空腹桁架圆弧拱只存在与弦杆垂直的腹杆，不设置传统桁架拱中常见的斜腹杆，因此具有更好的美观性和透光性。目前，空腹桁架拱在桥梁、体育场和机场等空间结构中具有较好的应用前景。本文对不带斜腹杆的矩形截面钢管空腹桁架圆弧拱的平面外弹性和弹塑性屈曲性能开展了系统研究，可为类似结构的稳定性设计提供参考。主要研究内容如下：

（1）推导了铰接空腹桁架圆弧拱截面弯曲刚度、截面剪切刚度和截面扭转刚度的理论公式。研究发现，加入杆件的扭转刚度可以显著提高空腹桁架拱的截面扭转刚度，其计算结果与有限元吻合较好。然后，利用平衡定理和数值拟合分别计算了铰接和固支空腹桁架拱的平面外弹性屈曲荷载。研究结果表明，在这两种情况下，平面外弹性屈曲荷载与腹杆-弦杆刚度比密切相关。此外，剪切变形对固支空腹桁架拱平面外整体屈曲荷载的影响不可忽视。

（2）研究了全跨均布径向荷载和两端纯弯荷载作用下空腹桁架圆弧拱平面外破坏机理。研究结果表明：全跨均布径向荷载作用下空腹桁架圆弧拱可采用中国钢结构设计规范（GB50017-2020）或欧洲规范3（Eurocode 3）中的a类稳定曲线进行平面外稳定设计。两端纯弯荷载作用下空腹桁架圆弧拱可采用欧洲钢结构协会ECCS中受弯梁的稳定曲线进行平面外稳定设计，设计过程中，亦可以直接将纯弯稳定系数取为1.0。

（3）研究了全跨均布竖向荷载和半跨均布竖向荷载作用下空腹桁架拱平面外破坏机理。研究结果表明：针对整体破坏模式，建议采用整体轴力与弯矩相关设计公式对空腹桁架拱进行平面外整体稳定承载力设计验算。针对局部破坏模式，选择弦杆局部破坏模式作为设计期望的局部破坏模式。此外，建议采用轴力与弯矩的强度设计公式对局部破坏进行稳定承载力验算，提出的设计公式经有限元验证具有很好的可靠性。

The Vierendeel truss circular arches has only a belly bar perpendicular to the string bar, and does not set the common inclined belly bar in the traditional truss arch, so it has better aesthetics and light transmission. At present, Vierendeel truss arches has a good application prospect in bridge, stadium, airport and other spatial structures. In this paper, the outside plane elastic and elastoplastic buckling properties of rectangular steel tubular Vierendeel truss circular arches structure without inclined belly bar are systematically studied, which can provide reference for the stability design of similar structures. The main research contents are as follows:

(1) The theoretical formulas of bending stiffness, shear stiffness and torsional stiffness of the circular arch section of pin-ended Vierendeel truss arches. It is found that the torsional rigidity of the Vierendeel truss arches can be significantly improved by adding the torsional rigidity of the bar, and the calculated results are in good agreement with the finite element method. Then, the external elastic buckling loads of pin-ended and end-fixed Vierendeel truss arches are calculated by means of equilibrium theorem and numerical fitting. The results show that in both cases, the out-of-plane elastic buckling load is closely related to the ratio of abdominal string-chord stiffness. In addition, the influence of shear deformation on the overall buckling load outside the plane of the end-fixed Vierendeel truss arches cannot be ignored.

(2) The out-of-plane failure mechanism of the arc arch of the Vierendeel truss arches under the uniform radial load of the whole span and the pure bending load at both ends is studied. The results show that the Vierendeel truss circular arches under the full span uniform radial load can be designed by using the Class a stability curve of the Chinese Code for Steel Structure Design (GB50017-2020) or Eurocode 3. Under the action of pure bending load at both ends, the Vierendeel truss circular arches can be designed with the stability curve of the bending beam in the ECCS, and the stability coefficient of pure bending can be taken as 1.0 directly during the design process.

(3) The failure mechanism of the Vierendeel truss arches outside plane under full-span uniform vertical load and half-span uniform vertical load is studied. The results show that: according to the overall failure mode, it is recommended to use the design formula of the overall axial force and bending moment to check the design calculation of the overall stability bearing capacity of the Vierendeel truss arches. In view of the local failure mode, string local failure mode is selected as the desired local failure mode. In addition, it is suggested to use the strength design formula of axial force and bending moment to check the stability bearing capacity of local failure. The proposed design formula has good reliability verified by finite element method.

[1] 郭彦林, 窦超. 现代拱形钢结构设计原理与应用[M]. 北京: 科学出版社, 2013.

[2] Sawyer M H. World's First Iron Bridge[J]. Civil Engineering—asce, 1979.

[3] 孟杰. 系杆拱桥结构体系研究[D]. 湖南大学, 2002.

[4] 陈宝春. 钢管混凝土拱桥设计与施工[M]. 钢管混凝土拱桥设计与施工, 1999.

[5] Thrall A P, Billington D P, Bréa K. The Maria Pia Bridge: A major work of structural art[J]. Engineering Structures, 2012, 40: 479–486.

[6] 云迪. 大跨中承式钢管混凝土拱桥静力及抗震性能[D]. 哈尔滨工业大学, 2007.

[7] Witcher T R. A Pioneering Structure: The Eads Bridge[J]. Civil Engineering Magazine Archive, 2017, 87(11): 42-45.

[8] 王福敏, 徐伟. 重庆朝天门长江大桥主桥结构体系研究[J]. 公路交通技术, 2005, S1: 23-28.

[9] German Code: Geman Buckling Specifications, DIN 18 800[M]. November: English version, 1992.

[10] 中华人民共和国规范: 钢结构设计规范(GB50017-2020)[M]. 北京: 中国计划出版社, 2020.

[11] 林冰. 钢拱平面内稳定性及稳定承载力设计方法研究[D]. 北京: 清华大学, 2007.

[12] 郭宇飞. 钢管桁架拱的平面内稳定性能及设计方法研究[D]. 北京: 清华大学, 2008.

[13] Timoshenko S P. Theory of elastic stability[M]. New York: McGraw- Hill, 1961.

[14] Vlasov V Z. Thin-walled elastic beam[M]. Jerusalem: Israel Program for Scientific Translation, 1961.

[15] Qaqish S, Haddadin A. Buckling Behavior of Elastic Circular Arches[J]. Journal of Engineering Mechanics, 1991, 117(11): 2659-2680.

[16] Papangelis J P, Trahair N S. Flexural-Torsional Buckling of Arches[J]. Journal of Structural Engineering, 1987, 113(7): 1433-1443.

[17] Pi Y L, Bradford M A, Uy B. In-plane stability of arches[J]. Int J Solids Struct, 2002, 39(1): 105-125.

[18] Pi Y L, Bradford M A. In-plane strength and design of fixed steel I-section arches[J]. Engineering Structures, 2004, 26(3): 291-301.

[19] 郭彦林, 郭宇飞, 窦超. 纯压圆弧形钢管桁架拱平面内稳定性能及设计方法[J]. 建筑结构学报, 2010(8): 9.

[20] Pi Y-L, Trahair N S. Non-linear buckling and postbuckling of elastic arches[J]. Engineering Structures, 1998, 20(7): 571-579.

[21] Bradford M A, Uy B, Pi Y L. In-Plane Elastic Stability of Arches under a Central Concentrated Load[J]. Journal of Engineering Mechanics, 2002, 128(7): 710-719.

[22] 魏德敏. 拱的非线性理论及其应用[M]. 拱的非线性理论及其应用, 2004.

[23] Pi Y-L, Trahair N S. In-Plane Buckling and Design of Steel Arches[J]. Journal of Structural Engineering, 1999, 125(11): 1291-1298.

[24] 林冰, 郭彦林. 纯压抛物线拱平面内稳定性及设计方法研究[J]. 建筑结构学报, 2009, 20(3): 9.

[25] 郭彦林, 林冰, 郭宇飞. 焊接工字形截面抛物线拱平面内稳定性试验研究[J]. 建筑结构学报, 2009, 30(3): 8.

[26] 郭彦林, 林冰, 郭宇飞. 压弯圆弧拱平面内稳定承载力设计方法的理论与试验研究[J]. 土木工程学报, 2011, 44(3): 8.

[27] 林冰, 郭彦林. 均匀受压工字型等截面三铰圆弧钢拱的平面内稳定性和极限强度[J]. 建筑结构, 2009(2): 4.

[28] Guo Y-L, Yuan X, Pi Y-L, et al. In-Plane Failure and Strength of Pin-Ended Circular Steel Arches Considering Coupled Local and Global Buckling[J]. Journal of Structural Engineering, 2017, 143(1): 04016157.

[29] 郭彦林, 黄李骥. 腹板开洞钢拱的平面内稳定极限承载力设计理论及方法[J]. 建筑结构学报, 2007, 28(3): 9.

[30] Guo Y-L, Chen H, Pi Y-L, et al. In-Plane Failure Mechanism and Strength of Pin-Ended Steel I-Section Circular Arches with Sinusoidal Corrugated Web[J]. Journal of Structural Engineering, 2016, 142(2): 04015121.

[31] 郭彦林, 郭宇飞, 窦超. 四边形截面圆弧空间钢管桁架拱平面内稳定性及试验研究[J]. 建筑结构学报, 2010, 31(8): 54-62.

[32] Guo Y L, Chen H, Pi Y L. In-plane failure mechanisms and strength design of circular steel planar tubular Vierendeel truss arches[J]. Engineering Structures, 2017, 151(15): 488-502.

[33] He H, Yuan B, Chen H, et al. In-plane failure mechanism and stability bearing capacity design of planar plate-tube-connected circular steel arches[J]. Mechanics Based Design of Structures and Machines, 2022, 50(1): 154-169.

[34] Shi M, Yuan B, Jiang T, et al. In-plane failure mechanisms and strength design of circular steel tubular Vierendeel truss arches with rectangular section[J]. Structures, 2021, 29: 1779-1790.

[35] Zhu B-L, Liu W-C, Guo Y-L, et al. In-plane stability behaviours and design of web-opened steel arch[J]. Journal of Constructional Steel Research, 2022, 194: 107326.

[36] Sakimoto T, Namita Y. Out-of-plane bucking of solid rib arches braced with transverse bars[J]. Proceedings of the Japan Society of Civil Engineers, 1971, 191: 109-116.

[37] Sakimoto T, Komatsu S. Ultimate Strength Formula for Steel Arches[J]. Journal of Structural Engineering, 1983, 109(3): 613-627.

[38] Stephenp. theory of elastic stability. /-2nd ed[M]. theory of elastic stability. /-2nd ed, 1961.

[39] Pi Y L, Papangelis J P, Trahair N S. Prebuckling Deformations and Flexural-Torsional Buckling of Arches[J]. Journal of Structural Engineering, 1995, 121(9): 1313-1322.

[40] 窦超. 钢拱平面外稳定性能及设计方法[D]. 北京: 清华大学, 2012.

[41] 窦超, 郭彦林. 单轴对称截面圆弧拱弹性弯扭屈曲临界荷载[J]. 建筑科学与工程学报, 2011, 28(4): 69-74.

[42] 窦超, 郭彦林. 圆弧拱平面外弹性弯扭屈曲临界荷载分析[J]. 工程力学, 2012, 29(3): 83-89.

[43] 许强. 薄壁曲梁线弹性理论和弹塑性稳定极限承载力分析[D]. 杭州: 浙江大学, 2002.

[44] 童根树. 钢结构的平面外稳定[M]. 北京: 中国建筑工业出版社, 2007.

[45] Pi Y L, Bradford M A. Elastic flexural-torsional buckling of fixed arches[J]. The Quarterly Journal of Mechanics and Applied Mathematics, 2004, 57(4).

[46] Yang Y, Kuo S, Cherng Y. Curved Beam Elements for Nonlinear Analysis[J]. Journal of Engineering Mechanics, 1989, 115(4): 840-855.

[47] Wen R K, Suhendro B. Nonlinear Curved-Beam Element for Arch Structures[J]. Journal of Structural Engineering, 1991, 117(11): 3496-3515.

[48] Pi Y L, Trahair N S. Three-dimensional nonlinear analysis of elastic arches[J]. Engineering Structures, 1996, 18(1): 49-63.

[49] 童根树, 许强. 工字形截面圆弧曲梁的非线性理论[J]. 土木工程学报, 2004, 37(4): 1-7.

[50] Palani G S, Rajasekaran S. Finite Element Analysis of Thin-Walled Curved Beams Made of Composites[J]. Journal of Structural Engineering, 1992, 118(8): 2039-2061.

[51] 刘磊. 大跨度混凝土桥梁的双非线性分析[D]. 北京: 北方交通大学, 2000.

[52] 程鹏. 两铰圆弧拱非线性弯曲理论和弹塑性稳定[D]. 上海: 同济大学, 2005.

[53] 杨永华. 弹性开口薄壁圆弧钢拱的稳定承载力研究[D]. 上海: 同济大学, 2006.

[54] 乔彩虹. 两铰纯压圆弧拱面外稳定性及设计方法研究[D]. 呼和浩特: 内蒙古工业大学, 2008.

[55] Sakimoto T, Komatsu S. Ultimate strength of steel arches with bracing systems [J]. Journal of Structural Division, 1982, 108(ST 5): 1064-1076.

[56] 乔彩虹, 郭彦林, 曹玉生. 圆管截面两铰圆弧纯压拱的平面外稳定性及设计方法[J]. 工业建筑, 2009(12): 5.

[57] Dou C, Guo Y L, Zhao S Y, et al. Elastic out-of-plane buckling load of circular steel tubular truss arches incorporating shearing effects[J]. Engineering Structures, 2013, 52(6): 697-706.

[58] 李睿, 黄政华, 黄勇. 平面钢管桁架拱平面外稳定承载力的参数分析[J]. 贵州大学学报：自然科学版, 2013, 30(4): 4.

[59] 黄政华, 李睿, 黄勇, et al. 平面钢管桁架拱平面外稳定试验研究与有限元分析[J]. 建筑结构学报, 2015, 36(9): 7.

[60] Guo Y-L, Zhao S-Y, Dou C, et al. Out-of-plane strength design of spatially trussed arches with a rectangular lattice section[J]. Journal of Constructional Steel Research, 2013, 88: 321-329.

[61] 赵思远, 郭彦林, 窦超. 倒三角截面空间桁架拱平面外弹塑性稳定设计方法研究[J]. 工程力学, 2014(9): 10.

[62] 赵思远, 王宏, 郭彦林. 平面桁架拱平面外弹性稳定及设计方法研究[J]. 建筑结构, 2016, 46(S1): 523-527.

[63] Wang S, Liu X, Yuan B, et al. Out-of-plane elastic buckling load and strength design of space truss arch with a rectangular section[J]. Frontiers of Structural and Civil Engineering, 2022, 16(9): 1141-1152.

[64] 项海帆, 刘光栋. 拱结构的稳定与振动[M]. 拱结构的稳定与振动, 1991.

[65] 陈绍蕃. 钢结构设计原理[M]. 北京: 科学出版社, 2009.

[66] Chen W K. Structural Stability-Theory and Implementation[M]. New York: Elsevier, 1987.

[67] Simitses G J, Hutchinson J W. An introduction to the elastic stability of structures[J]. Journal of Applied Mechanics, 1976, 43: 383-384.

[68] Kármán T, Tsien H S. The Buckling of Thin Cylindrical Shells under Axial Compression[J]. Journal of the Aeronantical Sciences, 1941, 8: 303-312.

[69] 陈骥. 钢结构稳定理论与设计[M]. 钢结构稳定理论与设计, 2001.

[70] ABAQUS Analysis User's Manual-Version 6.14-2. USA: ABAQUS Inc, 2018.

[71] 罗定安. 工程结构数值分析方法与程序设计[M]. 工程结构数值分析方法与程序设计, 1995.

[72] 项海帆. 高等桥梁结构理论[M]. 高等桥梁结构理论, 2001.

[73] 郭在田. 薄壁杆件的弯曲与扭转[M]. 薄壁杆件的弯曲与扭转, 1989.

[74] Pi Y L, Trahair N S. Out-of-Plane Inelastic Buckling and Strength of Steel Arches[J]. Journal of Structural Engineering, 1999, 124(2): 174-183.

[75] 杨永华, 陈以一. 双轴对称固支圆弧拱弯扭屈曲荷载的理论解[J]. 工程力学, 2008, 25(4): 5.

[76] 中华人民共和国国家质量监督检验检疫总局, 中华人民共和国建设部. 中华人民共和国国家标准(GB 50205-2001)：钢结构工程施工质量验收规范[M]. 中华人民共和国国家标准（GB 50205-2001）：钢结构工程施工质量验收规范 2002.

[77] Pi Y L, Bradford M A. Inelastic buckling and strengths of steel I-section arches with central torsional restraints[J]. Thin-Walled Structures, 2003, 41(7): 663-689.

[78] European convention for constructional steelwork[M]. ECCS, 1988.