﻿ 圆柱壳变形计算方法研究
 舰船科学技术  2016, Vol. 38 Issue (3): 1-4 PDF

1. 武汉第二船舶设计研究所, 湖北武汉 430064;
2. 中国舰船研究院, 北京 100192

Research on calculation methods of cylindrical shell Deformation
ZHOU Hai-bo1, PENG Yun-fei1, WU Hong-min1, JI Fang2
1. Wuhan Second Ship Design and Research Institute, Wuhan 430064, China;
2. China Ship Research and Development Academy, Beijing 100192, China
Abstract: Calculation on the deformation of cylindrical shell is a key problem, analytical method, numerical method and semi-analytic method become effective method to solve these problems. Because of the complexity of ring-stiffened cylindrical shell, it is difficult to use analytical method, FEM becomes the representation of numerical method by the developing of computer technology. The stress condition of typical cylindrical shell was anslysed in this paper, and the FEM model was established. According to the contrast of analytical method and numerical method, The influence of different methods to the cylindrical shell deformation was analysed.
Key words: cylindrical shelldeformation    analytical method    numerical method
0 引言

1 圆柱壳的无矩理论

 $\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial {F_{T1}}}}{{\partial \alpha }} + \frac{{\partial {F_{T12}}}}{{\partial \beta }} + {q_1} = 0}\ \\ {\frac{{\partial {F_{T2}}}}{{\partial \beta }} + \frac{{\partial {F_{T21}}}}{{\partial \alpha }} + {q_2} = 0}\ \\ {{F_{T2}} = R{q_3}} \end{array}} \right.$ (1)
 图 1 柱壳的无矩理论 Fig. 1 Non-distance theoretical of cylindrical shell

 $\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial u}}{{\partial \varepsilon }} = \frac{{{F_{T1}} - \mu {F_{T2}}}}{{E\delta }}}\ \\ {\frac{{\partial v}}{{\partial \beta }} + \frac{w}{R} = \frac{{{F_{T2}} - \mu {F_{T1}}}}{{E\delta }}}\ \\ {\frac{{\partial u}}{{\partial \beta }} + \frac{{\partial v}}{{\partial \alpha }} = \frac{{2(1 + \mu ){F_{T12}}}}{{E\delta }}}\ {} \end{array}} \right.$ (2)

 图 2 悬臂圆柱壳 Fig. 2 Cantilevered cylindrical shell

${F_{T2}} = R{q_3} = R{q_0}{\rm{cos}}\varphi$

${F_T}_{12} = - 2{q_0}\alpha {\rm{sin}}\varphi ,0 \le \alpha \le l/2 - {l_1}$

${F_T}_{{\rm{12}}} = {\rm{2}}{q_0}\left( {l - \alpha - {\rm{2}}{l_{\rm{1}}}} \right){\rm{sin}}\varphi ,l/2 - {l_1} \le \alpha \le l/2$

$\left\{ {\begin{array}{*{20}{l}} {{F_{T1}} = }&{\frac{{{q_0}\cos \varphi }}{R}\left[{{{\left( {\frac{l}{2} - {l_1}} \right)}^2} - {\alpha ^2}} \right]}\ \\ {}&{0 \le \alpha \le \frac{l}{2} - {l_1};}\ \\ {{F_{T1}} = }&{\frac{{{q_0}\cos \varphi }}{R}\left[{{\alpha ^2} - (l - 2{l_1})\alpha + 3{{\left( {\frac{l}{2} - {l_1}} \right)}^2}} \right]}\ \\ {}&{\frac{l}{2} - {l_1} \le \alpha \le \frac{l}{2}}\ {}&{} \end{array}} \right.$

 $\left\{ {\begin{array}{*{20}{l}} {u = }&{\frac{{{q_0}\alpha \cos \varphi }}{{E\delta R}}\left( {\frac{{{\alpha ^2}}}{3} - \alpha l + \mu {R^2} + {l^2}} \right),}\ \\ {v = }&{\frac{{{q_0}\sin \varphi }}{{E\delta {R^2}}}\left[{4(1 + \mu ){R^2}(l\alpha - \frac{1}{2}{\alpha ^2})} \right. + }\ \\ {}&{\left. {\frac{{{\alpha ^3}}}{3}(\frac{1}{4}\alpha - l) + \frac{1}{2}{\alpha ^2}(\mu {R^2} + {l^2})} \right],}\ \\ {w = }&{ - \frac{{{q_0}\cos \varphi }}{{E\delta {R^2}}}\left[{{R^4} + \mu {R^2}\left( {2l\alpha - {\alpha ^2} - {l^2}} \right) + D} \right].}\ {}&{} \end{array}} \right.$ (3)

 $\begin{array}{l} D = \left[{ - 2(1 + \mu ){R^2}{\alpha ^2} - \frac{{{\alpha ^4}}}{{12}} + \frac{{{\alpha ^3}l}}{3} - } \right.\ \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\frac{{{\alpha ^2}\mu {R^2}}}{2} + 4(1 + \mu ){R^2}l\alpha } \right] \end{array}$ (4)
2 圆柱壳的有矩理论

 图 3 底部简支圆柱壳 Fig. 3 Bottom simply supported cylindrical shell

A0 截面：X0 = RZ0 = 0；A2截面：X2 = 0，Z2 = RA 截面：X = RcosφZ = RsinφM2 为第Ⅰ象限 A2A 的圆柱壳自重作用于 A 截面的弯矩，P/2 是平台对右半圆圆柱壳的反作用力传递到 A0 截面的值。设第Ⅰ象限的圆柱壳于 A0 截面处固定之，则在 M2 作用下，A2 分别向下和向左位移[6]

 ${M_2} = {M_0} - {\pi ^2}{R^2}F\gamma (1 - \cos \phi )/2$ (5)

 ${M_1} = {M_0} - {\pi ^2}{R^2}F\gamma (1 - \cos \phi )/2$ (6)

 $M = {M_0} - {\pi ^2}{R^2}F\gamma (1 - \cos \phi )/2$ (7)

 $M = {R^2}F\gamma (\pi \cos /2 - 1)$ (8)

 ${U_1} = R\int_0^{ - {\rm{ }}\pi /2} {\omega \cos \phi {\rm{d}}\phi = {R^4}F\gamma ({\pi ^2}/8 - 1)} /EI$ (9)

 ${U_1} = R\int_0^{ - {\rm{ }}\pi /2} {\omega \cos \phi {\rm{d}}\phi = 2{R^4}F\gamma ({\pi ^2}/8 - 1)} /EI$ (10)

3 有限元仿真

1）理想圆柱壳底部简支状态的自重变形

 图 4 底部简支的理想圆柱壳垂直变形量随直径和壁厚变化曲线 Fig. 4 Vertical deformation of bottom simply supported cylindrical shell under different diameter and thickness

2）悬臂圆柱壳的自重变形

 图 5 悬臂圆柱壳端面变形量随直径和壁厚变化曲线 Fig. 5 Surface deformation of cantilevered cylindrical shell under different diameter and thickness

4 结语

1）无矩理论适用于一端固支，一端自由的圆柱壳自重变形计算，有矩理论仅适用于圆柱壳为理想圆柱壳，底部简支的情况；

2）对于一端固支，一端自由的圆柱壳，仿真结果表明，自由端的变形量随圆柱壳直径的增大而减小，与圆柱壳厚度无关，同一端面各点的变形量大致相同，即重力载荷对自由端面圆度影响不大，这与经典无矩理论规律一致；

3）对于理想圆柱壳底部简支的情况，仿真结果表明，圆柱壳在自身重力作用下，垂直位置的圆柱壳中径在垂直方向的减少量随直径的增大而增大，随厚度的增大而减小，变形量基本上与内径的四次方成正比，与厚度的二次方成反比，这与经典有矩理论规律一致；

4）对于复杂圆柱壳结构，圆柱壳的几何形状和所受载荷不再是连续可微函数，理论计算具有较大的局限性，限元方法较为合理准确。

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