Ch. 3 Torsion

1. Ch. 3 Torsion Aircraft Structures, EAS 4200C 9/17/2010 Robert Love

2. 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 ???

3. 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? This is what happens to an airplane that has wings with low torsional rigidity. Negative moment from the wing increases with velocity, so the wingtips just curl forward. That causes a decrease in angle of attack at the wingtips, which decreases the lift at the tips. The wings fold downwards. *SPLASH!*This is what happens to an airplane that has wings with low torsional rigidity. Negative moment from the wing increases with velocity, so the wingtips just curl forward. That causes a decrease in angle of attack at the wingtips, which decreases the lift at the tips. The wings fold downwards. *SPLASH!*

4. 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? Ways to increase Torsional Rigidity: Sheet the wing, add a carbon fiber tube as spar, cover wing with rigid film (like mylar/monocoat) From RC forums: The lower the airspeed the less torsional rigidity required. The lower the aspect ratio, the lower the torsional rigidity required. The lower the pitching moment coefficient of the airfoil, the lower the required torsional rigidity. The more the planform is tapered, the lower the torsional rigidity required. The torsional rigidity requirement at the wing tip is lowest and at the root end of the wing is highest. The more powerful the ailerons, the greater the torsional rigidity required. The closer the center of mass of the wing structure to the 25% chord line, the less the required torsional rigidity. When a wing produces lift, the wing tends to bend in the direction of the lift forces. The lift forces put the material on the lift side of the wing under compression and on the side away from the lift under tension. In other words, the material (part of the spar) on the lift side is compressed and is slightly shortened. The material on the side away from the lift forces are stretched under tension. The material in between the parts under tension and compression try to slide spanwise to accommodate the shortening on the compression side and the stretching on the tension side. This spanwise sliding is called shear. A spar that is being bent has internal forces of three kinds. There is compression on the inside of the bend, tension on the outside of the bend and shear in between. A typical spar structure consists of three pieces of material. They are a top spar cap, a bottom spar cap and shear webs in between connecting the two spar caps together. If the shear webs are left out of the spar structure, the top and bottom spar caps slide relative to each other and the wing looses bending stiffness.The top and bottom spar caps resist the tension and compression forces and the shear webs resist the sliding or shear forces internal to the spar structure. Resisting these internal spar structure forces, successfully, makes the spar stiff and strong.Then there are the torsional forces on a wing. They are caused by cambered airfoils and deflected control surfaces like flaps and ailerons. The structure of the wing that resists twisting is the tube like skin of the wing structure or a large tube inside the wing skin. When a tube is twisted, the lengthwise elements of the tube tend to slide relative to each other and experience shear forces. If a spanwise gap is introduced into the tube, there is no material there to resist the sliding or shear forces and the structure looses torsional stiffness. For stiffness, the tube must be closed around its circumference. The bigger the tube, the better it resists torsion. In a D-tube structure the tube consists of the top skin, the leading edge, the bottom skin and the shear web of the spar (which does double duty resisting both bending and torsional loads).In the case of a tube type spar, the tube resists both bending and twisting loads. The larger its diameter, the stiffer and stronger it will be, resisting both twisting and bending. When the tube consists of fibrous material the bending loads are best resisted with fibers that run length wise and the torsional loads are best resisted by fibers that run (spiral) at 45 degrees to the length of the tube.In a stressed skin wing structure, the tube has an airfoil cross sectional shape.Ways to increase Torsional Rigidity: Sheet the wing, add a carbon fiber tube as spar, cover wing with rigid film (like mylar/monocoat) From RC forums: The lower the airspeed the less torsional rigidity required. The lower the aspect ratio, the lower the torsional rigidity required. The lower the pitching moment coefficient of the airfoil, the lower the required torsional rigidity. The more the planform is tapered, the lower the torsional rigidity required. The torsional rigidity requirement at the wing tip is lowest and at the root end of the wing is highest. The more powerful the ailerons, the greater the torsional rigidity required. The closer the center of mass of the wing structure to the 25% chord line, the less the required torsional rigidity. When a wing produces lift, the wing tends to bend in the direction of the lift forces. The lift forces put the material on the lift side of the wing under compression and on the side away from the lift under tension. In other words, the material (part of the spar) on the lift side is compressed and is slightly shortened. The material on the side away from the lift forces are stretched under tension. The material in between the parts under tension and compression try to slide spanwise to accommodate the shortening on the compression side and the stretching on the tension side. This spanwise sliding is called shear. A spar that is being bent has internal forces of three kinds. There is compression on the inside of the bend, tension on the outside of the bend and shear in between. A typical spar structure consists of three pieces of material. They are a top spar cap, a bottom spar cap and shear webs in between connecting the two spar caps together. If the shear webs are left out of the spar structure, the top and bottom spar caps slide relative to each other and the wing looses bending stiffness.

5. 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

6. 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? On the OUTSIDE of the structure! On the OUTSIDE of the structure!

7. 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

8. 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! Torque may be Torque may be

9. 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 a=angle of rotation (twist angle) at z relative to end at z=0 ?=a/z=twist angle per unit length tyz and txz are only non-vanishing stress components Noodle vs. rag or paper when put in torsionNoodle vs. rag or paper when put in torsion

10. 3.2 Cont’d Torque and Torsion Constant Set Stress Function ?(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: d?/ds=0 or ?=constant Torque=integral of dT over entire cross section Torsion constant: J=T/(G?) Torsional Rigidity=GJ (Defined if find ?(x,y))

11. 3.3 Bars w/ Circular Cross-Sections Example (Assumed Stress Function, ?) 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)

12. 3.4 Bars w/Narrow Rectangular Cross Section Assumptions: Shear stress can’t be assumed to be perp. to radial direction, t not proportional to radial distance (Warping present) For Saint-Vernant: L > b, b>>t Find ?(x,y) Top/bottom face:traction free BC: tyz =0 Subst. into Stress Function: Assume: tyz ˜0 thru t Therefore ? independent of x Therefore compatibility equation reduces:

13. 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? At y=+/-t/2At y=+/-t/2

14. 3.4 Bars w/Narrow Rectangular Cross Section (Cont’d) Find Torque: Subst. ? 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 ß:

15. 3.5 Closed Single-Cell Thin-Walled Structures Wall thickness t >>length of wall contour Stress Free BC’s: d?/ds=0 on S0, S1 Integrate: ?=C0 on S0, ?=C1 on S1 Define (s,n) coordinate system Equilibrium Condition: Assume: change of tnz across t negligible Note: tnz=0 on S0, S1 so since t is small: tnz˜0 over entire wall section On lateral surface: no loads therefore stress vector (traction) vanishesOn lateral surface: no loads therefore stress vector (traction) vanishes

16. 3.5 Closed Single-Cell Thin-Walled Structures (Cont’d) Write ?(s,n), assume range of n small: Neglect HOTerms w/n to give linear function: Solve for ?0,?1 to get ?(s,n) Shear flow: q=force/contour length: constant along wall section irrespective of wall thickness Torque: Area enclosed by q: A=area enclosed by centerline wall section

17. Real Life Stress Testing Strain Gages and Point Loads Approximating Distributed Aerodynamic Loading Boeing 787: Bending Failure: Boeing 777: Compression Buckling Upper Panel: Aeronautics & Astronautics Engineering 331 Wing Box Project at the University of Washington. To design and build a wingbox with certain limitations (material, size, weight: no great than 3.5lbs) ours at 3.3lbs, took on torsional and bending loads of 530lbs with our unique corrugated spar design and wrapped skin. (james stoffel, david, doug, scott).Aeronautics & Astronautics Engineering 331 Wing Box Project at the University of Washington. To design and build a wingbox with certain limitations (material, size, weight: no great than 3.5lbs) ours at 3.3lbs, took on torsional and bending loads of 530lbs with our unique corrugated spar design and wrapped skin. (james stoffel, david, doug, scott).

18. 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? How much is the torsional rigidity of a solid cylinder reduced by hollowing out the center up to 70.7% of the radius? What is the error in the torsional rigidity for the section in Wednesday’s question if we use the thin-wall formula? How much is the torsional rigidity of a solid cylinder reduced by hollowing out the center up to 70.7% of the radius? What is the error in the torsional rigidity for the section in Wednesday’s question if we use the thin-wall formula?

19. 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