Stresses in column-supported hyperboloidal shells subject to seismic loading

The computation of stresses within a finite element displacement method analysis of a shell of revolution is considered. The common procedure of applying the kinematic and constitutive laws to the displacement functions is examined and justified for models where the displacements are represented by high-order polynomial expansions. Also, two alternative computational formats within this technique are explored. The influence of the column-supported base condition on a hyperboloidal shell of revolution is studied with respect to the stresses calculated from a response spectrum analysis. These studies emphasize the importance of accurately modelling the base region of a column-supported shell such as a hyperbolic cooling tower.     

地震作用下储罐与管道连接波纹管的动力响应

针对储罐与管道连接这个抗震薄弱环节进行研究，考虑了储罐与地基、管道与地基的相互弹性作用及流固耦合作用，使计算模型比较符合工程实际情况。将储罐罐壁看作为刚体，将波纹管部分和管道部分分别用旋转锥壳单元和空间梁单元离散化，通过分析得到波纹管与储罐连接接合面、波纹管与管道连接接合面不同单元之间的位移协调约束方程，并用罚函数法进行处理。根据流体力学速度势理论和有限元法的基本理论，利用哈密尔顿变分原理推导出储罐与管道连接波纹管系统动力分析方程，编制了系统动力分析有限元程序，计算了垂直地震激励不同场地土地基条件下储罐与管道连接波纹管位移响应。     

地下结构抗震分析的振动法与波动法对比研究

针对地下结构抗震分析中应用广泛的波动方法与振动方法,从物理方程和有限元格式出发,明确了两种方法的区别与联系。通过标准数值试验,证明了振动方法是波动方法采用下卧刚性基岩假设和地层水平无限延伸假设,仅考虑SV波垂直向上入射的一种特例。采用波动方法模拟弹性地基,对比振动方法结果,表明弹性地基和刚性地基上地表位移响应和隧道结构内力响应均有显著差异。建议应根据具体场地条件,选择振动或波动方法进行地下结构抗震设计研究。     

Analysis of unanchored liquid storage tanks under lateral loads

Many liquid storage tanks consist of a steel cylindrical shell, which is welded to a base plate, but not fixed to the foundation. When such an unanchored tank is subjected to lateral loads due to earthquake induced hydrodynamic pressures in the liquid, the tank wall tends to uplift locally, pulling the base plate up with it. The contact problem of the partially uplifted base plate and its interaction with the the cylindrical shell is solved in this paper using the finite difference energy method, and a Fourier decomposition of the displacements in the circumferential direction. Non-linearities due to contact, finite displacements and yield of the steel are included in the analysis. However, the equations for the shell are linearized. This uncouples the equations for the Fourier displacement coefficients in the cylindrical shell, and enables the degrees of freedom for the shell to be eliminated by static condensation at very little computational cost. Comparing the analytical results to (for the most part existing) experimental results, produces good agreement in some cases and not so good in others. A number of effects that could give rise to such differences are discussed. In most cases they represent experimental conditions that are not known or modelled in the analysis. The analysis results are also compared to those from a simplified analysis in which the hold-down action of the base plate is modelled by means of nonlinear Winkler springs.     

单层柱面网壳弹塑性地震反应特征

本文对网壳结构的弹塑性抗震性能进行了探讨。根据结构弹性有限元理论，结合网壳结构受力特点，推导出网壳结构单元的弹塑性矩阵表达式。然后对单层柱面网壳结构的弹塑性地震反应进行了具体分析，给出了节点位移和杆件应力变化规律，并讨论了矢跨比变化对网壳弹塑性性能的影响，从而揭示出这类网壳的弹塑性地震反应特点。     

Non-linear steady state vibration and dynamic snap through of shallow arch beams

The non-linear steady state vibration of shallow arch beams is studied by a finite element method based on the principle of virtual work. Both the free and forced periodic vibrations are considered. The axial and flexural deformations are coupled by the induced axial force along the beam element. The spatial discretization is achieved by the usual finite element method and the steady state nodal displacements are expanded into a Fourier series. The harmonic balance method gives a set of non-linear algebraic equations in terms of the vibrating frequency and the Fourier coefficients of nodal displacements. The non-linear algebraic equations are solved by the Newtonian algorithm iteratively. The combined algorithm is called the incremental harmonic balance method. The importance of the conditions of completeness and balanceability is presented. Since the non-linearity is essentially softening, different orders of internal resonances between two modes can occur repeatedly. Isolated response curves are possible and are connected to the bifurcation of a particular excited mode.     

Mass lumping models of the linear diffusion equation

The nodal domain integration method is used to develop a numerical model of the linear diffusion equation. The nodal domain integration approach is shown to represent an infinity of finite element mass matrix lumping schemes including the Galerkin and subdomain integration versions of the weighted residual method and an integrated finite difference method. Neumann, Dirichlet and mixed boundary conditions are accommodated analogous to the Galerkin finite element method. In order to reduce the overall integrated approximation relative error, a mass matrix lumping formulation is developed which is based on the Crank-Nicolson time advancement approximation. The optimum mass lumping factors are found to be strongly related to the model timestep size.     

Numerical simulations of shake-table experiment for dynamic soil-pile-structure interaction in liquefiable soils

A shake-table experiment on pile foundations in liquefi able soils composed of liquefi able sand and overlying soft clay is studied. A three-dimensional(3D) effective stress fi nite element(FE) analysis is employed to simulate the experiment. A recently developed multi-surface elasto-plastic constitutive model and a fully coupled dynamic inelastic FE formulation(u-p) are used to model the liquefaction behavior of the sand. The soil domains are discretized using a solid-fl uid fully coupled(u-p) 20-8 noded brick element. The pile is simulated using beam-column elements. Upon careful calibration, very good agreement is obtained between the computed and the measured dynamic behavior of the ground and the pile. A parametric analysis is also conducted on the model to investigate the effect of pile-pinning, pile diameter, pile stiffness, ground inclination angle, superstructure mass and pile head restraints on the ground improvement. It is found that the pile foundation has a noticeable pinning effect that reduces the lateral soil displacement. It is observed that a larger pile diameter and fi xed pile head restraints contribute to decreasing the lateral pile deformation; however, a higher ground inclination angle tends to increase the lateral pile head displacements and pile stiffness, and superstructure mass seems to effectively infl uence the lateral pile displacements.     

Stabilized finite elements for harmonic soil dynamics problems near the undrained-incompressible limit

Simple Finite Element models for soil dynamics and earthquake engineering problems in the frequency domain are a fast and valuable tool providing a first approximation before a full non-linear analysis in the time domain is performed.Quite often the problem concerns saturated soils with very small permeability and pore fluid of neglectable compressibility. In the limit, the permeability is assumed to be zero and the pore fluid incompressible. Here, engineers use standard finite element codes formulated in terms of displacements but incompressibility may result in volumetric locking of the mesh with a severe loss of accuracy.The purpose of this paper is to present a simple mixed finite element formulation in the frequency domain based on displacements and pore pressures as main variables. A suitable stabilization technique allowing for equal order interpolation of displacements and pressures has been introduced for incompressible and zero permeability limits.Of course, the range of application is limited to those problems in which the behaviour of the material can be approximated by linear models, and therefore modelling of phenomena such as liquefaction, cyclic mobility or cavitation occur is excluded.The paper shows as well an extremely simple way of coupling solid and water domains as it occurs for instance in quay walls under dynamic loading.     

Accuracy of finite difference representations for the transient response analysis of shells

The frequency versus wave number characteristics of four O(ΔS₂) finite difference formulations for one-dimensional linear shell (ring and axisymmetric) equations are investigated and compared with the exact continuum characteristics. It is found that three of the formulations give virtually identical results. These are half-spacing techniques with equilibrium in terms of displacements or resultants and whole-spacing with equilibrium in terms of displacements. The formulation based on whole-spacing with equilibrium in terms of resultants produces some dramatically different results. These discrepancies partially explain some late time instability problems and critical time step behaviour that have been reported by other investigators.     

Dynamic finite element analysis of three-dimensional soil models with a transmitting element

A method for the dynamic finite element analysis of a non-axisymmetric soil model with an axisymmetric boundary is presented. In the non-axisymmetric soil domain an arbitrary discretization with three-dimensional isoparametric solid elements is used. At the boundary a transmitting element is arranged. It is based on the semi-analytical element of Waas and Kausel. The transformation of the stiffness matrix of the Waas/Kausel element with cyclic symmetric displacements to general displacement fields is presented. For earthquake excitation the forces acting on the discretized domain are given. The method is illustrated by the dynamic analysis of an embedded box-type building. The distribution and magnitude of significant section forces are discussed.     

A practical numerical method for large strain liquefaction analysis of saturated soils

Accurate prediction of the liquefaction of saturated soils is based on strong coupling between the pore fluid phase and soil skeleton. A practical numerical method for large strain dynamic analysis of saturated soils is presented. The u–p formulation is used for the governing equations that describe the coupled problem in terms of soil skeleton displacement and excess pore pressure. A mixed finite element and finite difference scheme related to large strain analysis of saturated soils based on the updated Lagrangian method is given. The equilibrium equation of fluid-saturated soils is spatially discretized by the finite element method, whereas terms associated with excess pore pressure in the continuity equation are spatially discretized by the finite difference method. An effective cyclic elasto-plastic constitutive model is adopted to simulate the non-linear behavior of saturated soils under dynamic loading. Several numerical examples that include a saturated soil column and caisson-type quay wall are presented to verify the accuracy of the method and its usefulness and applicability to solutions of large strain liquefaction analysis of saturated soils in practical problems.     

Analytical and numerical analysis of pressure drawdown in a poroelastic reservoir with complete overburden effect considered

Starting from an analytical reservoir model that incorporates full interaction with an elastic overburden, a new hybrid mathematical approach is developed by combining two numerical discretization methods. A tabular reservoir (petroleum reservoir or an aquifer) in an infinite or semi-infinite domain is viewed as a macroscopic displacement discontinuity, allowing use of the efficient displacement discontinuity mathematical method to calculate stresses and displacements that arise because of pressure changes. A 3-D finite element method using a poroelastic formulation is used to discretize the reservoir itself. By coupling the displacement discontinuity and finite element methods, a 3-D large-scale poroelastic reservoir can be simulated within an infinite or semi-infinite domain. The numerical model has been verified through comparison to known solutions, and some time-dependent pressure drawdown problems are analyzed. Results indicate that including the complete overburden (reservoir surroundings) response has a significant effect on pressure drawdown in a poroelastic reservoir during pumping, and should be incorporated in appropriate applications such as well test equations and subsidence analyses.     

凯威特单层球面网壳结构隔震分析

将凯威特网壳结构的固定支座设置成隔震支座，分别采用弹簧支座和粘弹阻尼支座对结构进行了隔震控制。在有控和无控状态下，计算了凯威特网壳结构的自振频率，初步分析了凯威特网壳结构的隔震控制机理。分别在常遇和罕遇各三种地震作用下，对网壳结构进行了时程分析，对网壳结构的最大水平位移、隔震支座的最大水平位移、最大支座反力和网壳结构的杆件轴力进行了数值计算。研究结果表明：两种隔震装置均有效地控制了凯威特网壳结构的支座反力和杆件轴力，粘弹阻尼支座还明显地减少了隔震支座的水平位移和隔震结构的水平位移。隔震方法对于大跨网壳结构的减振是有效的。     

AN INVESTIGATION INTO THE BASE SLIDING RESPONSE OF RIGID CONCRETE GRAVITY DAMS TO DYNAMIC LOADING

A series of dynamic slip tests on a concrete gravity dam model was conducted on a shaking table. The aim of the experiments was to investigate the dynamically induced sliding and overturning characteristics of a typical low height gravity dam monolith cracked at its base. Tests indicated that downstream sliding is the main instability that could be expected during an earthquake. Dynamic, finite element analyses of the experimental model, using a Lagrangian contact surface algorithm, were also performed. A comparison of the experimental and analytical responses indicated that the seismically induced slip can be predicted reasonably by such a contact surface algorithm implemented in a standard finite element package. A comparison of observed displacements with Newmark's sliding block displacements indicated that a conservative estimate of seismic induced slip of a gravity dam could be obtained by using Newmark's sliding block concept, generally adopted for earth dams and embankments.     

层状介质对破裂过程的影响

震源动力学中破裂产生的地震动在层状介质中的传播模拟,是地震学以及地震工程学研究的前沿课题之一。本文通过建立精确的三维模型,选取具备灵活网格、高精度高效率计算性能的谱元法,利用有效抑制伪震荡的时间域离散方法——加权速度Newmark方法以及多次透射人工边界条件,进行了SCEC/USGS基准项目中TPV5模型的地震破裂过程模拟,得到基于层状介质模型和均匀介质模型（后者采用相同破裂模型）的埋深2km的震源参数结果。将二者进行对比,并具体分析破裂面位错、地震矩、破裂传播时间、上升时间和地表位移,发现层状介质对破裂过程的传播影响较为明显:① 层状介质的存在整体增加了破裂面上的位错,在层状介质模型下计算得到的地震矩约是均匀介质模型结果的1.3倍,因此认为层状介质增强了地震破裂过程中的能量释放;② 层状介质的存在使得破裂传播至地表的速度减慢,并缩短了地表各点的上升时间,增强了地表的地震动响应;③ 层状介质对于地表位移有着明显的增加作用,同时协同破裂面上的初始应力异常区域对位移峰值中心的改变有显著影响。④ 介质分异面附近地震动强烈。对结果进行整理后发现,在具有地下层状介质的地区要充分考虑层状介质产生的场地效应,否则可能会低估该地区的地震危险性。     

A displacement finite element modelling for convection-diffusion problems

The development of a displacement finite element formulation and its application to convective transport problems is presented. The formulation is based on the introduction of a generalized quantity defined as transport displacement. The governing equation is expressed in terms of this quantity and by using generalized coordinates a variational form of the governing equation is obtained. This equation may be solved by any numerical method, though it is of particular interest for application of the finite element method. Two finite element models are derived for the solution of convection-diffusion boundary value problems. The performance of the two element models is discussed and numerical results are given for different cases of convection and diffusion with two types of boundary conditions. The numerical results obtained show not only the efficiency of the numerical models in handling pure convection, pure diffusion and mixed convection-diffusion problems, but also good stability and accuracy. The applications of the developed numerical models are not limited to diffusion-convection problems but can also be applied to other types of problems such as mass transfer, hydrodynamics and wave propagation.     

Eigenrays in 3D heterogeneous anisotropic media,Part I: Kinematics

We present a new ray bending approach, referred to as the Eigenray method, for solving two‐point boundary‐value kinematic and dynamic ray tracing problems in 3D smooth heterogeneous general anisotropic elastic media. The proposed Eigenray method is aimed to provide reliable stationary ray path solutions and their dynamic characteristics, in cases where conventional initial‐value ray shooting methods, followed by numerical convergence techniques, become challenging. The kinematic ray bending solution corresponds to the vanishing first traveltime variation, leading to a stationary path between two fixed endpoints (Fermat's principle), and is governed by the nonlinear second‐order Euler–Lagrange equation. The solution is based on a finite‐element approach, applying the weak formulation that reduces the Euler–Lagrange second‐order ordinary differential equation to the first‐order weighted‐residual nonlinear algebraic equation set. For the kinematic finite‐element problem, the degrees of freedom are discretized nodal locations and directions along the ray trajectory, where the values between the nodes are accurately and naturally defined with the Hermite polynomial interpolation. The target function to be minimized includes two essential penalty (constraint) terms, related to the distribution of the nodes along the path and to the normalization of the ray direction. We distinguish between two target functions triggered by the two possible types of stationary rays: a minimum traveltime and a saddle‐point solution (due to caustics). The minimization process involves the computation of the global (all‐node) traveltime gradient vector and the traveltime Hessian matrix. The traveltime Hessian is used for the minimization process, analysing the type of the stationary ray, and for computing the geometric spreading of the entire resolved stationary ray path. The latter, however, is not a replacement for the dynamic ray tracing solution, since it does not deliver the geometric spreading for intermediate points along the ray, nor the analysis of caustics. Finally, we demonstrate the efficiency and accuracy of the proposed method along three canonical examples.     

Response of tanks to vertical seismic excitations

A method for analyzing the earthquake response of elastic, cylindrical liquid storage tanks under vertical excitations is presented. The method is based on superposition of the free axisymmetrical vibrational modes obtained numerically by the finite element method. The validity of these modes has been checked analytically and the formulation of the load vector has been confirmed by a static analysis. Two forms of ground excitations have been used: step functions and recorded seismic components. The radial and axial displacements are computed and the corresponding stresses are presented. Both fixed and partly fixed tanks are considered to evaluate the effect of base fixation on tank behaviour. Finally, tank response under the simultaneous action of both vertical and lateral excitations is calculated to evaluate the relative importance of the vertical component of ground acceleration on the overall seismic behaviour of liquid storage tanks.     

Time domain dynamic analysis of piles under impact loading

In this paper, time domain dynamic analysis of piles under impact loading is presented. For this purpose a hybrid boundary element technique is implemented. Linear beam column finite elements are used to model the piles and resulting governing equations are solved using an implicit integration scheme. The continuum is assumed to be elastic and an efficient step-by-step time integration scheme is implemented by using an approximate half space integral formulation. By enforcing displacement equilibrium conditions at each time step, a system of equations is generated which yields the solution. Results of this time domain formulation under linear material behavior are compared with Laplace domain results to validate the methods.