SOSlope is a hydro-mechanical model of slope stability that computes the factor of safety on a hillslope discretized into a two-dimensional array of blocks connected by bonds. Bonds between adjacent blocks represent mechanical forces acting across the blocks due to roots and soil . These forces can either be tensile or compressive depending on the relative displacements of the blocks. A digital elevation model (DEM) is used to divide the hillslope into squares in plan view, where the centers of the squares are points of the DEM (Fig. ). Three-dimensional blocks are created by extruding the squares to the bottom of the soil layer along the vertical. The center of mass of a block is connected to the four lateral blocks by four force bonds (Fig. ). Initially, bond forces between blocks are set to zero. Rainfall onto the slope will increase the mass and decrease the soil shear strength of the blocks. At each time step, the factor of safety is calculated for each block using a force balance (resistive force over active force; see equations below). If the factor of safety of one or more blocks is less than one, those blocks are moved in the direction of the local active force (defined below) by a predefined amount (usually 0.1 mm) and the factor of safety is recalculated for all blocks. Because of the relative motion between blocks that have moved and blocks that remain stationary, mechanical bond forces between blocks are no longer zero and the force balance changes. This relative motion triggers instantaneous force redistributions across the entire hillslope similar to a self-organized critical (SOC) system of which the spring-block model is a subset. Looping over blocks and moving those that are unstable is repeated until all blocks are either stable (factor of safety greater than or equal to 1) and the system reaches a new equilibrium or some blocks have failed (their displacements are greater than some set value, usually a few meters), triggering a landslide.

The factor of safety for each block is calculated as the ratio of resistive to active forces. Resistive forces include the soil basal shear strength and the strength of roots that cross the basal slip surface, assumed to be located at the bottom of the soil layer. The active forces include the gravitational driving force due to the soil mass and the push or pull forces between blocks that include the effects of soil and root tension and compression. These later forces are the bond forces between the blocks described above. Including all these forces in a force balance yields the factor of safety FOS=Fs+FrFd+∑j=14Fj, where Fs is the soil basal resistive force that includes soil cohesion and friction, Fr is the basal root resistance, Fd is the driving force vector due to gravity, and Fj, j=1,…,4, are the four bond vector forces that quantify soil and root tension or compression between the block and its four neighbors. The vertical bars in the denominator denote the norm of a vector. This factor of safety is calculated for each block but an index for the block number is not included so as not to clutter the equations.

Soil basal resistance is Fs=Aτb, where A is the surface area of the block along the failure surface and τb is the basal shear stress (described below). In the present model, we set Fr=0, focusing on lateral root reinforcement. This is justified in many cases where the depth of the slip surface is 1 m or greater and very few roots are present e.g.,. Basal root reinforcement can easily be added using a formulation similar to lateral root reinforcement (discussed below) with values of root reinforcement a function of the shear displacement and the density of roots crossing the slip surface.

The driving force is Fd=γDAt^, where γ is the specific weight of the wet soil, D is the depth to the shearing surface, perpendicular to slope, and t^ is the unit tangent to the slope in the direction of the maximum slope. The specific weight of the wet soil is calculated based on water content and solid fraction, i.e., γ=ρsϕs+ρwθg, where ρs and ρw are the solid (grain) and water densities, respectively, ϕs is the solid volumetric fraction, θ the volumetric water content, and g is gravity.

Bond forces are given by Fj=Fjsoil+Fjrootsb^j,j=1,…,4, where Fjsoil and Fjroots are the soil and root components of the four bond forces, respectively, and b^j are unit vectors along the bond axes pointing outward of the block. These quantities are detailed below.

The force in bond j between a block and its neighbor due to roots (Fjroot) depends on four factors: the root density and the root-diameter distribution at the bond center; the strength of roots, which depends on root diameter; and the change in length (elongation) of the bond with respect to its initial length. Changes in root density with depth e.g., are not taken into account. This force is computed using the Root Bundle Model (RBM) of with Weibull statistics, called RBMw. For the sake of completeness, the full details of the model are given below.

The soil bond force (Fjsoil, Eq. ) depends on whether the soil is in tension or in compression. For tension, we assume that resistance scales with soil apparent cohesion (including the effects of suction stress for unsaturated soils) as a function of displacement using a logarithmic function : Fjsoil,  T=caWD1-log⁡1+εjLjlog⁡1+εmaxTLj,εj<εmaxT,0,εj≥εmaxT, where ca is the apparent cohesion, εmaxT is a strain threshold above which soil loses any tensional resistance, and Lj is the length of bond j. In compression, following the work of we assume that the soil compressional resistance is mobilized across the shear plane that forms during the failure of a downslope wedge, similar to the earth pressure force in the geotechnical engineering literature that develops during the passive state when a retaining wall moves downslope toward the adjacent backfill e.g.,. According to , the mobilized force on the downslope wedge scales with the maximum passive earth pressure force Fp and with the displacement, i.e., Fjsoil,  C(xj)=-FpWPw1(xj)Sw2(xj), where Fp=KpγρgD22+Kpcc′D, and Kpγ and Kpc are the passive earth pressure coefficients due to soil weight and to cohesion, respectively, obtained from a fitting of equations given in ; c′ is effective soil cohesion; and Pw1 and Sw2 are the Weibull cumulative density and the Weibull survival functions, respectively, given by Pw1(xj)=1-exp⁡-xjμ1κ1 and Sw2(xj)=exp⁡-xjμ2κ2, with μ1, κ1, μ2, and κ2 four parameters determined from compression experiments. The first Weibull function, Pw1, serves to scale the maximum passive earth pressure force with displacement during initial block motion, while the second one, Sw2, reduces that same force as the wedge is overridden by the block and the failure surface area of the slip plane decreases seefor details. We neglect the active earth pressure force on upstream faces of cells because the magnitude of the active force is small in comparison to other forces.

Rainfall-triggered shallow landslides can fail under saturated conditions during increases of pore-water pressure and/or loss of suction under unsaturated conditions . Our objective here is not to reproduce the detailed physical mechanisms by which changes in subsurface hydrology trigger a landslide but to develop a simple empirical model that realistically mimics observed changes in pore-water pressure and water content during rainfall infiltration. Although diverse hydrologic triggers have been observed and described e.g.,, here we use, as a representative example for the hydrological conditions triggering a shallow landslide in our model, pore-pressure measurements during the artificial triggering of the Rüdlingen shallow landslide experiment in Switzerland . Data from indicate that high pore-water pressures were attained relatively quickly and remained steady across the slope long before failure occurred, and that the decrease in the standard deviation of the water saturation prior to failure indicated an increase in the connectivity of water-saturated regions that reduced soil shear strength across the full length of the slip surface leading to failure. Other data in different localities e.g., have also shown high, steady pore-water pressure prior to failure. Because our model focuses on the effects of roots and soil strength on slope stability rather than on the details of hydrologic triggering, we choose a simplified, empirical, dual-porosity model for our slope hydrology. Our objective is only to reproduce reasonable pore-water pressure distribution and water content evolution in both the matrix and the preferential flow domains, but not to model the physics of evolving subsurface hydrology. The model embodies the rapid increase in positive pore pressure in a preferential flow domain (representing macropores) and the slow decrease in suction in the soil matrix caused by slow water transfer from the macropores to the matrix. This decrease in suction is the equivalent of the increasing connections of water-saturated regions represented by the decrease in the standard deviation of water saturation observed by that eventually caused slope failure in the Rüdlingen experiment.

We assume that water flow in soils during a rainfall event is a combination of slow matrix flow (also called immobile water with capillary number lower than 1) and fast preferential flow (mobile water, capillary number higher than 1) . While slow matrix flow influences the change in suction stress, the fast preferential flow directly influences pore-water pressure in the macropores. Our formulation of this concept is empirical and is a simplification of the more common dual-porosity models that employ two flow equations (e.g., Richards' equation) that exchange moisture between the two domains, and mixture equations for water content, hydraulic conductivity, rainfall partitioning based on the volumetric ratio of the fast and slow flow domains e.g.,. In accord with continuum mixture theory for effective stress e.g.,, we write the mean pore-water pressure of the soil (matrix + macropores), p‾, as p‾=ψ1p‾1+ψ2p‾2, where ψ1 and ψ2 are the pore fractions along the potential failure surface of the landslide of the matrix and the macropores, respectively (volume of pore in matrix or macropores over total pore volume, with indices 1 for matrix and 2 for macropores) with ψ1+ψ2=1, and where p‾i,i=1,2 are the matrix and macropores intrinsic mean pore pressures. Pore fractions ψ1 and ψ2s are related to the partial porosities of the matrix and the macropores, ϕ1 and ϕ2, respectively, by ψi=ϕin,i=1,2, where ϕ1+ϕ2=n, with n being the total porosity of the soil. The solid volume fraction of the matrix (macropores have only pore space) is ϕs=1-ϕ1-ϕ2=1-n. The superscripts and subscripts in these equations and in equations below refer to partial and intrinsic quantities, respectively. Partial and intrinsic water content of the matrix and macropores are related as follows: θ1=ϕs+ϕ1θ1, θ2=ϕ2θ2, where θ1 and θ2 are the partial water contents of phase 1 and 2 (volumetric water content of phase 1 or 2 over total soil volume) and θ1 and θ2 are the intrinsic water contents of each phases (volumetric water content of phase i over volume of phase i, i=1,2). At saturation θ2=ϕ2 since the macropore phase contains only void space and thus θ2=1. The total water content of the soil is θ=θ1+θ2, and is used in Eq. () to compute the soil-specific weight. Equations similar to Eq. () can be written for saturated and residual water contents.

We assume that the time evolution of the intrinsic pore-water pressure in the macropores, p‾2, and of the partial water content in both the macropore (θ2) and the matrix phases (θ1) can be modeled using cumulative distribution functions. For the macropore phase, we write p‾2(t)=pmaxFt∗,μp,σp, and θ2(t)=ϕ2θ2r+(1-θ2r)Ft∗,μp,σp, where pmax is a constant here but ultimately depends on rainfall infiltration rate and upstream contributing area e.g.,, t∗ is a dimensionless time, F is the normal cumulative distribution function with mean μp and standard deviation σp, and θ2r is the intrinsic residual water content for the macropores (we have used the fact that the intrinsic saturated water content θ2s=1 since macropores have no solid fraction). For the water content in the matrix we assume that θ1(t)=θo-ϕ2+θs-θoFfoldt∗,μθ,σθ, where θo and θs are the soil initial and saturated water contents, respectively, and Ffold is the folded normal cumulative distribution with mean μθ and standard deviation σθ. The pore-water pressure in the matrix is given by p‾1(t)=Se1p1, where p1 is the intrinsic pore-water pressure in the matrix and Se is the equivalent degree of saturation (also called effective saturation) in the matrix. Following , we have used the equivalent degree of saturation (Se) in Eq. () instead of the more commonly used degree of saturation. Under unsaturated conditions, p‾1 is a matrix suction stress . The equivalent degree of saturation in the matrix is defined as Se1=θ1-θ1rθ1s-θ1r, where θ1r and θ1s are the intrinsic residual and saturated water content of the matrix phase with θ1s=ϕ1. Using Eqs. () and (), and equations for the residual and saturated water content equivalent to Eq. (), Eq. () can be rewritten as Se1=θ1-θrθs-ϕ2-θr, where θ1 is given by Eq. (). Using van Genuchten formulation , we can write the suction stress as p‾1(t)=-Se1αvgSe1nvg1-nvg-11nvg, where and αvg and nvg are the soil parameters.

Pore-water pressure in the macropores (Eq. ), matrix water content (Eq. ), matrix suction (Eq. ), and mean pore-water pressure (Eq. ) are computed at each block of the domain at each time step. The dimensionless time t∗ in these equations is time scaled with the characteristic time for reaching steady state (t∗=t/tss). Figure  illustrates the model behavior for parameters shown in Table . The standard deviations are chosen so that macropore water pressure reaches its maximum before matrix water content, to mimic, but not reproduce, the behavior observed by .

Basal shear resistance along the slip surface is calculated using the Mohr–Coulomb failure criterion including contributions from both the suction stress and the pore-water pressure using the mean pore-water pressure p‾ of Eq. (), i.e., τb=c′+σn-p‾tan⁡ϕ, where σn is the normal stress and ϕ is the soil friction angle.

Mechanical soil parameters from and other parameters used in simulations are listed in Table . Figure  shows the soil strength in tension and compression (positive and negative values of displacement, respectively) for different soil thicknesses.

Model parameters for roots (Table ) are taken from field and laboratory data of for Picea abies (Norway spruce). Figure  shows root reinforcement as a function of bond elongation (both in tension and compression) for four values of tree diameter (DBH, diameter at breast height) and for three distances (d) from the tree trunk (0.5, 1.5, and at 2.5 m). The maximum root reinforcement in tension occurs within the first 5 cm of displacement in both tension and compression. The magnitude is about 5 times higher in tension than in compression and depends strongly on the size of the tree. Small trees (i.e., DBH=0.1 m) provide negligible reinforcement at all displacements. For large trees (DBH>0.3 m) lateral root reinforcement upwards of tens of kilopascal is typical . In tension, root reinforcement becomes negligible once the bond has stretched over 0.1 m, regardless of the distance from the tree trunk. In compression, the bond elongation over which reinforcement is active depends on the distance from tree and range from 0.15 m close to the tree trunk to about 0.05 m at 2.5 m distance from the tree trunk.

Figure  illustrates the evolution of slope displacement and soil and root bond forces during loading (the rainfall event) at four different time steps, 900, 1200, 1358, and 1359 min after the start of loading. The last time step (1359 min) is when the slope (clearing) fails. Time step 1358 shows the slope at the time step immediately before failure. Until failure, all slope configurations are stable (factor of safety greater than 1 for all cells of the slope).

During loading, cells in the clearing move downhill more than cells in the stand (Fig. a–d). A discontinuity in displacement appears near the top of the clearing. This gap, 12 m long and slope parallel, occurs where the surface slope is about 0.62 (ca. 32∘). This gap represents the formation of a vertical tension crack at the upper edge of a soil slip that has yet to fail completely. With increasing loading, displacement across the crack grows to exceed 1 m prior to failure (Fig. c). Although this crack is in the clearing in a zone devoid of trees, a few small roots from trees above the crack are present and extend across this vertical tension crack (see Fig. b). Cells above the crack show barely perceptible displacements (<0.1 m). The situation is different in the forested area, where, up to failure, displacement is significantly smaller (about 10 times smaller), uniform (no discontinuity), and highest in the steepest portion of the slope (not visible in Fig. ), with no evidence of a crack forming in the upper part of the slope. The slope in the stand remains stable after the clearing fails for the remaining of the simulation (2200 min). In the forested area, cells that have undergone displacement extend further uphill than in the clearing. We attribute this effect to the connected root system of trees that activates tensional forces uphill and pulls rooted cells downhill. These tensional forces are absent in the clearing due to lack of roots and negligible soil tensional strength.

Figure e–h and i–l show the downslope (y axis) bond soil and root forces, respectively. During loading (Fig. e–g, i–k), soil compression forces increase near the bottom of the hillslope with significantly higher values in the clearing area (up to -30 kN in Fig. g, negative sign for compression). Soil tension is negligible owing to the soil minimal tensional resistance. In the forested area, roots of trees near the top of the hillslope are in tension with the tensional force increasing with increasing loading as the slope slowly slips downhill (Fig. i–k). Root tension perpendicular to slope is highest on both edges of the vertical crack. This is where the largest displacements are observed generating the highest tensional forces in the roots. In that zone, tension in roots reaches almost 20 kN just before the clearing fails (Fig. k). Simultaneously, some roots of trees in the lower part of the slope are in compression, relieving some of the compression in the soil.

Across-slope (also referred to as lateral or slope-parallel) root forces are shown in Fig. m–p. Downward motion of soil in the clearing causes a lateral tension in roots that span the transition zone from clearing to forested area. This zone is about 6–10 m wide. It is across this boundary that displacement gradients are high and across-slope root forces highest. The lateral tension increases up to about 6 kN with increasing downhill motion of the clearing and stays high after failure because the relative downslope displacement of cells across the slope remains.

Figure  yields additional insights into the dynamics and transfer of forces during loading. In that figure, values of displacement (Fig. a–d), downslope bond force (root + soil, Fig. e–h), and across-slope bond force (Fig. i–l) are shown for three sections perpendicular to slope, at the center line (x=0) that passes through the clearing, at x=-9 m near the left edge of the clearing, and at x=-12, which intersects the first row of trees next to the clearing (see black curves in Fig. a for location). Figure a–d clearly shows the formation of the vertical crack with discontinuous displacements across it at about y=14 m, initially only for the center line (black symbols), but with increasing time (or load) also at x=-9 m (pink symbols). The forested area (x=-12 m) never develops such a crack and the displacement there is always continuous. The bond force perpendicular to slope shown in Fig. e–h indicates how the main resistive forces holding the slope are redistributed during loading. Initially, except for the clearing, which cannot hold much tension because of a lack of roots, forces on the slope are in tension in the upper half and in compression in the lower part. The transition occurs halfway down the slope in the forested area (red symbols, y∼0), a little uphill at the edge of the clearing (pink symbols, y=5 m). With increasing load, both tension and compression in the slope increase. Tension is highest where root density is highest (x=-12) and slightly lower at the edge of the clearing (x=-9 m). At t=1200 min, the edge of the clearing has formed a crack and forces downhill of that crack are now in compression. Roots that cross the crack at the edge of the clearing are now broken and no longer provide any tensional resistance (pink symbols in Fig. f).

Bonds that were in tension in the upper part of the slope at t=900 min are now in compression owing to the failure of roots across the widening crack near the edges of the clearing. The clearing is now entirely held by compressive forces and by lateral (across-slope) tensile forces shown in Fig. i–l. These lateral forces are due to root tensile strength and are highest near the transition from forest to clearing (pink symbols, x=-9 m), where the relative downslope displacement between adjacent cells is highest. Along the first row of trees (red symbols), cells that host a tree have larger values of across-slope tensional forces than cells that do not giving rise to a saw-tooth pattern of tensional force. In the clearing (black symbols), positive lateral tensional forces are entirely due to the soil apparent cohesion, which reaches values of almost 1 kN. With increasing load and decreasing soil shear strength due to increasing mean pore-water pressure, the clearing eventually fails at t=1359 min but the forested area remains stable for the remainder of the simulation (up to 2200 min).

Results from this simulation demonstrate that maximum tensional and compressive forces in rooted slopes do not contribute simultaneously and equally to the stability of the slope during the initiation of a shallow landslide. Roots provide reinforcement in tension. This tensional root force can disappears once displacement across a vertical crack becomes sufficiently large. In our example, this occurs when the crack grows to about 0.1 m (see Fig. ). Compression is higher in the clearing (no roots) than in the vegetated area. Where present, when slope-perpendicular root tensional reinforcement is eliminated, soil stability is entirely accommodated by soil compressive resistance and by lateral tension held by roots. Lateral root forces provide additional stability to the clearing by redistributing slope-perpendicular forces laterally across the slope. The clearing fails when soil strength at the base can no longer be held by the combination of the lateral root bond forces and downslope soil compression, and compression in the soil exceeds the maximum strength.

We can summarize the redistribution of forces during the loading of a rooted hillslope into three distinct phases:

Increasing load and weakening of soil strength along the basal failure plane (not shown) without any soil motion (factor of safety above 1).

The timing and duration of these three phases will vary with soil mechanical properties, slope inclination, slope morphology, root distribution, and hydrology, resulting in an increase or decrease in the stability of the slope. These three phases of force redistribution are used as criteria to define the triggering of a landslide. In civil engineering, calculations using infinite slope analysis, for example, must yield a factor of safety greater than 1 for the slope to be deemed stable. Any values below 1 imply an unstable slope with the possibility of a landslide, even if slope motion subsequently stops with no occurrence of a runout. This definition of a landslide corresponds to the second phase of force redistribution where motion has initiated but complete failure has not yet occurred. Many such occurrences of a failed landslide (at least temporarily) exist; one is shown in Fig. . In risk analyses, or when studying geomorphological processes, a landslide occurs by definition only when the soil mass fails completely and is followed by a runout, corresponding to the third phase of our force redistribution process. In that case, the transition from phase 2 to phase 3 and the accompanying redistribution of forces, is the critical process.

Changes in the values of the factor of safety (FOS) over time help understand the processes of landslide triggering and illustrates the three phases of landslide initiation and force redistribution. Figure  shows the evolution of displacement, the factor of safety, and the mean pore-water pressure with time at the center of the clearing. Initially, FOS is larger than 1 and decreases with increasing mean pore-water pressure up until about 400 min. This corresponds to the phase 1 described above. Beyond 400 min, the value of FOS oscillates rapidly just above the value of 1. These oscillations, which last until failure, correspond to the critical state of a self-organized system before global failure e.g.,. Here, these oscillations correspond to our phases 2 and 3. During this critical period, the number of cell moves (not shown) increases dramatically as a result of force redistribution between bonds of connected cells and as the number of cells with factor of safety less than 1 increases. The increase in the number of redistributions with loading is similar to the process of avalanching in load-controlled self-organized systems like fiber bundle models . The increase in force redistribution across the slope corresponds to the progressive slope failure stage of coalescence of local failure surfaces that eventually leads to global failure . This is equivalent to our phase 3.

The decrease in the factor of safety is linked to the increase in mean pore-water pressure in the soil (Fig. ). A detailed analysis of how hydrology impacts slope stability is beyond the aim of this paper. Here we wish to point out that our simple dual-porosity model, with the coexistence of pore-water pressure in the macropores and suction stress in the matrix (see Fig. ), is realistic and can model a wide range of hydrological situations that can lead to shallow landslide triggering. In the simulation shown in Fig. , there is an imperceptible increase in the factor of safety during the first phase of the simulation until about 100 min. This increase is due to the increase (in absolute value) of the suction stress that increases the soil apparent cohesion (see Fig. ). The increase in pore pressure after about 200 min causes the soil to weaken with an associated decrease in the factor of safety eventually leading to the critical state (FOS close to 1). A decrease in matrix suction linked to flow of water from the macropores to the matrix increases the mean pore-water pressure (Fig. ) and eventually causes soil to weaken sufficiently for a landslide to occur. Depending on the application of the model and on the local hydrological properties, choices of different values of hydrological parameters than those used in this example could lead to different hydrological triggering. For example, triggering could be due to the rapid increase in macropore water pressure and the saturation of the soil from top to bottom with little time for changes in matrix pore pressure to occur. In our example, preferential flow paths lead to local increases of pore-water pressure that, in combination with a loss of suction stress in the soil matrix, result in a critical drop of soil shear strength typical of forested soils on compacted bedrock . Yet in another situation, high pore-water pressure can originate from ephemeral springs or water exfiltration from fractured bedrock .

Our results show that force mobilization and redistribution in the soil and in the root system during the triggering of a shallow landslide is a complex process. Our model can be used to investigate the effects of the various components of the bond force system (roots and soil) on the dominant reinforcement mechanisms (tension or compression, lateral or downslope) and how these forces control the stability of the slope. Understanding which of these forces control slope stability under certain conditions is important for making appropriate simplifications when the full level of details is not needed or not known.

Figure  shows the displacement for nine hillslope simulations where trees, spaced 3 m apart, cover the entire slope. In Fig. a, tree diameter (DBH) is 50 cm; in Fig. b, it is 40 cm; in Fig. c, it is 30 cm. In each of the three cases, three simulations are shown: trees with roots that have both tensile and compressive strength (the standard behavior), roots that only have tensile strength, and roots with only compressive strength. All simulations are run with the same hydrologic loading used in earlier simulations (see Fig. ).

The slope behaves differently depending on the tree size and the type of root reinforcement. Root reinforcement for the 50 cm diameter trees is sufficiently large that the slope does not fail regardless of the type of root reinforcement (tensile, compressive, or both). For the 40 cm diameter trees, there is a threshold: the tensile strength of roots is needed to keep the slope stable. Without root tensile strength (compression only), the slope fails (Fig. b). Finally, for the 30 cm diameter trees, all root reinforcement configurations lead to slope failure, but at different times, with compression-only roots failing first and roots with both compression and tension last.

Results indicate that roots with only tensile strength limit downward slope slip under loading and delay slope failure more than roots that have only compressive strength. Roots that have both tensile and compressive strength offer the best protection against slope motion and slope failure. Neglecting root compression in the simulations results in only a couple of centimeters' difference in slope displacement or less than 1 h in the timing of the landslide. Neglecting root tension, however, can result in predicting a false slope failure. Also, neglecting tension misses the jump in displacement during the early initiation of the landslide, when roots across the tension gap in the upper part of the slope fail under tension (see Fig. c). Note also that when roots across the vertical crack fail in tension (as is the case for the 30 cm diameter trees), the slope eventually fails. This appears to be the case for all simulations we tested. However, simulations with roots that do not break across the widening crack do not necessarily remain stable over the duration of the simulation (2200 min).

Figure  illustrates the conditions under which the slope fails for the different tree-size diameters and root-strength configurations. Each graph in Fig.  shows a bond force along the downslope section at the center of the slope (x=0). Downslope root bond force along that section (Fig. a, d, g) indicates that when roots have no tensile strength (C only), roots in the lower section of the slope bear a higher compressive load. Similarly, roots that have no compressive strength (T only) bear higher tensile loads in the upper part of the slope, but only slightly higher than roots that have both tensile and compressive strength (T+C). As expected, roots of larger trees can bear higher tensile and compressive forces owing to higher root densities and more roots of larger diameters. Downslope soil bond forces (Fig. b, e, h) indicate that soils in slopes covered by smaller trees must take more of the compressive force caused by the slope downhill motion. For the 30 cm diameter trees (Fig. h), the soil compressive force eventually reaches its maximum value and the slope fails, regardless of the root configuration, because roots hold only a small fraction of the tensile or compressive resistance that helps maintain the slope stable: roots are too few and too small for this tree size. This is also the case for the 40 cm diameter trees when roots have only compressive strength (Fig. d, e). Because roots do not hold any tension in the upper part of the slope, and root compression is insufficient to support much load, soil bond in compression eventually reaches a maximum and the slope fails at t=1960 min. Figure c, f, and i show the lateral root force across the slope. This force is 1 order of magnitude smaller than the downslope force and has only a limited role in the slope stability for the cases shown here. These simulations indicate that downslope root and soil forces control slope stability, which is regulated by the maximum soil compression. Roots can reduce soil compression by taking up some of the force in tension in the upper part of the slope, preventing or delaying failure. Root compression alone is insufficient to offset soil compression in the lower part of the slope.

The structure of the stand (dimension, density, and relative position of trees) plays an important role on root reinforcement and slope stability. found that susceptibility to landslide was higher in plots with longer downslope gaps in the tree stand and in locations where the distance to nearby trees was higher. Conversely, also found that susceptibility to landslide was smaller where root reinforcement, based on tree diameter and distance from tree, was high. Weak zones, zones with low values of root reinforcement, can serve as initiation points for slope movement and control the location and size of a landslide . An example of a weak zone where a soil slip initiated is shown in Fig. . Roots around a tree provide sufficient stiffness to make the soil around the tree behave as a rigid body. The zone in between the tree and its neighbors does not provide sufficient root reinforcement and a gap opens as a result of loading (here rainfall). Here, we explore independently how tree size (diameter) and tree spacing can affect landslide initiation and hillslope stability.

SOSlope was used to test the influence of the range of root diameter classes on the stability of a slope. Figure  shows the displacement at the center of the slope (x=y=0) as a function of time for six simulations with different maximum root diameter: 5, 7, 8, 10, 20, and 100 mm. The simulations with 20 and 100 mm maximum root size diameter have no landslide and are practically indistinguishable. This is because the number of roots larger than 20 mm is insignificant and contributes little additional strength to the root bundle. The 8 and 10 mm simulations also do not fail and have only slightly larger displacements (6 to 7 cm instead of 5 cm for the 20 and 100 mm simulations). The two simulations with a maximum root diameter class of 5 and 7 mm, however, fail at 1400 and 1500 min, respectively. The threshold for stability is thus obtained by including root size up to 8 mm in diameter. Root reinforcement that includes only smaller roots is significantly smaller than if the entire bundle is included. Not including large roots can yield incorrect predictions of slope behavior.

Figure  shows the downslope and across-slope forces at the center line for several simulations with different maximum root-size diameter. The 5 mm simulation has the smallest amount of downslope root force but the highest across-slope root force and downslope soil compression, explaining why this simulation fails while others (10, 20, and 100 mm maximum root diameter) do not. Insufficient root density and lack of large roots compromises the stability of the slope by offering little resistance to loading and declining shear strength of soil. Lateral root forces are small for all cases and has a negligible impact here on slope stability (Fig. d).