The objective of this book is to present a complete and up to date treatment of rectangular laminated plates with uniform cross sections. Dynamic Relaxation (DR) method is presented for the geometrically linear and nonlinear laterally loaded, rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse shear deformation. A computer program has been compiled. The convergence and accuracy of the DR solutions for elastic small and large deflection response are established by comparison with various exact and approximate solutions.
New numerical results are generated for uniformly loaded square laminated plates which serve to quantify the effects of shear deformation, material anisotropy, fiber orientation, and coupling between bending and stretching. It was found that linear analysis seriously over-predicts deflections of plates. The shear deflection depends greatly on a number of factors such as length/ thickness ratio, degree of anisotropy and number of layers. It was also found that coupling between bending and stretching can increase or decrease the bending stiffness of a laminate depending on whether it is positive or negative.
Table of Contents
CHAPTER (1) : Introduction
1.1 General introduction
1.2 Structure of composites
1.2.1 Mechanical behaviour of a fiber-reinforced lamina
1.2.2 Analytical modeling of composite laminates
1.3 Developments in the theories of laminated plates
1.4 The objectives of the present study
CHAPTER (2) : Mathematical modeling of plates
2.1 Linear theory
2.1.1 Assumptions
2.1.2 Equations of equilibrium
2.1.3 The strain-displacement equations
2.1.4 The constitutive equations
2.1.5 Boundary conditions
2.2 Non-linear theory
2.2.1 Assumptions
2.2.2 Equations of equilibrium
2.2.3 The strain-displacement relations
2.2.4 The constitutive equations
2.2.5 Boundary conditions
2.3 Transformation equations
2.3.1 Stress-strain equations
2.3.2 Transformation of stresses and strains
2.3.3 Transformation of the elastic moduli
CHAPTER (3) : Numerical technique
3.1 DR formulation
3.2 The plate equations
3.2.1 Dimensional plate equations
3.2.2 Non-dimensional plate equations
3.3 The finite difference approximation
3.3.1 Interpolating function F(x,y)
3.4 Finite difference form of plate equations
3.4.1 The velocity equations
3.4.2 The displacement equations
3.4.3 The stress resultants and couples equations
3.4.4 Estimation of the fictitious variables
3.5 The DR iterative procedure
3.6 The fictitious densities
3.7 Remarks on the DR technique
CHAPTER (4) : Verification of the computer program
4.1 Small deflection comparisons
4.2 Large deflection comparisons
CHAPTER (5) : Case Studies
5.1 Effect of load
5.2 Effect of length to thickness ratio
5.3 Effect of number of layers
5.4 Effect of material anisotropy
5.5 Effect of fiber orientation
5.6 Effect of reversing lamination order
5.7 Effect of aspect ratio
5.8 Effect of boundary conditions
5.9 Effect of lamination scheme
CHAPTER (6): Conclusions and Suggestions for Further Research
6.1 Conclusions
6.2 Suggestions for further research
Objectives and Topics
This work aims to provide a comprehensive analysis of rectangular laminated plates under uniform lateral loading by employing the Dynamic Relaxation (DR) method coupled with finite difference procedures, specifically focusing on the first-order shear deformation theory (FSDT) to investigate both linear and non-linear deflections.
- Analysis of small and large deflection responses in laminated plates.
- Investigation of factors influencing deflection, such as shear deformation, material anisotropy, fiber orientation, and coupling effects.
- Implementation and validation of a numerical DR program using finite difference approximations.
- Comparison of linear vs. non-linear analytical and numerical results for validation purposes.
Excerpt from the Book
1.1 General Introduction
Composites were first considered as structural materials a little more than half a century ago. And from that time to now, they have received increasing attention in all aspects of material science, manufacturing technology, and theoretical analysis.
The term composite could mean almost anything if taken at face value, since all materials are composites of dissimilar subunits if examined at close enough details. But in modern materials engineering, the term usually refers to a matrix material that is reinforced with fibers. For instance, the term "FRP" which refers to Fiber Reinforced Plastic usually indicates a thermosetting polyester matrix containing glass fibers, and this particular composite has the lion's share of today commercial market.
Many composites used today are at the leading edge of materials technology, with performance and costs appropriate to ultra-demanding applications such as space craft. But heterogeneous materials combining the best aspects of dissimilar constituents have been used by nature for millions of years. Ancient societies, imitating nature, used this approach as well: The book of Exodus speaks of using straw to reinforce mud in brick making, without which the bricks would have almost no strength. Here in Sudan, people from ancient times dated back to Merowe civilization, and up to now used zibala mixed with mud as a strong building material.
Summary of Chapters
CHAPTER (1) : Introduction: Provides an overview of composite materials, their structural applications, and outlines the specific objectives of the research.
CHAPTER (2) : Mathematical modeling of plates: Details the theoretical foundation, including linear and non-linear theories, constitutive equations, and boundary condition formulations for laminated plates.
CHAPTER (3) : Numerical technique: Describes the Dynamic Relaxation (DR) method, the derivation of DR formulae, and the application of finite difference approximations for solving plate equations.
CHAPTER (4) : Verification of the computer program: Presents validation efforts by comparing DR numerical results with known exact and approximate solutions for various plate configurations.
CHAPTER (5) : Case Studies: Investigates the influence of various parameters such as load, layer count, fiber orientation, and boundary conditions on plate behavior.
CHAPTER (6): Conclusions and Suggestions for Further Research: Summarizes the key findings regarding plate deflection behavior and proposes future research directions.
Keywords
Laminated plates, Dynamic Relaxation, Finite difference, FSDT, Shear deformation, Non-linear analysis, Composite materials, Structural mechanics, Bending, Deflection, Fiber reinforcement, Anisotropy, Numerical modeling, Orthotropic plates.
Frequently Asked Questions
What is the core focus of this publication?
The work focuses on the numerical analysis of rectangular laminated plates, specifically utilizing the Dynamic Relaxation (DR) method to solve problems related to small and large deflections.
What are the primary themes covered in the research?
Key themes include the mathematical modeling of plates using first-order shear deformation theory, the application of numerical iteration techniques, and the investigation of material and structural factors influencing plate performance.
What is the main objective of the study?
The primary objective is to develop and validate a theoretical model and a corresponding numerical DR program capable of predicting stresses and deformations in laminated plates under static lateral loads.
Which mathematical or computational methods are employed?
The study utilizes the Dynamic Relaxation (DR) method combined with finite difference approximations to discretize the governing equilibrium equations of the plates.
What topics are discussed in the main chapters?
The chapters cover material structure, theoretical modeling (linear and non-linear), detailed numerical techniques, verification against existing benchmarks, and extensive case studies on factors like fiber orientation and lamination schemes.
How can this work be characterized by its keywords?
The research is characterized by its focus on laminated composites, numerical stability, shear deformation theories, and the performance analysis of rectangular plates under various loading and boundary conditions.
Does the book address the limitations of traditional linear analysis?
Yes, it explicitly states that linear analysis often over-predicts plate deflections and demonstrates why non-linear theory is necessary for a more accurate representation, especially under large deformations.
How does fiber orientation affect plate deflection?
The study indicates that the angle of orientation of individual plies significantly influences the deflection, with specific trends observed for small versus large load magnitudes.
Why is the Dynamic Relaxation (DR) method preferred here over finite elements?
The author argues that the DR method is more efficient for this specific study as it requires less computer memory and fewer computations compared to traditional finite element methods for the analyzed geometries.
- Citation du texte
- Osama Mohammed Elmardi Suleiman Khayal (Auteur), 2003, Linear and Nonlinear Analysis of Laminated Plates using Small and Large Deflection Theory, Munich, GRIN Verlag, https://www.grin.com/document/464045
