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] 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。
θ
x
=
φ
(
x
)
{\displaystyle \theta _{x}=\varphi (x)}
d
w
/
d
x
{\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
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 的解耦系统：
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}}}
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
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后一项，得到有效的近似，式中
L
{\displaystyle L}
对于等截面均匀梁，合并以上两个方程，
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}}}}
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
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}
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]
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)}
其中因变量是梁的平移位移
w
(
x
,
t
)
{\displaystyle w(x,t)}
φ
(
x
,
t
)
{\displaystyle \varphi (x,t)}
ρ
{\displaystyle \rho }
密度 （而非线密度 ）；
A
{\displaystyle A}
E
{\displaystyle E}
弹性模量 ；
G
{\displaystyle G}
剪切模量 ；
I
{\displaystyle I}
轴惯性矩 ；
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}
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]
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}}}}
轴向影响 如果梁的位移由下式给出
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}
包含轴向力的梁方程合并为
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，除轴向力外，我们考虑与速度成正比的阻尼力，形如
η
(
x
)
∂
w
∂
t
{\displaystyle \eta (x)~{\cfrac {\partial w}{\partial t}}}
铁木辛柯梁的耦合控制方程成为
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)}
合并方程为
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}}}
切变系数 确定切变系数不是直接的，一般它必须满足：
∫
A
τ
d
A
=
κ
A
G
φ
{\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：
κ
=
10
(
1
+
ν
)
12
+
11
ν
{\displaystyle \kappa ={\cfrac {10(1+\nu )}{12+11\nu }}}
对于固态圆形截面：
κ
=
6
(
1
+
ν
)
7
+
6
ν
{\displaystyle \kappa ={\cfrac {6(1+\nu )}{7+6\nu }}}
参考文献
^ 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 ] . （原始内容存档 于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)
