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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 Timoshenko beam theory ： ウィキペディア英語版 Timoshenko beam theory The Timoshenko beam theory was developed by Stephen Timoshenko early in the 20th century.〔Timoshenko, S. P., 1921, ''On the correction factor for shear of the differential equation for transverse vibrations of bars of uniform cross-section'', Philosophical Magazine, p. 744.〕〔Timoshenko, S. P., 1922, ''On the transverse vibrations of bars of uniform cross-section'', Philosophical Magazine, p. 125.〕 The model takes into account shear deformation and rotational inertia effects, making it suitable for describing the behaviour of short beams, sandwich composite beams or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4th order but, unlike ordinary Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions. The latter effect is more noticeable for higher frequencies as the wavelength becomes shorter, and thus the distance between opposing shear forces decreases. If the shear modulus of the beam material approaches infinity - and thus the beam becomes rigid in shear - and if rotational inertia effects are neglected, Timoshenko beam theory converges towards ordinary beam theory. == Quasistatic Timoshenko beam == In static Timoshenko beam theory without axial effects, the displacements of the beam are assumed to be given by :$$ u_x(x,y,z) = -z~\varphi(x) ~;~~ u_y(x,y,z) = 0 ~;~~ u_z(x,y) = w(x) where $(x,y,z)$ are the coordinates of a point in the beam, $u\_x,\; u\_y,\; u\_z$ are the components of the displacement vector in the three coordinate directions, $\backslash varphi$ is the angle of rotation of the normal to the mid-surface of the beam, and $w$ is the displacement of the mid-surface in the $z$-direction. The governing equations are the following uncoupled system of ordinary differential equations: :$$ \begin & \frac\left(EI\frac\right) = q(x,t) \\ & \frac = \varphi - \frac \frac x}\left(EI\frac\right). \end The Timoshenko beam theory for the static case is equivalent to the Euler-Bernoulli theory when the last term above is neglected, an approximation that is valid when :$$ \frac \ll 1 where * $L$ is the length of the beam. * $A$ is the cross section area. * $E$ is the elastic modulus. * $G$ is the shear modulus. * $I$ is the second moment of area. * $\backslash kappa$, called the Timoshenko shear coefficient, depends on the geometry. Normally, $\backslash kappa\; =\; 5/6$ for a rectangular section. Combining the two equations gives, for a homogeneous beam of constant cross-section, :$$ EI~\cfrac = q(x) - \cfrac~\cfrac The bending moment $M\_$ and the shear force $Q\_x$ in the beam are related to the displacement $w$ and the rotation $\backslash varphi$. These relations, for a linear elastic Timoshenko beam, are: :$$ M_ = -EI~\frac \quad \text \quad Q_ = \kappa~AG~\left(-\varphi + \frac\right) \,. : = -z~\frac ~;~~ \varepsilon_ = \frac\left(\frac+\frac\right) = \frac\left(-\varphi + \frac\right) Since the actual shear strain in the beam is not constant over the cross section we introduce a correction factor $\backslash kappa$ such that :$$ \varepsilon_ = \frac~\kappa~\left(-\varphi + \frac\right) The variation in the internal energy of the beam is :$$ \delta U = \int_L \int_A (\sigma_\delta\varepsilon_ + 2\sigma_\delta\varepsilon_)~\mathrmA~\mathrmL = \int_L \int_A \left()~\mathrmL Integration by parts, and noting that because of the boundary conditions the variations are zero at the ends of the beam, leads to :$$ \delta U = \int_L \left()~\mathrmL = 0 The governing equations for the beam are, from the fundamental theorem of variational calculus, :$$ \frac - Q_x = 0 ~;~~ \frac + q = 0 For a linear elastic beam :$$ \begin M_ & = \int_A z~\sigma_~\mathrmA = \int_A z~E~\varepsilon_~\mathrmA = -\int_A z^2~E~\frac~\mathrmA = -EI~\frac \\ Q_ & = \int_A \sigma_~\mathrmA = \int_A 2G~\varepsilon_~\mathrmA = \int_A \kappa~G~\left(-\varphi + \frac\right)~\mathrmA = \kappa~AG~\left(-\varphi + \frac\right) \end Therefore the governing equations for the beam may be expressed as :$$ \begin \frac\left(EI\frac\right) + \kappa AG~\left(\frac-\varphi\right) & = 0 \\ \frac\left(AG\left(\frac - \varphi\right)\right ) + q & = 0 \end Combining the two equations together gives :$$ \begin & \frac\left(EI\frac\right) = q \\ & \frac = \varphi - \cfrac~\frac\left(EI\frac\right) \end |} 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「Timoshenko beam theory」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.