MT-1620 al.2002 Unit 4 Equations of Elasticity Readings R 2.3,2.6.28 T&G 84.85 B,M,P5.1-5.5,58,59 7.1-7.9 6.163,6.5-67 Jones (as background on composites Paul A Lagace, Ph. D Professor of aeronautics Astronautics and Engineering Systems Paul A Lagace @2001
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
MT-1620 al.2002 et's first review a bit from Unified, saw that there are 3 basic considerations in elasticity Equilibrium 2. Strain-Displacement 3. Stress-Strain Relations Constitutive Relations) Consider each 1. Equilibrium (3) ΣF=0,ΣM=0 Free body diagrams Applying these to an infinitesimal eleme yields 3 equilibrium equations Figure 4.1 Representation of general infinitesimal element dy z Paul A Lagace @2001 Unit 4-p. 2
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 Fi = 0, Σ Mi = 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
MT-1620 al.2002 0031 千1=0 dy 2 22 0032十 2=0(4-2) 2 13 do 0033 53=0(43) domn f=0 ym 2. Strain-Displacement(6) Based on geometric considerations Linear considerations (1.e, small strains only --we will talk about large strains later) (and infinitesimal displacements only) Paul A Lagace @2001 Unit 4-p.3
MIT - 16.20 Fall, 2002 ∂σ11 + ∂σ21 + ∂σ31 + f1 = 0 (4-1) ∂y1 ∂y2 ∂y3 ∂σ12 + ∂σ22 + ∂σ32 + f2 = 0 (4-2) ∂y1 ∂y2 ∂y3 ∂σ13 + ∂σ23 + ∂σ33 + f3 = 0 (4-3) ∂y1 ∂y2 ∂y3 ∂ ∂ + σmn 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
MT-1620 al.2002 (4-4) (4-5) 2 33 46) 3 12 7) 2 1 au du 31 13 十 (4-8) 3 2dy3 dv du 2 dyn dym Paul A Lagace @2001 Unit 4-p. 4
MIT - 16.20 Fall, 2002 ε11 = ∂ u1 (4-4) ∂ y1 ε22 = ∂ u2 (4-5) ∂ y2 ε33 = ∂ u3 (4-6) ∂ y3 ε21 = ε12 = 1 ∂ u1 + ∂ u2 2 ∂ y2 ∂ y1 ε31 = ε13 = 1 ∂ u1 + ∂ u3 2 ∂ y3 ∂ y1 ε32 = ε23 = 1 ∂ u2 + ∂ u3 2 ∂ y3 ∂ y2 (4-7) (4-8) (4-9) 1 ∂ um + ∂ un ε mn = 2 ∂ yn ∂ ym Paul A. Lagace © 2001 Unit 4 - p. 4
MT-1620 Fall 2002 3. Stress-Strain(6 Omn -mpg pg 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 → DYNAM|CS 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
MIT - 16.20 Fall, 2002 3. Stress - Strain (6) σmn = Emnpq εpq 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