What is the formula for the curve made when you bend a stiff card by pressing inwards at the edge between your

Question

Please help I am in need of this answer!

Answer 1

Edited answer: I was wrong to say that the curve is a parabola. Temporary brain fart.

Answer 2

If you get a stiff card or strip of spring steel and push the ends together, you get a wiggly curve called a sine generated curve. A meandering river, like the Thames flowing through London, has the same shape. So does a long string of railway carriages after a collision. What these all have in common is, for a fixed length and position of end points the total amount of bending is a minimum. Mathematically this curve is complicated and can't be expressed as an elementary function.

Answer 3

For small deflections this is the famous Euler buckling relation and the curve is a sine curve. The solution comes from a relatively simple second order differential equation. The limitation to the sine form is that the displacement is small relative to the length of the card. For large displacment the governing equation becomes non-linear and is not readily represented by ordinary functions.

I've read the below and I am adding a couple comments. The curve is not a parabola. You do get a parabola by applying a transverse load to the card that is uniformly distributed along the axis of the card. A suspension bridge is an example of this. It is also not a catenary. You do get a catenary from a transverse load that is uniformly distributed with respect to the card length (not the undefledted position). AN example of this is a chain hanging between two hosrizontal supports.

Answer 4

Hi. I don't think the stiffness matters much. You can measure the distance your fingers are apart and that should be the chord of the arc. Reading your first answer though gives me pause. Does the arc form a parabola?

Answer 5

I have not analyzed this fully, but without going and developing the model I will put my money that such a curve will be a catenary.

y = a cosh (x/a)

{By the way, easy to confuse with a parabola}

Answer 6

It is a parabola.

The general formula is y=Ax^2+Bx + C