Analysis of shear stress for curved composite steel box girders with corrugated webs based on theory of shells of revolution
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摘要:
为精确计算波形腹板钢箱组合弯梁的竖向剪应力并量化各分项效应贡献，提出一种同时考虑波形钢腹板抗弯、初始曲率及曲率沿梁宽变化的剪应力计算方法。基于旋转壳理论，推导出包含弯曲、自由扭转、约束扭转翘曲与畸变翘曲分量的剪应力解析公式；采用能量变分法建立控制微分方程，并通过有限差分法求解；通过结构平面曲率、加载位置及截面尺寸的参数分析，探明该类弯梁剪应力分布规律与关键表征指标。研究结果表明：解析值与文献试验值及有限元值吻合良好；与忽略曲率影响的常规方法相比，所提方法对自由扭转剪应力和畸变翘曲剪应力的计算准确度分别提高5.0%和13.8%；梁轴平曲线半径R与梁宽B之比(R/B)小于10时弯扭耦合效应显著，而R/B≥70时可忽略，建议将10和70作为该效应显著与可忽略的关键界限值。基于所提解析公式得到的剪应力偏载系数为1.33，表明偏载效应显著，设计中需重点考虑。若忽略约束扭转与畸变效应，外腹板总剪应力计算结果虽偏于保守，但平均误差将达3.3%~9.0%，精细化分析中应计入二者影响。所建立的解析公式及获得的规律，可为波形腹板钢箱组合弯梁的设计计算提供理论参考。
Abstract:To accurately calculate the vertical shear stress of curved composite steel box girders with corrugated webs and quantify the contribution of each component effect, a calculation method of shear stress was proposed, which simultaneously considered the flexural resistance of corrugated steel webs, initial curvature, and curvature variation along the girder width. Based on the theory of shells of revolution, analytical formulas for shear stress were derived, including the components of bending, free torsion, restrained torsional warping, and distortional warping. The governing differential equations were established by the energy variational method and solved by the finite difference method. Through parametric analysis of structural horizontal curvature, loading position, and cross-sectional dimensions, the distribution laws and key characteristic indices of shear stress for this type of curved girder were clarified. The results indicate that the analytical values agree well with the experimental values in the literature and the finite element values. Compared with conventional methods ignoring the influence of curvature, the proposed method improves the calculation accuracy of free torsional shear stress and distortional warping shear stress by 5.0% and 13.8%, respectively. When the ratio of girder horizontal curve radius to girder width (R/B) is less than 10, the bending-torsion coupling effect is significant, while it is negligible when R/B≥70. It is suggested that 10 and 70 be taken as the key thresholds to judge whether the bending-torsion coupling effect is significant or negligible. The eccentric load coefficient of shear stress obtained by the proposed formulas is 1.33, indicating that the eccentric load effect is significant and should be emphasized in design. If the restrained torsion and distortion effects are ignored, the calculated total shear stress of the outer web is conservative, but the average error reaches 3.3%-9.0%, indicating that the effects of both should be included in the refined analysis. The established analytical formulas and obtained laws can provide a theoretical reference for the design and calculation of curved composite steel box girders with corrugated webs.
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图 8 剪应力解析公式建立流程
Figure 8. Establishment process of analytical formulas of shear stress
图 9 试验梁三维示意(内偏载，单位：mm)
Figure 9. Three-dimensional schematic of test girder (under inner eccentric loading, unit: mm)
图 14 总剪应力随径宽比的变化曲线
Figure 14. Variation of total shear stress with radius-to-width ratio
图 15 总剪应力随腹板倾角的变化曲线
Figure 15. Variation of total shear stress with web inclination angle
图 18 总剪应力随截面高度的变化曲线
Figure 18. Variation curves of total shear stress with section height
图 20 剪应力比ζ随宽径比的变化曲线
Figure 20. Variation of shear stress ratio ζ with width-to-radius ratio
表 1 波形钢腹板几何参数值
Table 1. Geometric parameters of corrugated steel web
mm 类型 aw bw lw hhw tw rw 1200型 330 270 1 200 200 8~20 15tw 1600型 430 370 1 600 220 10~36 15tw 
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表 2 不同加载工况下截面A-A剪应力对比
Table 2. Shear stress comparison for section A-A under different loading conditions
测点 外偏载 内偏载 中心加载 试验值/MPa 解析值/MPa 相对差值/% 试验值/MPa 解析值/MPa 相对差值/% 试验值/MPa 解析值/MPa 相对差值/% 1 36.8 35.5 -3.5 20.6 20.0 -2.9 29.7 27.7 -6.7 2 33.7 35.7 5.9 18.7 20.2 8.0 26.2 27.9 6.5 3 33.7 35.7 5.9 18.7 20.2 8.0 26.5 27.9 5.3 4 32.4 35.7 10.2 18.8 20.2 7.4 26.3 27.9 6.1 5 32.3 35.5 9.9 19.1 20.0 4.7 26.7 27.7 3.7
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表 3 不同加载工况下截面B-B剪应力对比
Table 3. Shear stress comparison for section B-B under different loading conditions
测点 外偏载 内偏载 中心加载 试验值/MPa 解析值/MPa 相对差值/% 试验值/MPa 解析值/MPa 相对差值/% 试验值/MPa 解析值/MPa 相对差值/% 1 - 30.7 - - 15.8 - 20.9 23.3 11.5 2 31.2 30.8 -1.3 16.1 15.9 -1.2 23.4 23.4 0.0 3 29.7 30.9 4.0 15.7 16.0 1.9 23.3 23.4 0.4 4 27.8 30.9 11.2 14.6 16.0 9.6 23.2 23.4 0.9 5 27.5 31.0 12.7 14.4 16.1 11.8 21.9 23.6 7.8 注：“-”表示数据缺失。
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表 4 截面A-A剪应力对比
Table 4. Shear stress comparison for Section A-A
MPa 测点 总剪应力 弯曲剪应力 自由扭转剪应力 常规方法 本文方法 试验 常规方法 本文方法 常规方法 本文方法 1 28.6 27.7 29.7 15.2 15.3 13.6 12.5 2 28.8 27.9 26.2 15.4 15.5 13.6 12.5 3 28.8 27.9 26.5 15.4 15.5 13.6 12.5 4 28.8 27.9 26.3 15.4 15.5 13.6 12.5 5 28.6 27.7 26.7 15.1 15.3 13.6 12.5
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表 5 截面B-B剪应力对比
Table 5. Shear stress comparison for Section B-B
MPa 测点 总剪应力 弯曲剪应力 自由扭转剪应力 常规方法 本文方法 试验 常规方法 本文方法 常规方法 本文方法 1 23.3 23.3 20.9 14.9 15.0 9.6 8.8 2 23.4 23.4 23.4 15.1 15.2 9.6 8.8 3 23.4 23.4 23.3 15.1 15.2 9.6 8.8 4 23.4 23.4 23.2 15.1 15.2 9.6 8.8 5 23.6 23.6 21.9 14.8 14.9 9.6 8.8
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表 6 控制截面挠度对比
Table 6. Comparison of deflections at control sections
加载工况 挠度 控制截面 L/4 L/2 3L/4 外偏载 试验值/mm 2.0 3.1 2.0 有限元值/mm 1.8 2.8 1.8 相对差值/% -5.7 -8.9 -9.2 中心加载 试验值/mm 1.6 2.7 1.7 有限元值/mm 1.6 2.5 1.6 相对差值/% -0.1 -5.3 -4.9 内偏载 试验值/mm 1.3 2.1 1.3 有限元值/mm 1.4 2.2 1.4 相对差值/% 9.7 4.1 11.2
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表 7 外腹板剪应力对比
Table 7. Shear stress comparison for outer web
腹板位置/m 解析值/MPa 有限元值/MPa 相对差值/ % 弯曲 自由扭转 约束扭转 畸变 总应力 1.5 -5.2 -4.7 0.01 0.5 -9.3 -8.7 6.9 1.3 -5.3 -4.7 0.01 0.6 -9.4 -9.1 3.3 0.8 -5.3 -4.7 0.01 0.6 -9.4 -9.4 0.0 0.2 -5.2 -4.7 0.01 0.5 -9.4 -9.4 0.0 0.1 -5.0 -4.7 0.01 0.3 -9.4 -8.4 11.9 注：相对差值为总应力值与有限元值的相对差值。
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表 8 内腹板剪应力对比
Table 8. Shear stress comparison for inner web
腹板位置/m 解析值/MPa 有限元值/MPa 弯曲 自由扭转 约束扭转 畸变 总应力 1.5 5.3 -5.2 0.01 0.7 0.8 1.3 1.3 5.4 -5.2 0.01 0.7 0.9 0.6 0.8 5.4 -5.2 0.01 0.7 0.9 0.1 0.2 5.3 -5.2 0.01 0.7 0.9 1.3
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表 9 外偏载工况下剪应力对比
Table 9. Comparison of shear stresses under outer eccentric loading
腹板位置/m 解析值/MPa 有限元值/MPa 相对误差/ % 弯曲 自由扭转 约束扭转 畸变 总应力 1.5 -5.3 -7.2 0.01 0.4 -12.0 -11.7 2.6 1.3 -5.4 -7.2 0.01 0.4 -12.1 -12.2 -0.8 0.8 -5.4 -7.2 0.01 0.4 -12.1 -12.6 -4.0 0.2 -5.4 -7.2 0.01 0.4 -12.1 -12.5 -3.2 0.1 -5.1 -7.2 0.01 0.2 -12.1 -11.5 5.2
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表 10 内偏载工况下剪应力对比
Table 10. Comparison of shear stresses under inner eccentric loading
腹板位置/m 解析值/MPa 有限元值/MPa 相对误差/ % 弯曲 自由扭转 约束扭转 畸变 总应力 1.5 -5.0 -2.3 0.01 0.7 -6.6 -6.2 6.5 1.3 -5.1 -2.3 0.01 0.7 -6.7 -6.6 1.5 0.8 -5.2 -2.3 0.01 0.7 -6.7 -6.8 -1.5 0.2 -5.1 -2.3 0.01 0.6 -6.7 -6.8 -1.5 0.1 -4.9 -2.3 0.01 0.4 -6.8 -5.6 21.4
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