Torsion of beams with opened cross section

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Lecture 8 Torsion of beams with opened cross
section
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SHEAR STRESSES RELATED QUESTIONS

• shear flows due to the shear force, with no
  torsion
• shear center
• torsion of closed contour
• torsion of opened contour, restrained torsion
  and deplanation
• shear flows in the closed contour under combined
  action of bending and torsion
• twisting angles
• shear flows in multiple-closed contours.

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SHEAR CENTER AND TORSION - ILLUSTRATION
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TORSION IN MECHANICS OF MATERIALS
The measure of resistance to torsion is a polar
moment of inertia Ir . Relative twist
angle Maximal shear stress
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TORSION IN THIN-WALLED CROSS SECTIONS
The polar moment of inertia Ir is calculated as
a sum for rectangular portions of thin-walled
cross section. Relative twist angle Maximal
shear stress
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TORSION IN THIN-WALLED CROSS SECTION
Torsional moment is 1000 Nm. Material is steel
(G77 GPa). Ir 4.14 cm4. Shear stress
tmax241.3 MPa.
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CALCULATION OF DEPLANATIONS (WARPING)
Since the hypothesis of planar cross section is
not valid, the beam theory is not
applicable. Thus, specific theory developed by
Vlasov will be used.
Vlasovs theory is based on two main
hypotheses 1) The cross section keeps its shape
and rotates as a whole around the shear
center. 2) There are no shear strains and
stresses at the middle plane (gtz 0).
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CALCULATION OF DEPLANATIONS
where w displacement along z axis
(deplanation) u displacement along t axis.
where f angle of rotation of cross section
along z axis r lever from the shear center to
the direction of t axis at the given point.
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CALCULATION OF DEPLANATIONS
where w0 some displacement at the start
point.
If start point is set on the axis of symmetry, we
get where w sectorial coordinate (doubled
area covered by rotation of radius-vector)
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CALCULATION OF DEPLANATIONS
Analytical values Max is 0.67 mm At the corner
is 0.51 mm
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NORMAL STRESSES AT RESCTRICTED TORSION
Normal stresses could be found using the
formula where Iw sectorial moment of
inertia B bimoment (kind of scalar force
factor)
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NORMAL STRESSES AT RESCTRICTED TORSION
The distribution of normal stresses for real
structure is usually quite complex, so it is
usually wise to use FEA.
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COMPARISON OF OPENED AND CLOSED CONTOURS
For a tube of 25 mm diameter and thickness of 2
mm we get For the closed contour we get 1.227
cm4, while for opened only 0.021, which is 57
times smaller. If we would increase the
diameter, the difference will be increased
dramatically.
Lets take a thin-walled circle with radius 1 m
and thickness of 2 mm. For the closed contour we
get 628,300 cm4, while for opened only 1.67,
which is 375 thousands times smaller.
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WHERE TO FIND MORE INFORMATION?
Megson. An Introduction to Aircraft Structural
Analysis. 2010 Chapter 17.2
Internet is boundless
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