美国麻省理工大学：《结构力学》英文版 Unit 6 Plane Stress and Plane Strain

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

MT-1620 Fall 2002 4 4101-14202-146 G Note: can reduce these for orthotropic, isotropic (etc ) as before Strain-Displacement Primary ou 11 au 12 2 Paul A Lagace @2001 Unit

MIT - 16.20 Fall, 2002 1 ε4 = G4 [− η41 σ1 − η42 σ2 − η46σ6 ] 1 ε5 = G5 [−η51 σ1 − η52σ2 − η56σ6 ] Note: can reduce these for orthotropic, isotropic (etc.) as before. Strain - Displacement Primary ε11 = ∂u1 (6) ∂y1 ε22 = ∂u2 (7) ∂y2 ε12 = 1  ∂u1 + ∂u2  (8) 2  ∂y2 ∂y1  Paul A. Lagace © 2001 Unit 6 - p. 6

MT-1620 al.2002 Secondi 13 2 23 2(y30y2 33 Note: that for an orthotropic material 13 4.E5-0(due to stress-strain relations) Paul A Lagace @2001 Unit 7

MIT - 16.20 Fall, 2002 Secondary ε13 = 1  ∂u1 + ∂u3  2  ∂y3 ∂y1  ε23 = 1  ∂u2 + ∂u3  2  ∂y3 ∂y2  ε33 = ∂u3 ∂y3 Note: that for an orthotropic material (ε23 ) (ε13 ) ε4 = ε5 = 0 (due to stress-strain relations) Paul A. Lagace © 2001 Unit 6 - p. 7

MT-1620 al.2002 This further implies from above (since No in-plane variation du but this is not exactly true INCONSISTENCY Why? This is an idealized model and thus an approximation. There are, in actuality, triaxial (o,,etc. ) stresses that we ignore here as being small relative to the in-plane stresses we will return to try to define“sma∥) Final note: for an orthotropic material, write the tensorial stress-strain equation as x 2-D plane stress (a,β,,Y=12) Paul A Lagace @2001 Unit

⇒ MIT - 16.20 Fall, 2002 This further implies from above ∂ (since = 0) ∂y3 No in-plane variation ∂u3 = 0 ∂yα but this is not exactly true ⇒ INCONSISTENCY Why? This is an idealized model and thus an approximation. There are, in actuality, triaxial (σzz, etc.) stresses that we ignore here as being small relative to the in-plane stresses! (we will return to try to define “small”) Final note: for an orthotropic material, write the tensorial stress-strain equation as: 2-D plane stress σαβ = εσγ (, α β, σ, γ = 1 2 , ) αβσγ ∗ E Paul A. Lagace © 2001 Unit 6 - p. 8

MT-1620 There is not a 1-to-1 correspondence between the nmpg and-2, 002 the 2-D E aBor. The effect of E33 must be incorporated since E33 does not appear in these equations by using the (033=0)equation This gives Also, particularly in composites, another notation will be used in the case of plane stress in place of engineering notation subscript x=1=L(longitudinal).along major axis change y=2=T(transverse).along minor axis The other important"extreme"model is B. Plane strain This deals with long prismatic bodies Paul A Lagace @2001 Unit

MIT - 16.20 Fall, 2002 There is not a 1-to-1 correspondence between the 3-D Emnpq and the 2-D E* αβσγ. The effect of ε33 must be incorporated since ε33 does not appear in these equations by using the (σ33 = 0) equation. This gives: ε33 = f(εαβ) Also, particularly in composites, another “notation” will be used in the case of plane stress in place of engineering notation: subscript x = 1 = L (longitudinal)…along major axis change y = 2 = T (transverse)…along minor axis The other important “extreme” model is… B. Plane Strain This deals with long prismatic bodies: Paul A. Lagace © 2001 Unit 6 - p. 9

MT-1620 al.2002 Figure 6.3 Representation of Long Prismatic Body L 2 Dimension in z-direction is much much larger than in the x and y directions L>>X,y Paul A Lagace @2001 Unit 6-p. 10

MIT - 16.20 Fall, 2002 Figure 6.3 Representation of Long Prismatic Body Dimension in z - direction is much, much larger than in the x and y directions L >> x, y Paul A. Lagace © 2001 Unit 6 - p. 10