Tree roots have long been recognized to increase slope stability by reinforcing the strength of soils. Slope stability models usually include the effects of roots by adding an apparent cohesion to the soil to simulate root strength. No model includes the combined effects of root distribution heterogeneity, stress-strain behavior of root reinforcement, or root strength in compression. Recent field observations, however, indicate that shallow landslide triggering mechanisms are characterized by differential deformation that indicates localized activation of zones in tension, compression, and shear in the soil. Here we describe a new model for slope stability that specifically considers these effects. The model is a strain-step discrete element model that reproduces the self-organized redistribution of forces on a slope during rainfall-triggered shallow landslides. We use a conceptual sigmoidal-shaped hillslope with a clearing in its center to explore the effects of tree size, spacing, weak zones, maximum root-size diameter, and different root strength configurations. Simulation results indicate that tree roots can stabilize slopes that would otherwise fail without them and, in general, higher root density with higher root reinforcement results in a more stable slope. The variation in root stiffness with diameter can, in some cases, invert this relationship. Root tension provides more resistance to failure than root compression but roots with both tension and compression offer the best resistance to failure. Lateral (slope-parallel) tension can be important in cases when the magnitude of this force is comparable to the slope-perpendicular tensile force. In this case, lateral forces can bring to failure tree-covered areas with high root reinforcement. Slope failure occurs when downslope soil compression reaches the soil maximum strength. When this occurs depends on the amount of root tension upslope in both the slope-perpendicular and slope-parallel directions. Roots in tension can prevent failure by reducing soil compressive forces downslope. When root reinforcement is limited, a crack parallel to the slope forms near the top of the hillslope. Simulations with roots that fail across this crack always resulted in a landslide. Slopes that did not form a crack could either fail or remain stable, depending on root reinforcement. Tree spacing is important for the location of weak zones but tree location on the slope (with respect to where a crack opens) is as important. Finally, for the specific cases tested here, intermediate-sized roots (5 to 20 mm in diameter) appear to contribute most to root reinforcement. Our results show more complex behaviors than can be obtained with the traditional slope-uniform, apparent-cohesion approach. A full understanding of the mechanisms of shallow landslide triggering requires a complete re-evaluation of this traditional approach that cannot predict where and how forces are mobilized and distributed in roots and soils, and how these control shallow landslides shape, size, location, and timing.

Shallow landslides are hillslope processes that play a key role in shaping
landscapes in forested catchments

Basal root reinforcement acting on the basal shear surface of the landslide. This is the most efficient mechanism, if present. In many cases, however, this mechanism is absent because the position of the failure surface is deeper than the rooting zone.

Lateral root reinforcement acting on lateral surfaces of the landslide. The magnitude of the contribution of this mechanism depends on the type of deformation of the landslide mass. If the landslide behaves as a rigid mass, lateral reinforcement may act almost simultaneously along all the edges of the sliding mass (in tension, shear, and compression). In cases where there is differential deformation of the soil mass, this leads to the progressive activation of lateral reinforcement, first in tension at the top of the landslide, and then in compression at the toe at the end of the triggering. The magnitude of lateral root reinforcement depends on the spatial distribution of the root network.

Roots stiffening the soil mass. The presence of roots in the soil increases the macroscopic stiffness of the rooted soil mass, leading to a larger redistribution of forces at the scale of the hillslope through small deformations. This mechanism increases the effects of the previous two (basal and lateral root reinforcements).

The magnitude of root reinforcement (a function of forest structure and tree
species composition). Root reinforcement needs to reach values of the order
of a few kilopascal in order to be significant

The heterogeneity of root distribution (tree species, topography, local soil condition, etc.). Root reinforcement must be active in specific places and at specific times to have any effect on slope stability: mean values of apparent cohesion across the entire hillslope are not representative and not sufficient for considering the specifics of actual root reinforcement effects.

The depth of the landslide shear surface (effects of basal root reinforcement). The deeper the shear surface is, the less important the effect of basal root reinforcement is.

The length and volume of the landslide (lateral root reinforcement and buttressing/arching mechanisms and stiffening effects). The larger the landslide is, the lower are the effects of lateral root reinforcement. In order to characterize the efficacy of roots for slope stabilization, a spatiotemporal quantification of root reinforcement is needed.

Here we present a new model for shallow slope stability calculations that
specifically considers these important effects. To fully understand the
mechanisms of shallow landslide triggering, a complete re-evaluation of the
traditional apparent cohesion approach is required. To do so, it is important
to consider the forces held by roots in a way that is entirely different than
done thus far. Moreover, measurements and models indicate that the assumptions
of constant elasticity and homogeneous root properties, as applied in typical
finite element geotechnical model, cannot reproduce the mechanisms leading to
the triggering of forested slope failures

The SOSlope (for Self-Organized Slope) model presented here fills this gap by
developing a mechanistic model for predicting shallow landslide sizes across
landscapes, considering the effects of root reinforcement in a detailed
quantitative manner (spatiotemporal heterogeneity of root reinforcement).
The SOSlope model allows for exploring the activation of root reinforcement
during the triggering process and helps to shed light on the contribution of
roots to the slope stability. The SOSlope model is used in this work to test
the following main hypotheses:

Both tensional and compressional forces resulting from mobilization of forces in the roots and the soil are efficient in stabilizing slopes but have higher effectiveness when occurring simultaneously.

Weak zones in the root network

Coarse roots dominate reinforcement and its efficacy, when present.

In what follows we first present a general background on the importance of vegetation for geomorphic processes in the context of hillslopes and landslides (Sect. 2). We then describe the SOSlope model in detail (Sect. 3), present the data set for roots and soil used in simulations (Sect. 4), show and discuss results (Sect. 5), and synthesize a typical force redistribution process during landslide triggering (Sect. 6). Conclusions are given in Sect. 7.

Understanding the role of shallow landslides in the geomorphic evolution of
landscapes is of prime importance and motivates the present work. In some
regions, shallow landslides are the dominant regulating mechanisms by which
soil is delivered from the hillslope to steep channels or fluvial systems

On long timescales, shallow landslides are important
geomorphic processes shaping landscapes of both vegetated and
non-vegetated basins. For vegetated basins, the spatiotemporal distribution
of root reinforcement has a major impact on the dynamic of sediment transport
at the catchment scale

While soil as a resource is gaining increasing attention in the context of
global sustainable development

While sustainable resource management in forestry and in agriculture aims
to keep the frequency of shallow landslide events to pseudo-equilibrium
conditions at the catchment scale and to reduce the overall erosion rate

Risks due to shallow landslides are associated with different types of
phenomena ranging from hillslope debris flows (example of process causing a
direct risk to infrastructures and individuals) to various channel processes
such as large sediment transport during floods, wood debris transport,
channelized debris flows, etc. (examples of processes causing an indirect
risk to infrastructures and individuals). It is estimated that landslides
triggered by heavy rainfall cause damages upwards of several
billions each year and more than
600 fatalities per year

Next to the constellation of factors well known to influence the triggering of
shallow landslides, vegetation has been recognized to play an important role

In the long term, the presence of vegetation
(i) increases soil production rates through mechanical and chemical processes

In the short term, vegetation mainly influences root reinforcement and
regulates water fluxes. At the hillslope scale, the hydrological effects of
vegetation are assumed to play a small role on slope stability compared to the
contribution of root reinforcement

Root are considered the hidden half of plants due to the difficulties in
characterizing and quantifying their distribution and mechanical properties.
In slope stability, the process of root reinforcement remains hidden because
direct observations have not yet been made on steep hillslopes. Field and
laboratory experiments

Methods for the quantification of different types of root reinforcement
mechanisms have been through a succession of models in the last few decades,
starting with the assumption of the simultaneous breakage of all roots

Breakage versus slip-out. Field observations show that in tree-root
bundles, the dominant failure mechanism of roots is by breakage

The contribution of root reinforcement must be differentiated between
different types of stress conditions: tension, compression, and shearing.
While most of the literature has focused on the shear behavior of rooted soils

The mechanical interactions of neighboring roots in a bundle are usually
neglected.

The mechanical and geometrical variability in roots was recently
considered using survival functions

The spatial and temporal heterogeneity of root reinforcement is related
to several factors such as topography, soil water content, soil disturbances,
resistance and resilience of forest cover to disturbances, and animal browsing

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

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

Soil basal resistance is

The driving force is

Bond forces are given by

The force in bond

Roots are binned according to their diameters in 1 mm size bins from 0.5 mm
to an upper limit given by data. A bin is usually referred to as a
root-diameter class, with

Roots are assumed elastic in both tension

The variability in root bio-mechanical properties (e.g., maximum tensile or
compressive strength, elastic moduli in tension or compression) due to the
presence of biological or geometrical weak spots is handled probabilistically.
The probability of failure of a root in tension (or in compression) is
captured by multiplying the elastic force by a Weibull survival function (

The soil bond force (

Rainfall-triggered shallow landslides can fail under saturated conditions
during increases of pore-water pressure and/or loss of suction under
unsaturated conditions

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)

We assume that the time evolution of the intrinsic pore-water pressure in the
macropores,

Pore-water pressure in the macropores (Eq.

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

Time evolution of pore-water pressures and water content for the dual-porosity model.

Mechanical soil parameters from

Hydrological parameters used in all simulations.

Model parameters for roots (Table

Soil parameters used in all simulations.

Soil strength as a function of displacement for different soil
depths. Values of passive earth pressure coefficients for estimating soil
compressional strength are calculated using a surface slope of 40

Root reinforcement as a function of bond elongation for different
tree diameters (DBH) and different distances from the tree trunk (

Tree-covered sigmoid slope, 70 m

Time evolution of

To illustrate the capabilities of SOSlope to reproduce the triggering of
shallow landslides influenced by the presence of tree roots, we first present
simulations of a 70 m

Root parameters used in simulations.

Figure

During loading, cells in the clearing move downhill more than cells in the
stand (Fig.

Time evolution of

Figure

Across-slope (also referred to as lateral or slope-parallel) root forces are
shown in Fig.

Figure

Bonds that were in tension in the upper part of the slope at

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.

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).

Initiation of downward motion after some cells reach critical condition
(factor of safety equal to 1). Force redistributions (compression in soil,
tension and compression in roots) prevent the slope from failing. These forces
increase with increasing load and increasing mean pore-water pressure
(e.g., Fig.

Culmination of compressive forces leading to failure when exceeded
(

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.

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

Initiation of slip at Castel Vecchio, Italy, that did not result in a landslide in the geomorphic sense, but is considered as one in the engineering sense. See text for details.

Time series of the factor of safety (red), displacement (blue), and
mean pore-water pressure (

The decrease in the factor of safety is linked to the increase in mean
pore-water pressure in the soil (Fig.

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

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.

Effects of tensile and compressive strength of roots on slope
displacement and stability for trees of different diameters. Displacement at
the slope center (

Effects of tensile (

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.

Figure

The structure of the stand (dimension, density, and relative position of
trees) plays an important role on root reinforcement and slope stability.

Our base scenario is the simulation presented earlier with trees 50 cm in
diameter spaced 3 m apart on a sigmoid hillslope with a 20 m

Example of a weak zone in a forested area showing isolated tree stumps with a root system that behaved as a stiff island during the opening of a gap in a weak zone in between root systems of adjacent trees.

The time evolution of the factor of safety depends on the tree size inside the
clearing. Simulations 50/40 and 50/50 have values of factor of safety that
remain significantly higher than the remaining simulations, although their
values sometimes oscillate very close to 1. Although these two configurations
have undergone some downhill motion, it is limited to a few centimeters,
significantly less than the other cases. These two slopes with large trees
are in critical condition because their factors of safety is nearly equal to
1 (

Time series of

Effect of clearing tree size diameter on slope displacement and soil
and root bond forces. From left to right, slope displacement (

Effects of clearing tree size on

Figure

This behavior is also illustrated in Fig.

Results in that figure clearly show that soil bond forces along the slope
center (Fig.

Figure

Effects of tree spacing on slope stability also yielded some unexpected
results. Trees were spaced evenly on the slope using the center of the slope
(

Effects of tree spacing on slope displacement at the center
(

Effect of tree spacing on hillslope behavior.

To explore the effect of crack location on slope stability, trees in the 7 m spacing slope were offset 2 m uphill, so that a vertical crack
would form at a higher elevation than without the offset. This simulation is
shown with a dashed curve in Fig.

SOSlope was used to test the influence of the range of root diameter classes
on the stability of a slope. Figure

Effect of maximum root-size diameter (5 to 100 mm) on displacement
at the slope center (

Effect of maximum root diameter on

Figure

Figure

The largest force that contributes to slope stability is soil compression in
the area above the landslide toe. There, soil compression increases initially
rapidly until it plateaus at about 700 min. During this increase, root
tension across a growing crack increases and also plateaus. Root compression
downslope similarly increases and then plateaus but is significantly smaller than either root tension
upslope or soil compression downslope. This time period is defined as phase 2
of our landslide initiation process, which starts when many areas of the
slope have a factor of safety that has decreased to 1 (Fig.

Evolution of

At

The time span of the three phases varies with tree size, tree spacing, maximum root diameter, and of course soil and hydrological properties (here fixed for all simulations). Looking back at Figs. 13, 16, and 18, phase 2 can last from several hours to less than one. Sometimes, no crack forms, there is no crack-root failure, and phase 2 and 3 overlap. When the slope has no clearing (as in simulations shown in Figs. 17 and 18), these same three phases exist but lateral forces play no role. Force redistribution and force balance is dominated by soil compression, adjusted by root tension in the upslope area and to a lesser extent root compression downslope. Root forces modify the force balance significantly but soil compression, due to its magnitude, dominates and controls the slope stability and its time to failure. Simulations with smaller soil depth will change this balance: smaller depth will decrease the absolute values of soil compression (see Fig. 3) and tree roots will then support tensile and compressive forces equal or greater to soil compression. In such a situation, roots may be the main factor controlling slope stability.

There are growing evidences that the effects of root reinforcement on slope
stability are the results of complex interactions of different factors in
which individual contributions are difficult to isolate using classical
methods (e.g., infinite slope calculations). The model presented here,
SOSlope, is the final element of a series of related studies aiming to quantitatively
upscale the stress-strain behavior of rooted soils under tension, compression,
and shearing. In this framework, SOSlope represents the final module where
previously investigated aspects of root reinforcements are combined to
quantify the macroscopic influences of root reinforcement on slope stability
considering spatial heterogeneities of root distribution. The model can
produce a systematic analysis of the factors influencing the contribution of
root reinforcement on slope stability, yielding a quantitative basis for
discussion of root reinforcement mechanisms for slope stabilization and
support for the assumptions or simplifications needed to implement such
effects in simpler approaches for slope stability calculations

Maximum root reinforcement under tension and compression does not take place simultaneously.

Root tensile strength is more effective than root compressive strength in preventing or delaying a landslide.

The stabilization effect of roots depends on their spatial distribution: the presence of a “weak zone” leads to behavior similar to bare soils. With little or no root reinforcement, slope failure is more likely and occurs earlier.

Root reinforcement at the macroscopic scale is dominated by intermediate to coarse roots when present. For the species considered here and based on available data, roots between 5 and 20 mm contribute the most to root reinforcement.

Tree positions in the tension zone of a potential landslide influence the stability of the slope. In general, the effect of lateral root reinforcement in tension contributes most to stability along the transition between stable and unstable zones of the hillslope where a crack can form.

These observations indicate that the standard, slope-uniform, constant
apparent cohesion approach for rooted soil is often inappropriate, especially
for forested slopes, where roots contribute significantly to the balance of
forces. For example, our model shows that the specific locations of trees on a
slope (Fig.

To our knowledge, SOSlope is the first model to implement a new approach that
characterizes the force-displacement behavior of rooted soils under both
tension and compression. Including this fundamental behavior is key for
understanding and modeling shallow landslide triggering. Further work is
needed to extend the applicability of standard geotechnical methods

The SOSlope model can be applied at the hillslope scale to investigate the effect of single factors such as root distribution and root mechanical properties (species specific) on slope stability, and quantification of bio-engineering measures and protective effects of forests. An important application at the hillslope scale is the testing of hypotheses that would support the simplification of calculations in problem-specific applications, e.g., for slope stability model at a regional scale.

The use of the SOSlope model at the catchment scale will be useful for
studying the effects of vegetation on slope stability processes in the
short and long term. In the long term, root strength can vary by orders of
magnitude

This paper uses only previously published data; no new data have been obtained.

DC and MS contributed equally to the model development and to writing the manuscript.

The authors declare that they have no conflict of interest.

We acknowledge EcorisQ members for providing on-going motivation to develop the SOSlope model and support from EcorisQ. Massimiliano Schwarz was funded by the Swiss Office for the Environment grant WoodFlow, and by the New Zealand MBIE program Growing Confidence in Forestry's Future (C04X1306). We thank the two anonymous reviewers for their constructive comments. Edited by: Richard Gloaguen Reviewed by: two anonymous referees