Calculation method of post-tensioned prestressed anchorage loss considering influence of asymmetric friction
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摘要: 为改进后张预应力混凝土梁直线、曲线混合布束的预应力锚固损失计算方法，提高预应力锚固损失的理论计算精度，对预应力钢束微段建立静力平衡方程; 利用不同钢束形状间的变形协调关系和应力连续条件，考虑设计中预应力钢束直线、曲线混合分布的实际影响参数与正、反摩阻损失差异，建立了针对预应力混合布束锚固损失求解的分段逼近理论，推导了预应力锚固损失精确计算公式，并编制了Python程序，实现了自动求解与简化计算; 通过现场足尺模型试验比较了精确公式与现行公路和铁路桥梁设计规范中预应力锚固损失理论算法的计算误差。研究结果表明：后张预应力直线、曲线混合布束锚固时反摩阻效应小于张拉时的正摩阻效应，且实际中反摩阻影响长度与现行桥梁设计规范算法有较大偏差; 采用提出的方法推算的反摩阻影响范围总体与中国铁路桥梁设计规范更为接近，在精度和离散度方面比现行公路与铁路桥梁设计规范分别高出16.7%和14.9%，且与模型试验数据的相关度更高，变异性更小; 在后张预应力混凝土结构相关研究中应考虑实际预应力直线、曲线混合布束线形与不对称正、反摩阻效应的影响，采用分段逼近法计算钢束预应力锚固损失; 在进行后张预应力混凝土结构设计时，从简化计算的角度考虑，建议采用现行铁路桥梁设计规范计算反摩阻效应。Abstract: To improve the calculation method of prestressed anchorage loss of post-tensioned prestressed concrete beams with mixed straight lines and curves and enhance the theoretical calculation accuracy of the prestressed anchorage loss, an static equilibrium equation was established for the micro-segment of prestressed steel bundles. According to the deformation coordination relationship and stress continuity conditions between different steel bundle shapes, the actual influencing parameters of the mixed distribution of straight lines and curves of the prestressed steel bundles as well as the difference between the positive friction and anti-friction losses in design were considered. A piecewise approximation theory for calculating the anchorage loss of prestressed mixed bundles was established. The exact calculation formula of the prestressed anchorage loss was deduced, and a Python program was compiled to realize an automatic solution and simplified calculation. Through the field full-scale model test, the calculation errors of the exact formula and the theoretical algorithm for the prestressed anchorage loss in the current highway and railway bridge design codes were compared. Research results show that the anti-friction effect of post-tensioned prestressed bundles with mixed straight lines and curves for anchorage is smaller than the positive friction effect during the tensioning, and the actual anti-friction influence length greatly deviates from algorithms in the current bridge design codes. The anti-friction influence range calculated by the proposed method is generally closer to that in the Chinese railway bridge design code and is 16.7% and 14.9% higher than the current highway and railway bridge design codes in terms of accuracy and dispersion, respectively. Furthermore, it is highly correlated with the model test data, with small variability. In the related research on the post-tensioned prestressed concrete structures, the influences of the shape of actual prestressed mixed straight lines and curves, as well as the asymmetric positive friction and anti-friction effects, should be considered, and the piecewise approximation method should be used to calculate the prestressed anchorage loss of steel bundles. In the design of post-tensioned prestressed concrete structures, it is recommended to use the current railway bridge design code to calculate the anti-friction effect from the perspective of simplifying the calculation.
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Key words:
- bridge engineering /
- post-tensioning method /
- prestressed box beam /
- prestress loss /
- friction loss /
- anti-friction
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图 2 F位于AB段时钢束应力分布
Figure 2. Stress distribution of steel bundle when F is located in AB section
图 4 F位于BC段时钢束应力分布
Figure 4. Stress distribution of steel bundle when F is located in BC section
图 5 F位于CD段时钢束应力分布
Figure 5. Stress distribution of steel bundle when F is located in CD segment
图 6 F超过半跨钢束时应力分布
Figure 6. Stress distribution when F exceeds half-span of steel bundle
图 7 非对称反摩阻锚固损失计算流程
Figure 7. Calculation process asymmetric anti-friction anchorage loss
图 8 20 m跨径混凝土箱梁横截面(单位：mm)
Figure 8. Cross sections of concrete box beam with span of 20 m (Unit: mm)
图 9 20 m跨径混凝土箱梁预应力钢束布置(单位：mm)
Figure 9. Prestressed steel bundle arrangements of concrete box beam with span of 20 m (Unit: mm)
图 12 实测预应力正、反摩阻损失对比
Figure 12. Comparison of measured prestressed positive and negative friction losses
图 13 试验梁预应力锚固损失分布的不同算法对比
Figure 13. Comparison of different algorithms of prestress anchorage loss distributions in test beams
图 15 公路桥梁设计规范计算结果与实测值对比
Figure 15. Comparison between calculated results of highway bridge design code and measured values
图 16 铁路桥梁设计规范计算结果与实测值对比
Figure 16. Comparison between calculated results of railway bridge design code and measured values
图 17 提出算法的计算结果与实测值对比
Figure 17. Comparison between calculated results of proposed algorithm and measured values
表 1 试验梁结构尺寸
Table 1. Sizes of test beam structure
cm 跨径 位置 顶板厚 腹板宽 底板厚 梁高 梁宽 中梁 边梁 2 000 端部 18 30 30 120 120+120 165+120 跨中 18 18 18 120 120+120 165+120 2 500 端部 18 30 30 140 120+120 165+120 跨中 18 18 18 140 120+120 165+120 3 000 端部 18 30 30 160 120+120 165+120 跨中 18 20 18 160 120+120 165+120 
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表 2 试验梁测试钢束位置
Table 2. Positions of steel bundles in test beam
20 m梁 25 m梁 30 m梁 N1左腹板束 N1左腹板束 N1左腹板束 N3右腹板束 N3右腹板束 N3右腹板束 N4左底板束 N4左腹板束 N5左底板束
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表 3 预应力分布中各参数取值
Table 3. Parameter values in prestress distribution
试验梁跨径/m N1 N3 N4(N5) a/mm b/mm c/mm d/mm e/mm f/mm g/mm h/mm 20 8 625 1 175 19 600/2 5 425 4 375 19 600/2 9 800 19 600/2 25 9 874 2 426 24 600/2 6 453 5 847 24 600/2 12 300 24 600/2 30 12 080 2 720 29 600/2 9 223 5 577 29 600/2 14 800 29 600/2 30 12 080 2 720 29 600/2 9 223 5 577 29 600/2 14 800 29 600/2
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表 4 实测梁端钢束锚固损失
Table 4. Measured anchorage losses of steel bundles in beam end
测试梁 钢束号 锚索测力计位置 张拉控制应力/MPa 锚固后应力/MPa 瞬时损失比/% 锚固变形钢束收缩/mm 20 m中梁 N1 左 1 337.86 1 139.29 14.84 7.5 右 1 208.75 1 013.29 16.17 7.4 N3 左 1 499.29 1 269.82 15.31 7.2 右 1 395.54 1 164.82 16.53 7.4 N4 左 1 336.14 1 099.14 17.74 8.5 右 1 276.86 1 039.14 18.62 8.1 25 m边梁 N1 左 1 356.07 1 187.50 12.43 8.2 右 1 264.29 1 065.89 15.69 8.4 N3 左 1 315.71 1 143.57 13.08 6.0 右 1 229.57 1 073.00 12.73 5.6 N5 左 1 232.71 1 098.14 10.92 4.6 右 1 131.71 1 024.86 9.44 4.4 30 m中梁 N1 左 1 285.43 1 110.14 13.64 8.3 右 1 204.57 1 102.86 8.44 8.1 N3 左 1 404.52 1 201.19 14.48 9.5 右 1 283.81 1 105.83 13.86 8.0 N5 左 1 251.57 1 077.86 13.88 6.5 右 1 311.00 1 128.86 13.89 6.5 30 m边梁 N1 左 1 421.31 1 284.40 9.63 6.4 右 1 427.50 1 277.74 10.49 6.5 N3 左 1 347.86 1 112.50 17.46 9.3 右 1 273.93 1 070.48 15.97 9.1 N5 左 1 335.57 1 186.29 11.18 6.8 右 1 290.71 1 164.00 9.82 6.4
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表 5 实测梁内钢束锚固损失
Table 5. Measured anchorage losses of steel bundles inside beam
试验梁 测试阶段 磁通量测点编号 1# 2# 3# 4# 5# 20 m中梁 张拉阶段/MPa 1 250.54 1 420.00 1 396.43 1 310.00 1 271.00 锚固后/MPa 1 185.18 1 309.64 1 313.93 1 220.54 1 183.57 瞬时损失/% 5.23 7.77 5.91 6.83 6.88 25 m边梁 张拉阶段/MPa 1 263.39 1 260.71 1 185.57 1 227.00 1 156.86 锚固后/MPa 1 232.68 1 215.86 1 168.86 1 183.57 1 109.29 瞬时损失/% 2.43 3.56 1.41 3.54 4.11 30 m中梁 张拉阶段/MPa 1 216.29 1 340.36 1 321.67 1 312.38 1 222.14 锚固后/MPa 1 187.57 1 289.88 1 277.14 1 264.76 1 188.00 瞬时损失/% 2.36 3.77 3.37 3.63 2.79 30 m边梁 张拉阶段/MPa 1 349.88 1 289.29 1 268.45 1 251.19 1 236.86 锚固后/MPa 1 318.45 1 213.81 1 203.81 1 193.57 1 196.14 瞬时损失/% 2.33 5.85 5.10 4.61 3.29
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表 6 反摩阻影响长度比较
Table 6. Comparison of anti-friction influence length
试验箱梁 1/2计算跨径/cm 钢束号 锚固点位置 反摩阻影响长度/cm 公路桥规 铁路桥规 本文算法 20 m中梁 980 N1 左端 1 258 1 408 >l/2 右端 1 315 1 448 >l/2 N3 左端 1 070 1 349 >l/2 右端 1 124 1 348 >l/2 N4 左端 1 530 1 575 >l/2 右端 1 451 1 573 >l/2 25 m边梁 1 230 N1 左端 1 334 1 439 >l/2 右端 1 398 1 610 >l/2 N3 左端 1 079 1 333 >l/2 右端 1 078 1 333 >l/2 N5 左端 1 163 1 245 >l/2 右端 1 187 1 231 >l/2 30 m中梁 1 480 N1 左端 1 420 1 706 1 140 右端 1 447 1 450 1 160 N3 左端 1 397 1 624 >l/2 右端 1 341 1 559 >l/2 N5 左端 1 389 1 423 >l/2 右端 1 357 1 391 >l/2 30 m边梁 1 480 N1 左端 1 184 1 325 980 右端 1 190 1 258 940 N3 左端 1 411 1 614 >l/2 右端 1 435 1 594 >l/2 N5 左端 1 375 1 409 >l/2 右端 1 357 1 391 >l/2
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