CALCULATING THE DYNAMIC STIFFNESS MATRIX OF 2-D FOUNDATIONS BY DISCRETE WAVE NUMBER INDIRECT BOUNDARY ELEMENT METHODS

Apart from some special cases, calculating the dynamic stiffness matrix of foundations on a layered half-space, especially for embedded foundations, is computationally expensive. An efficient method for two-dimensional foundations in a horizontally layered soil media is presented in this paper. This method is based on indirect boundary element methods and uses discrete wave number solution methods for calculating Green's functions for displacements and analytical methods for the integrations over the boundary. For surface foundations, the present method applies at all frequencies. For embedded foundations or for constructing energy transmitting boundaries, because the free-field part is modelled by boundary elements and the excavated part is modelled by finite elements, the present method applies only at low frequencies for the spring coefficients (the real parts of the dynamic stiffness matrix) but applies at all frequencies for the damping coefficients (the imaginary part of the dynamic stiffness matrix) for undamped sites. The novelty of the method can be used for three-dimensional foundations. © 1997 by John Wiley & Sons, Ltd.     

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.     

Flexibility-based linear dynamic analysis of complex structures with curved-3D members

A flexibility-based formulation of a new mass matrix for the dynamic analysis of spatial frames consisting of curved elements with variable cross-sections is presented. The main characteristic of such formulations is the exact equilibrium of forces at any interior point, with no additional hypotheses about the distribution of displacements, strains or stresses. Accordingly, the derived element mass matrix takes into account the exact stiffness and mass distribution throughout each element. In validation tests, results obtained with this method are compared with those obtained by other numerical or analytical formulations, showing the accuracy of the proposed method. The comparison of experimental results for a multispan arch bridge subjected to a dynamic load with those achieved by means of the proposed method are finally included to illustrate its efficiency in the treatment of complex structures. © 1998 John Wiley & Sons, Ltd.     

基于紧凑存储的任意四边形网格有限元法地震波场数值模拟

采用有限元法精确模拟复杂介质条件下的地震波场,一般采用三角单元,但在节点数相同的情况下,三角单元的计算精度不如矩形单元高,采用三角单元模拟复杂界面时,编制确定结构刚度矩阵非零元素位置的程序也较为麻烦。采用矩形单元离散含有倾斜或起伏界面的地质模型时,无法避免绕射噪声,若加密网格又会增加计算量。为此,本文基于任意四边形单元模拟声波的传播,在倾斜或起伏界面条件下,可以有效避免因离散引起的“阶梯状”界面,在不增加计算量以及内存占用的前提下,有效地消除离散绕射噪声。采用对角的集中质量矩阵代替一致质量矩阵,避免矩阵的求逆运算,从而提高显式有限元法的计算效率;对结构刚度矩阵采用紧凑存储格式,每一行需要存储的元素最多为5个,同时零元素不参与运算,既减少内存的占用,又极大地提高计算效率。     

大型渡槽有限条法动力建模研究

根据渡槽结构特点，采用有限条划分渡槽槽身，推导出渡槽槽身结构的单元刚度矩阵、相容质量矩阵的显式表达式；渡槽支架采用空间梁单元模拟；联接槽身和支架的盆式橡胶支座采用弹性元件单元模拟。编程程序具体计算了某大型渡槽的模态，并用ANSYS对大型渡槽进行模态分析，计算结果表明：该模型计算的渡槽结构的固有频率和ANSYS计算结果接近。模型可用于大型渡槽的动力分析。     

Time‐harmonic response of non‐homogeneous elastic unbounded domains using the scaled boundary finite‐element method

The scaled boundary finite‐element method is extended to simulate time‐harmonic responses of non‐homogeneous unbounded domains with the elasticity modulus and mass density varying as power functions of spatial coordinates. The unbounded domains and the elasticity matrices are transformed to the scaled boundary coordinates. The scaled boundary finite‐element equation in displacement amplitudes are derived directly from the governing equations of elastodynamics. To enforce the radiation condition at infinity, an asymptotic expansion of the dynamic‐stiffness matrix for high frequency is developed. The dynamic‐stiffness matrix at lower frequency is obtained by numerical integration of ordinary differential equations. Only the boundary is discretized yielding a reduction of the spatial dimension by one. No fundamental solution is required. Material anisotropy is modelled without additional efforts. Examples of two‐ and three‐dimensional non‐homogeneous isotropic and transversely isotropic unbounded domains are presented. The results demonstrate the accuracy and simplicity of the scaled boundary finite‐element method. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamic non-linear analysis of reinforced concrete shear wall by finite element method with explicit analytical procedure

A numerical procedure for a dynamic non-linear finite element analysis is proposed here to analyse three-dimensional reinforced concrete shear wall structures subjected to earthquake motions. A shear wall is modelled as a quasi-three dimensional structure which is composed of plane elements considering the in-plane stiffness of orthogonal flange panels. The proposed constitutive model is based on the non-linearity of reinforcement and concrete in which the tension stiffening in tension and the degradation of stiffness and strength in compression of concrete after cracking are considered. The acceleration-pulse method, which is a kind of explicit analytical procedure, is employed to solve the non-linear dynamic equations, where the dynamic equation can be solved without stiffness matrix and so the iterative procedure is not necessary for descending portion of stress–strain relationship caused by cracking and softening after compressive strength in concrete. The damping effect is considered by assuming equivalent viscous damping which can give good cyclic behaviours of inertia force vs. displacement relationships. This analytical method was applied to a test specimen of a reinforced concrete shear wall with a H-shaped section which was vibrated up to failure by using a large-scale shaking table with high -performance in Japan. The test was performed as one of the dynamic model tests for evaluation of seismic behaviour of nuclear reactor buildings. The calculations were performed sequentially from the elastic range to failure. The comparison with the test results shows that this approach has good accuracy. © 1997 by John Wiley & Sons Ltd     

Soil–structure interaction in resonant railway bridges

This paper explores dynamic soil–bridge interaction in high speed railway lines. The analysis was conducted using a general and fully three-dimensional multi-body finite element–boundary element model formulated in the time domain to predict vibrations caused by trains passing over the bridge. The vehicle was modelled as a multi-body system, the track and the bridge were modelled using finite elements and the soil was considered as a half-space by the boundary element method. The dynamic response of bridges to vehicle passage is usually studied using moving force and moving mass models. However, the multi-body system allows to consider the quasi-static and dynamic excitation mechanisms. Soil–structure interaction was taken into account by coupling finite elements and boundary elements. The paper presents the results obtained for a simply supported short span bridge in a resonant regime under different soil stiffness conditions.     

Diffraction of SH-waves by topographic features in a layered transversely isotropic half-space

The scattering of plane SH-waves by topographic features in a layered transversely isotropic (TI) half-space is investigated by using an indirect boundary element method (IBEM). Firstly, the anti-plane dynamic stiffness matrix of the layered TI half-space is established and the free fields are solved by using the direct stiffness method. Then, Green’s functions are derived for uniformly distributed loads acting on an inclined line in a layered TI half-space and the scattered fields are constructed with the deduced Green’s functions. Finally, the free fields are added to the scattered ones to obtain the global dynamic responses. The method is verified by comparing results with the published isotropic ones. Both the steady-state and transient dynamic responses are evaluated and discussed. Numerical results in the frequency domain show that surface motions for the TI media can be significantly different from those for the isotropic case, which are strongly dependent on the anisotropy property, incident angle and incident frequency. Results in the time domain show that the material anisotropy has important effects on the maximum duration and maximum amplitudes of the time histories.     

时域谱元法的质量特性模型及其构建方法

研究了构建时域谱单元质量特性模型的数学机制,针对时域切比雪夫谱单元和勒让德谱单元建立了一种直接导出谱单元一致质量矩阵和集中质量矩阵的统一数学方法,对比分析两种谱单元质量特性模型的特征,并从物理角度探讨了谱单元质量特性模型的合理性.研究表明,数值积分点与谱单元节点选取是否一致是决定时域谱单元形成一致质量模型或集中质量模型...     

Dynamic transient behaviour of two- and three-dimensional structures including plasticity,large deformation effects and fluid interaction

The finite element method is employed in the prediction of the dynamic transient response of two- and three-dimensional solids exhibiting geometric (large deformations) and material (elasto-plastic) non-linearities. Explicit time marching schemes are adopted for integration of the dynamic equilibrium equation and a diagonal ‘lumped’ mass matrix is employed with a special procedure applicable to parabolic isoparametric elements. A variety of problems are presented including a solid/fluid interaction situation, and the method is shown to be able to solve economically many problems of dynamic or catastrophic nature which can occur in such structures as nuclear reactors, containment vessels, etc.     

A simplified model for lateral response of large diameter caisson foundations—Linear elastic formulation

The transient response of large embedded foundation elements of length-to-diameter aspect ratio D/B=2–6 is characterized by a complex stress distribution at the pier–soil interface that cannot be adequately represented by means of existing models for shallow foundations or flexible piles. On the other hand, while three-dimensional (3D) numerical solutions are feasible, they are infrequently employed in practice due to their associated cost and effort. Prompted by the scarcity of simplified models for design in current practice, we here develop an analytical model that accounts for the multitude of soil resistance mechanisms mobilized at their base and circumference, while retaining the advantages of simplified methodologies for the design of non-critical facilities. The characteristics of soil resistance mechanisms and corresponding complex spring functions are developed on the basis of finite element simulations, by equating the stiffness matrix terms and/or overall numerically computed response to the analytical expressions derived by means of the proposed Winkler model. Sensitivity analyses are performed for the optimization of the truncated numerical domain size, the optimal finite element size and the far-field dynamic boundary conditions to avoid spurious wave reflections. Numerical simulations of the transient system response to vertically propagating shear waves are next successfully compared to the analytically predicted response. Finally, the applicability of the method is assessed for soil profiles with depth-varying properties. The formulation of frequency-dependent complex spring functions including material damping is also described, while extension of the methodology to account for nonlinear soil behavior and soil–foundation interface separation is described in the conclusion and is being currently investigated.     

A lumped mass Chebyshev spectral element method and its application to structural dynamic problems

A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems, as the computational efforts can be greatly reduced in the process of mass matrix inversion. In this study, the nodal quadrature method is employed to construct a lumped mass matrix for the Chebyshev spectral element method (CSEM). A Gauss-Lobatto type quadrature, based on Gauss-Lobatto-Chebyshev points with a weighting function of unity, is thus derived. With the aid of this quadrature, the CSEM can take advantage of explicit time-marching schemes and provide an efficient new tool for solving structural dynamic problems. Several types of lumped mass Chebyshev spectral elements are designed, including rod, beam and plate elements. The performance of the developed method is examined via some numerical examples of natural vibration and elastic wave propagation, accompanied by their comparison to that of traditional consistent-mass CSEM or the classical finite element method (FEM). Numerical results indicate that the proposed method displays comparable accuracy as its consistent-mass counterpart, and is more accurate than classical FEM. For the simulation of elastic wave propagation in structures induced by high-frequency loading, this method achieves satisfactory performance in accuracy and efficiency.

     

Rocking response of flexible circular foundations on layered media

The analysis of the response of a flexible circular foundation on layered media due to an arbitrarily distributed vertical loading is presented. The analysis is based on the ‘ring method’ approach, i.e. discretization of the foundation in a set of concentric rings. The arbitrarily distributed loading is expanded in the circumferential direction in a Fourier series. The influence coefficient matrix of soil for each element of the series is evaluated utilizing the stiffness matrix approach. The stiffness matrix of the foundation is obtained from the finite difference energy method approach. Numerical examples illustrate the influence of several soil-foundation parameters on the rocking response of a foundation. Results are presented in terms of displacement and soil reaction distributions and impedance functions point to significantly different responses of flexible and rigid foundations.     

基于BP神经网络的空间索杆结构节点损伤识别研究

针对某实际空间索杆结构的节点损伤现象,采用BP神经网络与基于振动的损伤识别两步法对其进行了识别研究,即首先确定可能发生节点损伤的子区域,在此基础上利用对应子区域的子网络识别出具体的损伤位置和程度。识别过程中采用两个杆单元模拟发生节点损伤的杆件,用抗弯刚度降低的端部短杆单元模拟节点损伤。研究表明,虽然空间索杆结构的动力性能较为复杂,但基于结构固有频率和模态位移的组合指标对节点损伤仍较为敏感,利用它们进行节点损伤识别是有效的。     

无破碎带断层场地对SH波的散射

在层状半空间精确动力刚度矩阵和斜线荷载动力格林函数的基础上建立间接边界元方法,在频域内求解无破碎带断层场地对入射平面SH波的散射。为方便求解,将总波场分解为自由波场和散射波场,自由波场由直接刚度法求得,断层两侧的散射波场通过在断层面上分别对两侧施加均布斜线荷载产生的动力响应来模拟,虚拟荷载的密度可通过引入断层表面的边界条件确定,最后叠加自由波场和散射波场求得总波场。以有落差断层和无落差断层模型为例进行数值计算,分析断层落差、断层倾角以及断层两侧介质的刚度比对散射效应的影响。研究表明,断层落差与波长相当时,断层对SH波的放大作用最大;地表位移幅值随着断层倾角的增大逐渐增大;若断层无落差且其两侧刚度不同时,一般刚度较小一侧地表位移幅值较大且振荡更为剧烈,波从刚度较小一侧入射时位移幅值放大尤为显著。     

An efficient time-domain soil-structure interaction analysis based on the dynamic stiffness of an unbounded soil

An approximate method and a rigorous method are presented for the time-domain soil-structure interaction analysis, both of which use the stiffness of soil obtained numerically or experimentally in the limited frequency range. In the aseismic design of a large-scale structure, the approximate and rigorous methods are intended to be used in a preliminary analysis and a detailed study, respectively. Both methods are based on the approximation that the first few terms of the Fourier or Taylor expansions of a frequency-dependent function are used. The difference lies in the number of coefficients of each series, and also in the manner in which the coefficients can be determined. In order to demonstrate the validity of the proposed methods, a soil-structure system, whose soil has a complex profile under the foundation, is analysed. The dynamic stiffness of the soil is calculated by a 3-D hybrid approach that combines the finite element method and the boundary element method. As a result, the present methods are capable of evaluating the complexity of the soil more precisely and efficiently than conventional methods.     

Impedance of flexible suction caissons

The dynamic response of offshore wind turbines is affected by the properties of the foundation and the subsoil. The aim of this paper is to evaluate the dynamic soil–structure interaction of suction caissons for offshore wind turbines. The investigations include evaluation of the vertical and coupled sliding–rocking vibrations, influence of the foundation geometry and examination on the properties of the surrounding soil. The soil is simplified as a homogenous linear viscoelastic material and the dynamic stiffness of the suction caisson is expressed in terms of dimensionless frequency‐dependent coefficients corresponding to different degrees of freedom. The dynamic stiffness coefficients for the skirted foundation are evaluated using a three‐dimensional coupled boundary element/finite element model. Comparisons with known analytical and numerical solutions indicate that the static and dynamic behaviours of the foundation are predicted accurately using the applied model. The analysis has been carried out for different combinations of the skirt length, Poisson's ratio of the subsoil and the ratio of the soil stiffness to the skirt stiffness. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamic stiffness of deep foundations with inclined piles

The influence of inclined piles on the dynamic response of deep foundations and superstructures is still not well understood and needs further research. For this reason, impedance functions of deep foundations with inclined piles, obtained numerically from a boundary element–finite element coupling model, are provided in this paper. More precisely, vertical, horizontal, rocking and horizontal–rocking crossed dynamic stiffness and damping functions of single inclined piles and 2 × 2 and 3 × 3 pile groups with battered elements are presented in a set of plots. The soil is assumed to be a homogeneous viscoelastic isotropic half‐space and the piles are modeled as elastic compressible Euler–Bernoulli beams. The results for different pile group configurations, pile–soil stiffness ratios and rake angles are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Nonuniform earthquake input for arch dam–foundation interaction

Wave scattering and dam–foundation interaction are important aspects of a realistic earthquake analysis of arch dams. In the first part of this paper it is shown how the motion obtained from a two-dimensional scattering analysis can be used as an input for a three-dimensional dam–foundation analysis. In the second part, a method for calculating the scattered motion is explained. The scattered motion is obtained via the two-dimensional dynamic stiffness matrix. The dynamic stiffness matrix for an out-of-plane motion is calculated by the Complementary-Domain Method (CDM). Some examples are presented to verify the method and to show the influence of the scattering of the seismic ground motion.