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Analytical models for rock bolts.C.L*,StillborgAbstractThree analytical models have been developed for rock bolts: one for bolts subjected to concentrated pull load in pullout tests, one for bolts installed in uniformly deformed rock masses, and one for bolts subjected to the opening of individual rock joints. The development of the models has been based on the description of the mechanical coupling at the interface between the bolt and the grout medium for grouted bolts, or between the bolt and the rock for frictionally coupled bolts. For rock bolts in the pullout tests, the shear stress of the interfaces exponentially with increasing distance from the point of loading when the deformation is compatible across the interface. Decoupling may start first at the loading point when the applied load is large enough and then propagate towards the far end of the bolt with a further increase in the applied load. The magnitude of the shear stress on the decoupled bolt section depends on the coupling mechanism at the interface. For fully grouted bolts, the shear stress on the decoupled section is lower than the peak shear strength of the interface while for fully frictionally coupled bolts if is approximately the same as the peak shear strength. For rock bolts installed in uniformly deformed rock, the loading process of the bolts due to rock deformation has been taken into account in developing the model. Model simulations confirm the previous findings that a bolt in situ has a pick-up length, an anchor length and neutral point. It is also revealed that the face plate plays a significant role in enhancing the reinforcement effect. In jointed rock masses, several axial stress peaks may occur along the bolt because of the opening of rock joints intersecting the bolt.1. IntroductionRock bolts have been widely used for rock reinforcement in civil and mining engineering for a long time. Bolts reinforce rock masses through restraining the deformation within the rock masses. In order to improve bolting design, it is necessary: to have a good understanding of the behaviour of rock bolts in deformed rock masses. This can be acquired through field monitoring, laboratory tests, numerical modeling and analytical studies.Since the 1970s, numerous researchers have carried out field monitoring work on rock bolts installed in various rock formations. Freeman performed pioneering work in studying the performance of fully grouted rock bolts in the Kielder experimental runnel. He monitored both the loading process of the bolts and the distribution of his monitoring data, he proposed the concepts of “neutral point” “pick-up length” and “anchor length”. At the neutral point, the shear stress at the interface between the bolt and the grout medium is zero, while the tensile axial load of the bolt has a peak value. The pick-up length refers to the section of the bolt from the near end of the bolt (on the tunnel wall) to the neutral point. The shear stresses on this section of the bolt pick up the load from the rock and drag the bolt towards the tunnel. The anchor length refers to the section of the bolt from the neutral point to the far end of the bolt (its seating deep in the rock). The shear stresses on this section of the bolt anchor the bolt to the rock. These concepts clearly outline the behaviour of fully grouted rock bolts in a deformed rock formation. Bjonfot and Stephanssons work demonstrated that in jointed rock masses there may exist not only one but several neutral points along the bolt because of the opening displacement of individual joints.Pullout tests are usually used to examine the anchoring capacity of rock bolts. A great number of pullout tests have been conducted so far in various types of rocks. Farmer carried out fundamental work in studying the behaviour of bolts under tensile loading. His solution predicts that the axial stress of the bolt (also the shear stress at the bolt interface) will decrease exponentially from the point of loading to the far end of the bolt before decoupling occurs. Fig.1(a) illustrates the results of a typical pullout test. Curve a represents the distribution of the axial stress along the bolt under a relatively low applied load, at which the deformation is compatible on both sides of the bolt interface. Curve b represents the axial stress along the bolt at a relatively high applied load, at which decoupling has occurred at part of the bolt interface. Fig.1(b) shows the axial stress along a rock bolt installed in an underground mine drift. It is seen from this figure that the distribution of the axial stress along the section close to the borehole collar is completely different from that in pullout tests. However, along the section to the far end of the bolt, the stress varies similarly to that in pullout tests. The reason Fig.1 Distribution if the axial stress (a) along a grouted steel bar during pullout test, after Hawkes and Evan, and (b) along a grouted rock bolt in situ after sunfor these results is that bolts in situ have a pick-up length and an anchor length, while bolts in pullout tests only have an anchor length.It is thought that the relative movement between the rock and the bolt is zero at the neutral point. In the solution by Tao and Chen, the position of the neutral point depends only on the radius of the tunnel and the length of the bolt. That solution was implemented in the analytical models created by Indraratna and Kaiser and Hyett et.al. It seems that Tao and Chens solution is valid only when the deformation is compatible across the bolt interface. When decoupling occurs, the position of the neutral point is obviously also related to the shear strength of the interface. Field monitoring and pullout tests have indicated two facts concerning the loading of a rock bolt in situ: (1) rock deformation applied a load on the pick-up section of the bolt; (2) the load on the pick-up section drags the anchor section of the bolt towards the underground opening. These two facts must be taken into account in developing analytical models for rock bolts.The aim of this paper is to develop analytical models for fully coupled rock bolts. A model for rock bolts in pullout tests is introduced first, together with a description of the theoretical background, the development of the model and an illustrative example. Two models for rock bolts in situ are then presented, one in rock masses. The details of the development of the models are summarized in the appendices.2.Coupling between the bolt and the rockWindsor proposed the concept that a reinforcement system comprises four principal components: the rock, the reinforcing element, the internal fixture and the external fixture. For reinforcement with a bolt, the reinforcing element refers to the bolt and the external fixture refers to the face plate and nut. The internal fixture is either a medium, such as cement mortar or resin for grouted bolts, or a mechanical action like “friction” at the bolt interface for frictionally coupled bolts. The internal fixture provides a coupling condition at the interface. With reference to the component of internal fixture, Windsor classified the current reinforcement devices into three groups: “continuously mechanically coupled (CMC)”, “continuously frictionally coupled (CFC)”, “discretely mechanically or frictionally coupled (DMFC)” systems. According to this classification system, cement and resin-grouted bolts belong to the CMC system, while Split set and Swellex bolts belong to the CFC system.When fully grouted bolts are subjected to a pull load, failure may occur at the bolt grout interface, in the grout medium or at the grout rock interface depending on which one is the weakest. For fully frictionally coupled bolts, however, there is only one possibility if failure decoupling at the bolt rock interface. In this study we concentrate on the failure at the interface between the bolt and the coupling medium (either the grout medium or the rock).In general, the shear strength of an interface comprises three components: adhesion, mechanical interlock and friction. They are lost in sequence as the compatibility of deformation is lost across the interface. The result is a decoupling front that attenuates at an increasing distance from the point of the applied load. The decoupling front first mobilizes the adhesive component of strength, then the mechanical interlock component and finally the frictional component. The shear strength of the interface decreases during this process. The shear strength after the loss of some of the strength components is called the residual shear strength in this paper. For grouted rock bolts like rebar, all the three components of strength exist at the bolt interface. However, for the fully frictionally coupled bolt, the “Split set” bolt, only a friction component exists at the bolt interface. For Swelles bolts, mechanical interlock and friction comprise the strength of the interface.3. The theoretical background of rock bolts in pullout tests4.Concluding remarksAn analytical model has been established for rock bolts subjected to a pull load in pullout tests. Decoupling starts at the loading point and propagates along the bolt with an increasing applied load. The shear stress at the decoupled interface is lower than the ultimate shear stress strength of the interface and even drops to zero for fully grouted bolts, while it is approximately at the same magnitude as the ultimate shear stress strength for fully frictionally coupled interface decreases exponentially with increasing distance from the decoupling bolt.Two analytical models have been developed for rock bolts in situ, one for uniform rock deformation and another for discrete joint opening. For rock bolts in situ, the models confirm the previous findings: (i) in uniformly deformed rock masses, the bolt has a pick-up length, an anchor length and a neutral point;(ii) the face plate enhances the reinforcement effect through inducing a direct tensile load in the bolt and reducing the shear stress carried on the bolt surface;(iii) in jointed rock masses, the opening displacement of rock joint will induce axial stress peaks in the bolt.中文译文锚杆的分析模型C.Li*,B.Stillborg摘要：有三种锚杆的分析模型发展了起来：一种是在拉断试验中，易受到集中拉力载荷影响作用的锚杆，一种是安装在均匀变形岩体中的锚杆，另一种是易受到单个岩石节理影响作用的锚杆。这种分析模型是在注浆锚杆的锚杆与注浆之间或者是磨擦式锚杆的锚杆与岩石之间接触面上的机械耦合作用描述的基础上建立起来的。对于拉断试验中的锚杆，当接触面上的变形较小时，锚杆表面上的剪切应力随着距加载点距离的增加而成指数减小。如果施加的载荷足够大时，耦合首先发生加载点处，然后随着载荷的增加而逐渐向锚杆的深处传播。锚杆耦合部分的剪切应力的大小取决于接触面上的机械耦合作用。对于全长锚固锚杆来说，耦合阶段的剪切应力比接触面上的剪切强度的峰值要小，然而对于磨擦式锚杆，剪切应力大致和剪切强度的峰值相同。安装在均匀变形岩体中的锚杆，在建立锚杆分析模型时，锚杆的加载过程要考虑到岩体的变形情况。模型的模拟实验证实了先前的研究结果，在软岩中的锚杆有传感长度，锚固段长度，和一个中性点。这个实验也说明了锚杆托盘在围岩加固的效果中起着一个非常重要的作用。在有节理的岩体中，由于岩石节理的自由变形作用，锚杆轴向可能会有几个应力峰值发生在锚杆的延伸方向。1、前言在很长一段时间来，锚杆广泛的应用于民用建筑和矿业工程的岩石加固。锚杆通过在岩体中抑制岩体的变形来加固围岩。为了提高锚杆支护的结构，必须对在变形岩体中的锚杆的作用变化过程有一个良好的认识。这些认识可以通过现场监测、实验室的试验、数字模拟和研究分析来获得。自从20世纪70年代来，在不同的岩石地层中进行了大量的锚杆现场监测的研究工作。一个自由人士在Kielder的试验巷道中，进行了大量关于注浆锚杆特性的研究工作。他监测了锚杆的加载过程和应力沿锚杆的分布情况。在他所监测数据的基础上，他提出了关于“传感长度”、“锚固长度”、“中性点”的概念。在中性点上，锚杆和注浆之间的接触面上的剪切应力为零，然而在该点其轴向载荷的张力是一个峰值。传感长度指的是从接近锚杆末端的地方（在巷道壁上）到中性点的一段距离。在锚杆这部分是其剪切应力来自于岩石的载荷，并把锚杆向巷道方向进行拖拉。锚固长度指的是从锚杆的中性点到锚杆深处（固定在岩石深度）的一部分锚杆。在这部分上的剪切应力将锚杆锚固在岩石上。以上这些概念清楚的指出了安装在已变形岩层中的锚杆的作用变化过程。Bjornfot和Stephansson的研究工作证明，在已有节理的岩体中，由于单个节理的由自变形，在沿锚杆的方向上可能不仅存在一个中性点而且有可能存在多个中性点。锚杆的拉断试验通常用来监测锚杆的锚固能力，在不同种类的岩石中已经进行了大量的这种拉断试验工作测试。一著名人士进行了大量的基础工作来研究在拉力负荷的张力作用下锚杆的作用变化过程。他的解析方法指出：在锚杆发生耦合以前，锚杆的轴向应力（也可能是锚杆接触表面上的剪切应力）从加载点到锚杆的深处呈指数减小的趋势。图1（a）说明了这种典型拉断试验的结果，曲线a表示的是在相对较低的载荷情况下，沿锚杆方向轴向应力的分布情况，在这个图中可以看出，在锚杆锚固界面的两则，其变形是相等的。曲线b表示的是在相对较高的载荷下，沿锚杆方向轴向应力的分布，在此图上，锚杆接触面上已经发生了耦合作用。图1（b）表示的是安装在地下煤矿的主水平巷中的锚杆上的轴向应力分布情况。我们可以从这个图上看出，在接近钻孔口附近的轴向应力分布情况与在拉断试验中的分布情况完全不同。然而，锚杆深处阶段部分的的应力变化与拉断试验中的结果相似。出现这种情况的原因是，在软岩中的锚杆有传感长度和锚固长度，然而在拉断试验中的锚杆仅有锚固长度。图1在拉断试验中，（a）轴向应力沿在Hawkes和Evans之后的全锚固锚杆和(b)Sun之后的加固锚杆的分布我们认为在锚杆中性点上，岩石和锚杆之间的相对移动为零。在陶和陈的分析方法中，中性点的位置仅仅取决于巷道的半径和锚杆的长度。这种解决方法完善了由Kaiser和Hyett发明的分析模型。这看起来好在像陶和陈的解决方法只有当通过锚杆的界面点时，其变形量相互兼容时，才是有效的；当发生耦合后，中性点的位置与接触面的剪切应力强度有明显的关系。现场监测和拉断实验都表明在软岩中锚杆的载荷与两个因素有一定的关系：（1）当在锚杆的传感段施加一定的载荷时的岩石变形量；（2）把锚固段拉向地下巷道壁面的传感段的载荷。所以当建立锚杆分析的模型时，必须把这两个因素考虑进去。本论文的主要目的是建立一个耦合锚杆的分析模型。首先介绍的是一个在锚杆拉断实验中的锚杆模型，并且对其理论背景，模型的建立过程和说明的例子进行一下描述。然后说明两种在软岩中的锚杆的分析模型，一种是在均匀变形的岩体中，一种是在节理的岩体中。2、锚杆和岩石的联结Windsor指出锚杆的加固系统包含四个基本元件的概念：岩石、锚固构件、内部固定物和外部固定物。用锚杆进行加固围岩时，锚固构件是指锚杆；外部固定物是指锚杆托盘和螺冒。内部固定物是下面介质的两者或两者之一，例如锚注锚杆的水泥灰浆或树脂，或者是机械力学作用如摩擦式锚杆接触面上的摩擦力。内部固定物在锚杆的接触面上起到一种联结作用。由于上面所提到的内部固定物的构成不同，Windsor把目前的加固设施分为了三大类：“连续机械联结（CMC）”，“连续摩擦联结（CFC）”，“非连续机械或者摩擦联结（DMFC）”系统。通过这个分类，水泥赤浆和树脂锚固锚杆属于连续机械联结系统，而斯普利特（管缝）锚杆和斯韦莱克斯水胀锚杆属于连续摩擦式系统。当全长锚固锚杆受到拉力载荷的作用时，在注浆的接触面、注浆介质或是在注浆岩石的接触面上有可能会发生失效，这取决于它们之中那一个更加软弱。然而对于摩擦式锚杆，这里只有一种失效的可能性，即是发生在锚杆与岩石的耦合接触面上。在这项研究中，我们仅专注于锚杆与联结介质（或者是注浆介质或者是岩石）之间的耦合失效。通常，接触面的剪切应力强度包含三个方面的因素：粘附力、机械联结和摩擦。这些因素常在顺序上被忽视如锚杆的接触面的变形相等性被忽视等，结果使耦合面随着距加载点距离的增大而逐渐的衰减。这个耦合面首先能加强粘附元件的强度，然后就是机械联结元件，最后是摩擦元件。在些过程中，他们的剪切强度将会减小。当其中的一些强度元件失效后，在本论文中，其剪切强度叫做残余剪切强度。注浆锚杆如加固锚杆，其所有的三个强度元素均存在于锚杆的接触面上。然而，摩擦式锚杆、斯普利特锚杆仅有一个摩擦强度成分存在于锚杆的接触面上。斯韦莱克斯水胀锚杆中的机械联结力和摩擦力构成了其接触面的强度。3、锚杆拉断试验的理论背景4、结论一个锚杆在拉断试验中受到拉力作用的分析模型就这样建立起来了，耦合作用发生在锚杆的加载位置处，并且随着所加载荷的增加沿锚杆方向传播。全锚固锚杆在耦合界面的剪切应力小于最终接触面上的剪应力，甚至会降低到零。然而，摩擦式锚杆在此面上的剪切应力大致和最终的剪应力强度的大小相同。在没有耦合部分的锚杆上，其前应力随着距耦合界面的距离的增大而成指数方式减小。在软岩中建立了两种锚杆的分析模型，一种是在均匀变形的岩石中，另一种是不连续的节理面中。在软岩中的锚杆模型确定以先前的一个调查结果（1）在均匀变形的岩体中，锚杆有一个传感长度，一个锚固长度和一个中性点；（2）锚杆托盘通过增加锚杆的轴向拉力载荷和降低锚杆表面的剪应力来加固围岩的效果；（3）在有节理的岩体中，岩石处的节理的自由变形将会降低锚杆轴向的应力峰值。
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