Simulation of Climbing and Rescue Belays v3tmoyer/testing/... · • From current rope tension, calculate a = T/m +g • Calculate ∆v = a dt and ∆x = v dt • From new positions,

A Simulation of Climbing and Rescue Belays

Tom Moyer

2006 International Technical Rescue SymposiumThis presentation and the associated model can be downloaded at

http://www.xmission.com/~tmoyer/testing (© Tom Moyer)All images in this presentation were generated with RescueRigger (rescuerigger.com)

SALT LAKE COUNTYS H E R I F F ’ S O F F I C E

SEARCH RESCUE

1

How do TTRL belays compare to climbing belays?

2

46 N min209 N average 425 N max

No load above which 100% of the population can grip

Mauthner - Gripping Ability on Rope in Motion study

3

What gripping ability is required to hold the load statically?

F

f

F

f

F / f = force multiplication factor (FMF)

For a brake bar rack with 5 bars, FMF ≈ 20 with 6 bars, FMF ≈ 25

For an ATC, FMF ≈ 7.5

Force Multiplication Factors of Friction Devices

4

80 kg climber

66% efficiency over biner

ATC FMF = 7.5

Hand Force T0

80 kg rappeller

Rope tension T1

Hand Force T0

ATC FMF = 7.5

Rope tension T2

T1

Rappelling

T1 = 80 kg * 9.81 m/s2 = 785 N

T0 = 785 N / 7.5 = 105 N

Belaying

T2 = 80 kg * 9.81 m/s2 = 785 N

T1 = 785 N * 0.66 = 518 N

T0 = 518 N / 7.5 = 69 N

Climbing Scenarios –Static Loads

5200 kg load

50% efficiency over edge

Brake bar (5 bars) fmf = 20

Hand holds 49N

Vertical TTRL Belay

T2 = 200 kg * 9.81 m/s2 = 1,962 N

T1 = 1,962 N * 0.50 = 981 N

T0 = 981 N / 20 = 49 N

Vertical TTRL Scenario

6

35°

600 kg load

Brake bar (6 bars) fmf = 25

Rope tension T1

Hand holds 135N

Low Angle TTRL Belay

T1 = 600 kg * 9.81 * sin (35 °) = 3.38 kN or 760 lb

T0 = 3.38 kN / 25 = 135 N

Low Angle TTRL Scenario

7

Dynamic Models

Model dynamic events and compare to test data

Why model?• Repeatable• Cheaper than testing• Can study one variable at a time• Can study parameters that are difficult to test

Comparison Data

• No Hand• Weber - PMI drop tests• Moyer - cordelette tests• Manufacturer’s ratings

• With Hand• Petzl fall simulator• CMT test data & simulation (live belayers)• Rigging for Rescue TTRL tests (mechanical hand)

Lead Fall

0

1000

2000

3000

4000

5000

6000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Time (s)

Forc

e (N

)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

Dis

tanc

e (m

)

Runner load

Rope Load

Belay site load (N)

Climber position (m)

Slide Distance (m)

8

Gravitational potential energy = strain energy in the rope

Rope Modulus M = T/strain or TL/δPotential Energy = mg(h+δ)Strain Energy = ½T δ

Simple Linear ModelConservation of Energy

h

L

δ

m

FmgMmgmgT 21max ++=

where fall factor F = h/L Forc

e (N

)

Distance (m)

Strain Energy

δ

Fmax

9

Detailed ModelIterative Dynamic Motion Equations

Includes:• Nonlinear rope elasticity• Knots• Rope damping• Carabiner friction• Belay device friction• Slipping in belayer’s hand• Lifting of belayer

Iterative solution approach:• From current rope tension, calculate a = T/m +g• Calculate ∆v = a dt and ∆x = v dt• From new positions, calculate new rope strains ε = ∆L/L• From new strains, calculate rope tensions• Calculate slip distances at friction devices to limit tension ratios to allowed values• Calculate new rope strains and new rope tensions

10

Model ParametersFall Parameters

mc

L 1

L 2

h

d

FMFα

η

δs

Fall equationsFall height h = d + L2Fall factor F = h / L

Parameters at static loadLs = L + δshs = h + δsFs = hs / Ls

Climber mass (kg) mc

Belayer mass (kg) mb

Time step (s) dtRope span angle (deg) α

Runner angle (deg) βBelayer tie-in slack (m) b_slack

Belayer-biner distance (m) L1

Climber height above biner (m) dRope length (m) LBiner efficiency η

Rope 2nd order modulus (N/strain2) BRope 1st order modulus (N/strain) A

Damping coefficient (N/strain/s) λka/kb k_ratio

Number of knots #_knotsKnot 2nd order stiffness (N/m2) E

Knot 1st order stiffness (N/m) DKnot minimum force (N) C

Reaction time (s) Tr

Force multiplier FMFGrip (N) grip

Fall height (m) hFall factor F

δd

11

Three Components are Critical to Understand

The Rope

The Friction Device

The Hand

12

Rope Properties2nd order curve fits – Weber PMI Data

Elongation of Ropes2nd-order Curve Fits

y = 1,390,589x2 + 0x

y = 60023x2 + 4024x

y = 43072x2 + 4603x

y = 366035x2 + 318x

0

5,000

10,000

15,000

20,000

25,000

30,000

0% 10% 20% 30% 40% 50%

strain

Forc

e (N

)Weber PMI 12.5mm staticSterling Superstatic (TM)BD/PMI 10.5 dynamic (TM)Weber PMI 10.6

• Model results with nonlinear properties match Attaway’s analytical predictions

• Nonlinear rope still obeys fall factor rule. Impact force is a function of fall factor.

• Impact force for a zero ff drop on nonlinear rope is 3 x weight instead of 2 x weight.

13

Knot Properties2nd order curve fits – Weber PMI Data

Knot Elongation2nd-order Curve Fits

y = 260,435x2 + 7,462x + 783

y = 1,154,688x2 + 0x + 783

0

5,000

10,000

15,000

20,000

25,000

30,000

0 0.05 0.1 0.15 0.2 0.25Extension (m)

Forc

e (N

)PMI 12.5mm staticPMI 10.6mm dynamic

• Knots modeled as rope sources rather than compliance terms

• Knots are much more significant on short ropes

14

Rope Properties - Damping

What is damping?

• Elastic force is proportional to deflection (strain)

• Damping, or viscous force is proportional to velocity (strain rate)

• Elastic energy is returned on rebound.

• Damping energy is lost to heating in the rope.

• Damping causes oscillations to die out.

C. Zanantoni - CMT

15

Pavier Damping Model

• Spring in series with a spring/dashpot combination

• Simple spring/dashpot combo produces unrealistic results.

Initial impact forces too high.Damping values too low (too underdamped)

• Real ropes are close to critically damped.

• Damping values ka/kb and λ determined by trial and error to produce reasonable model behavior.

• Overall spring rate k from slow-pull testing

• Damping values could be determined experimentally with good force/deflection measurements in drop tests or fast pull-tests.

ka

kbλ

16

Comparison to Weber PMI Data Example Load Profile

• Drop-test values give maximum force, elongation, and energy.

• Data points are very close to the rope-only curve.

• Without damping, rope and rope + knots curves do not store sufficient strain energy.

• Therefore they over-predict both force and elongation.

Weber - PMI Drop Tests #92 & #13112.5mm PMI Static - hs = 5ft, Ls = 20ft

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Position (m)

Load

(N)

ModelWeber PMI Drop-testsSlow-pull, rope & knotsSlow pull, rope only

17

Impact ForcesPMI 12.5mm Static Rope, M = 80 kg, Ls* = 6.1m (20 ft)

0

5,000

10,000

15,000

20,000

25,000

0 1 2 3 4 5 6 7

Drop Distance* (m)

Forc

e (N

)

Second-Order Rope

Linear Rope k = 178 kN (40,000 lb)

Second Order Rope with Knots(Attaway)This Model (Second Order Rope withKnots & Damping)Weber Drop-Test Data

Linear Rope k = 67 kN (15,000 lb)

Comparison to Weber PMI Data

* Ls Length is at static resting point. Drop Distance is height above free (unstretched) length of rope.

18

UIAA Test

80 kg weightFall Factor 1.712.8 meter rope

Cordelette is at the direction change anchor

Black Diamond 10.5mm rope- rated impact force of8.4 kN (1888 lb)

Comparison to Moyer Cordelette Testing

19


20


UIAA Drop - 10/15/00 Boulder, CO

0

2000

4000

6000

8000

10000

12000

0.0 0.5 1.0 1.5 2.0

Time (s)

Forc

e (N

)

Runner load

Runner load (data)Rope load

Rope load (calc from data)Belay site load

Belay site load (data)

5mm Gemini Drop #1

UIAA drop, Boulder 80 m c80 m b

0.0005 dt36 α14 β

0 b_slack0.4 L1

1.96 d2.67 L0.78 η

62252 B2657 A2800 λ

6 k_ratio4 #_knots

260,435 E7,462 D

783 C0.00 Tr

6000.0 FMF1 grip

4.23 h1.584 F

21

UIAA Drop - 10/15/00 Boulder, CO

0

2000

4000

6000

8000

10000

12000

0.0 0.5 1.0 1.5 2.0

Time (s)

Forc

e (N

)

Runner load

Runner load (data)Rope load

Rope load (calc from data)Belay site load

Belay site load (data)

Same Drop –Damping Removed

The Effect of Damping

5mm Gemini Drop #1

1st peak slightly shifted

Bigger effect on 2nd peak

22


0

1000

2000

3000

4000

5000

6000

7000

8000

0.00 0.50 1.00 1.50

position

Load

(N)

ModelData

UIAA Drop – 10/15/00 Boulder Colorado5mm Gemini Drop #1

23

Drops with a Hand in the System

• Hand slipping makes rope properties relatively unimportant

Italian CMT has done extensive study of the behavior of the belay hand in climbing falls

• Force measurements in falls compared to slow-motion video of the belayer

• Three phases of belay-hand behavior identified• Inertial Phase• Muscular Phase• Slipping Phase

24

“INERTIAL” PHASEThe hand moves fast

THE UPPER BODY STANDS STILL

THE HAND MOVES FAST

25

“MUSCULAR” PHASEThe hand moves slowly

THE UPPER BODY MOVES

THE HAND MOVES SLOWLY

26

“HAND SLIPPING” PHASEPossible rope slipping in the operator’s hand

NOTE THE SLIPPAGE IN

THE HAND

27

FIX POINT BELAYFIX POINT BELAY

0

100

200

300

400

500

600

700

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4time ( s )

load

( d

aN )

0

1

2

3

4

5

6

7

8

9

10

disp

lace

men

t ( m

) -

spe

ed (

m/s

)

runner load (model) sliding length in the brake falling mass speed

fallig mass displacement hand speed

inertial phase

muscular phase

no more sliding in the brake

28

Comparison to CMT Belay Simulation and Data

CMT Fall Parameters given:• Mass m = 80 kg• Fall height h = 8 m• L1 = 7.15 m• Belay Device FMF = 7.5• Hand mass = 2.5 kg

CMT Lead Fall80 mc

0.0005 dt7.15 L1

4 d11.15 L0.58 η

0 B20920 A3000 λ

50 k_ratio0.15 Tr

7.5 FMF290 grip

8 h0.717 F

CMT Lead Fall

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Time (s)

Forc

e (N

)

0.00

0.40

0.80

1.20

1.60

2.00

2.40

2.80

3.20

3.60

Dis

tanc

e (m

)

Runner load

Rope Load

Belay site load (N)


Slide Distance (m)

29

CMT Conclusions on Belaying

• Hand acts as an inertial load for the first few hundred milliseconds.

• Slip distance is proportional to fall height, not fall factor. Confirmed.

• Peak force occurs at maximum hand acceleration, not at lowest climber position.

• Only a small amount of belayer lifting is helpful (~20 cm). More lifting increases fall distance and does not decrease peak force. Confirmed.

30

Comparison to Petzl Fall Simulator

Lead Fall - Comparison to Petzl

0

1000

2000

3000

4000

5000

6000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Time (s)

Forc

e (N

)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

Dis

tanc

e (m

)

Runner load

Rope Load

Belay site load (N)


Slide Distance (m)

Peak Force - on rope 3000 N

- on anchor 5000 N - on belayer 2000 N

- on belayer's hand 400 N Slide distance 4.95 m

Petzl Simulator values:• Hand Grip = 400N• Rope Burn Warning = 1800J• Reverso FMF = 5.0• Munter Hitch FMF = 7.5• Grigri FMF = ∞ (no slipping)

• 11mm rope modulus ≈ 44.1 kN• Carabiner efficiency = 66.6%• Knot elongation included

• No rope damping• No lifting of belayer

Petzl Comparison80 mc

0.0005 dt10 L1

8 d18 L

0.67 η0 B

44100 A0 λ

3.6 k_ratio1 #_knots

260,435 E7,462 D

783 C0.00 Tr

5.0 FMF400 grip16 h

0.889 F

31

Belay Device Details - FMF Values

240°

120°

120°

120°

120°

80°

Attaway Friction Analysis

T2/T1 = eµβ

Total = 800°4.4π

T1 = T2/31

T2

T1

Friction device properties are very important to the model predictions

32

Belay Device FMF ValuesBlack Diamond Testing

Mot

ion

Belay Device

33

Friction Device Force Multiplier Values (10.8mm rope)

0.0

5.0

10.0

15.0A

TC

Ato

ga

She

riff

Bug

Airb

rake

Airs

tyle

Mun

ter

Pyra

mid

Jaw

s

VC

Rap

tor

Cub

ik

Fixe

Forc

e M

ultip

lier

Low FrictionHigh Friction

Belay Device FMF ValuesBlack Diamond Test Data

34

Friction Device Force Multiplier ValuesVariation with Rope Diameter - (HF side of variable devices)

0.0

5.0

10.0

15.0

ATC

Ato

ga

She

riff

Bug

Airb

rake

Airs

tyle

Mun

ter

Pyr

amid

Jaw

s

VC

Rap

tor

Cub

ik

Fixe

Forc

e M

ultip

lier

10.8 mm9.9 mm9.4 mm8.1 mm


35

Friction Device Force Multiplier Values (10.8mm rope)Variation with Hand Force - (HF side of variable devices)

0.0

5.0

10.0

15.0

20.0

25.0

30.0

ATC

Ato

ga

Sher

iff

Bug

Airb

rake

Airs

tyle

Mun

ter

Pyr

amid

Jaw

s

VC

Rap

tor

Cub

ik

Fixe

Forc

e M

ultip

lier

1.0 lb18.4 lb37.7 lb


36

Comparison to Rigging for Rescue Drop-Test Data

• Measured values:5,626 N Peak Force, 184 cm slide distance, 231 cm FAS Extension

• Model values:3003 N Peak Force, 184 cm slide distance, 219 cm FAS extension

Rigging for Rescue Drop-Test #2

0

500

1,000

1,500

2,000

2,500

3,000

3,500

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Time (s)

Forc

e (N

)

0.0

0.5

1.0

1.5

2.0

2.5

Dis

tanc

e (m

)

Rope Load


Slide Distance (m)

• Brake Bar FMF determined by trial and error.

• FMF = 14.3 gives a slide distance equal to the measured value

• This underpredictsthe measured peak force

RFR Drop #21m drop3m rope length200 kg test mass210N hand setting7/16 Sterling SS rope

RFR Drop #2200 mc

90 θ0.001 dt

3.00 L1.00 h

366035 B318 A

10000 λ5 k_ratio1 #_knots

1,154,688 E0 D

783 C0.20 Tr

14.3 FMF210 grip

37

"Two Rope Mech Hand" set at 210 N 80kg mass 0 cm drop of 11 mm Sterling

Superstatic straightline pull through hand

0

100

200

300

400

500

0 0.4 0.8 1.2 1.6 2.0 2.4

Time (s)

Forc

e (N

)

Rigging for Rescue3/5/05: DT-62 Hand

Average 299 N

Comparison to Rigging for Rescue Drop-Test Data

• Slide distance is a function of the average mechanical hand force.

• Peak rope tension is a function of the peak mechanical hand force.

• Any spikes in the mechanical hand force will cause higher measured peak force values.

Rigging for Rescue Data – ITRS 2005

38

Brake Bar FMF - 5 bars - Function of Hand Force

0

10

20

30

40

50

60

0 100 200 300 400 500Hand Force (N)

FMF

RFR Drop Tests - peak force

RFR Drop Tests - slide distance

Comparison to Rigging for Rescue Drop-Test DataBrake Bar FMF varies with Hand Force

39

Brake Bar FMF Testing at Black Diamond

Mot

ion

40


Brake Bar FMF - 5 bars - Function of Hand Force

0

10

20

30

40

50

60

0 100 200 300 400 500Hand Force (N)

FMF

RFR Drop Tests - peak force

RFR Drop Tests - slide distance

SMC slow pull tests - 11mm

Slow Pull Tests

41

Brake Bar FMF - function of # of barsat 223N (50 lb) Hand Force

4.0

6.0

8.0

10.0

12.0

14.0

16.0

3 4 5 6 7# of bars

FMF


42

Back to the Original Question

How do TTRL belays compare to climbing belays?

43

Required Gripping Abilityfor Different Belay Scenarios

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 50 100 150 200 250 300 350 400

Gripping Ability (N)

Slid

e D

ista

nce

(m)

Vertical TTRL, 5 bars, 200 kg, h=1, L=3

Toprope, ATC, ff0, L=45

UIAA fall, ATC, ff1.71, L=2.8

Low Angle TTRL, 6 bars, 600 kg,preloaded drop, L=3Petzl Rope Burn

Gripping Ability Required for Climbing and Rescue Scenarios

How much slip is too much?

• BCCTR belay standard, 1m maximum total extension.

• Petzl rope burn warning, 1800J

• Some belay device slip is good -reduces peak force.

• Too much sliding increases chance of collisions.

• A reasonable limit might be slide distance less than fall height.

Average human gripping ability

44

Rope Stretch

• Rope stretch is very important at longer rope lengths• A preloaded rope is much better

Rope Elongation in Rescue Belay ScenariosLocked Belay - Sterling Superstatic Rope

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0 10 20 30 40 50 60

Rope Length (m)

Tota

l rop

e el

onga

tion

(m)

Low Angle BelayVertical BelayLow Angle TTRLVertical TTRL

Edge transition

1 m extension

45

• The hand is preloaded in a TTRL belay• A TTRL belayer can optimize brake bar setup

• Reaction time may be longer for a TTRL belay.• TTRL belay may already be sliding.• TTRL belayers typically wear gloves.• TTRL belayers are not expecting to catch falls.

Differences Between Rescue Belays and Climbing Belays

46

• TTRL grip requirements are similar to climbing.• Teams who prohibit manual devices should also

prohibit them for lead climbing and rappelling.• Brake bars are not very high friction devices.• Unlikely that TTRL belay would ever meet 1m

extension limit in the BCCTR test.

• The ideal rescue belay would be autolocking, force limiting and preloaded.

Conclusions

47

• Chuck Weber – PMI• Paul Tusting and Kolin Powick –

Black Diamond Equipment• Carlo Zanantoni - CMT• Mike Gibbs – Rigging for Rescue• Dave Custer – UIAA• Steve Achelis – RescueRigger• Garin Wallace – SMC• Marc Beverly and Steve Attaway

Thank You

Documents

Simulation of Climbing and Rescue Belays v3tmoyer/testing/... · • From current rope tension, calculate a = T/m +g • Calculate ∆v = a dt and ∆x = v dt • From new positions,