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A dynamic analysis of windthrow of
trees
A.H. ENGLAND1, C.J. BAKER2 AND S.E.T. SAUNDERSON1
1 Division
2 School
of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD, England
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
cD
E
Fm(u0)
–
Fm
GV
g
h
I0
–
I0
mT()
mR()
M0
Mm
–
Mm
Mg
u0
û
vn
Wm
r
Non-dimensional drag coefficient
Young’s modulus for the tree
Moment generated at the base by wind of speed u0
Non-dimensional moment Fm(u0)/M0
Gust factor û/u0
Acceleration due to gravity
Height of the tree
Moment of inertia of the tree about its base
Non-dimensional moment of inertia I0m/M02
Tipping moment at the base
Resisting moment at the base
Resisting moment at = 0
Maximum resisting moment
Non-dimensional moment Mm/M0
Tipping moment due to a gust
Constant horizontal wind speed
Extreme wind speed
Wind speed normal to the tree
Moment at the base due to the weight of the tree when = 90°
Scaled angle of inclination ( = /m)
Scaling parameter defined by equation (29)
Density of the trunk
© Institute of Chartered Foresters, 2000
Forestry, Vol. 73, No. 3, 2000
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a
/u
f
m
0
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F O R E S T RY
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 mT () and the resisting
moment by mR () and we suppose the tree can be
modelled as a rigid body, then the equation of
motion of the tree is
I0(t) = mT () – mR ()
(1)
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where I0 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
227
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 Mm at the point = m
and has the intercept mR(0) = M0 at = 0 (see
Figure 2). Hence the model for mR() is
mR() = Mm – (Mm – M0)( – m)2/m2
(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, M0).
Steady wind case
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
The aerodynamic drag on an element of the tree
is proportional to vn2dA where vn 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 u0. If we denote by Fm(u0) 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
Fm(u0) cos2 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 mR().
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F O R E S T RY
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 Wm sin , where
Wm 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 Fm(u0) due to the displaced mass
of the tree. This contribution is evaluated later.
Hence the equation of motion has the form
I0(t) = Fm(u0) cos2 + Wm sin – Mm + (Mm – M0) (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
(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
2(Mm – M0)
d
I0—– = Fm(u0) – M0 + Wm – ————— d
m
Mm – M0
+ ————–
– Fm(u0) 2
2m
(5)
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 = m
(8)
I02 = m[Fm(u0) – M0 – (Mm – M0)
+ (Mm – M0)2]
(9)
which yields
on neglecting terms of order m2.
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
(Mm – M0)2 < (Fm(u0) – M0)(Mm – M0)
I0(t) = Fm(u0)(1 – 2) + Wm – Mm
+ (Mm – M0)(1 – /m)2
equation (7) only holds when is of the same
order as m, so we put
or
Fm(u0) > (3Mm + M0)
(10)
Windthrow will occur when the wind speed u0 is
sufficiently large for the bending moment Fm(u0)
to satisfy the inequality (10).
Let us model the trunk of the tree as a uniform
circular cylinder of radius r0 and height h. Then
the drag on an element of height dy is
acDr0u02 dy
where a is the density of air, cD is the nondimensional drag coefficient (related to the trunk
Mm – M0
I02 = (Fm(u0) – M0) + Wm – ————– 2
m
Mm – M0
+ ————–
– Fm(u0) 3 + C
2m
(6)
for small angles . The condition that the tree
starts tipping with a zero angular velocity when is zero gives C = 0. Hence
Mm – M0
I02 = (Fm(u) – M0) + Wm – ————– m
Mm – M0
+ ————–
– Fm(u) 2
2m
(7)
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.
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radius of the tree) and u0 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
Fm(u0) =
h
acDr0 u02y dy = acDr0h2u02
(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 Fm(u0) 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 =
0.4gh3/(Er02), 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
cD 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 r0 = 0.0825) in a plantation
that the drag is 161 kg in a wind at 20 m s–1. This
corresponds to an equivalent drag coefficient cD
of 0.27. 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
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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
M0 = 8 kNm, Mm = 12.8 kNm,
m = 2.5° = 0.044 rads
(12)
and these correspond to trees with the following
(approximate) dimensions
h = 19 m, r0 = 0.1 m, = 900 kgm–3,
(13)
a = 1.2 kgm–3, cD = 0.6.
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
I0 = r02h3.
(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 I0 from the value given in
equation (14).
The minimum constant wind speed to cause
windthrow is given by
acDr0h2u02 = (3Mm + M0)
(15)
which yields, for this model
u0 = 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
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F O R E S T RY
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.
I0m
—————
2(M
– M )
m
1
0
3
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.
0
Fm(u0) – M0
– 2 + —————
Mm – M0
–
d
(17)
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 Fm(u0) 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
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 u0 acts on the tree
generating the constant moment Fm(u0) at the
base of the tree. Suppose that the impulsive
moment generated at the base of the tree by the
gust of wind is Mg and the reaction of the root
base generates the impulsive moment M0 . Then
the tree experiences the overall impulsive moment
(Mg – M0) 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
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t = 0, then the jump in the angular momentum of
the tree is
[I0]0+
0– = (Mg – M0)
so that the tree acquires the angular velocity
0 = (Mg – M0)/I0
(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
Mm – M0
I02 = (Fm(u0) – M0) + Wm – ————– 2
m
Mm – M0
+ ————–
– Fm(u0) 3 + C
2m
(19)
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 I002 and hence the kinetic energy is
Mm – M0
I02 = (Fm(u0) – M0) + Wm – ————– 2
m
Mm – M0
2
3+ (M – M )2 — (20)
+ ————–
–
F
(u
)
m
0
g
0
2m
I0
Now, as the impulse that the tree receives is due
to a gust, then the moment Mg 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 u0 by the gust factor
GV = û/u0
(21)
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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
[(Mm – M0)2 – 2(Mm – M0) + Fm(u0) – M0 = 0]
which is at the position = r where
Mm – Fm(u0)
r = 1 + ——————
Mm – M0
(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
2I0 (Mm – M0)m
M0
GV2 > ——— + ————————
(2r – 3)2r
2
3 Fm (u0)
Fm(u0)
(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
Fm(u0) < (3Mm + M0)
(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 GV is large
then the moment Mg can be expressed in terms of
the moment Fm(u0) due to the constant wind by
Mg = Fm(u0) GV2
(22)
Again, by putting = m and neglecting squared
terms in m, the angular velocity (20) is given by
I02 = [(Fm(u0) – M0) – (Mm – M0)
+ (Mm – M0)2]m
1
+ —- [Fm(u0)G2V – M0]22
2I0
(23)
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
Figure 5 Graph of the right-hand side of equation
(23).
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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 Mm/M0, Fm(u0)/M0 and the quantity
I0/M0 which has the dimension of time squared.
Note also that the condition (25) is independent of the quantity Wm, the tipping moment due
to the weight of the tree. If we define the nondimensional parameters by the following expressions
–
–
Fm = Fm(u0)/M0, Mm = Mm/M0,
–
I 0 = I0m/M02
(27)
then the gust factor for failure must satisfy
1
– –
2 GV2 > —
– 1 + [ I 0 (Mm – 1)(2r – 3)r]
Fm
Graphs of û against the constant wind speed u0
for different values of are given on Figure 6.
It will be seen that all of the curves terminate as
u0 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 u0 is of no practical interest,
so we are concerned with the region of the graph
above the line û = u0. 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
(28)
where
–
–
Mm – Fm
r = 1 + —————
–
Mm – 1
(29)
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.
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 u0 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
–
–
–
Fm = 1.62 u2010–3, Mm = 1.6, I 0 = 0.352 –2,
r = 1 + (2.67 – 2.71 u2010–3) (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 = 24.81 1 + 0.376 —(2r – 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.
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û
3600
—
ln ———
– = 1 + 0.42 —
u
u
f
(32)
where /u is the turbulence intensity and u– is the
mean wind speed (which we have taken to be the
constant wind speed u0 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
233
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 u0. 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.
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F O R E S T RY
Figure 8. The mean wind speed failure line with a
turbulence intensity of 0.2. Gust duration: 1 s,
2 s, 5 s.
1 s superimposed on a wind speed of about 21
m s–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 Fm about the base of the tree. The
– –
–
non-dimensional quantities Fm, Mm, 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 Mm
–
and in I 0. This means that the results obtained are
relatively insensitive to variations in the
maximum reaction moment Mm, and the angle m
at which it occurs. Similarly it is insensitive to the
moment of inertia I0 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 I0 is increased by 50 per cent.
The critical wind speed is quite sensitive to the
–
non-dimensional parameter Fm = Fm(u0)/M0.
Hence the moment Fm(u0) generated by the wind
and the initial reaction moment M0 generated by
the root base need to be accurately estimated. A
–
variation of 10 per cent in Fm will produce a variation of about 5 per cent in the estimated critical
wind speed. Hence if the reaction moment M0 of
the root plate is increased by (say) 20 per cent, the
–
non-dimensional quantity Fm 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 cD 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
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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 1.09. 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
0.352 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
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F O R E S T RY
–
–
–
Figure 10. Variation of the critical mean wind speed due to changes in Fm, Mm 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.
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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.
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