铁木辛柯梁理论

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。模型考虑了剪应力和转动惯性，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。 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。 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)} 是梁上一点的坐标， u x , u y , u z {\displaystyle u_{x},u_{y},u_{z}} 是位移矢量的三维坐标分量， φ {\displaystyle \varphi } 是对于梁的中性面的法向转角， 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} 是梁的长度。 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}}}} 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)} 是梁内一点的坐标， u x , u y , u z {\displaystyle u_{x},u_{y},u_{z}} 是位移矢量的三维坐标分量， φ {\displaystyle \varphi } 是对于梁的中性面的法向转角， w {\displaystyle w} 是中性面 z {\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： 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)} 和转角位移 φ ( x , t ) {\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， ρ {\displaystyle \rho } 是梁材料的密度（而非线密度）； A {\displaystyle A} 是截面面积； E {\displaystyle E} 是弹性模量； G {\displaystyle G} 是剪切模量； I {\displaystyle I} 是轴惯性矩； κ {\displaystyle \kappa } ，称作铁木辛柯剪切系数，由形状确定，通常矩形截面 κ = 5 / 6 {\displaystyle \kappa =5/6} ； q ( x , t ) {\displaystyle q(x,t)} 是载荷分布（单位长度上的力）； 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 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}} 是 x {\displaystyle x} 方向的附加位移，则铁木辛柯梁的控制方程成为 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)} 是外加轴向力。任意外部轴向力的平衡依靠应力 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}} 是轴向应力，梁的厚度设为 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}}}} 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}}} 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。 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 }}} 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.