Mitigation effects of trees on rockfall hazards: does rock shape matter?

Abstract

Does rock shape matter to the mitigation effects of trees on rockfall hazards? This question must be resolved in order to better quantify the protective role of mountain forests against rockfall. To probe this question, we investigate a single rock-tree interaction using non-smooth, hard-contact mechanics that allows us to consider rock shape at impact. The interaction of equant shaped rocks with cylinder-like tree stems is modeled. The equant shaped rocks are close to spherical but have a certain shape variability governed by the rock’s surface area ratio and aspect ratio. This work serves as an important follow-up study to the existing investigations from Toe et al. (Landslides 14: 1603-1614, 2017), where the effects of trees on block propagation are numerically investigated using spherical shaped rocks. The objective of our simulations is to understand how and to what extent, shape will influence energy dissipation and trajectory change. The primary results include: surface area ratio plays a more important role than aspect ratio in determining the rock’s post-impact dynamics. The primary parameters governing the rock kinematics after impact (i.e., block’s energy reduction, reflected rotational speed, and trajectory change) are impact velocity, impact eccentricity, and the tree stem diameter. The latter observation aligns well with previous findings and suggests that the shape factors, at least for nearly spherical rocks, can be integrated into the current block propagation models. However, from a statistical viewpoint, the anisotropic distribution of mass and hence the asymmetric moment of inertia of non-spherical rocks leads to stronger or weaker spin effects compared to mass- and volume-equivalent spheres. Apparently, the rotational motion of an irregular object serves as a kinetic energy reservoir leading to subsequent rock-tree impacts, and therefore significant differences in energy loss and trajectory in comparison to spherical shaped rocks. This effect must be further investigated using elongated and flattened blocks and underscores the importance of measuring rockfall rotation in experimental investigations.

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Abbreviations

A :

Closest point from tree stem to rock

B :

Rock-tree contact point

c d :

Drag coefficient (kg m−1)

C :

Rock-tree contact frame

\( {e}_x^I,{e}_y^I,{e}_z^I \) :

Ground inertial-frame coordinates

\( {e}_x^K,{e}_y^K,{e}_z^K \) :

Rock eigen-frame coordinates

d 0, d 1 :

Tree stem bottom, top diameter (m)

δE rot :

Change rotational energy (J)

δE tot :

Change total kinetic energy (J)

\( \delta {E}_{tot}^{\mathrm{max}} \) :

Maximum loss total kinetic energy (J)

δE tra :

Change translational energy (J)

ΔE red :

Percentage reduction total kinetic energy

ΔE rot :

Percentage change rotational energy

ΔE tot :

Percentage change total kinetic energy

ΔE tra :

Percentage change translational energy

E + :

Rock energy after impact (J)

E :

Rock energy before impact (J)

\( {F}_d^I \) :

Drag force (N)

\( {F}_g^I \) :

Gravitational force (N)

G :

Contact penetration depth (m)

h :

Tree stem height (m)

h c :

Rock-tree collision height (m)

I :

Ground inertial-frame

K :

Rock eigen-frame

l X, l Y, l Z :

Length of principal axis (m)

m :

Rock mass (kg)

n :

Rock-tree contact normal vector

N :

Number of simulations

p IK :

Rock orientation indicator

q :

Generalized coordinates

Q :

Tree stem

r 0, r 1 :

Tree stem bottom, top radius (m)

\( {r}_{OS}^I \) :

Tree stem positional vector

R :

Tree stem bottom center

S :

Rock center of mass

S :

Center of rock’s bounding sphere

t s :

Contact separating axis

\( {t}_s^{\prime } \) :

Reference axis

T K :

Torque (N m)

u :

Rock positional vector

v in :

Impact translational velocity (m s−1)

\( {V}_S^I \) :

Rock translational velocity (m s−1)

W(q) :

Matrix of generalized force directions

\( \hat{\mathcal{X}},\hat{\mathcal{Y}} \) :

PCE model input, output

β :

Error index

χ :

Vertical impact angle (rad)

ϵ n, ϵ t :

Normal, tangential restitution coefficient

γ :

Rock aspect ratio

ι :

Rock volume (m3)

κ :

Tree stem dimensional parameter

λ :

Contact force vector

\( {\lambda}_n^C,{\lambda}_t^C \) :

Normal, tangential contact force (N)

μ :

Frictional coefficient

ν :

Generalized velocities

ω in :

Impact rotational velocity (rad s−1)

ω out :

Reflected rotational velocity (rad s−1)

ΩK :

Rock rotational velocity (rad s−1)

ϕ :

Horizontal trajectory change (rad)

ψ :

Vertical trajectory change (rad)

ρ :

Rock density (kg m−3)

τ :

Sobol sensitivity index

\( {\theta}_S^K \) :

Rock inertia tensor

ϑ :

Horizontal impact angle (rad)

ξ :

Rock surface area ratio

Ξ(q, ν, t):

Term of gravitational, gyroscopic forces

\( {\zeta}_{BT},{\zeta}_{B{S}^{\prime }} \) :

Eccentricity indicators

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Correspondence to Guang Lu or Perry Bartelt.

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Lu, G., Ringenbach, A., Caviezel, A. et al. Mitigation effects of trees on rockfall hazards: does rock shape matter?. Landslides 18, 59–77 (2021). https://doi.org/10.1007/s10346-020-01418-2

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Keywords

  • Rockfall
  • Rock-tree interaction
  • Non-spherical rock
  • Hazard mitigation
  • Computational modeling
  • Simulation