Analytical analysis of the mechanical and hydrological effects of vegetation on shallow slope stability

Computers and Geotechnics 118 (2020) 103335

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Technical Communication

Analytical analysis of the mechanical and hydrological effects of vegetation on shallow slope stability S. Fenga,1, H.W. Liub,

⁎,2

T

, C.W.W. Ngc

a

College of Civil Engineering, Fuzhou University, Fuzhou City, Fujian Province, China College of Environment and Resources, Fuzhou University, Fuzhou City, Fujian Province, China c Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong b

A R T I C LE I N FO

A B S T R A C T

Keywords: Analytical solution Hydrological effect Mechanical effect Vegetation Slope stability

Analytical solutions are proposed considering the hydrological and mechanical effects of roots with different root architectures on shallow slope stability. Analytical parametric studies show that an exponential root has more prominent hydrological and mechanical effects than a uniform one. At drying conditions, the hydrological effect becomes more significant in gentler slopes, while the opposite is true for the mechanical effect. At a given slope angle, the hydrological and mechanical effects increase with transpiration rate and root tensile strength, respectively. After rainfall (with a 10-year return period), the hydrological effect vanishes almost entirely, and hence the mechanical effect dominates slope stability.

1. Introduction Hydrological and mechanical effects of vegetation could help improve the stability of civil infrastructure, including shallow slopes (i.e., those less than 2 m deep [1]), road/railway [2], small dams as well as levee embankments [3]. The hydrological effect refers to the reduction in pore-water pressure (PWP) through root water uptake, leading to a reduction in soil hydraulic conductivity but an increase in soil shear strength [4–11]. The mechanical effect refers to the enhanced soil shear strength attributed to the mechanical reinforcement of roots. This effect is often considered as an additional soil cohesive force (known as apparent root cohesion cr ) provided by roots [12–19]. Some studies have shown that the hydrological effect is more significant than the mechanical effect in enhancing shallow slope stability [8,20–24], while others have found the opposite [12,25]. It has been demonstrated that the relative importance of these two effects depends on weather conditions [26,27]. Recently, Ni et al. [28] investigated both the mechanical and hydrological effects of vegetation on slope stability through numerical simulations and revealed that the hydrological effect is more prominent at greater depths (> 1 m). This is similar to the findings of Liu et al. [1], who studied the hydrological effect of vegetation on shallow slope stability through an analytical approach. However, there is still a lack of quantitative studies that explicitly consider the factors determining the mechanical and

hydrological effects of vegetation. Moreover, to the best of the authors’ knowledge, analytical solutions considering both the hydrological and mechanical effects of vegetation on shallow slope stability are not available. This study aims to derive analytical solutions to investigate the mechanical and hydrological effects of roots on shallow slope stability. Analytical parametric studies are also carried out to explore the underlying factors. 2. Theoretical consideration 2.1. Modelling the hydrological effect of roots Fig. 1 shows an infinite vegetated slope considered in the present study. The water table is fixed at the base of the slope [29], while evaporation or rainfall is considered at the slope surface. Surface runoff is not modelled and hence the applied infiltration is equal to the difference between rainfall and surface runoff. Assuming that isolines of PWP are parallel to the slope, the water seepage in the infinite vegetated slope can be simplified as one-dimensional (1-D) water flow perpendicular to the slope [29,30]. According to water mass balance, unsaturated seepage in the vegetated infinite slope takes the following form [29]:

⁎

Corresponding author. E-mail addresses: [email protected] (S. Feng), [email protected] (H.W. Liu), [email protected] (C.W.W. Ng). 1 Formerly, Postdoctoral Fellow, Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Hong Kong. 2 Formerly, Doctoral Research Student, Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Hong Kong. https://doi.org/10.1016/j.compgeo.2019.103335 Received 8 August 2019; Received in revised form 21 October 2019; Accepted 3 November 2019 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.

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S (z ) =

L2

z z'

Fig. 1. Schematic diagram of a vegetated infinite slope. (after Liu [48]).

∂θw ∂ ⎛ ∂ψ ⎞ ∂k = cos β − S (z ) Tp H (z − L1) k + ∂t ∂z ⎝ ∂z ⎠ ∂z

(1)

L1 ⩽ z ⩽ (L1 + L2 ) within root zone 0 ⩽ z < L1 beyond root zone

(2)

where L1 represents the thickness of the region beyond the root zone (see Fig. 1); and L2 is the perpendicular root depth. According to Eq. ∂θ (1), the variation in soil moisture ∂tw is due to the water flow driven

( )

by the gravity

(

∂k ∂z

)

cos β and PWP gradient

(3)

k = ks exp(αψ)

(4)

θw = θr + (θs − θr ) exp (αψ)

(5)

2.2. Modelling the mechanical effect of roots

( (k ) ), and to root water ∂ ∂z

⎨⎡ exp(z − L1) − 1 ⎤ Exponential root ⎩⎣ exp(L2) − L2 − 1 ⎦

where ks denotes the saturated hydraulic conductivity of soil; α represents the desaturation coefficient of soil; θs and θr represent the saturated and residual volumetric water contents, respectively. It should be noted that the hydrological properties of unsaturated soils are highly nonlinear and may not be well captured by Eqs. (4) and (5). Nevertheless, for mathematical tractability, these two equations are adopted to derive analytical solutions. Moreover, the wetting-drying hysteresis is ignored. Root-induced changes in soil hydraulic properties (e.g., soil water permeability [33,34]) are also ignored due to mathematical difficulties encountered in deriving analytical solutions at transient state. Eqs. (1)–(5) describe the unsaturated seepage in a vegetated infinite slope. Analytical solutions of PWP are obtained for both the steady- and transient-state [29].

where θw and t are the volumetric water content (VWC) and time, respectively; Z denotes the coordinate perpendicular to the slope (see Fig. 1); k is the hydraulic conductivity function of soil; ψ is the PWP head; S (z ) represents the root architecture function; Tp is the transpiration rate; and H (z − L1) is defined as follows [30]:

1 H (z − L1) = ⎧ ⎨ ⎩0

Uniform root

To achieve a fair comparison, uniform and exponential roots spanning the same total area are considered in Eq. (3). It should be noted that the actual root architecture could be very complicated, rather than the idealized uniform and exponential root architectures considered in the present study. Furthermore, the root architecture could be affected by the underground seepage, while it could also influence the subsurface preferential flows [31]. Nevertheless, the interaction between root architecture and subsurface preferential flows is not considered in the present study. The unsaturated hydraulic conductivity function of soil and the soil water retention curve are described in the following forms, respectively [32]:

H0

L1

⎧ L2

∂ψ ∂z

The mechanical effects of vegetation on shallow slope (i. e. 2 m) stability are related to the so called “minimum rooting length” [35].When the root depth exceeds the “minimum rooting length”, the mechanical reinforcement of vegetation on shallow slope stability could be considered. According to Arnone et al. [27], the failure depth of

uptake (S (z ) Tp H (z − L1) ). Uniform and exponential root architectures (see the insets of Fig. 2) are considered in this study. Accordingly, the root architecture function S (z ) is given as follows [29]:

RAR (%) 0.00 0.0

0.02

0.04

0.06

0.08

Depth (m)

0.1

0.2

0.3

Uniform root

Exponential root

0.4

Exponential Uniform

0.5

Fig. 2. Two different root architectures and their root area ratio (RAR). 2

0.10

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1.00E+00

0.5

hydraulic conductivity (m/s)

0.4

SWRC 1.00E-04

0.3

1.00E-06 1.00E-08

0.2

1.00E-10

0.1

Volumetric water content (θw)

Hydraulic conductivity function

1.00E-02

1.00E-12 1.00E-14 0.01

0.1

1

10

100

0 1000

Suction (kPa) Fig. 3. Hydraulic properties of CDG.

breaking progressively as is the case in real life on soil shear strength. According to Preti and Schwarz [38], ζ equals 0.4 in the present study. Tr represents the root tensile strength. Experimental results reveal that the root tensile strength strongly depends on the diameters of roots [39,40]. Generally speaking, the smaller the diameter, the larger the root tensile strength. Accordingly, some studies in literature adopted statistical distribution functions of the diameters of roots to representatively estimate the increase of shear strength provided by roots [39,40]. Nevertheless, for simplicity, a constant root tensile strength (i.e., ignoring effects of root diameter distributions) is considered in the present study following the approach adopted by Ni et al. [33]. Rf represents the root orientation factor, which is defined as follows:

Table 1 Summary of parametric study. Series ID

Case IDa

Tr (MPa)

Tp (mm/ day)

Slope angle (°)

Analysis procedure

1

B E U U

0 10

0 4.5b

35

10

4.5

35 50 60 35

Drying steady sate → wetting statec Drying steady sate → wetting state Drying steady sate → wetting state Drying steady sate → wetting state

2

3

4

U

U

10

10 50 100d

0–4.5

4.5

35–60

transient

transient

transient

Rf = sinω + cos ω tan φ′

transient

(7)

where ω is the angle between a root and the failure surface at root breakage. In the present study, ω equals 90°, as roots are assumed to grow perpendicularly to the slope surface [7]. Accordingly, Rf equals one. It should be noted that root growth is affected by ambient conditions (e.g., gravity, nutrition, etc.). Hence, roots may not always grow perpendicularly to the slope surface in reality. Root area ratio (RAR in Eq. (6)), which is defined as the fraction of the cross-sectional area of the soil occupied by the roots, is determined as follows:

Note: a B, E and U represent bare soil, and soils reinforced by exponential and uniform root branching systems, respectively. b 4.5 mm/day is the average transpiration rate in the dry season in Hong Kong (Leung and Ng [21]). c Rainfall intensity of 181 mm/day with a duration of 24 h (i.e., a 10-year return period based on weather conditions in Hong Kong) is applied for all wetting transient cases, while the transpiration rate is ignored (Tp equals zero) during rainfall due to high humidity (Leung and Ng [21]; Allen et al. [49]). d Typical root tensile strength of shrubs (e.g., R. tomentosa) and trees (e.g. S. heptaphylla) ranges from 10 to 100 MPa, according to Leung [41].

n

A RAR = r = A

∑ πdi2 i=1

4A

(8)

most slopes is between 500 and 1000 mm, only when the depth of root reaches deeper soil layer, can the root system play a role on mechanical stability. During the last four decades, different theoretical models have been developed to estimate the mechanical effect of roots on slope stability, including (i) the Wu model (proposed by Wu [16] and later improved by Waldron [36]); and (ii) a fiber bundle model [37]. The Wu model assumes that all roots are mobilised and broken simultaneously, which is never true in practice. However, it is a popular model thanks to its simplicity and the fact that it requires a relatively small amount of input data. Therefore, the Wu model is adopted to describe the increase in soil shear strength induced by the mechanical effect of roots as follows:

The shear strength of an unsaturated vegetated soil (τf ) could be determined by incorporating the additional root-induced cohesion into the extended Mohr-Coulomb criterion [42] as follows:

cr = ζ × Tr × Rf × RAR

τf = cr + c′ + (σn − ua) tan φ′ + (ua − u w ) tan φb

where Ar and A represent the cross-sectional area of root and soil, respectively; di denotes the diameter of the ith root; and n is the total number of identified roots. Based on the measurements of RAR [41], the exponential and uniform root architectures are considered (see Fig. 2). 2.3. Hydrological and mechanical effects of roots on shallow slope stability

(6)

(9)

where c′ and φ′ denote the effective cohesion and effective friction angle, respectively; σn is the total normal stress; ua is the pore-air

where cr is the additional cohesion induced by the mechanical effect of roots; ζ is the correction factor, which accounts for the effects of roots 3

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Pore-water pressure (kPa) -80 0.0

-70

-60

-50

-40

-30

-20

-10

0

Root zone

Depth (m)

1.0

2.0 Bare Exponential

initial

Uniform Bare

3.0

Exponential

after 24-h rainfall

Uniform 4.0 Fig. 4. Pore-water pressure distributions during a 24-h rainfall (rainfall intensity of 181 mm/day) in a bare slope and in slopes reinforced by exponential and uniform root architecture.

shown in Fig. 1, an infinite vegetated slope is considered, which has a vertical root depth of 0.5 m and a vertical thickness (H0 ) of 5 m. A fixed water table is assigned at the bottom boundary. The top boundary is set as a zero-flux boundary at the drying steady state, while rainfall is applied at the wetting transient state. During the drying stage, a transpiration rate ranging from 1 to 4.5 mm/day is considered, while it is ignored during rainfall due to the relatively high humidity. During the wetting stage, a surface runoff of 40% of the rainfall intensity is adopted [46]. Completely decomposed granite (CDG) is adopted, which is commonly encountered in Hong Kong and is classified as silty sand. The hydraulic properties of CGD [29] are given in Fig. 3. Table 1 presents four series of parametric studies examining how (i) root architecture; (ii) slope angle; (iii) transpiration rate (Tp); and (iv) root tensile strength (Tr) determine the hydrological and mechanical effects of vegetation on shallow slope stability.

pressure (ua = 0 kPa in this study); u w denotes the PWP; and φb is the angle relating the increase in shear strength to the negative PWP. φb is known to vary nonlinearly with negative PWP [43]. However, as the negative PWP commonly encountered in slope stability analysis is less than 100 kPa, a constant φb is adopted for simplicity. Assuming that the potential slip surface is parallel to the slope [30], the factor of safety (FOS) of the vegetated slope is calculated by considering the force equilibrium parallel to the slope as follows:

Fs =

(c′ + ζ Rf Tr RAR − u w tan φb ) H0

⎡γ (H − z ′) + γ ∫ θ dz ′⎤ sin β cos β w W ⎢d 0 ⎥ Z′ ⎣ ⎦

+

tan φ′ tan β (10)

where γd and γW refer to the unit weight of dry soil and water, respectively; z ′ is the vertical coordinate (Fig. 1); and H0 is the vertical distance between the slope surface and the slope base (Fig. 1). Root water uptake affects both θw and u w (see Eq. (1)), both of which are calculated using the analytical solutions of Eq. (1) [29]. The main limits of the proposed analytical solutions include: (i) ignoring root diameter distribution and its influence on root tensile strength; (ii) surface runoff is not considered; (iii) the interaction between root architecture and the underground seepage is ignored. Vegetation would influence water infiltration, soil erodibility, and surface runoff. Liu et al. [44] reported that both aboveground parts of vegetation (e.g., shoots, leaves) and roots affected surface runoff and soil erodibility. Furthermore, both roots and aboveground parts of vegetation are important for rill and gully erosion, while the aboveground parts of vegetation plays the most important role in splash and interrill erosion [44,45]. Nevertheless, the aforementioned facts are ignored in the present study. The theoretical model may be further developed to consider the influence of vegetation (e.g., both aboveground parts and roots) on water infiltration, soil erodibility and surface runoff.

3.1. Role of root architecture in the hydrological and mechanical effects of roots Fig. 4 shows the responses of PWP in slopes reinforced by exponential and uniform root branching systems and in bare slopes during rainfall. As expected, the vegetated slopes have larger negative PWP near the slope surface before rainfall than that in the bare case, due to root water uptake. The difference in PWP between the vegetated and bare slopes diminishes with depth. The influence zone of root water uptake could be as large as 4–6 times the root depth, which is consistent with the findings of Ng et al. [29] and Woon [47]. Above the depth of 0.5 m, the slope reinforced by the exponential root branching system has larger negative PWP than the slope reinforced by the uniform root, due to the presence of more roots near the soil surface (Fig. 2), and hence it has a larger amount of root water uptake. After raining for 24 h, the negative PWP drops sharply for all cases due to rainfall infiltration. Nevertheless, the vegetated slopes still maintain the larger negative PWP by about 3 kPa than the bare case, due to the larger initial negative PWP. Moreover, the effects of root architecture on PWP distributions become almost negligible after rainfall.

3. Parametric study Analytical parametric study is conducted to analyse the hydrological and mechanical effects of roots on shallow slope stability. As 4

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FOS ratio 0.0

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Root zone

Depth (m)

1.0

E H+M

2.0

initial

EH E H+M 3.0

after 24-h rainfall

EH

4.0

(a) FOS ratio

1.0 0.0

1.2

1.4

1.6

1.8

2.0

2.2

Root zone

Depth (m)

1.0

2.0

U H+M

initial

UH 3.0

U H+M after 24- h rainfall UH

4.0

(b) Fig. 5. Comparison between the hydrological and mechanical effects of roots on the FOS ratio (the ratio of FOS in the vegetated slope to that in the bare slope) after a 24-h rainfall: (a) exponential-rooted slope; and (b) uniform-rooted slope. (E: exponential root architecture; U: uniform root architecture; and H and M indicate that hydrological and mechanical effects are considered respectively).

A similar trend of the FOS ratio could be observed in the slope reinforced by the uniform root branching system (Fig. 5(b)). Nevertheless, the FOS ratio within the root zone in this slope (Fig. 5(b)) before rainfall is lower than that in the slope reinforced by the exponential root branching system (Fig. 5(a)). This finding suggests that the hydrological and mechanical effects of roots are more significant in the slope with an exponential root architecture, as a result of a greater root presence near the slope surface and hence more root water uptake and more significant mechanical reinforcement.

Fig. 5(a) shows the hydrological and mechanical effects of the exponential root architecture on slope stability during rainfall. Before rainfall, ignoring the mechanical effect results in an FOS ratio between the vegetated slope and the bare slope that is about 0.2 lower than that when both the hydrological and mechanical effects are taken into account. This suggests that the hydrological effect of roots plays a dominant role in maintaining slope stability during the drying period. After rainfall, the FOS ratio decreases as the hydrological effect becomes less important due to rainfall infiltration. The difference between considering and ignoring the mechanical effect becomes more significant within the root zone, suggesting that the mechanical effect dominates slope stability inside the root zone during the wetting period. Similar outcomes have been found by Chirico et al. [26] and Arnone et al. [27], both demonstrating that during the wetting period the mechanical effect is more significant in slope stabilization.

3.2. Effects of slope angle on the stability of vegetated slopes Fig. 6(a) shows that at the drying steady state, the ratio of FOS in the slope with a uniform root architecture to that in the bare slope decreases as slope angle increases, when only the hydrological effect is 5

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FOS ratio 0.8 0.0

1.0

1.2

1.4

1.6

1.8

Root zone

Depth (m)

1.0

35° H+M 35° H

2.0

50° H+M 50° H 3.0

60° H+M 60° H

4.0

(a) FOS ratio

0.8 0.0

1.0

1.2

1.4

1.6

1.8

Root zone

Depth (m)

1.0

35° H+M 35° H

2.0

50° H+M 50° H 3.0

60° H+M 60° H

4.0

(b)

Fig. 6. The ratio of FOS in uniform-rooted slope to bare slope at different slope angles (35°, 50° and 60°) during rainfall; (a) t = 0; (b) t = 24 h.

considered. This decrease is caused by the reduction in the water flow ∂k induced by gravity ( ∂z cos β in Eq. (1); along the negative direction of the z coordinate in Fig. 1) at a larger slope angle, leading to a lower ∂ψ ∂ water flow induced by pore-water pressure gradients ( ∂z k ∂z in Eq. (1); along the positive direction of the z coordinate in Fig. 1) under a given root water uptake (S (z ) Tp H (z − L1) in Eq. (1)). Therefore, the hydrological effect is more prominent in a gentler slope. When considering both the hydrological and mechanical effects, the FOS ratio decreases as the slope angle increases from 35° to 50°. However, a further increase in the slope angle to 60° results in a slight increase in the FOS ratio. A larger difference in FOS ratio between the cases of considering and ignoring the mechanical effect could be observed in a steeper slope. This suggests that the mechanical effect of roots on shallow slope stability becomes more important in a steeper slope, as the hydrological effect decreases with slope angle. The FOS ratios under different slope angles after raining for 24 h are shown in Fig. 6(b). The FOS ratios are almost identical when only the hydrological effect of roots at different slope angles is considered. This is consistent with previous analytical studies performed by Liu et al.

[1], who revealed that the hydrological effect vanishes after rainfall. In contrast, a larger FOS ratio is observed in the steeper slope when the mechanical effect is considered. This is because the FOS is reduced to a greater extent with slope angle in the bare slope than in the vegetated slope (Eq. (10)). This goes to illustrate that the mechanical effect becomes more prominent in maintaining slope stability for a steeper slope, while the opposite is true for the hydrological effect.

( )

3.3. Role of TP in the stability of vegetated slopes Fig. 7(a) presents the relative importance of the hydrological and mechanical effects of a uniform root architecture on maintaining shallow slope stability under different Tp before rainfall. The uniform root architecture is selected as it represents a more critical case than the exponential root architecture (Fig. 5). For a given depth, the hydrological effect becomes more important as Tp increases, which is expected due to more root water uptake and hence larger negative PWP. When Tp exceeds 2 mm/day, the hydrological effect dominates shallow slope stability. Similarly, Chirico et al. [26] and Arnone et al. [27] found that the hydrological effect is more relevant during the dry 6

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1.8

1.8

1.4

1.6 D=0.1 m D=0.25 m D=0.5 m

1.2 1.0 0.8

FOS ratio (H/M)

FOS ratio (H/M)

1.6

D=0.1 m D=0.25 m

1.4

D=0.5 m

1.2 1.0 0.8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Tp (mm/day)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Tp (mm/day)

(b)

1.8

1.8

1.6

1.6

1.2

D=0.1 m D=0.25 m D=0.5 m

1.4

D=0.1 m D=0.25 m D=0.5 m

1.2 1.0

1.0 0.8

FOS ratio (H/M)

FOS ratio (H/M)

(a)

1.4

Fig. 7. The relative importance of hydrological and mechanical reinforcement of roots at different depths in the slope with uniform root architecture during rainfall: (a) t = 0; (b) t = 0.1 h; (c) t = 2 h; and (d) t = 24 h (FOS ratio (H/M) represents the ratio of FOS attributed to the hydrological effect (H) to that attributed to the mechanical effect (M); D represents depth).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Tp (mm/day)

0.8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Tp (mm/day)

(c)

(d)

hydrological effect becomes less significant in steeper slopes, while the mechanical effect plays a more significant role and eventually dominates the slope stability. Fig. 8(b) shows that the FOS ratio is almost close to one (i.e., the roots have negligible effects) and becomes independent of slope angle after raining for 24 h, when the mechanical effect of roots is ignored (Tr = 0). This is expected as the hydrological effect of roots almost vanishes after the 24-h rainfall (see Fig. 4). At larger Tr, a larger FOS ratio is observed, which increases with slope angle at an increasing rate. This finding is consistent with Kokutse et al. [12], who found that the FOS would increase more significantly with higher additional cohesion induced by larger root tensile strength (Tr). A more rapid increase in the FOS ratio with slope angle occurs at larger Tr. This is attributed to the mechanical effect of roots becoming more significant in steeper slopes as mentioned above. These results illustrate that the relative contribution of the hydrological and mechanical effects of roots depends on the tensile strength of root, in addition to the weather conditions as revealed in previous studies [26,27]. The mechanical effect could dominate slope stability even under drying conditions when Tr is large (e.g., 100 MPa). Vegetation with large root tensile strength should be selected to improve the stability of steep slopes.

summer season. The relative importance of the hydrological effect decreases as depth increases. This is due to a lower negative PWP at greater depths (Fig. 4). This finding illustrates that during the drying period, the hydrological effect on shallow slope stability becomes significant only when the transpiration rate is high (e.g., Tp > 2 mm/ day), which may occur in arid or semi-arid regions or in summer. Fig. 7(b) shows that the relative importance of the hydrological effect decreases at the depth of 0.1 m after raining for six minutes, while it is almost unchanged below the depth of 0.25 m. This can be explained by the rainfall infiltration at shallow depths. Fig. 7(c) shows that after raining for two hours, the mechanical effect dominates slope stability at shallow depths (i.e., the top 0.25 m). However, when Tp is higher than 2 mm/day, the hydrological effect is still larger than the mechanical effect at the bottom of the root zone (i.e. depth of 0.5 m). After raining for 24 h (Fig. 7(d)), the FOS ratio is smaller than 1 within the root zone, indicating that the mechanical effect dominates shallow slope stability after heavy rainfall (i.e., one with a 10-year return period). These findings show that during rainfall, the hydrological effect vanishes. Eventually, the mechanical effect is more important in maintaining shallow slope stability after heavy rainfall.

3.4. Effects of Tr on slope stability 4. Summary and conclusions Fig. 8(a) depicts the relationship between the FOS ratio and slope angle (β ) at different root tensile strengths (Tr) before rainfall (i.e., at the drying steady state). When Tr equals zero (i.e., only the hydrological effect of roots is considered), the FOS ratio decreases with slope angle, consistent with the earlier finding that the hydrological effect becomes less prominent in steeper slopes (Fig. 6(a)). When Tr is 10 MPa, a larger FOS ratio could be observed, which decreases with slope angle more gradually. As Tr further increases (Tr of 50 and 100 MPa), the FOS ratio reaches the minimum at a slope angle about 40°. This is because the

Analytical solutions are presented considering both the hydrological and mechanical effects of roots with different root architectures. Based on these analytical solutions, a series of parametric analyses are carried out to explore the factors that play a role in shallow slope stability. The hydrological and mechanical effects of roots are more significant in the slope reinforced by the exponential root branching system than in the one reinforced by the uniform root branching system. This is because the exponential root architecture has a greater 7

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4

51808125, 51908134 and 41861134011) provided by the National Natural Science Foundation of China. The first author (SF) would like to thank the Foundation of Key Laboratory of Soft Soils and Geoenvironmental Engineering (Zhejiang University), Ministry of Education, for supporting their project (2019P06). The authors would also like to thank Dr. Anthony Leung for meaningful discussions on the topic of the present study.

Tr=0 Tr=10 MPa 3

FOS ratio

Tr=50 MPa Tr=100 MPa

References 2

1 35

40

45

50

55

60

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Slope angle β (°)

(a) 4 Tr=0 Tr=10 MPa

FOS ratio

3

Tr=50 MPa Tr=100 MPa

2

1 35

40

45

50

Slope angle β (°)

(b) Fig. 8. The relationship between the FOS ratio (the ratio of FOS in the slope with uniform root architecture to that in the bare slope) and slope angle (β ) at different root tensile strengths (Tr ) in the uniform-rooted slope at a depth of 0.5 m, (a) before rainfall, and (b) after 24-h rainfall.

concentration of roots near the slope surface. The mechanical effect of vegetation can only enhance slope stability inside the root zone, while the hydrological effect can improve slope stability in the deeper zone (i.e. 3–4 times the root depth). At drying conditions, the hydrological effect of roots becomes less important as slope angle increases, while it is the opposite for the mechanical effect. Given the same conditions (e.g., the same soil type, root depth and slope angle), the relative importance between the hydrological and mechanical effects depends on the transpiration rate and root tensile strength. When the transpiration rate is small (i.e. Tp < 2 mm/day, such as under humid weather conditions), the mechanical effect of roots dominates slope stability even under drying conditions. After rainfall, the hydrological effect almost vanishes regardless of slope angle and hence the enhancement of slope stability mainly relies on the mechanical effect of roots. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors would like to acknowledge research grants (51778166, 8

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