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.中文譯文錨桿的分析模型C.Li*,B.Stillborg摘要：有三種錨桿的分析模型發(fā)展了起來(lái)：一種是在拉斷試驗(yàn)中，易受到集中拉力載荷影響作用的錨桿，一種是安裝在均勻變形巖體中的錨桿，另一種是易受到單個(gè)巖石節(jié)理影響作用的錨桿。這種分析模型是在注漿錨桿的錨桿與注漿之間或者是磨擦式錨桿的錨桿與巖石之間接觸面上的機(jī)械耦合作用描述的基礎(chǔ)上建立起來(lái)的。對(duì)于拉斷試驗(yàn)中的錨桿，當(dāng)接觸面上的變形較小時(shí)，錨桿表面上的剪切應(yīng)力隨著距加載點(diǎn)距離的增加而成指數(shù)減小。如果施加的載荷足夠大時(shí)，耦合首先發(fā)生加載點(diǎn)處，然后隨著載荷的增加而逐漸向錨桿的深處傳播。錨桿耦合部分的剪切應(yīng)力的大小取決于接觸面上的機(jī)械耦合作用。對(duì)于全長(zhǎng)錨固錨桿來(lái)說(shuō)，耦合階段的剪切應(yīng)力比接觸面上的剪切強(qiáng)度的峰值要小，然而對(duì)于磨擦式錨桿，剪切應(yīng)力大致和剪切強(qiáng)度的峰值相同。安裝在均勻變形巖體中的錨桿，在建立錨桿分析模型時(shí)，錨桿的加載過(guò)程要考慮到巖體的變形情況。模型的模擬實(shí)驗(yàn)證實(shí)了先前的研究結(jié)果，在軟巖中的錨桿有傳感長(zhǎng)度，錨固段長(zhǎng)度，和一個(gè)中性點(diǎn)。這個(gè)實(shí)驗(yàn)也說(shuō)明了錨桿托盤在圍巖加固的效果中起著一個(gè)非常重要的作用。在有節(jié)理的巖體中，由于巖石節(jié)理的自由變形作用，錨桿軸向可能會(huì)有幾個(gè)應(yīng)力峰值發(fā)生在錨桿的延伸方向。1、前言在很長(zhǎng)一段時(shí)間來(lái)，錨桿廣泛的應(yīng)用于民用建筑和礦業(yè)工程的巖石加固。錨桿通過(guò)在巖體中抑制巖體的變形來(lái)加固圍巖。為了提高錨桿支護(hù)的結(jié)構(gòu)，必須對(duì)在變形巖體中的錨桿的作用變化過(guò)程有一個(gè)良好的認(rèn)識(shí)。這些認(rèn)識(shí)可以通過(guò)現(xiàn)場(chǎng)監(jiān)測(cè)、實(shí)驗(yàn)室的試驗(yàn)、數(shù)字模擬和研究分析來(lái)獲得。自從20世紀(jì)70年代來(lái)，在不同的巖石地層中進(jìn)行了大量的錨桿現(xiàn)場(chǎng)監(jiān)測(cè)的研究工作。一個(gè)自由人士在Kielder的試驗(yàn)巷道中，進(jìn)行了大量關(guān)于注漿錨桿特性的研究工作。他監(jiān)測(cè)了錨桿的加載過(guò)程和應(yīng)力沿錨桿的分布情況。在他所監(jiān)測(cè)數(shù)據(jù)的基礎(chǔ)上，他提出了關(guān)于“傳感長(zhǎng)度”、“錨固長(zhǎng)度”、“中性點(diǎn)”的概念。在中性點(diǎn)上，錨桿和注漿之間的接觸面上的剪切應(yīng)力為零，然而在該點(diǎn)其軸向載荷的張力是一個(gè)峰值。傳感長(zhǎng)度指的是從接近錨桿末端的地方（在巷道壁上）到中性點(diǎn)的一段距離。在錨桿這部分是其剪切應(yīng)力來(lái)自于巖石的載荷，并把錨桿向巷道方向進(jìn)行拖拉。錨固長(zhǎng)度指的是從錨桿的中性點(diǎn)到錨桿深處（固定在巖石深度）的一部分錨桿。在這部分上的剪切應(yīng)力將錨桿錨固在巖石上。以上這些概念清楚的指出了安裝在已變形巖層中的錨桿的作用變化過(guò)程。Bjornfot和Stephansson的研究工作證明，在已有節(jié)理的巖體中，由于單個(gè)節(jié)理的由自變形，在沿錨桿的方向上可能不僅存在一個(gè)中性點(diǎn)而且有可能存在多個(gè)中性點(diǎn)。錨桿的拉斷試驗(yàn)通常用來(lái)監(jiān)測(cè)錨桿的錨固能力，在不同種類的巖石中已經(jīng)進(jìn)行了大量的這種拉斷試驗(yàn)工作測(cè)試。一著名人士進(jìn)行了大量的基礎(chǔ)工作來(lái)研究在拉力負(fù)荷的張力作用下錨桿的作用變化過(guò)程。他的解析方法指出：在錨桿發(fā)生耦合以前，錨桿的軸向應(yīng)力（也可能是錨桿接觸表面上的剪切應(yīng)力）從加載點(diǎn)到錨桿的深處呈指數(shù)減小的趨勢(shì)。圖1（a）說(shuō)明了這種典型拉斷試驗(yàn)的結(jié)果，曲線a表示的是在相對(duì)較低的載荷情況下，沿錨桿方向軸向應(yīng)力的分布情況，在這個(gè)圖中可以看出，在錨桿錨固界面的兩則，其變形是相等的。曲線b表示的是在相對(duì)較高的載荷下，沿錨桿方向軸向應(yīng)力的分布，在此圖上，錨桿接觸面上已經(jīng)發(fā)生了耦合作用。圖1（b）表示的是安裝在地下煤礦的主水平巷中的錨桿上的軸向應(yīng)力分布情況。我們可以從這個(gè)圖上看出，在接近鉆孔口附近的軸向應(yīng)力分布情況與在拉斷試驗(yàn)中的分布情況完全不同。然而，錨桿深處階段部分的的應(yīng)力變化與拉斷試驗(yàn)中的結(jié)果相似。出現(xiàn)這種情況的原因是，在軟巖中的錨桿有傳感長(zhǎng)度和錨固長(zhǎng)度，然而在拉斷試驗(yàn)中的錨桿僅有錨固長(zhǎng)度。圖1在拉斷試驗(yàn)中，（a）軸向應(yīng)力沿在Hawkes和Evans之后的全錨固錨桿和(b)Sun之后的加固錨桿的分布我們認(rèn)為在錨桿中性點(diǎn)上，巖石和錨桿之間的相對(duì)移動(dòng)為零。在陶和陳的分析方法中，中性點(diǎn)的位置僅僅取決于巷道的半徑和錨桿的長(zhǎng)度。這種解決方法完善了由Kaiser和Hyett發(fā)明的分析模型。這看起來(lái)好在像陶和陳的解決方法只有當(dāng)通過(guò)錨桿的界面點(diǎn)時(shí)，其變形量相互兼容時(shí)，才是有效的；當(dāng)發(fā)生耦合后，中性點(diǎn)的位置與接觸面的剪切應(yīng)力強(qiáng)度有明顯的關(guān)系?，F(xiàn)場(chǎng)監(jiān)測(cè)和拉斷實(shí)驗(yàn)都表明在軟巖中錨桿的載荷與兩個(gè)因素有一定的關(guān)系：（1）當(dāng)在錨桿的傳感段施加一定的載荷時(shí)的巖石變形量；（2）把錨固段拉向地下巷道壁面的傳感段的載荷。所以當(dāng)建立錨桿分析的模型時(shí)，必須把這兩個(gè)因素考慮進(jìn)去。本論文的主要目的是建立一個(gè)耦合錨桿的分析模型。首先介紹的是一個(gè)在錨桿拉斷實(shí)驗(yàn)中的錨桿模型，并且對(duì)其理論背景，模型的建立過(guò)程和說(shuō)明的例子進(jìn)行一下描述。然后說(shuō)明兩種在軟巖中的錨桿的分析模型，一種是在均勻變形的巖體中，一種是在節(jié)理的巖體中。2、錨桿和巖石的聯(lián)結(jié)Windsor指出錨桿的加固系統(tǒng)包含四個(gè)基本元件的概念：巖石、錨固構(gòu)件、內(nèi)部固定物和外部固定物。用錨桿進(jìn)行加固圍巖時(shí)，錨固構(gòu)件是指錨桿；外部固定物是指錨桿托盤和螺冒。內(nèi)部固定物是下面介質(zhì)的兩者或兩者之一，例如錨注錨桿的水泥灰漿或樹脂，或者是機(jī)械力學(xué)作用如摩擦式錨桿接觸面上的摩擦力。內(nèi)部固定物在錨桿的接觸面上起到一種聯(lián)結(jié)作用。由于上面所提到的內(nèi)部固定物的構(gòu)成不同，Windsor把目前的加固設(shè)施分為了三大類：“連續(xù)機(jī)械聯(lián)結(jié)（CMC）”，“連續(xù)摩擦聯(lián)結(jié)（CFC）”，“非連續(xù)機(jī)械或者摩擦聯(lián)結(jié)（DMFC）”系統(tǒng)。通過(guò)這個(gè)分類，水泥赤漿和樹脂錨固錨桿屬于連續(xù)機(jī)械聯(lián)結(jié)系統(tǒng)，而斯普利特（管縫）錨桿和斯韋萊克斯水脹錨桿屬于連續(xù)摩擦式系統(tǒng)。當(dāng)全長(zhǎng)錨固錨桿受到拉力載荷的作用時(shí)，在注漿的接觸面、注漿介質(zhì)或是在注漿巖石的接觸面上有可能會(huì)發(fā)生失效，這取決于它們之中那一個(gè)更加軟弱。然而對(duì)于摩擦式錨桿，這里只有一種失效的可能性，即是發(fā)生在錨桿與巖石的耦合接觸面上。在這項(xiàng)研究中，我們僅專注于錨桿與聯(lián)結(jié)介質(zhì)（或者是注漿介質(zhì)或者是巖石）之間的耦合失效。通常，接觸面的剪切應(yīng)力強(qiáng)度包含三個(gè)方面的因素：粘附力、機(jī)械聯(lián)結(jié)和摩擦。這些因素常在順序上被忽視如錨桿的接觸面的變形相等性被忽視等，結(jié)果使耦合面隨著距加載點(diǎn)距離的增大而逐漸的衰減。這個(gè)耦合面首先能加強(qiáng)粘附元件的強(qiáng)度，然后就是機(jī)械聯(lián)結(jié)元件，最后是摩擦元件。在些過(guò)程中，他們的剪切強(qiáng)度將會(huì)減小。當(dāng)其中的一些強(qiáng)度元件失效后，在本論文中，其剪切強(qiáng)度叫做殘余剪切強(qiáng)度。注漿錨桿如加固錨桿，其所有的三個(gè)強(qiáng)度元素均存在于錨桿的接觸面上。然而，摩擦式錨桿、斯普利特錨桿僅有一個(gè)摩擦強(qiáng)度成分存在于錨桿的接觸面上。斯韋萊克斯水脹錨桿中的機(jī)械聯(lián)結(jié)力和摩擦力構(gòu)成了其接觸面的強(qiáng)度。3、錨桿拉斷試驗(yàn)的理論背景4、結(jié)論一個(gè)錨桿在拉斷試驗(yàn)中受到拉力作用的分析模型就這樣建立起來(lái)了，耦合作用發(fā)生在錨桿的加載位置處，并且隨著所加載荷的增加沿錨桿方向傳播。全錨固錨桿在耦合界面的剪切應(yīng)力小于最終接觸面上的剪應(yīng)力，甚至?xí)档偷搅恪Ｈ欢?，摩擦式錨桿在此面上的剪切應(yīng)力大致和最終的剪應(yīng)力強(qiáng)度的大小相同。在沒(méi)有耦合部分的錨桿上，其前應(yīng)力隨著距耦合界面的距離的增大而成指數(shù)方式減小。在軟巖中建立了兩種錨桿的分析模型，一種是在均勻變形的巖石中，另一種是不連續(xù)的節(jié)理面中。在軟巖中的錨桿模型確定以先前的一個(gè)調(diào)查結(jié)果（1）在均勻變形的巖體中，錨桿有一個(gè)傳感長(zhǎng)度，一個(gè)錨固長(zhǎng)度和一個(gè)中性點(diǎn)；（2）錨桿托盤通過(guò)增加錨桿的軸向拉力載荷和降低錨桿表面的剪應(yīng)力來(lái)加固圍巖的效果；（3）在有節(jié)理的巖體中，巖石處的節(jié)理的自由變形將會(huì)降低錨桿軸向的應(yīng)力峰值。