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Volume 11, Number 2, February 2001

An improved version of the Element-free Galerkin method (EFGM) is presented here for
addressing the problem of transverse shear locking in shear-deformable beams with a high length over
thickness ratio. Based upon Timoshenko

Key Words
meshless methods; element-free Galerkin method; shear locking; thick beam.

W. Kanok-Nukulchai, W.J. Barry and K. Saran-Yasoontorn, School of Civil Engineering, Asian Institute of Technology, Pathumthani, 12120, Thailand

A high-fidelity model of a tracked vehicle traversing a flexible ground terrain with a varying
profile is presented here. In this work, we employed a recursive formulation to model the track
subsystem. This method yields a minimal set of coordinates and hence, computationally more efficient
than conventional approaches. Also, in the vehicle subsystem, the undercarriage frame is assumed to be
connected to the chassis by a revolute joint and a spring-damper unit. This increase in system mobility
makes the model more realistic. To capture the vehicle-ground interaction, a Winkler-type foundation with
springs-dampers is used. Simulation runs of the integrated tracked vehicle system for vibrations for four
varying ground profiles are provided.

Key Words
tracked vehicles; multibody dynamic models; flexible ground terrain; varying ground profile.

Ray P. S. Han, Department of Mechanical Engineering, The University of Iowa, Iowa City, IA 52242, USA
Brian S. Sander, Henderson Engineers Incorporated, Lenexa, KS 66214, USA
S.G. Mao, Department of Mechanical Engineering, The University of Iowa, Iowa City, IA 52242, USA

With increasing competition, the engineering industry is in need of optimization of designs
that would lead to minimum cost or weight. Recent developments in Genetic Algorithms (GAs) makes it
possible to model and obtain optimal solutions in structural design that can be put to use in industry. The
main objective of this paper is to illustrate typical applications of GAs to practical design of structural
systems such as steel trusses, towers, bridges, reinforced concrete frames, bridge decks, shells and layout
planning of buildings. Hence, instead of details of GA process, which can be found in the reported
literature, attention is focussed on the description of the various applications and the practical aspects that
are considered in Genetic Modeling. The paper highlights scope and future directions for wider applications
of GA based methodologies for optimal design in practice.

Key Words
optimization; structures; genetic algorithms; steel trusses; reinforced concrete frames; bridge structures; multi-storey buildings; shells; structural reliability; layout planning; knowledge modules.

C.S. Krishnamoorthy, Department of Civil Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

A direct discrete formulation suitable for the nonlinear analysis of masonry structures is
presented. The numerical approach requires a pair of dual meshes, one for describing displacement fields,
one for imposing equilibrium. Forces and displacements are directly used (instead of having to resort to a
model derived from a set of differential equations). Associated and nonassociated flow laws are dealt with
within a complementarity framework. The main features of the method and of the relevant computer code
are discussed. Numerical examples are presented, showing that the numerical approach is able to describe
plastic strains, damage effects and crack patterns in masonry structures.

Key Words
crack growth; damage mechanics; discrete formulation; linear complementarity problems; masonry; mathematical programming; nonlinear structural analysis; plasticity.

A. Nappi, Department of Civil Engineering, University of Trieste, P.le Europa 1, 34127 Trieste, Italy
F. Ti n- Loi, School of Civil and Environmental Engineering, The University of New South Wales, Sydney 2052, Australia

This paper presents the effect of axial stretching on large amplitude free vibration of an
extensible suspended cable supported at the same level. The model formulation developed in this study is
based on the virtual work-energy functional of cables which involves strain energy due to axial stretching
and work done by external forces. The difference in the Euler equations between equilibrium and motion
states is considered. The resulting equations govern the horizontal and vertical motion of the cables, and
are coupled and highly nonlinear. The solution for the nonlinear static equilibrium configuration is
determined by the shooting method while the solution for the large amplitude free vibration is obtained by
using the second-order central finite difference scheme with time integration. Numerical examples are
given to demonstrate the vibration behaviour of extensible suspended cables.

Key Words
cables; axial stretching; free vibration; large amplitude vibration; nonlinear vibration.

Somchai Chucheepsakul, Department of Civil Engineering, King Mongkut

Axially compressed circular cylinders repeat symmetry-breaking bifurcation in the postbuckling
region. There exist stable equilibria with all symmetry broken in the buckled configuration, and the
minimum postbuckling strength is attained at the deep bottom of closely spaced equilibrium branches. The
load level corresponding to such postbuckling stable solutions is usually much lower than the initial
buckling load and may serve as a strength limit in shell stability design. The primary concern in the
present paper is to compute these possible postbuckling stable solutions at the deep bottom of the
postbuckling region. Two computational approaches are used for this purpose. One is the application of
individual procedures in computational bifurcation theory. Path-tracing, pinpointing bifurcation points and
(local) branch-switching are all applied to follow carefully the postbuckling branches with the decreasing
load in order to attain the target at the bottom of the postbuckling region. The buckled shell configuration
loses its symmetry stepwise after each (local) branch-switching procedure. The other is to introduce the
idea of path jumping (namely, generalized global branch-switching) with static imperfection. The static
response of the cylinder under two-parameter loading is computed to enable a direct access to
postbuckling equilibria from the prebuckling state. In the numerical example of an elastic perfect circular
cylinder, stable postbuckling solutions are computed in these two approaches. It is demonstrated that a
direct path jump from the undeformed state to postbuckling stable equilibria is possible for an appropriate
choice of static perturbations.

Key Words
circular cylindrical shell; symmetry-breaking bifurcation; branch-switching; path jump; stable postbuckling solution.

Fumio Fujii, Department of Civil Engineering, Gifu University, Gifu 501-1193, Japan
Hirohisa Noguchi, Department of System Design Engineering, Keio University, Yokohama 223-8522, Japan

This paper presents the tangent stiffness method for 3-D geometrically nonlinear folding
analysis of a reversal arch. Experimental tests are conducted to verify the numerical analysis. The tangent
stiffness method can accurately evaluate the geometrical nonlinearity due to the element translating as a
rigid body, and the method can exactly handle the large rotation of the element in space. The arch in the
experiment is made from a thin flat bar, and it is found that the folding process of the arch may be
captured exactly by the numerical analysis with a model consisting of only 18 elements with the same

Key Words
large displacement analysis; geometrical nonlinearity; finite rotation; folding experiment.

Shin-ichi Iguchi and Shigeo Goto, FORUMEIGHT Ltd., 1-31 Tenya-machi, Hakata-ku Fukuoka, Japan
Katsushi Ijima and Hiroyuki Obiya, Department of Civil Engineering, Saga University, 1 Honjou Saga, Japan

A local point interpolation method (LPIM) is presented for the stress analysis of two-dimensional
solids. A local weak form is developed using the weighted residual method locally in two-dimensional
solids. The polynomial interpolation, which is based only on a group of arbitrarily distributed
nodes, is used to obtain shape functions. The LPIM equations are derived, based on the local weak form
and point interpolation. Since the shape functions possess the Kronecker delta function property, the
essential boundary condition can be implemented with ease as in the conventional finite element method
(FEM). The presented LPIM method is a truly meshless method, as it does not need any element or mesh
for both field interpolation and background integration. The implementation procedure is as simple as
strong form formulation methods. The LPIM has been coded in FORTRAN. The validity and efficiency
of the present LPIM formulation are demonstrated through example problems. It is found that the present
LPIM is very easy to implement, and very robust for obtaining displacements and stresses of desired
accuracy in solids.

Key Words
meshless method; stress analysis; interpolation function; weak form; strong form.

G.R. Liu and Y.T. Gu, Dept. of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

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