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Case 1:  applied and followed by  |
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…. (a) Work done when  is gradually applied |
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…. (b) Work done when  is gradually applied with  in place
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Total work done by the two forces for case 1 is
|
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Case2:  applied and followed by  |
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…. (c) Work done when  is gradually applied |
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…. (d) Work done when  is gradually applied with  in place |
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Total work done by the two forces for case 2 is |
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Since the final deflected position of the beam produced by the two cases of loads is the same regardless of the order in which the loads are applied. The total work done by the forces is also the same regardless of the order in which the loads are applied. Thus, equating the total work of Cases 1 and 2 give |
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If  , the equation (4.31) depicts the statement of the Maxwell-Betti law i.e. |
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The Maxwell-Betti theorem also holds for rotations as well as rotation and linear displacement in beams and frames. |
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