摘要
为研究桩基在桩-土界面连续条件下,桩顶受撞击等水平瞬态荷载作用时桩基位移响应、桩-土界面动力响应和桩周土的应力分布规律, 根据Biot理论和Novak平面应变假定,采用Euler梁建立桩-饱和土耦合振动的平面应变简化模型,利用Laplace变换和势函数分解求解系统动力控制方程. 针对桩-土系统在水平三角形冲击荷载作用下的振动状态,着重对桩基位移响应、桩-土界面和桩周土的动力响应开展时域分析. 研究发现:桩-土系统的位移场响应滞后于应力场响应;桩土模量比越小,桩-土界面有效径向应力、切应力及孔压响应越显著;土体渗透系数减小引起桩-土界面孔压增大,导致有效径向应力减小,桩-土界面处切应力几乎不受渗透系数变化的影响;渗透系数较大时,桩周土孔压分布较分散,最大有效径向应力出现在桩-土界面附近;渗透系数较小时,桩周土孔压分布较为集中,最大有效径向应力则出现在桩-土界面较远处.
桩基础因其抗震性能佳、沉降量小及能解决特殊地基承载力问题,常用于动力机械基础、交通设施和海洋工程. 在这些应用中,桩基础可能遭受动力机械启动的瞬时振动、车辆碰撞、船舶撞击等冲击荷载,由于冲击荷载的突发性和瞬时性,其作用机理相较于静载和简谐荷载更加难以预测.
关于桩-土动力接触问题,国内外学者已进行广泛研究,除了以有限元、边界元为代表的数值方法外,还有以动力Winkler地基梁模型和连续介质模型为主的解析和半解析方法. 动力Winkler地基梁模型采用并联弹簧和黏壶近似代替桩周土体的刚度和阻尼,考虑土体随荷载和深度的非线性变化,因其物理概念清晰、计算简单,成为分析桩-土相互作用的有效手段. Nogami
连续介质模型将土体视为连续体,Novak
由文献[
基于此,本文针对水平冲击荷载下的桩-土系统的振动状态,着重对桩基位移响应、桩-土界面和桩周饱和土的动力响应开展时域分析. 首先,采用Biot饱和多孔介质来描述饱和土层,构建严格桩-成层饱和土水平耦合振动的平面应变简化模型. 然后,通过Laplace变换、势函数分解及分离变量法求解桩-土系统的动力控制方程,得出Laplace变换域的桩身位移和桩周饱和土应力表达式. 最后,利用Laplace逆变换计算三角形冲击荷载下桩-土系统的时域响应,进行参数分析.
1 分析模型与基本假定
1.1 分析模型
如

图1 桩-土系统计算模型
Fig.1 Pile-soil system calculation model
对于桩顶冲击荷载形式的选择,三角形冲击荷载相较于正弦冲击荷载、矩形冲击荷载等具有短时间内达到峰值后迅速减小为零的特点. 因此,本文采用三角形冲击荷载来准确模拟桩基受到的短暂冲击动荷载. 三角形冲击荷载表达式为:
(1) |
式中:为荷载峰值,为荷载作用时长,(•)为Heaviside阶跃函数.
(2) |
式中:是的Laplace变换式,s为变换参数.
1.2 基本假定
1)桩周土体由两相饱和薄土层组成,薄土层之间相互独立且沿纵向忽略竖向连续.
2)桩-土系统水平振动时,桩侧薄土层之间保持充分接触,忽略土体竖向位移和桩侧摩阻力.
3)桩基为弹性圆柱体,垂直于半空间表面,采用Euler梁模型.
4)桩-土系统水平振动过程为小变形,桩-土界面不发生分离为完全接触,且桩-土接触面不透水.
2 控制方程
2.1 饱和土控制方程
针对桩-饱和土水平振动,采用Biot饱和多孔介质理论,忽略土体竖向位移,建立桩周饱和土控制方程:
(3) |
(4) |
(5) |
(6) |
式中:分别为土骨架和水相对于土骨架的径向位移;分别为土骨架和水相对于土骨架的切向位移;为Lamé常数;分别为土骨架和水的体应变;分别为反映土颗粒和水压缩性的Biot系数,,,;为土体总密度,;分别为固体颗粒和水的密度;为渗透系数;为重力加速度;为孔隙率;为Laplace算子.
2.2 桩基控制方程
采用Euler梁模型建立桩基控制方程:
(7) |
式中:up为桩基水平位移;分别为桩基弹性模量、截面惯性矩、截面半径、密度和桩-土相互作用的荷载集度.
2.3 定解条件
桩-土系统初始条件为:
(8) |
(9) |
土体在径向无穷远处边界条件为:
(10) |
(11) |
考虑桩土完全接触,则桩-土界面边界条件为:
(12) |
桩-土界面不透水条件为:
(13) |
桩顶边界条件为:
(14) |
水平荷载作用于桩顶通常对桩身上部影响显著,对桩身下部影响很小,所以桩底采用自由支撑边界条件:
(15) |
3 方程求解
3.1 饱和土控制方程求解
采用Helmholtz分解ur、wr:
(16) |
式中:均为势函数.
将
(17) |
式中:分别为的拉式变换,为时间因子的拉式变换.
将上式写成矩阵形式:
(18) |
(19) |
(20) |
(21) |
式中:,.
由算子分解理论可知,,且满足:
(22) |
(23) |
式中:,.
再分离变量,令带入
(24) |
对
(25) |
同理可得:
(26) |
式中:分别为第一类和第二类修正Bessel函数;A11、A12、B11、B12、C11、C12、D11、D12和n1、n2均为待定系数. 由边界条件
则有:
(27) |
同理可得:
(28) |
(29) |
(30) |
由之间的相关性可得:
(31) |
式中:为待定系数;,,,.
将式(27)~
(32) |
(33) |
式中:
,,, . |
饱和土体本构关系和渗流连续方程为:
(34) |
(35) |
式中:分别为有效径向应力、切应力和孔压.
(36) |
将式(32)~
(37) |
3.2 桩基控制方程求解
对
(38) |
化简整理得:
(39) |
式中:.
求解
(40) |
结合定解条件
其中,
. |
至此,桩-土系统的位移场和应力场均已确定,结合初始条件
(41) |
式中:、均为计算参数,且,计算中取,;为级数截取项数;为虚数单位. 试算表明,取50时,能满足精度要求.
4 方法验证
文献[
验证结果如

图2 与文献[
Fig.2 Comparison of calculation results with reference [
5 算例分析
采用
H/m | r0/m | EP/ GPa | ρp/ (kg· | Es/MPa | ρs/ (kg· | Ks/GPa | ρf/ (kg· |
---|---|---|---|---|---|---|---|
10 | 0.3 | 25.5 | 2 500 | 27 | 2 700 | 36 | 1 000 |
Kf/GPa | υs | n |
kd/ (m· | ζ | Pmax/kN | t0/s |
g/ (m· |
2 | 0.35 | 0.375 |
1×1 | 0.01 | 1 000 | 0.02 | 10 |
注: Es为土体弹性模量;Kf为孔隙水体积模量;υs为土体泊松比.

图3 桩顶位移响应
Fig.3 Pile top displacement response

图4 渗透系数对桩顶最大位移的影响
Fig.4 Influence of permeability coefficient on maximum displacement of pile top
由

(a) 有效径向应力

(b) 切应力

(c) 孔压
图5 荷载作用时间对桩-土界面应力的影响
Fig.5 Effect of loading time on the interface stress of pile-soil
结合

(a) 有效径向应力

(b) 切应力

(c) 孔压
图6 桩-土模量比对桩-土界面应力的影响
Fig.6 Effect of pile-soil modulus ratio on pile-soil interface stress

(a) 有效径向应力

(b) 切应力

(c) 孔压
图7 桩-土界面应力沿桩周分布
Fig.7 Distribution of pile-soil interface stress around the pile

(a) 有效径向应力

(b) 切应力

(c) 孔压
图8 渗透系数对桩-土界面应力的影响
Fig.8 Effect of permeability coefficient on pile-soil interface stress

图9 孔压沿径向分布
Fig.9 Radial distribution of pore pressure

(a) kd=1×1

(b) kd=1×1
图10 桩周土中孔压水平面分布()
Fig.10 Horizontal distribution of pore pressure in soil around pile

图11 有效径向应力沿径向分布
Fig.11 Radial distribution effective radial stress

(a) kd=1×1

(b) kd=1×1
图12 桩周土中有效径向应力水平面分布 ()
Fig.12 Horizontal distribution of effective radial stress in soil around pile
由
6 结 论
本文采用Euler梁建立桩-饱和土耦合振动的平面应变简化模型,研究三角形水平冲击荷载下桩顶位移、桩-土界面及桩周土的动力响应,得出以下结论:
1)三角形冲击荷载作用时间越长,桩顶位移响应越大,桩周土体有效径向应力峰值越大而孔压峰值越小;相同荷载时间下,桩基位移场反应因惯性效应滞后于桩周土体的应力场.
2)当桩周饱和土渗透系数时,孔隙水呈封闭状态;而渗透系数时,孔隙水处于敞开状态;这两种情况下,渗透系数变化对桩顶位移不会产生明显影响. 当渗透系数时,孔隙率越大,渗透系数对桩顶位移影响越显著.
3)桩周饱和土渗透系数不仅影响桩周土体孔压和有效径向应力大小,而且会改变孔压和有效径向应力分布. 在渗透系数较小的饱和土体中,可以通过增大桩径和桩基模量的方式来增大桩基刚度,避免桩周土体液化导致桩基失稳.
参考文献
NOGAMI T,KONAGAI K.Time domain axial response of dynamically loaded single piles[J].Journal of Engineering Mechanics, 1986, 112(11): 1241-1252. [百度学术]
NOGAMI T,KONAGAI K.Time domain flexural response of dynamically loaded single piles[J].Journal of Engineering Mechanics,1988,114(9):1512-1525. [百度学术]
NOVAK M.Dynamic stiffness and damping of piles[J].Canadian Geotechnical Journal,1974,11(4):574-598. [百度学术]
NOVAK M,NOGAMI T.Soil-pile interaction in horizontal vibration[J].Earthquake Engineering & Structural Dynamics,1977,5(3):263-281. [百度学术]
NOVAK M,ABOUL-ELLA F.Impedance functions of piles in layered media[J].Journal of the Engineering Mechanics Division,1978,104(3):643-661. [百度学术]
NOVAK M,ABOUL-ELLA F,NOGAMI T.Dynamic soil reactions for plane strain case[J].Journal of the Engineering Mechanics Division,1978,104(4):953-959. [百度学术]
GAZETAS G,DOBRY R.Horizontal response of piles in layered soils[J].Journal of Geotechnical Engineering,1984,110(1):20-40. [百度学术]
尚守平,余俊,王海东,等.饱和土中桩水平振动分析[J]. 岩土工程学报, 2007, 29(11): 1696-1702. [百度学术]
SHANG S P,YU J,WANG H D,et al. Horizontal vibration of piles in saturated soil[J]. Chinese Journal of Geotechnical Engineering, 2007, 29(11): 1696-1702.(in Chinese) [百度学术]
余俊, 尚守平,李忠,等.饱和土中端承桩水平振动动力响应分析[J].岩土工程学报,2009,31(3):408-415. [百度学术]
YU J,SHANG S P,LI Z,et al.Dynamical characteristics of an end bearing pile embedded in saturated soil under horizontal vibration[J]. Chinese Journal of Geotechnical Engineering, 2009,31(3): 408-415.(in Chinese) [百度学术]
余俊,尚守平,李忠,等.饱和土中桩水平振动引起土层复阻抗分析研究[J].岩土力学, 2009, 30(12): 3858-3864. [百度学术]
YU J,SHANG S P,LI Z,et al.Study of resistance factor of saturated soil caused by horizontal vibration of pile[J].Rock and Soil Mechanics, 2009, 30(12): 3858-3864.(in Chinese) [百度学术]
余俊,尚守平,黄娟,等.Biot动力固结方程简化模型在桩水平动力响应中适用性研究[J].岩土工程学报,2014,36(8):1558-1563. [百度学术]
YU J,SHANG S P,HUANG J,et al.Applicability of simplified model of Biot’s dynamic consolidation equation to response of horizontal vibration of piles[J].Chinese Journal of Geotechnical Engineering,2014,36(8):1558-1563.(in Chinese) [百度学术]
刘林超,闫启方,闫盼.考虑三维波动的饱和土中管桩群桩的水平振动研究[J].岩土力学,2017,38(10):2817-2825. [百度学术]
LIU L C,YAN Q F,YAN P. Horizontal vibration of pipe pile groups in saturated soil considering three-dimensional wave effects[J]. Rock and Soil Mechanics,2017,38(10):2817-2825.(in Chinese) [百度学术]
HU A F,FU P,XIA C Q,et al.Lateral dynamic response of a partially embedded pile subjected to combined loads in saturated soil[J].Marine Georesources & Geotechnology,2017,35(6):788-798. [百度学术]
赵密, 黄义铭, 王丕光, 等. 桩顶水平动荷载作用下水-桩-土相互作用的解析解[J]. 岩土工程学报, 2022,44(5): 907-915. [百度学术]
ZHAO M,HUANG Y M,WANG P G,et al.Analytical solution for water-pile-soil interaction under horizontal dynamic loads on pile head[J].Chinese Journal of Geotechnical Engineering,2022,44(5):907-915.(in Chinese) [百度学术]
范小雪,李原,吴文兵,等.饱和土中大直径缺陷桩水平振动响应研究[J].岩石力学与工程学报,2020,39(2):413-423. [百度学术]
FAN X X, LI Y, WU W B, et al. Horizontal vibration response of defective large-diameter piles embedded in saturated soils[J].Chinese Journal of Rock Mechanics and Engineering, 2020, 39(2): 413-423.(in Chinese) [百度学术]
郑长杰, 刘汉龙, 丁选明, 等. 饱和黏性土地基中现浇大直径管桩水平振动响应解析解[J]. 岩土工程学报, 2014, 36(8):1447-1454. [百度学术]
ZHENG C J,LIU H L,DING X M,et al.Analytical solution of horizontal vibration of cast-in-place large-diameter pipe piles in saturated soils[J].Chinese Journal of Geotechnical Engineering,2014, 36(8): 1447-1454.(in Chinese) [百度学术]
郑长杰,丁选明,栾鲁宝.黏弹性地基中管桩水平动力特性分析[J]. 岩土力学, 2017, 38(1): 26-32. [百度学术]
ZHENG C J,DING X M,LUAN L B.Analysis of lateral dynamic response of pipe pile in viscoelastic soil layer[J].Rock and Soil Mechanics, 2017, 38(1): 26-32.(in Chinese) [百度学术]
HU A F,FU P,XIA C Q,et al.Horizontal impedances of saturated soil layer with radially inhomogeneous boundary zone[J]. Soil Dynamics and Earthquake Engineering,2018,111:184-192. [百度学术]
ZHENG C J, LUAN L B,QIN H Y, et al.Horizontal dynamic response of a combined loaded large-diameter pipe pile simulated by the Timoshenko beam theory[J]. International Journal of Structural Stability and Dynamics, 2020, 20(2): 2071003. [百度学术]
章敏, 王星华, 冯国瑞. 非饱和土中端承桩水平振动特性研 究[J].岩土力学,2015,36(2):409-422. [百度学术]
ZHANG M,WANG X H,FENG G R.Horizontal vibration of an end-bearing pile in unsaturated soil[J].Rock and Soil Mechanics, 2015, 36(2): 409-422.(in Chinese) [百度学术]
郭诚.SH波作用下非饱和土中管桩水平动力特性的理论及试验研究[D]. 太原: 太原理工大学, 2018: 27-28. [百度学术]
GUO C. Theoretical and experimental study on horizontal dynamic characteristics of pipe pile in unsaturated soil under SH wave[D].Taiyuan:Taiyuan University of Technology,2018:27-28 (in Chinese). [百度学术]
杨紫健, 吴文兵, 张云鹏, 等. 考虑竖向荷载影响的非饱和 地基中桩的水平振动[J]. 岩石力学与工程学报, 2022, 41 (增刊1): 2979-2990. [百度学术]
YANG Z J,WU W B,ZHANG Y P,et al.Horizontal vibration of piles in unsaturated foundation considering vertical load[J].Chinese Journal of Rock Mechanics and Engineering, 2022, 41(Sup.1): 2979-2990.(in Chinese) [百度学术]
MAMOON S M,BANERJEE P K.Time-domain analysis of dynamically loaded single piles[J].Journal of Engineering Mechanics, 1992, 118(1): 140-160. [百度学术]
刘圆圆, 王星华, 章敏,等.饱和土中单桩水平瞬态响应研 究[J].岩土力学, 2013, 34(9): 2699-2706. [百度学术]
LIU Y Y,WANG X H,ZHANG M,et al.Transient response of single pile under horizontal load in saturated soil[J].Rock and Soil Mechanics, 2013, 34(9): 2699-2706.(in Chinese) [百度学术]
尚微.饱和土中管桩水平瞬态动力特性研究[D].太原: 太原理工大学, 2018. [百度学术]
SHANG W.Study on horizontal transient dynamic characteristics of pipe pile in saturated soil[D].Taiyuan:Taiyuan University of Technology, 2018.(in Chinese) [百度学术]
赵仓龙,章敏,李亚楠,等.层状饱和土中管桩瞬态扭转振动研究[J].应用力学学报, 2022, 39(3): 516-526. [百度学术]
ZHAO C L,ZHANG M,LI Y N,et al.Transient torsional response of a pipe pile embedded in layered saturated soil[J].Chinese Journal of Applied Mechanics, 2022, 39(3): 516-526.(in Chinese) [百度学术]
CRUMP K S. Numerical inversion of Laplace transforms using a Fourier series approximation[J]. Journal of the ACM, 1976, 23(1):89-96. [百度学术]