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ASEN 3112 - Structures
Torsion OfOpen Thin Wall (OTW) Sections
8
ASEN 3112 Lecture 8 – Slide 1
Classification of Thin Wall (TW) Cross Sections Under Torque
ASEN 3112 - Structures
Closed Thin Wall (CTW) Sections at least one cell shear flow circuit can be established Single Cell: just one cell Multicell: more than one cell
Open Thin Wall (OTW) Sections no cell shear flow circuit can be established
Hybrid Thin Wall (HTW) Sections contains both OTW and CTW components
ASEN 3112 Lecture 8 – Slide 2
Sample OTW Cross Sections
ASEN 3112 - Structures
ASEN 3112 Lecture 8 – Slide 3
Prototype OTW Cross Section:Solid Rectangle
ASEN 3112 - Structures
b
b >= t is called the long dimensiont is the wall thickness or simply thickness
z
y
tRectangle is called thin (or narrow) if b >> t, usually b > 10 t
ASEN 3112 Lecture 8 – Slide 4
Solid Rectangle Torque Behavior:Deformation
ASEN 3112 - Structures
z
y
b
t
Pictured rectangular cross section is not narrow, to facilitate visualization
Note that cross sections warp as torque is appliedThis happens for all non-circular cross sections(warping => plane sections do not remain plane)
Before After
ASEN 3112 Lecture 8 – Slide 5
Solid Rectangle Torque Behavior:Shear Stresses
ASEN 3112 - Structures
τ
b
t
τ occurs at the midpoint of the long side
max This shear stressmust be zero
ASEN 3112 Lecture 8 – Slide 6
Solid Rectangle Torque Behavior:Shear Stresses For Thin Rectangle
ASEN 3112 - Structures
y
y
z
z
τ = T t /Jαmax
τ = T t /Jαmax
t
b
ASEN 3112 Lecture 8 – Slide 7
Basic Formulas for Single RectangularCross Section (Not Necessarily Thin)
ASEN 3112 - Structures
These formulas are exact in the sense that they are provided by the Theory of Elasticity.They are obtained by solving a Partial Differential Equation (PDE) of Poisson's type
τ = T t T
Jmaxφ' = =
d φdx G Jα
α
β
βin which J = α b t and J = β b t 33
Let T = applied torque, b = longest rectangle dimension, t = shortest rectangle dimension, G = shear modulus.Then the stress and twist angle formulas are
The dimensionless coefficients α and β are functions of theaspect ratio b/t, and are tabulated in the next slide
ASEN 3112 Lecture 8 – Slide 8
Coefficients α and β for Single RectangularCross Section as Functions of Aspect Ratio
ASEN 3112 - Structures
b/t 1.0 1.2 1.5 2.0 2.5 3.0 4.0 5.0 6.0 10.0 ∞α 0.208 0.219 0.231 0.246 0.258 0.267 0.282 0.291 0.299 0.312 1/3
β 0.141 0.166 0.196 0.229 0.249 0.263 0.281 0.291 0.299 0.312 1/3
Interpolation formulas valid for all aspect ratios are givenin the Lecture 8 Notes. If b/t > 3, α ∼ β within 1%
If the section is sufficiently thin so that b/t > 5 (say)one can take α = β ~ 1/3, which is easy to remember.
ASEN 3112 Lecture 8 – Slide 9
Rectification of Curved TW Profiles
ASEN 3112 - Structures
Simple curved TW sections can be "rectified" into an equivalent rectangle. For example the slitted section of Lab 1:
t
t
Rb
thin longitudinal cut Equivalent rectangle
z
y
b = 2 π R
This is valid only if curved wall is truly thin so b/t > 5 (say)
ASEN 3112 Lecture 8 – Slide 10
Decomposition Into Rectangles:T Section
ASEN 3112 - Structures
flange
web
bf
tf
twbw= +
flange
web
ASEN 3112 Lecture 8 – Slide 11
T Section Torsion AnalysisASEN 3112 - Structures
flange
web
bf
tf
twbw
f w
= = = =
fJ JJ
J
J
Jw
ww
max f max fmax w max max w f
f f
f
α f
α
From statics, decompose the total applied torque T into portions taken by the flange and web:
That is one equation for two unknowns. Twist angle rate compatibility provides a second equation
Solving (1) and (2) we get (G drops out)
in which J = J + J . To obtain the maximum shear stress, apply the stress formula to the flange and web in turn, and pick the largest (note that J is now used)
T = T + T (1)
(2)
βf
βf
βf
βf βf
β
β β
βw
βw βw
βw
βw
τ = , τ = => τ = max(τ , τ ) T tJ
w w
α w
T tJ
T = T = T, T = T = T J + J βf βw J + J
d φ d φ T d φ Tdx dx G J dx G J
ASEN 3112 Lecture 8 – Slide 12
Decomposition Into Rectangles:Z Section
ASEN 3112 - Structures
buf
tuf
tw
bw =
+
+lower flange
upper flange
web
tlf
blflower flange
upper flange
web
ASEN 3112 Lecture 8 – Slide 13
