**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。模型考虑了剪应力和转动惯性，使其适于描述短梁、层合梁以及波长接近厚度的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。

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
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### 准静态铁木辛柯梁

铁木辛柯梁的变形。{\displaystyle \theta \_{x}=\varphi (x)}不等于{\displaystyle dw/dx}。
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

{\displaystyle u\_{x}(x,y,z)=-z~\varphi (x)~;~~u\_{y}(x,y,z)=0~;~~u\_{z}(x,y)=w(x)}
式中(x,y,z)是梁上一点的坐标，{\displaystyle u\_{x},u\_{y},u\_{z}}是位移矢量的三维坐标分量，\varphi 是对于梁的中性面的法向转角，w是中性面的在z方向的位移。

控制方程是以下常微分方程的解耦系统：

{\displaystyle {\begin{aligned}&{\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{aligned}}}
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

{\displaystyle {\frac {EI}{\kappa L^{2}AG}}\ll 1}
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L是梁的长度。 

对于等截面均匀梁，合并以上两个方程，

{\displaystyle EI~{\cfrac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}=q(x)-{\cfrac {EI}{\kappa AG}}~{\cfrac {\mathrm {d} ^{2}q}{\mathrm {d} x^{2}}}}

### 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

{\displaystyle 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)}
式中(x,y,z)是梁内一点的坐标，{\displaystyle u\_{x},u\_{y},u\_{z}}是位移矢量的三维坐标分量，\varphi 是对于梁的中性面的法向转角，w是中性面z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：

{\displaystyle \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]}
{\displaystyle \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)}
其中因变量是梁的平移位移{\displaystyle w(x,t)}和转角位移{\displaystyle \varphi (x,t)}。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，

* \rho 是梁材料的密度（而非线密度）；
* A是截面面积；
* E是弹性模量；
* G是剪切模量；
* I是轴惯性矩；
* \kappa，称作铁木辛柯剪切系数，由形状确定，通常矩形截面{\displaystyle \kappa =5/6}；
* {\displaystyle q(x,t)}是载荷分布（单位长度上的力）；
* {\displaystyle m:=\rho A}
* {\displaystyle J:=\rho I}

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

{\displaystyle EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}+m~{\cfrac {\partial ^{2}w}{\partial t^{2}}}-\left(J+{\cfrac {EIm}{kAG}}\right){\cfrac {\partial ^{4}w}{\partial x^{2}~\partial t^{2}}}+{\cfrac {mJ}{kAG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}=q(x,t)+{\cfrac {J}{kAG}}~{\cfrac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {EI}{kAG}}~{\cfrac {\partial ^{2}q}{\partial x^{2}}}}

#### 轴向影响

如果梁的位移由下式给出

{\displaystyle 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)}
其中u\_{0}是x方向的附加位移，则铁木辛柯梁的控制方程成为

{\displaystyle {\begin{aligned}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{aligned}}}
其中{\displaystyle J=\rho I}，{\displaystyle N(x,t)}是外加轴向力。任意外部轴向力的平衡依靠应力

{\displaystyle N\_{xx}(x,t)=\int \_{-h}^{h}\sigma \_{xx}~dz}
式中{\displaystyle \sigma \_{xx}}是轴向应力，梁的厚度设为2h。

包含轴向力的梁方程合并为

{\displaystyle 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 AG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}}
#### 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

{\displaystyle \eta (x)~{\cfrac {\partial w}{\partial t}}}
铁木辛柯梁的耦合控制方程成为

{\displaystyle 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)}
{\displaystyle 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)}
合并方程为

{\displaystyle {\begin{aligned}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 AG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}\end{aligned}}}

切变系数
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确定切变系数不是直接的，一般它必须满足：

{\displaystyle \int \_{A}\tau dA=\kappa AG\varphi \,}
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。

对于固态矩形截面：

{\displaystyle \kappa ={\cfrac {10(1+\nu )}{12+11\nu }}}
对于固态圆形截面：

{\displaystyle \kappa ={\cfrac {6(1+\nu )}{7+6\nu }}}

参考文献
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1. ↑  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.
2. ↑  Timoshenko, S. P. 1922, *On the transverse vibrations of bars of uniform cross-section*, Philosophical Magazine, p. 125.
3. ↑ .  [2013-03-22]. （原始内容存档于2007-10-15）.
4. ↑  Thomson, W. T. 1981, **Theory of Vibration with Applications**
5. ↑  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.
6. ↑  Stephen Timoshenko, James M. Gere. Mechanics of Materials. Van Nostrand Reinhold Co. 1972. Pages 207.

* Stephen P. Timoshenko.. Verlag von Julius Springer. 1932.