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Ministry of Higher Education
And Scientific Research
University of Babylon
College of Engineering
Department of Mechanical Engineering
Using of Unit Cell Method in Micromechanical Stress
Analysis of Fiber-Reinforced Composite Beam
A Thesis
Submitted to the College of Engineering
of the University of Babylon in partial
Fulfilment of the degree of Master
of Science in Mechanical
Engineering
By
Luay Muhammad Ali Ismaeel
B.sc. 1986
May 2007
Thoul Ki'dah. 20, 38
Abstract
The good design of a structural composite part requires a thorough
and deep understanding of the behaviors of that composite on both macroand microscopic levels. In order to meet some specific requirements among of
which, the durability, reliability, economy and safety, the reliable design
should meet, base on and satisfy both precise macro- and micromechanical
analyses of that composite material.
The unit cell method in micromechanical analysis based on the finite
element method is one of the important, beneficial and effective means to
achieve the purpose referred to above; this is why this work adopted it.
Hexagonal fiber-matrix packing system is idealized for the material from
which the structural part of the present work is comprised, and selected to be
a unidirectional fiber-reinforced composite. The reasons beyond these
selections can be summarized as the unidirectional fiber-reinforced
composites are the most widely used composites; from the other hand the
beams are also the most vastly employed engineering structures in various
industrial and engineering fields. Two types of analyses are made on the
unidirectional fiber-reinforced composite beam under consideration. The first
is carried out in the longitudinal direction of the beam to determine the
vertical displacements (deflections) at the various sections of the beam,
stresses and strains. These field parameters (displacements, stresses & strains)
are studied under different types of fibers and matrices with various fiber
volume fractions at two different boundary conditions (simply supported and
built–in beam cases). The effect of load variations is also studied on the beam
behavior and stresses induced. The pertinent equations of the analysis are
formulated using the finite element method based on the displacement
approach and minimum potential energy principle. A matlab program is
implemented to solve these equations. The second analysis made in a
transverse direction to define the state and distribution of the stresses induced
In the cross-section. A failure criterion can be used to check the stress level
with respect to the beam materials strength. In the unit cell method, the field
quantities are found at a certain point then; they are mapped to the whole
cross-section domain according to the bending theory rules to determine their
values wherever required. The package of ANSYS 5.4 is used to solve for the
stresses, strains and displacements of the beam of interest throughout the
current work. The most prominent results obtained can be summarized as
follows:
1. The method of unit cell can be efficiently applied to the bending state of
stress for the unidirectional fiber-reinforced composite beam.
2. In case of clamped beam, the stresses induced are more than those of
simply supported case.
3. The deflections are not highly affected by the type of support.
4. The bending stresses are not considerably influenced by the variation of
fiber volume fraction and the applied load due to high elastic moduli of the
considered composite.
The method of solution of the transversely isotropic problem in
macroscopic analysis is applied to an isotropic counterpart. The results are
then compared; good agreement was distinguished between them. As well as
the unit cell macro- and micromechanical results are compared with those of
isotropic case, the same convergence is also noticed verifying the validity of
the analysis method adopted by the current work to such a problem.