membrane analogy and torsion of thin walled tube in advanced solid mechanics for finding torsion in different cross section

1. MEMBRANE ANALOGY
AND
TORSION OF THIN-WALLED
TUBES
Presented by,
ROLWYN MARIAN CARDOZA
1RV18MMD15
MTech MACHINE DESIGN
RV COLLEGE OF ENGINEERING.

2. MEMBRANE ANALOGY
• The analytical solutions are difficult for bar with
complicated cross- sections.
• Hence we search for some other techniques—
experimental or otherwise.
• The membrane analogy introduced by Prandtl is
one such technique that allows the stress
distribution on any cross section to be determined
experimentally.
• We connect this analogy to Torsion of general
prismatic bars–solid sections

3. • Let a thin homogeneous membrane like a thin rubber sheet be
stretched with uniform tension and fixed at its edge, which is a given
curve
• When the membrane is subjected to a uniform lateral pressure p, it
undergoes a small displacement z where z is a function of x and y.
• Consider the equilibrium of an infinitesimal element ABCD of the
membrane after deformation.
• Let F be the uniform tension per unit length of the membrane. The
value of the initial tension F is large enough to ignore its change when
the membrane is blown up by the small pressure p.

6. • Now, if we adjust the membrane tension F or the air
pressure p such that p/F becomes numerically equal to 2Gϴ,
then
of the membrane becomes
identical to of the torsion stress function φ.
• Further, if the membrane height z remains zero at the
boundary contour of the section, then the height z of the
membrane becomes numerically equal to the torsion stress
function .
• The slopes of the membrane are then equal to the shear
stresses and these are in a direction perpendicular to that of
the slope. The twisting moment is numerically equivalent to
twice the volume under the membrane

7. TORSION OF THIN-WALLED TUBES
• Consider a thin-walled tube
subjected to torsion.
• The thickness of the tube need
not be uniform
• Since thickness is small the
shear stress will be parallel to
boundary
• Let ‘τ ‘ be shear stress ‘t’ be
thickness.

8. • Consider the equilibrium of an element
• The areas of cut faces AB and CD are respectively.
• For equilibrium in z direction
1
2
3

9. Multiplying and dividing by t
4
5

11. Quiz
1. The membrane analogy was developed by ___________(Prandtl)
2. The membrane analogy is ____________technique. (experimental)
3. The twisting moment is numerically equivalent to ________ the volume
under the membrane.(twice)
4. The total elastic strain energy of a thin walled tube subjected to torsion is
given by _________
5. The twist of thin walled tube subjected to torsion is given by __________
Questions
1. Explain Prandtl’s membrane analogy in detail ?
2. Discuss in detail, the torsion of thin walled tubes ?

12. Reference
• Advances Mechanics of Solids – L S Srinath