鐵木辛柯梁理論

鐵木辛柯梁是20世紀早期由美籍俄裔科學家與工程師斯蒂芬·鐵木辛柯提出並發展的力學模型。^[1][2]模型考慮了剪應力和轉動慣性，使其適於描述短梁、層合梁以及波長接近厚度的高頻激勵時梁的表現。結果方程有4階，但不同於一般的梁理論，如歐拉-伯努利梁理論，還有一個2階空間導數呈現。實際上，考慮了附加的變形機理有效地降低了梁的剛度，結果在一穩態載荷下撓度更大，在一組給定的邊界條件時預估固有頻率更低。後者在高頻即波長更短時效果更明顯，反向剪力距離縮短時也有同樣效果。

鐵木辛柯梁（藍）的變形與歐拉-伯努利梁（紅）的對比

如果梁材料的剪切模量接近無窮，即此時梁為剪切剛體，並且忽略轉動慣性，則鐵木辛柯梁理論趨同於一般梁理論。

控制方程

准靜態鐵木辛柯梁

鐵木辛柯梁的變形。${\displaystyle \theta _{x}=\varphi (x)}$不等於${\displaystyle dw/dx}$。

在靜力學中鐵木辛柯梁理論沒有軸向影響，假定梁的位移服從於

${\displaystyle u_{x}(x,y,z)=-z~\varphi (x)~;~~u_{y}(x,y,z)=0~;~~u_{z}(x,y)=w(x)}$

式中${\displaystyle (x,y,z)}$是梁上一點的坐標，${\displaystyle u_{x},u_{y},u_{z}}$是位移矢量的三維坐標分量，${\displaystyle \varphi }$是對於梁的中性面的法向轉角，${\displaystyle w}$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統：

${\displaystyle {\begin{aligned}&{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\left(EI{\frac {\mathrm {d} \varphi }{\mathrm {d} x}}\right)=q(x,t)\\&{\frac {\mathrm {d} w}{\mathrm {d} x}}=\varphi -{\frac {1}{\kappa AG}}{\frac {\mathrm {d} }{\mathrm {d} x}}\left(EI{\frac {\mathrm {d} \varphi }{\mathrm {d} x}}\right).\end{aligned}}}$

靜態條件下的鐵木辛柯梁理論，若在以下條件成立時，等同於歐拉-伯努利梁理論

${\displaystyle {\frac {EI}{\kappa L^{2}AG}}\ll 1}$

此時，可忽略上面控制方程的最後一項，得到有效的近似，式中${\displaystyle L}$是梁的長度。

對於等截面均勻梁，合併以上兩個方程，

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

動態鐵木辛柯梁

在鐵木辛柯梁理論中若不考慮軸向影響，則給出梁的位移

${\displaystyle u_{x}(x,y,z,t)=-z~\varphi (x,t)~;~~u_{y}(x,y,z,t)=0~;~~u_{z}(x,y,z,t)=w(x,t)}$

式中${\displaystyle (x,y,z)}$是梁內一點的坐標，${\displaystyle u_{x},u_{y},u_{z}}$是位移矢量的三維坐標分量，${\displaystyle \varphi }$是對於梁的中性面的法向轉角，${\displaystyle w}$是中性面${\displaystyle z}$方向的位移.

從以上假設，鐵木辛柯梁，考慮到振動，要用線性耦合偏微分方程描述：^[3]

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

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

其中因變量是梁的平移位移${\displaystyle w(x,t)}$和轉角位移${\displaystyle \varphi (x,t)}$。注意不同於歐拉-伯努利梁理論，轉角位移是另一個變量而非撓度斜率的近似。此外，

• ${\displaystyle \rho }$是梁材料的密度（而非線密度）；
• ${\displaystyle A}$是截面面積；
• ${\displaystyle E}$是彈性模量；
• ${\displaystyle G}$是剪切模量；
• ${\displaystyle I}$是軸慣性矩；
• ${\displaystyle \kappa }$，稱作鐵木辛柯剪切係數，由形狀確定，通常矩形截面${\displaystyle \kappa =5/6}$；
• ${\displaystyle q(x,t)}$是載荷分布（單位長度上的力）；
• ${\displaystyle m:=\rho A}$
• ${\displaystyle J:=\rho I}$

這些參數不一定是常數。

對於各向同性的線彈性均勻等截面梁，以上兩個方程可合併成^[4][5]

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

軸向影響

如果梁的位移由下式給出

${\displaystyle u_{x}(x,y,z,t)=u_{0}(x,t)-z~\varphi (x,t)~;~~u_{y}(x,y,z,t)=0~;~~u_{z}(x,y,z)=w(x,t)}$

其中${\displaystyle u_{0}}$是${\displaystyle x}$方向的附加位移，則鐵木辛柯梁的控制方程成為

${\displaystyle {\begin{aligned}m{\frac {\partial ^{2}w}{\partial t^{2}}}&={\frac {\partial }{\partial x}}\left[\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)\right]+q(x,t)\\J{\frac {\partial ^{2}\varphi }{\partial t^{2}}}&=N(x,t)~{\frac {\partial w}{\partial x}}+{\frac {\partial }{\partial x}}\left(EI{\frac {\partial \varphi }{\partial x}}\right)+\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)\end{aligned}}}$

其中${\displaystyle J=\rho I}$，${\displaystyle N(x,t)}$是外加軸向力。任意外部軸向力的平衡依靠應力

${\displaystyle N_{xx}(x,t)=\int _{-h}^{h}\sigma _{xx}~dz}$

式中${\displaystyle \sigma _{xx}}$是軸向應力，梁的厚度設為${\displaystyle 2h}$。

包含軸向力的梁方程合併為

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

阻尼

如果，除軸向力外，我們考慮與速度成正比的阻尼力，形如

${\displaystyle \eta (x)~{\cfrac {\partial w}{\partial t}}}$

鐵木辛柯梁的耦合控制方程成為

${\displaystyle m{\frac {\partial ^{2}w}{\partial t^{2}}}+\eta (x)~{\cfrac {\partial w}{\partial t}}={\frac {\partial }{\partial x}}\left[\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)\right]+q(x,t)}$

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

合併方程為

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

切變係數

確定切變係數不是直接的，一般它必須滿足：

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

切變係數由泊松比確定。更嚴格的表達方法由多位科學家完成，包括斯蒂芬·鐵木辛柯、雷蒙德·明德林（Raymond D. Mindlin）、考珀（G. R. Cowper）和約翰·哈欽森（John W. Hutchinson）等。工程實踐中，斯蒂芬·鐵木辛柯的表達一般狀況下足夠好。^[6]

對於固態矩形截面：

${\displaystyle \kappa ={\cfrac {10(1+\nu )}{12+11\nu }}}$

對於固態圓形截面：

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