the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Dynamic magnification factors for tree blow-down by powder snow avalanche air blasts

### Perry Bartelt

### Peter Bebi

### Thomas Feistl

### Othmar Buser

### Andrin Caviezel

We study how short duration powder avalanche blasts can break and overturn tall trees. Tree blow-down is often used to back-calculate avalanche pressure and therefore constrain avalanche flow velocity and motion. We find that tall trees are susceptible to avalanche air blasts because the duration of the air blast is near to the period of vibration of tall trees, both in bending and root-plate overturning. Dynamic magnification factors for bending and overturning failures should therefore be considered when back-calculating avalanche impact pressures.

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In this paper we develop a simple method to determine the dynamic response of trees to impulsive loads. This is an important problem in natural hazards engineering where historical evidence of forest destruction or tree breakage is often used to evaluate the potential avalanche hazard. Any indication of forest damage is particularly valuable to avalanche engineers because it helps define the destructive reach of an extreme and infrequent event. Fallen tree stems delineate the spatial extent of an avalanche and create a natural vector field indicating the primary flow direction of the movement (Fig. 1). The age of the destroyed trees can be additionally used to link the historical observations to the avalanche return period (Reardon et al., 2008; Schläppy et al., 2014; Gadek et al., 2017). In many cases observations of forest destruction are the only data the engineer has to quantify avalanche danger.

The problem with using evidence of tree destruction for avalanche mitigation planning is that a simple relationship between avalanche impact pressure and tree failure is difficult to establish. Tree-breaking depends on both the avalanche loading and tree strength. Trees fall if the bending stress exerted by the avalanche exceeds the bending strength of the tree stem (Johnson, 1987; Mattheck and Breloer, 1994; Peltola et al., 1997, 1999) or if the applied torque overcomes the strength of the root-soil plate, leading to uprooting and overturning (Coutts, 1983; Mattheck and Breloer, 1994; Jonsson et al., 2006). Both mechanisms depend on the local flow height of the avalanche. Recent observations by Feistl et al. (2015b) suggest that the magnitude of the avalanche impact pressure is strongly related to the avalanche flow regime. Although long recognised that dense flowing avalanches can easily break, overturn and uproot trees (Bartelt and Stöckli, 2001; Feistl et al., 2015a), tree destruction by powder avalanche air blasts has received less attention. A mechanical understanding of how trees are blown-down by powder avalanche blasts would allow engineers to quantify powder avalanche pressures from case studies and historical records.

Here we develop a mechanical model to predict the natural frequency of trees
subject to full-height air-blasts of powder snow or ice avalanches. We assume
two deformation modes: stem bending and root-plate overturning, see Figs. 2
and 3. The ratio of the natural tree frequency to the frequency of the
avalanche air-blast defines the dynamic magnification factor *D*
(Clough and Penzien, 1975). This value is used to magnify the non-impulsive loadings
*D* > 1 to account for the increase in stress under an impulsive load. The
eigenfrequency of the tree is a function of the tree height, stiffness and
mass distribution between the stem and branches. It therefore depends on
forest age and tree species. We show that dynamic magnification factors for
fully grown trees are large indicating that mature forests are especially
vulnerable to powder snow avalanches. As we shall see, an error of up to 25 %
can be made when back-calculating avalanche velocities. For example, an
avalanche travelling at 35 m s^{−1} exerts the same pressure as an avalanche
travelling at 50 m s^{−1} if the impulsive nature of the loading is considered.
These are significant differences in hazard mitigation studies.

Measurements on real avalanches reveal that the air-blast is intermittent and
of short duration, lasting only a few seconds
(Grigoryan et al., 1982; Sukhanov and Kholobaeva, 1982; Sukhanov, 1982). When a powder avalanche hits a
forest the ice-dust cloud is typically moving at velocities in excess of
50 m s^{−1} (similar to extreme wind gusts). The height of the cloud is equal, if not
larger, than the height of the tree, i.e. *H* > 20 m. The pressure blast thus
acts over the entire width and height of the tree, producing large bending
moments in the stem and straining the root base plate. The impulsive
character of the powder avalanche air-blast, however, magnifies the static
stress state (Clough and Penzien, 1975). The fallen tree stems often point in the
direction of the flow, indicating that the trees had little time to sway and
react to blast and that the inertial effects are of considerable importance.

To calculate the dynamic magnification factor *D* we first make three
simplifying assumptions. Firstly, the air blast can be expressed as a sine
wave impulse with duration time *t*_{0}. Moreover,

where $\stackrel{\mathrm{\u203e}}{\mathit{\omega}}$ is the circular frequency of the loading
$\stackrel{\mathrm{\u203e}}{\mathit{\omega}}$ = *π*∕*t*_{0}. The magnitude of the force *F*_{0} is
as follows:

where *p*_{0} is the amplitude of the avalanche pressure given by the density
of the powder cloud *ρ*, the form drag coefficient of the tree *c*_{d} and
the maximum velocity of the blast *U*_{max} (Bozhinskiy and Losev, 1998; Feistl et al., 2015a).
The tree area over which the blast acts is denoted *A*, typically given by
the tree height *H* and effective tree width *W*. Thus, if the cloud density
and velocity are known as well as the tree geometry, the magnitude of the
applied blast force *F*_{0} is given.

After the loading time *t*_{0}, the tree vibrates freely with natural
frequency *ω*. The natural frequency is found using the Rayleigh
quotient method (Clough and Penzien, 1975), which assumes the deflected form is known
(but not the magnitude of deformation). The assumption of a deflected shape
reduces the tree to a single degree of freedom system. The frequency is found
by equating the maximum strain energy *V*_{max} to the maximum kinetic
energy *T*_{max} developed during the tree response. By calculating the strain and
kinetic energy produced by the avalanche blast, we find the generalised
stiffness *K* and generalised mass *M* of the tree:

The natural frequency for two different deformation modes, stem bending *ω*_{sb}
and root-overturning *ω*_{ro} will be determined in the next sections.

In both cases the total tree height is *H*. Tree mass is divided into two
parts: the stem mass *m*_{s} (a mass per unit length of the tree kg m^{−1}) and the
total mass of the branches *M*_{b} (kg). The branch mass, including the mass of
needles, is lumped at the tree centre-of-mass. The mass *M*_{b} can include the
mass of snow held by the branches and thus, like the tree elasticity, have
some seasonal variation. As we assume a constant stem diameter *d* the stem
mass per unit length is,

with

The density of the stem wood is *ρ*_{t}. For both the bending and
overturning cases, the concentrated load *F*_{0} acts at the tree
centre-of-mass, which is located a distance *a* from the ground (see Figs. 2 and 3).

Finally, the third assumption, the maximum response of the tree, will be reached before the damping forces can absorb the energy of the air blast. Only the undamped response to a short duration blast is considered.

## 2.1 Eigenfrequency: tree bending mode

For the case of tree bending, the deformation *x*(*z*) at height *z* is given by (see
Fig. 2):

and

where *E* is the modulus of elasticity of the tree stem and *I* is the moment
of inertia. The functions *ψ*_{1}(*z*) and *ψ*_{2}(*z*) represent interpolation
functions for the deformation field. These equations for lateral tree
deformation are found by assuming the tree is a statically determinate
cantilever-type structure fixed at the base to the ground (see Fig. 2 and
Clough and Penzien, 1975). The largest bending moment in the tree is found at the
tree base, *z* = 0. The quantity *X*_{0} is the static deformation under the
blast load *F*,

The moment of inertia is taken for circular stem sections,

The maximum potential strain energy in bending is as follows (Clough and Penzien, 1975)

In the bending case, the tree is firmly rooted in the ground and strain energy
is stored in the tree stem between the ground and the point of load
application *z* = *a*. The tree stem above *z* > *a* is stress free, swaying back
and forth as a rigid body. The maximum kinetic energy *T*_{max} is composed
of two parts containing the stem energy ${T}_{\mathrm{max}}^{\mathrm{s}}$ and the branch
energy ${T}_{\mathrm{max}}^{\mathrm{b}}$ of the tree, *T*_{max} = ${T}_{\mathrm{max}}^{\mathrm{s}}$ + ${T}_{\mathrm{max}}^{\mathrm{b}}$
(Clough and Penzien, 1975):

and

The eigenfrequency ${\mathit{\omega}}_{\mathrm{sb}}^{\mathrm{2}}$ is found by equating *T*_{max} = *V*_{max}:

## 2.2 Eigenfrequency: tree overturning mode

For the tree overturning case,

where *k* is the overturning stiffness of the root-plate. This equation is
found by assuming the lateral tree deformation is governed by a torsional
spring, representing the stiffness of the root-plate (see Fig. 3 and
Chajes, 1974). The maximum potential strain energy (overturning) is then

Similar to the bending case, the maximum kinetic energy is found by considering the stem and branch energies separately:

and

The eigenfrequency ${\mathit{\omega}}_{\mathrm{ro}}^{\mathrm{2}}$ is found by equating *T*_{max} = *V*_{max}:

The equation of motion for an undamped system subjected to a harmonic loading is as follows:

which has the general solution for 0 ≤ *t* ≤ *t*_{0},

and for *t* > *t*_{0}:

where *β* = $\frac{\stackrel{\mathrm{\u203e}}{\mathit{\omega}}}{\mathit{\omega}}$ is the ratio between the
frequency of the avalanche blast and eigenfrequency of the tree. The
magnitude of the dynamic response therefore depends on the ratio of the load
duration to the period of vibration of the tree. For the case when *β* < 1
the maximum deformation occurs when the impulsive load is active. It can
be shown (see Clough and Penzien, 1975) that the time to this peak response *t*_{max}
is:

which can be substituted into the general solution to find the dynamic magnification factor for a long duration impulse:

It can likewise be shown that the maximum response for the free vibration
case occurs when *β* > 1, *t* > *t*_{0}. For this case, the dynamic
magnification factor for a short duration impulse is:

For the resonance case *β* = 1

To demonstrate how the dynamic magnification factor *D* can be found, we
consider the following problem: a powder snow avalanche enters a spruce
forest with considerable speed (> 50 m s^{−1}) and exerts a short duration
air-blast with frequency $\stackrel{\mathrm{\u203e}}{\mathit{\omega}}$. The duration of the blast is on the
order of a few seconds. The height of the trees is between 25 and 30 m, which
is also the height of the powder cloud. The cloud has decoupled from the
avalanche core which has stopped before reaching the forest. Moreover, the
only loading on the trees is the air-blast.

Using the measured mass values tabulated in Table 1, we set the total
branch and needle mass of a single tree to be *M*_{b} = 540 kg. The stem mass
per length is approximately 60 kg m^{−1} (wood density 480 kg m^{−3}). The total
force of the avalanche impact acts at the tree's centre-of-mass which is
located *a* = 16.5 m above ground. This allows us to define the natural
frequency in bending of the tree by Eq. (13), *ω*_{sb} = 1.48 rad s^{−1}
(0.24 Hz), see Table 2. This value is in very good agreement with the measurements (see
Jonsson et al., 2007). The modulus of elasticity was set to *E* = 10 GPa based
on experimental measurements (Haines et al., 1996). For the calculations, a tree
diameter somewhat smaller than the diameter at breast height (DBH) is
selected. In this case *d* = 0.2 m, which is 1∕2 of the DBH diameter (this
provides the best match to the experimental frequencies).

Consider first a duration sine impulse lasting 2.50 s ($\stackrel{\mathrm{\u203e}}{\mathit{\omega}}$ = *π*∕6).
In this case *β* = 0.699; that is, the maximum deformation occurs during
the time the load is acting. For this case, application of Eq. (23),
we find *D* = 1.76, a rather large magnification factor. For a shorter duration
impulse lasting 1.66 s, *β* = 1.27 and from Eq. (24), we find
*D* = 1.36. The primary conclusion to draw from this analysis is that the
natural frequency in bending of tall trees is close to the frequency of the
applied avalanche air-blast. Measurements of air-blast duration times
reported by Russian researchers are within this range, lasting only a few
seconds (see Grigoryan et al., 1982; Sukhanov and Kholobaeva, 1982; Sukhanov, 1982).

Measurements of root plate stiffness are rare; however, values for 10–14 m
high spruce reported in Neild and Wood (1998) vary between *k* = 80 kN m
(*H* = 10 m) and *k* = 1200 kN m (*H* = 14 m). These values suggest a large variation
in *k* depending on growth conditions. The application of these *k* stiffness
values for spruce trees predicts natural frequencies for root-plate
overturning in *ω*_{o} > 2 Hz (Eq. 18), see Table 2. The calculated
*β* factors for overturning are typically *β* < 1. This result suggests that
large dynamic magnification factors can only be generated by very short
duration impulses (less than *t* < 0.5 s). Tall trees (*H* > 20 m) with low
root plate stiffness (*k* ≈ 100 kN m) are vulnerable to powder avalanche air-blasts.

We draw several conclusions from our analysis. Firstly, the natural frequency of tall trees – in bending and overturning – is
close to the loading frequency of powder avalanches,
*ω* ≈ $\stackrel{\mathrm{\u203e}}{\mathit{\omega}}$. Thus, tall trees are susceptible to powder avalanche blow-down. When
using tree blow-down to estimate avalanche impact pressures (and therefore
speed and density of the powder cloud) a dynamic magnification factor should
be applied in the analysis. Moreover, powder avalanches can knock down trees
with lower velocity than is presently assumed. This result is also valid for
other types of tall structures, including power pylons, or buildings with
long over-hanging roofs.

Secondly, both tree bending and root-plate overturning are possible tree
failure modes when hit by a powder avalanche. Interestingly, the natural
frequencies of tree bending and root-plate overturning are similar, when the
root-plate stiffness is low (*k* < 100 kN m) and the tree is tall
(*H* > 20 m). Although there is considerable data available to constrain the value of
the modulus of elasticity of wood *E*, there is less information available to
constrain the root-plate stiffness. In the future, field investigations that
document forest destruction should clearly separate bending and overturning
failures. This would help understand the variability of tree anchorage on
mountain slopes. The field examinations should also quantify the stem
diameter *d* at more than one location as this is necessary to accurately
determine the bending eigenfrequency.

Finally, the fact that tall trees can be broken in bending and overturning indicates the nature of the avalanche air blast. It appears to be a high velocity, short duration pulse of flowing material (ice-dust), similar to a high-density gust of wind. It is not a compression wave travelling at the speed of sound.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

This work was performed within the framework of the joint Austrian-Swiss
project bDFA, a study of avalanche motion beyond the dense flow avalanche
regime. We thank the Austrian Academy of Science (ÖAW) for their financial
support as well as the Austrian research partners (Austrian Research Centre
for Forests, Torrent and Avalanche Control and the University of Innsbruck).

Edited by: Oded Katz

Reviewed by: two anonymous referees

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