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
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 • ???
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?
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?
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
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?
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
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!
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
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))
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)
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: --->Integrate!
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?
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:
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
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
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.youtube.com/watch?v=sA9Kato1CxA http://www.buzzhumor.com/videos/7668/Boeing_777_Wing_Stress_Test
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?
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