Perforation of Flexible Rockfall Barriers by Normal Block Impact

Abstract

Flexible rockfall barriers are a common form of protection against falling blocks of rock and rock fragments (rockfall). These barriers consist of a system of cables, posts, and a mesh, and their capacity is typically quantified in terms of the threshold of impact (kinetic) energy at which the barrier fails. This threshold, referred to here as the “critical energy,” is often regarded as a constant. However, several studies have pointed out that there is no single representative value of critical energy for a given barrier. Instead, the critical energy decreases as the block size decreases, a phenomenon referred to as the “bullet effect.” In this paper, we present a simple analytical model for determining the critical energy of a flexible barrier. The model considers a block that impacts normally and centrally on the wire mesh, and rather than incorporate the structural details of the cables and posts explicitly, the supporting elements are replaced by springs of representative stiffness. The analysis reveals the dependence of the critical energy on the block size, as well as other relevant variables, and it provides physical insight into the impact problem. For example, it is shown that bending of the wire mesh during impact reduces the axial force that can be sustained within the wires, thus reducing the energy that can be absorbed. The formulas derived in the paper are straightforward to use, and the analytical predictions compare favorably with data available in the literature.

Appendix: Combined Tension and Bending of a Circular Member

This appendix is devoted to deriving the maximum tension that can be sustained by a member with a solid circular cross section in the presence of an applied moment. Although the general procedure for evaluating the capacity of members in combined tension and bending is well known (e.g., Jirásek and Bazant 2002), a reference pertaining specifically to a circular member could not be found, and the full derivation is therefore presented here.

Figure 14 shows the distribution of plastic stresses in a circular member in combined tension and bending, assuming a fully plastic stress state (i.e., yield stress σ_y) everywhere in the section. The neutral axis is located at a distance z _n above the centroid of the section, and it divides the section into a region of tensile stress with area A ⁺ and a region of compressive stress with area A ⁻ (see Fig. 14). Areas A ⁺ and A ⁻ are given by

$$ A^{ + } = \frac{1}{8}D^{2} \left( {2\pi - \omega + \sin \omega } \right),\,\,\,\,\,A^{ - } = \frac{1}{8}D^{2} \left( {\omega - \sin \omega } \right) $$

(A.1)

where D is the diameter of the section and central angle ω (see Fig. 14) is related to z _n by

$$ \omega = 2\cos^{ - 1} \left( {\frac{{2z_{n} }}{D}} \right) $$

(A.2)

Fig. 1

Schematic of a flexible rockfall barrier

Fig. 2

Flow of forces during block impact on flexible barrier

Fig. 3

Deformed configuration from numerical simulation of block impact on a wire mesh supported by springs. Contours show major principal stresses in the wires (σ ₁) normalized by yield strength (σ _y ). At the instant shown, the block moves into the mesh with some velocity, and perforation has not yet occurred

Fig. 4

Cross-shaped region of wire mesh sustaining highest stress levels during block impact

Fig. 5

Schematic of two-dimensional block impact

Fig. 6

Sequence of images from high-speed camera showing local mesh deformation during block impact. Subfigures a, b show the intact mesh moments prior to perforation of the mesh, and c, d show the post-failure response

Fig. 7

Comparison of critical velocities from Spadari et al. (2012) and analytical predictions assuming uniaxial tension with the wire mesh

Fig. 8

Relative error in analytical predictions of critical velocity v _c as a function of the critical mesh deflection angle θ _c

Fig. 9

Comparison of critical velocities from Spadari et al. (2012) and predictions from analytical model with bending

Fig. 10

Relationship between dimensionless groups $E^{*}$ and $S^{*}$ for various values of $G^{*}$ (Spadari et al. 2012)

Fig. 11

Relationship between dimensionless groups $E^{*}$ and $S^{*}$ as determined from the analytical model

Fig. 12

Critical velocity versus block diameter from Cazzani et al. (2002) with trends corresponding to the analytical model and constant critical energy E _c

Fig. 13

Critical energy versus block diameter from Cazzani et al. (2002) with trends corresponding to the analytical model and constant critical energy E _c

Fig. 14

Plastic stress distribution for a circular member in combined tension and bending

The stress distribution must be in equilibrium with the applied tension F and applied moment M; it follows that

$$ F = \sigma_{y} \left( {A^{ + } - A^{ - } } \right) $$

(A.3)

$$ M = \sigma_{y} \left( {A^{ + } z^{ + } + A^{ - } z^{ - } } \right) $$

(A.4)

In Eq. (A.4), z ⁺ and z ⁻ are the distances from the section’s centroid to the centroids of the regions of tensile and compressive stress, respectively, and they are given by

$$ z^{ + } = \frac{{2D\sin^{3} \left( {\frac{\omega }{2}} \right)}}{6\pi - 3\omega + 3\sin \omega },\,\,\,\,\,z^{ - } = \frac{{2D\sin^{3} \left( {\frac{\omega }{2}} \right)}}{3\omega - 3\sin \omega } $$

(A.5)

Expressions for the applied tension and moment can be obtained by combining Eqs. (A.1)–(A.5). In terms of the central angle ω, they are

$$ F = \frac{1}{4}\sigma_{y} D^{2} \left( {\pi - \omega + \sin \omega } \right) $$

(A.6)

$$ M = \frac{1}{6}\sigma_{y} D^{3} \sin^{3} \left( {\frac{\omega }{2}} \right) $$

(A.7)

After some manipulation, the angle ω can be eliminated from Eqs. (A.6) and (A.7) to arrive at the following expression, which gives tensile force F in terms of the moment M

$$ F = \frac{1}{2}\sigma_{y} D^{2} \left\{ {\left( {\frac{6M}{{\sigma_{y} D^{3} }}} \right)^{\frac{1}{3}} \sqrt {1 - \left( {\frac{6M}{{\sigma_{y} D^{3} }}} \right)^{\frac{2}{3}} } + \cos^{ - 1} \left[ {\left( {\frac{6M}{{\sigma_{y} D^{3} }}} \right)^{\frac{1}{3}} } \right]} \right\} $$

(A.8)

It is noted that Eq. (A.6) with $\omega = 0$ gives the maximum tensile force, corresponding to M = 0, as

$$ F_{y} = \frac{\pi }{4}\sigma_{y} D^{2} $$

(A.9)

Similarly, Eq. (A.7) with $\omega = \pi$ gives the maximum moment, corresponding to F = 0, as

$$ M_{y} = \frac{1}{6}\sigma_{y} D^{3} $$

(A.10)

Given Eqs. (A.9) and (A.10), Eq. (A.8) can be rewritten as

$$ \frac{F}{{F_{y} }} = \frac{2}{\pi }\left\{ {\left( {\frac{M}{{M_{y} }}} \right)^{\frac{1}{3}} \sqrt {1 - \left( {\frac{M}{{M_{y} }}} \right)^{\frac{2}{3}} } + \cos^{ - 1} \left[ {\left( {\frac{M}{{M_{y} }}} \right)^{\frac{1}{3}} } \right]} \right\} $$

(A.11)

Last, it is noted that Eq. (A.11) can be approximated by the much simpler expression

$$ \frac{F}{{F_{y} }} = \sqrt {1 - \frac{M}{{M_{y} }}} $$

(A.12)

The normalized axial force F/F _y evaluated from Eq. (A.12) is smaller than the value obtained from Eq. (A.11), but the maximum difference between the two expressions over the full range of possible moments, 0 ≤ M/M _y  ≤ 1, is less than 4 %. For comparison, both curves are plotted in Fig. 15.

Fig. 15

Exact and approximate failure envelopes for a circular member in combined tension and bending