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Volume 26, Number 1, May10 2007

Because of complexity, the literature regarding the free vibration analysis of a Timoshenko beam carrying ?multiple? spring-mass systems is rare, particular that regarding the ?exact? solutions. As to the ?exact? solutions by further considering the joint terms of shear deformation and rotary inertia in the differential equation of motion of a Timoshenko beam carrying multiple concentrated attachments, the information concerned is not found yet. This is the reason why this paper aims at studying the natural frequencies and mode shapes of a uniform Timoshenko beam carrying multiple intermediate spring-mass
systems using an exact as well as a numerical assembly method. Since the shear deformation and rotary inertia terms are dependent on the slenderness ratio of the beam, the shear coefficient of the cross-section, the total number of attachments and the support conditions of the beam, the individual and/or combined effects of these factors on the result are investigated in details. Numerical results reveal that the effect of the shear deformation and rotary inertia joint terms on the lowest five natural frequencies of the combined vibrating system is somehow complicated.

Key Words
Timoshenko beam; shear deformation; rotary inertia; spring-mass systems; shear coefficient; natural frequency.

Jee-Ray Wang: Institute of Mechatronoptic Systems, ChienKuo Technology University, Changhua City, Taiwan 500, Republic of China
Tsung-Lung Liu and Der-Wei Chen: Department of Naval Architecture and Marine Engineering, Chung Cheng Institute of Technology, National Defense University, Dashi, Taoyuan, Taiwan 335, Republic of China

In the paper, one focuses on the problem of duality in non-linear programming, applied to the solution of no-tension problems by means of Limit Analysis (LA) theorems for Not Resisting Tension (NRT) models. In details, one demonstrates that, starting from the application of the duality theory to the non-linear program defined by the static theorem approach for a discrete NRT model, this procedure results in the definition of a dual problem that has a significant physical meaning: the formulation of the kinematic theorem.

Key Words
structural analysis; operational research; NRT model; duality; non-linear programming; limit analysis.

A. Baratta and O. Corbi: Dept. of Structural Engineering, University of Naples ?Federico II?, via Claudio 21, 80125 Naples, Italy

The unsymmetric finite element formulation has been proposed recently to improve predictions from distorted finite elements. Studies have also shown that this special formulation using parametric functions for the test functions and metric functions for the trial functions works surprisingly well because the former satisfy the continuity conditions while the latter ensure that the stress
representation during finite element computation can retrieve in a best-fit manner, the actual variation of stress in the metric space. However, a question that remained was whether the unsymmetric formulation
was variationally correct. Here we determine that it is not, using the simplest possible element to amplify the principles.

Key Words
mesh distortion; unsymmetric formulation; best-fit paradigm; parametric-metric element; projection theorem; three-node bar element.

G. Prathap: CSIR Centre for Mathematical Modelling and Computer Simulation, Bangalore 560 037, India
S. Manju: National Aerospace Laboratories, Bangalore 560 017, India
V. Senthilkumar: CSIR Centre for Mathematical Modelling and Computer Simulation, Bangalore 560 037, India

Using the Mindlin-Reissner plate theory, many quadrilateral plate bending elements have been developed so far to analyze thin and moderately thick plate problems via displacement based finite element
method. Here new formulation has been made to analyze thin and moderately thick plate problems using force based finite element method called Integrated Force Method (IFM). The IFM is a novel matrix
formulation developed in recent years for analyzing civil, mechanical and aerospace engineering structures. In this method all independent/internal forces are treated as unknown variables which are calculated by
simultaneously imposing equations of equilibrium and compatibility conditions. In this paper the force based new bilinear quadrilateral plate bending element (MQP4) is proposed to analyze the thin and
moderately thick plate bending problems using Integrated Force Method. The Mindlin-Reissner plate theory has been used in the formulation of this element which accounts the effect of shear deformation. Standard plate bending benchmark problems are analyzed using the proposed element MQP4 via Integrated Force Method to study its performance with respect to accuracy and convergence, and results are compared with those of displacement based 4-node quadrilateral plate bending finite elements available in the literature. The results are also compared with the exact solutions. The proposed element MQP4 is free from shear locking and works satisfactorily in both thin and moderately thick plate bending situations.

Key Words
stress-resultant fields; flexibility matrix; equilibrium matrix; displacement fields; Mindlin-Reissner theory; Integrated Force Method.

H. R. Dhananjaya: Dept. of Civil Engineering, Manipal Institute of Technology, Manipal - 576 104, India
J. Nagabhushanam: Dept. of Aerospace Engineering, Indian Institute of Science, Bangalore - 560 012, India
P. C. Pandey: Dept. of Civil Engineering, Indian Institute of Science, Bangalore - 560 012, India

A four-noded plate bending quadrilateral (PBQ4) and an eight-noded plate bending quadrilateral (PBQ8) element based on Mindlin plate theory have been adopted for modeling the thick plates on elastic foundations using Winkler model. Transverse shear deformations have been included, and the stiffness matrices of the plate elements and the Winkler foundation stiffness matrices are developed using Finite Element Method based on thick plate theory. A computer program is coded for this purpose. Various loading and boundary conditions are considered, and examples from the literature are solved for
comparison. Shear locking problem in the PBQ4 element is observed for small value of subgrade reaction and plate thickness. It is noted that prevention of shear locking problem in the analysis of the thin plate is
generally possible by using element PBQ8. It can be concluded that, the element PBQ8 is more effective and reliable than element PBQ4 for solving problems of thin and thick plates on elastic foundations.

Key Words
finite element method; thick plate theory; elastic foundation; Winkler model.

K. Ozgan: Yuksel Proje Uluslararas A. . Birlik Mah. 9. Cad. No:41 Cankaya 06610 - Ankara, Turkey
Ayse T. Daloglu: Dept. of Civil Engineering, Karadeniz Technical University, Trabzon, 61080, Turkey

Some mixed mode fracture criterion may be converted in to elliptical or ellipsoidal formula with the aid of mathematical translation. Hence, the crack initiation in mixed mode fracture I+II emanating from notches, has been studied using notched circular ring specimens. On the basis of Irwin (1957) theory, a new criteria in mixed mode fracture I+II, based fracture elliptic criterion and notch stress intensity factors has been developed.

Key Words
notch effect; angled crack problem; maximum tangential stress criterion; effective tangential stress; effective tangential distance; relative tangential stress gradient; weight function; notch stress intensity factors KI? and KII?.

H. El Minor: Laboratoire d?Energie, Ressources Naturelles et Modelisation, Faculte Polydisciplinaire de Taza_Universite Sidi Mohemed Ben Abdelah Fes, Morocco
G. Pluvinage: Universite de Metz, Laboratoire de Fiabilite Mecanique, Ile du Saulcy Metz 57045, France
Z. Azari: Universite de Picardie Jules Verne, Faculte de Mathematiques et d\'Informatique, 33 rue Saint-Leu, 80039 Amiens, France

A cracked composite specimen, comprised of an epoxy and an aluminium plate, was fractured under a tensile load. In this paper, two crack configurations were investigated. The first was an artificial center crack positioned in the epoxy plate parallel to the material interface. The other was for two edge cracks in the epoxy plate, again, parallel to the interface. A tensile test was carried out by gradually increasing the applied load and it was verified that the cracks always moved suddenly in an outward direction from the interface. The d/a ratio was gradually reduced to zero, and it was confirmed
that the maximum stress intensity factor value for the artificial center crack, , approached that of an artificial interface crack, (where: 2a is the crack length and d is the offset between the crack and
interface). The same phenomenon was also verified for the edge cracks. Specifically, when the offset, d, was reduced to zero, the maximum stress intensity factor value, , approached that of an artificial
interface edge crack.

Key Words
stress intensity factor; parallel crack; interface crack; epoxy-aluminium composites; tensile test.

S. Itou: Dept. of Mechanical Engineering, Kanagawa University, Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan

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