# Scaling and ballast calculations

Discussion in 'Boat Design' started by Tussock, Nov 19, 2014.

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### TussockJunior Member

Laws of similtude suggest scaling up a design by a factor of 1.2 would increase the displacement by a factor of 1.73 for an equivalent loaded water line. How do I allow for ballast in the above scaling and keep the ballast ratio constant?

Thanks!

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### daiquiriEngineering and Design

Do you have a particular reason for wanting to keep the ballast ratio constant?

Everything else being equal, the righting moment scales as K^4 and the heeling moment due to wind forces scales as K^3, where K is the scale factor.
So a K=1.2 gives 1.73 times higher heeling momeent but also a 2.07 times higher righting moment. In other words, for the same ballast ratio, same construction materials used and the same wind speed, the scaled-up model will heel considerably less. Hence, for the same heel angle it will actually require less ballast.

Things get even better if the heeling moment is not due to wind but due to the crew weight, assumed constant. In that case the heeling moment scales as K, and the righting moment as K^4 - and the scaled-up boat will settle at an even smaller list angle than the original one.

Cheers

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### TussockJunior Member

Thanks.... the reasons for scaling are to raise the righting moment for safety offshore, and a secondary but important consideration is to gain a little extra space below decks. The AVS is great (>155 degrees, and the moment is zero at 180 degrees) but the aim was to double the 'area under the graph', or the force required to tip it over.

If I understand correctly, the righting moment scaling at K^4 is contingent on the ballast ratio being constant, which holds the LWL constant... which means scaling the ballast by K^3... Does that hold in the real world, or is it too much of a simplification?

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### daiquiriEngineering and Design

From the mathematical point of view, it is correct.

But, be very aware of the following fact. Scaling-up boat's dimensions by 20%, although it might sound as a small number, is not a trivial task. Raising the righting moment implies increasing all the forces acting on the boat, and also the mechanical stress in the structural members. For example, it can be demonstrated through dimensional analysis that both bending-moment and axial-force induced stress scale as K. Hence, the factor of safety (FOS) decreases as 1/K.

So, you don't just scale your model. You have to re-engineer it.

Cheers

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### TussockJunior Member

Thanks Daiquiri.

The designer has given his blessings for the scaling, so apparently the scantlings will hold. It's sheath strip planked, and all linear dimensions (floors, knees, breastplates, strip size etc. gets the *1.2 treatment. I'd scale it further if it weren't for the need to keep to the scantlings.

Thanks for the input - much appreciated.

Edit: thinking further, what's the relationship between material strength and K? In other words, if you raise the dimensions of, say, floor thickness, what happens to it's strength - presumably it's better than linear?

Later edit: I assume strength in compression varies as K^2?

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### HJSMember

Check with this Excel file.

js

#### Attached Files:

• ###### Scaling by Barkla.xls
File size:
24.5 KB
Views:
488
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### philSweetSenior Member

LWL is usually taken to mean Length Water Line. I would suggest you write out "Scaled water line" or some such.

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### daiquiriEngineering and Design

Where do the formulae in that excel file come from?

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### HJSMember

H M Barkla, University of St Andrews, Scotland

js

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### daiquiriEngineering and Design

Thanks. Do you know where can I find the paper in which these formulae are derived?
I tried to google it, but couldn't find any reference.

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### TussockJunior Member

It appears to be as you would expect - linear dimensions are multiplied by the scaling factor, areas are squared, volumes are cubed.

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### daiquiriEngineering and Design

I am afraid they are not. Linear dimensions in that Excel sheet appear to be power functions with coefficients which very with boat size and dimension which is being measured.
It is ok as long as we know where these formulae come from and how they were derived. But that paper by H M Barkla was pretty hard to retrieve so far, so I can't figure out up to which point is it valid. Which makes the usage of that sheet without questioning its output dangerous.

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### TussockJunior Member

Sorry, I was going from the 'm' column, which gave standard values (m, m^2, m^3), and I didn't crunch the numbers. Yes, I'd like to know more too!

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### HJSMember

I can not find it at present.
I think it is somewhere in my AYRS sediments. 1955-1986

Larsson & Eliasson have it in there book.

js

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