unit7_beamenergy

unit7_beamenergy - 16.21 Techniques of Structural Analysis...

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Unformatted text preview: 16.21 Techniques of Structural Analysis and Design Spring 2004 Unit #7 (continued)- Concepts of work and energy Strain energy and potential energy of a beam Ra´ul Radovitzky March 2, 2005 Figure 1: Kinematic assumptions for a beam ¯ Kinematic assumptions for a beam: From the figure: AA = u 3 ( x 1 ). ¯ ¯ Assume small deflections: B B , BB = u 3 + du 3 . CC = u 3 ( x ) + ∼ 1 u 1( x 1 , x 3 ). Assume planar sections normal to the neutral axis remain planar after deformation . Then: u 3 = u 3 ( x 1 ) (1) du 3 u 1 ( x 1 , x 3 ) = − x 3 (2) dx 1 u 3 ( x 1 ) is the only primary unknown of the problem (3) From these kinematic assumptions we can derive a theory for beams. Strains : du 1 d 2 u 3 11 = = − x 3 (4) dx 1 dx 2 1 22 = 33 = − ν 11 , plane stress (5) 13 = 1 2 du 1 dx 3 + du 3 dx 1 = 1 2 − du 3 dx 1 + du 3 dx 1 = 0 (6) Constitutive : σ 11 = E 11 = − Ex 3 d 2 u 3 dx 2 1 (7) Equilibrium : Apply equilibrium (in the undeformed configuration) to in- tegral quantities (moment...
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This note was uploaded on 11/07/2011 for the course AERO 16.21 taught by Professor Dffs during the Fall '10 term at MIT.

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unit7_beamenergy - 16.21 Techniques of Structural Analysis...

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