Estimating the mechanical effects of riparian vegetation on stream bank stability using a fiber bundle model

Transcription

1 WATER RESOURCES RESEARCH, VOL. 41,, doi: /2004wr003801, 2005 Estimating the mechanical effects of riparian vegetation on stream bank stability using a fiber bundle model Natasha Pollen and Andrew Simon Channel and Watershed Processes Research Unit, National Sedimentation Laboratory, ARS, USDA, Oxford, Mississippi, USA Received 9 November 2004; revised 7 April 2005; accepted 27 April 2005; published 23 July [1] Recent research has suggested that the roots of riparian vegetation dramatically increase the geomechanical stability (i.e., factor of safety) of stream banks. Past research has used a perpendicular root reinforcement model that assumes that all of the tensile strength of the roots is mobilized instantaneously at the moment of bank failure. In reality, as a soil-root matrix shears, the roots contained within the soil have different tensile strengths and thus break progressively, with an associated redistribution of stress as each root breaks. This mode of progressive failure is well described by fiber bundle models in material science. In this paper, we apply a fiber bundle approach to tensile strength data collected from 12 riparian species and compare the root reinforcement estimates against direct shear tests with root-permeated and non-root-permeated samples. The results were then input to a stream bank stability model to assess the impact of the differences between the root models on stream bank factor of safety values. The new fiber bundle model, RipRoot, provided more accurate estimates of root reinforcement through its inclusion of progressive root breaking during mass failure of a stream bank. In cases where bank driving forces were great enough to break all of the roots, the perpendicular root model overestimated root reinforcement by up to 50%, with overestimation increasing an order of magnitude in model runs where stream bank driving forces did not exceed root strength. For the highest bank modeled (3 m) the difference in factor of safety values between runs with the two models varied from 0.13 to 2.39 depending on the riparian species considered. Thus recent work has almost certainly overestimated the effect of vegetation roots on mass stability of stream banks. Citation: Pollen, N., and A. Simon (2005), Estimating the mechanical effects of riparian vegetation on stream bank stability using a fiber bundle model, Water Resour. Res., 41,, doi: /2004wr Introduction [2] The role of vegetation in the stability of hillslopes was largely ignored until the 1960s when researchers in the United States, the Soviet Union, and Japan began to focus attention on the effects of root reinforcement [Greenway, 1987]. Application of these effects on stream bank stability and other channel processes did not occur until the 1990s [Thorne, 1990] when tensile strength testing and mapping of roots of woody, riparian species were conducted and quantified as an additional cohesion [Abernethy and Rutherford, 1998; Simon and Collison, 2002]. Although the effects of vegetation on soil moisture and matric suction have been recognized for some time [Brenner, 1973; Biddle, 1983], it was not until the pioneering work in unsaturated soil mechanics by Fredlund et al. [1978] and the application of these techniques to vegetation-covered stream banks [Simon and Collison, 2002] that the hydrologic effects of vegetation on shearing resistance and stream bank stability were quantified. The generally stabilizing effect of vegetation on soils is currently the subject of field and experimental studies attempting to explain channel stabilization, This paper is not subject to U.S. copyright. Published in 2005 by the American Geophysical Union. morphology and patterns in fluvial systems over a wide range of temporal and spatial scales [e.g., Nanson et al., 1995; Gran and Paola, 2001; Tal et al., 2004]. Empirical studies have shown that vegetated channels erode more slowly, and are deeper and narrower than similar nonvegetated banks [Hickin, 1984; Hey and Thorne, 1986]. However, tools for quantifying the interactions between vegetation and bank processes are necessary if we are to move away from general discussion of vegetation effects to more accurate prediction of rates of channel widening for vegetated stream banks. The research presented in this paper focuses on improved quantification of mechanical reinforcement of soil by roots. [3] Stream bank instability and associated sedimentation are major issues of environmental concern due to their detrimental effects on agricultural land, infrastructure, and the overall ecological health of waterways. Vegetated buffer strips are widely advocated as a best management practice (BMP) in riparian environments because of their ability to act as a sink for contaminated water and sediment, whilst also providing diverse ecological niches. In addition to the ecological benefits of vegetated stream banks, it is commonly recognized that the root networks of plants and trees act to increase the apparent cohesion of soil through a combination of mechanical and hydrologic effects. However, 1of11

2 POLLEN AND SIMON: MODELING ROOT REINFORCEMENT OF STREAM BANKS quantification of these reinforcing effects is often difficult due to the invasive nature of investigations of root systems, and the impact of environmental variability on root architecture. The limited amount of data available on the physical properties of roots and the way they interact with a soil as it shears have therefore tended to limit the way in which the reinforcement of roots to a soil matrix has been modeled. Attempts to quantify root reinforcement of soil have been dominated by the use of simple perpendicular root models such as those developed by Waldron [1977] and Wu et al. [1979], which simply require knowledge of the tensile strength of the roots, and the cross-sectional area of fibers crossing the shear plane. Perpendicular root models can be considered to be static models, in that they estimate maximum root reinforcement at a single instance in time, when all of the roots contained in the soil matrix have reached their maximum tensile strength. [4] However, in reality as a soil-root matrix shears the roots contained within the soil have different tensile strengths and thus break progressively, with an associated redistribution of stress as each root breaks. The assumptions made by static models of root reinforcement therefore overestimate the increased apparent cohesion provided to the soil by a root network, as essentially they simply sum the tensile strengths of all of the roots. This paper discusses the results of fieldwork and laboratory studies carried out to test the assumptions of static root reinforcement models, and the root reinforcement model (RipRoot) constructed subsequently to include progressive root breaking during soil shearing. Root reinforcement values estimated by RipRoot are compared to those calculated using the perpendicular root model of Wu et al. [1979], and these values are used to establish the effect of using static versus progressive breaking root reinforcement models on predictions of stream bank stability. Results of the validation of the RipRoot model estimates are also presented. 2. Theory [5] In order to quantify the effects of root reinforcement on soil strength, two methods have commonly been used. In the first method, the values collected from in situ shear tests of root-permeated soils have been used to replace the value of the soil strength alone [e.g., Wu et al., 1988a]. However, in situ shear tests present a number of problems. Whilst they do provide a direct assessment of the amount of increased cohesion provided to the soil by the roots, isolating a block of root permeated soil to shear is not an easy task, and the soil and the anchoring of the roots may be disturbed before shearing is undertaken. In addition to this problem, Wu et al. [1988b] comment that when carrying out in situ tests of root permeated soil, the forces developed in the roots and therefore their contribution to soil shear strength, are dependent on the dimensions of the shear box. [6] The second method involves the development of physically based formulae to establish the relationships between the root and soil properties that cause the roots to increase the shear strength of the soil. These physical formulae generally form the basis for static root reinforcement models in which the forces acting in the root-soil matrix are considered at an instant in time, usually taken as being when all of the roots reach their maximum tensile stress. One such formula is the simple perpendicular root model developed by Waldron [1977]. This root reinforcement model is based on the Coulomb equation in which soil shearing resistance is calculated from cohesive and frictional forces: S ¼ c þ s N tan f where S is soil shearing resistance, s N is the normal stress on the shear plane, f is soil friction angle, and c is the cohesion. [7] To extend equation (1) for root-permeated soils, Waldron [1977] assumed that all roots extended vertically across a horizontal shearing zone, and the roots act like laterally loaded piles, so tension is transferred to them as the soil is sheared. The modified Coulomb equation therefore became S ¼ c þ DS þ s N tan f where DS is increased shear strength due to roots (kn m 2 ). [8] In this simple root model, the tension developed in the root as the soil is sheared is resolved with a tangential component resisting shear and a normal component increasing the confining pressure on the shear plane and DS can therefore be represented by ð1þ ð2þ DS ¼ T r ðsin q þ cos q tan fþ ð3þ where T r is average tensile strength of roots per unit area of soil (kpa). [9] Gray [1974] reported that the results of several studies on root permeated soil showed that the angle of internal friction of the soil appeared to be affected little by the presence of roots, and sensitivity analyses carried out by Wu et al. [1979] showed that the value of the bracketed term in (3) is fairly insensitive to normal variations in q and f (40 90 and respectively) with values ranging from 1.0 to 1.3. A value of 1.2 was therefore selected by Wu et al. [1979] to replace the bracketed term and the simplified equation became DS ¼ T r ða R =AÞ1:2 ð4þ where A R /A = is root area ratio (no units). [10] Thus, according to the simple perpendicular root model of Wu et al. [1979], the magnitude of reinforcement simply depends on the amount of roots present in the soil, and the strength of those roots Assumptions and Limitations of the Equation of Wu et al. [1979] [11] A number of assumptions are made in the root reinforcement model of Wu et al. [1979]. The simple model of Wu et al. [1979] assumes that the roots are perpendicular to the slip plane. However, the angles of the roots in relation to the direction of the force applied to the soil are important, as this dictates the distribution of stresses within the root volume and hence the maximum tensile strength reached before failure of the root occurs [Niklas, 1992]. Extended models allowing for inclined roots have been developed, but Gray and Ohashi [1983] have shown from laboratory tests that perpendicular orientations of reinforcing fibers provide 2of11

3 POLLEN AND SIMON: MODELING ROOT REINFORCEMENT OF STREAM BANKS slip out of the soil the soil-root bond must be broken [Ennos, 1990], and the slipping term is hence a function of the surface area of the root, and the soil properties [Waldron and Dakessian, 1981, p. 428]: F P ¼ 2tL=d ð5þ Figure 1. Graph showing estimated pullout versus breaking forces for river birch roots in a soil with strength of 6 kpa. where F P is pullout force (kpa), t` is maximum tangential stress (kpa), L is root length (m), and d is root diameter (m). [14] However, roots may not be able to withstand this pullout force, and may break. For each soil type and moisture content a threshold root surface area therefore exists above which the frictional bond between the root and soil is stronger than the tensile strength of the root; the full tensile strength of the root can thus be realized and the root breaks. Below the threshold, the roots will be pulled out of the soil as their frictional bonds with the soil are weaker than their tensile strength; in this case the frictional bonds are not strong enough to allow the uptake of tension in the roots and they are simply pulled out of the soil (Figure 1). comparable reinforcement to randomly orientated fibers. This lends support to the use of the simple perpendicular root model, where it may be assumed that the roots are randomly orientated in the soil. [12] An important assumption of equations (3) and (4) is that the full tensile strength of all the roots is mobilized when the soil shears. However, laboratory and field strength testing of stream bank materials and riparian roots show that root strength is typically mobilized at much larger displacements than soil strength [Pollen et al., 2002, 2004]. In a soil, peak strength is typically mobilized in the first few millimeters of strain, and then decreases to a residual value reflecting particle realignment that minimizes shear resistance. However, in the case of some roots, straightening of the root to remove its tortuosity may occur before strain is taken up. Whether or not straightening occurs is a function of the strength of the frictional bond between the roots and soil. This bond is largely dependent on soil type and soil moisture, and therefore in some cases the force required to break a root may be greater than that required to straighten it. Roots which have been straightened before breaking will generally break at greater displacements than the displacement of soil at its peak strength. This suggests that peak root strengths may not be fully mobilized at the time of maximum soil instability, and that the banks may fail before the full theoretical contribution from roots is achieved [Pollen et al., 2004]. Over prediction of the increased soil shear strength may therefore occur [Waldron and Dakessian, 1981], but analysis of the stress displacement characteristics of roots has been limited to date so it is unknown how significant this overestimation may be. [13] The model of Wu et al. [1979] also assumes that the roots are well anchored and do not pull out when tensioned. Laboratory and field shear tests have shown that, as with the failure of other composite materials, root failure occurs by two mechanisms: pullout (slipping due to bond failure) or rupture (tension failure) [Coppin and Richards, 1990]. The predominant failure mechanism, as with other composite materials is a function of the variations in material properties, and the geometries of the fibers and the matrix [Beaudoin, 1990]. As tension is applied to the root, for it to 3of Dynamic Fiber Bundle Models [15] The field of materials science has seen a large increase in understanding and modeling of systems over the last 60 years. One of the most important developments has been the construction of fiber bundle models (FBMs) to aid in the understanding of composite materials. The first of these models was proposed by Daniels [1945] with many versions and developments on this first model having since ensued. The basic principle of a standard FBM is that the maximum load withstood by the bundle of fibers is less than the sum of each of their individual strengths. This is because when a load is applied to the bundle the fibers will not all break at the same time; as noted earlier, it is this very assumption that causes the Wu et al. [1979] model to overestimate the additional cohesion provided by roots in a soil. FBMs therefore take this into account by following simple rules: an initial load is added to the bundle, containing a number, n, of parallel fibers. Although at first the load is distributed equally between the n fibers, once the load is increased sufficiently for a fiber to break, the load that was carried by the fiber is redistributed to the remaining (n 1) intact roots, each of which then bears a larger share of the load than before and is hence more likely to give way itself. If this redistribution of load causes any further roots to break, further redistribution of load occurs (in this type of model this is known as an avalanche effect), and so on until no more breakages occur. Another increment of load is then added to the system, and the process is repeated until either all of the fibers have been broken, or the matrix the fibers are contained in fails. [16] As stated above, the simplest of these FBMs assume that the load from the broken fibers is distributed evenly between the remaining intact fibers, according to their diameters. This type of load redistribution is known as global load sharing (GLS) [Hidalgo et al., 2002]. The alternative to GLS is a local load sharing (LLS) approach. In an LLS FBM, the load is redistributed according to the proximity of intact fibers to the broken fibers [Hidalgo et al., 2002]. GLS FBMs assume that the interactions between the fibers in the bundle are long-range, whereas the LLS FBMs assume that short-range interactions between the

4 POLLEN AND SIMON: MODELING ROOT REINFORCEMENT OF STREAM BANKS Figure 2. Views of Root-Puller (a) showing winching of attached root and (b) detailed view of the connection between the root and load cell. fibers are more dominant. Hidalgo et al. [2002] state that in reality in homogenous materials, the load sharing rules should fall somewhere between GLS and LLS, since an important fraction of the load is redistributed to the entire matrix, even though the concentration of redistribution remains in the locality of the breakage. [17] As is necessary with all models, some assumptions about the behavior of the fibers are made. In the case of the simple FBMs described so far, the main assumptions made are that the elastic properties of the fibers are the same, and that the fibers are all parallel to each other and the direction of loading. Further extensions to FBMs include timedependent models in which fibers break after being exposed to a particular stress for a certain time [Gomez et al., 1998], and models that take into account the possible nonlinear behavior of materials. Despite their simplicity FBMs are easily parameterizable and incorporate the most important aspects of soil-root interactions, using an approach that removes the assumption that all of the fibers in a matrix break simultaneously. 3. Methods 3.1. Field Data Collection [18] Root tensile strengths and stress displacements were measured using a device called the Root-Puller (based on a design by Abernethy [1999]) (Figure 2). This is composed of a metal frame, with a winch attached to a load cell and displacement transducer, both connected to a Campbell CR510 data logger. The Root-Puller was attached to the side of the trench by tying a rope around the stem of the tree, and different size roots were attached to the load cell and displacement transducer using u bolts of varying sizes. The winch was then cranked at a steady rate until the root broke, and the diameter of each root was recorded. Table 1 shows a list of the species tested and the field site locations. Each of the species was chosen because of their common occurrence throughout riparian zones across the United States. Table 1 also shows the number of roots of each species tested for tensile strength. The number tested varied widely between species, largely due to variations in the number of roots available at a particular site, but a minimum of 30 roots of each species were tested so as to be able to distinguish any trends in the data from natural variability. [19] The roots to be tested were selected based on a number of factors. First, roots of isolated trees were preferable to those in dense stands as it was easier to determine which specimen and thus species each root belonged to; color and smell also helped root species identification in some cases. Second, roots intersecting the wall profile perpendicular to the bank face were tested first, but where the number of suitable roots to be tested was lower, roots intersecting the bank face at angles were also sampled. Third, the roots tested were all within a particular diameter range (0.1 to 20 mm for the tree species and 0.1 to 5 mm for the grass species). The upper diameter limit for the roots of the tree species was necessary because of the load limit of the load cell used. The upper diameter limit for the grass species was simply a cause of the smaller variation in root sizes for these species. In all cases the trenches were dug on the upslope side of the trees, parallel to the bank face. Although the roots of plants commonly grow asymmetrically on sloping planes [Chiatante et al., 2002], the gentle gradients at the locations of trenching meant that any asymmetry in rooting architecture, root shape and tissue organization [Chiatante et al., 2002] that could have affected root tensile strengths and the distribution of mechanical forces within the soil should be minimal. [20] The maximum load applied to each root before breaking, and its diameter at the point of rupture (the point of root clamping for all tests carried out here) were used to calculate the tensile strength of each root. Root tensile Table 1. Species Tested for Root Tensile Strength With the Number of Roots Tested for Each Species and the a and b Values for the Parameters of the Tensile Strength Curves Attained From Field Data Common Name Latin Name Location Number of Roots Tested for T R a b r 2 Black willow Salix nigra Mississippi Cottonwood Populus fremontii Oregon, Kansas Douglas Spirea Spirea douglasii Oregon Eastern sycamore Plantanus occidentalis Mississippi Gamma grass Tripsacum dactyloides Mississippi Himalayan blackberry Rubus discolor Oregon Longleaf pine Pinus palustris miller Mississippi Oregon ash Fraxinus latifolia Oregon River birch Betula nigra Mississippi Sandbar willow Salix exigua Kansas Sweetgum Liquidamber stryaciflua Mississippi Switchgrass Panicum virgatum Mississippi of11

5 POLLEN AND SIMON: MODELING ROOT REINFORCEMENT OF STREAM BANKS Figure 3. Flowchart of the simple rules used to construct the RipRoot model. strength-diameter relations were developed for each species to be used as input to both the perpendicular root model of Wu et al. [1979] and the fiber bundle model, RipRoot Laboratory Tests [21] Saturated direct shear tests were run with and without roots to establish the magnitude of actual root reinforcement compared to the estimates obtained using the Wu et al. [1979] equation and the RipRoot model. The saturation, consolidation and running speed of the direct shear tests were all selected for the soil type, using the guidelines of Lambe [1951]. Normal forces were applied to the tests to match the bulk density (15 kn/m 3 ) and depth at which the samples were taken (0.5 m). The direct shear tests were all performed on a silt soil chosen as representative of upper bank profile soils in the main area of this study (northern Mississippi). It should be noted here that soil properties and vegetation communities are not independent of each other; particular species may be associated with certain soil properties and growing conditions. Root growth, tensile strengths of roots and soils, and the interaction of roots and soil during shearing may therefore vary for other species and growing conditions to those tested here. [22] Ten control tests without roots were carried out first to obtain a mean shear strength value for the soil alone. Twenty samples containing switchgrass roots were then tested to obtain values for shear strength with roots. Switchgrass was selected as the test species because its roots only range in size from <1 mm to approximately 5 mm in diameter; the benefit of this small range of root sizes compared to the tree species studied was that the diameters of the roots remained small relative to the diameter of the shear box sample, minimizing the edge effects that could be caused by the use of samples containing large roots. The mean shear strength for the soil was subtracted from each of the root-permeated samples to obtain an estimate of the root reinforcement. After each root-permeated test, the sample was removed and carefully cut along the shear plane to 5of11 expose any roots that dissected it. All of the root diameters were measured and recorded and used as input data in the Wu et al. [1979] equation and RipRoot model RipRoot Model [23] The main source of error in the estimates of DS produced using the Wu et al. root model comes from the assumption that all of the roots break at the same time, and that their full tensile strength is mobilized at breaking point. As an alternative, a fiber bundle model (FBM) was constructed for the riparian species under investigation. The fiber bundle model, RipRoot, takes into account the fact that roots within the soil matrix have different maximum strengths, and therefore break at different points as a load is applied to the soil. In addition to this, the model redistributes the load from the broken roots at each step in the model, to the remaining intact roots crossing the shear surface. Although RipRoot follows just a few simple rules, it removes the assumptions made by the simple perpendicular root models and should therefore reduce overestimations included in estimates from the Wu et al. [1979] model. [24] The flowchart in Figure 3 shows the simple rules used to develop RipRoot. Within the model, an initial increment of load is added to the bundle, containing a number (n) of parallel fibers. The stress is distributed equally across the roots crossing the shear plane so that smaller fibers hold a smaller proportion of the load than larger fibers. The loading of the shear plane continues incrementally until the load is increased sufficiently for a fiber to break, following which the load that was carried by the fiber is redistributed to the remaining (n 1) intact roots, each of which then bears a larger share of the load. If this redistribution of load causes any further roots to break, further redistribution of load occurs, and so on until no more breakages occur. Another increment of load is then added to the system, and the process is repeated until all of the fibers have been broken, the matrix containing the fibers fails, or the full driving force acting on the soil-root matrix is supported by the root bundle. The most important rule in RipRoot relates to the way that the model calculates how the load is to be redistributed to the remaining intact roots once one or more roots break. In this case a global load sharing (GLS) approach was taken, in which the stress was redistributed equally to the remaining intact roots, therefore assuming long-range interactions [Moreno et al., 2001] among the soil-root matrix. The use of a simple GLS approach can be justified for the case of the stream bank failure planes that are being modeled; as failure blocks of stream banks tend to be relatively small compared to failures on slopes [Abernethy and Rutherford, 1998, 2000], it is acceptable to assume that long-range interactions exist across the entire failure block. In the case of slope stability it would be better to use a local load sharing (LLS) approach because redistribution of stresses would be more unlikely to act equally over the potentially larger shear plane surface. An additional assumption of the RipRoot model is that the elastic properties of the roots are all the same. Although this assumption simplifies reality, it is difficult to include the complexities of root tortuosity and variations in uptake of strain, with the data available thus far. Finally, in RipRoot all roots are assumed to break rather than pull out of the soil. It is reasonable to make this assumption as Pollen et al. [2004] demonstrated that for the soil conditions they

6 POLLEN AND SIMON: MODELING ROOT REINFORCEMENT OF STREAM BANKS Figure 4. Tensile strength data for (a) longleaf pine (high degree of scatter) and (b) black willow (low degree of scatter). investigated, similar forces were required to break roots and pull roots out of the soil. [25] In classic FBMs developed in the field of materials science, the strengths of the fibers, are distributed according to a probability distribution. The most commonly used distribution for this type of model is the Weibull distribution. However, in RipRoot the strengths of the roots are determined using the field data pertaining to the frequency of different size roots, collected during Root Area Ratio investigations [Pollen et al., 2004]. The strength of these roots is then calculated from the diameter strength curves established from the tensile strength data. In this case the use of the field data collected provides a more conceptually appropriate basis for the distribution of root tensile strengths than the Weibull distribution. [26] The model allows the user to select the riparian species to be considered, the number of roots within the soil matrix, the increments of load increase (a value between 0 and the driving force acting on the bank), and the length of bank the roots are embedded in. The driving force acting on the bank and the length of the shear plane the roots act over, vary for stream banks with different geometries and geotechnical properties. These values are obtained automatically by RipRoot from the bank stability model of Simon et al. [1999]. [27] As in the equation of Wu et al. [1979], the final output from the RipRoot model is an estimate of root reinforcement (DS) that is added to the shear strength of the soil. The interaction of the roots and soil they are embedded in are not considered in this version of the RipRoot model, so all of the driving force acting on the bank is assumed to be transferred to the roots, along with any redistributed stresses from broken roots. Future work will introduce the effects of soil moisture and soil type, but the aim of this paper was to investigate the differences in root reinforcement estimates between situations with simultaneous and progressive root breaking Stream Bank Stability Model Runs [28] Stream banks of heights 1, 2, and 3 m were simulated using the bank stability model of Simon et al. [1999]. The bank was constructed entirely of silt (friction angle = 25, cohesion = 5 kpa) with the bank angle set to 80. Inall model runs the flow depth was set to be 30% of the bank height, representing a relatively low flow condition typical of the recession of storm flows when bank stability is at its lowest. Model runs were carried out for scenarios with (1) no vegetation (2) root reinforcement estimates from the RipRoot model and (3) root reinforcement estimates calculated using the Wu et al. [1979] equation. All estimates of root reinforcement were based on calculations involving 200 roots for each tree species and 1000 roots for Switchgrass as this species tends to have denser root networks [Simon and Collison, 2002]. These values were based on field data presented in two earlier papers [Simon and Collison, 2002; Pollen et al., 2004], which showed that the number of roots crossing a shear plane ranged from approximately for 3 12 year old riparian trees and roots for clumps of 5 year old switchgrass. 4. Results and Discussion 4.1. Root Tensile Strength [29] Root tensile strengths decreased nonlinearly with increasing root diameter for all twelve of the riparian species tested; curves of best fit for the data sets were of the form y = ax b (Table1)[Pollen et al., 2004]. The force required to break a root increases linearly with increasing root diameter, but tensile strength is calculated per unit area. Therefore smaller roots are stronger per unit area than large roots, resulting in decreasing root tensile strength with increasing root diameter (Figure 4). This decreasing tensile strength with increasing root diameter is a trend that has been found in many previous studies of root strength [Greenway et al., 1984; Wu, 1976; Burroughs and Thomas, 1977]. [30] The data sets show a considerable degree of scatter, with the r 2 values (Table 1) ranging from just 0.14 for longleaf pine, to 0.75 for black willow (Figure 4), but in all cases the regressions were significant (p < 0.01). The fact that the tensile strengths of the roots of different riparian species show large scatter is perhaps unsurprising given the variability in a number of controlling factors other than just diameter, for example, moisture content and root bark roughness and thickness. Plants adapt dramatically to their 6of11

7 POLLEN AND SIMON: MODELING ROOT REINFORCEMENT OF STREAM BANKS Figure 5. Results of direct shear tests compared to estimates of root reinforcement calculated using (a) the Wu et al. [1979] equation and the RipRoot model and (b) a detail of Figure 5a using a constrained y axis to show only the RipRoot results compared to the direct shear tests. local environmental conditions, such as soil moisture, texture, and nutrient status. As such, variations in growth rates occur not only between species, but also between individual specimens of the same species. These variations in root growth can affect some of the variables determining root tensile strength, for example, root tortuosity, elasticity, and root moisture [Collins, 2001]. [31] The angle of the root relative to the face of the trench and the Root-Puller may also have affected the tensile strength measurements taken. Although an attempt was made to make sure the roots were perpendicular to the trench face, in some cases the roots were pulled at slight angles, hence developing a slight bending stress (tension on one side of the root and compression on the other side) rather than a true tensile stress [Wu et al., 1988a]. These slight variations in root angles may therefore have added to the variations in tensile strength seen in the results. In addition to variations in root tensile strength with root diameter, variations in root wood strength have even been found to occur along roots with a fairly constant diameter, with the degree of variation being related to the root architecture of the system [Stokes and Mattheck, 1996]. Plate root systems tend to be dominated by shallow lateral roots with little branching. Loading forces in this type of root network are usually transmitted into the soil further 7of11 away from the stem, so the high root wood strength observed along the entire length of roots of plate root systems may therefore be an adaptation to resist mechanical stress. In contrast, heart root systems are more highly branched with a network extending both laterally and vertically through the soil. In heart root and taproot systems, tensile strengths decline with increasing distance away from the plant stem, possibly because the strength provided by the main root means that a high investment in strength further along the roots in unnecessary [Stokes and Mattheck, 1996]. It is therefore possible that some of the scatter in the tensile strength data sets is associated with changes in strength along the length of the roots, independent of soil diameter. [32] Despite the high degree of scatter within some of the data sets, a nonlinear decreasing trend does exist for all of the 12 species, and whilst the fit of the curve to the data varies between species, the use of these curves is the best way to compare the tensile strength data collected (Table 1). The tensile strength curves all lie within the same range of tensile strength values, but some differentiation between species can be seen, particularly for small diameter roots where variations in the b parameter of the curves has the greatest effect on tensile strength values Measured Versus Estimated Root Reinforcement Values [33] The results from the direct shear tests without roots gave mean soil shear strength of 6.69 kpa for the silt soil tested, with a small standard deviation of This mean soil shear strength value was subtracted from the shear strengths of each of the root-permeated samples to give the root reinforcement for each sample. These root reinforcement values were then compared to the corresponding Wu et al. [1979] and RipRoot estimates for each sample. [34] For the example of switchgrass roots and the representative silt soil, the values for root reinforcement provided by the Wu et al. equation are significantly different to the data set from the direct shear tests (Mann-Whitney U test, p < 0.001), with Wu et al. [1979] estimates values ranging from 619% to 1433% higher than the direct shear tests (Figure 5). Overestimation by the Wu et al. equation can be seen to increase as the number of roots in the sample increased. The large discrepancy between the two data sets is a result of the simplistic nature of the root reinforcement equation of Wu et al.; the equation assumes that all of the roots in each direct shear test broke, but in reality the driving forces exerted on each soil sample were not sufficient to break all of the roots. This is known because when each soil sample was cut along the shear plane after testing, in all cases at least some of the roots had to be cut to separate the two halves of the sample. These direct shear tests therefore show another deficiency of the simple root model, which is that the driving force acting on the shear surface is assumed to be great enough to shear both the soil, and the entire root network. [35] The results of the direct shear tests with and without Switchgrass roots were used to validate the DS estimates produced with RipRoot. The number of roots crossing the shear plane of each shear test, and their diameters were input to the model. For a stream bank, the driving force acting on a failure block is the downward force acting on the failure block. For the direct shear tests the driving force

8 POLLEN AND SIMON: MODELING ROOT REINFORCEMENT OF STREAM BANKS for each model run was calculated from the maximum normal force applied to each shear test. The results of this validation are seen in Figures 5a and 5b. Figure 5a shows the difference between the control direct shear tests and those with roots and compares these values to those estimated using the Wu et al. equation and the RipRoot model. [36] It can be seen from Figure 5a that the estimates of root reinforcement from RipRoot are much more accurate than those calculated using the Wu et al. equation. Whereas the Wu et al. values vary from +640 to +1430% of the actual root reinforcement measured in the direct shear tests, the estimates from RipRoot vary from 60 to +110% of the direct shear tests. Although this variation from the actual measured values is still relatively large, the estimates are considerably more accurate than those calculated using the Wu et al. equation. Statistical tests (Mann-Whitney Rank Sum tests) carried out to determine whether the data were from significantly different populations showed that the direct shear values and the Wu et al. values were significantly different (p < 0.001), but that the direct shear values and RipRoot estimates were not significantly different (p = 0.113, critical p = 0.05). [37] Figure 5b shows just the direct shear root reinforcement values and the estimates from RipRoot. An interesting feature of this graph is that the root reinforcement values measured in the direct shear tests are only linear up to a particular threshold of root density within the shear box samples. Above the threshold of approximately 20 roots in each sample, root reinforcement increased exponentially, suggesting a threshold above which the load is more optimally distributed across the root system. The presence of this threshold supports other work carried out on reinforced matrices, for example, Schmidt et al. [2001]. They note that the strength of densely reinforced soil is not linearly related to reinforcement concentration, although the threshold reported therein (RAR = 10 2 ) was a magnitude higher than the threshold in this study (RAR = 10 3 ). The most likely explanation for this difference in the threshold RAR is that the relative strengths of the matrix and fibers differed between studies. The RAR of 10 2 noted by Schmidt et al. [2001] was for tests carried out in fiberreinforced sand, whereas the direct shear in this study had higher silt content. The relative strengths of the fibers used in each study is not known, but even the difference in matrix properties is likely to be sufficient to alter the rates of uptake of stress in the fibers, hence affecting the threshold RAR of fibers above which root reinforcement increased nonlinearly. [38] RipRoot overestimated root reinforcement at low root densities but underestimated root reinforcement at higher root densities. At low root densities it appears that the assumption made by RipRoot that all of the force acting on the shear plane is directed to the roots, underestimates the importance of the shear strength of the soil, with less force being transferred to the small cross-sectional area of the roots than expected. In contrast, at higher root densities the uptake of stress by the roots is greater than predicted by RipRoot. This may be due to the assumption that all of the roots contained in the sample have the same elastic properties. The load displacement characteristics of the roots tested, showed a random distribution of rates of uptake of load; this was due in part to the tortuosity of the roots which were measured in situ, causing part of each load displacement curve to simply show root straightening rather than uptake of stress. As uptakes of stress vary even for roots of the same diameter, it is possible that the underestimation of root reinforcement by RipRoot at higher root densities is a result of loading of roots with higher tensile modulus values in the direct shear tests, which cannot be predicted within the current version of RipRoot. Further development of the model may allow a root elasticity function to be added. It should be noted here that there is one limitation associated with testing the RipRoot model using direct shear tests. In a stream bank failure, the movement of the failure block is caused by an initial stress, whereas in the direct shear tests the shearing is controlled by a constant strain. This difference in controlling variables may be a limiting factor in the validation of the RipRoot model. [39] From the view point of practical stream management it is preferable that any inaccuracies in root reinforcement are underestimates rather than overestimates of potential values. This is because it is better to be conservative in the prediction of any vegetative reinforcing effects, and resulting stream bank factor of safety values so that stabilities are presented in the form of a worst-case scenario. Overestimates in root reinforcement and stream bank stabilities may lead to overconfidence in the stability of a particular stream bank, with potentially disastrous consequences in critical cases where for example, a structure such as a bridge is at risk. In this regard the underestimation by RipRoot at higher root densities is slightly less concerning than the overestimation at low root densities. It is important to stress though that both the under and overestimations by RipRoot discussed here are orders of magnitude smaller than the inaccuracies produced with the Wu et al. equation. [40] The results of RipRoot compare favorably to validation data in the switchgrass example. However, tree root diameters have a much greater range of root diameters and tensile strengths. Further validation for tree species was not possible as the sample diameter for the direct shear apparatus was just 6.5 cm. As long as the tensile strength diameter curve accurately reflects the variations in tensile strength with root diameter, RipRoot estimates for tree species should be as accurate as for the switchgrass investigated here. However, additional experiments need to be carried out to test this expectation. [41] In addition to the more accurate estimates of root reinforcement from RipRoot compared to the Wu et al. [1979] equation qualitative evidence also supports the way in which RipRoot models soil-root interactions during shearing; during the direct shear tests only some, if any of the roots actually broke, with some having to be cut to expose the failure surface so the root diameters could be measured. As part of the model output from RipRoot, the number of roots predicted to be left intact is given. In the case of all of the validation tests run, the RipRoot model predicted that some, or in a few cases, all, of the roots would remain intact under the driving forces applied Estimates of DS Using the Wu et al. [1979] Equation Versus RipRoot [42] The root reinforcement estimates calculated for the direct shear tests containing switchgrass roots showed a significant difference between the Wu et al. equation values 8of11

9 POLLEN AND SIMON: MODELING ROOT REINFORCEMENT OF STREAM BANKS Table 2. Comparison of RipRoot and Wu et al. [1979] Root Reinforcement Estimates for Roots of Each Species Acting on a 10 m Long Reach, With a Shear Surface Length of 1.15 m Species Number of Roots RipRoot DS, kpa Wu et al. DS, kpa RipRoot/ Wu et al. Cottonwood Cottonwood Cottonwood Cottonwood Cottonwood Sycamore Sycamore Sycamore Sycamore Sycamore River birch River birch River birch River birch River birch Pine Pine Pine Pine Pine Switchgrass Switchgrass Switchgrass Switchgrass Switchgrass Black willow Black willow Black willow Black willow Black willow and those of RipRoot. To establish the magnitude of differences between the two root reinforcement models, for the remaining species included in RipRoot, estimates were obtained for hypothetical simulations with the number of roots varying from 200 to 1000 for each species, at intervals of 200 roots. It should be noted here that differences between species for the hypothetical model runs are simply a function of variations in tensile strength between species because the root size distributions were all set to be identical. [43] The results show that for all species the Wu et al. [1979] estimates are greater than those of the RipRoot model (Table 2). The difference in root cohesion values varies for each species, but the ratio between the models remains fairly constant for each species over the range of roots tested. The shape of the root tensile strength diameter curves seems to have an influence on the amount of overestimation produced by the Wu et al. equation, with species with similar b parameter values showing ratios of RipRoot/Wu et al. in the same range. For example, cottonwood and river birch have b parameter values of 0.66 and 0.64 respectively and have similar values for the RipRoot/Wu et al. ratio of approximately 0.61 (Table 2). Similarly eastern sycamore and black willow have values for the ratio of approximately 0.75 (Table 2) with b parameter values of 0.94 and 1.1 respectively. It therefore appears that for the tree species, the more negative the exponent of the tensile strength curve, the lower the overestimation by the Wu et al. equation. [44] The higher overestimation for cottonwood and river birch results from the fact that their negative exponents produce tensile strength curves with weak small roots compared to eastern sycamore and black willow. These weak small roots therefore break fairly early in the model run before the peak load is attained. The Wu et al. equation assumes that these small roots break at the same time as the large roots hence producing the overestimation. In the case of black willow and eastern sycamore, their smaller roots are stronger and so fewer have broken by the time the peak load is attained, leading to less overestimation by the Wu et al. equation. [45] For switchgrass we must consider not only the tensile strength curve, but also the different distribution of root diameters and therefore tensile strengths compared to the tree species. The distribution of roots is also smaller, ranging from just 0 to 5 mm instead of 0 to 10 mm for the tree species. During each run of RipRoot, the number of roots in each diameter size class is used to distribute root diameters evenly within each diameter range. Therefore as the number of roots in a particular size class increases, the diameters and resulting tensile strengths of two consecutive roots become closer together in value. However, the narrower distribution of root diameters for grass model runs (0 to 5 mm instead of 0 to 10 mm) means that each of the smaller diameter size classes contains more roots than in the tree species model runs with the same number of roots. Tensile strengths calculated for the grass roots are therefore closer together in value than values for the tree species. During the model runs for grass the result of this is a gradual breaking of roots as the increments of load are added to the system, producing a large overestimation by the Wu et al. [1979] equation of approximately 52%. [46] It can be seen that there is an order of magnitude difference between overestimation of root reinforcement by the Wu et al. [1979] equation in Figure 5 (approximately %) and Table 2 (approximately 20 50%). This difference in overestimation is due to the driving forces involved in the direct shear tests compared to the model runs carried out to obtain the values in Table 2. As stated previously, in the direct shear tests the driving forces were not great enough to break all of the roots. The root reinforcement measured during the shear tests, and the accompanying values calculated by RipRoot (Figure 5) take the driving forces into account, and root reinforcement values therefore do no represent the maximum value that could be attained under higher driving forces. The equation of Wu et al. [1979] assumes that all roots break, regardless of driving forces, which is why overestimation is so large in Figure 5. For the model runs in Table 2 the driving forces used were great enough to break all of the roots, and overestimation by the Wu et al. [1979] equation was thus smaller Variations in Stream Bank Factor of Safety Using Static and Progressive Breaking Root Reinforcement Estimates [47] Results of the bank stability model runs carried out are shown in Table 3. In all cases the addition of root reinforcement (DS) resulted in higher F S values than nonvegetated banks. The root reinforcement estimated using RipRoot and the equation of Wu et al. [1979] varied for banks of different heights, with the two root models 9of11