MT-1620 al.2002 E1) 0(0p +V|+ dy axd ao dv d'o dv es) 0(0 E2) 0=0? dx dxdy dy( dx dφ0p,dv a_0 (yes =Equilibrium automatically satisfied using Airy stress function Does that mean that any function we pick for o(x, y) will be valid? No, it will satisfy equilibrium, but we still have the strain-displacement and stress-strain equations If we use these we can get to the governing equation Paul A Lagace @2001 Unit 8-p. 6
∂ ∂ ∂ ∂ MIT - 16.20 Fall, 2002 2 ∂ ∂2φ ( ) E1 : 2 +V + ∂ − ∂ φ − ∂V = 0 ? ∂x ∂y ∂y ∂ ∂x y ∂x ∂3φ + ∂V − ∂3φ − ∂ V = 0 ⇒ (yes) ∂ ∂x y x y 2 ∂x ∂ ∂ 2 ∂x 2 ∂ − ∂ φ ∂ ∂2φ (E2) : ∂x ∂ ∂y + ∂y ∂x 2 + V − ∂V = 0 ? x ∂y ∂3φ + ∂3φ + ∂ V − ∂ V ⇒ = 0 (yes) 2 2 xy xy ∂y ∂y ⇒Equilibrium automatically satisfied using Airy stress function! Does that mean that any function we pick for φ (x, y) will be valid? No, it will satisfy equilibrium, but we still have the strain-displacement and stress-strain equations. If we use these, we can get to the governing equation: Paul A. Lagace © 2001 Unit 8 - p. 6
MT-1620 al.2002 Step 1 Introduce o into the stress-strain equations(compliance form) E3) E (E4) 2(1+ E (E5) So 十 V E E3) oX E y EaX E4) E 2(1+y)a2φ (E5) e day Paul A Lagace @2001 Unit 8-p. 7
(E MIT - 16.20 Fall, 2002 Step 1: Introduce φ into the stress-strain equations (compliance form): 1 εxx = E3 E (σxx − νσyy ) () 1 εyy = (−νσxx + σyy ) (E4) E εxy = 2 (1 E + ν) σxy (E5) So: εxx = 1 ∂2φ − ν ∂2φ + (1 − ν) V (E3′) E ∂y2 ∂x2 E εyy = 1 ∂2φ − ν ∂2φ + (1 − ν) V (E4′) E ∂x2 ∂y2 E εxy = − 2 (1 + ν) ∂2φ (E5′) E ∂ ∂x y Paul A. Lagace © 2001 Unit 8 - p. 7
MT-1620 al.2002 Step 2: Use these in the plane stress compatibility equation E6) dX we get quite a mess! After some rearranging and manipulation this results in 02(△T)02(△T) Eat ax temperature term we a= coefficient of thermal expansion haven't yet considered AT= temperature differential This is the basic equation for isotropic plane stress in stress function form Reca:φ is a scalar Paul A Lagace @2001 Unit 8-p. 8
MIT - 16.20 Fall, 2002 Step 2: Use these in the plane stress compatibility equation: ∂2εxx ∂2εyy ∂2εxy + = ∂y x y 2 ∂x2 ∂ ∂ (E6) ⇒ we get quite a mess! After some rearranging and manipulation, this results in: V V ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ = − ∂ ( ) ∂ + ∂ ( ) ∂ − ( ) ∂∂ + ∂∂ 4 4 4 2 2 4 2 2 2 2 2 2 2 2 1 φ φ α x y y E T x T y y 2 ∆ − 2 φ ν x x ∆ (*) temperature term we haven’t yet considered α = coefficient of thermal expansion ∆T = temperature differential This is the basic equation for isotropic plane stress in Stress Function form Recall: φ is a scalar Paul A. Lagace © 2001 Unit 8 - p. 8
MT-1620 al.2002 If we recall a little mathematics, the Laplace Operator in 2-D is x4+20 This is the biharmonic operator(also used in fluids) So the f)equation can be written Ey4AT)-(1=y2v() Finally, in the absence of temperature effects and body forces this becomes v4=0 homogeneous form What happened to E, v?? Paul A Lagace @2001 Unit 8-p. 9
MIT - 16.20 Fall, 2002 If we recall a little mathematics, the Laplace Operator in 2-D is: ∇2 ∂2 ∂2 = + ∂x2 ∂y2 ∂4 ∂4 ∂4 ⇒ ∇2∇2 = ∇4 = ∂x4 + 2 2 2 + ∂y4 ∂ ∂ x y This is the biharmonic operator (also used in fluids) So the (*) equation can be written: ∇4φ = −Eα∇2 (∆T) − (1 − ν) ∇2 V (*) Finally, in the absence of temperature effects and body forces this becomes: ∇4φ = 0 homogeneous form What happened to E, ν?? Paul A. Lagace © 2001 Unit 8 - p. 9
MT-1620 Fall 2002 this function, and accompanying governing equation, could be defined in any curvilinear system(we'l see one such example later) and in plane strain as well But. what's this all useful for??? This may all seem like"magic". Why were the os assumed as they were? This is not a direct solution to a posed problem, per se, but is known as The Inverse method In general, for cases of plane stress without body force or temp (V#o =O) A stress function o(x, y) is assumed that satisfies the biharmonic equation 2. The stresses are determined from the stress function as defined in equations (8-1)-(8-3) 3. Satisfy the boundary conditions (of applied tractions ☆☆4. Find the( structural problem that this satisfies Paul A Lagace @2001 Unit 8-p. 10
MIT - 16.20 Fall, 2002 ⇒ this function, and accompanying governing equation, could be defined in any curvilinear system (we’ll see one such example later) and in plane strain as well. But…what’s this all useful for??? This may all seem like “magic”. Why were the σ’s assumed as they were? This is not a direct solution to a posed problem, per se, but is known as… The Inverse Method In general, for cases of plane stress without body force or temp (∇4φ = 0): 1. A stress function φ (x, y) is assumed that satisfies the biharmonic equation 2. The stresses are determined from the stress function as defined in equations (8-1) - (8-3) 3. Satisfy the boundary conditions (of applied tractions) 4. Find the (structural) problem that this satisfies Paul A. Lagace © 2001 Unit 8 - p. 10