Exact finite element method for time-dependent analysis of steel-concrete composite beam considering shear deformation
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摘要:
为了改善传统位移型有限元方法在分析钢-混组合梁长期力学行为时易产生的曲率闭锁问题，提高计算精度和效率，提出了一种同时考虑双层梁界面滑移效应、梁层剪切变形以及混凝土收缩与徐变影响的钢-混组合梁精确有限元法。基于弹性力学基本方程，结合混凝土线性黏弹性徐变本构模型，推导得到了组合梁单元的控制微分方程并进行了解析求解；采用直接刚度法进一步推导得到了组合梁的精确单元刚度矩阵和等效荷载矩阵，并开发了相应的数值计算程序；通过3个典型算例对所提出的方法进行了验证，并开展了参数分析。分析结果表明：提出的有限元法在同时考虑梁层剪切变形和界面滑移影响的情况下，能够准确预测钢-混组合梁的时变力学响应，即使在单元划分数量较少、收缩与徐变分析时间步数较小的条件下，仍可获得较高精度的计算结果；与解析法相比，365 d的挠度误差不超过3.2%，但计算效率和通用性大大提升；与不考虑剪切效应的有限元法相比，误差减小了10%以上。该方法可为钢-混组合梁长期性能分析及工程设计提供一种高效、可靠的计算手段。
Abstract:To overcome curvature locking in conventional displacement-based finite element methods for analyzing the long-term mechanical behavior of steel-concrete composite beams, and to improve computational accuracy and efficiency, an exact finite element method that accounts for interfacial slip between the two beam layers, shear deformation of beam layers, and the effects of concrete shrinkage and creep was proposed. Based on the fundamental equations of elasticity and a linear viscoelastic constitutive model for concrete creep, the governing differential equations for the composite beam element were derived and solved analytically. The exact element stiffness matrix and equivalent load matrix were formulated using the direct stiffness method, and a corresponding numerical program was developed. The proposed method was validated using three representative examples, followed by a parametric study. The results indicate that the proposed finite element method can accurately predict the time-dependent mechanical response of steel-concrete composite beams while accounting for both shear deformation and interfacial slip. High accuracy can still be achieved with a relatively coarse mesh and a limited number of time steps in shrinkage and creep analysis. Compared with the analytical solution, the error in deflection at 365 d is less than 3.2%, while the computational efficiency and applicability are significantly improved. In comparison with a finite element method that neglects shear deformation, the proposed method reduces the error by more than 10%. The proposed method therefore provides an efficient and reliable computational tool for the long-term performance analysis and engineering design of steel-concrete composite beams.
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图 1 钢-混组合梁微段受力状态
Figure 1. Stress state of an infinitesimal segment of a steel-concrete composite beam
图 2 钢-混组合梁微段变形
Figure 2. Deformation of an infinitesimal segment of a steel-concrete composite beam
图 3 组合梁单元节点力与节点位移
Figure 3. Nodal forces and nodal displacements of composite beam element
图 10 承受均布荷载与集中荷载的组合梁（单位：mm）
Figure 10. Composite beam subjected to uniformly distributed and concentrated loads (unit: mm)
图 11 时变影响下跨中挠度与跨高比关系
Figure 11. Relationship between mid-span deflection and L/H under time-dependent effects
图 12 时变影响下界面端部滑移与跨高比关系
Figure 12. Relationship between interface end slip and L/H under time-dependent effects
图 13 时变影响下跨中挠度与剪力连接刚度关系
Figure 13. Relationship between mid-span deflection and shear connection stiffness under time-dependent effects
表 1 不同时期梁跨中挠度
Table 1. Mid-span deflection at different time stages
mm 来源 瞬时 30 d 50年 本文 -21.86 -23.09 -31.26 文献[37] -21.33 -22.62 -30.80 
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表 2 不同单元数划分下计算结果对比
Table 2. Comparison of calculation results under different element mesh divisions
mm 单元数 瞬时 30 d 50年 2 -21.859 333 3 -23.087 610 7 -31.264 224 4 4 -21.859 333 3 -23.087 610 7 -31.264 224 4 6 -21.859 333 3 -23.087 610 7 -31.264 224 4 10 -21.859 333 3 -23.087 610 7 -31.264 224 4 20 -21.859 333 3 -23.087 610 7 -31.264 224 4
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表 3 不同时期计算结果对比
Table 3. Comparison of calculation results at different time stages
mm 步数 30 d 2年 10年 50年 5 -23.087 61 -29.024 99 -30.730 55 -31.264 22 7 -23.087 61 -29.009 82 -30.724 32 -31.261 96 9 -23.087 61 -28.999 37 -30.712 22 -31.261 35 11 -23.087 61 -28.997 30 -30.652 70 -31.235 32
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表 5 L/H=3时跨中挠度以及端部滑移值
Table 5. Mid-span deflections and end slips at L/H = 3
mm 时期 跨中挠度 端部滑移 计算值 参考值 计算值 参考值 瞬时 0.261 88 0.203 55 0.100 68 0.100 56 31 d 0.323 02 0.250 35 0.127 01 0.124 88 500 d 0.515 93 0.419 26 0.226 21 0.218 09 25 550 d 0.570 57 0.467 58 0.255 29 0.245 01
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表 6 L/H=12时跨中挠度以及端部滑移值
Table 6. Mid-span deflections and end slips at L/H = 12
mm 时期 跨中挠度 端部滑移 计算值 参考值 计算值 参考值 瞬时 15.027 6 14.652 2 1.884 73 1.882 48 31 d 17.856 7 17.381 7 2.254 20 2.243 84 500 d 24.730 1 23.945 3 3.205 25 3.157 74 25 550 d 26.532 3 25.651 0 3.458 21 3.398 17
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表 7 不同时期及跨高比下跨中挠度计算值与参考值之比
Table 7. Ratios of calculated value to reference value of mid-span deflection under different time periods and span-to-depth ratios
时期 不同跨高比（L/H）时的计算值与参考值之比/% 10 7 6 5 4 3 瞬时 3.3 7.0 9.2 12.7 18.4 28.6 500 d 2.6 6.9 8.6 11.2 15.3 23.1 25 550 d 2.5 6.8 8.5 10.9 14.8 22.0
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