空间钢框架高等分析方法研究
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摘要
传统的分析和设计方法存在很多局限性,随着计算机技术和结构分析设计理论的发展,高等分析和设计方法将是未来结构分析和设计实践的一个重要发展方向。本论文建立了基于梁柱理论稳定函数和精细塑性铰模型的空间梁单元。单元能够考虑几何非线性、材料非线性、剪切变形、弯曲缩短、残余应力以及连接柔性等非线性效应的影响。单元程序利用商用有限元ANSYS软件的二次开发技术编写,并在ANSYS软件平台上运行。单元能够应用于空间钢结构的静力或动力高等分析。论文的主要工作详述如下。
推导了考虑剪切影响的转角位移方程,并利用它计算单元内力,其中的稳定函数可以准确考虑单元P ?δ效应。讨论了单元考虑弯曲缩短影响(弓形效应)的方法,算例分析表明,非迭代近似方法能够有效考虑弯曲缩短的影响。利用虚位移原理和稳定插值函数推导了空间薄壁梁单元的刚度方程。考察了不同形式的几何刚度矩阵对单元精度的影响,在非线性分析中若采用相同的方法计算单元内力,则几何刚度矩阵的形式和精度对分析结果几乎没有影响。采用节点方向矩阵和单元刚体转角两种不同方法更新单元坐标转换矩阵,对两种方法进行了对比分析;静力分析时,两种方法计算的结果比较接近;但进行动力分析时,前者的稳定性和收敛性要优于后者。
进行弹性分析时,采用总量算法计算单元内力要优于采用增量算法计算单元内力。进行几何非线性分析时,本论文梁单元的精度总体上要优于ANSYS软件中的BEAM4单元,与BEAM189单元的精度相当甚至略优于BEAM189单元。
采用精细塑性铰模型考虑材料非线性,单元能够较好地模拟平面和空间钢框架结构的弹塑性承载力极限状态,单元分析结果与塑性区模型BEAM189单元分析的结果吻合良好。单元具有良好的精度,结构若只承受节点荷载,则每根构件通常只需用1个单元模拟;若构件承受分布荷载,则每根构件只需用2个单元模拟即可。而采用ANSYS软件中的BEAM189单元进行弹塑性分析时,每根构件通常要用3～5个单元模拟才能获得较准确的结果。本论文精细塑性铰梁单元不需要进行数值积分运算,因此单元计算效率很高;进行大型结构分析时,与使用数值积分单元相比,使用本论文的梁单元可以节省大量计算时间。
采用修正梁单元方法考虑连接柔性的影响,单元分析结果与相关文献的试验结果吻合较好。采用ANSYS软件中的COMBIN39单元也能有效考虑连接柔性的影响,COMBIN39组合BEAM189单元分析的结果与本论文梁单元分析的结果吻合很好。采用本论文梁单元考虑连接半刚性时,单元使用方便,可以采用与刚性连接框架相同的模型;而采用COMBIN39单元模拟连接时,建模过程和单元实常数的确定比较复杂,节点、单元数量多。
利用ANSYS软件的用户可编程特性(UPFs)采用FORTRAN语言编写了精细塑性铰梁单元程序,通过编译以及与ANSYS程序的连接获得了用户定制ANSYS程序。运行定制的ANSYS程序可以方便地调用用户梁单元,用于平面和空间钢结构的静力或动力分析。进行结构分析时用户梁单元能够与ANSYS中的标准单元联合使用,因此可充分利用ANSYS软件丰富的单元、强大的求解器和便捷的前后处理功能。
针对精细塑性铰梁单元和数值积分梁单元两种不同类型的单元,对刚性和半刚性钢框架进行了弹性及弹塑性地震时程响应对比分析。对于塑性发展不严重或者冗余度较高(例如框架支撑结构)的刚性或半刚性钢结构,采用本论文精细塑性铰梁单元进行动力时程分析能够获得比较准确的结果。而对于塑性发展严重可能导致局部楼层侧向刚度发生急剧削弱的结构,则应采用数值积分单元进行分析,才能获得准确的结果。
在地震激励作用下,与刚性框架相比,半刚性框架将具有更小的基底总剪力响应。随着连接刚度的减小,同一地震激励引起的最大基底总剪力也减小;但半刚性框架的最大侧移和层间侧移并不总是随着连接刚度的减小而增大。
Traditional analysis and design methods have many limitations. With the development of computer technology and theories of structural analysis and design, the advanced analysis and design method must be an important trend in future analysis and design practices. A three-dimensional beam element, which based on stability functions of beam-column theory and refined plastic hinge model, has been developed. Nonlinear effects, such as geometic nonlinearity, material nonlinearity, shear deformation, effect of curvature shortening, residual stress and connection flexibility, can be considered in element formulation.
Element routines are developed using the redevelopment technique of commercial finite element ANSYS software, and run on the platform of ANSYS software. Element can be used for static or dynamic advanced analysis of three-dimensional steel structures. The followings are the mian contents in detail. The slope-deflection equations including shear effect are derived and used to compute element internal forces, in which the element P ?δeffect can be accurately captured by the stability functions. Methods to consider the effect of curvature shortening (bowing effect) are discussed. Results of analysis examples show that the bowing effect can be taken into account effectively by an approximate non-iterative method. Using virtual displacement principle and stability interpolation functions, stiffness equation of a three-dimensional thin-walled beam element is derived. Influences of different types of element stiffness matrixes on the accuracy of element are observed. It is found that the form and accuracy of the geometric stiffness matrix used in a nonlinear analysis almost have no effect on the accuracy of the final solution when the same expressions are used to calculate the internal forces. The element transformation matrix is updated either by joint orientation matrix or by element rigid body rotation, and difference between two methods is analysed. It is found that the results of two methods are close in static analysis, whereas in dynamic analysis the former has better robustness and convergence than the latter.
In elastic analysis, algorithm for internal forces calculation based on the total equilibrium condition is better than that based on the incremental equilibrium condition. As far as geometric nonlinear analysis is concerned, the performance of proposed beam element is better than that of BEAM4 element, and is comparable to, to some extent even better than, that of BEAM189 element in ANSYS program.
Material nonlinearity is modeled using refined plastic hinge approach. Inelastic strength limit state of planar and space steel frames can be captured by proposed element. The analytical results of proposed element are compared fairly well with those of plastic zone based beam element BEAM189. In structural inelastic analysis, only one or two proposed elements for each member loaded by nodal loads or by distributed loads, respectively, in contrast to three to five BEAM189 elements for each member, are needed to achieve acceptable accuracy. The proposed beam element has excellent efficiency since it does not do any numerical integration operation. A lot of computing time can therefore be saved in the analysis of large-scale structures by using proposed element, as compared to using numerically integrated element.
Modified beam element method is used to account for the effect of connection flexibility on the structural behavior. The analytical results of proposed element are compared reasonably well with the test results. Effect of connection flexibility can also be taken into account in analysis by connection element, such as COMBIN39 element in ANSYS program. The analytical results by use of COMBIN39 and BEAM189 elements are very close to those by proposed element. Using proposed element is quite convenient, because the same structural model as the rigid frame can be used. However, using COMBIN39 element is more complicated since the creation of structural model and the determination of real constants are troublesome, and the number of nodes and elements in finite element model is large.
Routines of the refined plastic hinge based beam element are written in FORTRAN language using User Programmable Features (UPFs) of ANSYS program. Customized ANSYS program has been obtained after these routines are compiled and linked into ANSYS program. The user beam element can be easily activated in customized ANSYS program and then can be used for static or dynamic analysis of planar and space steel structures. The user beam element can be used with standard elements in ANSYS program, therefore, the advantages of ANSYS program, such as the abundant elements, the powerful solution tools and the convenient pre- and post-processor can be readily and well used in the structural analysis.
Elastic and plastic comparison analyses between two types of elements, namely the refined plastic hinge based beam element and the numerically integrated beam element, are performed for rigid and semi-rigid steel frames under earthquake excitations. High accuracy results can be obtained using proposed beam element in the time-history analysis for rigid and semi-rigid structures where spread-of-plasticity is not significant or where redundancy is high, such as for braced frames. However, numerically integrated elements should be employed in the time-history analysis to obtain accurate results for structures where spread-of-plasticity is significant and abrupt reduction in lateral stiffness of some story may be induced.
In comparison with rigid frames, semi-rigid frames may have lesser total base shear response under earthquake excitations. As the stiffness of connection decreases, the maximum total base shear resulting from the same earthquake excitation decreases. However, the maximum lateral drifts and the maximum inter-story drifts of the semi-rigid frames do not always increase with the decreasing of connection stiffness.
推导了考虑剪切影响的转角位移方程,并利用它计算单元内力,其中的稳定函数可以准确考虑单元P ?δ效应。讨论了单元考虑弯曲缩短影响(弓形效应)的方法,算例分析表明,非迭代近似方法能够有效考虑弯曲缩短的影响。利用虚位移原理和稳定插值函数推导了空间薄壁梁单元的刚度方程。考察了不同形式的几何刚度矩阵对单元精度的影响,在非线性分析中若采用相同的方法计算单元内力,则几何刚度矩阵的形式和精度对分析结果几乎没有影响。采用节点方向矩阵和单元刚体转角两种不同方法更新单元坐标转换矩阵,对两种方法进行了对比分析;静力分析时,两种方法计算的结果比较接近;但进行动力分析时,前者的稳定性和收敛性要优于后者。
进行弹性分析时,采用总量算法计算单元内力要优于采用增量算法计算单元内力。进行几何非线性分析时,本论文梁单元的精度总体上要优于ANSYS软件中的BEAM4单元,与BEAM189单元的精度相当甚至略优于BEAM189单元。
采用精细塑性铰模型考虑材料非线性,单元能够较好地模拟平面和空间钢框架结构的弹塑性承载力极限状态,单元分析结果与塑性区模型BEAM189单元分析的结果吻合良好。单元具有良好的精度,结构若只承受节点荷载,则每根构件通常只需用1个单元模拟;若构件承受分布荷载,则每根构件只需用2个单元模拟即可。而采用ANSYS软件中的BEAM189单元进行弹塑性分析时,每根构件通常要用3～5个单元模拟才能获得较准确的结果。本论文精细塑性铰梁单元不需要进行数值积分运算,因此单元计算效率很高;进行大型结构分析时,与使用数值积分单元相比,使用本论文的梁单元可以节省大量计算时间。
采用修正梁单元方法考虑连接柔性的影响,单元分析结果与相关文献的试验结果吻合较好。采用ANSYS软件中的COMBIN39单元也能有效考虑连接柔性的影响,COMBIN39组合BEAM189单元分析的结果与本论文梁单元分析的结果吻合很好。采用本论文梁单元考虑连接半刚性时,单元使用方便,可以采用与刚性连接框架相同的模型;而采用COMBIN39单元模拟连接时,建模过程和单元实常数的确定比较复杂,节点、单元数量多。
利用ANSYS软件的用户可编程特性(UPFs)采用FORTRAN语言编写了精细塑性铰梁单元程序,通过编译以及与ANSYS程序的连接获得了用户定制ANSYS程序。运行定制的ANSYS程序可以方便地调用用户梁单元,用于平面和空间钢结构的静力或动力分析。进行结构分析时用户梁单元能够与ANSYS中的标准单元联合使用,因此可充分利用ANSYS软件丰富的单元、强大的求解器和便捷的前后处理功能。
针对精细塑性铰梁单元和数值积分梁单元两种不同类型的单元,对刚性和半刚性钢框架进行了弹性及弹塑性地震时程响应对比分析。对于塑性发展不严重或者冗余度较高(例如框架支撑结构)的刚性或半刚性钢结构,采用本论文精细塑性铰梁单元进行动力时程分析能够获得比较准确的结果。而对于塑性发展严重可能导致局部楼层侧向刚度发生急剧削弱的结构,则应采用数值积分单元进行分析,才能获得准确的结果。
在地震激励作用下,与刚性框架相比,半刚性框架将具有更小的基底总剪力响应。随着连接刚度的减小,同一地震激励引起的最大基底总剪力也减小;但半刚性框架的最大侧移和层间侧移并不总是随着连接刚度的减小而增大。
Traditional analysis and design methods have many limitations. With the development of computer technology and theories of structural analysis and design, the advanced analysis and design method must be an important trend in future analysis and design practices. A three-dimensional beam element, which based on stability functions of beam-column theory and refined plastic hinge model, has been developed. Nonlinear effects, such as geometic nonlinearity, material nonlinearity, shear deformation, effect of curvature shortening, residual stress and connection flexibility, can be considered in element formulation.
Element routines are developed using the redevelopment technique of commercial finite element ANSYS software, and run on the platform of ANSYS software. Element can be used for static or dynamic advanced analysis of three-dimensional steel structures. The followings are the mian contents in detail. The slope-deflection equations including shear effect are derived and used to compute element internal forces, in which the element P ?δeffect can be accurately captured by the stability functions. Methods to consider the effect of curvature shortening (bowing effect) are discussed. Results of analysis examples show that the bowing effect can be taken into account effectively by an approximate non-iterative method. Using virtual displacement principle and stability interpolation functions, stiffness equation of a three-dimensional thin-walled beam element is derived. Influences of different types of element stiffness matrixes on the accuracy of element are observed. It is found that the form and accuracy of the geometric stiffness matrix used in a nonlinear analysis almost have no effect on the accuracy of the final solution when the same expressions are used to calculate the internal forces. The element transformation matrix is updated either by joint orientation matrix or by element rigid body rotation, and difference between two methods is analysed. It is found that the results of two methods are close in static analysis, whereas in dynamic analysis the former has better robustness and convergence than the latter.
In elastic analysis, algorithm for internal forces calculation based on the total equilibrium condition is better than that based on the incremental equilibrium condition. As far as geometric nonlinear analysis is concerned, the performance of proposed beam element is better than that of BEAM4 element, and is comparable to, to some extent even better than, that of BEAM189 element in ANSYS program.
Material nonlinearity is modeled using refined plastic hinge approach. Inelastic strength limit state of planar and space steel frames can be captured by proposed element. The analytical results of proposed element are compared fairly well with those of plastic zone based beam element BEAM189. In structural inelastic analysis, only one or two proposed elements for each member loaded by nodal loads or by distributed loads, respectively, in contrast to three to five BEAM189 elements for each member, are needed to achieve acceptable accuracy. The proposed beam element has excellent efficiency since it does not do any numerical integration operation. A lot of computing time can therefore be saved in the analysis of large-scale structures by using proposed element, as compared to using numerically integrated element.
Modified beam element method is used to account for the effect of connection flexibility on the structural behavior. The analytical results of proposed element are compared reasonably well with the test results. Effect of connection flexibility can also be taken into account in analysis by connection element, such as COMBIN39 element in ANSYS program. The analytical results by use of COMBIN39 and BEAM189 elements are very close to those by proposed element. Using proposed element is quite convenient, because the same structural model as the rigid frame can be used. However, using COMBIN39 element is more complicated since the creation of structural model and the determination of real constants are troublesome, and the number of nodes and elements in finite element model is large.
Routines of the refined plastic hinge based beam element are written in FORTRAN language using User Programmable Features (UPFs) of ANSYS program. Customized ANSYS program has been obtained after these routines are compiled and linked into ANSYS program. The user beam element can be easily activated in customized ANSYS program and then can be used for static or dynamic analysis of planar and space steel structures. The user beam element can be used with standard elements in ANSYS program, therefore, the advantages of ANSYS program, such as the abundant elements, the powerful solution tools and the convenient pre- and post-processor can be readily and well used in the structural analysis.
Elastic and plastic comparison analyses between two types of elements, namely the refined plastic hinge based beam element and the numerically integrated beam element, are performed for rigid and semi-rigid steel frames under earthquake excitations. High accuracy results can be obtained using proposed beam element in the time-history analysis for rigid and semi-rigid structures where spread-of-plasticity is not significant or where redundancy is high, such as for braced frames. However, numerically integrated elements should be employed in the time-history analysis to obtain accurate results for structures where spread-of-plasticity is significant and abrupt reduction in lateral stiffness of some story may be induced.
In comparison with rigid frames, semi-rigid frames may have lesser total base shear response under earthquake excitations. As the stiffness of connection decreases, the maximum total base shear resulting from the same earthquake excitation decreases. However, the maximum lateral drifts and the maximum inter-story drifts of the semi-rigid frames do not always increase with the decreasing of connection stiffness.
引文
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43 S. E. Kim, J. Lee. Improved Refined Plastic-Hinge Analysis Accounting for Lateral Torsional Buckling. Journal of Constructional Steel Research. 2002, 58(11):1431~1453
44 S. E. Kim, M. H. Park, S. H. Choi. Direct Design of Three-Dimensional Frames Using Practical Advanced Analysis. Engineering Structures. 2001, 23(11):1491~1502
45 S. E. Kim, S. H. Choi. Practical Advanced Analysis for Semi-Rigid Space Frames. International Journal of Solids and Structures. 2001, 38(50-51): 9111~9131
46 J. Y. R. Liew, L. K. Tang. Advanced Plastic Hinge Analysis for the Design of Tubular Space Frames. Engineering Structures. 2000, 22(7):769~783
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55 L. H. Teh, M. J. Clarke. Co-Rotational and Lagrangian formulations forElastic Three-Dimensional Beam finite Elements. Journal of Constructional Steel Research. 1998, 48(2-3):123~144
56 S. Rajasekaram, D. W. Murray. Finite Element Solution of Inelastic Beam Equations. Journal of the Structural division. 1973, 99(ST6):1025~1041
57 J. L. Meek, W. J. Lin. Geometric and Material Nonliear Analysis of Thin-Walled Beam-Columns. Journal of Structural Engineering. 1990, 116(6): 1473~1490
58 Y. L. Pi, N. S. Trahair. Nonlinear Inelastic Analysis of Steel Beam-Columns.Ⅰ: Theory. Journal of Structural Engineering. 1994, 120(7):2041~2061
59 Y. L. Pi, N. S. Trahair. Nonlinear Inelastic Analysis of Steel Beam-Columns.Ⅱ: Applications. Journal of Structural Engineering. 1994, 120(7):2062~2085
60 A. Conci, M. Gattass. Natural Approach for Thin-Walled Beam-Columns with Elastic-Plasticity. International Journal for Numerical Methods in Engineering. 1990, 29(8):1653~1679
61 G. Shi, S. N. Atluri. Elasto-Plastic Large Deformation Analysis of Space-Frames: A Plastic-Hinge and Stress-Based Explicit Derivation of Tangent Stiffnesses. International Journal for Numerical Methods in Engineering. 1988, 26(3):589~615
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63 郑廷银, 赵惠麟. 空间钢框架结构的改进双重非线性分析. 工程力学. 2003, 20(6):202-208
64 J. Y. R. Liew, H. Chen, N. E. Shanmugam, W. F. Chen. Improved Nonlinear Plastic Hinge Analysis of Space Frame Structures. Engineering Structures. 2000, 22(10):1324-1338
65 X. M. Jiang, H. Chen, J. Y. R. Liew. Spread-of-Plasticity Analysis of Three-Dimensional Steel Frames. Journal of Constructional Steel Research. 2002, 58(2):193-212
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70 A. Neuenhofer, F. C. Filippou. Geometrically Nonlinear Flexibility-Based Frame Finite Element. Journal of Structural Engineering. 1998, 124(6): 704~711
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36 S. E. Kim, J. Lee, J. S. Park. 3-D Second-Order Plastic-Hinge Analysis Accounting for Local Buckling. Engineering Structures. 2003, 25(1):81~90
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42 K. Wongkaew, W. F. Chen. Consideration of Out-of-Plane Buckling in Advanced Analysis for Planar Steel Frame Design. Journal of Constructional Steel Research. 2002, 58(5-8):943~965
43 S. E. Kim, J. Lee. Improved Refined Plastic-Hinge Analysis Accounting for Lateral Torsional Buckling. Journal of Constructional Steel Research. 2002, 58(11):1431~1453
44 S. E. Kim, M. H. Park, S. H. Choi. Direct Design of Three-Dimensional Frames Using Practical Advanced Analysis. Engineering Structures. 2001, 23(11):1491~1502
45 S. E. Kim, S. H. Choi. Practical Advanced Analysis for Semi-Rigid Space Frames. International Journal of Solids and Structures. 2001, 38(50-51): 9111~9131
46 J. Y. R. Liew, L. K. Tang. Advanced Plastic Hinge Analysis for the Design of Tubular Space Frames. Engineering Structures. 2000, 22(7):769~783
47 K. J. Bathe, S. Bolourchi. Large Displacement Analysis of Three-Dimensional Beam Structures. International Journal for Numerical Methods in Engineering. 1979, 14(7):961~986
48 R. S. Barsoum, R. H. Gallagher. Finite Element Analysis of Torsional and Lateral Stability Problems. International Journal for Numerical Methods in Engineering. 1970, 2(3):335~352
49 Y. B. Yang, W. McGuire. Stiffness Matrix for Geometric Nonlinear Analysis. Journal of Structural Engineering. 1986, 112(4):853~877
50 Y. B. Yang, W. McGuire. Joint Rotation and Geometric Nonlinear Analysis. Journal of Structural Engineering. 1986, 112(4):879~905
51 S. Kitipornchai, S. L. Chan. Nonlinear Finite Element Analysis of Angle and Tee Beam-Columns. Journal of Structural Engineering. 1987, 113(4): 721~739
52 F. G. A. Al-Bermani, S. Kitipornchai. Nonlinear Analysis of Thin-Walled Structures Using Least Element/Member. Journal of Structural engineering. 1990, 116(1):215~234
53 H. Chen, G. E. Blandford. Thin-Walled Space Frames.Ⅰ: Large-Deforation Analysis Theory. Journal of Structural Engineering. 1991, 117(8):2499~2520
54 A. Conci, M. Gattass. Natural Approach for Geometric Nonlinear Analysis of Thin-Walled Frames. International Journal for Numerical Methods in Engineering. 1990, 30(2):207~231
55 L. H. Teh, M. J. Clarke. Co-Rotational and Lagrangian formulations forElastic Three-Dimensional Beam finite Elements. Journal of Constructional Steel Research. 1998, 48(2-3):123~144
56 S. Rajasekaram, D. W. Murray. Finite Element Solution of Inelastic Beam Equations. Journal of the Structural division. 1973, 99(ST6):1025~1041
57 J. L. Meek, W. J. Lin. Geometric and Material Nonliear Analysis of Thin-Walled Beam-Columns. Journal of Structural Engineering. 1990, 116(6): 1473~1490
58 Y. L. Pi, N. S. Trahair. Nonlinear Inelastic Analysis of Steel Beam-Columns.Ⅰ: Theory. Journal of Structural Engineering. 1994, 120(7):2041~2061
59 Y. L. Pi, N. S. Trahair. Nonlinear Inelastic Analysis of Steel Beam-Columns.Ⅱ: Applications. Journal of Structural Engineering. 1994, 120(7):2062~2085
60 A. Conci, M. Gattass. Natural Approach for Thin-Walled Beam-Columns with Elastic-Plasticity. International Journal for Numerical Methods in Engineering. 1990, 29(8):1653~1679
61 G. Shi, S. N. Atluri. Elasto-Plastic Large Deformation Analysis of Space-Frames: A Plastic-Hinge and Stress-Based Explicit Derivation of Tangent Stiffnesses. International Journal for Numerical Methods in Engineering. 1988, 26(3):589~615
62 舒兴平, 沈蒲生. 空间钢框架结构的非线性全过程分析. 工程力学. 1997, 14(3):36~45
63 郑廷银, 赵惠麟. 空间钢框架结构的改进双重非线性分析. 工程力学. 2003, 20(6):202-208
64 J. Y. R. Liew, H. Chen, N. E. Shanmugam, W. F. Chen. Improved Nonlinear Plastic Hinge Analysis of Space Frame Structures. Engineering Structures. 2000, 22(10):1324-1338
65 X. M. Jiang, H. Chen, J. Y. R. Liew. Spread-of-Plasticity Analysis of Three-Dimensional Steel Frames. Journal of Constructional Steel Research. 2002, 58(2):193-212
66 K. J. Bath. Finite Element Procedures. Prentice Hall, 1996:522~528
67 M. A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures. Volume 1: Essentials. John Wiley & Sons, 1991:77~80 211~221
68 M. A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures. Volume 2: Advanced Topics. John Wiley & Sons, 1991:213~226 188~212
69 A. Chajes, J. E. Churchill. Nonlinear Frame Analysis by Finite Element Methods. Journal of Structural Engineering. 1987, 113(6):1221~1235
70 A. Neuenhofer, F. C. Filippou. Geometrically Nonlinear Flexibility-Based Frame Finite Element. Journal of Structural Engineering. 1998, 124(6): 704~711
71 K. D. Hjelmstad, E. Taciroglu. Mixed Methods and Flexibility Approaches for Nonlinear Frame Analysis. Journal of Constructional Steel Research. 2002, 58(5-8): 967~993
72 陈骥. 钢结构稳定理论与设计. 科学出版社, 2001:67~89
73 王勖成, 邵敏. 有限单元法基本原理和数值方法. 第 2 版. 清华大学出版社, 1997:531~567
74 C. Oran. Tangent Stiffness in Space Frames. Journal of the Structural Division. 1973, 99(ST6):987~1001
75 A. Kassimali, R. Abbasnia. Large Deformation Analysis of Elastic Space Frames. Journal of Structural Engineering. 1991, 117(7):2069~2087
76 李国强, 沈祖炎. 钢结构框架体系弹性及弹塑性分析与计算理论. 上海科学技术出版社, 1998:7~21 23~41
77 S. Timoshenko, J. Gere. 材料力学. 胡人礼. 科学出版社, 1978:159~169 466~472
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