Load Analysis of Statically Indeterminate Support Systems for Reaction Vessel Wheels
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摘要:
转轮支撑结构是后处理反应容器的重要组成部分，为提高反应容器的安全强度，该支撑系统采用超静定结构，针对支撑辊子的强度和系统的均载性问题，本文基于能量法，引入拉格朗日乘数，结合能量守恒定律解算各辊子上的法向力，进而求解出弯曲应力和接触应力；建立简易三维模型，使用有限元软件分析各辊子的应力分布和变化规律，并通过实验对结果进行验证；定义载荷分配系数，通过SPSSAU分析得出各制造误差对载荷分配系数的影响规律. 结果表明：辊子的最大弯曲应力为42 MPa，远小于材料屈服强度；有限元结果与理论结果的最大误差约为8%，并通过辊子的应变实验验证该方法的准确性；载荷分配系数与辊子弧面曲率半径呈负相关，与转轮弧面曲率半径呈正相关.
Abstract:The support structure of wheels is an important component of post-treatment reaction vessels. To improve the safety strength of the reaction vessels, the support system adopts a statically indeterminate structure. In view of the strength of supporting rollers and load-bearing characteristics of the system, based on the energy method, Lagrange multipliers were introduced and the law of energy conservation was combined to calculate the normal forces on each roller, thus solving the bending stress and contact stress. A simple 3D model was built, and by adopting finite element software, the stress distribution and variation laws of each roller were analyzed, with the results verified via experiments. The load distribution factor was defined, and the influence laws of manufacturing errors on the load distribution factor were analyzed by utilizing SPSSAU. The results show that the maximum bending stress of the roller is 42 MPa, which is much smaller than the yield strength of the material. The maximum error between the finite element simulation results and theoretical results is about 8%, and the accuracy of the proposed method was verified by strain experiments on rollers. The load distribution factor negatively correlates with the curvature radius of the roller surface, and positively correlates with the curvature radius of the wheel surface.
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Key words:
- statically indeterminate structure /
- energy method /
- Lagrange multiplier /
- stress /
- finite element /
- load distribution
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图 5 不同网格数量下的接触应力平均值
Figure 5. Average value of contact stress under different grid numbers
图 13 大转轮运转一圈时辊子的应变曲线
Figure 13. Strain curves of rollers during one rotation cycle of large wheels
图 14 完全支撑情况下实验与理论结果对比
Figure 14. Comparison of experimental and theoretical results under full support
图 15 制造误差对载荷分配系数的影响曲线
Figure 15. Influence curves of manufacturing errors on load distribution factors
表 2 转轮与辊子的尺寸参数
Table 2. Size parameters of wheel and roller
参数 数值 辊子接触截面半径 R11/mm 90 转轮内径 R12/mm 1865 辊子弧面曲率半径 R21/mm 162.5 转轮弧面曲率半径 R22/mm 162.9 转轮重力 Gr/N 49000 物料总重力 Gw/N 7840 辊子最大弯曲应力力臂 l/mm 124 辊子最大弯曲应力处截面直径 d/mm 120 接触长度 B/mm 80 转轮与辊子弹性模量 E/MPa 210000 摩擦系数 μ 0.003 泊松比 ν 0.3 转轮外径 R3/mm 1975 转轮偏心圆半径 Rh/mm 1275 
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表 3 静止工况下各辊子受力
Table 3. Forces on each roller in static working conditions
辊子编号 1# 8# 2# 7# 3# 6# 4# 5# 支反力/N 4985 4985 7885 7885 10282 10282 11851 11851
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表 4 运转工况下各辊子受力
Table 4. Forces on each roller in operating conditions
辊子编号 1# 2# 3# 4# 支反力/N 5192 8061 10578 12352 辊子编号 5# 6# 7# 8# 支反力/N 11351 9883 7580 4780
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表 5 工况列表
Table 5. Working conditions
工况编号 支撑状态 加载状况 1 1#、8#辊子承载 静止工况 2 运转工况 3 4#、5#辊子承载 静止工况 4 运转工况 5 4#、8#辊子承载 静止工况 6 运转工况 7 1#～8#辊子承载 静止工况 8 运转工况
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表 6 解析解和有限元解的应力对比
Table 6. Stress comparison between analytical and finite element solutions
工况编号 辊子编号 弯曲应力/MPa 接触应力/MPa 解析解 有限元解 解析解 有限元解 1 1# 41.546 42.220 567.9 577.3 8# 41.546 42.240 567.9 576.7 2 1# 41.722 42.220 568.7 577.6 8# 41.372 41.760 567.1 573.6 3 4# 21.506 22.800 456.0 457.3 5# 21.506 22.900 456.0 456.1 4 4# 21.868 22.500 458.6 461.7 5# 21.145 21.360 453.4 451.5 5 4# 37.249 36.86 547.6 545.0 8# 11.132 16.99 366.1 379.7 6 4# 37.355 37.79 548.2 552.1 8# 10.930 11.37 363.9 361.4
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表 7 部分支撑情况下实验与理论结果对比
Table 7. Comparison of experimental and theoretical results under partial support
工况编号 辊子编号 弯曲应力/MPa 辊子编号 弯曲应力/MPa 1（理论结果） 1# 41.55 8# 41.55 1（实验结果） 41.98 30.12 2（理论结果） 1# 41.72 8# 41.37 2（实验结果） 3（理论结果） 4# 21.51 5# 21.51 3（实验结果） 28.39 20.97 4（理论结果） 4# 21.87 5# 21.15 4（实验结果） 31.91 29.95 5（理论结果） 4# 37.25 8# 11.13 5（实验结果） 26.13 14.04 6（理论结果） 4# 37.36 8# 10.93 6（实验结果） 35.31 14.72
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表 8 制造误差因素水平表
Table 8. Factors and levels of manufacturing errors
水平编码 R11/mm R12/mm R21/mm R22/mm ∆l/mm ∆d/mm −4 −0.046 −0.260 −0.057 −0.057 −0.040 −0.040 4 0.046 0.260 0.057 0.057 0.040 0.040 间隔 0.01150 0.06500 0.01425 0.01425 0.01000 0.01000
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表 9 多元线性回归分析结果
Table 9. Multiple linear regression analysis results
参数 未标准化系数 B 标准化系数 β B的95%置信区间 t p VIF 取值 标准错误 下限 上限 常数项 1.274 0.006 1.261 1.287 207.242 <0.001 x1 −0.006 0 −0.709 −0.006 −0.006 −237.045 <0.001 1.000 x2 0.006 0 0.705 0.006 0.006 235.855 <0.001 1.000
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