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越顶破坏模式下沉埋桩受荷段及沉埋段推力算法

闫玉平

闫玉平. 越顶破坏模式下沉埋桩受荷段及沉埋段推力算法[J]. 水文地质工程地质,2023,50(3): 76-84. DOI: 10.16030/j.cnki.issn.1000-3665.202211007
引用本文: 闫玉平. 越顶破坏模式下沉埋桩受荷段及沉埋段推力算法[J]. 水文地质工程地质,2023,50(3): 76-84. DOI: 10.16030/j.cnki.issn.1000-3665.202211007
YAN Yuping. Calculation method of thrust force of the embedded stabilizing piles under the overtop-sliding failure mode[J]. Hydrogeology & Engineering Geology, 2023, 50(3): 76-84. DOI: 10.16030/j.cnki.issn.1000-3665.202211007
Citation: YAN Yuping. Calculation method of thrust force of the embedded stabilizing piles under the overtop-sliding failure mode[J]. Hydrogeology & Engineering Geology, 2023, 50(3): 76-84. DOI: 10.16030/j.cnki.issn.1000-3665.202211007

越顶破坏模式下沉埋桩受荷段及沉埋段推力算法

基金项目: 国家自然科学基金项目(41831290)
详细信息
    作者简介:

    闫玉平(1989-),男,博士,工程师,主要从事铁路设计、边坡稳定性分析、边坡加固等方面的研究工作。E-mail:2420907827@qq.com

  • 中图分类号: P642.23

Calculation method of thrust force of the embedded stabilizing piles under the overtop-sliding failure mode

  • 摘要:

    滑坡推力的确定对于抗滑桩设计极其重要,沉埋桩作为对传统抗滑桩的优化,其受荷段推力的研究目前主要借助于模型试验和数值模拟,缺乏深入的理论分析。为了建立沉埋桩后侧受荷段及桩顶沉埋段滑坡推力计算方法,针对沉埋桩加固的基岩-覆盖层式滑坡,基于潜在越顶破坏模式,由桩顶位置将越顶滑面分为顶部、底部两段,其中,顶部滑面上水平方向合力即为沉埋段推力,可由积分求得,底部滑面上各个方向的力求解方法与此类似;在此基础上,利用刚体极限平衡理论对底部滑面与桩受荷段所围滑体进行受力分析,进而可得受荷段推力计算公式。实例分析表明:理论算法所得沉埋段与受荷段推力值与FLAC3D结果非常接近,其中,受荷段推力、沉埋段推力、设桩位置处总推力随沉埋比的增大而分别非线性减小、增大、减小;沉埋比位于0~0.67范围内时,沉埋段与受荷段推力之比由0缓慢增大到0.30~0.50,随着沉埋比增大到0.8,该比值急剧增大到1.47~2.12;一般沉埋深度下,沉埋段推力小于受荷段推力。沉埋桩推力的理论研究对于桩体内力优化、沉埋深度确定具有重要的现实意义,将有助于该桩型的进一步推广应用。

    Abstract:

    Embedded piles act as an optimization structure compared with the traditional stabilizing pile. The determination of the thrust on the loading section is based mainly on the model test and numerical simulation, and there is a lack of in-depth theoretical analysis. For the bedrock-talus landslide reinforced by embedded piles, according to the potential overtop-sliding failure mode, the slide surface can be divided into top and bottom sections by the position of pile top, and the horizontal resultant force of the top section can be obtained by integration, which is the so-called thrust of the embedded section. Similarly, the force on the bottom section of the overtop-sliding surface can also be obtained. Based on the limit equilibrium theory, the force analysis of the sliding mass enclosed by the bottom sliding surface and the load section of the pile can be carried out, and the thrust on the loaded section can also be obtained. Example analyses show that the thrust of the embedded section and the loaded section obtained by the theoretical method are very consistent with the results of FLAC3D, the resultant force of the loading section decreases nonlinearly with the increase of the ratio, while the resultant force of the embedded section presents an opposite trend. With the increase of the embedded ratio from 0 to 0.67, the thrust ratio of embedded section and loading section increases slowly from 0 to 0.3−0.5. With the increase of embedded ratio from 0.67 to 0.8, the ratio increases sharply to 1.47−2.12. Generally, the thrust of embedded section is less than that of loaded section. The theoretical research of the thrust of the embedded pile is of great practical significance for the optimization of the pile internal force and the determination of the pile embedded depth, which will promote the further application of this structure.

  • 抗滑桩至今已有70多年的实践历史,其中运用最多的结构形式是单排桩[1-3]。随着滑(边)坡治理规模的需要及设计理论的完善,抗滑桩逐渐朝着大直径[4-5]、多排桩[6-7]的方向发展,治理方案趋于综合化[8],结构型式趋于多样化[9],抗滑桩受力趋于更加合理[10-12]

    抗滑桩加固坡体设计主要包括桩体受力分析与加固后坡体稳定性评价[13-16]。传统抗滑桩,桩顶一般位于坡面,所以桩体需要承担后侧滑体传来的全部滑坡推力,这样桩体前侧滑体抗力无法有效发挥且桩身受力偏大[617-18]。沉埋桩是在桩位不变的情况下,将桩顶埋入坡面一定深度,已有研究表明,该桩结构可以有效克服全长桩的不足[17-18]。目前针对沉埋桩桩体受力的研究主要如下:熊治文[19]通过模型试验与数值模拟,对沉埋桩受力分布规律、沉埋深度做了探讨,得出沉埋桩较全长桩受力更小的结论;雷文杰等[20-21]利用模型试验与数值模拟分析了设桩位置不变时,不同桩顶埋深所对应设桩位置的滑坡总推力和桩身内力,发现沉埋深度增加可以有效降低后侧推力与桩身内力,且沉埋深度对设桩截面处总推力影响不大;雷文杰等[22]采用有限元法对沉埋桩加固后的实际滑坡做了分析,得到沉埋段土体推力随其深度增加而增大,桩顶土体自身可以承受一定的推力,沉埋桩桩身受力较全长桩明显降低,桩身内力和桩后推力分布随沉埋深度增加变得更为合理等结果;宋雅坤等[23]结合大型物理模型试验和数值方法分析了沉埋段和受荷段滑坡推力,初步得出加固后坡体稳定系数随沉埋段的增加而减小,受荷段与沉埋段推力随埋深变化趋势与雷文杰等[22]的结论一致。但是针对沉埋桩受力的理论研究,目前鲜有文献报道。

    在加固后坡体稳定性评价方面,Xiao[17]在假定滑体为无黏性土的基础上,利用极限平衡理论和传递系数法得出了后排桩沉埋深度的计算方法,该方法可将桩顶埋深与坡体稳定系数建立关系,同样得出加固后坡体稳定系数随着沉埋深度的增加而降低,同时桩身内力也在减小的结论。Yan等[18]基于沉埋桩加固滑坡可能发生的越顶破坏模式,结合传递系数法与极限平衡理论框架下的变分法,推导了对黏性土和无黏性土都适用的后排桩顶沉埋深度的解析解,得出与Xiao[17]一致的结论。

    综上,目前针对加固后坡体稳定性分析已经有了较为系统全面的认识,而桩体受力特别是受荷段推力的研究主要借助于模型试验和数值模拟,缺乏深入的理论研究。鉴于此,本文在Yan等[18]所提越顶破坏模式的基础上,由桩顶位置将越顶滑面分为顶部、底部两段,通过积分求得两段滑面上合力,其中,顶部滑面上水平方向合力即为沉埋段推力;然后借助刚体极限平衡理论对底部段滑面与桩体受荷段所围滑体进行受力分析,求得受荷段推力。

    桩体后缘传来的滑坡推力,一部分作用在桩上,不妨称此部分桩体为受荷段(图1);另一部分越过桩顶滑体继续向前传递,通过前侧滑体自身稳定性提供抗力[17],桩顶距坡面深度部分不妨称之为沉埋段。

    图  1  沉埋桩加固滑坡横截面示意图
    Figure  1.  Sketch map of the cross section of a landslide reinforced with embedded pile

    Yan等[18]提出的越顶破坏模式如图2所示,其中,极坐标系下越顶滑面BDC方程ρ(θ)及其正应力函数σ(θ)分别如式(1)(2)所示[18]

    图  2  沉埋桩越顶破坏模式
    Figure  2.  Overtop-sliding failure mode of the embedded piles
    ρ(θ)=AetanφFsθ (1)
    σ(θ)=Aγ1+9(tanφFs)2etanφFsθ(sinθ3tanφFscosθ)+Be2tanφFsθctanφ (2)

    式中:θ——极角/(°);

    Fs——与越顶破坏模式对应的稳定系数;

    γ——滑体重度/(kN·m−3);

    c、φ——黏聚力/kPa、内摩擦角/(°);

    AB——积分常数。

    越顶滑面段BDDC,受荷段BDHHDE块的受力分析如图3所示。做如下假定:

    图  3  各块体受力分析模型
    Figure  3.  Mechanical analysis model of each typical rigid block

    (1)越顶滑面与桩体受荷段之间的初始滑面BHE处于极限状态且满足库仑强度准则。

    (2)初始滑面BHE上只有一个折点H,无折点情况为其特解,多折点求解方法类似一个折点。

    (3)BDHHDE块都为刚体,除越顶滑面段BD外其他边界上力作用点都位于相应作用面中点。

    下面对图3所示的越顶滑面BDC、刚性块体BDHHDE分别进行受力分析。

    1)越顶滑面BD段上合力求解

    图3(a)中越顶滑面BDC上各点处的正应力函数如式(2)所示,根据摩尔-库仑强度理论,切向应力函数τ(θ)可表示为式(3)。滑面BDC上长度增量记为dl,任意一点切向力和法向力增量分别记为dT和dN,则dl、dT和dN可分别表示为式(4)、式(5)、式(6)。

    τ(θ)=σ(θ)tanφ+c=AγtanφFs1+9(tanφFs)2etanφFsθ(sinθ3tanφFscosθ)+BtanφFse2tanφFsθ (3)
    dl=ρ(θ)cosφdθ=AetanφFsθcosφdθ (4)
    dT=τ(θ)dl=[AγtanφFs1+9(tanφFs)2etanφFsθ(sinθ3tanφFscosθ)+BtanφFse2tanφFsθ]AetanφFsθcosφdθ (5)
    dN=σ(θ)dl=[Aγ1+9(tanφFs)2etanφFsθ(sinθ3tanφFscosθ)+Be2tanφFsθctanφ]AetanφFsθcosφdθ (6)

    令滑面BDCxy方向上力的增量分别为dFx和dFy,则有:

    {dFx=dNcos(θφ)+dTsin(θφ)dFy=dNsin(θφ)dTcos(θφ) (7)

    O’点为矩心,可求得滑面BD段上的水平向合力FxBD、竖向合力FyBD及对O’点的力矩MBD,即:

    {FxBD=θDθBdFx=θDθB[σ(θ)cos(θφ)+τ(θ)sin(θφ)]ρ(θ)cosφdθFyBD=θDθBdFy=θDθB[σ(θ)sin(θφ)τ(θ)cos(θφ)]ρ(θ)cosφdθMBD=cA2θDθBe2tanφFsθdθ (8)

    2)块体BDH受力分析

    BDH块受力分析模型如图3(b)所示,B、D、H各点坐标分别记为(xByB)、(xDyD)、(xHyH),同时令BH中点坐标为(x1y1),HD中点坐标为(x2y2),BHHD长度记为lBHb。由静力平衡条件(以O’为矩心)及假定(1)可得:

    {Fx=0Fy=0MO=0T1=N1tanφmFs+cmFslBH (9)

    式中:cmϕm——原滑面BHE上的黏聚力、内摩擦角;

    T1N1——BH面上的切向力、法向力/kN。

    为简化计算,曲线BD按直线边计算,则其重力(G1)可表示为如下形式:

    G1=12γlBHbsin(πηω) (10)

    BDH重心到O’点的水平距离为d1,则可求得如下BH线方程以及经过BH中点的垂线方程:

    {yHyBxHxBxy+xHyBxByHxHxB=0xHxByHyBx+yy1xHxByHyBx1=0 (11)

    O’到线段BH及其过中点垂线的距离分别记为l1l2,其表达式如式(12)所示:

    {l1=|yHyBxHxBx0y0+xHyBxByHxHxB|(yHyBxHxB)2+1l2=|xHxByHyB(x0x1)+y0y1|(xHxByHyB)2+1 (12)

    同理,可得DH线方程及经过DH中点的垂线方程(为了节省篇幅,在此不再给出),记O’到线段DH及其过中点垂线的距离分别记为l3l4,将式(10)—(12)代入式(9),则有:

    {FxBD+N1sinηT1cosηT2cosωN2sinω=0FyBD+G1N1cosηT1sinη+T2sinωN2cosω=0MBDG1d1+N1l2T1l1T2l2N2l4=0T1=N1tanφmFs+cmFslBH (13)

    式中:η、ω——BH边、DH边与水平面的夹角/(°);

    T2N2——DH面上的切向力、法向力/kN。

    求解方程组(13),可解得各个边界上的力如式(14)所示:

    {N1=[FxBDsinω+(FyBD+G1)cosω](l3cosω+l4sinω)+[l3sinη+l1sinω(l3cosω+l4sinω)sin(ω+η)]cmFslBH(MBDG1d1)sinω+(FyBD+G1)l3l2sinωl3cosη+(l3cosω+l4sinω)[cos(ω+η)+sin(ω+η)tanφmFs]T1=N1tanφmFs+cmFslBHN2=FxBDsinω+(FyBD+G1)cosωcmFslBHsin(ω+η)[cos(ω+η)+sin(ω+η)tanφmFs]N1T2=FxBDcosω(FyBD+G1)sinωcmFslBHcos(ω+η)+[sin(ω+η)cos(ω+η)tanφmFs]N1 (14)

    3)块体HDE受力分析

    HDE块受力分析模型如图3(c)所示,记HEDE长度分别为jk。由静力平衡条件(以DE中点为矩心)及假定(1)可得式(15)。求解式(15),则可解得边界HEDE上的力如式(16)(17)所示。

    {N2cosξ+T2sinξ+N3sinδT3cosδNp=0N2sinξT2cosξN3cosδT3sinδTp+G2=0(N3sinδT3cosδ)b2cosξ(N3cosδ+T3sinδ)b2sinξ+(N2sinξT2cosξ)j2cosδ(N2cosξ+T2sinξ)j2sinδ+j3G2sin(π2δ)=0T3=N3tanφmFs+cmFsj (15)
    {N3=scmFsj+N2jtan(δξ)+T2j2G2jsin(π2δ)3cos(δξ)b[tan(δξ)tanφmFs]T3=[N2jtan(δξ)+T2j2G2jsin(π2δ)3cos(δξ)]tanφmFs+bcmFsjtan(δω)b[tan(δω)tanφmFs] (16)
    {Np=N2cosξ+T2sinξ+bcmFs+N2tan(δξ)+T22G2sin(π2δ)3cos(δξ)b[tan(δξ)tanφmFs]jsinδ[N2tan(δξ)+T22G2sin(π2δ)3cos(δξ)]tanφmFs+bcmFstan(δξ)b[tan(δξ)tanφmFs]jcosδTp=N2sinξT2cosξbcmFs+N2tan(δξ)+T22G2sin(π2δ)3cos(δξ)b[tan(δξ)tanφmFs]jcosδ[N2tan(δξ)+T22G2sin(π2δ)3cos(δξ)]tanφmFs+bcmFstan(δξ)b[tan(δξ)tanφmFs]jsinδ+G2 (17)

    式中:T3N3——HE面上切向力、法向力/kN;

    TpNp——DE面上切向力、受荷段推力/kN;

    δ——HE边与水平面的夹角/(°);

    ξ——HD边与竖向的夹角/(°);

    G2——块体HDE的重力/kN。

    4)沉埋段推力求解

    图3(a)中越顶滑面DC段水平方向合力即为沉埋段推力,记为FxDC,求解方法同滑面BD段,表达式如下:

    FxDC=θCθDdFx=θCθD[σ(θ)cos(θφ)+τ(θ)sin(θφ)]ρ(θ)cosφdθ (18)

    上述求解过程给出了越顶破坏模式下,沉埋段与受荷段推力的严格解答,该求解过程可以借助MATLAB完成[24]

    四川境内宝成铁路沿线某中风化大理岩上覆碎石土的基覆式滑坡主横断面如图4所示,通过现场地质勘查与室内试验,坡体主要物理力学参数如表1所示。利用传递系数法[25-26]求得自然坡体稳定系数为1.06。设计安全系数为1.20,拟采用沉埋桩对该坡体加固,桩体截面尺寸为2 m×3 m,平面外桩间距为5 m,设桩位置见图4,该位置滑体厚度为15.1 m。

    图  4  沉埋桩加固宝成铁路沿线某滑坡横断面图
    Figure  4.  Cross section of a practical landslide reinforced with embedded stabilizing piles
    表  1  抗滑桩及坡体主要物理力学参数
    Table  1.  Main physical and mechanical parameters of the landslide and piles
    材料类型土体重度
    /(kN·m–3
    黏聚力
    /kPa
    内摩擦角
    /(°)
    弹性模量
    /MPa
    泊松比
    碎石土221120.6500.33
    大理岩2345037.06000.25
    抗滑桩2530 0000.22
    注:抗滑桩视为弹性体,故不考虑其黏聚力和内摩擦角。
    下载: 导出CSV 
    | 显示表格

    为了便于表述,引入Yan等[18]对沉埋比(η)的定义,即:η=桩顶埋深/设桩点滑体厚度。通过Yan等[18]的方法可得,设计安全系数为1.20时,η=0.364,即桩顶最大沉埋深度为5.5 m。利用本文方法可得此沉埋深度下沉埋段推力及受荷段推力分别为439.9 kN/m和2949.6 kN/m,其比值为0.149。

    为了进一步验证本文方法的合理性,利用FLAC3D进行数值模拟。数值模型含47436个8节点六面体单元(图5),坡体采用服从摩尔-库仑屈服准则和关联流动法则的理想弹塑性本构模型,桩体视为弹性材料,利用结构单元模拟。模型前后左右4个边界采用水平位移约束,底面采用水平和竖向位移约束。采用强度折减法进行数值模拟,得到沉埋深度为5.5 m时,该滑坡的稳定系数为1.21,此时临界滑动面如图6所示,该结果与Yan等[18]的计算结果非常相近,在此不再赘述。

    图  5  沉埋桩加固滑坡数值模型
    Figure  5.  Numerical model of the practical landslide reinforced with embedded pile
    图  6  数值模拟得到的临界滑动面
    Figure  6.  Critical slip surface obtained with the shear strength reduction method

    图7所示为数值模拟得到的越顶破坏模式下滑坡的水平应力分布云图,由此易得沉埋段与受荷段推力分别为428.9 kN/m和2 788.9 kN/m,其比值为0.154。可见,数值结果与理论计算结果非常吻合,这说明了本文方法的合理性。

    图  7  沉埋桩加固滑坡的水平应力云图
    Figure  7.  Contour of horizontal stress of the reinforced landslide using the embedded pile

    下面对受荷段推力、沉埋段推力、设桩位置处总推力与沉埋比的关系逐一讨论。不同沉埋比对应不同的越顶滑面与稳定系数,为了论述的完整性,首先利用文献[18]与FLAC3D分别获得加固后坡体稳定系数(Fs)随沉埋深度的变化情况,如图8所示,可见沉埋比(η)越大,坡体稳定系数越小。

    图  8  加固后坡体稳定系数随η的变化情况
    Figure  8.  Variation of stability coefficient of reinforced landslide with the embedded ratio

    图9为理论方法与FLAC3D得出的受荷段推力及沉埋段推力随η的变化情况。可见,FLAC3D结果与理论计算结果非常接近,随着η由0增大到1,沉埋段推力由0非线性增大到2000 kN/m左右,受荷段推力则由4300 kN/m非线性减小到0,同时,η位于0.7~0.8区间时,受荷段与沉埋段推力分别急剧减小与增加。

    图  9  受荷段及沉埋段推力随η的变化情况
    Figure  9.  Variation of the thrust behind the pile and the thrust in the submerged section with the embedded ratio

    将沉埋段推力与受荷段推力叠加可得不同沉埋比时设桩位置处总推力的变化曲线(图10),可见2种方法求得的设桩位置处总推力非常接近,且变化趋势一致。η为0时,桩位处总滑坡推力明显大于η为1时的情况,前者为4350 kN/m,后者为2040 kN/m,两者比值为2.13,这可由图8解释,全长桩时,加固后坡体稳定系数最大,而无桩情况下,坡体稳定系数最小。

    图  10  设桩处总滑坡推力随η的变化情况
    Figure  10.  Variation of the total thrust at the section of pile position with the embedded ratio

    进一步将2种方法得到的沉埋段与受荷段推力之比随η的变化情况绘于图11,可见2种方法求得的比值随η非线性增大,且变化趋势非常一致。当η位于0~0.67时,沉埋段推力与桩后受荷段推力之比由0缓慢增大到0.30~0.50,当η在0.67~0.80范围内时,该比值急剧增大到1.47~2.12,此时抗滑桩作用效果已经非常小,说明了该沉埋深度在工程上需要避免。此外,图11中也标出了沉埋段推力与桩后受荷段推力之比为1时,η为0.72~0.74。综合分析可得,沉埋深度合理时,沉埋段推力小于受荷段推力。

    图  11  沉埋段、受荷段推力之比与η的关系曲线
    Figure  11.  Variation curve of thrust ratio of the submerged section and the loading section with the embedded ratio

    (1)针对沉埋桩加固基覆式滑坡,基于潜在越顶破坏模式,由桩顶位置将越顶滑面分为顶部、底部两段。对顶部滑面上的力进行积分可得沉埋段推力,利用刚体极限平衡理论对底部滑面与受荷段所围滑体进行受力分析可得受荷段推力,进而可得受荷段与沉埋段推力随桩顶埋深的变化规律、不同沉埋深度下设桩位置处滑坡推力、受荷段与沉埋段推力比值等。

    (2)理论计算所得沉埋段与受荷段推力值与FLAC3D结果非常接近,其中,受荷段推力、沉埋段推力、设桩位置处总推力随沉埋比的增大而分别非线性减小、增大、减小。

    (3)沉埋比为0和1时,桩位处总设计滑坡推力分别为4350 kN/m与2040 kN/m,前者与后者比值为2.13;沉埋比为0.7~0.8时,受荷段与沉埋段推力分别急剧减小与增加。沉埋比为0~0.67时,沉埋段与受荷段推力之比由0缓慢增大到约0.3~0.5,沉埋比由0.67增大到0.80时,该比值急剧增大到1.47~2.12。

    (4)为了充分发挥沉埋桩加固效果,一般沉埋深度下,沉埋段推力应小于受荷段推力。

  • 图  1   沉埋桩加固滑坡横截面示意图

    Figure  1.   Sketch map of the cross section of a landslide reinforced with embedded pile

    图  2   沉埋桩越顶破坏模式

    Figure  2.   Overtop-sliding failure mode of the embedded piles

    图  3   各块体受力分析模型

    Figure  3.   Mechanical analysis model of each typical rigid block

    图  4   沉埋桩加固宝成铁路沿线某滑坡横断面图

    Figure  4.   Cross section of a practical landslide reinforced with embedded stabilizing piles

    图  5   沉埋桩加固滑坡数值模型

    Figure  5.   Numerical model of the practical landslide reinforced with embedded pile

    图  6   数值模拟得到的临界滑动面

    Figure  6.   Critical slip surface obtained with the shear strength reduction method

    图  7   沉埋桩加固滑坡的水平应力云图

    Figure  7.   Contour of horizontal stress of the reinforced landslide using the embedded pile

    图  8   加固后坡体稳定系数随η的变化情况

    Figure  8.   Variation of stability coefficient of reinforced landslide with the embedded ratio

    图  9   受荷段及沉埋段推力随η的变化情况

    Figure  9.   Variation of the thrust behind the pile and the thrust in the submerged section with the embedded ratio

    图  10   设桩处总滑坡推力随η的变化情况

    Figure  10.   Variation of the total thrust at the section of pile position with the embedded ratio

    图  11   沉埋段、受荷段推力之比与η的关系曲线

    Figure  11.   Variation curve of thrust ratio of the submerged section and the loading section with the embedded ratio

    表  1   抗滑桩及坡体主要物理力学参数

    Table  1   Main physical and mechanical parameters of the landslide and piles

    材料类型土体重度
    /(kN·m–3
    黏聚力
    /kPa
    内摩擦角
    /(°)
    弹性模量
    /MPa
    泊松比
    碎石土221120.6500.33
    大理岩2345037.06000.25
    抗滑桩2530 0000.22
    注:抗滑桩视为弹性体,故不考虑其黏聚力和内摩擦角。
    下载: 导出CSV
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