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Unformatted text preview: Chapter 11 Energy Method  Utilize the Energy Method to solve engineering mechanics problems. Set aside the Equations of quilibrium 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 11.2 Strain Energy (11.1) dU Pdx = 1 x U Pdx = 1 energy x Strain U Pdx = = (11.2) If the material response is elastic: (11.3) P kx = 1 2 1 1 2 x U kxdx kx = = 1 1 1 2 U P x = 11.3 StrainEnergy Density (11.4) 1 x U P dx V A L = 1 x x U d V = 1 energy density x x Strain u d = = Modulus of Toughness = Toughness = area under the  curve. Modulus of Resilience: (11.5) x x E = 1 2 1 2 x x E u E d = = 2 1 2 u E = 2 2 Y Y u E = (11.6) (11.7) (11.8) 11.4 Elastic strain Energy for Normal Stresses (11.9) lim V U u V = dU u dV = 1 x x u d = 2 2 1 1 1 2 2 2 x x x x u E E = = = (11.9) x x E = 2 2 1 1 1 2 2 2 x x x x u E E = = = 2 2 x U dV E = 2 2 2 P U dV EA = (11.12) (11.11) (11.13) 2 2 2 P U dV EA = 2 2 L P U dx AE = dV Adx = 2 2 P L U AE = 11.5 Elastic Strain Energy for Shearing Stresses (11.18) xy xy xy u d = 2 2 1 1 2 2 2 xy xy xy xy u G G = = = dU u or U udV dV = = 2 2 xy U dV G = (11.19) (11.20) (11.21) Strain Energy in Torsion (11.22) 2 2 2 2 2 2 xy T U dV dV G GJ = = dV dAdx = 2 2 2 2 ( ) L T U dA dx GJ = 2 2 L T U dx GJ = 2 2 T L U GJ = (11.21) (11.19) 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 = + + + + + 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 = + + + + + + + (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 = + =  + =  + = = = (2.38) From Eq. (2.38) 2 2 2 2 2 2 1 1 2 2 2 ( ( )] ( ) x y z x y y...
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This note was uploaded on 12/03/2011 for the course EMA 3702 taught by Professor Wu during the Fall '11 term at FIU.
 Fall '11
 Wu

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