Live Oak Mechanics

Reports and materials related to the Live Oak Project. The project was initiated by Larry Tucei to document the great live oak tree ((Quercus virginia) found in the southern United States.

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edfrank
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Live Oak Mechanics

Post by edfrank » Fri Apr 02, 2010 9:03 pm

Oak Trees are awesome
Posted on: April 2, 2010 8:00 AM, by Rhett Allain

http://scienceblogs.com/dotphysics/2010 ... wesome.php

In this part of the world, we have oak trees. Technically they are called live oaks - but I don't get it. Of course they are alive. I was at a soccer game and this is the tree I always look at.
2010-03-30_i_photo.jpg
Look how far those limbs extend horizontally. That branch is about 12 meters long. Why is this amazing? Have you ever tried to hold an 8 foot 2 x 4 board horizontally by holding one end? Pretty tough. How about I calculate the forces needed to hold that branch in place? I will do a simple model and then maybe later I can make it more complicated. Suppose I replace that limb with one straight uniform limb that looks like this:
2010-03-30_untitled.jpg
2010-03-30_untitled.jpg (46.71 KiB) Viewed 29781 times
In this replacement limb, I am going to say it is a cylinder that is 9 meters long and 30 cm in diameter. Let me assume that this limb is connected at two points to the tree (the white dots). So, for this limb to stay there, the following must be true:
2010-03-30_la_te_xi_t_1_12.jpg
2010-03-30_la_te_xi_t_1_12.jpg (16.72 KiB) Viewed 29781 times

The first two equations say that the total force must be zero. The last one says that the torque about any point must be zero (since it is in rotational equilibrium about any point). First for the forces. There is the gravitational force. This pulls on all parts of the limb, but I can represent this as one force pulling on the limb at the center of mass (long ago, I said I would explicitly show this - but I haven't yet). Then there are two other forces. Let me pretend like there are two pins that hold the limb to the tree. Each of these pins can exert a force in the vertical and horizontal direction I will call these F1-y F2-y etc...where the top pin will be 1. That is 5 forces.

For these forces and the first two equations, I get:
2010-03-31_la_te_xi_t_1.jpg
2010-03-31_la_te_xi_t_1.jpg (13.02 KiB) Viewed 29781 times

So, already I have some constraints. The horizontal components of the forces from the two pins must be equal and opposite. The vertical components of the forces from the pins have to add up to the weight of the log. Now for the torque, I am going to add up the torques about the lower pin. Let me draw a distorted view of the log so that the important distances can be seen.
2010-03-31_untitled.jpg
2010-03-31_untitled.jpg (18.86 KiB) Viewed 29781 times

What is torque? Torque is like a rotational force. Here is an example, what if you try to open a door by pushing near the hinges? It is much harder than pushing near the handle, right? When rotating about some axis, the torque is:
2010-04-01_la_te_xi_t_1_5.jpg
2010-04-01_la_te_xi_t_1_5.jpg (10.76 KiB) Viewed 29781 times

Here F is the applied force, r is the distance from the point where the force is applied to the axis. Theta is the angle between F and r. I will call torques that would make a rotation counterclockwise positive (really, torque is a vector). So, what is the some of torques about axis O (that passes through point O)? First, there are some forces that have zero torque. Both of the vertical pin forces have either theta = 0 or r = 0 so that the torque is zero. The same is true for the horizontal force on the bottom pin. This just leave two forces that have non-zero torques:
2010-04-01_la_te_xi_t_1_2.jpg
2010-04-01_la_te_xi_t_1_2.jpg (12.95 KiB) Viewed 29781 times

Now that I have the horizontal force on the top pin, the bottom pin has the same value (but in the opposite direction). I don't have an expression for the two vertical pin forces. Let me just say that each has a force equal to have the weight of the limb.

How about some values? First, I need the mass of the limb. If this is a cylinder of wood (with a density rho) then the mass is:
2010-04-01_la_te_xi_t_1_3.jpg
2010-04-01_la_te_xi_t_1_3.jpg (8.87 KiB) Viewed 29781 times

So the magnitude of the two horizontal forces on the pins would be:
2010-04-01_la_te_xi_t_1_4.jpg
2010-04-01_la_te_xi_t_1_4.jpg (11.19 KiB) Viewed 29781 times

If I use my values from above and an estimation of the density of wood, I get:
2010-04-01_la_te_xi_t_1_5.jpg
2010-04-01_la_te_xi_t_1_5.jpg (10.76 KiB) Viewed 29781 times

Wow. Oh, I know I made some estimations but even 50,000 Newtons would be huge. Impressive, most impressive. I salute you mighty tree.



TrackBack URL for this entry: http://scienceblogs.com/mt/pings/135269
"I love science and it pains me to think that so many are terrified of the subject or feel that choosing science means you cannot also choose compassion, or the arts, or be awe by nature. Science is not meant to cure us of mystery, but to reinvent and revigorate it." by Robert M. Sapolsky

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Steve Galehouse
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Re: Oak Trees are awesome

Post by Steve Galehouse » Fri Apr 02, 2010 9:57 pm

Ed-

I know intuitively what you're describing, but I have to admit I'm sort of a vegan when it comes to math or physics. Tension wood is under a lot of tension, especially in live oaks.

Steve
every plant is native somewhere

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edfrank
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Re: Oak Trees are awesome

Post by edfrank » Fri Apr 02, 2010 10:16 pm

Steve,

This is not my math, but that of the author of the blog I copied here. It does sound quite impressive however.

Ed
"I love science and it pains me to think that so many are terrified of the subject or feel that choosing science means you cannot also choose compassion, or the arts, or be awe by nature. Science is not meant to cure us of mystery, but to reinvent and revigorate it." by Robert M. Sapolsky

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Larry Tucei
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Re: Oak Trees are awesome

Post by Larry Tucei » Sat Apr 03, 2010 8:08 am

Ed, Fasinating stuff thats way cool! Lets ask the math Guru a question. Bob, at 55 lbs a Cubic Ft., a limb say 12 Cir., at the tree tapering to 1 ft., 75' long would way what? Tons of force where the limb meets the Trunk. Some of the Live Oak limbs are even larger. I'll go back and review some of the trees I've measured and post the results. Perhaps the twisting of the grain on Live Oak limbs is the secret to there strength. Great topic, Ed! I'm attaching a photo for a good example the limb in the photo left is over 12' Cir, and 75' long. At almost 90 degrees to the trunk the forces here are awesome! Larry
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The Ruskin Oak
The Ruskin Oak

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dbhguru
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Re: Oak Trees are awesome

Post by dbhguru » Sat Apr 03, 2010 4:40 pm

Larry,

Using the frustum of a cone formula to model the form and the density you gave, the total weight would be 17,179 lbs.

Bob
Robert T. Leverett
Co-founder, Native Native Tree Society
Co-founder and President
Friends of Mohawk Trail State Forest
Co-founder, National Cadre

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Larry Tucei
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Re: Oak Trees are awesome

Post by Larry Tucei » Sat Apr 03, 2010 6:26 pm

Bob, Wow! Way cool. Trees are truly amazing. Thats a lot of weight to hold horizontally. I wonder how would you do a PSI count where the limb meets the Trunk? Bob any idea! Larry

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Jenny
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Re: Oak Trees are awesome

Post by Jenny » Mon Apr 05, 2010 9:55 am

I know I've posted this poem several times, but never has it been more apt! Seeing this post describing the impossible feats of the oak while cherry blossoms are blooming madly makes it too tempting:

The oak tree.
Not interested
In cherry blossoms.

-Basho (1644 - 1694, Japan)
Suburbia is where the developer bulldozes out the trees, then names the streets after them. ~Bill Vaughn

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Steve Galehouse
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Re: Oak Trees are awesome

Post by Steve Galehouse » Mon Apr 05, 2010 9:06 pm

The cherry tree.
Needs not
the weighty limb.
every plant is native somewhere

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Jenny
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Re: Oak Trees are awesome

Post by Jenny » Tue Apr 06, 2010 7:09 am

Nice riposte from the cherry. Is that a Galehouse?

Jenny
Suburbia is where the developer bulldozes out the trees, then names the streets after them. ~Bill Vaughn

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Steve Galehouse
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Re: Oak Trees are awesome

Post by Steve Galehouse » Tue Apr 06, 2010 8:36 am

Jenny wrote:Nice riposte from the cherry. Is that a Galehouse?

Jenny
Gēru-hausu, 2010
every plant is native somewhere

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