MT-1620 al.2002 resul tant Ory COSY +Oxx siny geometrically.cosγ d s siny Thus: τds ds to ds dy ozx dx We know that W G kx dy W G-ky Paul A Lagace @2001 Unit 12-6
MIT - 16.20 Fall, 2002 τresultant = σzy cos γ + σzx sin γ geometrically: cos γ = dy ds dx sin γ = ds Thus: dx τ ds = ∫ dyds σzy ds + σzx ds ∫ ds = ∫ σzy dy + σzx dx We know that: ∂w σzy = G kx + ∂y σzx = G −ky + ∂w ∂x Paul A. Lagace © 2001 Unit 12 - 6
MT-1620 al.2002 W G k Gi-ky dx Gfow dx dx We further know that dw =w=0 around closed contour So we re left with τds=Gkd{xd Paul A Lagace @2001 Unit 12-7
MIT - 16.20 Fall, 2002 ⇒ = ∫ τ ds ∫ G kx + + ∫ dy ∂w G −ky + ∂wdx ∂y ∂x + + ∫ dx dy Gk ∫ ∂w ∂w = G {xdy − ydx} ∂x ∂y = dw We further know that: ∫ dw = w = 0 around closed contour So we’re left with: ∫ τds = Gk ∫ {xdy − ydx} Paul A. Lagace © 2001 Unit 12 - 7