08 2Energy Deflection

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DEFLECTIONS: ENERGY METHODS ! ! ! ! External Work and Strain Energy Principle of Work and Energy Principle of Virtual Work Method of Virtual Work: ! ! Trusses Beams and Frames Trusses Beams and Frames ! Castigliano's Theorem ! ! 1 External Work and Strain Energy Most energy methods are based on the conservation of energy principle, which states that the work done by all the external forces acting on a structure, Ue, is transformed into internal work or strain energy, Ui. Ue = Ui ã Externa
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  1 ! External Work and Strain Energy ! Principle of Work and Energy ! Principle of Virtual Work ! Method of Virtual Work: ! Trusses ! Beams and Frames ! Castigliano ' s Theorem ! Trusses ! Beams and Frames DEFLECTIONS: ENERGY METHODS  2 U  e Eigen work  External Work and Strain Energy Most energy methods are based on the conservation of energy principle, which statesthat the work done by all the external forces acting on a structure, U  e , is transformedinto internal work or strain energy, U  i . U  e = U  i  L F ∆  x F  P   x P  F  ∆= As the magnitude of  F is  gradually increasedfrom zero to some limiting value  F  =  P  , the finalelongation of the bar becomes ∆ .ã  External Work-Force. ∆ Eigen work   FdxdU  e = ∫ =  xe FdxU  0 ∫ ∆ ∆= 0 )( dx x P U  e ∆=∆= ∆  P  x P U  e 21)2( 02  3 P  L F 'Displacement work   x F  ∆  P    E  i g  e  n   w o  r  k  ( U  e ) Total  = (Eigen Work)  P    + (Eigen Work)  F     + (Displacement work)  P  ∆∆ '  L ∆ ´  F  ' +  P  )'()')('( 21))((21)( ∆+∆+∆= P  F  P U  Total e  4 10 mm  L 20 kN  L x (m)  F  0.01 m20 kN m N U  e ã=×= 100)1020)(01.0( 21 3
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