MT-1620 al.2002 Unit 6 Plane stress and plane strain Readings T&G 8.9,.10.11,12.14.15,16 Paul A Lagace, Ph. D Professor of aeronautics Astronautics and Engineering Systems Paul A Lagace @2001
MIT - 16.20 Fall, 2002 Unit 6 Plane Stress and Plane Strain Readings: T & G 8, 9, 10, 11, 12, 14, 15, 16 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001
MT-1620 al.2002 There are many structural configurations where we do not have to deal with the full 3-d case First lets consider the models Lets then see under what conditions we can apply them A. Plane stress This deals with stretching and shearing of thin slabs Figure 6.1 Representation of Generic Thin Slab Paul A Lagace @2001 Unit
MIT - 16.20 Fall, 2002 There are many structural configurations where we do not have to deal with the full 3-D case. • First let’s consider the models • Let’s then see under what conditions we can apply them A. Plane Stress This deals with stretching and shearing of thin slabs. Figure 6.1 Representation of Generic Thin Slab Paul A. Lagace © 2001 Unit 6 - p. 2
MT-1620 al.2002 The body has dimensions such that h<<a b (Key: where are limits to"<<??? We'll consider later) Thus, the plate is thin enough such that there is no variation of displacement(and temperature)with respect to y3(z) Furthermore, stresses in the z-direction are zero(small order of magnitude) Figure 6.2 Representation of Cross-Section of Thin Slab 2y,ix Paul A Lagace @2001 Unit
MIT - 16.20 Fall, 2002 The body has dimensions such that h << a, b (Key: where are limits to “<<“??? We’ll consider later) Thus, the plate is thin enough such that there is no variation of displacement (and temperature) with respect to y3 (z). Furthermore, stresses in the z-direction are zero (small order of magnitude). Figure 6.2 Representation of Cross-Section of Thin Slab Paul A. Lagace © 2001 Unit 6 - p. 3
MT-1620 al.2002 Thus we assume 0000 So the equations of elasticity reduce to Equilibrium 01 0012 22 2=0(2) dy1 dy (3rd equation is an identity) 0=0 In general: do ga +f=0 Paul A Lagace @2001 Unit
MIT - 16.20 Fall, 2002 Thus, we assume: σzz = 0 σyz = 0 σxz = 0 ∂ = 0 ∂z So the equations of elasticity reduce to: Equilibrium ∂σ11 + ∂σ21 + f1 = 0 (1) ∂y1 ∂y2 ∂σ12 + ∂σ22 + f2 = 0 (2) ∂y1 ∂y2 (3rd equation is an identity) 0 = 0 (f3 = 0) In general: ∂σβα + fα = 0 ∂yβ Paul A. Lagace © 2001 Unit 6 - p. 4
MT-1620 al.2002 Stress-Strain (fully anisotropic) Primary(in-plane) strains 2 V,101+0 o6」(4 2 Ek= 6 0,+0 6 Invert to get aB= eaBoy& Secondary(out-of-plane)strains =( they exist, but they are not a primary part of the problem) 3 311 Paul A Lagace @2001 Unit 6-p. 5
MIT - 16.20 Fall, 2002 Stress-Strain (fully anisotropic) Primary (in-plane) strains 1 ε1 = E1 [σ1 − ν12σ2 − η16 σ6 ] (3) 1 ε 2 = E2 [− ν21 σ1 + σ2 − η26 σ6 ] (4) 1 ε6 = G6 [−η61 σ1 − η62σ2 + σ6 ] (5) Invert to get: σ * αβ = Eαβσγ εσγ Secondary (out-of-plane) strains ⇒ (they exist, but they are not a primary part of the problem) 1 ε3 = E3 [− ν31σ1 − ν32σ2 − η36σ6 ] Paul A. Lagace © 2001 Unit 6 - p. 5