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。
鐵木辛柯梁的變形。
θ
x
=
φ
(
x
)
{\displaystyle \theta _{x}=\varphi (x)}
不等於
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)}
是梁上一點的坐標,
u
x
,
u
y
,
u
z
{\displaystyle u_{x},u_{y},u_{z}}
是位移矢量的三維坐標分量,
φ
{\displaystyle \varphi }
是對於梁的中性面的法向轉角,
w
{\displaystyle w}
是中性面的在
z
{\displaystyle z}
方向的位移。
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}
是梁的長度。
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}}}}
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)}
是梁內一點的坐標,
u
x
,
u
y
,
u
z
{\displaystyle u_{x},u_{y},u_{z}}
是位移矢量的三維坐標分量,
φ
{\displaystyle \varphi }
是對於梁的中性面的法向轉角,
w
{\displaystyle w}
是中性面
z
{\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]
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)}
和轉角位移
φ
(
x
,
t
)
{\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,
ρ
{\displaystyle \rho }
是梁材料的密度 (而非線密度 );
A
{\displaystyle A}
是截面面積;
E
{\displaystyle E}
是彈性模量 ;
G
{\displaystyle G}
是剪切模量 ;
I
{\displaystyle I}
是軸慣性矩 ;
κ
{\displaystyle \kappa }
,稱作鐵木辛柯剪切係數,由形狀確定,通常矩形截面
κ
=
5
/
6
{\displaystyle \kappa =5/6}
;
q
(
x
,
t
)
{\displaystyle q(x,t)}
是載荷分布(單位長度上的力);
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}}}}
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}}
是
x
{\displaystyle x}
方向的附加位移,則鐵木辛柯梁的控制方程成為
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)}
是外加軸向力。任意外部軸向力的平衡依靠應力
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}}
是軸向應力,梁的厚度設為
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}}}}
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}}}
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 }}}
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.
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.