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] 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。
θ
x
=
φ
(
x
)
{\displaystyle \theta _{x}=\varphi (x)}
d
w
/
d
x
{\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
u
x
(
x
,
y
,
z
)
=
−
z
φ
(
x
)
;
u
y
(
x
,
y
,
z
)
=
0
;
u
z
(
x
,
y
)
=
w
(
x
)
{\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
)
{\displaystyle (x,y,z)}
u
x
,
u
y
,
u
z
{\displaystyle u_{x},u_{y},u_{z}}
φ
{\displaystyle \varphi }
w
{\displaystyle w}
z
{\displaystyle z}
控制方程是以下常微分方程 的解耦系統：
d
2
d
x
2
(
E
I
d
φ
d
x
)
=
q
(
x
,
t
)
d
w
d
x
=
φ
−
1
κ
A
G
d
d
x
(
E
I
d
φ
d
x
)
.
{\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
E
I
κ
L
2
A
G
≪
1
{\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
L
{\displaystyle L}
對於等截面均勻梁，合併以上兩個方程，
E
I
d
4
w
d
x
4
=
q
(
x
)
−
E
I
κ
A
G
d
2
q
d
x
2
{\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
u
x
(
x
,
y
,
z
,
t
)
=
−
z
φ
(
x
,
t
)
;
u
y
(
x
,
y
,
z
,
t
)
=
0
;
u
z
(
x
,
y
,
z
,
t
)
=
w
(
x
,
t
)
{\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
)
{\displaystyle (x,y,z)}
u
x
,
u
y
,
u
z
{\displaystyle u_{x},u_{y},u_{z}}
φ
{\displaystyle \varphi }
w
{\displaystyle w}
z
{\displaystyle z}
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]
ρ
A
∂
2
w
∂
t
2
−
q
(
x
,
t
)
=
∂
∂
x
[
κ
A
G
(
∂
w
∂
x
−
φ
)
]
{\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]}
ρ
I
∂
2
φ
∂
t
2
=
∂
∂
x
(
E
I
∂
φ
∂
x
)
+
κ
A
G
(
∂
w
∂
x
−
φ
)
{\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
w
(
x
,
t
)
{\displaystyle w(x,t)}
φ
(
x
,
t
)
{\displaystyle \varphi (x,t)}
ρ
{\displaystyle \rho }
密度 （而非線密度 ）；
A
{\displaystyle A}
E
{\displaystyle E}
彈性模量 ；
G
{\displaystyle G}
剪切模量 ；
I
{\displaystyle I}
軸慣性矩 ；
κ
{\displaystyle \kappa }
κ
=
5
/
6
{\displaystyle \kappa =5/6}
q
(
x
,
t
)
{\displaystyle q(x,t)}
m
:=
ρ
A
{\displaystyle m:=\rho A}
J
:=
ρ
I
{\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]
E
I
∂
4
w
∂
x
4
+
m
∂
2
w
∂
t
2
−
(
J
+
E
I
m
k
A
G
)
∂
4
w
∂
x
2
∂
t
2
+
m
J
k
A
G
∂
4
w
∂
t
4
=
q
(
x
,
t
)
+
J
k
A
G
∂
2
q
∂
t
2
−
E
I
k
A
G
∂
2
q
∂
x
2
{\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
u
x
(
x
,
y
,
z
,
t
)
=
u
0
(
x
,
t
)
−
z
φ
(
x
,
t
)
;
u
y
(
x
,
y
,
z
,
t
)
=
0
;
u
z
(
x
,
y
,
z
)
=
w
(
x
,
t
)
{\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
{\displaystyle u_{0}}
x
{\displaystyle x}
m
∂
2
w
∂
t
2
=
∂
∂
x
[
κ
A
G
(
∂
w
∂
x
−
φ
)
]
+
q
(
x
,
t
)
J
∂
2
φ
∂
t
2
=
N
(
x
,
t
)
∂
w
∂
x
+
∂
∂
x
(
E
I
∂
φ
∂
x
)
+
κ
A
G
(
∂
w
∂
x
−
φ
)
{\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
J
=
ρ
I
{\displaystyle J=\rho I}
N
(
x
,
t
)
{\displaystyle N(x,t)}
N
x
x
(
x
,
t
)
=
∫
−
h
h
σ
x
x
d
z
{\displaystyle N_{xx}(x,t)=\int _{-h}^{h}\sigma _{xx}~dz}
式中
σ
x
x
{\displaystyle \sigma _{xx}}
2
h
{\displaystyle 2h}
包含軸向力的梁方程合併為
E
I
∂
4
w
∂
x
4
+
N
∂
2
w
∂
x
2
+
m
∂
2
w
∂
t
2
−
(
J
+
m
E
I
κ
A
G
)
∂
4
w
∂
x
2
∂
t
2
+
m
J
κ
A
G
∂
4
w
∂
t
4
=
q
+
J
κ
A
G
∂
2
q
∂
t
2
−
E
I
κ
A
G
∂
2
q
∂
x
2
{\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
η
(
x
)
∂
w
∂
t
{\displaystyle \eta (x)~{\cfrac {\partial w}{\partial t}}}
鐵木辛柯梁的耦合控制方程成為
m
∂
2
w
∂
t
2
+
η
(
x
)
∂
w
∂
t
=
∂
∂
x
[
κ
A
G
(
∂
w
∂
x
−
φ
)
]
+
q
(
x
,
t
)
{\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)}
J
∂
2
φ
∂
t
2
=
N
∂
w
∂
x
+
∂
∂
x
(
E
I
∂
φ
∂
x
)
+
κ
A
G
(
∂
w
∂
x
−
φ
)
{\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
E
I
∂
4
w
∂
x
4
+
N
∂
2
w
∂
x
2
+
m
∂
2
w
∂
t
2
−
(
J
+
m
E
I
κ
A
G
)
∂
4
w
∂
x
2
∂
t
2
+
m
J
κ
A
G
∂
4
w
∂
t
4
+
J
η
(
x
)
κ
A
G
∂
3
w
∂
t
3
−
E
I
κ
A
G
∂
2
∂
x
2
(
η
(
x
)
∂
w
∂
t
)
+
η
(
x
)
∂
w
∂
t
=
q
+
J
κ
A
G
∂
2
q
∂
t
2
−
E
I
κ
A
G
∂
2
q
∂
x
2
{\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：
∫
A
τ
d
A
=
κ
A
G
φ
{\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：
κ
=
10
(
1
+
ν
)
12
+
11
ν
{\displaystyle \kappa ={\cfrac {10(1+\nu )}{12+11\nu }}}
對於固態圓形截面：
κ
=
6
(
1
+
ν
)
7
+
6
ν
{\displaystyle \kappa ={\cfrac {6(1+\nu )}{7+6\nu }}}
參考文獻
^ 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.
^ Timoshenko, S. P., 1922, On the transverse vibrations of bars of uniform cross-section , Philosophical Magazine, p. 125.
^ Timoshenko's Beam Equations . [2013-03-22 ] . （原始內容存檔 於2007-10-15）. ^ Thomson, W. T., 1981, Theory of Vibration with Applications
^ 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.
^ 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. 
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ae3e2d14c1b5e59f0e4cd53c7a506b7d1111f8)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c35d8e346f27ef78d0a9c29ac3449ed4fb94b21)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f72225eea89af02d344a6877126ab80e276b33c9)
