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# lec25 - 2.001 MECHANICS AND MATERIALS I Lecture 27 Prof...

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2.001 - MECHANICS AND MATERIALS I Lecture # 12/13/2006 Prof. Carol Livermore Example Problems Q: What is the force in the spring? FBD 4 unknowns 3 equilibrium equations statically indeterminate. Force-Deformation F = d 2 v ( x ) = M ( x ) dx 2 EI Compatibility: 1 27

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δ = u B y B v (2 a )= u y v M ( x M D x ( x 2 3 Lx +2 L 2 ) , 0 x L 6 EIL v P ( x Fbx ( L 2 b 2 x 2 ) , 0 x d 6 v P ( x Fd ( L x ) [ L 2 d 2 ( L x ) 2 ] ,d x L 6 v total = v M D ( x )+ v P ( x ) To Find F : v M 0 ( x =2 a M 0 (2 a ) ± (3 a ) 2 3(3 a )(2 a )+2(3 a ) 2 ² 6 EI (3 a ) 4 M 0 a 2 = 9 v F ( x a Fa (2 a ) ± (3 a ) 2 a 2 (2 a ) 2 ² 6 (3 a ) 4 3 = 9 4 a 2 ±² v total (2 a + M 0 9 Recall: v total (2 a δ 2
±² F 4 a 2 = Fa + M 0 k 9 EI 14 a 3 4 a 2 M 0 F += k 9 9 4 a 2 M 0 F = ± 9 ² 1 + 4 a 3 k 9 Example: Seamed Pressure Vessel If max tensile stress on a seam is 0 . 8 max of the rest of the structure, what is max W ? Note: 3

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± ± F x =0 PπR 2 =2 πRtσ xx pR σ xx = 2 t F θ P (2 R x = σ θθ (2 t Δ x ) pR σ θθ = t a tan φ = 2 πR W cos φ = a W sin φ 4
±² Draw Mohr’s Circle (2 φ ) = 90 + α 0 . 05 pR 1 t sin α = α =s in 1 . 25 pR 5 t sin 1 1 5 φ =45 + 2 ± sin 1 1 ² 5 W =2 πR sin 45 + 2 Example: What is tip deﬂection? What is max stress? d 2 v ( x ) dx 2 = M ( x ) EI ( x ) I = 1

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lec25 - 2.001 MECHANICS AND MATERIALS I Lecture 27 Prof...

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