简支楔形变截面工字钢梁的弹性弯扭屈曲
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摘要
本文首先对腹板的高度线性变化的变截面工字钢梁的剪切中心位置进行研究分析，发现只要保持截面的其他尺寸不变，每个截面的剪切中心连线仍是一条直线，弯扭失稳是以截面剪心的位移为准的，因此仍可以取传统的直线坐标系统。
其次，本文通过对大量不同截面尺寸的双轴对称和单轴对称简支等截面梁在不等端弯矩作用下的情况进行有限元分析后，发现我国规范提供的等效弯矩系数公式β_b=1.75-1.05k+0.3k~2≤2.3(k=M_0/M_1)精度欠佳，尤其用在双轴对称截面梁产生异向曲率时过分保守，丧失了经济性。本文根据有限元计算结果重新提出β_b的计算公式，且在计算单轴对称等截面梁时，认为拟合公式的适用范围在-I_(yB)/I_(yT)≤k≤1内的误差较小，对双轴对称截面则自动退化到-1≤k≤1。
最后，考虑变截面梁的楔率影响，按能量法推导了上述情况下梁的临界荷载公式，保留公式形式，用有限元结果进行计算机回归分析，调整公式中的各项系数，得到物理意义明确，且形式与等截面梁一致，并便于在工程设计应用的梁平面外弹性弯扭屈曲临界荷载表达式，发现拟合公式比《门式刚架轻型房屋钢结构技术规程CECS102：2002》中公式精度高，并且将CECS102：2002中的公式与《钢结构设计规范》(GB 50017-2003)中对等截面钢梁的稳定系数公式进行对比，发现CECS102：2002中所提供的计算变截面梁平面外屈曲临界荷载公式的不足之处，并指明了该公式不合理的原因。
At first, this thesis studied the position of shear center of tapered I-beam. If only the web is linearly varied along the member, and the other dimensions are kept constant, and it can be shown that the shear center axis of the tapered I-beam is still a straight line. Flexural-torsional buckling may be still studied using the shear center line as a reference axis to define the lateral displacement and the rotation.
Secondly, the elastic lateral buckling of simple supported I-beams with doubly-symmetrical and mono-symmetrical cross-sections under moment gradient is
investigated by FEA method, it is found that the formula for betab (equivalent moment
coefficient) currently used in Steel Structures Code is not accurate enough in some cases, e.g. when beams are bent in double curvature. An improved design formula is proposed on the basis of results of FEA. Furthermore, we thought the application to
mono-symmetry section should be confined and -1≤k≤1 to
doubly-symmetry section.
At last, considering the effects of taper ratio, we derived critical load of tapered I-beam by energy method. Using the formulation thus obtained, but adjusting the coefficients in this formulation based on the results of FEA, we obtained a new formula for the Critical moment for lateral-torsional buckling of tapered I-beams, it has a similar formulation as those for beams with constant cross-sections,, with clear meaning and easy to application in practice. We found the precision of the new formula is better than CECS 102:2002. Moreover, we point out the formula in CECS 102:2002 not to be exact because it considered the monosymmetry twice.
其次，本文通过对大量不同截面尺寸的双轴对称和单轴对称简支等截面梁在不等端弯矩作用下的情况进行有限元分析后，发现我国规范提供的等效弯矩系数公式β_b=1.75-1.05k+0.3k~2≤2.3(k=M_0/M_1)精度欠佳，尤其用在双轴对称截面梁产生异向曲率时过分保守，丧失了经济性。本文根据有限元计算结果重新提出β_b的计算公式，且在计算单轴对称等截面梁时，认为拟合公式的适用范围在-I_(yB)/I_(yT)≤k≤1内的误差较小，对双轴对称截面则自动退化到-1≤k≤1。
最后，考虑变截面梁的楔率影响，按能量法推导了上述情况下梁的临界荷载公式，保留公式形式，用有限元结果进行计算机回归分析，调整公式中的各项系数，得到物理意义明确，且形式与等截面梁一致，并便于在工程设计应用的梁平面外弹性弯扭屈曲临界荷载表达式，发现拟合公式比《门式刚架轻型房屋钢结构技术规程CECS102：2002》中公式精度高，并且将CECS102：2002中的公式与《钢结构设计规范》(GB 50017-2003)中对等截面钢梁的稳定系数公式进行对比，发现CECS102：2002中所提供的计算变截面梁平面外屈曲临界荷载公式的不足之处，并指明了该公式不合理的原因。
At first, this thesis studied the position of shear center of tapered I-beam. If only the web is linearly varied along the member, and the other dimensions are kept constant, and it can be shown that the shear center axis of the tapered I-beam is still a straight line. Flexural-torsional buckling may be still studied using the shear center line as a reference axis to define the lateral displacement and the rotation.
Secondly, the elastic lateral buckling of simple supported I-beams with doubly-symmetrical and mono-symmetrical cross-sections under moment gradient is
investigated by FEA method, it is found that the formula for betab (equivalent moment
coefficient) currently used in Steel Structures Code is not accurate enough in some cases, e.g. when beams are bent in double curvature. An improved design formula is proposed on the basis of results of FEA. Furthermore, we thought the application to
mono-symmetry section should be confined and -1≤k≤1 to
doubly-symmetry section.
At last, considering the effects of taper ratio, we derived critical load of tapered I-beam by energy method. Using the formulation thus obtained, but adjusting the coefficients in this formulation based on the results of FEA, we obtained a new formula for the Critical moment for lateral-torsional buckling of tapered I-beams, it has a similar formulation as those for beams with constant cross-sections,, with clear meaning and easy to application in practice. We found the precision of the new formula is better than CECS 102:2002. Moreover, we point out the formula in CECS 102:2002 not to be exact because it considered the monosymmetry twice.
引文
[1] Amirikian,A., "Wedge-Beam Framing," Trans,ASCE, 117,596(1952)。
[2] Kerensky, O.A., Flint, A.R., and Brown, W.C., "the Basis for Design of Beams and Plate Giriders in the Revised British Standard 153," Proceeding, I.C.E. Part Ⅲ, Vol.5, August, 1956, p396。
[3] Lee, G . C., Morrell, M. L. and Ketter, R. L., "Design of Tapered Members", Welding Research Council Bulletin, No. 173, June,1972
[4] 门式刚架轻型房屋钢结构技术规程CECS102：2002。
[5] Kitipornchai, S., and Trahair, N. S., "Elastic Stability of Tapered I-Beams", Journal of Structural Division ,Vol.98, No.ST3, March, 1972, ASCE
[6] Kitipornehai, S., and Trahair, N. S., "Elastic Behavior of Tapered Monosymmetric I-Beams", Journal of the Structural Division ,Vol.101, No.ST8, August, 1975, ASCE
[7] Brown, T.G., "Lateral-Torsional Buclding of Tapered I-Beams", Journal of the Structural Division, Vol. 107, No.ST4, April, 1981,ASCE
[8] Bradford, M. A., Cuk, P. E., "Elastic Buckling of Tapered Monosymmetric I-Beams", Journal of Structural Engineering, Vol.114, No.5, May, 1988, ASCE
[9] Wekezer, J. W., "Instability of Thin Walled Bars", Journal of Engineering Mechanics, Vol. 111, No.7, July, 1985, ASCE
[10] Wekezer, J. W., "Elastic Torsion of Thin Walled Bars of Variable Cross Sections", Computers & Structures, Vol.19, No.3, 1984
[11] 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
[12] 苏庆田，王俊平，焊接工字形楔形梁的稳定设计研究，昆明理工大学学报
[13] 纪平，《楔形变截面工字钢构件的平面外稳定性研究》，浙江大学硕士学位论文。
[14] 纪平，童根树，变截面楔形工字钢简支构件的平面外弹性稳定计算，上海农学院学报
[15] 中华人民共和国国家规范《钢结构设计规范》GBJ50017-2003。
[16] 李德滋等译，欧洲建筑钢结构协会《钢结构设计手册》。
[17] 夏志斌、潘有昌，《结构稳定理论》，高等教育出版社，北京 1988。
[18] 陈骥，《钢结构稳定理论与设计》，科学出版社，北京 2001。
[19] 陈绍藩，《钢结构设计原理》，科学出版社，北京 2000。
[20] 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。
[21] 钟善桐，《钢结构稳定设计》 中国建筑工业出版社，北京 1991。
[22] 唐家祥、王仕统、裴若绢，《结构稳定理论》，中国铁道出版社，北京 1989。
[23] 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。
[24] 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。
[25] Sritawat Kitipornchai, Chien Ming Wang, and Nicholas S., "Buckling of Monosymmetric I-Beam under Moment Gradient"
[26] N.S Trahair, Flexural-Torsional Buckling of Structure E&FN SPON., 1993
[27] Buckling Properties of Monosymmetric I-Beams, Journal of the Structural Division,By Sritawat Kitipornchai and S.Trahair, Members,ASCE,1980,941-957
[28] Chien Ming Wang and Sritawat Kitipornchai, "Buckling Capacities of Monosymmetric I-Beams
[29] 童根树，张磊，薄壁钢梁稳定性计算的争议及其解决，建筑结构学报
[30] S.Kitipornchai and C.M.Wang, "On Stability of Monosymmetric Cantilevers" Eng.Struct, 1986,Vol.8,July 169-180
[31] Kitipornchai, S., and Trahair, N. S., " Inelastic Buckling of Simply Supported Steel I-Beams", Journal of Structural Division ,Vol.101, No.ST7, July, 1975, ASCE
[32] 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
[33] Lee, G. C., Chen, Y. C., and Hsu, T. L., "Allowable Axial Stress of Restrained Multi-Segment Tapered Roof Girders", Welding Research Council Bulletin, No.248, May,1979
[34] Lee, G. C. and Szabo, B. A., "Torsional Response of Tapered I-Girders", Journal of the Structumal Division, Vol.93, No.ST5, 1967, ASCE
[35] 郭耀杰，《悬臂构件稳定性理论及其应用》。
[36] Morrell, M. L., and Lee, G. C., "Allowable Stress for Web-Tapered Beams with Lateral Restraints", Welding Research Council Bulletin, No. 192, Feb., 1974
[37] Chong, K. P., and Swanson, W. D., "Shear Analysis of Tapered Beams", Journal of the Structural Division, Vol. 102, No. ST9, 1976, ASCE
[38] Davies, G., Lamb, R. S. and Snell, C., "Stress Distribution in Beams of Varying Depth", the Structure Engineer, Vol.51, No. 11,1973
[39] Russo, E.P. and Garic, G, "Shear-Stress Distribution in Symmetrically Tapered Cantilever Beam", Journal of Structual Engineering, Vol. 118, No. 11, 1992,ASCE
[40] Prawel, S. P., et al., "Bending and Buckling Strength of Tapered Structural Members", Welding Journal, Vol.53, No.2, Res. Suppl., Feb., 1974
[41] Rajasekaran, S., and Murray, D. W., "Finite Element Solution of Inelastic Beam Equations", Journal of Structural Division ,Vol.99, No.ST6, June, 1973, ASCE
[42] Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, McGraw-Hill, New York, 1961
[43] D.A.Nethercot, "the effective lengths of cantilevers as governed by lateral buckling"
[44] Siu Lai Chan, " Buckling analysis of structures composed of tapered members"p 1893-1906
[45] Stanley Poley, J.M.ASCE, "Lateral Buckling of Cantilevered I-Beam Under uniform load" , Paper No.2821
[2] Kerensky, O.A., Flint, A.R., and Brown, W.C., "the Basis for Design of Beams and Plate Giriders in the Revised British Standard 153," Proceeding, I.C.E. Part Ⅲ, Vol.5, August, 1956, p396。
[3] Lee, G . C., Morrell, M. L. and Ketter, R. L., "Design of Tapered Members", Welding Research Council Bulletin, No. 173, June,1972
[4] 门式刚架轻型房屋钢结构技术规程CECS102：2002。
[5] Kitipornchai, S., and Trahair, N. S., "Elastic Stability of Tapered I-Beams", Journal of Structural Division ,Vol.98, No.ST3, March, 1972, ASCE
[6] Kitipornehai, S., and Trahair, N. S., "Elastic Behavior of Tapered Monosymmetric I-Beams", Journal of the Structural Division ,Vol.101, No.ST8, August, 1975, ASCE
[7] Brown, T.G., "Lateral-Torsional Buclding of Tapered I-Beams", Journal of the Structural Division, Vol. 107, No.ST4, April, 1981,ASCE
[8] Bradford, M. A., Cuk, P. E., "Elastic Buckling of Tapered Monosymmetric I-Beams", Journal of Structural Engineering, Vol.114, No.5, May, 1988, ASCE
[9] Wekezer, J. W., "Instability of Thin Walled Bars", Journal of Engineering Mechanics, Vol. 111, No.7, July, 1985, ASCE
[10] Wekezer, J. W., "Elastic Torsion of Thin Walled Bars of Variable Cross Sections", Computers & Structures, Vol.19, No.3, 1984
[11] 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
[12] 苏庆田，王俊平，焊接工字形楔形梁的稳定设计研究，昆明理工大学学报
[13] 纪平，《楔形变截面工字钢构件的平面外稳定性研究》，浙江大学硕士学位论文。
[14] 纪平，童根树，变截面楔形工字钢简支构件的平面外弹性稳定计算，上海农学院学报
[15] 中华人民共和国国家规范《钢结构设计规范》GBJ50017-2003。
[16] 李德滋等译，欧洲建筑钢结构协会《钢结构设计手册》。
[17] 夏志斌、潘有昌，《结构稳定理论》，高等教育出版社，北京 1988。
[18] 陈骥，《钢结构稳定理论与设计》，科学出版社，北京 2001。
[19] 陈绍藩，《钢结构设计原理》，科学出版社，北京 2000。
[20] 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。
[21] 钟善桐，《钢结构稳定设计》 中国建筑工业出版社，北京 1991。
[22] 唐家祥、王仕统、裴若绢，《结构稳定理论》，中国铁道出版社，北京 1989。
[23] 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。
[24] 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。
[25] Sritawat Kitipornchai, Chien Ming Wang, and Nicholas S., "Buckling of Monosymmetric I-Beam under Moment Gradient"
[26] N.S Trahair, Flexural-Torsional Buckling of Structure E&FN SPON., 1993
[27] Buckling Properties of Monosymmetric I-Beams, Journal of the Structural Division,By Sritawat Kitipornchai and S.Trahair, Members,ASCE,1980,941-957
[28] Chien Ming Wang and Sritawat Kitipornchai, "Buckling Capacities of Monosymmetric I-Beams
[29] 童根树，张磊，薄壁钢梁稳定性计算的争议及其解决，建筑结构学报
[30] S.Kitipornchai and C.M.Wang, "On Stability of Monosymmetric Cantilevers" Eng.Struct, 1986,Vol.8,July 169-180
[31] Kitipornchai, S., and Trahair, N. S., " Inelastic Buckling of Simply Supported Steel I-Beams", Journal of Structural Division ,Vol.101, No.ST7, July, 1975, ASCE
[32] 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
[33] Lee, G. C., Chen, Y. C., and Hsu, T. L., "Allowable Axial Stress of Restrained Multi-Segment Tapered Roof Girders", Welding Research Council Bulletin, No.248, May,1979
[34] Lee, G. C. and Szabo, B. A., "Torsional Response of Tapered I-Girders", Journal of the Structumal Division, Vol.93, No.ST5, 1967, ASCE
[35] 郭耀杰，《悬臂构件稳定性理论及其应用》。
[36] Morrell, M. L., and Lee, G. C., "Allowable Stress for Web-Tapered Beams with Lateral Restraints", Welding Research Council Bulletin, No. 192, Feb., 1974
[37] Chong, K. P., and Swanson, W. D., "Shear Analysis of Tapered Beams", Journal of the Structural Division, Vol. 102, No. ST9, 1976, ASCE
[38] Davies, G., Lamb, R. S. and Snell, C., "Stress Distribution in Beams of Varying Depth", the Structure Engineer, Vol.51, No. 11,1973
[39] Russo, E.P. and Garic, G, "Shear-Stress Distribution in Symmetrically Tapered Cantilever Beam", Journal of Structual Engineering, Vol. 118, No. 11, 1992,ASCE
[40] Prawel, S. P., et al., "Bending and Buckling Strength of Tapered Structural Members", Welding Journal, Vol.53, No.2, Res. Suppl., Feb., 1974
[41] Rajasekaran, S., and Murray, D. W., "Finite Element Solution of Inelastic Beam Equations", Journal of Structural Division ,Vol.99, No.ST6, June, 1973, ASCE
[42] Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, McGraw-Hill, New York, 1961
[43] D.A.Nethercot, "the effective lengths of cantilevers as governed by lateral buckling"
[44] Siu Lai Chan, " Buckling analysis of structures composed of tapered members"p 1893-1906
[45] Stanley Poley, J.M.ASCE, "Lateral Buckling of Cantilevered I-Beam Under uniform load" , Paper No.2821
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