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Edge effects in the membrane analogy

Investigation indicates that edge effect on the membrane analogy is adequately explained by surface tension, although gravity and drainage might play some part in the case of wet membranes

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Abstract

This paper explains the origin and magnitude of an edge effect which causes errors in stress analyses made with the use of the membrane analogy.

Although Griffith and Taylor noted this effect in their classical paper on the use of soap films in solving torsion problems, it was not explained, nor were the magnitudes of the errors estimated there.

The errors caused by the edge effect cannot be eliminated, but can be reduced by proper design, and compensated for once their origin and behavior are known.

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Abbreviations

A 0 :

cross-sectional area of membrane, sq in.

A 1,A 2 :

areas defined in text

C :

boundary curve

D :

diameter, in.

E :

percentage error

G :

modulus of rigidity, psi

H :

maximum height of loaded membrane, in.

h :

depression of unloaded membrane, in.

K 1,K 2 :

constants defined in text

k,k 1,k 2,k 3 :

constants defined in text

p :

pressure, psi

R, r :

radius, in.

S :

specific gravity of soap solution

T :

surface tension, lb/in.

V :

volume, cu in.

\(\bar w\) :

average weight of membrane, gms

x, y :

rectangular coordinates, in.

α:

angle of contact, membrane to solid

β:

chamfer angle

γ:

apparent edge angle of membrane

ζ:

angle subtended by top fillet, loaded membrane

Θ:

unit angle of twist

θ:

angle subtended by top fillet, unloaded membrane

\(\rho w\) :

density of water, gms/cu in.

ϕ:

stress function

ψ:

height of membrane, in.

ω:

elevation of bounding solid

\(\frac{\partial }{{\partial n}}\) :

normal derivative

t :

subscript for top membrane

b :

subscript for bottom membrane

c :

subscript for main part of membrane

References

  1. Prandtl, L., “Zur Torsion von prismatischen Stäben,”Physikalische Zeitschrift, 4, 758–759 (1903).

    MATH  Google Scholar 

  2. Taylor, G. I. andA. A. Griffith, “The Use of Soap Films in Solving Torsion Problems,”Great Brit. Rep. and Mem. Aeronautical Research Comm. (Lond.) Reports and Memoranda, 3 (333),920–970 (1917-18).

    Google Scholar 

  3. Hetényi, M., “Handbook of Experimental Stress Analysis,” John Wiley and Sons, 727 (1954).

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Authors and Affiliations

  1. Kansas State University, Manhattan, Kansas

    P. G. Kirmser (Professor of Applied Mechanics) & F. J. McCormick (Professor of Applied Mechanics)

Authors
  1. P. G. Kirmser
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  2. F. J. McCormick
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Kirmser, P.G., McCormick, F.J. Edge effects in the membrane analogy. Experimental Mechanics 5, 11–15 (1965). https://doi.org/10.1007/BF02320898

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  • DOI: https://doi.org/10.1007/BF02320898

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