Proportions based on physics...statics

[Mark Singer]

Many things in nature get their perfect proportions from physical properties. For years I have felt that when the stresses in an object are balanced then you are suporting it in a natural way.....the way it would have evolved if it were flora or fauna .... Nature seeks a balance. Architects have for centuries tried to create formulas to establish "good proportions" the Golden Section.....Corbu's Modulator....the Tatami module used in Japan...the root five rectangle that was the basis of Eudoxus's mathetmatics used in the Mastery of the Parthenon and the Acropolis in Athens dating to 700 BC.

My simple theory....In statics ( the part of engineering that deals with forces in still objects) there is a way to caclulate Moments. Moments are the bending forces that act on a beam or other component. This is expressed in ft-pounds usually. In the design of a table with 2 points of support, there is a place where the negative cantilevered moments and the positive or centered (sagging moments) are in perfect balance. The proportions that represent this balance are shown in the following diagran.
Here the positive and negative moments as taken a bout the points of support are equal. The cantelever reduces the stresses of the center and the center portion reduces the stresses that are acting on the cantiliver.
A lot of the tables and other furniture I have made follows this basic theory. I may adjust a little for the with of a chair at a dining table....but the .207 (round off to .21) is a common ratio that I use.
For one thing it makes the object stronger! The second thing is esthetics....this proportion seems to be in balance....by nature it is ...it looks balanced and you should never argue with mother nature ...:rolleyes:

http://www.engineersedge.com/beam_bending/beam_bending5.htm

[John Schreiber]

Thanks Mark, It's great to be able to pick up some of the basic rules from someone who knows them well.

I can often tell when something just looks right or doesn't, but I don't know why. I also like to start with the idea that form follows function, but sometimes it doesn't or sometimes you want to go in a different direction.

There is a lot to the basics of design.

[Mark Singer]

Here is one example
http://www.sawmillcreek.org/showthread.php?t=5510&highlight=wood+movement

[Alex Yeilding]

For one thing it makes the object stronger! The second thing is esthetics....this proportion seems to be in balance....by nature it is ...it looks balanced and you should never argue with mother nature ...:rolleyes:

VERY interesting. My untrained intuition would have told me that the balance point depended on the stiffness of the beam. Wish I had more knowledge to inform my intuition in that area!

The calculations assume a load evenly distributed over the beam. Do you adjust the overhang percentage when the weight (either physical or visual) is largely concentrated at the middle or ends?

Root five rectangle? What is that? I'm familiar with the root 5 in the calculation of golden mean. Is this related?

[Mark Singer]

Alex,
It assumes a uniformly distributed load. For tables a great deal of the loading is the dead load of the table itself....the dishes even a Thanksgiving Turkey and a Magnum of Opus I don't add all that much...

I think on the root 5 we are saying the same thing....its been a while for me and it is out of my lecture realm:confused: Square root of 5 is 2.23 golden section is 1.6 approx..
This is based on te formula....http://upload.wikimedia.org/math/7/4/e/74e543eab217fa991c70f52cc9b7b30b.png

http://www.anselm.edu/homepage/dbanach/pyth4.htm

[Greg Cole]

Mark,
I admire your work and woud like to say thanks for the time you take to explain things for us all, many thanks for the "pro bono work ". By the way, I love the jonts on the corners of that Princess Table. Guys such as yourself are a wealth of information & inspiration. Knowing you craft what you design makes the delivery of info GREAT.
The design portion is something I admit to well, maybe not avoiding but taking an idea or concept and bending it to my taste & style.
I have read some of the rules/guidelines for design but often find there is not one rule that applies to everything. The numbers you give for the table design are simple, rational and as you said natural. I have been sitting on a big pile of 8/4 & 6/4 curly maple for a Trestle Table for sometime as I can't seem to settle on a design & refuse to start until I know I "have it right". I think I will try the ".21 method" on paper at least.
Personally, I can't wrap my mind around a rendition on any CAD, Sketch-up or computerized format.
Then again I look at alot of CAD & Solid Egde 3D stuff at work... and much of my tolerances at work are in the +/- 0.001 mm area.

Thanks again.

Greg

[Ken Garlock]

I guess I am just getting old.:) And that I am most likely missing, or mis-reading, a detail in your post.

It seems to me that if I were to make a cut, front to back, through the exact center of your bench, it would collapse into the center. If the ratio was 25/50/25, the center of the bench would have no stress and a cut through the midpoint would leave each half in balance on its respective support. This would be based upon a knife edge support.


I admit to not knowing any of the considerations of eye-appeal. My idea of design eye-appeal are the models on the runway.;)

[Chris Friesen]

It seems to me that if I were to make a cut, front to back, through the exact center of your bench, it would collapse into the center.

That's correct. But then you've changed the system and so the result will also be different.

The fact that the middle section is supported by legs on each side changes the way the forces are distributed.

[Mark Singer]

Chris is correct te bending stresses are distributed through the beam and the stress at the point of support (negative bending stress) at the cantilever equals the bending stress at the center which is a positive bending stress (holds water)....The opposing moments complement each other.....the cantilever reduces the stress at the center....and cinversely the stress at the cantilever is reduced by the center span.... Balanced!

[Ken Garlock]

OK, as I said I thought I mis-read something.

I agree that the cantilever reduces the downward moment by the length of the cantilever on each end, but it does not balance them back to zero.. There would always be a downward moment equal to the total moment between the supports minus two time the moment of a cantilever.
I thought you were telling me that the cantilever in your drawing would totally compensate for the beam between the supports. That is where I went awry.

As I recall, David Marks made a bench much like your example on one of his shows.

[Mark Singer]

OK, as I said I thought I mis-read something.

I agree that the cantilever reduces the downward moment by the length of the cantilever on each end, but it does not balance them back to zero.. There would always be a downward moment equal to the total moment between the supports minus two time the moment of a cantilever.
I thought you were telling me that the cantilever in your drawing would totally compensate for the beam between the supports. That is where I went awry.

As I recall, David Marks made a bench much like your example on one of his shows.
Ken,
It cannot bring the moment to zero since the areas we are considering are where maxium moment occurs. At the maxium monent the shear force is minium often zero....at maxium shear the moment is minium...we use to draw moment and shear diagrams when I studied engineering...the peaks are in different locations...

[Mark Singer]

Here is a diagram that shows the beam diagrams....Free body....loading diagram, shear diagram and moment diagram...
This is really a bit technical for most woodworkers....the concept is simple let the structural aspects suggest an esthetic solution....very often what works well to support also looks very good!


http://www.uoregon.edu/~struct/courseware/461/461_lectures/461_lecture36/461_lecture36.html

[Chris Friesen]

VERY interesting. My untrained intuition would have told me that the balance point depended on the stiffness of the beam. Wish I had more knowledge to inform my intuition in that area!

To a certain extent it does. Imagine fastening a piece of rope across the supports--the cantilevered bits would collapse under their own weight, while the middle could withstand quite a bit of weight assuming the legs are braced somehow.

I suspect this case isn't covered by the sandard beam deflection formulas in the link Mark gave. In fact, if you reduce the moment of inertia then the amount of deflection increase is the same for each case (in between the supports or on the cantilevers). This obviously doesn't match the real-world scenario of a rope fastened to rigid supports.

[Ken Garlock]

I was reading today in the latest Discover magazine an article about Fibonacci numbers. The short article observed that given a number 'n' of the Fibonacci sequence, if you divide it by the preceding number, it will approximate the 'golden ratio.' The larger the number in the sequence, the closer the ratio will be. The article didn't offer any supporting mathematics so it might just be a coincidence.

EX:
Fib nbr.
#10/#9 - 55/34 = 1.6176
#29/#28 - 514229/317811 = 1.61803

A Fibonaci number is the sum of the two previous numbers.
1 1 2 3 5 8 13 21, ad infinitum.

[Phil Thien]

This is a fantastic post, because it answers a question that has often plauged me. And it also tells me "why." And finally, it provides insight that allows me to look at other designs and nature's designs in a hole new light.

And I love your work.

And I've been practicing hand-cut dovetails and I really suck at it. So far.

[Mark Singer]

Here are the calculations for a 10ft long table (beam) with a uniformly distributed load of 10 lbs/foot or 100 pounds total.
Firs taking Momentsa about R1=21.42
Then taking moments about the center of the table(beam) =21.5

So it is clear that these proportions yeild balanced moments and equal positive and negative stresses at the centerline and cantilever respectively


any engineers out there to check my calcs??? It been a few light years for me ...as they say in quantum physics:rolleyes:

[Ken Garlock]

I can agree with your calculations, although I had to study the picture for a minute.:) I had to figure out that you were considering the load, although uniform, to be concentrated at the midpoint of each section and therefore giving an effective lever arm of 1/2 the actual length.

If I were building the bench I would have placed the supports at a distance of 1/4 from each end. Now I can see that the .21 is a better solution. I had always thought that the design and engineering were to separate and apart items, but your discussion shows that they are essentially one in the same.

Disclaimer: I am not an engineer, but I did know many in college;) I did have both a statics and dynamics course in college over 45 years ago, and long forgotten.:o

[Mark Singer]

I can agree with your calculations, although I had to study the picture for a minute.:) I had to figure out that you were considering the load, although uniform, to be concentrated at the midpoint of each section and therefore giving an effective lever arm of 1/2 the actual length.

If I were building the bench I would have placed the supports at a distance of 1/4 from each end. Now I can see that the .21 is a better solution. I had always thought that the design and engineering were to separate and apart items, but your discussion shows that they are essentially one in the same.

Disclaimer: I am not an engineer, but I did know many in college;) I did have both a statics and dynamics course in college over 45 years ago, and long forgotten.:o
Ken,
With your backround and mine we can send a cofee table to the moon:eek:

[Stephen Stokes]

Interesting thread. One question. How did you originally arrive at 0.207?

Stokes

[Mark Singer]

Interesting thread. One question. How did you originally arrive at 0.207?

Stokes

I used to work for a company that specialized in form and precast concrete design engineering many years ago when I was going to engieering school . This was long before I was an Architect... At this firm we analyized stress and designed reinforcing for different elements. One common problem is a tilt-up panel being lifted by a crane. Steel inserts are cast in the panel for erection by a crane. To equalize the sresses we used this and many other formulas. You can solve this formula for the distance (.207) that would make the positive center moment egual the cantilever moment. You use X for the distance of the cantilever and solve for X...the other distance is expressed as L minu(the entire distance)s x

[David Weaver]

My wife and i just went through kentuck knob a couple of weeks ago ( a frank lloyd wright house), and the entire roof is built like that. It creates quite an overhang on the outside of the house - to the point that all of the porches all the way around are covered. It's a great way to construct things and take advantage of the natural compression strength of wood in the direction of the grain.

[Tom Scott]

Mark,
Great thread. Everything you said seems to be right on.

To others, the stiffness won't affect the moments. The typical design proceedure for any structural element would be to analyze the load conditions and get the moments and shears within the element. Then with this information, find the section properties that satisfy all of your criteria (deflection, stress, etc.). So to carry this a step further, once you have the proportions of your furniture piece (a table in this instance) one should also proportion the structural element (the table top) accordingly. For instance, a really thin top would not look natural on a long table...and similarly, a thick top would not look correct for a really short table.

I just wish more architects I work with had an engineering background. Too many seem to think that a 20' cantilever from a free-standing column or wall (with no back-span) looks natural. :eek:

Tom (structural engineer by day)

[Rob Blaustein]

I was reading today in the latest Discover magazine an article about Fibonacci numbers. The short article observed that given a number 'n' of the Fibonacci sequence, if you divide it by the preceding number, it will approximate the 'golden ratio.' The larger the number in the sequence, the closer the ratio will be. The article didn't offer any supporting mathematics so it might just be a coincidence.


Ken--it's not a coincidence at least in the mathematical sense. Here is a site that describes why the ratios tend towards phi (the golden ratio): http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html Scroll down to the section entitled "The Ratio of neighbouring Fibonacci Numbers tends to Phi" for a mathematical derivation. --Rob