Mechanics 1K

# Mechanics 1K - 2.001 MECHANICS AND MATERIALS I Lecture#8...

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± ± 2.001 - MECHANICS AND MATERIALS I Lecture #8 10/4/2006 Prof. Carol Livermore Recall from last lecture: Find: u ( x ) ,F A B C 1. Equilibrium F u =0 P F A F B F C M A Pa F c L F B L 2 2. Force-Deformation F A = A 1

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F B = B F C = C 3. Compatibility δ A = u A y A δ B = u y + L tan ϕ A z L AA δ C = u y +t a n ϕ z 2 New this lecture: Small Angle Assumption: sin ϕ A ϕ A zz tan ϕ A = = ϕ A z z cos ϕ A 1 z For small ϕ A z : Arc length a straight line displacemtn in y. Rewrite compatibility. A δ A = u y A δ B = u y + A z A L ϕ A δ A = u y + z 2 Substitute compatibility into force-deformation F A = ku y A A F B = k ( u y + z A ) 2
±² ± ² A L ϕ A F C = k ( u y + z ) 2 Substitute this result into equilibrium equations: AA P ku A y k ( u y + A z ) k ( u y + L ϕ A z )=0 2 Pa L 2 k ( u A y + L 2 ϕ A z ) Lk ( u A y + A z Solve: 3 P =3 A + Lkϕ A yz 2 35 = Lku A + L 2 A 2 y 4 z Divide by k. P A 3 u + A k y 2 z Divide by Lk/ 2.

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Mechanics 1K - 2.001 MECHANICS AND MATERIALS I Lecture#8...

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