MT-1620 a.2002 With this established, We get 0 +】0=B=8mb+N+m w d dy w dw+d wdx dx dx dx Eliminating like terms and canceling out dxdy gives d21 P1+ N N d-w Governing partia Differential x 2 dy n\ Equ uation 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 With this established, we get: ∂w ∂w ∂2 w + ∑Fz = 0 ⇒ pi dxdy − N dx + N + 2 dydx ∂y ∂y ∂y ∂w ∂w ∂2 w − N dy + N + 2 dxdy = 0 ∂x ∂x ∂x Eliminating like terms and canceling out dxdy gives: ∂2 w ∂2 w pi + N ∂y2 + N ∂x2 = 0 Governing Partial ⇒ ∂ ∂ ∂ ∂ 2 2 2 2 w x w y p N 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
MT-1620 al.2002 Torsion Membrane Parti Differential V2φ=2GkVW=-p/N Equation Boundary Condition p=0 on contour W=0 on contour Membrane Torsion Analogy dw 0 zy 0 Volume ∫ waxy Paul A Lagace @2001 Unit 11-p. 7
MIT - 16.20 Fall, 2002 Torsion Membrane Partial Differential ∇2 φ = 2Gk ∇2 w = – pi / N Equation Boundary φ = 0 on contour w = 0 on contour Condition Analogy: Membrane Torsion w → φ p → - k i N → 1 2G → ∂ ∂ w x ∂ ∂ = φ σ x zy → ∂ ∂ = − φ σ y zx ∂ ∂ w y Volume = ∫∫ wdxdy → − Τ2 Paul A. Lagace © 2001 Unit 11 - p. 7
MT-1620 al.2002 Note: for orthotropic, would need a membrane to give different N's in different directions in proportion to G and g Membrane analogy only applies to isotropic materials This analogy gives a good physical picture for o Easy to visualize deflections of membrane for odd shapes Figure 11.5 Representation of o 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. 1 Paul A Lagace @2001 Unit 11-p 8
φ MIT - 16.20 Fall, 2002 Note: for orthotropic, would need a membrane to give different N’s in different directions in proportion to Gxz and Gyz ⇒ 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