ANALYTICAL METHOD OF INTERFACIAL STRESS TRANSFER AND BEARING CAPACITY OF PRESSURE-TYPE ANCHORAGE SYSTEM AT EARTHEN SITE WITH CRACK SEALED BY GROUTING
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摘要: 土遗址锚固工程中, 压力型锚杆相比于全长黏结拉力型锚杆而言具有高承载力和耐易溶盐侵蚀的优势, 但由于此类锚固系统传力机理尚不明确, 导致其在实际工程中的应用受到严重制约. 本文将遗址稳定体内锚固段分为弹性压缩段和黏结−滑移段两部分, 分别基于线性弹簧和浆体/土体界面黏结−滑移强化型本构建立简化力学模型, 对界面黏结−滑移全过程, 即弹性阶段、弹性−强化阶段和强化阶段进行理论解析, 推导了各阶段对应的位移、应变以及剪应力分布等计算公式, 给出了压力型锚杆极限抗拔承载力解析解. 结果表明, 峰值载荷前载荷−位移曲线理论值与试验值吻合较好; 弹性压缩段占比与锚固长度对载荷−位移关系的影响主要体现在弹性−强化阶段. 参数敏感度分析表明, 忽略弹性压缩段影响时, 锚固长度与极限承载力线性相关; 浆体弹性模量主要影响界面应力随载荷增加时的传递进程, 对承载力影响有限; 黏结−滑移模型的剪应力峰值对承载力有显著影响. 该解析方法对土遗址压力型锚杆锚固系统传力过程分析具有良好适用性.Abstract: In the anchorage engineering of earthen site, the pressure-type anchor compared with full-range grouted tension-type anchor possesses better bearing capacity and corrosion resistance. However, the application and development of pressure-type anchor in practical anchorage engineering of earthen site are seriously restricted, because the stress transmission mechanism of this type of anchorage system is not clear. Anchorage section in stable soil of earthen site is divided into two parts including elastic compression section and bond-slip section. The mechanical model of elastic compression section is simplified based on linear spring, while the mechanical model of bond-slip section is simplified based on hardened constituent of bond-slip at grout-soil interface. The whole process of bond-slip at the interface under the reinforcement model, which can be divided into elastic stage, elastic-hardened stage and hardened stage, was analyzed. The displacement of pressure plate, strain of grout, distribution of shear stress of grout-soil interface at each stage are derived, while the analytical solution of ultimate pull-out capacity of pressure-type anchor is given. The results show that the values which were calculated with theory of this paper of the load-displacement curve before peak load are in line with the values which were obtained from experiment. The influences of the ratio of elastic compression section is mainly reflected in elastic-hardened stage. From the parametric analysis, anchorage length is linearly related to ultimate bearing capacity, with the neglect of elastic compression section. The elastic modulus of grout mainly has influence on the stress transmission progress at interface, and has a limited impact on bearing capacity. Compared with the elastic modulus of grout, the peak value of shear stress of bond-slip model has a much greater influence on bearing capacity. This analytical method has good applicability to the analysis of the stress transmission process of pressure-type anchor used in earthen site.
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引 言
土遗址是以土为主要材料的历史建筑遗迹, 包括城墙、烽燧、长城和民居等, 是我国文化遗产的重要组成部分[1]. 然而不幸的是, 由于地震、极端降雨等自然营力作用[2], 现存土遗址普遍存在内、外部裂隙发育、基础掏蚀等病害, 大幅削弱了结构整体性, 易引发滑移、倾覆和坍塌等破坏[3-4], 造成不可挽回的巨大损失.
针对上述问题, 相关学者依据“安全第一、最小干预、修旧如旧”原则[5], 逐渐达成了采用锚固[6]方式隐蔽地增强土遗址稳定性的共识. 由于土遗址兼具建筑与文物双重属性, 为保证锚固系统的变形协调、良好相容和高耐久性, 目前主要采用竹、木锚杆[7]配合改性浆体[8]的全长黏结锚固方式进行加固.
全长黏结锚固系统的作用原理是利用浆体与孔壁的黏结摩阻力产生抗拔力, 由锚杆通过杆体/浆体界面(第一界面)和浆体/土体界面(第二界面)将载荷最终传递至土体中[9-13], 为典型的拉力型锚杆受力模式. 此类锚杆施工简便, 传力路径明确, 在土遗址锚固工程中被广泛应用. 然而, 随着土遗址锚固工程的规模化开展, 传统拉力型锚固系统也呈现出一定弊端, 如: (1) 拉力型锚固系统主控面为杆体/浆体界面, 黏结面积小, 承载力有限, 不适用于城墙等大型遗址锚固; (2) 锚固界面应力分布极不均匀, 始端附近剪应力较大, 易因剪胀效应引发土体开裂松动; (3) 始端附近土体开裂后, 在降雨和风蚀等作用下, 易因易溶盐侵蚀、土体强度劣化等造成锚固系统承载力的损失, 不利于锚固力的长久保持. 压力型锚固系统可有效解决上述问题. 压力型锚固系统杆体与浆体之间通过套管分隔, 载荷直接传递至承压板并对浆体产生挤压作用, 其典型破坏模式为浆体/土体界面的滑移失效. 由于压力型锚杆第二界面接触面积较第一界面大幅增加[14], 且主要受力区域位于锚杆末端(承压板附近), 因此压力型锚杆在承载力与抗腐蚀性能上较拉力型锚杆更优[15].
可见, 压力型锚杆根据其应力传递特征在受力和耐久性方面具有优势, 尤其适用于大型遗址锚固工程. 然而, 现有土遗址锚固机理方面的研究主要集中于拉力型锚杆方面, 相关学者通过拉拔试验建立了杆体/土体界面的双线型黏结滑移模型[16]、三线型黏结滑移模型[17]以及考虑完全脱黏[6,18]的三线型黏结滑移模型, 在此基础上, 基于锚固微段的受力平衡关系, 采用载荷传递法[19]和传递矩阵法[20]等方法对杆体/浆体界面的滑移失效全过程进行了理论解析, 同时基于非线性弹簧单元建立了锚杆滑移失效的有限元模拟方法[21-22]. 上述研究虽然是针对拉力型锚杆进行的, 但相关方法在压力型锚杆研究中同样适用, 能为压力型锚杆承载力及浆体/土体界面的应力传递分析提供借鉴.
虽然土遗址锚固领域对压力型锚杆锚固机理的研究有限, 但在传统岩土边坡锚固领域的研究已经较为深入, 主要手段包括拉拔实验[23]、有限元模拟[24-25]和理论解析等. 刘超等[26-27]和廖军等[28]在浆体/土体界面剪应力推导过程中引入泊松效应因素, 即考虑浆体压缩过程中由于径向膨胀引起的剪应力增量, 推导了临界锚固长度与极限锚固力的相关公式, 但是由于推导过程基于完全线弹性假定, 与实际的界面弹塑性转变情况存在差异. 易梅辉等[29]考虑界面剪切流变, 分析推导了压力型锚杆周围土体全部处于弹性、局部处于塑性、局部处于滑移时界面位移与轴力分布公式, 但该方法计算较为复杂, 主要原因是考虑浆体的泊松效应使得界面黏结−滑移关系的解析较困难. 单婷婷等[30]、张景科等[31]和Wang等[32]通过浆体推出实验, 得到光滑型和粗糙型两类浆体/土体界面的黏结−滑移关系, 其可简化为双线强化型或三线软化型黏结−滑移模型, 这为压力型锚杆浆体/土体界面的应力传递和承载力解析研究奠定了基础.
综上, 已建立的锚固界面黏结−滑移模型为锚固系统传力机理的解析研究提供了极大的便利. 本文基于单婷婷等[30]提出的土遗址第二界面(浆体/土体界面)强化型黏结−滑移模型, 利用载荷传递法, 对压力型锚杆浆体/土体界面黏结−滑移全过程进行分阶段解析, 推导得到每个阶段的位移、应变以及剪应力分布、载荷−位移关系, 给出极限承载力以及临界锚固长度的计算公式, 为大型土遗址压力型锚固系统的性能预测和参数设计提供依据.
1. 简化力学模型
鉴于城墙等大型土遗址普遍存在纵向宽大裂隙, 主要通过锚固方法将裂隙临空侧的危险土体与对侧的稳定土体牢固拉结, 并对裂隙部位进行灌浆封护, 以增强结构整体性, 如图1所示.
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图 1 含纵向裂隙土遗址压力型锚杆锚固示意Figure 1. Schematic of earthen site with longitudinal crack reinforced with pressure-type anchors由于裂隙左侧危险体的位移受到锚板的限制, 杆体与浆体间的相对滑移可忽略, 故可将其简化为集中质量, 在受到水平地震作用时, 该集中质量在惯性力作用下对锚杆产生轴向载荷[33-35], 由承压板将轴向载荷传递至浆体中, 最终由浆体传递至土体中, 如图1所示. 对于裂隙右侧稳定体内锚固段, 裂隙内灌浆后结石体强度较高, 且群锚能够固定其与遗址本体的相对位置, 故可将注浆后的裂隙处视为固定端约束. 当单根压力型锚杆受到轴向载荷作用时, 由于杆体/浆体界面被套管隔离, 该界面无黏结力, 其载荷直接传递至承压板, 对其前部浆体施加压力, 浆体/土体界面随即产生滑移; 然而, 工程实测表明, 当锚固深度较大时, 由于界面剪应力自承压板向裂隙端传递, 靠近裂隙端的锚孔界面所受剪应力较小, 主要发生弹性压缩变形, 加之受到裂隙处灌浆结石体的约束, 其相对土体的滑移量非常有限, 故可将该部分简化为线性弹簧. 后续研究中, 该区域长度根据试验结果假定为稳定体内锚固段长度的10% ~ 15%.
综上, 本文基于以下基本假定建立力学模型:
(1) 忽略危险体内锚固段的相对滑移, 将其视为整体, 简化为集中质量;
(2) 忽略裂隙注浆封护后结石体与遗址本体的相对位移, 将其视为固定端约束;
(3) 忽略套管厚度, 且假定套管与浆体间无相对滑移;
(4) 假定杆体始终处于弹性变形范围, 且杆材的抗拉强度足够(杆体在拉拔过程中不发生拉断破坏);
(5) 假定浆体不发生受压破坏, 锚固系统主要因浆体/土体界面的滑移失效而破坏;
(6) 稳定体内锚固段力学模型划分为两部分考虑, 即弹性压缩段和黏结−滑移段;
(7) 弹性压缩段力学行为由线性弹簧表征, 黏结−滑移段力学行为由浆体/土体界面黏结−滑移模型表征.
据此建立的土遗址单根压力型锚杆锚固系统结构及其各部分的简化力学模型如图2所示.
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图 2 土遗址加固工程中压力型锚杆计算模型简化示意Figure 2. Simplification of calculation model of pressure-type anchor in anchorage engineering of earthen site2. 控制方程
锚杆加载端受轴向拉拔载荷P作用时, 由于杆体与浆体间无黏结, 锚杆所受载荷将直接传递至承压板, 承压板随即对浆体产生压力, 承压板范围以钻孔土壁为限. 解析时忽略套管厚度. 压力型锚杆黏结−滑移锚固微段受力如图3所示. 其中R为杆体截面圆心至遗址土体外边缘的距离; r为杆体截面圆心至浆体/土体界面的距离; r0为杆体截面圆心至浆体内壁的距离.
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图 3 压力型锚杆浆体/土体界面微段受力平衡关系Figure 3. Stress balance between grout/soil interface of pressure-type anchor取浆体与土体界面附近长度为dx的微元进行分析.
由土体微段受力平衡得
$$ {\mathrm{d}}{\sigma _s} \cdot \text{π} \left( {{R^2} - {r^2}} \right) = \tau \left( x \right) \cdot 2\text{π} r \cdot {\rm{d}}x $$ (1) 整理得
$$ \frac{{{\mathrm{d}}{\sigma _s}}}{{{\mathrm{d}}x}} = \frac{{2r}}{{{R^2} - {r^2}}} \cdot \tau \left( x \right) $$ (2) 由浆体微段受力平衡得
$$ {\mathrm{d}}{\sigma _J} \cdot \text{π} \left( {{r^2} - {r_0}^2} \right) = \tau \left( x \right) \cdot 2\text{π} r \cdot {\mathrm{d}}x $$ (3) 整理得
$$ \frac{{{\mathrm{d}}{\sigma _J}}}{{{\mathrm{d}}x}} = \frac{{2r}}{{{{r^2} - {r_0}^2} }} \cdot \tau \left( x \right) $$ (4) 由浆体/土体系统微段平衡有
$$ {\sigma _s}\left( x \right) \cdot \text{π} \left( {{R^2} - {r^2}} \right) + {\sigma _J} \cdot \text{π} \left( {{r^2} - {r_0}^2} \right) = 0 $$ (5) 土体与浆体材料在弹性变形时的应力−应变关系可由以下两式表示
$$\qquad\quad {\sigma _s}\left( x \right) = {E_s} \cdot \frac{{{\mathrm{d}}{u_s}\left( x \right)}}{{{\mathrm{d}}x}} = {E_s} \cdot {\varepsilon _s} $$ (6) $$\qquad\quad {\sigma _J}\left( x \right) = {E_J} \cdot \frac{{{\mathrm{d}}{u_J}\left( x \right)}}{{{\mathrm{d}}x}} = {E_J} \cdot {\varepsilon _J} $$ (7) 式中, Es, EJ分别为遗址土体与浆体的弹性模量, $\varepsilon _s $, $\varepsilon _J $分别为遗址土体与浆体的轴向线应变.
浆体与土体的相对位移通常采用二者间的相对滑移量进行表示, 故浆体与土体间的相对滑移量s(x)可表示为
$$ s\left( x \right) = {u_J}\left( x \right) - {u_s}\left( x \right) $$ (8) 式中, uJ(x), us(x)分别为浆体与土体的位移.
式(8)两边对x求一阶微分得
$$ \frac{{{\mathrm{d}}s\left( x \right)}}{{{\mathrm{d}}x}} = \frac{{{\mathrm{d}}{u_J}\left( x \right)}}{{{\mathrm{d}}x}} - \frac{{{\mathrm{d}}{u_s}\left( x \right)}}{{{\mathrm{d}}x}} $$ (9) 将式(6)、式(7)代入式(9)可得
$$ \frac{{{\mathrm{d}}s\left( x \right)}}{{{\mathrm{d}}x}} = \frac{{{\sigma _J}\left( x \right)}}{{{E_J}}} - \frac{{{\sigma _s}\left( x \right)}}{{{E_s}}} $$ (10) 式(10)两边对x求一阶微分得
$$ \frac{{{{\mathrm{d}}^2}s\left( x \right)}}{{{\mathrm{d}}{x^2}}} = \frac{1}{{{E_J}}} \cdot \frac{{{\mathrm{d}}{\sigma _J}\left( x \right)}}{{{\mathrm{d}}x}} - \frac{1}{{{E_s}}} \cdot \frac{{{\mathrm{d}}{\sigma _s}\left( x \right)}}{{{\mathrm{d}}x}} $$ (11) 将式(2)和式(4)代入式(11), 同时考虑到浆体与土体界面的黏结−滑移关系可由滑移量s(x)与剪应力τ(x)确定, 得到下式
$$ \frac{{{{\mathrm{d}}^2}s\left( x \right)}}{{{\mathrm{d}}{x^2}}} - {\lambda ^2} \cdot \tau \left( x \right) = 0 $$ (12) 式中
$$ {\lambda ^2} = \frac{{2r}}{{{E_J} \cdot \left( {{r^2} - r_0^2} \right)}} - \frac{{2r}}{{{E_J} \cdot \left( {{R^2} - {r^2}} \right)}} $$ (13) 通常情况下, 土体厚度远大于浆体厚度, 因此$ \dfrac{{2 r}}{{{E_J} \cdot \left( {{r^2} - r_0^2} \right)}} $通常远大于$ \dfrac{{2 r}}{{{E_J} \cdot \left( {{R^2} - {r^2}} \right)}} $, 且当遗址土体厚度足够大时, $ \dfrac{{2 r}}{{{E_J} \cdot \left( {{R^2} - {r^2}} \right)}} $将趋近于0, 因此式(13)可简写为
$$ {\lambda ^2} = \frac{{2r}}{{{E_J} \cdot \left( {{r^2} - r_0^2} \right)}} $$ (14) 将式(5)和式(8)代入式(7)得
$$ \frac{{{\mathrm{d}}s(x)}}{{{\mathrm{d}}x}} = \left[\frac{1}{{{E_J}}} + \frac{{{r^2} - {r_0}^2}}{{({R^2} - {r^2}) \cdot {E_s}}}\right] \cdot {\sigma _J}(x) $$ (15) 同理可得$ \dfrac{1}{{{R^2} - {r^2}}} $趋近于0, 故式(15)简化为
$$ \frac{{{\mathrm{d}}s(x)}}{{{\mathrm{d}}x}} = \frac{1}{{{E_J}}} \cdot {\sigma _J}(x) = {\varepsilon _J}(x) $$ (16) 3. 浆体/土体界面黏结−滑移模型
当浆体/土体界面黏结−滑移模型确定后, 可将界面黏结−滑移模型各阶段所确定的剪应力表达式代入式(12), 依据边界条件确定剪应力沿锚杆长度的分布规律.
根据张景科等[31]的研究, 土遗址锚固工程中, 基于烧料姜石的改性浆液与土体间界面的黏结−滑移关系主要可分为两种: (1) 应变强化型(光滑界面); (2) 应变软化型(螺纹类粗糙界面).
考虑当前土遗址锚固时钻孔内表面一般较为光滑, 难以保证内螺纹等均匀的粗糙界面, 因此本文将浆体/土体间的黏结−滑移关系按照应变强化模型(光滑界面)进行简化, 即将界面的黏结−滑移行为分为2段: (1)弹性段, 此段滑移量0 ≤ s ≤ se; (2)强化段, 此段界面黏结应力不随滑移量的增大而改变, 其值恒定为τe, 滑移量se ≤ s. 令界面峰值剪应力τf与残余剪应力τs之比为k, 则τs = k·τf . 该应变强化型模型可由下式表达
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图 4 浆体/土体界面应变强化型黏结−滑移模型Figure 4. Strain hardening bond-slip model of grout-soil interface$$\qquad\qquad\quad \tau = \frac{{\tau }_{e}}{{s}_{e}},\quad 0\leqslant s\leqslant {s}_{e}$$ (17) $$\qquad\qquad\quad \tau = k\cdot {\tau }_{e},\quad {s}_{e}\leqslant s,\;k = 1$$ (18) 图4所示为依据表达式绘制的黏结−滑移模型曲线.
4. 黏结−滑移全过程分析
依据简化的浆体/土体光滑界面强化型黏结−滑移模型, 通过求解控制方程, 可得到浆体/土体界面的滑移量以及剪应力、剪应变沿锚固深度分布的封闭解.
考虑危险体锚固段内靠近裂隙一侧一定范围的浆体/土体界面所传递的剪应力较小, 相对滑移量有限, 主要发生浆体的弹性压缩变形, 故将该段浆体简化为长度为lF总刚度为A·K的线性弹簧, 即单位截面积上的弹簧刚度为K
$$ K = \frac{{{E_J}}}{{{l_F}}} $$ (19) 式中, lF为浆体弹性压缩段长度.
图5所示为浆体/土体界面黏结−滑移全过程中各个阶段的剪应力分布示意图, 其中界面黏结−滑移全过程分为3个阶段, 即: 弹性阶段(elastic state)、弹性−强化阶段(elastic-hardened state)和强化阶段(hardened state).
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图 5 压力型锚杆拉拔过程界面传力的3个阶段Figure 5. Three states of force transfer in pull-out process of pressure-type anchor4.1 弹性阶段
黏结−滑移段界面处于弹性阶段, 在承压板附近黏结应力取得最大值, 如图5(b)所示. 将式(17)代入控制方程式(12), 可得
$$ \frac{{{{\mathrm{d}}^2}s\left( x \right)}}{{{\mathrm{d}}{x^2}}} - {\alpha _1}^2 \cdot s\left( x \right) = 0 $$ (20) 其中
$$ \alpha _1^2 = {\lambda ^2} \cdot \frac{{{\tau _e}}}{{{s_e}}} = \frac{{2r}}{{{E_J} \cdot \left( {{r^2} - {r_0}^2} \right)}} \cdot \frac{{{\tau _e}}}{{{s_e}}} $$ (21) 考虑边界条件
$$ {\varepsilon }_{J} = \frac{s(0)}{{l}_{F}}\text{, }{\sigma }_{J} = K\cdot s(0)\quad (x=0) $$ (22) $$ {\varepsilon }_{J} = \frac{P}{{E}_{J}\cdot \text{π} \left({r}^{2}-{r}_{0}{}^{2}\right)}\text{, }{\sigma }_{J} = \frac{P}{\text{π} \left({r}^{2}-{r}_{0}{}^{2}\right)}\quad (x=l) $$ (23) 由边界条件式(22)和式(23)求解微分方程式(20), 可得
$$ s(x) = \frac{P}{{\text{π} {E_J} \cdot ({r^2} - {r_0}^2) \cdot {\alpha _1}}} \cdot \frac{{\cosh ({\alpha _1} \cdot x) + \beta \cdot \sinh ({\alpha _1} \cdot x)}}{{\beta \cdot \cosh ({\alpha _1} \cdot l) + \sinh ({\alpha _1} \cdot l)}} $$ (24) 式中
$$ \beta = \frac{K}{{{E_J} \cdot {\alpha _1}}} $$ (25) 将式(24)代入式(4)和式(16)得
$$ {\varepsilon _J}(x) = \frac{P}{{\text{π} {E_J} \cdot ({r^2} - {r_0}^2)}} \cdot \frac{{\sinh ({\alpha _1} \cdot x) + \beta \cdot \cosh ({\alpha _1} \cdot x)}}{{\sinh ({\alpha _1} \cdot l) + \beta \cdot \cosh ({\alpha _1} \cdot l)}} $$ (26) $$ \tau (x) = \frac{P}{{2\text{π} r}} \cdot {\alpha _1} \cdot \frac{{\cosh ({\alpha _1} \cdot x) + \beta \cdot \sinh ({\alpha _1} \cdot x)}}{{\beta \cdot \cosh ({\alpha _1} \cdot l) + \sinh ({\alpha _1} \cdot l)}} $$ (27) 当x = l处的界面剪切应力达到τe时, 此时界面滑移量s = se, 将其代入式(24)可得最大拉拔载荷P1
$$ {P_1} = \text{π} {E_J} \cdot ({r^2} - {r_0}^2) \cdot {\alpha _1} \cdot {s_e} \cdot \frac{{\beta + \tanh ({\alpha _1} \cdot l)}}{{1 + \beta \cdot \tanh ({\alpha _1} \cdot l)}} $$ (28) 当锚固长度足够长, 即l趋近于无穷, 则tanh(a1·l) 趋近于1, 式(28)可化简为
$$ {P_1} = \text{π} {E_J} \cdot ({r^2} - {r_0}^2) \cdot {\alpha _1} \cdot {s_e} $$ (29) 对于强化模型, 其弹性阶段的临界锚固长度可由tanh(a1·l) = 1[18]求得
$$ {l_{{\mathrm{cri}}}} = \frac{2}{{{\alpha _1}}} = {s_e} \cdot \sqrt {\frac{{2{E_J} \cdot ({r^2} - r_0^2)}}{{r \cdot {\tau _e} \cdot {s_e}}}} $$ (30) 4.2 弹性−强化阶段
当x = l处的界面剪切应力达到τe后(s = se), 承压板附近浆体/土体界面开始进入应变强化, 并将黏结−滑移段界面分为两个部分, 即处于弹性阶段的部分与处于强化阶段的部分, 如图5(c)所示. 处于强化阶段的部分, 其所产生的界面剪应力值均恒等于τe.
将式(18)代入控制方程式(12), 可得
$$ \frac{{{{\mathrm{d}}^2}s\left( x \right)}}{{{\mathrm{d}}{x^2}}} = {\lambda ^2} \cdot {\tau _e} $$ (31) 此阶段中边界条件式(22)和式(23)同样适用, 并存在以下边界条件和连续条件
$$ s = {s}_{e}\text{, }\tau = {\tau }_{e} \quad (x=l-l_{h,1})$$ (32) $$ {\varepsilon _J}{\text{ is continuous at }}x = l - {l_{h,1}} $$ (33) 由边界条件式(22)和式(29)求解微分方程式(20)可得处于弹性阶段部分(0≤x≤l − lh,1)的浆体滑移量s(x)
$$ s(x) = {s_e} \cdot \frac{{\cosh ({\alpha _1} \cdot x) + \beta \cdot \sinh ({\alpha _1} \cdot x)}}{{\cosh [{\alpha _1} \cdot (l - {l_{h,1}})] + \beta \cdot \sinh [{\alpha _1} \cdot (l - {l_{h,1}})]}} $$ (34) 将式(34)代入式(4)和式(16)得
$$ {\varepsilon _J}(x) = {s_e} \cdot {\alpha _1} \cdot \frac{{\sinh ({\alpha _1} \cdot x) + \beta \cdot \cosh ({\alpha _1} \cdot x)}}{{\beta \cdot \sinh [{\alpha _1} \cdot (l - {l_{h,1}})] + \cosh [{\alpha _1} \cdot (l - {l_{h,1}})]}} $$ (35) $$\begin{split} &\tau (x) = \frac{{{E_J} \cdot ({r^2} - {r_0}^2)}}{{2r}} \cdot {s_e} \cdot {\alpha _1}^2\cdot \\ &\qquad \frac{{\cosh ({\alpha _1} \cdot x) + \beta \cdot \sinh ({\alpha _1} \cdot x)}}{{\cosh [{\alpha _1} \cdot (l - {l_{h,1}})] + \beta \cdot \sinh [{\alpha _1} \cdot (l - {l_{h,1}})]}}\end{split} $$ (36) 由式(32)、式(33)和式(23)求解微分方程式(31)可得处于强化段(l − lh,1≤x≤l)的浆体滑移量s(x)
$$ \begin{split} & s\left( x \right) = \frac{{{s_e} \cdot {\alpha _1}^2}}{2} \cdot {x^2} + \left[\frac{P}{{\text{π} {E_J} \cdot ({r^2} - {r_0}^2)}} - l \cdot {s_e} \cdot {\alpha _1}^2\right] \cdot x +\\ &\qquad {s_e} - \frac{{{s_e} \cdot {\alpha _1}^2}}{2} \cdot {(l - {l_{h,1}})^2}- \Bigr[\frac{P}{{\text{π} {E_J} \cdot ({r^2} - {r_0}^2)}} - l \cdot {s_e} \cdot \\ &\qquad {\alpha _1}^2\Bigr] \cdot (l - {l_{h,1}})\\[-12pt] \end{split} $$ (37) 将式(37)代入式(4)和式(16)得
$$ {\varepsilon _J}\left( x \right) = {s_e} \cdot {\alpha _1}^2 \cdot x + \frac{P}{{\text{π} {E_J} \cdot ({r^2} - {r_0}^2)}} - l \cdot {s_e} \cdot {\alpha _1}^2 $$ (38) $$ \tau \left( x \right) = {\tau _e} $$ (39) 由x = l − lh,1处应变连续可得
$$ \begin{split} &P = \text{π} {E_J} \cdot ({r^2} - {r_0}^2) \cdot {s_e} \cdot {\alpha _1} \cdot \\ &\qquad \left\{ {{\alpha _1} \cdot {l_{h,1}} + \frac{{\tanh [{\alpha _1} \cdot (l - {l_{h,1}})] + \beta }}{{1 + \beta \cdot \tanh [{\alpha _1} \cdot (l - {l_{h,1}})]}}} \right\}\end{split} $$ (40) 当lh,1 = l时, 载荷P达到临界拉拔载荷P2
$$ \begin{split} & {P_2} = \text{π} {E_J} \cdot ({r^2} - {r_0}^2) \cdot {s_e} \cdot {\alpha _1} \cdot ({\alpha _1} \cdot l + \beta )= \\ &\qquad 2\text{π} r \cdot {\tau _e} \cdot l + \text{π} ({r^2} - r_0^2) \cdot K \cdot {s_e} \end{split} $$ (41) 当x = 0处τ = τe, s = se, 即lh,1 = l, P = P2, 弹性−强化阶段结束. 此时浆体x = l处的位移S1, 可由边界条件式(22)以及上述条件代入式(37)求得
$$ {S_1} = \left(1 + \frac{K}{{{E_J}}} \cdot l + \frac{{{l^2} \cdot {\alpha _1}^2}}{2}\right) \cdot {s_e} $$ (42) 4.3 强化阶段
当x = 0处的界面剪切应力达到τe时(s = se), 黏结−滑移段界面进入强化阶段, 该阶段界面处剪切应力由残余摩擦力提供, 即全界面剪切应力值均等于τe, 如图5(d)所示.
该阶段, 两段浆体均存在压缩变形. 将黏结−滑移段浆体压缩视为浆体/土体界面长度减小, 可得
$$ (1 - \delta ) \cdot {E_J} = K \cdot \varDelta + \frac{{r \cdot {\tau _e} \cdot {l^2} \cdot \delta }}{{({r^2} - r_0^2) \cdot (l - {S_1} + {s_e})}} $$ (43) $$ P = \text{π} ({r^2} - {r_0}^2) \cdot K \cdot \varDelta + 2\text{π} r \cdot {\tau _e} \cdot {l^2} \cdot \frac{\delta }{{l - {S_1} + {s_e}}} $$ (44) 式中, D为弹性压缩段浆体长度的压缩量; d为黏结−滑移段浆体的压缩后长度与压缩前长度的比值.
联立求解式(43)与式(44), 得D与d的关系以及浆体的滑移量S
$$ \varDelta = \left[{E_J} - \frac{{r \cdot {\tau _e} \cdot {l^2}}}{{({r^2} - r_0^2) \cdot (l - {S_1} + {s_e})}}\right] \cdot \frac{\delta }{K} - \frac{{{E_J}}}{K} $$ (45) $$ S = (1 - \delta ) \cdot l - \varDelta $$ (46) 最终弹性压缩段浆体压缩量达到lF, 强化阶段结束, 此时有
$$ \varDelta = {l}_{F}\text{, }\delta = 0 $$ (47) $$ {P_3} = \text{π} ({r^2} - {r_0}^2) \cdot K \cdot {l_F} $$ (48) $$ {S_3} = l + {l_F} $$ (49) 5. 载荷−位移曲线
根据第4节关于强化模型下锚固系统黏结−滑移全过程的阶段划分, 载荷−位移曲线中的3个特征值点如图6(a)所示, 分别为Ⅰ(u1,h, P1,h), Ⅱ(u2,h, P2,h), Ⅲ(u3,h, P3,h), 分别对应着弹性阶段的结束、弹性—强化阶段的结束和黏结−滑移段浆体完全推出. 特征点Ⅰ, Ⅱ所对应的界面剪应力分布如图6(b)所示, 可以看出剪应力在承压板附近达到最大值后, 黏结−滑移段界面强化段不断向始端延伸, 直至特征点II, 此时黏结−滑移段界面全部进入强化阶段.
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图 6 锚固系统黏结−滑移全过程载荷-位移关系以及界面剪应力分布变化Figure 6. Load-slip relationship and variation in distribution of shear stress of feature point during bond-slip process由式(29)、式(41)、式(42)、式(48)和式(49)可得
$$ {u}_{1,h} = {s}_{e} $$ (50) $$ {P}_{1,h} = \text{π} \sqrt{2r\cdot {E}_{J}\cdot \left({r}^{2}-{r}_{0}{}^{2}\right)\cdot {\tau }_{e}\cdot {s}_{e}}$$ (51) $$ {u}_{2,h} = \left(1 + \frac{K}{{E}_{J}}\cdot l\right)\cdot {s}_{e} + \frac{{l}^{2}\cdot r\cdot {\tau }_{e}}{{E}_{J}\cdot \left({r}^{2}-{r}_{0}{}^{2}\right)} $$ (52) $$ {P}_{2,h} = 2\text{π} r\cdot {\tau }_{e}\cdot l + \text{π} ({r}^{2}-{r}_{0}^{2})\cdot K\cdot {s}_{e} $$ (53) $$ {u}_{3,h} = l + {l}_{F} $$ (54) $$ {P}_{3,h} = \text{π} ({r}^{2}-{r}_{0}{}^{2})\cdot K\cdot {l}_{F} $$ (55) 强化段长度在弹性−强化阶段中随载荷的变化关系曲线如图7(a)所示. 此类情况下剪应力于强化段处取得最大值, 且后续剪应力值保持恒定. 这使得锚固系统的极限承载力于特征值点Ⅱ处取得, 即当强化段长度等于黏结−滑移段长度时, 锚杆达到承载力峰值. 当锚固长度一定时, 弹性压缩段占比增大, 即简化弹簧的刚度以及黏结−滑移段界面长度减小, 将导致锚杆承载力降低, 如图7(b)所示. 当弹性压缩段占比一定时, 随着锚固长度增加, 锚杆承载力峰值以及界面滑移量逐渐增加, 且越靠近锚杆末端(承载板)滑移量增幅越显著, 如图7(c)所示. 结合图7(b)和图7(c)可以看出, 锚固长度与弹性压缩段占比对于载荷−位移关系的影响均体现在弹性阶段结束后, 弹性段占比的增加和锚固长度的减小均会降低锚杆承载力.
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图 7 浆体/土体界面在黏结−滑移过程中强化段长度与载荷的变化关系以及锚固长度和弹性压缩段占比对载荷−位移关系的影响 (续)Figure 7. Variations of hardened lengths and influences of proportion of elastic compression section and anchorage length on load-slip relationship during bond-slip process of grout-soil interface (continued)6. 理论结果验证
本节将张景科等[31]的浆体推出试验结果和本课题组前期进行的压力型锚杆拉拔试验结果, 与本文解析方法所得的峰值前载荷−位移关系进行对比, 验证本文方法的可靠性. 由于峰值后的的载荷−位移关系很难通过试验获得, 因此本文只对比了峰值前二者的载荷−位移曲线差异.
第1组采用G-C-60[31]的试验结果. 材料物理力学参数如下: 浆体直径60 mm, 弹性模量647 MPa, 锚固段长度300 mm, 无弹性压缩段(K取0); 黏结−滑移模型参数为: se = 6.15 mm, τe = 0.152 MPa. 图8(a)所示为G-C-60推出过程载荷−位移关系曲线实验值与理论值的对比. 结果显示: 载荷小于4 kN时, 试验值与理论值吻合较好; 载荷大于4 kN小于7 kN时, 位移的理论值较试验值略微偏大, 可能是由于界面在实际试验中存在浆液不饱满、局部开裂等问题, 使得各锚固微段间的力学性能存在差异; 载荷大于7 kN时, 即载荷接近峰值时, 理论解析所得最大位移与试验值相比偏小, 但载荷峰值吻合较好.
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图 8 载荷−位移曲线理论值与试验值对比Figure 8. Comparison between the analytical solution and experimental data of the load-displacement curve第2组采用G-C-70[31]的试验结果. 材料物理力学参数如下: 浆体直径60 mm, 弹性模量212 MPa, 锚固段长度300 mm, 无弹性压缩段(K取0); 黏结−滑移模型参数为: se = 6.20 mm, τe = 0.132 MPa. 图8(b)所示为G-C-70推出过程载荷−位移关系曲线实验值与理论值的对比, 结果显示: 载荷小于4 kN时, 试验值与理论值吻合较好; 载荷超过4 kN后, 位移的理论值与试验值相比略微偏大, 原因与第1组相似; 总体而言二者变化趋势与吻合程度均较好.
第3组采用课题组开展的压力型锚杆拉拔试验结果. 试样夯土与浆体均参照文献[31]的有关要求准备, 夯土体尺寸500 mm(长) × 500 mm(宽) × 450 mm(高), 锚孔直径50 mm, 锚杆(白蜡杆)直径20 mm, 浆体弹性模量667 MPa, 锚固段长度400 mm, 黏结−滑移模型参数取se = 3.54 mm, τe = 0.112 MPa. 值得说明的是, 本实验采用DIC观测方法, 锚杆设置于观测面处, 因此浆体/土体界面并非完整圆形, 而需依据实际接触面积进行换算, 本实验中取接触平面圆弧长度为99.12 mm (图9中红色线段). 结果表明, 该锚固系统的典型破坏模式为浆体/土体界面的滑移破坏, 如图9; 载荷−位移曲线与理论值的对比(图10)结果表明, 当载荷小于5 kN时, 试验值与理论值吻合较好; 载荷大于5 kN时, 即载荷接近峰值时, 理论解析所得载荷和位移, 与试验值相比偏大, 但误差均在10%以内.
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图 9 压力型锚杆拉拔试验与破坏模式示意图Figure 9. Schematic of pull-out test and destruction mode of pressure-type anchor![]()
图 10 压力型锚杆拉拔试验载荷−位移曲线理论值与试验值对比Figure 10. Comparison between the analytical solution and experimental data of the load-displacement curve of pressure-type anchor pull-out test综合对比上述3组试验结果, 锚杆承载力最大差异为13.0%; 位移最大差异为30.0%, 且位于载荷峰值附近, 主要由于实际试验过程中界面可能存在接触不充分、局部浆体开裂等情况, 界面力学行为难以完全符合强化模型, 同时, 在实际推出过程中, 浆体可能产生应变堆积, 导致相同载荷下浆体滑移量的增大. 但整体而言, 本文解析方法能够较为准确地预测浆体/土体界面光滑接触时的黏结−滑移过程和极限承载力.
7. 参数敏感度分析
对于一般岩土体锚固工程, 裂隙通常较小且不会进行专门注浆处理, 即存在本文假定的固定端约束的情况较少, 此外, 考虑弹性压缩段影响时, 计算结果仅对特征点II处的载荷和位移产生一定的增量. 因此, 为使本文解析方法具有更广泛适用性, 本节忽略弹性压缩段及前端固定约束对黏结−滑移段浆体的影响(即取K = 0, lF趋近于无穷), 着重分析锚固长度、浆体弹性模量以及黏结−滑移模型对锚固系统性能的影响规律. 浆体/土体界面黏结−滑移模型特征值及相关参数取值参见表1.
表 1 黏结−滑移模型及相关参数取值Table 1. Values needed to define the bond-slip modelBond-slip model ${\tau _e}$/MPa ${s_e}$/mm ${E_J}$/MPa ${\alpha _1}$/10−2 mm−1 #1 0.4 5 300 0.422 200 0.516 100 0.730 #2 0.4 10 300 0.298 200 0.365 100 0.516 #3 0.2 10 300 0.211 200 0.258 100 0.365 7.1 锚固长度
本部分分析所用黏结−滑移模型取自表1中的#1模型, 分别取锚固长度1. 0, 0. 8和0. 6 m进行对比分析, 不同锚固长度下的载荷−位移关系曲线、强化段长度随载荷的变化关系曲线和极限拉拔载荷的变化趋势如图11所示. 所选锚固长度均满足式(30)中锚固长度足够长的条件, 即锚固长度超过弹性阶段临界锚固长度.
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图 11 锚固长度对界面力学性能的影响Figure 11. Influence of anchorage length on mechanical properties of interface图11(a)中3条曲线在弹性阶段基本重合, 锚固长度对载荷−位移曲线的影响主要在弹性阶段结束后(Ⅰ点后)产生, 且随着锚固长度的增加, 载荷峰值以及达到峰值载荷所需的位移量明显增加. 结合图11(a)和图11(b)可知, 当载荷相同时, 锚杆位移和强化段长度均随锚固长度的增加而减小. 此外, 锚固长度的增加也为界面剪应力的传递和演化提供了更大空间, 按照本文假定和解析方法, 土遗址压力型锚杆极限拉拔载荷与锚固长度近似线性相关, 如图11(c)所示.
7.2 浆体的弹性模量
本部分分析所用黏结−滑移模型取自表1中的#1模型, 假定锚固长度为1.0 m, 分别取浆体弹性模量EJ为300, 200和100 MPa进行对比分析, 3种不同弹性模量下的载荷−位移关系曲线及强化段长度随载荷的变化关系曲线如图12所示.
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图 12 弹性模量对界面力学性能的影响Figure 12. Influence of elastic modulus on mechanical properties of interface如图12(a)中所示, 随浆体弹性模量的增加, 相同载荷下浆体/土体滑移量越小, 但3种弹性模量下所能承担的峰值载荷相同, 只是达到最大载荷时对应的滑移量有所差异. 图12(b)为浆体强化段长度随载荷的变化趋势, 当浆体弹性模量越大时, 界面最初进入强化所需的载荷越大, 即更难进入强化阶段, 但随载荷的增加, 强化段长度增长趋势相近, 当界面均进入强化阶段后, 锚固系统达到极限承载力. 因此, 相同锚固长度下的极限载荷与浆体强度无关. 结合图12(a)和图12(b)可知, 相同位移量下以及相同强化段长度下的耗能(即载荷−位移曲线和强化段长度随载荷变化曲线向横轴的投影区面积)将随弹性模量的增加而增加.
7.3 黏结−滑移模型
为分析不同主要耗能段对理论模型中各关系曲线的影响, 设置了3种不同的黏结−滑移模型, 相关特征值如表1 (#1, #2, #3)所示, 据此绘制的黏结−滑移模型曲线如图13所示.
由于本文采用的强化模型无软化段和脱黏段, 即强化段为无限长, 因此, 实际黏结−滑移模型中强化段的耗能(曲线与坐标轴围闭面积)要远大于图13中所体现出的耗能, 故#1组和#2组强化段耗能的差异可忽略不计, 可通过#1组与#2组的对比分析弹性段耗能的变化对极限承载力和黏结−滑移过程的影响. #1组与#3组弹性段的耗能相同, 强化段峰值剪应力相差一倍, 可用来分析强化段耗能变化对极限承载力和黏结−滑移过程的影响.
采用1.0 m的锚固长度, 分别以#1, #2和#3中的数据进行计算并对比分析3种不同黏结−滑移模型下的载荷−位移关系曲线、强化段长度随载荷变化关系, 如图13所示.
图14(a)所示为不同黏结−滑移模型对应的载荷−位移曲线, #1组与#2组载荷位移曲线变化趋势差异主要体现在加载初期, 主要由于二者弹性段耗能能力有所差异, 随载荷进一步增加, 界面逐渐进入强化阶段, 二者变化趋势相近, 且最终达到的峰值载荷相同; #1组的界面黏结−滑移模型剪应力峰值是3#组的2倍, 当界面均进入强化阶段时, 其载荷峰值也接近3#组的2倍, 表明此类锚固系统的极限承载力主要由强化段控制. 图14(b)所示为强化段长度随载荷的变化趋势, #1组与#2组曲线变化趋势相近, 与图14(a)相对应, 且载荷最大时二者强化段长度相等, 即均为锚固长度; 而#3组由于黏结−滑移模型峰值剪应力较小, 使得强化段长度随载荷的增加迅速增长, 即锚固系统极限承载力与黏结−滑移模型的峰值剪应力密切相关, 剪应力峰值越大, 极限承载力越大, 如图14(c)所示.
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图 14 黏结−滑移模型对载荷−位移曲线与强化段长度随载荷变化关系的影响Figure 14. Influence of bond-slip model on load-slip curve and relationship between load and hardened length8. 结论
(1) 本文解析方法能对土遗址压力型锚固系统界面应力分布与传递过程进行准确分析, 峰值载荷前的载荷−位移关系理论解析结果与试验结果吻合较好.
(2) 锚杆极限承载力主要与锚固长度、弹性压缩段比例、黏结−滑移模型峰值剪应力等因素密切相关, 而浆体弹性模量对承载力影响有限.
(3) 弹性压缩段占比与锚固长度对载荷−位移关系的影响主要体现在弹性−强化阶段, 弹性段占比的增加和锚固长度的减小均会降低锚杆承载力.
(4) 当忽略弹性压缩段影响时, 锚固长度与极限承载力近似线性相关; 浆体弹性模量的增加主要影响界面应力随载荷增加时的传递进程, 弹性模量越大, 则需要更大的载荷才能使界面进入强化阶段.
(5) 强化模型下压力型锚杆滑移失效全过程受黏结−滑移模型影响较大, 模型强化阶段耗能增加会使得锚固系统延性以及总耗能显著提升; 锚固系统承载力主要由强化段控制, 黏结−滑移模型的峰值剪应力对承载力影响显著, 二者近似线性相关.
本文解析方法主要适用于土遗址压力型锚固系统的承载力计算, 对传统岩土体锚固工程亦有一定借鉴意义.
数据可用性声明
支撑本研究的科学数据已在中国科学院科学数据银行(Science Data Bank) ScienceDB平台公开发布, 访问地址为https://cstr.cn/31253.11.sciencedb.12914或 https://doi.org/10.57760/sciencedb.12914.
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图 1 含纵向裂隙土遗址压力型锚杆锚固示意
Figure 1. Schematic of earthen site with longitudinal crack reinforced with pressure-type anchors
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图 2 土遗址加固工程中压力型锚杆计算模型简化示意
Figure 2. Simplification of calculation model of pressure-type anchor in anchorage engineering of earthen site
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图 3 压力型锚杆浆体/土体界面微段受力平衡关系
Figure 3. Stress balance between grout/soil interface of pressure-type anchor
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图 4 浆体/土体界面应变强化型黏结−滑移模型
Figure 4. Strain hardening bond-slip model of grout-soil interface
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图 5 压力型锚杆拉拔过程界面传力的3个阶段
Figure 5. Three states of force transfer in pull-out process of pressure-type anchor
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图 6 锚固系统黏结−滑移全过程载荷-位移关系以及界面剪应力分布变化
Figure 6. Load-slip relationship and variation in distribution of shear stress of feature point during bond-slip process
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图 7 浆体/土体界面在黏结−滑移过程中强化段长度与载荷的变化关系以及锚固长度和弹性压缩段占比对载荷−位移关系的影响 (续)
Figure 7. Variations of hardened lengths and influences of proportion of elastic compression section and anchorage length on load-slip relationship during bond-slip process of grout-soil interface (continued)
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图 8 载荷−位移曲线理论值与试验值对比
Figure 8. Comparison between the analytical solution and experimental data of the load-displacement curve
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图 9 压力型锚杆拉拔试验与破坏模式示意图
Figure 9. Schematic of pull-out test and destruction mode of pressure-type anchor
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图 10 压力型锚杆拉拔试验载荷−位移曲线理论值与试验值对比
Figure 10. Comparison between the analytical solution and experimental data of the load-displacement curve of pressure-type anchor pull-out test
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图 11 锚固长度对界面力学性能的影响
Figure 11. Influence of anchorage length on mechanical properties of interface
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图 12 弹性模量对界面力学性能的影响
Figure 12. Influence of elastic modulus on mechanical properties of interface
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图 14 黏结−滑移模型对载荷−位移曲线与强化段长度随载荷变化关系的影响
Figure 14. Influence of bond-slip model on load-slip curve and relationship between load and hardened length
表 1 黏结−滑移模型及相关参数取值
Table 1 Values needed to define the bond-slip model
Bond-slip model ${\tau _e}$/MPa ${s_e}$/mm ${E_J}$/MPa ${\alpha _1}$/10−2 mm−1 #1 0.4 5 300 0.422 200 0.516 100 0.730 #2 0.4 10 300 0.298 200 0.365 100 0.516 #3 0.2 10 300 0.211 200 0.258 100 0.365 
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