哈尔滨工业大学学报  2020, Vol. 52 Issue (11): 113-119  DOI: 10.11918/201904050 0

### 引用本文

CUI Chunyi, MENG Kun, LIANG Zhimeng, ZHAO Huijie. Vertical vibration impedance of pipe pile embedded in radially and vertically inhomogeneous layered soil with viscous damping[J]. Journal of Harbin Institute of Technology, 2020, 52(11): 113-119. DOI: 10.11918/201904050.

### 文章历史

Vertical vibration impedance of pipe pile embedded in radially and vertically inhomogeneous layered soil with viscous damping
CUI Chunyi, MENG Kun, LIANG Zhimeng, ZHAO Huijie
School of Transportation Engineering, Dalian Maritime University, Dalian 116026, Liaoning, China
Abstract: The construction of pipe pile will cause the inhomogeneity of the surrounding soil. By considering the influence of the radial inhomogeneity effect and the vertically layer characteristic on the vertical vibration dynamic impedance of pipe pile, a multi-circle plane strain model based on soil viscous damping was proposed, and a simplified model for vertical vibration analysis of pipe pile in radially and vertically layered soil was established. Based on the Laplace transformation and the transitivity of impedance function, the analytical solution of the dynamic impedance of the pile head was derived. Computational results show that the vertical soft and hard interlayer had a significant effect on the amplitude of the dynamic impedance curve in a certain low frequency range, where the softer (harder) the interlayer was, the higher (lower) the amplitude of the dynamic impedance curve at the resonance frequency became. The radial inhomogeneity caused by construction disturbance had a significant influence on the complex stiffness curve of the pile head. In the analysis of vertical vibration characteristics of pipe piles, the analytical solution derived by the proposed soil-pile interaction model was more reasonable.
Keywords: pipe pile    radially inhomogeneous    vertically layer    complex stiffness transfer    dynamic impedance

Novak等[4]利用滞回阻尼模型考虑土体黏性，并采用平面应变理论建立了存在单层内部扰动区域的桩身-土体动力相互作用模型，初步分析了土体径向非均质效应对桩身纵向振动特性的影响.在此基础上，EI Naggar[5]为更合理地考虑桩周土的径向不均匀性，将内部区域进一步划分为多个圈层，分析了施工扰动效应对桩基纵向振动特性的影响.王奎华等[6-7]指出EI Naggar模型与实测曲线存在较大差别，Wang等[8-9]基于此对EI Naggar模型进行修正，土体材料采用滞回阻尼模型，提出了较为严格的平面应变径向多圈层模型，进行了桩顶振动特性频域解析和时域半解析研究.

1 桩土动力相互作用模型 1.1 力学简化模型

 ${G_{ij}}(r) = \left\{ {\begin{array}{*{20}{c}} {{G_{i1}},}&{r = {r_{i1}},}\\ {{G_{i(n + 1)}} \times {f_i}(r),}&{{r_{i1}} < r < {r_{i(n + 1)}},}\\ {{G_{i(n + 1)}},}&{r \ge {r_{i(n + 1)}},} \end{array}} \right.$ (1)
 ${\eta _{ij}}(r) = \left\{ {\begin{array}{*{20}{c}} {{\eta _{i1}},}&{r = {r_{i1}},}\\ {{\eta _{i(n + 1)}} \times {f_i}(r),}&{{r_{i1}} < r < {r_{i(n + 1)}},}\\ {{\eta _{i(n + 1)}},}&{r \ge {r_{i(n + 1)}}.} \end{array}} \right.$ (2)

 图 1 桩土动力相互作用模型 Fig. 1 Dynamic interaction model for pile-soil system
1.2 定解问题

 $\begin{array}{*{20}{c}} {{G_{ij}}\frac{{{\partial ^2}w_{ij}^1(r,t)}}{{\partial {r^2}}} + {\eta _{ij}}\frac{{{\partial ^3}w_{ij}^1(r,t)}}{{\partial t\partial {r^2}}} + \frac{{{G_{ij}}}}{r}\frac{{{\partial ^2}w_{ij}^1(r,t)}}{{\partial {t^2}}} + }\\ {\frac{{{\eta _{ij}}}}{r}\frac{{{\partial ^3}w_{ij}^1(r,t)}}{{\partial t\partial r}} = {\rho _{ij}}\frac{{{\partial ^2}w_{ij}^1(r,t)}}{{\partial {t^2}}},} \end{array}$ (3)
 $\begin{array}{*{20}{c}} {{G_{i0}}\frac{{{\partial ^2}w_i^0(r,t)}}{{\partial {r^2}}} + {\eta _{i0}}\frac{{{\partial ^3}w_i^0(r,t)}}{{\partial t\partial {r^2}}} + \frac{{{G_{i0}}}}{r}\frac{{\partial w_i^0(r,t)}}{{\partial r}} + }\\ {\frac{{{\eta _{i0}}}}{r}\frac{{{\partial ^3}w_i^0(r,t)}}{{\partial t\partial r}} = {\rho _{i0}}\frac{{{\partial ^2}w_i^0(r,t)}}{{\partial {t^2}}},} \end{array}$ (4)

 $\frac{{{\partial ^2}w_i^p}}{{\partial {z^2}}} - \frac{{2\pi {r_{i1}}f_i^{{S_1}}}}{{E_i^PA_i^P}} - \frac{{2\pi {r_{i0}}f_i^{{S_0}}}}{{E_i^PA_i^P}} = \frac{{\rho _i^P}}{{E_i^P}}\frac{{{\partial ^2}w_i^p}}{{\partial {t^2}}},$ (5)

 ${\mathop {{\rm{lim}}}\limits_{r \to 0} w_i^0(r,t) = {\rm{ 有限值 }},}$ (6)
 ${w_i^0({r_{i0}},t) = w_i^p({r_{i0}},t),}$ (7)
 ${f_i^{{S_0}} = \tau _i^{{S_0}}({r_{i0}}).}$ (8)

 ${\mathop {{\rm{lim}}}\limits_{r \to \infty } w_{i(n + 1)}^1(r,t) = 0,}$ (9)
 ${w_i^1({r_{i1}},t) = w_i^p({r_{i1}},t),}$ (10)
 $f_i^{{S_1}} = \tau _i^{{S_1}}({r_{i1}}).$ (11)

 ${{{\left. {E_m^PA_m^P\frac{{\partial w_m^P}}{{\partial z}}} \right|}_{z = 0}} = - p(t),}$ (12)
 ${E_1^P\frac{{\partial w_1^P}}{{\partial z}} + {{\left. {({k_{\rm{p}}}w_1^P + {\delta _{\rm{p}}}\frac{{\partial w_1^P}}{{\partial t}})} \right|}_{z = H}} = 0.}$ (13)
2 解析推导过程 2.1 桩周土解析解

 $\begin{array}{l} {G_{ij}}\frac{{{\partial ^2}}}{{\partial {r^2}}}W_{ij}^1(r,s) + {\eta _{ij}}s\frac{{{\partial ^2}}}{{\partial {r^2}}}W_{ij}^1(r,s) + \frac{{{G_{ij}}}}{r}\frac{\partial }{{\partial r}}W_{ij}^1(r,s) + \\ \frac{{{\eta _{ij}}s}}{r}\frac{\partial }{{\partial r}}W_{ij}^1(r,s) = {\rho _{ij}}{s^2}W_{ij}^1(r,s), \end{array}$ (14)

 $\frac{{{\partial ^2}}}{{\partial {r^2}}}W_{ij}^1 + \frac{1}{r}\frac{\partial }{{\partial r}}W_{ij}^1 = {(q_{ij}^{{S_1}})^2}W_{ij}^1,$ (15)

 $W_{ij}^1 = A_{ij}^1{K_0}(q_{ij}^1r) + B_{ij}^1{I_0}(q_{ij}^1r).$ (16)

 $FF_{ij}^1 = 2\pi {r_{ij}}({G_{ij}} + {\eta _{ij}}s)q_{ij}^1\frac{{C_{ij}^1 + M_{ij}^1FF_{i(j + 1)}^1}}{{D_{ij}^1 + N_{ij}^1FF_{i(j + 1)}^1}}.$ (17)

 $\begin{array}{*{20}{l}} {C_{ij}^1 = 2\pi {r_{i(j + 1)}}({G_{ij}} + {\eta _{ij}}s)q_{ij}^1[{I_1}(q_{ij}^1{r_{i(j + 1)}}){K_1}(q_{ij}^1{r_{ij}}) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_1}(q_{ij}^1{r_{i(j + 1)}}){I_1}(q_{ij}^1{r_{ij}})],} \end{array}$
 $\begin{array}{*{20}{l}} {D_{ij}^1 = 2\pi {r_{i(j + 1)}}({G_{ij}} + {\eta _{ij}}s)q_{ij}^1[{I_1}(q_{ij}^1{r_{i(j + 1)}}){K_0}(q_{ij}^1{r_{ij}}) + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_1}(q_{ij}^1{r_{i(j + 1)}}){I_0}(q_{ij}^1{r_{ij}})],} \end{array}$
 ${M_{ij}^1 = {K_0}(q_{ij}^1{r_{i(j + 1)}}){I_1}(q_{ij}^1{r_{ij}}) + {K_1}(q_{ij}^1{r_{ij}}){I_0}(q_{ij}^1{r_{i(j + 1)}}),}$
 ${N_{ij}^1 = - {K_0}(q_{ij}^1{r_{i(j + 1)}}){I_0}(q_{ij}^1{r_{ij}}) + {K_0}(q_{ij}^1{r_{ij}}){I_0}(q_{ij}^1{r_{i(j + 1)}}).}$
2.2 桩芯土解析解

 $\begin{array}{*{20}{l}} {{G_{i0}}\frac{{{\partial ^2}}}{{\partial {r^2}}}W_i^0(r,s) + {\eta _{i0}}s\frac{{{\partial ^2}}}{{\partial {r^2}}}W_i^0(r,s) + \frac{{{G_{i0}}\partial }}{r}\frac{\partial }{{\partial r}}W_i^0(r,s) + }\\ {\frac{{{\eta _{i0}}s}}{r}\frac{\partial }{{\partial r}}W_i^0(r,s) = {\rho _{i0}}{s^2}W_i^0(r,s),} \end{array}$ (18)

 $\frac{{{\partial ^2}}}{{\partial {r^2}}}W_i^0 + \frac{1}{r}\frac{\partial }{{\partial r}}W_i^0 = {(q_i^0)^2}W_i^0,$ (19)

 $\begin{array}{*{20}{l}} {FF_i^0 = - \frac{{2\pi {r_{i0}}\tau _i^{{S_0}}({r_0})}}{{W_i^P}} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 2\pi {r_{i0}}q_i^0({G_{i0}} + {\eta _{i0}}s)\frac{{{I_1}(q_i^0{r_{i0}})}}{{{I_0}(q_i^0{r_{i0}})}}.} \end{array}$ (20)
2.3 桩振动方程求解

 $\frac{{{\partial ^2}W_i^P}}{{\partial {z^2}}} - \alpha _i^2W_i^P = 0.$ (21)

 $W_i^P = M_i^P{{\rm{e}}^{{{\bar \alpha }_i}z/{l_i}}} + N_i^P{{\rm{e}}^{ - {{\bar \alpha }_i}z/{l_i}}}.$ (22)

 $Z_m^P{|_{z = {h_m} = 0}} = \frac{{ - E_m^PA_m^P{{\bar \alpha }_m}({\beta _m} - 1)}}{{{l_m}({\beta _m} + 1)}} = \frac{{ - E_m^PA_m^P}}{{{l_m}}}Z_m^{{P^\prime }}.$ (23)

3 算例分析

3.1 解析解对比验证

 图 2 本文解析解退化与已有解[18]对比验证 Fig. 2 Comparison of the proposed analytical solution with the solution in Ref.[18]
 图 3 本文解析解退化与已有解[6]对比验证 Fig. 3 Comparison of the proposed analytical solution with Wang's solution in Ref.[6]
3.2 参数化分析

 图 4 纵向软、硬夹层对桩顶复刚度曲线的影响 Fig. 4 Effect of vertical soft and hard interlayer on complex stiffness curves of pile head

 图 5 桩周土软化程度对桩顶复刚度曲线的影响 Fig. 5 Effect of softening degree of pile surrounding soil on complex stiffness curves of pile head
 图 6 桩周土硬化程度对桩顶复刚度曲线的影响 Fig. 6 Effect of hardening degree of pile surrounding soil on complex stiffness curves of pile head

 图 7 桩周土体软化范围对桩顶复刚度曲线的影响(软化工况S1) Fig. 7 Effect of softening range of pile surrounding soil on complex stiffness curves of pile head (softening condition S1)
 图 8 桩周土体硬化范围对桩顶复刚度曲线的影响(硬化工况H4) Fig. 8 Effect of hardening range of pile surrounding soil on complex stiffness curves of pile head (hardening condition H4)
4 结论

1) 纵向软硬夹层对一定低频区间内动力阻抗曲线振幅水平影响显著.夹层越软(硬)，共振频率处对应的动力阻抗曲线振幅水平越高(低).当夹层剪切波速比为1.8时，动力阻抗共振幅值减小约25%，当夹层剪切波速比为0.6时，动力阻抗共振幅值增大约19%.

2) 桩周土软(硬)化程度对管桩桩顶复刚度曲线的共振幅值及共振频率均有显著影响，当桩周土软化到未扰动土的60%时，桩顶动力阻抗曲线共振幅值增大约38%；当桩周土比未扰动土硬60%时，桩顶动力阻抗曲线共振幅值减小约17%.说明在对管桩纵向振动特性进行分析时，不考虑桩周土体的软化和硬化效应计算所得桩顶纵向振动特性存在较大误差，将对管桩抗振防振设计产生不利影响.

3) 施工扰动土体软(硬)化范围主要影响管桩桩顶复刚度曲线的共振幅值，即使桩周土体扰动范围很小(0.1r1=0.05 m)，管桩施工仅影响桩身附近几厘米厚度范围内土体性质，其对桩顶动力阻抗特性的影响同样不可忽略，这就说明考虑施工扰动效应的影响对管桩纵向振动特性进行研究的必要性.