查看“︁铁木辛柯梁理论”︁的源代码

<!--{{Continuum mechanics|cTopic=[[Solid mechanics]]}}-->
'''铁木辛柯梁'''是20世纪早期由美籍俄裔科学家与工程师[[斯蒂芬·铁木辛柯]]提出并发展的力学模型。<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>模型考虑了[[剪应力]]和[[转动惯量|转动惯性]]，使其适于描述短梁、层合梁以及[[波长]]接近厚度的[[頻率 (物理學)|高频]]激励时梁的表现。结果方程有4阶，但不同于一般的梁理论，如[[歐拉-伯努力棟樑方程|欧拉-伯努利梁理论]]，还有一个2阶空间导数呈现。实际上，考虑了附加的变形机理有效地降低了梁的[[刚度]]，结果在一稳态载荷下[[挠度]]更大，在一组给定的边界条件时预估[[固有频率]]更低。后者在高频即波长更短时效果更明显，反向剪力距离缩短时也有同样效果。
[[Image:TimoshenkoBeam.svg|thumb|400px|铁木辛柯梁（蓝）的变形与欧拉-伯努利梁（红）的对比]]

如果梁材料的[[剪切模量]]接近无穷，即此时梁为剪切[[刚体]]，并且忽略转动惯性，则铁木辛柯梁理论趋同于一般梁理论。

== 控制方程 ==

=== 准静态铁木辛柯梁 ===
[[Image:Plate theory.svg|thumb|200px|铁木辛柯梁的变形。<math>\theta_x = \varphi(x)</math>不等于<math>dw/dx</math>。]] 
在[[静力学]]中铁木辛柯梁理论没有轴向影响，假定梁的位移服从于
:<math>
  u_x(x,y,z) = -z~\varphi(x) ~;~~ u_y(x,y,z) = 0 ~;~~ u_z(x,y) = w(x)
</math>
式中<math>(x,y,z)</math>是梁上一点的坐标，<math>u_x, u_y, u_z</math>是位移矢量的三维坐标分量，<math>\varphi</math>是对于梁的中性面的法向转角，<math>w</math>是中性面的在<math>z</math>方向的位移。

控制方程是以下[[常微分方程]]的解耦系统：
:<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>

静态条件下的铁木辛柯梁理论，若在以下條件成立時，等同于欧拉-伯努利梁理论
:<math>
\frac{EI}{\kappa L^2 A G} \ll 1
</math>
此時，可忽略上面控制方程的最后一项，得到有效的近似，式中<math>L</math>是梁的长度。 

对于等截面均匀梁，合并以上两个方程，
:<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>
<!--
:{| 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>
|}
-->

=== 动态铁木辛柯梁 ===
在铁木辛柯梁理论中若不考虑轴向影响，则给出梁的位移
:<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>
式中<math>(x,y,z)</math>是梁内一点的坐标，<math>u_x, u_y, u_z</math>是位移矢量的三维坐标分量，<math>\varphi</math>是对于梁的中性面的法向转角，<math>w</math>是中性面<math>z</math>方向的位移.

从以上假设，铁木辛柯梁，考虑到振动，要用线性耦合[[偏微分方程]]描述：<ref>{{Cite web |url=http://ccrma.stanford.edu/~bilbao/master/node163.html |title=Timoshenko's Beam Equations<!-- Bot generated title --> |accessdate=2013-03-22 |archive-date=2007-10-15 |archive-url=https://web.archive.org/web/20071015100642/http://ccrma.stanford.edu/~bilbao/master/node163.html |dead-url=no }}</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>

其中因变量是梁的平移位移<math>w(x,t)</math>和转角位移<math>\varphi(x,t)</math>。注意不同于欧拉-伯努利梁理论，转角位移是另一个变量而非挠度斜率的近似。此外，

* <math>\rho</math>是梁材料的[[密度]]（而非[[线密度]]）；
* <math>A</math>是截面面积；
* <math>E</math>是[[弹性模量]]；
* <math>G</math>是[[剪切模量]]；
* <math>I</math>是[[截面二次轴矩|轴惯性矩]]；
* <math>\kappa</math>，称作铁木辛柯剪切系数，由形状确定，通常矩形截面<math>\kappa = 5/6</math>；
* <math>q(x,t)</math>是载荷分布（单位长度上的力）；
* <math>m := \rho A</math>
* <math>J := \rho I</math>

这些参数不一定是常数。

对于各向同性的线弹性均匀等截面梁，以上两个方程可合并成<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>
|}
-->
==== 轴向影响 ====
如果梁的位移由下式给出
:<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>
其中<math>u_0</math>是<math>x</math>方向的附加位移，则铁木辛柯梁的控制方程成为
:<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>
其中<math>J = \rho I</math>，<math>N(x,t)</math>是外加轴向力。任意外部轴向力的平衡依靠应力
:<math>
   N_{xx}(x,t) = \int_{-h}^{h} \sigma_{xx}~dz
 </math>
式中<math>\sigma_{xx}</math>是轴向应力，梁的厚度设为<math>2h</math>。

包含轴向力的梁方程合并为
:<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>

==== 阻尼 ====
如果，除轴向力外，我们考虑与速度成正比的阻尼力，形如
:<math>
   \eta(x)~\cfrac{\partial w}{\partial t}
 </math>
铁木辛柯梁的耦合控制方程成为
:<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>
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)
</math>
合并方程为
:<math>
  \begin{align}
   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} \\
  & -\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>

== 切变系数 ==
确定切变系数不是直接的，一般它必须满足：

:<math>\int_A \tau dA = \kappa A G \varphi\,</math>

切变系数由[[泊松比]]确定。更严格的表达方法由多位科学家完成，包括[[斯蒂芬·铁木辛柯]]、雷蒙德·明德林（Raymond D. Mindlin）、考珀（G. R. Cowper）和约翰·哈钦森（John W. Hutchinson）等。工程实践中，斯蒂芬·铁木辛柯的表达一般状况下足够好。<ref>Stephen Timoshenko, James M. Gere. Mechanics of Materials. Van Nostrand Reinhold Co., 1972. Pages 207.</ref>

对于固态矩形截面：
:<math>
\kappa = \cfrac{10(1+\nu)}{12+11\nu}
</math>

对于固态圆形截面：
:<math>
\kappa = \cfrac{6(1+\nu)}{7+6\nu}
</math>

==参考文献==
{{reflist|colwidth=30em}}

*{{cite book| author=Stephen P. Timoshenko| title=Schwingungsprobleme der technik| publisher=Verlag von Julius Springer| year=1932 | id=}}

[[Category:固体力学]]