Analysis method for relaxation effect of prestressed anchor cable tension based on a series rheological model
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摘要: 加固坡体的预应力锚索结构存在锚索拉力随时间松弛问题,为了合理预测分析锚索拉力松弛,基于锚索-边坡体系中滑床、锚索、滑体、坡面抑制构件之间的相互作用与锚固系统受力、变形基本机理,建立了一种采用虎克体模拟锚索或坡面抑制构件、开尔文体模拟滑床、开尔文体或广义开尔文体模拟土质或岩质滑体的锚索-滑床-滑体-坡面抑制件的四体串联式流变模型,推导了锚索拉力松弛的计算方程。实例分析表明:所提出模型的锚索拉力松弛计算值与试验或实测结果的误差小于既有模型的计算误差,计算得到的锚索拉力松弛收敛值的最大计算误差约为11%,松弛历时的最大误差约为10%;锚索拉力松弛率随锚索的直径和弹性模量的增大呈线性增大,随锚孔间距、滑床和滑体的滞后弹性模量与黏滞系数(尤其初期阶段)的增大呈非线性减弱,而滑床和滑体的瞬时弹性模量、坡面抑制构件的弹性模量均对锚索松弛效应影响很小。所建立的方法可用于定量评估预测预应力锚索加固坡体的锚拉力松弛效应,进而可用于分析锚固边坡长期稳定性。Abstract: Anchor cable tension relaxation with time is a typical problem in the prestressed anchor cable structure used to reinforce slopes. In order to reasonably predict the tension relaxation of anchor cable, based on the interaction among the stable layer, anchor cable, slide mass and constraint components on the slope face in the anchor cable-slope system and the basic loading and deformation mechanism of the anchorage system, a four-body series rheological model composed of anchor cable, slide bed, slide body, and constraint components is established, in which the anchor cable and constraint components are simulated with Hooke body, the slide mass is simulated with Kelvin body or generalized Kelvin body, and the stable layer is simulated with generalized Kelvin body. The calculation equation of the anchor cable tension relaxation is accordingly derived. Some examples show that the error between the proposed value of cable tension relaxation and the observed results is smaller than those using the existing calculation methods. The maximum error of cable tension relaxation convergence value using the proposed method is about 11%, and the maximum error of the relaxation duration is about 10%. The relaxation rate of the anchor cable tension increases linearly with the diameter and elastic modulus of the anchor cable, and decreases nonlinearly with the increasing anchor hole spacing, the hysteresis elastic modulus and the viscosity coefficient (particularly in the initial stage) of the stable layer and slide mass. The instantaneous elastic modulus of the stable layer and the slide mass as well as the elastic modulus of the constraint components have little effect on the tension relaxation of the anchor cable. The proposed method can be used to quantitatively evaluate the anchor tension relaxation of slopes reinforced with prestressed anchor cables in practical engineering, which naturally allows to analyze the long-term stability of the anchored slopes.
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Keywords:
- slope /
- prestressed anchor cable /
- anchor tension /
- relaxation effect /
- series rheological model
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预应力锚索结构是工程边坡或滑坡的一种有效加固措施[1 − 4],其锚索的拉力存在随时间松弛衰减的现象[5 − 7],锚索拉力的损失程度直接影响着预应力锚索结构加固效果和所加固边坡的长期稳定性。
已有很多学者对这一问题进行了研究。丁多文等[8]基于广义Kelvin体(K体),即K体与Hooke体(H体)的串联模型,分析了预应力锚索加固岩体的锚拉力随时间变化规律;Shi等[9]将模拟锚索的H体与模拟锚固段地层的K-H体并联,与模拟坡面框架梁的K体串联组成并-串联模型,并基于现场试验,建立了预应力锚索的预应力损失计算方法;陈安敏等[10]通过室内模型试验模拟了软岩在锚索拉力作用下的蠕变作用并讨论了锚索拉力随时间的变化规律;杨栋等[11]通过锚索单根屈服套室试验获得了3种稳定输出恒阻的屈服套,并基于室内张拉试验和极限承载试验发现增加了屈服套装置的锚索的吨位和行程有了较大提升;李涛等[12]基于室内模型试验研究了不同含水率下锚索拉力的损失规律,并建立了由H体和Bingham模型并联组成的耦合模型;李安润等[13]采用三轴压缩蠕变试验发现在不同含水率下,水-岩作用对软岩的各蠕变阶段均有不利的影响,并在Burgers模型基础上进行改进,提出了水-岩作用劣化的损伤蠕变本构模型;郭长宝等[14]通过在不同围压下的岩石三轴压缩和三轴蠕变试验,得到了岩石力学参数和蠕变力学特征,在西原模型的基础上结合试验测试建立了新的蠕变模型;陈沅江等[15]、朱晗迓等[16]、王清标等[17]分别基于模拟锚固段地层流变力学性质的K-H体和模拟锚索的H体组成的并联模型,对锚索拉力松弛与岩土体流变力学性质之间的耦合作用进行了理论分析;邓东平等[18]将锚索等效成H体和K-H体2种模式,岩土体等效为H+nK(n≤3)体,建立了锚索和岩体并联的耦合模型;高大水等[19]对三峡大坝永久船闸高边坡锚索拉力变化监测值进行了整理和分析,总结出此工程中预应力锚索拉力损失的规律;王军等[20]采用H体模拟锚索体,K-H体模拟岩土体,并添加Mohr-Coulomb剪切塑性元件模拟土体的加速蠕变特性,建立了能够反映预应力锚索结构蠕变三阶段变形特征的塑性加速元件串-并联模型;徐毅青等[21]采用H体模拟锚索体,H-K体模拟岩土体,建立了H-K、H-2K2种锚索体与岩土体耦合的并联模型;冯忠居等[22]基于广义胡克定律、松弛率时程响应方程与采用H体并联的西原体模型[23],建立了考虑3种要素的并联模型,推导出了三因素损失理论计算方程;陈拓等[24]基于改进的西原流变模型模拟边坡岩土体和H体模拟锚索体,建立了锚固段岩土体和锚索体组成的并联模型。王国富等[25]将锚索体等效为黏弹性体,采用K-H体模拟锚固段岩土体,建立了一种新的锚索体与锚固段岩土体组成的并联模型。
上述研究均认为在岩土体蠕变和锚索松弛的耦合作用中,岩土体和锚索的变形是相等的,因而所建立的耦合模型均为并联模型;实际上,从整体力学作用来看,锚索和岩土体虽然所受作用力大小相等,但两者变形并不相等。肖世国等[26]采用模拟锚索、锚固段岩土体流变力学性质的H体、H-K体组成的串联模型,推导出了锚拉力松弛方程。以往这2类模型均没有考虑锚索自由段岩土体的流变特性引起的锚拉力损失,导致理论计算值和实测值有时相差较大。董旭光等[27]将锚索自由段与滑体用H|K并联模型模拟、锚索锚固段与滑床用H-K串联模型模拟,锚固段与自由段串联组成的并-串联模型反映预应力锚索拉力变化与岩土蠕变的关系;该模型将作为弹性体(H体)的锚索分为了自由段锚索和锚固段锚索,且认为自由段锚索与锚固段锚索所受作用力、变形均不相等,这又与实际锚固工程中锚索受力作用和变形特征有矛盾之处。
综合而言,以往所建立的理论模型能够在一定程度上反映锚拉力随时间的变化规律,但这些模型并没有完全体现实际的锚固边坡中锚索与自由段滑体和锚固段地层岩土体的相互作用机制,且大多都没有考虑坡面抑制构件的影响。因此,本文从预应力锚索加固边坡的锚固系统受力及变形基本机理出发,全面考虑锚索、锚固段地层、自由段滑体、坡面抑制构件这4个要素,将其体现于同一耦合分析模型中,建立反映“锚索-锚固段地层-自由段滑体-坡面抑制构件”式锚固系统中锚索拉力松弛的四体串联流变分析模型。基于此模型推导出锚索拉力松弛的理论计算公式,与已有的试验和理论模型进行对比,验证本文所提出模型的合理性,并进一步分析本模型中各主要参数对锚索拉力松弛效应的影响特征。
1. 分析模型
1.1 锚固系统受力及变形基本机理
预应力锚索是一种主要承受拉力的杆状构件,通过钻孔及注浆体将钢绞线固定于锚固段稳定地层(滑床)中,在被加固体表面对钢绞线(锚索)张拉产生预应力,从而达到使被加固体稳定和限制其变形的目的。在实际的预应力锚索加固边坡所形成的加固系统中,主要包括4个组成部分:锚索、锚固段稳定地层(滑床)、自由段所在的滑体以及坡面抑制构件(如地梁、垫墩等)。从预应力锚索结构加固边坡的整体力学作用效果来看,对锚索所施加的张拉力(P)使锚固段产生轴向位移,从而引起锚固段侧表面(锚固体与滑床接触界面)摩阻力,如图1(a)所示。由锚固段轴向静力平衡可知,此侧摩阻力的合力即等于锚索拉力,该摩阻力的反作用力则作用于滑床,从而引起滑床产生相应的位移。锚索拉力引起滑床移动对潜在滑体产生一定的挤压而使潜在滑体产生位移,从而对坡面抑制件继续产生挤压,进一步引起坡面抑制件的位移。这样,对于锚固体-边坡体系,相当于在锚索拉力作用下,引起了锚索、滑床、潜在滑体和坡面抑制构件的渐次位移及其间挤压协调性,如图1(b)所示(虚线表示移动后)。由于边坡岩土体(包括滑床和滑体)自身具有流变属性,在预应力锚索的拉力作用下,岩土体发生蠕变,其位移与蠕变特性有关,因而使锚索拉力与该蠕变特性相关联,而锚索材料自身又有松弛特性,于是两者相互影响,使得锚索拉力的松弛和岩土体的蠕变存在耦合作用关系。
1.2 模型建立
对于预应力锚索加固边坡系统,根据岩土体及锚固结构的一般流变特征[23],锚固段地层常采用通用开尔文体(K-H体)模拟;对于滑体为较完整岩体、类土质地层的情况,可以分别采用K-H体、K体模拟;锚索及其坡面抑制构件均可采用H体模拟。因此,根据上述锚固结构与边坡岩土体的相互作用特征以及锚固系统变形基本机理,可建立如图2所示的锚固结构-边坡体系的整体串联式流变模型。其中,模型I、II分别针对滑体为较完整岩体、类土质地层的情况。
2. 公式推导
考虑到实际工程中锚索间距一般远大于锚孔直径,故忽略群锚效应。同时,本文主要针对实践中常见的非全长粘结型锚索,为简化分析,假设砂浆性质与锚固段滑床围岩相同,且不考虑砂浆的蠕变与外部其他环境因素的影响。根据图2所示的分析模型I和II,在天然工况下,可分别建立锚索拉力(P)和模型总拉应变(ε)之间的关系。
2.1 模型Ⅰ—较完整岩质滑体
由图2(a)所示的分析模型,可得出P和ε之间的关系表达式为:
PAa1Ea1+PAr1(Ek1+Dη1)+PAr1Eh1+PAa2Ea2+PAr2(Ek2+Dη2)+PAr2Eh2=ε (1) 式中:Aa1——单孔锚索坡面抑制件的等效横截面积/m2,记作Aa1=Asd/a,其中As是抑制件的实际截面积/m2,d是锚孔间距/m,a是抑制件的截面宽度/m;
Aa2——单孔锚索截面积/m2;
Ar1——单孔锚索的有效作用范围/m2,可视为相 邻锚索“中-中”间距的范围;
Ar2——单孔锚索对岩体有效影响范围的面积/m2, 其影响范围可视为圆形区域且该圆形区域 的直径可取为锚孔中心间距;
D——与偏微分相关的计算符号,记作D=∂/∂t。
对于松弛问题,有ε为常量,即Dε=0。对式(1)两侧同乘以Aa1Aa2Ar1Ar2Ea1Ea2Eh1Eh2Ek1Ek2,再经通分后可化简为:
k1P″ (2) 其中,k1、k2、k3、k4均为系数,分别可表示为:
{k_1} = {\eta _1}{\eta _2}\left( \begin{gathered} {A_{{\text{a1}}}}{A_{{\text{r1}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{{\text{h1}}}}{E_{{\text{h2}}}} + {A_{{\text{a1}}}}{A_{{\text{a2}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{{\text{h2}}}} + \\ {A_{{\text{a2}}}}{A_{{\text{r1}}}}{A_{{\text{r2}}}}{E_{{\text{a2}}}}{E_{{\text{h1}}}}{E_{{\text{h2}}}} + {A_{{\text{a1}}}}{A_{{\text{a2}}}}{A_{{\text{r1}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{{\text{h1}}}} \\ \end{gathered} \right); \begin{split} {k_2} = &\left( \begin{gathered} {A_{{\text{a1}}}}{A_{{\text{r1}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{{\text{h1}}}}{E_{{\text{h2}}}} + {A_{{\text{a1}}}}{A_{{\text{a2}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{{\text{h2}}}} + \\ {A_{{\text{a2}}}}{A_{{\text{r1}}}}{A_{{\text{r2}}}}{E_{{\text{a2}}}}{E_{{\text{h1}}}}{E_{{\text{h2}}}} + {A_{{\text{a1}}}}{A_{{\text{a2}}}}{A_{{\text{r1}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{{\text{h1}}}} \\ \end{gathered} \right) \cdot \\ &\left( {{E_{{\text{k1}}}}{\eta _2} + {E_{{\text{k2}}}}{\eta _1}} \right) + {A_{{\text{a1}}}}{A_{{\text{a2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{{\text{h1}}}}{E_{{\text{h2}}}}\left( {{A_{{\text{r1}}}}{\eta _1}{\text{ + }}{A_{{\text{r2}}}}{\eta _2}} \right); \\ \end{split} \begin{split} {k_3} =& {E_{{\text{k1}}}}{E_{{\text{k2}}}}\left( \begin{gathered} {A_{{\text{a1}}}}{A_{{\text{r1}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{{\text{h1}}}}{E_{{\text{h2}}}} + {A_{{\text{a1}}}}{A_{{\text{a2}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{{\text{h2}}}} + \\ {A_{{\text{a2}}}}{A_{{\text{r1}}}}{A_{{\text{r2}}}}{E_{{\text{a2}}}}{E_{{\text{h1}}}}{E_{{\text{h2}}}} + {A_{{\text{a1}}}}{A_{{\text{a2}}}}{A_{{\text{r1}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{{\text{h1}}}} \\ \end{gathered} \right) +\\ & {A_{{\text{a1}}}}{A_{{\text{a2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{{\text{h1}}}}{E_{{\text{h2}}}}\left( {{A_{{\text{r1}}}}{E_{{\text{k1}}}}{\text{ + }}{A_{{\text{r2}}}}{E_{{\text{k2}}}}} \right) ;\\ \end{split} {k_4} = {A_{{\text{a1}}}}{A_{{\text{a2}}}}{A_{{\text{r1}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{{\text{h1}}}}{E_{{\text{h2}}}}{E_{{\text{k1}}}}{E_{{\text{k2}}}}。 求解式(2)所示的微分方程,可得:
P\left( t \right) = {b_1}\varepsilon + {b_2}{{\text{e}}^{{n_1}t}} + {b_3}{{\text{e}}^{{n_2}t}} (3) 式中:t——时间/h;
n1、n2——方程k1n2+k2n+k3=0的2个实根;
b1、b2、b3——积分常数,其中b1=k4/k3。
由锚索拉力初始条件P(0)=P0,式(3)可表达为:
P\left( t \right) = \frac{{{k_4}}}{{{k_3}}}\varepsilon + \left( {{P_0} - \frac{{{k_4}}}{{{k_3}}}\varepsilon - {C_1}} \right){{\text{e}}^{{n_1}t}} + {C_1}{{\text{e}}^{{n_2}t}} (4) 其中,C1为任意常数,可根据典型时刻锚拉力监测值确定。
由于ε等于初始应变,可根据图2(a)由锚拉初始状态确定,即:
\varepsilon = \frac{{{P_0}}}{{{A_{{\text{a1}}}}{E_{{\text{a1}}}}}} + \frac{{{P_0}}}{{{A_{{\text{r1}}}}{E_{{\text{h1}}}}}} + \frac{{{P_0}}}{{{A_{{\text{a2}}}}{E_{{\text{a2}}}}}} + \frac{{{P_0}}}{{{A_{{\text{r2}}}}{E_{{\text{h2}}}}}} (5) 2.2 模型II—类土质滑体
由图2(b)所示的分析模型,可得出P和ε之间的关系表达式为:
\begin{split} &\frac{P}{{{A_{{\text{a1}}}}{E_{{\text{a1}}}}}} + \frac{P}{{{A_{{\text{r1}}}}\left( {{E_{{\text{k1}}}} + {\text{D}}{\eta _1}} \right)}} + \frac{P}{{{A_{{\text{a2}}}}{E_{{\text{a2}}}}}} + \frac{P}{{{A_{{\text{r2}}}}{E_{\text{h}}}}} + \\ &\frac{P}{{{A_{{\text{r2}}}}\left( {{E_{{\text{k2}}}} + {\text{D}}{\eta _2}} \right)}} = \varepsilon \end{split} (6) 式(6)两侧同乘Aa1Aa2Ar1Ar2Ea1Ea2Eh2Ek1Ek2,可化简得:
{\lambda _1}P'' + {\lambda _2}P' + {\lambda _3}P = {\lambda _4}\varepsilon (7) 其中,λ1、λ2、λ3、λ4均为系数,可分别表示为:
{\lambda _1}{\text{ = }}{A_{{\text{r1}}}}{\eta _1}{\eta _2}\left( {{A_{{\text{a1}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{\text{h}}} + {A_{{\text{a2}}}}{A_{{\text{r2}}}}{E_{{\text{a2}}}}{E_{\text{h}}} + {A_{{\text{a1}}}}{A_{{\text{a2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}} \right); \begin{split} {\lambda _2}{\text{ = }}&{A_{{\text{r1}}}}\left( {{E_{{\text{k1}}}}{\eta _2} + {E_{{\text{k2}}}}{\eta _1}} \right)\left( \begin{gathered} {A_{{\text{a1}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{\text{h}}} + {A_{{\text{a2}}}}{A_{{\text{r2}}}}{E_{{\text{a2}}}}{E_{\text{h}}}+ \\ {A_{{\text{a1}}}}{A_{{\text{a2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}} \\ \end{gathered} \right)+ \\ &{A_{{\text{a1}}}}{A_{{\text{a2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{\text{h}}}\left( {{A_{{\text{r1}}}}{\eta _1}{\text{ + }}{A_{{\text{r2}}}}{\eta _2}} \right); \\ \end{split} \begin{split} {\lambda _3} = &{A_{{\text{r1}}}}{E_{{\text{k1}}}}{E_{{\text{k2}}}}\left( \begin{gathered} {A_{{\text{a1}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{\text{h}}} + {A_{{\text{a2}}}}{A_{{\text{r2}}}}{E_{{\text{a2}}}}{E_{\text{h}}} + \\ {A_{{\text{a1}}}}{A_{{\text{a2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}} \\ \end{gathered} \right)+ \\ &{A_{{\text{a1}}}}{A_{{\text{a2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{\text{h}}}\left( {{A_{{\text{r1}}}}{E_{{\text{k1}}}}{\text{ + }}{A_{{\text{r2}}}}{E_{{\text{k2}}}}} \right) ;\\ \end{split} {\lambda _4} = {A_{{\text{a1}}}}{A_{{\text{a2}}}}{A_{{\text{r1}}}}{A_{{\text{r2}}}}{E_{{\text{a1}}}}{E_{{\text{a2}}}}{E_{\text{h}}}{E_{{\text{k1}}}}{E_{{\text{k2}}}}。 求解式(7)所示的微分方程,可得:
P\left( t \right) = {d_1}\varepsilon + {d_2}{{\text{e}}^{{r_1}t}} + {d_3}{{\text{e}}^{{r_2}t}} (8) 式中:r1、r2——方程λ1r2+λ2r+λ3=0的2个实根;
d1、d2、d3——积分常数,其中d1=λ4/λ3。
由锚索拉力初始条件P(0)=P0,式(8)可表达为:
P\left( t \right) = \frac{{{\lambda _4}}}{{{\lambda _3}}}\varepsilon + \left( {{P_0} - \frac{{{\lambda _4}}}{{{\lambda _3}}}\varepsilon - {C_2}} \right){{\text{e}}^{{r_1}t}} + {C_2}{{\text{e}}^{{r_2}t}} (9) 其中,C2为任意常数,可根据典型时刻锚拉力监测值 确定。
初始应变ε可根据图2(b)由锚拉初始状态确定,即:
\varepsilon = \frac{{{P_0}}}{{{A_{{\text{a1}}}}{E_{{\text{a1}}}}}} + \frac{{{P_0}}}{{{A_{{\text{a2}}}}{E_{{\text{a2}}}}}} + \frac{{{P_0}}}{{{A_{{\text{r2}}}}{E_{\text{h}}}}} (10) 由此,可根据式(4)、式(9)分别确定模型I和II的锚索拉力随时间的松弛变化规律。
对于实际存在一系列锚索拉力监测值的情况,为了更加合理地确定式(4)、式(9)中的常数C1和C2,可根据不同时刻ti所监测的锚索拉力Pi,求解出一系列C1或C2值,再分别将相应的计算曲线与实测曲线进行比较,二者相关系数最高时的C1或C2值即为所求的该常数值。
3. 试验验证
3.1 室内模型试验
以陈安敏等[10]的室内模型试验(图3)为实例,其模型岩体尺寸为80 cm×80 cm×80 cm,均布4根锚索。锚索模拟材料用ϕ6 mm×2 mm的铜管,抗拉承载力为1 080 N,铜管的弹性模量为1.32×105 MPa;模型岩体采用黄黏砂土材料模拟,含水率为15%,内摩擦角为19°,重度为20 kN/m3。锚索的自由段、锚固段长度分别为35,25 cm。锚索垫墩弹性模量取2.04×104 MPa。根据其试验结果[10],可用K体描述黄黏砂土,其蠕变参数反演结果见表1。
表 1 模型岩体的流变参数Table 1. Rheological parameters of model rock mass组号 Ek/MPa η/(MPa∙h−1) 2 9.187 582.101 3 9.340 597.862 平均值 9.264 589.982 以初始拉力为54,83 N的2#、3#锚索为例进行计算,取表1中的平均值代入计算,由于试验模型的锚索自由段和锚固段“围岩”均为黄黏砂土,故Ek1=Ek2=Ek,η1=η2=η。对于2#、3#锚索,t=20 h时具有相对最优计算值,可分别确定出常数C2为8.656 N和22.591 N。将这些参数代入式(9),可得2#、3#的锚索拉力表达式分别为:
P\left( t \right) = 28.538 + 16.806{{\text{e}}^{{-}0.015\;7t}} + 8.656{{\text{e}}^{{-}0.029\;7t}} (11) P\left( t \right) = 43.864 + 16.545{{\text{e}}^{{-}0.015\;7t}} + 22.591{{\text{e}}^{{-}0.029\;7t}} (12) 锚索拉力试验值与不同方法计算值对比如图4所示。可见,对于2#锚索,本文方法得到的锚索松弛后拉力收敛值与试验结果的误差约为6%。若定义锚索拉力松弛率为δ(t)=1−P(t)/P0(P0为锚索拉力初始值),则锚索拉力松弛率的本法计算值约为47%,而试验值约为44%,二者吻合较好,且与朱晗迓等[16]、肖世国等[26]、董旭光等[27]的计算结果也较为接近,但与Shi等[9]的计算结果存在显著差异。Shi等[9]的锚索拉力松弛率几乎没有变化,这是因为其模型适用于边坡岩体是较为完整且坚硬的条件,而本室内试验的“模型岩体”为黄黏砂土。
3.2 现场测试
(1)较完整岩质高边坡
陈沅江等[15]对湖南西部常张高速公路K123边坡工程中预应力锚索拉力进行了现场监测,边坡岩层以风化泥质页岩为主,在边坡支护中取了2根预应力锚索作为试验锚索,张拉吨位为600 kN,其中锚索体长25~30 m,锚索由4根高强度低松弛无黏结钢绞线组成,极限抗拉强度1860 MPa,直径15.24 mm,锚孔直径110 mm,锚索弹性模量195 GPa,锚孔间距4 m,框架梁截面尺寸0.4 m×0.5 m,框架梁的弹性模量3.0×104 MPa。测点布设如图5所示。
根据边坡岩体蠕变试验结果,陈沅江等[15]给出了锚固段岩体的K-H体参数,即:H体弹性模量、K体弹性模量、黏滞系数分别为15 790 MPa、46 MPa、657 MPa/d。由于坡体整体为页岩岩质边坡,可视为锚索自由段所在滑体与锚固段岩体的流变性质近似相同,均可取此试验值。
取 t = 4 d时(计算值相对最优)的锚索拉力监测值,可算得锚索2-4-A、2-4-B相应的计算常数C1分别为−41.312,−37.871 kN。把相关参数代入式(4)可得锚拉力的表达式分别为:
P\left( t \right) = 448.797{\text{ + }}192.515{{\text{e}}^{{-}0.070\;0t}}{-}41.312{{\text{e}}^{{-}0.093\;6t}} (13) P\left( t \right) = 448.792{\text{ + }}189.079{{\text{e}}^{{-}0.070\;0t}}{-}37.871{{\text{e}}^{{-}0.093\;6t}} (14) 图6给出了现场实测值与不同方法计算值的对比。可见,本文模型整体上呈现出的锚索拉力变化规律更接近于试验值,锚拉力初期迅速下降、中期下降速率变缓、后期逐渐趋于稳定,其中初期第0~18 天锚拉力损失104.27 kN,约占锚拉力稳定后总损失量的69%。本文方法相对现场实测得到的锚索拉力收敛值的误差为11%,计算与实测锚索拉力松弛率分别约为25.2%、15.3%,本法计算误差相对较小。朱晗迓等[16]、肖世国等[26]和Shi等[9]计算的锚索拉力松弛率分别为4.6%、5.8%和50.6%,与实测结果的误差相对较大;而董旭光等[27]的计算锚拉力松弛率为35.3%,与现场实测值差异也较显著,且其锚拉力松弛曲线异常(甚至出现负值)。导致这种现象的原因是其模型适用于岩土体瞬时弹性模量相对较小和较为软弱或较为破碎的边坡,但本例边坡岩体较为完整,瞬时弹性模量相对黏滞弹性模量较大,导致其计算结果相对实测值误差较大。
(2)碎裂岩质高边坡
朱晗迓等[16]对采用预应力锚索框架梁加固的金丽温高速公路K81高边坡的锚索拉力进行了监测。该边坡潜在滑体呈碎裂状态,松动变形迹象明显。边坡区出露的地层为上侏罗统熔结凝灰岩,岩性单一。锚索张拉吨位为850 kN,锚索由6根高强度低松弛无黏结钢绞线组成,极限抗拉强度为1 860 MPa,直径15.24 mm,锚孔直径为110 mm,锚索弹性模量取195 GPa,锚孔间距横向为4 m,纵向为6 m。钢筋混凝土框架梁截面尺寸为横梁0.45 m×0.45 m,纵梁为0.60 m×0.45 m,其弹性模量为30 GPa。
这里以长度约为22.5 m的BZ16-23号锚索为例,朱晗迓等[16]根据K-H模型的蠕变公式,通过对现场位移及锚拉力监测数据反分析,得到了边坡滑体的H体弹性模量为60 MPa、K体弹性模量为46 MPa、黏滞系数为657 MPa/d。对于较为完整的滑床岩体,借鉴相关室内三轴蠕变试验结果[28],其K-H体流变参数取为:H体弹性模量550 MPa、K体弹性模量4 680 MPa、黏滞系数734 400 MPa/d。当取t=13 h时计算值相对最优,可算得常数C1=106.397 kN。由此,将相关参数代入式(4)中得到BZ16-23锚索的拉力随时间变化的表达式为
P\left( t \right) = 753.434{-}9.831{{\text{e}}^{{-}0.006\;4t}} + 106.397{{\text{e}}^{{-}0.078\;8t}} (15) 锚索拉力的现场实测值与不同方法计算值的对比如图7所示。可见,本文计算结果与实测值、朱晗迓等[16]计算结果都相当接近,可以很好地模拟锚索拉力随时间衰减的规律。对于本例,锚索拉力初期迅速下降,中期缓慢递减,后期逐渐收敛于稳定值,其中初期第0~14天锚拉力损失70.26 kN,约占锚拉力稳定收敛后总损失量的72%;本文模型计算值和现场实测值的锚索拉力收敛值误差小于0.1%。现场实测的锚索拉力松弛率约为11.5%,朱晗迓等[16]的并联模型、肖世国等[26]的串联模型、董旭光等[27]的考虑滑体的并-串联模型、Shi等[9]的考虑框架梁的并-串联模型计算得到的锚索拉力松弛率分别为11.8%、13.7%、10.5%、0.5%,与实测值的误差分别为0.3%、2.6%、1.1%、12.5%。因此,本文方法计算结果相对更符合实测值。
表2给出了试验得到的上述各例的锚索拉力松弛历时(锚索张拉锁定后达到拉力稳定状态的经历时长)及本文方法的计算误差。可见,本法计算的最大误差仅约为9.8%,具有较好的合理性。
表 2 锚索拉力松弛历时及本法计算误差Table 2. Relaxation duration of anchor tension and proposed calculation errors锚索编号 锚拉力初始值/kN 松弛历时/d 确定方法 本法误差/% 2# 0.054 5.1 试验值 9.8 5.6 本文值 3# 0.083 5.2 试验值 9.6 5.7 本文值 2-4-A 600 52 试验值 7.7 56 本文值 2-4-B 600 50 试验值 8.0 54 本文值 BZ16-23 850 62 试验值 3.2 60 本文值 (3)堆积层高边坡
四川省汶川至马尔康高速公路K93+940—K94+145左侧堆积层高边坡,采用预应力锚索框架加固。由工程地质测绘及钻探揭露,其堆积层主要为块石、碎石和角砾构成的半成岩,基岩由深灰色千枚岩为主,夹薄层状石英砂岩。取其中的2根预应力锚索A1H和C1H作为现场测试锚索,张拉吨位分别为480,620 kN,其锚索长度为60 m,锚索由6根高强度低松弛无黏结钢绞线组成,极限抗拉强度1860 MPa,弹性模量195 GPa,直径15.24 mm;锚孔直径110 mm,锚孔间距横向4 m、纵向5 m。钢筋混凝土框架梁截面尺寸为0.50 m×0.50 m,其弹性模量为30 GPa。
根据蠕变试验,得到可描述该边坡堆积层流变特性的K-H模型参数为:Eh1=60 MPa、Ek1=46 MPa、η1=657 MPa/d;坡体滑床岩体的K-H体模型参数为:Eh2=480 MPa、Ek2=6580 MPa、η2=1483 MPa/d。采用本文算法,当取t=10 d时计算值相对最优,可算得两锚索的计算常数C1分别为−13.146,−18.804 kN。由此,将相关参数代入式(4)中可得锚拉力随时间变化的表达式为:
P\left( t \right) = 415.698{\text{ + }}77.448{{\text{e}}^{{-}0.080\;7t}} - 13.146{{\text{e}}^{{-4.444\;0}t}} (16) P\left( t \right) = 536.943{\text{ + }}101.861{{\rm{e}}^{{-}0.080\;7t}} - 18.804{{\rm{e}}^{{{ - 4.444\;0}}t}} (17) 锚索拉力的现场实测值与不同方法计算值的对比如图8所示。可见,本文计算结果与现场实测值相当接近,可以很好地模拟锚索拉力随时间衰减的规律。以图8(a)为例,本文模型计算值和现场实测值的锚索拉力收敛值误差约为1.3%。现场实测的锚索拉力松弛率约为14.6%,朱晗迓等[16]的并联模型、肖世国等[26]的串联模型、董旭光等[27]的考虑滑体的并-串联模型、Shi等[9]的考虑框架梁的并-串联模型计算得到的锚索拉力松弛率分别为16%、18.9%、11.7%、0.3%,与实测值的误差分别为1.7%、5.1%、3.3%、16.6%。因此,本文方法计算结果相对更贴近实测值,相对误差更小。
4. 参数讨论
由式(4)、式(9)可见,锚索直径、锚孔间距、锚索弹性模量、锚固段地层与自由段滑体的流变参数、坡面抑制构件的弹性模量等因素均对锚索拉力松弛有影响。下面以第3.2节中实例(1)的相关参数作为基本参数,基于本文分析模型采用控制变量法具体讨论这些因素对锚索拉力松弛率的影响特征。
4.1 锚索直径
图9为锚索直径对锚索拉力松弛率的影响曲线。可见,锚索直径为9.5~17.8 mm时,锚索拉力松弛率随锚索直径线性增大,且随着时间的增长松弛率的变化速率也愈大,同时锚拉力达到稳定状态的时间也随之延长。
4.2 锚孔间距与锚索弹性模量
图10、图11分别给出了锚孔间距、锚索弹性模量对锚索拉力松弛率的影响曲线。可见,随着锚孔间距(1~7 m)的增大,锚索的松弛率呈显著非线性减弱,锚拉力达到稳定状态的时间也随之缩短;而随着锚索弹性模量的增大(160~220 GPa),锚索的松弛率呈线性增强,锚拉力达到稳定状态的时间也随之延长。
4.3 锚固段地层的流变参数
图12(a)是锚固段地层的H体弹性模量对锚索拉力松弛率的影响曲线。可见,随着H体弹性模量由8~24 GPa增大,锚索拉力松弛率基本不变。
图12(b)是锚固段地层的K体弹性模量对锚索拉力松弛率的影响曲线。可见,随着K体弹性模量由25~65 MPa增大,锚索拉力松弛率整体呈非线性减弱,且随着时间的增长锚拉力松弛率变化速率愈大,同时锚拉力达到稳定状态的时间也愈短。
图12(c)是锚固段地层的黏滞系数对锚索拉力松弛率的影响曲线。可见,随着黏滞系数由150~950 MPa/d增大,在初期阶段锚索拉力松弛率呈较显著的非线性减弱,经过较长历时后,锚索拉力松弛率不再受黏滞系数影响。
4.4 坡面抑制构件的弹性模量
图13为坡面抑制构件的弹性模量对锚索拉力松弛率的影响曲线。可见,当坡面抑制构件的混凝土材料为C20~C40(弹性模量为25.5~32.5 GPa)时,随着坡面抑制构件弹性模量的增大,其对锚索拉力松弛率几乎没有影响。
4.5 自由段滑体的流变参数
这里以Ea1=1 GPa为例,以Eh1=1 GPa、Ek1=30 MPa和η1=300 MPa/d为自由段滑体流变参数的基本值进行讨论。
图14给出了锚索拉力松弛率随着自由段滑体流变参数的变化特征。可见,随着自由段滑体的H体弹性模量由0.25~4.00 GPa增大,锚索拉力松弛率几乎不产生变化;随着自由段滑体的K体弹性模量由15~45 MPa增大,锚索拉力松弛率呈非线性减弱,且随着时间的增长锚拉力松弛率变化速率愈大;随着自由段滑体的黏滞系数由150~450 MPa/d增大,锚索拉力松弛率呈非线性减弱,且随着时间的增长锚拉力松弛率变化速率愈小,经过较长历时后,锚索松弛效应不再受黏滞系数影响。
5. 结论
(1)对于加固边坡的预应力锚索的拉力松弛问题,根据锚固结构-边坡体系中锚固段地层、锚索、自由段滑体、坡面抑制构件之间的相互作用特征与锚固系统受力、变形基本机理,可采用2种不同的H体模拟锚索、坡面抑制构件,采用K体或K-H体模拟自由段滑体,采用K-H体模拟锚固段地层,建立“锚固段地层-锚索-自由段滑体-坡面抑制构件”式锚固系统的四体串联流变模型,以此推导出锚索拉力松弛方程,可以预测锚索拉力随时间的变化规律。
(2)本文方法所计算的锚索拉力随时间松弛特征表现为:初期迅速下降、中期下降速率渐缓和后期收敛于稳定状态,与室内模型试验和现场实测所呈现的规律基本一致;本文方法计算得到的锚索拉力初期(张拉锁定后15 d左右)松弛率占锚索拉力总松弛率的70%左右,锚索拉力松弛收敛值及松弛历时的最大计算误差分别约为11%、10%。
(3)锚索拉力松弛率随锚索的直径和弹性模量的增大呈线性增大,随锚孔间距、锚固段地层和自由段滑体的滞后弹性模量的增大呈非线性减弱;在初期阶段,锚索拉力松弛率随锚固段地层和自由段滑体的黏滞系数的增大呈非线性减弱,但随时间增长锚索松弛率逐渐不再受影响;锚固段地层和自由段滑体的瞬时弹性模量、坡面抑制构件的弹性模量均对锚索拉力松弛率影响很小。
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表 1 模型岩体的流变参数
Table 1 Rheological parameters of model rock mass
组号 Ek/MPa η/(MPa∙h−1) 2 9.187 582.101 3 9.340 597.862 平均值 9.264 589.982 表 2 锚索拉力松弛历时及本法计算误差
Table 2 Relaxation duration of anchor tension and proposed calculation errors
锚索编号 锚拉力初始值/kN 松弛历时/d 确定方法 本法误差/% 2# 0.054 5.1 试验值 9.8 5.6 本文值 3# 0.083 5.2 试验值 9.6 5.7 本文值 2-4-A 600 52 试验值 7.7 56 本文值 2-4-B 600 50 试验值 8.0 54 本文值 BZ16-23 850 62 试验值 3.2 60 本文值 -
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