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Chapter11EnergyMethods

Chapter11EnergyMethods - Chapter 11 Energy Method Utilize...

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Chapter 11 Energy Method -- Utilize the Energy Method to solve engineering mechanics problems. -- Set aside the Equations of quilibrium
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1. Introduction The relations between forces and deformation : Fundamental concept of We will learn: Strain Ch 2 Strain Energy – Ch 11 Stress -- Ch 1 1. Modulus of Toughness 2. Modulus of resilience 3. Castigliano Theorem
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11.2 Strain Energy (11.1) dU Pdx = 1 0 x U Pdx = 1 0 energy x Strain U Pdx = = (11.2)
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If the material response is elastic: (11.3) P kx = 1 2 1 0 1 2 x U kxdx kx = = 1 1 1 2 U P x =
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11.3 Strain-Energy Density (11.4) 1 0 x U P dx V A L = 1 0 x x U d V ε σ ε = 1 0 energy density x x Strain u d ε σ ε = = Modulus of Toughness = Toughness = area under the σ - ε curve.
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Modulus of Resilience: (11.5) x x E σ ε = 1 2 1 0 2 x x E u E d ε ε ε ε = = 2 1 2 u E σ = 2 2 Y Y u E σ = (11.6) (11.7) (11.8)
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11.4 Elastic strain Energy for Normal Stresses (11.9) 0 lim V U u V = dU u dV = 1 0 x x u d ε σ ε = 2 2 1 1 1 2 2 2 x x x x u E E σ ε σ ε = = =
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(11.9) x x E σ ε = 2 2 1 1 1 2 2 2 x x x x u E E σ ε σ ε = = = 2 2 x U dV E σ = 2 2 2 P U dV EA = (11.12) (11.11)
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(11.13) 2 2 2 P U dV EA = 2 0 2 L P U dx AE = dV Adx = 2 2 P L U AE =
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11.5 Elastic Strain Energy for Shearing Stresses (11.18) 0 xy xy xy u d γ τ γ = 2 2 1 1 2 2 2 xy xy xy xy u G G τ γ τ γ = = = dU u or U udV dV = = 2 2 xy U dV G τ = (11.19) (11.20) (11.21)
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Strain Energy in Torsion (11.22) 2 2 2 2 2 2 xy T U dV dV G GJ τ ρ = = dV dAdx = 2 2 2 0 2 ( ) L T U dA dx GJ ρ = 2 0 2 L T U dx GJ = 2 2 T L U GJ = (11.21) (11.19)
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11.6 Strain Energy for a General State of Stress (11.25) 1 2 ( ) x x y y z z xy xy yz yz zx zx u σ ε σ ε σ ε τ γ τ γ τ γ = + + + + + 2 2 2 2 2 2 1 1 2 2 2 ( ( )] ( ) x y z x y y z z x xy yz zx u E G σ σ σ ν σ σ σ σ σ σ τ τ τ = + + - + + + + + (11.26) y z X x y z X y y z X z xy yz zx xy yz zx E E E E E E E E E G G G υσ υσ σ ε σ υσ υσ ε υσ σ υσ ε τ τ τ γ γ γ = + - - = - + - = - - + = = = (2.38) From Eq. (2.38)
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2 2 2 2 2 2 1 1 2 2 2 ( ( )] ( ) x y z x y y z z x xy yz zx u E G σ σ σ ν σ σ σ σ σ σ τ τ τ = + + - + + + + + 2 2 2 1 2 2 ( ( )] a b c a b b c c a u E σ σ σ ν σ σ σ σ σ σ = + + - + + (11.26) (11.27) If the principal stresses are used: Where σ a , σ b , σ c = the principal stresses
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