University of FloridaFlight Controls/Visualization Laboratory 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 FloridaFlight 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 FloridaFlight 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 FloridaFlight 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 FloridaFlight 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) – – Aeroservoelastic Phenomena –Flapping Wings –Limit Cycle Oscillations

University of FloridaFlight 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 FloridaFlight 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 FloridaFlight 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 FloridaFlight 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 τ yz and τ xz are only non-vanishing stress components

University of FloridaFlight Controls/Visualization Laboratory 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: t z =0: d ϕ /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))

University of FloridaFlight Controls/Visualization Laboratory 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)

University of FloridaFlight 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: τ yz =0 Subst. into Stress Function: Assume: τ yz ≈0 thru t –Therefore ϕ independent of x –Therefore compatibility equation reduces: --->Integrate!

University of FloridaFlight Controls/Visualization Laboratory 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?

University of FloridaFlight Controls/Visualization Laboratory 3.4 Bars w/Narrow Rectangular Cross Section (Cont’d) Find Torque: Subst. ϕ into torque definition: Assume torsion constant J=bt 3 /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:

University of FloridaFlight Controls/Visualization Laboratory 3.5 Closed Single-Cell Thin-Walled Structures Wall thickness t >>length of wall contour Stress Free BC’s: d ϕ /ds=0 on S 0, S 1 –Integrate: ϕ =C 0 on S 0, ϕ =C 1 on S 1 Define (s,n) coordinate system Equilibrium Condition: Assume: change of τ nz across t negligible –Note: τ nz =0 on S 0, S 1 so since t is small: –τ nz ≈0 over entire wall section

University of FloridaFlight 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: Solve for ϕ 0, ϕ 1 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

University of FloridaFlight Controls/Visualization Laboratory Real Life Stress Testing Strain Gages and Point Loads Approximating Distributed Aerodynamic Loading Boeing 787: Bending Failure: Boeing 777: Compression Buckling Upper Panel:

University of FloridaFlight 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 FloridaFlight Controls/Visualization Laboratory References All Reference figures and Theory: C.T. Sun, Mechanics of Aircraft Structures, 2 nd Edition, DBF: Boeing Dreamliner Wing Flex: Boeing Wing Break: Rectangular Torsion: Wing w/Aero Contours: Wing Flex: Wrights: Stress Concentration in Torsion: Helicopter blade twist: Sensorcraft: X-29 Composite Tailoring: Torsional mode: LCO: Boeing Wing Box: