Can Weight Ratio Be Used to Solve Area Problems in Engineering?

[maskerwsk]

Hi guys, hope I am posting in the correct place.

At work I am creating a excel spreadsheet that will work out the cross sectional areas of a manufactured bi metal tape'.
And i can't figure out the domed part.

ive uploaded an image of the shape that needs calculating.

any idea if it can be done or does it require more infomation?

Thanks.

 

[Mentallic]

These are quite troubling problems. Are you expecting exact answers, or approximations?
Anyway, I believe I could flesh out the first problem because it does have enough information, but for the second, you've got the circular part labelled as 9.00, while the angle it makes at the end is 7^o, but the "diameter" is 3.36.
How is it possible that the circular part is 9.00 (or is that counting the slanted edges as well?), but even so, if we made the slanted edges and circular part longer by turning the figure into a rectangle with length 3.36 and height 0.50, then all edges excluding the bottom will add to 4.36, which is much less than 9.00, so your numbers just don't add up.

 

[maskerwsk]

i was hoping for exact answers.
im not sure why you think the numbers don't add up.

the tape is 3.36mm wide with an overall height of 0.50mm there is a taper of 7° (either side) and a radius of 9.00mm

we produce this tape at work millions of times aday and many others relativly similar.

if you could figure out the top problem that would be great as i could use that and subtract that area from a theoretical trapezoid (factoring out the radius of the tape)

hope that makes sense, I've uploaded another drawing if not :)

 

[arildno]

So, just to make this clear:
The circular arc is a segment from a circle with a 9.00 mm radius?

 

[maskerwsk]

So, just to make this clear:
The circular arc is a segment from a circle with a 9.00 mm radius?

exactly :)

 

[haruspex]

First problem:
(W+h tan(θ))²+(R-h)²=R²
You can turn that into a quadratic in h and solve.
Second problem:
Consider one end of the curved part. Let X be its horizontal distance from the centre line and Y be its height above the base line.
X = W/2 -Y tan(θ)
R² = X² + (Y+R-H)²
From those, get a quadratic in X or Y and solve.
Let the angle the arc subtends at the centre of its circle be 2ψ.
Cos(ψ) = (Y+R-H)/R
Solve for ψ.
Area = ψR²-R²sin(ψ)cos(ψ)+2XY+Y² tan(θ)

 

[arildno]

BTW:
I don't see why an engineer would bother with finding analytical formulae for arbitrary area shapes.
A much simpler approach in general would be to work out the area ratio of two pieces of the same material by calculating their weight ratio. By controlling the area of one of the pieces, you'll readily find the other.