多点地震动激励下的高效反应谱方法

王君杰 郭进

王君杰,郭进. 2022. 多点地震动激励下的高效反应谱方法. 地震学报,44(5):810−823 doi: 10.11939/jass.20220093
引用本文: 王君杰,郭进. 2022. 多点地震动激励下的高效反应谱方法. 地震学报,44(5):810−823 doi: 10.11939/jass.20220093
Wang J J,Guo J. 2022. An efficient seismic response spectrum method under multi-support excitations. Acta Seismologica Sinica,44(5):810−823 doi: 10.11939/jass.20220093
Citation: Wang J J,Guo J. 2022. An efficient seismic response spectrum method under multi-support excitations. Acta Seismologica Sinica44(5):810−823 doi: 10.11939/jass.20220093

多点地震动激励下的高效反应谱方法

doi: 10.11939/jass.20220093
基金项目: 国家自然科学基金(52078384)和国家重点研发计划课题(2018YFC1504306)联合资助
详细信息
    通讯作者:

    王君杰,博士,教授,主要从事桥梁抗震研究,e-mail:jjwang@tongji.edu.cn

An efficient seismic response spectrum method under multi-support excitations

  • 摘要: 在多点地震动激励下,结构的反应谱分析计算非常耗时。结构的地震谱响应可以用若干个相关系数来表示,如果相关系数使用解析形式来表示,可以大大减少计算时间。本文提出了空间相干函数的近似表达式,并对其系数进行积分,得到了相关系数的解析式。该解析表达式根据克拉夫-彭津(Clough-Penzien)和胡聿贤自功率谱密度函数模型推导得出。案例桥梁的计算结果表明,相关系数的近似解析表达式具有足够的工程精度,用于多点地震反应谱计算具有极高的效率。

     

  • 图  1  空间相干函数模型(Qu et al,1996

    (a) ρrs-空间距离关系;(b) ρrs-频率关系

    Figure  1.  Spatial coherence function model (Qu et al,1996

    (a) ρrs-spatial distance relationship;(b) ρrs-frequency relationship

    图  2  空间相干函数模型(Harichandran,Vanmarcke,1986

    (a) ρrs-空间距离关系;(b) ρrs-频率关系

    Figure  2.  Spatial coherence function model (Harichandran,Vanmarcke,1986

    (a) ρrs-spatial distance relationship;(b) ρrs-frequency relationship

    图  3  自功率谱密度函数${S_{rr}} ( \omega ) $的特征(ζg为场地土的阻尼比,下同)

    (a) 胡聿贤自功率谱密度模型;(b) 克拉夫−彭津自功率谱密度模型

    Figure  3.  The characteristic for auto-power spectral density function (ζg is the damping ratio of site soil,the same below)

    (a) Hu’s auto-power spectral density model;(b) Clough-Penzien’s auto-power spectral density model

    图  4  系数$ \;{\rho _{{\rm{g}}r{\text{g}}s}} $的精确值与近似值的比较

    Figure  4.  The comparison between exact value and approximate value of coefficient $ \;{\rho _{{\rm{g}}r{\text{g}}s}} $

    图  6  系数$ \;{\rho _{irjs}} $$ {\zeta _i} = {\zeta_j} = 0.05 $)精确值与近似值的比较

    Figure  6.  The comparison between exact value and approximate value of coefficient $\; {\rho _{irjs}} $$ {\zeta _i} = {\zeta_j} = 0.05 $

    (a) $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 300\;{\rm{m}} $,ωi=0.1 Hz;(b) $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 1\;000\;{\rm{m}} $,ωi=0.1 Hz;(c) $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 300\;{\rm{m}} $,ωi=1.0 Hz; (d) $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 1\;000\;{\rm{m}} $,ωi=1.0 Hz;(e) $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 300\;{\rm{m}} $,ωi=4.0 Hz;(f) $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 1\;000\;{\rm{m}} $,ωi=4.0 Hz

    图  5  $\left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 300\;{\rm{m}}$ (a)和1 000 m (b)时系数$\; {\rho _{{\rm{g}}rjs}} $的精确值与近似值的比较(${\zeta _j} = 0.05$

    Figure  5.  The comparison between exact value and approximate value of coefficient $\; {\rho _{{\rm{g}}rjs}} $${\zeta _j} = 0.05$) with $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 300\;{\rm{m}} $ (a) and 1 000 m (b)

    图  7  系数$\;{\rho _{{\rm{g}}r{\rm{g}}s}}$的精确值与近似值的比较

    Figure  7.  The comparison between exact value and approximate value of coefficient $\;{\rho _{{\rm{g}}r{\rm{g}}s}}$

    图  8  $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 300\;{\rm{m}} $ (a)和1 000 m (b)时系数$\; {\rho _{{\rm{g}}rjs}} $${\zeta _j} = 0.05$)精确值与近似值的比较

    Figure  8.  The comparison between exact value and approximate value of coefficient $ \;{\rho _{{\rm{g}}rjs}} $${\zeta _j} = 0.05$) with $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 300\;{\rm{m}} $ (a) and 1 000 m (b)

    图  9  系数$ \;{\rho _{irjs}} $${\zeta _i} = {\zeta _j} = 0.05$)精确值与近似值的比较

    Figure  9.  The comparison between exact value and approximate value of coefficient $\; {\rho _{irjs}} $${\zeta _i} = {\zeta _j} = 0.05$

    (a) $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 300\;{\rm{m}} $,ωi=0.1 Hz;(b) $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 1\;000\;{\rm{m}} $,ωi=0.1 Hz;(c) $\left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 300\;{\rm{m}}$,ωi=1.0 Hz; (d) $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 1\;000\;{\rm{m}} $,ωi=1.0 Hz;(e) $\left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 300\;{\rm{m}}$,ωi=4.0 Hz;(f) $ \left| {\Delta {{\boldsymbol{r}}_{rs}}} \right| = 1\;000\;{\rm{m}} $,ωi=4.0 Hz

    图  10  算例桥梁的有限元模型

    Figure  10.  Finite element model of the example bridge

    表  1  Qu等(1996)相干模型下的地震位移响应及相对误差

    Table  1.   Seismic displacement response and its relative error under the coherence model of Qu et al1996

    胡聿贤的自功率谱密度模型 克拉夫−彭津的自功率谱密度模型
    L1/cmR1/cmL2/cmR2/cmL3/cmR3/cmL1/cmR1/cmL2/cmR2/cmL3/cmR3/cm
    A 7.44 7.40 6.27 5.79 7.02 6.23 7.42 7.38 6.22 5.68 7.03 6.08
    B 7.44 7.40 6.27 5.79 7.02 6.23 7.42 7.38 6.22 5.68 7.03 6.08
    C 7.53 7.49 6.24 5.77 6.97 6.17 7.48 7.44 6.21 5.67 7.02 6.04
    eBA 0 0 0 0 0 0 0 0 0 0 0 0
    eCA1.22%1.22%−0.48%−0.35%−0.71%−0.96% 0.81%0.81%−0.16%−0.18%−0.14%−0.66%
    下载: 导出CSV

    表  4  Harichandran和Vanmarcke (1986)相干模型下的地震力或弯矩响应及相对误差

    Table  4.   Seismic force or moment response and its relative error under the coherence model of Harichandran and Vanmarcke (1986

    胡聿贤的自功率谱密度模型 克拉夫−彭津的自功率谱密度模型
    L4/kNR4/kNL5/kNR5/kNL6/kN·mR6/kN·mL4/kNR4/kNL5/kNR5/kNL6/kN·mR6/kN·m
    A 719 691 1111 1069 29656 27697 712 685 1109 1066 29377 27307
    B 719 691 1107 1066 29617 27655 712 685 1106 1064 29355 27284
    C 720 693 1111 1073 29688 27794 712 686 1106 1068 29393 27393
    eBA 0 0 −0.36% −0.28% −0.13% −0.15% 0 0 −0.27% −0.19% −0.07% −0.08%
    eCA0.14%0.29%0 0.37% 0.11% 0.35% 00.15%−0.27% 0.19% 0.05% 0.31%
    下载: 导出CSV

    表  2  Harichandran和Vanmarcke (1986)相干模型下的地震位移响应及相对误差

    Table  2.   Seismic displacement response and its relative error under the coherence model of Harichandran and Vanmarcke (1986

    胡聿贤的自功率谱密度模型 克拉夫−彭津的自功率谱密度模型
    L1/cmR1/cmL2/cmR2/cmL3/cmR3/cmL1/cmR1/cmL2/cmR2/cmL3/cmR3/cm
    A 7.27 7.24 6.24 5.81 7.08 6.40 7.26 7.22 6.19 5.69 7.10 6.27
    B 7.27 7.24 6.24 5.81 7.09 6.41 7.25 7.22 6.19 5.69 7.10 6.27
    C 7.36 7.32 6.21 5.78 7.05 6.36 7.31 7.28 6.18 5.68 7.09 6.25
    eBA 0 0 0 0 0.14% 0.16% −0.14% 0 0 0 0 0
    eCA1.24%1.10%−0.48%−0.52%−0.42%−0.63% 0.69%0.83%−0.16%−0.18%−0.14%−0.32%
    下载: 导出CSV

    表  3  Qu等(1996)相干模型下的地震力或弯矩响应及相对误差

    Table  3.   Seismic force or moment response and its relative error under the coherence model of Qu et al1996

    胡聿贤的自功率谱密度模型 克拉夫−彭津的自功率谱密度模型
    L4/kNR4/kNL5/kNR5/kNL6/kN·mR6/kN·mL4/kNR4/kNL5/kNR5/kNL6/kN·mR6/kN·m
    A 730 699 1058 1011 29192 27002 723 692 1057 1010 28931 26618
    B 730 699 1054 1008 29159 26966 723 692 1055 1008 28915 26601
    C 731 702 1057 1016 29219 27107 723 694 1054 1012 28944 26714
    eBA 0 0 −0.38% −0.30% −0.11% −0.13% 0 0 −0.19% −0.20% −0.06% −0.06%
    eCA0.14%0.43%−0.09% 0.49% 0.09% 0.39% 00.29%−0.28% 0.20% 0.04% 0.36%
    下载: 导出CSV

    表  5  计算时间比较(单位:s)

    Table  5.   Comparison of time consumption of computation (Unit:s)

    胡聿贤的自功率谱密度模型克拉夫−彭津的自功率谱密度模型
    Qu等(1996)相干模型Harichandran和Vanmarcke (1986
    相干模型
    Qu等(1996)相干模型Harichandran和Vanmarcke (1986
    相干模型
    ACRACRACRACR
    6273 93 67.5 6280 93 67.5 5833 111 52.5 5842 111 52.6
    注:R为计算精确解与近似解用时的比值,下同。
    下载: 导出CSV

    表  6  计算时间比较(单位:s)

    Table  6.   Comparison of time consumption of computation (Unit:s)

    胡聿贤的自功率谱密度模型克拉夫−彭津的自功率谱密度模型
    Qu等(1996)相干模型Harichandran和Vanmarcke (1986
    相干模型
    Qu等(1996)相干模型Harichandran和Vanmarcke (1986
    相干模型
    ACR ACR ACR ACR
    309 5 61.8 310 5 62.0 278 5 55.6 279 6 46.5
    1112 16 69.5 1119 15 74.6 1031 19 54.3 1031 19 54.3
    6273 93 67.5 6280 93 67.5 5833 111 52.5 5842 111 52.6
    下载: 导出CSV
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  • 收稿日期:  2022-06-07
  • 修回日期:  2022-07-13
  • 网络出版日期:  2022-09-01
  • 刊出日期:  2022-09-15

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