Report

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 – ??? University of Florida 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? University of Florida 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: – – – – – – 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? University of Florida 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 University of Florida 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? 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 University of Florida 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! University of Florida 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 University of Florida 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)) University of Florida 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) University of Florida 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 University of Florida 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? University of Florida 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 University of Florida 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 University of Florida 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 ) n1 ( 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 University of Florida 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 • 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 University of Florida Flight Controls/Visualization Laboratory