lec21 - 2.001 - MECHANICS AND MATERIALS I Lecture #23...

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2.001 - MECHANICS AND MATERIALS I Lecture 11/29/2006 Prof. Carol Livermore Recall Moment-Curvature Equation E ( x ) I ( x ) M ( x )= ρ ( x ) or EI eff for composite beams. 1 = 2 v = M ( x ) ρ ( x ) ∂x 2 E ( x ) I ( x ) Approach: Integrate, get v ( x ), θ ( x dv ( x ) . Use boundary conditions to get dx constants of integration. Library of Solutions (Thus Far): Mx 2 v ( x 2 Px 2 ± x ² v ( x L 2 3 PL 3 v tip ( x = L 3 Recall example from last time: 1 #23
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±² ±²± ² a M ( x )= P 1 x, 0 x a L a M ( x P 1 x Px a, a x L L For a = L 2 (symmetric special case) L M ( x , 0 x 22 PLL M ( x + x L 2 Left: d 2 vP x = dx 2 2 EI dv P x 2 =+ c 1 dx 4 3 v c 1 x + c 2 12 Boundary Conditions: v ( x =0)=0 c 2 =0 θ ( x = L 2 )=0 2 PL 2 0= + c 1 c 1 = 4 21 6 So: Left: PP L 2 L v ( x x 3 x, 0 x 12 16 2 Right: 2
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PP L 2 L v ( x )= ( L x ) 3 ( L x ) , 0 x 12 EI 16 2 Note: The right is the same as the left but starting at x = L and moving left. This is due to symmetry. This situation is called 3-point bending.
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This note was uploaded on 05/05/2010 for the course MSE 2.001 taught by Professor Carollivermore during the Fall '06 term at MIT.

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lec21 - 2.001 - MECHANICS AND MATERIALS I Lecture #23...

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