Linear and Nonlinear Analysis of Laminated Plates using Small and Large Deflection Theory

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

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.