# Clamped spherical pressure vessel

I am struggling to find the right terms to search my questions, so even just pointing me to reference material would be appreciated.

1) Suppose I have a spherical cap (thin), clamped at the edges, and pressure is applied from underneath to inflate it.

a) I know the spherical cap approximation model for this, but it neglects the clamped boundaries. From my understanding, the material undergoes equibiaxial stretching only at the pole, and not everywhere else. For instance, the clamped parts would be unable to stretch along the axis of clamping. Is my understanding correct? (first question)

b) My second question is - in this case, how can I analyze stress distribution across the surface? Is there a model or equation for this?

c) My third question is about my understanding of "principal strain". Suppose I am looking at my pressure vessel in terms of a 2D cross section, as an arc with a minimal thickness. If I get strain at a particular location along the arc, but in terms of x and y directions, but what I really want is strain along the arc. - Is principal strain the correct term for this? My understanding is that principal strain would involve rotating my element until the shear component is zero, and there is pure tension or compression (which is what I would expect in my pressure vessel -- there is no shear involved). Is this correct?

2) As the thickness of the vessel increases, how do I account for this? a) My searching led me to "Lame equations" for stress across the thickness, but that still doesn't account for the clamped boundary. b) Furthermore, if I was trying to calculate how strain varies through the thickness, there is now a shear component, correct? So I could not use principal strain?

I would appreciate it if someone could let me know if my understanding is correct or completely wrong. If there is a reference I could look at to learn more about clamped boundary conditions for spherical caps or something along those lines, that would be great too. Even partially answering or confirming/denying my understanding of bits and pieces would be much appreciated.

Thank you.

• Are you modeling a particular system? Is your "cap" a hemisphere then? – nicoguaro Aug 6 '19 at 23:12
• The cap is a hemisphere, yes. – Confusedstudent Aug 7 '19 at 3:47
• Before analyzing the thick-wall version of this problem, you should first solve the thin walled version. That is a much easier problem, and it will give you a good start. Plus, you will never be able to solve the thick walled version if you can't solve the thin walled version. At the clamped end, the longitudinal strain are zero, but not the latitudinal strain. You need to develop some background in small-deformation solid mechanics, based on Hooke's law in 3D, and based on good deformational kinematics basics. Get a book on Strength of Materials. – Chet Miller Aug 7 '19 at 11:45
• That is very helpful. Thank you! – Confusedstudent Aug 7 '19 at 20:54