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浙江大学学报(工学版)
浙江大学学报(工学版)  2023, Vol. 57 Issue (11): 2254-2265    DOI: 10.3785/j.issn.1008-973X.2023.11.013
环境与土木工程     
基于平衡化谱聚类算法的高拱坝结构地震易损性研究
苏扬1,2(),张程3,4,胡恩良1,2,*()
1. 云南师范大学 数学学院，云南 昆明 650500
2. 云南省现代分析数学及其应用重点实验室，云南 昆明 650500
3. 长江水利委员会长江科学院，湖北 武汉 430010
4. 流域水资源与生态环境科学湖北省重点实验室，湖北 武汉 430010
Seismic fragility study of high arch dam based on balanced spectral clustering algorithm
Yang SU1,2(),Cheng ZHANG3,4,En-liang HU1,2,*()
1. College of Mathematics, Yunnan Normal University, Kunming 650500, China
2. Yunnan Key Laboratory of Modern Analytical Mathematics and Applications, Kunming 650500, China
3. Changjiang River Scientific Research Institute of Changjiang Water Resources Commission, Wuhan 430010, China
4. Key Laboratory of Basin Water Resource and Eco-Environmental Science in Hubei Province, Wuhan 430010, China
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摘要：

提出融入无监督聚类的地震易损性分析方法?将隶属度矩阵的近似正交约束与谱聚类相结合的平衡化谱聚类算法. 该算法基于图Laplacian矩阵的重表示矩阵，从给定的所有地震记录中筛选出具有代表性的样本子集，缓解由所选地震动强度指标过多引起的“维数灾难”问题；并通过隶属度矩阵的近似正交约束，解决应用传统谱聚类算法筛选地震波时产生的“均匀效应”问题. 以实际工程为例，开展以“拱坝-地基结构”为整体体系的地震易损性研究，分别建立以15条规范反应谱样本、109条整体样本为参照基准的地震易损性模型进行聚类算法效果验证. 结果表明，聚类算法筛选的样本与整体样本的地震易损性结果接近，在损伤体积比、坝顶位移、横缝最大开度性能指标下，两者易损性概率最大误差分别为4.39%、3.84%、6.64%，误差不超过5%的最小概率分别为92.24%、99.19%、81.75%，表明该算法在筛选典型地震样本方面的有效性.
关键词： 高拱坝;  地震易损性;  多样条带分析法(MSA);  谱聚类;  聚类隶属度矩阵;  正交约束    
Abstract:

A seismic fragility analysis method integrating unsupervised clustering was proposed, which was a balanced spectral clustering algorithm that combined approximate orthogonal restrictions of the membership matrix with spectral clustering. A representative subset of samples was selected from the provided seismic waves based on the re-representation of the Laplacian matrix, which alleviated the "curse of dimensionality" generated by too many selected ground motion intensity measures. The problem of "uniform effect" resulting from the use of the traditional spectral clustering algorithm was solved by incorporating the approximate orthogonal restrictions of the membership matrix. As a practical case, the seismic fragility of a "arch dam-foundation structure" as a whole system was studied. The seismic fragility model with 15 samples of the normative response spectrum and 109 overall samples as the reference was established to evaluate the effect of clustering model. Results showed that the seismic fragility findings from clustered samples were similar to those of the overall samples. Under the performance indicators of the damage volume ratio, dam crest displacement, and the maximum joint opening, the maximum errors of their fragility probability were 4.39%, 3.84%, and 6.64%, respectively, and the minimum probabilities of the errors not exceeding 5% were 92.24%, 99.19%, and 81.75%, respectively, which illustrating the effectiveness of the algorithm in selecting typical seismic waves.
Key words: high arch dam    seismic fragility    multiple stripe analysis (MSA)    spectral clustering    clustering membership matrix    orthogonal constraint
收稿日期: 2022-09-19 出版日期: 2023-12-11
CLC:  TU 312  
基金资助: 国家自然科学基金资助项目(62266055)
通讯作者: 胡恩良     E-mail: 18261786736@163.com;el.hu@ynnu.edu.cn
作者简介: 苏扬（1995—），男，硕士生，从事机器学习研究. orcid.org/0000-0001-6552-5746. E-mail： 18261786736@163.com
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引用本文:

苏扬,张程,胡恩良. 基于平衡化谱聚类算法的高拱坝结构地震易损性研究[J]. 浙江大学学报(工学版), 2023, 57(11): 2254-2265.

Yang SU,Cheng ZHANG,En-liang HU. Seismic fragility study of high arch dam based on balanced spectral clustering algorithm. Journal of ZheJiang University (Engineering Science), 2023, 57(11): 2254-2265.

链接本文:

https://www.zjujournals.com/eng/CN/10.3785/j.issn.1008-973X.2023.11.013    或    https://www.zjujournals.com/eng/CN/Y2023/V57/I11/2254

图 1  地震动样本反应谱曲线
IM 定义
1）注： $ {t_{{\rm{tot}}}} $为地震持时指标，PGA为峰值加速度指标，均为基本说明指标.
地震持时、有效持时、重要持时、主导周期、平均周期 ${t_{ {\text{tot} } } } ^{1)}$、 ${t_{ {\text{ed} } } } = {t_{ {\text{IA} } = 0.125\;{\rm{m}}/{\rm{s}}} } - {t_{ {\text{IA} } = 0.01\;{\rm{m}}/{\rm{s}}} }$、 $ {t_{{\text{sd}}(5 - 95)}} = {t_{0.95{\text{IA}}}} - {t_{0.05{\text{IA}}}} $、 $ {T_{\text{P}}} $、 $ {T_{\text{M}}} $
震级、震源距、震中距、土体类型、峰值加速度、速度、位移 $ M $、 $ {R_{{\text{hypo}}}} $、 $ {R_{{\text{epi}}}} $、 $ {\text{VS}} $、 $ {\text{PGA}} $、 $ {\text{PGV}} $、 $ {\text{PGD}} $
谱加速度、速度、位移、峰值谱加速度、速度、位移 ${ {{S} }_a}(T,\zeta )$、 $ {S_v}(T,\zeta ) $、 $ {{\text{S}}_d}(T,\zeta ) $、 $ {\text{P}}{{\text{S}}_a} $、 $ {\text{P}}{{\text{S}}_v} $、 $ {\text{P}}{{\text{S}}_d} $
A95加速度、有效设计加速度、持续最大加速度、速度 ${\text{A} }95 、 {\text{EDA} }、{\text{SMA} }、{\text{SMV} }$
Arias强度、特征强度、 ${\rm{C}} {\text{-} } {\rm{M}}$强度 ${\text{IA} } = \dfrac{{\text{π} } }{ {2g} }\displaystyle \int_0^{ {t_{ {\text{tot} } } }} {\mathop {\left[ {\ddot u(t)} \right]}\nolimits^2 } {\text{d} }t$、 $ {\text{IC}} = {\left( {{A_{{\text{rms}}}}} \right)^{3/2}}{\left( {{t_{{\text{sd}}}}} \right)^{1/2}} $、 ${\text{ID} } = \dfrac{ {2g} }{{\text{π} } }{\text{IA} }{\left( { {\text{PGA} } } \right)^{ - 1} }{\left( { {\text{PGV} } } \right)^{ - 1} }$
有效峰值加速度、速度、位移 $ {\text{EPA}} = \dfrac{1}{{2.5}}\left( {\left. {{\text{S}}{{\text{a}}_{{\text{avg}}}}\left( {{T_i},\zeta } \right)} \right|_{{T_{{\text{down}}}} = 0.1}^{{T_{{\text{up}}}} = 0.5}} \right) $、 $ {\text{EPV}} = \dfrac{1}{{2.5}}\left( {\left. {{\text{S}}{{\text{v}}_{{\text{avg}}}}\left( {{T_i},\zeta } \right)} \right|_{{T_{{\text{down}}}} = 0.8}^{{T_{{\text{up}}}} = 2.0}} \right) $、

$ {\text{EPD}} = \dfrac{1}{{2.5}}\left( {\left. {{\text{S}}{{\text{d}}_{{\text{avg}}}}\left( {{T_i},\zeta } \right)} \right|_{{T_{{\text{down}}}} = 2.5}^{{T_{{\text{up}}}} = 4.0}} \right) $
均方加速度、速度、位移 ${ {{P} }_a} = \dfrac{1}{ { {t_{ {\text{tot} } } } } }\displaystyle \int_0^{ {t_{ {\text{tot} } } }} {\mathop {\left[ {\ddot u(t)} \right]}\nolimits^2 } {\text{d} }t$、 ${ {{P} }_v} = \dfrac{1}{ { {t_{ {\text{tot} } } } } }\displaystyle \int_0^{ {t_{ {\text{tot} } } }} {\mathop {\left[ {\dot u(t)} \right]}\nolimits^2 } {\text{d} }t$、 ${ {{P} }_d} = \dfrac{1}{ { {t_{ {\text{tot} } } } } }\displaystyle \int_0^{ {t_{ {\text{tot} } } }} {\mathop {\left[ {\dot u(t)} \right]}\nolimits^2 } {\text{d} }t$
均方根加速度、速度、位移 $ {A_{{\text{rms}}}} = {\left( {\dfrac{1}{{{t_{{\text{tot}}}}}}\displaystyle \int_0^{{t_{{\text{tot}}}}} {\mathop {\left[ {\ddot u(t)} \right]}\nolimits^2 } {\text{d}}t} \right)^{1/2}} $、 $ {V_{{\text{rms}}}} = {\left( {\dfrac{1}{{{t_{{\text{tot}}}}}}\displaystyle \int_0^{{t_{{\text{tot}}}}} {\mathop {\left[ {\dot u(t)} \right]}\nolimits^2 } {\text{d}}t} \right)^{1/2}} $、

$ {D_{{\text{rms}}}} = {\left( {\dfrac{1}{{{t_{{\text{tot}}}}}}\displaystyle \int_0^{{t_{{\text{tot}}}}} {\mathop {\left[ {u(t)} \right]}\nolimits^2 } {\text{d}}t} \right)^{1/2}} $
平方加速度、速度、位移 ${ {{E} }_a} = \displaystyle \int_0^{ {t_{ {\text{tot} } } }} {\mathop {\left[ {\ddot u(t)} \right]}\nolimits^2 } {\text{d} }t$、 ${ {{E} }_v} = \displaystyle \int_0^{ {t_{ {\text{tot} } } }} {\mathop {\left[ {\dot u(t)} \right]}\nolimits^2 } {\text{d} }t$、 ${ {{E} }_d} = \displaystyle \int_0^{ {t_{ {\text{tot} } } }} {\mathop {\left[ {u(t)} \right]}\nolimits^2 } {\text{d} }t$
平方根加速度、速度、位移 $ {A_{{\text{rs}}}} = {\left( {\displaystyle \int_0^{{t_{{\text{tot}}}}} {\mathop {\left[ {\ddot u(t)} \right]}\nolimits^2 } {\text{d}}t} \right)^{1/2}} $ 、 $ {V_{{\text{rs}}}} = {\left( {\displaystyle \int_0^{{t_{{\text{tot}}}}} {\mathop {\left[ {\dot u(t)} \right]}\nolimits^2 } {\text{d}}t} \right)^{1/2}} $ $ {D_{{\text{rs}}}} = {\left( {\displaystyle \int_0^{{t_{{\text{tot}}}}} {\mathop {\left[ {u(t)} \right]}\nolimits^2 } {\text{d}}t} \right)^{1/2}} $
伪谱加速度、速度 ${P_{ {\text{seu} } } }{S_a} = \dfrac{ {2{\text{π} } } }{T}{S_v}(T,\zeta ) = \dfrac{ {4{ {\text{π} } ^2} } }{ { {T^2} } }{S_d}(T,\zeta )$、 ${P_{ {\text{seu} } } }{S_v} = \dfrac{ {2{\text{π} } } }{T}{S_d}(T,\zeta )$
速度强度、加速度谱强度、速度 ${ {{I} }_v} = { { {E} }_v}{\left( { {\text{PGV} } } \right)^{ - 1} }$、 $ {\text{ASI}} = \displaystyle \int_{0.1}^{0.5} {{S_a}} (T,\zeta ){\text{d}}t $、 $ {\text{VSI}} = \displaystyle \int_{0.1}^{2.5} {{S_v}} (T,\zeta ){\text{d}}t $
累积绝对速度、Fajfar强度、复合强度 $ {\text{CAV}} = \displaystyle \int_0^{{t_{{\text{tot}}}}} {\left| {\ddot u(t)} \right|} {\text{d}}t $、 ${ {\text{I} }_{\rm{F}}} = {\text{PGV} }{\left( { {t_{ {\text{sd} } } }} \right)^{1/4} }$、 $ {I_v} = {\left( {{\text{PGV}}} \right)^{2/3}}{\left( {{t_{{\text{sd}}}}} \right)^{1/3}} $、 $ {I_d} = {\text{PGD}}{\left( {{t_{{\text{sd}}}}} \right)^{1/3}} $
频率比 $ {I_{v/a}} = {\text{PGV}}{\left( {{\text{PGA}}} \right)^{ - 1}} $、 $ {I_{{v^2}/a}} = {\left( {{\text{PGA}}} \right)^{ - 1}}{\left( {{\text{PGV}}} \right)^2} $、 $ {I_{d/v}} = {\text{PGD}}{\left( {{\text{PGV}}} \right)^{ - 1}} $
表 1  地震动强度指标及定义
图 2  聚类样本数与算法收敛性
图 3  各类地震波空间分布效果
材料 E/GPa ρ/（kg·m?3） ν
坝体 31.2 2400 0.167
基础 26.0 2800 0.240
钢筋 260.0 7800 0.300
表 2  拱坝结构各项材料参数
图 4  拱坝有限元模型及模态振型图
损伤描述 判别准则 样本类别 DVR/% DT/m $ C $/m
坝踵与坝肩均无明显宏观开裂，拱坝下游表面出现
损坏或损坏较小仅存在细微开裂 		基本完好
No Damage
（ND） 		整体样本 <0.031 <0.072 <0.050
反应谱样本 <0.032 <0.076 <0.057
聚类样本 <0.030 <0.066 <0.051
坝踵与坝肩开裂，下游面中部与上部开始发生宏观裂缝，
上游面暂未发生宏观开裂 		轻微破坏
Slight Damage（SD） 		整体样本 0.031~0.099 0.072~0.106 0.050~0.080
反应谱样本 0.032~0.109 0.076~0.111 0.057~0.089
聚类样本 0.030~0.107 0.066~0.096 0.051~0.079
坝体下游产生大面积宏观开裂，
并且延伸至上游发生宏观开裂 		中等破坏
Moderate Damage（MD） 		整体样本 0.099~0.236 0.106~0.154 0.080~0.123
反应谱样本 0.109~0.229 0.111~0.167 0.089~0.132
聚类样本 0.107~0.245 0.096~0.146 0.079~0.123
坝体下游中上部开裂延伸至上游直至贯穿，
上游也已开始产生大面积宏观开裂 		严重破坏
Extensive Damage（ED） 		整体样本 >0.236 >0.154 >0.123
反应谱样本 >0.229 >0.167 >0.132
聚类样本 >0.245 >0.146 >0.123
表 3  高拱坝性能水平划分
图 5  各类样本的有限元计算点统计结果
图 6  各类样本的变异系数统计图
图 7  各类样本地震易损性曲线图
图 8  各类样本地震易损性曲线误差分析图
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