I Board Breaks: Small or Large Deflections?

[makamo66]

I'm trying to figure out if the case of martial arts wooden board breaking is a case of small deflection or large deflection. When I look it up with artificial intelligence, the results show both possibilities and it's thus inconclusive. If it's a case of large deflection, then I would be required to use non-linear equations to solve the problem and that's more complicated than the case for small deflections. Small deflections means maximum deflection is small in comparison with the thickness of the board and that the midplane remains unstrained subsequent to breaking, large deflections mean displacements greater than the board thickness
For small deflections we can use the governing differential equation for deflection of thin plates/ biharmonic equation This equation, also known as the Kirchhoff plate theory, is a fourth-order partial differential equation that describes how a thin plate bends under applied loads.

 

[Dale]

Looks like pretty small deflections to me:



Unfortunately, my Google-Fu was not sufficient to find a really fast video where it could be measured explicitly

 

[makamo66]

Thanks for the video Dale. That looks like large deflections to me!

 

[makamo66]

I also wanted to know if the boundary conditions for supporting the edges of the board are for 2 sides clamped, 2 sides free or 2 sides simply supported and the other 2 sides free.

 

[makamo66]

Maybe I can use the formula given below
AI Overview
Cantilever Beam Calculations: Formulas, Loads & Deflections
To calculate the deflection of a breaking board, you need to consider the board's material properties, dimensions, and the way it's supported and loaded. For a simple scenario like a board supported at both ends and loaded in the middle, you can use the formula: Δ = (PL³)/(48EI), where Δ is the deflection, P is the applied force, L is the length of the board, E is the material's modulus of elasticity, and I is the moment of inertia of the board's cross-section

 

[makamo66]

I googled how to calculate the deflection of a breaking board but all of the formulas shown were for beams not boards/plates.

 

[Dale]

Maybe I can use the formula given below
AI Overview

I wouldn’t trust anything physics-related from a modern LLM. They all hallucinate frequently.

That looks like large deflections to me!

Only after the board is broken. To me the deflection seems small immediately before the break.

 

[makamo66]

What I really need is an academic paper that shows experimentally how big the deflection is.

 

[makamo66]

I found this https://members.itkd.co.nz/reference/essays/13-Power_required_to_break_Fuji_Mae_Polar_boards.pdf

 

[Baluncore]

Once the first board breaks, it focusses the continuing force onto a narrow strip along the board below, so less energy needs to be invested into bending a board, before that board breaks and wastes that energy.

The focussing effect is amplified if spacers are used between the ends of the stack of boards or bricks.

By first scoring a line across the underside of the top board, the stress causing failure under tension will be concentrated and reduced.

 

[makamo66]

Yes, I found this source that confirms your reply
https://sciencedemonstrations.fas.harvard.edu/presentations/karate-blow
The boards are stacked on top of one another and kept slightly separate with pencils placed between them; rupture can proceed successively through the boards with each rupture involving a smaller force than if a single thick board were used. This also has the effect of the momentum of the downward-moving broken pieces of the top board helping to break the board beneath it, and so on down the stack. Thus the peak force to break, say, eight boards is less than eight times the force needed to break one board.

 

[makamo66]

I was so happy that the deflections are small because that means I don't have to use nonlinear equations but then I looked at the book "Theories and Applications of Plate Analysis" by Rudolph Szilard the capital with the title "Classical Small Deflection Theory of Thin Plates" and one of the assumptions is that there is a maximum deflection of one-tenth the thickness that is considered the limit of small-deflection theory so now I'm at a loss of what to do.

Board Deflection required to break 18 mm pine = 10 mm
SOURCE: Richard Burr,Power Required to Break Boards, A thesis presented to iTKD by Richard Burrin preparation for grading to IV Dan

 

[makamo66]

I've attached the capitol about small deflection theory

 

[makamo66]

Maybe I can find a different source for the board deflection required to break wood other than Richard Burr

 

[makamo66]

Maybe I just need a source with a thicker board so that the deflection decreases as described here:
A thicker wooden breaking board will generally require a greater force to achieve the same amount of deflection as a thinner board, and will also exhibit less deflection under the same force. This is because thicker boards have a higher resistance to bending due to increased stiffness

 

[makamo66]

I found this source https://www.academia.edu/36316365/Chapter_13_Flat_Plates which had a more generous definition of a small deflection than the book by Szilard. For this paper, the deflection is less than half of the plate thickness. With a thickness t of 18 mm over a maximum deflection w_max of 10 mm, this is 0.55 which is close to 0.5 but I've attached a graph from Ugural's book which shows that the small and large deflection theories begin to diverge at w_max/t = 0.3 against the load so there's that to consider.

At this source https://ia600803.us.archive.org/31/items/PlateBendingTheory/Plate Bending Theory.pdf I get the less forgiving thickness to deflection ratio and I get the definition of the thickness t to width ratio less than 1/20 that I also can't confirm for a typical breaking board of 10" x 12" x 0.75": t/width = 0.75/10 = 0.075 but 1/20 = 0.05.

The analyses of plates are categorized into two types based on thickness to breadth ratio: thick plate and thin plate analysis. If the thickness to width ratio of the plate is less than 1/20 and the maximum deflection is less than one tenth of thickness, then the plate is classified as thin plate.

 

[Dale]

How accurate do you need this to be? I mean, what are the consequences if you are off by say 20%?

 

[makamo66]

There are no consequences, I'm just making a model. But I would like it to be accurate and 20% error sounds like a lot to me.

 

[Dale]

If there are no consequences then it doesn’t matter.

But since the whole point is the process, why not do it both ways? That will get you the full experience of the model building and give you a better perspective on both the size of the errors (I have no idea if it is 20%) and the difference in difficulty.

 

[makamo66]

I was hoping I would only need to calculate for small deflections because at first glance, the calculations for large deflections look a lot more complicated and I haven't been able to understand them yet. They involve non-linear equations that have to solved numerically.

 

[Dale]

So do that first, for practice and education. Once you have worked through the easy approximation then you will have the experience and background to do the more complicated calculations. You can then also use your approximation to check your results for the more difficult calculations: they should match for small deflections

 

[makamo66]

I can't even use the Kirchhoff thin plate theory by itself anyway because the breaking boards are borderline moderately thick plates not thin ones. For example, a google search for wooden boards to break in karate shows boards with the dimensions: 6" x 12" x 0.5" so the ratio of the thickness over the shortest span is 0.5/6 = 0.83 so it's only borderline thin. Moderately thick plates have an approximate thickness-to-span ratio h/L ≈ 1/10 - 1/5 = 0.1 - 0.2 (Szilard* p 45) They are moderately thick plates not thick plates because Szilard defines a thick plate as (p 453) when the thickness-to-span ratio h/L exceeds approximately 2.0. For moderately thick and thick plates you must consider additional deformations of the plate caused by transverse shearing (Szilard* p 446).
*Theories and Applications of Plate Analysis
Classical, Numerical and Engineering Methods
Rudolph Szilard, Dr.-Ing., P.E.

 

[Dale]

you must consider additional deformations of the plate caused by transverse shearing

That makes sense. And with wood the shearing strength will probably be different across the grain from along the grain.

 

[makamo66]

Thank you, Dale, that is a useful insight. I will have to account for that.