Ch. 3 Torsion
Aircraft Structures, EAS 4200C
9/17/2010
Robert Love
Organizational:
Turn in Project Part 1 at Front of Class
Pick up HW #2 as it Goes Around
University of Florida
Flight Controls/Visualization Laboratory
Examples of Importance of Torsional Analysis
• Past
– Wright Brothers (wing warping)
• Recent Past
–
–
–
–
–
Active Aeroelastic Wing F-18
Boeing Dreamliner
Helicopter Rotors
HALE Aircraft
Wind Turbine Blades
• Future?
– AFRL Joined Wing Sensor Craft
– Active Wing Morphing/Flapping
Wings
– ???
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Flight Controls/Visualization Laboratory
Why Do You Need to Know How to Design For Torsional Loads?
• AIAA DBF 2003: Wings Torsional Rigidity is Too Low!
• What could they potentially have done to fix this?
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Flight Controls/Visualization Laboratory
When is Wing Torsional Strength Really Important?
• Where on the wing are your
torsional loads the most?
• Trends: What happens to the
required torsional rigidity as:
–
–
–
–
–
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Airspeed decrease
AR decrease
Pitching Moment decrease
Aileron power decrease
Move from root to tip
Move cg of wing closer to ¼ chord
• Practicality: how do you increase
torsional rigidity by wing design?
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Flight Controls/Visualization Laboratory
More Complex Situations Torsional Strength Is Needed
• Structural Tailoring w/Composites
– Bend/Twist Coupling
• Aeroelastic Phenomena
– Bending Flutter (induces torsion)
– Torsional Flutter (rare)
– http://www.youtube.com/watch?v=8D7YCCLGu5Y
– http://www.youtube.com/watch?v=ca4PgyBJAzM
• Aeroservoelastic Phenomena
– Flapping Wings
– Limit Cycle Oscillations
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Flight Controls/Visualization Laboratory
Efficiency in Torsional Design
• Where is the material most efficiently used?
– Red=High Stress, Blue=Low Stress
• What would be the most efficient torsional member? Why?
• Why can’t we always use that type of member?
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Flight Controls/Visualization Laboratory
3.1 Saint Venant’s Principle: Static Equivalence
• Stresses or strains at a point sufficiently far from two applied
loads don’t differ significantly if the loads have the same
resultant force and moment (loads are statically equivalent)
• Distance req. ≈ 3x size of region of load application
• Ex: ≈ valid beyond 3x height of three stringer panel from the
load application end
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Flight Controls/Visualization Laboratory
3.2 Torsion of Uniform Bars
• Torque: a moment (N m) which acts about longitudinal
axis of a shaft
– NOT a bending moment! These act perpendicular to
longitudinal axis of shaft
– Shafts of thin sections under torsion, watch boundary layer
• Know Your Assumptions! Mechanics of materials: torsion
in prismatic shaft, isotropic, linearly elastic solid
– Deformation and stress fields generated, assume:
• Plane sections of shaft remain plane, circular after
deformation produced by torque
• Diameters in plane sections remain straight after
deformation
• Therefore: shear strain & shear stress = linear function of
radial distance from point of interest to center of section
• Not valid for shafts of noncircular cross section!
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Flight Controls/Visualization Laboratory
3.2 Cont’d Classical Approaches to Torsion of Solid
Shafts, Non-Circular Cross Section
• Approaches
– Prandtl’s Stress Function Method
– St. Venant’s Warping Function Method
• Set origin of CS at center of twist of cross section (unknown?)
– COT: where in-plane displacements=0, sometimes shear center
• α=angle of rotation (twist angle) at z relative to end at z=0
• θ=α/z=twist angle per unit length
• τyz and τxz are only non-vanishing stress components
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Flight Controls/Visualization Laboratory
3.2 Cont’d Torque and Torsion Constant
• Set Stress Function ϕ(x,y) such that:
• Compatibility Equation for Torsion
 xz 

y
2 
,  yz  

yz
x

 
 2G  
y
2
x
  xz
y
2
• Using Stress Strain Relations:

 
2

x
2
• Torsion Problem: Find Stress Function, Satisfy Boundary Cond.
• Traction Free BC’s: tz =0:
dϕ/ds=0
or
ϕ=constant
• Torque=integral of dT over entire cross section T  2    d xd y
A
• Torsion constant: J=T/(Gθ)
• Torsional Rigidity=GJ (Defined if find ϕ(x,y))
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Flight Controls/Visualization Laboratory
3.3 Bars w/ Circular Cross-Sections
• Example (Assumed Stress Function, ϕ)
2
 x2

y
  C  2  2  1
a
a

• Substitutions (Torque, Shear Stress): See book
• Only non-vanishing component of stress vector:
• Tangential shear stress on z face:
t z   G r
 
• Observe this is result for torsion of circular bars (Torque
magnitude proportional to r)!
Tr
J
• Therefore for bars w/ circular cross sections under torsion, there
is no warping (w=0)
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Flight Controls/Visualization Laboratory
3.4 Bars w/Narrow Rectangular Cross Section
• Assumptions:
– Shear stress can’t be assumed to be perp.
to radial direction, τ not proportional to
radial distance (Warping present)
– For Saint-Vernant: L > b, b>>t
• Find ϕ(x,y)
• Top/bottom face:traction free BC: τyz =0
• Subst. into Stress Function:
• Assume: τyz ≈0 thru t
yz


x
0
– Therefore ϕ independent of x
– Therefore compatibility equation reduces:
2
 
 2G  
--->Integrate!
2
y
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Flight Controls/Visualization Laboratory
3.4 Bars w/Narrow Rectangular Cross Section (Cont’d)
• Integration Gives Stress Function:
   G y  C1 y  C 2
2
B C 1, 2 :   0
at
y   / t / 2
2
t
C 1  0, C 2  G 
4
• Shear Stress from Def. Stress Function:
 xz 

y
  2 G  y ,  yz  

x
0
• Where is max shear stress?
• What is max shear stress?
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Flight Controls/Visualization Laboratory
3.4 Bars w/Narrow Rectangular Cross Section (Cont’d)
• Find Torque: Subst. ϕ into torque definition: T  2    d xd y
A
• Assume torsion constant J=bt3/3
• Find Warping: (show linear lines on model)
w
  xz   y 
x
w   xy
 xz
  y   y
G
– Note: w=0 at centerline of sheet!
• Ex: Can also use to address multiple thin walled
sheets!
• Note: If b>>t need to correct J with β:
J   bt / 3
3
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Flight Controls/Visualization Laboratory
3.5 Closed Single-Cell Thin-Walled Structures
• Wall thickness t >>length of wall contour
• Stress Free BC’s: dϕ/ds=0 on S0, S1
– Integrate: ϕ=C0 on S0, ϕ=C1 on S1
• Define (s,n) coordinate system
• Equilibrium Condition:
 sz  

n
,  nz 

s
• Assume: change of τnz across t negligible
– Note: τnz=0 on S0, S1 so since t is small:
– τnz≈0 over entire wall section
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Flight Controls/Visualization Laboratory
3.5 Closed Single-Cell Thin-Walled Structures (Cont’d)
• Write ϕ(s,n), assume range of n small:
– Neglect HOTerms w/n to give linear
function:
 ( s , n )   0 ( s )  n1 ( s )
B C 1 :  ( s , t / 2)   0  ( t / 2)1  C 0 on S 0
B C 2 :  ( s ,  t / 2)   0  ( t / 2)1  C 1 on S 1
• Solve for ϕ0,ϕ1 to get ϕ(s,n)
• Shear flow: q=force/contour length:
– constant along wall section irrespective of
wall thickness
q   t  C1  C 0
• Torque: Area enclosed by q:
T 
 2q d A  2 q A
– Ā=area enclosed by centerline wall section
A
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Flight Controls/Visualization Laboratory
Real Life Stress Testing
• Strain Gages and Point Loads Approximating Distributed
Aerodynamic Loading
• Boeing 787: Bending Failure:
http://www.youtube.com/watch?v=sA9Kato1CxA
• Boeing 777: Compression Buckling Upper Panel:
http://www.buzzhumor.com/videos/7668/Boeing_777_Wing_Stress_Test
University of Florida
Flight Controls/Visualization Laboratory
What now?
• Your boss comes in and says “Find out if the material we are
using here will fail due to torsional loads”? What do you do?
University of Florida
Flight Controls/Visualization Laboratory
References
•
All Reference figures and Theory: C.T. Sun, Mechanics of Aircraft Structures, 2nd Edition, 2006
•
2003 DBF: http://www.youtube.com/watch?v=iD_xHeHkuXc
•
Boeing Dreamliner Wing Flex: http://www.youtube.com/watch?v=ojMlgFnbvK4
•
Boeing Wing Break: http://www.youtube.com/watch?v=sA9Kato1CxA&feature=related
•
Rectangular Torsion: http://www.bugman123.com/Engineering/index.html
•
Wing w/Aero Contours: http://www.cats.rwth-aachen.de/research/cae
•
Wing Flex: http://www.youtube.com/watch?v=gvBiu71l6d4&NR=1
•
Wrights: http://www.gravitywarpdrive.com/Wright_Brothers_Images/First_in_Flight.gif
•
Stress Concentration in Torsion: http://www.math.chalmers.se/Math/Research/Femlab/examples/examples.html
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Helicopter blade twist: http://www.onera.fr/dads-en/rotating-wing-models/active-helicopter-blades.php
•
Sensorcraft: http://www.flightglobal.com/articles/2005/07/05/200103/over-the-horizon.html
•
X-29 Composite Tailoring: http://www.pages.drexel.edu/~garfinkm/Spar.html
•
Torsional mode: http://en.wikipedia.org/wiki/File:Beam_mode_2.gif
•
LCO: http://aeweb.tamu.edu/aeroel/gallery1.html
•
Boeing Wing Box: http://www.mae.ufl.edu/haftka/structures/Project-Givens.htm
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Flight Controls/Visualization Laboratory