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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 Macaulay's method ： ウィキペディア英語版 Macaulay's method Macaulay’s method (the double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. Use of Macaulay’s technique is very convenient for cases of discontinuous and/or discrete loading. Typically partial uniformly distributed loads (u.d.l.) and uniformly varying loads (u.v.l.) over the span and a number of concentrated loads are conveniently handled using this technique. The first English language description of the method was by Macaulay.〔(W. H. Macaulay, "A note on the deflection of beams", Messenger of Mathematics, 48 (1919), 129. )〕 The actual approach appears to have been developed by Clebsch in 1862.〔J. T. Weissenburger, ‘Integration of discontinuous expressions arising in beam theory’, AIAA Journal, 2(1) (1964), 106–108.〕 Macaulay's method has been generalized for Euler-Bernoulli beams with axial compression,〔W. H. Wittrick, ‘A generalization of Macaulay’s method with applications in structural mechanics’, AIAA Journal, 3(2) (1965), 326–330.〕 to Timoshenko beams,〔A. Yavari, S. Sarkani and J. N. Reddy, ‘On nonuniform Euler–Bernoulli and Timoshenko beams with jump discontinuities: application of distribution theory’, International Journal of Solids and Structures, 38(46–7) (2001), 8389–8406.〕 to elastic foundations,〔A. Yavari, S. Sarkani and J. N. Reddy, ‘Generalised solutions of beams with jump discontinuities on elastic foundations’, Archive of Applied Mechanics, 71(9) (2001), 625–639.〕 and to problems in which the bending and shear stiffness changes discontinuously in a beam〔Stephen, N. G., (2002), "Macaulay's method for a Timoshenko beam", Int. J. Mech. Engg. Education, 35(4), pp. 286-292.〕 ==Method== The starting point for Macaulay's method is the relation between bending moment and curvature from Euler-Bernoulli beam theory :$$ \pm EI\dfrac = M Where $w$ is the curvature and $M$ is the bending moment. This equation〔The sign on the left hand side of the equation depends on the convention that is used. For the rest of this article we will assume that the sign convention is such that a positive sign is appropriate.〕 is simpler than the fourth-order beam equation and can be integrated twice to find $w$ if the value of $M$ as a function of $x$ is known. For general loadings, $M$ can be expressed in the form :$$ M = M_1(x) + P_1\langle x - a_1\rangle + P_2\langle x - a_2\rangle + P_3\langle x - a_3\rangle + \dots where the quantities $P\_i\backslash langle\; x\; -\; a\_i\backslash rangle$ represent the bending moments due to point loads and the quantity $\backslash langle\; x\; -\; a\_i\backslash rangle$ is a Macaulay bracket defined as :$$ \langle x - a_i\rangle = \begin 0 & \mathrm~ x < a_i \\ x - a_i & \mathrm~ x > a_i \end Ordinarily, when integrating $P(x-a)$ we get :$$ \int P(x-a)~dx = P\left(- ax\right ) + C However, when integrating expressions containing Macaulay brackets, we have :$$ \int P\langle x-a \rangle~dx = P\cfrac + C_m with the difference between the two expressions being contained in the constant $C\_m$. Using these integration rules makes the calculation of the deflection of Euler-Bernoulli beams simple in situations where there are multiple point loads and point moments. The Macaulay method predates more sophisticated concepts such as Dirac delta functions and step functions but achieves the same outcomes for beam problems. 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「Macaulay's method」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.