摘要
本文对门式刚架中常用到的楔形变截面构件，以腹板高度线性渐变的工字钢楔形构件为研究对象，对约束问题和压弯杆的平面外临界荷载等进行一定的研究。主要有四个方面的内容：
一是按照虚功原理，用数值方法拟合出楔形构件在无侧移和有侧移情况下的转角位移的近似计算方法，并提出单段变截面梁的四个约束刚度。分析了门式刚架发生有侧移、无侧移失稳时，由单段、两段及三段变截面构件组成的钢梁对钢柱的转动约束问题，给出精度较高公式。
二是从增量变分理论出发，研究了适合薄壁构件非线性分析的有限元计算方法。所推导的有限单元法，考虑了构件的内应力和初变形，以便实际计算中的应用。对于不同的约束、荷载情况，都可应用这个有限单元法，作为特例研究。该方法为下文做弹性分析和以后的弹塑性分析做准备。
三是对理想条件下腹板高度线性渐变的工字钢楔形压弯构件进行了弹性稳定分析，利用有限单元法进行数值计算，结合能量法求得楔形压弯构件临界荷载近似计算式，得到物理意义明确，精度较高的扭转、弯扭屈曲荷载表达式。公式形式与等截面杆屈曲荷载的计算公式形式相同，从而将变截面构件问题转化为等截面构件问题。文章还给出梁柱P-M关系，便于在工程中应用。
四是分析由若干段等截面或者变截面构件组成的压杆，平面内稳定问题。先研究在两端铰接下，由三段等截面构件组成的压杆平面内稳定问题，并对两端为变截面构件，中间为等截面构件组成的压杆平面内稳定问题进行分析，得到形式类似的屈曲荷载计算公式。最后还对两端固接的情况进行研究分析。
Firstly, this paper presented an explicit rotation-deflection relation of tapered member based on numerical analysis, this relation is used to derive the rotational restraints provided to columns by the rafters in portal frames, the rafters may be composed of multiple segments of tapered members. These restraints are intended for employment in place of the APPENDIX D of the Code of Light-Weight Steel Portal-Frame(CECS 102:2002).
Secondly, a finite-element formulation is presented for analysis of the inelastic flexural-torsional instability of non-prismatic beam-columns. The formulation represents a general approach to this problem.
Thirdly,this paper reports an investigation into the elastic torsional buckling under axial load and elastic flexural-torsional buckling under eccentric axial force loads of web tapered I-section beam-columns.Several formulae for calculating buckling loads are presented to provide simple means of checking the out of plane stability of these kinds of members.
Finally, research is conducted on the in-plane elastic stability of columns composed of several prismatic or tapered members under different boundary conditions,the results may be used for checking the stability of rafters when large axial forces exist in these members.
引文
[1] 夏志斌、潘有昌，《结构稳定理论》高等教育出版社，北京 1988。
[2] 陈骥，《钢结构稳定理论与设计》科学出版社，北京 2001。
[3] 陈绍藩，《钢结构设计原理》科学出版社，北京 2000。
[4] 吕烈武、沈世钊等，《钢结构构件稳定理论》中国建筑工业出版社，北京 1983。
[5] 中华人民共和国国家规范《钢结构设计规范》GBJ50017-2003。
[6] 门式刚架轻型房屋钢结构技术规程CECS 102：98。
[7] 中华人民共和国国家规范，冷弯薄壁型钢结构技术规程 GBJ18-87。
[8] 钟善桐，《钢结构稳定设计》中国建筑工业出版社，北京 1991。
[9] 唐锦春、孙炳楠、郭鼎康，《计算结构力学》浙江大学出版社，杭州 1989。
[10] 王元汉等，《有限元方法基础与程序设计》华南理工大学出版社，广州 2001
[11] 王勖成、劭敏，《有限单元法基本原理和数值方法》清华大学出版社，北京 1997
[12] 李开禧，钢压杆的柱子曲线，重庆建筑工程学院学报，1985．1。
[13] 陈绍藩，轻型钢结构变截面门式刚架的稳定计算，建造结构，1998．8。
[14] 李开禧、肖允徽，逆算单元长度法计算单轴失稳时钢压杆的临界力，重庆建筑工程学院学报，1982．4
[15] 纪平、童根树、许钧陶，楔形变截面工字钢构件的平面外稳定性研究，硕士论文，浙江大学，1999．1
[16] 陈婷、童根树，变截面构件的应力分布和平面内弹塑性稳定研究，硕士论文浙江大学，2003．1
[17] 饶芝英、童根树，变截面门式刚架的平面内弹性稳定计算，建筑结构，2000．4。
[18] 朱群红、童根树，简支楔形工字钢梁平面外弹性稳定计算，建筑结构，已录用
[19] 童根树、张磊，薄壁构件弯扭失稳的一般理论，建筑结构学报，2003．6
[20] 李国强、李进军，Timoshenko-Euler楔形梁有限元，力学季刊，2002．3
[21] 郭彦林、王文明、赵熙元，变截面门式刚架结构的有限元方法，建筑结构，1999．10
[22] 郭彦林、王文明、石永久，变截面门式刚架结构的非线性性能，工程力学，2000．8
[23] Washizu. K, Variational Methods in elasticity and plasticity,3rd Edition,Pergamon Press, 1981
[24] Timoshenko, S. P., and Gere, V. N., Mechanics of Materials, Van Nostrand, New York, N. Y., 1972
[25] Goldberg, J. E., and Bogdanoff, J. L., "Inelastic Buckling of Nonuniform Columns", Trans., Vol. 122, 1957, ASCE
[26] Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, McGraw-Hill, New York, 1961
[27] Gere, J. M., and Carter, W.O., "Critical Buckling Loads for Tapered Columns", Journal of the Structural Divison, Vol.88, No.ST1, February, 1962, ASCE
[28] Butler, D. J. and Anderson, G. B., "the Elastic Buckling of Tapered Beam-Columns", Welding Journal, Vol.42, No. 1, Res. Suppl., Jan., 1963
[29] Miranda, C. and Ojalve, M., "Inelastic Lateral-Torsional Buckling of Beam Columns", Journal of Engineering Mechanics Division, Vol.91, No.6, 1965, ASCE
[30] Lee, G. C. and Szabo, B. A., "Torsional Response of Tapered I-Girders", Journal of the Structurnal Division, Vol.93, No.ST5, 1967, ASCE
[31] Appl, F.J. and Smith, J.O., "Buckling of Inelastic, Tapered, Pin-Ended Columns",Journal of Engineering Mechanics Division, Vol.94, No.EM2, 1968, ASCE
[32] Culver, C.G.. and Preg, S.M., "Elastic Stability of Tapered Beam-Columns", Journal of the Struetual Division, Vol.94, No.ST2,1968,ASCE
[33] G irijavallahan, C. V "Buckling Loads of Nonuniform Columns", Journal of the Structural
Division ,Vol.95,No.ST11,1969,ASCE
[34] Kitipornchai, S., and Trahair, N. S., "Elastic Stability of Tapered Ⅰ-Beams", Journal of Structural Division,Vol.98, No.ST3, March, 1972, ASCEm
[35] Lee, G. C., Morrell, M. L. and Ketter, R. L., "Design of Tapered Members", Welding Research Council Bulletin, No. 173, June, 1972
[36] Rajasekaran, S., and Murray, D. W., "Finite Element Solution of Inelastic Beam Equations", Journal of Structural Division ,Vol.99, No.ST6, June, 1973, ASCE
[37] Yang, Y. B., and Nimeiri, M. E., "Large-Deflection Spatial Buckling of Thin-Walled Beams and Frames", Journal of Engineering Mechanics Division, Vol.99, No.EM6, December, 1973, ASCE
[38] Kitipornchai, S., and Trahair, N. S., "Elastic Behavior of Tapered Monosymmetric Ⅰ-Beams", Journal of Structural Division ,Vol. 101, No.STS, August, 1975, ASCE
[39] O'Rourke, M., and Zebrowsk, T., "Budding Load for Nonuniform Columns", Computers & Structures, Vol.7,1977
[40] Sankaran, G. V. and Rao, G. V., "Stability of Tapered Cantilever Columns Subjected to Follower Forces", Computers and Structures, Vol.6, 1976
[41] Salter, J. B., Anderson, D., and May, I. M., "Tests on Tapered Steel Columns", the Structural Engineering, Vol.58, No.6, June, 1980
[42] Brown, T.G., "Lateral-Torsional Buckling of Tapered Ⅰ-Beams", Journal of the Structural Division, Vol. 107, No.ST4, April, 1981,ASCE
[43] Wekezer, J. W., "Elastic Torsion of Thin Walled Bars of Variable Cross Sections", Computers & Structures, Vol.19, No.3, 1984
[44] Wekezer, J. W., "Instability of Thin Walled Bars", Journal of Engineering Mechanics, Vol. 111, No.7, July, 1985, ASCE
[45] Yang,Y.B and Mcguire,W, "Stiffness Matrix for Geometric Nonlinear Analysis",Journal of Structural Engineering,Vol. 112,No.4,1986,ASCE
[46] Ermopoulos,J.C "Buckling of Tapered Bars Under Stepped Axial Loads",Journal of Structural Engineering Vol. 112,No.6,1986,ASCE
[47] Sakiyama, T., "A Method of Analyzing the Elastic Buckling of Tapered Columns", Computers & Structures, Vol.23, No. 1, 1986
[48] Alghamdi, S. A., "Minimum Weight Design of Linearly Tapered Columns", Computers & Structures, Vol.27, No.4, 1987
[49] Yang, Y. B., and Yau, J. D., "Stability of Beams with Tapered Ⅰ-Sections", Journal of Engineering Mechanics Division, Vol. 113, No.9, September, 1987, ASCE
[50] Bradford, M. A., Cuk, P. E., "Elastic Buckling of Tapered Monosymmetric Ⅰ-Beams", Journal of Structural Engineering, Vol. 114, No.5, May, 1988, ASCE
[51] Smith, W. G., "Analytic Solutions for Tapered Columns Buckling", Computers & Structures, Vol.28, No.5, 1988
[52] Bradford,M.A "Lateral Stability of Tapered Beam-Columns with Elastic Restraints"Structural Engineer;Vol.66,No.22,1988
[53] Liew, J. Y., Shanmugam, N. E., and Lee, S. L., "Tapered Box Columns under Biaxial Loading", Journal of Structural Engineering, Vol.115, No.7, July, 1989, ASCE
[54] Takabatake, H, "Cantilevered and Linearly Tapered Thin-Walled Members", Journal of Engineering Mechanics Division, Vol. 116, No.4, April, 1990, ASCE
[55] Clarke,M.J and Hancock,G.J "A Study of Incremental-Iterative Strategies for Non-Linear Analysis",International Journal for Numerical Methods in Engineering Vol.29 1990
[56] Lee,S.Y and Kuo,Y.H "Elastic Stability of Non-Uniform Columns",Journal of Sound and Vibration, 1991
[57] Ronagh,H.R and Bradford,M.A "A Rational Model for the Distortional Buckling of Tapered Members",Computer Methods in Applied Mechanics and Engineering, 1996
[58] Al-Gahtani,H.J,"Exact Stiffness for Tapered Members",Journal of the Structual Engineering, Vol.122,No.10,1996,ASCE
[59] Kim, M. C., Chang, K. C., and Lee, G. C., "Elastic and Inelastic Buckling Analysis of Thin-Walled Tapered Members", Journal of Engineering Mechanics,. Vol. 123, No.7, July, 1997, ASCE
[60] Baptista, A. M. and Muzeau, J. P., "Design of Tapered Compression Members According to Eurecode 3", J. Construct. steel Res., Vol. 46, 1998
[61] Ronagh,H.R and Bradford,M.A,"Nonlinear Analysis of Thin-Walled Members of Variable Cross-Section .Part Ⅰ: Theory", Computers and Structures,Vol.77,No.3,2000