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AERSP 301 Torsion of closed and open section beams

AERSP 301 Torsion of closed and open section beams. Jose L. Palacios July 2008. REMINDERS. IF YOU HAVE NOT TURN IN HW# 4 PLEASE DO SO ASAP TO AVOID FURTHER POINT PENALTIES. HW #5 DUE FRIDAY, OCTOBER 3 HW #6 (FINAL HW from me) DUE FRIDAY OCTOBER 10

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AERSP 301 Torsion of closed and open section beams

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  1. AERSP 301Torsion of closed and open section beams Jose L. Palacios July 2008

  2. REMINDERS • IF YOU HAVE NOT TURN IN HW# 4 PLEASE DO SO ASAP TO AVOID FURTHER POINT PENALTIES. • HW #5 DUE FRIDAY, OCTOBER 3 • HW #6 (FINAL HW from me) DUE FRIDAY OCTOBER 10 • EXAM: OCTOBER 20 – 26 HOSLER – 8:15 – 10:15 PM • REVIEW SESSION: OCTOBER 19 – 220 HAMMOND – 6 – 9 PM

  3. To simultaneously satisfy these, q = constant Thus, pure torque  const. shear flow in beam wall A closed section beam subjected to a pure torque T does not in the absence of axial constraint, develop any direct stress, z Torsion of closed section beams • Now look at pure torsion of closed c/s

  4. Torque produced by shear flow acting on element s is pqs Since q = const. & Torsion of closed section beams [Bredt-Batho formula] Hw # 3, problem 3

  5. Torsion of closed section beams • Already derived warping distribution for a shear loaded closed c/s (combined shear and torsion) • Now determine warping distribution from pure torsion load • Displacements associated with Bredt-Batho shear flow (w & vt): 0 = Normal Strain

  6. Torsion of closed section beams No axial restraint • In absence of direct stress, • Recall

  7. Torsion of closed section beams • To hold for all points around the c/s (all values of ) c/s displacements have a linear relationship with distance along the beam, z

  8. Torsion of closed section beams Twist and Warping of closed section beams Lecture • Earlier, • For const. q Also Needed for HW #5 problem 3

  9. Torsion of closed section beams • Starting with warping expression: • For const. q • Using

  10. Determine warping distribution in doubly symmetrical, closed section beam shown subjected to anticlockwise torque, T. From symmetry, center of twist R coincides with mid-point of the c/s. When an axis of symmetry crosses a wall, that wall will be a point of zero warping. Take that point as the origin of S. Twisting / Warping sample problem

  11. Sample Problem • Assume G is constant From 0 to 1, 0 ≤ S1≤ b/2 and Find Warping Distribution

  12. Sample Problem • Warping Distribution 0-1 is:

  13. Sample Problem • The warping distribution can be deduced from symmetry and the fact that w must be zero where axes of symmetry intersect the walls. • Follows that: w2 = -w1, w3 = w1, w4 = -w1 What would be warping for a square cross-section? What about a circle?

  14. Sample Problem • Resolve the problem choosing the point 1 as the origin for s. • In this case, we are choosing an arbitrary point rather than a point where WE KNEW that wowas zero.

  15. Sample Problem • In the wall 1-2

  16. Sample Problem • Similarly, it can be show that a s2 b

  17. Sample Problem • Thus warping displacement varies linearly along wall 2, with a value w’2at point 2, going to zero at point 3. • Distribution in walls 34 and 41 follows from symmetry, and the total distribution is shown below: Now, we calculate w0 which we had arbitrary set to zero

  18. Sample Problem We use the condition that for no axial restraint, the resultant axial load is zero:

  19. Sample Problem Substituting for w’12and w’23 and evaluating the integral: Offset that need to be added to previously found warping distributions

  20. Torsion / Warping of thin-walled OPEN section beams • Torsion of open sections creates a different type of shear distribution • Creates shear lines that follow boundary of c/s • This is why we must consider it separately Maximum shear located along walls, zero in center of member

  21. Torsion / Warping of thin-walled OPEN section beams • Now determine warping distribution, Recall: • Referring tangential displacement, vt, to center or twist, R:

  22. On the mid-line of the section wall zs = 0, Integrate to get warping displacement: Torsion / Warping of thin-walled OPEN section beams Distance from wall to shear center AR, the area swept by a generator rotating about the center of twist from the point of zero warping where

  23. Torsion / Warping of thin-walled OPEN section beams The sign of wsis dependent on the direction of positive torque (anticlockwise) for closed section beams. For open section beams, pris positive if the movement of the foot of pralong the tangent of the direction of the assumed positive s provides a anticlockwise area sweeping AR R S = 0 (W = 0) ρR

  24. Torsion / Warping Sample Problem • Determine the warping distribution when the thin-walled c-channel section is subjected to an anti-clockwise torque of 10 Nm G = 25 000 N/mm2 SideNote:

  25. BEGINNING SIDENOTE

  26. SideNote: Calculation of torsional constant J(Chapter N, pp 367 Donaldson, Chapter 4 Megson) • Torsional Constants Examples and Solutions

  27. Stresses for Uniform Torsion • Assumptions: • Constant Torque Applied • Isotropic, Linearly Elastic • No Warping Restraint y Mt Mt x z All Sections Have Identical Twist per Unit Length: No Elongation No Shape Change

  28. St. Venant’s Constant For Uniform Torsion (or Torsion Constant) z y Mt Φ F

  29. Torsion Constant • J is varies for different cross-sections #1 #2 #3

  30. EXAMPLE #1 (ELLIPSE) • Find S. Torsion Constant For Ellipse: • Find Stress Distribution (σxyσxz) z 1) Eq. Boundary: 2b y 2) Ψ = 0 on Boundary: 2a 3) Substitute Ψinto GDE:

  31. EXAMPLE # 1 Area Ellipse: Polar Moment of Inertia: 4) J: z 2b y 2a 5) Substitute into Ψ(y,z) 6) Differentiate 5)

  32. EXAMPLE #2 (RECTANGLE) • Find S. Torsion Constant For Ellipse: • Find Stress Distribution (σxyσxz) z • Eq. Boundary: Simple Formulas • Do Not Satisfy GDE and BC’s • NEED TO USE SERIES • For Orthogonality use Odd COS Series • (n & m odd) b y a 2) Following the procedure in pp 391 and 392

  33. Stress and Stiffness Parametersfor Rectangular Cross-Sections (pp 393)

  34. a>>b Rectangle z b y No variation in Ψin y BC’s: Integrating Differentiating Ψ

  35. Similarly: Open Thin Cross-Sections S t S is the Contour Perimeter

  36. Extension to Thin Sections with Varying Thickness (pp 409) η Thickness b(ξ) z ξ By analogy to thin section y

  37. Torsional Constants for an Open and Closed CS

  38. END SIDENOTE

  39. Torsion / Warping Sample Problem • Determine the warping distribution when the thin-walled c-channel section is subjected to an anti-clockwise torque of 10 Nm G = 25 000 N/mm2 Side Note:

  40. Torsion / Warping Sample Problem Origin for s (and AR) taken at intersection of web and axis of symmetry, where warping is zero Center of twist = Shear Center, which is located at: (See torsion of beam open cross-section lecture) Positive pR In wall 0-2: Since pRis positive

  41. Torsion / Warping Sample Problem Warping distribution is linear in 0-2 and:

  42. Torsion / Warping Sample Problem In wall 2-1: pR21 -25 mm Negative pR The are Swept by the generator in wall 2-1 provides negative contribution to AR

  43. Torsion / Warping Sample Problem Again, warping distribution is linear in wall 2-1, going from -0.25 mm at pt.2 to 0.54 mm at pt.1 The warping in the lower half of the web and lower flange are obtained from symmetry

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