combined with an infinite slope model to investigate the effect of vegetated slope stability (Liu et al. 2016;Feng et al. 2020).
The Fredlund strength equation is used to analyze the unsaturated soil strength and slope stability (Fredlund et al. 1978).Although the soil tensile strength is neglected and affected by some factors (Chen 1975), it is a critical factor affecting unsaturated vegetated slope.Currently, the tensile strength of unsaturated soil is obtained by extending the Fredlund strength envelope.Fredlund strength equation leads to an overestimation of the tensile strength (Li et al. 2020), resulting in a larger safety factor.Based on M-C criterion, the concept of tensile strength cut-off is proposed (Paul 1961), which is widely used in the stability analysis of clay slopes (Michalowski 2013(Michalowski , 2018;;Abd and Utili 2017).Nevertheless, few studies have reported on the effect of vegetated slope stability using the tensile strength cut-off criterion (C-F criterion).
This study propose a new method for analyzing vegetated slopes stability based on root water absorption and C-F criterion.Section 2 introduces the analytical solution of pore-water pressure of vegetated slopes with different root architectures at steady transpiration state.Section 3 discusses the strength criterion of the root-soil composite based on C-F criterion.In Section 4, a stability model for vegetated slopes is established.Parametric studies are performed to investigate the effects of vegetation roots on slope stability.Conclusions are drawn in the end.

Vegetation roots in unsaturated soils
Figure 1 shows the one-dimensional water infiltration into homogeneous unsaturated soil.The contour lines of pore-water pressure head in the slope are parallel to the slope surface.The underground water level is considered as the datum plane.The coordinate transformation relationship in Fig. 1 is outlined below (Wu et al. 2016) where β is the slope angle; x and z are the reference coordinate axes; x * and z * are the transformed coordinates.( ) where k is the permeability coefficient;  is the pressure head; t is the infiltration time; θ is the volumetric water content; L * 1 is the root depth; L * 2 is the thickness outside root zone; (z * -L2 * ) is the Heaviside function (Polyanin 2002): where T represents the total transpiration rate, and g(z * ) refers to the shape function of roots at a depth of z * .Figure 2 displays various root architectures, including uniform (Lynch 1995), triangular (Lynch 1995), exponential (Ghestem et al. 2011), and parabolic (Leung 2015).The uniform root architecture is considered in this study.At steady-state stage, Eq. ( 3) can be simplified as: ( ) The hydraulic conductivity and the volumetric water content of unsaturated soil are expressed in terms of pressure head () (Wu and Zhang 2009), as follows: where ks is the saturated permeability coefficient; θs is the saturated volumetric water content; θr is the residual volumetric water content; α is the desaturation coefficient.
Substituting Eqs. ( 7) and (8) into Eq.( 6) leads to (Wu and Zhang 2009;Ng et al. 2015): Assuming that the groundwater level is fixed, the boundary condition in slope bottom is (Zhu et al. 2021): The boundary condition of the soil surface satisfies the flow boundary condition which can be expressed as: where q0 represents the rainfall intensity (a constant), and L * =L * 1+ L * 2 represents the soil height.
The following variables are defined as: Substituting Eq. ( 12) into Eq.( 9) leads to: The analytical solution of pore-water pressure head can be obtained using Green's function method: The pressure head can be calculated as follows:

C-F criterion of root-soil composite
The classical Mohr Coulomb (M-C) condition for soil shear strength includes two parameters: internal friction angle and effective cohesion.In unsaturated soil, the Fredlund strength equation is expressed as follows (Fredlund et al. 1978): where c' represents the effective cohesion, and φ' denotes the effective internal friction angle; The total normal stress is indicated by σ , and ϕ b is the internal friction angle related to the matric suction.
In addition, ua and uw refer to the pore air pressure and pore-water pressure, respectively.When the matrix suction is 0, the Fredlund strength equation is classical M-C strength criterion.
According to the Fredlund strength equation, the tensile strength could be overestimated (Li and cδ is cohesion of the tensile shear zone (cδ=OT).In Fig. 3, cδ is one of the key factors in establishing the C-F strength criterion, and cδ is expressed as: where ft= OQ is the tensile strength of the root-soil composite, and r is the radius of the strength envelope QP.They are expressed as: In Eq. ( 18), the tensile strength of root-soil composite is neglected when ζ = 0.However, when ζ=1, the tensile strength predicted by the C-F criterion is consistent with Fredlund strength equation.
c is the apparent cohesion of root-soil composite, which reflects the combined effects of the soil intrinsic cohesion and the root additional cohesion.More precisely, c can be expressed: This study employs the WW model to determine root cohesion (Wu 1976;Waldron 1977), namely: where ξ is the ratio of root cross-sectional area, which is 0.00025 (Leung 2014); Ts is the root tensile strength .
When the effective stress is computed by Eq. ( 26), the relationship between φ' and φ b in unsaturated soil is expressed by effective saturation (Sr): Substituting Eq. ( 28) into Eq.( 25) leads to: In summary, the C-F criterion of unsaturated root-soil composite is: Substituting Eq. ( 29) into Eq.( 30) leads to:

Stability analysis and discussions
In practice, infinite slope model are used to assess the safety of slopes.The soil mass is moving parallel to a plane failure surface (Cornforth 2016).Infinite slope can be regarded as a half-space problem (Fig. 4a).Every point in half-space needs to satisfy the differential equations (Eq.( 32)) (Michalowski 2018).In accordance with the assumptions in Section 2, an unsaturated vegetated slope is shown in Fig. 4b.D is distance between the potential sliding surface and the ground.Dw is distance between the groundwater level line and the ground.In Fig. 4b, the normal and shear stress in sliding surface can be obtained through direct integration of Eq. ( 32): where γ is the average weight of the root-soil composite.
If the sliding surface is situated above the groundwater level, the safety factor (Fs) can be determined using C-F criterion and limit equilibrium method, namely: According to the Fredlund strength equation, Fs can be expressed as follows: ( ) r tan tan tan sin sin The C-F strength criterion is utilized in this study to develop a stability model.The hydraulic parameters in this study are from Vahedifard et al (2016), Liu et al (2016a), Li et al (2020) and Wu et al (2022).The parameters of vegetation roots are determined based on the statistics of shrubs (Leung et al. 2013;Feng et al. 2020), and are listed in Table 1.

Effect of root depth
In this context, Fig. 5(a) illustrates the impact of uniform root depth on uw.Eqs. ( 35) and ( 36) are utilized to compute Fs.Meanwhile, Fig. 5(b) exhibits the relationship between Fs and root depth, with root depth of 0.5m, 1.0m, and 1.5m, and slope angle of 30°.Other parameters are presented in Table 1.The following conclusions can be drawn from Fig. 5

Effect of slope angle
The study aims to investigate the effect of slope angle on uw and Fs, with fixed root depth of 1m and slope angles of 15 °, 30 °, and 45 °.Table 1 list

Effect of rain intensity
In this study, we investigate the effect of rainfall intensity on uw and Fs with rainfall intensities of 0.1ks, 0.5ks, and 0.9ks, respectively.Table 1 list the other parameters.The following conclusions can be obtained from Fig. 7(a) and (b): a) uw decrease rapidly in the shallow layer of soil slope with increasing rainfall intensity (Ng et al. 2014).As rainfall increase, the soil transitions from unsaturated to saturated state (Wu et al. 2017); b) Fs decrease as the rainfall intensity increase, indicating a increase likelihood of slope instability; c) As in sections 4.1 and 4.2, the C-F criterion is suitable for calculating Fs in shallow slopes, while the Fredlund strength equation is suitable for deep slopes; d) Fs increase rapidly within the root zone, indicating that roots reinforce slopes through both hydrological and mechanical effects.

Effect of transpiration rate
Transpiration rate is an important index of vegetation water absorption, which reflects the hydrological effect of roots on slope stability.Three kinds of root parameters, including transpiration rate, tensile strength, and depth, were determined based on the statistics of shrub roots (Leung and Ng 2013;Feng et al. 2020).Figure 8(a) and (b) report uw and Fs of the slope with transpiration rates of 2.77, 4.50 and 6.23 mm/d, respectively.The root depth and slope angle are set as 1m and 30 °, and the other parameters are listed in Table 1.The results indicate that a low transpiration rate leads to a small uw at the same position.This is because root water uptake decrease soil moisture and increase soil matric suction (Wu et al. 2022).As the transpiration rate increase, Fs increase, indicating that the slope is more stable.Furthermore, uw inside the root zone is observed to be greater than outside the root zone.

Effect of root tensile strength
The tensile strength of roots is an essential factor that influence the mechanical effect of vegetated slope stability.The tensile strength test of roots was carried out (Fig. 9).Tensile strength is computed as follows: where P is tensile strength (MPa), F is tensile force (N), and D is the diameter of the root (mm).Figure 10 shows the variation of tensile forces and tensile strength of roots with diameter.The tensile forces and tensile strength of dry roots were the highest, followed by fresh and saturated roots.
Because root water absorption reduces the binding strength between organic polymers of cell wall (Zhang et al. 2019;Hales et al. 2013).Landslides usually occur during rainfall, and the reinforcement effect of roots will be greatly weakened.Based on the tensile strength of dry roots and fresh roots could overestimate the slope stability, but it is reasonable to use saturated roots (Zhang et al. 2019).The tensile strength of fresh root, saturated root, and dry root increase in power law with increasing of diameter, and the tensile strength decrease in power law with increasing of diameter.

Discussions
The hydrologic and mechanical effects of roots influence slope stability.The sink term in Eq. ( 3) reflects the hydrological effect, and Eq. ( 31) reflects the mechanical effect produced by the root architecture.The calculated results based on infinite slopes reveal that the root architecture can have a significant impact on slope stability.However, this model is only suitable for slopes with uniform root architectures.The C-F criterion is best suited for evaluating the shallow slope stability, while the Fredlund strength equation is more advantageous for deep slopes.Nonetheless, the hydrologic and mechanical effects of different roots architectures are different.Therefore, appropriate slope stability models incorporating different root architecture functions should be developed in future research works.
As pointed out above, Eqs. ( 7) and ( 8) used to calculate hydraulic conductivity and the volumetric water content in the current study are approximate formulas, which may not accurately reflect the soil-water characteristic curve of unsaturated soils.To obtain more accurate solutions, the Richards equation, defined by the Van-Genuchten model, can be solved using numerical methods such as the finite difference method.Finally, the stability charts in Figs.5-9 can be utilized directly by engineers based on interpolation methods.

Yang 2019 ;
Li et al. 2020), resulting in a potential overestimation of slope stability.To solve this limitation, a new criterion (C-F criterion) has been developed based on the Fredlund strength equation.The C-F criterion considers the partial truncation of the tensile strength in Fig.3, where the strength envelope of the tensile stress zone is PQ.δ is internal friction angle of the tensile shear zone.

Fig. 3
Fig. 3 Strength envelope of the C-F criterion.

Fig. 4
Fig. 4 Infinite slope diagram: (a) Infinite slope as a half-space; (b) Force analysis (a) and (b): a) uw decrease with increasing root depth.Fs increase with increasing root depth.b) uw of outside the root zone shows no significant difference when the root depth is determined; c) During rainfall, uw of shallow slopes is more sensitive than that of deep slopes; d) The variation in Fs obtained by C-F criterion and Fredlund strength criterion is consistent, indicating that the C-F criterion is reliable; e) Fs computed by C-F criterion is lower than Fredlund strength criterion in shallow slopes.Therefore, the C-F criterion is more suitable for analyzing shallow slopes stability, while the Fredlund strength criterion is more sensitive to deep slopes.Effect of root depth on (a) uw and (b) Fs.
the other parameters.Results are presented in Fig. 6(a) and (b).The following conclusions can be drawn: a) uw increases with decreasing slope angle, owing to the positive effect of slope angle on rainfall infiltration (Wu et al. 2022); b) Fs decrease as slope angle increase, indicating a decrease of slope instability; c) Compared with deep slopes, shallow slopes have a higher safety factor due to root reinforcement.Fs rapidly increase at z = 1 m; d) With increasing of slope angle, the effect of roots on Fs decreases.
Effect of slope angle on (a) uw and (b) Fs.
Effect of rainfall intensity on (a) uw and (b) Fs.
Effect of transpiration rate on (a) uw and (b) Fs.

Fig. 9
Fig. 9 Tensile test device Figure 11 depicts the Fs-depth curves of vegetated slopes with root tensile strength (Ts) of 10

Fig. 11
Fig. 11 Effect of root tensile strength on Fs.

Table 1
Input parameters