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Ch_23 - Two-Dimensional Elasticity 2D 3D elem only 2D Plane...

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Chapter 23 Page 1 K. Haghighi Two-Dimensional Elasticity 2D elem. only 3D 2D Plane Stress Plane Strain Plane Stress: A state of stress in which we have a very thin elastic body and no load in the direction parallel to the thickness exist.
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Chapter 23 Page 2 K. Haghighi along t are very small and assumed to be zero when applied loads lie in the xy plane. s ' σ ( ) 0 = σ = σ = σ zy zx zz Stress vector reduces to { } [ ] xy yy xx T σ σ σ = σ t A thin body in a state of plane stress.
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Chapter 23 Page 3 K. Haghighi and using { } [ ] { } σ = ε C we have ( ) 0 σ + σ µ = ε yy xx zz E 0 = ε = ε yz xz not indep. though! also { } [ ] xy yy xx T ε ε ε = ε elastic strain vector
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Chapter 23 Page 4 K. Haghighi { } [ ] xy yy xx T e e e e = while { } [ ] 0 T T T T αδ αδ = ε The matrix now reduces to [ ] { } [ ] { } [ ] ε = σ D ; D [ ] µ µ µ µ = 2 1 0 0 0 1 0 1 1 2 E D
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Chapter 23 Page 5 K. Haghighi Plane Strain: Is a state of 2D stress in members that are not free to expand in the direction normal to the plane of applied loads. If loads are in the x-y plane, then w disp. in the z-direction is zero and u & v are functions of x & y only. 0 = = z w e zz 0 = + = x w z u e xz 0 = + = y w z v e yz (note: ) ! zz 0 σ
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Chapter 23 Page 6 K. Haghighi Then strain vectors are { } [ ] xy yy xx T e e e e = { } [ ] xy yy xx T ε ε ε = ε and { } [ ] 0 T T T T αδ αδ = ε while { } [ ] xy yy xx T σ σ σ = σ Similar to the state of plane stress!
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Chapter 23 Page 7 K. Haghighi to get the stress vector: { } [ ] { } , D ε = σ Hooke’s law Subst. [ ] µ + = c c c b b b b d E D 0 0 0 1
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Chapter 23 Page 8 K. Haghighi and also 0 = ε = ε = ε yz xz zz to get = σ zz ( ) 0 2 1 1 2 1 1 αδ µ µ σ + σ µ µ µ + T E yy xx (not indep. though!)
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Chapter 23 Page 9 K. Haghighi The resulting [D] for this state of stress is [ ] µ + = 2 1 0 0 0 0 1 d b b d E D where ( ) µ µ = µ µ = 2 1 2 1 1 b & d
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Chapter 23 Page 10 K. Haghighi The Displacement Equations: 2-D prob: 2DOF @ each node or 2 unknown disp’s w = 0 plane strain = f (u, v) plane stress disp. in the z direction u & v model k U 2 1 2 k U 1 2 j U 1 2 i U j U 2 i U 2 i j k Model: The nodal displacements for a triangular elasticity element.
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Chapter 23 Page 11 K. Haghighi using a linear variation for each of u & v w/in the elem., we may write ( ) 1 2 1 2 1 2 + + = k k j j i i U N U N U N y , x u ( ) k k j j i i U N U N U N y , x v 2 2 2 + + = or in general 1 2 2 1 2 0 + + = j j i i i U N U U N u k k k j j U N U U N 2 1 2 2 0 + + + k k k j U U N U 2 1 2 2 0 0 + + + 1 2 2 1 2 0 0 + + = j i i i U U N U v
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Chapter 23 Page 12 K. Haghighi using matrix notation: ( ) ( ) = k k j j i i k j i k j i U U U U U U N N N N N N y , x v y , x u 2 1 2 2 1 2 2 1 2 0 0 0 0 0 0 or in a compact form ( ) ( ) [ ] ( ) { } 1 6 6 2 × × = e U N y , x v y , x u elem. nodal disp’s.
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Chapter 23 Page 13 K. Haghighi Strain-disp. relations now reduce to x v y u e & y v e x u e xy yy
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