mechanics of chain-link wire nets

Engineering Structures 101 (2015) 68–87

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Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

Mechanics of chain-link wire nets with loose connections

http://dx.doi.org/10.1016/j.engstruct.2015.07.0050141-0296/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: WSL Institute for Snow and Avalanche Research SLF,Flueelastrasse 11, CH-7260 Davos Dorf, Switzerland.

E-mail addresses: [email protected] (J.P. Escallón), [email protected](v. Boetticher), [email protected] (C. Wendeler), [email protected] (E. Chatzi), [email protected] (P. Bartelt).

J.P. Escallón a,b,⇑, v. Boetticher b, C. Wendeler c, E. Chatzi b, P. Bartelt a

a WSL Institute for Snow and Avalanche Research SLF, Flueelastrasse 11, CH-7260 Davos Dorf, Switzerlandb ETH Swiss Federal Institute of Technology Zurich, Wolfgang-Pauli-Str. 15, CH-8093 Zurich, Switzerlandc Geobrugg AG – Geohazard Solutions, Aachstrasse 11, CH-8590 Romanshorn, Switzerland

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 December 2014Revised 7 May 2015Accepted 2 July 2015

Keywords:Rockfall protection barrierFinite Element modelChain-link wire netsNatural hazardsGeneral contact

Chain-link wire nets are used for slope stabilization, natural hazard protection systems, mine and tunnelsafety and many other important applications. In rockfall protection barriers the nets are designed towithstand dynamic, impulsive loadings. As they are composed of ultra-high strength steel wires withloose three-dimensional connections, the high resistance nets are very flexible and serve to efficientlydistribute loads throughout the structure. Rockfall barrier design requires accurate numerical simula-tions. In this work, a Finite Element model of chain-link nets is developed. To treat the complex contactinteractions among chain-link elements and rockfall barrier components we develop a computationalscheme relying on a general contact algorithm. The non-linear force displacement response of the netobtained in tensile quasi-static laboratory tests is successfully reproduced by the numerical model.The model parameters are obtained by optimization techniques. The calibrated chain-link model withcontact is shown to successfully simulate a full-scale test of a flexible rockfall protection barrier. Thecomputational schemes allow us to accurately model the mechanical behaviour of chain-link wire netswith loose connections.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Flexible steel wire nets are essential components of protectionsystems. Three specific applications in rockfall hazard mitigationare: interception barriers [1–7], drapes [8] and attenuator systems[9], see Figs. 1 and 2. A general feature of these systems is the useof flexible steel wire nets with different geometries (hexagonal,rhomboidal, ring) and connection types (loose, double twisted).Manufactures of wire net systems require accurate numericalsimulation techniques to develop new systems and reduce devel-opment costs. This is challenging because the range of wire nettypes demand robust numerical algorithms treating complex con-tact interactions efficiently and realistically.

Commonly used wire nets include chain-link and double-twistedhexagonal nets. The double-twisted nets are arranged in repeatinghexagonal mesh geometries [10]. Loose connection chain link netsare arranged in repeating rhomboidal patterns (Fig. 3), composedof a single wire or a twisted triple wire. The combination of flexibilityand high resistance make loose connection chain-link nets ideal for

rockfall barriers, slope stabilization, and temporary tunnel supportwhere protection measures must withstand large forces. In looseconnection nets, wires are bent to form the chain-to-chain connec-tions. This construction method generates non-linear deformationbehaviour thanks to the shape and out-of-plane dimension (eccen-tricity) of the loose wire connections. It is these properties thatmake for highly efficient impact interception structures. This workdeals with flexible chain-link nets composed of ultra-high strengthsteel (UHSS) twisted triple wire with loose connections.

Numerical Finite Element (FE) and Discrete Element (DE) modelshave been developed to simulate rockfall barriers containingdouble-twisted, hexagonal nets [8,10,11]. Hexagonal nets behavestiffer because the double twisted connections restrictwire-to-wire sliding and thus friction can be neglected.Furthermore, load eccentricities have a negligible effect on theirbehaviour, because effectively the net lies on a bi-dimensionalplane. Therefore, a two-dimensional approximation of the netgeometry is sufficient to capture their response to loading.Modelling loose connection chain-link nets on the other handrequires that eccentric connections and contact with slidingfriction is modelled. Thus, the three-dimensionality and the truecontact interactions of the chain-link net elements need to be takeninto account to capture the soft, non-linear response to tensile load-ing. A first attempt to model such nets was developed by [12] whoapplied a DE model to reproduce the force–displacement behaviour

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Fig. 1. Rockfall protection barriers: (a) wire-rope chain link net as part of a rockfall barrier impacted by a rock avalanche and (b) double chain-link interception structure aspart of a rockfall barrier.

Fig. 2. Rockfall protection attenuators: (a) sketch drawing of a rockfall attenuator system and (b) filmed perspective of rock impact on an attenuator system [17].

J.P. Escallón et al. / Engineering Structures 101 (2015) 68–87 69

obtained from laboratory and field tests. The model was simplifiedsuch that the 3D geometry effects were replaced by a 2D geometryin combination with a non-linear material law that could accountfor the three-dimensional geometric effects. Axial elastic–plasticsprings are located at the connections. The spring stiffness and resis-tance varies according to the mesh opening angle [12]. This is in facta consequence of the complex mechanical behaviour as a function ofthe net’s three-dimensional geometry. This model was imple-mented in a DE code for rockfall barrier simulations [2]. This mod-elling approach is restricted by the large testing requirement tocalibrate the angle dependent stiffness and failure load when geom-etry changes are considered.

This paper deals with the development and improvement of amodelling scheme to treat chain-link contact with sliding friction[6] which models complex mechanical net-connection behaviourrespecting the three-dimensional geometric effects. The approachrelies on general contact (GC) in which the hard contact beha-viour is approximated by a penalty contact method [13]. Thismethod approximates contact enforcement using penalty stiff-ness. The frictional contact behaviour is considered using aCoulomb-type model. The basic idea behind this scheme is tomodel the contact interactions between structural componentsas close as possible to those of real systems. Ring-net rockfall

systems have already been modelled with this approach [5,7].Initial studies on the use of GC to model a single chain-link wiresystem have been carried out [6]. In this work the modelling ofcontact interactions includes a number of model extensions. Thefirst is an improved model of the end knot connections that closechain-link elements (Fig. 3). The second models the morecomplex twisted triple wire strand with an equivalent circularwire including ductile damage. The third addition is the applica-tion of optimization techniques to calibrate the model parameters(equivalent wire area and the elastic–plastic constitutiveparameters).

A problem addressed in this work is the appropriate elementsize for the FE discretization. Local FE mesh refinement isrequired in the area of connections to accurately represent itsgeometry and mechanical behaviour. However, in the explicitcalculations performed herein this local refinement decreasedthe element-by-element stability limit [7,13]. In this paper amass-scaling approach [14,15] is proposed to deal with this prob-lem. The scaling is only applied to the elements needed to modelthe connections. The scaling is small enough that it does not mod-ify the overall dynamic behaviour during impact calculations.Criteria are suggested for an acceptable level of mass scaling forboth quasi-static and dynamic analysis.

Fig. 3. Chain-link panel.

Fig. 4. Quasi-static tension case: (a) line load c approximation of the contact forces,(b) cut of the wire and equilibrium of the cut face, and (c) N;V and M deriveddiagrams.

N

(a)(b)(c)

70 J.P. Escallón et al. / Engineering Structures 101 (2015) 68–87

The accuracy of the investigated approach is finally verified bycomparing simulation and field test results of a maximum energylevel (MEL) test (2000 kJ) on a rockfall barrier according toEuropean standards [16].

M

Fig. 5. Sketch of the bending moment–normal force interaction diagram for a steel-wire section close to the contact region: (a) plasticity domain, (b) elasticity domain,and (c) N–M path for increasing Fs and Fsþ1.

2. Mechanics of flexible chain-link wire nets with looseconnections

The central problem in understanding the mechanics of flexiblechain-link nets is to describe how loads are distributed throughoutthese systems. In a flexible chain-link net, a wire or a bundle ofwires is bent into a zig-zag pattern (Fig. 3). Each ‘‘zig’’ (a bend inone direction) hooks with the element immediately above it, andeach ‘‘zag’’ (a bend in the opposite direction) with the elementbelow it, forming a rhomboidal pattern. The chain-linked zigzagelements are in loose contact at the connections. The ends of theelements are knotted to prevent the nets from unravelling.

A closed-form analytical mechanical model of flexiblechain-link nets with loose connections was developed to gaininsight into its mechanical behaviour [12]. In this model, the distri-bution of the contact force at the connections immediately beforefailure is approximated by an elliptic distribution of a line load ofmagnitude c distributed over the contact area Ac (Fig. 4a). The con-tact area is assumed to be elliptically shaped because of the patternof scratches and deformations found on the experimentally testedchain-link net samples [12]. Assuming an elastic-perfectly plasticbehaviour of the steel wire, the maximum contact pressure equalsthe yield stress of the wire. By knowing the force resultant Fs þ Fsþ1

(see Fig. 4a) that is balanced by the contact pressure, the area Ac isobtained. Furthermore, the contact length is known from thethree-dimensional geometry of the corresponding ellipse of thecontact area, so that the magnitude of the equivalent line load ccan be calculated [12]. In the case of quasi-static tension loading,Fs ¼ Fsþ1, the line load c represents the distribution of the reactionforces balancing the tension loads Fs and Fsþ1, which act on thechain-link element parts that are straight. The equilibrium offorces at a cutting plane (Fig. 4b) leads to the normal (N), shear(V) and bending moment (M) diagrams shown in Fig. 4c. This

determines the distribution of N and V along the chain-link ele-ment. The integration of V over the element length provides thebending moment distribution M. This analytical procedure revealsthe importance of defining a realistic contact area for the looseconnections for determining the overall loading capacity of the net.

Another result of the analytical mechanical model is the stronginteraction between the bending moment M and the axial force Nat the connection due to the load eccentricity e. The axial andbending behaviour is elastic until �60% of the deformation capac-ity is mobilized, before becoming plastic. The extent of the regionof plastic bending increases until the bending limit of the elementis reached. For the applied normal force Fs to increase, the loadingeccentricity, which defines the bending, must be reduced so thatthe bending remains below its limit capacity. This complex interac-tion between the axial and bending behaviour therefore influencesthe overall stiffness of a chain-link net.

In the beginning of the loading process, the load eccentricity e isrelatively large. Small increments in tension thus cause large bend-ing deformations, leading to large segment elongations. However,with increasing load, the increments in tensile loading reduce theincrements of the bending deformations. The ongoing bendingdeformations cause progressively smaller segment elongations asthe lever arm of e is reduced. The mesh thus becomes stiffer.However, the bending moment–normal force interaction withincreasing load reduces the bending resistance of the wire, produc-ing the opposite effect on the overall mesh stiffness. This leads tothe loading path N–M, as shown in Fig. 5, resulting in the meshbecoming stiffer at the beginning of the loading path and less stiff

Fig. 6. Triple twisted wire chain-link: (a) net sample and (b) mesh dimensions.

Table 1Summary of the mesh characteristics and dimensions as indicated in Fig. 6.

Wire net producer GeobruggS (mm) 180 (+/�5 %)H (mm) 300 (+/�5 %)Mesh width DL (mm) 8.6Mesh angle e (�) 47

Fig. 7. Tensile quasi-static test set-up.

1 For interpretation of color in Figs. 7 and 22, the reader is referred to the webversion of this article.


at the end of the loading path [12]. Between the ranges of initialloading and failure load, the two opposing processes counterbal-ance each other, leading to the mesh stiffness behaving nearly lin-early. This behaviour is described in Section 3.

For more details on the mechanical behaviour of chain-link netsand alternative implementations using the DE method one canrefer to [12]. The results obtained by this closed-form analyticalmodel reflect the fact that a numerical model requires considera-tion of the coupling between bending and axial behaviour, includ-ing the non-linear geometric effects of the connection deformation.

3. Quasi-static tensile experiments

To develop the numerical scheme to model chain-link nets withloose connections, a series of quasi-static tensile tests was per-formed in the Geobrugg testing facility in Romanshorn (SG),Switzerland. The tests were used to help quantify (1) the frictionbetween the mesh elements, (2) the onset of material damageand (3) the change in load eccentricity during deformation.

In this particular investigation a chain-link net consisting of athree-strand twisted wire was investigated (Fig. 6). Each of thethree wires has a 2 mm radius. The geometric dimensions of therhomboidal mesh elements are given in Table 1.

A tensile test bed machine was used to perform the displace-ment controlled quasi-static tests. This machine uses a combina-tion of a tension type load cell and a cylinder with a rotary pulsegenerator to measure both the tensile force and the net sampleelongation. For more details regarding a tensile test bed apparatusone can refer to [18]. A rectangular net panel of 1100 � 900 mm isconnected at three sides via shackles to a fixed frame (in blue1

Fig. 7). The upper side is connected via shackles to a moving frame(in red Fig. 7). The fixed frame is composed of steel beams whichare attached to each other via bolts. The moving frame is composedof two steel plates which are attached by pins and screws. Steel slid-ing connections are inserted in between the frame beams and pull-ing machine plates. Two wire-rope cables (Fig. 7) connect themoving frame with the cylinder [18]. Prior to the test, the net isslightly pre-tensioned to avoid any possible sag. Fig. 8a shows theinitial configuration of the net panel. After applying the pretension,the cables pull the moving frame at a constant velocity of100 mm/min. The chain-link net strain rate is approximately0.002 s�1, which can be considered as quasi-static. A high-speedcamera was placed above the net panel to capture the net deforma-tion in time. The high-speed camera recorded 250 frames per secondduring the last 1.5 s of the test. Two tests lasting approximately115 s were performed. In both tests damage occurred only withinthe last 0.5 s. Fig. 9a–d shows four frames from the high-speed cam-era, depicting the damage sequence in the zone highlighted in

Fig. 8. Tensile quasi-static test: (a) initial set-up and (b) net sample failure.

Fig. 10. Typical force vs. displacement behaviour to quasi-static tensile tests oftriple twisted wire chain-link nets.


Fig. 8b. The pulling force reaches a peak and then drops as aconsequence of damage (Fig. 8b). Failure in both tests took placeat different inner connections. This is in agreement with other tests[19] that show that in repeated tests failure occurs at different innerconnections.

The measured non-linear force–displacement response curvesexhibit a progressive stiffening (Fig. 10) up to 100–120 mm defor-mation (stage 1) at which point the stiffness is approximately con-stant until failure (stage 2). The connections wear because of

Fig. 9. Damage sequence of a connection: (a) failure of an element, (b) the eleme

friction (Fig. 11). As a result, tangential contact forces should beconsidered in the numerical model.

4. Finite Element modelling of chain-link nets

4.1. FE model with chain-to-chain contact

The FE model of the triple twisted wire chain-link net withloose connections was implemented in the commercially availableFinite Element code Abaqus/Explicit 6.13. The developed modeluses constitutive material data such as (1) yield curve(strain-hardening metal plasticity), (2) Young’s modulus andPoisson ratio (elasticity), (3) equivalent plastic displacement atfailure and (4) fracture strain (ductile damage). The ductile damagemodel uses the stress–displacement concept to decrease meshdependency [13]. The numerical approach relies on the generalcontact algorithm of Abaqus/Explicit. The penalty method is usedto approximate hard contact. Additionally, Coulomb type slidingfriction is included in the contact model.

nt moves downward, (c) the element slides to the left, and (d) rest position.

Fig. 11. Wearing out of the connections because of friction.

Fig. 13. Modelling of the knotted connections: (a) knotted connection at the initialconfiguration, and (b) deformed configuration of the knot, (c) connection typeCARTESIAN [13], and (d) connection type REVOLUTE [13].


In addition to metal plasticity, ductile damage evolution hasbeen taken into account. A linear damage evolution law in combi-nation with a ductile damage initiation criterion is applied. Theimplementation of the stress–displacement concept in a FEMrequires the definition of the fracture energy per unit area, Gf .More details about damage for ductile metals can be found in[7,13].

The twisted triple wire is idealized as a single equivalent wire ofcircular area. However, the area of the equivalent wire is assumedto be smaller than the sum of the areas of the three wires compos-ing the real structure. This is due to the fact that the cross-sectionalarea of the single wires is not perpendicular to the axial load, as itis the case for the equivalent area. Furthermore, the bending stiff-ness is overestimated if the area is not reduced. In order to takeinto account these effects, a dimensionless area reduction factorv is introduced into the FEM. This approach has been appliedand verified for ring net systems [5,7].

The input geometry of the chain-link elements and their con-nectivity (Fig. 12a) is modelled according to the measurements ofthe manufactured net panel. A finer mesh (Fig. 12b) is needed inthe connections between chain-link elements to reproduce theaxial, shear and bending behaviour (Fig. 4c) obtained by the analyt-ical chain-link model discussed in Section 2. Mesh convergencestudies leaded to the use of nine elements per connection. Threeelements in the middle of the connection are required to capturethe shear transition while the remaining six elements (three to

Fig. 12. Modelling of inner connections: (a) connection three-dimensio

each side of the centre elements) are needed to capture the bend-ing deformations of the connection, see Fig. 12b. These six ele-ments were necessary to obtain an accurate pattern of the slidingbetween chain-link elements. Four elements (two to each side ofthe centre elements) cannot correctly approximate the curvedgeometry of the contact zone, causing unrealistic sliding patternsbetween chain-link elements.

The knotted connections between the ends of the chain-linkelements are simplified. The connections between the border knotsof the chain-link net elements are placed at the same coordinate(Fig. 13a). To avoid initial over-closures [13], contact betweenchain-link elements is excluded in the region of the knots.Actions are transmitted from one ‘‘zig-zag’’ element to its neigh-bour at the knots by means of the so-called connector elementsavailable in Abaqus [13].

Cartesian and Revolute connections are used to model the bor-der ‘‘zig-zag’’ chain-link element connections or knots (Fig. 13b).The Cartesian component provides a connection between the ver-tices of adjacent ‘‘zig-zag’’ chain-link elements that allows inde-pendent behaviour in three local Cartesian directions that followthe system at the vertex a (Fig. 13c). Rigid behaviour is specifiedin the local 3-direction (z-local), while node b is allowed to changeposition along 1- and 2-directions (x- and y-local). In this model,the z-global direction is opposite to gravity, and x- and y-globallay on the chain-link panel plane. The z-local direction is directedopposite to gravity as well. The relative positions of node b with

nality with rendered beam profiles and (b) beam element nodes.

Fig. 14. Bi-dimensional chain-link model: (a) tie contact conditions between chain-link elements (in red), and (b) boundary conditions applied to chain-link element nodes ina simplified bi-dimensional case. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Von Mises stress contours (MPa) at maximum elongation: (a) overall view,and (b) zoom-in view.


respect to node a in the local 1- and 2-directions is fixed in a work-ing range by means of the stop option available in Abaqus/Explicit[13]. A close observation of a physical net panel reveals that theavailable gap for translational movement in the knotted connec-tions is not constant. Such irregularities influence the macroscopicbehaviour of the net. It is necessary to take them into account toreproduce the soft response of the net panel to tensile loading. Inthis work, the connector stop in the 1-direction is assumed to bea random number between 5 mm and 6 mm (with a seed of0.1 mm). A possible random variation of the gap in the2-direction has a negligible effect on the net panel response.Therefore, it is assumed to remain constant and equal to 6 mm.The Revolute connection type (Fig. 13d) is used to constrain therotations around the 2- and 3-directions, while the 1-directionrotational component is free. Further clarifications to Fig. 13c andd can be found in [13].

4.2. Model of the tensile quasi-static test

To achieve the highly detailed model described in the previoussection, simpler models were first explored. Initial modellingattempts considered a bi-dimensional geometry of the chain-linkelements and tie constraints (Fig. 14a) between the contact nodesbetween chain-link elements [13]. This contact condition ensurescontinuity in the translational degrees of freedom between the ver-tices of the chain-link elements in contact, while the rotations arefree. Boundary conditions applied to the chain-link elements repre-sented an approximation to the test true boundary conditions(Fig. 14b). In this model, the total elongation that the test panelreached (until failure) UðtÞ was applied as a boundary condition(Fig. 14b). This displacement was applied using a smooth function[13], so that the test is simulated quasi-statically. This modelshows that higher stresses are concentrated in the beam elementsin contact (Fig. 15a and b). However, a model taking into accountbending should show that the highest stresses occur in the beamelements adjacent to the elements in contact. Moreover, it is thisnumerical panel is more resistant that the test panel, and failuredoes not take place. To account for bending in a simplified manner,the bi-dimensional chain-link elements were assembled with anoffset in the z-plane with a distance equal to that of the real con-nection. The aforementioned Cartesian plus Revolute connectorswere used to model the connections where contact between neigh-bour chain-link elements take place (Fig. 16a). When approxi-mately 60% of the deformation capacity is mobilized in themodel, the elements bend in an unrealistic manner, i.e., the ele-ments adjacent to the vertices suffer from large nodal rotationrates with respect to the local 1-axis (Fig. 16b). This is because a

Revolute connector behaves as a rigid beam when the vertices ofthe connected elements do not coincide [13] (Fig. 16b).

Finally, an improved approximation to the true contact condi-tions was considered to ensure that the chain-link parameterswere not influenced by significant contact simplifications. Underthis approximation, the rigid and moving frames, as well as thesliders were omitted, similar to approach used in the first mod-elling attempts. However, in the new model, the steel pins con-necting the net to the sliders were included as well as theshackles. The presence of the sliders was taken into account byconstraining the movement in the z-direction of all the shacklesin contact with the sliders. Furthermore, the movement in thez-direction of some beam elements was also constrained to simu-late the presence of the sliders. The test velocity V was applied tothe pins that are attached to the moving frame. Fig. 17 show theboundary conditions applied in the model. Fig. 17a shows thezones in the chain-link net where the boundary condition UZ ¼ 0was applied to take into account the movement restriction causedby the sliders. Fig. 17b shows the shackles where boundary condi-tions were applied, and the constrained degrees of freedom in thepins. The test velocity was applied to the upper pins, and can beexpressed as:

VðtÞ ¼ Vf ð1Þ

where V equals two times the maximum test displacement u, and fis an Abaqus defined smooth function [13]. The definition of f

Fig. 16. Three-dimensional chain-link model with simplified connections: (a) Cartesian plus Revolute connectors joining the chain-link elements, and (b) large rotations ofthe elements adjacent to the contact nodes.

Fig. 17. Model boundary conditions: (a) applied to the net, and (b) applied to pins and shackles.

Table 2Chain-link optimization parameters.

Parameter Lower Upper Optimal

ry (MPa) 1500 1700 1637ru (MPa) 2800 3000 2893E (MPa) 100,000 110,000 100,355req (mm) 2.4 2.5 2.4578


requires three points: (0,0), (0.5tstep,1), and (tstep, 0), where tstep is thestep duration. The integral of V is equal to the maximum test dis-placement. The use of the smooth function allows performing thetest in a quasi-static manner. The numerical simulation of the ten-sile test was carried out using an explicit time integration scheme.The calculation time was decreased by artificially increasing thedensity of the net using the mass-scaling approach [14,15,13]. Itis verified that the ratio of the total kinetic energy to the total inter-nal energy is less than 1% at the end of the simulation [7]. Hence,the induced dynamic effects were negligible, thus maintaining thequasi-static condition of the test.

All shackles and pins were modelled as solid three dimensionalrigid bodies, because they are one order of magnitude stiffer thanthe chain-link net. The rigid bodies were still discretized by meansof linear hexahedral elements (pins) and linear tetrahedral ele-ments (shackles), in order to have contact facets for the contactcalculations.

4.3. Parameter optimization

To determine the best-fit parameters for the chain-link netmodel, a FEM based optimization scheme was implemented. Thesoftware Isight 5.8 [20] is used for this analysis due to its compat-ibility to the Abaqus software. For the bi-dimensional and simpli-fied three-dimensional models, the absolute area differencebetween the simulation and target plot was chosen as the objectivefunction to be minimized, which can be expressed as:

Objective Function ¼X

i

jDAj ð2Þ

where DA is the area difference between the experimental and thesimulation curve within successive intersecting points. For moredetails about the selected objective function one can refer to [7].

The parameters to optimize in the bi-dimensional model wereset as: (1) the yield curve described by two points (0, ry) and(ep

u;ru), (2) the Young Modulus, (3) the area reduction factor v,and (4) the damage parameters �ep

D (plastic strain at the onset ofdamage), and Gf (fracture energy per unit area).

The optimum Latin Hypercube Sampling (LHS) technique [21]was applied to perform the Design of Experiments (DoE) for allthree FE models. In each of the three cases, the objective functionwas evaluated at each parameter combination chosen by the LHStechnique. The evaluation of the objective function for thebi-dimensional model revealed that this model was not suitablefor data fitting because the response was significantly more rigidthan the measured response.

In the case of the simplified three-dimensional model the offsetbetween the bi-dimensional chain-link elements was introducedas an additional parameter. In reality, however, the out of planeconstruction distance is fixed. Within a sensitivity analysis

Fig. 18. Radial basis function surface approximation of the objective function vs.yield stress and cross-section radius.

Fig. 19. Force–displacement response of the different models.


framework, we investigated the effect of varying this offset.Including this offset as a parameter in addition to considering dam-age results in a highly multi-modal optimization surface.Therefore, the Multi-Island Genetic Algorithm (MIGA) to optimizeparameters was used, as it was performed in the case of ring nets[7]. The optimization results revealed that the fitted offset washigher than the real value. Moreover, the fitted elasticity moduluswas an unrealistically low (<50,000 Mpa). Finally, the simulationsoften suffered from large rotation rates (Fig. 16c) in the elementsadjacent to the contact nodes, causing some of the simulations ofthe optimization cycle to fail before completion due tosingularities.

These results demonstrated the importance of accurately mod-elling the bending of the contact connections. As a consequence wedecided to model the three-dimensional geometry of the netsaccording to the real measurements of the manufacturedchain-link net. Furthermore, as explained in the previous section,we decided to adopt a significantly improved approximation tothe true contact conditions of the simulated experiment.

To increase the efficiency of the optimization of thethree-dimensional detailed model, a two stage optimization pro-cess was performed. In the first stage, damage was excluded, andto enhance the stability of the model, the ultimate plastic strainep

u was assumed to be significantly large (1.5). Damage could beexcluded from the first stage of the optimization since it isobserved to occur at the last 0.5 s of the 115 s long test. The chosenobjective function in this case consists in the difference betweenthe numerical and experimental maximum force, which can beexpressed as:

Objective Function ¼ jFmax;exp � Fmax;modj ð3Þ

where Fmax;exp is the maximum experimental force, while Fmax;mod isthe maximum force of a model optimization run. The parameters tobe optimized are: (1) the equivalent cross-section radius req, (2) theelastic modulus E, (3) the yield and ultimate stresses (ry;ru). Thecorresponding v factor of the chain-link net is obtained as:

v ¼ ðreq=rwÞ2=3 ð4Þ

where req is the radius of the equivalent wire modelling thethree-strand twisted wire, and rw is the radius of each of the threewires used in the macroscopic structure.

In this case the optimization surface is a smooth function. Thisoptimization scheme allowed us to use a Downhill SimplexTechnique (DST), which converges significantly faster than GAs. Alarge enough step size allowing finding the global minimum wasselected after first optimization trials. For more details on the gen-eral workings of a DST one can refer to previous work by [22,23], aswell as fundamental literature on this topic [24].

Two DoEs were performed to define the search space regardingthe three-dimensional detailed model. The evaluation of the objec-tive function was used to define the range of parameter variation(search space) where the global minimum exists (Table 2). A smal-ler search space with respect to the initial assumption resultedfrom the DoE analyses.

Finally, the optimization process is formulated as the problemof minimizing the specified objective function. Table 2 illustratesthe optimization parameters for the chain-link net, the lower andupper search space bounds, and the final optimized values.

Fig. 18 depicts the radial basis approximation of the objectivefunction vs. the yield stress ry and the cross-section radius rsection.The optimization function surface is smooth and exhibits severallocal maxima and a region where the global minimum value isfound.

The simulation results of the model with the best fit parame-ters, served to select the plastic strain at the onset of damage �ep

0.

The value of Gf allowing damage to occur at the end of the exper-iment was then optimized in the second stage of the optimizationcycle.

Fig. 19 displays the results from one of the quasi-static tests andof its numerical simulation with the detailed three-dimensionalmodel including ductile damage parameters. To demonstrate thesignificant improvement with respect to the two preliminarymodels they are also shown. The three models are run with theoptimized parameters of the detailed three-dimensional model.

4.4. Response features of the FE model with chain-to-chain contact

The chain-link net FEM with realistic contact conditions is ableto reproduce the mechanical behaviour tensile response obtainedwith the closed form analytical model. Fig. 20a–c shows the axial,bending, and shear diagrams immediately before failure along acoordinate system that follows the path illustrated in Fig. 21aand b. The action diagrams obtained from the numerical simula-tion exhibit a similar behaviour to what was obtained from theprocedure explained in Section 2.

Contact thickness reductions cause the apparent contact pene-trations seen in Fig. 21b. Contact between beam segments is onlydetected as edge-to-edge contact in Abaqus/Explicit. The Abaquscontact algorithm assigns a contact thickness to each beam contactnode, which is represented with a sphere centred at the node witha radius equal to the cross-section radius. The GC algorithmrequires that the contact thickness does not exceed a certain

Fig. 20. Action diagrams along the connection: (a) axial force, (b) shear force 2-local axis, and (c) bending moment around the 1-local axis.

Fig. 21. Path plot: (a) general view, and (b) close-up view.

Fig. 22. Path plot: (a) general view, and (b) axial-bending interaction in the contact area (element 69).


fraction of the element length [13]. The GC algorithm will reducethe contact thickness where necessary, but will use the correctcross-section area for element calculations. In the developedchain-link model, a very fine discretization at the connections isneeded. This causes the contact thickness of the nodes located atthe connections to be reduced. In this study the reduction factoris small enough to avoid undesired effects on the contact orchain-link behaviour. Further issues related to contact modellingwill be provided in Section 5.1.

Fig. 22a shows the nine elements (in red) used to discretize thechain-link equivalent wire in the contact zone. Fig. 22b showsthe loading M–N path from the onset of plastic behaviour untilthe onset of damage, taking place at the element subjected tohigher contact forces (number 69). Using the experimental data,a parabolic shaped failure domain and the elastic region can bedetermined (Fig. 22b) as:

ðM=MRÞ2 þ ðN=NRÞ2 ¼ 1 ð5Þ

Fig. 23. Damage progression: (a) first sequence of failure, and (b) the net loses continuity.

Fig. 24. GBE-2000A rockfall system at the end of the impact process.


ðM=MelÞ2 þ ðN=NelÞ2 ¼ 1 ð6Þ

where MR and Mel are the cross-section resistance for bending andthe maximum elastic bending respectively, and NR and Nel are thecross-section resistance for tension and the maximum elastic ten-sile force respectively.

Fig. 23a and b shows the progression of the chain-link failure asthe damage variable d [13,7] equals unity d = 1 in one of the con-nections. All inner contact zones reach approximately the samevalue of damage variable d. This shows that different failure mech-anisms are possible.

In the simulation an inner connection close to a knotted con-nection fails (Fig. 23a), and subsequently unravelling takes placesuntil the net loses continuity (Fig. 23b) and its ability to carry moreload. However, analogous laboratory tests on larger chain-link netsamples have shown that when a wire is cut, a hole is created but it

Fig. 25. Connection of chain-link net to cables: (a) free meshes rule, and (b) connection bkit manual.

tends to stabilize and stop its growth, allowing the net to continuecarrying load. The chain-link net stiffness decreases on the bordersof the hole, leading to a load redistribution that causes the holegrowth to stop [12]. This shows that the redundancy ofchain-link nets is a function of its size. A small sample behavesas a non-redundant structural system, while a large sample tendsto behave as a redundant system [25].

5. Application in natural hazards protection: rockfall barrier

This investigation is a continuation of previous work [5–7] inwhich a new modelling scheme to simulate rockfall barriers wasdeveloped. The numerical scheme relies on the general contact(GC) algorithm of the Abaqus/Explicit FE code [13]. The hard con-tact behaviour is approximated by using a penalty method toenforce the contact constraint. The frictional behaviour at contactis modelled using a Coulomb-type friction. Rockfall barriers withring nets have been simulated using this scheme. To match theexperimental results, an accurate geometric representation of thestructural components and the contact interactions is required [7].

In this investigation a rockfall barrier with chain-link netting ismodelled. This barrier obtained the European Technical Approvalaccording to the ETAG027 guideline in 2010. Full scale test dataare available to verify the FE model [26]. The system is namedGBE-2000A indicating that it is able to stop a falling rock with amaximum kinetic energy of 2000 kJ, and that it belongs to categoryA in the residual height classification [16]. Fig. 24 depicts the finaldeformed state of the barrier at the end of the maximum energylevel (MEL) test.

A special mechanical feature of this particular system is the lowforce transmitted to the up-slope anchors. By interrupting the con-tact between the chain-link meshes near the posts and the supportcables, stress concentrations are avoided. This is accomplished by

etween net panel and between net and cables (bottom) according to the installation

Fig. 26. Post model: (a) post foot to I-beam tie constraint, (b) post head to I-beamtie constraint, and (c) post-to-base plate connection via a bolt.


not threading the support cables through seven meshes on eachside of the inner posts and on one side of each border post(Fig. 25a). There are free meshes in the bottom part as well(Fig. 25b) following this same pattern. Three of the ‘‘free’’ meshesboth located on the top and bottom of the net are attached tothe support cables by means of round clips (Fig. 25a). The roundclips are made of galvanized steel wire with a diameter of 3 mmand a tensile strength of 1770 N/mm2. Their maximum breakingforce is 13,760 N. The clips have negligible structural importance.All clips will fail in the case of an impact with the serviceabilitykinetic energy level (730 kJ). Cable-to-net openings are created asthese round clips fail progressively. The openings grow until allclips have failed. These gaps alleviate the stresses on the retainingcables and on the chain-link elements in the vicinity of the posts.

5.1. Contact modelling

The challenges in the use of GC to model rockfall barriers are:

1. Large contact thickness reductions when using structural ele-ments (beams and trusses), and local fine mesh discretizationsare needed. In this work, the contact reductions were notsignificant.

2. Excessive penetrations when nodes involved in contact do nothave adequate mass. The reasons for this problem are discussedin [13,27]. In order to efficiently deal with this problem,non-structural masses were added to components which expe-rienced excessive penetration. Other practical solutions to thisproblems are proposed in [13]. The influence of mass incre-ments on the dynamic behaviour of simulations performedherein is negligible.

3. Initial contact over-closures. To avoid physically unreasonablecontact interactions, some regions of the model are excludedfrom GC (e.g., two cables sharing a common vertex at an anchorpoint).

The possibility of defining surface-to-surface or node-to-surfacetie constraints in Abaqus [13] allowed the development of a newpost model, in which true contact conditions and metal plasticityare efficiently considered. The posts were divided into three differ-ent parts: (1) the I-beam (HE-160A), (2) the post foot connectingthe I-beam to the base plate and (3) the post head that connectsthe I-beam to cables. The inner posts are only connected to theup-slope cables at the post head, while the border posts are addi-tionally connected to the lateral cables. The long I-beam was dis-cretized into linear beam elements and assigned elastic–plasticproperties corresponding to the steel grade S355 [28]. Both thepost foot and head were modelled as three-dimensional parts.The post foot is connected to the base plate with a bolt. The posthead is connected to the cables with shackles. These componentsare treated as rigid bodies because during tests they do not exhibitdeformation. Tie constraints are used to connect the I-beam withthe post foot and head. This shows an improvement of the postmodel with respect to previous work [7]. Fig. 26a and b showsthe master surfaces belonging to the post foot and head includingthe slave nodes of the I-beam. The post foot and head assume themaster role for the tie constraint because they are modelled asrigid bodies. The applied modelling techniques allow the post torotate as it does in reality (Fig. 26c).

Fig. 27a and b is CAD images from barrier installation manual[29] showing the top and bottom parts of the ‘‘left’’ border postprior to the complete connection of the chain-link net. Fig. 27cand d shows how the FE model of these system parts capture thecontact conditions in high detail. Higher stresses tend to be con-centrated in the cable-to-posts connections. With the aim ofdecreasing these stress concentrations, running wheels are used

to connect the support cables to the posts. Double clips are usedto fix the vertical cables to the support cables. These componentshave a torque resistance of 120,000 N mm. The large scale testshave shown that the forces transmitted to the vertical cables arenot large enough to cause an acting torque higher than the speci-fied resistance; thus, the vertical cables do not slide through thedouble clips. This fact permitted the implementation of asurface-to-surface tie constraint between the vertical cable surfaceand the double clip circular surfaces inside the holes (Fig. 28). Thedouble clips and running wheels were treated as rigid bodies in theFEM. To minimize contact search and contact calculations, otherconnection components such as shackles and clevis are treated asrigid bodies, and discretized using beam elements.

Fig. 27. Connection of cables to posts: (a) border post (top) according to the installation kit manual, (b) border post (bottom) according to the installation kit manual, (c)modelled border post (top), and (d) modelled border post (bottom).

Fig. 28. Tie constraint between the vertical cable and double clip surface aroundthe holes.


In the FE model the threading connection of the cable throughthe mesh openings are simplified by attaching the cable to thenet with idealised rigid-body clips (Fig. 29a and b). Fig. 30a andb shows the lateral anchors (left side of the barrier) to which theenergy dissipating devices (Section 5.3) are connected.Furthermore, the two bottom and the two support cables are con-nected to independent energy dissipating devices (two per side ofthe barrier). The energy dissipating devices are attached to steelanchors with flexible heads (Fig. 30a and b). Fig. 30a shows theconnection of the top support cables to the left energy dissipatingdevice. Moreover, the kit installation manual gives two options foranchoring the lateral and bottom support rope cables: (1) connec-tion of both bottom support cables and lateral cables to commonflexible head anchors (Fig. 30b), and (2) the lateral cables are con-nected to independent flexible head anchors. This second optionwas taken for the large scale tests (Fig. 30c), and was implementedin the FE model (Fig. 30d). Fig. 31a shows the up-slope anchor con-figuration of the rockfall barrier. There are two up-slope cablesattached to each single steel post head, which are also connectedto flexible head steel anchors. There are two up-slope cables con-nected to each intermediate anchor (Fig. 31b), while at the borderanchors, there is only one cable attached. In the FE model, theanchor is substituted by applying a pin (UX = UY = UZ = 0) bound-ary condition to each up-slope cable. Furthermore, the cables areanchored at the same coordinate; therefore, cable-to-cable contactis excluded to avoid making unnecessary corrections of contactover-closures [7,13].

For the sake of model set-up efficiency, Python scripts areapplied to generate the FE model. The scripts reproduce

components following repeatable patterns. This approach neglectsinstallation tolerances and deviations from the patterns within a0–50 mm range.

5.2. Large scale test

The large scale test site is located in ‘‘Lochezen’’ Walenstadt(SG), Switzerland. Large scale tests of three span rockfall barriers

Fig. 29. Connection of chain-link net to cables: (a) zone where the top supportcables are threaded through the net in the ETA test installation, and (b) modelledconnections via rigid clips.


and single spanned nets supported in a steel frame are performedin this test site [30] (Fig. 32b and c). An overview plan of the site isshown in Fig. 32a. Installation of the rockfall protection barrier inthe terrain is carried out by using a crane to lift and attach the bar-rier system to the rock face (Fig. 32a). The crane is also used to liftthe block to a specified height, before it is dropped into the net. Theblock trajectory is completely vertical. The position of the impactblock was determined according to the annex A.3 of theETAG027 [16]. The block is made of concrete and its shape is apolyhedron defined in the annex A.2 of the ETAG027. The blockmass was 4700 kg. The block was dropped from a height of

Fig. 30. Flexible anchors: (a) top support cables-to-energy dissipating device-to-flexibllateral cables-to-energy dissipating device-to-flexible anchor connection according to t(Field), and (d) modelled anchor configuration.

44.26 m and reached a speed of 29.46 m/s at net impact, providingimpact energy of 2040 kJ. The slope inclination is almost vertical(82�); therefore, the trajectory inclination satisfies the ETAG027requirements. For further information about the testing require-ments, please refer to [16]. The ETA approval of this system wasissued by the Building Testing and Research Institute (TSUS).

Fig. 33 depicts the positions of the load cell as well as the anchorand energy dissipating devices in the rockfall barrier.

5.3. Frictional interactions

In addition to plastic dissipation, energy is dissipated by fric-tion. The assumed general dynamic friction coefficient is ld = 0.1.However, different dynamic friction coefficients were assumedfor three contact interactions: (1) a dynamic friction coefficientof ld = 0.35 is used to model the concrete block-chain-link netinteraction. This friction coefficient corresponds to measuredconcrete-steel l values 0.3–0.4 [31]. (2) A friction coefficient ofl = 0.2 is used to model the interaction between shackles, clips,and cables. The sliding friction data published in [32] was consid-ered to define the steel-steel friction coefficients. For comprehen-sive study concerning the influence of different steel-steel andblock-steel friction coefficients one can refer to [2,33]. (3) A frictioncoefficient of ld = 0.25 is used to model the interaction betweenchain-link elements, which was selected via a numerical simula-tion which matched the deformation pattern of the experimentalresults (Section 4.2).

The typology of energy dissipating device used in the modelledrockfall system is explained in [7]. In this barrier model, the energydissipating device behaviour is assessed by means of a 1D model.Coulomb-like friction is defined in the axial connectors [13] usedto model the energy dissipating devices. In this way, the tangentialforce causing sliding friction in the connector can be defined.

e anchor connection according to the installation manual, (b) bottom support andhe installation manual, (c) lateral anchor configuration used in the large scale test

Fig. 31. Up-slope flexible anchors: (a) anchor configuration, and (b) intermediate up-slope anchors connected to two retaining cables.

Fig. 32. Rockfall test site in Walenstadt (Switzerland): (a) overview plan of the testssite, (b) three span system and (c) single spanned nets supported in a steel frame[30].


Fig. 34a and b shows the relative movement of node a with respectto node b. This relative movement is given by the non-linear piece-wise tangential contact force versus relative displacement alongthe connector (Fig. 35). For more details regarding this

Fig. 33. Load cell p

one-dimensional scheme to model energy dissipating devices oneis referred to [7].

Typically the resulting contact tangential force (T) – relative dis-placement (u) law is evaluated by means of quasi-static tests [7].However, for high energies it can occur that the forces transmittedby the energy dissipating devices in the dynamic cases are largerthan those transmitted in quasi-static conditions. This occurredin one energy dissipating devices (no. 3) attached to the top sup-port cables (Fig. 33). This aspect deserves a detailed investigation,including laboratory, field tests and 3D numerical simulations. Onepossible explanation for the difference in this particular casebetween the quasi-static and dynamic loadings is the existenceof stress concentrations. To temporarily tackle this issue, the Tvs. u law was modified in this energy dissipating device (Fig. 35).

5.4. Quasi-static calculation

Prior to simulating the actual block impact, we applied thegravitational load in a quasi-static calculation step. In this step,we increased gravity from 0 to g in 500 ms according to a smoothfunction [7,13]. The mass-scaling technique is applied throughoutthis step only to the elements of the chain-link connections byusing an automatic controlled scheme [13]. The cumulated massincrease in the model at the end of the quasi-static step corre-sponds to about 4%. This is due to the fact that the selected targettime increment [13] was slightly higher than the modelelement-by-element stability limit [13,27]. An increase of about

ositions [26].

Fig. 34. Axial connectors that model the energy dissipating devices: (a) axialconnectors at 0 ms of the impact step, and (b) axial connectors at 420 ms of theimpact step.

Fig. 35. Tangential versus relative displacement law according to quasi-staticlaboratory experiments, and law obtained from the behaviour of a device in thelarge scale test.

Fig. 36. Rockfall barrier model: (a) beginning of the quasi-static step, (b) end of thequasi-static step, (c) beginning of the impact step, and (d) meaning of the BCsymbols.


4% is small enough to ensure accurate initial conditions for thedynamic analysis.

During the static step a constant velocity boundary condition isgiven to the block’s centre-of-mass (Fig. 36a and b). This velocityequals the one measured during the test at impact. The simulatedblock height is proportional to the duration of the static step(tstatic ¼ 500 ms). It takes into account the static sag. Further detailsabout the calculation of the falling height are provided in [7].

The support cables are pre-tensioned in the model during thequasi-static step. Neumann boundary conditions are imposed [7].

The rotational velocities of the anchor points are constrained dur-ing pre-tensioning, while the translational velocities are free(Fig. 36a). The magnitude of the pretension force corresponds tothose applied in the certification test. These forces are increasedfrom 0 at tstatic ¼ 0 ms to the predefined values at tstatic ¼ 500 msaccording to the same smooth function applied to increase thegravitational acceleration. For stability reasons the shackles con-necting the net to the vertical cables were restrained to rotatearound all axes (Fig. 36a–c).

5.5. Impact calculation

The duration of the impact simulation is set equal to 600 ms,which approximately represents the impact duration. The concrete

Fig. 37. Experimental video frames (first column), simulation video frames (second column), and block movement at t1 = 0.0 ms, t2 = 220 ms, t3 = 420 ms and t4 = 600 ms(third column).


block is treated as a rigid body [7]. The block is discretized with lin-ear tetrahedral elements. The velocity boundary conditionimparted to the block is removed at this step [7].

The support cables are fully pre-tensioned at the end of thequasi-static step. As a result, the boundary conditions corre-sponding to anchors are imposed at the beginning of the impactstep (Fig. 36c). For more details regarding this procedure, referto [7].

The model mass was re-initialized for the impact calculation.This means that the amount of mass increase by mass-scalingin the quasi-static step was removed. During the impact step,

the applied target time increment approach [13] was set-up atto an optimal value which is equal to the element-by-elementstability limit. The elements that discretize the connections werestabilized with this method. The total mass increase in the impactcalculation due to mass-scaling was about 1%. The external workby mass-scaling [13] with respect to the frictional and plastic dis-sipation was about 1% and 2%, respectively. Thus, the effectsinduced by mass-scaling were negligible during the impactcalculation.

A summary of the quasi-static step, and the initial conditionsfor the dynamic step can be seen in Fig. 36a–d.

Fig. 38. Load cell histories: (a)–(d) lateral anchors, and (e)–(h) uphill anchors.


6. Numerical approach assessment: numerical versusexperimental results

This section presents the comparison between the experimentaland the numerical simulation results of the large scale barrier test.The force cell devices and high speed cameras recorded data

450 ms before impact. The actual dimensions and shape of theforce cell are not considered in the FEM analysis. For simplicityeach force cell is composed of a single beam element with addi-tional non-structural nodal masses and steel elastic properties.The total mass of the modelled force cell is equal to the real com-ponent mass.


6.1. Barrier elongation

Fig. 37 compares the experimental measurements (first col-umn) and numerical simulation (second column) of the barrierduring the MEL test. The third column depicts the z-location ofthe concrete block in the simulation. The experimental andnumerical results are shown at the same scale. The maximumnet elongation during the MEL test (evaluated according to theETAG No. 027) was 8960 mm, while the calculated elongation is9013 mm (a difference of 53 mm). The deformation pattern,including the shape of the net surrounding the block, is also ingood agreement with the experiments. The openings betweenthe net and support cables around the posts have similardimensions as in the experiments.

6.2. Load cell measurements

The sampling frequency of both the experimental and numeri-cal force history data is 1/0.5 ms (2 kHz). The force histories atthe load cells (MEL test) are shown in Fig. 38. The experimentalforce histories (dark) [26] are compared to the FEM force histories(light).

The force time histories can be described as a fluctuatingincrease until the block kinetic energy is entirely dissipated.After reaching this point, the block bounces back and the systemis progressively unloaded. A progressive decrease of the force his-tories is observed. A transition zone between the load increase andload decrease is observed in the force cells installed near theenergy dissipating devices. This transition zone lasts approxi-mately 300 ms. In this transition zone, the energy dissipatingdevices elongate under an approximate constant load. The timeneeded to dissipate the total kinetic energy of the block is the brak-ing time, which is approximately 430 ms (Fig. 37c).

The comparison between peak forces PF of calculated and mea-sured values in the support cables (load cells 3, 4, and 10) showthat the absolute difference jPFdiff j is between 8% and 13%. Thenumerical response of these cables is dominated by the piecewiseT vs. u relationship used to model the energy dissipating devices(Section 5.3). The peak values are given by the maximum contactforce that is transmitted to the energy dissipating devices. A goodmatch is observed between the increasing and decreasing experi-mental and simulation parts of the force histories.

The model prediction of the up-slope and lateral cable anchorforce histories depend on the accuracy of the model approximationof the contact interactions. Modelling true contact conditions ofhighly flexible systems composed of many structural items inter-acting with each other is challenging. In these flexible systemsthe positions of the connecting components have a tolerance thatis much higher with respect to rigid structures. In the cables with-out energy dissipating devices, the installation tolerances mightincrease the difference between the calculated and measured val-ues. The difference in the calculated and measured jPFdiff j for cableswithout energy dissipating devices is between 7% and 38% in fourof the instrumented cables (load cells 1, 5, 9 and 13), while anextreme difference of 69% (load cell 6) occurs in the remainingmeasured cable.

An alternative to measure the quality of the simulation resultsis to consider the entire time history, instead of the peak values.The force–time integral (FTI) can be evaluated numerically as

FTIm ¼ZðFmÞdt ¼

Xn

i¼1

�FmiDti ð7Þ

FTIc ¼ZðFcÞdt ¼

Xn

i¼1

�FciDti ð8Þ

where �Fmi and �Fci are the mean measured and calculated forces dur-ing the sampling time step; Dti is the sampling time step whichcoincides with the experimental sampling. The value of the absolutedifference FTIdiff , expressed as a percentage of the measured FTIm, is

jFTIdiff j ¼FTIm � FTIc

FTIm

�� ð9Þ

The value of jFTIdiff j for cables with energy dissipating devices (loadcells 3, 4, and 10) is between 2% and 9%. In the case of cables with-out energy dissipating devices (load cells 1, 5, 9 and 13) jFTIdiff j isbetween 5% and 40% for three cables. However, the remaining cable(load cell 6) exhibits a difference of 70%. This clearly shows thatthere are cables that are more sensitive to installation tolerancesdutol in the FEM analysis.

7. Concluding remarks and discussion

In this paper a FE model to account for contact interactions inflexible chain-link wire nets with loose connections was developedwhich could be applied to simulate a concrete-block impact into arockfall barrier. A key feature of the developed FEM in this workwas that it could include the salient mechanical properties of theloose chain-link connections that were identified in theclosed-form analytical model of [12]. The analytical procedurerevealed the features that must be taken into account to accuratelyreproduce the complex mechanical behaviour of the net. The fea-tures are:

1. Realistic modelling of (1) the contact area and (2) the frictionalcontact interactions between chain-link elements.

2. Modelling the three-dimensional input geometry of thechain-link elements and their connectivity according to precisemeasurements of the manufactured net.

3. Using a fine Finite Element discretization at the connectionsbetween chain-link elements.

The most important response characteristics of the model are:

1. Progressive bending of the connection.2. Reduction of the load eccentricity at the connections as tensile

forces increase.3. Coupling between the bending and axial behaviour of the

chain-link elements.4. Modelling of the shear contact forces at the connections

between chain-link elements.

Quasi-static tensile laboratory tests performed on chain-linknet samples served to identify these behavioural features. The testsprovided additional insight into the damage process and the role offriction at the contact between chain-link elements.

Initially simplified bi- and three-dimensional FE models wereassessed for their agreement with quasi-static tensile laboratorytests of chain-link nets. The poor agreement of these simplifiedmodels highlighted the importance of taking into account theaspects included in the analytical model. As a result, a detailedFEM was developed accounting for the contact conditions betweenchain-link elements. The twisted triple wire cross-section of thechain-link was substituted by an equivalent wire of circularcross-section. The Von Mises plasticity model and a ductile damagelaw were used to model the chain-link material. The equivalentwire cross-section radius, and constitutive parameters were foundby using parameter optimization techniques. This detailed FEMwas able to reproduce the behaviour of the shear (V), axial (N),and bending (M) diagrams obtained with the closed form analyticalmodel. Furthermore, the FEM predicted the axial (N) and bending


(M) interaction obtained with the conceptual model. The modelreproduced the M–N path in the plastic range until failure.Finally, the overall force–displacement response of the net waswell approximated.

The approach was applied to model the impact of aconcrete-block into a flexible rockfall barrier consisting ofchain-link nets with loose connections. Full scale test data wereavailable to verify the FE model. The rockfall barrier is able to stopa falling rock with a maximum kinetic energy of 2000 kJ andobtained the European Technical Approval according to theETAG027 guideline in 2010. This flexible rockfall barrier allowslarge deformations of the chain-link net and uses special deviceswith large energy dissipation capacity. In particular an additionalfeature of this system is that the contact between the supportcables and the chain-link meshes in the vicinity of the posts isinterrupted. This design, therefore, avoids stress concentrationsin the chain-link meshes near the posts, and hence reduces theforces transmitted to the up-slope anchors.

Another important improvement in modelling rockfallbarriers through this work is the application of GC to simulatethe complex interaction between barrier components such as:(1) flexible net and wire-rope cables, (2) shackle-to-wire-ropecables, (3) shackle-to-flexible net, (4) post-to-shackles and (5)block-to-flexible net. These interactions include friction and con-tribute to the dissipation of block kinetic energy. Modelling contactinteractions in an accurate manner allowed us to obtain a force dis-tribution within the barrier that matches the measured data.Furthermore, it serves to dissipate the impact energy of the blockin a manner that matches the images from high speed camerasused in large field tests.

Simpler FE models consider the overall net as a macroscopicstructure where contact between chain-link or ring elements isneglected. Simplified procedures do not capture frictional dissipa-tion within the wire net and therefore must compensate withnon-physical plastic deformations to dissipate the kinetic energyof the block. Careful use of mass-scaling, rigid body assumptionsand parallel computing makes this approach efficient and suitablefor many practical applications. This is particularly important inparametrizing and standardizing the design of flexible steel wirenets.

Acknowledgements

The authors thank the Commission of Technology andInnovation CTI for the financial support for this project under theauspices of 12385.1 PFIW-IW. JPE would also like to thankDr. Axel Reichert from the Dessault Systemes office in Munich forintellectual support regarding this investigation.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.engstruct.2015.07.005.These data include MOL files and InChiKeys of the mostimportant compounds described in this article.

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