Homogenization framework for three‐dimensional elastoplastic finite element analysis of a grouted pipe‐roofing reinforcement method for tunnelling

This paper deals with the grouted pipe‐roofing reinforcement method that is used in the construction of tunnels through weak grounds. This system consists on installing, prior to the excavation of a length of tunnel, an array of pipes forming a kind of ‘umbrella’ above the area to be excavated. In some cases, these pipes are later used to inject grout to strengthen the ground and ‘connect’ the pipes. This system has proven to be very efficient in reducing tunnel convergence and water inflow when tunnelling through weak grounds. However, due to the geometrical and mechanical complexity of the problem, existing finite element frameworks are inappropriate to simulate tunnelling using this method. In this paper, a mathematical framework based on a homogenization technique to simulate ‘grouted pipe‐roofing reinforced ground’ and its implementation into a 3‐D finite element programme that can consider stage construction situations are presented. The constitutive model developed allows considering the main design parameters of the problem and only requires geometrical and mechanical properties of the constituents. Additionally, the use of a homogenization approach implies that the generation of the finite element mesh can be easily produced and that re‐meshing is not required as basic geometrical parameters such as the orientation of the pipes are changed. The model developed is used to simulate tunnelling with the grouted pipe‐roofing reinforcement method. From the analyses, the effects of the main design parameters on the elastic and the elastoplastic analyses are considered. Copyright © 2004 John Wiley & Sons, Ltd.     

A homogenization approach for evaluating the longitudinal shear stiffness of reinforced soils: column versus cross trench configuration

The macroscopic linear elastic behaviour of inclusion‐reinforced soils, regarded as periodic composite media, is investigated by means of the homogenization theory. Special attention is given here to the determination of their longitudinal shear stiffness properties, which strongly govern the reinforced ground response under lateral loading. Combining the use of analytical, variational and numerical methods, we thoroughly examined three particular engineering‐relevant configurations: single trench, column and cross trench reinforcements. Fairly accurate closed‐form expressions are thus obtained, giving the value of the reinforced soil longitudinal shear stiffness as a function of the individual components shear moduli and reinforcement volume fraction. It is shown in particular that adopting a cross trench reinforcement layout instead of the classical column configuration results in a much higher improvement of the longitudinal shear stiffness. The results are then applied to assessing the reduction of soil liquefaction risk, which can be attributed to the presence of the reinforcing inclusions. Again, they clearly demonstrate the excellent performance of the cross trench configuration as compared with the complete inefficiency of the column reinforcement technique. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability analysis of membrane‐reinforced curved earth retaining walls,using a multiphase approach

The stability analysis of curved earth retaining walls, stabilized by reinforcing membranes, is investigated by means of a multiphase model developed in the framework of the yield design approach. This model is an extension of that previously developed for soils reinforced by linear inclusions. It combines the advantage of a homogenization approach in terms of improved computational efficiency, with its capability to account for a specific soil–reinforcement failure condition, in a rational and systematic way. Application of this model is performed on the illustrative example of a cylindrical‐reinforced retaining wall by means of the kinematic approach of yield design, which provides upper bound estimates for the retaining wall stability factor. Nondimensional charts are finally presented assessing the influence of relevant parameters such as the curvature of the wall, the length of the reinforcing membranes or the reinforcement pull‐out resistance. Copyright © 2009 John Wiley & Sons, Ltd.     

Extrusion movements of a tunnel head reinforced by finite length bolts—a closed‐form solution using homogenization approach

An analytical model on the general behaviour of a tunnel head, reinforced by finite length bolts is proposed. This model is based on the homogenization method and spherical symmetry assumption. Despite its simplicity, and in consequence its limits of validity, it does allow a quick estimation of the key design parameters: frontal displacement, extension of decompressed zone, ground stresses and bolt tension, and constitutes thereby a very useful and handy tool for design engineers. In particular, the influence of the reinforcement length, as well as other important design parameters, are studied by the proposed model. The charts resulting from the parametric studies are directly applicable. Otherwise, the comparison to a 3D numerical model is also presented in this paper. The first results provide the validation of the analytical solution, at least in terms of average extrusion movements. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element simulations of the behavior of piled‐raft foundations using a multiphase model

Finite element simulations of the behavior of a piled raft foundation have been carried out using a multiphase model conceived as an improved homogenization approach. According to this model, the ground reinforced by a group of piles is treated as a homogeneous continuous medium. In this approach, no specific interface elements are necessary to account for the mechanical interaction between the piles and the ground: this interaction is described by means of two scalar parameters, one stiffness parameter and one which can easily be derived from the maximum ground‐pile friction. The implementation of the model into a finite element code provides an efficient tool for the analysis of the influence of the pile number or length on the settlement and bearing capacity of a square piled raft foundation and of the way the total applied load is shared between the raft and the piles. Results are compared with a standard 3D finite element analysis. The comparison highlights the fact that the proposed approach remains to be improved to account for tip resistance. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite element analysis of the effect of corrosion on the behavior of reinforced earth walls

A simple model for the corrosion‐induced loss of stiffness and strength of the steel strips of earth‐reinforced walls was introduced in a finite element simulation of the long‐term behavior of the wall, in which the backfill‐strips interactions are taken into account by means of a generalized homogenization procedure (called a multiphase model). The results show an initial phase of slow displacements induced by the loss of stiffness, followed after a few decades by a steep acceleration of the displacements, leading to wall failure. The influences of the parameter controlling corrosion, the backfill cohesion and the heterogeneity of the corrosion process are discussed. Results are used to discuss a strategy for reinforced earth wall surveillance. Copyright © 2011 John Wiley & Sons, Ltd.     

Estimating the degree of reinforcement of horizontally bedded roofs strengthened by non-tensioned,full-column grouted bolts

Summary Roof sag or convergence during mining of horizontally bedded strata is accompanied by relative shear displacements along the separation planes. These displacements and therefore the roof convergence can be reduced by mobilizing the shear stiffness and resistance of full-column, non-tensioned bolts installed across the layers. Ultimately, if sufficient bolts are installed, there would be a negligible difference between the deflection of the bolted layers and that of a single layer of the same total thickness.In this paper a limiting equilibrium method is presented for determining the degree to which ideal reinforcement is reached.A number of examples are used to illustrate the use of the method for estimating the degree of reinforcement of a given rock bolt pattern, or for determining the bolt pattern required to attain a required degree of reinforcement. Both uniform and non-uniform loadings of the roof strata are considered.     

Multiphase model for inclusion‐reinforced geostructures: Application to rock‐bolted tunnels and piled raft foundations

A multiphase model is proposed to describe the mechanical behaviour of geomaterials reinforced by linear inclusions. This macroscopic approach considers the reinforced soil or rock mass as the superposition of continuous media. Equations of motion and constitutive laws of the model are first derived. Its implementation in a finite element computer code is then detailed. A modified implicit algorithm for elastoplastic problems is proposed. The model and its implementation are fully validated for rock‐bolted tunnels (comparison with scale model experiments) and piled raft foundations (comparison with the classical ‘hybrid method’). The Messeturm case history is finally presented to assess the handiness of the approach for real structures. Copyright © 2001 John Wiley & Sons, Ltd.     

Stability analysis of an embankment resting upon a column‐reinforced soil

The design against failure of an embankment resting upon a soft soil improved by a group of columns is investigated with the help of the yield design homogenization approach. Assuming that both constituents of the reinforced ground are purely cohesive materials (‘lime column’ technique), an upper bound estimate for the macroscopic strength condition of the reinforced soil as a homogenized medium is first obtained, providing definite evidence of a shear strength anisotropy associated with the reinforcement preferential orientation. The kinematic method of yield design is then performed on the basis of such a criterion, making use of rotational failure mechanisms involving slip circles in the reinforced ground. Upper bound estimates are finally obtained for the embankment stability factor, as functions of the degree of reinforcement and relative thickness of the soil layer. These results are compared with those derived from a simplified analysis, where the reinforced soil is assumed to exhibit an averaged isotropic cohesion. This comparison clearly indicates that the latter simplified analysis may produce quite unsafe estimates for the embankment stability, which can be attributed to the fact that it fails to capture the inherent strength anisotropy of the reinforced soil. Copyright © 2010 John Wiley & Sons, Ltd.     

Modeling strategy for jointed rock masses reinforced by rock bolts in tunneling practice

This paper deals with the modeling of jointed rock masses reinforced by rock bolts. It is well known that rock bolts are extremely effective in reinforcing jointed rocks. However, if a continuum approach is adopted for modeling jointed rock masses, it is often misleading to evaluate the effectiveness of the rock bolts by numerical analyses such as the finite element method. This may be due to the fact that since no more joints exist in the continuum, the effectiveness of the rock bolts in constraining the relative displacements along the joints cannot be evaluated properly. In order to investigate the reinforcement effect of rock bolts, physical model tests were performed in the laboratory. The test results revealed that jointed rock masses should be modeled as an equivalent continuum after the installation of rock bolts and that the mechanical parameters of the equivalent continuum should be evaluated by considering the reinforcement effect of the rock bolts. Therefore, the values of the mechanical parameters differ from place to place in accordance with the relation between joint orientation and rock bolt direction, even though joint systems are homogeneous. In conclusion, in the continuum approach for modeling jointed rock masses, it should be emphasized that rock bolts and jointed rock masses should not be modeled separately but should be modeled simultaneously by considering the reinforcement effect of the rock bolts in constraining joint movement. The modeling of shotcrete reinforced by steel ribs is also discussed in comparison to the modeling by rock bolts.     

大变形锚杆-围岩耦合模型及其计算方法

在矿山、水利、交通等工程领域中大变形锚杆已经获得了广泛的认可和应用,而相应理论水平落后于工程实践的现状则限制了大变形锚杆支护技术的进一步发展。针对大变形锚杆的力学及变形特性,提出了一种锚杆-围岩耦合作用结构模型,并基于塑性增量理论,从锚杆-围岩相互作用的角度,提出了大变形锚杆加固岩体的求解方法,推导了加锚岩体的平衡方程、位移协调方程和锚杆响应方程。在Visual Basic开发环境中编制有限差分计算程序,利用龙格库塔法求得了大变形锚杆-围岩耦合模型的半解析解,并通过数值模拟验证了理论模型的有效性。基于上述理论模型和计算方法,不仅可以计算得到大变形锚杆轴力和剪应力分布情况,定量分析其支护效应,还可以系统地求得加锚岩体的地层响应曲线及锚杆自身的响应曲线,对于地下工程中大变形锚杆的支护设计有基础性指导意义。     

锚杆支护圆形隧洞的等效强度参数及可靠性分析

通过采用均匀化方法,研究了圆形隧洞的锚杆支护特性,将高密度支护模式下的岩石和锚杆复合体考虑成均匀、连续、强度参数增强的等效材料,简化了岩石和锚杆间复杂的力学耦合问题。通过定义锚杆密度参数来反映不同支护模式的特性,建立锚杆密度参数与Mohr-Coulomb屈服准则中主要参数之间的关系,推导出等效弹性模量、等效黏聚力和等效内摩擦角的表达式,并分析比较了隧洞在支护前后的位移情况。结合可靠性理论,采用容许极限位移量作为失稳判据,分析了隧洞在支护前后的可靠性指标与破坏概率,结果表明,文中提出的方法简单可靠,锚杆支护对隧洞的位移限制效果明显,可显著提高隧洞的可靠性。     

锚杆长度和间距对洞室抗爆性能影响研究

根据Froude相似理论,通过模型试验,对普通锚杆、中密锚杆、长密锚杆、短密锚杆、长短相间密锚杆加固洞室的抗爆性能进行了研究,分析了不同长度、间距的全长粘结式锚杆加固洞室在围岩应力、洞壁加速度、洞室拱顶底板相对位移、洞壁环向应变、洞室破坏形态等方面的不同特点。采用FLAC3D对模型试验进行了模拟,并利用参数化建模方法分析了洞顶位移随锚杆长度、间距的变化规律。研究结果表明：减小锚杆间距比增加锚杆长度更能有效地提高洞室抗爆能力,且锚杆间距必须达到一定密度时才可以阻止围岩裂缝进入锚固区;在锚杆长度相同的情况下,密锚杆能明显减小洞室拱顶加速度峰值、拱顶底板相对位移峰值、残余值和拱脚部位压应变峰值、残余值;当锚杆长度增加到一定程度后,加固的效果并不明显,而且带来了底板加速度峰值、拱顶拉应变峰值的增加;锚杆的最佳长度可取为1/3洞室跨度,锚杆的最佳间距可取为1/15洞室跨度。     

Analytical model for the design of grouted rock bolts

An analytical model which represents the behaviour of a reinforced rock mass near a circular underground opening in a homogeneous, uniform stress field has been developed. The theory adopts the concepts of elastoplasticity and considers a proper interaction mechanism between the ground and the grouted (or friction) bolts. It highlights the influence of the bolt pattern on the extent of the yield zone and tunnel deformation. A dimensionless parameter is introduced as a design tool which relates the tunnel convergence to the bolt spacing for a given bolt length. This publication contains the derivation of the analytical model and an illustration of the effect of bolts on the stress and displacement field near an opening. Its application to tunnel design is discussed briefly. The verification of the theory by laboratory simulation and field measurements will be presented, in detail, in a future publication.     

Elastoplastic multiphase model for simulating the response of piled raft foundations subject to combined loadings

A multiphase model and corresponding computational time‐saving finite element code is proposed in this paper for predicting the settlements experienced by a piled raft foundation when subject to the combined action of vertical and lateral loadings. This model, which is formulated in the framework of an elastoplastic behaviour for the soil and the reinforcing piles as well, explicitly accounts for the shear and flexural behaviour of the latter. Starting from a simple analytical example where all the concepts attached to this model are clearly illustrated, the main stages leading to its finite element implementation are then presented. The numerical tool thus elaborated, is applied to the simulation of a pile‐reinforced strip foundation submitted to a horizontally applied seismic load in addition to a permanent vertical load. One of the key results of such a simulation in terms of design recommendation, lies in the conclusion that, while the shear and flexural contributions of the reinforcement play quite a negligible role in the case of a vertical load (as compared with their axial resistance), they remain absolutely essential for withstanding the seismic lateral loading. Copyright © 2006 John Wiley & Sons, Ltd.     

加筋材料动态松弛有限元模型的建立及应用

为分析加筋材料的抗弯刚度对加筋性能的影响,加筋材料采用梁单元形式。基于动态松弛法,通过定义梁单元的刚度矩阵,求解内力矢量,随后定义虚拟质量密度而建立总质量矩阵,将加筋材料的梁单元有限元模型嵌入到已有的动态松弛法求解程序中。通过对简支梁的简单加载模拟验证了该梁单元模型的准确性能。随后,将该有限元模型与已有的动态松弛法计算程序结合（含砂土本构及弱面单元模型）,对加筋砂土地基室内模型试验进行了数值模拟。将梁单元的模拟结果与杆单元（梁单元的特例）模拟结果进行了比较,并分别探讨了抗拉刚度和抗弯刚度对加筋砂土地基承载性能的影响。结果表明：抗拉刚度对承载能力的影响较小;抗弯刚度对承载力的影响程度与加筋材料的布置形式有关,特别是当加筋砂土中出现剪切带以后,其影响逐渐增大。因此,在分析加筋砂土结构的增强机制时,建议采用梁单元（具有一定的抗弯刚度）对加筋材料进行模拟。     

FE analysis of grid reinforced embankment system on soft Bangkok clay

The behavior of a reinforced embankment on soft Bangkok clay has been analyzed by plane strain finite element method. The finite element analysis considers the selection of proper soil/reinforcement properties according to the relative displacement pattern of upper and lower interface elements. The large deformation phenomenon is simulated by updating the node coordinates, including those of the embankment elements above the current construction level, which ensures that the applied fill thickness simulates the actual field value. A full scale test reinforced embankment with a vertical face (wall) on Bangkok clay has been analyzed by the proposed finite element method, and the numerical results are compared with the field data. The response of a reinforced embankment on soft ground is principally controlled by the interaction between the reinforced soil mass and soft ground and the interaction between the grid reinforcement and the backfill soil. The tension in reinforcement and lateral displacement of the wall face varied during consolidation of foundation soil. The maximum tension force occurred in the reinforcement layer placed at the base of reinforced mass, due to bending of the reinforced mass resulting from differential settlements. It is considered necessary to account for the permeability variation of the soft ground foundation in the finite element analysis.     

锚杆对围岩的加固效果和动载响应的数值分析

利用数值分析软件LS-DYNA3D程序,对锚杆对围岩的加固效果和动态力学性能进行了显示动力分析,比较了洞室围岩是否用锚杆加固时,爆炸波引起的岩体中垂直应力、拱顶位移以及锚杆本身轴向应变的变化特点。结果发现：同毛洞相比较,加固洞室围岩应力较大,拱顶位移较小;通过分析锚杆轴向应变-时程曲线,发现不同安装角的锚杆其轴向应变对动载的响应不同,拱部锚杆先受压后受拉,直墙部锚杆全受拉,两部位锚杆均产生轴向拉应变;将模拟的锚杆应变-时程曲线同模型试验的相应曲线进行了比较,两者一致性较好,其结果为动载作用下坑道围岩锚杆加固的布置方法提供了参考。     

Mechanism and countermeasures of preceding tunnel distortion induced by succeeding EPBS tunnelling in close proximity

Twin tunnels are frequently used to address the increasing transportation demands in large cities. To ensure the safety of twin tunnels in close proximity, it is often necessary to take protective measures that have not been well studied. Field monitoring was conducted for a project of twin earth pressure balance shield (EPBS) tunnels in typical soft ground. The preceding tunnel was reinforced by various measures, including trailer bracing, compensation grouting, artificial freezing and scaffold bracing. The entire deformation of the reinforced tunnel was recorded during the succeeding tunnelling process. A three dimensional finite-element method (FEM) model was established to simulate the entire process of twin EPBS tunnelling, particularly the reinforcement measures. The computed deformations of the reinforced tunnel were consistent with the measured data. Furthermore, the stress history and pore pressure of the surrounding soil were analysed to investigate the deformation mechanism of the tunnel. Both the measured and computed results indicate that although the face pressure of the succeeding tunnel was smaller than the earth pressure at rest, the preceding tunnel could still experience an inward horizontal convergence and a deflection away from the succeeding tunnel. These distortion modes were caused by the squeezing effect of the horizontal soil arch in front of the succeeding tunnel face. Finally, convergence and deflection indices were proposed to quantify and assess the effectiveness of the reinforcement measures. The trailer bracing, as an “in-tunnel” reinforcement technique, was found to be the most effective method for controlling tunnel convergence. However, artificial freezing as an “out-tunnel” reinforcement technique led to the largest reductions in tunnel deflection. A combination of both “in-tunnel” and “out-tunnel” reinforcements was recommended.