ONE-DIMENSIONAL FINITE ELEMENT ANALYSIS OF WARPING TORSION FOR THIN-WALLED MEMBERS WITH OPEN-CLOSED CROSS SECTIONS BASED ON COMPATIBLE WARPING FIELD
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摘要: 开口及闭口薄壁杆件约束扭转问题已由经典Timoshenko和Benscoter理论解决。然而，开闭口混合薄壁截面杆件约束扭转分析必须考虑开、闭口部分翘曲能力的差异，翘曲剪流形成机理有待进一步研究。该文假定开、闭口截面翘曲分别满足Vlasov和Umanskii假定，考虑开、闭口截面公共节点翘曲连续性要求，建立含有待定翘曲参数的协调翘曲模型。由截面受力平衡，确定翘曲参数显式列式，提出开闭口混合薄壁截面杆件约束扭转分析的一维有限元模型。算例及参数分析结果表明，基于Umanskii第二理论的I类方法在悬臂板及闭口周边引入附加剪流，影响翘曲剪应力精度。基于Umanskii第二理论的II类方法只能计算截面板件平均剪应力，无法反映真实翘曲剪流分布。基于Vlasov约束扭转假定的Beam-189单元忽略闭口周边约束效应产生的附加翘曲及剪流，影响翘曲正应力和剪应力精度。该文方法与Shell-63单元能得到基本吻合的变形与应力结果，说明一维梁元能正确反映开闭口混合薄壁截面杆件约束扭转及翘曲刚度。
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关键词:
- 开闭口混合薄壁截面杆件 /
- 协调翘曲场 /
- 约束扭转 /
- 翘曲应力 /
- 一维有限元分析
Abstract: The warping torsion of thin-walled members with open and closed cross sections was well addressed by the classical theories presented by Timoshenko and Benscoter, respectively. However, the restrained torsional behavior of thin-walled members with open-closed profile cannot be correctly accounted for without considering the distinctive warping properties between open and closed parts of the cross section. The development of warping shear flows within the mid-plane of the cross section needs further elaborations. This work assumes that the torsional warping of open and closed segments correspondingly adheres to the classical assumptions of Vlasov and Umanskii. The warping displacements are required to coincide at the common points of the open and closed segments, leading to a compatible warping field which contains undetermined warping parameter. The warping parameter is explicitly obtained based on the equilibrium requirements. A one-dimensional finite element model is naturally developed for warping torsion analysis of a thin-walled member with open-closed cross section. It is shown by numerical investigations and parametric studies that the type I method based on the Second Umanskii theory can artificially introduce the additional shear flows that reduce the accuracy of shear stresses. As an alternative, the type II method based on the Second Umanskii theory can only obtain the average shear stress of each segment of thin-walled section which fails to provide the correct distribution of shear flows. The Beam-189 element model based on the Vlasov assumption unfortunately neglects the induced warping displacements and shear flows by the constrained effects of the closed contour. Hence, the use of Beam-189 element will reduce the accuracies of both normal and shear stresses. Close agreement can be observed between the Shell-63 element model and the present method for calculating the torsional deformation and stresses, which demonstrates the capability of the one-dimensional finite element model to describe the torsional and warping stiffness of the thin-walled members with open-closed cross section. -
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图 4 简支箱梁跨中截面翘曲正应力随悬臂板宽度变化曲线
Figure 4. Correlations between the warping normal stresses at mid-span and the width of the cantilever slab of a simply- supported box girder
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图 5 简支箱梁端截面(x=30)剪应力随悬臂板宽度变化曲线
Figure 5. Correlations between the shear stresses at the right end and the width of the cantilever slab of a simply-supported box girder
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图 6 简支箱梁跨中截面翘曲正应力随梁高变化曲线
Figure 6. Correlations between the warping normal stresses at mid-span and the height of a simply-supported box girder
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图 7 简支箱梁端截面(x=30)剪应力随梁高变化曲线
Figure 7. Correlations between the shear stresses at the right end and the height of a simply-supported box girder
表 1 悬臂箱梁约束扭转位移和正应力
Table 1 Displacements and normal stresses for warping torsion of a cantilever box girder
计算模型 自由端扭转角/(×10−5 rad) 自由端翘曲/(×10−5 m)/初等梁理论翘曲/(×10−5 m) 固定端正应力/MPa/初等梁理论应力/MPa ③ ② ① ③ ② ① 本文一维梁元 −4.95 −1.838/−1.84 −10.72/−10.71 −14.21/−14.24 −0.091/−0.092 −0.545/−0.536 −0.687/−0.712 Shell-63单元 −5.05 −1.7/−1.84 −11.0/−10.71 −14.0/−14.24 −0.092/−0.092 −0.54/−0.536 −0.68/−0.712 文献[7] − − − − −0.091/−0.092 −0.598/−0.536 −0.55/−0.712 注：翘曲位移和应力的正负号与Ox坐标轴指向保持一致。 
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表 2 均布扭矩作用下简支箱梁约束扭转位移和应力
Table 2 Displacements and stresses for warping torsion of a simply-supported box girder under the action of uniformly distributed torques
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表 3 跨中集中偏载作用下简支箱梁正应力和扭转角
Table 3 Normal stresses and angle of twist of a simply-supported box girder subjected to an eccentric transverse load applied at mid-span
计算模型 跨中扭转角/
(×10−2rad)跨中正应力/(×108Pa)/
弯曲正应力/(×108Pa)⑥ ⑤ ④ 本文一维梁元 2.802 −2.69/−2.93 −3.14/−2.93 3.69/3.57 Beam-189模型 2.678 −2.70/−2.93 −3.12/−2.93 3.67/3.57 Shell-63模型 2.783 −2.82/−2.93 −3.32/−2.93 4.01/3.57 Umanskii第二理论 2.812 −2.79/−2.93 −3.05/−2.93 3.63/3.57
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