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Article(id=1243226197765763149, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, articleNumber=null, orderNo=null, doi=10.7511/jslx20240323001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1711123200000, receivedDateStr=2024-03-23, revisedDate=1715356800000, revisedDateStr=2024-05-11, acceptedDate=null, acceptedDateStr=null, onlineDate=1774337823573, onlineDateStr=2026-03-24, pubDate=1761580800000, pubDateStr=2025-10-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774337823573, onlineIssueDateStr=2026-03-24, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774337823573, creator=13701087609, updateTime=1774337823573, updator=13701087609, issue=Issue{id=1243226190786441246, tenantId=1146029695717560320, journalId=1242798230522609684, year='2025', volume='42', issue='5', pageStart='699', pageEnd='888', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1774337821909, creator=13701087609, updateTime=1774338282025, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1243228120724128564, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1243228120724128565, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=883, endPage=888, ext={EN=ArticleExt(id=1243226198952751197, articleId=1243226197765763149, tenantId=1146029695717560320, journalId=1242798230522609684, language=EN, title=Vertical vibration of end-bearing pile in fractional rheological soil, columnId=1243226196507471936, journalTitle=Chinese Journal of Computational Mechanics, columnName=Research Notes, runingTitle=null, highlight=null, articleAbstract=

Using fractional derivatives to modify the Zener standard rheological solid model and considering the instantaneous rheological effect of the soil around the pile, a vertical coupled vibration model of the pile-soil system is constructed. The frequency-domain analytical solution of the system dynamic control equation is derived using Laplace transform and potential function decomposition methods. The time domain response under instantaneous excitation at the pile top is obtained through numerical Laplace inversion. Then, numerical examples are used to analyze the frequency domain characteristics of displacement and dynamic stiffness, and dynamic damping of end-bearing pile vertical vibration in a rheological clay layer, as well as the wave response under instantaneous excitation at the pile top. Research has found that the rheological effect of soil reduces the amplitude of pile top displacement and dynamic stiffness and the rheological effect of soil causes a decrease in the amplitude of the pile top response and a weakening of the reflected wave signal under instantaneous excitation.

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利用分数阶导数修正Zener标准流变固体模型，考虑桩周土的瞬时流变效应构建桩-土系统竖向耦合振动模型。采用Laplace变换和势函数分解方法推导系统动力控制方程的频域解析解，再通过Laplace数值反演得到桩顶瞬时激励下的时域响应。利用退化模型和有限元模型对本文方法的有效性进行验证，而后通过数值算例分析流变性黏土地层中端承桩竖向振动的位移、动刚度和动阻尼的频域特征，以及桩顶瞬时激励下的波动响应。研究发现，土体流变效应使桩顶位移、动刚度和动阻尼在频域中的幅值减小；土体流变效应造成瞬时激励下桩顶响应幅值减小，反射波信号减弱。

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陈振宏*（1985-），男，高级工程师（E-mail：）.

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陈振宏*（1985-），男，高级工程师（E-mail：）.

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陈振宏*（1985-），男，高级工程师（E-mail：）.

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figureFileBig=k6P/ap39IhiQHLomapSBSg==, tableContent=null), ArticleFig(id=1243226253919105065, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226197765763149, language=EN, label=Tab. 1, caption=

Calculation parameters

, figureFileSmall=null, figureFileBig=null, tableContent=
参数量纲数值
桩基弹性模量EpGPa25
桩基密度ρpkg•m-32500
桩长Hm6
桩径r0m0.3~0.6
土体静弛豫模量E0MPa5.5
土体泊松比v/0.2
土体密度ρskg•m-32700
), ArticleFig(id=1243226254070100011, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226197765763149, language=CN, label=表1, caption=

计算参数

, figureFileSmall=null, figureFileBig=null, tableContent=
参数量纲数值
桩基弹性模量EpGPa25
桩基密度ρpkg•m-32500
桩长Hm6
桩径r0m0.3~0.6
土体静弛豫模量E0MPa5.5
土体泊松比v/0.2
土体密度ρskg•m-32700
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分数阶流变土层中端承桩的竖向振动特性研究
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陈振宏 1 , 王立安 2 , 郁园园 1 , 马飞 3
计算力学学报 | 研究简报 2025,42(5): 883-888
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计算力学学报 | 研究简报 2025, 42(5): 883-888
分数阶流变土层中端承桩的竖向振动特性研究
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陈振宏1 , 王立安2, 郁园园1, 马飞3
作者信息
  • 1.甘肃省矿产资源综合勘查利用与保护工程研究中心（甘肃省地质矿产勘查开发局第一地质矿产勘查院），天水　741020
  • 2.兰州交通大学　土木工程学院，兰州　730070
  • 3.宁夏回族自治区固原市彭阳县自然资源局，固原　756500

    • 陈振宏*（1985-），男，高级工程师（E-mail：）.

Vertical vibration of end-bearing pile in fractional rheological soil
Zhenhong CHEN1 , Lian WANG2, Yuanyuan YU1, Fei MA3
Affiliations
  • 1.Gansu Province Engineering Research Center for Comprehensive Exploration, Utilization and Protection of Mineral Resources (No. 1 Institute of Geology and Mineral Resources Exploration, Gansu Provincial Bureau of Geology and Mineral Resources Exploration and Development), Tianshui 741020, China
  • 2.School of Railway Technology, Lanzhou Jiaotong University, Lanzhou 730070, China
  • 3.Pengyang County Natural Resources Bureau of Ningxia Hui Autonomous Region, Guyuan 756500, China
出版时间: 2025-10-28 doi: 10.7511/jslx20240323001
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摘要
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利用分数阶导数修正Zener标准流变固体模型，考虑桩周土的瞬时流变效应构建桩-土系统竖向耦合振动模型。采用Laplace变换和势函数分解方法推导系统动力控制方程的频域解析解，再通过Laplace数值反演得到桩顶瞬时激励下的时域响应。利用退化模型和有限元模型对本文方法的有效性进行验证，而后通过数值算例分析流变性黏土地层中端承桩竖向振动的位移、动刚度和动阻尼的频域特征，以及桩顶瞬时激励下的波动响应。研究发现，土体流变效应使桩顶位移、动刚度和动阻尼在频域中的幅值减小；土体流变效应造成瞬时激励下桩顶响应幅值减小，反射波信号减弱。

流变性土  /  分数阶导数  /  端承桩  /  竖向振动

Using fractional derivatives to modify the Zener standard rheological solid model and considering the instantaneous rheological effect of the soil around the pile, a vertical coupled vibration model of the pile-soil system is constructed. The frequency-domain analytical solution of the system dynamic control equation is derived using Laplace transform and potential function decomposition methods. The time domain response under instantaneous excitation at the pile top is obtained through numerical Laplace inversion. Then, numerical examples are used to analyze the frequency domain characteristics of displacement and dynamic stiffness, and dynamic damping of end-bearing pile vertical vibration in a rheological clay layer, as well as the wave response under instantaneous excitation at the pile top. Research has found that the rheological effect of soil reduces the amplitude of pile top displacement and dynamic stiffness and the rheological effect of soil causes a decrease in the amplitude of the pile top response and a weakening of the reflected wave signal under instantaneous excitation.

rheological soil  /  fractional derivative  /  end-bearing pile  /  vertical vibration
陈振宏, 王立安, 郁园园, 马飞. 分数阶流变土层中端承桩的竖向振动特性研究. 计算力学学报, 2025 , 42 (5) : 883 -888 . DOI: 10.7511/jslx20240323001
Zhenhong CHEN, Lian WANG, Yuanyuan YU, Fei MA. Vertical vibration of end-bearing pile in fractional rheological soil[J]. Chinese Journal of Computational Mechanics, 2025 , 42 (5) : 883 -888 . DOI: 10.7511/jslx20240323001
正文
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1　引言
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桩基竖向振动特性是小应变检测的理论基础，利用桩顶反射波检测桩身完整性在桩基施工和质量检测方面发挥着至关重要的作用。桩基础作为一种深基础，利用桩体穿过软弱土层将荷载直接传至执力层，而桩身将部分或完全埋置于软弱土层中。
自然界中的土体普遍具有流变性（松弛和蠕变），在应力作用下土颗粒的重新排列和骨架错动具有明显的时间效应[1]。为描述土体流变行为，Taylor等[2]最早利用Kelvin线性流变模型分析了土体的固结变形，发现流变对土体次固结阶段的变形产生明显影响。之后，Tan[3]采用Maxwell模型考虑黏性土的次时间效应，并提出黏弹性固结理论。学者们通过与实验结果对比，发现传统整数阶形式的Kelvin模型、Maxwell模型及Zener标准线性固体模型在模拟黏土流变时不能满足精度要求，需引入更多拟合参数才能实现有效模拟[4]。为改善传统整数阶流变模型的适用性，文献[5，6]提出了分数阶导数流变模型。分数阶微分算子为卷积积分，能够准确描述力学与物理过程中的历史记忆性和空间全域相关性。最近几年，大量学者引入Caputo分数阶导数来修正Kelvin模型、Maxwell模型及Zener标准线性固体模型，重新研究了饱和黏土的流变行为，已取得显著成果[7,8]。
针对单相弹性土和两相饱和土中桩基的振动动响应，学者们已开展了大量研究。马文杰等[9]考虑土体的三相组成研究了非饱和土中桩基的纵向波动响应；余云燕等[10]利用Winkler地基和Euler梁模型，研究了大直径桩的自振特性；崔春义等[11]采用径向多圈层平面应变模型，推导了非均匀土中桩顶动阻抗的解析解。孟坤等[12,13]考虑桩底土的支撑刚度研究了浮沉桩的纵向振动。上述研究考虑了桩周土的固-流多相组成和渗流作用，但并未考虑土体的流变效应。实际工程中，黏性土具有显著的流变性，当桩基埋置于黏性土层中时，其竖向振动特性将受到流变性的影响。
本文针对流变性饱和黏土地层中桩基的小应变纵向波动特性开展理论研究。利用分数阶导数修正Zener标准固体流变模型，构建桩-土系统的竖向耦合振动模型。利用Laplace变换和势函数分解推导桩-土系统的频域解析解，并通过数值反演得到时域解。经过频域和时域的算例分析，研究了流变性黏土地层中端承桩竖向振动响应。
2　分析模型
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2.1　分数阶导数流变固体模型
Zener标准固体流变模型由一个Kelvin单元和一个Hooke弹簧串联而成，既能反映松弛也能反映蠕变。因此，本文利用Zener标准固体流变模型进行分数阶导数修正。将经典Zener标准固体流变模型中的Newton黏壶替换为Abel黏壶[14]，则得到分数阶导数流变固体模型，如图1所示。图中σ和ε为总应力和总应变，ε1为单个弹簧元件的应变，ε2为Kelvin单元的应变；k1和k2为弹簧元件的弹性模量，η和a为Abel黏壶的黏度和黏性阶数（分数阶导数的阶数，0≤a≤1）。分数阶导数流变固体模型的应力-应变关系为
式中τσ=η/（k1+k2）为应力松弛的弛豫时间，τε=η/k2为蠕变弛豫时间；E0=k1k2/（k1+k2）为静弛豫模量（频率为0时的弛豫模量）；t为时间变量。从式（1）可以看出，当a=0时模型退化为线弹性固体，a=1时模型退化为传统整数阶流变模型。
对任意时域函数f（t）引入如下Lapace变换：
式中为函数f（t）的Laplace变换，s为时间变量t的变换参数，上标表示对应物理量已进入Laplace变换域；ζ为任意实数，为虚数单位。对式（1）执行Laplace变换，得
由式（3）得出复模量的表达式为
将式（3）描述的分数阶导数流变本构模型拓展到三维应力状态，写为张量形式为
式中，，为Laplace变换域中的应力、应变张量和体应变；I为三阶单位矩阵；λ，μ为Laplace变换域中的Lamé常数，根据其定义有
式中v为泊松比。
2.2　桩-土竖向耦合振动模型
如图2所示，流变性黏土地层中埋置端承桩，桩底为固定端，桩顶受竖向瞬时荷载P（t），桩长H，桩径r0。采用三维柱坐标系（r-φ-z）构建桩-土纵向耦合振动模型。
桩-土纵向振动为轴对称问题，柱坐标系下流变性黏土的运动方程张量表达式为
式中u=[ur，uz]，σ=[σr，σz，τrz]分别为土体的位移和应力张量；为Hamilton算子；ρs为土体密度。
将式（5）代入式（7），进行Laplace变换后得到位移形式的运动方程为
采用Euler杆模拟桩基，其纵向振动方程为
式中up为桩基竖向位移，Ep，Ap，ρp分别为桩基的弹性模量、截面面积和密度；q为桩周土作用于桩基的抗力。
地表为自由边界，边界条件为
土体在径向无穷远处的边界条件为
土体底部为固定边界，边界条件为
桩-土界面的接触耦合边界条件为
桩顶自由端边界条件为
对于端承桩，桩底为固定端：
3　控制方程求解
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对土体位移矢量引入如下Helmholtz分解：
式中ϕ为标量势函数，ψ为矢量势函数。
将式（15）代入式（8）得
式中，为复化压缩波速和剪切波速。
根据张量运算性质，式（17）可分解为
将矢量势函数ψ分解为两个标量势函数为
联合式（18，19），可得
利用分离变量法分别求解式（20）得出ϕ，φ和χ的通解为
式中I1（）、K1（）分别为第一类和第二类1阶虚宗量Bessel函数，A1~A12为待定系数。
对式（9）进行Laplace变换得
式中为桩基位移up的Laplace变换。式（22）的通解为
式中，，B1—B7为待定系数。利用边界条件式（10—15）构建线性方程组确定出待定系数A1—A12，则得到桩-土系统的频域解。根据桩基动刚度和动阻尼的定义，取s=iω（ω为角频率）得出流变性黏土地层中桩基的动刚度和动阻尼的频域解为
式中Kd，Nd分别为桩基动刚度和动阻尼，Re表示取实部，Im表示取虚部。通过对式（23）取时间的一阶导数可进一步得到桩顶速度，。定义桩顶瞬时荷载P（t）为半正弦脉冲激励，其时域函数和对应的Laplace变换为
式中P0为荷载幅值，t0为脉冲激励的时间宽度，H（t）为Heaviside阶跃函数
4　方法验证
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4.1　退化模型验证
将本文流变性黏土模型退化为线弹性弹性土，与文献[15]的结果进行对比验证。对桩顶位移和频率进行无量纲化处理，u*=u/H，。将桩顶瞬时荷载定义为半正弦脉冲激励，将桩顶速度的时域响应与文献[16]的理论结果和文献[17]的测试结果进行对比，模型计算参数列入表1。图3和图4显示，本文退化解与文献结果吻合，而且与现场实测曲线也非常相似。
4.2　有限元模型验证
针对本文模型建立相应的有限元模型进行对比验证，基于COMSOL多物理场分析软件，在固体力学模块构建桩周土的有限元模型，采用分数阶导数标准线性固体模型描述土体应力应变关系，用Euler梁模拟桩基，并在径向边界处采用无限元域，如图5所示。图6为本文解与有限元解的对比，显示本文结果与有限元计算结果基本吻合。
5　算例分析
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在表1基础上补充黏滞系数η=1×104 Pa•s，计算流变性黏土地层中桩基纵向振动的频域响应。图7为不同黏性阶数a取值下桩顶位移的幅频响应，可见桩顶位移幅值随频率振荡下降，在共振频率处出现极大值，土体黏性阶数越大，共振频率处的极值越小。图8和图9为不同黏性阶数a下桩顶动刚度和动阻尼的频域曲线，显示土体黏性阶数越大，动刚度和动阻尼的幅值越小。
图10给出了半正弦瞬时激励下桩顶速度的时域响应，可见土体黏性阶数越大，桩顶反射波的速度幅值越小。计算发现，当黏性阶数为1时，第一反射波的速度幅值减小幅度达到23.7%，这在实际工程检测中将导致接收到的反射波信号减弱，不利于桩基缺陷的辨识。
6　结论
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本文利用分数阶导数修正Zener标准流变固体模型，考虑桩周土的瞬时流变效应构建桩-土系统纵向耦合振动模型，研究了流变性饱和黏土地层中端承桩动刚度、动阻尼的频域特征，以及桩顶瞬时激励下的竖向波动响应，得出以下几点结论。
（1）土体流变效应造成桩基动刚度和动阻尼在频域中的波动幅值减小，土体流变性越强，桩基动刚度和动阻尼沿频率的增长速率越缓慢。
（2）土体流变效应导致桩顶反射波的速度幅值减小，在实际工程检测中造成接受到的反射波信号减弱，不利于桩基缺陷辨识。
（3）大直径桩的反射波信号弱于小直径桩，土体流变效应使得信号进一步减弱。本文结果显示，桩基长细比小于15时不宜采用反射波检测。
基金
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  • 甘肃省科技计划(25CXGA038)
文献
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参考文献 引证文献
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2025年第42卷第5期
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文章信息
doi: 10.7511/jslx20240323001
  • 接收时间：2024-03-23
  • 首发时间：2026-03-24
  • 出版时间：2025-10-28
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  • 收稿日期：2024-03-23
  • 修回日期：2024-05-11
基金
甘肃省科技计划(25CXGA038)
作者信息
    1.甘肃省矿产资源综合勘查利用与保护工程研究中心（甘肃省地质矿产勘查开发局第一地质矿产勘查院），天水　741020
    2.兰州交通大学　土木工程学院，兰州　730070
    3.宁夏回族自治区固原市彭阳县自然资源局，固原　756500
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