﻿ 含固支边矩形叠层厚板的状态空间新解法
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 力学学报  2015, Vol. 47 Issue (5): 762-771  DOI: 10.6052/0459-1879-15-033 0

### 引用本文 [复制中英文]

[复制中文]
Hu Wenfeng, Liu Yihua. A NEW STATE SPACE SOLUTION FOR RECTANGULAR THICK LAMINATES WITH CLAMPED EDGES Hu Wenfeng Liu Yihua[J]. Chinese Journal of Ship Research, 2015, 47(5): 762-771. DOI: 10.6052/0459-1879-15-033.
[复制英文]

### 文章历史

2015-01-22收稿
2015-06-15录用
2015–06–19网络版发表

1 单层板的状态方程及其解

 $\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\sigma _x}\\ {\sigma _y}\\ {\sigma _z}\\ {\tau _{yz}}\\ {\tau _{zx}}\\ {\tau _{xy}} \end{array} \end{array}} \right\} = \left[{\begin{array}{*{20}{c}} {{C_{11}}}&{{C_{12}}}&{{C_{13}}}&0&0&0\\ {{C_{12}}}&{{C_{22}}}&{{C_{23}}}&0&0&0\\ {{C_{13}}}&{{C_{23}}}&{{C_{33}}}&0&0&0\\ 0&0&0&{{C_{44}}}&0&0\\ 0&0&0&0&{{C_{55}}}&0\\ 0&0&0&0&0&{{C_{66}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} \frac{{\partial u}}{{\partial x}}\\ \frac{{\partial v}}{{\partial y}}\\ \frac{{\partial w}}{{\partial z}}\\ \frac{{\partial w}}{{\partial y}} + \frac{{\partial v}}{{\partial z}}\\ \frac{{\partial u}}{{\partial z}} + \frac{{\partial w}}{{\partial x}}\\ \frac{{\partial v}}{{\partial x}} + \frac{{\partial u}}{{\partial y}} \end{array} \end{array}} \right\}$ (1)
 图 1 单层矩形板 Fig. 1 Rectangular single plate

 $\frac{{\partial r}}{{\partial z}} = Dr$ (2)

 $r = {\left[{u\;\;v\;\;{\sigma _z}\;\;{\tau _{zx}}\;\;{\tau _{yz}}{\rm{ }}w} \right]^{\rm{T}}}$
 $D = \left[{\begin{array}{*{20}{c}} 0&0&0&{{C_8}}&0&{ - \alpha }\\ 0&0&0&0&{{C_9}}&{ - \beta }\\ 0&0&0&{ - \alpha }&{ - \beta }&0\\ { - {C_2}{\alpha ^2} - {C_6}{\beta ^2}}&{ - ({C_3} + {C_6})\alpha \beta }&{{C_1}\alpha }&0&0&0\\ { - ({C_3} + {C_6})\alpha \beta }&{ - {C_6}{\alpha ^2} - {C_4}{\beta ^2}}&{{C_5}\beta }&0&0&0\\ {{C_1}\alpha }&{{C_5}\beta }&{{C_7}}&0&0&0 \end{array}} \right]$

 $\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\sigma _x}\\ {\sigma _y}\\ {\tau _{xy}} \end{array} \end{array}} \right\} = \left[{\begin{array}{*{20}{c}} {{C_2}\alpha }&{{C_3}\beta }&{ - {C_1}}\\ {{C_3}\alpha }&{{C_4}\beta }&{ - {C_5}}\\ {{C_6}\beta }&{{C_6}\alpha }&0 \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} u\\ v\\ {\sigma _z} \end{array} \end{array}} \right\}$ (3)

 $\left. {\begin{array}{*{20}{l}} \begin{array}{l} x = 0{\mkern 1mu} ,\;a{\mkern 1mu} ;\;\;u = v = w = 0\\ y = 0{\mkern 1mu} ,\;b{\mkern 1mu} ;\;\;u = w = 0{\mkern 1mu} ,{\sigma _y} = 0 \end{array} \end{array}} \right\}$ (4)

 $u = \bar u + {f_1}(x){u^{(0)}} + {f_2}(x){u^{(a)}}$ (5)

 ${f_1}(x) = 1 - \frac{x}{a}{\mkern 1mu} ,\;\;{f_2}(x) = \frac{x}{a}{\rm{ }}$ (6)

 $\frac{{\partial \overline r }}{{\partial z}} = D\overline r + B$ (7)

 $B = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} - \partial /\partial z\\ 0\\ 0\\ - {C_2}{\alpha ^2} - {C_6}{\beta ^2}\\ - ({C_3} + {C_6})\alpha \beta \\ {C_1}\alpha \end{array} \end{array}} \right\}\left[{{f_1}(x){u^{(0)}} + {f_2}(x){u^{(a)}}} \right]$

 $\left\{ {\begin{array}{*{20}{c}} {\bar uv{\sigma _z}{\tau _{zx}}{\tau _{yz}}w} \end{array}} \right\} = \sum\limits_m {\sum\limits_n {\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {{\bar U}_{mn}}(z)\cos \zeta x\sin \eta y\\ {V_{mn}}(z)\sin \zeta x\cos \eta y\\ {Z_{mn}}(z)\sin \zeta x\sin \eta y\\ {X_{mn}}(z)\cos \zeta x\sin \eta y\\ {Y_{mn}}(z)\sin \zeta x\cos \eta y\\ {W_{mn}}(z)\sin \zeta x\sin \eta y \end{array} \end{array}} \right\}} }$ (8)
 ${u^{(0)}} = \sum\limits_{n = 1}^\infty {U_n^{(0)}} (z)\sin \eta y{\mkern 1mu} ,\;{u^{(a)}} = \sum\limits_{n = 1}^\infty {U_n^{(a)}} (z)\sin \eta y$ (9)
 ${f_1}(x) = \frac{1}{2} + \sum\limits_{m = 1}^\infty {\xi \cos \zeta x} ,\;{f_2}(x) = \frac{1}{2} - \sum\limits_{m = 1}^\infty {\xi \cos \zeta x}$ (10)

 $\begin{array}{l} \left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{l}} {{\sigma _x}}\\ {{\sigma _y}}\\ {{\tau _{xy}}} \end{array}} \end{array}} \right\} = \sum\limits_n {\sum\limits_m {\left\{ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {{S_{xmn}}(z)\sin \zeta x\sin \eta y}\\ {{S_{ymn}}(z)\sin \zeta x\sin \eta y}\\ {{S_{xymn}}(z)\cos \zeta x\cos \eta y} \end{array}} \end{array}} \right\}} } + \\ \frac{1}{a}\sum\limits_n {\left\{ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} { - {C_2}\left[ {U_n^{(0)}(z) - U_n^{(a)}(z)} \right]\sin \eta y}\\ { - {C_3}\left[ {U_n^{(0)}(z) - U_n^{(a)}(z)} \right]\sin \eta y}\\ {{C_6}\eta \left[ {(a - x)U_n^{(0)}(z) + xU_n^{(a)}(z)} \right]\cos \eta y} \end{array}} \end{array}} \right\}} \end{array}$ (11)

 $\frac{d}{{dz}}{\overline R _{mn}}(z) = {D_{mn}}{\overline R _{mn}}(z) + {B_{mn}}(z)$ (12)

 $\begin{array}{l} {D_{mn}} = \left[{\begin{array}{*{20}{c}} 0&0&0&{{C_8}}&0&{ - \zeta }\\ 0&0&0&0&{{C_9}}&{ - \eta }\\ 0&0&0&\zeta &\eta &0\\ {{C_2}{\zeta ^2} + {C_6}{\eta ^2}}&{({C_3} + {C_6})\zeta \eta }&{{C_1}\zeta }&0&0&0\\ {({C_3} + {C_6})\zeta \eta }&{{C_6}{\zeta ^2} + {C_4}{\eta ^2}}&{{C_5}\eta }&0&0&0\\ { - {C_1}\zeta }&{ - {C_5}\eta }&{{C_7}}&0&0&0 \end{array}} \right]\\ {B_{mn}}(z) = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} B_{0n}^*\left[{U_n^{(0)}(z) + U_n^{(a)}(z)} \right]\;\;(m = 0)\\ B_{mn}^*\left[{U_n^{(0)}(z) - U_n^{(a)}(z)} \right]\;\;(m \ne 0) \end{array} \end{array}} \right. \end{array}$

$B_{mn}^*{\left[{ - \xi \frac{d}{{dz}}\;\;0\;\;0\;\;{C_6}{\eta ^2}\xi \;\;({C_3} + {C_6})\zeta \eta \xi \;\; - {C_1}\zeta \xi } \right]^{\rm{T}}}$

 $\left. {\begin{array}{*{20}{l}} \begin{array}{l} \sum\limits_{n = 1}^\infty {\left[{U_n^{(0)}(z) + \sum\limits_{m = 0}^\infty {{{\bar U}_{mn}}(z)} } \right]} \sin \eta y = 0\\ \sum\limits_{n = 1}^\infty {\left[{U_n^{(a)}(z) + \sum\limits_{m = 0}^\infty {{{( - 1)}^m}{{\bar U}_{mn}}(z)} } \right]} \sin \eta y = 0 \end{array} \end{array}} \right\}$ (13)

 $\left. {\begin{array}{*{20}{l}} \begin{array}{l} U_n^{(0)}(z) + \sum\limits_{m = 0}^\infty {{{\bar U}_{mn}}(z) = 0} \\ U_n^{(a)}(z) + \sum\limits_{m = 0}^\infty {{{( - 1)}^m}{{\bar U}_{mn}}(z) = 0} \end{array} \end{array}} \right\}$ (14)

 $\begin{array}{l} \frac{d}{{dz}}{\overline R _{mn}}(z) + B_{mn}^{(0)}\frac{d}{{dz}}U_n^{(0)}(z) + B_{mn}^{(a)}\frac{d}{{dz}}U_n^{(a)}(z) = \\ {D_{mn}}{\overline R _{mn}}(z) + \overline B _{mn}^{(0)}U_n^{(0)}(z) + \overline B _{mn}^{(a)}U_n^{(a)}(z) \end{array}$ (15)

 $B_{mn}^{(0)} = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {\left[{1/2\;\;0\;\;0\;\;0\;\;0\;\;0} \right]^{\rm{T}}}\;\;(m = 0)\\ {\left[{\xi \;\;0\;\;0\;\;0\;\;0\;\;0} \right]^{\rm{T}}}\;\;(m \ne 0) \end{array} \end{array}} \right.$
 $B_{mn}^{(a)} = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {\left[{1/2\;\;0\;\;0\;\;0\;\;0\;\;0} \right]^{\rm{T}}}\;\;(m = 0)\\ {\left[{ - \xi \;\;0\;\;0\;\;0\;\;0\;\;0} \right]^{\rm{T}}}\;\;(m \ne 0) \end{array} \end{array}} \right.$
 $\overline B _{mn}^{(0)} = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {\left[{0\;\;0\;\;0\;\;{C_6}{\eta ^2}/2\;\;0\;\;0} \right]^{\rm{T}}}\;(m = 0)\\ {\left[{0\;\;0\;\;0\;\;{C_6}{\eta ^2}\xi \;\;({C_3} + {C_6})\zeta \eta \xi \;\; - {C_1}\zeta \xi } \right]^{\rm{T}}}\\ (m \ne 0) \end{array} \end{array}} \right.$
 $\overline B _{mn}^{(a)} = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {\left[{0\;\;0\;\;0\;{C_6}{\eta ^2}/2\;\;0\;\;0} \right]^{\rm{T}}}\;\;(m = 0)\\ {\left[{0\;\;0\;\;0\;\; - {C_6}{\eta ^2}\xi \;\; - ({C_3} + {C_6})\zeta \eta \xi \;\;{C_1}\zeta \xi } \right]^{\rm{T}}}\\ (m \ne 0) \end{array} \end{array}} \right.$ (09)

 $\left. {\begin{array}{*{20}{l}} \begin{array}{l} \frac{d}{{dz}}U_n^{(0)}(z) + \sum\limits_{m = 0}^\infty {\frac{d}{{dz}}{{\bar U}_{mn}}(z) = 0} \\ \frac{d}{{dz}}U_n^{(a)}(z) + \sum\limits_{m = 0}^\infty {{{( - 1)}^m}\frac{d}{{dz}}{{\bar U}_{mn}}(z) = 0} \end{array} \end{array}} \right\}$ (16)

 ${\theta _n}\frac{d}{{dz}}{\Phi _n}(z) = {\overline \theta _n}{\Phi _n}(z)$ (17)

 ${\Phi _n} = {\left[{{{\overline R }_{0n}}\;\;{{\overline R }_{1n}}\;\; \cdots \;\;{{\overline R }_{mn}}\;U_n^{(0)}\;\;U_n^{(a)}} \right]^{\rm{T}}}$
 ${\theta _n} = \left[{\begin{array}{*{20}{c}} I&{\bf{0}}& \cdots &{\bf{0}}&{B_{0n}^{(0)}}&{B_{0n}^{(a)}}\\ {\bf{0}}&I& \cdots &{\bf{0}}&{B_{1n}^{(0)}}&{B_{1n}^{(a)}}\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ {\bf{0}}&{\bf{0}}& \cdots &I&{B_{mn}^{(0)}}&{B_{mn}^{(a)}}\\ \theta &\theta & \cdots &\theta &1&{\bf{0}}\\ {{\theta _0}}&{{\theta _1}}& \cdots &{{\theta _m}}&{\bf{0}}&1 \end{array}} \right]$
 ${\overline \theta _n} = \left[{\begin{array}{*{20}{c}} {{D_{0n}}}&{\bf{0}}& \cdots &{\bf{0}}&{\overline B _{0n}^{(0)}}&{\overline B _{0n}^{(a)}}\\ {\bf{0}}&{{D_{1n}}}& \cdots &{\bf{0}}&{\overline B _{1n}^{(0)}}&{\overline B _{1n}^{(a)}}\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ {\bf{0}}&{\bf{0}}& \cdots &{{D_{mn}}}&{\overline B _{mn}^{(0)}}&{\overline B _{mn}^{(a)}}\\ {\bf{0}}&{\bf{0}}& \cdots &{\bf{0}}&{\bf{0}}&{\bf{0}}\\ {\bf{0}}&{\bf{0}}& \cdots &{\bf{0}}&{\bf{0}}&{\bf{0}} \end{array}} \right]$

 $U_n^{(0)}(z) = - U_n^{(a)}(z)$ (18)

 ${\theta _n}\frac{d}{{dz}}{\Phi _n}(z) = {\overline \theta _n}{\Phi _n}(z)$ (19)

 ${\Phi _n} = {\left[{{{\overline R }_{1n}}\;\;{{\overline R }_{3n}}\;\; \cdots \;\;{{\overline R }_{mn}}\;U_n^{(0)}} \right]^{\rm{T}}}$
 ${\theta _n} = \left[{\begin{array}{*{20}{c}} I&{\bf{0}}& \cdots &{\bf{0}}&{{B_{1n}}}\\ {\bf{0}}&I& \cdots &{\bf{0}}&{{B_{3n}}}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {\bf{0}}&{\bf{0}}& \cdots &I&{{B_{mn}}}\\ \theta &\theta & \cdots &\theta &1 \end{array}} \right]$
 ${\overline \theta _n} = \left[{\begin{array}{*{20}{c}} {{D_{1n}}}&{\bf{0}}& \cdots &{\bf{0}}&{{{\overline B }_{1n}}}\\ {\bf{0}}&{{D_{3n}}}& \cdots &{\bf{0}}&{{{\overline B }_{3n}}}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {\bf{0}}&{\bf{0}}& \cdots &{{D_{mn}}}&{{{\overline B }_{mn}}}\\ {\bf{0}}&{\bf{0}}& \cdots &{\bf{0}}&{\bf{0}} \end{array}} \right]$

 ${B_{mn}} = {\left[{2\xi \;\;0\;\;0\;\;0\;\;0\;\;0} \right]^{\rm{T}}}$
 ${\overline B _{mn}} = {\left[{0\;\;0\;\;0\;\;2{C_6}{\eta ^2}\xi \;\;2({C_3} + {C_6})\zeta \eta \xi \;\; - 2{C_1}\zeta \xi } \right]^{\rm{T}}}$

 $\frac{d}{{dz}}{\Phi _n}(z) = {\Pi _n}{\Phi _n}(z)$ (20)

 ${\Phi _n}(z) = {{\rm{e}}^{{\Pi _n} \cdot z}}{\Phi _n}(0)$ (21)

 ${\Phi _n}(h) = {{\rm{e}}^{{\Pi _n} \cdot h}}{\Phi _n}(0)$ (22)

 $\left. {\begin{array}{*{20}{l}} \begin{array}{l} U_n^{(0)}(0) + \sum\limits_{m = 0}^\infty {{{\bar U}_{mn}}(0) = 0} \\ U_n^{(a)}(0) + \sum\limits_{m = 0}^\infty {{{( - 1)}^m}{{\bar U}_{mn}}(0) = 0} \end{array} \end{array}} \right\}$ (23)

 ${\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\sigma _x}\\ {\sigma _y}\\ {\sigma _z} \end{array} \end{array}} \right\}_{x = 0,a}} = - \frac{1}{a}\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {C_{11}}\\ {C_{12}}\\ {C_{13}} \end{array} \end{array}} \right\}\sum\limits_{n = 1}^\infty {\left[{U_n^{(0)}(z) - U_n^{(a)}(z)} \right]} \sin \eta y$ (24)

2 叠层板的状态方程及其解

 $\frac{d}{{d{z_j}}}\Phi _n^{(j)}({z_j}) = \Pi _n^{(j)}\Phi _n^{(j)}({z_j})$ (25)
 图 2 叠层矩形板 Fig. 2 Rectangular laminated plate

 $\Phi _n^{(j)}({z_j}) = {{\rm{e}}^{{\rm{ }}\Pi _n^{(j)} \cdot {z_j}}}\Phi _n^{(j)}(0)$ (26)

 $\Phi _n^{(j)}({h_j}) = {{\rm{e}}^{\Pi _n^{(j)} \cdot {h_j}}}\Phi _n^{(j)}(0)$ (27)

 $\Phi _n^{(j)}({h_j}) = \Psi _n^{(j)}\Phi _n^{(1)}(0)$ (28)

 $\Phi _n^{(p)}({h_p}) = \Psi _n^{(p)}\Phi _n^{(1)}(0)$ (29)

3 算例与分析

 图 3 固支边位移$u$随$z$的变化情况$(h/a=0.1)$ Fig. 3 Variation of the displacement $u$ with $z$ on clamped boundary of single plate $(h/a=0.1)$

 图 4 固支边附近$\sigma _z$随$x$和$z$的变化情况($h/a =0.1$) Fig. 4 Variation of the stress $\sigma _z$ with $x$ and $z$ near clamped boundary of single plate ($h/a =0.1$)
 图 5 固支边附近$\tau _{xz}$随$x$和$z$的变化情况($h/a=0.1$) Fig. 5 Variation of the shear stress $\tau _{xz}$ with $x$ and $z$ near clamped boundary of single plate ($h/a=0.1$)

4 结 论

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A NEW STATE SPACE SOLUTION FOR RECTANGULAR THICK LAMINATES WITH CLAMPED EDGES
Hu Wenfeng, Liu Yihua
School of Civil Engineering, Hefei University of Technology, Hefei 230009, China
Abstract: Based on the Refs.[1-3], boundary displacement functions are treated as state variables and introduced into the state equation. The exact solutions are presented for the rectangular thick single and laminated plates with clamped edges. In the solving process, the state equation becomes a homogeneous equation from a nonhomogeneous one and the intermediate process to solve undetermined constants is omitted. The displacement boundary conditions are satisfied strictly on clamped edges, and it is not necessary to divide the layer with same material into some sub-layers. Therefore the obtained solutions are more accurate. Additionally, a new calculating method of the stresses on the clamped edges is proposed and more exact boundary stresses can be obtained. The results of two examples show that the present solutions have better constringency and higher precision, and are approaching to FEM solutions well than the existing exact solutions, especially on the clamped edges.
Key words: rectangular thick laminate    clamped boundary    boundary displacement function    state variable    state equation    exact solution