VM137

VM137
Large Deflection of a Circular Membrane

Overview

Reference:	S. Timoshenko, S. Woinowsky-Knieger, Theory of Plates and Shells, McGraw-Hill Book Co., Inc., New York, NY, 1959, pg. 404, eq. 236.
Analysis Type(s):	Static Analysis (ANTYPE = 0) with Large Deflection and Stress Stiffening
Element Type(s):	2-Node Finite Strain Axisymmetric Shell (SHELL208)
Input Listing:	vm137.dat

Test Case

A circular membrane (of radius a and thickness t), clamped around its outer rim, is loaded with a uniform pressure p. Determine the deflection δ at the center, the radial stress σ_o at the center, and the radial stress σ_a at the rim.

Figure 191: Circular Membrane Problem Sketch

Material Properties

Geometric Properties

E = 30 x 10⁶ psi

α = 1 x 10^-5 in/in-°F

υ = 0.25

a = 10 in

t = 0.0001 in

p = 0.1 psi

Analysis Assumptions and Modeling Notes

Since there is no significant bending stiffness, the membrane is cooled by -50°F (arbitrary) in the first load step to induce a thermal prestress for stability. The pressure load is applied in the second load step. The cooling load is removed in the third load step. Moment convergence is removed by specifying force convergence, since the moments are not significant to the solution, thereby speeding convergence.

Results Comparison

Load Step 3	Target	Mechanical APDL	Ratio
Deflection, in (UY @ Node 1)	-0.459	-0.464	1.010
Stress_o , psi	61,010	61,583.479 [1]	1.009
Stress_a , psi	47,310	48,142.557 [2]	1.018

Near center at x = 0.5 in (SM stress at MID for element 1)
Near rim at x = 9.5 in (SM stress at MID for element 10)