S T R U C T U R A L A N A L Y S I SFUNDAMENTALFacul t y of Engi neer i ngChul al ongkor n Uni ver si t yJar oon Rungamor nr atZubi zur i Br i dge Bi l bao. Bi scay, Euskadi , Spai nCMYCMMYCYCMYK J. Rungamornrat, Ph.D. Department of Civil Engineering Faculty of Engineering Chulalongkorn University Copyright 2011 J. Rungamornrat FUNDAMENTAL STRUCTURALANALYSIS Dedication To My parents, my wife and my beloved son FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Preface Copyright 2011 J. Rungamornrat P-1 PREFACE The book entitled Fundamental Structural Analysis is prepared with the primary objective to provide complete materials for a fundamental structural analysis course (i.e., 2101310 Structural Analysis I) in civil engineering at Chulalongkorn University. This basic course is offered every semester and is a requisite for the third year undergraduate students with a major in civil engineering. Materials contained in this book are organized into several chapters which are arranged in an appropriate sequence easy to follow. In addition, for each analysis technique presented, underlying theories and key assumptions are considered very crucial and generally outlined at the very beginning of the chapter, so readers can deeply understand its derivation, capability and limitations. To clearly demonstrate the step-by-step analysis procedure involved in each technique, various example problems supplemented by full discussion are presented. Structural analysis has been recognized as an essential component in the design of civil engineering and other types of structures such as buildings, bridges, dams, factories, airports, vehicle parts, machine components, aerospace structures, artificial human organs, etc. It concerns primarily the methodology to construct an exact or approximate solution of an existing or newly developed mathematical model, i.e., a representative of the real structure known as an idealized structure. A process to construct an appropriate model or idealized structure, commonly known as the structural idealization, is considered very crucial in the structural modeling (due to its significant influence on the accuracy of the representative solution to describe responses of the real structure) and must be carried out before the structural analysis can be applied. However, this process is out of scope of this text; a brief discussion is provided in the first chapter only to emphasize its importance and remind readers about the difference between idealized and real structures. A term structure used throughout this text therefore means, unless state otherwise, the idealized structure. Nowadays, many young engineers have exposed to various user-friendly, commercial software packages that are capable of performing comprehensive analysis of complex and large-scale structures. Most of them have started to ignore or even forget the basic background of structural analysis since classical hand-based calculations have almost been replaced by computer-based analyses. Due to highly advances in computing devices and software technology, those available tools have been well-designed and supplemented by user-friendly interfaces and easy-to-follow user-manuals to draw attention from engineers. In the analysis, users are only required to provide complete information of (idealized) structures to be analyzed through the input channel and to properly interpret output results generated by the programs. The analysis procedure to determine such solutions has been implemented internally and generally blinded to the users. Upon the existence of powerful analysis packages, an important question concerning the necessity to study the foundation of structural analysis arises. Is only learning how to use available commercial programs really sufficient? If not, to what extent should the analysis course cover? Answers to above questions are still disputable and depends primarily on the individual perspective. In the authors view, having the background of structural analysis is still essential for structural engineers although, in this era, powerful computer-aided tools have been dominated. The key objective is not to train engineers to understand the internal mechanism of the available codes or to implement the procedure into a code themselves as a programmer, but to understand fundamental theories and key assumptions underlying each analysis technique; the latter is considered crucial to deeply recognize FUNDAMENTAL STRUCTURAL ANALYSIS Preface Jaroon Rungamornrat Copyright 2011 J. Rungamornrat P-2 their capability and limitations. In addition, during the learning process, students can gradually accumulate and finally develop sense of engineers through the problem solving strategy. An engineer fully equipped with knowledge and engineering sense should be able to recognize obviously wrong or unreasonable results, identify sources of errors, and verify results obtained from the analysis. Fully trusting results generated by commercial analysis packages without sufficient verification and check of human errors can lead to dramatic catastrophe if such information is further used in the design. The author has attempted to gather materials from various valuable and reliable resources (including his own experience accumulated from the undergraduate study at Chulalongkorn University, the graduate study at the University of Texas at Austin, and, more importantly, a series of lectures in structural analysis classes at Chulalongkorn University for several years) and put them together in a fashion, hopefully, easy to digest for both the beginners and ones who would like to review what they have learned before. The author anticipate that this book should be valuable and useful, to some extent, for civil engineering students, as supplemented materials to those covered in their classes and a source full with challenging exercise problems. The ultimate goal of writing this book is not only to transfer the basic knowledge from generations to generations but also to provide the motive for students, when start reading it, to gradually change their perspective of the subject from very tough to not as tough as they thought. Surprisingly, from the informal interview of several students in the past, their first impression about this subject is quite negative (this may result from various sources including exaggerated stories or scary legends told by their seniors) and this can significantly discourage their interest since the first day of the class. This book is organized into eleven chapters and the outline of each chapter is presented here to help readers understand its overall picture. The first chapter provides a brief introduction and basic components essential for structural modeling and analysis such as structural idealization, basic quantities and basic governing equations, classification of structures, degree of static and kinematical indeterminacy, stability of structures, etc. The second chapter devotes entirely to the static analysis for support reactions and internal forces of statically determinate structures. Three major types of idealized structures including plane trusses, beams, and plane frames are the main focus of this chapter. Chapter 3 presents a technique, called the direct integration method, to determine the exact solution of beams (e.g. deflections, rotations, shear forces, and bending moment as a function of position along the beam) under various end conditions and loading conditions. Chapter 4 presents a graphical-based technique, commonly known as the moment or curvature area method, to perform displacement and deformation analysis (i.e. determination of displacements and rotations) of statically determinate beams and frames. Chapter 5 introduces another method, called the conjugate structure analogy, which is based on the same set of equations derived in the Chapter 4 but such equations are interpreted differently in a fashion well-suited for analysis of beams and frames of complex geometry. Chapter 6 is considered fundamental and essential for the development of modern structural analysis techniques. It contains various principles and theorems formulated in terms of works and energies and having direct applications to structural analysis. The chapter starts by defining some essential quantities such as work and virtual work, complimentary work and complimentary virtual work, strain energy and virtual strain energy, complimentary strain energy and complimentary virtual strain energy, etc., and then outlines important work and energy theorems, e.g., conservation of work and energy, the principle of virtual work, the principle of complementary virtual work, the principle of stationary total potential energy, the principle of stationary total complementary potential energy, reciprocal theorem, and Castiglianos 1st and 2nd theorems. Chapter 7 presents applications of the conservation of work and energy, or known as the method of real work, to the displacement and deformation analysis of statically determinate structures. Chapter 8 clearly demonstrates applications of the principle of complimentary virtual work, commonly recognized as the unit load method, to the displacement and deformation analysis of statically determinate trusses, beams and frames. Chapter 9 consists of two parts; the first part FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Preface Copyright 2011 J. Rungamornrat P-3involves the application of Castiglianos 2nd theorem to determine displacements and rotations of statically determinate structures whereas the second part presents its applications to the analysis of statically indeterminate structures. Chapter 10 devotes entirely to the development of a general framework for a force method, here called the method of consistent deformation, for analysis of statically indeterminate structures. Full discussion on how to choose unknown redundants, obtain primary structures and set up a set of compatibility equations is provided. The final chapter introduces the concept of influence lines and their applications to the analysis for various responses of structures under the action of moving loads. Both a direct procedure approach and those based on the well-known Mller-Breslau principle are presented with various applications to both statically determinate and indeterminate structures such as beams, floor systems, trusses, and frames. Jaroon Rungamornrat, Ph.D. Department of Civil Engineering Faculty of Engineering Chulalongkorn University Bangkok Thailand FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Table of Contents Copyright 2011 J. Rungamornrat T-1 TABLE OF CONTENTS PREFACE P-1 TABLE OF CONTENTS T-1 Chapter 1 INTRODUCTION TO STRUCTURAL ANALYSIS 1 1.1 Structural Idealization 1 1.2 Continuous Structure versus Discrete Structure Models 10 1.3 Configurations of Structure 10 1.4 Reference Coordinate Systems 11 1.5 Basic Quantities of Interest 14 1.6 Basic Components for Structural Mechanics 19 1.7 Static Equilibrium 21 1.8 Classification of Structures 24 1.9 Degree of Static Indeterminacy 31 1.10 Investigation of Static Stability of Structures 41 Exercises 45 Chapter 2 ANALYSIS OF DETERMINATE STRUCTURES 49 2.1 Static Quantities 49 2.2 Tools for Static Analysis 50 2.3 Determination of Support Reactions 54 2.4 Static Analysis of Trusses 60 2.5 Static Analysis of Beams 78 2.6 Static Analysis of Frames 113 Exercises 139 Chapter 3 DIRECT INTEGRATION METHOD 143 3.1 Basic Equations 143 3.2 Governing Differential Equations 148 3.3 Boundary Conditions 149 3.4 Boundary Value Problem 154 3.5 Solution Procedure 156 3.6 Treatment of Discontinuity 176 3.7 Treatment of Statically Indeterminate Beams 198 Exercises 207 FUNDAMENTAL STRUCTURAL ANALYSIS Table of Contents Jaroon Rungamornrat Copyright 2011 J. Rungamornrat T-2Chapter 4 METHOD OF CURVATURE (MOMENT) AREA 211 4.1 Basic Assumptions 211 4.2 Derivation of Curvature Area Equations 213 4.3 Interpretation of Curvature Area Equations 215 4.4 Applications of Curvature Area Equations 218 4.5 Treatment of Axial Deformation 249 Exercises 256 Chapter 5 CONJUGATE STRUCTURE ANALOGY 261 5.1 Conjugate Structure Analogy for Horizontal Segment 261 5.2 Conjugate Structure Analogy for Horizontal Segment with Hinges 271 5.3 Conjugate Structure Analogy for Inclined Segment 278 5.4 Conjugate Structure Analogy for General Segment 287 Exercises 300 Chapter 6 INTRODUCTION TO WORK AND ENERGY THEOREMS 303 6.1 Work and Complimentary Work 303 6.2 Virtual Work and Complimentary Virtual Work 307 6.3 Strain Energy and Complimentary Strain Energy 309 6.4 Virtual Strain Energy and Complimentary Virtual Strain Energy 312 6.5 Conservation of Work and Energy 313 6.6 Principle of Virtual Work (PVW) 315 6.7 Principle of Complimentary Virtual Work (PCVW) 321 6.8 Principle of Stationary Total Potential Energy (PSTPE) 323 6.9 Principle of Stationary Total Complimentary Potential Energy (PSTCPE) 330 6.10 Reciprocal Theorem 337 6.11 Castiglianos 1st and 2nd Theorems 338 Exercises 344 Chapter 7 DEFORMATION/DISPLACEMENT ANALYSIS BY PRW 347 7.1 Real Work Equation 347 7.2 Strain Energy for Various Effects 348 7.3 Applications of Real Work Equation 352 7.4 Limitations of PRW 361 Exercises 362 Chapter 8 DEFORMATION/DISPLACEMENT ANALYSIS BY PCVW 365 8.1 PCVW with Single Virtual Concentrated Load 365 8.2 Applications to Trusses 367 8.3 Applications to Flexure-Dominating Structures 376 FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Table of Contents Copyright 2011 J. Rungamornrat T-3Exercises 403 Chapter 9 APPLICATIONS OF CASTIGLIANOS 2nd THEOREM 407 9.1 Castiglianos 2nd Theorem for Linearly Elastic Structures 407 9.2 Applications to Statically Determinate Structures 409 9.3 Applications to Statically Indeterminate Structures 422 Exercises 432 Chapter 10 METHOD OF CONSIST DEFORMATION 437 10.1 Basic Concept 437 10.2 Choice of Released Structures 441 10.3 Compatibility Equations for General Case 443 Exercises 475 Chapter 11 INFLUENCE LINES 479 11.1 Introduction to Concept of Influence Lines 479 11.2 Influence Lines for Determinate Beams by Direct Method 485 11.3 Influence Lines by Mller-Breslau Principle 501 11.4 Influence Lines for Beams with Loading Panels 516 11.5 Influence Lines for Determinate Floor Systems 524 11.6 Influence Lines for Determinate Trusses 536 11.7 Influence Lines for Statically Indeterminate Structures 563 Exercises 594 REFERENCE R-1 INDEX I-1 ACKNOWLEDGEMENT A-1 FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 1CHAPTER 1 INTRODUCTION TO STRUCTURAL ANALYSIS This first chapter provides a brief introduction of basic components essential for structural analysis. First, the concept of structural modeling or structural idealization is introduced. This process involves the construction of a mathematical model or idealized structure to represent a real structure under consideration. The structural analysis is in fact a subsequent process that is employed to solve a set of mathematical equations governing the resulting mathematical model to obtain a mathematical solution. Such solution is subsequently employed to characterize or approximate responses of the real structure to a certain level of accuracy. Conservation of linear and angular momentum of a body in equilibrium is also reviewed and a well known set of equilibrium equations that is fundamentally important to structural analysis is also established. Finally, certain classifications of idealized structures are addressed. 1.1 Structural Idealization A real structure is an assemblage of components and parts that are integrated purposely to serve certain functions while withstanding all external actions or excitations (e.g. applied loads, environmental conditions such as temperature change and moisture penetration, and movement of its certain parts such as foundation, etc.) exerted by surrounding environments. Examples of real structures mostly encountered in civil engineering application include buildings, bridges, airports, factories, dams, etc as shown in Figure 1.1. The key characteristic of the real structure is that its responses under actions exerted by environments are often very complex and inaccessible to human in the sense that the real behavior cannot be known exactly. Laws of physics governing such physical or real phenomena are not truly known; most of available theories and conjectures are based primarily on various assumptions and, as a consequence, their validity is still disputable and dependent on experimental evidences. Figure 1.1: Schematics of some real structures FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 2 Since behavior of the real structure is extremely complex and inaccessible, it necessitates the development of a simplified or approximate structure termed as an idealized structure. To be more precise, an idealized structure is a mathematical model or a mathematical object that can be used to approximate behavior or responses of the real structure to certain degree of accuracy. The main characteristic of the idealized structure is that its responses are accessible, solvable, and can be completely determined using available laws of physics and mathematics. The process for obtaining the idealized structure is called structural idealization or structural modeling. This process generally involves imposing various assumptions and simplifying the complexity embedded in the real structure. The idealized structure of a given real structure is in general not unique and many different idealized structures can be established via use of different assumptions and simplifications. The level of idealization considered in the process of modeling depends primarily on the required degree of accuracy of (approximate) responses of the idealized structure in comparison with those of the real structure. The idealization error is an indicator that is employed to measure the discrepancy between a particular response of the real structure and the idealized solution obtained by solving the corresponding idealized structure. The acceptable idealization error is an important factor influences the level of idealization and a choice of the idealized structure. While a more complex idealized structure can characterize the real structure to higher accuracy, it at the same time consumes more computational time and effort in the analysis. The schematic indicating the process of structural idealization is shown in Figure 1.2. For brevity and convenience, the term structure throughout this text signifies the idealized structure unless stated otherwise. Some useful guidelines for constructing the idealized structure well-suited for structural analysis procedure are discussed as follows. Figure 1.2: Diagram indicating the process of structural idealization 1.1.1 Geometry of structure It is known that geometry of the real structure is very complex and, in fact, occupies space. However, for certain classes of real structures, several assumptions can be posed to obtain an idealized structure possessing a simplified geometry. A structural component with its length much larger than dimensions of its cross section can be modeled as a one-dimensional or line member, e.g. truss, beam, frame and arch shown in Figure 1.3. A structural component with its thickness much smaller than the other two dimensions can properly be modeled as a two-dimensional or surface member, e.g. plate and shell structures. For the case where all three dimensions of the Assumptions + Simplification Governing Physics Idealization error Idealized solutionStructural analysis Response interpretation Real structure Complex & Inaccessible Idealized structure Simplified & Solvable FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 3structure are comparable, it may be obligatory to be modeled as a three-dimensional member, e.g. dam and a local region surrounding the connections or joints. Figure 1.3: Schematics of idealized structures consisting of one-dimensional members 1.1.2 Displacement and deformation Hereby, the term deformation is defined as the distortion of the structure while the term displacement is defined as the movement of points within the structure. These two quantities have a fundamental difference, i.e., the former is a relative quantity that measures the change in shape or distortion of any part of the structure due to any action while the latter is a total quantity that measures the change in position of individual points resulting from any action. It is worth noting that the structure undergoing the displacement may possess no deformation; for instance, there is no change in shape or distortion of the structure if it is subjected to rigid translation or rigid rotation. This special type of displacement is known as the rigid body displacement. On the contrary, the deformation of any structure must follow by the displacement; i.e. it is impossible to introduce non-zero deformation to the structure with the displacement vanishing everywhere. For typical structures in civil engineering applications, the displacement and deformation due to external actions are in general infinitesimal in comparison with a characteristic dimension of the structure. The kinematics of the structure, i.e. a relationship between the displacement and the deformation, can therefore be simplified or approximated by linear relationship; for instance, the linear relationship between the elongation and the displacement of the axial member, the linear relationship between the curvature and the deflection of a beam, the linear relationship between the rate of twist and the angle of twist of a torsion member, etc. In addition, the small discrepancy between the undeformed and deformed configurations allows the (known) geometry of the undeformed configuration to be employed throughout instead of using the (unknown) geometry of the deformed configuration. It is important to remark that there are various practical situations where the small displacement and deformation assumption is not well-suited in the prediction of structural responses; for instance, structures undergoing large displacement and deformation near their collapse state, very flexible structures whose configuration is sensitive to applied loads, buckling and post-buckling behavior of axially dominated components, etc. Various investigations concerning structures undergoing large displacement and rotation can be found in the literature (e.g. Rungamornrat et al, 2008; Tangnovarad, 2008; Tangnovarad and Rungamornrat, 2008; Tangnovarad and Rungamornrat, 2009; Danmongkoltip, 2009; Danmomgkoltip and Rungamornrat, 2009; Rungamornrat and Tangnovarad, 2011; Douanevanh, 2011; Douanevanh et al, 2011). Frame Truss Beam Arch FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 4 1.1.3 Material behavior The behavior of a constituting material in real structures is extremely complex (i.e. it is generally nonlinear, nonhomogeneous, anisotropic and time and history dependent) and, as a consequence, construction of a suitable constitutive model is both theoretically and computationally challenging. In constitutive modeling, the behavior of materials is generally modeled or approximated via the relationship between the internal force measure (e.g. axial force, torque, bending moment, shear force, and stress) and the deformation (e.g. elongation, rate of twist, curvature, and strain). Most of materials encountering in civil engineering applications (e.g. steel and concrete) are often modeled as an idealized, simple material behavior called an isotropic and linearly elastic material. The key characteristics of this class of materials are that the material properties are directional independent, its behavior is independent of both time and history, and stress and strain are related through a linear function. Only two material parameters are required to completely describe the material behavior; one is the so-called Youngs modulus denoted by E and the other is the Poissons ratio denoted by v. Other material parameters can always be expressed in terms of these two parameters; for instance, the shear modulus, denoted by G, is given by ) 1 ( 2EGv += (1.1) The Youngs modulus E can readily be obtained from a standard uniaxial tensile test while G is the elastic shear modulus obtained by conducting a direct shear test or a torsion test. The Poissons ratio can then be computed by the relation (1.1). Both E and G can be interpreted graphically as a slope of the uniaxial stress-strain curve (o-c curve) and a slope of the shear stress-strain curve (t- curve), respectively, as indicated in Figure 1.4. The Poissons ratio v is a parameter that measures the degree of contraction or expansion of the material in the direction normal to the direction of the normal stress. Figure 1.4: Uniaxial and shear stress-strain diagrams 1.1.4 Excitations All actions or excitations exerted by surrounding environments are generally modeled by vector quantities such as forces and moments. The excitations can be divided into two different classes depending on the nature of their application; one called the contact force and the other called the remote force. The contact force results from the idealization of actions introduced by a direct E 1 c o Uniaxial stress-strain curveG 1 t Shear stress-strain curveFUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 5contact between the structure and surrounding environments such as loads from occupants and wind while the remote force results from the idealization of actions introduced by remote environments such as gravitational force. The contact or remote force that acts on a small area of the structure can be modeled by a concentrated force or a concentrated moment while the contact or remote force that acts over a large area can properly be modeled by a distributed force or a distributed moment. Figure 1.5 shows an example of an idealized structure subjected to two concentrated forces, a distributed force and a concentrated moment. Figure 1.5: Schematic of a two-dimensional, idealized structure subjected to idealized loads 1.1.5 Movement constraints Interaction between the structure and surrounding environments to maintain its stability while resisting external excitations (e.g. interaction between the structure and the foundation) can mathematically be modeled in terms of idealized supports. The key function of the idealized support is to prevent or constrain the movement of the structure in certain directions by means of reactive forces called support reactions. The support reactions are introduced in the direction where the movement is constrained and they are unknown a priori; such unknown reactions can generally be computed by enforcing static equilibrium conditions and other necessary kinematical conditions. Several types of idealized supports mostly found in two-dimensional idealized structures are summarized as follows. 1.1.5.1 Roller support A roller support is a support that can prevent movement of a point only in one direction while provide no rotational constraint. The corresponding unknown support reaction then possesses only one component of force in the constraint direction. Typical symbols used to represent the roller support and support reaction are shown schematically in Figure 1.6. Figure 1.6: Schematic of a roller support and the corresponding support reaction FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 6 1.1.5.2 Pinned or hinged support A pinned or hinged support is a support that can prevent movement of a point in both directions while provide no rotational constraint. The corresponding unknown support reaction then possesses two components of force in each direction of the constraint. Typical symbols used to represent the pinned or hinged support and the support reactions are shown schematically in Figure 1.7. Figure 1.7: Schematic of a pinned or hinged support and the corresponding support reactions. 1.1.5.3 Fixed support A fixed support is a support that can prevent movement of a point in both directions and provide a full rotational constraint. The corresponding unknown support reaction then possesses two components of force in each direction of the translational constraint and one component of moment in the direction of rotational constraint. Typical symbols used to represent the fixed support and the support reactions are shown schematically in Figure 1.8. Figure 1.8: Schematic of a fixed support and the corresponding support reactions 1.1.5.4 Guided support A guided support is a support that can prevent movement of a point in one direction and provide a full rotational constraint. The corresponding unknown support reaction then possesses one component of force in the direction of the translational constraint and one component of moment in the direction of rotational constraint. Typical symbols used to represent the guided support and the support reactions are shown schematically in Figure 1.9. Figure 1.9: Schematic of a guided support and the corresponding support reactions FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 71.1.5.5 Flexible support A flexible support is a support that can partially prevent translation and/or rotational constraints. The corresponding unknown support reaction is related to the unknown displacement and/or rotation of the support. Typical symbols used to represent the flexible support and the support reactions are shown schematically in Figure 1.10. Figure 1.10: Schematic of a flexible support and the corresponding support reactions 1.1.6 Connections Behavior of a local region where the structural components are connected is very complicated and this complexity depends primarily on the type and details of the connection used. To extensively investigate the behavior of the connection, a three dimensional model is necessarily used to gain accurate results. For a standard, linear structural analysis, the connection is only modeled as a point called node or joint and the behavior of the node or joint depends mainly on the degree of force and moment transfer across the connection. 1.1.6.1 Rigid joint A rigid joint is a connection that allows the complete transfer of force and moment across the joint. Both the displacement and rotation are continuous at the rigid joint. This idealized connection is usually found in the beam or frame structures as shown schematically in Figure 1.11. Figure 1.11: Schematic of a real connection and the idealized rigid joint 1.1.6.2 Hinge joint A hinge joint is a connection that allows the complete transfer of force across the joint but does not allow the transfer of the bending moment. Thus, the displacement is continuous at the hinge joint while the rotation is not since each end of the member connecting at the hinge joint can rotate freely from each other. This idealized connection is usually found in the truss structures as shown schematically in Figure 1.12. FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 8 Figure 1.12: Schematic of a real connection and the idealized hinge joint. 1.1.6.3 Partially rigid joint A partially rigid joint is a connection that allows the complete transfer of force and a partial transfer of moment across the joint. For this particular case, both the displacement is continuous at the joint while rotation is not. The behavior of the flexible joint is more complex than the rigid joint and the hinge joint but it can better represent the real behavior of the connection in the real structure. The schematic of the partially rigid joint is shown in Figure 1.13. Figure 1.13: Schematic of an idealized partially rigid joint 1.1.7 Idealized structures In this text, it is focused attention on a particular class of idealized structures that consist of one-dimensional and straight components, is contained in a plane, and is subjected only to in-plane loadings; these structures are sometimes called two-dimensional or plane structures. Three specific types of structures in this class that are main focus of this text include truss, beam and frame. Figure 1.14: Schematic of idealized trusses FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 91.1.7.1 Truss Truss is an idealized structure consisting of one-dimensional, straight structural components that are connected by hinge joints. Applied loads are assumed to act only at the joints and all members possess only one component of internal forces, i.e. the axial force. Examples of truss structures are shown in the Figure 1.14. 1.1.7.2 Beam Beam is an idealized structure consisting of one-dimensional, straight members that are connected in a series either by hinge joints or rigid joints; thus, the geometry of the entire beam must be one-dimensional. Loads acting on the beam must be transverse loadings (loads including forces normal to the axis of the beam and moments directing normal to the plane containing the beam) and they can act at any location within the beam. The internal forces at a particular cross section consist of only two components, i.e., the shear force and the bending moment. Examples of beams are shown in Figure 1.15. Figure 1.15: Schematic of idealized beams 1.1.7.3 Frame Frame is an idealized structure consisting of one-dimensional, straight members that are connected either by hinge joints or rigid joints. Loads acting on the frame can be either transverse loadings or longitudinal loadings (loads acting in the direction parallel to the axis of the members) and they can act at any location within the structure. The internal forces at a particular cross section consist of three components: the axial force, the shear force and the bending moment. It can be remarked that when the internal axial force identically vanishes for all members and the geometry of the structure is one dimensional, the frame simply reduces to the beam. Examples of frame structures are shown in Figure 1.16. Figure 1.16: Schematic of idealized frames FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 10 1.2 Continuous Structure versus Discrete Structure Models A continuous structure is defined as an idealized structure where its responses at all points are unknown a priori and must be determined as a function of position (i.e. be determined at all points of the structure) in order to completely describe behavior of the entire structure. The primary unknowns of the continuous structure are in terms of response functions and, as a result, the number of unknowns counted at all points of the structure is infinite. Analysis of such continuous structure is quite complex and generally involves solving a set of governing differential equations. In the other hand, a discrete structure is a simplified idealized structure where the responses of the entire structure can completely be described by a finite set of quantities. This type of structures typically arises from a continuous structure furnishing with additional assumptions or constraints on the behavior of the structures to reduce the infinite number of unknowns to a finite number. A typical example of discrete structures is the one that consists of a collection of a finite number of structural components called members or elements and a finite number of points connecting those structural components to make the structure as a whole called nodes or nodal points. All unknowns are forced to be located only at the nodes by assuming that behavior of each member can be completely determined in terms of the nodal quantities quantities associated with the nodes. An example of a discrete structure consisting of three members and four nodes is shown in Figure 1.17. Figure 1.17: An example of a discrete structure comprising three members and four nodes 1.3 Configurations of Structure There are two configurations involve in the analysis of a deformable structure. An undeformed configuration is used to refer to the geometry of a structure at the reference state that is free of any disturbances and excitations. A deformed configuration is used to refer to a subsequent configuration of the structure after experiencing any disturbances or excitations. Figure 1.18 shows both the undeformed configuration and the deformed configuration of a rigid frame. Figure 1.18: Undeformed and deformed configurations of a rigid frame under applied loads Node 4 Node 3 Node 2 Node 1 Member 1 Member 2 Member 3 v u u X Y Deformed configuration Undeformed configuration FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 111.4 Reference Coordinate Systems In structural analysis, a reference coordinate system is an indispensable tool that is commonly used to conveniently represent quantities of interest such as displacements and rotations, applied loads, support reactions, etc. Following subsections provide a clear notion of global and local coordinate systems and a law of coordinate transformation that is essential for further development. 1.4.1 Global and local coordinate systems There are two types of reference coordinate systems used throughout the development presented further in this book. A global coordinate system is a single coordinate system that is used to reference geometry or involved quantities for the entire structure. A choice of the global coordinate system is not unique; in particular, an orientation of the reference axes and a location of its origin can be chosen arbitrarily. The global reference axes are labeled by X, Y and Z with their directions strictly following the right-handed rule. For a two-dimensional structure, the commonly used, global coordinate system is one with the Z-axis directing normal to the plane of the structure. A local coordinate system is a coordinate system that is used to reference geometry or involved quantities of an individual member. The local reference axes are labeled by x, y and z. This coordinate system is defined locally for each member and, generally, based on the geometry and orientation of the member itself. For plane structures, it is typical to orient the local coordinate system for each member in the way that its origin locates at one of its end, the x-axis directs along the axis of the member, the z-axis directs normal to the plane of the structure, and the y-axis follows the right-handed rule. An example of the global and local coordinate systems of a plane structure consisting of three members is shown in Figure 1.19. Figure 1.19: Global and local coordinate systems of a plane structure 1.4.2 Coordinate transformation In this section, we briefly present a basic law of coordinate transformation for both scalar quantities and vector quantities. To clearly demonstrate the law, let introduce two reference coordinate systems that possess the same origin: one, denoted by {x1, y1, z1}, with the unit base vectors {i1, j1, k1} and the other, denoted by {x2, y2, z2}, with the unit base vectors {i2, j2, k2} as indicated in Figure 1.20. Now, let define a matrix R such that X Y x y y x y x FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 12 ((((

u u u u u u u u u=((((

=33 23 1332 22 1231 21 112 1 2 1 2 12 1 2 1 2 12 1 2 1 2 1cos cos coscos cos coscos cos cosk k k j k ij k j j j ii k i j i iR (1.2) where {u11, u21, u31} are angles between the unit vector i2 and the unit vectors {i1, j1, k1}, respectively; {u12, u22, u32} are angles between the unit vector j2 and the unit vectors {i1, j1, k1}, respectively; and {u13, u23, u33} are angles between the unit vector k2 and the unit vectors {i1, j1, k1}, respectively. Figure 1.20: Schematic of two reference coordinate systems with the same origin 1.4.2.1 Coordinate transformation for scalar quantities Let be a scalar quantity whose values measured in the coordinate system {x1, y1, z1} and to the coordinate system {x2, y2, z2} are denoted by 1 and 2, respectively. Since a scalar quantity possesses only a magnitude, its values are invariant of the change of reference coordinate systems and this implies that 2 1 = (1.3) 1.4.2.2 Coordinate transformation for vector quantities Let v be a vector whose representations with respect to the coordinate system {x1, y1, z1} and the coordinate system {x2, y2, z2} are given by 22z 22y 22x 11z 11y 11xv v v v v v k j i k j i v + + = + + = (1.4) where {1z1y1xv , v , v } and {2z2y2xv , v , v } are components of a vector v with respect to the coordinate systems {x1, y1, z1} and {x2, y2, z2}, respectively. To determine the component 2xv in terms of the components {1z1y1xv , v , v }, we take an inner product between a vector v given by (1.4) and a unit vector i2 to obtain ) ( v ) ( v ) ( v v2 11z 2 11y 2 11x2xi k i j i i + + = (1.5) y1 y2 x1 x2 z1 z2 i1 i2 j1 j2 k1 k2 FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 13Similarly, by taking an inner product between a vector v given by (1.4) and a unit vector j2 and k2, it leads to ) ( v ) ( v ) ( v v2 11z 2 11y 2 11x2yj k j j j i + + = (1.6) 2 1 1 1z x 1 2 y 1 2 z 1 2v v ( ) v ( ) v ( ) = + + i k j k k k (1.7) With use of the definition of the transformation matrix R given by (1.2), equations (1.5)-(1.7) can be expressed in a more concise form as )`=)`((((

=)`1z1y1x1z1y1x2 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12z2y2xvvvvvvvvvRk k k j k ij k j j j ii k i j i i (1.8) The expression of the components {1z1y1xv , v , v } in terms of the components {2z2y2xv , v , v } can readily be obtained in a similar fashion by taking a vector inner product of the vector v given by (1.4) and the unit base vectors {i1, j1, k1}. The final results are given by )`=)`((((

=)`2z2y2xT2z2y2x1 2 1 2 1 21 2 1 2 1 21 2 1 2 1 21z1y1xvvvvvvvvvRk k k j k ij k j j j ii k i j i i (1.9) where RT is a transpose of the matrix R. Note that the matrix R is commonly termed a transformation matrix. 1.4.2.3 Special case Let consider a special case where the reference coordinate system {x2, y2, z2} is simply obtained by rotating the z1-axis of the reference coordinate system {x1, y1, z1} by an angle |. The transformation matrix R possesses a special form given by ((((

| | | |=1 0 00 cos sin0 sin cosR (1.10) The coordinate transformation formula (1.8) and (1.9) therefore reduce to )`((((

| | | |=)`1z1y1x2z2y2xvvv1 0 00 cos sin0 sin cosvvv (1.11) )`((((

| | | |=)`2z2y2x1z1y1xvvv1 0 00 cos sin0 sin cosvvv (1.12) FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 14 This clearly indicates that the component along the axis of rotation is unchanged and is independent of the other two components. The laws of transformation (1.11) and (1.12) can also be applied to the case of two vectors v and w where v is contained in the x1-y1 plane (and the x2-y2 plane) and w is perpendicular to the x1-y1 plane (and the x2-y2 plane). More precisely, components of both vectors v and w in the {x1, y1, z1} coordinate system and in the {x2, y2, z2} coordinate system are related by 2 1x x2 1y y2 1z zv cos sin 0 vv sin cos 0 vw 0 0 1 w | | ( (= | | ` ` ( ( ) ) (1.13) 1 2x x1 2y y1 2z zv cos sin 0 vv sin cos 0 vw 0 0 1 w | | ( (= | | ` ` ( ( ) ) (1.14) 1.5 Basic Quantities of Interest This section devotes to describe two different classes of basic quantities that are involved in structural analysis, one is termed kinematical quantities and the other is termed static quantities. 1.5.1 Kinematical quantities Kinematical quantities describe geometry of both the undeformed and deformed configurations of the structure. Within the context of static structural analysis, kinematical quantities can be categorized into two different sets: one associated with quantities used to measure the movement or change in position of the structure and the other is associated with quantities used to measure the change in shape or distortion of the structure. Displacement at any point within the structure is a quantity representing the change in position of that point in the deformed configuration measured relative to the undeformed configuration. Rotation at any point within the structure is a quantity representing the change in orientation of that point in the deformed configuration measured relative to the undeformed configuration. For a plane structure shown in Figure 1.18, the displacement at any point is fully described by a two-component vector (u, v) where u is a component of the displacement in X-direction and v is a component of the displacement in Y-direction while the rotation at any point is fully described by an angle u measured from a local tangent line in the undeformed configuration to a local tangent line at the same point in the deformed configuration. It is important to emphasize that the rotation is not an independent quantity but its value at any point can be computed when the displacement at that point and all its neighboring points is known. A degree of freedom, denoted by DOF, is defined as a component of the displacement or the rotation at any node (of the discrete structure) essential for describing the displacement of the entire structure. There are two types of the degree of freedom, one termed as a prescribed degree of freedom and the other termed as a free or unknown degree of freedom. The former is the degree of freedom that is known a priori, for instance, the degree of freedom at nodes located at supports where components of the displacement or rotation are known while the latter is the degree of freedom that is unknown a priori. The number of degrees of freedom at each node depends primarily on the type of nodes and structures and also the internal releases and constraints present within the structure. In general, it is equal to the number of independent degrees of freedom at that node essential for describing the displacement of the entire structure. For beams, plane trusses, FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 15space truss, plane frames, and space frames containing no internal release and constraint, the number of degrees of freedom per node are 2 (a vertical displacement and a rotation), 2 (two components of the displacement), 3 (three components of the displacement), 3 (two components of the displacement and a rotation) and 6 (three components of the displacement and three components of the rotation), respectively. Figure 1.21 shows examples of both prescribed degrees of freedom and free degrees of freedom of beam, plane truss and plane frames. The number of degrees of freedom of a structure is defined as the number of all independent degrees of freedom sufficient for describing the displacement of the entire structure or, equivalently, it is equal to the sum of numbers of degrees of freedom at all nodes. For instance, a beam shown in Figure 1.21(a) has 6 DOFs {v1, u1, v2, u2, v3, u3} consisting of 3 prescribed DOFs {v1, u1, v3} and 3 free DOFs {v2, u2, u3}; a plane truss shown in Figure 1.21(b) has 6 DOFs {u1, v1, u2, v2, u3, v3} consisting of 3 prescribed DOFs {u1, v1, v2} and 3 free DOFs {u2, u3, v3}; and a plane frame shown in Figure 1.21(c) has 9 DOFs {u1, v1, u1, u2, v2, u2, u3, v3, u3} consisting of 3 prescribed DOFs { u1, v1, v3} and 6 free DOFs {u1, u2, v2, u2, u3, u3}. It is evident that the number of degrees of freedom of a given structure is not unique but depending primarily on how the structure is discretized. As the number of nodes in the discrete structure increases, the number of the degrees of freedom of the structure increases. Figure 1.21: (a) Degrees of freedom of a beam, (b) degrees of freedom of a plane truss, and (c) degrees of freedom of a plane frame X v1=0 u1=0 u3 v3 u2 v2=0 Y Node 1 Node 2 Node 3 (b) (c) (a) v1 = 0 u1 = 0 v2 u2 v3 = 0 u3 Y X Node 1 Node 2 Node 3 v1=0 u1=0 u1 u3 u3 v3=0 u2 v2 u2 Y X Node 1 Node 2 Node 3 FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 16 Deformation is a quantity used to measure the change in shape or the distortion of a structure (i.e. elongation, rate of twist, curvature, strain, etc.) due to disturbances and excitations. The deformation is a relative quantity and a primary source that produces the internal forces or stresses within the structure. For continuous structures, the deformation is said to be completely described if and only if the deformation is known at all points or is given as a function of position while, for discrete structures, the deformation of the entire structure is said to be completely described if and only if the deformation of all members constituting the structure are known. The deformation for each member of a discrete structure can be described by a finite number of quantities called the member deformation (this, however, must be furnished by certain assumptions on kinematics of the member to ensure that the deformation at every point within the member can be determined in terms of the member deformation). The quantities selected to be the member deformation depend primarily on the type and behavior of such member. For instance, the elongation, e, or a measure of the change in length of a member is commonly chosen as the member deformation of a truss member as shown in Figure 1.22(a); the relative end rotations {|s, |e} where |s and |e denotes the rotations at both ends of the member measured relative to a chord connecting both end points as shown in Figure 1.22(b) are commonly chosen as the member deformation of a beam member; and the elongation and two relative end rotations {e, |s, |e} as shown in Figure 1.22(c) are commonly chosen as the member deformation of a frame member. It is remarked that the deformation of the entire discrete structure can fully be described by a finite set containing all member deformation. Figure 1.22: Member deformation for different types of members: (a) truss member, (b) beam member, and (c) frame member A Rigid body motion is a particular type of displacement that produces no deformation at any point within the structure. The rigid body motion can be decomposed into two parts: a rigid translation and a rigid rotation. The rigid translation produces the same displacement at all points while the rigid rotation produces the displacement that is a linear function of position. Figure 1.23 shows a plane structure undergoing a series of rigid body motions starting from a rigid translation in the X-direction, then a rigid translation in the Y-direction, and finally a rigid rotation about a point A. L L= L + e y x L L= L y x |e L L= L+ e y x |e (a) (b)(c) |s |s FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 17Within the context of static structural analysis, the structure under consideration must sufficiently be constrained to prevent both the rigid body motion of the entire structure and the rigid body motion of any part of the structure. The former is prevented by providing a sufficient number of supports and proper directions against movement and the latter is prevented by the proper arrangement of members and their connections. A structure shown in Figure 1.24(a) is a structure in Figure 1.23 after prevented all possible rigid body motions by introducing a pinned support at a point A and a roller support at a point B. A structure shown in Figure 1.24(b) indicates that although many supports are provided but in improper manner, the structure can still experience the rigid body motion; for this particular structure, the rigid translation can still occur in the X-direction. Figure 1.23: An unconstrained plane structure undergoing a series of rigid body motions Figure 1.24: (a) A structure with sufficient constraints preventing all possible rigid body motions and (b) a structure with improper constraints 1.5.2 Static quantities Quantities such as external actions and reactions in terms of forces and moments exerted to the structure by surrounding environments and the intensity of forces (e.g. stresses and pressure) and theirs resultants (e.g. axial force, bending moment, shear force, and torque, etc.) induced internally at any point within the structure are termed as static quantities. Applied load is one of static quantities referring to the prescribed force or moment acting to the structure. Support reaction is a term referring to an unknown force or moment exerted to the structure by idealized supports (representatives of surrounding environments) in order to prevent its movement or to maintain its Y X (a) (b) Y X Y X A A B FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 18 stability. Support reactions are generally unknown a priori. There are two types of applied loads; one called a nodal load is an applied load acting to the node of the structure and the other called a member loads is an applied load acting to the member. An example of applied loads (both nodal loads and member loads) and support reactions of a plane frame is depicted in Figure 1.25. Stress is a static quantity used to describe the intensity of force (force per unit area) at any plane passing through a point. Internal force is a term used to represent the force or moment resultant of stress components on a particular surface such as a cross section of a member. Note again that a major source that produces the stress and the internal force within the structure is the deformation. The distribution of both stress and internal force within the member depends primarily on characteristics or types of that member. For standard one-dimensional members in a plane structure such as an axial member, a flexural member, and a frame member, the internal force is typically defined in terms of the force and moment resultants of all stress components over the cross section of the member a plane normal to the axis of the member. Figure 1.25: Schematic of a plane frame subjected to external applied loads An axial member is a member in which only one component of the internal force, termed as an axial force and denoted by f a force resultant normal to the cross section, is present. The axial force f is considered positive if it results from a tensile stress present at the cross section; otherwise, it is considered negative. Figure 1.26 shows an axial member subjected to two forces {fx1, fx2} at its ends where fx1 and fx2 are considered positive if their directions are along the positive local x-axis. The axial force f at any cross section of the member can readily be related to the two end forces {fx1, fx2} by enforcing static equilibrium of both parts of the member resulting from an imaginary cut; this gives rise to f = fx1 = fx2. Such obtained relation implies that {f, fx1, fx2} are not all independent but only one of these three quantities can equivalently be chosen to fully represent the internal force of the axial member. Figure 1.26: An axial member subjected to two end forces A flexural member is a member in which only two components of the internal force, termed as a shear force denoted by V a resultant force of the shear stress component and a bending moment denoted by M a resultant moment of the normal stress component, are present. The shear y x fx1 fx2 y x fx1 fx2 f f Node 1 Node 2 Node 3 Node 4 FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 19force V and the bending moment M are considered positive if their directions are as shown in Figure 1.27; otherwise, they are considered negative. Figure 1.27 illustrates a flexural member subjected to forces and moments {fy1, m1, fy2, m2} at its ends where fy1 and fy2 are considered positive if their directions are along the positive local y-axis and m1 and m2 are considered positive if their directions are along the positive local z-axis. The shear force V and the bending moment M at any cross section of the member can readily be related to the end forces and moments {fy1, m1, fy2, m2} by enforcing static equilibrium of both parts of the member resulting from an imaginary cut. It can be verified that only two quantities from a set {fy1, m1, fy2, m2} are independent and the rest can be obtained from equilibrium of the entire member. This implies in addition that two independent quantities from {fy1, m1, fy2, m2} can be chosen to fully represent the internal force of the flexural member; for instance, {m1, m2} is a common choice for the internal force of the flexural member. Figure 1.27: A flexural member subjected to end forces and end moments. A frame member is a member in which three components of the internal force (i.e. an axial force f, a shear force V, and a bending moment M) are present. The axial force f, the shear force V and the bending moment M are considered positive if their directions are as indicated in Figure 1.28; otherwise, they are considered negative. Figure 1.28 shows a frame member subjected to a set of forces and moments {fx1, fy1, m1, fx2, fy2, m2} at its ends where fx1 and fx2 are considered positive if their directions are along the positive local x-axis, fy1 and fy2 are considered positive if their directions are along the positive local y-axis and m1 and m2 are considered positive if their directions are along the positive local z-axis. The axial force can readily be related to the end forces {fx1, fx2} by a relation f = fx1 = fx2 and the internal forces {V, M} at any cross section of the member can be related to the end forces and end moments {fy1, m1, fy2, m2} by enforcing static equilibrium to both parts of the member resulting from a cut. It can also be verified that only three quantities from a set {fx1, fy1, m1, fx2, fy2, m2} are independent and the rest can be obtained from equilibrium of the entire member. This implies that two independent quantities from {fy1, m1, fy2, m2} along with one quantity from {f, fx1, fx2} can be chosen to fully represent the internal forces of the frame member; for instance, {f, m1, m2} is a common choice for the internal force of the frame member. Figure 1.28: A frame member subjected to a set of end forces and end moments. 1.6 Basic Components for Structural Mechanics There are four key quantities involved in the procedure of structural analysis: 1) displacements and rotations, 2) deformation, 3) internal forces, and 4) applied loads and support reactions. The first two quantities are kinematical quantities describing the change of position and change of shape or x y x fy1 y x m1 fy2 m2 fy1 fy2 M V M V m1 m2 y x fy1 y m1 fy2 m2 fy1 m1 fy2 m2 M V M V fx1 fx2 fx1 fx2 f f FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 20 distortion of the structure under external actions while the last two quantities are static quantities describing the external actions and the intensity of force introduced within the structure. It is evident that the displacement and rotation at any constraint points (supports) and the applied loads are known a priori while the rest are unknown a priori. As a means to solve such unknowns, three fundamental laws are invoked to establish a set of sufficient governing equations. 1.6.1 Static equilibrium Static equilibrium is a fundamental principle essential for linear structural analysis. The principle is based upon a postulate: the structure is in equilibrium if and only if both the linear momentum and the angular momentum conserve. This postulate is conveniently enforced in terms of mathematical equations called equilibrium equations equations that relate the static quantities such as applied loads, support reactions, and the internal force. Note that equilibrium equations can be established in several forms; for instance, equilibrium of the entire structure gives rise to a relation between support reactions and applied loads; equilibrium of a part of the structure resulting from sectioning leads to a relation between applied loads, support reactions appearing in that part, and the internal force at locations arising from sectioning; and equilibrium of an infinitesimal element of the structure resulting from the sectioning results in a differential relation between applied loads and the internal force. 1.6.2 Kinematics Kinematics is a basic ingredient essential for the analysis of deformable structures. The principle is based primarily upon the geometric consideration of both the undeformed configuration and the deformed configuration of the structure. The resulting equations obtained relate the kinematical quantities such as the displacement and rotation and the deformation such as elongation, rate of twist, curvature, and strain. 1.6.3 Constitutive law A constitutive law is a mathematical expression used to characterize the behavior of a material. It relates the deformation (a kinematical quantity that measures the change in shape or distortion of the material) and the internal force (a static quantity that measures the intensity of forces and their resultants). To be able to represent behavior of real materials, all parameters involved in the constitutive modeling or in the material model must be carried out by conducting proper experiments. 1.6.4 Relation between static and kinematical quantities Figure 1.29 indicates relations between the four key quantities (i.e. displacement and rotation, deformation, internal force, and applied loads and support reactions) by means of the three basic ingredients (i.e. static equilibrium, kinematics, and constitutive law). This diagram offers an overall picture of the ingredients necessitating the development of a complete set of governing equations sufficient for determining all involved unknowns. It is worth noting that while there are only three basic principles to be enforced, numerous analysis techniques arise in accordance with the fashion they apply and quantities chosen as primary unknowns. Methods of analysis can be categorized, by the type of primary unknowns, into two central classes: the force method and the displacement method. The former is a method that employs static quantities such as support reactions and internal forces as primary unknowns while the latter is a method that employs the displacement and rotation as primary unknowns. FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 21 Figure 1.29: Diagram indicating relations between static quantities and kinematical quantities 1.7 Static Equilibrium Equilibrium equations are of fundamental importance and necessary as a basic tool for structural analysis. Equilibrium equations relate three basic static quantities, i.e. applied loads, support reactions, and the internal force, by means of the conservation of the linear momentum and the angular momentum of the structure that is in equilibrium. The necessary and sufficient condition for the structure to be in equilibrium is that the resultant of all forces and moments acting on the entire structure and any part of the structure vanishes. For three-dimensional structures, this condition generates six independent equilibrium equations for each part of the structure considered: three equations associated with the vanishing of force resultants in each coordinate direction and the other three equations corresponding to the vanishing of moment resultants in each coordinate direction. These six equilibrium equations can be expressed in a mathematical form as 0 F ; 0 F ; 0 FZ Y X (1.15) 0 M ; 0 M ; 0 MAZ AY AX (1.16) where {O; X, Y, Z} denotes the reference Cartesian coordinate system with origin at a point O and A denotes a reference point used for computing the moment resultants. DISPLACEMENTMETHODApplied Loads & Support Reactions (Known and unknown) Internal Forces (Unknown) Deformation (Unknown) Displacement & Rotation (Known and unknown) Static Equilibrium Constitutive Law Kinematics FORCEMETHOD FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 22 For two-dimensional or plane structures (which are the main focus of this text), there are only three independent equilibrium equations: two equations associated with the vanishing of force resultants in two directions defining the plane of the structure and one associated with the vanishing of moment resultants in the direction normal to the plane of the structure. The other three equilibrium equations are satisfied automatically. If the X-Y plane is the plane of the structure, such three equilibrium equations can be expressed as 0 M ; 0 F ; 0 FAZ Y X (1.17) It is important to emphasize that the reference point A can be chosen arbitrarily and it can be either within or outside the structure. According to this aspect, it seems that moment equilibrium equations can be generated as many as we need by changing only the reference point A. But the fact is these generated equilibrium equations are not independent of (1.15) and (1.16) and they can in fact be expressed in terms of a linear combination of (1.15) and (1.16). As a result, this set of additional moment equilibrium equations cannot be considered as a new set of equations and the number of independent equilibrium equations is still six and three for three-dimensional and two-dimensional cases, respectively. It can be noted, however, that selection of a suitable reference point A can significantly be useful in several situation; for instance, it can offer an alternative form of equilibrium equations that is well-suited for mathematical operations or simplify the solution procedures. To clearly demonstrate the above argument, let consider a plane frame under external loads as shown in Figure 1.30. For this particular structure, there are three unknown support reactions {RA, RBX, RBY}, as indicated in the figure, and three independent equilibrium equations (1.17) that provide a sufficient set of equations to solve for all unknown reactions. It is evident that if a point A is used as the reference point, all three equations FX = 0, FY = 0 and MAZ = 0 must be solved simultaneously in order to obtain {RA, RBX, RBY}. To avoid solving such a system of linear equations, a better choice of the reference point may be used. For instance, by using point B as the reference point, the moment equilibrium equation MBZ = 0 contains only one unknown RA and it can then be solved. Next, by taking moment about a point C, the reaction RBX can be obtained from MCZ = 0. Finally the reaction RBY can be obtained from equilibrium of forces in Y-direction, i.e. FY = 0. It can be noted, for this particular example, that the three equilibrium equations MBZ = 0, MCZ = 0 and FY = 0 are all independent and are alternative equilibrium equations to be used instead of (1.17). Note in addition that an alternative set of equilibrium equations is not unique and such a choice is a matter of taste and preference; for instance, {MBZ = 0,FX = 0,FY = 0}, {MBZ = 0,FY = 0,MDZ = 0}, {MBZ = 0,MCZ = 0,MDZ = 0} are also valid sets. Figure 1.30: Schematic of a plane frame indicating both applied loads and support reactions A B RA RBY RBX X Y C D FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 23 The number of independent equilibrium equations can further be reduced for certain types of structures. This is due primarily to that some equilibrium equations are satisfied automatically as a result of the nature of applied loads. Here, we summarize certain special systems of applied loads that often encounter in the analysis of plane structures. 1.7.1 A system of forces with the same line of action Consider a body subjected to a special set of forces that have the same line of action as shown schematically in Figure 1.31. For this particular case, there is only one independent equilibrium equation, i.e. equilibrium of forces in the direction parallel to the line of action. The other two equilibrium equations are satisfied automatically since there is no component of forces normal to the line of action and the moment about any point located on the line of action identically vanishes. Truss members and axial members are examples of structures that are subjected to this type of loadings. Figure 1.31: Schematic of a body subjected to a system of forces with the same line of action 1.7.2 A system of concurrent forces Consider the body subjected to a system of forces that pass through the same point as shown in Figure 1.32. For this particular case, there are only two independent equilibrium equations (equilibrium of forces in two directions defining the plane containing the body, i.e. FX = 0 and FY = 0). The moment equilibrium equation is satisfied automatically when the two force equilibrium equations are satisfied; this can readily be verified by simply taking the concurrent point as the reference point for computing the moment resultant. An example of structures or theirs part that are subjected to this type of loading is the joint of the truss when it is considered separately from the structure. Figure 1.32: Schematic of a body subjected to a system of concurrent forces 1.7.3 A system of transverse loads Consider the body subjected to a system of transverse loads (loads consisting of forces where their lines of action are parallel and moments that direct perpendicular to the plane containing the body) Line of action F1 F2 F3 F1 F2 F3 F4 X Y FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 24 as shown schematically in Figure 1.33. For this particular case, there are only two independent equilibrium equations (equilibrium of forces in the direction parallel to any line of actions and equilibrium of moment in the direction normal to the plane containing the body, i.e. FY = 0 and MAZ = 0). It is evident that equilibrium of forces in the direction perpendicular to the line of action is satisfied automatically since there is no component of forces in that direction. Examples of structures that are subjected to this type of loading are beams. Figure 1.33: Schematic of a body subjected to a system of transverse loads An initial step that is important and significantly useful for establishing the correct equilibrium equations for the entire structure or any part of the structure (resulting from the sectioning) is to sketch the free body diagram (FBD). The free body diagram simply means the diagram showing the configuration of the structure or part of the structure under consideration and all forces and moments acting on it. If the supports are involved, they must be removed and replaced by corresponding support reactions, likewise, if the part of the structure resulting from the sectioning is considered, all the internal forces appearing along the cut must be included in the FBD. Figure 1.34(b) shows the FBD of the entire structure shown in Figure 1.34(a) and Figure 1.34(c) shows the FBD of two parts of the same structure resulting from the sectioning at a point B. In particular, the fixed support at A and the roller support at C are removed and then replaced by the support reactions {RAX, RAY, RAM, RCY}. For the FBD shown in Figure 1.34(c), the internal forces {FB, VB, MB} are included at the point B of both the FBDs. 1.8 Classification of Structures Idealized structures can be categorized into various classes depending primarily on criteria used for classification; for instance, they can be categorized based on their geometry into one-dimensional, two-dimensional, and three-dimensional structures or they can be categorized based on the dominant behavior of constituting members into truss, beam, arch, and frame structures, etc. In this section, we present the classification of structures based upon the following three well-known criteria: static stability, static indeterminacy, and kinematical indeterminacy. Knowledge of the structural type is useful and helpful in the selection of appropriate structural analysis techniques. 1.8.1 Classification by static stability criteria Static stability refers to the ability of the structure to maintain its function (no collapse occurs at the entire structure and at any of its parts) while resisting external actions. Using this criteria, idealized structures can be divided into several classes as follows. 1.8.1.1 Statically stable structures A statically stable structure is a structure that can resist any actions (or applied loads) without loss of stability. Loss of stability means the mechanism or the rigid body displacement (rigid translation F2 F1 F4 F3 M1 M2 X Y FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis Copyright 2011 J. Rungamornrat 25and rigid rotation) develops on the entire structure or any of its parts. To maintain static stability, the structure must be properly constrained by a sufficient number of supports to prevent all possible rigid body displacements. In addition, members constituting the structure must be arranged properly to prevent the development of mechanics within any part of the structure or, in the other word, to provide sufficient internal constraints. All desirable idealized structures considered in the static structural analysis must fall into this category. Examples of statically stable structures are shown in Figures 1.3, 1.5 and 1.14-1.16. Figure 1.34: (a) A plane frame subjected to external loads, (b) FBD of the entire structure, and (c) FBD of two parts of the structure resulting from sectioning at B. 1.8.1.2 Statically unstable structures A statically unstable structure is a structure that the mechanism or the rigid body displacement develops on the entire structure or any of its parts when subjected to applied loads. Loss of stability in this type of structures may be due to i) an insufficient number of supports as shown in Figure 1.35(a), ii) inappropriate directions of constraints as shown in Figure 1.35(b), iii) inappropriate P M MB FB VB MB FB VB (c) B A C RAY RAX RAM RCY (a) (b) X Y RAY RAX RAM RCY P M P M FUNDAMENTAL STRUCTURAL ANALYSIS Introduction to Structural Analysis Jaroon Rungamornrat Copyright 2011 J. Rungamornrat 26 arrangement of member as shown in Figure 1.35(c), and iv) too many internal releases such as hinges as shown in Figure 1.35(d). This class of structures can be divided into three sub-classes based on how the rigid body displacement develops. 1.8.1.2.1 Externally, statically unstable structures An externally, statically unstable structure is a statically unstable structure that the mechanism or the rigid body displacement develops only on the entire structure when subjected to applied loads. Loss of stability of this type structure is due to an insufficient number of supports provided or an insufficient number of constraint directions. Examples of externally, statically unstable structures are shown in Figure 1.35(a) and 1.35(b). 1.8.1.2.2 Internally, statically unstable structures An internally, statically unstable structure is a statically unstable structure that the mechanism or the rigid body displacement develops only on a certain part of the structure when subjected to applied loads. Loss of stability of this type of structure is due to inappropriate arrangement of member and too many internal releases. Examples of internally, statically unstable structures are shown in Figure 1.35(c) and 1.35(d). Figure 1.35: Schematics of statically unstable structures 1.8.1.2.3 Mixed, statically unstable structures A mixed, statically unstable structure is a statically unstable structure that the mechanism or the rigid body displacement can develop on