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| <!--{{Continuum mechanics|cTopic=[[Solid mechanics]]}}-->
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| The '''Timoshenko beam theory''' was developed by Ukrainian-born scientist and engineer [[Stephen Timoshenko]] early in the 20th century.<ref name=Timo1>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.</ref><ref name=Timo2>Timoshenko, S. P., 1922, ''On the transverse vibrations of bars of uniform cross-section'', Philosophical Magazine, p. 125.</ref> The model takes into account [[Shear stress|shear deformation]] and rotational [[inertia]] effects, making it suitable for describing the behaviour of short beams, [[Sandwich structured composite|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 beam theory - i.e. [[Euler–Bernoulli beam theory]] - there is also a second order spatial 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 [[eigenfrequency|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.
| |
| [[Image:TimoshenkoBeam.svg|thumb|400px|Deformation of a Timoshenko beam (blue) compared with that of an Euler-Bernoulli beam (red).]]
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| | |
| 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]].
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| | |
| == Quasistatic Timoshenko beam ==
| |
| [[Image:Plate theory.svg|thumb|200px|Deformation of a Timoshenko beam. The normal rotates by an amount <math>\theta_x = \varphi(x)</math> which is not equal to <math>dw/dx</math>.]]
| |
| In [[statics|static]] Timoshenko beam theory without axial effects, the displacements of the beam are assumed to be given by
| |
| :<math>
| |
| u_x(x,y,z) = -z~\varphi(x) ~;~~ u_y(x,y,z) = 0 ~;~~ u_z(x,y) = w(x)
| |
| </math>
| |
| where <math>(x,y,z)</math> are the coordinates of a point in the beam, <math>u_x, u_y, u_z</math> are the components of the displacement vector in the three coordinate directions, <math>\varphi</math> is the angle of rotation of the normal to the mid-surface of the beam, and <math>w</math> is the displacement of the mid-surface in the <math>z</math>-direction.
| |
| | |
| The governing equations are the following uncoupled system of [[ordinary differential equation]]s:
| |
| :<math>
| |
| \begin{align} | |
| & \frac{\mathrm{d}^2}{\mathrm{d} x^2}\left(EI\frac{\mathrm{d} \varphi}{\mathrm{d} x}\right) = q(x,t) \\
| |
| & \frac{\mathrm{d} w}{\mathrm{d} x} = \varphi - \frac{1}{\kappa AG} \frac{\mathrm{d}}{\mathrm{d} x}\left(EI\frac{\mathrm{d} \varphi}{\mathrm{d} x}\right).
| |
| \end{align}
| |
| </math>
| |
| | |
| The Timoshenko beam theory for the static case is equivalent to the [[Euler-Bernoulli beam equation|Euler-Bernoulli theory]] when the last term above is neglected, an approximation that is valid when
| |
| :<math> | |
| \frac{EI}{\kappa L^2 A G} \ll 1
| |
| </math>
| |
| where <math>L</math> is the length of the beam.
| |
| | |
| Combining the two equations gives, for a homogeneous beam of constant cross-section,
| |
| :<math>
| |
| EI~\cfrac{\mathrm{d}^4 w}{\mathrm{d} x^4} = q(x) - \cfrac{EI}{\kappa A G}~\cfrac{\mathrm{d}^2 q}{\mathrm{d} x^2}
| |
| </math>
| |
| | |
| The bending moment <math>M_{xx}</math> and the shear force <math>Q_x</math> in the beam are related to the displacement <math>w</math> and the rotation <math>\varphi</math>. These relations, for a linear elastic Timoshenko beam, are:
| |
| :<math>
| |
| M_{xx} = -EI~\frac{\partial \varphi}{\partial x} \quad \text{and} \quad
| |
| Q_{x} = \kappa~AG~\left(-\varphi + \frac{\partial w}{\partial x}\right) \,.
| |
| </math>
| |
| | |
| :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
| |
| !Derivation of quasistatic Timoshenko beam equations
| |
| |-
| |
| |From the kinematic assumptions for a Timoshenko beam, the displacements of the beam are given by
| |
| :<math>
| |
| u_x(x,y,z,t) = -z~\varphi(x,t) ~;~~ u_y(x,y,z,t) = 0 ~;~~ u_z(x,y,z) = w(x,t)
| |
| </math>
| |
| Then, from the strain-displacement relations for small strains, the non-zero strains based on the Timoshenko assumptions are
| |
| :<math>
| |
| \varepsilon_{xx} = \frac{\partial u_x}{\partial x} = -z~\frac{\partial \varphi}{\partial x} ~;~~
| |
| \varepsilon_{xz} = \frac{1}{2}\left(\frac{\partial u_x}{\partial z}+\frac{\partial u_z}{\partial x}\right)
| |
| = \frac{1}{2}\left(-\varphi + \frac{\partial w}{\partial x}\right)
| |
| </math>
| |
| Since the actual shear strain in the beam is not constant over the cross section we introduce a correction factor <math>\kappa</math> such that
| |
| :<math>
| |
| \varepsilon_{xz} = \frac{1}{2}~\kappa~\left(-\varphi + \frac{\partial w}{\partial x}\right)
| |
| </math>
| |
| The variation in the internal energy of the beam is
| |
| :<math>
| |
| \delta U = \int_L \int_A (\sigma_{xx}\delta\varepsilon_{xx} + 2\sigma_{xz}\delta\varepsilon_{xz})~\mathrm{d}A~\mathrm{d}L
| |
| = \int_L \int_A \left[-z~\sigma_{xx}\frac{\partial (\delta\varphi)}{\partial x} + \sigma_{xz}~\kappa\left(-\delta\varphi + \frac{\partial (\delta w)}{\partial x}\right)\right]~\mathrm{d}A~\mathrm{d}L
| |
| </math>
| |
| Define
| |
| :<math>
| |
| M_{xx} := \int_A z~\sigma_{xx}~\mathrm{d}A ~;~~ Q_x := \kappa~\int_A \sigma_{xz}~\mathrm{d}A
| |
| </math>
| |
| Then
| |
| :<math>
| |
| \delta U = \int_L \left[-M_{xx}\frac{\partial (\delta\varphi)}{\partial x} + Q_{x}\left(-\delta\varphi + \frac{\partial (\delta w)}{\partial x}\right)\right]~\mathrm{d}L
| |
| </math>
| |
| Integration by parts, and noting that because of the boundary conditions the variations are zero at the ends of the beam, leads to
| |
| :<math>
| |
| \delta U = \int_L \left[\left(\frac{\partial M_{xx}}{\partial x} - Q_x\right)~\delta\varphi - \frac{\partial Q_{x}}{\partial x}~\delta w\right]~\mathrm{d}L
| |
| </math>
| |
| The variation in the external work done on the beam by a transverse load <math>q(x,t)</math> per unit length is
| |
| :<math>
| |
| \delta W = \int_L q~\delta w~\mathrm{d}L
| |
| </math>
| |
| Then, for a quasistatic beam, the principle of virtual work gives
| |
| :<math>
| |
| \delta U = \delta W \implies
| |
| \int_L \left[\left(\frac{\partial M_{xx}}{\partial x} - Q_x\right)~\delta\varphi - \left(\frac{\partial Q_{x}}{\partial x} + q\right)~\delta w\right]~\mathrm{d}L = 0
| |
| </math>
| |
| The governing equations for the beam are, from the fundamental theorem of variational calculus,
| |
| :<math>
| |
| \frac{\partial M_{xx}}{\partial x} - Q_x = 0 ~;~~ \frac{\partial Q_{x}}{\partial x} + q = 0
| |
| </math>
| |
| For a linear elastic beam
| |
| :<math>
| |
| \begin{align}
| |
| M_{xx} & = \int_A z~\sigma_{xx}~\mathrm{d}A = \int_A z~E~\varepsilon_{xx}~\mathrm{d}A =
| |
| -\int_A z^2~E~\frac{\partial \varphi}{\partial x}~\mathrm{d}A = -EI~\frac{\partial \varphi}{\partial x} \\
| |
| Q_{x} & = \int_A \sigma_{xz}~\mathrm{d}A = \int_A 2G~\varepsilon_{xz}~\mathrm{d}A =
| |
| \int_A \kappa~G~\left(-\varphi + \frac{\partial w}{\partial x}\right)~\mathrm{d}A = \kappa~AG~\left(-\varphi + \frac{\partial w}{\partial x}\right)
| |
| \end{align}
| |
| </math>
| |
| Therefore the governing equations for the beam may be expressed as
| |
| :<math>
| |
| \begin{align}
| |
| \frac{\partial }{\partial x}\left(EI\frac{\partial \varphi}{\partial x}\right) + \kappa AG~\left(\frac{\partial w}{\partial x}-\varphi\right) & = 0 \\
| |
| \frac{\partial }{\partial x}\left[\kappa AG\left(\frac{\partial w}{\partial x} - \varphi\right)\right] + q & = 0
| |
| \end{align}
| |
| </math>
| |
| Combining the two equations together gives
| |
| :<math>
| |
| \begin{align}
| |
| & \frac{\partial^2 }{\partial x^2}\left(EI\frac{\partial \varphi}{\partial x}\right) = q \\
| |
| & \frac{\partial w}{\partial x} = \varphi - \cfrac{1}{\kappa AG}~\frac{\partial }{\partial x}\left(EI\frac{\partial \varphi}{\partial x}\right)
| |
| \end{align}
| |
| </math>
| |
| |}
| |
| | |
| === Boundary conditions ===
| |
| The two equations that describe the deformation of a Timoshenko beam have to be augmented with [[boundary condition]]s if they are to be solved. Four boundary conditions are needed for the problem to be [[well-posed problem|well-posed]]. Typical boundary conditions are:
| |
| * '''Simply supported beams''': The displacement <math>w</math> is zero at the locations of the two supports. The bending moment <math>M_{xx}</math> applied to the beam also has to be specified. The rotation <math>\varphi</math> and the transverse shear force <math>Q_x</math> are not specified.
| |
| * '''Clamped beams''': The displacement <math>w</math> and the rotation <math>\varphi</math> are specified to be zero at the clamped end. If one end is free, shear force <math>Q_x</math> and bending moment <math>M_{xx}</math> have to be specified at that end.
| |
| | |
| === Example: Cantilever beam ===
| |
| [[File:TimoCantBeamPointLoad.svg|thumb|350px|A cantilever Timoshenko beam under a point load at the free end.]]
| |
| For a [[cantilever beam]], one boundary is clamped while the other is free. Let us use a [[Orientation (vector space)|right handed coordinate system]] where the <math>x</math> direction is positive towards right and the <math>z</math> direction is positive upward. Following normal convention, we assume that positive forces act in the positive directions of the <math>x</math> and <math>z</math> axes and positive moments act in the clockwise direction. We also assume that the sign convention of the [[stress resultants]] (<math>M_{xx}</math> and <math>Q_x</math>) is such that positive bending moments compress the material at the bottom of the beam (lower <math>z</math> coordinates) and positive shear forces rotate the beam in a counterclockwise direction.
| |
| | |
| Let us assume that the clamped end is at <math>x=L</math> and the free end is at <math>x=0</math>. If a point load <math>P</math> is applied to the free end in the positive <math>z</math> direction, a [[free body diagram]] of the beam gives us
| |
| :<math>
| |
| -Px - M_{xx} = 0 \implies M_{xx} = -Px
| |
| </math>
| |
| and
| |
| :<math> P + Q_x = 0 \implies Q_x = -P\,.
| |
| </math>
| |
| Therefore, from the expressions for the bending moment and shear force, we have
| |
| :<math>
| |
| Px = EI\,\frac{d\varphi}{dx} \qquad \text{and} \qquad -P = \kappa AG\left(-\varphi + \frac{dw}{dx}\right) \,.
| |
| </math>
| |
| Integration of the first equation, and application of the boundary condition <math>\varphi = 0</math> at <math>x = L</math>, leads to
| |
| :<math>
| |
| \varphi(x) = -\frac{P}{2EI}\,(L^2-x^2) \,.
| |
| </math>
| |
| The second equation can then be written as
| |
| :<math>
| |
| \frac{dw}{dx} = -\frac{P}{\kappa AG} - \frac{P}{2EI}\,(L^2-x^2)\,.
| |
| </math>
| |
| Integration and application of the boundary condition <math>w = 0</math> at <math>x = L</math> gives
| |
| :<math>
| |
| w(x) = \frac{P(L-x)}{\kappa AG} - \frac{Px}{2EI}\,\left(L^2-\frac{x^2}{3}\right) + \frac{PL^3}{3EI} \,.
| |
| </math>
| |
| The axial stress is given by
| |
| :<math>
| |
| \sigma_{xx}(x,z) = E\,\varepsilon_{xx} = -E\,z\,\frac{d\varphi}{dx} = -\frac{Pxz}{I} = \frac{M_{xx}z}{I} \,.
| |
| </math>
| |
| | |
| == Dynamic Timoshenko beam ==
| |
| In Timoshenko beam theory without axial effects, the displacements of the beam are assumed to be given by
| |
| :<math>
| |
| u_x(x,y,z,t) = -z~\varphi(x,t) ~;~~ u_y(x,y,z,t) = 0 ~;~~ u_z(x,y,z,t) = w(x,t)
| |
| </math>
| |
| where <math>(x,y,z)</math> are the coordinates of a point in the beam, <math>u_x, u_y, u_z</math> are the components of the displacement vector in the three coordinate directions, <math>\varphi</math> is the angle of rotation of the normal to the mid-surface of the beam, and <math>w</math> is the displacement of the mid-surface in the <math>z</math>-direction.
| |
| | |
| Starting from the above assumption, the Timoshenko beam theory, allowing for vibrations, may be described with the coupled linear [[partial differential equations]]:<ref>[http://ccrma.stanford.edu/~bilbao/master/node163.html Timoshenko's Beam Equations<!-- Bot generated title -->]</ref>
| |
| | |
| :<math>
| |
| \rho A\frac{\partial^{2}w}{\partial t^{2}} - q(x,t) = \frac{\partial}{\partial x}\left[ \kappa AG \left(\frac{\partial w}{\partial x}-\varphi\right)\right]
| |
| </math>
| |
| | |
| :<math>
| |
| \rho I\frac{\partial^{2}\varphi}{\partial t^{2}} = \frac{\partial}{\partial x}\left(EI\frac{\partial \varphi}{\partial x}\right)+\kappa AG\left(\frac{\partial w}{\partial x}-\varphi\right)
| |
| </math>
| |
| | |
| where the dependent variables are <math>w(x,t)</math>, the translational displacement of the beam, and <math>\varphi(x,t)</math>, the angular displacement. Note that unlike the [[Euler-Bernoulli beam equation|Euler-Bernoulli]] theory, the angular deflection is another variable and not approximated by the slope of the deflection. Also,
| |
| | |
| * <math>\rho</math> is the [[density]] of the beam material (but not the [[linear density]]).
| |
| * <math>A</math> is the cross section area.
| |
| * <math>E</math> is the [[elastic modulus]].
| |
| * <math>G</math> is the [[shear modulus]].
| |
| * <math>I</math> is the [[second moment of area]].
| |
| * <math>\kappa</math>, called the Timoshenko shear coefficient, depends on the geometry. Normally, <math>\kappa = 5/6</math> for a rectangular section.
| |
| * <math>q(x,t)</math> is a distributed load (force per length).
| |
| * <math>m := \rho A</math>
| |
| * <math>J := \rho I</math>
| |
| | |
| These parameters are not necessarily constants.
| |
| | |
| For a linear elastic, isotropic, homogeneous beam of constant cross-section these two equations can be combined to give<ref name=Thomson>Thomson, W. T., 1981, '''Theory of Vibration with Applications'''</ref><ref name=Rosinger>Rosinger, H. E. and Ritchie, I. G., 1977, ''On Timoshenko's correction for shear in vibrating isotropic beams,'' J. Phys. D: Appl. Phys., vol. 10, pp. 1461-1466.</ref> | |
| :<math>
| |
| EI~\cfrac{\partial^4 w}{\partial x^4} + m~\cfrac{\partial^2 w}{\partial t^2} - \left(J + \cfrac{E I m}{k A G}\right)\cfrac{\partial^4 w}{\partial x^2~\partial t^2} + \cfrac{m J}{k A G}~\cfrac{\partial^4 w}{\partial t^4} = q(x,t) + \cfrac{J}{k A G}~\cfrac{\partial^2 q}{\partial t^2} - \cfrac{EI}{k A G}~\cfrac{\partial^2 q}{\partial x^2}
| |
| </math>
| |
| :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
| |
| !Derivation of combined Timoshenko beam equation
| |
| |-
| |
| |The equations governing the bending of a homogeneous Timoshenko beam of constant cross-section are
| |
| :<math> | |
| \begin{align}
| |
| (1) & & \quad m~\frac{\partial^2 w}{\partial t^2} & = \kappa AG~\left(\frac{\partial^2 w}{\partial x^2} - \frac{\partial \varphi}{\partial x}\right) + q(x,t) ~;~~ m := \rho A \\
| |
| (2) & & \quad J~\frac{\partial^2 \varphi}{\partial t^2} & = EI~\frac{\partial^2 \varphi}{\partial x^2} + \kappa AG~\left(\frac{\partial w}{\partial x} - \varphi\right) ~;~~ J := \rho I
| |
| \end{align}
| |
| </math>
| |
| From equation (1), assuming appropriate smoothness, we have
| |
| :<math>
| |
| \begin{align}
| |
| (3) & & \quad \frac{\partial \varphi}{\partial x} & = -\cfrac{m}{\kappa AG}~\frac{\partial^2 w}{\partial t^2} + \frac{\partial^2 w}{\partial x^2} + \cfrac{q}{\kappa AG} \\
| |
| (4) & & \quad \frac{\partial^2 q}{\partial t^2} & = m~\cfrac{\partial^4 w}{\partial t^4} - \kappa AG~\left(\cfrac{\partial^4 w}{\partial x^2\partial t^2} - \cfrac{\partial^3\varphi}{\partial x\partial t^2}\right)
| |
| \end{align}
| |
| </math>
| |
| From (3), assuming appropriate smoothness,
| |
| :<math>
| |
| (5) \qquad \cfrac{\partial^3\varphi}{\partial x^3} = -\cfrac{m}{\kappa AG}~\cfrac{\partial^4 w}{\partial x^2\partial t^2} + \cfrac{\partial^4 w}{\partial x^4} + \cfrac{1}{\kappa AG}~\frac{\partial^2 q}{\partial x^2}
| |
| </math>
| |
| Differentiating equation (2) gives
| |
| :<math>
| |
| (6) \qquad \cfrac{\partial^3\varphi}{\partial x \partial t^2} = \cfrac{EI}{J}~\cfrac{\partial^3 \varphi}{\partial x^3} + \cfrac{\kappa AG}{J}~\left(\frac{\partial^2 w}{\partial x^2} - \frac{\partial \varphi}{\partial x}\right)
| |
| </math>
| |
| From equations (4) and (6)
| |
| :<math>
| |
| (7) \qquad
| |
| \cfrac{1}{\kappa AG}~\frac{\partial^2 q}{\partial t^2} -\cfrac{m}{\kappa AG}~\cfrac{\partial^4 w}{\partial t^4} + \cfrac{\partial^4 w}{\partial x^2\partial t^2}
| |
| = \cfrac{EI}{J}~\cfrac{\partial^3 \varphi}{\partial x^3} + \cfrac{\kappa AG}{J}~\left(\frac{\partial^2 w}{\partial x^2} - \frac{\partial \varphi}{\partial x}\right)
| |
| </math>
| |
| From equations (3) and (7)
| |
| :<math>
| |
| (8) \qquad
| |
| \cfrac{1}{\kappa AG}~\frac{\partial^2 q}{\partial t^2} -\cfrac{m}{\kappa AG}~\cfrac{\partial^4 w}{\partial t^4} + \cfrac{\partial^4 w}{\partial x^2\partial t^2}
| |
| = \cfrac{EI}{J}~\cfrac{\partial^3 \varphi}{\partial x^3} + \cfrac{m}{J}~\frac{\partial^2 w}{\partial t^2} - \cfrac{q}{J}
| |
| </math>
| |
| Plugging equation (5) into (8) gives
| |
| :<math>
| |
| (9) \qquad
| |
| \cfrac{J}{\kappa AG}~\frac{\partial^2 q}{\partial t^2} -\cfrac{mJ}{\kappa AG}~\cfrac{\partial^4 w}{\partial t^4} + J~\cfrac{\partial^4 w}{\partial x^2\partial t^2}
| |
| = -\cfrac{mEI}{\kappa AG}~\cfrac{\partial^4 w}{\partial x^2\partial t^2} + EI~\cfrac{\partial^4 w}{\partial x^4} + \cfrac{EI}{\kappa AG}~\frac{\partial^2 q}{\partial x^2}
| |
| + m~\frac{\partial^2 w}{\partial t^2} - q
| |
| </math>
| |
| Rearrange to get
| |
| :<math>
| |
| EI~\cfrac{\partial^4 w}{\partial x^4} + m~\frac{\partial^2 w}{\partial t^2} - \left(J+\cfrac{mEI}{\kappa AG}\right)~\cfrac{\partial^4 w}{\partial x^2 \partial t^2} + \cfrac{mJ}{\kappa AG}~\cfrac{\partial^4 w}{\partial t^4} = q + \cfrac{J}{\kappa AG}~\frac{\partial^2 q}{\partial t^2} - \cfrac{EI}{\kappa A G}~\frac{\partial^2 q}{\partial x^2}\quad\square
| |
| </math>
| |
| |}
| |
| | |
| === Axial effects ===
| |
| If the displacements of the beam are given by
| |
| :<math>
| |
| u_x(x,y,z,t) = u_0(x,t)-z~\varphi(x,t) ~;~~ u_y(x,y,z,t) = 0 ~;~~ u_z(x,y,z) = w(x,t)
| |
| </math>
| |
| where <math>u_0</math> is an additional displacement in the <math>x</math>-direction, then the governing equations of a Timoshenko beam take the form
| |
| :<math>
| |
| \begin{align}
| |
| m \frac{\partial^{2}w}{\partial t^{2}} & = \frac{\partial}{\partial x}\left[ \kappa AG \left(\frac{\partial w}{\partial x}-\varphi\right)\right] + q(x,t) \\
| |
| J \frac{\partial^{2}\varphi}{\partial t^{2}} & = N(x,t)~\frac{\partial w}{\partial x} + \frac{\partial}{\partial x}\left(EI\frac{\partial \varphi}{\partial x}\right)+\kappa AG\left(\frac{\partial w}{\partial x}-\varphi\right)
| |
| \end{align}
| |
| </math>
| |
| where <math>J = \rho I</math> and <math>N(x,t)</math> is an externally applied axial force. Any external axial force is balanced by the stress resultant
| |
| :<math>
| |
| N_{xx}(x,t) = \int_{-h}^{h} \sigma_{xx}~dz
| |
| </math>
| |
| where <math>\sigma_{xx}</math> is the axial stress and the thickness of the beam has been assumed to be <math>2h</math>.
| |
| | |
| The combined beam equation with axial force effects included is
| |
| :<math>
| |
| EI~\cfrac{\partial^4 w}{\partial x^4} + N~\cfrac{\partial^2 w}{\partial x^2} + m~\frac{\partial^2 w}{\partial t^2} - \left(J+\cfrac{mEI}{\kappa AG}\right)~\cfrac{\partial^4 w}{\partial x^2 \partial t^2} + \cfrac{mJ}{\kappa AG}~\cfrac{\partial^4 w}{\partial t^4} = q + \cfrac{J}{\kappa AG}~\frac{\partial^2 q}{\partial t^2} - \cfrac{EI}{\kappa A G}~\frac{\partial^2 q}{\partial x^2}
| |
| </math>
| |
| | |
| === Damping ===
| |
| If, in addition to axial forces, we assume a damping force that is proportional to the velocity with the form
| |
| :<math>
| |
| \eta(x)~\cfrac{\partial w}{\partial t}
| |
| </math>
| |
| the coupled governing equations for a Timoshenko beam take the form
| |
| :<math>
| |
| m \frac{\partial^{2}w}{\partial t^{2}} + \eta(x)~\cfrac{\partial w}{\partial t} = \frac{\partial}{\partial x}\left[ \kappa AG \left(\frac{\partial w}{\partial x}-\varphi\right)\right] + q(x,t)
| |
| </math>
| |
| | |
| :<math>
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| J \frac{\partial^{2}\varphi}{\partial t^{2}} = N\frac{\partial w}{\partial x} + \frac{\partial}{\partial x}\left(EI\frac{\partial \varphi}{\partial x}\right)+\kappa AG\left(\frac{\partial w}{\partial x}-\varphi\right)
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| </math>
| |
| and the combined equation becomes
| |
| :<math>
| |
| \begin{align}
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| EI~\cfrac{\partial^4 w}{\partial x^4} & + N~\cfrac{\partial^2 w}{\partial x^2} + m~\frac{\partial^2 w}{\partial t^2} - \left(J+\cfrac{mEI}{\kappa AG}\right)~\cfrac{\partial^4 w}{\partial x^2 \partial t^2} + \cfrac{mJ}{\kappa AG}~\cfrac{\partial^4 w}{\partial t^4} + \cfrac{J \eta(x)}{\kappa AG}~\cfrac{\partial^3 w}{\partial t^3} \\
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| & -\cfrac{EI}{\kappa AG}~\cfrac{\partial^2}{\partial x^2}\left(\eta(x)\cfrac{\partial w}{\partial t}\right) + \eta(x)\cfrac{\partial w}{\partial t} = q + \cfrac{J}{\kappa AG}~\frac{\partial^2 q}{\partial t^2} - \cfrac{EI}{\kappa A G}~\frac{\partial^2 q}{\partial x^2}
| |
| \end{align}
| |
| </math>
| |
| A caveat to this Ansatz damping force (resembling viscosity) is that, whereas viscosity leads to a frequency-dependent and amplitude-independent damping rate of beam oscillations, the empirically measured damping rates are frequency-insensitive, but depend on the amplitude of beam deflection.
| |
| | |
| == Shear coefficient ==
| |
| Determining the shear coefficient is not straightforward (nor are the determined values widely accepted, i.e. there's more than one answer); generally it must satisfy:
| |
| | |
| :<math>\int_A \tau dA = \kappa A G \varphi\,</math>
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| | |
| The shear coefficient depends on the Poisson's ratio. The attempts to provide precise expressions were made by many scientists, including [[Stephen Timoshenko]], [[Raymond D. Mindlin]], G. R. Cowper, [[John W. Hutchinson]], etc. In engineering practice, the expressions by [[Stephen Timoshenko]]<ref>Stephen Timoshenko, James M. Gere. Mechanics of Materials. Van Nostrand Reinhold Co., 1972. Pages 207.</ref> are sufficient in most cases.
| |
| | |
| For solid rectangular cross-section,
| |
| :<math>
| |
| \kappa = \cfrac{10(1+\nu)}{12+11\nu}
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| </math>
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| | |
| For solid circular cross-section,
| |
| :<math>
| |
| \kappa = \cfrac{6(1+\nu)}{7+6\nu}
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| </math>
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| | |
| ==See also==
| |
| | |
| * [[Bending moment]]
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| * [[Bending]]
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| * [[Euler–Bernoulli beam theory]]
| |
| * [[Sandwich theory]]
| |
| * [[Plate theory]]
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| | |
| ==References==
| |
| {{reflist|colwidth=30em}}
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| | |
| *{{cite book| author=Stephen P. Timoshenko| title=Schwingungsprobleme der technik| publisher=Verlag von Julius Springer| year=1932 | id=}}
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| | |
| [[Category:Continuum mechanics]]
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| [[Category:Structural analysis]]
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