lec19 - 2.001 - MECHANICS AND MATERIALS I Lecture #21...

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± ² 2.001 - MECHANICS AND MATERIALS I Lecture 11/21/2006 Prof. Carol Livermore Recall from last time: Beam Bending y = 0 on neutral axis ± xx = ρ y (Note: purely geometric, no material properties) σ xx = ± xx E (All other σ are equal to 0) So: Ey σ xx = ρ Force Equilibrium: F x =0 σ xx dA A 1 #21
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± ² ³ ± ± ± Ey dA =0 ρ A If E is constant in y then A ydA 0. Moment Equilibrium M z M = σ xx ydA A 2 M = dA ρ A Special case: E constant: 1 M = EI ρ I = y 2 dA A New this time: Recall: σ xx = ρ For constant E (special case): M = ρ So: EM σ xx == ρI y My σ xx = I EXAMPLE: Find location of neutral axis for rectangular beam 2
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± ± ± ± ± E is constant across cross-section. Recall force equilibrium. Ey dA =0 ρ A E ydA ρ A E ± 2 ± h a b ydydz ρ b 2 a 2 ² 2 b y biggl ] h a a dz b 2 2 b 2 1 [( h a ) 2 a 2 ] dz b 2 2 b 2 1 ( h 2 2 ha ) dz b 2 2 b 1 ²³ 2 ( h 2 2 ha ) z 2 b 2 1 bb ( h 2 2 ha )( + ) = 0 22 2 b ( h 2 2 ha )=0 2 h 2 =2 ha h a = 2 So the neutral axis is in the center of the beam.
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lec19 - 2.001 - MECHANICS AND MATERIALS I Lecture #21...

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