Basic Finite Element Method as Applied to Injury Biomechanics

By King-Hay Yang

Basic Finite Element Method as Applied to Injury Biomechanics provides a unique introduction to finite element methods. Unlike other books on the topic, this comprehensive reference teaches readers to develop a finite element model from the beginning, including all the appropriate theories that are needed throughout the model development process.

In addition, the book focuses on how to apply material properties and loading conditions to the model, how to arrange the information in the order of head, neck, upper torso and upper extremity, lower torso and pelvis and lower extremity. The book covers scaling from one body size to the other, parametric modeling and joint positioning, and is an ideal text for teaching, further reading and for its unique application to injury biomechanics.

With over 25 years of experience of developing finite element models, the author's experience with tissue level injury threshold instead of external loading conditions provides a guide to the "do’s and dont's" of using finite element method to study injury biomechanics.

• Covers the fundamentals and applications of the finite element method in injury biomechanics
• Teaches readers model development through a hands-on approach that is ideal for students and researchers
• Includes different modeling schemes used to model different parts of the body, including related constitutive laws and associated material properties

• Language: English
• Publisher: Academic Press
• Release date: Sep 22, 2017
• ISBN: 9780128098325

King-Hay Yang

Dr. King-Hay Yang received his B.S. degree from the National Taiwan University in 1976 and Ph.D. degree from Wayne State University in 1985. He served as an Assistant Professor at the Department of Orthopedic Surgery, West Virginia University from 1985 to 1988. Presently, Dr. Yang is a Professor of the Department of Biomedical Engineering and the Director of the Bioengineering Center at Wayne State University. Since 2014, he has been serving as a member of the Human Factors Panel of the National Academies. Dr. Yang’s research interests include injury biomechanics, contact impact biomechanics, and bone fracture biomechanics. His most recent research involves detailed modeling of the human body from head to toe in efforts to investigate mechanisms of injury related to motor vehicle collisions, contact sports, and blast-induced injuries. His team developed a human brain model that is now on a permanent display at the Computer World of the Smithsonian Institution. Dr. Yang received the 1984 Volvo award in Biomechanics, The Creators: Metro Detroit’s Inventors award by Crane’s Detroit Business; the Best Biomechanics Paper Award of the 2001 International Congress on Whiplash Associated Disorders; the 2001, 2007, and 2008 John Paul Stapp Best Paper Awards; and the 2009 Ralph H. Isbrandt Award for the best paper in automotive safety at the Society of Automotive Engineers (SAE) World Congress. He is a Fellow of the American Institute for Medicine and Biological Engineering (AIMBE) and a Fellow of SAE. He has served as a Visiting Professors at the Swiss Federal Institute of Technology in Zurich, Switzerland, in 1994; Monash University in Melbourne, Australia from 2001 to 2007; the National Cheng-Kung University in Tainan, Taiwan in 2003; the Honk Kong Polytech University in Kowloon, Hong Kong, in 2005; and the Taipei Medical University in 2011. He has also been a Chang-Jiang (Yangtze River) Professor of Hunan University in Changsha, Hunan, China since 2005. Dr. Yang is a member of the Stapp Advisory Committee of the Stapp Car Crash Conference, the premier forum for presentation of research in impact biomechanics, human injury tolerance, and related fields that advance the knowledge of land-vehicle crash-injury protection. Additionally, he is a council member of the International Research Council on Biomechanics of Injury (IRCOBI), which aims at improving the scientific basis on which safety and crashworthiness standards are formulated. Lastly, he has been serving on the Advisory Committee of the Life Science Advisory Board, Lawrence Technical University, since 2007.

Book preview

Basic Finite Element Method as Applied to Injury Biomechanics

Editor

King-Hay Yang

Table of Contents

Cover image

Title page

Copyright

List of Contributors

Foreword

Preface

Part 1. Basic Finite Element Method and Analysis as Applied to Injury Biomechanics

Chapter 1. Introduction

1.1. Finite Element Method and Analysis

1.2. Calculation of Strain and Stress From the FE Model

1.3. Sample Matrix Structural Analysis

1.4. From MSA to a Finite Element Model

Chapter 2. Meshing, Element Types, and Element Shape Functions

2.1. Structure Idealization and Discretization

2.2. Node

2.3. Element

2.4. Formation of Finite Element Mesh

2.5. Element Shape Functions and [B] Matrix

Chapter 3. Isoparametric Formulation and Mesh Quality

3.1. Introduction

3.2. Natural Coordinate System

3.3. Isoparametric Formulation of 1D Elements

3.4. Isoparametric Formulation of 2D Element

3.5. Isoparametric Formulation of 3D Element

3.6. Transfer Mapping Function for 2D Element

3.7. Jacobian Matrix and Determinant of Jacobian Matrix

3.8. Element Quality (Jacobian, Warpage, Aspect Ratio, etc.)

3.9. Saint-Venant Principle and Patch Test

Chapter 4. Element Stiffness Matrix

4.1. Introduction

4.2. Direct Method

4.3. Strong Formulation

4.4. Weak Formulation

4.5. Derive Element Stiffness Matrix From Shape Functions

4.6. Method of Superposition

4.7. Coordinate Transformation

4.8. Chapter Summary Using a Numerical Example

Chapter 5. Material Laws and Properties

5.1. Material Laws

5.2. Material Test Strategy and Associated Property

5.3. Building Laboratory-Specific Material Property Library

Chapter 6. Prescribing Boundary and Loading Conditions to Corresponding Nodes

6.1. Essential and Natural Boundary Conditions

6.2. Nodal Constraint and Prescribed Displacement

6.3. Natural Boundary/Loading Conditions

Chapter 7. Stepping Through Finite Element Analysis

7.1. Introduction

7.2. Iterative Procedures Versus Gaussian Elimination

7.3. Verification and Validation

7.4. Response Variables

Chapter 8. Modal and Transient Dynamic Analysis

8.1. Introduction

8.2. Element Mass Matrix

8.3. Modal Analysis

8.4. Damping

8.5. Direct Integration Methods

8.6. Implicit and Explicit Solvers

Part 2. Modeling Human Body for Injury Biomechanics Analysis

Introduction

Introduction

Chapter 9. Developing FE Human Models From Medical Images

9.1. Introduction

9.2. Biomedical Images for Finite Element Mesh Development

9.3. Physics Behind 3D Segmentation of Medical Images

9.4. Meshing Human Body

9.5. Exemplary Whole Body FE Mesh Development

Chapter 10. Parametric Human Modeling

10.1. Introduction

10.2. Current State-of-The-Art FE, Whole-Body, Human Models

10.3. How to Build a Parametric Human Model

10.4. How to Validate a Parametric Human Model

10.5. Chapter Conclusion

Chapter 11. Modeling Passive and Active Muscles

11.1. Introduction

11.2. Methods for Modeling Passive Muscle

11.3. Methods for Modeling Muscular Activation

11.4. Application of Muscle Models

11.5. Chapter Conclusion

Chapter 12. Modeling the Head for Impact Scenarios

12.1. Why Is Numerical Modeling of the Human Head Essential?

12.2. Introduction of Corresponding Anatomy

12.3. Injury Mechanism

12.4. Material Models

12.5. Material Properties

12.6. Test Data Available for Model Validation

12.7. Brief Overview of Human Head Models

12.8. Discussion

12.9. Concluding Remarks

Chapter 13. Modeling the Neck for Impact Scenarios

13.1. Introduction

13.2. Anatomy of the Neck

13.3. Neck Anthropometrics

13.4. Neck Injury

13.5. Material Models and Properties for Tissues

13.6. Test Data for Computational Model Verification and Validation

13.7. Computational Neck Models

13.8. Closure

Chapter 14. Modeling the Thorax for Impact Scenarios

14.1. Introduction and Corresponding Anatomy

14.2. Injury Types and Mechanisms

14.3. Factors Affecting the Thorax Modeling

14.4. FE Thorax Models

14.5. Concluding Remarks

Chapter 15. Modeling the Lower Torso for Impact Scenarios

15.1. Introduction and Corresponding Anatomy

15.2. Injury Severity and Experimentally Derived Material Properties

15.3. Computational Abdomen Models

15.4. Test Data Available for Model Validation

15.5. Concluding Remarks

Chapter 16. Modeling the Spine and Upper and Lower Extremities for Impact Scenarios

16.1. Introduction and Corresponding Anatomy

16.2. Injury Types

16.3. Factors May Affect Spine and Extremity Modeling

16.4. Finite Element Spine and Extremity Models

16.5. Concluding Remarks

Chapter 17. Modeling of Vulnerable Subjects

17.1. Introduction and Background

17.2. Modeling of Pediatric Subjects

17.3. Modeling of Elderly Female Subjects

17.4. Chapter Conclusion

Chapter 18. Modeling of Blast Wave and Its Effect on the Human/Animal Body

18.1. Basic Blast Physics

18.2. Blast Wave Modeling Strategies in the Numerical Simulations

18.3. Simulations of Blast Wave Effect on the Body—Case Studies

Concluding Remarks

Index

Copyright

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

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List of Contributors

Jeffery Barker,     The University of Waterloo, Waterloo, Ontario, Canada

Duane S. Cronin,     The University of Waterloo, Waterloo, Ontario, Canada

Donata Gierczycka,     The University of Waterloo, Waterloo, Ontario, Canada

Jingwen Hu,     University of Michigan, Ann Arbor, Michigan, United States

Masami Iwamoto,     Toyota Central Research & Development Laboratories, Inc., Nagakute-city, Japan

Xin Jin,     Wayne State University, Detroit, Michigan, United States

Anil Kalra,     Ford Motor Company, Dearborn, Michigan, United States

Haojie Mao,     Western University, London, Ontario, Canada

Barbara R. Presley,     Wayne State University, Detroit, Michigan, United States

David Shen,     The University of Waterloo, Waterloo, Ontario, Canada

Dilaver Singh,     The University of Waterloo, Waterloo, Ontario, Canada

King H. Yang,     Wayne State University, Detroit, Michigan, United States

Feng Zhu,     Embry-Riddle Aeronautical University, Daytona Beach, Florida, United States

Foreword

It is both an honor and a pleasure to write this foreword to Dr. King Yang's book on finite element analysis applied to injury biomechanics. I have worked closely with Dr. Yang for over 30  years on a large number of injury biomechanics projects involving experimentation or modeling or both. Dr. Yang is a pioneer in the modeling of impact biomechanics, and together with his students and postdoctoral fellows, he has formulated models of the entire human body, including head and brain, neck, shoulder, thorax, abdomen, pelvis, and lower extremities. He has also been teaching courses in finite element methods for many years. This book is a culmination of his research and teaching experience, containing a vast amount of information related to the theory of the finite element method and its application to impact injury to the human.

The first part of the book is for students who have not been introduced to the finite element method of structural analysis, which is usually a semester-length course by itself. The material is presented succinctly but with rigor. It also serves as an excellent review resource for those who have taken a first course in finite element methods. For the second part, Dr. Yang collaborated with his colleagues and former students to describe the state-of-the-art research on modeling of human and animal responses to impact, including details of the formulated models. To the best of my knowledge, the material presented in Part II of the book is unique and up to date.

Although he may not have specifically stated this in his book, it is a fact that every model he has ever published was validated in some fashion against available experimental data. It has been his intent right from the start of his career that he will publish only validated models. This is now a policy in Dr. Yang's Advanced Human Modeling Lab. However, not all journals require model validation, and it becomes very difficult for the reader to ascertain the validity and reliability of an unvalidated model.

The Stapp Car Crash Journal will only publish models that have been validated. This not only raises the quality of the journal, but also gives the reader and user the confidence that results from simulations using these models will be, at the very least, reasonable and fairly accurate. As the complexity of the models increase along with the cost of doing impact testing, it is the responsibility of all modelers to validate their models and to include validation as part of their publication.

Finally, let me echo the same sentiments expressed by Dr. Yang and acknowledge with gratitude those who have donated their bodies to science, and specifically for impact biomechanics research. Their generosity has made it possible for the development of crash-test dummies and for the validation of impact models. Looking to the future, we can envision the day when the human computer model becomes more humanlike than the crash dummies. It is my hope that the biomechanics community will design cars that are safe for people and not dummies, and opt to use modeling as a replacement for all crash testing, including the crash dummies.

Albert I. King,     Member, US National Academy of Engineering

Preface

The purpose of this book is to provide basic principles behind the finite element method for static and dynamic analyses, and to augment this material with practical applications in the modeling of biological tissues, organs, and the whole body. The aim is to aid senior undergraduate students and beginning graduate students, especially those in civil engineering, mechanical engineering, bioengineering, or biomedical engineering programs, who use the finite element method to analyze their designs for setting up research projects. In order to encourage self-study, a brief refresher in fundamental principles taught in earlier engineering curriculum is included; therefore students in this targeted audience do not need review their previous textbooks.

In recent years, there has been a change in what knowledge is necessary to perform finite element modeling. Published research results and our own experiences indicate that, as a result of increasingly fast computational speeds, it is more advantageous to use a large number of the simplest types of elements than a smaller number of elements based on higher-order interpolation functions. Additionally, the need to teach students how to write finite element analysis code is no longer as critically needed as it was in the 1970s, because advanced finite element software packages are readily available at little or no cost to university students. As such, only those theories deemed necessary for understanding finite element formulations and interpretations of analysis results are presented.

Part I of this book covers basic engineering principles pertinent to the finite element method. Basic components forming a finite element model and the concepts of strain and stress relevant to finite element analysis are reviewed in Chapter 1. The rest of Part I is organized to follow the sequential order of developing a finite element model: (1) idealizing the geometry to develop a finite element mesh and establishing various element types and element shape functions (Chapter 2), (2) formulating element shape functions based on isoparametric formulation and ensuring high-quality mesh (Chapter 3), (3) setting up element and global stiffness matrices (Chapter 4), (4) implementing material laws and properties (Chapter 5), (5) setting up proper boundary and loading conditions (Chapter 6), (6) stepping through static solutions using the finite element method (Chapter 7), and (7) describing issues related to dynamic solutions (Chapter 8).

In Part II, examples of component and subsystem models of biological tissues are presented. This part is specially designed for readers who need to develop finite element models to solve impact biomechanics-related problems. Chapter 9 covers the general procedures for converting medical images to finite element meshes. In Chapter 10, methods related to parametric modeling of human body models are addressed. The methods for modeling passive and active muscles are introduced in Chapter 11. From Chapters 12 to 16, injury mechanisms, material laws, and material properties relevant for developing finite element models of various human body regions are discussed. The need for modeling the most vulnerable populations is covered in Chapter 17. Due to the recent prevalence of blast-induced injuries, fundamentals of blast modeling are presented lastly in Chapter 18.

The first part of this book is partially based on lecture notes developed by the author over the past 30  years. Additional contributions have come from several outstanding colleagues with whom I have had the privilege of collaborating over the years. We are very grateful for the support of Altair, Engineering Technology Associates, ESI Group, ESTECO, LSTC, and Materialise for providing education and research software packages, and to graduate students at Wayne State University who have provided feedback through the development of this textbook. In particular, I would like to thank Mr. Dominic Isopi for making exercise problems, and Dr. Clifford C. Chou for his valuable suggestions. Last but not least, I am greatly indebted to Mrs. Jane Yang, my wife and best friend, for her excellent illustration work, and Ms. Barbara R. Presley for her editorial assistance and critique.

King H. Yang,     Detroit, Michigan

Part 1

Basic Finite Element Method and Analysis as Applied to Injury Biomechanics

Outline

Chapter 1. Introduction

Chapter 2. Meshing, Element Types, and Element Shape Functions

Chapter 3. Isoparametric Formulation and Mesh Quality

Chapter 4. Element Stiffness Matrix

Chapter 5. Material Laws and Properties

Chapter 6. Prescribing Boundary and Loading Conditions to Corresponding Nodes

Chapter 7. Stepping Through Finite Element Analysis

Chapter 8. Modal and Transient Dynamic Analysis

Chapter 1

Introduction

King H. Yang     Wayne State University, Detroit, Michigan, United States

Abstract

We first describe the origin of the finite element method and its applications in injury biomechanics. This is followed by determining the segmental stiffness matrix using the direct method as part of the matrix structural analysis. Definitions of different types of strain and stress are provided for readers who require memory refreshers. Several example problems are used to illustrate the derivation of the segmental stiffness matrix. Next, similarities between the matrix structural analysis method and the finite element method are highlighted. A simple truss-bridge problem is then used to demonstrate the solution processes after all the boundary and loading conditions are included. Lastly, the concept of Gauss elimination is introduced to find nodal displacements.

Keywords

Constitutive equation; Finite element; Injury biomechanics; Matrix structural analysis; Normal and shear strains

1.1. Finite Element Method and Analysis

The finite element (FE) method comprises a set of numerical procedures for obtaining solutions to many continuum mechanics problems, with an accuracy acceptable to engineers. In classical continuum mechanics, problems are described with partial differential equations. As long as the geometry can be described with a simple equation, the chance of finding the exact solution through the use of the classical method is reasonably high. Unfortunately, real-world problems tend to involve complex geometry and loading conditions. As such, most real-world problems cannot be solved analytically. In contrast, the FE method provides answers to almost all structural mechanics problems. However, the accuracy of the results depends on how the FE model is set up to represent the problem. In general, an FE model that consists of a large number of interconnected subregions (to be described in Chapter 2 as elements) will yield better accuracy, but a model with more elements will require more computing resources. Thus, engineers need to balance the acceptable extent of accuracy with computational costs, which should include the cost of downtime while waiting for the results.

The word continuum is defined in the Cambridge dictionary as something that changes in character gradually or in very slight stages without any clear dividing points. In contrast, discrete is defined as having an independent existence or form apart from other similar things; separate. The field of continuum mechanics deals with the analysis of mechanical behaviors of a material, which can be represented as a continuous mass, as opposed to discrete particles. As such, materials that cannot be represented as a continuous mass cannot be analyzed readily with continuum mechanics methods.

The continuum requirement does not mean that only one material can be analyzed at one instance. We will use trabecular bone, which consists of trabeculae and marrow, to illustrate different situations for applying the concept of continuum. An FE model made of 3  mm or larger typical size elements representing the trabecular bone will not allow the modeler to distinguish trabeculae and marrow within one element. In this case, it would be appropriate for analyzing the trabecular bone with this model. This is because the element size is about 20 times that of a typical trabecula (around 150  μm). At this relatively large ratio, each element consists of approximately the same volume of trabeculae and marrow, and hence can be considered as a continuum.

If a small section of the same trabecular bone is observed with a very powerful magnifying glass (such as a microcomputed tomography (CT) scanner with a resolution of 10  μm), all features of bone marrow, trabeculae, and void spaces will be prominently visible. To represent this small section, each trabecula and bone marrow will need to be explicitly represented, because material properties for these two components are significantly different. In this case, a 20  μm or smaller typical element size FE model will be needed to properly analyze the structure at this scale without violating the continuum assumption. In other words, we can model a structure consisting of two (or more) materials, as long as each material satisfies its own continuum assumption. Obviously, a model with such detailed information would be computationally very costly and would provide little benefit over modeling with continuum mechanics methods for understanding the overall behaviors of trabecular bones.

There are two predecessors to the FE method, and the first was the slope-deflection method. George A. Maney (1888–1947, according to the American Society of Civil Engineers) derived the slope-deflection method to analyze beam and bending frame responses (Maney, 1914), and this method was considered to be the predecessor of the matrix structural analysis (MSA). The MSA method was initiated in the 1930s and became mature by the 1970s. At that time, this type of analysis was mostly conducted manually, with assistance of a slide rule and a simple digital calculator. According to Felippa (2001), major contributors in this field include Duncan and Collar (1934) who formulated discrete aeroelasticity in matrix form, Argyris who analyzed structural responses using energy theorems (Argyris and Kelsey, 1960), and Turner (1959) who proposed the direct stiffness method.

The FE method, derived from the MSA method, has become more popular than MSA in recent years, because MSA is limited to solving one-dimensional (1D) truss, beam, and frame (or a combination of truss and beam) problems, whereas the FE method can be used for two-dimensional (2D) area and three-dimensional (3D) volume elements in addition to 1D elements. Initially, the FE method was mainly used in the aerospace industry, where there were sufficient financial resources to afford large-scale mainframe computers for engineering analysis. In the past five decades, numerous research articles have been published addressing various fundamental formulations associated with the FE method. Based on these new theoretical developments, numerous FE software packages have been developed through either public institutions or commercial entities. More importantly, these FE solution packages are available for college students at little or no cost.

Finite element analysis (FEA) is now a standard practice that is used routinely in numerous fields of engineering. Compared to a couple of decades ago, general purpose FE software packages (solvers) are now readily available for use in both academic institutions and engineering industries. These packages are frequently updated to incorporate the newest advancements in the FE method. Considering the availability of these powerful FE solvers, along with the ever-increasing computational power at very low costs, this book is written with the assumption that students will be using an FE software package for problem solving while concurrently learning the theoretical background of the FE method. Thus, in Part I (Chapters 1–8) of this book, we will use a step-by-step approach to describe theories behind the FE method. This will be done in accordance with the steps required to create an FE model, including identifying material properties, applying boundary and loading conditions, and finding solutions for FEA using the FE method.

In Chapters 2–6, the fundamental knowledge needed to develop a static FE model is presented. To enhance true understanding of the knowledge, conceptual and real-world practical examples are illustrated to show how the theories are applied. For the remaining chapters in Part I, emphasis is placed on solving static and dynamic problems, validating models, and analyzing results. We hope that the study of this book will lead not only to the capability of developing high-quality FE models, but also to greater awareness of basic theories and drawbacks behind the FE method. If you are an advanced student who wishes to develop new fundamental theories in order to augment the capabilities of existing FE solution packages, we recommend that you read other publications dedicated to in-depth theoretical aspects of the FE method.

A real-world example that emulates an FE mesh is a truss bridge, as shown in Fig. 1.1. The MSA method is frequently used to solve problems related to truss-bridge designs. A member of an FE mesh possesses connecting points, or nodes, and is considered an element. A truss is a structure in which only axial forces are relevant. A 1D truss element contains two nodes. Connections of truss members (elements) to form a bridge can be numerically idealized as an FE mesh. This mesh consists of a number of 1D straightline elements interconnected at nodes. Because the number of elements must be limited in order for the problem to be solved by a digital computer, Professor Ray Clough at UC Berkeley proposed the terminology finite element method in 1957, and this terminology is still used (Clough and Wilson, 1999).

In an idealized truss bridge, all joints are pinned together (i.e., free to rotate) and the weight of each truss member is neglected, because external forces are typically much larger than body forces induced by gravity. For MSA or FEA, all external loads are applied at the nodes, because any nonnodal loads cannot be calculated by either of the methods. Upon loading, only axial forces are developed within each truss member. In the real world, truss members are either bolted or welded together, and hence each truss member may be subjected to bending moments and shear forces. Because the bending moments and shear forces are much smaller than the axial forces that are developed, these bending moments and shear forces can be neglected during analysis.

Figure 1.1  Ravenswood bridge, a truss bridge over Ohio River on highway US 33.

The FE method has capabilities of finding solutions to 2D and 3D element types that are not available in the MSA method. In addition, the FE method is excellent for solving problems that involve complex geometries, multiple material compositions, and complex boundary and loading conditions; none of which can be easily calculated using analytical methods. Additional advantages include capabilities for displaying displacement, strain, and stress contours, reducing the time of design cycles, and eliminating or reducing developmental prototypes by allowing new designs to be evaluated on a computer prior to creating a physical prototype. It should be noted that only in very limited cases (such as a simple plane truss problem) can the FE method be used to find exact solutions. However, if the model is correctly formulated and a sufficient number of elements are used to develop the FE model, the solutions will be very close to exact solutions.

As mentioned previously, FEA is now routinely used for analyzing problems related to a number of engineering fields, such as structural mechanics, biomechanics, heat transfer, fluid dynamics, and fluid–solid interface problems. For simplicity, illustrations in this book will be limited to those relevant to structural mechanics and injury biomechanics, unless otherwise specified. For these two fields, FEA comprises a set of discrete numerical procedures that are used to solve for strain and stress distributions, and other response variables, such as displacement, velocity, acceleration, rotation, von Mises stress, and strain rate. Some applications of such analyses include examining risks of injury in human bodies subjected to impacts, designing new parts or systems of parts, modifying existing products, reducing weight while increasing the load-carrying capacity, and determining if a structure is safe prior to manufacturing. Because a high quantity of numerical calculations is frequently involved, a multicore, high-performance computer is needed for real-world FEA. Also, several software packages are needed to develop meshes, calculate corresponding stresses/strains, and analyze the results.

The toll of traffic accidents on human suffering and associated costs are substantial, and the aforementioned characteristics of the FE method make it a perfect tool for studying risks and prevention in this important area of unintentional injury and death. Traffic accidents are among the top 10 causes of death in the United States. In 2010 alone, 32,999 people died and 3.9  million people were injured in such accidents in the United States (Blincoe et al., 2015). Further increasing the significance of this type of loss is that victims of fatal automotive crashes are more likely to be in their productive years than are people who die from other causes. Further, the total social and economic costs, which include lost productivity, medical costs, legal and court costs, emergency service costs, insurance administration costs, traffic congestion costs, property damage, and workplace losses, from these crashes were estimated to be $242  billion, or about 1.6% of the annual gross domestic product.

In the fields of injury biomechanics and impact biomechanics, principles of mechanics are applied to study the impact responses and injury tolerances of biological materials or systems under extreme loading conditions. A good understanding of this discipline is required if one wishes to design protective devices or countermeasures to prevent or reduce the severity of injuries due to automotive crashes or other types of impact. Using experimental methods, researchers in the discipline of injury biomechanics study overall responses, such as globally or locally measurable impact forces or accelerations at regions of interest (RoIs) within a body. These data are used to establish injury thresholds, or the safety (or tolerance) limits, for those body regions. However, the risk of injury is governed by internal responses of the tissue or body part in response to the impact, not the impact force or acceleration. Currently, FEA is the most suitable way to study internal responses, such as stress and strain within a body part. The same set of procedures that are used for studying the risk of injury can be applied for studying the integrity of orthopedic implant designs, determining the efficacy of protective equipment for sports activities, designing an age- and gender-specific personal protection system for minimizing injuries resulting from local impacts, automotive crashes, among other applications.

1.2. Calculation of Strain and Stress From the FE Model

When a material is loaded with a force, a stress is produced. At the same time, the stress will deform the material and induce a strain. Finding stress/strain distributions is the ultimate goal for most structural mechanics problems, because failure of a material depends on internal responses (e.g., reaching the failure stress at a specific point in a specified direction), not the overall external input (force, acceleration, etc.). In the FE method, we calculate the nodal displacement first, and then the strain is calculated using the strain–displacement relationship. Finally, we compute the stress by using the corresponding constitutive equation(s) specific to that material. In the following sections, we define different components of strain and stress.

1.2.1. Average Strain and Point Strain

Axial loading is defined as applying a force on a structure directly along an axis of the structure. As an example, we start with a one-dimensional (1D) truss member formed by points P1 and P2, with an initial length of L (Fig. 1.2) and a deformed length of L′, after axial loading is applied. The average axial engineering strain is defined as the total amount of deformation (L′  −  L) divided by the initial length (L) of the truss member (Eq. 1.1). Typically, the Greek letter ε (Epsilon) is used to denote strain. Based on this definition, the average axial strain (also known as the normal strain or extensional strain) has a unit of measure of inch/inch or m/m, or is simply dimensionless. The word normal may cause confusion, because we use the term normal vector to define a vector that is perpendicular to the surface at a specific point in mathematics. In mechanics, the term normal stress/strain indicates that the stress/strain component is along the axial direction in a one-dimensional (1D) problem, or the stress/strain components are along x-, y-, or z-direction of the coordinate system in a 3D problem.

Figure 1.2  The original (undeformed) and deformed configurations of a truss member inline with the x -axis. The truss member is loaded at point P 2 toward the right side, while point P 1 is restrained from any movement. Upon axial loading, the deformed length of the truss member becomes L ′.

When the axial loading is in tension, the deformed length L′ is greater than the original length L. Hence, the tensile (extensional) strain is defined to have a positive value. Conversely, a compressive loading results in a negative strain.

(1.1)

The concept of an average strain over the entire truss member given in Eq. (1.1) is different from that of a point strain, which is defined as the strain measured at a particular point inside the truss member. Fig. 1.3 illustrates a truss member with two internal points (P and Q) that are separated by a small distance dx in the undeformed configuration. These two points are deformed to P′ and Q′ after loading.

We will begin introducing the concept of the FE method by gradually adding some FE terminology. In Fig. 1.3, we will consider that the entire truss member is the whole structure to be analyzed, P and Q represent two nodes within the structure, and element P–Q is one of the constituent elements that form the structure. We will now analyze the strain for element P–Q from the displacements measured at nodes P and Q. In other words, instead of measuring the overall deformed and original lengths (L′ and L) of the truss member shown in Fig. 1.2 to determine the average strain, we will quantify the displacements (u) at points P and Q given in Fig. 1.3 to define the point strain.

From the bottom part of Fig. 1.3, point P is axially displaced by uP to P′ and point Q is displaced by uQ to Q′. We define Δu as the difference in the axial displacements uQ and uP, that is, Δu  =  uQ  −  uP. We will now prove that Δu (the difference in point displacement) is the same as LP′–Q′  −  LP–Q (i.e., the difference in lengths formed by P′–Q′ and P–Q). From the figure, we depict that LP–Q  =  Δx and LP′–Q′  =  Δx  +  uQ  −  uP  =  Δx  +  Δu. Thus, LP′–Q′  −  LP–Q  =  Δu.

Figure 1.3  The point strain at point P can be calculated from the difference in point displacements ( u P and u Q ) of the deformed configuration and the original length (Δ x , length between points P and Q ).

When the distance Δx between points P and Q approaches zero, the point strain εP at point P is defined by taking the limit of the average strain over the segment P–Q, as shown in the following equation.

(1.2)

This equation is called the 1D strain–displacement equation. As its name implies, this equation describes the relationship between the difference in nodal displacements and the point strain for a 1D truss member. For FE terminology, the term point becomes node, the expression truss member or segment becomes element, the phrase point displacements becomes nodal displacements, and the corresponding nodal strains can be computed from nodal displacements by using the strain–displacement equation.

In contrast to an engineering strain in which the magnitude of deformation is divided by the original length, a true strain in the axial direction is defined as the magnitude of deformation divided by the current (original plus deformation) length. Because large strains cannot be reached instantaneously, the overall true strain is approximated by summing the true strain at each step, which is calculated by dividing the change in length by the current length. If the loading is continuous (i.e., step size is infinitesimally small), integration rather than summation should be used to determine the true strain.

where ln designates the natural log.

Using different step sizes, we demonstrate the differences between calculating true strain based on summation and integral methods, respectively. Here the word step could mean the time-step needed in a dynamic problem for advancing the time, or the load-step related to solving a nonlinear problem in a step-by-step manner. For a step size of x in engineering strain using the summation method, the true strain can be approximated as

where i is the step number and it is assumed that (εT)0  =  0 (i.e., no initial strain exists). Using the summation method, a step increment of 5% in compressive engineering strain (x  =  −0.05) results in a true strain of −5% (−0.05/1  =  −0.05) in the first step, and −10.26% (−0.05  +  (−0.05)/(1  −  0.05)  =  −0.1026) in the second step. Note that directional changes (i.e., from compression to tension) result in different absolute magnitudes when the summation method is used for determining engineering strain. For example, a constant increment of 5% in tensile strain (x  =  0.05) results in a true strain of 5% in the first step and 9.76% in the second step. In contrast, the same increment of 5% in compressive deformation creates a true strain of −5% and −10.26%, respectively, for the first and second steps.

When loading is continuous, the integration method provides an accurate calculation of the true strain. The difference in results between the integration and summation methods becomes smaller as the step size becomes smaller. For example, a 5% shortening in the original length results in εT  =  ln(0.95/1)  =  −5.13% using the integration method, which represents a difference of 0.13% compared to that calculated from the first step of the summation method. Calculating compressive engineering strain using five steps in which each step is 1%, rather than one step at 5% in the previous example, results in an approximate true strain of −5.10% (see Table 1.1). This represents a difference of only 0.03% compared to that calculated from the integration method.

Because strains are calculated from nodal displacements at each step in the FE method, such strains must be true strains in nature. Fig. 1.4 shows strain contours computed by an FE model representing a rectangular block subjected to uniform compressive displacement loading on the left side while the right side is restrained from any movement. The prescribed 4-step loading conditions are that the block is compressed by 5%, 10%, 15%, and 20%, respectively. By the definition of engineering strain, the strain at each load step is the ratio of the change in length (i.e., −5%, −10%, −15%, and −20%) and the original length (100%), with compression being negative. As such, the engineering strains should be −5%, −10%, −15%, and −20%, respectively, for the four steps.

Table 1.1

Comparison of Engineering Strain Versus True Strains Calculated by Summation and Integration Methods for 1%–5% Engineering Strain With 1% Increments and 5%–25% Engineering Strain With 5% Increments in Compression

Figure 1.4  FE model computations of strain contours subjected to uniformly distributed displacement loading on the left edge with up to 20% engineering strain in four steps. The contours clearly show that the strains calculated with the FE model are approximately  − 5%,  − 10.5%,  − 16%, and  − 21.5%, respectively, for steps 1–4. This exercise demonstrates that strains calculated with an FE model are true strains in nature, not engineering strain.

In the first step (the upper-left block), the calculated strain for the entire FE model is −5% (observed from the contour legend). For the second step (the block on the upper right), it is seen that the strain is −10.5%. For the third (the block on the lower left) and fourth (the block on the lower right) steps, the respective strains are −16% and −21.5%. These values indicate that strains computed using an FE software package are not engineering strain. Rather, they are closer to true strains computed from the summation method as only four steps are prescribed. Compared to Table 1.1, the contour values shown in Fig. 1.4 are not accurate, because of the limited number of contour brackets and large overall range (2%) selected when drawing the contour plots. Note that when a homogeneous block is subjected to infinitesimal strain, the differences among the engineering strain, point strain, and true strain are negligible.

1.2.2. Normal and Shear Strain

Normal strain (also known as extensional strain) is different from shear strain in that normal strain is the ratio of the change in length to the original length along the x-, y-, or z-direction, whereas shear strain is a measure of a change in angle in an x-y, y-z, or z-x plane. Normal strain is dimensionless, whereas the unit of measure for shear strain is radians. For axially loaded 1D elements, only normal strains exist. For 2D or 3D problems, strains need to be decomposed into normal and shear components.

We shall consider two sidelines of a 2D infinitesimal rectangular element that are formed by nodal points P0 and P2 in the x-direction and P0 and P1 in the y-direction. The sizes of this element are dx along the x-direction and dy along the y-direction (Fig. 1.5). The angles α and β . In this figure, u represents the displacement along the x-axis and v represents the displacement along the y-axis (i.e., the displacements needed to move from point P). Although not shown in the figure, the next letter in the alphabet, w, is chosen to represent the displacement along the zcan be calculated using the following equation:

Figure 1.5  Two sidelines P 0 – P 2 and P 0 – P 1 of an infinitesimal element with dimensions of dx and dy . The displacement along the x -axis is denoted by u while the displacement along the y -axis is denoted by v . The angles α and β are greatly exaggerated to better highlight the angle changes.

(1.3)

In the y, because the angle β . Because the normal strain along the x-direction (εxx) is defined as the change in length along the x-axis divided by the original length, εxx can be calculated as

(1.4)

It is fairly common to see εxx written as εx in other engineering books. Because shear strain requires two subindices (e.g., εxy) to fully describe its association with the axes involved, a double x is chosen for consistency. Using the same approach as given in , where v is the displacement along the y, where w is the displacement along the zare the normal strain–displacement equations for a 3D element.

A simple shear can be thought of as a change that would happen to a rectangular element if the top surface was pushed lightly toward the right, as demonstrated in Fig. 1.6. We can see from this figure that the volume of the element remains constant (i.e., isochoric). In this configuration, the change in angle (θ) is defined as the average shear strain, γavg, that is, γavg  =  θ. A positive average shear strain means that there is a clockwise rotation of the vertical sideline. This simple shear configuration is a special case of the general shear configuration, which involves two angle changes (one with respect to the x-axis and one with respect to the y-axis), as shown in Fig. 1.5.

The total shear strain (also known as the engineering shear strain) is generally denoted by the Greek letter γ (Gamma). The xy component of the total shear strain, γxy, is the sum of the changes in angles with respect to both x- and y-directions. In other words, γxy  =  α  +  β, as shown in Fig. 1.5.

From trigonometry, the small angle approximation states that for a very small angle, the angle and the tangent of the angle are approximately the same. Thus, from Fig. 1.5, β can be approximated as tan β(i.e., normal strain is much smaller than the axial length), we can derive β from Fig. 1.5 as given in Eq. (1.5).

Figure 1.6  A rectangular element ( solid lines ) is gently pushed on the top surface to the right side ( dashed lines ). The magnitude of the average shear strain ( θ ( vertical line ) to a smaller angle (clockwise rotation).

(1.5)

Similarly, α can be derived as

(1.6)

and γxy is

(1.7)

where the shear strain εxy , which is equivalent to one half (1/2) of the engineering shear strain γxy. The shear strain εxy discussed here is also known as Cauchy's shear strain and is applicable to problems involving only small deformations. Students may wonder why we need to have two different definitions (εxy, γxy) for shear strain. This is because engineering strain (γxy) does not possess the quality of tensor while Cauchy's strain is a tensor, which is needed in FE formulation. For materials that may exhibit large deformations, such as soft biological tissues, the constitutive law (see Section 1.2.3) based on Cauchy's strain will not be suitable. As such, you are encouraged to study Green strain and Almansi strain, which are suitable for both small and large (finite) deformations, but are not in the scope of this book.

For a 3D element, the shear strain–displacement equations due to small deformations are illustrated in the equation below:

(1.8)

Based on the principle of symmetry, we can show that γxy  =  γyx, γyz  =  γzy, and γzx  =  γxz. If we combine all normal and shear components, the strain–displacement equation can be written in matrix form, as shown in Eq. (1.9) for a 1D element, Eq. (1.10) for a 2D element, and Eq. (1.11) for a 3D element, where u, v and w are displacements along the x-, y-, and z-direction, respectively.

(1.9)

(1.10)

(1.11)

1.2.3. Calculation of Stress

As described in entry level mechanics courses, the intensity of a force distributed over a given cross-sectional area is defined as the stress on that section. Stress is generally denoted by the Greek letter σ (sigma). Typically, a positive sign is used to indicate a tensile stress (member in tension), and a negative sign to indicate a compressive stress (member in compression). As shown in the previous section, stresses can also have normal and shear components just as strains can have normal and shear components. In the FE method, stresses are calculated from strains based on the constitutive equation(s) rather than from calculations in which the force components are divided by corresponding cross-sectional areas.

In accordance with System International (SI) metric units, stress is expressed in units of Pascals (Pa).

1  Pascal (Pa)  =  1  N/m²

1  kPa  =  10³  Pa  =  10³  N/m²

1  MPa  =  10⁶  Pa  =  10⁶  N/m²  =  1 N/mm²

1  GPa  =  10⁹  Pa  =  10⁹  N/m²

In accordance with U.S. customary imperial units, stress is expressed in pounds per square in (psi).

1  psi  =  1  lb/in²  =  6.895  kPa, and

1  ksi  =  1  kilopounds/in²  =  6.895  MPa

A constitutive equation describes relationships among two or more physical measures (such as stress vs. strain, stress vs. strain with strain rate effect, etc.) that are specific to a material. In the field of mechanics of materials, the relationship between the stress and the strain of a specific type of material is described as the constitutive relationship of that particular material. While it is commonly implied that constitutive equations can be derived mathematically, the reality is that we must rely on obtaining experimental data and using curve fitting procedures for finding the material constants associated with these equations.

Assuming that all stress and strain components are linearly related for a material, the constitutive equation, or the stress–strain relationship of this material, has the general form

(1.12)

where Eijkl is a fourth-order tensor of the material (also known as the material stiffness matrix) and i, j, k, and l range from 1 to 3 for 3D problems and from 1 to 2 for 2D problems. We can determine that Eijkl possesses 81 (3  ×  3  ×  3  ×  3) separate components for a 3D problem. Obviously, finding the 81 constants needed to fully describe such a material is a daunting task. As a result, this material type is seldom used. Note that the terms material stiffness matrix expressed here and element stiffness matrix, to be introduced in Section 1.3 and Chapter 4, are different but related. The material stiffness matrix describes the stress–strain relationship, while the element stiffness matrix is associated with the size and material properties of the element.

An important distinction between standard FE procedures and classical mechanics is that in FE methods stresses are calculated from strains as shown in Eq. (1.12), whereas in classical mechanics strains are calculated from stresses. A similar expression in terms of the strain-stress (as opposed to stress–strain) relationship is shown in Eq. (1.13), where C is a symmetric tensor of a material compliance matrix. By multiplying [E]−¹ to both sides of Eq. (1.12), the relationship between the material stiffness and compliance matrices can be easily depicted. Note that in standard FE procedures, strains are calculated from nodal displacements before stresses are computed from strains, and in classical mechanics it is taught that strain is induced by stress, and stress is the result of force. Therefore, Eq. (1.13) is seldom used in FE methods.

(1.13)

We can show that both shear stresses and shear strains are symmetric in nature. In other words, σxy  =  σyx, σyz  =  σzy, and σzx  =  σxz, and likewise εxy  =  εyx, εyz  =  εzy, and εzx  =  εxz. Due to the symmetry of the tensor, the indices for the stress (i and j) can be swapped, and so can the indices for the strain (k and l): Cijkl  =  Cjikl  =  Cijlk  =  Cjilk. This symmetrical condition reduces the number of independent elastic components from 81 to only 36 for full description of the stiffness and compliance tensors. The 36 independent components can be expressed by assigning σxx  =  σ1, σyy  =  σ2, σzz  =  σ3, σyz  =  σ4, σxz  =  σ5, σxy  =  σ6 and εxx  =  ε1, εyy  =  ε2, εzz  =  ε3, γyz  =  2εyz  =  ε4, γxz  =  2εxz  =  ε5, γxy  =  2εxy  =  ε6, and then Eq. (1.11) can be written as

(1.14)

where i and j each ranges from 1 to 6. We write Eq. (1.14) in component form as

(1.15)

The advantage of using the tensor format is that one single tensor equation can be used to represent a system of equations. Additionally, equations written in tensor notation can be directly incorporated into computer code with little effort. Considering the limited scope of this book, and that the targeted readers are unlikely to know much about tensors, descriptions of the tensor notation and how to manipulate tensors are not covered. Unless a tensor is specifically needed, we use matrix notation to describe the stress–strain relationship.

Using the complicated constitutive equations, shown in Eq. (1.15), is not the only means for describing the stress–strain relationship. Commonly available engineering materials can be expressed with much simpler forms of constitutive equations. For example, the most widely used isotropic linear elastic material was derived from Hooke's law. Robert Hooke (Jul. 1635–Mar. 1703) found that the extension of a spring was proportional to the force applied to it. Hooke's birth and death dates are based on the MacTutor History of Mathematics Archive, a wonderfully organized database compiled by O'Connor and Robertson (2017). Looking at the birth and death dates in this database gives you an appreciation that the FE method is not at all a new branch of engineering.

Because extension is related to strain and stress is related to force, Hooke's law became the constitutive equation for a 1D, linear elastic material. To expand from the 1D Hooke's law to a perfectly isotropic elastic 3D material, two nonzero components are required to fully describe the relationship between stresses and strains as shown in Eq. (1.16), where λ and μ are the Lamé's constants, named after a French mathematician Gabriel Lamé (Jul. 1795–May 1870).

(1.16)

The dimensions (units) for μ and λ are the same as pressure (e.g., Pascal). In the theory of elasticity, the second Lamé constant (μ) has the same definition as that of the shear modulus (G). However, the first Lamé constant (λ) has no specific physical meaning, although μ and λ are related to the speed of elastic wave. For this reason, the Lamé constants are not directly measurable. As such, Young's (elastic) modulus (E) and Poisson's ratio (ν), which can be directly measured from experiments, are more common than the Lamé constants when describing the stress–strain relationship for an isotropic, linear elastic material. The Lamé constants are related to Young's modulus and Poisson's ratio as:

(1.17)

and

(1.18)

From Eq. (1.16), we find that the summation of three normal stresses, which is a quantity related to the overall volume change, is

(1.19)

Fluid-like materials, such as water, brain tissues, or the nucleus of an intervertebral disc, provide no resistance to shear loading. This material type has a Poisson's ratio of 0.5, which is known to be an incompressible material to express the fact that it exhibits zero volume change when loaded. Because normal stresses are equal (σxx  =  σyy  =  σzz) for fluid-like materials in static equilibrium, pressure induced by external forces is used to determine the response.

Recall that hydrostatic pressure in liquid is the pressure due to the force of gravity. Thus, hydrostatic pressure increases in proportion to the depth of the fluid. In most real-world problems, the magnitude of the hydrostatic pressure is much smaller than the force-induced pressure. Hence, pressure within a fluid-like material is the key response to study.

Based on Eq. (1.18), the corresponding λ value for an incompressible fluid-like material (ν  =  0.5) would be infinitely large. Thus, it is not physically possible to solve such a problem using the FE method. To avoid this difficulty, many FE modelers have used a Poisson's ratio of 0.4999, which represents a nearly incompressible fluid, when modeling materials such as water, brain, or nucleus. As a result of setting Poisson's ratio very close but not equal to 0.5, the corresponding normal stress components are not equal, that is (σxx  ≠  σyy  ≠  σzz). In this case, the pressure is derived from Eq. (1.19) as

(1.20)

Through when compared with the zero loading state. We will introduce a term bulk modulus (K) to refer to a property that is defined as the ratio of pressure change (ΔP) to the magnitude of volume change (ΔV) divided by the original volume (V, if all second order and higher terms are neglected. Thus, the bulk modulus can be expressed as a function of the Lamé's constants. Eq. (1.21) also describes K in terms of the more commonly known elastic modulus and Poisson's ratio, which will be discussed next.

(1.21)

The bulk modulus is measurable and frequently listed in engineering handbooks for fluid-like materials, such as oil, honey, and gasoline. As such, it is a commonly prescribed material property for fluid-like materials. Eqs. (1.17) and (1.18) express the two Lamé constants as functions of the elastic modulus (E) and Poisson's ratio (ν) for isotropic, linear elastic materials. These two properties are discussed in almost all fundamental textbooks and corresponding values for many materials are reported in numerous engineering handbooks and online sources. Hence, these two constants are used frequently in setting up FE models for structural analysis.

For 2D analysis, there are two cases, plane stress and plane strain, that deserve special mention. For a plane stress problem, no stresses are perpendicular to the cross section. An example is a flat plate that lies in the x-y plane with loading forces only along the direction of the plane. In this loading condition, the assumption is that σzz  =  τyz  =  τzx  =  0. Notice that the shear stress τyz is used interchangeably as σyz displayed in Eq. (1.16), because the Greek letter τ emphasizes the origin of the stress is due to shear, while σ underscores the fact that it is a component of the stress tensor. Eq. (1.22) shows the stress–strain equation for this plane stress problem.

(1.22)

For a plane strain problem, no strains normal to the cross section are allowed. An example problem that is commonly solved using this method involves a long dam with its long axis lying along the z-direction. This dam has the same cross section throughout the entire length, and is loaded by water along the surface. In this loading configuration, all strain components involving the z-axis must vanish, that is, εzz  =  γyz  =  γzx  =  0. Eq. (1.23) shows the stress–strain relationships of a plane strain problem.

(1.23)

For a general 3D problem, the constitutive equation for an isotropic linear elastic material in terms of E and ν has the form of

(1.24)

Eq. (1.24) describes an idealized isotropic linear material, which does not exist in the real world, since there are always microscopic differences. Nevertheless, this material remains the most commonly used because of its simplicity. Other materials are more complex than the isotropic linear elastic material model can describe. The corresponding constitutive equations of those special materials will be listed when they become relevant in later chapters.

1.3. Sample Matrix Structural Analysis

For this section, we shall work through several examples to appreciate the similarities and differences between the MSA and FEA methods. The MSA method, also known as the direct stiffness method, is in essence a tool for computing forces and displacements within each truss or frame element of the structure. This direct stiffness method is the most commonly used to describe the fundamental theory behind the FE method. By knowing the stiffness-force relationship of each of the elements that make up an entire structure, a matrix form of the global stiffness equations can be assembled from the stiffness matrices of the individual elements. The equilibrium relationship between the global stiffness matrix and externally applied forces can then be solved to determine any unknown displacements and reaction forces of the structure.

1.3.1. Element Stiffness Matrix of a Linear Spring

Consider a weightless and linear spring element, aligned along the x-direction, having a spring constant k (Fig. 1.7). This spring element is formed by two nodes (P1 and P2), and each node allows only one axial displacement for a total of two degrees of freedom (DOFs). It is further assumed that a positive value indicates that both the force and displacement are pointing toward the right side. Based on Hook's law, the static equilibrium equation for a linear spring can be written as

(1.25)

where F is the applied force and Δx is the axial displacement of the spring.

Figure 1.7  The free-body diagram of a linear spring element. P 1 and P 2 represent the two nodes, and F 1 and F 2 are nodal forces. The corresponding nodal displacements are u 1 and u 2 .

Assuming that the spring is fixed at the left-hand side, that is, u1  =  0, we can write the following equation based on Hooke's law:

(1.26)

We then calculate the reaction force from the static equilibrium equation ∑F  =  0 as:

(1.27)

Rewriting these two equations in matrix form yields:

(1.28)

Similarly, if the spring is fixed at the right side, the three corresponding equations are:

(1.29)

(1.30)

(1.31)

Because both u1 and u2 need not be zero, Eqs. (1.28) and (1.31) can be assembled , where the numbers in the exponent position do not represent an exponent, but is "u1 = 0 for Eq. (1.28) and u. However, this assembly process is not totally a summation process. If it were, the results on the right-hand side of Eq. (1.32) would be 2F1 and 2F2. The reason we use the term assemble is that only the stiffness portions of the equations are summed at each respective matrix location, while terms related to vectors [forces F1 and F2 from Eqs. 1.28 and 1.31] are not summed to form Eq. (1.32).

Another way to look at