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A Simulation of Climbingand 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
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How do TTRL belays compareto climbing belays?
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46 N min209 N average425 N max
No load abovewhich 100% of
the populationcan grip
Mauthner - Gripping Ability on Rope in Motion study
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What gripping ability is required tohold the load statically?
F
f
F
f
F / f = force multiplication factor(FMF)
For a brake bar rack with 5bars, FMF ≈ 20 with 6 bars,FMF ≈ 25
For an ATC, FMF ≈ 7.5
Force Multiplication Factors of Friction Devices
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80 kgclimber
66% efficiency overbiner
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
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200 kg load
50% efficiencyover edge
Brake bar (5 bars) fmf = 20
Hand holds49N
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
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35°
600 kgload
Brake bar(6 bars)fmf = 25
Rope tension T1
Hand holds 135N
Low Angle TTRL BelayT1 = 600 kg * 9.81 * sin (35 °)= 3.38 kN or 760 lb
T0 = 3.38 kN / 25 = 135 N
Low Angle
TTRLScenario
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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)
F o r c e ( N )
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
D i s t a n c e ( m )
Runner load
Rope LoadBelay site load (N)
Climber position (m)
Slide Distance (m)
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Gravitational potential energy
= strain energy in the rope
Rope Modulus M = T/strain or TL/ δ
Potential Energy = mg(h+δ)
Strain Energy = ½T δ
Simple Linear Model
Conservation of Energy
h
L
δ
m
F
mg
M mgmgT
21max ++=
where fall factor F = h/L F o r c e ( N )
Distance (m)
StrainEnergy
δ
Fmax
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Detailed Model
Iterative 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
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Model ParametersFall Parameters
mc
L1
L 2
h
d
FMFα
η
δs
Fall equations
Fall height h = d + L2
Fall factor F = h / L
Parameters at static load
Ls = L + δs
hs = h + δs
F s = hs / Ls
Climber mass (kg) mc
Belayer mass (kg) mb
Time step (s) dt
Rope span angle (deg) α
Runner angle (deg) βBelayer tie-in slack (m) b_slack
Belayer-biner distance (m) L1
Climber height above biner (m) d
Rope length (m) L
Biner efficiency η
Rope 2nd order modulus (N/strain ) B
Rope 1st order modulus (N/strain) A
Damping coefficient (N/strain/s) λ
ka /kb k_ratio
Number of knots #_knots
Knot 2nd order stiffness (N/m2) E
Knot 1st order stiffness (N/m) D
Knot minimum force (N) CReaction time (s) Tr
Force multiplier FMF
Grip (N) grip
Fall height (m) h
Fall factor F
δd
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Three Components areCritical to Understand
The Rope
The Friction Device
The Hand
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Rope Properties2nd order curve fits – Weber PMI Data
Elongation of Ropes
2nd-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
F o r c 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.
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Knot Properties2nd order curve fits – Weber PMI Data
Knot Elongation
2nd-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 0.25
Extension (m)
F o r c 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
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Rope Properties - Damping
What is damping?
• Elastic force is proportional
to deflection (strain)
• Damping, or viscous forceis proportional to velocity
(strain rate)
• Elastic energy is returnedon rebound.
• Damping energy is lost toheating in the rope.
• Damping causes
oscillations to die out.
C. Zanantoni - CMT
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Pavier Damping Model
• Spring in series with a spring/dashpot combination
• Simple spring/dashpot combo produces unrealisticresults.
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 trialand error to produce reasonable model behavior.
• Overall spring rate k from slow-pull testing
• Damping values could be determinedexperimentally with good force/deflectionmeasurements in drop tests or fast pull-tests.
ka
kb
λ
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Comparison to Weber PMI Data
Example Load Profile
• Drop-test values give
maximum force,
elongation, and energy.
• Data points are veryclose to the rope-only
curve.
• Without damping,rope and rope + knotscurves do not store
sufficient strain energy.
• Therefore they over-
predict both force andelongation.
Weber - PMI Drop Tests #92 & #131
12.5mm 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)
L o a d ( N )
ModelWeber PMI Drop-tests
Slow-pull, rope & knots
Slow pull, rope only
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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)
F o r c e ( N )
Second-Order Rope
Linear Rope k = 178 kN (40,000 lb)
Second Order Rope with Knots
(Attaway)
This Model (Second Order Rope with
Knots & 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. DropDistance is height above free (unstretched)length of rope.
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UIAA Test
80 kg weightFall Factor 1.71
2.8 meter rope
Cordelette is at the directionchange anchor
Black Diamond 10.5mm rope
- rated impact force of 8.4 kN (1888 lb)
Comparison to Moyer Cordelette Testing
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Comparison to Moyer Cordelette Testing
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Comparison to Moyer Cordelette Testing
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)
F
o r c 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, Boulder80 m c
80 m b
0 .0005 d t36 α14 β
0 b_s lack
0.4 L11.96 d2.67 L0.78 η
62 25 2 B2657 A2800 λ
6 k_ra tio
4 #_kno ts260 ,435 E
7 ,4 62 D783 C
0.00 r
6000.0 FMF1 grip
4.23 h
1 .5 84 F
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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)
F o r c 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 peakslightlyshifted
Biggereffect on2nd peak
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Comparison to Moyer Cordelette Testing
0
1000
2000
3000
4000
5000
6000
7000
8000
0.00 0.50 1.00 1.50
position
L
o a d ( N )
Model
Data
UIAA Drop – 10/15/00 Boulder Colorado5mm Gemini Drop #1
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Drops with a Hand in the System
• Hand slipping makes rope properties relatively unimportant
Italian CMT has done extensive study of the behavior of thebelay hand in climbing falls
• Force measurements in falls compared to slow-motionvideo of the belayer
• Three phases of belay-hand behavior identified
• Inertial Phase• Muscular Phase• Slipping Phase
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“INERTIAL” PHASE
The hand moves fast
THE UPPER BODY STANDS STILL
THE HAND MOVESFAST
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“MUSCULAR” PHASE
The hand moves slowly
THE UPPER BODYMOVES
THE HAND MOVESSLOWLY
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“HAND SLIPPING” PHASE
Possible rope slipping in the operator’s hand
NOTE THESLIPPAGE IN
THE HAND
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FIX POINT BELAY
FIX POINT BELAY
0
100
200
300
400
500
600
700
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4
time ( s )
l o a d
( d a N
)
0
1
2
3
4
5
6
7
8
9
10
d i s p l a c e m e n t ( m )
- s
p e e d
( 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
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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 Fallmc
0.0005 dt7.15 1
4 d11.15 L0.58 η
0 B20920 A3000 λ
_ra o
0.15 r
7.5 FMF290 grip
8 h
0.717 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)
F o r c e ( N )
0.00
0.40
0.80
1.20
1.60
2.00
2.40
2.80
3.20
3.60
D i s t a n
c e ( m )
Runner load
Rope Load
Belay site load (N)
Climber position (m)
Slide Distance (m)
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CMT Conclusions on Belaying
• Hand acts as an inertial load for the first few hundredmilliseconds.
• Slip distance is proportional to fall height, not fall factor.Confirmed.
• Peak force occurs at maximum hand acceleration, notat lowest climber position.
• Only a small amount of belayer lifting is helpful (~20cm). More lifting increases fall distance and does notdecrease peak force. Confirmed.
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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 2.2Time (s)
F o r c e ( N )
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
D i s t a n c e
( m )
Runner load
Rope Load
Belay site load (N)
Climber position (m)
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 Comparisonmc
0.0005 dt
10 L1
8 d18 L
0.67 η
0 B44100 A
0 λ . _rat o
1 #_knots260,435 E
7,462 D783 C0.00 r
5.0 FMF
400 grip16 h
0.889 F
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Belay Device Details - FMF Values
240°
120°
120°
120°
120°
80°
Attaway Friction
Analysis
T2 /T1 = e
µ β
Total = 800°
4.4π
T1 = T2 /31
T2
T1
Friction deviceproperties arevery important to
the modelpredictions
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Belay Device FMF ValuesBlack Diamond Testing
M o t i o n
BelayDevice
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Friction Device Force Multiplier Values (10.8mm rope)
0.0
5.0
10.0
15.0
A T C
A t o g a
S h e r i f f
B u g
A i r b r a k e
A i r s t y l e
M u n t e r
P y r a m i d
J a w s
V C
R a p t o r
C u b i k
F i x e
F o r c e M
u l t i p l i e r
Low Friction
High Friction
Belay Device FMF ValuesBlack Diamond Test Data
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Friction Device Force Multiplier Values
Variation with Rope Diameter - (HF side of variable devices)
0.0
5.0
10.0
15.0
A T C
A t o g a
S h e r i f f
B u g
A i r b r a k e
A i r s t y l e
M u n t e r
P y r a m i d
J a w s
V C
R a p t o r
C u b i k
F i x e
F o r c e M u
l t i p l i e r
10.8 mm
9.9 mm9.4 mm
8.1mm
Belay Device FMF ValuesBlack Diamond Test Data
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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
A T C
A t o g a
S h e r i f f
B u g
A i r b r a k e
A i r s t y l e
M u n t e r
P y r a m i d
J a w s
V C
R a p t o r
C u b i k
F i x e
F o r c e M
u l t i p l i e r
1.0 lb
18.4 lb
37.7 lb
Belay Device FMF ValuesBlack Diamond Test Data
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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)
F o r c e ( N )
0.0
0.5
1.0
1.5
2.0
2.5
D i s t a n c e ( m )
Rope Load
Climber position (m)
Slide Distance (m)
• Brake BarFMF determinedby trial anderror.
• FMF = 14.3gives a slidedistance equalto the measuredvalue
• Thisunderpredictsthe measuredpeak force
RFR Drop #2
1m drop3m rope length
200 kg test mass
210N hand setting
7/16 Sterling SS rope
RFR Drop #2200 mc
90 θ0.001 dt
3.00 L1.00 h366035 B
318 A10000 λ
_ra o
1 #_knots1,154,688 E
0 D783 C0.20 r
14.3 FMF
210 grip
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"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)
F o
r c e
( N )
Rigging for Rescue
3/5/05: DT-62 Hand
Average 299 N
Comparison to Rigging for Rescue Drop-Test Data
• Slide distance is afunction of the average
mechanical hand force.
• Peak rope tension is afunction of the peakmechanical hand force.
• Any spikes in themechanical hand forcewill cause highermeasured peak forcevalues.
Rigging for Rescue Data – ITRS 2005
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Brake Bar FMF - 5 bars - Function of Hand Force
0
10
20
30
40
50
60
0 100 200 300 400 500
Hand Force (N)
F M F
RFR Drop Tests - peak force
RFR Drop Tests - slide distance
Comparison to Rigging for Rescue Drop-Test DataBrake Bar FMF varies with Hand Force
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Brake Bar FMF Testing at
Black Diamond
M
o t i o n
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Brake Bar FMF Testing at Black Diamond
Brake Bar FMF - 5 bars - Function of Hand Force
0
10
20
30
40
50
60
0 100 200 300 400 500
Hand Force (N)
F M F
RFR Drop Tests - peak force
RFR Drop Tests - slide distance
SMC slow pull tests - 11mm
Slow Pull Tests
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Brake Bar FMF - function of # of bars
at 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
F M
F
Brake Bar FMF Testing at Black Diamond
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Back to the Original Question
How do TTRL belays compareto climbing belays?
Gripping Ability Required for
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Required Gripping Ability
for 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)
S l i d e D i s t a n c
e ( 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 forClimbing and Rescue Scenarios
How much slip is too much?
• BCCTR belay standard, 1mmaximum total extension.
• Petzl rope burn warning, 1800J
• Some belay device slip is good -
reduces peak force.
• Too much sliding increaseschance of collisions.
• A reasonable limit might be slidedistance less than fall height.
Averagehuman
gripping ability
Rope Stretch
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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)
T o t a l r o
p e e l o n g a t i o n ( m )
Low Angle Belay
Vertical Belay
Low Angle TTRL
Vertical TTRL
Edgetransition
1 m extension
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• 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 Belaysand Climbing Belays
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• TTRL grip requirements are similar to climbing.
• Teams who prohibit manual devices should alsoprohibit them for lead climbing and rappelling.
• Brake bars are not very high friction devices.
• Unlikely that TTRL belay would ever meet 1mextension limit in the BCCTR test.
• The ideal rescue belay would be autolocking,force limiting and preloaded.
Conclusions
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• 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