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hw32solutions - AAE 221 Aerospace Structures II Spring 2004...

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AAE 221 – Aerospace Structures II – Spring 2004 Chapter 3.2 – Analytical Solutions of Static Problems Using Energy Methods – Solution 1a. Starting with the GDE, ( 29 - - = 3 2 ' ' ' ' L x P x EIw δ Boundary conditions: ( 29 ( 29 ( 29 ( 29 ( 29 0 ' ' ' 0 ' ' 0 ' 0 ' ' 0 0 = = = = L EIw L EIw Kw EIw w Integrating, ( 29 ( 29 ( 29 ( 29 4 3 2 2 3 1 3 3 2 2 1 2 2 1 1 2 6 3 2 6 3 2 2 3 2 2 3 2 ' 3 2 3 2 ' ' 3 2 ' ' ' C x C x C x C L x stp P L x x EIw C x C x C L x stp P L x x EIw C x C L x Pstp L x x EIw C L x Pstp x EIw + + + + - - - = + + + - - - = + + - - - = + - - = Applying the boundary conditions, we get The first BC above gives 0 4 = C . The forth BC above gives P C C P = + - = 1 1 0 . The third BC above gives 3 2 3 2 0 2 PL C PL P L L - + + - - = . The second BC above gives ( 29 3 2 0 ' PL Kw - = , but since ( 29 ( 29 EI C w C EIw 3 3 0 ' 0 ' = = , α 3 2 3 2 3 2 2 3 3 PL K PLEI C PL EI C K - = - = - = . ( 29 x PL PLx Px L x stp L x P x EIw α 3 2 3 1 6 1 3 2 3 2 6 2 2 3 3 - - + - - - = In non-dimensional form, ( 29 - - + - - - = L x L x L x L x stp L x PL x EIw α 3 2 3 1 6 1 3 2 3 2 6 1 2 3 3 3 When x = L , the right-hand-side of the non-dimensionalized equation is a constant, i.e., ( 29 - - = α 3 2 81 14 3 EI PL L w Therefore, doubling the length increases the deflection by a factor of 8.
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1b. Applying Castigliano’s theorems, ( ( 29 K M dx EI x M U y L y 0 2 1 2 1 2 0 2 + = where ( 29 ( 0 ' 0 Kw M y = . We can apply a dummy load D P at the end of the beam and then inspect the free body diagram at various sections of the beam. For L x L < 3 2 , ( ( 29 x L P x M D y - = For 3 2 0 L x , ( 29 ( 29 - - - = x L P x L P x M D y 3 2 Substituting, ( 29 ( 29 2 3 / 2 2 2 3 / 2 0 2 3 2 2 1 2 1 3 2 2 1 - + - + - - - = L P L P K dx x L P EI dx x L P x L P EI U D L L D L D Integrating, ( 29 2 3 2 3 3 3 3 3 2 2 1 162 1 2 3 81 1 81 1 2 1 - + + + - + - + + - = PL L P K EI L P P P P P L P P L P EI U D D D D D D
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