A dynamic analysis of windthrow of trees

Transcription

1 A dynamic analysis of windthrow of trees A.H. ENGLAND 1, C.J. BAKER 2 AND S.E.T. SAUNDERSON 1 1 Division of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD, England 2 School of Civil Engineering, University of Birmingham, Birmingham B15 2TT, England Summary This paper considers the dynamics of the tipping of a tree due to a gust of wind superimposed on a steady wind. The resistance of the root system to the rotational motion of the tree is known from tree-pulling experiments. A gust of wind generates an impulsive load on the system and hence the dynamics of the tree may be modelled. An empirical formula which relates the speed of a gust of wind and its duration to the mean wind speed is used to determine the mean wind speed necessary to overturn a tree. Realistic mean wind speeds are found for trees such as Sitka spruce. Notation c D Non-dimensional drag coefficient E Young s modulus for the tree F m (u 0 ) Moment generated at the base by wind of speed u 0 F m Non-dimensional moment F m (u 0 )/M 0 G V Gust factor û/u 0 g Acceleration due to gravity h Height of the tree I 0 Moment of inertia of the tree about its base I 0 Non-dimensional moment of inertia I 0 m /M 0 2 m T () Tipping moment at the base m R () Resisting moment at the base M 0 Resisting moment at = 0 M m Maximum resisting moment M m Non-dimensional moment M m /M 0 M g Tipping moment due to a gust u 0 Constant horizontal wind speed û Extreme wind speed v n Wind speed normal to the tree W m Moment at the base due to the weight of the tree when = 90 Scaled angle of inclination ( = / m ) r Scaling parameter defined by equation (29) Density of the trunk Institute of Chartered Foresters, 2000 Forestry, Vol. 73, No. 3, 2000

2 226 FORESTRY a /u f m 0 Density of the air Turbulence intensity of the wind Short time interval Time interval for filtering the time series Angle of inclination of the tree to the vertical Angle at which the maximum resistance occurs Angular velocity of the tree Initial angular velocity Introduction The understanding of the phenomenon of tree fall in high winds is a problem of some considerable practical importance that has been investigated extensively in the past from a number of different viewpoints field experiments of both tree behaviour in high winds and tree rooting behaviour, wind tunnel experiments of model canopies, laboratory testing to obtain the mechanical properties of trees and parts of trees and so on. The interested reader is referred to Coutts and Grace (1995) for details of work in this field. One area of work that has not been considered in any great detail to date has been the mathematical modelling of tree behaviour in high winds. Such work is potentially of some use in providing a coherent framework for the consideration of field and laboratory results, and for the deeper understanding of the problem. Peltola (1995) modelled Scots pines as single beams in the ground and considered their stability in steady wind conditions, making calculations of the wind speeds necessary for trees to blow over. Gardiner (1992) developed a model of plantation conifers that represented them as tapered cantilever beams, with a lumped mass at 70 per cent of the tree height to allow for the presence of the canopy. He carried out a dynamic analysis of the system to obtain the natural frequencies, and the mean and fluctuating displacements. Very recently Kerzenmacher and Gardiner (1998) examined the dynamic response of a spruce tree to the wind. Baker (1995) developed a model of trees and crops based on a lumped mass on top of a weightless elastic stem, and through a frequency domain approach was able to calculate tree and crop failure wind speeds for different ground and meteorological conditions. Finnegan and Mulhearn (1978) used a similar approach to develop scaling criteria for the wind tunnel testing of model crops. Blackwell et al. (1990) developed a model of the root system of Picea sitchensis, by modelling the various components of root plate resistance as simple mechanical systems, and investigating the behaviour of this model to a dynamic input. Papesch (1974) gave a simplified theoretical analysis of the factors that influence the windthrow of trees. These contributions being noted, however, it would seem that there is a need for a model of tree behaviour in high winds that integrates the below and above ground components in a logical way, and more adequately models the dynamic behaviour of the complex system. Saunderson (1997), in the course of his doctoral study, has developed detailed mathematical models of a number of aspects of the aerodynamic loading of trees. The dynamic behaviour of a Sitka spruce in high winds has been considered by Saunderson et al. (1999). The present paper considers the windthrow of an isolated tree due to a steady wind and also due to a gusting wind. Realistic wind speeds are predicted by regarding the tree as a rigid body which rotates by the tilting of its root plate. Some implications of this idealization are discussed in the last section. Other papers concerning the vibrations of a branched tree and the modelling of the root plate of a tree are in preparation (see Saunderson et al., 1999). Dynamic analysis When the wind acts on a tree the aerodynamic forces on its trunk and branches generate a tipping moment at its base which is resisted by the moments generated by the root base. If we denote the tipping moment by m T () and the resisting moment by m R () and we suppose the tree can be modelled as a rigid body, then the equation of motion of the tree is I 0 (t) = m T () m R () (1)

3 A DYNAMIC ANALYSIS OF WINDTHROW OF TREES 227 where I 0 is the moment of inertia of the tree about its base and is the angle of inclination of the tree to the vertical. Coutts (1986) has measured the resisting moment of the root base for Picea sitchensis and found that it depends on the resistance of the hinge at the base of the trunk, the soil tension, the soil shear, the strength of the windward roots, the weight of the root soil plate and so on, giving rise to a graph of the form shown in Figure 1 (see also Blackwell et al., 1990). This experiments. Note that the resistance grows rapidly for very small values of, reaches a maximum between 2 and 5 and drops to about 50 per cent of its maximum value at about 10. For the purpose of this paper we shall model the resisting moment by a quadratic curve which has its maximum value of M m at the point = m and has the intercept m R (0) = M 0 at = 0 (see Figure 2). Hence the model for m R () is m R () = M m ( )( m ) 2 / 2 m (2) Note that the very rapidly rising part of the resisting moment curve over the range between 0 and 0.5 on Figure 1 has been replaced by a curve which intersects the axes at the point (0, M 0 ). Figure 1. Total resisting moment as a function of the angle of deflection. shows the combined resisting moment as a function of the angle of inclination to the vertical. This curve is typical of many found in tree-pulling Steady wind case The aerodynamic drag on an element of the tree is proportional to v n 2 da where v n is the component of the velocity of the wind normal to the element and da is the area of the exposed crosssection. Let us suppose the wind has a constant horizontal velocity u 0. If we denote by F m (u 0 ) the moment generated at the base of the tree by the drag due to this particular wind when the tree is vertical, then this moment will be reduced to F m (u 0 ) cos 2 when the tree is inclined at angle to the vertical. This is because the normal component of the wind to the inclined tree depends on cos. In addition, there will be a tipping moment proportional to the weight of the tree Figure 2. The model for the resisting moment m R ().

4 228 FORESTRY due to the displacement of its centre of gravity. The moment at the base of the tree due to the rotation of the tree has the form W m sin, where W m may be thought of as the moment generated at the base when the tree is rotated to a horizontal position. However, since the tree bends under the wind loading, there is an additional contribution to the moment F m (u 0 ) due to the displaced mass of the tree. This contribution is evaluated later. Hence the equation of motion has the form I 0 (t) = F m (u 0 ) cos 2 + W m sin M m + ( ) (1 / m ) 2 (3) where is measured in radians. As the resistance of the roots is small when is greater than about 10 (or 0.17 radians), we can make the small angle approximations that sin, cos 1 2 and reduce the equation of motion to I 0 (t) = F m (u 0 )(1 2 ) + W m M m + ( )(1 / m ) 2 (4) for small values of. The angular acceleration (t) can be expressed in terms of the angular velocity = (t) in the form (t) = (t) = d/d. Hence equation (4) becomes d 2( ) I 0 = F m (u 0 ) M 0 + W m d m + F m (u 0 ) 2 (5) 2 m This equation may be integrated to yield the kinetic energy of rotation of the tree, and hence the angular velocity of the tree, as a function of equation (7) only holds when is of the same order as m, so we put = m (8) which yields I 0 2 = m [F m (u 0 ) M 0 ( ) + ( ) 2 ] (9) on neglecting terms of order 2 m. Now, if the right-hand side of equation (9) remains positive for positive values of and does not drop to zero, the angular velocity will stay positive and the tree will continue to tip as increases. As the signs of the coefficients in equation (9) are known, it can be shown that the cubic (9) has the shape of the curves shown on Figure 3. The condition that (9) has no positive roots reduces to ( ) 2 < (F m (u 0 ) M 0 )( ) or F m (u 0 ) > (3M m + M 0 ) (10) Windthrow will occur when the wind speed u 0 is sufficiently large for the bending moment F m (u 0 ) to satisfy the inequality (10). Let us model the trunk of the tree as a uniform circular cylinder of radius r 0 and height h. Then the drag on an element of height dy is a c D r 0 u 2 0 dy where a is the density of air, c D is the nondimensional drag coefficient (related to the trunk I 0 2 = (F m (u 0 ) M 0 ) + W m 2 m + F m(u 0 ) 3 + C (6) 2 m for small angles. The condition that the tree starts tipping with a zero angular velocity when is zero gives C = 0. Hence I 0 2 = (F m (u) M 0 ) + W m m + F m(u) 2 (7) 2 m The angle m is small ( m < 0.1 radians) and Figure 3. Graph of equation (9); (i) when condition (10) is not satisfied, (ii) when condition (10) is satisfied.

5 A DYNAMIC ANALYSIS OF WINDTHROW OF TREES 229 radius of the tree) and u 0 is the constant (mean) wind speed acting on the tree. Hence the moment at the base due to the drag over an isolated tree is F m (u 0 ) = h a c D r 0 u 2 0 y dy = a c D r 0 h 2 u 2 0 (11) 0 Dr B.A. Gardiner has pointed out that, since the tree bends under the wind loading, there is an additional contribution to the moment F m (u 0 ) due to the displaced mass of the tree. The extra contribution has the effect of multiplying the righthand side of equation (11) by 1 + where = 0.4gh 3 /(Er 2 0 ), where the tree has a density, a Young s modulus E and the tree is bent by a uniform wind. We can either take this term into account explicitly or regard the drag coefficient c D in equation (11) as the actual drag coefficient enhanced by the factor 1 +. We adopt the latter approach and discuss the effect of this enhanced moment when examining the sensitivity of the solution in the sensitivity analysis section. Note that the model assumes a trunk of constant radius with a constant drag coefficient. The effect of the canopy is taken into account by an enhanced drag coefficient. Gardiner (1989) has shown for his standard tree (with h = 15 and r 0 = ) in a plantation that the drag is 161 kg in a wind at 20 m s 1. This corresponds to an equivalent drag coefficient c D of However, the drag on an isolated tree will be greater than that on a plantation tree and it seems reasonable to take the equivalent drag coefficient to be 0.6 for the purposes of this calculation. Mayhead (1973) in a series of wind tunnel tests on a range of isolated trees has calculated the true drag coefficients and found them to decrease from about 0.7 with increasing wind speed. He proposed a drag coefficient of 0.35 for use in critical height determinations for Sitka spruce. This coefficient has, of course, to be multiplied by the exposed cross-sectional area of the canopy and the height of the centre of pressure to evaluate the actual moment acting at the base. As the crosssectional area for the present model is based on the dimensions of the trunk and the centre of pressure is at h, it seems sensible to select a larger drag coefficient than that proposed by Mayhead. The sensitivity of results to the value of the turning moment is examined in the sensitivity analysis section. Coutts (1986) has performed winching tests on a range of Sitka spruce trees and Blackwell et al. (1990) have modelled the root anchorage of a shallow-rooted tree of this type and calibrated it with Coutts results. Figure 4 is taken from their paper. The values for the resisting moment may be estimated to be M 0 = 8 knm, M m = 12.8 knm, m = 2.5 = rads (12) and these correspond to trees with the following (approximate) dimensions h = 19 m, r 0 = 0.1 m, = 900 kgm 3, a = 1.2 kgm 3, c D = 0.6. (13) Coutts (1986) has commented on the variability of the root-strength of trees indicating that the uprooting moments for trees of this size and type can vary from 10 to 50 knm. The numbers selected correspond to the lower end of this scale. If we model the tree as a uniform cylinder of density then its moment of inertia about its base is I 0 = r 0 2 h 3. (14) It should be noted that this does not take into account the moment of inertia of the root plate or the branches of the tree. These considerations indicate that the estimates we make of the wind speed to induce windthrow will yield underestimates of the critical wind speed for failure, assuming the turning moment is broadly correct. The sensitivity analysis section considers the effect of increasing I 0 from the value given in equation (14). The minimum constant wind speed to cause windthrow is given by a c D r 0 h 2 u 2 0 = (3M m + M 0 ) (15) which yields, for this model u 0 = 29.9 m s 1 (16) or approximately 66 m.p.h., which agrees well with the suggestion by Mayhead (1973) that windthrow is likely to occur at about this speed. It is convenient at this stage to estimate the length of time it will take for a tree to start to topple. The angular velocity = d/dt is given by equation (9), and hence the length of time it takes the tree to fall from the vertical to the angle = m (at which the root resistance is greatest) is

6 230 FORESTRY Figure 4. Blackwell, Rennolls and Coutts model of root resistance: a, net total resistive turning moment; b, tension in roots; c, weight of root soil plate; d, bending of leeside roots; e, tension in soil; f, weight of stem and crown. 0 I 0 m 2( ) F m (u 0 ) M d (17) 1 This integral can be evaluated numerically for wind speeds greater than the critical wind speed of 29.9 m s 1. At the critical wind speed it will take less than 2 s for the tree to reach the angle m (for the numerical values given in equations (13) and (14)) and about 4.5 s for the tree to reach the angle where the angular velocity takes its minimum value. Hence, in summary, if the tree is subject to a constant wind which generates a moment F m (u 0 ) at the base of the tree which satisfies the condition (10), then the tree will be overturned due to this constant wind. Obviously this is a rather artificial situation and trees are much more likely to be thrown by strong gusts of wind superimposed on a wind of constant velocity. This will be considered in the next section. Impulsive loading Suppose the tree receives a strong gust of wind which lasts for a very short time and thereafter a wind of constant velocity u 0 acts on the tree generating the constant moment F m (u 0 ) at the base of the tree. Suppose that the impulsive moment generated at the base of the tree by the gust of wind is M g and the reaction of the root base generates the impulsive moment M 0. Then the tree experiences the overall impulsive moment (M g M 0 ) over a very short time interval. If we model the effect of the gust as an impulsive moment of this magnitude that acts at the time

7 A DYNAMIC ANALYSIS OF WINDTHROW OF TREES 231 t = 0, then the jump in the angular momentum of the tree is [I 0 ] 0+ 0 = (M g M 0 ) so that the tree acquires the angular velocity 0 = (M g M 0 )/I 0 (18) at the time t = 0+. The subsequent motion of the tree is governed by the constant velocity case, so that the equation of motion (4) has the solution (6), namely I 0 2 = (F m (u 0 ) M 0 ) + W m 2 m + F m(u 0 ) 3 + C (19) 2 m where C is a constant of integration. As the tree has gained the angular velocity 0 at the time t = 0+ due to the gust, C must be chosen to have the value I and hence the kinetic energy is I 0 2 = (F m (u 0 ) M 0 ) + W m 2 m 2 + F m(u 0 ) 3 + (M g M 0 ) 2 (20) 2 m I 0 Now, as the impulse that the tree receives is due to a gust, then the moment M g on the tree that this gust generates will depend on the square of the extreme -second wind speed. If we denote the ratio of the extreme wind speed û to the steady wind speed u 0 by the gust factor G V = û/u 0 (21) then the moment M g can be expressed in terms of the moment F m (u 0 ) due to the constant wind by M g = F m (u 0 ) G 2 V (22) Again, by putting = m and neglecting squared terms in m, the angular velocity (20) is given by I 0 2 = [(F m (u 0 ) M 0 ) ( ) + ( ) 2 ] m [F m (u 0 )G 2 V M 0] 2 2 (23) 2I 0 As in the previous section, the right-hand side of equation (23) must remain positive for the tree to continue to tip. As the signs of all of the coefficients are known it can be confirmed that this cubic in has the general form shown in Figure 5. The stationary points are found by setting the gradient of (23) to zero, which yields a quadratic equation for. The position of the local minimum (at = r ) corresponds to the highest root of [( ) 2 2( ) + F m (u 0 ) M 0 = 0] which is at the position = r where M m F m (u 0 ) r = 1 + (24) Hence the angular velocity will remain non-zero if the right-hand side of equation (23) is positive at the minimum point = r, i.e. if M 0 2I 0 ( ) m G 2 V > + (2 r 3)2 r F m (u 0 ) 3 F m 2 (u 0 ) (25) The square-root term (in (25)) remains real provided the position of the minimum r is greater than. From (24) this corresponds to the condition F m (u 0 ) < (3M m + M 0 ) (26) which is the complement of the condition (10). Condition (10) implied that if the wind velocity was large enough, then the tree would be blown over by the constant wind. Condition (26) indicates that if the constant wind alone is not sufficient to uproot the tree, then the tree will be overturned by a combination of the gust and the constant wind provided the gust factor G V is large Figure 5 Graph of the right-hand side of equation (23).

8 232 FORESTRY enough and the length of the gust long enough to satisfy condition (25). Note that condition (25) depends on the non-dimensional ratios of the moments M m /M 0, F m (u 0 )/M 0 and the quantity I 0 /M 0 which has the dimension of time squared. Note also that the condition (25) is independent of the quantity W m, the tipping moment due to the weight of the tree. If we define the nondimensional parameters by the following expressions F m = F m (u 0 )/M 0, M m = M m /M 0, I 0 = I 0 m /M 0 2 (27) then the gust factor for failure must satisfy 1 G 2 V > 1 + [ I 0 (M m 1)(2 r 3) 2 r ] (28) F m where M m F m r = 1 + (29) M m 1 These results are general and based only on the assumption that the resisting moment at the root base has the form given in equation (2), where m is small. The application of them to the case of a Sitka spruce modelled as a uniform cylinder is given in the next section. Graphs of û against the constant wind speed u 0 for different values of are given on Figure 6. It will be seen that all of the curves terminate as u 0 tends to the critical wind speed of 29.9 m s 1. The case where the gust velocity û is less than the constant wind speed u 0 is of no practical interest, so we are concerned with the region of the graph above the line û = u 0. It is also necessary to have an estimate of the time interval over which a gust will blow and this will clearly depend on the gust velocity. If an hour-long wind-velocity time series is divided into time steps of length f, then the maximum wind velocity û which can be expected to act for this time can be found, using an empirical expression due to Wieringa (1973), to be Mean wind speed failure criterion In the previous section the failure condition (28) was derived and found to depend on the nondimensional parameters defined in (27) and (29), which, in turn, are functions of the constant wind speed u 0 and the duration of the gust. If we adopt the numerical values for a Sitka spruce which were used in equations (12) and (13) the non-dimensional parameters become F m = 1.62 u , M m = 1.6, I 0 = , r = 1 + ( u ) (30) Windthrow will take place provided the gust velocity û and its duration satisfy the inequality (28), which implies that windthrow is just possible if r û = (2 r 3) (31) Figure 6. The gust velocity required to cause windthrow as a function of the constant wind speed for gust durations of = 0.2, 0.5, 1, 2, 5, 10 s.

9 A DYNAMIC ANALYSIS OF WINDTHROW OF TREES 233 û 3600 = ln (32) u u f where /u is the turbulence intensity and u is the mean wind speed (which we have taken to be the constant wind speed u 0 in this paper). For an isolated tree /u is approximately 0.2 with higher values in a forest environment. Our initial calculations are based on a turbulence intensity of 0.2 and results for a higher turbulence intensity are given later. It must be noted that the f used in this expression is not strictly the same as, the duration of the gust. More sophisticated expressions than (32) are available (see Baker (1995)) but an empirical formula such as (32) provides an adequate model for this analysis. Plotting this expression for various values of f will give a family of straight lines passing through the origin. The intersection of each line of this family with the corresponding curve given by equation (31) for the same value of and f will produce a wind speed failure line as shown by the thick line on Figure 7. Values of the gust velocity û would not be expected to occur above this failure line. Alternatively we can eliminate the gust duration = f from equations (31) and (32) and solve numerically for û as a function of u 0. The result of this calculation is shown on Figure 8 and has been plotted as the heavy line on Figure 7. It can be seen by inspection of these graphs that failure can occur at a minimum mean wind speed of about 17.7 m s 1 or 39 m.p.h. with a gust speed of about 27.5 m s 1 or 61 m.p.h., with a gust duration of about 5 s. Although a gust duration of 5 s seems a very short time to give an impulse to the tree, the time needed to tip the tree to its position of maximum resistance is of the order of 2 s. So, for consistency, we should look for gusts of shorter duration. A gust of speed 35.1 m s 1 (78 m.p.h.) of duration Figure 7. The mean wind speed failure line when the turbulence intensity is 0.2.

10 234 FORESTRY Figure 8. The mean wind speed failure line with a turbulence intensity of 0.2. Gust duration: 1s, 2 s, 5s. 1 s superimposed on a wind speed of about 21 ms 1 or 46 m.p.h. will cause windthrow. In a more turbulent environment with a greater value of the turbulence intensity the straight lines on Figure 7 are steeper so that failure can occur at a lower mean wind speed. Repeating the calculation above with a turbulence intensity of 0.4 yields the graph in Figure 9. This indicates that failure can occur with a minimum mean wind speed of about 13.2 m s 1 with a gust speed of about 27.8 m s 1 for a duration of 5 s. Note that the gust speed is almost the same for both levels of turbulence intensity. A more conservative estimate based on the duration of 1 s yields the mean wind speed of 15.6 m s 1 (34.6 m.p.h.) approximately and a gust speed of 37 m s 1 (82 m.p.h.). Sensitivity analysis Although we have thought of the tree as a rigid cylindrical body, the analysis applies for any shape of body for which the wind loading is proportional to the wind velocity squared and generates a moment F m about the base of the tree. The non-dimensional quantities F m, M m, I 0 defined in (27) determine the mean wind speed for which windthrow is possible, as derived above. It is interesting to determine how sensitive this critical wind speed is to changes in these quantities. If each non-dimensional quantity in turn is multiplied by parameter p while keeping the others constant, the critical mean-wind speed for failure may be calculated as a function of the multiplier p, as p varies over a given range such as 0.5 p 1.5. To be explicit, the wind speed was calculated corresponding to a gust duration of 1 s which should yield a good approximation to the minimum wind speed. The results of these perturbations are shown in Figure 10. It will be seen that the critical wind speed is largely independent of variations in M m and in I 0. This means that the results obtained are relatively insensitive to variations in the maximum reaction moment M m, and the angle m at which it occurs. Similarly it is insensitive to the moment of inertia I 0 and the duration of the wind gust. We have noted that the moment of inertia used in the calculation did not include any allowance for the presence of the root plate or the canopy, but the results indicate that the critical wind speed only increases by about 5 per cent when the value of I 0 is increased by 50 per cent. The critical wind speed is quite sensitive to the non-dimensional parameter F m = F m (u 0 )/M 0. Hence the moment F m (u 0 ) generated by the wind and the initial reaction moment M 0 generated by the root base need to be accurately estimated. A variation of 10 per cent in F m will produce a variation of about 5 per cent in the estimated critical wind speed. Hence if the reaction moment M 0 of the root plate is increased by (say) 20 per cent, the non-dimensional quantity F m is decreased by 17 per cent and the critical wind speed necessary to cause windthrow is increased by about 8 per cent. Similarly the calculation rests on an estimated value of the equivalent drag coefficient c D as 0.6. If this is increased by 10 per cent, corresponding to a revised estimate of the moment due to the wind loading on the canopy, then the critical wind speed will be reduced by 5 per cent. It has been suggested that since the tree bends under the wind loading the extra moment due to

11 A DYNAMIC ANALYSIS OF WINDTHROW OF TREES 235 Figure 9. The mean wind speed failure line with a turbulence intensity of 0.4. the weight of the bent tree should be taken into account. Using the dimensions of the Sitka spruce given in equation (13), the moment is enhanced by a factor of 1.4. This factor results from modelling the tree as a uniform cylinder and is clearly an overestimate, since the weight distribution along the tree is not uniform. Alternatively, if the tapering stem model proposed by Gardiner (1989) is used, the enhancement factor due to the stem is Gardiner has suggested the overall enhancement factor should be about 20 per cent which will reduce the critical mean wind speed to about 19 m s 1. This model depends directly upon the gust model introduced in the failure criterion section. If different aerodynamic gust models are introduced the calculations must be revised starting from equations (28) and (29). If I 0 is fixed at corresponding to a gust duration of = 1 s, the critical mean wind speed decreases with an increasing gust factor as shown in Figure 11. The wind speeds corresponding to Wieringa s model with turbulence intensities of 0.2 and 0.4 are shown in the figure. Conclusion The behaviour of a tree under a strong aerodynamic loading is governed by the resistance of the root plate. Using a simple model of the root plate resistance, the steady wind speed to cause windthrow of the tree can be estimated. The effect of a gust on the tree has been modelled as an impulsive loading on the tree. This, together with a model of the gustiness of the wind, enables us to estimate the critical mean wind speed at which windthrow of trees is likely to occur. The results

12 236 FORESTRY Figure 10. Variation of the critical mean wind speed due to changes in F m, M m and I 0. Figure 11. Graph of the critical mean wind speed as a function of the gust factor for a constant value of the non-dimensional parameter I 0. TI is the turbulence intensity in Wieringa s (1973) model.

13 A DYNAMIC ANALYSIS OF WINDTHROW OF TREES 237 for an isolated Sitka spruce tree yield realistic wind speeds and depend on comparatively few non-dimensional parameters. Acknowledgement The Authors would like to thank Dr B.A. Gardiner of the Forestry Commission for his assistance and helpful advice during the course of this research. References Baker, C.J The development of a theoretical model for the windthrow of plants. J. Theoret. Biol. 175, Blackwell, P.G., Rennolls, K. and Coutts, M A root anchorage model for shallowly rooted Sitka spruce. Forestry 63, Coutts, M Components of tree stability in Sitka spruce on peaty gley soil. Forestry 59, Coutts, M. and Grace, J. (eds) 1995 Wind and Trees. Cambridge University Press, Cambridge. Finnegan, J.J. and Mulhearn, P.J Modelling waving crops in a wind tunnel. Boundary-Layer Meteorol. 14, Gardiner, B.A Mechanical characteristics of Sitka spruce. Forestry Commission Occasional Paper 24, Gardiner, B.A Mathematical modelling of the static and dynamic characteristics of plantation trees. In Mathematical Modelling of Forest Ecosystems. J.Franke and A.Roeder (eds). Sauerlanders Verlag, Frankfurt am Main, Kerzenmacher, T. and Gardiner, B.A A mathematical model to describe the dynamic response of a spruce tree to the wind. Trees: their Structure and Function 12, Mayhead, G.J Some drag coefficients for British forest trees derived from wind tunnel studies. Agric. Meteorol. 12, Papesch, A.J.G A simplified theoretical analysis of the factors that influence windthrow of trees. In Fifth Australasian Conference on Hydraulics and Fluid Mechanics, A.J. Sutherland and D. Lindley (eds). University of Canterbury, Christchurch, New Zealand Peltola, H Studies on the mechanism of wind induced damage of Scots pine. Ph.D. thesis, University of Joenssu, Finland. Saunderson, S.E.T The aerodynamic behaviour of trees in high winds. Ph.D. thesis, University of Nottingham. Saunderson, S.E.T., England, A.H. and Baker, C.J A dynamic model of the behaviour of Sitka spruce in high winds J. Theoret. Biol. 200, Saunderson, S.E.T., England, A.H. and Baker, C.J Modelling the failure of trees in high winds. Paper presented at the 10th International Conference on Wind Engineering, Copenhagen. Wieringa, J Gust factors over open water and built-up country. Boundary-Layer Meteorol. 3, Received 27 November 1998

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