Influence of PGV and PGD on Structural Nonlinear Seismic Response of A 6-story Steel Building
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摘要: 深入揭示地震动峰值特性影响是推进地震动工程特性研究的有效手段。地震动峰值速度和峰值位移特性对结构弹塑性地震反应的影响规律尚需要探索。本文基于窄带时程叠加方法，人工合成具有相同加速度反应谱但峰值速度和峰值位移不同的4个序列地震动时程。其中第1、2序列地震动峰值速度为0.20 m/s，峰值位移分别为0.20 dm和0.40 dm，而第3、4序列地震动峰值位移为0.30 dm，峰值速度分别为0.15 m/s和0.30 m/s。将地震动峰值加速度分别标定至400 cm/s2和800 cm/s2，并以此作为输入开展建设地震观测系统的6层钢结构弹塑性地震反应分析，使得结构发生不同弹塑性地震反应，对比分析在不同序列地震动作用下层间位移角和延性系数等结构工程需求参数差别，探索峰值位移和峰值速度对结构弹塑性地震反应的影响规律。分析表明，在非线性反应阶段后，结构层间位移角和延性系数的变异系数随着输入地震动峰值的增加而增大，地震动峰值特性对结构层间位移角和延性系数等参数有一定影响，影响幅度随输入地震动增加而增大，且峰值速度较峰值位移的影响更为显著。在进行结构设计地震动参数选取时，应重视地震动速度和位移峰值特性的影响。Abstract: Revealing the influences of amplitude of strong motions is an effective method to push forward the earthquake engineering research. Influence pattern to structural inelastic response of the peak velocity and peak displacement is required to carry out. Based on a specific acceleration response spectrum, 4 sets of artificial ground motions are synthesized with 30 samples. The peak accelerations of the 1st and 2nd sets are identical with value of 1 m/s2 and their peak velocities are similarly the same with the value of 0.20 m/s. Their peak displacements differ from each other, with the values at 0.2 dm for the 1st set and 0.4 dm for the 2nd set. The peak accelerations and peak displacements are consistent with the peak velocity of 0.15 m/s for the 3rd set and 0.30 m/s for the 4th set. The ground motions are separately scaled to 400cm/s2 and 800cm/s2 and they are introduced as the input motions to a 6-story steel building to render certain different degree of non-linear seismic response. Engineering demand parameters including inter-story drift ratio and ductility coefficient are compared for the response under different sets of ground motions. It is inferred from the comparsions that the influence appears a tendency to increase with the amplitude of input ground motions. Analyses also indicate peak velocities have large influences on nonlinear responses and the influence of peak displacement is relatively weaker. Therefore, when ground motions are selected for structural seismic design, it is not appropriate to merely consider spectral characteristics, ignoring the influence of amplitude characteristics.
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引言
随着城市地下空间的不断开发和利用，基坑工程数量日益增多，为分析基坑引起的变形(李平等，2010)，数值模拟方法得到广泛应用。目前，针对基坑开挖分析的岩土工程软件较多，如ABAQUS、ANSYS、FELAC、ADINA等，如果对软件适用性缺乏了解、应用不当，将造成分析结果不可靠，甚至产生错误结论。Hashash等(2010)、Grande(1998)、Potts等(2001)及徐中华等(2010)根据不同本构模型下基坑开挖变形结果与实测资料的对比分析，得出具有实用价值的研究成果，但这些成果多适用于特定土体条件下的工程，普遍使用性较差。Potts等(2002)指出，用于基坑工程数值模拟的土体本构模型既应能刻画分析问题特点，又应简单易实现。蒋明镜等(2012)基于PFC2D和FLAC3D软件对Mohr-Coulomb模型、Drucker-Prager模型在基坑分析中的适用性进行探讨。秦会来等(2012)对比分析基于ABAQUS软件修正剑桥模型和Mohr-Coulomb模型的基坑开挖二维数值模拟结果，认为Mohr-Coulomb模型不能反映加卸载模量差异和土体压硬性，故不适用于基坑开挖引起的变形模拟。但目前鲜有开展ABAQUS软件内置本构模型(Mohr-Coulomb模型、Drucker-Prager模型、修正剑桥模型)的基坑开挖变形三维数值模拟可靠性分析。笔者以ABAQUS软件为平台，分析内置本构模型对基坑开挖变形数值分析结果的影响，讨论内置本构模型数值模拟结果的合理性，在此基础上分析基于修正剑桥模型的开挖方式和二维简化分析模型的影响，并对比三维模型和二维模型模拟结果，论证基坑稳定性分析二维简化分析模型的适用条件，本文相关分析结论对基于ABAQUS软件的基坑开挖变形数值分析具有一定参考价值。
1. 模型建立
为简化分析，在弹性均匀半空间内，建立长120m、宽120m、高20m的分析模型，顶面为自由边界，侧面与底面均为约束边界(图 1)。考虑边界效应对基坑开挖变形静力分析的影响，基坑开挖于分析模型地表中央，且基坑平面长20m、宽20m，设计开挖深度5m，可确保消除边界效应(管俊峰等，2017)。分析模型介质为黏性土，其中土体弹性模量取25.2MPa，粘聚力30kPa，密度1.8g/cm3，对数体积模量k取0.04，泊松比v取0.35，λ取0.2，应力比M取1.2，初始屈服面а0取0，β默认为1，流应力比K取1。初始孔隙比取1。(李广信，2004)。
分析模型采用八节点六面体(费康等，2013)单元(C3D8、全局尺寸5m×5m)进行离散，由于基坑处及附近土体变形较大，采用较小尺寸单元(C3D8、尺寸1m×1m×1m)进行离散，离散过程中采用单精度偏移措施将小尺寸单元逐渐过渡到大尺寸单元，分析模型有限元离散大样如图 2。
2. 数值模拟结果分析
根据开挖基坑实际自应力状态，采用以下分析步骤模拟整个施工过程：①首先对模型施加重力荷载，使模型在自重应力作用下稳定建立自重应力场，并清除历史上自重作用造成的位移；②采用ABAQUS软件中的单元生死功能(Model change)开挖相应土体，并进行基坑开挖变形分析。变形分析结果分别为基坑开挖引起的地表沉降变形、基坑侧壁水平位移及坑底隆起变形，地表沉降变形、基坑侧壁水平位移及坑底隆起变形沿地表路径DE、坑底路径BC、基坑侧壁路径AB给出(图 3)。
2.1 本构模型对基坑变形模拟结果的影响
不同本构模型下基坑开挖产生的地表沉降变形、基坑侧壁水平位移及坑底隆起变形分别如图 4-图 6所示。
图 4 地表路径DE沉降模拟结果(不同本构模型)Figure 4. DE settlement simulation results of ground pit surface path (Different constitutive models)
图 5 基坑侧壁路径AB水平位移模拟结果(不同本构模型)Figure 5. Simulation results of horizontal displacement of foundation pit sidewall path AB (Different constructive models)
图 6 坑底路径BC隆起模拟结果(不同本构模型)Figure 6. Simulation results of BC uplift of the bottom pit path (Different constructive models)由图 4-图 6可知，在基坑开挖未支护的情形下，Mohr-Coulomb本构模型与Drucker- Prager本构模型模拟结果基本一致，基坑侧壁地表边缘在开挖后出现隆起变形，侧壁水平变形形态为内胀腹形，坑底隆起形态呈拱形，底角处隆起量最小，中部隆起量最大，这主要因为开挖基坑周边的土体未出现塑性变形区，Mohr-Coulomb模型与Drucker-Prager模型模拟基坑开挖在弹性变形范围内的变形量相等；修正剑桥模型模拟得到的地表变形在基坑侧壁地表边缘也为隆起变形，但侧壁顶隆起量小于Mohr-Coulomb本构模型与Drucker-Prager本构模型模拟结果，最大值出现在边缘附近，基坑侧壁水平位移变化表现为向基坑倾覆，坑底隆起形态呈拱形，底角处隆起量最小，中部隆起量最大，这是因为以Mohr-Coulomb为屈服或破坏准则的理想弹塑性本构模型不能反映加卸载模量的差异及土体压硬性，由此可知，修正剑桥模型沉降区域大于Mohr-Coulomb本构模型与Drucker-Prager本构模型模拟结果，坑底隆起差异不明显，且基坑侧壁水平位移变形特征较合理，表明修正剑桥模型较Mohr-Coulomb本构模型与Drucker-Prager本构模型更适用于模拟基坑开挖变形分析。
2.2 三维模型与二维简化分析模型对比
为简便计算，实际分析中常对大型基坑问题进行简化处理，将实际的三维问题简化为二维问题。为检验此处理方法的可靠性，笔者建立长300m、宽120m、高20m的分析模型，应用ABAQUS软件进行基坑开挖变形分析时，设计的开挖宽度、深度、荷载边界条件、有限单元网格划分与前文模型一致，本构模型均采用修正剑桥模型模拟，改变基坑模型长度，使长宽比分别为1:1、3:1、5:1、7:1及10:1。变形分析结果分别为基坑开挖引起的地表沉降变形、基坑侧壁水平位移及坑底隆起变形，观测路径与前文路径相同(图 3)。长宽比为10:1的分析模型如图 7。
为简化分析，常取基坑长边中轴线截面(图 7(a)中A′B′C′)为二维简化分析模型。分析模型采用四节点四边形单元(CPE4、近似全局布种取5)进行离散，由于基坑处及附近土体变形较大，采用较小尺寸单元(CPE4、近似单元布种取1)进行离散，离散过程中采用单精度偏移措施将小尺寸单元逐渐过渡到大尺寸单元。
采用ABAQUS软件对不同长宽比的三维基坑模型及二维简化分析模型进行基坑开挖变形分析，结果如图 8-图 10。
图 8 地表路径DE沉降模拟结果(不同分析模型)Figure 8. DE settlement simulation results of ground pit surface path (Different constitutive models)
图 9 基坑侧壁路径AB水平位移模拟结果(不同分析模型)Figure 9. Simulation results of horizontal displacement of foundation pit sidewall path AB (Different constitutive models)
图 10 坑底路径BC隆起模拟结果(不同分析模型)Figure 10. Simulation results of BC uplift of the bottom pit path由图 8-图 10可知，当基坑长宽比小于5:1时，二维简化分析模型基坑变形模拟结果明显区别于三维模型模拟结果；长宽比大于等于5:1时，二维简化分析模型基坑变形模拟结果与三维模型模拟结果相近；基坑长宽比达10:1时，二维简化分析模型基坑变形模拟结果与三维模型模拟结果一致。由此可认为，当基坑长宽比不小于5:1时，可采用二维简化分析模型模拟基坑变形。
3. 结论
本文通过分析基坑开挖引起的地表沉降变形、基坑侧壁水平位移及坑底隆起变形，研究ABAQUS软件内置本构模型及二维简化分析模型对基坑开挖数值模拟结果的影响，并给出基坑开挖三维模型简化成二维模型的适用条件，主要研究结论如下：
(1) 使用Mohr-Coulomb模型和Drucker-Prager模型进行三维基坑模拟时，如果土体未发生屈服变形，这两种本构模型模拟结果一致。
(2) 修正剑桥模型沉降区域大于Mohr-Coulomb本构模型与Drucker-Prager本构模型模拟结果，坑底隆起差异不明显，且基坑侧壁水平位移变形特征较合理，表明修正剑桥模型较Mohr-Coulomb本构模型与Drucker-Prager本构模型更适用于模拟基坑开挖变形分析。
(3) 当长宽比不小于5:1时，可采用二维简化分析模型进行基坑开挖变形分析。
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图 3 6层钢结构地震反应观测系统传感器布设位置
Figure 3. The sensor location of the structural seismic response observation system in the 6-story steel building
图 4 模态分析计算出的结构前9阶自振振型
Figure 4. The first 9 vibration modes of the building identified from modal analysis
图 5 基于自振频率的瑞雷阻尼比计算
Figure 5. Rayleigh damping ratio based on the first 9 natural frequencies
图 6 结构数值模拟结果与观测结果的加速度时程对比
Figure 6. Comparison of acceleration time history between simulated results and observation results
图 7 结构数值模拟结果与观测结果的位移反应对比
Figure 7. Comparison of displacement time history between simulated results and observation results
图 8 地震动输入为400 cm/s2时4个序列地震动作用下的结构层间位移角分布
Figure 8. Inter-story drift distributions for four sets of ground motions when the input acceleration amplitude is 400 cm/s2
图 9 不同地震动输入下的结构层间位移角平均值
Figure 9. Average inter-story drift ratio for four sets of ground motions with different input acceleration amplitudes
图 10 不同地震动输入下的结构延性系数平均值
Figure 10. Average ductility coefficient for four sets of ground motions with different input acceleration amplitudes
表 1 人造地震动特征
Table 1. The characteristics of peak values of artificial ground motion
合成地震动
序列峰值加速度
PGA/ （m·s−2）峰值速度
PGV/（m·s−1）峰值位移
PGD/dm第1序列 1.0 0.20 0.20 第2序列 1.0 0.20 0.40 第3序列 1.0 0.15 0.30 第4序列 1.0 0.30 0.30 
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表 2 结构梁柱截面配置表
Table 2. Section configuration of the beams and columns
楼层 梁截面 柱截面 6 W24×68 W14×90 5 W24×84 W14×90 3～4 W24×84 W14×132 2 W27×102 W14×176 1 W30×116 W14×176
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表 3 楼层加速度反应统计
Table 3. Amplitude statistics of floor acceleration response
楼层 加速度/（cm·s−2） 东西向 南北向 顶层 270.7 441.1 2 195.2 279.8 1 208.6 293.0
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表 4 结构自振频率对比分析
Table 4. Comparative analysis of natural frequency of the building
振型
编号振型
特性系统识别
频率/HzOpenSees计算
频率/Hz1 y方向 1.462 1.456 2 x方向 1.557 1.537 3 扭转 1.679 1.574 4 y方向 4.556 4.531 5 x方向 5.502 5.177 6 扭转 5.475 5.459 7 y方向 9.319 9.312 8 x方向 10.281 10.277 9 扭转 10.976 10.868
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