STUDY ON ANALYTICAL MODEL AND ROBUSTNESS RANKING INDEX OF RC FRAMES WITH UNEQUAL SPANS AGAINST PROGRESSIVE COLLAPSE
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摘要: 现有钢筋混凝土(RC)框架抗连续倒塌的理论分析模型仅适用于等跨结构。通过理论推导将现有等跨结构的分析模型推广至不等跨结构，并提出了一个新的修正三折线模型，可以考虑不同工况下结构的抗弯、压拱、悬梁线和拉膜效应机制。利用所提理论分析模型建立了一个鲁棒性评判指标，用于危险工况的快速判断。数值与理论结果对比表明：所提出的修正三折线理论分析模型准确性高，能够为不等跨RC框架抗连续倒塌设计提供参考；所建立的鲁棒性评判指标可用于快速确定出RC框架的最危险柱子失效工况。Abstract: The existing analytical models of reinforced concrete (RC) frames against progressive collapse are only available to structures with equal spans. Therefore, a new modified tri-linear model is proposed for RC frames with unequal spans, which can consider the flexural action, compressive arch action, catenary action and tensile membrane action of the structure under different single column loss. A robustness ranking index is also established based on the proposed model to quickly identify the dangerous column loss scenarios. The results show that the proposed model is accurate and can provide a reference for the design of RC frames with unequal spans against progressive collapse. And the robustness ranking index can be used to quickly identify the worst column loss scenarios of RC frames.
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Key words:
- progressive collapse /
- RC frame /
- unequal span /
- analytical model /
- robustness ranking index
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图 1 结构抗连续倒塌承载力理论分析模型
Figure 1. Analytical models of progressive collapse resistance of structure
图 2 RC框架结构的6种典型失效工况
Figure 2. Six typical column removal scenarios of RC frame structure
图 3 不等跨结构的假定板屈服线和梁塑性铰分布
Figure 3. Assumed plastic hinges and yield-line patterns of structure with unequal spans
图 8 不同失效工况下的结构竖向位移时程响应
Figure 8. Time history of vertical displacement of different column loss
图 10 1层和7层柱失效下的荷载因子对比
Figure 10. Comprisions of load factor of 1st and 7th story column loss
图 11 不同工况下的结构竖向位移云图
Figure 11. Vertical displacement contours of structure under different scenarios
图 12 结构荷载因子曲线数值的与理论结果对比
Figure 12. Comparisons of analytical and numerical load factors
表 1 结构最大荷载因子的理论与数值结果对比
Table 1. Comparisons of analytical and numerical maximum load factor
工况 数值 模型1 相对误差R/(%) 模型2 相对误差R/(%) 模型3 相对误差R/(%) A1 1.90 1.41 25.9 1.56 18.0 1.56 18.0 A2 1.89 1.44 23.9 1.77 6.4 1.90 −0.3 A3 2.07 1.51 27.3 1.84 11.3 1.96 5.2 A4 2.79 1.91 31.4 2.30 17.6 2.47 11.4 A5 2.23 1.49 33.1 1.82 18.2 1.95 12.4 A6 1.53 1.18 22.9 1.44 5.9 1.53 −0.5 B1 4.05 2.58 36.2 3.05 24.7 3.32 18.0 B2 3.52 1.99 43.4 3.00 14.8 3.20 9.0 B3 3.57 2.14 40.1 3.15 12.0 3.35 6.3 B4 4.19 2.34 44.3 3.46 17.4 3.68 12.1 B5 3.45 1.91 44.6 2.89 16.2 3.05 11.4 B6 2.91 1.66 43.1 2.56 12.0 2.72 6.5 
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表 2 结构鲁棒性评判指标的参数表达式
Table 2. Expression of parameters for structural robustness evaluation index
工况 A B U CC $ \dfrac{{3( {{\chi _y}/{\chi _x} + {\chi _x}/{\chi _y}} ) + 2.5( {1/{\chi _x} + 1/{\chi _y}} )}}{{8{\chi _x}{\chi _y} + 2{\gamma _{\rm{S}}}( {{\chi _x} + {\chi _y}} )}} $ $\dfrac{ { { {\chi _y}/{\chi _x} + {\chi _x}/{\chi _y} } } }{ {3.6[ { {\chi _x}{\chi _y}{\rm{ + } }{\gamma _{\rm{S} } }( { {\chi _x} + {\chi _y} } )/6} ]} }$ 0.02L SC $ \dfrac{{( {8{\chi _y}/{\chi _x} + 2{\chi _x}/{\chi _y}} ) + ( {23/{\chi _x} + 2.5{\alpha _x}/{\chi _y}} )}}{{8{\chi _x}{\chi _y} + 2{\gamma _{\rm{S}}}( {{\chi _x} + {\alpha _x}{\chi _y}} )}} $ $\dfrac{ {8( { {\chi _y}/{\chi _x} } ) + 5/{\chi _x} } }{ {3.6[ { {\chi _x}{\chi _y}{\rm{ + } }{\gamma _{\rm{S} } }( { {\chi _x} + {\chi _y} } )/6} ]} }$ 0.03L PEC $ \dfrac{{( {6{\chi _y}/{\chi _x} + 2{\chi _x}/{\chi _y}} ) + ( {23/{\chi _x} + 2.5{\alpha _x}/{\chi _y}} )}}{{8{\chi _x}{\chi _y} + 2{\gamma _{\rm{S}}}( {{\chi _x} + {\alpha _x}{\chi _y}} )}} $ IC $ \dfrac{{8( {{\chi _y}/{\chi _x} + {\chi _x}/{\chi _y}} ) + 23( {{\alpha _y}/{\chi _x} + {\alpha _x}/{\chi _y}} )}}{{8{\chi _x}{\chi _y} + 2{\gamma _{\rm{S}}}( {{\alpha _y}{\chi _x} + {\alpha _x}{\chi _y}} )}} $ $\dfrac{ {8( { {\chi _y}/{\chi _x} + {\chi _x}/{\chi _y} } ) + 5( { {\alpha _y}/{\chi _x} + {\alpha _x}/{\chi _y} } )} }{ {3.6[ { {\chi _x}{\chi _y}{\rm{ + } }{\gamma _{\rm{S} } }( { {\chi _x} + {\chi _y} } )/6} ]} }$ 0.05L PSC $ \dfrac{{6{\chi _y}/{\chi _x} + 8{\chi _x}/{\chi _y} + 23( {{\alpha _y}/{\chi _x} + {\alpha _x}/{\chi _y}} )}}{{8{\chi _x}{\chi _y} + 2{\gamma _{\rm{S}}}( {{\alpha _y}{\chi _x} + {\alpha _x}{\chi _y}} )}} $ PIC $ \dfrac{{6( {{\chi _y}/{\chi _x} + {\chi _x}/{\chi _y}} ) + 23( {{\alpha _y}/{\chi _x} + {\alpha _x}/{\chi _y}} )}}{{8{\chi _x}{\chi _y} + 2{\gamma _{\rm{S}}}( {{\alpha _y}{\chi _x} + {\alpha _x}{\chi _y}} )}} $
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表 3 参数
$ \gamma $ R、hs1和$ \lambda $ 的敏感性分析Table 3. Sensitivity analysis against
$ \gamma $ R, hs1 and$ \lambda $ 参数 取值 $ \rho $ 建议值 − 0.990 $ \gamma $R 10.00 0.977 25.00 0.988 hs1 0.10 0.976 0.18 0.994 $ \lambda $ 3.00 0.982 5.00 0.993
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符号表
. List of symbols
符号 含义 符号 含义 Ab, As 梁截面和板单位宽度截面的受拉钢筋面积 P, P' 结构抗力和外荷载 $f_{\rm{c}}' $ 混凝土抗压强度 qs, qb 板和梁的设计荷载 fy 钢筋屈服强度 Tx, Ty 沿x向和y向板单位宽度的轴拉力 Fx, Fy 沿x向和y向梁的轴拉力 u1 屈服点位移 hb0, hs0 梁和板截面有效高度 u2 过渡段位移 hb1, hs1 梁和板截面受拉钢筋合力点到受压区混凝土中心的高度 u3 失效位移 K 结构抗力刚度 $W_{ {{\rm e}x} }^{\rm{n} }$ 结构设计荷载所做外力虚功 KL 结构荷载因子刚度 $ {W_{{\rm{in}}}} $ 结构内力虚功 L 等效最短梁跨长 αx, αy 沿x轴和y轴的短跨与中心跨之间的比值 Lx, Ly 沿x向和y向梁的总跨长 χx, χy 沿x轴和y轴的梁总跨长与标准跨长L0的比值 LF1, $ {LF_1'} $ 修正前和修正后屈服点对应的荷载因子 γR, γS 描述梁板的抗弯强度比和荷载比的参数 LFD 结构动荷载因子 $ \kappa $ 梁屈服承载力修正因子 Mb 梁的极限抵抗弯矩 λ 描述梁板截面高度比的参数 ms 板的单位宽度极限抵抗弯矩 $ \prod $ 鲁棒性快速评判指标
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