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Volume 11, Number 4, April 2001

A new class of semi-active tuned mass dampers, named as

Key Words
tuned mass damper; tuned vibration absorber; semi-active damper; vibration control.

Mehdi Setareh, Department of Architecture, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A.

In this paper, cantilevered tall structures are treated as cantilever bars with varying cross-section
for the analysis of their free longitudinal (or axial) vibrations. Using appropriate transformations,
exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one step
non-uniform bar are derived by selecting suitable expressions, such as exponential functions, for the
distributions of mass and axial stiffness. The frequency equation of a multi-step bar is established using
the approach that combines the transfer matrix procedure or the recurrence formula and the closed-form
solutions of one step bars, leading to a single frequency equation for any number of steps. The Ritz
method is also applied to determine the natural frequencies and mode shapes in the vertical direction for
cantilevered tall structures with variably distributed stiffness and mass. The formulae proposed in this
paper are simple and convenient for engineering applications. Numerical example shows that the
fundamental longitudinal natural frequency and mode shape of a 27-storey building determined by the
proposed methods are in good agreement with the corresponding measured data. It is also shown that the
selected expressions are suitable for describing the distributions of axial stiffness and mass of typical tall

Key Words
longitudinal vibration; tall buildings; high-rise structures; natural frequencies; mode shapes.

Q.S. Li, Department of Building and Construction, City University of Hong Kong
Tat Chee Avenue, Kowloon, Hong Kong
J.Y. Xu and G.Q. Li, Department of Civil Engineering, Wuhan University of Technology, Wuhan 430070, China

The main idea behind the paper is to present two alternative methods of homogenization of
the heat conduction problem in composite materials, where the heat conductivity coefficients are assumed
to be random variables. These two methods are the Monte-Carlo simulation (MCS) technique and the second
order perturbation second probabilistic moment method, with its computational implementation known as
the Stochastic Finite Element Method (SFEM). From the mathematical point of view, the deterministic
homogenization method, being extended to probabilistic spaces, is based on the effective modules approach.
Numerical results obtained in the paper allow to compare MCS against the SFEM and, on the other hand,
to verify the sensitivity of effective heat conductivity probabilistic moments to the reinforcement ratio. These
computational studies are provided in the range of up to fourth order probabilistic moments of effective
conductivity coefficient and compared with probabilistic characteristics of the Voigt-Reuss bounds.

Key Words
homogenization method; stochastic second order perturbation; stochastic finite element method; Monte-Carlo simulation; composites.

Marcin Kamioski, Division of Mechanics of Materials, Technical University of Lodo, Al. Politechniki 6, 93-590 Lodo, Poland

The present paper discusses the behavior of the reinforced concrete beams subjected to
torsion by applying the endochronic plastic model in conjunction with the softened truss model. The
endochronic constitutive equations are developed to describe the behavior of concrete. The mechanical
behavior of concrete is decomposed into hydrostatic part and deviatoric part. New definition of the bulk
modulus and the shear modulus are defined in terms of compressive strength of concrete. Also, new
deviatoric hardening function is developed. Then, the endochronic constitutive equations of concrete are
applied with the softened truss model for the behavior of the reinforced concrete beams subjected to
torsion. The theoretical results obtained based on the present model are compared with the experimental
data. The present model has shown the ability to describe the behavior of reinforced concrete beams
subjected to torsion.

Key Words
endochronic model for concrete; reinforced concrete beam; softened truss model.

Jun-Kai Lu and Wen-Hsiung Wu, Department of Civil Engineering, National Pingtung University of Science and Technology, Pingtung, Taiwan 912, ROC.

The linear constitutive relations and the failure criteria of composite materials made of
thermoviscoelastic solids are presented. The post-failure material behavior is proposed and the dynamic
finite element equations are formulated. However, a nonlinear term is kept in the energy equation because
it represents the effect of the second law of thermodynamics. A general purpose nonlinear three-dimensional
dynamic finite element program COMPASS is upgraded and employed in this work to investigate the
interdependence among stress wave propagation, stress concentration, failure progression and temperature
elevation in composite materials. The consequence of truthfully incorporating the second law of thermodynamics is
clearly observed: it will always cause temperature rise if there exists a dynamic mechanical process.

Key Words
dynamic finite element analysis; second law of thermodynamics; thermo-mechanical coupling; thermoviscoelasticity; composite materials; wave propagation.

Siyuan J. Shen, Jens C. Pfister and James D. Lee, Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, DC 20052, U.S.A.

This paper presents a method of elastic-plastic analysis for planar steel frames that provides
the accuracy of distributed plasticity methods with the computational efficiency that is greater than that of
distributed plasticity methods but less than that of plastic-hinge based methods. This method accounts for
the effect of spread of plasticity accurately without discretization through the cross-section of a beam-column
element, which is achieved by the following procedures. First, nonlinear equations describing the
relationships between generalized stresses and strains of the cross-section are derived analytically. Next,
nonlinear force-deformation relationships for the beam-column element are obtained through lengthwise
integration of the generalized strains. Elastic-plastic flexibility coefficients are then calculated by
differentiating the above element force-deformation relationships. Finally, an elastic-plastic stiffness matrix
is obtained by making use of the flexibility-stiffness transformation. Adding the conventional geometric
stiffness matrix to the elastic-plastic stiffness matrix results in the tangent stiffness matrix, which can
readily be used to evaluate the load carrying capacity of steel frames following standard nonlinear
analysis procedures. The accuracy of the proposed method is verified by several examples that are
sensitive to the effect of spread of plasticity.

Key Words
second-order analysis; steel frames; spread of plasticity; flexibility matrix.

Liang-Jenq Leu and Ching-Huei Tsou, Department of Civil Engineering, National Taiwan University, Taipei, 10617 Taiwan, R.O.C.

This paper presents an effective application of a variable-node axisymmetric solid element
designated as AQV (Axisymmetric Quadrilateral Variable-node element). The variable-node element with
physical midside nodes helps to overcome some problems in connecting the different layer patterns on a
quadrilateral mesh in the adaptive h-refinement. This element alleviates the necessity of imposing
displacement constraints on irregular (hanging) nodes in order to enforce the inter-element compatibility.
Therefore, the elements with variable mid-side nodes can be used effectively in the local mesh refinement
for the axisymmetric structures which have stress concentrations. A modified Gaussian quadrature should
be adopted to evaluate the stiffness matrices of the variable-node elements mainly because of the slope
discontinuity of assumed displacement within the elements. Some numerical examples show the usefulness
of variable-node axisymmetric elements in the practical application.

Key Words
quadrilateral axisymmetric element; irregular node; variable-node element; transition element; adaptive h-refinement.

Chang-Koon Choi and Eun-Jin Lee, Department of Civil Engineering, KAIST, Taejon 305-600, Korea
Wan-Hoon Lee, Department of Civil & Environment Engineering, Chungwoon University, Chungnam 350-701, Korea

The aim of this work is to establish the coefficient that defines the critical buckling load for
isotropic annular plates of variable thickness whose outer boundary is simply supported and subjected to
uniform pressure. It is assumed that the plate thickness varies in a continuous way, according to an
exponential law. The eigenvalues are determined using an optimized Rayleigh-Ritz method with
polynomial coordinate functions which identically satisfy the boundary conditions at the outer edge. Good
engineering agreement is shown to exist between the obtained results and buckling parameters presented
in the technical literature.

Key Words
critical buckling load; annular plates; variable thickness.

P.M. Ciancio and J.A. Reyes, Department of Civil Engineering, Universidad Nacional del Centro de la Provincia de Buenos Aires. Av. Del Valle 5737, 7400 Olavarria, Argentina

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