Easy kN

You would expect that if Hooke's law was truly accurate at all loads. But the effect of things like sheath friction leads to deviations---not unexpected---at low loads such as the ones used for "static" elongation. The actual graph is not a straight line but rather an s-curve, a significant portion of which is pretty straight.

Perhaps it should be added that the next step in modeling is to assume viscous damping in the presence of Hooke's law. Since viscous damping depends on velocity, there would be no effect at static loads and maximal effect when velocity is maximal, which, I think, is at the instant the rope begins to catch the fall. The CAI model uses a system of such equations, one for each of multiple segments in the belay chain.

Kedron Silsbee wrote:That seems not to be a very good approximation based on these data. One would expect static elongation/.8kn = dynamic elongation/impact force (assuming a test mass of 80 kg for the static elongation). These ratios seem to disagree by a factor of two in some cases. I guess the maximum impact force varies roughly as the square root of the spring constant in the hooke's law model, so perhaps this uncertainty isn't as dire as it first seems, but it seems like no model is going to make much better than factor of 2 level predictions without going beyond the simple spring model.

Using any type of linear model for predicting rope elongation and impact force is going to be inaccurate. Ropes are far from idea springs and not only are their stress-strain profiles not linear, they change depending on many variables including age and environmental factors. In many cases, there is a very noticeable difference in rope elongation and impact force when comparing a brand-new rope of a particular model to a well-used rope of the same model even if the rope was used exclusively in the gym on draws that were spit shined and cleaned hourly.

First thing's first: get out out of US and into metric units. This is coming from an American, mind you. You're talking about feet and pounds, but your gear is in meters and kilonewtons... Aside from the debate over switching to the metric system as a whole, it's at least helpful to work in similar units for simplicity.

There is some great advice mentioned above, and the main point is the answer is not that simple. Differences in climber and belayer weight (and even experience), length and design of rope, rope drag, and other frictional forces all work together to determine the ultimate impact force. For example, if there's a 20 kg difference in climber and belayer (~196 newtons), and the leader takes a fall, it could be enough to lift the belayer off the ground, decreasing the impact force on the top piece. Most drop tests are a worst case scenario, which are negated in real life due to the dynamic nature of the belay. Generally you should expect around 70% or less of the force from those drop tests in real world climbing. The factor that matters is the abruptness of the catch.

That being said, don't rely on marginal pro just because the numbers add up. People have been saved by RURPs and micros, and fallen on bolts and "bomber" pro.

20 kN wrote: Using any type of linear model for predicting rope elongation and impact force is going to be inaccurate. Ropes are far from idea springs and not only are their stress-strain profiles not linear, they change depending on many variables including age and environmental factors. In many cases, there is a very noticeable difference in rope elongation and impact force when comparing a brand-new rope of a particular model to a well-used rope of the same model even if the rope was used exclusively in the gym on draws that were spit shined and cleaned hourly.

Well, yes, but perhaps you go too far. I suspect you are using "linear model" in a far more restricted sense than the term actually implies. One can include damping and one can use a system of linear models on each segment of the belay chain. There are linear models, such as the CAI one, that are claimed to match experimental data, including data obtained using real belayers, fairly well.

You also suggest that the "inaccuracy" is somehow connected to the "linearity," but your examples simply mention factors (eg rope age) that were perhaps not incorporated into the model, but might be, sometimes with ease. This has nothing to do with linearity.

The standard equation is indeed just the equation for undamped simple harmonic motion, and so is a crude approximation for a climbing rope. That said, climbing ropes under dynamic loading do exhibit linear behavior over a surprising range of loads, the deviations from linear happening at very low and very high loads.

rgold wrote: The standard equation is indeed just the equation for undamped simple harmonic motion, and so is a crude approximation for a climbing rope. That said, climbing ropes under dynamic loading do exhibit linear behavior over a surprising range of loads, the deviations from linear happening at very low and very high loads.

Damping isn't completely irrelevant for this system. If you model it as an over-damped oscillator, that would help to explain the additional forces.

Andrew Morrison wrote: Damping isn't completely irrelevant for this system. If you model it as an over-damped oscillator, that would help to explain the additional forces.

You seem to have misunderstood my point. Damping is clearly relevant, and it can be incorporated in the form of viscous damping while keeping the equation (or system of equations) linear. It isn't particularly clear why viscous damping should be the mode of damping experienced by climbing ropes, but the CAI equations use it to apparently good effect. Ropes may be nearly overdamped, but are certainly not actually overdamped, by the way.

Other rope models use combinations of damped harmonic oscillators in series and parallel to better capture low-load and high-load behavior.

Yes, that's correct. I misunderstood your point.