桩波动性状若干问题的研究
摘要
本文用弹性动力学方程研究了无限长埋入桩的波动性状的若干问
题。针对轴对称情况，推导了纵向模态的频率方程。频率方程代表了
无量纲频率和无量纲波数之间的超越关系。频率方程的解为无究多个，
并且形成无穷多支频散曲线（无量纲频率-无量纲波数曲线）。本文用
数值方法计算了在软土与硬土中混凝土桩的前5支频散曲线。敏感性
分析表明，无量纲波数实部与桩周土的剪切模量和密度无关，并与自
由杆很一致。而波数虚部则，随着桩周土剪切模量与密度的增大而增
大。
导波模态单位长度的衰减可以直接由波数虚部来代表。相速度和
群速度随频率或波数而变化，这点与一维波动理论的假设相反。随着
曲线分支阶次和频率的提高，导波模态的能量及位移分布曲线的起伏
程度也相应增大。当无量纲频率低于2时，L（0，1）模态衰减最小，
容易被激发，这点与实际中广泛采用的频响函数测试法（导纳法）相
印证。当无量纲频率高于12时，模态L（0，2）衰减最小，其模态相对
比较简单，这意味着这些模态在实际中也可能被激发。
Guideci wave propagatjon in an infinitely long cylindrical pile embedded
in soil is developed from dynamic equations ot?elasticity. Considering
axisymmetric motion in the pile, the frequency equation for longitudinal
modes is derived herein. The frequency equation represents a transcendental
relationship between the non梔imensional frequency, Q, and non梔imensional
wave number, Ja .The solution of the frequency equation is satisfied by an
infinite number of modes that form branches in Q梸a plane. The first five
branches of the longitudinal family of modes in a concrete pile embedded in
sottjloose and hard dense soils were numerically evaluated respectively.
Sensitivity analyses show that the real branches in the Q梸tplane are
essentially independent of the shear modulus and density of the surrounding
soil, and correspond closely to the branches of longitudinal modes in a
free梥tanding?pile. However, the imaginary components of the branches in
the Q梸~ plane are higher for soils with increased shear modulus and
density.
The attenuation of the longitudinal guided wave modes is represented
directly by the imaginary part of the wave number, in nepers per length. The
phase and group velocities vary with the frequency or wave number, contrary
to the assumptions made in the theory of the one梔imensional approach. The
power and displacement profiles of a given guided wave mode generally become
more oscillatory as the order o~ the branch increases and as the frequency
increases. For non梔imensional frequencies less than 2, the L (0, 1) modes
display the lowest attenuation and are easily induced in a pile, as evidenced
by the popularity of the impulse response test in practice. Modes on the
L(0,2)hranch have the least attenuation compared to all other modes for
non梔imensional frequencies above 12, and their relative simple mode shapes
suggest that they are likely to be induced in practice.
题。针对轴对称情况，推导了纵向模态的频率方程。频率方程代表了
无量纲频率和无量纲波数之间的超越关系。频率方程的解为无究多个，
并且形成无穷多支频散曲线（无量纲频率-无量纲波数曲线）。本文用
数值方法计算了在软土与硬土中混凝土桩的前5支频散曲线。敏感性
分析表明，无量纲波数实部与桩周土的剪切模量和密度无关，并与自
由杆很一致。而波数虚部则，随着桩周土剪切模量与密度的增大而增
大。
导波模态单位长度的衰减可以直接由波数虚部来代表。相速度和
群速度随频率或波数而变化，这点与一维波动理论的假设相反。随着
曲线分支阶次和频率的提高，导波模态的能量及位移分布曲线的起伏
程度也相应增大。当无量纲频率低于2时，L（0，1）模态衰减最小，
容易被激发，这点与实际中广泛采用的频响函数测试法（导纳法）相
印证。当无量纲频率高于12时，模态L（0，2）衰减最小，其模态相对
比较简单，这意味着这些模态在实际中也可能被激发。
Guideci wave propagatjon in an infinitely long cylindrical pile embedded
in soil is developed from dynamic equations ot?elasticity. Considering
axisymmetric motion in the pile, the frequency equation for longitudinal
modes is derived herein. The frequency equation represents a transcendental
relationship between the non梔imensional frequency, Q, and non梔imensional
wave number, Ja .The solution of the frequency equation is satisfied by an
infinite number of modes that form branches in Q梸a plane. The first five
branches of the longitudinal family of modes in a concrete pile embedded in
sottjloose and hard dense soils were numerically evaluated respectively.
Sensitivity analyses show that the real branches in the Q梸tplane are
essentially independent of the shear modulus and density of the surrounding
soil, and correspond closely to the branches of longitudinal modes in a
free梥tanding?pile. However, the imaginary components of the branches in
the Q梸~ plane are higher for soils with increased shear modulus and
density.
The attenuation of the longitudinal guided wave modes is represented
directly by the imaginary part of the wave number, in nepers per length. The
phase and group velocities vary with the frequency or wave number, contrary
to the assumptions made in the theory of the one梔imensional approach. The
power and displacement profiles of a given guided wave mode generally become
more oscillatory as the order o~ the branch increases and as the frequency
increases. For non梔imensional frequencies less than 2, the L (0, 1) modes
display the lowest attenuation and are easily induced in a pile, as evidenced
by the popularity of the impulse response test in practice. Modes on the
L(0,2)hranch have the least attenuation compared to all other modes for
non梔imensional frequencies above 12, and their relative simple mode shapes
suggest that they are likely to be induced in practice.
引文
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[2]张何舟，赵淳生，桩基故障诊断理论分析(一)——理想桩的自由振动特性，振动测试与诊断。1991．Vol．ii(1)：1—8页。
[3]工奎华，基桩完整性动测定量化及拟合分析(硕士学位论文)，河海大学，1993。
[4]汪凤泉，王奎华等，非完整桩土耦合系统动力学模型及定解问题，东南大学学报，1993．Vol．23(增刊)：1—6页。
[5]王奎华，谢康和，曾国熙，有限长桩受迫振动问题解析解及应用，岩土工程学报，。
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