MT-1620 al.2002 Unit 11 Membrane analogy for Torsion) Readings Rivello 83.8.6 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 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
MT-1620 Fa.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, 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 x需 membrane covering a CL Paul A Lagace @2001 Unit 11-p 2
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 pi “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
MT-1620 al.2002 n= constant tension force per unit length [lbs/in[/ ook at this from the side igure 11.2 side view of membrane under pressure over cutout [ib/:] [N/m] Assume: lateral displacements( w) are small such that no appreciable changes in n occur We want to take equilibrium of a small element dwdW assume small angles ax'ay Paul A Lagace @2001 Unit 11-p 3
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
MT-1620 al.2002 Figure 11.3 Representation of deformation of infinitesimal element of membrane Z X Look at side view(one side Figure 11.4 Side view of deformation of membrane under pressure Z A hdy- Note: we have similar picture in the x-z plane Paul A Lagace @2001 Unit 11-p 4
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
MT-1620 al.2002 We look at equilibrium in the z direction Take the z-components of N e. g + dw z-component=-Nsin hy note +z direction for small angle dw dw sin dy dy dw →z- component=-N dy(acts over dx face Paul A Lagace @2001 Unit 11-p 5
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