铁木辛柯梁理论

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，

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

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]

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

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

${\displaystyle EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}+N~{\cfrac {\partial ^{2}w}{\partial x^{2}}}+m~{\frac {\partial ^{2}w}{\partial t^{2}}}-\left(J+{\cfrac {mEI}{\kappa AG}}\right)~{\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}+{\cfrac {mJ}{\kappa AG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}=q+{\cfrac {J}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {EI}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}}$

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

${\displaystyle {\begin{aligned}EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}&+N~{\cfrac {\partial ^{2}w}{\partial x^{2}}}+m~{\frac {\partial ^{2}w}{\partial t^{2}}}-\left(J+{\cfrac {mEI}{\kappa AG}}\right)~{\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}+{\cfrac {mJ}{\kappa AG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}+{\cfrac {J\eta (x)}{\kappa AG}}~{\cfrac {\partial ^{3}w}{\partial t^{3}}}\\&-{\cfrac {EI}{\kappa AG}}~{\cfrac {\partial ^{2}}{\partial x^{2}}}\left(\eta (x){\cfrac {\partial w}{\partial t}}\right)+\eta (x){\cfrac {\partial w}{\partial t}}=q+{\cfrac {J}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {EI}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}\end{aligned}}}$