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The areas occupied by diameters and squares are different in size. For simplicity take a 4" by 4" square and draw a 4" circle in the center. You will find that the circle will fit neatly inside, touching on all the sides but leaving the corners of the square open. The volume of area the circle takes up is less than the area of the square. The circle is smaller than the square in area. Notice the open corners of the square not occupied by the circle.
There is a considerable difference in the area's of the circle and the square. Figuring the differences in area is easy, as that difference in area is always proportionately and constantly the same.
Accordingly a cord of wood 4' by 4' by 8' = 128 sq. ft of wood.
128 sq. ft in Live Oak would weigh @ 76 lbs = 9,728 lbs.
The difference in the volume in a square and a circle is constantly the same ratio. The Circle is 0.7854 of the square.
To figure the 4' dia. log 8' long. The taper leaves it different diameters on each end. For accuracy one should know or correctly estimate what those diameters are on each end and what the distance is from one end to the other.
I ran across a posting of a man who apparently had some considerable experience with working with large trees and he had made the following statement.
>I estimate by the cord, of which Euc weighs about 3 tons. Oak weighs 2 1/2, but Euc is in a category all by itself. This has to be a log, because a 12 footer would be over a hundred-fifty feet tall, not 40'. Next, we don't know how much the log tapers. In a city, this can vary, depending if the tree has been limbed up from the ground quite a ways. We can figure as if there is a relatively quick taper, the small end measuring about 6'.
>estimate of a 8 foot dia. Log 40 feet long is 20 cords
>An 8 foot diameter log is 4 cord. So, if there was an average diameter of 8 feet X 40' long, the log would contain >20 cord of wood. This is 40,000 pounds. The biggest Sikorsky SkyCrane can lift 25,000 pounds.
His above statements concerned me as in my experience the figures he was using were well out of balance.
Using the same dimensions as he did I attempted to reconstruct the way I thought the calculations should be made so that an accurate weight could be achieved
A log with a 12 foot dia. On the big end, 40 foot of length, and 6 foot dia on the small end.
Using the frustum method from the second edition of Related Mathematics for Carpenters. By the way its no longer available its about 23 years old but you may be able to secure a Third edition of Practical Mathematics published by the American Technical Society.
To figure the volume of a evenly tapered log.
If the top of a cone is cut off by a plane parallel to the base the part remaining is a frustum.
Formula =The volume of the frustum of a cone is equal to the sum of the squares of the diameters of the bases, plus the product of the diameters of the bases with this sum multiplied by .2618 times the altitude.
Formula
You will need to substitute in the figures for the letters.
V = volume of log
D = big diameter end of log
d = small diameter end of log
H = Height or distance between ends of log
V = .2618H ( D2 + d2 + Dd ) = .2618 @ H @ ( D2 + d2 + Dd )
D = 12 = foot diameter on big end
d = 6 = foot diameter on small end
H = 40 = height or distance between ends
V = .2618 by 12 ( 12 to 2nd power + 6 to 2nd power +12 by 6 )
V= .2618 by 12 by ( 12 to 2nd power + 6 to 2nd power + 12 by 6 )
V = 3.1416 (144 + 36 + 72)
V = 3.1416 by 252
V = 791.6832 cu. feet
If the logs volume is 791.6832 cu feet and there are 128 cu feet in a cord than 791.6832 divided by 128 should equal how many cords are in the log.
791.6832 divided by 128 = 6.185025 cords.
The logs weight can now be determined by multiplying the actual weight of the species of wood by the number of cubic feet determined to be in the log.
Since Oak weighs 72 lbs per cu foot than the log would weigh 72 lbs. Times 791.6832 cu. ft. and the log in Live Oak would weigh 57,001.1904 lbs
One thing that bothers me is that a cord is equal to 128 cu feet and with Live Oak at72 lbs. A cord of Live Oak should weigh 72 by 128 = 9,216 lbs. Per cord. But he used only 5,000 lbs per cord one time and only 2,000 per cord another and he says Euc is heavier than live Oak by about 20%. In presuming he is wrong and the chart is right for the weight of Live Oak some discrepancies appear to be in his calculations.
He estimated the weight of the log to be 40,000 lbs
Using the chart that shows live Oak to weigh 76 lbs per cubic foot. And the frustum formula to get a volume of the log of 791.6832 cu. ft.
A Live Oak log would weigh 57,001.1904 lbs.
Since in the believe that he could be right about Euc weighing 20% more than Live Oak.
Adding 20% to 57,001.1904 = using his observation as to the weight for Euc the log could weigh 68,401.42848 lbs
Personally I would suspect the specie weight per cu foot to be about the same as sycamore or Cottonwood as they are two of the biggest trees I know and their weight is about 50 lbs per cu foot.
At 50 lbs per cu. ft. The log could weigh as little as 39,584.16 lbs.
Which if true leaves his original estimate extremely accurate at 40,000 lbs.
I think he made a good call but his figuring is all over the place and suspect that it was his exceptional work experience that came up with a figure that was respective with his experiences.
Its amazing how things can be wrong but still turn out right.
However I think it would be a whole lot more comforting to have made an accurate estimation from using known methods of establishing volumes and weights.
Besides he happened to be giving this information to a person who was not using a crane but a helicopter. Makes me wonder if that log ever left the ground. Bet there may have been a lot of sawdust left behind.
George schrader@beaches.net
ISA Certified Arborist SO-0448
American Tree Services 904 769 4060
Panama City, Fl.