Stiffness by pressure calculations

[Stormer]

Inflatable SUP boards, LEI kites, Inflatable kayaks, and so on get their stiffness by having two layers of airtight materials separated by pressurized air. And it gets stiffer the higher pressure you can contain, or by increasing the separation between the materials. The simplest way to look at this is probably with the flat structure of inflatable SUP boards that is constructed by two airtight membranes and "drop stitching" between to maintain the flat shape.
But how can you calculate the stiffness of the board based on pressure and layer separation?

 

[jrmichler]

Compressive stiffness. When local pressure, such as from your foot standing on it, exceeds the internal pressure, it will deform.

Bending stiffness. The top and bottom sheets have separation and stiffness. The material stiffness has units of strain per force/unit width. Multiply by the width to get sheet stiffness in strain per total force.

The board / kite / whatever has a bending moment distribution that can be calculated. The bending moment at any location results in tensile and compressive stresses in the sheets. Those tensile and compressive stresses are added to the tensile stress from the inflation pressure. The calculation is a little simpler if you neglect the internal pressure, and assume simple tensile and compressive stress. The sketch below shows the top sheet, neutral axis, and bottom sheet with a bending moment. Internal pressure is ignored. The neutal axis has a length L. The bottom sheet has a tensile force, therefore a positive strain, and a length L + $\delta$. The top sheet has a compressive force, therefore a negative strain, and a length L - $\delta$. The bending radius can be calculated from the strains and the sheet separation. Any basic strength of materials book has a very similar analysis, in greater detail, in the chapter on bending of beams.

 

[Stormer]

The calculation is a little simpler if you neglect the internal pressure, and assume simple tensile and compressive stress.

Well that neglects the whole problem... Because the fabric in a inflatable like that has pretty much zero compressive (buckeling) strength in the lengt direction. It all comes from the internal pressure. Remove the pressure and it's a floppy piece of fabric.

 

[Baluncore]

Remove the pressure and it's a floppy piece of fabric.

Obviously correct. But the flexibility is a function of the fabric, not the pressure. The internal pressure moves the operating point from zero to a point where there is always tension in the fabric. The flexibility is then material dependent.

Buckling failure will occur when the sheet collapses due to loss of tension in the fabric, which is dependent on internal pressure.
The prestressing of the reinforcing in concrete operates on a similar principle.

 

[Stormer]

But the flexibility is a function of the fabric, not the pressure. The internal pressure moves the operating point from zero to a point where there is always tension in the fabric. The flexibility is then material dependent.

The fabric can have as much tensile strength and as little elongation as you want but it is still going to bend a lot when the internal pressure is low. And it will bend a lot before buckling failure. Because even if the outside fabric does not stretch, the tube can still bend by shrinking the inside fabric (of the bend direction) against the pneumatic spring determined by the pressure in the tube.

 

[Baluncore]

But how can you calculate the stiffness of the board based on pressure and layer separation?

You cannot. The pressure decides the point where it will buckle. If the skin tension falls below zero it will collapse. Once the collapse begins, that is another game, and all bets are off.

With sufficient internal pressure, only the fabric surfaces will be important in determining the small deflections due to small loads. The length of the skin surface will change to be longer or shorter as the loads change. You must model the elasticity of the vessel as a whole, as two separated elastic skins.
https://en.wikipedia.org/wiki/Elastic_modulus
The stress~strain line is straight, only the slope is important. Changes in pressure simply move the operating point along the line.

 

[Stormer]

You cannot. The pressure decides the poinhttps://lh3.googleusercontent.com/proxy/UmOLbozQAjf82jKJkG448jyGqtSU-Q5K0G-l5fSgwG-q_9rYTHQ6CT_ltcE6Hqud7lYtDX7pouW1wr-Hem6qW0doZmcs8Zw59DQi-FYNYJ1zU02u9jnZcAkC7wt where it will buckle. If the skin tension falls below zero it will collapse. Once the collapse begins, that is another game, and all bets are off.

Obviously the fabric layer on the inside of the curve will get small wrinkles to allow it to shrink lengthwise to be able to bend the board. But i don't consider this as a buckling failure. When i think of a buckling failure in this context i think of when the board start creasing in one spot allowing the board to fold over in that spot. But you can bend the board a lot before that happens. And for most fabrics i think most of the bending comes from shrinking the inside layer (trough wrinkles, against the pneumatic spring pressure) rather than trough stretching the outer fabric. And if you have ever tried to bend an inflatable like this you know that the pressure you have pumped it up to is affecting how stiff it is when trying to bend it.

 

[Baluncore]

Crinkles will form if the surface ceases to be under tension. That is a form of surface buckling.

I think you are misinterpreting the term “stiffness”. Stiffness is a measure of the relation between small forces and small deflections of a structure. When bending a double skin structure, the length of the inner skin is shortened by the same proportion as the outer skin is lengthened. That holds until a crinkle forms on the inner skin when tension is no longer positive.

Until you accept the boundary between the structure deflecting and the structure buckling, you will be unable to analyse the system. Two separate models must be used.

 

[hutchphd]

Inflatable SUP boards, LEI kites, Inflatable kayaks, and so on get their stiffness by having two layers of airtight materials separated by pressurized air.

As an aside you have left out of this list my favorite, the Inflatoplane from Goodyear:
https://en.wikipedia.org/wiki/Goodyear_InflatoplaneCan you not get a good number by assuming zero shear or compressive strength and infinite tensile strength for all materials. Then for a simple box beam lxwxh carrying a point load at the center of force F $$\frac {Fl} 2=Pwh^2$$ and so the pressure needs to be $$P=\frac {Fl} {2wh^2}$$ For a box 30x3x1 (ft) F=200lb this gives a pressure of 7 psi which is the spec for inlatoplane

 

[Baluncore]

And if you have ever tried to bend an inflatable like this you know that the pressure you have pumped it up to is affecting how stiff it is when trying to bend it.

Are you trying to calculate the point of buckling collapse, or the stiffness of the inflated structure? As the internal pressure is increased, the load at which buckling collapse occurs is increased.
Stiffness is not a function of internal pressure. It is a function of materials and structure.