Coupled vibration of tall building structures

Free vibration analysis is presented for general tall building structures, which may consist of any combination of frames, shear walls, structural cores and coupled walls. Emphasis of the analysis is placed on the coupled lateral–torsional vibration characteristic of the structures. Based on the continuum technique and D'Alembert's principle, the governing equation of free vibration and corresponding eigenvalue problem are derived. By applying the Galerkin technique, a generalized method of solution is proposed for the analysis of coupled vibration of general tall building structures. Based on the proposed method, a computation procedure is presented for predicting the natural frequencies and associated mode shapes of the structures in coupled vibration. Numerical investigation is conducted to validate the simplicity and accuracy of the proposed method. It has been shown that the proposed analysis provides an effective way, particularly at the preliminary design stage, for evaluating the vibration behaviour of tall buildings. Copyright © 2004 John Wiley & Sons, Ltd.     

Free longitudinal vibrations of tall buildings and high-rise structures

In this paper, tall buildings and high-rise structures are considered as cantilever bars with variable cross-section for the analysis of their free vibrations. The differential equations of free longitudinal vibrations of bars with variable cross-section are reduced to Bessel's equations by selecting suitable expressions, such as power functions and exponential functions, for the distribution of stiffness and mass. An approach is proposed for determining the natural frequencies and mode shapes in the vertical direction for tall buildings and high-rise structures with variably distributed stiffness and variably distributed mass. The derived solutions are expressed in terms of Bessel functions. A numerical example shows that the value of the natural frequency computed by the proposed method is close to full scale measured data. It is shown that the selected expressions are suitable for describing the distributions of stiffness and mass of tall buildings and high-rise structures. It is demonstrated that the proposed method has practical significance for free longitudinal vibration analysis. © 1998 John Wiley & Sons, Ltd.     

An analytical model for dynamic response analysis of tubular tall buildings

In this paper, on the basis of the D'Alembert's principle, approximate formulas for dynamic response of tubular tall building structures are presented. Using D'Alembert's principle and applying the compatibility conditions on deformation of the tubes, the governing dynamic equation of the tubular structure's motion is derived. Then, natural boundary conditions of the parallel cantilevered flexural–shear beams are derived, and by using Rayleigh–Ritz method, value problem is solved, and trivial and nontrivial solutions are derived, which can be used for calculating natural frequencies and mode shapes of tubular structures. By solving numerically the frequency equation, a design chart and graph are given for the first five nondimensional natural frequencies of tubular tall buildings. The proposed mathematical model gives dynamic characteristics and provides a simple, efficient and reasonably accurate algorithm for free vibration studies that are needed to be quick at the preliminary design stages of tall buildings with tubular systems. Copyright © 2012 John Wiley & Sons, Ltd.     

高层混凝土结构重力二阶效应的影响分析

传统的高层混凝土结构重力二阶效应分析大多是近似方法,对于实际高层结构,这种近似分析方法会带来较大差错。本文通过在单元几何方程中引入二次项,较为精确地考虑了重力二阶效应对高层结构受力特性的影响。根据高层结构的特点,建立了梁柱单元、单片墙单元和筒体墙单元三种非线性宏观单元,并编制了三维推覆分析程序。单元截面分析采用高斯数值积分方法,可对任意不规则截面进行分析,在非线性方程组的求解算法中引入了弧长算法,能够搜索到结构反应的全过程。对两栋不同结构体系的高层混凝土建筑进行了推覆分析,详细讨论了二阶效应对结构顶点侧移和基底倾覆力矩的影响。研究结果表明:对高度100m左右的高层混凝土结构,多遇地震作用下的强度和变形验算可不考虑重力二阶效应的影响,罕遇地震作用下的弹塑性变形验算一般应考虑重力二阶效应的影响。     

Approximate analysis of tall buildings using sandwich beam models with variable cross‐section

In this paper an approximate method is carried out for determining the natural periods of multistory buildings subjected to earthquake. The building resists lateral forces through a combination of lateral resisting systems. These systems could be replaced by a cantilever Timoshenko or a sandwich beam with varying cross‐section that characterizes three kinds of stiffnesses: the global bending stiffness, the local bending stiffness and the shear stiffness. Using appropriate transformations, the differential equations for flexural and shear free vibration of a cantilever beam with variably distributed mass and stiffness are reduced to Bessel's equations and ordinary differential equations. The frequency equations can be solved by selecting suitable expressions such as exponential and power functions for stiffness and mass distribution along the height of the building. The calculated frequencies are combined appropriately by using approximate methods. Based on the fact that shear and bending deformations are all considered, the free vibration frequency of the structure could be calculated. The capability and accuracy of the proposed method are demonstrated by a numerical example in which finite element results are compared with the proposed methodology and other approximate methods. Copyright © 2007 John Wiley & Sons, Ltd.     

悬臂梁的静力分析及自由振动分析

大跨度及高层空间结构是目前发展最快的结构类型。大跨度建筑及作为其核心的高层空间结构技术的发展状况是代表一个国家建筑科技水平的重要标志之一。本文主要研究变截面均质悬臂梁的结构体系在水平均布荷载作用下的静力分析,以及EULER悬臂梁的自由振动分析。根据哈密顿原理推导出控制方程和边界条件,然后用常微分求解器COLSYS进行求解。     

Parameter study of framed‐tube structures with outriggers using genetic algorithm

In this paper, the analysis of optimal locations of framed‐tube structures with outriggers is conducted with the uniform and nonuniform core and peripheral columns using genetic algorithm, aiming to minimize the interstory drift. Also, comparison and difference of the results between the uniform and nonuniform structures are carried out and discussed. Besides, several parameters which influence the behavior of the structure are identified and analyzed, such as different objective functions, segments of outriggers, thickness of core wall, stiffness of outriggers, and grade of concrete strength. In addition, a Matlab program is written to perform the parameter analysis of optimal location of outriggers. Take a 260‐m high‐rise building as a target building, the optimal locations of one to two sets of outriggers subjected to three kinds of horizontal loadings (uniform, parabolic, and triangular) are obtained and can be utilized for the structural preliminary design of tall buildings.     

Dynamic analysis of combined system of framed tube and shear walls by Galerkin method using B‐spline functions

In this article, dynamic parameters (natural frequencies and mode shapes) of tall buildings that consist of framed tube and shear walls are obtained using a simple approximate method. The three‐dimensional structure is replaced by an equivalent cantilever beam, considering both bending and shear deformations. On the basis of dynamic equilibrium, the governing differential equation of motion is obtained and converted to its corresponding weak form. B‐spline functions are then utilized to approximate the weak form and to obtain the final matrix form of the problem. Finally, by applying essential boundary conditions, the natural frequencies and corresponding mode shapes are calculated. To demonstrate the accuracy of the proposed method, numerical examples are solved, and the results are compared with those obtained from SAP2000 computer analysis. The results show that the proposed method is efficient and accurate enough to be used in preliminary design. Copyright © 2014 John Wiley & Sons, Ltd.     

Vibration analysis of tall buildings with narrow rectangular plane configuration

In this paper, tall buildings of narrow rectangular plane configuration are treated as one-step or multi-step cantilever flexural-shear plates for the analysis of their free vibrations. The governing differential equations for the vibration of flexural-shear plates considering the effects of both flexural and shear deformation are established. The general solutions for one-step flexural-shear plates are derived and used to obtain the eigenvalue equation for multi-step cantilever flexural-shear plates. A new exact approach is presented which combines the transfer matrix method and a closed-form solution for a uniform flexural-shear plate. A numerical example demonstrates that the calculated natural frequencies and mode shapes of a tall building are in good agreement with the experimentally measured data. It is also shown that the effect of shear deformation on the fundamental natural frequency can be ignored, but its effect on the higher natural frequencies should be considered. Copyright © 1998 John Wiley & Sons, Ltd.     

Optimum seismic design of steel framed‐tube and tube‐in‐tube tall buildings

The relatively large number of structural elements and the variety of design code requirements complicate the design process of tall buildings. This process is exacerbated when the target is to obtain the seismic code‐compliant optimal design with minimum weight. The present paper aims at providing a practical methodology for the optimal design of steel tall building structures considering the constraints imposed by typical building codes. The applicability of the proposed approach is demonstrated through the determination of the optimal seismic design for 20‐, 40‐, and 60‐story buildings with a framed tube as well as a tube‐in‐tube system. Such a design gives rise to a basis for the fair comparison of the behavior of the framed tube with that of the tube‐in‐tube system under applied loads. The optimal weight of the buildings with the tube‐in‐tube system turns out to be slightly less than that of the buildings with the conventional framed‐tube system.     

Vibration characteristics of tall buildings braced by shear walls and thin‐walled open‐section structures

An analytical method is presented for the three‐dimensional frequency analysis of buildings braced by shear walls and thin‐walled open‐section structures. Owing to the asymmetry of the structure, the centre of gravity and the centre of flexural rigidity of the floor plan do not coincide, and hence the flexural vibration in two mutually perpendicular directions and the warping torsional vibrations are all coupled. Based on the continuum approach and D'Alembert's principle, the governing differential equation of free vibration and its corresponding eigenvalue problem for asymmetric shear walls and thin‐walled open‐section structures are derived. Based on the theory of differential equations, an analytical method of solution is proposed to solve the eigenvalue problem and a general solution is derived for determining the natural frequencies of the structures. Results obtained from the proposed method for the example structure show good agreement with those of finite element analysis. It is also shown that the proposed analysis is efficient and accurate enough to be used both at the concept design stage and for final analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Timoshenko悬臂梁自由振动特性的近似分析方法

本文讨论了用模态摄动法分析Timoshenko悬臂梁的自由振动特性。Timoshenko梁是目前较为流行的梁模型之一,Timoshenko梁方程一般较复杂,很难给出解的解析形式。利用模态摄动法,将Timoshenko梁方程转化为一组非线性代数方程组的求解,不仅可以简化计算过程,而且计算结果具有较高的精度,对任意边界条件下的Timoshenko梁都适用。通过算例,讨论了剪切变形和转动惯量对悬臂梁的固有频率的影响。     

Free vibration analysis of horizontally curved steel I-girder bridges

Presented herein is a finite element formulation for free vibration analysis of horizontally curved steel I-girder bridges. Stiffness as well as mass matrices of the curved and the straight beam elements is formulated. Each node of both of them possesses seven degrees of freedom including the warping degree of freedom. The curved beam element is derived based on the Kang and Yoo's thin-walled curved beam theory in 1994. A computer program is developed to carry out free vibration analyses of the various bridges. Comparing with the frequencies using the general purpose program ABAQUS, the validity of the presented numerical formulation is shown. The numerical formulation is extensively applied to investigate free vibration characteristics of the bridges considering effects of the initial curvature, boundary condition, modeling method, and degrees of freedom of cross frame. Invaluable information which help practicing engineers better understand the vibration characteristics is provided.     

Simplified dynamic analysis of eccentric buildings with a setback. 1: the effect of mass irregularity

The dynamic response of eccentric buildings with a setback is investigated and a simple method is proposed for assessing vibration frequencies and modal base shears and torques. With the proposed methodology the aforementioned basic dynamic quantities are determined by analysing first the symmetrical counterpart structure and then an equivalent one‐storey eccentric system. The type of buildings examined represents the class of tall structures with a significant reduction in the size of floor plans above a certain level, but without any modification on the structural system, which is considered to be uniform over the height of the building, but eccentric to the mass axis. It is shown that the vertical mass irregularity thus produced increases substantially the contribution of the higher modes of vibration to the total response of such structures and this contribution increases with the height of the tower structure above the setback level. A number of common buildings with a mixed‐bent‐type structural system are analysed by the proposed method and the results are compared with more accurate results obtained by three‐dimensional dynamic analyses. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamic P‐Δ effects for multistorey steel columns in long‐span tube‐in‐tube tall buildings

An Euler–Bernoulli beam–column model is employed to investigate the dynamic P‐Δ effects of steel columns located at the external tube of a long‐span tube‐in‐tube tall building. A steel column with multistorey lateral open‐web beam supports is modelled as a beam–column element supported by elastic springs, from which a governing partial differential equation is established for the vibration of the column subjected to an axial compression force, lateral beam forces and base excitations. The problem is solved by a modal superposition method for both transient and steady‐state solutions. A numerical example is illustrated and the results indicate that the secondary moment of a steel column induced from dynamic multistorey P‐Δ effects is essential and should not be ignored in a practical design. The influences of the factors including flexural rigidity of the column, axial stiffness of the open‐web beams and the magnitude of the compressive axial force are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

Free vibration of shear beams with finite rotational inertia

Free vibration of shear beams is studied when rotational motion is taken into account, while classical shear beams do not consider rotational motion. From a single governing equation of Timoshenko beams, we analytically derive Rayleigh beams and shear beams as two limiting cases of the ratio of reduced shear stiffness to bending stiffness being sufficiently large and small, respectively. Emphasis is placed on the analysis of free vibration of nonclassical shear beams without damping effect. Under the condition of general end restraints, a characteristic equation for nonclassical shear beams with finite rotational inertia is derived in explicit form. A condition that the nonclassical shear beams reduce to the classical ones is found, and classical shear beams may be understood as nonclassical ones with infinite large rotational inertia. Nonclassical natural frequencies and mode shapes are calculated for a standing shear beam on an elastic foundation. Previous results of pinned-free, and free-free shear beams can be taken as special cases of the present analysis. The effects of finite rotational inertia, material properties, geometrical conditions and end restraints on the natural frequencies of shear beams are discussed.     

Evaluation of wind loads on super‐tall buildings from field‐measured wind‐induced acceleration response

With the nonstationary wind‐induced acceleration data from full‐scale measurements, an approach for estimation of the wind‐induced overturning bending moments for super‐tall buildings was proposed in this paper. The empirical mode decomposition was employed to decompose the measured acceleration data into a set of intrinsic mode functions and a residual component. To remove the baseline offset, the residual component and the intrinsic mode function components with long‐period were eliminated before their integrations into velocity and displacement components. Then, the intrinsic mode function components, which have the same dominant periods as the natural periods of the studied tall buildings, were extracted from the original signals, and the natural frequency and damping ratio for the first vibration mode of the building were identified. Finally, the wind‐induced overturning bending moments of the building were obtained from the generalized wind loads for the first vibration mode, which could be obtained from the time history analysis of dynamic equation. The Hilbert spectrum of wind‐induced overturning bending moments was utilized to observe its characteristics in both time and frequency domains, and the Strouhal number was thus identified. The proposed scheme and some selected results may be helpful for further understanding of wind effects on super‐tall buildings. Copyright © 2012 John Wiley & Sons, Ltd.     

Damage detection method for large structures using static and dynamic strain data from distributed fiber optic sensor

A method for damage detection applicable to large slender steel structures such as towers of large-scale wind turbines, long-span bridges, and high-rise buildings is presented. This method is based on continuous strain data obtained by distributed fiber optic sensor (FOS) and neural network (NN) analysis. An analytical model for cracked beam based on an energy balance approach was used to train a NN. The continuous static strains and the natural frequencies obtained from the distributed FOSs were used as the input to the trained NN to estimate the crack depths and locations. An experimental study was carried out on a cracked cantilever beam to verify the present method for damage identification. The cracks were inflicted on the beam, and static and free vibration tests were performed for the intact case and the damage cases. The distributed FOSs were used to measure the continuous strains. The damage estimation was carried out for the 5 damage cases using the NN technique. It has been found that the identified crack depths and locations agree reasonably well with the inflicted cracks on the structure.     

Coupled flexural–torsional dynamic stiffness matrix of an elastically supported axially loaded beam with shear resistant in‐fill

The dynamic stiffness matrix of an axially loaded elastically supported uniform beam with doubly asymmetric cross‐section that exhibits coupling between flexural and torsional motions is developed and subsequently used to investigate its free vibration characteristics. The beam comprises a thin‐walled outer section that encloses, and works compositely with, a core of shear resistant in‐fill material. The outer layer provides flexure, warping and Saint–Venant rigidity, while the inner layer provides both Saint–Venant and shear rigidity. A three‐parameter Winkler model is used to describe the distributed elastic support. Hamilton's principle is used to derive the partial differential equations governing the free vibration of the beam, together with the associated natural boundary conditions. This gives rise to three coupled equations that are subsequently combined into a single, 12th order, ordinary differential equation. Throughout the process, the uniform distribution of mass in the member is accounted for exactly and thus necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick–Williams algorithm, which enables the required natural frequencies to be converged upon to any required accuracy with the certain knowledge that none have been missed. Finally, in order to verify the accuracy of the presented theory, the numerical solutions are given and compared with the results that are available in the literature and finite element solutions using abaqus software. Copyright © 2014 John Wiley & Sons, Ltd.     

A simple approximate method for free vibration analysis of framed tube structures

A simple approximate method is developed for determining natural frequency of tall buildings. Timoshenko's beam model, which considers the influence of shear and flexural deformation, was used in modelling framed tube structures. In this paper, natural frequency and mode shape of framed tube structures were calculated based on the flexural and shear rigidities along with the effects of rotational inertia. Dynamic model of Timoshenko's beam can be obtained by writing equilibrium equations of forces acting on an infinite element. The solution of the dynamic model was obtained by first applying the separation of variables to time and space, followed by the assumption of harmonic motion in time, the steady state eigen system was obtained. Natural frequencies of framed tube structures were calculated by solving the eigenproblem. A numerical example has been presented to demonstrate the ease of application and accuracy of the proposed method. The structure's fundamental frequency was computed using ETABS V9.0.0 (Computer and Structures, Berkeley, California, USA) and compared with the result obtained from proposed method, which shows that the percentage of error was low and acceptable. The proposed method can be adopted as an alternative procedure to evaluate the natural frequency of framed tube in the preliminary stages of structural design. Copyright © 2010 John Wiley & Sons, Ltd.