experimental and theoretical simulations of climbing...

Experimental and theoretical simulations of climbing falls

Martyn Pavier

Department of Mechanical Engineering, University of Bristol, Bristol, BS8 1TR

AbstractA modern rock climber attempting dif®cult routes will expect to take a number of falls.Typically the falls will be from the same point on the climb, for example at a particularlydif®cult series of moves, therefore the same part of the climbing rope will be damagedby each fall. Since the consequences of rope failure are serious, it is important that theclimber should be aware of the number of falls the rope can withstand.

The work described in this paper was carried out to understand the mechanics ofclimbing falls so as to enable predictions to be made of the tension developed in therope. Once the value for this tension is known, an estimation can be made of the life ofthe rope measured as the number of falls to failure.

A theoretical simulation of climbing falls has been developed which includes thenonlinearity of the rope behaviour and friction where the rope runs through`karabiners,' the metal clips used to anchor the climbers to the rock. The results ofthe simulation have been compared to experimentally measured rope tensions.

The number of falls to failure has been measured experimentally for various lengths offall and fall factors (the ratio of the length of fall to the length of rope). Failure of therope invariably occurs where the rope runs over the most heavily loaded karabiner;therefore, the karabiner edge radius has also been varied. The experimental data havebeen used to derive a failure curve for the rope that may be combined with thetheoretical simulation to predict the number of falls to failure.

Keywords: Climbing; failure analysis; mechanics; theoretical simulation.

Notation

d Vertical distance above last runnerg Acceleration due to gravityk Modulus of rope for linear elastic model

k1, k2 Moduli of rope for visco-elastic modelli Length of rope segment in Number of rope segmentsr Runner radiussi Total slip at runner i

xi, yi Co-ordinates of runner iC Matrix relating incremental slips to incremental strains

Ó 1998 Blackwell Science Ltd · Sports Engineering (1998) 1, 79±91 79

Correspondence Address:Martyn Pavier, Department of Mechanical Engineering,University of Bristol, Bristol BS8 1TR, U.K.E-mail: [email protected]

_E Rate of energy dissipation at a runner due to frictionF Fall factorI Unit matrixK Matrix relating incremental strains to incremental tensionsL Length of rope run-outL Matrix of slip conditionsM Mass of leaderS Critical tension in rope segment 1 for slip at belayTi Total tension in rope segment iT Vector of total tensionsei Total strain in rope segment igi Maximum tension ratio at runner ik Viscous coef®cient of rope for visco-elastic modelli Coef®cient of friction at runner ihi Angle of lap at runner iri Slip condition at runner i

Dsi Incremental slip at runner iDs Vector of incremental slipsDt Duration of the increment of time

DTi Incremental tension in rope segment iDT Vector of incremental tensionsDei Incremental strain in rope segment iDe Vector of incremental strains

Introduction

The modern rock climber puts considerable trustin the rope: falls while climbing are commonplace,particularly for experienced climbers attemptingdif®cult routes, and the consequences of rope

failure are usually very serious. Consequently,climbing ropes are of a sophisticated construction,combining low mass with high strength and energyabsorbing properties. The modern climbing rope isof a kernmantle type construction as shown inFig. 1, consisting of a twisted nylon kern (the load-

Figure 1 Construction of a kernmantleclimbing rope.

Simulation of climbing falls · M. Pavier

80 Sports Engineering (1998) 1, 79±91 · Ó 1998 Blackwell Science Ltd

bearing part of the rope) and a woven nylon mantleto protect the kern from abrasion. The kern ismade up from a number of cords, the cords in turnbeing made from several strands twisted together.Each strand is itself formed from nylon threads,each of about 25 lm diameter. A typical rope hasapproximately ®ve million threads. The rope isdesigned to decelerate a falling climber withoutsubjecting the body to very high forces.

Climbers should understand the effect on theirrope of any climbing falls so that they are able todecide whether the strength of the rope has beenreduced suf®ciently for it to be retired from service.Currently, climbers rely on rule-of-thumb methodsto assess the effects of falls on their ropes, but theunderstanding of the physical processes involvedduring falls is often poor. Fortunately, climbingrope failures are not common, and when they dooccur are usually because of other factors, forexample the rope running over the sharp edge of arock. Nevertheless, studies have shown (Microys1983) that even experienced climbers may havelittle appreciation of the current state of theirropes, and in some cases their ropes may be close tothe end of their useful lives.

The research described in this paper aims toprovide a more precise description of a climbingfall and therefore give a better appreciation of thetension developed in the rope. Such an appreciationmay be combined with a model for the degradationof the rope so as to provide a procedure for the lifeprediction assessment of a climbing rope.

Little previous work of this nature exists; how-ever, Schubert (1986) provides some data ontensions developed in climbing ropes and a de-scription of what in¯uences these tensions, al-though the work is largely descriptive. In addition,there have been some previous reports of experi-mental measurements of tensions developed inclimbing ropes during falls (see for example Perkins1987).

In this paper, the procedures involved in climb-ing and the circumstances leading to a fall are ®rstdescribed. A simple analysis of a climbing fall isprovided as an introduction to the subsequenttheoretical analysis. The theoretical analysis is then

presented with predictions of the tensions devel-oped in the rope which are compared with experi-mental tests. Finally, a series of experimental testsare carried out to derive a failure curve for the ropethat can be used to predict the number of falls tofailure for a particular fall situation.

Background to rock climbing

Smith (1996) presented an interesting account ofthe development of the procedures involved in rockclimbing. Usually climbers climb as a pair: forexample, a female leading climber (or leader) and asecond male climber (or second). The secondbelays himself to the rock face, by attachinghimself, via his harness, to anchors ®xed to therock. These anchors take various forms, but typi-cally they may be nuts (metal wedges) jammed intocracks, pitons (metal spikes) hammered into cracksor slings of rope draped over a rock spike. Theleader climbs above the second to the top of therock face, or until she can ®nd a safe place to belayherself. Once the leader is belayed, the secondremoves his belay and climbs to rejoin the leader.As the second climbs, the leader takes in the rope soas to protect the second should he slip.

The critical part of the climbing procedure is asthe leader climbs above her second. To make thisstage safer, the leader will run the rope throughkarabiners (metal clips) attached to additionalanchors ®xed to the rock. These additionalanchors are known as running belays (or runners).A situation during a climb where the leader has®xed three running belays may then be as shownin Fig. 2. Should the leader fall from the rock, thedistance she falls until the rope begins to decel-erate her will be twice the distance from the lastrunning belay, assuming this was vertically be-neath her. As the leader climbs, the second paysthe rope out through a belay device attached tohis harness. The belay device is designed to makeit easy for the second to hold the rope tight in theevent of a fall. The rope from second to leader isreferred to as the live rope. The remaining part ofthe rope yet to be paid out by the second isknown as the dead rope.

M. Pavier · Simulation of climbing falls


Experienced climbers undertaking a dif®cultclimb may expect to take a number of falls duringthe climb. These falls inevitably lead to a deteri-oration in the rope and it is therefore useful for theclimber to have an appreciation of the factorscontrolling the magnitude of the tensions devel-oped in the rope. The traditional measure of theseverity of a fall has been the fall factor, the ratio ofthe length of the fall to the length of rope betweenthe leader and the second. The fall factor variesfrom zero when the leader falls just after placing arunner, to two when the leader falls before she hasbeen able to place any runners. This fall factorapproach implicitly assumes the rope behaves in alinear elastic manner and takes no account of theslip through the belay device and friction of therope over the runners.

Later in this paper, a theoretical procedure willbe developed for taking into account these effects.First an analytical expression for the maximumtension in the rope will be found using the standardfall factor approach (Wexler 1950) to demonstratethe basic mechanics of climbing falls.

Figure 3 shows a simple climbing fall where theleader of mass M, falls from a distance d above herlast runner. The fall factor F is given by

F � 2d

L�1�

where L is the length of live rope. The leader falls adistance 2d before the rope begins to tighten plus afurther distance d before coming to a halt due tostretch in the rope. The loss of potential energy ofthe leader due to the loss of height is Mg(2d + d).This potential energy is transferred into strainenergy stored in the rope of kd2/(2L). Here, g is theacceleration due to gravity and k is the elasticmodulus of the rope. From the principle ofconservation of energy the further distance d iscalculated to be

d �M gL

k1�

��1� 2kF

Mg

s" #

and hence the maximum tension in the rope isfound to be

T �Mg 1��1� 2kF

Mg

s" #�2�

This analysis leads to the conclusion that themaximum tension in the rope is controlled by the

Figure 3 Geometry of a simple climbing fall.

Figure 2 Description of terms used in climbing.



fall factor, the mass of the climber and the stiffnessof the rope, but not by the absolute length of thefall.

When the modulus of elasticity of the rope canbe assumed to be much larger than the weight ofthe falling climber, a simpli®ed calculation for themaximum tension may be written as:

T ��2kFMg

p�3�

Theoretical analysis of climbing falls

A theoretical analysis of climbing falls is nowpresented which includes slip at the belay, frictionat runners and a more realistic model for thebehaviour of the rope. The analysis uses anincremental approach, necessitating the use of acomputer, but allowing the analysis of a fallsituation with an arbitrary number of runners.The results give tensions in individual rope seg-

ments, loads at runners, slip through the belaydevice and the position of the falling climber.

First, the geometry of a pitch is described using atwo-dimensional Cartesian co-ordinate system withthe second at the origin (Fig. 4). The leader hasplaced n ) 1 runners, dividing the rope into nsegments. Runner i is located at xi, yi. The leaderfalls from a point, distance d, vertically above herlast runner. Calculation provides the originallength li of rope segment i between runners i ) 1and i, the angle of lap hi at runner i and the totallength of rope run out L. A maximum tension ratiogi is calculated for each runner from

gi � elihi �4�

where li is the coef®cient of friction at runner i.The tension ratio relates the tensions for ropesegments on each side of a runner when the ropeslips over the runner. Experimental measurementsof the tension in the rope segments on each side of

Figure 4 Description of pitch geometryused in the theoretical simulation.



the runner have been made for different angles oflap. These measurements con®rm the exponentialform of eqn (4) and show a coef®cient of friction of0.18±0.22 between steel and rope.

The relationship between the strain in each ropesegment and the total slip at each runner is nowconsidered. Taking segment i (Fig. 5):

si�1� ei� ÿ siÿ1�1� ei� � eili �5�where ei is the strain in rope segment i and si is thetotal slip at runner i, the slip being positive in thedirection from the second to the leader andmeasured with zero strain in the rope. An incre-mental form of eqn (5) may be derived:

Dsi�1� ei� ÿ Dsiÿ1�1� ei� � Dei�li � siÿ1 ÿ si��6�

where Dei is the incremental strain in rope segmenti and Dsi the incremental slip at runner i. Eqns (5)and (6) are also valid for the ®rst rope segment if s0and Ds0 are taken to be the slip and incremental slipthrough the belay device. The ®nal rope segment,between the leader and her last runner, gives amodi®ed version of eqns (5) and (6):

sn ÿ snÿ1�1� en� � end �7�

Dsn ÿ Dsnÿ1�1� en� � Den�d � snÿ1� �8�

where sn and Dsn are the position and incrementalposition of the leader measured positive downwardsrelative to the point at which the rope becomestight. That is, the origin for sn is a distance dvertically below the last runner. For small incre-ments, Dsn is given by

Dsn � _snDt �9�where _sn is the velocity of the leader and Dt theduration of the increment of time.

A matrix equation can now be formulated givingthe vector of incremental strains in each ropesegment De in terms of the vector of incrementalslips Ds

CDs � De �10�

where DsT � �Ds0 . . . Dsi . . . Dsn� and DeT � �De0 . . .Dei . . . Den� Note that matrix C has dimension n byn� 1. A typical row of matrix C relating theincremental strain in rope segment 4 to theincremental slips at runners 3 and 4 has the form:

0 0

..

. ...

0 . . . ÿ1ÿe4l4�s3ÿs4

1�e4l4�s3ÿs4

. . . 0

..

. ...

0 0

2666666664

3777777775

Ds0

..

.

Ds3

Ds4

..

.

Dsn

266666666664

377777777775�

De1

..

.

De4

..

.

Den

266666664

377777775The incremental strain in any rope segment may berelated to the incremental tension by the use of aconstitutive model for the rope. In this work avisco-elastic model is used with a spring in serieswith an in-parallel combination of another springand dashpot (Fig. 6). In general, the quantitiesde®ning the rope model may be made a function ofstrain, but here these quantities have been set toconstant values. Therefore, for rope segment i

DTi � k1Dei � Dt

kk1k2ei ÿ Ti k1 � k2� ��

� kiDei � DT0i

�11�

where k1, k2 and k are the constants in the materialmodel. Any model for the rope may be used,

Figure 5 Notation used for tensions and slips in an individualrope segment.



provided it can be written in a form relatingincrements of tension to increments of strain [as ineqn (11)].

A matrix equation is written giving the vector ofincremental tensions DT in terms of the vector ofincremental strains

KDe � DT ÿ DT0 �12�

where DTT � �DT1 . . . DTi . . . DTn� and DT0T ��DT0

1 . . . DT0i . . . DT0

n �. Matrix K is of dimension nby n and is given by K � k1 I, where I is the n by nunit matrix.

At each runner slip may occur depending on themagnitude of the tension in the rope segmentseither side of the runner. If slip does occur theincremental tensions vary according to the tensionratio for the runner. These conditions for runner iare expressed as:

DTi � giDTi�1; Dsi < 0; ri � ÿ1 if Ti � giTi�1

Dsi � 0; ri � 0; if giTi�1 < Ti <Ti�1

gi�13�

DTi � DTi�1

gi; Dsi > 0; ri � �1 if Ti � Ti�1

gi

where ri is used to specify the slip condition.It is assumed that slip occurs at the belay if the

tension in the ®rst rope segment is equal to acritical value. The condition for slip at the belay isexpressed as:

Ds0 � 0; r0 � 0 if T1 < S

DT1 � 0; Ds0 > 0; r0 � �1 if T1 � S�14�

where S is the critical tension in the ®rst ropesegment.

Equations (13) and (14) can be used to derive amatrix equation relating the incremental tensions

LDT � 0 �15�

Each row of matrix L corresponds to a slipcondition where ri is nonzero. For example, if theslip condition at runner 3, r3, is equal to �1, thecorresponding row of matrix L has the form:

0 0... ..

.

0 . . . 1 ÿ1=g3 . . . 0

..

. ...

0 0

2666664

3777775

DT1

..

.

DT3

DT4

..

.

DTn

266666664

377777775 �0...

0...

0

2666664

3777775It has been found necessary to use a modi®ed formof eqn (15) to avoid instabilities when the size ofincrement is other than very small:

LDT � ÿLT �16�

where the vector of rope tensions TT � �T1 . . .Ti . . . Tn�. Equations (15) and (16) are identicalprovided eqn (15) has been satis®ed exactly for allprevious increments.

Equations (10), (12) and (16) may now be com-bined to give a set of linear equations that can besolved to ®nd the vector of incremental slips, Ds.

LKCDs � ÿL�T � DT0� �17�

At the end of each increment, the incrementalstrains are calculated from the incremental slips byeqn (10). The incremental tensions are then found

Figure 6 Visco-elastic model of a rope.



by eqn (12) and the current rope tensions incre-mented, taking account that the rope tensions mustbe always greater than or equal to zero. Finally thenew velocity of the leader m can be found from

m � _sn � �g ÿ Tn=M�Dt:

A computer program has been written to carry outthe calculations described above. The duration ofthe time increment Dt is set small enough toachieve good accuracy, but it is found that thenumerical calculations are well-behaved and largelyinsensitive to the duration of the increment,provided it is small enough.

Comparison of theoretical and experimentalsimulations

The theoretical analysis of climbing falls presentedabove are now used to simulate climbing falls,providing predictions of the tension developed inthe rope. These predictions are compared with

measurements of tension taken from experimentalsimulations of climbing falls.

The experimental apparatus consisted of atrolley running on a vertical steel column asshown in Fig. 7. Extra lead bars could be added toalter the mass of the trolley up to a maximum of80 kg. A catch at the top of the steel column heldthe trolley in the raised position until a test wascarried out.

Two belay mounts equipped with strain gaugeswere used, one to represent a running belay boltedto the column, the other to represent the second'sbelay, bolted to the ¯oor. The belay mounts weremanufactured so as to have similar cross sections tostandard karabiners. Strain gauges were attached tothe belay mounts in a full-bridge con®guration andconnected to Fylde type FE-492-BBS bridge con-ditioners and FE-254-GA ampli®ers. The outputfrom the strain gauge ampli®ers was recordedduring a test using a Datalab DL1200 dataloggerwith a sampling interval of 500 ls. The tensions in

Figure 7 Details of experimental appara-tus.



the two rope segments during the test were derivedfrom the loads measured by the belay mounts.

The rope used was Edelweiss M/W V.31 ofdiameter 8.7 mm. The rope was tied to the trolley,fed through the running belay then back to thebelay device. The grip of the second's hand on therope wass simulated using a pneumatic gripper asshown in Fig. 7. The gripper allowed differenttensions to be applied to the rope running throughthe belay device and different angles to be set forthe rope either side of the belay device. The gripperwas designed so as to be able to apply a higher loadthan a human second would ®nd possible.

Before a theoretical simulation could beundertaken, the parameters used to model themechanical behaviour of the climbing rope had tobe derived by experiment. The behaviour of therope could not be measured using a conventionaltensile test machine since the rope can suffer largestrains and exhibits a strong strain rate dependency.The method used here wass to load the ropedynamically using the falling trolley and record thetension in the rope as a function of time. Norunning belay was used and the rope was tied off atthe belay so that the measurements related purelyto the behaviour of the rope. The acceleration ofthe trolley could be calculated from the tension inthe rope and hence the position of the trolley andtherefore the strain in the rope could be derived bya process of double integration.

The tension vs. strain response for a typical testis shown in Fig. 8. This test used a 2.1-m length ofrope, a fall factor of 1.3 and a trolley mass of 40 kg.One test provided the complete response since therope was loaded to the maximum tension and thenrelaxed to zero as the trolley returned upwards.The rope behaved in a signi®cantly nonlinearfashion and exhibited considerable hysterisis. Thenonlinearity of the rope was most noticeable at lowstrains due to the ®bres in the rope aligningthemselves with the loading direction. The hyster-isis was a result of the friction between adjacent®bres and the damping behaviour of the nylon®bres themselves. The experimental trace shows asuperimposed oscillation which is believed to be aresult of tension waves in the rope.

For the purposes of the theoretical analysis ofclimbing falls conducted here, it was felt to besuf®cient to model the rope using the linear visco-elastic model shown in Fig. 6, but including aninitial strain offset so that the rope only begins totake tension once a certain value of strain has beenexceeded. This provides a straightforward methodof including the signi®cant nonlinearity of the ropeat low strains. Values for the spring and dashpotparameters in the model were chosen to give anadequate match with the measured tension vs.strain curve in Fig. 8. No direct procedure exists toderive values directly for the various parameters,since the strain rate varies during the test. Thevalues for the chosen parameters are:

k1 � 35.0 kNk2 � 20.0 kNk � 3.0 kN s)1

e0 � 0.05where e0 is the initial strain offset. The strain offsetcan be included in the existing theoretical simula-tion, merely as a factor on the initial length of therope.

The theoretical technique may now be usedto provide predictions of tension vs. time that

Figure 8 Tension vs. strain from a dynamic test on a ropecompared with the model.



can be compared with the experimental results.Figure 9(a,b) shows two such comparisons. Thetension given refers to the segment of rope betweenthe running belay and the trolley. The mass oftrolley, length of rope and fall factor for thesecomparisons are given in Table 1.

A typical trace of tension for the case of the ropetied off at the belay is shown in Fig. 9(a). Thetension rises quickly to a maximum of 2.6 kN, thendecays more gradually to zero. Two smaller peaksthen occur, after which the tension remains con-stant at a value corresponding to the weight of thetrolley. The theoretical simulation agrees well withthe experimental measurements, particularly forthe maximum tension.

Figure 9(b) shows a typical trace with a Stitchplate, a common belay device. The tension risesquickly to a maximum of 1.9 kN, then dropsslightly to a small plateau of 1.6 kN before fallingto zero. There is then a small second peak afterwhich the tension remains at a constant value equalto the weight of the trolley. The cause of the smallinitial peak is the behaviour of the pneumaticgripper where the tension to initiate slip throughthe gripper is higher than that for slip to continue.The theoretical simulation assumes a constantvalue for the critical tension for slip at the belayand is therefore unable to predict the small drop inload after the initial peak that occurs in the testresults. The theory is thus unable to provide awholly accurate simulation when slip through thebelay occurs.

Prediction of rope failure

The theoretical simulation of climbing falls pro-vides predictions of the tension in rope segmentsand the slip at running belays. A useful simulationwould provide additional information concerningthe damage to the rope that could be used to assessthe lifetime of a rope. The precise mechanisms ofrope damage are largely unknown, therefore aprogramme of experimental work has been under-taken using the same apparatus as has already beendescribed.

Figure 9 Comparison of experimental and theoretical simula-tions of the tension in the rope (a) for the case of a fall with therope tied off at the belay and (b) for the case of a fall with aStitch plate at the belay.

Table 1 Test details for comparisons between experimental andtheoretical simulations

Figure Belay Length of Fall Mass Runner

condition rope (m) factor (kg) radius (mm)

9(a) Tied off 4.0 0.50 40 6.0

9(b) Stitch plate 4.9 0.54 40 6.0



Rope failure inevitably occurs where the roperuns through the running belay. Typically, severalfalls are required to cause rope failure dependingon the fall factor, length of fall, mass of climber andwhether a belay device was used or the rope tiedoff. A critical factor was identi®ed as the edgeradius of the belay mount used to represent therunning belay. Belay mounts were manufacturedwith different radii varying from 2 mm to 6 mm.The smaller radius was designed to simulate therope running over a sharp rock edge, whereas thelarger radii were intended to match the crosssection of standard karabiners.

It is proposed that the damage to the rope as itruns over the running belay depends on the rate ofenergy dissipation through frictional losses. Thetension T2 in the rope segment due to frictionbetween the running belay and falling climber isgreater than the tension T1 in the segment betweenthe running and ®xed belays (Fig. 3). These tensionsare related by the tension ratio g for the runningbelay so that T2 � gT1. The rate of energy dissi-pation _E at the running belay therefore is given by

_E � _sT2 1ÿ 1

g

� ��18�

where _s is the rate of slip. However, any one sectionof rope dissipates an amount of energy proportionalto T2(1 ) 1/g) since, although the rate of energydissipation increases with _s, so too does the rate atwhich the rope runs through the belay. Therefore, itis proposed that the damage to the rope depends onthe maximum value of tension generated during afall. However, since the energy dissipation by anysection of rope is increased by a smaller radius ofrunner this dependency on tension will be alteredwith a different radius of runner.

Table 2 provides results of the number of falls-to-failure and the maximum tension recorded inthe rope between the running belay and the trolley.A measurement of four falls-to-failure, for example,means that the rope failed during the fourth fall.For severe falls, the rope mantle and kern of therope fail together as seen in Fig. 10(a). For lesssevere falls, however, the mantle of the rope failsbefore the kern [Fig. 10(b)]. Following mantle

failure, a number of extra falls are required tocause failure of the kern. The number of falls-to-failure in Table 2 relates to complete failure of therope: kern and mantle. Various conditions for a fallwere considered: different trolley mass, runnerradius, fall factor and length of rope. Most of thetests were carried out with the rope tied off at thebelay, but for two tests a Stitch plate was used inconjunction with the pneumatic gripper. The twotest results for the Stitch plate were derived usingdifferent gripper pressures.

The test results support the implications of thesimple analytical model that the maximum tension,and hence the number of falls-to-failure, principal-ly depend on the fall factor, mass of climber andrunner radius. It is particularly noticeable how theuse of a belay device reduces the maximum tensionand greatly increases the number of falls to failure.

Table 2 Number of falls to failure for various fall conditions

Belay Runner Mass Fall Length of Falls to Maximum

condition radius

(mm)

(kg) factor rope (m) failure tension

(kN)

Tied off 6.0 60 0.86 2.50 15 5.7

Tied off 6.0 60 1.32 2.50 5 7.1

Tied off 6.0 60 1.77 2.50 4 8.2

Tied off 6.0 70 0.40 2.50 18 4.3

Tied off 6.0 70 0.80 2.50 10 5.9

Tied off 6.0 70 0.86 2.50 9 6.2

Tied off 6.0 70 0.87 1.50 4 6.1

Tied off 6.0 70 0.90 1.50 4 6.4

Tied off 6.0 70 0.90 1.50 9 6.2

Tied off 6.0 70 0.90 1.50 7 6.3

Tied off 6.0 70 0.97 2.50 8 6.5

Tied off 6.0 70 1.00 1.00 11 6.5

Tied off 6.0 70 1.00 1.33 9 6.6

Tied off 6.0 70 1.00 2.00 9 6.8

Tied off 6.0 70 1.32 2.50 5 7.6

Tied off 6.0 70 1.77 2.50 3 8.8

Tied off 6.0 80 0.86 2.50 7 6.6

Tied off 6.0 80 0.97 2.50 8 7.0

Tied off 6.0 80 1.32 2.50 4 8.2

Tied off 6.0 80 1.77 2.50 3 9.4

Tied off 4.0 70 0.40 2.50 11 4.2

Tied off 4.0 70 0.40 2.50 10 4.3

Tied off 4.0 70 0.90 1.50 7 6.3

Tied off 4.0 70 0.90 1.50 3 6.2

Tied off 4.0 70 1.33 2.00 4 7.7

Tied off 2.0 70 0.40 2.50 5 4.3

Tied off 2.0 70 1.33 2.00 2 7.8

Stitch plate 6.0 70 0.22 4.50 43 2.6

Stitch plate 6.0 70 0.22 4.50 27 4.0



Figure 11 shows the data of Table 2 plotted as themaximum tension vs. the number of falls to failurefor the three different runner radii considered in thetests. Error bars of plus or minus one fall areincluded since it is not known how close to failurethe rope is after each fall. The data is plotted in asimilar manner to a conventional stress amplitude vs.cycles to failure fatigue curve. It can be appreciatedthat reducing the radius of the runner for the samemaximum tension generated in the fall substantiallyreduces the number of falls-to-failure.

If the proposed mechanism of rope failure is valid,the data for one radius of runner should be shrunkonto one curve. The data is reasonably consistentwith this proposed mechanism; a line is superim-posed on the graph showing a linear ®t to the data fora runner radius of 6 mm. Fall conditions below thisline can be thought of as safe, while for those abovethis line, rope failure is likely to occur.

Discussion

The theoretical simulation of climbing falls thathas been developed in this paper includes themajor phenomena in¯uencing the tension devel-oped in a climbing rope. In the case of the ropetied off at the belay, the theoretical predictionsmatch closely the observed experimental behav-iour. Predictions are less good when slip at thebelay occurs, largely because a simple model hasbeen used for the behaviour of the belay device.Certainly a more sophisticated model could beused, for example by including a critical tension toinitiate slip at the belay followed by a lower valuefor slip to continue.

The visco-elastic model used for the rope ischaracterized by three constant parameters, with astrain offset to allow some account to be taken ofnonlinearity at low strains. The derivation of theparameters of the rope model is currently achievedusing an ad hoc approach. A more systematicmethod is required to derive a rope behaviourmodel, but this is likely to require special purposetest equipment to allow dynamics tests to be carriedout under constant strain rate conditions.

Since the mass of the rope itself was not includedin the analysis, the model is unable to predict the

Figure 10 Ropes showing (a) kern failure and (b) mantle failure.

Figure 11 Maximum tension vs. the number of falls to failure.



size of tension waves in the rope. For a linear elasticrope the magnitude of these tension waves can bepredicted from Timoshenko & Goodier (1982):

T � vcm �19�where T is the magnitude of the wave, v is thevelocity of the climber just before the rope becomestight, c is the velocity of propagation of waves in therope given by c � ��

k=mp

and m is the mass per unitlength of the rope. Using typical values, themagnitude of the tension waves may be found tobe much less than the predicted tensions neglectingthe mass of the rope. Conversely, tension waves inpolymer ropes used for mooring ships are signi®-cant, but the behaviour of these ropes is verydifferent from a climbing rope (Leeuwen 1981).

It has been proposed that the failure of a rope isrelated to the dissipation of friction as the rope runsover the running belay. Certainly, evidence existsthat melting and abrasion of the mantle does occur.Experimental data of the maximum tension vs. thenumber of falls-to-failure does support this pro-posed failure mechanism and enables a safe workingenvelope to be constructed to ensure rope failuredoes not occur. However, no account is taken ofenvironmental factors such as grit working its wayinto the kern. Tests have been carried out to examinesuch effects, demonstrating considerable differencesbetween new and environmentally damaged ropes inthe number of falls to failure (Pavier 1998).

The severity of damage to the rope has beenassessed by the number of falls to failure. Althoughthis approach appears to work for the case of anumber of identical falls it is dif®cult to extend it tothe more general situation where one rope experi-ences a number of different falls. Not only dodifferent falls create different magnitudes of ten-sion, but also different falls do not damage the samesection of the rope. Although in principle the effectof different falls could be taken into account, it isunlikely that a climber would record the history ofthe rope in suf®cient detail to allow an accurateassessment of remaining life to be made. The workthat has been described in this paper is likely to beof most practical use in giving a better qualitative

understanding of the factors that determine thelifetime of a rope.

Conclusions

A theoretical simulation of climbing falls has beendeveloped, allowing good predictions to be made ofthe tension generated in the rope. The simulationincludes slip of the rope through the belay device,friction at runners and a visco-elastic model for therope. The simulation can be extended to account formore general nonlinearity of the rope behaviour anda more sophisticated simulation of the belay device.

Experimental testing has provided informationon the number of falls-to-failure for different fallsituations. The data show that the number of fallscan be related to the tension developed at therunning belay. The theoretical simulation maytherefore be used to enable predictions to be madeof the number of falls-to-failure for a general fallsituation.

References

Leeuwen, J.H.V. (1981) Dynamic behaviour of syntheticropes, Proc 13th Offshore Technology, 453±463, Texas.

Microys, H.F. (1983) Summit, 31,.Pavier, M.J. (1996) Derivation of a rope behaviour model

for the analysis of forces developed during a rockclimbing leader fall. Proceedings of the 1st InternationalConference on the Engineering of Sport (ed. S.J. Haake),271±279, A.A. Balkema, Rotterdam.

Pavier, M.J. (1998) Failure of climbing ropes resulting frommultiple leader falls. Proceedings of the 2nd InternationalConference on the Engineering of Sport (ed. S.J. Haake),415±422, Blackwell Science, Oxford.

Perkins, A. (1987) Development of Fall Arrest EquipmentUsing Textiles, PhD Thesis, University of Leeds.

Schubert, P. (1986). Ausrustung, Sicherung, Sicherheit. BLV-Verlagsgesellschaft, Munich.

Smith, R.A. (1996) The development of protection systemsfor rock climbing. Proceedings of the 1st InternationalConference on the Engineering of Sport (ed. S. Haake), 229±238, A.A. Balkema, Rotterdam.

Timoshenko, S.P. & Goodier, J.N. (1982) Theory ofElasticity, 3rd edn. McGraw-Hill, New York.

Wexler, A. (1950) The Theory of Belaying, American AlpineJournal, 7, 379±405.



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