Dynamics of a fall

You often see people quote fall factors in climbing articles  (distance of fall/total length of rope out). For example, they’ll state that a fall factor of 2 (e.g. 4m fall with 2m of rope out) directly onto a belay is bad, but a fall factor of 0.1 (1m fall with 10m of rope out) is OK.  But what do fall factors actually mean in the real world?  The answer it turns out is “not a lot”.  What is important is the stretch in the rope, the friction in the karabiners (conveniently ignored by most people), the strength of the runners and/or belay anchors, as well as the fall factor ,which it turns out is simply the geometry of the fall.   We’ll have a look at some of these things using real world measured data obtained from simple tests in my garage. 

Rope stretch

In a fall a climbing rope performs a rather vital function of absorbing the energy generated in a fall by stretching.  This stretching reduces the deceleration on a falling climber at the expense of increased fall distance.  Remember, a fall never hurt anyone, it’s the sudden deceleration on landing that does the damage and the lower the rate of deceleration the less the damage, unless of course the rope stretches so much you hit the deck first.  Climbing ropes are sometimes referred to as dynamic ropes as they are designed to stretch when loaded.  Static ropes e.g. abseil ropes, blue polypropylene rope from B&Q, are not designed to stretch, (well, not as much) and are therefore unsuitable for lead climbing. The figure below shows the force on an 11mm climbing rope versus strain.  Strain is simply the ratio of the amount of stretch to the original length.  If for example you stretched 10m of rope by 1m, then the strain would be 0.1.  The dotted line on the graph shows the unloading of the rope.  It can be seen that the rope does not return to its original length straight away after unloading.  This is because nylon is known as a visco-elastic material and it takes time to slowly creep back to its original length.  Half an hour after these readings were taken the rope had returned to its original length.  The “stretchiness” of the rope is the average slope of the curve, however, if this rope was to be loaded, unloaded, then immediately loaded again, it would re-load back along the dotted line.  This unloading/reloading line has a steeper slope than the original loading line and so higher forces would be generated in the event of a second fall occurring before the rope had had time to return to its original length.  Visco-elastic behaviour is why you’re supposed to wait several minutes before climbing again after taking a big fall. 

The graph shows that at a load of 800N (~80kg), the strain is 0.055.  If you weighed 80kg and sat on the end of 50m of this rope, it would stretch 2.75m. 

Other ropes have different stretch characteristics and although there is some variation between makes due to different constructions, the main difference is rope diameter. The skinnier the rope the more stretchy it is.  If we scale this rope down to 8.5 mm diameter by taking the ratio of the cross sectional areas, then the same 80kg weight on the end of 50m of 8.5mm rope would cause the rope to stretch by 4.6m.

This stretch can be put to your advantage.  If you throw your ropes down for an abseil and they don’t quite reach the bottom, then you’ll get an extra couple of metres from the rope stretch as you go down.

Friction

In a fall the loaded rope runs over the karabiner attached to the top runner.  There is friction between the rope and the karabiner and this has a huge effect on the forces involved.  I measured the friction coefficient of a rope over a karabiner, it was 0.72. If it was frictionless then the coefficient would be 0.0.  Replacing the karabiner with a pulley reduced the friction coefficient to 0.49 (even a pulley has a lot of friction in the bearing).

The top karabiner (attached to the gear that held the fall) can be considered as a simple pulley. 

Consider the case with no friction.  The climber’s weight is W and the belayer needs to hold the weight W to stop the climber falling.  The force on the gear is 2W, i.e. the weight of the climbing plus that of the belayer.

Now consider the case where there is so much friction in the system the belayer feels no load at all (friction coefficient = 1.0).  The climber exerts a weight W on the karabiner but the belayer exerts no weight on the karabiner so the force on the gear is W, i.e. friction in the runners reduces the load on the gear.

Using my measured coefficient of friction of 0.72, the climber still exerts a force of W but the belayer only feels a force of 0.28W. In this case the load on the gear is 1.28W.

If the friction in the karabiner was reduced by using say a DMM Revolver karabiner, one would expect the friction coefficient to be similar to a pulley, i.e. 0.49.  In this case the force on the climber is still W, but the force on the belayer is 0.51W and the force on the gear is 1.51W.  This is really important.  It will be shown that the forces involved in even a relatively short fall can be more than sufficient to break anchors.  The use of low friction karabiners such as DMM Revolver significantly increases the load on the gear, thereby increasing the risk of gear failure.  The revolver is designed to reduce rope drag on routes that zigzag, and that it will do effectively.  However, if you clip one into a marginal piece of gear e.g. a small nut, and then fall off onto that bit of gear, then you are increasing the load on that piece of gear and therefor increasing the risk of it failing.  Beware using trendy new gear without knowing the physics of how it works.

Some maths

A falling climber can be considered as a mass-spring-damper system.

Using Newton’s laws of motion it is possible to construct an equation that describes the motion of the mass (a.k.a. falling climber):

For a rope this needs modifying as the damping is hysteretic (frictional) rather than viscous:

 m is the mass, c is the viscous damping,  is the damping ratio and k is the spring (rope) force.

This is a non-linear 2^nd order differential equation and it is necessary to use iterative techniques to solve it.  I turned to an excellent piece of Microsoft software – Excel.  With Excel it is quite easy to use numerical techniques to solve the equation (and allow for karabiner friction as well) and calculate the forces on the belayer, the climber and most importantly, the gear.

Gear

Even if your placements are perfect, not all protection is created equally.  Somewhat unsurprisingly a big fat #10 hex has a much higher failure load than a #00 micro nut.  Here are some of the failure loads for gear.  Note that they are the minimum failure loads and in practice the gear should be able to take a slightly greater load.

Calculations

For these calculations I assumed a 75 kg climber (~11½ stones) and both 11mm and 8.5mm ropes.  A coefficient of friction of 0.72 was assumed at the karabiner.  The maximum forces on the climber, belayer and gear are calculated, as is the amount of stretch in the rope that the fall will generate.  It is also assumed that the climber falls into free space.  If this is not the case (e.g. a slab) then the forces generated will be less as some of the fall energy will be absorbed by the climber’s body, a.k.a. gravel rash!  It was also assumed that the belayer locks the rope of without any movement.  This is a worst case assumption as if the belayer were to somehow pay a little rope out rather than lock the belay off then the forces generated would be less.  The assumption of instant belay lock is however valid.  Go down the local climbing wall and watch a belayer as a climber takes a lead fall.  Assuming the belayer is awake and paying attention, he/she will see the climber fall and instantly lock the belay off and at the same time lean backwards so as to minimise inevitable jerk forwards - every single time. Give a dynamic belay?  No way, it just doesn’t happen in the real world, the belayer always locks the belay plate off straight away.

Results

The first fall is typical of a lead fall at an indoor climbing wall – falling off 1m above the third clip.  A maximum force of 1.51kN is applied to the belayer.  Albeit momentarily, this is equivalent to a 154kg mass and will certainly get your belayer’s attention, in fact there’s a good chance you’ll meet in mid air!

The next two cases are falls directly onto the belay (no runners in) of 2 and 5m.  For these falls the fall factor is identical yet the force on the belay and climber varies from 4.76 to 7.68 kN.  This goes to show that fall factor is not  a particularly useful concept in fall dynamics as it over-simplifies things.

The final two cases show the effects of a typical lead fall (runner at 10m and fall off 1m above runner) and a real whipper (runner at 20m and fall off 2.5m above the runner).  In both cases the falls generate sufficient force to snap small nuts. When you look at a #1 nut and see 4kN written on it, you do the sums and calculate that its equivalent to 407 kg and no matter how fat you are you not THAT heavy.  This shows that falls can easily generate very large loads in the gear, sufficient to break it.   

The difference between climbing on a big fat 11mm rope and a skinny 8.5mm half rope is evident: the half rope generates less force but at the expense of increased fall distance.

Sometimes you don’t have any choice other than to use micro-nuts for protection but if you do have to place them consider putting in two and equalising them.  The sums clearly show that you don’t have to go very far above them in order to generate enough force to snap them.