基于高阶剪切变形理论的四边形求积元板单元及其应用

申志强; 夏军; 宋殿义; 程盼

doi:10.6052/0459-1879-18-225

基于高阶剪切变形理论的四边形求积元板单元及其应用

doi: 10.6052/0459-1879-18-225

申志强^*,
夏军^*,
宋殿义^*2), ,,
程盼^†

^*(国防科技大学军事基础教育学院, 长沙 410072)
^†(国防科技大学空天科学学院, 长沙 410072)

基金项目: 1)国家自然科学基金(51508562,51809271)和学校科研计划(ZK2017-03-40)资助项目.

详细信息

作者简介:
2)宋殿义,副教授,主要研究方向：新型结构抗冲击与抗爆研究. E-mail：changshasong@nudt.edu.cn

通讯作者:
宋殿义

中图分类号: O343.1;
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出版历程
- 收稿日期: 2018-07-10
- 刊出日期: 2018-09-18

A QUADRILATERAL QUADRATURE PLATE ELEMENT BASED ON REDDY'S HIGHER-ORDER SHEAR DEFORMATION THEORY AND ITS APPLICATION

Shen Zhiqiang^*,
Xia Jun^*,
Song Dianyi^{*2)
, ,},
Cheng Pan^†

^*(College of Military Education and Training, National University of Defense Technology,Changsha)
^†(College of Aerospace Science and Engineering, National University of Defense Technology,Changsha)

摘要

摘要: 近年来由各类新型复合材料或功能梯度材料构成的板结构在工程领域得到了广泛应用,其显著特点是材料性能沿板厚变化.为合理考虑横向剪切应变,许多学者基于Reddy高阶剪切变形理论,构建了不同的有限元单元对该类板结构进行分析,但其中满足$C^{1}$连续条件的单元相对较少.本文基于Reddy高阶剪切变形理论,采用求积元方法,建立了$C^{1}$连续的四边形板单元.利用该单元对均质材料、复合材料、功能梯度材料构成的等厚度矩形板、变厚度矩形板及等厚度斜板的线弹性弯曲和自由振动问题进行了计算分析,并与现有文献中的相应计算结果进行了对比.研究表明：基于高阶剪切变形理论的四边形求积元板单元具有较高的计算效率和良好的适应性,文中各类材料构成的等变厚度矩形板及等厚度斜板均只需1个单元即可得到理想的计算结果.对于等/变厚度矩形板,可仅使用9$\times$9个积分点,而对于等厚度斜板,随着斜角的增大,所需积分点的数目逐渐增多至15$\times $15.该四边形求积元板单元可进一步用于新型复合材料板的非线性分析.
- 高阶剪切变形理论 /
- 四边形板单元 /
- 弱形式求积元法 /
- 复合材料板 /
- 变厚度板
Abstract: Plate structures made of advanced composite materials or functionally graded materials have been widely used in engineering practice recently, which is characterized by the variation of material properties along the plate thickness. Several plate elements have been presented utilizing the finite element formulation based on Reddy's higher-order shear deformation theory which yields more accurate transverse shear strain distributions of these structures. However, the $C^{1}$ continuous plate elements is very limited. Based on Reddy's higher-order shear deformation theory, a $C^{1}$ continuous quadrilateral plate element is established using the weak form quadrature element method in this work. The element presented here is then used for linear flexural and free vibrational analyses of the rectangular and skew plates made of homogenous or composite materials with constant thickness as well as the homogenous rectangular plates with variable thickness. The numerical results of quadrature element formulation are compared with those of other numerical method from the open literatures in order to validate the correctness and efficiency of the presented quadrature plate element. It is shown that only one quadrature element is fully competent for linear analysis of a quadrilateral plate regardless of its thickness variation and component materials. As for rectangular plates with constant or variable thickness, one quadrature element with only 9$\times $9 numerical integration points is needed. And for skew plates, the number of numerical integration points required for acceptable accuracy gradually increases to 15$\times $15 with the skew angle enlargement. As a completive numerical formulation, the quadrilateral quadrature plate element can be further applied in nonlinear and transient analyses of composite material plate structures.
- higher-order shear deformation theory /
- quadrilateral plate element /
- weak form quadrature element method /
- composite material plate /
- plate with variable thickness

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参考文献(1)

[1]	Jones RM.Mechanics of composite materials. CRC Press, 2014
[2]	刘人怀, 薛江红. 复合材料层合板壳非线性力学的研究进展. 力学学报, 2017, 49(3): 487-506
[2]	(Liu Renhuai, Xue Jianghong.Development of nonlinear mechanics for laminated composite plates and shells. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(3): 487-506 (in Chinese))
[3]	段铁城, 李录贤. 厚板的高阶剪切变形理论研究. 力学学报, 2016, 48(5): 1096-1113
[3]	(Duan Tiecheng, Li Luxian.Study on higher-order shear deformation theories of thick-plate. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(5): 1096-1113 (in Chinese))
[4]	Reddy JN.A refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures, 1984, 20(9-10): 881-896
[5]	Singh G, Rao GV, Iyengar NGR.Large deflection of shear deformable composite plates using a simple higher-order theory. Composites Engineering, 1993, 3(6): 507-525
[6]	Liu IW.An element for static, vibration and buckling analysis of thick laminated plates. Computers & Structures, 1996, 59(6): 1051-1058
[7]	Phan ND, Reddy JN.Analysis of laminated composite plates using a higher-order shear deformation theory. International Journal for Numerical Methods in Engineering, 1985, 21(12): 2201-2219
[8]	Ren JG, Hinton E.The finite element analysis of homogeneous and laminated composite plates using a simple higher order theory. Communications in Applied Numerical Methods, 1986, 2(2): 217-228
[9]	Kant T, Pandya BN.A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates. Composite Structures, 1988, 9(3): 215-246
[10]	Pervez T, Seibi AC, Al-Jahwari FKS.Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory. Composite Structures, 2005, 71(3-4): 414-422
[11]	Shankara CA, Iyengar NGR.A C0element for the free vibration analysis of laminated composite plates. Journal of Sound and Vibration, 1996, 191(5): 721-738
[12]	Ansari MI, Kumar A, Barnat-Hunek D, et al.Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates. Applied Sciences, 2018, 8(5): 834
[13]	Nayak AK, Moy SSJ, Shenoi RA.Free vibration analysis of composite sandwich plates based on Reddy's higher-order theory. Composites Part B: Engineering, 2002, 33(7): 505-519
[14]	Nayak AK, Moy SSJ, Shenoi RA.Quadrilateral finite elements for multilayer sandwich plates. The Journal of Strain Analysis for Engineering Design, 2003, 38(5): 377-392
[15]	Sheikh AH, Chakrabarti A.A new plate bending element based on higher-order shear deformation theory for the analysis of composite plates. Finite Elements in Analysis and Design, 2003, 39(9): 883-903
[16]	Taj MNAG, Chakrabarti A, Sheikh AH.Analysis of functionally graded plates using higher order shear deformation theory. Applied Mathematical Modelling, 2013, 37(18-19): 8484-8494
[17]	Putcha NS, Reddy JN.A mixed shear flexible finite element for the analysis of laminated plates. Computer Methods in Applied Mechanics and Engineering, 1984, 44(2): 213-227
[18]	Serdoun SMN, Hamza Cherif SM.Free vibration analysis of composite and sandwich plates by alternative hierarchical finite element method based on Reddy's C1 HSDT. Journal of Sandwich Structures & Materials, 2016, 18(4): 501-528
[19]	Asadi E, Fariborz SJ.Free vibration of composite plates with mixed boundary conditions based on higher-order shear deformation theory. Archive of Applied Mechanics, 2012, 82(6): 755-766
[20]	王伟, 伊士超, 姚林泉. 分析复合材料层合板弯曲和振动的一种有效无网格方法. 应用数学和力学, 2015, 36(12): 1274-1284
[20]	(Wang Wei, Yi Shi-Chao, Yao Linquan.An effective meshfree method for bending and vibration analyses of laminated composite plates. Applied Mathematics & Mechanics, 2015, 36(12):1274-1284 (in Chinese))
[21]	Selim BA, Zhang LW, Liew KM.Impact analysis of CNT-reinforced composite plates based on Reddy's higher-order shear deformation theory using an element-free approach. Composite Structures, 2017, 170: 228-242
[22]	赵寿根, 王静涛, 黎康等. 考虑辐射散热叠层板热诱发振动的有限元分析. 力学学报, 2010, 42(5): 978-982
[22]	(Zhao Shougen, Wang Jingtao, Li Kang, et al.Finite element method analysis of thermally induced vibration of laminated plates considering radiation. Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(5): 978-982 (in Chinese))
[23]	Zhong H, Yu T.Flexural vibration analysis of an eccentric annular Mindlin plate. Archive of Applied Mechanics, 2007, 77(4): 185-195
[24]	Shen Z, Zhong H.Static and vibrational analysis of partially composite beams using the weak-form quadrature element method. Mathematical Problems in Engineering, 2012, 2012: 974023
[25]	Xiao N, Zhong H.Non-linear quadrature element analysis of planar frames based on geometrically exact beam theory. International Journal of Non-Linear Mechanics, 2012, 47(5): 481-488
[26]	Zhong HZ, Yue ZG.Analysis of thin plates by the weak form quadrature element method. Science China Physics, Mechanics and Astronomy, 2012, 55(5): 861-871
[27]	Zhong H, Wang Y.Weak form quadrature element analysis of Bickford beams. European Journal of Mechanics-A/Solids, 2010, 29(5): 851-858
[28]	Wang X, Yuan Z, Jin C.Weak form quadrature element method and its applications in science and engineering: a state-of-the-art review. Applied Mechanics Reviews, 2017, 69(3): 030801
[29]	申志强, 夏军, 吴克刚等. 楔形变截面钢$\!$-$\!$-$\!$混凝土组合梁弯曲和自由振动的求积元分析. 国防科技大学学报, 2018, 40(1): 42-48
[29]	(Shen Zhiqiang, Xia Jun, Wu Kegang, et al.Flexural and free vibrational analysis of tapered partially steel-concrete composite beams using the weak form quadrature element method. Journal of National University of Defense Technology, 2018, 40(1): 42-48 (in Chinese))
[30]	Davis PJ, Rabinowitz P.Methods of Numerical Integration. Courier Corporation, 2007
[31]	Wang X.Differential Quadrature and Differential Quadrature Based Element Methods: Theory and Applications. Butterworth-Heinemann, 2015
[32]	Reddy JN.A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics, 1984, 51(4): 745-752
[33]	Shufrin I, Eisenberger M.Stability and vibration of shear deformable plates---first order and higher order analyses. International Journal of Solids and Structures, 2005, 42(3-4): 1225-1251
[34]	Reddy JN, Phan ND.Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. Journal of Sound and Vibration, 1985, 98(2): 157-170
[35]	Shufrin I, Eisenberger M.Vibration of shear deformable plates with variable thickness---first-order and higher-order analyses. Journal of Sound and Vibration, 2006, 290(1-2): 465-489
[36]	Bacciocchi M, Eisenberger M, Fantuzzi N, et al.Vibration analysis of variable thickness plates and shells by the generalized differential quadrature method. Composite Structures, 2016, 156: 218-237
[37]	Wang ZX, Shen HS.Nonlinear vibration of nanotube-reinforced composite plates in thermal environments. Computational Materials Science, 2011, 50(8): 2319-2330