工程地质学报  2018, Vol. 26 Issue (5): 1311-1317   (1943 KB)    
基于湛江组结构性黏土的桩土接触面模型及解析解
汤斌, 殷建风, 费建武    
① 桂林理工大学, 土木与建筑工程学院 桂林 541004;
② 武汉科技大学, 城市建设学院 武汉 430065
摘要:湛江组结构性黏土因其特有的结构特性而使该土层中桩基承载性状复杂,传统的桩基设计计算理论和方法应用于该土层中桩基承载力计算时其理论计算值与工程实测值或多或少存在一定误差而存在适用性问题。基于湛江组结构性黏土直剪试验及无侧限抗压强度试验,在剪切位移法的基础上,提出了桩土接触面软化模型,即桩土接触面处于弹性阶段时,桩侧摩阻力随位移线性增加;桩土接触面处于软化阶段时,桩侧摩阻力随位移线性减小,且减小速率与土体灵敏度、桩土接触面法向压力有关;桩土接触面处于滑移阶段时,桩侧摩阻力不随位移变化。依据该模型推导了湛江组结构性黏土中竖向荷载作用下单桩位移及轴力解析解,并比较了计算与试验结果,验证了所提模型的合理性。
关键词湛江组结构性黏土    桩基础    竖向承载力    剪切位移法    土体灵敏度    
PILE-SOIL INTERFACE MODEL AND ANALYTICAL SOLUTION BASED ON STRUCTURAL CLAY OF ZHANJIANG GROUP
TANG Bin, YIN Jianfeng, FEI Jianwu    
① College of Civil Engineering and Architecture, Guilin University of Technology, Guilin 541004;
② School of Urban Construction, Wuhan University of Science and Technology, Wuhan 430065
Abstract: Due to the unique structural property of the structural clay of Zhanjiang group, the bearing capacity of pile is complex in the clay. If the traditional pile foundation design calculation theory and method are applied to calculate the bearing capacity of the pile foundation, the deviation between the theoretical calculation value and the measured value of the pile is more or less. Based on direct shear test and unconfined compression strength test of Zhanjiang group structural clay, as well as the shear displacement method, the model of softening of pile-clay interface is put forward. When the pile-soil interface is in the elastic stage, the side friction of pile increases linearly with displacement. In the stage of softening, the side friction of pile decreases linearly with the displacement. The reduction rate is related to the soil sensitivity and normal pressure of the pile-soil interface. In the stage of sliding, the side friction of pile can not change with the displacement. Therefore, the displacement and axial force analysis of single pile under the vertical load in Zhanjiang group structural clay are derived according to the proposed model. Furthermore, the calculation and test results are compared and the feasibility of the proposed method is verified.
Key words: Zhanjiang group structural clay    Pile foundation    Vertical bearing capacity    Shear displacement method    Soil sensitivity    

0 引言

桩基础因其承载力高、沉降小、适用范围广等优点成为土木工程中应用广泛的一种基础形式,其设计计算理论也随着桩基实践的发展而发展。在竖向受荷桩理论研究方面,Seed et al. (1957)首次提出了荷载传递法,Coyle et al. (1966)运用荷载传递法对竖向受荷单桩荷载传递机理进行了研究。Cooke(1974)针对桩基的沉降及桩土荷载传递问题首先提出了剪切位移法,认为桩基沉降的主要因素为桩周土体的剪切变形。吕军(2016)通过单桩模型试验验证了剪切位移法的可行性。Kraft et al. (1981)推导了非线性桩周土体的单桩承载解析解。聂更新等(2005)用线性弹性和塑性线性强化模型模拟桩周土体应力-应变非线性关系,将剪切位移法由弹性阶段推广到了塑性阶段,建立了广义剪切位移法理论。陈洪胜(2005)用三折线模型模拟了桩周土体的非线性,用广义剪切位移法研究了两桩相互作用关系。江杰等(2008)对于桩端采用线性的荷载传递函数,运用剪切位移法推导了基于弹塑性模型的单桩竖向荷载沉降的解析解。李波等(2013)推导了单桩及桩帽下土体的荷载传递矩阵,提出了层状地基中带帽单桩的等效剪切位移法。张瑞坤等(2013)利用一维杆系结构有限单元法与剪切位移法相耦合的混合法分析了大直径超长灌注桩竖向受荷单桩沉降问题。湛江组结构性黏土因其特有的结构特性而使该土层中桩基承载性状复杂,上述桩基设计计算理论和方法应用于计算该土层中桩基承载力时其理论计算值与工程实测值或多或少存在一定误差而存在适用性问题。本文依据湛江组结构性黏土直剪试验数据和剪切位移法,提出了桩土接触面软化模型,用以模拟湛江组结构性黏土桩基桩周土体从完好到逐渐破坏直至残余强度这一过程,并基于能量守衡原理,推导出湛江组结构性黏土中桩基承载性状解析解,且通过与试验数据的对比验证了该方法的合理性。

1 侧摩阻力软化模型与桩端受力模型
1.1 侧摩阻力软化模型

为计算湛江组结构性黏土中竖向荷载作用下单桩桩身截面位移及轴力解析解,本文作如下假设:

桩周土处于弹性阶段时,侧摩阻力随桩身截面位移线性增加,线性增加斜率定义为k1;桩周土处于软化阶段时,侧摩阻力随桩身截面位移线性减少,线性减少斜率定义为k2;桩周土处于滑移阶段时,侧摩阻力不变(图 1)。

图 1 桩身侧摩阻力-位移模型 Fig. 1 Friction-displacement model of pile

对于斜率k1,可引入传统剪切位移法确定:

$ {k_1} = G/\left[ {{r_0}\ln \left( {{r_m}/{r_0}} \right)} \right] $ (1)

式中,G为土体剪切模量(MPa);rm为桩顶荷载影响的桩周土半径,且Randolph et al. (1979)指出,rm=2.5L(1-υ)ρ,单位(m);υ为土体泊松比;L为桩长(m);ρL/2深度的剪切模量与桩底处土的剪切模量之比。

对于斜率k2(图 2),取桩体埋深z处微段,并编号为0单元,取0号单元相临环形土体微段,并编号为1单元,取1号单元相临环形土体微段,并编号为2单元,对单元0、1、2接触面受力分析,如图 2右下角。F12表示单元2对单元1的剪力,F21表示单元1对单元2的剪力,其他依此类推。不难看出,F12F21是一对相互作用力,其大小相等,方向相反,作用在一条直线上;F01F10也是一对相互作用力;不计土单元重力,F10F12是一对平衡力,由于土单元1静力平衡,所以F10=F12,由于F10F01也是一对相互作用力,所以F01=F12F01为桩周土体对桩单元的阻力,即侧摩阻力,F12为土体单元间的剪力。当土单元非常小,F01F12位置接近时,桩周土体剪应力等于侧摩阻力,设土体处于软化阶段时剪应力随位移的软化斜率为k2,则有k2=k2

图 2 桩周土体微段受力分析 Fig. 2 Stress analysis of soil around pile

考虑土体结构性及桩侧表面法向压力对桩侧摩阻力软化阶段的影响,对土体进行无侧限抗压强度试验和不同垂直压力下自然放置方向竖直破坏面固结不排水直剪试验(桩侧表面法向压力σx对应于直剪土样垂直压力p,即σx=p),研究土体灵敏度St、直剪土样垂直压力p与软化斜率k2的数量相关性。

在广东省湛江市东海岛宝钢湛江钢铁项目场地内不同位置取3组湛江组结构性黏土原状土试样(分别命名为A、B、C),取土深度分别为6.25~7.05 m(A)、40.25~41.05 m(B)、19.25~20.05 m(C),其中,土样C的钻孔位置经纬度为北纬021°03′ 50.160 531″,东经110°30′ 33.692 196″。采用应变控制式直剪仪,使每组土样在0.1 MPa、0.2 MPa、0.4 MPa、0.8 MPa垂直压力下自然放置方向竖直面剪坏,得到土体剪应力τ与剪位移Δl非线性关系如图 3所示。由图 3可知,湛江组结构性黏土为应变软化型黏土,剪应力皆随剪位移的增大呈先增大后减小最后趋于稳定状态。

图 3 不同垂直压力下剪应力-剪位移曲线 Fig. 3 Shear stress and shear displacement curves of soil under different normal stresses a.土A;b.土B;c.土C

图 3中剪应力-剪位移曲线软化阶段进行线性拟合,得到侧摩阻力软化斜率(表 1)。

表 1 软化斜率k2、土样垂直压力p一览表 Table 1 Softening slope k2 and vertical pressure of soil

对每组土样原状土和重塑土进行无侧限抗压强度试验,可得土样A、B、C灵敏度(表 2)。

表 2 土体灵敏度表 Table 2 Soil sensitivity table

依据表 1表 2中数据,基于MATLAB拟合工具,以灵敏度Stx轴,以土样垂直压力py轴,以软化斜率k2z轴,进行空间拟合。拟合相关系数R2=0.9166,说明斜率k2与土体灵敏度St和土样垂直压力p之间具有一定数量关系,拟合结果(图 4),拟合公式如下:

图 4 系数k2Stp拟合结果 Fig. 4 The fitting results of pile of k2Stp

$ {k_2} = \frac{a}{{p + b}}{S_t} $ (2)

式中,a=-4.849 01 MPa2·m-1b=0.367 58 MPa,a/b为垂直压力为0时单位灵敏度软化斜率(MPa·m-1);p为直剪土样垂直压力(MPa)。

假设自然放置方向竖直破坏面直剪试验垂直压力p等效为桩土接触面法向压力,式(2)又可写成:

$ {k_2} = \frac{a}{{{\sigma _x} + b}}{S_t} $ (3)

式中,σx为桩侧表面法向压力(MPa)。

若桩周土体先后进入弹性、软化和滑移阶段,与之相对应,桩土接触面侧摩阻力-桩身截面位移曲线先后进入线性增加阶段、线性减小阶段和线性不变阶段,其数学表达式如下:

$ {\tau _z} = \left\{ \begin{array}{l} {k_1}{w_z},{w_z} \le {w_1};\\ {k_1}{w_1} + {k_2}\left( {{w_z} - {w_1}} \right),{w_1} < {w_z} \le {w_2}\\ {\tau _2},{w_2} < {w_z} \end{array} \right. $ (4)

式中,τz为埋深z处侧摩阻力(MPa);k1=G/[r0ln(rm/r0)](MPa·m-1);G为土体弹性阶段剪切模量(MPa);r0为桩身半径(m);rm为受桩顶荷载影响的桩周土半径(m);k2=aSt/(σx+b)(MPa·m-1);a=-4.84901 MPa2·m-1b=0.367 58 MPa,a/b为垂直压力为0时单位灵敏度软化斜率(MPa·m-1);σx为桩侧表面法向压力。wz为埋深z处桩身截面位移(m);w1w2为侧摩阻力软化阶段开始点、结束点位移(m),可由试验数据反分析得到;τ2为残余侧摩阻力(MPa)。

1.2 桩端阻力模型

可采用双折线模型模拟桩端阻力与桩端位移间的关系(张乾青等,2012)。双折线模型示意图(图 5)。其数学表达式为:

图 5 桩端阻力双折线模型 Fig. 5 Double-line model of pile tip resistance

$ \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {Q_b} = {\lambda _1}{s_b}\\ {Q_b} = {\lambda _1}{s_{b1}} + {\lambda _2}\left( {{s_b} - {s_{{b_1}}}} \right) \end{array}&\begin{array}{l} {s_b} \le {s_{b1}}\\ {s_b} > {s_{b1}} \end{array} \end{array}} \right. $ (5)

式中,Qb为桩端阻力(MN);sb为桩端位移(m);λ1为弹性阶段桩端法向刚度系数(MN·m-1),其值可按Randolph的建议,λ1=2.35Esr0/(1-υ2)(Randolph et al., 1979);Es为土体压缩模量(MPa);r0为桩身半径(m);υ为土体泊松比;λ2塑性阶段桩端法向刚度系数(MN·m-1),其值可近似通过现场实测荷载沉降曲线反分析得到;sb1为桩端弹性阶段到塑性阶段的界限位移(m),可由试验数据反分析得到。

2 控制方程的建立与求解

由于桩顶承受较小荷载时,桩身截面位移和轴力随深度的增加而减小,当桩顶荷载逐渐增大后,桩周土体由浅至深依次进入弹性、软化和滑移阶段。图 6为竖向荷载作用下桩周土体进入滑移阶段后受力状态分析图。设桩顶荷载为Q,沉降为s,桩端所受土体提供的反力为Qb,沉降为sbL1为桩周土体弹性区与软化区接触面,L2为桩周土体软化区与滑移区接触面,桩体埋深z处桩身截面轴力为Qz,位移为wz。由静力平衡和狭义胡克定律可得桩土接触面荷载传递微分控制方程:

图 6 桩身受力分析 Fig. 6 Stress analysis of body-pile

$ \frac{{{{\rm{d}}^2}{w_z}}}{{{\rm{d}}{z^2}}} = \frac{U}{{WA}}{\tau _z} $ (6)

式中,E为桩身弹性模量(MPa);A为桩的横截面积(m2);U为桩的周长(m)。

2.1 桩周土全部处于弹性阶段

桩周土处于弹性阶段时,桩土接触面无相对位移产生,侧摩阻力τz=k1wz,将其代入式(6),可得通解为:

$ \left\{ \begin{array}{l} {w_z} = {C_1}\cosh \left[ {{m_1}\left( {L - z} \right)} \right] + {C_2}\sinh \left[ {{m_1}\left( {L - z} \right)} \right]\\ {Q_z} = {m_1}EA\left[ {{C_1}\sinh \left[ {{m_1}L - z} \right)} \right] + {C_2}\cosh \left. {\left[ {{m_1}\left( {L - z} \right)} \right]} \right] \end{array} \right. $ (7)

式中,${m_1} = \sqrt {U{k_1}/EA} $ (m-1);C1C2为通解未知系数。

将边界条件wz=L=sbQz=L=Qb代入式(7)可得:C1=sb(m);C2=Qb/(m1EA)(m)。

C1C2代入式(7),可得式(7)矩阵形式为:

$ \left[ {\begin{array}{*{20}{c}} {{w_z}}\\ {{Q_z}} \end{array}} \right] = {T_1}\left( z \right)\left[ {\begin{array}{*{20}{c}} {{s_b}}\\ {{Q_b}} \end{array}} \right] $ (8)

式中

$ {T_1}\left( z \right) = \left[ {\begin{array}{*{20}{c}} \begin{array}{l} \cosh \left[ {{m_1}\left( {L - z} \right)} \right]\\ {m_1}EA\sinh \left[ {{m_1}\left( {L - z} \right)} \right] \end{array}&\begin{array}{l} \frac{1}{{{m_1}EA}}\sinh \left[ {{m_1}\left( {L - z} \right)} \right]\\ \cosh \left[ {{m_1}\left( {L - z} \right)} \right] \end{array} \end{array}} \right] $

z=0,则桩顶位移及沉降为:

$ \left[ \begin{array}{l} s\\ Q \end{array} \right] = {T_1}\left( 0 \right)\left[ \begin{array}{l} {s_b}\\ {Q_b} \end{array} \right] $ (9)

由于此阶段桩周土体全部处于弹性阶段,所以,此阶段桩端土体也处于弹性阶段,桩端土体刚度系数为λ1,桩端阻力为:

$ {Q_b} = {\lambda _1}{s_b} $ (10)

联立式(9)、式(10)可得:

$ Q = \frac{{{m_1}EA\sinh \left( {{m_1}L} \right) + {\lambda _1}\cosh \left( {{m_1}L} \right)}}{{\cosh \left( {{m_1}L} \right) + {\lambda _1}{{\left( {{m_1}EA} \right)}^{ - 1}}\sinh \left( {{m_1}L} \right)}}s $ (11)

2.2 桩周土部分进入软化阶段

桩周土体处于软化阶段时,桩土接触面无相对位移产生,侧摩阻力τz=k1w1+k2(w-w1),将其代入式(6),可得通解为:

$ \left\{ \begin{array}{l} {w_z} = {C_1}\cos \left[ {{m_2}\left( {{L_1} - z} \right)} \right] + {C_2}\sin \left[ {{m_2}\left( {{L_1} - z} \right)} \right] - {w_1}\left( {{k_1} - {k_2}} \right)/{k_2}\\ {Q_z} = - {m_2}EA\left[ {{C_1}\sin \left[ {{m_2}\left( {{L_1} - z} \right)} \right] - {C_2}\cos \left[ {{m_2}\left( {{L_1} - z} \right)} \right]} \right] \end{array} \right. $ (12)

式中,${m_2} = \sqrt { - U{k_2}/EA} $ (m-1);C1C2为通解未知系数。

将边界条件wz=L1=sL1Qz=L1=QL1代入式(12)可得:C1=sL1+w1(k1-k2)/k2(m);C2=QL1/(m2EA) (m);

C1C2代入式(12),可得式(12)矩阵形式为:

$ \left[ {\begin{array}{*{20}{c}} {{w_z}}\\ {{Q_z}} \end{array}} \right] = {T_2}\left( z \right)\left[ {\begin{array}{*{20}{c}} {{s_{{L_1}}}}\\ {{Q_{{L_1}}}} \end{array}} \right] + {N_2}\left( z \right) $ (13)

式中

$ {T_2}\left( z \right) = \left[ {\begin{array}{*{20}{c}} \begin{array}{l} \cos \left[ {{m_2}\left( {{L_1} - z} \right)} \right]\\ - {m_2}EA\sin \left[ {{m_2}\left( {{L_1} - z} \right)} \right] \end{array}&\begin{array}{l} \frac{1}{{{m_2}EA}}\sin \left[ {{m_2}\left( {{L_1} - z} \right)} \right]\\ \cos \left[ {{m_2}\left( {{L_1} - z} \right)} \right] \end{array} \end{array}} \right] $

$ {N_2}\left( z \right) = \frac{{{w_1}\left( {{k_1} - {k_2}} \right)}}{{{k_2}}}\left[ \begin{array}{l} - 1 + \cos \left[ {{m_2}\left( {{L_1} - z} \right)} \right]\\ - {m_2}EA\sin \left[ {{m_2}\left( {{L_1} - z} \right)} \right] \end{array} \right] $

z=0,则桩顶位移及沉降为:

$ \left[ {\begin{array}{*{20}{c}} s\\ Q \end{array}} \right] = {T_2}\left( 0 \right)\left[ {\begin{array}{*{20}{c}} {{s_{{L_1}}}}\\ {{Q_{{L_1}}}} \end{array}} \right] + {N_2}\left( 0 \right) $ (14)

L1zL时,桩身z处土体处于弹性阶段,桩侧摩阻力τz=k1w,桩体埋深z=L1处截面位移及轴力按式(8)计算:

$ \left[ {\begin{array}{*{20}{c}} {{s_{{L_1}}}}\\ {{Q_{{L_1}}}} \end{array}} \right] = {T_1}\left( {{L_1}} \right)\left[ {\begin{array}{*{20}{c}} {{s_b}}\\ {{Q_b}} \end{array}} \right] $ (15)

联立式(14)、式(15),可得:

$ \left[ {\begin{array}{*{20}{c}} s\\ Q \end{array}} \right] = {T_2}\left( 0 \right){T_1}\left( {{L_1}} \right)\left[ {\begin{array}{*{20}{c}} {{s_b}}\\ {{Q_b}} \end{array}} \right] + {N_2}\left( 0 \right) $ (16)

对于L1的确定,sb < w1时,L1可根据连续条件(弹性区的结束位置即为软化区的开始位置),由下式经MATLAB中solve函数确定:

$ \begin{array}{l} {s_i} = {w_1} = {s_b}\cosh \left[ {{m_1}\left( {L - {L_1}} \right)} \right] + {Q_b}{\left( {{m_1}EA} \right)^{ - 1}} \cdot \\ \;\;\;\;\;\;\sinh \left[ {{m_1}\left( {L - {L_1}} \right)} \right]\\ \left[ {\begin{array}{*{20}{c}} s\\ Q \end{array}} \right] = {T_2}\left( 0 \right){T_1}\left( {{L_1}} \right)\left[ {\begin{array}{*{20}{c}} 1\\ {{\lambda _1}} \end{array}} \right]{s_b} + {N_2}\left( 0 \right) \end{array} $ (17)

上式中存在Qssb3个未知数,根据等式左右矩阵相等可列出两个等式,待定一系列sb可确定Q-s关系。

sbw1时,L1=L。其Q-s关系如下:

(1) 当桩端阻力处于弹性阶段时,其Q-s关系同式(17)。

(2) 当桩端阻力处于塑性阶段时,其Q-s关系为:

$ \left[ {\begin{array}{*{20}{c}} s\\ Q \end{array}} \right] = {T_2}\left( 0 \right){T_1}\left( {{L_1}} \right)\left[ {\begin{array}{*{20}{c}} 1\\ {{\lambda _1}{s_{{b_1}}} + {\lambda _2}\left( {{s_b} - {s_{{b_1}}}} \right)} \end{array}} \right]{s_b} + {N_2}\left( 0 \right) $ (18)

2.3 桩周土部分进入滑移阶段

桩周土体处于滑移阶段时,桩土接触面产生相对位移,侧摩阻力τz=τ2,将其代入式(6),可得通解为:

$ \left\{ \begin{array}{l} {w_z} = {C_1}{\left( {{L_2} - z} \right)^2} + {C_2}\left( {{L_2} - z} \right) + {C_3}\\ {Q_z} = EA\left[ {2{C_1}\left( {{L_2} - z} \right) + {C_2}} \right] \end{array} \right. $ (19)

式中,C1C2C3为通解未知系数。

将边界条件wz=L2=sL2Qz=L2=QL2代入式(19)可得:C2=QL2/(EA)(m);C3=sL2(m)。

由等式wz=2/EA可得:

$ {C_1} = U{\tau _2}/\left( {2EA} \right)\left( {\rm{m}} \right) $

C1C2C3代入式(19),得式(19)的矩阵形式为:

$ \left[ {\begin{array}{*{20}{c}} {{w_z}}\\ {{Q_z}} \end{array}} \right] = {T_3}\left( z \right)\left[ {\begin{array}{*{20}{c}} {{s_{{L_1}}}}\\ {{Q_{{L_1}}}} \end{array}} \right] + {N_3}\left( z \right) $ (20)

式中

$ {T_3}\left( z \right) = \left[ {\begin{array}{*{20}{c}} 1&{\frac{{{L_2} - z}}{{EA}}}\\ 0&1 \end{array}} \right] $

$ {N_3}\left( z \right) = U{\tau _1}\left[ \begin{array}{l} \frac{{{{\left( {{L_2} - z} \right)}^2}}}{{2EA}}\\ {L_2} - z \end{array} \right] $

z=0,则桩顶位移及沉降为:

$ \left[ {\begin{array}{*{20}{c}} s\\ Q \end{array}} \right] = {T_3}\left( 0 \right)\left[ \begin{array}{l} {s_{{L_2}}}\\ {Q_{{L_2}}} \end{array} \right] + {N_3}\left( 0 \right) $ (21)

L2z < L1时,桩周z处土体处于软化阶段,τz=k1w1+k2(w-w1),桩体埋深z=L2处截面位移及轴力按式(13)计算:

$ \left[ \begin{array}{l} {s_{{L_2}}}\\ {Q_{{L_2}}} \end{array} \right] = {T_2}\left( {{L_2}} \right)\left[ \begin{array}{l} {s_{{L_1}}}\\ {Q_{{L_1}}} \end{array} \right] + {N_2}\left( {{L_2}} \right) $ (22)

L1+L2zL时,桩身z处土体处于弹性阶段,τz=k1w,桩体埋深z=L1处截面位移及轴力按式(8)计算:

$ \left[ \begin{array}{l} {s_{{L_1}}}\\ {Q_{{L_1}}} \end{array} \right] = {T_1}\left( {{L_1}} \right)\left[ \begin{array}{l} {s_b}\\ {Q_b} \end{array} \right] $ (23)

联立式(21)、式(22)、式(23)可得:

$ \begin{array}{l} \left[ \begin{array}{l} s\\ Q \end{array} \right] = {T_3}\left( 0 \right){T_2}\left( {{L_2}} \right){T_1}\left( {{L_1}} \right)\left[ \begin{array}{l} {s_b}\\ {Q_b} \end{array} \right] + \\ \;\;\;\;\;\;\;\;\;\;\;\;{T_3}\left( 0 \right){N_2}\left( {{L_2}} \right) + {N_3}\left( 0 \right) \end{array} $ (24)

对于L2的确定,当sb < w2时,L2可根据连续条件(软化区的结束位置即为滑移区的开始位置),由下式经MATLAB中solve函数确定:

$ \begin{array}{l} {s_j} = {w_2} = \left( {{s_{{L_1}}} + {w_1}\left( {{k_1} - {k_2}} \right)/{k_2}} \right)\cos \left[ {{m_2}\left( {{L_1} - {L_2}} \right)} \right] + \\ \;\;\;\;\;\;\left( {{Q_{{L_1}}}/\left( {{m_2}EA} \right)} \right)\sin \left[ {{m_2}\left( {{L_1} - {L_2}} \right)} \right] - {w_1}\left( {{k_1} - {k_2}} \right)/{k_2} \end{array} $

其中,sL1QL1由式(15)确定。

$ \begin{array}{l} \left[ \begin{array}{l} s\\ Q \end{array} \right] = {T_3}\left( 0 \right){T_2}\left( {{L_2}} \right){T_1}\left( {{L_1}} \right)\left[ {\begin{array}{*{20}{c}} 1\\ {{\lambda _1}} \end{array}} \right]{s_b} + \\ \;\;\;\;\;\;\;\;\;\;\;\;{T_3}\left( 0 \right){N_2}\left( {{L_2}} \right) + {N_3}\left( 0 \right) \end{array} $ (25)

sbw2时,L2=L1=L,此时桩周受力全部进入滑移阶段。其Q-s关系如下:

(1) 当桩端阻力处于弹性阶段时,其Q-s关系同式(25)。

(2) 当桩端阻力处于塑性阶段时,其Q-s关系为:

$ \begin{array}{l} \left[ {\begin{array}{*{20}{c}} s\\ Q \end{array}} \right] = {T_3}\left( 0 \right){T_2}\left( {{L_2}} \right){T_1}\left( {{L_1}} \right)\left[ {\begin{array}{*{20}{c}} 1\\ {{\lambda _1}{s_{b1}} + {\lambda _2}\left( {{s_b} - {s_{b1}}} \right)} \end{array}} \right]{s_b}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + {T_3}\left( 0 \right){N_2}\left( {{L_2}} \right) + {N_3}\left( 0 \right) \end{array} $ (26)

3 方法验证与应用

为了验证本文所提方法的合理性,将上述所提解析解与湛江组结构性黏土中模型试验(汤斌等,2016)结果进行对比分析。试验过程中,桩长为0.6 m、1.2 m和1.8 m,桩径均为0.03 m,弹性模量均取E=2.06×105 MPa。土体压缩模量Es=13.4 MPa,灵敏度St=1.6,泊松比经验值取υ=0.35,剪切模量G=Es/(2(1+υ))=5.0 MPa。桩侧表面法向压力σx=9cu=0.550 08 MPa,λ1=2.35Esr0/(1-ν2)=0.394464 MN·m-1,其他参数如表 3w1w2λ2sb1由试验数据反分析得到,Q-s计算结果如图 7所示,计算公式如式(24)。

表 3 参数表 Table 3 Parameter table

图 7 桩顶竖向荷载-沉降曲线对比分析 Fig. 7 Vertical load-settlement curve analysis of top-pile

式(24)中,当桩周土全为弹性阶段时,L1=L2=0,上式可化简后为式(9);当桩周土体部分进入软化阶段时,L2=0,上式可化简为式(16)。

湛江组结构性黏土中单桩极限承载力计算值与实测值平均误差为:

$ \begin{array}{l} \bar \delta = \left( {\left( {3.73 - 3.6} \right)/3.6 + \left( {6.22 - 5.52} \right)/5.52} \right.\\ \;\;\;\;\;\;\left. { + \left( {8.12 - 7.59} \right)/7.59} \right)/3 \times 100\% = 7.76\% \end{array} $

4 结论

(1) 假设桩周土处于弹性阶段时,侧摩阻力随桩身截面位移线性增加,线性增加斜率定义为k1;桩周土处于软化阶段时,侧摩阻力随桩身截面位移线性减少,线性减少斜率定义为k2;桩周土处于滑移阶段时,侧摩阻力不变。根据以上假设,并基于一定的边界条件和连续条件,利用力的平衡及胡克定律建立竖向受荷单桩微分控制方程,推导出湛江组结构性黏土中桩基位移轴力解析解。

(2) 考虑湛江组结构性黏土中桩基承载时侧摩阻力的软化现象,建立了软化斜率k2与土体灵敏度和桩侧表面法向压力间的数量关系。

(3) 用本文方法计算得到的湛江组结构性黏土中单桩极限承载力与湛江组结构性黏土中桩基模型试验所得单桩极限承载力实测值平均误差为7.76%。

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