铁木辛柯梁理论

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。^[1][2]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

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${\displaystyle (x,y,z)}$ 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，${\displaystyle \varphi }$ 是对于梁的中性面的法向转角，${\displaystyle w}$ 是中性面的在${\displaystyle z}$ 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${\displaystyle L}$ 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${\displaystyle (x,y,z)}$ 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，${\displaystyle \varphi }$ 是对于梁的中性面的法向转角，${\displaystyle w}$ 是中性面${\displaystyle z}$ 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：^[3]

${\displaystyle \rho A{\frac {\partial ^{2}w}{\partial t^{2}}}-q(x,t)={\frac {\partial }{\partial x}}\left[\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)\right]}$ 

${\displaystyle \rho I{\frac {\partial ^{2}\varphi }{\partial t^{2}}}={\frac {\partial }{\partial x}}\left(EI{\frac {\partial \varphi }{\partial x}}\right)+\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)}$ 

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，

• ${\displaystyle \rho }$ 是梁材料的密度（而非线密度）；
• ${\displaystyle A}$ 是截面面积；
• ${\displaystyle E}$ 是弹性模量；
• ${\displaystyle G}$ 是剪切模量；
• ${\displaystyle I}$ 是轴惯性矩；
• ${\displaystyle \kappa }$ ，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^[4][5]

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

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${\displaystyle u_{0}}$ 是${\displaystyle x}$ 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${\displaystyle 2h}$ 。

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

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。^[6]

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

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. Timoshenko's Beam Equations. [2013-03-22]. （原始内容于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. Schwingungsprobleme der technik. Verlag von Julius Springer. 1932.