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# lec24 - 2.001 MECHANICS AND MATERIALS I Lecture#26 Prof...

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± 2.001 - MECHANICS AND MATERIALS I Lecture # 12/11/2006 Prof. Carol Livermore Energy Methods 3 Basic Ingredients of Mechanics: 1. Equilibrium 2. Constitutive Relations σ ±... Stress-Strain F δ Force-Deformation 3. Compatibility Dealing with geometric considerations Castigliano’s Theorem Start with work: W = F ² · d²s Conservative Forces: EX: Gravity 1 26

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u = mgh Potential Energy, Path Independent ±± h W = F ± · d±s = mgdz = mgh 0 Note: Elastic systems are conservative. Plastic deformation is not conservative. EX: Springs Energy = 1 / 2 2 ± δ ± δ 1 W = Pdδ = kδdδ = 2 2 00 Stored Energy = Work Put In U = W What if the spring were stretched a little further? 2
dU = Pdδ dU P = Complementary Energy ( U ) ± U = δdP Note: U = U due to linearity = U What if the load were stretched a little further? U ± if this is non-linear 3

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±± dU = δdP dU δ = dP Recall, for linear elastic material dU δ = dP Castigliano’s Theorem: Express complementary energy in terms of the loads
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lec24 - 2.001 MECHANICS AND MATERIALS I Lecture#26 Prof...

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