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Title:  SuperConvergent Finite Elements For Analysis Of Higher Order Laminated Composite Beams 
Authors:  Murthy, MVVS 
Advisors:  Gopalakrishnan, S 
Keywords:  Laminated Composite Beams Composite Materials Finite Element Analysis Secong Order Shear Deformation Theory (SSDT) SuperConvergent Formulation Interlaminar Stresses Wave Propagation Beam Theory Deformations Composite Beam Finite Element Formulation EulerBernoulli Beam Theory (EBT) First Order Shear Deformation Theory (FSDT) Poisson's Contraction 
Submitted Date:  Jan2007 
Series/Report no.:  G20957 
Abstract:  Advances in the design and manufacturing technologies have greatly enhanced the utility of fiber reinforced composite materials in aircraft, helicopter and space
craft structural components. The special characteristics of composites such as high
strength and stiffness, lightweight corrosion resistance make them suitable sub
stitute for metals/metallic alloys. However, composites are very sensitive to the anomalies induced during their fabrication and service life. Also, they are suscepti
ble to the impact and high frequency loading conditions because the epoxy matrix is
atleast an order of magnitude weaker than the embedded reinforced carbon fibers.
On the other hand, the carbon based matrix posses high electrical conductivity which
is often undesirable. Subsequently, the metal matrix produces high brittleness. Var
ious forms of damage in composite laminates can be identified as indentation, fiber
breakage, matrix cracking, fibermatrix debonding and interply disbonding (delam
ination). Among all the damage modes mentioned above, delamination has been
found to be serious for all cases of loading. They are caused by excessive interlaminar shear and normal stresses. The interlaminar stresses that arise in the case of composite materials due to the mismatch in the elastic constants across the plies.
Delamination in composites reduce it’s tensile and compressive strengths by consid
erable margins. Hence the knowledge of these stresses is the most important aspect to be looked into. Basic theories like the EulerBernoulli’s theory and Timoshenko beam theory are based on many assumptions which poses limitation to determine these stresses accurately. Hence the determination of these interlaminar stresses accurately requires higher order theories to be considered.
Most of the conventional methods of determination of the stresses are through
the solutions, involving the trigonometric series, which are available only to small
and simple problems. The most common method of solution is by Finite Element (FE) Method. There are only few elements existing in the literature and very few in the commercially available finite element software to determine the interlaminar
stresses accurately in the composite laminates. Accuracy of finite element solution depends on the choice of functions to be used as interpolating polynomials for the field variable. Inappropriate choice will manifest in the form of delayed convergence. This delayed convergence and accuracy in predicting these stresses necessiates a formulation of elements with a completely new concept. The delayed convergence is sometimes attributed to the shear locking phenomena, which exist in most finite element formulation based on shear deformation theories. The present work aims
in developing finite elements based on higher order theories, that alleviates the slow convergence and achieves the solutions at a faster rate without compromising on the
accuracy. The accuracy primarily depends on the theory used to model the problem. Thus the basic theories (such as Elementary Beam theory and Timoshenko Beam theory) does not suffice the condition to accuratley determine the interlaminar stresses through the thickness, which is the primary cause for delamination in composites. Two different elements developed on the principle of superconvergence has been presented in this work. These elements are subjected to several numerical experiments and their performance is assessed by comparing the solutions with those available in literature.
Spacecraft and aircraft structures are light in weight and are also lightly damped because of low internal damping of the material of construction. This increased exibility may allow large amplitude vibration, which might cause structural instability. In addition, they are susceptible to impact loads of very short
duration, which excites many structural modes. Hence, structural dynamics and wave propagation study becomes a necessity. The wave based techniques have found appreciation in many real world problems such as in Structural Health Monitoring
(SHM). Wave propagation problems are characterized by high frequency loads, that
sets up stress waves to propagate through the medium. At high frequency, the wave
lengths are small and from the finite element point of view, the element sizes should be of the same order as the wave lengths to prevent free edges of the element to act as a free boundary and start reflecting the stress waves. Also longer element size makes the mass distribution approximate. Hence for wave propagation problems, very large finite element mesh is an absolute necessity. However, the finite element problems size can be drastically reduced if we characterize the stiffness of the structure accurately. This can accelerate the convergence of the dynamic solution significantly. This can be acheived by the superconvergent formulation. Numerical results are presented to illustrate the efficiency of the new approach in both the cases of dynamic studies viz., the free vibration study and the wave propagation study.
The thesis is organised into five chapters. A brief organization of the thesis is
presented below,
Chapter1 gives the introduction on composite material and its constitutive law. The details of shear locking phenomena and the interlaminar stress distribution across
the thickness is brought out and the present methods to avoid shear locking has been presented.
Chapter2 presents the different displacement based higher order shear deformation theories existing in the literature their advantages and limitations.
Chapter3 presents the formulation of a superconvergent finite element formulation,
where the effect of lateral contraction is neglected. For this element static and
free vibration studies are performed and the results are validated with the solution
available in the open literature.
Chapter4 presents yet another superconvergent finite element formulation, wherein the higher order effects due to lateral contraction is included in the model. In addition to static and free vibration studies, wave propagation problems are solved to demonstrate its effectiveness. In all numerical examples, the superconvergent property is emphasized.
Chapter5 gives a brief summary of the total research work performed and presents further scope of research based on the current research. 
URI:  http://etd.iisc.ernet.in/handle/2005/587 
Appears in Collections:  Aerospace Engineering (aero)

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