Stability evaluation of karst roof of pile foundation based on variable weight fuzzy comprehensive theory
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摘要:
常权层次分析法在评价桩基下岩溶顶板稳定性中存在准确性较低的问题。基于变权理论对评价指标的结构模型进行改进，构建一种有效的判定稳定性的数学评价方法。以岩溶发育区深圳市轨道交通三号线停车场为研究区，利用故障树分析理论得到影响桩基下岩溶顶板稳定性的主要因素，通过解析结构模型理论确定稳定性评价指标的结构模型，依据模糊数学提出的层次分析法、变权理论的综合评价方法等手段对稳定性进行评价，并与常权下的稳定性评价结果对比。结果表明：（1）基于故障树分析理论得到的稳定性评价因素，可以系统评估稳定性，确定可能发生的事件路径，识别高风险区域，克服评价因素不全面的问题；（2）引入的变权模糊综合评价方法从“惩罚”变权的机制确定了需要变权的评价因素，通过放大评价因素的影响达到准确评价现场实际情况的目的；（3）变权模型下研究区稳定性等级为“不稳定”，相较于与常权模型下“较不稳定”的评价，变权模糊综合评价模型计算结果更符合现场勘查结果。研究表明基于变权理论的桩基下岩溶顶板稳定性评价方法能反映研究区整体稳定性水平，对类似岩溶工程的前期稳定性判别具有重要的参考意义。
Abstract:Regarding the lower accuracy of the Analytic Hierarchy Process (AHP) in assessing the stability of karst cave roof under pile foundations, an improved structural model of evaluation indicators based on variable weight theory was developed to establish an effective mathematical assessment method for stability determination. Focusing on the parking lot of Shenzhen Rail Transit Line 3 within a karst development area, the main factors influencing the stability of the karst cave roof under pile foundations were identified utilizing Fault Tree Analysis theory. The structural model of stability assessment indicators was delineated through theoretical analysis. Subsequently, based on Fuzzy Mathematics, a comprehensive evaluation method of stability integrated the AHP and the assessment method under variable weights was conducted to contrast the outcomes with stability assessment results under constant weights. Results indicate: (1) Factors obtained from Fault Tree Analysis theory can be used to evaluate stability systematically, identify potential event pathways, and mitigate incomplete evaluation factors. (2) The comprehensive evaluation method introduced variable weight fuzzy, employing a “penalty” mechanism for weight variation to determine the need for weight variation in assessment factors, achieves accurate on-site assessments through amplifying their influences. (3) The stability grade within the study area under the variable weight model was “unstable”, contrasting the evaluation of “relatively unstable” under the constant weight model, demonstrating that the variable weight fuzzy comprehensive evaluation model aligns more closely with on-site survey results. This study underscores that the variable weight theory-based assessment method for karst cave roof stability under pile foundations reflects the overall stability level within the research area, offering crucial insights for preliminary stability assessment in similar karst engineering endeavors.
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表 1 常见故障树名称
Table 1. Common fault tree names
符号 名称 定义 
逻辑与门 所有输入事件发生输出事件才发生 
底事件 不能再进行展开的事件 
逻辑或门 有一个输入事件发生输出事件就发生 
结果事件 顶事件：位于故障树顶端
中间事件：除顶事件外的其他可展开事件
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表 2 连续性指标分级标准
Table 2. Criteria for the classification of continuity indicators
评价指标名称 稳定性状态 稳定 较稳定 较不稳定 不稳定 顶板岩层厚度
与桩径比值≥4.5 3.0～<4.5 1.5～<3.5 <1.5 岩石单轴饱
和抗压强度/MPa≥100 70～<100 40～<70 <40 岩芯完整情况/% ≥90 70 ~ <90 50～<70 <50 溶洞宽度与桩径比值 <1 1～<2 2～<4 ≥4 溶洞高度与桩径比值 <1 1～<2 2～<4 ≥4 桩长与桩径比值 ≥40 25～<40 10～<25 <10 基桩设计荷载/kN < 3000 3000 ～<8000 8000 ~<12000 ≥ 12000
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表 3 离散性指标分级标准
Table 3. Criteria for the classification of discrete indicators
影响因素 稳定 欠稳定 岩石层状构造 厚层块状、强度高的岩石 泥质岩、白云质灰岩、薄层状有互层且岩性软化 裂隙发育情况 无断裂、裂隙不发育或胶结好 有断层通过；裂隙发育，岩体被两组以上裂隙切割，裂隙张开，岩体呈平砌状 岩石风化程度 基本无风化现象或微风化 中等至强风化 顶板情况 顶板岩层厚度与洞径比值大，顶板呈板状或拱状，可见钙质沉积 顶板岩层厚度与洞径比值小，有悬挂岩体，被裂隙切割且未胶结 溶洞内充填情况 密实沉积物填满且无水冲蚀的可能 未充填或半充填，水流冲蚀充填物，洞中见有近期塌落物
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表 4 离散性指标因素隶属度取值
Table 4. Values of the membership degree of discrete index factors
离散型
指标因素指标值 稳定 较稳定 较不稳定 不稳定 裂隙发育及
填充情况裂隙不发育或发育
但胶结良好0.50 0.30 0.15 0.05 裂隙发育但多数胶结充填 0.40 0.40 0.15 0.05 裂隙发育且少数充填 0.20 0.30 0.40 0.10 裂隙极发育且无充填 0.05 0.15 0.35 0.45 岩石风化
程度微风化 0.50 0.30 0.15 0.05 弱风化 0.10 0.30 0.40 0.20 强风化 0.05 0.15 0.30 0.50 岩石层状
构造巨厚 0.40 0.30 0.20 0.10 厚层 0.20 0.40 0.30 0.10 薄层 0.10 0.20 0.25 0.45 溶洞充填
情况全充填 0.40 0.30 0.20 0.10 半充填 0.15 0.20 0.35 0.30 无充填 0.05 0.20 0.35 0.40
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表 5 桩基下岩溶顶板稳定性评价模糊关系矩阵
Table 5. Fuzzy relationship matrix for the stability evaluation of karst roof of pile foundation
评价指标 实际评价值 模糊关系矩阵 相对权重 稳定 较稳定 较不稳定 不稳定 顶板岩层厚度 43.2 m 1.00 0 0 0 0.318 裂隙发育情况 节理、裂隙只在大理岩中发育且胶结良好 0.50 0.30 0.15 0.05 0.1031 岩石风化 中等风化 0.10 0.30 0.40 0.20 0.0959 岩石单轴饱和抗压强度 34.40 MPa 0 0 0.25 0.75 0.3896 岩石层状构造 中厚层岩石：0.3 m±0.05 m 0.20 0.40 0.30 0.10 0.0439 岩芯完整情况 87.65% 0.20 0.40 0.20 0.20 0.0494 溶洞宽度 3.45 m 0 0 0.30 0.70 0.6833 溶洞高度 3.07 m 0 0.17 0.83 0 0.1998 溶洞充填情况 溶洞充填少量粉质黏土 0.15 0.20 0.35 0.30 0.1168 桩长 23.2 m 0 0 0.25 0.75 0.250 0 基桩设计荷载 8007 kN0 0.10 0.90 0 0.750 0
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表 6 A-B 判断矩阵
Table 6. A-B judgment matrix
评价指标 判断矩阵 常权权重 $ A $ $ {B}_{1} $ $ {B}_{2} $ $ {B}_{3} $ 顶板特征 $ {B}_{1} $ 1 1 2 0.4 溶洞特征 $ {B}_{2} $ 1 1 2 0.4 基桩特征 $ {B}_{3} $ 0.5 0.5 1 0.2
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表 7
$ {B}_{1} $ -C判断矩阵Table 7.
$ {\mathit{B}}_{1} $ -C judgment matrix评价指标 判断矩阵 常权权重 $ {B}_{1} $ $ {C}_{1} $ $ {C}_{2} $ $ {C}_{3} $ $ {C}_{4} $ $ {C}_{5} $ $ {C}_{6} $ 顶板岩层厚度 $ {C}_{1} $ 1 4 6 0.5 4 7 0.3180 裂隙发育及充填情况 $ {C}_{2} $ 0.25 1 1 0.25 3 3 0.1031 岩石风化 $ {C}_{3} $ 0.1667 1 1 0.2 3 3 0.0959 岩石单轴饱和抗压强度 $ {C}_{4} $ 2 4 5 1 6 6 0.3896 岩石层状构 $ {C}_{5} $ 0.25 0.3333 0.3333 0.1667 1 0.5 0.0439 岩芯完整程度 $ {C}_{6} $ 0.1429 0.3333 0.3333 0.1667 2 1 0.0494
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表 8
$ {B}_{2} $ -C判断矩阵Table 8.
$ {\mathit{B}}_{2} $ -C judgment matrix评价指标 判断矩阵 常权权重 $ {B}_{2} $ $ {C}_{7} $ $ {C}_{8} $ $ {C}_{9} $ 溶洞宽度 $ {C}_{7} $ 1 4 5 0.6833 溶洞高度 $ {C}_{8} $ 0.25 1 2 0.1998 溶洞充填情况 $ {C}_{9} $ 0.2 0.5 1 0.1168
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表 9
$ {B}_{3} $ -C判断矩阵Table 9.
$ {\mathit{B}}_{3} $ -C judgment matrix评价指标 判断矩阵 常权权重 $ {B}_{3} $ $ {C}_{10} $ $ {C}_{11} $ 桩长 $ {C}_{10} $ 1 0.3333 0.25 基桩设计荷载 $ {C}_{11} $ 3 1 0.75
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表 10 层次分析法结果
Table 10. Analytic hierarchy results
排序层 H $ \lambda_{\mathrm{max}} $ C.I C.R A-B [0.4，0.4，0.2] 3 0 0 $ {B}_{1} $ -C[0.318， 0.1031 ，0.0959 ，0.3896 ，0.0439 ，0.0494 ]6.5574 0.07182 0.057 $ {B}_{2} $ -C[ 0.6833 ，0.1998 ，0.1168 ]3 0.0082 0.015769 $ {B}_{3} $ -C[0.25，0.75] 2 0 0
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表 11 不同隶属度阈值及其调权水平表
Table 11. Different affiliation thresholds and their level of weighting
$ {b}_{j} $ 值0 0.2 0.2 0.4 0.6 0.8 1 调权水平 0.09 0.12 0.15 0.18 0.21 0.24 0.27 稳定性评价结果 稳定 较稳定 较稳定 较不稳定 不稳定 不稳定 不稳定
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表 12 桩基下岩溶顶板稳定性评价指标变权权重
Table 12. Variable weighting of pile foundation karst top stability evaluation index
评价指标 模糊关系矩阵 相对权重 变权权重 稳定 较稳定 较不稳定 不稳定 顶板岩层厚度 1.00 0 0 0 0.3180 0.318 0 裂隙发育及充填情况 0.50 0.30 0.15 0.05 0.1031 0.0822 岩石风化 0.10 0.30 0.40 0.20 0.0959 0.0892 岩石单轴饱和抗压强度 0 0 0.25 0.75 0.3896 0.4432 岩石层状构 0.20 0.40 0.30 0.10 0.0439 0.0311 岩芯完整情况 0.20 0.40 0.20 0.20 0.0494 0.0363 溶洞宽度 0 0 0.30 0.70 0.6833 0.7140 溶洞高度 0 0.17 0.83 0 0.1998 0.1657 溶洞充填情况 0.15 0.20 0.35 0.30 0.1168 0.1023 桩长 0 0 0.25 0.75 0.250 0 0.3076 基桩设计荷载 0 0.10 0.90 0 0.750 0 0.6925
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表 13 基于变权理论的一级模糊综合评判矩阵
Table 13. One-level fuzzy comprehensive judgment matrix based on variable weight theory
一级评价指标 模糊综合评判矩阵 稳定 较稳定 较不稳定 不稳定 顶板特征 0.3815 0.0784 0.1754 0.3610 溶洞特征 0.0153 0.0486 0.3875 0.5305 基桩特征 0 0.06925 0.7002 0.2307
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表 14 基于常权理论的一级模糊综合评判矩阵
Table 14. One-level fuzzy comprehensive judgment matrix based on constant power theory
一级评价指标 模糊综合评判矩阵 稳定 较稳定 较不稳定 不稳定 顶板特征 $ {u}_{1} $ 0.3978 0.0970 0.1743 0.3262 溶洞特征 $ {u}_{2} $ 0.0175 0.0573 0.4117 0.5134 基桩特征 $ {u}_{3} $ 0 0.075 0 0.7375 0.1875
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