MIT 1 050 - Lecture 27 Introduction: Energy bounds in linear elasticity

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1.050 Engineering Mechanics I Lecture 27 Introduction: Energy bounds in linear elasticity 11.050 – Content overview I. Dimensional analysis 1. On monsters, mice and mushrooms 2. Similarity relations: Important engineering tools II. Stresses and strength 3. Stresses and equilibrium 4. Strength models (how to design structures, foundations.. against mechanical failure) III. Deformation and strain 5. How strain gages work? 6. How to measure deformation in a 3D structure/material? IV. Elasticity 7. Elasticity model – link stresses and deformation 8. Variational methods in elasticity V. How things fail – and how to avoid it 9. Elastic instabilities 10. Plasticity (permanent deformation) 11. Fracture mechanics Lectures 1-3 Sept. Lectures 4-15 Sept./Oct. Lectures 16-19 Oct. Lectures 20-31 Oct./Nov. Lectures 32-37 Dec. 21.050 – Content overview I. Dimensional analysis II. Stresses and strength III. Deformation and strain IV. Elasticity Lecture 20: Introduction to elasticity (thermodynamics) Lecture 21: Generalization to 3D continuum elasticity Lecture 22: Special case: isotropic elasticity Lecture 23: Applications and examples Lecture 24: Beam elasticity Lecture 25: Applications and examples (beam elasticity) Lecture 26: … cont’d and closure Lecture 27: Introduction: Energy bounds in linear elasticity (1D system) Lecture 28: Introduction: Energy bounds in linear elasticity (1D system), cont’d … V. How things fail – and how to avoid it Lectures 32..37 3Convexity of a function f (x) ∂f | (b − a) ≤ f (b) − f (a)∂xx=a secant f (x) tangent x ab 4Example system: 1D truss structure We will use this example to illustrate all key concepts 5Total external work vr vr Wd =ξ⋅ Fd +ξd ⋅ R Work done by Work done by prescribed prescribed forces displacements, Displacements force unknown unknown 6Total internal work Ni * iψComplementary free energy State equations Ni = ∂ψi ∂δi δi = ∂ψi *ψi ∂Ni Free energy δi ∑δiNi =ψi *(Ni ) +ψi (δi ) i 7Combining it… vr vr ! Wd =ξ⋅ Fd +ξd ⋅ R =ψ+ψ* * d d− (ψ −ξ v ⋅ R r)= ! ψ−ξ v ⋅ F r Complementary Complementary energy energy =: ε=: εcom pot Solution to elasticity problem −εcom =εpot 8Quiz II – Monday Nov. 19 • Focus on material presented in lectures 16-26 • Preparation: Problem sets, old quizzes, lecturematerial • Deformation and strain, isotropic elasticity, beamdeformation (beam bending and beamstretching), forensic beam elasticity, sketchsolution of beam problems, concept ofsuperposition (frame structures)


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MIT 1 050 - Lecture 27 Introduction: Energy bounds in linear elasticity

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