一种新型后张拉整体成形穹顶的成形和承载力性能研究
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摘要
后张拉整体成形空间网壳结构是一种新型的空间结构体系，与传统网壳结构相比，这种新型网壳结构体系可以使网壳的拼装过程完全在地面上实现，而且平面外腹杆的使用有效的改善了单层网壳结构的稳定性性能，所以这是一种具有很好工程应用前景的新型网壳结构体系。基于对后张拉整体成形穹顶结构形式的研究，本文提出了一种具有新型组成形式的后张拉整体成形穹顶的单弦杆平板网架模型，通过试验和非线性有限元分析，对穹顶的成形过程和承载力性能进行了研究。其主要内容有：
通过非线性有限元模拟成形分析，对新型单弦杆平板网架中的后张拉方法、节点连接形式以及穹顶成形形状进行了研究，为试验模型的建立奠定了理论基础。
设计制作了一个试验模型，进行了穹顶的成形试验和承载力试验。试验结果表明：本文提出的单弦杆平板网架模型具有很好的空间变形能力，可以张拉成形具有较大矢跨比的穹顶；穹顶的成形形状可以通过张拉下弦杆内的预留缝隙值进行很好的控制；后张拉力和成形初应力均较小，成形初应力中以腹杆弯曲应力为主，杆件轴向应力均较小；成形后的单弦杆穹顶具有很好的承载力性能，破坏形式为延性破坏，未出现穹顶失稳问题。
试验模型中，为了实现纯铰接连接，提出了一种新型的单向铰节点连接形式；成形试验中，由于后张拉力较小，提出了一种新型拧螺母抽钢丝的加载方法，很好的解决了千斤顶易超张拉和加载速率不均匀等问题；承载力试验中，为了解决加载点较多、单点加载较大的问题，提出了一种新型多点定滑轮竖向加载系统。试验结果表明，本文提出的各种改进方法是合理正确的，它们保证了试验过程的顺利完成，采集到了较理想的试验数据。
在考虑几何非线性和材料非线性的基础上，采用多种有限元模型对穹顶的成形试验和承载力试验进行了较系统的有限元模拟分析，主要考虑节点连接刚度、节点域大小、节点和支座偏心、成形初应力等因素的影响。得出了一系列重要结论。
有限元分析中，采用节点增设独立梁单元的方法，研究了鸭嘴半刚性连接和上弦节点域大小对穹顶成形过程的影响；提出了统一连续的分析方法，很好的解决了成形初应力向穹顶承载力有限元分析模型中的施加问题。
对单弦杆穹顶的非线性稳定性进行了较为系统的分析研究，考虑了穹顶网壳厚度、矢跨比、张拉下弦杆与下弦节点的连接形式和上弦张拉钢丝补杆刚度以及杆件受力对单弦杆穹顶
西安建筑科技大学博士学位论文
稳定性性能的影响。得到了一些重要的结论。
最后总结了本文的研究工作并提出了对以后研究工作的展望。
The post-tensioned and shaped space trusses is a new space structural system. Compared with the traditional space trusses, the assembly of the new space trusses can be finished wholly at ground level and it improves validly the stability behavior of single-layer shells because of the use of out-of plane web members, so the new space structrual system will have the important advantage in the future application. Based on the research on the post-tensioned and shaped dome, a new single-chorded plane space truss formed dome with post-tension is put forward firistly in this paper, through model tests and nonlinear finite element analysis, the shape forming process and load behaviors is studied. The paper mainly does work as follows:
Though simulating the shape forming process of post-tensioned and shaped dome by nonlinear finite element analysis, the post-tensioning method and joint connections in the new single-chorded plane space truss as well as the forming shape of dome are studied. It lays a theoretical foundation for the design and manufacture of experimental model.
A single-chorded plane space truss experimental model of post-tensioned and shaped dome is finished and the shape forming test and ultimate load test are carried out. Results showed that the single-chorded plane space truss model in this paper has very good deformability and it can form dome with more rise-to-span ratio by means of a post-tensioning technique, the forming shape of dome can be controled by the gaps which are given in certain bottom chords, the post-tensioning forces and member forces produced in the shape forming process are small and the bending stress is much larger in comparison with the axial stress in the member forces, the post-tensioned and shaped single-chorded dome has satisfactory utimate load capacity with ductility-type failure and the instability failure has not happened.
In the test model, a new type joint connection of one-way hinge is put forward for achieving pure-hinge connection; In the shape forming test, because of the little post-tensioning forces, a new type loading method of twisting nut for drawing wire is put forward in stead of jack tensioning for solving the questions, such as super-tensioning and the nonuniform rate of loading; In the load
test, a new type vertical loading system of multiple-point fixed pulley is put forward for solving the questions, such as more loading points and larger vertical force at one loading point. Results showed that these improvement methods that are put forward in this paper are rational and correct, they ensured the test processes to be successful and gathering the fine test data.
Then an thorough computer modeling test using various finite element models with geometry and material nonlinear taken into account is carried out on a set of influencing factors, such as the stiffness of joint connections, the size of joint, the eccentricty of joint and support connections, the member forces produced in the shape forming process, etc. Lots of useful conclusions are drawn form computer modeling.
In the finite element analysis, by adding individual beam elements to model the connector between top joints and chords in finite element model, based on the experimental model, the factors affecting the shape forming process of dome, such as the bending stiffness of duckbilled connector and the size of top joints, can be taken into account; A whole and continuous analysis method is put forward for solving the question of applying the member forces produced in the shape forming process to the finite element models of load analysis.
The nonlinear stability of single-chorded domes is analyzed. The factors affecting the stability of single-chorded domes are taken into account, such as the thickness of shells, the ratio of rise to span, the connections between botttom chords and bottom joints, the stiffness of chords takeing the place of top post-tensioning wires, members forces, etc. Lots of useful conclusions are drawn.
Last, the overall conclusion is drawn and the prospect of f
通过非线性有限元模拟成形分析，对新型单弦杆平板网架中的后张拉方法、节点连接形式以及穹顶成形形状进行了研究，为试验模型的建立奠定了理论基础。
设计制作了一个试验模型，进行了穹顶的成形试验和承载力试验。试验结果表明：本文提出的单弦杆平板网架模型具有很好的空间变形能力，可以张拉成形具有较大矢跨比的穹顶；穹顶的成形形状可以通过张拉下弦杆内的预留缝隙值进行很好的控制；后张拉力和成形初应力均较小，成形初应力中以腹杆弯曲应力为主，杆件轴向应力均较小；成形后的单弦杆穹顶具有很好的承载力性能，破坏形式为延性破坏，未出现穹顶失稳问题。
试验模型中，为了实现纯铰接连接，提出了一种新型的单向铰节点连接形式；成形试验中，由于后张拉力较小，提出了一种新型拧螺母抽钢丝的加载方法，很好的解决了千斤顶易超张拉和加载速率不均匀等问题；承载力试验中，为了解决加载点较多、单点加载较大的问题，提出了一种新型多点定滑轮竖向加载系统。试验结果表明，本文提出的各种改进方法是合理正确的，它们保证了试验过程的顺利完成，采集到了较理想的试验数据。
在考虑几何非线性和材料非线性的基础上，采用多种有限元模型对穹顶的成形试验和承载力试验进行了较系统的有限元模拟分析，主要考虑节点连接刚度、节点域大小、节点和支座偏心、成形初应力等因素的影响。得出了一系列重要结论。
有限元分析中，采用节点增设独立梁单元的方法，研究了鸭嘴半刚性连接和上弦节点域大小对穹顶成形过程的影响；提出了统一连续的分析方法，很好的解决了成形初应力向穹顶承载力有限元分析模型中的施加问题。
对单弦杆穹顶的非线性稳定性进行了较为系统的分析研究，考虑了穹顶网壳厚度、矢跨比、张拉下弦杆与下弦节点的连接形式和上弦张拉钢丝补杆刚度以及杆件受力对单弦杆穹顶
西安建筑科技大学博士学位论文
稳定性性能的影响。得到了一些重要的结论。
最后总结了本文的研究工作并提出了对以后研究工作的展望。
The post-tensioned and shaped space trusses is a new space structural system. Compared with the traditional space trusses, the assembly of the new space trusses can be finished wholly at ground level and it improves validly the stability behavior of single-layer shells because of the use of out-of plane web members, so the new space structrual system will have the important advantage in the future application. Based on the research on the post-tensioned and shaped dome, a new single-chorded plane space truss formed dome with post-tension is put forward firistly in this paper, through model tests and nonlinear finite element analysis, the shape forming process and load behaviors is studied. The paper mainly does work as follows:
Though simulating the shape forming process of post-tensioned and shaped dome by nonlinear finite element analysis, the post-tensioning method and joint connections in the new single-chorded plane space truss as well as the forming shape of dome are studied. It lays a theoretical foundation for the design and manufacture of experimental model.
A single-chorded plane space truss experimental model of post-tensioned and shaped dome is finished and the shape forming test and ultimate load test are carried out. Results showed that the single-chorded plane space truss model in this paper has very good deformability and it can form dome with more rise-to-span ratio by means of a post-tensioning technique, the forming shape of dome can be controled by the gaps which are given in certain bottom chords, the post-tensioning forces and member forces produced in the shape forming process are small and the bending stress is much larger in comparison with the axial stress in the member forces, the post-tensioned and shaped single-chorded dome has satisfactory utimate load capacity with ductility-type failure and the instability failure has not happened.
In the test model, a new type joint connection of one-way hinge is put forward for achieving pure-hinge connection; In the shape forming test, because of the little post-tensioning forces, a new type loading method of twisting nut for drawing wire is put forward in stead of jack tensioning for solving the questions, such as super-tensioning and the nonuniform rate of loading; In the load
test, a new type vertical loading system of multiple-point fixed pulley is put forward for solving the questions, such as more loading points and larger vertical force at one loading point. Results showed that these improvement methods that are put forward in this paper are rational and correct, they ensured the test processes to be successful and gathering the fine test data.
Then an thorough computer modeling test using various finite element models with geometry and material nonlinear taken into account is carried out on a set of influencing factors, such as the stiffness of joint connections, the size of joint, the eccentricty of joint and support connections, the member forces produced in the shape forming process, etc. Lots of useful conclusions are drawn form computer modeling.
In the finite element analysis, by adding individual beam elements to model the connector between top joints and chords in finite element model, based on the experimental model, the factors affecting the shape forming process of dome, such as the bending stiffness of duckbilled connector and the size of top joints, can be taken into account; A whole and continuous analysis method is put forward for solving the question of applying the member forces produced in the shape forming process to the finite element models of load analysis.
The nonlinear stability of single-chorded domes is analyzed. The factors affecting the stability of single-chorded domes are taken into account, such as the thickness of shells, the ratio of rise to span, the connections between botttom chords and bottom joints, the stiffness of chords takeing the place of top post-tensioning wires, members forces, etc. Lots of useful conclusions are drawn.
Last, the overall conclusion is drawn and the prospect of f
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1.60 Ellen, P. E., Post-tensioning of steel structures, Proc. of IASS Symposium, Osaka, Elsevier Science Publishers, Amsterdam, Vol. 2, 1986, 289-296.
1.61 Clarke, M. J. and Hancock, G. J., The behavior and design of Stressed-Arch(Strarch) frames, Spatial, Lattice and Tension Structures, Proc. of IASS-ASCE Int. Symposium, Atlanta, 1994, 200-209.
1.62 Duncan, J.P. and Duncan, J. L., Folded developables, Proc. of R. Soc. Lond., U. K., A 383,1982, 191-205.
1.63 Schmidt, L. C., Erection of Space Trusses of Different Forms by Tensioning of Near Mechanisms, IASS Congress, Madrid, Spain, 1989, Vol. 4: 12P.
1.64 Calladine, C. R., Buckminster Fuller's tensegrity structures and Clerk Maxwell's rule for the construction of stiff frames, Int. J. Solids Struct., 1978, 14(3): 161-172.
1.65 Tan, T. H. and Schmidt, L. C., Ultimate load behaviour of a barrel vault space truss, Intl. J. Space structure, 1986/87, 2(1): 1-10.
1.66 陈务军，付功义等，带肋局部双层网壳的失稳特征研究，工程力学，2002，19(4)：61-66．
1.67 任小强，陈务军等，ANSYS 在分析局部双层网壳几何非线性时的应用，第十届空间结构学术会议论文集，北京，中国建材工业出版社，2002，226-230．
1.68 陈务军，付功义等，大矢跨比局部双层球面网壳的形式及稳定性分析，空间结构，2002，8(1)：19-28．
1.69 陈务军，付功义等，带肋局部双层球面网壳稳定性分析，空间结构，2001，7(3)：33-39．
1.70 夏开全，董石麟，局部双层网壳结构的几何非线性分析，空间结构，1999，5(2)：29-37．
1.71 陈东，董石麟，局部双层柱面网壳的几何非线性稳定分析，空间结构，1999，5(3)：33-39．
1.72 Dehdashti, G., and Schmidt, L. C., Dome-Shaped Space Trusses Formed by means of post-tensioning, J. Struct. Engrg.., ASCE, 1996, 122(10): 1240-1245.
1.73 Schmidt, L. C., and Dehdashti, G., Curved space trusses formed from single-chorded planar space trusses, ACMSM, 13 :Proc., 13th Australasian Conf. on the Mech. of Struct. and Materials, L. C. Schmidt, ed., Univ. of Wollongong, Australia, 1993, 777-784.
1.74 Schmidt, L. C., Li, H. and Chua, M., Posttensioned and Shaped Hexagonal Grid Dome:Test and Analysis, J. of Structural Engineering, ASCE, 1998, 124(6): 696-703.
1.75 Li, H. and Schmidt, L. C., Ultimate Load Test and Analysis of a Retrofitted Model Steel Dome, Int. J. Space Structrues, 1998, 13(2):53-63.
1.76 Schmidt, L. C. and Selby, S., Domical space trusses and braced domes:Shaping, Ultimate Strength and Stiffness, Int. J. Space Structrues, 1999, 14(1):17-23.
1.77 Schmidt, L. C., and Li, H. ,Shape formation of deployable metal domes, Int. J. Space structure, 1995, 10(4): 189-194.
1.78 Kim Jin Woo, Hao Jiping and Lee Kang-Woon, New attachment device for post-tensioning of full size scale space truss, The Eighth East Asia-Paci c Conference on Structural Engineering and Construction 5-7 December 2001, Nanyang Technological Univesity, Singapore, 1034-1039.
1.79 Li, H., and Ding, J., A Micro-computer Optimisation Program for Tower Crane Beam, J. of Qingdao Institude Architecture and Engineering, 1990,11 (3): 10-16(Chinese).
1.80 Kim Jin Woo and Hao Jiping, Behavior Characteristic of a Full-Size Scale Pyramidal Space Truss Unit, Journal of Civil Engineering, KSCE, 2002, 6(1):33-38.
1.81 Li, H., Chua, M. and Schmidt, L. C., Shape prediction of a post-tensioned and shaped dome, Advances in Steel Structures, Eds. S. L. Chan and J. G. Tong, Elsevier Sci. Ltd., Oxford, 1996:309-314.
1.82 Li, H. and Schmidt, L. C., Posttensioned and Shaped Hypar space Trusses, J. Struct. Engrg., ASCE, 1997,123(2): 130-137.
1.83 Punniyakotty, N. M. and Richard Liew J. Y. and Shanmugam N. E., Nonlinear analysis of Self-Erecting framework by Cable-Tensioning Technique, J. Struct. Engrg., ASCE, 2000, 126(3): 361-370.
1.84 Livesley. R. K. and Chandler, D. B., Stability functions for structural frameworks, Manchester University, UK. 1956.
1.85 Smith, E. A., Space truss nonlinear analysis, J. Struct. Engrg., ASCE, 1984, 110(4): 688～705.
1.86 张汝清，詹先义，非线性有限元分析，重庆：重庆大学出版社，1990．
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1.88 董石麟，钱若军，空间网格结构分析理论与计算方法，北京：中国建筑工业出版社，2000．
1.89 雷晓燕，有限元法，北京：中国铁道出版社，2000．
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1.10 尹德钰，肖炽，20年来中国空间结构的施工与质量问题，第十届空间结构学术会议论文集，北京：中国建材工业出版社，2002，53-63．
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1.15 罗尧治，胡宁，董石麟，肖炽，网壳结构“折叠展开式”计算机同步控制整体提升施工技术的工程应用，第十届空间结构学术会议论文集，北京：中国建材工业出版社，2002，678-685．
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1.22 崔恒忠，曹资等，可展开折叠式空间结构模型试验研究，空间结构，1997，3(1)：43-47．
1.23 陈务军，关富玲等，一个空间伸展臂方案设计、展开过程与受力分析，空间结构，1997，3(2)：18-24．
1.24 陈向阳，关富玲等，复杂剪式铰结构的几何分析和设计，空间结构，1998，4(1)：45-51．
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1.29 张京街，关富玲等，带弹簧节点的大型构架式展开天线结构的设计和研究，空间结构，2000，6(2)：30-37．
1.30 陈向阳，关富玲等，可展星载抛物面天线结构设计，空间结构，2000，6(4)：41-46．
1.31 胡其彪，关富玲等，四变形单元构造旋转抛物面结构的研究—基于可变对角杆技术，空间结构，2001，7(1)：51-58．
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1.38 郭璐，蓝天，膜结构的初始形状判定与静力分析及其模型试验研究，第七届空间结构学术会议论文集，山东-文登，1994，392-397．
1.39 李中利，吴健生，黄大，膜结构找形受荷与剪裁分析，第八届空间结构学术会议论文集，开封，1997，299-304．
1.40 曹立岭，罗尧治等，索膜结构预张力取值的研究，第十届空间结构学术会议论文集，北京，中国建材工业出版朴，2002，268-274．
1.41 蔡兴东，袁英战，成都野生动物世界主表演场膜屋盖设计及施工，第十届空间结构学术会议论文集，北京，中国建材工业出版社，2002，719-722．
1.42 王洪军，刘锡良，陈志华，动力松驰法在索穹顶施工过程分析中的应用，第十届空间结构学术会议论文集，北京，中国建材工业出版社，2002，392-399．
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1.44 曹喜，刘锡良，张拉整体结构的预应力优化设计，空间结构，1998，4(1)：32-37．
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1.46 袁行飞，董石麟，索穹顶结构施工控制反分析，建筑结构学报，2001，22(2)：75-79．
1.47 袁行飞，董石麟，索穹顶结构整体可行预应力概念及其应用，土木工程学报，2001，34(2)：33-37．
1.48 黄呈伟，邓宜等，索穹顶结构模型试验研究，空间结构，1999，5(3)：40-46．
1.49 刘锡良，夏定武，索穹顶与张拉整体穹顶，空间结构，1997，3(2)：10-17．
1.50 吕晶，徐国彬，鞍山体育中心劲柔索张拉穹顶屋盖设计与施工，第十届空间结构学术会议论文集，北京，中国建材工业出版社，2002，759-763．
1.51 白正仙，刘锡良，李义生，新型空间结构形式—张弦梁结构，空间结构，2001，7(2)：33-38．
1.52 周观根，俞水其等，张弦立体桁架结构施工新技术，第十届空间结构学术会议论文集，北京，中国建材工业出版社，2002，842-847．
1.53 陈以一，沈祖炎等，上海埔东国际机场候机楼R2钢屋架足尺试验研究，建筑结构学报，1999，20(2)：9-17．
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1.55 Cuoco,D.A.,State-of-the-art of apace frame roof structures,Long Span Roof Structures,Proc. of ASCE Symposium,Missouri,1981:1-47．
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1.58 Ellen,P.E.,The design and development of post-tensioning steel structures,Proc. of lSt Pacific Structural
Steel Conf.,Auckland, 1986, Vol. 3:69-74.
1.59 Saar, O. S., Self-erecting two-layer steel prefabricated arch, Proc. of 3rd Conf. on Space Structures, Ed. H. Nooshihn,Elsevier Applied Science Publishers, England, 1984, 823-827.
1.60 Ellen, P. E., Post-tensioning of steel structures, Proc. of IASS Symposium, Osaka, Elsevier Science Publishers, Amsterdam, Vol. 2, 1986, 289-296.
1.61 Clarke, M. J. and Hancock, G. J., The behavior and design of Stressed-Arch(Strarch) frames, Spatial, Lattice and Tension Structures, Proc. of IASS-ASCE Int. Symposium, Atlanta, 1994, 200-209.
1.62 Duncan, J.P. and Duncan, J. L., Folded developables, Proc. of R. Soc. Lond., U. K., A 383,1982, 191-205.
1.63 Schmidt, L. C., Erection of Space Trusses of Different Forms by Tensioning of Near Mechanisms, IASS Congress, Madrid, Spain, 1989, Vol. 4: 12P.
1.64 Calladine, C. R., Buckminster Fuller's tensegrity structures and Clerk Maxwell's rule for the construction of stiff frames, Int. J. Solids Struct., 1978, 14(3): 161-172.
1.65 Tan, T. H. and Schmidt, L. C., Ultimate load behaviour of a barrel vault space truss, Intl. J. Space structure, 1986/87, 2(1): 1-10.
1.66 陈务军，付功义等，带肋局部双层网壳的失稳特征研究，工程力学，2002，19(4)：61-66．
1.67 任小强，陈务军等，ANSYS 在分析局部双层网壳几何非线性时的应用，第十届空间结构学术会议论文集，北京，中国建材工业出版社，2002，226-230．
1.68 陈务军，付功义等，大矢跨比局部双层球面网壳的形式及稳定性分析，空间结构，2002，8(1)：19-28．
1.69 陈务军，付功义等，带肋局部双层球面网壳稳定性分析，空间结构，2001，7(3)：33-39．
1.70 夏开全，董石麟，局部双层网壳结构的几何非线性分析，空间结构，1999，5(2)：29-37．
1.71 陈东，董石麟，局部双层柱面网壳的几何非线性稳定分析，空间结构，1999，5(3)：33-39．
1.72 Dehdashti, G., and Schmidt, L. C., Dome-Shaped Space Trusses Formed by means of post-tensioning, J. Struct. Engrg.., ASCE, 1996, 122(10): 1240-1245.
1.73 Schmidt, L. C., and Dehdashti, G., Curved space trusses formed from single-chorded planar space trusses, ACMSM, 13 :Proc., 13th Australasian Conf. on the Mech. of Struct. and Materials, L. C. Schmidt, ed., Univ. of Wollongong, Australia, 1993, 777-784.
1.74 Schmidt, L. C., Li, H. and Chua, M., Posttensioned and Shaped Hexagonal Grid Dome:Test and Analysis, J. of Structural Engineering, ASCE, 1998, 124(6): 696-703.
1.75 Li, H. and Schmidt, L. C., Ultimate Load Test and Analysis of a Retrofitted Model Steel Dome, Int. J. Space Structrues, 1998, 13(2):53-63.
1.76 Schmidt, L. C. and Selby, S., Domical space trusses and braced domes:Shaping, Ultimate Strength and Stiffness, Int. J. Space Structrues, 1999, 14(1):17-23.
1.77 Schmidt, L. C., and Li, H. ,Shape formation of deployable metal domes, Int. J. Space structure, 1995, 10(4): 189-194.
1.78 Kim Jin Woo, Hao Jiping and Lee Kang-Woon, New attachment device for post-tensioning of full size scale space truss, The Eighth East Asia-Paci c Conference on Structural Engineering and Construction 5-7 December 2001, Nanyang Technological Univesity, Singapore, 1034-1039.
1.79 Li, H., and Ding, J., A Micro-computer Optimisation Program for Tower Crane Beam, J. of Qingdao Institude Architecture and Engineering, 1990,11 (3): 10-16(Chinese).
1.80 Kim Jin Woo and Hao Jiping, Behavior Characteristic of a Full-Size Scale Pyramidal Space Truss Unit, Journal of Civil Engineering, KSCE, 2002, 6(1):33-38.
1.81 Li, H., Chua, M. and Schmidt, L. C., Shape prediction of a post-tensioned and shaped dome, Advances in Steel Structures, Eds. S. L. Chan and J. G. Tong, Elsevier Sci. Ltd., Oxford, 1996:309-314.
1.82 Li, H. and Schmidt, L. C., Posttensioned and Shaped Hypar space Trusses, J. Struct. Engrg., ASCE, 1997,123(2): 130-137.
1.83 Punniyakotty, N. M. and Richard Liew J. Y. and Shanmugam N. E., Nonlinear analysis of Self-Erecting framework by Cable-Tensioning Technique, J. Struct. Engrg., ASCE, 2000, 126(3): 361-370.
1.84 Livesley. R. K. and Chandler, D. B., Stability functions for structural frameworks, Manchester University, UK. 1956.
1.85 Smith, E. A., Space truss nonlinear analysis, J. Struct. Engrg., ASCE, 1984, 110(4): 688～705.
1.86 张汝清，詹先义，非线性有限元分析，重庆：重庆大学出版社，1990．
1.87 王勖成，邵敏，有限单元法基本原理和数值方法(第2版)[M]，北京：清华大学出版社，1997．
1.88 董石麟，钱若军，空间网格结构分析理论与计算方法，北京：中国建筑工业出版社，2000．
1.89 雷晓燕，有限元法，北京：中国铁道出版社，2000．