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Research Papers

# Symmetrical Solutions for Edge-Loaded Annular Elastic Membranes

[+] Author and Article Information
Weiwei Yu

Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109

Dale G. Karr

Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109dgkarr@umich.edu

J. Appl. Mech 76(3), 031010 (Mar 10, 2009) (7 pages) doi:10.1115/1.3005568 History: Received August 14, 2007; Revised September 30, 2008; Published March 10, 2009

## Abstract

The Föppl–Hencky nonlinear membrane theory is employed to study the axisymmetric deformation of annular elastic membranes. The general solutions for displacements and stresses are established for arbitrary edge boundary conditions. New exact solution results are developed for central loading and edge forcing conditions. Both positive and negative radial stress solutions are found. Comparisons are made for special cases to previously known solutions with excellent agreement.

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## Figures

Figure 1

Geometry and notation for a center-loaded membrane. The inner portion provides a net vertical force P. The resulting edge-loaded annular membrane ri<r<ra is analyzed.

Figure 2

Vertical force P, applied at the inner boundary versus vertical displacement at inner boundary w(ri) for different Poisson’s ratios with prescribed boundary conditions w(ra)=u(ri)=u(ra)=0, and (ri/ra)=0.5

Figure 3

Horizontal displacement u(r) for different Poisson’s ratios with prescribed vertical displacements w(ri)=0.1, w(ra)=u(ri)=u(ra)=0, and (ri/ra)=0.5

Figure 4

Integration constant a2 versus horizontal displacement at inner boundary u(ri) with prescribed vertical edge displacement w(ri)=0.1, w(ra)=u(ra)=0

Figure 5

Integration constant a2 versus horizontal displacement at inner boundary u(ri) for various vertical displacements w(ri) with Poisson’s ratio ν=13

Figure 6

Integration constant a2 versus horizontal displacement at inner boundary u(ri), with vertical displacement w(ri)=0.1, Poisson’s ratio ν=13 and (ri/ra)=0.5

Figure 7

Vertical displacements w(r) of points A and B in Fig. 6 versus radial position r+u(ri)

Figure 8

Nondimensional radial and circumferential stresses of points A and B of Fig. 6 versus radial position r

Figure 9

Vertical force P, applied with frictionless indentor at the inner boundary, versus vertical displacement at inner loaded boundary w(rc) for different Poisson’s ratios with u(rc)+rc=ri

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