令式(4)中β=1和β=2可得分数导数Kelvin粘弹性桩周土和桩芯土的扭转振动方程分别为

对方程式(5)和式(6)两端分别进行Fourier变换可得

式中：${\tilde u}$_θ1和${\tilde u}$_θ2分别为u_θ1、u_θ2的Fourier变换。令$\bar r = \frac{r}{{{r_1}}}$，$\bar z = \frac{z}{{{r_1}}}$，$\bar \omega = \frac{{{r_1}\omega }}{{{v_{s1}}}}$，${v_{s1}} = \sqrt {{\mu _1}/{\rho _1}} $，${{\bar u}_{\theta 1}} = \frac{{{{\tilde u}_{\theta 1}}}}{{{r_1}}}$，${{\bar u}_{\theta 2}} = \frac{{{{\tilde u}_{\theta 2}}}}{{{r_1}}}$，T_a1=a₁v_s1/r₁，T_b1=b₁v_s1/r₁，$\mu = \frac{{{\mu _2}}}{{{\mu _1}}}$，$\rho = \frac{{{\rho _2}}}{{{\rho _1}}}$，T_a2=a₂v_s1/r₁，T_b2=b₂v_s1/r₁，则式(7)和式(8)整理为

式中：${q_1} = \frac{{{{\bar \omega }^2}\left[{1 + {{\left( {{\rm{i}}\bar \omega {\mathit{T}_{a1}}} \right)}^{{\alpha _1}}}} \right]}}{{1 + {{\left( {{\rm{i}}\bar \omega {T_{b1}}} \right)}^{{\alpha _1}}}}}$，${q_2} = \frac{{\rho {{\bar \omega }^2}\left[{1 + {{\left( {{\rm{i}}\bar \omega {\mathit{T}_{a2}}} \right)}^{{\alpha _2}}}} \right]}}{{\mu \left[{1 + {{\left( {{\rm{i}}\bar \omega {\mathit{T}_{b2}}} \right)}^{{\alpha _2}}}} \right]}}$。

为了求解式(9)和式(10)，令${\bar u}$_θ1=R₁(${\bar r}$)Z₁(${\bar z}$)、${\bar u}$_θ2=R₂(${\bar r}$)Z₂(${\bar z}$)，对式(9)和式(10)分离变量可得

式中：β₁²=m₁²－q₁，β₂²=m₂²－q₂。由式(11)~式(14)可将桩周和桩芯粘弹性土的环向位移分别表示为

式(11)和式(14)中，I₁(·)、K₁(·)分别为1阶第1和第2类虚宗量贝塞尔函数，A₁、B₁、C₁、D₁、A₂、B₂、C₂、D₂为待定系数。

考虑问题的边界条件，由${\bar u}$_θ1| _{${\bar r}$→∞}=0，可得B₁=0；当${\bar r}$→0时，${\bar u}$_θ2为有限值，所以A₂=0；由于桩周土和桩芯土体上表面自由，则σ_zθ|_z=0=0、σ_zθ2|_z=0=0，由此可得C₁=0、C₂=0；当假设管桩桩底端为基岩时，则有${\bar u}$_θ1(δ)=0，${\bar u}$_θ2(δ)=0，这里δ=H/r₁，由此可得

由式(17)可得到

则分数导数Kelvin粘弹性桩周土和桩芯土的环向位移可以分别用级数表示为

式中：A_2n、B_2n为待定系数。考虑分数导数Kelvin粘弹性体本构关系式(3)，并对其进行Fourier变换，则有

式中：${\zeta _1} = \frac{{1 + {{\left( {{\rm{i}}\bar \omega {\mathit{T}_{b1}}} \right)}^{{\alpha _1}}}}}{{1 + {{\left( {{\rm{i}}\bar \omega {\mathit{T}_{a1}}} \right)}^{{\alpha _1}}}}}$，${\zeta _2} = \frac{{1 + {{\left( {{\rm{i}}\bar \omega {\mathit{T}_{b2}}} \right)}^{{\alpha _2}}}}}{{1 + {{\left( {{\rm{i}}\bar \omega {\mathit{T}_{a2}}} \right)}^{{\alpha _2}}}}}$，${{\bar \sigma }_{r\theta 1}} = \frac{{{{\tilde \sigma }_{r\theta 1}}}}{{{\mu _1}}}$，${{\bar \sigma }_{r\theta 2}} = \frac{{{{\tilde \sigma }_{r\theta 2}}}}{{{\mu _1}}}$，${\tilde \sigma }$_rθ1、${\tilde \sigma }$_rθ2分别为应力σ_rθ1、σ_rθ2的Fourier变换。由式(21)和式(22)可知，在管桩-土接触面处桩周土和桩芯土对管桩的切向作用力分别为

式中：${P_1} = \frac{{{{\bar P}_1}}}{{{\mu _1}}}$，${P_2} = \frac{{{{\bar P}_2}}}{{{\mu _1}}}$，${\bar P}$₁和${\bar P}$₂分别为${\tilde P}$₁和${\tilde P}$₂的Fourier变换，${\tilde P}$₁和${\tilde P}$₂分别为管桩-土接触面处桩周土和桩芯土对管桩的切向作用力。

3 分数导数Kelvin粘弹性土端承管桩的扭转振动求解

以管桩桩身微元作为研究对象可以建立分数导数Kelvin粘弹性土中管桩的扭转振动控制方程为

$ {{\tilde G}_{\rm{p}}}{J_{\rm{p}}}\frac{{{\partial ^2}\tilde \theta }}{{\partial {z^2}}} - 2\pi r_1^2{{\tilde P}_1} - 2\pi r_2^2{{\tilde P}_2} = {\rho _{\rm{p}}}{J_{\rm{p}}}\frac{{{\partial ^2}\tilde \theta }}{{\partial {t^2}}} $

(25)

式中：${\tilde G}$_p为管桩桩身的剪切模量，ρ_p为管桩桩身的密度，${\tilde \theta }$为管桩的扭转角，J_p为管桩桩身截面的极惯性矩，且J_p=π(r₁⁴－r₂⁴)/2，${\tilde P}$₁、${\tilde P}$₂为桩周土和桩芯土对管桩桩身的剪切作用力。对式(25)两端进行Fourier变换，两端同除以μ₁，同时考虑桩周土和桩芯土对管桩的剪切作用力式(23)和式(24)，得

$ \begin{array}{*{20}{c}} {\frac{{{\partial ^2}\bar \theta }}{{\partial {{\bar z}^2}}} - {\lambda ^2}\bar \theta = \frac{{4{\zeta _1}}}{{{G_{\rm{p}}}\left( {1 - \bar r_2^4} \right)}}\sum\limits_{n = 1}^\infty {{A_{1n}}\left[ {{\beta _{1n}}{K_0}\left( {{\beta _{1n}}} \right) + } \right.} }\\ {\left. {2{K_1}\left( {{\beta _{1n}}} \right)} \right]\cos \left( {{\gamma _n}\bar z} \right) + \frac{{4\bar r_2^2\mu {\zeta _2}}}{{{G_{\rm{p}}}\left( {1 - \bar r_2^4} \right)}}\sum\limits_{n = 1}^\infty {{B_{2n}}} } \end{array} $

$ \left[ {{\beta _{2n}}{I_0}\left( {{\beta _{2n}}{{\bar r}_2}} \right) - \frac{2}{{{{\bar r}_2}}}{I_1}\left( {{\beta _{2n}}{{\bar r}_2}} \right)} \right]\cos \left( {{\gamma _n}\bar z} \right) $

(26)

式(19)中，${G_{\rm{p}}} = \frac{{{{\tilde G}_{\rm{p}}}}}{{{\mu _1}}}$，${{\bar \rho }_{\rm{p}}} = \frac{{{\rho _{\rm{p}}}}}{{{\rho _1}}}$，${\lambda ^2} =-\frac{{{{\bar \rho }_{\rm{p}}}{{\bar \omega }^2}}}{{{G_{\rm{p}}}}}$，${\bar \theta }$为${\tilde \theta }$的Fourier变换。端承管桩桩底和桩顶以及管桩与桩芯土和桩周土接触面处的边界条件为

$ \bar \theta \left( {\bar z} \right){|_{\bar z = \delta }} = 0,{\left. {\frac{{\partial \bar \theta \left( {\bar z} \right)}}{{\partial \bar z}}} \right|_{\bar z = 0}} = - \frac{{\bar T}}{{{G_{\rm{p}}}\left( {1 - \bar r_2^4} \right)}} $

(27)

$ {{\bar u}_{1\theta }} = \bar \theta ,{{\bar u}_{2\theta }} = {{\bar r}_2}\bar \theta $

(28)

式中：${\bar T}$=2${\tilde T}$/μ₁πr₁³，${\tilde T}$为T(t)的Fourier变换。由边界条件式(27)可求得

$ \begin{array}{*{20}{c}} {\bar \theta = - \frac{{{{\bar T}_0}{{\rm{e}}^{\lambda \bar z}}}}{{\lambda {{\bar G}_{\rm{p}}}\left( {1 - \bar r_2^4} \right)\left( {1 + {{\rm{e}}^{2\lambda \delta }}} \right)}} + \frac{{{{\rm{e}}^{2\lambda \delta }}{{\bar T}_0}{{\rm{e}}^{ - \lambda z}}}}{{\lambda {{\bar G}_{\rm{p}}}\left( {1 - \bar r_2^4} \right)\left( {1 + {{\rm{e}}^{2\lambda \delta }}} \right)}} - }\\ {\frac{{4{\zeta _1}}}{{{G_{\rm{p}}}\left( {1 - \bar r_2^4} \right)}}\sum\limits_{n = 1}^\infty {{A_{1n}}\left[ {{\beta _{1n}}{K_0}\left( {{\beta _{1n}}} \right) + 2{K_1}\left( {{\beta _{1n}}} \right)} \right] \cdot } }\\ {\frac{{\cos \left( {{\gamma _n}\bar z} \right)}}{{\gamma _n^2 + {\lambda ^2}}} - \frac{{4\bar r_2^2\mu {\zeta _2}}}{{{G_{\rm{p}}}\left( {1 - \bar r_2^4} \right)}}\sum\limits_{n = 1}^\infty {{B_{2n}}\left[ {{\beta _{2n}}{I_0}\left( {{\beta _{2n}}{{\bar r}_2}} \right) - } \right.} }\\ {\left. {\frac{2}{{{{\bar r}_2}}}{I_1}\left( {{\beta _{2n}}{{\bar r}_2}} \right)\frac{{\cos \left( {{\alpha _n}\bar z} \right)}}{{\gamma _n^2 + {\lambda ^2}}}} \right]} \end{array} $

(29)

考虑管桩与桩周土和桩芯土接触面边界条件式(28)，由式(19)、式(20)可得

$ \begin{array}{*{20}{c}} {\sum\limits_{n = 1}^\infty {{\beta _{2n}}{I_1}\left( {{\beta _{2n}}{{\bar r}_2}} \right)\cos \left( {{\gamma _n}\bar z} \right) = } }\\ {{{\bar r}_2}\sum\limits_{n = 1}^\infty {{A_{1n}}{K_1}\left( {{\beta _{1n}}} \right)\cos \left( {{\gamma _n}\bar z} \right)} } \end{array} $

(30)

$ \begin{array}{*{20}{c}} {\sum\limits_{n = 1}^\infty {{A_{1n}}{K_1}\left( {{\beta _{1n}}} \right)\cos \left( {{\alpha _n}\bar z} \right) = - \frac{{{{\bar T}_0}{{\rm{e}}^{\lambda \bar z}}}}{{\lambda {{\bar G}_{\rm{p}}}\left( {1 - \bar r_2^4} \right)\left( {1 + {{\rm{e}}^{2\lambda \delta }}} \right)}}} }\\ {\frac{{{{\rm{e}}^{2\lambda \delta }}{{\bar T}_0}{{\rm{e}}^{ - \lambda z}}}}{{\lambda {{\bar G}_{\rm{p}}}\left( {1 - \bar r_2^4} \right)\left( {1 + {{\rm{e}}^{2\lambda \delta }}} \right)}} - \frac{{4{\zeta _1}}}{{{G_{\rm{p}}}\left( {1 - \bar r_2^4} \right)}}\sum\limits_{n = 1}^\infty {{A_{1n}}} }\\ {\left[ {{\beta _{1n}}{K_0}\left( {{\beta _{1n}}} \right) + 2{K_1}\left( {{\beta _{1n}}} \right)} \right]\frac{{\cos \left( {{\gamma _n}\bar z} \right)}}{{\gamma _n^2 + {\lambda ^2}}} - }\\ {\frac{{4\bar r_2^2\mu {\zeta _2}}}{{{G_{\rm{p}}}\left( {1 - \bar r_2^4} \right)}}\sum\limits_{n = 1}^\infty {{B_{2n}}\left[ {{\beta _{2n}}{I_0}\left( {{\beta _{2n}}{{\bar r}_2}} \right) - } \right.} }\\ {\left. {\frac{2}{{{{\bar r}_2}}}{I_1}\left( {{\beta _{2n}}{{\bar r}_2}} \right)\frac{{\cos \left( {{\gamma _n}\bar z} \right)}}{{\gamma _n^2 + {\lambda ^2}}}} \right]} \end{array} $

(31)

式(30)和式(31)两端分别同乘以cos(α_n${\bar z}$)，进行三角函数正交性运算, 然后求解方程组有

$ {B_{2n}} = \frac{{{{\bar r}_2}{K_1}\left( {{\beta _{1n}}} \right)}}{{{I_1}\left( {{\beta _{2n}}{{\bar r}_2}} \right)}}{A_{1n}} $

(32)

$ {A_{1n}} = \frac{{2{I_1}\left( {{\beta _{2n}}{{\bar r}_2}} \right)}}{{{\psi _n}}}{{\bar T}_0} $

(33)

式中:ψ_n=δ${\bar G}$_p(1－${\bar r}$₂⁴)(λ²+α_n²)K₁(β_1n)I₁(β_2n${\bar r}$₂)+4ζ₁δI₁(β_2n${\bar r}$₂)[β_1nK₀(β_1n)+2K₁(β_1n)]+4${\bar r}$₂μζ₂δK₁(β_1n)[β_2n${\bar r}$₂I₀(β_2n${\bar r}$₂)－2I₁(β_2n${\bar r}$₂)]。

由此可得分数导数Kelvin粘弹性土中管桩扭转振动的扭转角为

$ \bar \theta \left( 0 \right) = {{\bar u}_{1\theta }}{|_{\bar r = 1}} = \sum\limits_{n = 1}^\infty {\frac{{2{I_1}\left( {{\beta _{2n}}{{\bar r}_2}} \right){K_1}\left( {{\beta _{1n}}} \right)}}{{{\psi _n}}}} {{\bar T}_0} $

(34)

进而可以得到频率域内分数导数Kelvin粘弹性土管桩桩顶的扭转复刚度

$ K = \frac{{{{\bar T}_0}}}{{{{\bar \theta }_0}}} = \frac{1}{{ - \sum\limits_{n = 1}^\infty {\frac{{2{I_1}\left( {{\beta _{2n}}{{\bar r}_2}} \right){K_1}\left( {{\beta _{1n}}} \right)}}{{{\psi _n}}}} }} $

(35)

4 数值算例分析与讨论

为了研究分数导数Kelvin粘弹性土中管桩的扭转振动响应特性，根据式(35)得到的管桩桩顶扭转复刚度编写计算程序并进行数值分析，图 2—图 11为：管桩与桩周土剪切模量比G_p=2 000，分数导数阶数α₁=α₂=0.5，管桩长径比δ=20，管桩与桩周土密度比ρ_p/ρ₁=2.0，土体分数导数本构模型参数T_a1=T_a2=2.0，T_b1=T_b2=4.0，桩芯土与桩周土剪切模量比μ=1.0，以上参量的取值是在桩身混凝土和土体实际常用参量值的基础上确定的，土体分数导数模型参数可以通过实验数据拟合的方法得到^[22]。图 2给出了管桩扭转振动的分数导数粘弹性解、经典粘弹性解和弹性解的对比，可以看出，分数导数粘弹性解可退化到经典粘弹性和弹性解的情况，间接说明采用分数导数粘弹性模型分析管桩扭转振动的可行性和正确性，且弹性解的扭转复刚度要较粘弹性解大。图 3和图 4分别给出了桩周土和桩芯土本构关系中分数导数的阶数α₁、α₂不同时对管桩扭转复刚度的影响，可以看出，桩周土分数导数的阶数α₁对扭转复刚度的实部和虚部的影响相对较大，且分数导数的阶数越大，扭转复刚度实部和虚部随频率变化曲线的峰值越小；而桩芯土分数导数的阶数α₂对扭转复刚度的影响与频率有关，频率较低时α₂几乎没有影响，而高频时有一定的影响。图 5和图 6给出了参数T_b1和T_b2不同时复刚度随频率的变化曲线，T_b1和T_b2对扭转复刚度的影响与分数导数的阶数α₁和α₂的影响规律类似，即T_b1的影响较大且T_b1越大扭转复刚度的实部和虚部随频率变化曲线的峰值越小，而T_b2的影响与频率有关。由图 7和图 8可知，当桩芯土与桩周土剪切模量比μ大于1时，其对管桩桩顶扭转复刚度几乎没有影响，这可能是因为桩芯土与管桩接触面相对较小，导致增大桩芯土剪切模量时其对管桩的剪切作用力增大不明显；当μ小于1且较小时剪切模量比对管桩复刚度有一定的影响但不是太大，但当μ较大时扭转复刚度实部和虚部随频率变化曲线波动则较大。管桩壁厚对扭转复刚度的影响较大(如图 9)，随着桩周内半径的增大，即管桩壁厚的减小，扭转复刚度实部和虚部随频率变化曲线峰值对应的频率越小，曲线波动越大。与实芯桩一样，管桩长径比(图 10)和管桩与土体的剪切模量比(图 11)对管桩的影响也较大，管桩长径比越大复刚度的实部和虚部越小，桩长较长时长径比的影响程度将减小；随着管桩与土体剪切模量比的增大，管桩桩顶复刚度实部和虚部越大，这可能是因为管桩模量较大时管桩抗扭刚度大，管桩扭转角小而导致复刚度增大。

图 2

图 2 α13种模型对比分析 Fig. 2 Contrastive analysis of three models

图 3

图 3 α1不同时桩顶扭转复刚度随频率变化曲线 Fig. 3 Curves of the torsional complex stiffness varying with frequency for different α1

图 4

图 4 α2不同时桩顶扭转复刚度随频率变化曲线 Fig. 4 Curves of the torsional complex stiffness varying with frequency for different α2

图 5

图 5 Tb1不同时桩顶扭转复刚度随频率变化曲线 Fig. 5 Curves of the torsional complex stiffness varying with frequency for different Tb1

图 6

图 6 Tb2不同时桩顶扭转复刚度随频率变化曲线 Fig. 6 Curves of the torsional complex stiffness varying with frequency for different Tb2

图 7

图 7 μ不同时桩顶扭转复刚度随频率变化曲线 Fig. 7 Curves of the torsional complex stiffness varying with frequency for different μ

图 8

图 8 μ不同时桩顶扭转复刚度随频率变化曲线 Fig. 8 Curves of the torsional complex stiffness varying with frequency for different μ

图 9

图 9 r2/r1不同时桩顶扭转复刚度随频率变化曲线 Fig. 9 Curves of the torsional complex stiffness varying with frequency for different r2/r1

图 10

图 10 δ不同时桩顶扭转复刚度随频率变化曲线 Fig. 10 Curves of the torsional complex stiffness varying with frequency for different δ

图 11

图 11 Gp不同时桩顶扭转复刚度随频率变化曲线 Fig. 11 Curves of the torsional complex stiffness varying with frequency for different Gp

5 结论

考虑桩周土和桩芯土的粘弹性特性，并借助分数导数Kelvin粘弹性模型描述土体的应力-应变关系，研究了分数导数粘弹性土中管桩的扭转振动。得到以下主要结论：

1) 桩周土本构模型参数α₁和T_b1对管桩的振动的影较大，而桩芯土本构模型参数T_b2和和α₂对管桩的扭转振动的影响与频率有关，即高频时有影响低频时几乎没有影响。

2) 桩芯土与桩周土剪切模量比μ大于1和小于1时对管桩桩顶扭转复刚度的影响规律不同，当桩芯土剪切模量较小时需要考虑桩周土和桩芯土力学性质的差异的影响。

3) 管桩内外半径比和长径比等几何特性和桩土模量比等力学特性对管桩扭转振动影响较大，需要重点考虑。

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