外文翻译锚杆的力学分析及工程应用

英文原文Rock bolt mechanical analysis and its application to engineering Abstract Combined with the 3D FEM, end-anchored anchorage bolts were simulated by implicit anchorage bolt element. Implicit anchorage bolt elements hide in the elements of rock mass and extremely simplify the element subdivision. The calculated value of anchorage bolt stress is larger than the measured one for the most time. we further analyzed the reciprocity of anchorage bolt and rock mass, and then deduced the analytical equations of anchorage bolt stress and rock mass deformation under elasto-plastic state. The results indicate that it is essential to revise the anchorage bolts stress by using the formulas deduced when rock mass is softened or significantly deformed. Finally, a case study indicates that the calculated results agree with those measured. Some helpful methods are offerd for more accurate simulation of the support effect and anchorage bolts real stress state. Keywords implicit element of bolt , mechanical analysis on bolt , elasto-plastic analysisIntroduction Load transfer mechanism and strengthening effect of anchor bolts are the two major areas of anchor theory at home and abroad 1. Geotechnical investigators have achieved a great deal of helpful datum through numerical analysis, analog test and in-situ survey method 2,3. Reference 4 put forward implicit bolt element method, applied it to numerical calculation of large underground engineering and has gained good effect. References 5,6 used similar method to solve idiographic problems and developed it. By using Mindlins displacement solution, Reference 7 deduced the shearing force distribution along anchorage length of anchorage bolt which was pulled. The researches above just partially emphasize on either load transfer mechanism or strengthening effect. Few band the two together to study effectively. Using the case of large underground power station, we analyze the reciprocity of end-anchored bolt and surrounding rock deformation under elastoplastic state, and get some helpful conclusions.1 Basic analysis and simplified hypotheses The method of implicit bolt element extremely simplifies the element subdivision of FEM, further-more simulates the support effect of rock bolts preferably. The entire process of calculability is provided in Reference 4. The method of implicit bolt element distributes support effect averagely into surrounding rock elements. Many literatures take anchorage bolt and surrounding rock as complete com-patible deformation. However, the local influence of rock bolt ought to be taken into consideration in fact. In order to help deduce, the simplified hypotheses are settled as follows. (1) The anchored rock mass is elasto-plastic material which is double polygonal linear hardened. Shown in the Fig.1, E1 is the elastic modulus of rock mass; E2 is the deformation modulus of rock mass under plastic state. (2) The rock bolt is long enough to thread the plastic zone of surrounding rock, and the anchorage segment is also long enough.Fig.1 Bilinear constitutive model of anchored rock mass(3) The bolt is relatively long, and the interaction of two end-points of a rock bolt is weak. The deformation of rock bolt and surrounding rock under elasto-plastic stage are shown in Fig.2. S1 is the critical elastic displacement of external rock mass which is far from the bolt. With the stress releasing, when the displacement value of external rock nearby the bolt body reaches S1, the displacement value of external rock far from the bolt body should reach .Fig.2 Deformation of rock bolt and half-infinite body of surrounding rock2 Surrounding rock is under elastic stage The force of anchorage pallet is annular uniformed load on the external rock mass of half-infinite body. Supposing F is the axis force of anchor bolt, according to elasticity mechanics analysis, the center settlement of rock bolt is: （1）where u is the Poissons radio of surrounding rock; q is the converted uniformed load; a1 is the radius of drilling hole; a2 is the radius of anchorage pallet; E1 is the elastic modulus of surrounding rock. You Chuna8 simplified it as infinite body, utilized Kelvins solution, then deduced the shearing forces distribution of internal anchorage segment: （2）where, a is the radius of anchorage bolt; ,here Es is the modulus elasticity of anchorage bolt.According to the load-displacement reciprocal theorem, the displacement of outer anchorage endpoint brought by the arbitrary point force P along anchorage segment is: （3）Integrating Eqs.(2), (3), the displacement of outer anchorage end-point is （4）Supposing the relative displacement of external anchorage point and outer anchorage point is u by FEM calculation, so the real relative displacement considered anchorage bolt effect is (u-wf-wi). Constitute the balance equation of axial force of rock bolt as follows: （5）Displace the counterpoint of Eq. (5) by Eq.(1) and Eq.(4), can attain the real axial force of anchorage bolt: （6）3 Surrounding rock is under elasto-plastic stage It is a complicated process that the surrounding rock changes from critical elastic stage to critical plastic stage. We try to work out the two values of critical state, and get other middle values by linear interpolation. Under satisfying the condition of engineering precision, it can reflect that the axial force of anchorage bolt changes and anchorage rock mass de-forms under elasto-plastic stage. When surrounding rock is under critical elastic stage, the displacement value of external rock mass which is far from the bolt body is S1; the relative displacement of external anchorage point and outer anchorage point is u1. The value of S1 and u1 can be obtained by FEM calculation. Considering the local effect of rock bolt, the real relative displacement of external anchorage point and outer anchorage point is: （7）where, wf1 and wi1 are the displacement of external and outer anchorage point respectively, which are all aroused by the anchorage force of rock bolt. The value of wf1 and wi1 can be derived from Eqs. (1), (4), (6). With the load releasing, when surrounding rock deform further to critical plastic stage, the real relative displacement of two points of bolt is: （8）where, wi2 is the displacement of outer anchorage point, which is brought by the force of anchorage bolt Fs2 when surrounding rock is under critical plastic stage. Simplifying Eq. (8) and build the balance equation of the axis force and strain of anchorage bolt: （9）According to Eq. (2), wi2 of Eq. (9) can be given by Eq. (4). When surrounding rock is under critical plastic stage, the axis force of anchorage bolt can be derived from Eqs. (4), (9): （10）In fact, it is impossible that the whole rock mass is under plastic stage. However, according to the analysis of elasto-plastic mechanism, when the thickness of external rock mass is over one meter (this is easy to satisfy in actual engineering), the error of simplifying it as plastic stage is very small. So, displacing E1 by E2 in Eq. (1), S2 is given by: （11）When surrounding rock is under elasto-plastic stage, supposing the FEM calculating displacement of external rock mass is S, corresponding the relative displacement of two points of anchorage bolt is up, the settlement displacement of external anchorage point can be given by interpolation using Eq. (12): （12）Resembling with the second section, when surrounding rock is under elasto-plastic stage, the axis force of anchorage bolt can be given by: （13）4 Engineering case study Situated in the Wutai Country of Shanxi Province, Xilongchi pumped-storage hydropower station is a large project, its capacity is 1 200 MW, the caves of underground power station are made up of main and auxiliary powerhouse, main transformer hall, diversion tunnel, tailrace tunnel. Main powerhouse dimension is 150 m23 m50 m (length width height). Main transformer hall dimension is 130.4 m16.4 m17.5 m (length width height). The surrounding rock grade is , deformation modulus E1=9 GPa, E2=0.9 GPa, Poissons ratio u=0.26, cohesion C=0.95, friction angle=46.7. The 3D FEM model is shown in Fig.3.End-anchorage bolts are partially used on the top arch of main powerhouse and trans-former, the elastic module of rock bolts E1=210 GPa, the diameter =32mm, long-short alternate with 5 m and 7 m, its arrangement is shown in Fig.4. Fig.3 Opening elements of 3D FEM modelFig.4 Rock bolts arrangement of around cavesAfter excavation, the axis forces of anchorage bolts are shown in Table 1, the crack states of rock mass around caves without and with rock bolts are shown in Fig.5.Table 1 Comparison of calculation and measuring for the force of rock boltsSerial number of rock boltsCalculation value of complete compatible deformationCalculation value of improved methodReal value by measure1258996988175977872835862796254764349Fig.5 Crack state of surrounding rockFrom Table 1, we know that the axis forces of rock bolts by old calculation method are larger than real measured value, and the calculation values by this method fit better with the real ones. This dis-plays the method is correct. From Fig.5, we know that damage range is confined in small scope with anchorage bolts, it agrees with those measured, implicit anchorage bolts element simulates the support effect of rock bolts preferably.5 Conclusions (1) Implicit anchorage bolts method can extremely simplify the model subdivision of FEM model, effectively simulate the support effect of rock bolts. (2) By the mechanics analysis of anchorage bolt with surrounding rock under elastic and plastic stage, from Eqs. (6), (13), we know that the local influence effect is great when surrounding rock is soft rock or under plastic stage, so its necessary to revise the axis forces of rock bolts by the formula. (3) From theoretical analysis of surrounding rock deformation and anchorage bolt bearing forces, we give out the reason why the anchorage force of end-anchorage bolts are often smaller than that in the real engineering; from the practical structure of endanchorage bolt, with the augmentation of axis force, it is always the main factor that the medium of anchorage contact interface between bolt body and rock mass crack or glide.中文译文锚杆的力学分析及工程应用 摘 要 结合三维有限元法，锚固螺栓末端进行了模拟的隐式锚固螺栓的因素。隐式锚固螺栓的内容隐藏在内部岩体和极其简化的因素细分。计算出的价值锚固螺栓应力大于衡量一个大部分时间。我们进一步分析了相互作用锚固螺栓和岩体，然后推导出的分析方程锚固螺栓应力和岩体变形弹塑性状态。结果表明，有必要修改锚固螺栓应力公式推导使用时岩体软化或明显变形。最后，个案研究表明，计算结果与这些标准相吻合。一些有用的方法是提供支撑效果和锚固螺栓实际应力状态更加准确地模拟。关键词 隐式锚杆，力学分析螺栓，弹塑性分析 导 言在国内和国外荷载传递机理和加强锚杆锚固效果是两个主要领域的理论学说 1 。土力工程调查已取得了大量有益的数据，通过数值分析，模拟试验和现场调查方法 2,3 。参考 4 提出了隐式锚杆元方法，应用它的数值计算大型地下工程，并已取得良好的效果。参考文献 5,6 用类似的方法来解决具体问题和发展它。通过使用明德林的位移解，文献 7 推导出的剪切力分布锚固长度锚固螺栓被引用。上述研究只是部分强调任何荷载传递机理或加强锚杆锚固效果。一些结合两个理论共同进行有效研究。大型地下电站的使用情况下，我们分析了在弹塑性状态下相互作用锚杆和围岩变形，并得到一些有益的结论。1 基本分析和简化假设隐式的方法极为锚杆单元，简化了内容细分的有限元，况且模拟支持作用锚杆最好。参考文献提供了整个可计算的过程 4 。该方法的隐螺栓因素的影响，平均分配支撑效果到围岩要素。许多文献采取锚固锚杆与围岩完整毁坏变形。然而，当地的影响锚杆应该考虑到的事实。为了帮助推断，简化假设是解决如下。（ 1 ）锚岩体弹塑性材料，双多边形线性硬化。如图1显示， E1是弹性模量岩体; E2是变形模量的岩体塑性状态下。（ 2 ）锚杆是足够长绪塑性区围岩和锚固段也足够长的时间。图1 双线性本构关系锚岩体（ 3 ）螺栓相对较长，并相互作用的两个端点的锚杆薄弱。变形锚杆与围岩下弹塑性阶段如图2所示 。关键是一弹性位移外部岩体是远离螺栓。随着应力释放，当位移值的外部岩石附近的螺栓机构达到一位移值的外部岩石远离螺栓机构应达到。图2 变形锚杆和半无限体围岩2 围岩弹性阶段该部分的锚地托盘是环形单轴负载外部岩体半无限体。假设F是力锚杆，根据弹性力学分析，该中心解决锚杆是： （1）其中u是泊松围岩系数; q是转换的单轴的负荷; a1是钻孔半径;素a2是锚地托盘的半径; E1是弹性模量围岩。 尤春安 8 简化为无限体，利用开尔文的解决方案，然后推导出的剪切力分布内部锚地部分： （2）其中，a是锚固螺栓半径; ,Es是锚固螺栓弹性模量。根据负荷位移互等定理，位移外锚地端点所带来的力量任意点P沿锚固段： （3）根据公式（2），（3）得出，取代外部下锚结果是 （4）假如相对位移的外部锚地点和外点是的有限元计算，所以真正的相对位移认为锚固螺栓效果（u-wf-wi）。构成平衡方程的轴向力的锚杆如下： （5）根据公式（ 5 ）、（ 1 ）和公式（ 4 ），才能实现真正的轴向力锚固螺栓： （6）3 下围岩弹塑性阶段这是一个复杂的过程，围岩变化的关键阶段，关键的弹性塑料阶段。我们尝试了两个值的临界状态，并获得其他中间值线性插值。满足条件下的工程精度，它可以反映，轴向力的变化和锚固螺栓锚固岩体去形式下弹塑性阶段。 当围岩弹性正在关键阶段，位移值的外部岩体是远离螺栓机构中一;相对位移的外部锚地停泊点和外点是U1路。价值的S1和U1路可以通过有限元