﻿ 计算曲面壳体等效节点热载荷的新方法
«上一篇
 文章快速检索 高级检索

 力学学报  2015, Vol. 47 Issue (2): 328-336  DOI: 10.6052/0459-1879-14-136 0

### 引用本文 [复制中英文]

[复制中文]
Liu Yunfei, Lü Jun, Bai Ruixiang, Gao Xiaowei. A NEW METHOD TO COMPUTE THERMAL EQUIVALENTNODAL LOADS ON CURVED SURFACES[J]. Chinese Journal of Ship Research, 2015, 47(2): 328-336. DOI: 10.6052/0459-1879-14-136.
[复制英文]

### 文章历史

2014-05-15收稿
2014-07-09录用
2014-09-17网络版发表

 图 1 用折板代替曲面壳体 Fig.1 Folded plate simulating curved surface shells

1 曲面壳体结构的等效节点热载荷计算

1.1 平面应力问题的等效节点热载荷

 $P_{{\varepsilon _0}}^e = \int \int {} {B^{\rm{T}}}D{\varepsilon _0}t{\rm{d}}x'{\rm{d}}y'$ (1)

 $B = \left[{{B_1}\;\;{B_2}\;\;{B_3}\;\;{B_4}} \right]$ (2)
 ${B_i} = \left[{\begin{array}{*{20}{c}} {\frac{{\partial {N_i}}}{{\partial x'}}}&0\\ 0&{\frac{{\partial {N_i}}}{{\partial y'}}}\\ {\frac{{\partial {N_i}}}{{\partial y'}}}&{\frac{{\partial {N_i}}}{{\partial x'}}} \end{array}} \right]$ (3)

 ${N_i}\left( {\xi ,\eta } \right) = \frac{1}{4}\left( {1 + {\xi _i}\xi } \right)\left( {1 + {\eta _i}\eta } \right)$ (4)

 $\left[{\begin{array}{*{20}{c}} {{\xi _1}}&{{\eta _1}}\\ {{\xi _2}}&{{\eta _2}}\\ {{\xi _3}}&{{\eta _3}}\\ {{\xi _4}}&{{\eta _4}} \end{array}} \right] = \left[{\begin{array}{*{20}{c}} { - 1}&{ - 1}\\ 1&{ - 1}\\ 1&1\\ { - 1}&1 \end{array}} \right]$ (5)

$D$为平面应力问题的弹性刚度矩阵,其表达式为

 $D = \frac{E}{{1 - {\mu ^2}}}\left[{\begin{array}{*{20}{c}} 1&\mu &0\\ \mu &1&0\\ 0&0&{\frac{{1 - \mu }}{2}} \end{array}} \right]$ (6)

 ${\varepsilon _0} = \alpha \Delta T{\left[{1\;\;1\;\;0} \right]^{\rm{T}}} = \alpha \left( {T - {T_0}} \right){\left[{1\;\;1\;\;0} \right]^{\rm{T}}}$ (7)

 $T = \sum\limits_{i = 1}^n {{N_i}} \left( {\xi ,\eta } \right){T_i}$ (8)

 $\begin{array}{l} \left\{ {{P_{{\varepsilon _0}}}} \right\}_i^e = \left\{ {\begin{array}{*{20}{l}} {{P_{{\varepsilon _0}x'}}}\\ {{P_{{\varepsilon _0}y'}}} \end{array}} \right\}_i^e = \\ \qquad \frac{{E\alpha }}{{1 - \mu }}\int \smallint \left\{ {\begin{array}{*{20}{l}} {{N_{i,x'}}}\\ {{N_{i,y'}}} \end{array}} \right\}t\Delta T{\rm{d}}x'{\rm{d}}y' = \\ \qquad \frac{{E\alpha }}{{1 - \mu }}\int_{ - 1}^1 {\int_{ - 1}^1 {\left\{ {\begin{array}{*{20}{l}} {{N_{i,x'}}}\\ {{N_{i,y'}}} \end{array}} \right\}} } t\Delta T\left| {{\rm{ }}J} \right|{\rm{d}}\xi {\rm{ d}}\eta \end{array}$ (9)

 $\left\{ \begin{array}{l} \frac{{\partial {N_i}}}{{\partial \xi }}\\ \frac{{\partial {N_i}}}{{\partial \eta }} \end{array} \right\} = \left\{ {\begin{array}{*{20}{c}} {\frac{{\partial x'}}{{\partial \xi }}}&{\frac{{\partial y'}}{{\partial \xi }}}\\ {\frac{{\partial x'}}{{\partial \eta }}}&{\frac{{\partial y'}}{{\partial \eta }}} \end{array}} \right\}\left\{ \begin{array}{l} \frac{{\partial {N_i}}}{{\partial x'}}\\ \frac{{\partial {N_i}}}{{\partial y'}} \end{array} \right\} = J\left\{ \begin{array}{l} \frac{{\partial {N_i}}}{{\partial x'}}\\ \frac{{\partial {N_i}}}{{\partial y'}} \end{array} \right\}$ (10)

 $\left\{ \begin{array}{l} \frac{{\partial {N_i}}}{{\partial x'}}\\ \frac{{\partial {N_i}}}{{\partial y'}} \end{array} \right\} = {J^{ - 1}}\left\{ \begin{array}{l} \frac{{\partial {N_i}}}{{\partial \xi }}\\ \frac{{\partial {N_i}}}{{\partial \eta }} \end{array} \right\}$ (11)
 $\left| J \right| = \left| {\begin{array}{*{20}{c}} {\frac{{\partial x'}}{{\partial \xi }}}&{\frac{{\partial y'}}{{\partial \xi }}}\\ {\frac{{\partial x'}}{{\partial \eta }}}&{\frac{{\partial y'}}{{\partial \eta }}} \end{array}} \right|$ (12)

1.2 计算壳单元等效节点热载荷的新方法 1.2.1 建立壳单元局部坐标系

 图 2 传统方法矩形单元的总体和局部坐标系 Fig.2 Local orthogonal set of axes over a rectangular element and global orthogonal set of axes in the traditional way

 图 3 新方法建立的单元的局部坐标系 Fig.3 Local orthogonal set of axes over a surface element in the new way
1.2.2 形函数对局部坐标的导数计算式

 $\left\{ \begin{array}{l} \frac{{\partial {N_i}}}{{\partial x'}}\\ \frac{{\partial {N_i}}}{{\partial y'}} \end{array} \right\} = \left\{ \begin{array}{l} \frac{{\partial {N_i}}}{{\partial x{'_1}}}\\ \frac{{\partial {N_i}}}{{\partial x{'_2}}} \end{array} \right\} = \left[{\begin{array}{*{20}{c}} {\frac{{\partial \xi }}{{\partial {{x'}_1}}}}&{\frac{{\partial \eta }}{{\partial x{'_1}}}}\\ {\frac{{\partial \xi }}{{\partial {{x'}_2}}}}&{\frac{{\partial \eta }}{{\partial x{'_2}}}} \end{array}} \right]\left\{ \begin{array}{l} \frac{{\partial {N_i}}}{{\partial \xi }}\\ \frac{{\partial {N_i}}}{{\partial \eta }} \end{array} \right\}$ (13)

 $\left. {\begin{array}{*{20}{l}} {\frac{{\partial \xi }}{{\partial {{x'}_1}}} = \frac{1}{{\left| {{m_1}} \right|}}}\\ {\frac{{\partial \xi }}{{\partial {{x'}_2}}} = \frac{{ - \cos \theta }}{{\left| {{m_1}} \right|\sin \theta }}}\\ {\frac{{\partial \eta }}{{\partial {{x'}_1}}} = 0}\\ {\frac{{\partial \eta }}{{\partial {{x'}_2}}} = \frac{1}{{\left| {{m_2}} \right|\sin \theta }}} \end{array}\;\;\;} \right\}$ (14)

 $\left. {\begin{array}{*{20}{l}} {\left| {{m_1}} \right| = \sqrt {{{\left( {\frac{{\partial {x_1}}}{{\partial \xi }}} \right)}^2} + {{\left( {\frac{{\partial {x_2}}}{{\partial \xi }}} \right)}^2} + {{\left( {\frac{{\partial {x_3}}}{{\partial \xi }}} \right)}^2}} }\\ {\left| {{m_2}} \right| = \sqrt {{{\left( {\frac{{\partial {x_1}}}{{\partial \eta }}} \right)}^2} + {{\left( {\frac{{\partial {x_2}}}{{\partial \eta }}} \right)}^2} + {{\left( {\frac{{\partial {x_3}}}{{\partial \eta }}} \right)}^2}} } \end{array}\;\;\;} \right\}$ (15)
 $\cos \theta = \frac{1}{{\left| {{m_1}} \right|\left| {{m_2}} \right|}}\frac{{\partial {x_i}}}{{\partial \xi }}\frac{{\partial {x_i}}}{{\partial \eta }}\;\;(i = 1,3)$ (16)

1.2.3 壳单元中从局部到自然坐标系积分转换的雅可比计算式

 $\left. {\begin{array}{*{20}{l}} {{r_\xi } = \frac{{\partial {x_1}\left( {\xi ,\eta } \right)}}{{\partial \xi }}i + \frac{{\partial {x_2}\left( {\xi ,\eta } \right)}}{{\partial \xi }}j + \frac{{\partial {x_3}\left( {\xi ,\eta } \right)}}{{\partial \xi }}k}\\ {{r_\eta } = \frac{{\partial {x_1}\left( {\xi ,\eta } \right)}}{{\partial \eta }}i + \frac{{\partial {x_2}\left( {\xi ,\eta } \right)}}{{\partial \eta }}j + \frac{{\partial {x_3}\left( {\xi ,\eta } \right)}}{{\partial \eta }}k} \end{array}\;\;\;} \right\}$ (17)
 $\begin{array}{l} {n^ * } = {r_\xi } \times {r_\eta } = \\ \;\left( {\frac{{\partial {x_2}\left( {\xi ,\eta } \right)}}{{\partial \xi }} \cdot \frac{{\partial {x_3}\left( {\xi ,\eta } \right)}}{{\partial \eta }} - \frac{{\partial {x_3}\left( {\xi ,\eta } \right)}}{{\partial \xi }} \cdot \frac{{\partial {x_2}\left( {\xi ,\eta } \right)}}{{\partial \eta }}} \right)i + \\ \;\left( {\frac{{\partial {x_3}\left( {\xi ,\eta } \right)}}{{\partial \xi }} \cdot \frac{{\partial {x_1}\left( {\xi ,\eta } \right)}}{{\partial \eta }} - \frac{{\partial {x_1}\left( {\xi ,\eta } \right)}}{{\partial \xi }} \cdot \frac{{\partial {x_3}\left( {\xi ,\eta } \right)}}{{\partial \eta }}} \right)j + \\ \;\left( {\frac{{\partial {x_1}\left( {\xi ,\eta } \right)}}{{\partial \xi }} \cdot \frac{{\partial {x_2}\left( {\xi ,\eta } \right)}}{{\partial \eta }} - \frac{{\partial {x_2}\left( {\xi ,\eta } \right)}}{{\partial \xi }} \cdot \frac{{\partial {x_1}\left( {\xi ,\eta } \right)}}{{\partial \eta }}} \right) \end{array}$ (18)
 图 4 单元面的法向量 Fig.4 Normal vector to a surface element

 ${\left| J \right|^\prime } = \left| {{n^ * }} \right| = \left| {{r_\xi } \times {r_\eta }} \right|$ (19)

 $n = \frac{{{n^ * }}}{{\left| {{n^ * }} \right|}} = {n_1}i + {n_2}j + {n_3}k$ (20)

1.3 壳单元局部到整体坐标系转换矩阵计算法

 $\left. {\begin{array}{*{20}{c}} {u{'_i} = {L_{ij}}{u_j}}\\ {P{'_i} = {L_{ij}}{P_j}} \end{array}\;\;(i,j = 1,3)} \right\}$ (21)

 $\left. {\begin{array}{*{20}{l}} {{u_j}\left( {\xi ,\eta } \right) = \sum\limits_{\alpha = 1}^M {{N_\alpha }\left( {\xi ,\eta } \right)} u_j^\alpha }\\ {{P_j}\left( {\xi ,\eta } \right) = \sum\limits_{\alpha = 1}^M {{N_\alpha }\left( {\xi ,\eta } \right)} P_j^\alpha } \end{array}\;\;\left( {j = 1,3} \right)} \right\}$ (22)

 ${L_{1j}} = \frac{1}{{\left| {{m_1}} \right|}}\frac{{\partial {x_j}}}{{\partial \xi }}\;\;\left( {j = 1,3} \right)$ (23)
 $\left. {\begin{array}{*{20}{l}} {{L_{21}} = {n_2}{L_{13}} - {n_3}{L_{12}}}\\ {{L_{22}} = {n_3}{L_{11}} - {n_1}{L_{13}}}\\ {{L_{23}} = {n_1}{L_{12}} - {n_2}{L_{11}}} \end{array}\;\;\;} \right\}$ (24)
 ${L_{3j}} = {n_j}\;\;(j = 1,3)$ (25)

 $\left. {\begin{array}{*{20}{c}} {{{U'}^e} = L{U^e}}&{{U^e} = {L^{\rm{T}}}{{U'}^e}}\\ {{{P'}^e} = L{P^e}}&{{P^e} = {L^{\rm{T}}}{{P'}^e}} \end{array}\;\;\;} \right\}$ (26)
2 算例分析

 图 5 柱面壳体模型 Fig.5 The cylindrical shell body

 图 6 映射方式划分矩形网格 Fig.6 Meshing model in mapped method
 图 7 自由方式划分自由网格 Fig.7 Meshing model in free method

 图 8 任意不规则四边形曲面模型单元 Fig.8 The formed irregular surface shell elements
 图 9 平面壳体模型单元 Fig.9 The plane shell elements

3 结 论

 [1] 孙博华. 壳体简史. 力学与实践, 2013, 35(5): 104-106 (Sun Bohua. A brief history of shell. Mechanics in Engineering, 2013, 35(5): 104-106 (in Chinese)) [2] 张志田, 葛耀君. 悬索桥静动力特性分析的有限板壳单元法. 结构工程师, 2003, (4): 21-25 (Zhang Zhitian, Ge Yaojun. Static and dynamic analysis of suspension bridges based on an orthotropic shell finite element method. Structural Engineers, 2003, (4): 21-25 (in Chinese)) [3] 高铁军, 王忠金, 高洪波, 等. 涡扇发动机异形曲面壳体零件整体成形工艺. 推进技术, 2007, 28(4):437-440 (Gao Tiejun, Wang Zhongjin, Gao Hongbo, et al. Integral forming for special-shaped curved surface shell parts of turbofan engine. Journal of Propulsion Technology, 2007, 28(4): 437-440 (in Chinese)) [4] Lü Jun, Zhang Hongwu, Yang Dongsheng. Multiscale method for mechanical analysis of heterogeneous materials with polygonal microstructures. Mechanics of Materials, 2013(56): 38-52 [5] 吕军, 张洪武, 陈飙松. 基于扩展多尺度有限元法的含液闭孔材料拓扑优化. 固体力学学报, 2013, 34(4): 342-351 (Lü Jun, Zhang Hongwu, Chen Biaosong. Topology optimization of closed liquid cell materials based on extended multiscale finite element method. Acta Mechanica Solida Sinica, 2013, 34(4): 342-351 (in Chinese)) [6] 张洪武, 王鲲鹏. 材料非线性微-宏观分析的多尺度方法研究. 力学学报, 2004, 36(3): 359-363 (Zhang Hongwu, Wang Kunpeng. Multiscale methods for nonlinear analysis of composite materials. Acta Mechanica Sinica, 2004, 36(3): 359-363 (in Chinese)) [7] 王勖成. 有限单元法. 北京: 清华大学出版社, 2003 (Wang Xucheng. Finite Element Method. Beijing: Tsinghua University Press, 2003 (in Chinese)) [8] Clough R W, Johncon C P. A finite element approximation for the analysis of thin shell. Int. J. Solids Structures, 1968(4): 43-60 [9] Ahmad S, Irons B M, Zienkiewicz O C. Analysis of thick and thin shell structures by curved elements. Int. J. Numer. Methods. Eng, 1970(2): 419-451 [10] 白瑞祥, 郭兆璞, 陈浩然, 等. 复合材料层合板、壳胶-铆混合连接件有限元分析. 大连理工大学学报, 2000, 40(3): 280-284 (Bai Ruixiang, Guo Zhaopu, Chen Haoran, et al. Finite element analysis of adhesive-bolted joint for composite plate and rotary shell structures. Journal Of Dalian University of Technology, 2000, 40(3): 280-284 (in Chinese)) [11] 张志峰, 陈浩然, 白瑞祥. 一种分析AGS结构的三角形加筋板壳单元. 工程力学, 2006, 23(1): 203-208 (Zhang Zhifeng, Chen Haoran, Bai Ruixiang. A new triangular stiffened plate/shell elment for composite grid structure analysis. Engineering Mechanics, 2006, 23(1): 203-208 (in Chinese)) [12] 许磊平, 秦顺全, 马润平. 基于平面壳单元的分阶段成形结构平衡方程.西南交通大学学报, 2013, 48(5): 857-862 (Xu Leiping, Qin Shunquan, Ma Runping. Equilibrium equation derivation of structures formed by stages based on plane shell element. Journal of Southwest Jiaotong University, 2013, 48(5): 857-862 (in Chinese)) [13] 高蕊, 曹汝龙, 王岩, 等.板式结构体系的箱形平板薄壳单元.济南大学学报, 2013, 27(3): 315-319 (Gao Rui, Cao Rulong, Wang Yan, et al. A new box-type plate and shell element used in panel structures. Journal of University of Jinan, 2013, 27(3): 315-319 (in Chinese)) [14] 尹进, 杨东生, 张盛, 等. 几种基于组合方法的复合材料层合板壳单元.航空学报, 2013, 34(1): 86-96 (Yin Jin, Yang Dongsheng, Zhang Sheng, et al. Laminated composite plate and shell elements based on combination method. Acta Aeronautica et Astronautica Sinica, 2013, 34(1): 89-96 (in Chinese)) [15] 文颖, 戴公连, 曾庆元.基于带面内转角自由度四节点平板壳单元的板壳非线性分析.中南大学学报, 2013, 44(4): 1525-1531 (Wen Ying, Dai Gonglian, Zeng Qingyuan. 4-node flat shell element with drilling degrees of freedom for nonlinear analysis of plates and shells. Journal of Central South University, 2013, 44(4): 1525-1531 (in Chinese)) [16] 李维特, 黄保海, 毕仲波. 热应力理论分析及应用. 北京: 中国电力出版社, 2004 (Li Weite, Huang Baohai, Bi Zhongbo. The Theory Analysis and Application of Thermal Stress. Beijing: China Electric Power Press, 2004 (in Chinese)) [17] 邵红艳. 结构温度场和温度应力场分析. [硕士论文].西安: 西北工业大学, 2001 (Shao Hongyan. The structure temperature field and thermal stress field analysis. [Master's Thesis]. Xi'an: Northwestern Polytechnical University, 2001 (in Chinese)) [18] 侯建国, 安旭文. 有限单元法程序设计. 第二版.武汉: 武汉大学出版社. 2012 (Hou Jianguo, An Xuwen. Finite Element Method Program Design. Second edition. Wuhan: Wuhan University Press. 2012 (in Chinese)) [19] Gao XW, Davies TG. Boundary Element Programming in Mechanics. London: Cambridge University Press, 2002 [20] Lachat JC. A further development of the boundary integral technique for elastostatics. [PhD Thesis]. UK: University of Southampton, 1975 [21] Lachat JC, Watson JO, et al. A second generation boundary integral program for three dimensional elastic analysis. In: Cruse TA, Rizzo FJ eds. Boundary Integral Equation Method: Computational Applications in Applied Mechanics. New York: ASME, 1975 [22] Becker AA. The Boundary Element Method in Engineering. London: McGraw -Hill Book Co, 1992
A NEW METHOD TO COMPUTE THERMAL EQUIVALENTNODAL LOADS ON CURVED SURFACES
Liu Yunfei, Lü Jun, Bai Ruixiang, Gao Xiaowei
State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China
Fund: The project was supported by the National Natural Science Foundation of China (11172055, 11302040) and China Postdoctoral Science Foundation (2014T70244).
Abstract: A new method using flat shell element to compute thermal equivalent nodal loads on curved surface is presented. Firstly, the local Cartesian coordinate system is established on the tangential plane to the element surface. Then, the unknowns included in the integrand about thermal equivalent nodal loads are computed in terms of the theory proposed in this paper, which include the derivative of shape functions with respect to the variables of local coordinate system and the Jacobian of the transformation from the local three-dimensional coordinate system to the intrinsic two-dimensional coordinate system of the surface patch. Finally, a matrix transformation formulation is derived from the local Cartesian coordinate system to the global one, based on which the thermal equivalent nodal loads in the global Cartesian coordinate system can be found. Comparison with the results from FEM analysis shows that the proposed method in this paper is correct and accurate.
Key words: thermal equivalent nodal loads    tangential plane    Jacobian    thermal stress