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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Betti's theorem ： ウィキペディア英語版 Betti's theorem Betti's theorem, also known as Maxwell-Betti reciprocal work theorem, discovered by Enrico Betti in 1872, states that for a linear elastic structure subject to two sets of forces i=1,...,m and , j=1,2,...,n, the work done by the set P through the displacements produced by the set Q is equal to the work done by the set Q through the displacements produced by the set P. This theorem has applications in structural engineering where it is used to define influence lines and derive the boundary element method. Betti's theorem is used in the design of compliant mechanisms by topology optimization approach. ==Proof== Consider a solid body subjected to a pair of external force systems, referred to as $F^P\_i$ and $F^Q\_i$. Consider that each force system causes a displacement fields, with the displacements measured at the external force's point of application referred to as $d^P\_i$ and $d^Q\_i$. When the $F^P\_i$ force system is applied to the structure, the balance between the work performed by the external force system and the strain energy is: :$$ \frac\sum^n_F^P_id^P_i = \frac\int_\Omega \sigma^P_\epsilon^P_\,d\Omega The work-energy balance associated with the $F^Q\_i$ force system is as follows: :$$ \frac\sum^n_F^Q_id^Q_i = \frac\int_\Omega \sigma^Q_\epsilon^Q_\,d\Omega Now, consider that with the $F^P\_i$ force system applied, the $F^Q\_i$ force system is applied subsequently. As the $F^P\_i$ is already applied and therefore won't cause any extra displacement, the work-energy balance assumes the following expression: :$$ \frac\sum^n_F^Q_id^Q_i + \sum^n_F^P_id^Q_i = \frac\int_\Omega \sigma^Q_\epsilon^Q_\,d\Omega + \int_\Omega \sigma^P_\epsilon^Q_\,d\Omega Conversely, if we consider the $F^Q\_i$ force system already applied and the $F^P\_i$ external force system applied subsequently, the work-energy balance will assume the following expression: :$$ \frac\sum^n_F^P_id^P_i + \sum^n_F^Q_id^P_i = \frac\int_\Omega \sigma^P_\epsilon^P_\,d\Omega + \int_\Omega \sigma^Q_\epsilon^P_\,d\Omega If the work-energy balance for the cases where the external force systems are applied in isolation are respectively subtracted from the cases where the force systems are applied simultaneously, we arrive at the following equations: :$$ \sum^n_F^P_id^Q_i = \int_\Omega \sigma^P_\epsilon^Q_\,d\Omega :$$ \sum^n_F^Q_id^P_i = \int_\Omega \sigma^Q_\epsilon^P_\,d\Omega If the solid body where the force systems are applied is formed by a linear elastic material and if the force systems are such that only infinitesimal strains are observed in the body, then the body's constitutive equation, which may follow Hooke's law, can be expressed in the following manner: :$$ \sigma_=D_\epsilon_ Replacing this result in the previous set of equations leads us to the following result: :$$ \sum^n_F^P_id^Q_i = \int_\Omega D_\epsilon^P_\epsilon^Q_\,d\Omega :$$ \sum^n_F^Q_id^P_i = \int_\Omega D_\epsilon^Q_\epsilon^P_\,d\Omega If we subtracting both equations then we obtain the following result: :$$ \sum^n_F^P_id^Q_i = \sum^n_F^Q_id^P_i 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Betti's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.