Static solution of a crack degenerated from dynamic solution of a propagating crack

ＳｔａｔｉｃｓｏｌｕｔｉｏｎｏｆａｃｒａｃｋｄｅｇｅｎｅｒａｔｅｄｆｒｏｍｄｙｎａｍｉｃｓｏｌｕｔｉｏｎｏｆａｐｒｏｐａｇａｔｉｎｇｃｒａｃｋＳｈｉ－ＹｕＬＩ（李世愚）ａｎｄＹｕｎ－ＴａｉＣＨＥＮ（陈运泰）（ＩｎｓｔｉｔｕｔｅｏｆＧｅｏｐｈｙｓｉｃｓ...     

考虑地基辐射阻尼的结构静动综合响应计算

推导了便于工程应用的底边界同时入射三向地震波时地基底边界和四个侧边界相应于黏弹性边界的地震动输入公式;对目前静动力分析中边界条件不统一的解决方法进行了综述,分析了其合理性;由工程实例可以看出,由基本方法求得的结构最大主应力较直接用黏弹性动力边界求解静力问题的方法更加接近实际,而最大主应力往往控制着材料的损伤破坏,因此,虽然操作过程比较麻烦,但仍建议用基本方法求解结构的静动综合响应。     

Forced vertical vibration of rigid circular disc buried in an arbitrary depth of a transversely isotropic half space

This paper is concerned with the investigation of the vertical vibration of a rigid circular disc buried at an arbitrary depth in a transversely isotropic half space in such a way the axis of material symmetry of the half space is normal to the surface of it and parallel to the vibration direction. By using the Hankel integral transforms, the mixed boundary-value problem is transformed to a pair of integral equations called dual integral equations, which generally can be reduced to a Fredholm integral equation of the second kind. With the aid of complex variable or contour integration, the governing integral equation is numerically solved in the general dynamic case. Two degenerated cases (i) the disc is buried in a transversely isotropic full space, and (ii) rigid circular disc is attached on the surface of the half space are discussed. The reduced static case of the dual integral equations is solved analytically and the vertical displacement, the contact pressure and the static impedance/compliance function are explicitly found. It is shown that the vertical pressure and the compliance function reduced for isotropic half space are identical to the previous solutions reported in the literature. The dynamic contact pressure under the disc and the impedance function are numerically evaluated in general dynamic case and graphically shown that the singularity exists in the contact pressure at the edge of the disc is the same as the static case. In addition, the impedance functions evaluated here for the isotropic domain are collapsed on the solution given by Luco and Mita. To show the effect of different material anisotropy, the numerical evaluations are given for some different transversely isotropic materials and compared.     

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.     

Method for solving dynamic equilibrium equations based on displacement excitation

Mode superposition is a widely used method for solving the dynamic equilibrium equation in structural dynamic analysis. However, the accuracy of this method may be reduced when the dynamic equilibrium equations are set up using displacement excitation. A new method for developing solutions for dynamic equilibrium equations based on displacement excitation is introduced. The dynamic equilibrium equation is decomposed into two parts, namely displacement excitation and velocity excitation, and precise integration and mode superposition methods are combined to solve the equation. Ritz vectors are then used to calculate the static response of the truncated modes of the structure, and a method for determining the number of participating modes is obtained. Using multi-degree-of-freedom systems as two computational examples, the differences in the structural responses obtained from the displacement excitation and acceleration excitation are compared and analyzed. It is shown that the new solution method generates consistent accuracy between the displacement excitation and acceleration excitation.     

Response of piles and pile groups to travelling SH-waves

The response of single piles and pile groups under vertically and obliquely incident seismic waves is obtained using the hybrid boundary element (BEM) formulation. The piles are represented by compressible beam-column elements and the soil as a hysteretic viscoelastic half-space. A recently developed Green function corresponding to the dynamic Mindlin problem is implemented in the numerical formulation. Exact analytical solutions for the differential equations for the piles under distributed harmonic excitations are used. Treating the half-space as a three-dimensional elastic continuum, the interaction problem is formulated by satisfying equilibrium and displacement compatibility along the pile-soil interface. Solutions adopted for the seismic waves are obtained by direct integration of the differential equations in terms of amplitudes. Salient features of the seismic response are identified in several non-dimensional plots. Results of the analyses compare favourably with the limited data available in the literature.     

Non-linear finite element dynamic analysis of axisymmetric solids

The subject of this paper is the finite element analysis of axisymmetric solids subjected to axisymmetric static and dynamic loading, and taking into account material as well as geometric non-linearities. A general Lagrangian formulation forms the basis for the incremental equations of motion which are solved using direct integration methods. Solution accuracy is improved by applying equilibrium correction loads at each step. Finite element discretization is achieved through the use of quadrilateral plane stress and axisymmetric elements with incompatible modes added for improvement of the element flexural characteristics. Several numerical examples are presented to demonstrate the effectiveness of the developed computer program.     

Loading Rate Dependence of Tensile Strength Anisotropy of Barre Granite

Granitic rocks usually exhibit strongly anisotropy due to pre-existing microcracks induced by long-term geological loadings. The understanding of the rock anisotropy in mechanical properties is critical to a variety of rock engineering applications. In this paper, Brazilian tests are conducted statically with a material testing machine and dynamically with a split Hopkinson pressure bar system to measure both static and dynamic tensile strength of Barre granite. To understand the anisotropy in tensile strength, samples are cored and labelled using the three principle directions of Barre granite to form six sample groups. For dynamic tests, a pulse shaping technique is used to achieve dynamic equilibrium in the samples during the dynamic test. The finite element method is then implemented to formulate equations that relate the failure load to the material tensile strength by employing an orthotropic elastic material model. For samples in the same orientation group, the tensile strength shows clear loading rate dependence. The tensile strengths also exhibit clear anisotropy under static loading while the anisotropy diminishes as the loading rate increases, which may be due to the interaction of pre-existing microcracks.     

Theory and application of nonlinear river dynamics

A theoretical model for river evolution including riverbed formation and meandering pattern formation is presented in this paper.Based on nonlinear mathematic theory,the nonlinear river dynamic theory is set up for river dynamic process.Its core content includes the stability and tropism characteristics of flow motion in river and river selves＇ evolution.The stability of river dynamic process depends on the response of river selves to the external disturbance,if the disturbance and the resulting response will eventually attenuate,and the river dynamics process can be restored to new equilibrium state,the river dynamic process is known as stable;otherwise,the river dynamic process is unstable.The river dynamic process tropism refers to that the evolution tendency of river morphology after the disturbance.As an application of this theory,the dynamical stability of the constant curvature river bend is calculated for its coherent vortex disturbance and response.In addition,this paper discusses the nonlinear evolution of the river peristaltic process under a large-scale disturbance,showing the nonlinear tendency of river dynamic processes,such as river filtering and butterfly effect.     

Linear stability analysis for the differentially heated rotating annulus

We use linear stability analysis to approximate the axisymmetric to nonaxisymmetric transition in the differentially heated rotating annulus. We study an accurate mathematical model that uses the Navier–Stokes equations in the Boussinesq approximation. The steady axisymmetric solution satisfies a two-dimensional partial differential boundary value problem. It is not possible to compute the solution analytically, and thus, numerical methods are used. The eigenvalues are also given by a two-dimensional partial differential problem, and are approximated using the matrix eigenvalue problem that results from discretizing the linear part of the appropriate equations.

A comparison is made with experimental results. It is shown that the predictions using linear stability analysis accurately reproduce many of the experimental observations. Of particular interest is that the analysis predicts cusping of the axisymmetric to nonaxisymmetric transition curve at wave number transitions, and the wave number maximum along the lower part of the axisymmetric to nonaxisymmetric transition curve is accurately determined. The correspondence between theoretical and experimental results validates the numerical approximations as well as the application of linear stability analysis.

A linear stability analysis is also performed with the effects of centrifugal buoyancy neglected. Along the lower part of the transition curve, the results are significantly qualitatively and quantitatively different than when the centrifugal effects are considered. In particular, the results indicate that the centrifugal buoyancy is the cause of the observation of a wave number maximum along the transition curve, and is the cause of a change in concavity of the transition curve.     

Horizontal vibrations of a disk on a poroelastic half-space

This paper analytically examines the horizontal vibration of a rigid disk on a saturated poroelastic half-space. The pressure-solid displacement form of the harmonic equations of motion for asymmetric dynamic problem are developed from the form of the equations originally presented by Biot. Making use of a new method the solution of the above equations is obtained. According to the mixed boundary -value conditions, the dual integral equations of the horizontal vibration of a rigid disk on a saturated poroelastic half-space are established. By appropriate transforms, it is shown that the dual integral equations can be reduced to a pair of Fredholm integral equations of the second kind, whose solutions are then computed. Numerical results for the horizontal dynamic compliance coefficient are given at the end of this paper.     

Seismic response of an embedded pile in a transversely isotropic half-space under incident P-wave excitations

A rigorous mathematical formulation is presented for the analysis of a thin cylindrical shell embedded in a transversely isotropic half-space under vertically incident P-wave excitation. By virtue of a set of ring-loads Green's functions for the shell and a group of dynamic fundamental solutions for the half-space under arbitrary interfacial dynamic loads, the problem is shown to be reducible to a pair of Fredholm integral equations. By utilizing an adaptive-gradient family capable of capturing regular-to-singular solution transitions smoothly, an accurate numerical procedure is developed. To assess the effect of material anisotropy on the dynamic load-transfer process, a set of comprehensive numerical results presented for various material and geometrical conditions. The accuracy of the proposed numerical scheme is confirmed by its comparison with a benchmark solution for the corresponding isotropic problem.     

Scattering of vertically-incident P-waves by an embedded pile

An exact theoretical formulation is presented for the analysis of a thin-walled pile embedded in an elastic half-space under vertically-incident P-wave excitation. In the framework of three-dimensional elastodynamics and a shell theory, the axisymmetrical wave-scattering problem is shown to be reducible to a set of Fredholm boundary integral equations. With the incorporation of the singular characteristics of the wave-induced contact load distributions into the solution scheme, a computational boundary element method is developed for a rigorous treatment of the seismic soil-structure interaction problem. Typical results for the dynamic contact load distributions, displacements, complex-valued foundation input motion functions, and resonant pile foundation response are included for direct engineering applications.     

On the stability of columns subjected to non-stationary random or deterministic support motion

The dynamic stability of an elastic column subjected to random or deterministic support motion is studied via the Liapunov method. The definitions of mean square stability and the related theorems are presented. Several criteria regarding the stability of the equilibrium state of the column are established. It is shown that if the static axial loading is less than the Euler critical load and the base acceleration is absolute integrable in the mean for random motion of the support or simply absolute integrable for deterministic motion of the base, the equilibrium state of the column is stable.     

微分求积法在结构动力分析中的应用

微分求积法（DQM）是1种求解微分方程初（边）值问题的数值方法,通常以较小的计算工作量即可获得较高的数值精度。这种方法应用于工程领域时多用来解决梁、板等结构的静力分析或结构特征值分析等问题,即对边值问题的微分方程的求解。结构动力分析属于初值问题,荷载和结构反应都具有特殊性,直接套用DQM求解边值问题并不能获得问题的解。本文尝试利用微分求积原理建立求解结构动力反应的具体方法。借鉴单元法的思想,将荷载持时划分为若干个时步,在每个时步内对动态荷载和结构反应进行离散,然后用DQM对时步逐个进行求解,得到体系在整个时域内的反应过程。通过对3种不同自振周期的线弹性单自由度体系在不同频率简谐激励下反应的计算,阐释了本文方法的可行性以及高精度、高效率的特点,通过数值试验确定了时步内相对较优的节点数,并为时步长度的选取提供了建议。     

边坡地震稳定性分析探讨

传统的拟静力法和安全系数时程分析法在评价边坡地震稳定性时存在一定的局限性。在提出准确的评价边坡地震稳定性必需因素的基础上,建议对边坡地震稳定性分析方法重新进行分类。根据动力分析得到的边坡在地震作用下的破坏机制和破裂面的性质和位置,提出基于拉-剪破坏的动力时程分析法和强度折减动力分析法。第一种方法将FLAC计算得到破坏时刻的动应力施加到静力情况下边坡上,采用动力分析得到的拉-剪破裂面,结合极限平衡法求解边坡地震安全系数,是一种改进的动力有限元时程分析法;第二种方法考虑了拉-剪破坏的FLAC强度折减动力分析法,是完全动力的方法。最后通过算例分析验证了新方法的可行性,为边坡地震安全系数计算提供了一种新的思路。     

Dynamic response of a flexible plate on saturated soil layer

An analytical approach is developed to study the dynamic response of a flexible plate on single-layered saturated soil. The analysis is based on Biot's two-phased theory of poroelasticity and also on the classical thin-plate theory. First, the governing differential equations for saturated soil are solved by the use of Hankel transform. The general solutions of the skeleton displacements, stresses, and pore pressures, derived in the transformed domain, are subsequently incorporated into the imposed boundary conditions, which leads to a set of dual integral equations describing the corresponding mixed boundary value problem. These governing integral equations are finally reduced to the Fredholm integral equations of the second kind and solved by standard numerical procedures. The accuracy of the present solution is validated via comparisons with existing solutions for an ideal elastic half-space. Furthermore, some numerical results are presented to show the influences of the layer depth, the plate flexibility, and the soil porosity on the dynamic compliances.     

Dynamic stability of liquid-filled cylindrical shells under vertical excitation,Part II: Theoretical results

Theoretical analyses are presented for the dynamic stability of a clamped-free cylindrical shell partially filled with liquid, under vertical excitation. In the analyses, the dynamic version of the Donnell equations and the velocity potential theory were used for the motions of the shell and the liquid, respectively. The problem was solved by using the modified Galerkin method so as to satisfy the boundary conditions. The equations of motion coupling the shell and the liquid were derived from a type of coupled Mathieu's equation. It is found that the parametric principal resonance could occur, as well as the parametric combination resonance of the sum type, involving two natural vibrations with the same circumferential wave number but with different axial mode number. The latter type of parametric resonance apparently has not been previously studied. The instability regions where parametric resonance occurs were determined by using Hsu's method.¹⁷ To compare with the experimental results which had been stated in a companion paper,¹⁵ detailed numerical calculations were carried out for the two test cylinders partially filled with water. Excellent agreement between theory and experiment was demonstrated for the instability regions.