Perhaps you can explain it to us so we'll all know.....
The math for calculating loads is way over my head, and it isn't all that accurate anyhow. If you've got plenty of spare time have a read of this;
RR668: Evaluation of current rigging and dismantling practices used in arboriculture
It's a hefty 370 pages, and it's the most comprehensive study available on rigging methods used in our industry. The sections on math are pretty dry and not especially practical, but there is a lot of good info in there on tree safety assesment, how much load a branch can handle, best practices for rigging etc. There are some very good tests that measure forces in the text, and they make reccomendations on how to reduce your loads. Basically it comes down to reducing the size of the piece, reducing the distance of fall when negative blocking, and letting it run.
The test data on page 356 shows that tops typically reach a force (measured at the block) of 5-6 times their own mass, though it can be less. bear in mind that all forces are measured at the block where forces are doubled. This means that a branch typically creates a force of not much more than double it's own weight. The mechanical disadvantage created by the block doubles that force, for a net shock of 5-6 times the mass of the branch.
Logs however typically generate close to 10 times their own mass in shock force, measured at the block. This can be reduced by letting it run, but you can't bank on that. You have to assume worst case scenario which is 10 times. You may have the best groundie, but what if the rope jams?
There isn't much value in trying to calculate the force of any given take mathematically, but using these rules of thumb will create safe rigging scenarios.
In order to apply this info, we need to understand cycles to failure (CTF) as it applies to ropes, as well as other strength losses. Ropes lose their strength over time. Partly from abrasion and picking, but partly due to the stresses. Samson publishes a good chart that shows CTF for different ropes, to give you some idea, some ropes will have a CTF of 1000 cycles before failing if they are shock loaded at 20% of their breaking strain, but only 4 or 5 cycles if shock loaded at their full breaking strain. The short end of the stick is that if you apply heavy loads continuously to your rope, it will have a short life, and fail quickly. There are other strength losses due to knots also.
Now we are understanding that we need to keep the total 'shock' felt by the rope under 1/5 of its breaking strain. Some manufacturers tell you the Working load limit which is usually 1/5 of the breaking strain. Some tell you the breaking strain. Some say both.
So lets bring it home....
For a typical 5/8" bull rope, the breaking strain will be somewhere around 18,000lbs.
To make our rope last a long time, we need to give it a shock of no more than 1/5 of that. Which gives us a working load limit (WLL) of 3,600lbs
For branches (which we know can generate up to 5x their own mass at the pulley) this means we shouldn't be taking branches than more than around 700bs
For logs, which we know can generate up to 10x their own mass at the pulley, this means we shouldn't be taking picks much larger than 350lbs.
You can use these same rough formulas to work out the size of picks you ought to be taking on your own ropes. All you need to know is your ropes tensile strength or breaking strain. You only have to work it out once. You should of course also be using pulleys rather than natural crotching.
Some rough guides are;
typical 3/4" breaking 23,000lbs
WLL 4,600lbs
max branch 920lbs
max log 460lbs
typical 5/8" breaking 18,000lbs
WLL 3,600lbs
max branch 720lbs
max log 360lbs
typical 9/16" breaking 13,000lbs
WLL 2,600lbs
max branch 520lbs
max log 260 lbs
typical 1/2" breaking 10,000lbs
WLL 2,000lbs
max branch 400lbs
max log 200lbs
As you can see, you've got to get some pretty serious rope before you can go smashing down 1,000lb logs, unless you want to be replacing your rope frequently.
Shaun