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unit4 - MIT 16.20 Fall 2002 Unit 4 Equations of Elasticity...

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MIT - 16.20 Fall, 2002 Unit 4 Equations of Elasticity Readings : R 2.3, 2.6, 2.8 T & G 84, 85 B, M, P 5.1-5.5, 5.8, 5.9 7.1-7.9 6.1-6.3, 6.5-6.7 Jones (as background on composites) Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001
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MIT - 16.20 Fall, 2002 Let’s first review a bit … from Unified, saw that there are 3 basic considerations in elasticity: 1. Equilibrium 2. Strain - Displacement 3. Stress - Strain Relations (Constitutive Relations) Consider each: 1. Equilibrium (3) Σ F i = 0, Σ M i = 0 Free body diagrams Applying these to an infinitesimal element yields 3 equilibrium equations Figure 4.1 Representation of general infinitesimal element Paul A. Lagace © 2001 Unit 4 - p. 2
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MIT - 16.20 Fall, 2002 ∂σ 11 + ∂σ 21 + ∂σ 31 + f 1 = 0 (4-1) y 1 y 2 y 3 ∂σ 12 + ∂σ 22 + ∂σ 32 + f 2 = 0 (4-2) y 1 y 2 y 3 ∂σ 13 + ∂σ 23 + ∂σ 33 + f 3 = 0 (4-3) y 1 y 2 y 3 + σ mn m n y f 0 = 2. Strain - Displacement (6) Based on geometric considerations Linear considerations ( I.e., small strains only -- we will talk about large strains later ) (and infinitesimal displacements only) Paul A. Lagace © 2001 Unit 4 - p. 3
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MIT - 16.20 Fall, 2002 ε 11 = u 1 (4-4) y 1 ε 22 = u 2 (4-5) y 2 ε 33 = u 3 (4-6) y 3 ε 21 = ε 12 = 1 u 1 + u 2 2 y 2 y 1 ε 31 = ε 13 = 1 u 1 + u 3 2 y 3 y 1 ε 32 = ε 23 = 1 u 2 + u 3 2 y 3 y 2 (4-7) (4-8) (4-9) 1 u m + u n ε mn = 2 y n y m Paul A. Lagace © 2001 Unit 4 - p. 4
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MIT - 16.20 Fall, 2002 3. Stress - Strain (6) σ mn = E mnpq ε pq we’ll come back to this … Let’s review the “4th important concept”: Static Determinance There are there possibilities (as noted in U.E.) a. A structure is not sufficiently restrained (fewer reactions than d.o.f.) degrees of freedom DYNAMICS b. Structure is exactly (or “simply”) restrained (# of reactions = # of d.o.f.) STATICS (statically determinate) Implication : can calculate stresses via equilibrium ( as done in Unified ) Paul A. Lagace © 2001 Unit 4 - p. 5
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MIT - 16.20 Fall, 2002 c. Structure is overrestrained (# reactions > # of d.o.f.) STATICALLY INDETERMINATE …must solve for reactions simultaneously with stresses, strains, etc. in this case, you must employ the stress-strain equations --> Overall, this yields for elasticity: 15 unknowns and 15 equations 6 strains = ε mn 3 equilibrium ( σ ) 6 stresses = σ mn 6 strain-displacements ( ε ) 3 displacements = u m 6 stress-strain ( σ - ε ) IMPORTANT POINT: The first two sets of equations are “universal” (independent of the material) as they depend on geometry (strain-displacement) and equilibrium (equilibrium). Only the stress-strain equations are dependent on the material. Paul A. Lagace © 2001 Unit 4 - p. 6
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MIT - 16.20 Fall, 2002 One other point : Are all these equations/unknowns independent? NO Why? --> Relations between the strains and displacements (due to geometrical considerations result in the Strain Compatibility Equations (as you saw in Unified) General form is: 2 ε nk + 2 ε m l 2 ε n
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