Fundamental Structural Analysis.8765.1408344506

8/17/2019 Fundamental Structural Analysis.8765.1408344506

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S T R U C T U R A LA N A L Y S I S

Faculty of EngineeringChulalongkorn University

Jaroon Rungamornrat

Zubizuri BridgeBilbao. Biscay, Euskadi, Spain


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J. Rungamornrat, Ph.D.

Department of Civil Engineering

Faculty of EngineeringChulalongkorn University

Copyright © 2011 J. Rungamornrat

FUNDAMENTAL

STRUCTURAL

ANALYSIS


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Dedication

To

My parents my wife

and my beloved son


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FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Preface


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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 toemphasize 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 andto 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

author’s 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


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FUNDAMENTAL STRUCTURAL ANALYSIS Preface Jaroon Rungamornrat


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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 sufficientverification 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 providethe 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 mainfocus 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. Thechapter 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 Castigliano’s 1st and 2

nd

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


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involves the application of Castigliano’s 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 Müller-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


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FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Table of Contents


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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 492.1 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 1433.1 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


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Chapter 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 2154.4 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 287Exercises 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 Castigliano’s 1st and 2

nd 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


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Exercises 403

Chapter 9 APPLICATIONS OF CASTIGLIANO’S 2nd

THEOREM 407

9.1 Castigliano’s 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 Müller-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


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FUNDAMENTAL STRUCTURAL ANALYSIS Jaroon Rungamornrat Introduction to Structural Analysis


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CHAPTER 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


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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 becompletely 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 morecomplex 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 analysisRes onse inter retation

Real structure

Complex

&

Inaccessible

Idealized structure

Simplified

&

Solvable


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structure 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 notingthat 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).

FrameTruss

Beam Arch


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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 Young’s modulus denoted by E and the other is

the Poisson’s ratio denoted by . 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(2

EG

(1.1)

The Young’s 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 Poisson’s 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 (- curve) and a slope of the shear stress-strain curve (- curve),respectively, as indicated in Figure 1.4. The Poisson’s ratio is a parameter that measures thedegree 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

Uniaxial stress-strain curve

G

1

Shear stress-strain curve


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contact 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 largearea 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 idealizedsupport 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 onlyone 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


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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 inthe 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


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1.1.5.5 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 supportreactions 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 jointA 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.


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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. Theschematic 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 structuresIn 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


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1.1.7.1 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 areshown 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 connectedeither 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


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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 primaryunknowns 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 completelydetermined 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

vu

X

Y

Deformed configuration

Undeformed configuration


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11

1.4 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 coordinatesystems 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. Alocal 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,

k 1} and the other, denoted by {x2, y2, z2}, with the unit base vectors {i2, j2, k 2} as indicated in

Figure 1.20. Now, let define a matrix R such that

X

Y

x

y

y

x

y

x


28/629



12

332313

322212

312111

212121

212121

212121

coscoscos

coscoscos

coscoscos

k k k jk i

jk j j ji

ik i jii

R (1.2)

where {11, 21, 31} are angles between the unit vector i2 and the unit vectors {i1, j1, k 1},respectively; {12, 22, 32} are angles between the unit vector j2 and the unit vectors {i1, j1, k 1},respectively; and {13, 23, 33} are angles between the unit vector k 2 and the unit vectors {i1, j1,k 1}, 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 thecoordinate 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

21 (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

2

2

z2

2

y2

2

x1

1

z1

1

y1

1

x vvvvvv k jik jiv (1.4)

where { 1z1

y

1

x v,v,v } and {2

z

2

y

2

x v,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 component2

xv in terms of the

components { 1z

1

y

1

x v,v,v }, we take an inner product between a vector v given by (1.4) and a unit

vector i2 to obtain

)(v)(v)(vv 211

z21

1

y21

1

x

2

x ik i jii (1.5)

y1

y2

x1

x2

z1 z2

i1

i2 j1

j2 k

1

k 2


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13

Similarly, by taking an inner product between a vector v given by (1.4) and a unit vector j2 and k 2,

it leads to

)(v)(v)(vv 211

z21

1

y21

1

x

2

y jk j j ji (1.6)

2 1 1 1

z 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

1

z

1

y

1

x

1

z

1

y

1

x

212121

212121

212121

2

z

2

y

2

x

v

v

v

v

v

v

v

v

v

R

k k k jk i

jk j j ji

ik i jii

(1.8)

The expression of the components { 1z1

y

1

x v,v,v } in terms of the components {2

z

2

y

2

x v,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, k 1}. The final results are given by

2

z

2

y

2

x

T

2

z

2

y

2

x

121212

121212

121212

1

z

1

y

1

x

v

v

v

v

v

v

v

v

v

R

k k k jk i

jk j j ji

ik i jii

(1.9)

where R T 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 transformationmatrix R possesses a special form given by

100

0cossin

0sincos

R (1.10)

The coordinate transformation formula (1.8) and (1.9) therefore reduce to

1

z

1

y

1

x

2

z

2

y

2

x

v

v

v

100

0cossin

0sincos

v

v

v

(1.11)

2

z

2

y

2

x

1

z

1

y

1

x

v

v

v

100

0cossin

0sincos

v

v

v

(1.12)


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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 1

x x

2 1

y y

2 1

z z

v cos sin 0 v

v sin cos 0 v

w 0 0 1 w

(1.13)

1 2

x x

1 2

y y

1 2

z z

v cos sin 0 v

v sin cos 0 v

w 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 thechange 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 measured from a local tangent line in the undeformed configuration toa 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 thedisplacement 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,


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15

space 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 freedomand 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,

1, v2, 2, v3, 3} consisting of 3 prescribed DOFs {v1, 1, v3} and 3 free DOFs {v2, 2, 3}; a planetruss 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, 1, u2, v2, 2, u3, v3, 3} consisting of 3 prescribed DOFs { u1, v1, v3} and 6 free DOFs {1,u2, v2, 2, u3, 3}. It is evident that the number of degrees of freedom of a given structure is notunique 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=0u1=0

u3

v3

u2

v2=0

Y

Node 1 Node 2

Node 3

(b)

(c)

(a)

v1 = 0

1 = 0

v2

2 v3 = 0

3

Y

X

Node 1 Node 2 Node 3

v1=0u1=0

1

3 u3 v3=0u2

v2

2

Y

X

Node 1

Node 2 Node 3


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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 positionwhile, 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} wheres 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 Figure1.22(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

xL

L´= Ly

x

e

L

L´= L+ e y

x

e

(a) (b)

(c)

s

s


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17

Within 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 inFigure 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 motionsand (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

XA

A´ B


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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 { f x1, f x2} at its

ends where f x1 and f x2 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 { f x1,

f x2} by enforcing static equilibrium of both parts of the member resulting from an imaginary cut;

this gives rise to f = – f x1 = f x2. Such obtained relation implies that { f , f x1, f x2} 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 f x1 f x2

y

x f x1 f x2 f f

Node 1

Node 2

Node 3

Node 4


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19

force 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 { f y1, m1, f y2, m2} at its ends where f y1 and f y2 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 anycross section of the member can readily be related to the end forces and moments { f y1, m1, f y2, 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 { f y1, m1, f y2, m2} are independent and the rest can be

obtained from equilibrium of the entire member. This implies in addition that two independent

quantities from { f y1, m1, f y2, 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 { f x1, f y1, m1, f x2, f y2, m2} at its ends where f x1 and f x2 are considered positive if

their directions are along the positive local x-axis , f y1 and f y2 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{ f x1, f x2} by a relation f = – f x1 = f x2 and the internal forces {V , M } at any cross section of the

member can be related to the end forces and end moments { f y1, m1, f y2, 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 { f x1, f y1, m1, f x2, f y2, m2} are independent and the rest can be obtained from

equilibrium of the entire member. This implies that two independent quantities from { f y1, m1, f y2,

m2} along with one quantity from { f , f x1, f x2} 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

f y1

y

xm1

f y2

m2

f y1 f y2

M V M V m1 m2

y

x

f y1

y

m1

f y2

m2

f y1

m1

f y2

m2 M V M V f x1

f x2 f x1

f x2 f f


36/629



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 internalforce 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 quantitiesFigure 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.


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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 structurevanishes. 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

0ΣF ; 0ΣF ; 0ΣF ZYX (1.15)

0ΣM ; 0ΣM ; 0ΣM AZAYAX (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.

D I S P L A C E

M E N T M E T H O D

Applied Loads & Support Reactions

(Known and unknown)

Internal Forces(Unknown)

Deformation

(Unknown)

Displacement & Rotation(Known and unknown)

Static Equilibrium

Constitutive Law

Kinematics

F O R C E

M E T H O D


38/629



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, suchthree equilibrium equations can be expressed as

0ΣM ; 0ΣF ; 0ΣF AZYX (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-dimensionalcases, 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

{R A, R BX, R BY}, 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 {R A, R BX, R BY}. 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 thereference point, the moment equilibrium equation MBZ = 0 contains only one unknown R A and it

can then be solved. Next, by taking moment about a point C, the reaction R BX can be obtained from

MCZ = 0. Finally the reaction R BY 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 equationsMBZ = 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

R A

R BY

R BX

X

Y

CD


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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 inFigure 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 actionF

1 F

2 F

3

F1

F2

F3

F4

X

Y


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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 ofstructures 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 thesupport reactions {R AX, R AY, R AM, R CY}. 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


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and 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 staticstructural 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

R AY

R AX

R

AM

R CY

(a) (b)

X

Y

R AY

R AX

R

AM

R CY

P

M

P

M


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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.

Fundamental Structural Analysis.8765.1408344506

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