University of Florida Flight Controls/Visualization Laboratory

Ch. 3 TorsionAircraft 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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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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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:– 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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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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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?

University of Florida 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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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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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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3.2 Cont’d Torque and Torsion Constant

• Set Stress Function ϕ(x,y) such that:

• Compatibility Equation for Torsion

• Using Stress Strain Relations:

• 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

• Torsion constant: J=T/(Gθ)

• Torsional Rigidity=GJ (Defined if find ϕ(x,y))

,xz yzy x

2 yz xz

x y

2 2

2 22G

y x

2A

T dxdy

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3.3 Bars w/ Circular Cross-Sections

• Example (Assumed Stress Function, ϕ)

• Substitutions (Torque, Shear Stress): See book

• Only non-vanishing component of stress vector:

• Tangential shear stress on z face:

• Observe this is result for torsion of circular bars (Torque

magnitude proportional to r)!

• Therefore for bars w/ circular cross sections under torsion, there

is no warping (w=0)

Tr

J

zt G r

2 2

2 21

x yC

a a

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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 – Therefore ϕ independent of x– Therefore compatibility equation reduces:

0yz x

2

22G

y

--->Integrate!

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3.4 Bars w/Narrow Rectangular Cross Section (Cont’d)

• Integration Gives Stress Function:

• Shear Stress from Def. Stress Function:

• Where is max shear stress?

• What is max shear stress?

21 2

1,22

1 2

: 0 / / 2

0,4

G y C y C

BC at y tt

C C G

2 , 0xz yzG yy x

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3.4 Bars w/Narrow Rectangular Cross Section (Cont’d)

• Find Torque: Subst. ϕ into torque definition:

• Assume torsion constant J=bt3/3

• Find Warping: (show linear lines on model)

– 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:

2A

T dxdy

xzxz

wy y y

x Gw xy

3 / 3J bt

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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:

• Assume: change of τnz across t negligible– Note: τnz=0 on S0, S1 so since t is small:– τnz≈0 over entire wall section

,sz nzn s

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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:

• Solve for ϕ0,ϕ1 to get ϕ(s,n)

• Shear flow: q=force/contour length: – constant along wall section irrespective of

wall thickness

• Torque: Area enclosed by q:

– Ā=area enclosed by centerline wall section

0 1

0 1 0 0

0 1 1 1

( , ) ( ) ( )1: ( , / 2) ( / 2)2 : ( , / 2) ( / 2)

s n s n sBC s t t C on SBC s t t C on S

1 0q t C C

2 2A

T qdA qA

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Real Life Stress Testing

• Strain Gages and Point Loads Approximating Distributed

Aerodynamic Loading

• Boeing 787: Bending Failure:

• Boeing 777: Compression Buckling Upper Panel:

http://www.buzzhumor.com/videos/7668/Boeing_777_Wing_Stress_Test

http://www.youtube.com/watch?v=sA9Kato1CxA

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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?

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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

• 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