CVEN9802 - Energy Formulation 3.pdf

CVEN9802 - Energy Formulation 3.pdf - CVEN9802 Structural...

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CVEN 9802 Stability Page 1 CVEN9802 Structural Stability Energy Method Numerical Formulation 3 University of New South Wales
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CVEN 9802 Stability Page 2 Outline Stepped Column Problem - Raleigh-Ritz Method Curvature Diagram Finite Element Method Beam Stability Matrix
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CVEN 9802 Stability Page 3 EI Stepped Column Problem 0.4 EI P 0.7L 0.3L 0.7 2 2 2 1 1 1 , , , 2 2 2 0 0.7 0 0.4 L L L xx xx x L EI v dx EI v dx P v dx  
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CVEN 9802 Stability Page 4 Raleigh-Ritz Approximation General Formulation  2 1 2 , , 2 2 4 , , 2 ...... dim , 2 ...... dim 1 1 1 i i i i j i j i x x i i j i i j i j i xx xx i i j x v x a i L ija a ia v x v x i j L L ij i j a a a v i i x v x L L    
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CVEN 9802 Stability Page 5 Stability of a Column  3 1 2 3 2 1 1 0.4 0.6*0.7 3 1 i j i j ij i j i j EI a a L PL ij EI i j    
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CVEN 9802 Stability Page 6 Stability of a Column    3 1 1 1 0.4 0.6*0.7 3 1 1 1 1 1 0.4 0.6*0.7 1 1 1 , 1 .... dim 1 1 i j ij ij n m nm nm ij i j ij K S i j i j n i m j nm n m K n m n m S n m n m  
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CVEN 9802 Stability Page 7 Stability of a Column - Results dim % Error 1 2.46 6.5 2 2.3656 2.5 10 2.3292 0.8 20 2.3223 0.6 c.f. 2.3090 2 PL EI
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CVEN 9802 Stability Page 8 % Error Error more pronounced because of jump in curvature We are trying to use a smooth continuous function for curvature
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CVEN 9802 Stability Page 9 Stepped Column Problem EI 0.4 EI P Buckled shape Bending Moment Diagram
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CVEN 9802 Stability Page 10 Curvature Diagram EI 0.4 EI Curvature Diagram , xx M v EI
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CVEN 9802 Stability Page 11 Curvature Approximation Curvature Diagram , xx M v EI We are trying to use: (Very Difficult !!)   2 , 1 i i xx i a v i i x L
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CVEN 9802 Stability Page 12 Stability of a Column 2 2 1 , , 2 0 L xx x EI v P v dx   Lets look at expression for second variation of total potential: Let’s break up the column into segments:
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CVEN 9802 Stability Page 13 Finite Elements P Buckled shape segment 1 segment 2 , x v Has to be a smooth continuous function
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CVEN 9802 Stability Page 14 Finite Elements P segment 1 segment 2 , xx v Does not necessarily have to be a smooth continuous function for to have meaning.
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