Dynamic analysis of flexible rectangular fluid containers subjected to horizontal ground motion

In this paper, an analytical method is proposed to determine the dynamic response of 3‐D rectangular liquid storage tanks with four flexible walls, subjected to horizontal seismic ground motion. Fluid–structure interaction effects on the dynamic responses of partially filled fluid containers, incorporating wall flexibility, are accounted for in evaluating impulsive pressure. The velocity potential in which boundary conditions are satisfied is solved by the method of separation of variables using the principle of superposition. The impulsive pressure distribution is then computed. Solutions based on 3‐D modeling of the rectangular containers are obtained by applying the Rayleigh–Ritz method using the vibration modes of flexible plates with suitable boundary conditions. Trigonometrical functions that satisfy boundary conditions of the storage tank such that the flexibility of the wall is thoroughly considered are used to define the admissible vibration modes. The analysis is then performed in the time domain. Moreover, an analytical procedure is developed for deriving a simple formula that evaluates convective pressure and surface displacements in a similar rigid tank. The variation of dynamic response characteristics with respect to different tank parameters is investigated. A mechanical model, which takes into account the deformability of the tank wall, is developed. The parameters of such a model can be obtained from developed charts, and the maximum seismic loading can be predicted by means of a response spectrum characterizing the design earthquake. Accordingly, a simplified but sufficiently accurate design procedure is developed to improve code formulas for the seismic design of liquid storage tanks. Copyright © 2013 John Wiley & Sons, Ltd.     

Filtering action of embedded massive foundations: New analytical expressions and evidence from 2 instrumented buildings

This paper deals with the effect of the foundation mass on the filtering action exerted by embedded foundations. The system under examination comprises a rigid rectangular foundation embedded in a homogeneous isotropic viscoelastic half‐space under harmonic shear waves propagating vertically. The problem is addressed both theoretically and numerically by means of a hybrid approach, where the foundation mass is explicitly included in the kinematic interaction between the foundation and the surrounding soil, thus referring to a “quasi‐kinematic” interaction problem. Based on the results of an extensive parametric study, it is shown that the filtering problem depends essentially on three dimensionless parameters, i.e.: the dimensionless frequency of the input motion, the foundation width‐to‐embedment depth ratio, and the foundation‐to‐soil mass density ratio. In complements to the translational and rotational kinematic interaction factors that are commonly adopted to quantify the filtering effect of rigid massless foundations on the free‐field motion, an additional kinematic interaction factor is introduced, referring to the horizontal motion at the top of a rigid massive foundation. New analytical expressions for the above kinematic interaction factors are proposed and compared with foundation‐to‐free‐field transfer functions computed from available earthquake recordings on two instrumented buildings in LA (California) and Thessaloniki (Greece). Results indicate that the foundation mass can have a strong beneficial effect on the filtering action with increasing foundation‐to‐soil mass density and foundation width‐to‐embedment depth ratios.     

A fundamental dual‐zone continuum theory for dynamic soil–structure interaction

A continuum theory for an improved characterization of dynamic soil–structure interaction in the framework of three‐dimensional elastodynamics is presented. Effective in demonstrating the importance of integrating free‐field and near‐field effects under general soil and foundation conditions, a compact two‐zone delineation of the soil medium is proposed as a quintessential mechanics perspective for this class of problems. Sufficient to deliver a practical resolution of some perennial analytical and experimental conflicts, a fundamental formulation commensurate to a gradated unification of the homogenization approach and any sole free‐field inhomogeneous representation is developed and implemented computationally. Specialized to the problem of a rigid circular footing on sand, a nominal set of dynamic contact stress distributions and related impedance functions by the dual‐zone theory is included for theoretical and engineering evaluation. Through its comparison with benchmark analytical solutions and relevant physical measurements, the usage of the underlying conceptual platform as an advanced yet practical foundation for general dynamic soil–structure interaction is illustrated. Copyright © 2010 John Wiley & Sons, Ltd.     

Hydroelastic vibration and buckling of rectangular Mindlin plates on Pasternak foundations under linearly varying in-plane loads

Hydroelastic vibration and buckling analysis of horizontal rectangular plates resting on Pasternak foundation and subjected to linearly varying in-plane loads is investigated for different boundary conditions. Fluid is assumed to be inviscid and incompressible. To analyze the interaction of the Mindlin plate with the elastic foundation and fluid system, three displacement components of the plate are expressed in the Ritz method by adopting a set of static Timoshenko beam functions satisfying geometric boundary conditions. Convergence study is first carried out to confirm the stability of the present method. Then, a few comparisons are made for limited cases between present results and the available data in the literature. Finally, parametric study is conducted to highlight the effects of loading factors, fluid levels, foundation stiffness parameters, aspect ratios, thickness to width ratios and different boundary conditions on the critical buckling loads and wet natural frequencies of rectangular plates.     

Investigation of sloshing damping in baffled rectangular tanks subjected to the dynamic excitation

In this research, an analytical model is developed to estimate the hydrodynamic damping ratio of liquid sloshing for wall bounded baffles using the velocity potential formulation and linear wave theory. Here, an analytical solution approach and experimental investigations are conducted for describing the hydrodynamic damping which is provided by vertical and horizontal baffles in partially filled rectangular liquid tanks. In order to evaluate the accuracy of the analytical solution which is developed in present work, a series of experiments are carried out with a rectangular liquid tank excited by harmonic oscillation. The parametric study is conducted on the damping efficiencies of both vertical and horizontal baffles with various dimensions and locations. According to the results of the present investigations, the hydrodynamic damping is significantly affected by the size and location of baffles. Furthermore, the validity of the developed analytical approach as well as the effectiveness of various baffle configurations are discussed. Finally, a simple approach is proposed for estimating the damping ratios of the baffles during earthquake motions.     

A three‐dimensional transmitting boundary formulated in Cartesian co‐ordinate system for the dynamics of non‐axisymmetric foundations

A three‐dimensional transmitting boundary is formulated in the Cartesian co‐ordinate system. It is developed for the dynamic soil–structure interaction problems of arbitrary shape foundations in laterally heterogeneous strata overlying rigid bedrock. Dynamics of a rectangular rigid surface foundation on a homogeneous stratum is analysed by a hybrid approach in which the finite region including foundation is modelled by the conventional finite element method and the surrounding infinite region by the newly developed transmitting boundary. To demonstrate its strength, the present method is applied to a rectangular foundation in a horizontally heterogeneous ground consisting of two distinct regions divided by and welded along a vertical plane. Copyright © 2000 John Wiley & Sons, Ltd.     

Free vibration and transient response of thick and thin plates using the finite element method

An isoparametric quadrilateral plate bending element is used for the free and forced vibration analysis of both thick and thin plates. Plates of rectangular, circular and triangular planform are analysed and excellent results are obtained. The element performance is assessed by comparison with analytical solutions based on Mindlin's thick plate theory, three-dimensional elasticity solutions and solutions based on thin plate theory.     

Out-of-plane (SH) soil-structure interaction: Semi-circular rigid foundation revisited

The model studied in this paper presents an extension of previous work for a shear wall on a semi-circular rigid foundation in an isotropic homogeneous and elastic half-space. The objective is to develop a soil-structure interaction model that can later be applied to the case of a flexible foundation. As shown in the Introduction below, Luco considered the case of a rigid foundation subjected to vertical incident plane SH waves, and Trifunac extended the solution for the same rigid foundation subjected to SH waves but for arbitrary angles of the incidence. In this paper, a new approach and model are presented for the same semi-circular rigid foundation with a tapered-shape (instead of rectangular) superstructure. The analytical expression for the deformation of the semi-circular rigid foundation below this tapered shear wall with soil-structure interaction in an isotropic homogeneous and elastic half-space is thus derived. Results are then compared with those of Trifunac discussed in the section below. This problem formulation can and will later be extended in the case of a flexible foundation that is semi-circular or arbitrarily shaped.     

Wave damping by a vertical porous structure placed near and away from a rigid vertical wall

Damping of water waves by a vertical porous structure placed at some distance from a vertical wall is investigated within the framework of linear water wave theory. The rectangular porous structure is placed on a small rectangular elevation. An incident wave of small amplitude propagates through the structure – some portion gets reflected back while some portion gets transmitted to a third region bounded vertically by a rigid wall which is considered to be at a distance near the porous structure, and also away from the wall at a large distance as a separate case. Boundary value problems are set up in all three regions and, by using the matching conditions along the vertical boundaries, a system of linear equations is deduced. The roots of the relevant dispersion relation are used in setting up the system of equations. The overall scattering phenomenon is studied with respect to different relevant parameters. The dependence of the coefficients on the thickness (width) of the porous structure is investigated for different numbers of modes and porosity. It is observed that, except for the case when the porous structure is thin, the reflection and transmission coefficients give rise to values as expected. In the case, when the rigid wall is nearer to the structure, the reflection coefficient decreases rapidly for a thin structure and converges for all numbers of evanescent modes afterwards. The transmission coefficient also decreases as the width increases, ultimately converging and vanishing for a wide structure. When the wall is at a large distance away from the structure, the behavior of both the reflection and transmission coefficients remain the same. For both cases of the wall being nearer and away from the structure, higher porosity gives rise to lower reflection coefficients and higher transmission coefficients. However, the transmission coefficients converge and vanish when the porous structure is very wide. We also discuss the energy loss against the width of the porous structure for different values of number of modes and porosity. Irrespective of the positioning of the rigid wall, we observe that for higher values of porosity, energy loss is more pronounced when the structure is not thin, whereas energy loss is same for all numbers of modes. All our observations are supported by graphs. Good agreement of our result with earlier results justifies our model.     

钢框架-带缝钢板剪力墙结构受力性能分析

本文对4种钢框架、6种带缝(两排)钢板剪力墙片(四周与构件无连接)和6种固接的钢框架-带缝钢板剪力墙结构在3种不同竖向荷载作用下的抗侧能力和往复荷载下的滞回性能进行了研究,并对比分析。结果表明:前两种结构的侧移刚度、抗侧能力相对较低,屈曲后刚度退化快;钢框架-带缝钢板剪力墙结构的侧移刚度、抗侧能力和耗能能力比前两种结构有明显的提高,说明钢框架与带缝钢板剪力墙片固接后工作协调性能良好。带缝钢板剪力墙片与钢框架-带缝钢板剪力墙结构的整体设计参数宽高比W/H,开缝设计参数开缝墙肢的高宽比h/b、宽厚比b/t、开缝墙肢与剪力墙的高宽比h/H对结构的抗侧能力和滞回性能有很大影响。W/H增大,结构的抗侧能力增强,滞回性能降低;h/b、b/t、h/H增大,结构的抗侧能力降低,滞回性能提高。     

ANALYTICAL EXPRESSIONS FOR GRAVITY DUE TO HOMOGENEOUS REVOLUTIONAL COMPARTMENTS IN THE GAUSSIAN DIVERGENCE APPROACH*

Complete analytical expressions are developed for the first and second derivatives of the Newtonian gravitational potential in arbitrary directions due to the homogeneous revolutional compartment with a polygonal vertical section by applying the Gaussian divergence theorem in the cylindrical coordinate system. Elementary solutions presented can easily be translated into magnetic anomalies caused by a uniformly magnetized body. The divergence approach in the polar coordinate system is also described, and gravity attractions in the radial direction are presented in closed form associated with a homogeneous cap compartment. The explicit solutions are tested against well-known formulae for a cylinder, cone, infinite plate, and sphere.     

Dynamic stiffness of rigid rectangular foundations on the half-space

The dynamic soil–structure interaction of a rigid rectangular foundation with the subsoil represents a mixed-boundary value problem. This problem is formulated in terms of a system of coupled Fredholm integral equations of the first kind. The subsoil is modelled by a homogeneous, linear-elastic and isotropic half-space which is perfectly bonded to the rigid, rectangular foundation. An approximate solution for the resultant loads between the foundation and the half-space due to a unit forced displacement or rotation is obtained using the Bubnov–Galerkin method. Using this method the displacement boundary value conditions are exactly satisfied and the contact stress distributions between the foundation and the half-space are approximated by series expansions of Chebyshev polynomials. This method provides a simple means of studying the soil-structure interaction of rectangular foundations with different inertia properties.     

Numerical study of harbor oscillations induced by water surface disturbances within harbors of constant depth

Oscillations within an enclosed rectangular harbor and a set of partially opened rectangular harbors with various widths and locations of entrance induced by cubic water surface disturbances with different initial heights and locations are simulated using FUNWAVE 2.0 model. The height and the location of the cubic water surface disturbance refer to its thickness and its relative horizontal position inside the harbor, respectively. The water depth inside and outside all harbors is set to be constant. The aim of this paper is to investigate how different heights and locations of the water surface disturbance and various widths and locations of harbor entrance affect the oscillations inside the harbor. Results show that for the given harbors and the range of the initial height of water surface disturbance studied in this paper, all the response amplitudes of various eigenfrequencies increase linearly with the initial height of water surface disturbance. The variations of the initial location of water surface disturbance along the backwall and sidewall of the harbor can significantly change the transverse and longitudinal oscillation patterns of various modes, respectively. The effects of the variations of the width and location of the harbor entrance on the response amplitudes of various resonant modes both depend on the relative positions of their node lines and antinode lines to the harbor entrance.     

Some exact solutions of unsteady boundary layer equations-II

Summary Some exact solutions are presented for the unsteady boundary layer flows of a homogeneous, viscous, incompressible fluid bounded by (i) an infinite rigid oscillating flat plate or (ii) two parallel rigid oscillating flat plates. An explicit representation of the velocity fields for both the configurations has been given. The structures of the associated periodic boundary layers are determined with physical interpretations. Several results of interest have been recovered as special cases of this general theory. The Heaviside operational calculus along with the theory of residues of analytic functions is adopted in finding the solutions.     

用广义协调元分析复合式多层无孔和开孔矩形薄板的振动

复合式多层无孔和开孔矩形薄板是桥梁结构和建筑等结构中的最重要的组成部分。当研究它们的振动问题时,用解析法求解自振频率十分困难,而采用通用的有限元法求解自由度数目多,收敛也比较慢。本文利用三角形薄板广义协调元分析了复合式多层四边简支、四边固定无孔和开孔矩形薄板的自由振动,求出了前几阶频率系数,其结果与用ANSYS软件算出的结果基本接近,当为单层无孔和开孔板时,与文献[8]的结果完全相同,且与文献[10]的解析解吻合较好,证明了本文理论推导及程序编制的正确性,表明广义协调元与有限元相比具有自由度少、精度高、程序简便以及收敛快等优点。     

Approximate analysis methods for asymmetric plan base‐isolated buildings

An approximate method for linear analysis of asymmetric‐plan, multistorey buildings is specialized for a single‐storey, base‐isolated structure. To find the mode shapes of the torsionally coupled system, the Rayleigh–Ritz procedure is applied using the torsionally uncoupled modes as Ritz vectors. This approach reduces to analysis of two single‐storey systems, each with vibration properties and eccentricities (labelled ‘effective eccentricities’) similar to corresponding properties of the isolation system or the fixed‐base structure. With certain assumptions, the vibration properties of the coupled system can be expressed explicitly in terms of these single‐storey system properties. Three different methods are developed: the first is a direct application of the Rayleigh–Ritz procedure; the second and third use simplifications for the effective eccentricities, assuming a relatively stiff superstructure. The accuracy of these proposed methods and the rigid structure method in determining responses are assessed for a range of system parameters including eccentricity and structure flexibility. For a subset of systems with equal isolation and structural eccentricities, two of the methods are exact and the third is sufficiently accurate; all three are preferred to the rigid structure method. For systems with zero isolation eccentricity, however, all approximate methods considered are inconsistent and should be applied with caution, only to systems with small structural eccentricities or stiff structures. Copyright © 2001 John Wiley & Sons, Ltd.     

Earthquake analysis of concrete gravity dams including dam-water-foundation rock interaction

A general procedure for analysis of the response of concrete gravity dams, including the dynamic effects of impounded water and flexible foundation rock, to the transverse (horizontal) and vertical components of earthquake ground motion is presented. The problem is reduced to one in two dimensions, considering the transverse vibration of a monolith of the dam. The system is analysed under the assumption of linear behaviour for the concrete, foundation rock and water. The complete system is considered as composed of three substructures—the dam, represented as a finite element system, the fluid domain, as a continuum of infinite length in the upstream direction, and the foundation rock region as a viscoelastic half-plane. The structural displacements of the dam are expressed as a linear combination of Ritz vectors, chosen as normal modes of an associated undamped dam-rock system. The effectiveness of this analytical formulation lies in its being able to produce excellent results by considering only a few Ritz vectors. The generalized displacements due to earthquake motion are computed by synthesizing their complex frequency responses using Fast Fourier Transform procedures. The stress responses are calculated from the displacements. An example analysis is presented to illustrate results obtained from this analytical procedure. Computation times for several analyses are presented to illustrate the effectiveness of the procedure.     

Dynamic plate-soil interaction - finite and infinite, flexible and rigid plates on homogeneous, layered or Winkler soil

An integral method to calculate the solution of a homogeneous or layered soil due to a harmonic point load is described. An infinite plate at the surface of the soil can be introduced in this integration in wavenumber domain, too. Finite structures on the soil are calculated by a combined finite element and boundary element method, which makes use of the point load solution of the soil. The compliance functions for a vertical point load and some vibration modes are calculated for realistic parameters of the plate and the soil and for a wide range of frequencies. The influence of the stiffness of the soil and the foundation is investigated, showing that the soil mainly affects the low-frequent response whereas the structural properties are more important at higher frequencies. A rigid approximation of flexible plates is only found at low frequencies, if the elastic length is used as the radius of a rigid disk. At higher frequencies, a characteristic behaviour of the flexible plate of approximately is observed, what is in clear contrast to the compliance of rigid foundations. A plate on a visco-elastic support (Winkler soil) shows similar displacements as a plate on a homogeneous half-space, but the maximal stresses between the plate and the soil are considerably smaller which is found to be more realistic for a plate on a layered soil. For practical applications, a normalized diagram and some explicit formulas of the exact and the approximate solutions of an infinite plate on a homogeneous half-space are given, which is a useful model to represent the soil-structure interaction of flexible foundations.     

Analytical solution of specific energy and specific force equations: Trapezoidal and triangular channels

The specific energy and specific force equations have many applications in open-channel flow problems. At present, these equations have analytical solution only for rectangular channels. Trial and error procedure also graphical solutions are the existing methods of solving these equations. No analytical solutions are available in the technical literature for these equations in trapezoidal and triangular channels because it is presumed that these equations are quintic equations. The inversion of such equations consists of finding the roots of quintic equations. In the current study for a given channel geometry and discharge, the subcritical (supercritical) depth is analytically found in terms of the other supercritical (subcritical) depth. For this purpose, by considering physically meaningful domains, a quintic equation has been reduced to a quartic equation. In the next step, this quartic equation has been converted to a resolvent cubic equation and two quadratic equations. This research shows these steps clearly to reach an acceptable physical analytic solution for water depth in trapezoidal and triangular channels.