A Timoshenko finite element straight beam with internal degrees of freedom

We present hereafter the formulation of a Timoshenko finite element straight beam with internal degrees of freedom, suitable for nonlinear material problems in geomechanics (e.g., beam type structures and deep pile foundations). Cubic shape functions are used for the transverse displacements and quadratic for the rotations. The element is free of shear locking, and we prove that one element is able to predict the exact tip displacements for any complex distributed loadings and any suitable boundary conditions. After the presentation of the virtual power and the weak form formulations, the construction of the elementary stiffness matrix is detailed. The analytical results of the static condensation method are provided. It is also proven that the element introduced by Friedman and Kosmatka in 1 , with shape functions depending on material properties, is derived from the new beam element. Validation is provided using linear and material nonlinear applications (reinforced concrete column under cyclic loading) in the context of a multifiber beam formulation. Copyright © 2015 John Wiley & Sons, Ltd.     

活动性断裂带错动下隧道纵向响应的解析解

震害调研表明,活动性断裂带区域的隧道灾害最为严重。针对错动作用下穿越活动性断裂带隧道的纵向响应进行了研究,推导了隧道纵向力学响应的解析解并进行了验证。考虑断裂破碎带围岩力学性质较差且处于错动变形的主要影响区,将隧道沿纵向进行分区,包括错动影响区、过渡影响区和非影响区。采用Pasternak双参数弹性地基梁,假定不同分区的地基参数和计算模式不同,建立了满足变形和内力连续的隧道纵向力学解析模型并进行了求解。解析计算结果与数值模拟结果、室内试验数据基本一致,验证了解析解的正确性。结果表明:错动作用下,活动性断裂带区域的隧道内力和变形发生了显著变化;隧道纵向挠曲变形与错动方向一致,但在断裂带与上下盘交界区域发生了反向的挠曲;在正断层错动下,纵向弯矩在断裂带与上下盘交界区域达到最大值,且上、下盘区域的隧道拱顶分别出现受拉和受压区域;断裂带区域内的剪力远大于其他区域,且受到较大弯矩,隧道结构易发生破坏。上述计算结果与实际调研结果相一致,表明了提出的解析计算方法可用于活动性断裂带错动下的隧道纵向响应分析。最后,针对地基系数和断裂带宽度两个关键参数进行了敏感性分析,得到了有益规律,可为该类区域的隧道设...     

Simplified method for evaluating the response of existing tunnel induced by adjacent excavation

Adjacent excavation may have a negative influence on the existing tunnel underneath. Thus, it is important to evaluate the response of the tunnel due to adjacent excavation. However, there is little report about using the Kerr foundation model to simulate the tunnel-soil interaction. Meanwhile, the Timoshenko beam, which can take the tunnel shearing effect into consideration, is more suitable to estimate the behavior of the tunnel. To simulate the interaction between soil and tunnel, the existing tunnel is simplified as a Timoshenko beam lying on the Kerr foundation model, and a simplified theoretical method is proposed to calculate the response of the existing tunnel induced by adjacent excavation. The proposed method is validated by two field case studies. Results indicate that the predictions given by the proposed method show great agreement with field measurements and it is more accurate to evaluate the tunnel-soil interaction compared with the previous method. The further parametric study shows that the relative position between excavation and tunnel, the ground Young's modulus, the depth of existing tunnel centerline, and length and width of excavation are both significant factors governing the tunnel response induced by adjacent excavation, while the influence of tunnel shear stiffness and skew between tunnel and excavation are slight. The proposed method can be applied to predict the potential risk of existing tunnels induced by adjacent excavation in relevant engineering projects.     

Load–Deflection response of transversely isotropic piles under lateral loads

In general, pile materials are assumed to be isotropic during the analysis of the load–deflection response of piles under lateral loads. However, commonly used materials such as reinforced concrete and timber as well as potentially promising new pile materials such as fiber reinforced polymers are typically transversely isotropic materials. Experimental studies have shown that transversely isotropic materials have a high ratio of section longitudinal modulus to the section in‐plane shear modulus (E_zz/G_xz) compared to the value for isotropic materials. The high modulus ratio leads to a more significant shear deformation effect in beam bending. To account for the shear deformation effect, the Timoshenko Beam Theory has been adopted in deriving the solutions for the load–deflection response of transversely isotropic piles under lateral loads instead of the Classical (Euler–Bernoulli) Beam Theory. The load–deflection responses depend on the shear effect coefficient, the lateral soil resistance, the embedment ratio, and the boundary conditions. The deflection of the pile, if the shear deformation effect is considered, is always larger than if it is neglected. Copyright © 2000 John Wiley & Sons, Ltd.     

An analytical solution for deforming twin‐parallel tunnels in an elastic half plane

The interaction between twin‐parallel tunnels affects the tunnelling‐induced ground deformation, which may endanger the nearby structures. In this paper, an analytical solution is presented for problems in determining displacements and stresses around deforming twin‐parallel tunnels in an elastic half plane, on the basis of complex variable theory. As an example, a uniform radial displacement was assumed as the boundary condition for each of the two tunnels. Special attention was paid to the effects of tunnel depth and spacing between the two tunnels on the surface movement to gain deep insight into the effect of the interaction between twin‐parallel tunnels using the proposed analytical approach. It is revealed that the influence of twin tunnel interaction on surface movements diminishes with both the increase of the tunnel depth and the spacing between the two tunnels. The presented analytical solution manifests that, similar to most of the existing numerical results, the principle of superposition can be applied to determine ground deformation of twin‐parallel tunnels with a certain large depth and spacing; otherwise, the interaction effect between the two tunnels should be taken into account for predicting reliable ground movement. Copyright © 2014 John Wiley & Sons, Ltd.     

Analytical solution for longitudinal seismic response of long tunnels subjected to Rayleigh waves

The longitudinal seismic response of a long tunnel subjected to Rayleigh waves is investigated in this paper. The tunnel is assumed to be infinitely long, has a uniform cross section, and rests on a viscoelastic foundation. The free-field deformation under Rayleigh waves traveling parallel to the tunnel axis is decomposed into two directions, namely, the axial motion and the vertical motion, and transformed into dynamic loads imposed on the tunnel. Based on the Fourier and Laplace integral transform techniques, the governing equations of tunnels are simplified into algebraic equations, and the analytical solutions are obtained with the convolution theorem. The final solutions of the tunnel responses in terms of deflection, velocity, acceleration, axial force, bending moment, and shear force are investigated. The proposed solution is verified by comparison of its results and those from the finite element program ABAQUS. Further parametric analysis is carried out to investigate the influence of soil-structure relative stiffness ratio and wave frequency on dynamic longitudinal responses of the tunnel.     

Analytical solutions for Euler–Bernoulli beam on visco‐elastic foundation subjected to moving load

Analytical solutions for the steady‐state response of an infinite beam resting on a visco‐elastic foundation and subjected to a concentrated load moving with a constant velocity are developed in this paper. The beam responses investigated are deflection, bending moment, shear force and contact pressure. The mechanical resistance of the foundation is modeled using two parameters k_s and t_s — k_s accounts for soil resistance due to compressive strains in the soil and t_s accounts for the resistance due to shear strains. Since this model represents the ground behavior more accurately than the Winkler spring model, the developed solutions produce beam responses that are closer to reality than those obtained using the existing solutions for Winkler model. The dynamic beam responses depend on the damping present in the system and on the velocity of the moving load. Based on the study, dynamic amplification curves are developed for beam deflection. Such amplification curves for deflection, bending moment, shear force and contact pressure can be developed for any beam‐foundation system and can be used in design. Copyright © 2012 John Wiley & Sons, Ltd.     

Moving load response of an axially loaded Timoshenko beam on a multilayered transversely isotropic half-space comprising different media

This paper investigates the dynamic response of an axially loaded Timoshenko beam coupled with a multilayered transversely isotropic (TI) half-space subjected to a moving load. An axial force induced by the thermal expansion is taken into account in the Timoshenko beam. The half-space considers the alternate distribution of an arbitrary number of TI elastic and poroelastic layers to model foundation soils with different properties and moisture conditions. To solve the governing equations, Fourier transform is adopted. The stratified foundation is formulated by extending an “adapted stiffness matrix method” to a more general scenario with an arbitrary number of layers. The beam is then coupled with the foundation to derive solutions to the system in the frequency-wavenumber domain. The final results in the time-spatial domain are recovered by the inverse Fourier transform. After confirming the accuracy of the method in this study, the influences of the pore water existence, the transverse isotropy of different parameters, and the axial force are investigated. It can be observed that the effect of pore water existence on the maximum beam deflection can reach 22% in this study. The transverse isotropy of the elastic and shear moduli influences the critical speed of the beam deflection by altering the phase velocity of the first wave propagation mode of the beam-foundation system. The vertical permeability coefficient is more important than the horizontal one in determining the excess pore pressure. The rise of the beam temperature (axial force) decreases the critical speed and magnifies the vibrations.     

考虑椭圆化地层变形影响的浅埋隧道弹性解

研究了浅埋隧道周围地层最终变形的计算问题,提出了考虑椭圆化变形影响的计算模型和浅埋隧道位移边界条件,并利用Airy应力函数得到了该问题弹性解析解。在此基础上,通过算例分析了椭圆化变形对竖向位移和水平位移的影响。结果表明,椭圆化变形对水平位移的影响远大于竖向位移。     

Deformation analysis of tunnel excavation below existing pipelines in multi‐layered soils based on displacement controlled coupling numerical method

The effect of tunneling on surrounding environments, especially on existing buried pipelines is a problem that engineers designing and practicing in urban geotechnical environments encounter more frequently than in the past. However, previous studies are usually based on the assumption that the soil is homogeneous. How to reflect soil stratification is the main focus for the problem of tunneling in multi‐layered soils. A displacement controlled coupling numerical method is presented for the displacement analysis of tunnel excavation below existing pipelines in multi‐layered soils. On the basis of the layered soil model, to consider the soil nonhomogeneous characteristic, the finite element method and boundary element method are coupled to simulate the deformation of existing pipelines induced by tunneling. The solutions indicate that good agreements are obtained between the proposed coupling numerical method and the commercial software. The accuracy of the proposed numerical method is better than the two stages method based on the existing closed‐form solutions. Moreover, the results discussed in this paper show that the error obtained by the previous method of weighted average on the basis of homogeneous half space converted from layered soils is not negligible for the obvious difference of elastic parameters among successive layers. Copyright © 2012 John Wiley & Sons, Ltd.     

Axial kinematic response of end‐bearing piles to P waves

Kinematic pile–soil interaction under vertically impinging seismic P waves is revisited through a novel continuum elastodynamic solution of the Tajimi type. The proposed model simulates the steady‐state kinematic response of a cylindrical end‐bearing pile embedded in a homogeneous viscoelastic soil stratum over a rigid base, subjected to vertically propagating harmonic compressional waves. Closed‐form solutions are obtained for the following: (i) the displacement field in the soil and along the pile; (ii) the kinematic Winkler moduli (i.e., distributed springs and dashpots) along the pile; (iii) equivalent, depth‐independent, Winkler moduli to match the motion at the pile head. The solution for displacements is expressed in terms of dimensionless transfer functions relating the motion of the pile head to the free‐field surface motion and the rock motion. It is shown that (i) a pile foundation may significantly alter (possibly amplify) the vertical seismic excitation transmitted to the base of a structure and (ii) Winkler moduli pertaining to kinematic loading differ from those for inertial loading. Simple approximate expressions for kinematic Winkler moduli are derived for use in applications. Copyright © 2013 John Wiley & Sons, Ltd.     

Vertical dynamic response of pile in a radially heterogeneous soil layer

An analysis of a pile vertical response considering soil inhomogeneity in the radial direction under dynamic loads is presented. The solution technique is based on a three‐dimensional axisymmetric model, which includes the consideration of the vertical displacement of the soil. The soil domain is subdivided into a number of annular vertical zones, and the continuity of the displacements and stresses are imposed at both the interface of pile–soil and the interfaces of adjacent soil zones to establish the dynamic equilibrium equations of the pile–soil interaction. Then, the equations of each soil zone and of the pile are solved one by one to obtain the analytical and semi‐analytical dynamic responses at the top of the pile in the frequency domain and time domain. Parametric studies have been performed to examine the influence of soil parameters' variations in the radial direction caused by the construction effect on the dynamic responses of pile. The results of the studies have been summarized and presented in figures to illustrate the influences of the soil parameters as they change radially. The effect of the radius of the disturbed soil zone caused by construction is also studied in this paper. Copyright © 2008 John Wiley & Sons, Ltd.     

横观各向同性地基中管桩扭转振动响应解析解

考虑地基沉积过程中产生的竖向和水平向力学性质的差异,对横观各向同性地基中管桩扭转振动频域响应进行了理论研究。基于横观各向同性材料的本构关系以及桩-土耦合扭转振动,建立了桩土系统定解问题,通过Laplace变换和分离变量法求得了桩周土和桩芯土扭转振动位移形式解。通过桩-土接触面的连续条件,求得了管桩扭转频域响应解析解,并得到了桩顶复动刚度和速度导纳的表达式。将所得解退化到横观各向同性地基中实心桩解以及均匀地基中管桩解,并与已有文献进行了对比,验证了解的合理性。通过数值算例,分析了桩周土和桩芯土的横观各向同性力学参数对桩顶扭转复刚度及速度导纳的影响。     

Embedded beam element with interaction surface for lateral loading of piles

This paper presents a numerical formulation of a three dimensional embedded beam element for the modeling of piles, which incorporates an explicit interaction surface between soil and pile. The formulation is herein implemented for lateral loading of piles but is able to represent soil–pile interaction phenomena in a general manner for different types of loading conditions or ground movements. The model assumes perfect adherence between beam and soil along the interaction surface. The paper presents a comparison of the results obtained by means of the present formulation and by means of a previously formulated embedded pile element without interaction surface, as well as reference semi‐analytical solutions and a fully 3D finite element (FE) model. It is seen that the proposed embedded element provides a better convergence behavior than a previously formulated embedded element and is able to reproduce key features of a full 3D FE model. Copyright © 2015 John Wiley & Sons, Ltd.     

1000－7598－(2003)增1―0001―08

随着地下工程的发展,对于埋置较深和地质环境较为复杂的隧道工程,高地温问题已经成为一个重要的工程问题并引起了国内外学者的广泛关注。研究隧道的温度场分布问题对于解决寒区隧道冻融问题、高温隧道问题、隧道火灾灾害等是十分必要的。针对高地温环境中的圆形隧道,假定其具有一定长度且承受轴对称的温度荷载和地应力作用,建立两端简支的二维稳态的热传导方程和平衡方程。利用无量纲化和微分方程级数求解的方法,得到了包含温度场、位移场及应力场的热弹性理论解。依据上述研究结果,以某包含衬砌的圆形断面隧道为例进行了热弹性分析,并得到了由于隧道开挖而引起的围岩的温度变化范围。研究所得结论可为隧道施工及运营过程中隧道的保温隔热提供重要的理论依据。     

锚杆与土体界面渐进破坏的解析解

利用点荷载,作用于半无限空间的Mindlin位移解,考虑锚杆与土体界面的渐进破坏过程,推导出界面摩阻力的微分方程解析解。编制了相应的计算程序,把计算结果和现场试验值进行比较,结果较为吻合,二者的数据都表明,随着锚头拉力的增加,土-锚杆摩阻力峰值逐渐向锚杆末端转移,而锚杆前端则发生部分范围的滑移。然后,利用所求得的解析解,研究了土体弹性模量、锚杆孔径对锚杆摩阻力分布的影响,可见土体弹性模量越大,则界面上的摩阻力越容易达到峰值,从而产生破坏,而锚杆孔径越大,则界面上的摩阻力上升越慢,可延缓破坏过程。     

Exact solution for granular flows

In this paper, we present the exact solution of the Riemann problem for the nonlinear one‐dimensional so‐called shallow‐water or Saint‐Venant equations with friction proposed by SAVAGE and HUTTER to describe debris avalanches. This model is based on the depth‐averaged thin layer approximation of granular flows over sloping beds and takes into account a Coulomb type friction law with a constant friction coefficient. A particular configuration of the Riemann problem corresponds to a dam of infinite length in one direction from which granular material is released from rest at a given time over an inclined rigid or erodible bed. We solve analytically and numerically the depth‐averaged long‐wave equations derived in a topography‐linked coordinate system for all the possible Riemann problems. The detailed mathematical proof of the derivation of the analytical solutions and the analysis of their structure and properties is intended, first of all, for geophysicists, mathematicians, and physicists because of the possible extension of this study to more complex problems (geometries, friction laws, …). The numerical solution of the first‐order finite‐volume method based on a Godunov‐type scheme is compared with the proposed exact Riemann problem solution. This solution is used to solve the dam‐break problem and analyze the influence of the thickness of the erodible bed on the speed of the granular front. Comparison with existing experimental results shows that, for an erodible bed, the equations lack fundamental physical significance to reproduce the observed dynamics of erosive granular flows. Copyright © 2012 John Wiley & Sons, Ltd.     

A revised solution for the horizontal vibration of an end‐bearing pile in viscoelastic soil

This note presents a new method to derive closed‐form expressions describing the horizontal response of an end‐bearing pile in viscoelastic soil subjected to harmonic loads at its head. The soil surrounding the pile is assumed as a linearly viscoelastic layer. The propagation of waves in the soil and pile is treated mathematically by three‐dimensional and one‐dimensional theories, respectively. Unlike previous studies of the problem, the formulation presented allows the governing equations of the soil to be solved directly, eliminating the need to introduce potential functions. Accordingly, the dynamic response of the pile is obtained by means of the initial parameter method, invoking the requirement for continuity at the pile–soil interface. It is demonstrated that the derived compact solution matches exactly an existing solution that utilises potential functions to formulate the problem. Copyright © 2016 John Wiley & Sons, Ltd.     

饱和土一维简谐响应解析解的求解和应用:I求解

对于饱和土一维简谐响应的u-w和u-p方程，通过消元分别将二阶微分方程组变为一个四阶的微分方程。求解此微分方程则得到一维简谐响应的通解。对于特定的边值问题，通过求解描述边界条件的线性方程组得到满足边界条件的解析解。此解析解可用于饱和土动力响应研究的诸多方面。     

A new closed‐form solution for analysis of unlined pressure tunnels under seepage forces

In this study, a simplified analytical closed‐form solution, considering plane strain and axial symmetry conditions, for analysis of a circular pressure tunnel excavated underwater table, is developed. The method accounts for the seepage forces with the steady‐state flow and is based on the generalized effective stress law. To examine the effect of pore pressure variations and also the boundary conditions at the ground surface, the formulations are derived for different directions around the tunnel. The proposed method can be applied for analysis and design of pressure tunnels. Illustrative examples are given to demonstrate the performance of the proposed solution and also to examine the effect of seepage forces on the stability of tunnels. The simplified analytical solution derived in this study is compared with numerical analyses. It is concluded that the classic solutions (Lame's thick‐walled solution), considering the internal pressure as a mechanical load applied to the tunnel surface, are not applicable to pervious media and can result in an unsafe design. Copyright © 2012 John Wiley & Sons, Ltd.