Application To Thin-walled, Open Cross Sections
While the membrane analogy allows the stress distribution on any cross section to be determined experimentally, it also allows the stress distribution on thin-walled, open cross sections to be determined by the same theoretical approach that describes the behavior of rectangular sections. Using the membrane analogy, any thin-walled cross section can be "stretched out" into a rectangle without affecting the stress distribution under torsion. The maximum shear stress, therefore, occurs at the edge of the midpoint of the stretched cross section, and is equal to, where T is the torque applied, b is the length of the stretched cross section, and t is the thickness of the cross section.
It can be shown that the differential equation for the deflection surface of a homogeneous membrane, subjected to uniform lateral pressure and with uniform surface tension and with the same outline as that of the cross section of a bar under torsion, has the same form as that governing the stress distribution over the cross section of a bar under torsion.
This analogy was originally proposed by Ludwig Prandtl in 1903.
Read more about this topic: Membrane Analogy
Famous quotes containing the words application to, application, open, cross and/or sections:
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“They never open their mouths without subtracting from the sum of human knowledge.”
—Thomas Brackett Reed (1839–1902)
“Expecting me to grovel,
she carefully covers both feet
with the hem of her skirt.
She pretends to hide
a coming smile
and won’t look straight at me.
When I talk to her,
she chats with her friend
in cross tones.
Even this slim girl’s rising anger
delights me,
let alone her deep love.”
—Amaru (c. seventh century A.D.)
“Childhood lasts all through life. It returns to animate broad sections of adult life.... Poets will help us to find this living childhood within us, this permanent, durable immobile world.”
—Gaston Bachelard (1884–1962)