# Deflection of Shallow Spherical Shells

Many papers and books, such as Roark's Formulas for Stress and Strain cite Reissner, E.: "Stresses and Small Displacements of Shallow Spherical Shells", Journal of Mathematics and Physics, vol. 25, No. 4, 1947. Part 1, pp. 80-85, Part 2 pp pp 279-300. This paper is not online, and the 66 y.o. journal is not on most shelves. Dr. Reissner's article is somewhat difficult to understand, but I will attempt to do so here. Corrections welcomed!

Reissner considers three cases:

- (1) Point load at center, edges fixed in angle position and angle
- (2) Point load at center, edges constrained vertically but free to turn and splay outwards
- (3) Distributed uniform load in disk area around center, edges constrained vertically but free to turn and splay outwards

Curved-surface deployment resembles case (3). Not exactly - thinsats are not circular and a detailed finite element model will be needed someday - but an estimate helps.

v |
Poisson's ratio - typically 0.33 for aluminum |

R |
radius of curved surface in meters |

MoreLater about Reissner paper.

## Guesstimates based on geometry

There will be two kinds of thinsat:

(1)

**overcurved**- curved more than the stack average(2)

**undercurved**- curved less than the stack average

When an overcurved thinsat is flattened into a stack, the outer edges are stretched and the inner disk is compressed. The opposite happens when an undercurved thinsat is curved extra onto the stack. Assume the thinsats are round, radius r_t , and that the boundary between compression and stretch is somewhere between 0.5 \times r_t (equal radial distance) and 0.7071 \times r_t (equal areas).