Chapter 2 Review of Elasticity

Chapter 2 Review of Elasticity - Chapter 2. Review of...

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Unformatted text preview: Chapter 2. Review of Elasticity 1. Stress h Plane-Strain - The plane-strain distribution is based on the assumption that z u z = = (2-1) where z represents the lengthwise direction of an elastic elongated body of constant cross section subjected to uniform loading (in the case of uniaxial problems, / y y u = = ). xy , , x y u v u v x y y x = = = + (2-2) are functions of x and y only and the strain components xz yz , , z w w u w v z x z y z = = + = + (2-3) vanish. The corresponding strain-stress relationship becomes ( 29 ( 29 ( 29 1 1 1 1 2 1 x x y y y x xy xy xy E E G E + =-- + =-- + = = (2-4) Hence, ( 29 ( 29 29,000 11,154 2 1 2 1 0.3 E G = = = + + (2-5) 1 The stress-strain relationship is ( 29 ( 29 ( 29 1 1 1 1 2 1 2 / 2 x x y y xy xy E - =- +- - (2-6) ( 29 z x y = + (2-7) P Plane-Stress - The plane stress distribution is based on the assumption that z yz xz = = = . (2-8) The stress-strain relationship is ( 29 2 1 1 1 1 / 2 x x y y xy xy E = - - (2-9) and the strain-stress relationship is ( 29 1 1 1 2 1 x x y y xy xy E - =- + (2-10) If we assume a state of plane stress, we need not have a corresponding plane strain and conversely, if we assume a state of plane strain, we need not have a corresponding plane stress state. These ar e entirely different assumptions. Home Work #3 2 Using the generalized Hookes law, prove that the maximum value of Poissons ratio, , cannot exceed 0.5. P For an analytical determination of the distribution of static or dynamic displacements and stresses in a structure (body) under prescribed external loading and temperature, we must obtain a solution to the basic equations of the theory of elasticity, satisfying the imposed boundary conditions on forces and/or displacements. P For general 3-D structures, we have 6: strain-displacement relationships; ( 29 , , f u v w = 6: stress-strain relationship, generalized Hookes law 3 : Equations of equilibrium, linear momentum balance 15 P Thus, there are 15 equations available for 15 unknowns, namely 3: displacements: u, v, w 6: stresses 6 : strains 15 3 In the case of rigid body mechanics problems, we may use 6 equations of equilibrium including three equations of linear momentum balance and three equations of angular momentum balance. Since the three equations of angular momentum balance are used up in establishing the...
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This note was uploaded on 09/24/2011 for the course CIVL 7690 taught by Professor Staff during the Summer '10 term at Auburn University.

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Chapter 2 Review of Elasticity - Chapter 2. Review of...

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