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Department of Aerospace Engineering AERSP 301 Torsion of closed and open section beams Jose L. Palacios July 2008

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Department of Aerospace Engineering 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

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Department of Aerospace Engineering Torsion of closed section beams 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 Now look at pure torsion of closed c/s

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

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Department of Aerospace Engineering 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 & v t ): 0 = Normal Strain

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Department of Aerospace Engineering Torsion of closed section beams In absence of direct stress, Recall No axial restraint

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Department of Aerospace Engineering 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

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

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Department of Aerospace Engineering Torsion of closed section beams Starting with warping expression: For const. q Using

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Department of Aerospace Engineering Twisting / Warping sample problem 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.

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Department of Aerospace Engineering Sample Problem Assume G is constant From 0 to 1, 0 ≤ S 1 ≤ b/2 and Find Warping Distribution

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Department of Aerospace Engineering Sample Problem Warping Distribution 0-1 is:

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Department of Aerospace Engineering 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: w 2 = -w 1, w 3 = w 1, w 4 = -w 1 What would be warping for a square cross-section? What about a circle?

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Department of Aerospace Engineering 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 w o was zero.

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Department of Aerospace Engineering Sample Problem In the wall 1-2

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Department of Aerospace Engineering Sample Problem Similarly, it can be show that b a s2s2

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Department of Aerospace Engineering Thus warping displacement varies linearly along wall 2, with a value w ’ 2 at point 2, going to zero at point 3. Distribution in walls 34 and 41 follows from symmetry, and the total distribution is shown below: Sample Problem Now, we calculate w 0 which we had arbitrary set to zero

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Department of Aerospace Engineering Sample Problem We use the condition that for no axial restraint, the resultant axial load is zero:

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Department of Aerospace Engineering Sample Problem Substituting for w’ 12 and w’ 23 and evaluating the integral: Offset that need to be added to previously found warping distributions

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Department of Aerospace Engineering 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

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Department of Aerospace Engineering Torsion / Warping of thin-walled OPEN section beams Now determine warping distribution, Recall: Referring tangential displacement, v t, to center or twist, R:

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

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Department of Aerospace Engineering Torsion / Warping of thin-walled OPEN section beams S = 0 (W = 0) ARAR R ρRρR The sign of w s is dependent on the direction of positive torque (anticlockwise) for closed section beams. For open section beams, p r is positive if the movement of the foot of p r along the tangent of the direction of the assumed positive s provides a anticlockwise area sweeping

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Department of Aerospace Engineering 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 SideNote: G = 25 000 N/mm 2

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Department of Aerospace Engineering BEGINNING SIDENOTE

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Department of Aerospace Engineering SideNote: Calculation of torsional constant J SideNote: Calculation of torsional constant J (Chapter N, pp 367 Donaldson, Chapter 4 Megson) Torsional Constants Examples and Solutions

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Department of Aerospace Engineering Stresses for Uniform Torsion z x y MtMt MtMt Assumptions: 1)Constant Torque Applied 2)Isotropic, Linearly Elastic 3)No Warping Restraint All Sections Have Identical Twist per Unit Length: No Elongation No Shape Change

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Department of Aerospace Engineering St. Venant’s Constant For Uniform Torsion (or Torsion Constant) F MtMt z y Φ

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Department of Aerospace Engineering Torsion Constant J is varies for different cross-sections #1#2 #3

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Department of Aerospace Engineering EXAMPLE #1 (ELLIPSE) Find S. Torsion Constant For Ellipse: Find Stress Distribution (σ xy σ xz ) 2b 2a 1) Eq. Boundary: 2) Ψ = 0 on Boundary: 3) Substitute Ψ into GDE: y z

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

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Department of Aerospace Engineering EXAMPLE #2 (RECTANGLE) b a 1)Eq. Boundary: Simple Formulas Do Not Satisfy GDE and BC’s NEED TO USE SERIES For Orthogonality use Odd COS Series (n & m odd) 2) Following the procedure in pp 391 and 392 y z Find S. Torsion Constant For Ellipse: Find Stress Distribution (σ xy σ xz )

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Department of Aerospace Engineering Stress and Stiffness Parameters for Rectangular Cross-Sections (pp 393)

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Department of Aerospace Engineering a>>b Rectangle b y z No variation in Ψ in y BC’s: Integrating Differentiating Ψ

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Department of Aerospace Engineering Similarly: Open Thin Cross-Sections t S S is the Contour Perimeter

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Department of Aerospace Engineering Extension to Thin Sections with Varying Thickness (pp 409) Thickness b(ξ) η ξ z y By analogy to thin section

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Department of Aerospace Engineering Torsional Constants for an Open and Closed CS

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Department of Aerospace Engineering END SIDENOTE

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Department of Aerospace Engineering 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 Side Note: G = 25 000 N/mm 2

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Department of Aerospace Engineering Torsion / Warping Sample Problem Origin for s (and A R ) 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) In wall 0-2: Since p R is positive Positive p R

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Department of Aerospace Engineering Torsion / Warping Sample Problem Warping distribution is linear in 0-2 and:

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Department of Aerospace Engineering Torsion / Warping Sample Problem In wall 2-1: p R21 -25 mm Negative p R The are Swept by the generator in wall 2-1 provides negative contribution to A R

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Department of Aerospace Engineering 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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