MIT - 16.20 Fall, 2002 Unit 11 Membrane Analogy (for Torsion) Readings : Rivello 8.3, 8.6 T & G 107, 108, 109, 110, 112, 113, 114 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace ? 2001 MIT - 16.20 Fall, 2002 For a number of cross-sections, we cannot find stress functions. However, we can resort to an analogy introduced by Prandtl (1903). Consider a membrane under pressure p i “ Membrane ” : structure whose thickness is small compared to surface dimensions and it (thus) has negligible bending rigidity (e.g. soap bubble) ? membrane carries load via a constant tensile force along itself. N.B. Membrane is 2-D analogy of a string (plate is 2-D analogy of a beam) Stretch the membrane over a cutout of the cross-sectional shape in the x-y plane: Figure 11.1 Top view of membrane under pressure over cutout membrane covering a cutout Paul A. Lagace ? 2001 Unit 11 - p . 2 MIT - 16.20 Fall, 2002 N = constant tension force per unit length [lbs/in] [N/M] Look at this from the side: Figure 11.2 Side view of membrane under pressure over cutout Assume : lateral displacements (w) are small such that no appreciable changes in N occur. We want to take equilibrium of a small element: ? w ? w (assume small angles ? x , ? y ) Paul A. Lagace ? 2001 Unit 11 - p . 3 MIT - 16.20 Fall, 2002 Figure 11.3 Representation of deformation of infinitesimal element of membrane y x z Look at side view (one side): Figure 11.4 z y Side view of deformation of membrane under pressure Note : we have similar picture in the x-z plane Paul A. Lagace ? 2001 Unit 11 - p . 4 MIT - 16.20 Fall, 2002 We look at equilibrium in the z direction. Take the z-components of N: e. g. w z-component = ? N sin ? ? y note +z direction for small angle: sin ? w ≈ ? w ? y ? y ? w ? z-component = ? N ? y (acts over dx face) Paul A. Lagace ? 2001 Unit 11 - p . 5 MIT - 16.20 Fall, 2002 With this established, we get: ? w ? ? w ? 2 w ? + ∑ F z = 0 ? p i dxdy ? N dx + N ? + 2 dy ? dx ? y ? ? y ? y ? ? w ? ? w ? 2 w ? ? N dy + N ? + 2 dx ? dy = 0 ? x ? ? x ? x ? Eliminating like terms and canceling out dxdy gives: ? 2 w ? 2 w p i + N ? y 2 + N ? x 2 = 0 Governing Partial ? ? ? ? ? 2 2 2 2 w x w y pN i + ? = Differential Equation for deflection, w, of a membrane Boundary Condition : membrane is attached at boundary, so w = 0 along contour ? Exactly the same as torsion problem: Paul A. Lagace ? 2001 Unit 11 - p . 6 MIT - 16.20 Fall, 2002 Torsion Membrane Partial Differential ? 2 φ = 2Gk ? 2 w = – p i / N Equation Boundary φ = 0 on contour w = 0 on contour Condition Analogy: Torsion Membrane φ → w - k → p i → N 1 2G → ? ? w x ? ? = φ σ x zy → ? ? = ? φ σ y zx ? ? w y → Volume wdxdy = ∫∫ ? Τ 2 Paul A. Lagace ? 2001 Unit 11 - p . 7 φφφ MIT - 16.20 Fall, 2002 Note : for orthotropic , would need a membrane to give different N ’ s in different directions in proportion to G xz and G yz ? Membrane analogy only applies to isotropic materials ? This analogy gives a good “physical ” picture for φ ? Easy to visualize deflections of membrane for odd shapes Figure 11.5 Representation of φ and thus deformations for various closed cross-sections under torsion etc. Can use (and people have used) elaborate soap film equipment and measuring devices (See Timoshenko , Ch. 11) Paul A. Lagace ? 2001 Unit 11 - p . 8 MIT - 16.20 Fall, 2002 From this, can see a number of things: ? Location of maximum shear stresses (at the maximum slopes of the membrane) ? Torque applied (volume of membrane) ? “External ” corners do not add appreciability to the bending rigidity (J) ? eliminate these: Figure 11.6� Representation of effect of external corners external corner ? about the same ? Fillets (i.e. @ internal corners) eliminate stress concentrations Paul A. Lagace ? 2001 Unit 11 - p . 9 MIT - 16.20 Fall, 2002 Figure 11.7 Representation of effect of internal corners relieved stress high stress concentration concentration To illustrate some of these points let ’s consider specifically… Paul A. Lagace ? 2001 Unit 11 - p . 1 0 MIT - 16.20 Fall, 2002 Torsion of a Narrow Rectangular Cross-Section Figure 11.8 Representation of torsion of structure with narrow rectangular cross-section Cross-Section b >> h Paul A. Lagace ? 2001 Unit 11 - p . 1 1 MIT - 16.20 Fall, 2002 Use the Membrane Analogy for easy visualization: Figure 11.9 Representation of cross-section for membrane analogy Consider a cross-section in the middle (away from edges): Figure 11.10 Side view of membrane under pressure Paul A. Lagace ? 2001 Unit 11 - p . 1 2 MIT - 16.20 Fall, 2002 The governing Partial Differential Equation. is: ? 2 w + ? 2 w = ? p i ? x 2 ? y 2 N Near the middle of the long strip (away from y = ± b/2), we would ? 2 w expect 2 to be small. Hence approximate via: ? y ? 2 w ≈? p i ? x 2 N To get w, let ’ s integrate: ? w ≈? p i x + C 1 ? x N w ≈? p i x 2 + C 1 x + C 2 2 N Now apply the boundary conditions to find the constants: h @ x =+ , w = 0 2 Paul A. Lagace ? 2001 Unit 11 - p . 1 3 MIT - 16.20 Fall, 2002 ? 0 = ? p i h 2 + C 1 h + C 2 2 N 4 2 h @ x = ? , w = 0 2 ? 0 = ? p i h 2 ? C 1 h + C 2 2 N 4 2 This gives: C 1 = 0 2 i C = ph 2 8 N Thus: w ≈ p i ?? h 2 ? x 2 ? ? 2 N ? 4 ? Check the volume: Volume = ∫∫ w dxdy Paul A. Lagace ? 2001 Unit 11 - p . 1 4 MIT - 16.20 Fall, 2002 integrating over dy: h 2 ? = b ∫ ? 2 h p i ?? h 2 ? xd x ? 2 2 N ? 4 ? h pb ? h 2 x 3 ? 2 = 2 i N ? ? 4 x ? 3 ? ? ? h 2 pb ? h 2 2 h ? 2 h 3 ? i = 2 N ? ? 4 2 38 ? ? 3 i ? Volume = pb h N 12 Using the Membrane Analogy: p i = ? k 1 N = 2 G 3 Volume = ? T = p i b h 2 N 1 2 Paul A. Lagace ? 2001 Unit 11 - p . 1 5 MIT - 16.20 Fall, 2002 ? kb h 3 2 G T = ? 12 2 3 T ? k = ? Gbh 3 (k - T relation) d α where: k = dz So: d α T = dz GJ bh 3 where: J = 3 To get the stress: σ = ? w = ? p i x = 2 kGx yz ? x N σ yz = 2 T x (maximum stress is twice J that in a circular rod) Paul A. Lagace ? 2001 Unit 11 - p . 1 6 MIT - 16.20 Fall, 2002 ? w σ xz = = 0 ( away from edges ) ? y Near the edges, σ xz ≠ 0 and σ yz changes: Figure 11.11 Representation of shear stress “flow ” in narrow rectangular cross-sections 2 Τ at these σ yz = J x points different here (generally, these are the maximum stresses) Need formulae to correct for “finite ” size dependent on ratio b/h. This is the key in b >> h. Paul A. Lagace ? 2001 Unit 11 - p . 1 7 MIT - 16.20 Fall, 2002 Other Shapes Through the Membrane Analogy, it can be seen that the previous theory for long, narrow rectangular sections applies also to other shapes. Figure 11.12 Representation of different thin open cross-sectional shapes for which membrane analogy applies Slit tube Channel I-beam Consider the above (as well as other similar shapes) as a long, narrow membrane → consider the thin channel that then results …. Paul A. Lagace ? 2001 Unit 11 - p . 1 8 MIT - 16.20 Fall, 2002 Figure 11.13 Representation of generic thin channel cross-section Volume = ? T 2 p i ? bh 3 b h 3 b h 3 ? T 11 + 22 + 33 = ? N ? ? 12 12 12 ? ? 2 (from solution for narrow rectangle) This gives: Paul A. Lagace ? 2001 Unit 11 - p . 1 9 MIT - 16.20 Fall, 2002 ? bh 3 b h 3 b h 3 ? T ? kG ? 11 + 22 + 33 = ? 2 ? 12 12 12 ? ? 2 T ? k = ? k - T relation GJ where: J = 1 b h 3 + 1 b h 3 + 1 b h 3 = ∑ 1 b h 3 3 11 3 22 3 33 i 3 ii For the stresses: σ = ? w = ? p i x = k 2 G x = 2 T x yz ? x N J ? maximum (“local ” x) 2 T h 1 σ yz = J 2 in section 1 2 T h 2 σ yz = J 2 in section 2 2 T h 3 σ yz = J 2 in section 3 Paul A. Lagace ? 2001 Unit 11 - p . 2 0 MIT - 16.20 Fall, 2002 Figure 11.14 Representation of shear stress “flow ” in thin channel under torsion 2 Τ h 2 σ xz = J 2 Actually have shear concentrations at corners (large slopes ? w ? w ) , ? y ? x ? make “ fillets ” there Figure 11.15 Channel cross-section with “ fillets” at inner corners decrease slope Paul A. Lagace ? 2001 Unit 11 - p . 2 1 MIT - 16.20 Fall, 2002 Use the Membrane Analogy for other cross-sections for example: variable thickness (thin) cross-section Figure 11.15 Representation of wing cross-section (variable thickness thin cross-section) Using the Membrane Analogy: 1 y T 3 2 T h J ≈ 3 ∫ y L h d y σ zy ≈ J 2 etc. Now that we ’ve looked at open, walled sections; let ’ s consider closed (hollow) sections. (thick, then thin) Paul A. Lagace ? 2001 Unit 11 - p . 2 2