凯威特型带肋局部双层网壳形状优化及稳定性分析
|
摘要
网壳具有优美的建筑造型,不论在建筑平面、立面或型体都能给设计师以充分的创造自由。薄壳与网架结构不能实现的形态,网壳结构几乎都可以实现。这种结构形式既能表现静态美,又可通过平面与立面切割以及网格、支承与杆件等变化表现动态美。现在,网壳结构已广泛应用于体育建筑、机场建筑、展览馆、工业厂房等各类建筑中。
带肋局部双层球面网壳是单双层球面网壳的一种,这种网壳能够将单层和双层网壳各自的优点很好结合在一起。近些年来,出现了不少这种网壳形式的应用。凯威特型网壳结构网格均匀,刚度大,是一种应用最为广泛的网壳结构形式之一。但凯威特型单层球面网壳的稳定性能较差；凯威特型双层球面网壳杆件、节点数量大,杆件视觉效果不好。为了克服这两种网壳的缺点,根据网壳结构主要传力路线特点、建筑美观以及经济指标的要求,在凯威特型单层球面网壳主肋处增加下弦杆和腹杆,由此形成凯威特型带肋局部双层网壳。这是一种受力合理、结构性能优异、形式简洁美观、经济实用的局部双层网壳结构形式。目前,国内外的研究重点主要集中在单层网壳的稳定性和极限承载力以及双层网壳的优化和应用等方面,而有关带肋局部双层网壳体系的研究相对较少,特别是形状优化方面的研究更是少见。
本文主要研究的是凯威特型带肋局部双层网壳的形状优化。首先运用大型有限元分析软件ALGOR对不同跨度、双层区厚度下的凯威特型带肋局部双层网壳进行静力分析,研究网壳结构随矢跨比和厚跨比变化时的受力特性和位移特性,为下面凯威特型带肋局部双层网壳结构形状优化初始宏观曲面参数和杆件截面的选取提供依据。这将有助于减少网壳结构形状优化的迭代次数。
然后针对球面网壳的特点,本文提出了网壳结构形状优化的两级算法。采用FORTRAN语言编写网壳形状优化两级算法设计程序,并与大型有限元分析软件ALGOR连接。对不同跨度、不同类型的凯威特型单层球面网壳进行优化并分析验证此算法是否具有有效性和可靠性。在此基础上利用上述算法对凯威特型带肋局部双层网壳进行形状优化设计,探究网壳最优矢跨比、厚跨比的分布范围及网壳单层区网格划分密度对网壳优化结果的影响。在带肋局部双层网壳的选型过程中,若已确定使用凯威特型带肋局部双层网壳,这些结果将对其宏观曲面参数、网格尺寸及杆件截面的选择具有一定的参考价值。
随着结构优化的发展,网壳优化的结果总的趋势是跨度越来越大,厚度越来越薄；但与此同时,网壳结构稳定性问题也变的突出起来。因此本文利用ALGOR软件对优化后的凯威特型带肋局部双层网壳进行屈曲分析(特征值屈曲分析、考虑初始缺陷的非线性屈曲分析),确定优化后凯威特型带肋局部双层网壳的屈曲模态及优化后网壳的屈曲荷载与设计荷载的比值。以此作为评价本文网壳优化结果优劣的依据。
Dome has a beautiful architectural style, whether in architecture, facade, or type body. It can give designers the creative freedom to the full. Shape can not be achieved with the grid structure, almost all the dome structure can do. This structure not only can performance static beauty,but also dynamic beauty by plane and cross cutting and the grid, supporting such changes and pole.Now, dome structure has been widely used in sports buildings, airport buildings, exhibition halls, industrial plants and other kinds of buildings.
Partial double-ribbed dome is a single layer of a spherical dome,and it can combine the strengths of single-dome and double dome. In recent years,many forms of application of this dome appear. Because of uniform grid dome structure and rigidity, Kiewitt dome is the most widely used form of dome structure.But, Kiewitt spherical dome has poor stability and Kiewitt double spherical dome with a large number of bars and nodes has poor visual effects bar. According to the main power line transmission characteristics, aesthetic and economic indicators of construction demand of Dome structure,adding chords and belly bars in the bottom of the main rib of Kiewitt spherical dome, Kiewitt dome with partial double rib can be formed.It is a Double-dome local structure which is Reasonable force,Structure-performance, Simple and beautiful form,and ecomomical. Currently, Research priorities at home and abroad focus on optimization and application dome of Double dome and Stability and ultimate load of Single-layer dome,but the study of Partial double-ribbed dome System is seldom, especially in respect to the optimization of the shape.
This paper studies primarily the partial double Kiewitt ribbed shell shape optimization. First, we use finite element analysis software ALGOR to analysis Static characteristics of the Kiewitt partial double ribbed dome,which has different span and the thickness of double area. With span ratio and thick span ratio changes, Force characteristics and displacement characteristics of dome can be obtained.It can provide the basis for selection of dome structure for the following initial macroscopic surface shape optimization parameters.
Then, shape optimization of dome structure of the two algorithms accounted for the characteristics of spherical dome in this paper. FORTRAN language is adopted for writing program about the Sequential two-level algorithm of dome shape optimization,and connect to finite element analysis software named ALGOR. Using the optimized results of different span and types of dome show the validity and reliability of the algorithm. Then Shape Optimization of the dome, using the above algorithm,can get the optimal span ratios, the ratio of thickness to span the scope and parameters of surface macro-dome dome optimization results. In the double-dome ribbed local selection process, if the designer has identified to use Kiewitt partial double ribbed dome, these results can provide a reference value for selection macro-surface parameters for the dome, mesh size and bar section.
With the development of structural optimization, the trend of optimization results is that the span is growing and Thickness is getting thin. At the same time, Stability of dome structures also become prominent.This paper use ALGOR to analysis of stability of optimized Kiewitt dome with partial double rib, including Eigenvalue buckling analysis、Nonlinear buckling of a rational analysis of dome and Initial defect of nonlinear buckling analysis. Buckling mode and Buckling load and the ratio of design load of optimized Kiewitt dome with partial double rib can be obtained. So,it can be used to evaluate the advantage and disadvantages of Dome optimization results.
带肋局部双层球面网壳是单双层球面网壳的一种,这种网壳能够将单层和双层网壳各自的优点很好结合在一起。近些年来,出现了不少这种网壳形式的应用。凯威特型网壳结构网格均匀,刚度大,是一种应用最为广泛的网壳结构形式之一。但凯威特型单层球面网壳的稳定性能较差；凯威特型双层球面网壳杆件、节点数量大,杆件视觉效果不好。为了克服这两种网壳的缺点,根据网壳结构主要传力路线特点、建筑美观以及经济指标的要求,在凯威特型单层球面网壳主肋处增加下弦杆和腹杆,由此形成凯威特型带肋局部双层网壳。这是一种受力合理、结构性能优异、形式简洁美观、经济实用的局部双层网壳结构形式。目前,国内外的研究重点主要集中在单层网壳的稳定性和极限承载力以及双层网壳的优化和应用等方面,而有关带肋局部双层网壳体系的研究相对较少,特别是形状优化方面的研究更是少见。
本文主要研究的是凯威特型带肋局部双层网壳的形状优化。首先运用大型有限元分析软件ALGOR对不同跨度、双层区厚度下的凯威特型带肋局部双层网壳进行静力分析,研究网壳结构随矢跨比和厚跨比变化时的受力特性和位移特性,为下面凯威特型带肋局部双层网壳结构形状优化初始宏观曲面参数和杆件截面的选取提供依据。这将有助于减少网壳结构形状优化的迭代次数。
然后针对球面网壳的特点,本文提出了网壳结构形状优化的两级算法。采用FORTRAN语言编写网壳形状优化两级算法设计程序,并与大型有限元分析软件ALGOR连接。对不同跨度、不同类型的凯威特型单层球面网壳进行优化并分析验证此算法是否具有有效性和可靠性。在此基础上利用上述算法对凯威特型带肋局部双层网壳进行形状优化设计,探究网壳最优矢跨比、厚跨比的分布范围及网壳单层区网格划分密度对网壳优化结果的影响。在带肋局部双层网壳的选型过程中,若已确定使用凯威特型带肋局部双层网壳,这些结果将对其宏观曲面参数、网格尺寸及杆件截面的选择具有一定的参考价值。
随着结构优化的发展,网壳优化的结果总的趋势是跨度越来越大,厚度越来越薄；但与此同时,网壳结构稳定性问题也变的突出起来。因此本文利用ALGOR软件对优化后的凯威特型带肋局部双层网壳进行屈曲分析(特征值屈曲分析、考虑初始缺陷的非线性屈曲分析),确定优化后凯威特型带肋局部双层网壳的屈曲模态及优化后网壳的屈曲荷载与设计荷载的比值。以此作为评价本文网壳优化结果优劣的依据。
Dome has a beautiful architectural style, whether in architecture, facade, or type body. It can give designers the creative freedom to the full. Shape can not be achieved with the grid structure, almost all the dome structure can do. This structure not only can performance static beauty,but also dynamic beauty by plane and cross cutting and the grid, supporting such changes and pole.Now, dome structure has been widely used in sports buildings, airport buildings, exhibition halls, industrial plants and other kinds of buildings.
Partial double-ribbed dome is a single layer of a spherical dome,and it can combine the strengths of single-dome and double dome. In recent years,many forms of application of this dome appear. Because of uniform grid dome structure and rigidity, Kiewitt dome is the most widely used form of dome structure.But, Kiewitt spherical dome has poor stability and Kiewitt double spherical dome with a large number of bars and nodes has poor visual effects bar. According to the main power line transmission characteristics, aesthetic and economic indicators of construction demand of Dome structure,adding chords and belly bars in the bottom of the main rib of Kiewitt spherical dome, Kiewitt dome with partial double rib can be formed.It is a Double-dome local structure which is Reasonable force,Structure-performance, Simple and beautiful form,and ecomomical. Currently, Research priorities at home and abroad focus on optimization and application dome of Double dome and Stability and ultimate load of Single-layer dome,but the study of Partial double-ribbed dome System is seldom, especially in respect to the optimization of the shape.
This paper studies primarily the partial double Kiewitt ribbed shell shape optimization. First, we use finite element analysis software ALGOR to analysis Static characteristics of the Kiewitt partial double ribbed dome,which has different span and the thickness of double area. With span ratio and thick span ratio changes, Force characteristics and displacement characteristics of dome can be obtained.It can provide the basis for selection of dome structure for the following initial macroscopic surface shape optimization parameters.
Then, shape optimization of dome structure of the two algorithms accounted for the characteristics of spherical dome in this paper. FORTRAN language is adopted for writing program about the Sequential two-level algorithm of dome shape optimization,and connect to finite element analysis software named ALGOR. Using the optimized results of different span and types of dome show the validity and reliability of the algorithm. Then Shape Optimization of the dome, using the above algorithm,can get the optimal span ratios, the ratio of thickness to span the scope and parameters of surface macro-dome dome optimization results. In the double-dome ribbed local selection process, if the designer has identified to use Kiewitt partial double ribbed dome, these results can provide a reference value for selection macro-surface parameters for the dome, mesh size and bar section.
With the development of structural optimization, the trend of optimization results is that the span is growing and Thickness is getting thin. At the same time, Stability of dome structures also become prominent.This paper use ALGOR to analysis of stability of optimized Kiewitt dome with partial double rib, including Eigenvalue buckling analysis、Nonlinear buckling of a rational analysis of dome and Initial defect of nonlinear buckling analysis. Buckling mode and Buckling load and the ratio of design load of optimized Kiewitt dome with partial double rib can be obtained. So,it can be used to evaluate the advantage and disadvantages of Dome optimization results.
引文
[1]尹德钰,刘善维,钱若军.网壳结构设计[M].北京：中国建筑工业出版社,1996：54-375
[2]刘锡良.国外建筑钢结构应用概况[J].金属世界,2004,4：29-32
[3]曹品然,张海荣.上海国际会议中心大球壳设计研究[J].建筑钢结构进展.2001.3(3)：13-18
[4]董石麟,罗尧治,赵阳.大跨度空间结构的工程实践与学科发展[J].空间结构.2005.11(4)：3-10
[5]张毅刚.大跨度空间结构[M].机械工业出版社,2005：15-55
[6]刘锡良,韩庚华.网格结构设计与施工[M].天津：天津大学出版社,2004
[7]许素强,夏人伟.结构优化方法研究综述[J].航空学报,1995,16(4)：385-396
[8]张炳华,侯昶.土建结构优化设计[M].上海：同济大学出版社,1998：5-30
[9]汪树玉,刘国华,包志仁.结构优化设计的现状与进展[J].基建优化.1999
[10]石连栓.离散变量结构优化设计算法研究综述[J].天津职业技术师范学院学报.2001,Vol.11, No.1
[11]Vanderplatts G N, Moses F. Automated design of trusses for optimum geometry[J]. Struct. Div, ASCE, 1972,98(ST3):671-690
[12]王希成,钱令希.多层次联合的结构优化设计[J].计算结构力学及其应用,1991,7(4)
[13]Fleury C. Reconcilation of mathematical programming and optimality criteria methods. In:Morres A J (Eds). Foundation of Structural Optimization-A Unified Approach. New York:Wiley interscione, Chapter,1982; 10
[14]沈祖炎,陈扬骥.网架与网壳[M].上海：同济大学出版社,1996
[15]刘锡良,韩庚华.网格结构设计与施工[M].天津：天津大学出版社,2004：108-114
[16]JGJ61-2003网壳结构技术规程[S].北京：中国建筑工业出版社,2003
[17]GB 50017-2003钢结构设计规范[S].北京：中国建筑工业出版社,2003
[18]GB 50009-2001建筑结构荷载规范[S].北京：中国建筑工业出版社,2001
[19]GB 50011-2001建筑抗震设计规范[S].北京：中国建筑工业出版社,2001
[20]丁芸孙,刘罗静等.网架网壳设计与施工[M]..北京：中国建筑工业出版社,2006.8
[21]尚凌云.基于离散变量的网壳结构截面与形状优化设计[D].山东建筑工程学院,2002
[22]J L M eek, S Loganathan. Geometric and material nonlinear behavior of structures comprising circular rectangular hollow sections[A]. Progress in Shell and Spatial Structures[C], IASS,1989
[23]M Kashani, J G A Croll. Non-linear buckling response of spherical space-domes [A]. Progress in Shell and Spatial Structures[C], IASS,1989,4
[24]E.Salajegheh. Optimum design of structures with high-quality approximation of frequency constraints[J]. Advances in Engineering Software 31(2000):381-384
[25]W.M.Jenkins. On the application of natural algorithms to structural design optimization [J]. Engineering Structures. Vol.19,No 4,pp:302-308,1997
[26]刘锡良,王建明,周学军.拉索预应力网壳结构的优化设计[J].工业建筑.2001
[27]李萍,王守波,赵晓伟,鹿晓阳.双层肋环式球面网壳结构预应力优化设计[J].山东建筑工程学院学报.2003.18(3)：6-9
[28]尚凌云,鹿晓阳,周学军.基于离散变量的网壳结构截面优化设计[J].山东建筑工程学院学报.2002.17(2)：74-77
[29]沈世钊.网壳结构的稳定性[J].土木工程学报.1999.Vol.32,No.6
[30]曹正罡,范峰,沈世钊.单层球面网壳的弹塑性稳定性[J].土木工程学报.2006.39(10)：6-10
[31]吴剑国,张其林.网壳结构稳定性的研究进展[J].空间结构.2002.Vol.8,No.1
[32]张爱林,刘学春,张晓峰等.2008年奥运会羽毛球馆预应力张弦穹顶结构整体稳定分析[J]建筑结构.2006.Vol.36增刊
[33]刘博,刘桂臣,冯东利.大跨空间网壳结构界限跨度值研究[J].水利与建筑工程学报.2007.Vol.5.No.4：102-104
[34]蓝倜恩,钱若军.网架的最优网格与高度及其选型[J].建筑结构,1987,3：32-38
[35]D T W right. Membrance force and buckling in reticulated shells[J]. J. St ruct.Div, ASCE,1965,91(1).
[36]D Dubina. Computation models and numerical solution procedures for nonlinear of single layer lattice shells[J]. Int. J. Space Structures,1993,8 (3):3212333.
[37]Robert Levy, Ariel Hanaor&Nancy Rizzuto. Experimental Investigation of Prestressing in Double-Layer Grid[J]. International Journal of Space Structures.1994.1:21-26
[38]万党水.考虑几何非线性的结构形状优化设计及其工程应用研究[D].湖南大学.2006
[39]张年文,董石麟等.考虑几何非线性影响的单层网壳优化设计[J].空间结构.2003.Vol.9,No.1：31-34
[40]刘宗发,李正良,晏致涛.单层球面网壳优化分析[J].重庆建筑大学学报.2005.27(1)：67-70
[41]贺拥军.混沌优化法在双层圆柱面网壳结构优化中的应用[J].煤炭学报.2001.26(6)：663-666
[42]王法武,唐敢.考虑整体稳定的单层网壳截面优化设计[J].空间结构.2006.12(3)：31-34
[43]Schmit L A, Kicher T P. Synthesis of material and configuration selection[J]. J.Struc.Div. ASCE, 88:79-102
[44]Lipson S L, Gwin L B. Discrete sizing of trusses for optimal geometry [J]. J.Struc.Div. ASCE,1977, 103:1031-1046
[45]Jenkins W M. Towards structural optimization via the genetic algorithm. Comp[J]. Struc.,1991, 40(5):1321-1327
[46]Salajegheh E, Vanderplaats G N. Optimum design of trusses with sizing and shape variables[J]. Stru.Opt.,1993,6:79-85
[47]Robert Levy,Ariel Hanaor&Nancy Rizzuto. Experimental Investigation of Prestressing in Grid[J]. International Journal of Space Structures,1994,1:21-26
[48]马克俭,李晓红.预应力钢网壳结构的研究和应用.贵州工学院勘察设计研究院空间结构研究所,1994
[49]Rajan S.D. (Arizona State Univ). Sizing, shape, and topology design optimization of trusses using genetic algorithm.Source:Journal of Structural Engineering 12110 Oct 1995 ASCE p 1480-1487 0733-9445
[50]孙焕纯,王跃方,黄吉锋.离散变量桁架的形状优化设计[J].大连理工大学学报,1995，35(1)
[51]柴山,孙焕纯.求解一类(0,1)规划问题的相对差商法[J].系统工程学报,1996,Vol.11,No.1
[52]柴山,孙焕纯.求解一类(0,1,2)规划问题的二级定界组合算法[J].大连理工大学学报,1996,Vol.36,No.3
[53]石连拴,孙焕纯,柴山.两类变量综合处理的结构形状优化设计方法[J].计算力学学报.1999,Vol.16, No.4
[54]邓华,董石麟.空间网壳结构的形状优化[J].浙江大学学报(工学版),1999,33(4)：71-375
[55]邓华.空间网格结构基于离散变量的优化设计[J].空间结构,2000,6(3)：26-32
[56]孙焕纯,柴山,王跃方.离散变量结构优化设计[M].大连：大连理工大学出版社,1995
[57]柴山等.空间网架结构离散变量优化设计的研究[J].山东：山东工程学院学报,1995,8(4).
[58]聂文雅.基于离散变量球面网壳的形状优化设计[D].山东：山东建筑大学申请硕士学位论文,2008
[59]刘宗发.单层球面网壳的选型优化设计[D].重庆：重庆大学,2005：47-48
[60]唐可,田彩军,寇晓东.ALGOR结构分析高级教程[M].北京：清华大学出版社,2008
[61]郭玉霞.单层球面网壳选型优化[D].山东：山东建筑大学申请硕士学位论文,2007
[62]宋秀卿.双层柱面网壳选型优化[D].山东：山东建筑大学申请硕士学位论文,2009
[63]曲晓东,赵丽红.离散变量结构优化设计方法的研究进展与展望[J].辽宁工学院学报.2004
[64]郭鹏飞.离散变量结构优化的拟满应力设计方法[J].工程力学,2000,(1)
[65]郭鹏飞.离散变量结构优化的斐波那契遗传算法[J].辽宁工学院学报,2003,23(2)：2-4
[66]杨冬梅.基于实验设计法的桁架结构几何优化[J].力学与实践,1999,(6)：38-39
[67]陈建军等.多工况下具有可靠性约束的桁架结构拓扑优化设计[J].固体力学学报,2000,(1)：12-17
[68]王跃方等.离散变量结构拓扑优化设计的研究[J].固体力学学报,1998,(1)：61-62
[69]徐菁,杨松森,刁延松.单层球面网壳的优化设计[J].空间结构.2006.12(3)：35-37
[2]刘锡良.国外建筑钢结构应用概况[J].金属世界,2004,4：29-32
[3]曹品然,张海荣.上海国际会议中心大球壳设计研究[J].建筑钢结构进展.2001.3(3)：13-18
[4]董石麟,罗尧治,赵阳.大跨度空间结构的工程实践与学科发展[J].空间结构.2005.11(4)：3-10
[5]张毅刚.大跨度空间结构[M].机械工业出版社,2005：15-55
[6]刘锡良,韩庚华.网格结构设计与施工[M].天津：天津大学出版社,2004
[7]许素强,夏人伟.结构优化方法研究综述[J].航空学报,1995,16(4)：385-396
[8]张炳华,侯昶.土建结构优化设计[M].上海：同济大学出版社,1998：5-30
[9]汪树玉,刘国华,包志仁.结构优化设计的现状与进展[J].基建优化.1999
[10]石连栓.离散变量结构优化设计算法研究综述[J].天津职业技术师范学院学报.2001,Vol.11, No.1
[11]Vanderplatts G N, Moses F. Automated design of trusses for optimum geometry[J]. Struct. Div, ASCE, 1972,98(ST3):671-690
[12]王希成,钱令希.多层次联合的结构优化设计[J].计算结构力学及其应用,1991,7(4)
[13]Fleury C. Reconcilation of mathematical programming and optimality criteria methods. In:Morres A J (Eds). Foundation of Structural Optimization-A Unified Approach. New York:Wiley interscione, Chapter,1982; 10
[14]沈祖炎,陈扬骥.网架与网壳[M].上海：同济大学出版社,1996
[15]刘锡良,韩庚华.网格结构设计与施工[M].天津：天津大学出版社,2004：108-114
[16]JGJ61-2003网壳结构技术规程[S].北京：中国建筑工业出版社,2003
[17]GB 50017-2003钢结构设计规范[S].北京：中国建筑工业出版社,2003
[18]GB 50009-2001建筑结构荷载规范[S].北京：中国建筑工业出版社,2001
[19]GB 50011-2001建筑抗震设计规范[S].北京：中国建筑工业出版社,2001
[20]丁芸孙,刘罗静等.网架网壳设计与施工[M]..北京：中国建筑工业出版社,2006.8
[21]尚凌云.基于离散变量的网壳结构截面与形状优化设计[D].山东建筑工程学院,2002
[22]J L M eek, S Loganathan. Geometric and material nonlinear behavior of structures comprising circular rectangular hollow sections[A]. Progress in Shell and Spatial Structures[C], IASS,1989
[23]M Kashani, J G A Croll. Non-linear buckling response of spherical space-domes [A]. Progress in Shell and Spatial Structures[C], IASS,1989,4
[24]E.Salajegheh. Optimum design of structures with high-quality approximation of frequency constraints[J]. Advances in Engineering Software 31(2000):381-384
[25]W.M.Jenkins. On the application of natural algorithms to structural design optimization [J]. Engineering Structures. Vol.19,No 4,pp:302-308,1997
[26]刘锡良,王建明,周学军.拉索预应力网壳结构的优化设计[J].工业建筑.2001
[27]李萍,王守波,赵晓伟,鹿晓阳.双层肋环式球面网壳结构预应力优化设计[J].山东建筑工程学院学报.2003.18(3)：6-9
[28]尚凌云,鹿晓阳,周学军.基于离散变量的网壳结构截面优化设计[J].山东建筑工程学院学报.2002.17(2)：74-77
[29]沈世钊.网壳结构的稳定性[J].土木工程学报.1999.Vol.32,No.6
[30]曹正罡,范峰,沈世钊.单层球面网壳的弹塑性稳定性[J].土木工程学报.2006.39(10)：6-10
[31]吴剑国,张其林.网壳结构稳定性的研究进展[J].空间结构.2002.Vol.8,No.1
[32]张爱林,刘学春,张晓峰等.2008年奥运会羽毛球馆预应力张弦穹顶结构整体稳定分析[J]建筑结构.2006.Vol.36增刊
[33]刘博,刘桂臣,冯东利.大跨空间网壳结构界限跨度值研究[J].水利与建筑工程学报.2007.Vol.5.No.4：102-104
[34]蓝倜恩,钱若军.网架的最优网格与高度及其选型[J].建筑结构,1987,3：32-38
[35]D T W right. Membrance force and buckling in reticulated shells[J]. J. St ruct.Div, ASCE,1965,91(1).
[36]D Dubina. Computation models and numerical solution procedures for nonlinear of single layer lattice shells[J]. Int. J. Space Structures,1993,8 (3):3212333.
[37]Robert Levy, Ariel Hanaor&Nancy Rizzuto. Experimental Investigation of Prestressing in Double-Layer Grid[J]. International Journal of Space Structures.1994.1:21-26
[38]万党水.考虑几何非线性的结构形状优化设计及其工程应用研究[D].湖南大学.2006
[39]张年文,董石麟等.考虑几何非线性影响的单层网壳优化设计[J].空间结构.2003.Vol.9,No.1：31-34
[40]刘宗发,李正良,晏致涛.单层球面网壳优化分析[J].重庆建筑大学学报.2005.27(1)：67-70
[41]贺拥军.混沌优化法在双层圆柱面网壳结构优化中的应用[J].煤炭学报.2001.26(6)：663-666
[42]王法武,唐敢.考虑整体稳定的单层网壳截面优化设计[J].空间结构.2006.12(3)：31-34
[43]Schmit L A, Kicher T P. Synthesis of material and configuration selection[J]. J.Struc.Div. ASCE, 88:79-102
[44]Lipson S L, Gwin L B. Discrete sizing of trusses for optimal geometry [J]. J.Struc.Div. ASCE,1977, 103:1031-1046
[45]Jenkins W M. Towards structural optimization via the genetic algorithm. Comp[J]. Struc.,1991, 40(5):1321-1327
[46]Salajegheh E, Vanderplaats G N. Optimum design of trusses with sizing and shape variables[J]. Stru.Opt.,1993,6:79-85
[47]Robert Levy,Ariel Hanaor&Nancy Rizzuto. Experimental Investigation of Prestressing in Grid[J]. International Journal of Space Structures,1994,1:21-26
[48]马克俭,李晓红.预应力钢网壳结构的研究和应用.贵州工学院勘察设计研究院空间结构研究所,1994
[49]Rajan S.D. (Arizona State Univ). Sizing, shape, and topology design optimization of trusses using genetic algorithm.Source:Journal of Structural Engineering 12110 Oct 1995 ASCE p 1480-1487 0733-9445
[50]孙焕纯,王跃方,黄吉锋.离散变量桁架的形状优化设计[J].大连理工大学学报,1995，35(1)
[51]柴山,孙焕纯.求解一类(0,1)规划问题的相对差商法[J].系统工程学报,1996,Vol.11,No.1
[52]柴山,孙焕纯.求解一类(0,1,2)规划问题的二级定界组合算法[J].大连理工大学学报,1996,Vol.36,No.3
[53]石连拴,孙焕纯,柴山.两类变量综合处理的结构形状优化设计方法[J].计算力学学报.1999,Vol.16, No.4
[54]邓华,董石麟.空间网壳结构的形状优化[J].浙江大学学报(工学版),1999,33(4)：71-375
[55]邓华.空间网格结构基于离散变量的优化设计[J].空间结构,2000,6(3)：26-32
[56]孙焕纯,柴山,王跃方.离散变量结构优化设计[M].大连：大连理工大学出版社,1995
[57]柴山等.空间网架结构离散变量优化设计的研究[J].山东：山东工程学院学报,1995,8(4).
[58]聂文雅.基于离散变量球面网壳的形状优化设计[D].山东：山东建筑大学申请硕士学位论文,2008
[59]刘宗发.单层球面网壳的选型优化设计[D].重庆：重庆大学,2005：47-48
[60]唐可,田彩军,寇晓东.ALGOR结构分析高级教程[M].北京：清华大学出版社,2008
[61]郭玉霞.单层球面网壳选型优化[D].山东：山东建筑大学申请硕士学位论文,2007
[62]宋秀卿.双层柱面网壳选型优化[D].山东：山东建筑大学申请硕士学位论文,2009
[63]曲晓东,赵丽红.离散变量结构优化设计方法的研究进展与展望[J].辽宁工学院学报.2004
[64]郭鹏飞.离散变量结构优化的拟满应力设计方法[J].工程力学,2000,(1)
[65]郭鹏飞.离散变量结构优化的斐波那契遗传算法[J].辽宁工学院学报,2003,23(2)：2-4
[66]杨冬梅.基于实验设计法的桁架结构几何优化[J].力学与实践,1999,(6)：38-39
[67]陈建军等.多工况下具有可靠性约束的桁架结构拓扑优化设计[J].固体力学学报,2000,(1)：12-17
[68]王跃方等.离散变量结构拓扑优化设计的研究[J].固体力学学报,1998,(1)：61-62
[69]徐菁,杨松森,刁延松.单层球面网壳的优化设计[J].空间结构.2006.12(3)：35-37