Nwamarah Uche

ANALYSIS OF

STRUCT

TECHNICHQUES

Digitally Signed by: Content manager’s Name

DN : CN = Weabmaster’s name

O= University of Nigeria, Nsukka

Nwamarah Uche

FACULTY OF ENGINEE

DEPARTMENT OF CIVIL ENGINEERING

ANALYSIS OF RIGID JOINTED PLANE TRUSS

STRUCTURES USING FINITE ELEMENT ANALYSIS TECHNICHQUES

EBU, ANIEBIET IME

PG/M.ENG/09/51772

Digitally Signed by: Content manager’s

DN : CN = Weabmaster’s name

O= University of Nigeria, Nsukka

ENGINEERING

CIVIL ENGINEERING

PLANE TRUSS

RES USING FINITE ELEMENT ANALYSIS

EBU, ANIEBIET IME PG/M.ENG/09/51772

ANALYSIS OF RIGID JOINTED PLANE TRUSS

STRUCTURES USING FINITE ELEMENT ANALYSIS TECHNICHQUES

BY

EBU, ANIEBIET IME PG/M.ENG/09/51772

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

AWARD OF MASTERS OF ENGINEERING DEGREE (M.ENG) IN CIVIL

ENGINEERING (STRUCTURAL ENGINEERING)

SUPERVISOR ENGR. PROF. N. N. OSADEBE

DEPARTMENT OF CIVIL ENGINEERING, UNIVERSITY OF

NIGERIA, NSUKKA

MARCH, 2013 CERTIFICATION

I, EBU, ANIEBIET IME, (PG/M.ENG/51772) hereby certify that this research work “Analysis of Rigid Jointed Plane Truss Structures using Finite Element analysis Techniques” is original to me and has not been submitted elsewhere for the award of a diploma or degree. EBU, ANIEBIET IME DATE (PG/M.ENG/09/51772)iii

APPROVAL This work “Analysis of Rigid Jointed Plane Truss Structures using Finite Element analysis Techniques” is hereby approved as a satisfactory research work for the award of a Master Degree (M.Eng) in the department of Civil Engineering. ENGR. PROFESSOR N.N. OSADEBE DATE PROJECT SUPERVISOR ENGR. PROFESSOR O.O.Ogwu DATE HEAD, CIVIL ENGINEERING DEPARTMENT, UNIVERSITY NIGERIA NSUKKA. ENGR. PROFESSOR T. C. MABUEME DATE DEAN, FACULTY OF ENGINEERING, UNIVERSITY OF NIGERIA NSUKKA. iv

DEDICATION To my daughter, Mfonabasi Aniebiet Ebu who at her birth, solution to this dissertation work was carried out. v

ACKNOWLEDGEMENT I wish to express my profound gratitude to the almighty God, the creator of heaven and earth who has given me the enablement to carry out this work. Secondly, the painstaking supervision, intellectual assistance and encouragement of my dear supervisor: Engr. Prof. N. N. Osadebe throughout the course of this work is deeply appreciated. Let me also appreciate the effort of my lecturers and other staff of Civil Engineering department, University of Nigeria Nsukka for providing me with the adequate academic base in course of my program, to name but a few, the head of Civil Engineering Department, University of Nigeria Nsukka in the person of Engr.prof.O.O. Ogwu, Engr. Prof. N. N. Osadebe, Prof J. C. Agunwamba, Engr. Dr. F. O. Okafor, Engr. Dr. C. U. Nwoji, Engr. Dr. Onah, Engr. A. Adamu and many others. You all have contributed immensely to knowledge especially in Civil Engineering, may God continue to reward you abundantly. My joy will not complete if I fail to mention Engr. U. C. Anya of Civil Engineering Department, Federal University of Technology owerri, who is the first to teach me how to write programs to solve Civil Engineering problems in my undergraduate days. Sir , thanks for your contribution. Also, I would like to express my love to my parents Elder/Mrs. Ime Akpanebu, my love- Ntiense and my siblings- Abasiofon, Emaediong, Mmekemeabasi, mfonmma, and Imekan for their wonderful support and encouragement. Lastly, let me express my gratitude to my boss and counselor Engr, Camillus Umoh of Shell Petroleum Development company of Nigeria(SPDC) for his good advises that spurred me up on the course of this work. May God continually bless you and your family. Amen. vi

LIST OF SYMBOLS A: Sectional area of element Ax: Amplitude of vibration D: Flexural Rigidity E : Young’s modulus. Fx: Force vector Ix: Moment of inertia K: Stiffness Matrix L: Length of member M: Mass matrix Mx: Resultant moment N: Finite Element Interpolation Function q: Distributed load intensity T: Kinetic Energy of bar U: Displacement vector �: Density of material vii

�: Natural Frequency of vibration w: Deflection W: Shape function Π: Functional σx: Bending stresses ε Axial stresses viii

TABLE OF CONTENTS Title Page i Certification Page ii Approval Page - iii Dedication iv Acknowledgment- v List of symbols vi Abstract vii Table of contents viii List of figures - xi List of Tables xii CHAPTER ONE 1.1.0 General introduction- 1 1.1.2 Static analysis 2 1.1.3 Modal analysis 2 1.1.4 Natural frequency 3 1.1.4 Modes 3 1.1.5 Mode shape 3 1.1.6 Computer programs 4 1.2 Statement of problem 5 1.3 Objectives of study 6 1.4 General methodology 6 ix

1.5 Significance of study 7 1.6 Scope of work 7

1.7 Limitation of study 8 CHAPTER TWO: LITERATURE REVIEW 2.1 History of finite element method 9 2.2 Basic steps on the finite element method. 10 2.3 Definition of truss 11 2.4 Finite element interpolation functions for general element formulation 12 2.6 Matrix methods 15 2.7 Direct stiffness approach 15 2.8Finite element formulation of plane truss using the concept of direct stiffness method 20 2.9 Direction cosine 20 2.10 Assembly of global stiffness matrix 21 2.11 Static analysis of plane truss structures 22 2.12 Boundary condition, constraint forces 22 2.13 Element stress and strain 24 2.14 Modal analysis of truss structures 25 2.15 Modal analysis of plane frames 25 2.16 Consistent mass matrix in plane truss structures assuming pin connection 26 2.17 Degrees of freedom 30 2.18 Natural frequency 30 2.19 Resonance 30 x

2.20 Equation of motion for free undamped vibrations of a plane truss structure system using finite element method 31 2.21 Determination of natural frequency of a plane truss 32 2.22 Eigenvalue and eigenvector in dynamic analysis 32 2.23 Definition of eigenvalue and eigenvector 32 2.24 Positive, negative complex eigenvalues 33 2.25 Solution of eigenvalue problem 35 2.26 Determination of eigenvectors 35 2.27 Mode shapes 36 2.28 Commercial software 36 2.29 Matlab computing language 36 2.30 Writing functions subroutne in matlab 37 CHAPTER THREE: METHODOLOGY 3.1 Elementary beam theory 39 3.2 Analysis of flexure beam element 41 3.3 Flexure element stiffness matrix 46 3.4 Stiffness matrix for rigid jointed plane truss member 49 3.5 Equivalent nodal load vector of rigid jointed plane truss subjected to out of joint tranverse loading 53 3.6 Bending stresses in rigid jointed plane truss member 59 3.7 Axial stresses in rigid jointed plane truss member 60 xi

3.8 Modal analysis (natural frequency) 60 3.81 Formation of lump mass matrix 60 3.82 Transformation from local to global co-ordinate system 61 3.83 Calculation of eigen value and eigen vector 61 3.84 Natural frequencies 61 3.9 Orthogonality of principal mode 62 CHAPTER FOUR: ANALYSIS AND RESULTS 4.1 Analysis 66 4.2 Results 171 4.3 Discussion of results 172 4.31 Static analysis 173 4.311 Displacements 173 4.312 Axial stresses 173 4.313 Plane trusses with uniformly distributed loads and out of joint loadings 173 4.314 Bending stresses 174 4.32 Modal analysis 174 CHAPTER FIVE Conclusions and Recommendations 175 References 176 xii

APPENDICES Appendix 1:Matlab program users’ manual 179 Appendix 2: m-files 181 Apendix 3: Matlab program for the analysis pin jointed plane truss structures (static and modal analysis) 188 Apendix 4: Matlab program for the analysis of rigid jointed plane truss structures (static and modal analysis) 190 xiii

ABSTRACT Analysis of plane truss structures based on the assumptions that (1) members are connected at joints by frictionless pins and (2) loads are applied at joints only have become a norm not minding the facts that our practical trusses are always constructed by connecting members to gusset plates by welds, rivets, or high-strength bolts and loads may not necessarily be applied only at the nodes, have made such assumptions not to yield quite precise results. By the nature of practical nodes connections, joints in our trusses are rigid and not frictionless pins and as such analyzing plane trusses as rigidly connected nodes yields more precise results. This project aims at studying plane truss analysis with rigid joints as compared to frictionless pin joints in both static and dynamic (modal) forms. Finite element analysis techniques using stiffness method as its basis for formulating stiffness and mass matrices are to be employed in the analysis. Computer program using MATLAB codes will be developed to aid the analysis. xiv

LIST OF FIGURES Figure 1.10 Structural Analysis Computational Flow. 5 Figure 2.1 A simple truss subjected to a load. 11 Figure 2.2 Statically Indeterminate and Determinate Structure 12 Figure 2.3 Nodal forces 16 Figure 2.4: Relationship between local and global coordinates. 18 Figure 2.5: The relation of element and global displacements at a single node. 28 Figure 2.6 Plane truss 33 Figure 3.1 (a) Simply supported beam subjected to arbitrary (negative) distributed load. (b) Deflected beam element. (c) Sign convention for shear force and bending moment. 39 Figure 3.3 (a) and (b) Beam elements with identical end deflections but quite different deflection characteristics. (c) Physically unacceptable discontinuity at the connecting node. 42 Figure 3.4 Bending moment diagram for a flexure element. Sign convention per the strength of materials theory. 44 Figure 3. 4 Nodal displacements of rigid jointed plane truss member 49 Figure 3.5 (a) Nodal displacements in the element coordinate system. (b) Nodal displacements in the global coordinate system. 51 Figure 3.6: Flexure element with node at the point of application of a point load along the element Recalling stiffness matrix of a flexure element as: 53 Figure 3.7: work equivalent nodal forces and moment for a uniformly distributed load 57 Figure 4.1 Warren truss for Railway Bridge. 67 xv

Figure 4.2 Nodal coordinates and element numbering. 67 Fundamental mode shape Pin jointed plane truss structures 174 Fundamental mode shape Rigid jointed plane truss structures 174 xvi

LIST OF TABLES Table 4.1 Reaction Forces(KN) 171 Table 4.2 Member Stresses(MPA) 171 Table 4.3 Nodal Displacements(mm) 172 Table 4.4 Natural Frequencies(Hetz) 172 CHAPTER ONE INTRODUCTION 1.1.1 GENERAL INTRODUCTION Fundamentally, the behavior of all types of structures – framework, plates, shells or trusses is described by means of differential equations. In practice, the writing of differential equations for truss structures is rarely necessary. It has been long established that such structures may be treated as assemblage of one-dimensional members. Exact or approximate solutions to the ordinary differential equations for each member are well-known. These solutions can be cast in the form of relationships between the forces and the displacements at the ends of the members. Proper combinations of these relationships with the equations of equilibrium and compatibility at the joints and supports yield a system of algebraic equations that describes the behavior of the structure. Structures consisting of two- or three- dimensional components- plates, trusses, membrane shells, solids are more complicated in that rarely do exact solutions exist for applicable partial differential equations as said before. One approach to obtaining practical, numerical solutions is the finite element method. The basic concept of the method is that a continuum (the total structure) can be modeled analytically by its subdivision into regions (the finite elements) [20], ‗each of the behavior is described by a set of assumed functions representing the stresses or displacements in that region‘. This permits the problem formulation to be altered to one of the establishment of a system of algebraic equations. Therefore, the high speed precise computing and increased memory of the computers have made it possible to solve complex models. Finite element method and matrix methods are the two methods which show great compatibility for computing processes and have become the most powerful tools in many engineering branches. Therefore, analysis that was considered cumbersome and consequently avoided can now be carried out easily using finite element method which has a great potential of being easily programmed especially with such computing language like MATLAB (used in this work) that has codes which can easily manipulate and solve mathematical problems. This has made actual structures like plane trusses to be analyzed based on its true service conditions (nodes connections and loadings) without 2

assuming conditions (frictionless pins connection and loads only acting at the joints) for the purpose of simplifying the analysis that this can lead to results that are not precise or uneconomical in designs. This work, analysis of rigid-jointed plane truss structures using finite element analysis technique tries to understudy analysis of plane trusses using the assumptions of frictionless pins joints and loads acting only at the joints to develop plane truss analysis with rigid joints and no restriction of loading conditions. It modeled rigid jointed plane truss as plane frames, then uses stiffness method to formulate the stiffness and mass matrices. Finite element interpolation functions for truss element formulation was used to obtain formula for strain and stresses and equivalent nodal loads in the case of uniformly distributed loads and point loads. The work was carried out in two phases namely: (1) static analysis and (2) modal analysis. 1.1.2 STATIC ANALYSIS Static analysis of plane truss structures using finite element analysis technique is achieved by converting nodal coordinates of the truss structure from local coordinates to global coordinates system [2]. The stiffness matrix of individual members were calculated and then transformed into the global stiffness matrix using conversions outlined in chapters two and three for both frictionless pin joints and rigid joints respectively. The elements by elements global stiffness matrix are then assembled into the structure stiffness matrix by direct combination procedure [8]. This was then followed by providing constraints to the finite element equation of the plane truss problem: KU = F . . . . . . . . . . . . . . . . . .. . 1.1 Where K is the stiffness matrix, U is the displacement vector and F is the force vector. Displacements are first sort as the primary variables and the values are substituted back into the equation1.1 to find secondary variables such as supports reactions, strains and stresses. The procedure is the same for both assumptions of pin and rigid joints, only that in addition to axial stresses bending stresses were also evaluated in the case of rigid joints. 1.1.3 MODAL ANALYSIS 3

Modal analysis is the study of the dynamic properties of structures under vibrational excitation [24]. When a structure undergoes an external excitation, its dynamic responses are measured and analyzed. This process of measuring and analyzing is called modal analysis. Modal analysis can be used to measure the response of a car‘s body to a vibration when vibration of an electromagnetic shaker is attached or the pattern created by noise of a loudspeaker which acts as excitation [27] In structural engineering, modal analysis is applied to find the various periods that the structure will naturally resonate at, by using the structural overall mass and stiffness. The modal analysis is very important in earthquake engineering, because the periods of vibration evaluated helps in checking that a building‘s natural frequency does not coincide with the frequency of earthquakes prone region where the building is to be constructed. In case a structure‘s natural frequency coincidentally equals an earthquake‘s frequency, the structure suffers severe structural damage due to resonance [36] The frequency and mode shape of a model is determined by modal analysis. When the models are subjected to cyclic or vibration loads, the dynamic response of structures due to these external loads acting, which include resonance frequencies (natural frequencies), mode shape and damping, are estimated. 1.1.4 NATURAL FREQUENCY All models have a natural frequency. If a model is subjected to dynamic load that is close to its natural frequency, the model oscillates to large extent than in normal condition. The results of a modal analysis help to determine whether a model requires more or less damping to prevent failure. Modal analysis can be used to find frequency at which resonance occurs under specific constraints. 1.1.5 MODES Modes measure the vibration of an object at a particular frequency. Each mode is assigned a number. The lowest speed at which a structure vibrates after all external loads are removed is assigned mode (1). This mode is called the free vibration mode because it has no external forcing function. 4

1.1.6 MODE SHAPE In the study of vibration in engineering, the expected curvature (or displacement of a surface at a particular mode due to vibration is the mode shape. To determine vibration of a system, multiplying the mode shape by a time-dependent function, the vibration of a system is found out. Thus the mode shape always describes the time-to-time curvature of vibration where the magnitude of the curvature will change. The mode shape depends on two factors: (1) the shape of the surface

(2) the boundary condition of the supports. 1.1.7 COMPUTER PROGRAMS The equations of the finite element approach are of a form so generally applicable that it is possible in theory to write a single computer programs that will solve an almost limitless variety of problem in structural mechanics. Many commercial available general purpose programs such as ROBOT structural analysis program attempts to achieve this objective although on a restricted scale. The advantage of general-purpose programs is not merely this compatibility but the unity afforded in the instruction of prospective users input and output data interpretation procedures and documentations. The four components in the flow chart of fig. 1.1 are common to virtually all general- purpose, finite element analysis programs. As a minimum, the input phase should require of the user no information beyond that relating to the material of construction, geometric description of the finite element representation (including support conditions), and conditions of loading. The most sophisticated general-purpose programs facilitates this input process through such features as pre-stored material property schedules and graphical displays of the finite element idealization so that errors in input can be detected prior to performance of calculations. In the phase comprised of the library of finite elements is the coded formulated process for the individual elements. The solution phase operates on the equation of the problem formed in the prior phase. In the case of a linear static analysis program this may mean no more than the single solution of a set of linear algebraic equations for a known right-hand side. Solutions for dynamic response may require very extensive computations over a time-history of applied loads. In still other cases, particularly where 5

the unknown is very large, it may be advantageous to divide the total structure into several substructures. A finite element idealization of each substructure is made and analyzed, and the results are properly combined into solution for the full structure. Finally, the output phase presents the analyst with a numerical or pictorial record of the solution upon which engineers can base decisions regarding the proportioning of the structure and other design questions. In this work, the only element considered is a plane truss element. The structural analysis computational flow is as follows: Input Definition of physical model Lib. of Element Geo. Mat., loading & B. C Generation of maths models For elem. And loading Solution Output Construction $ solution Display of Predicted Of maths model of system Results Where, Geo. is geometric description of the finite element representation, Mat is the material properties of the element and B.C is the boundary condition. Figure 1.10 Structural Analysis Computational Flow Chart. 1.1 STATEMENT OF PROBLEM The quest for precise and economic designs of structures has encouraged review of conventional assumptions used in analyzing structures. As it is well known, most of the assumptions used in simplifying analysis of structures are not based on the true service conditions of such structures, therefore results obtained by such assumptions may not necessarily be the true representative of such structures. This may add to or reduce actual values of derived variables like stresses, strains, natural frequencies and even mode shapes of the designed structure and could consequently lead to either under design or over design of structures. With the use of analysis method like finite element method which can be easily programmed with a computing language such as MATLAB, highly indeterminate structures can be analyzed with ease. This 6

drive the quest for this work, to analyze plane truss structures based on its true service conditions so as to achieve more actual results. 1.3 OBJECTIVES OF STUDY The main objective of this dissertation work is to analyze plane truss structures as rigidly connected joints and compare its results with those obtained using the assumption of frictionless pin connections using finite element analysis techniques. It is however realized that the fulfillment of the following sub-objectives would in-turn fulfill the main objective. i. To understand the use of finite element analysis techniques in the analysis of plane truss structures.

ii. To compare results achieved in static analysis of plane truss structures with the condition of rigid joints and those obtained with the assumption of frictionless joints connections.

iii. To use finite element analysis technique in formulating equations of strain and stresses in plane truss structures.

iv. To use finite element techniques interpolation functions to derive equivalent nodal forces for plane trusses subjected to uniformly distributed loads and point loads acting at any portion of the truss member other than the joints.

v. To compare results achieved in modal analysis for both frictionless pin joints connections and rigid joint connections

vi. To develop and run MATLAB program for both static and modal analysis of plane truss structures for both frictionless pin connections and rigid connections. 1.4 GENERAL METHODOLOGY This work ―Analysis of Rigid Jointed Plane Truss Structures using Finite Element Analysis Techniques� is imminent in structural engineering as trusses are commonly used in our everyday structures. With the knowledge of finite element analysis of plane truss structures with the assumptions: (1) members are connected together by frictionless pins (2) loads are applied only at the nodes. Stiffness and mass matrices for rigid jointed plane trusses were established. Finite element interpolation functions 7

were used to find expressions for equivalent nodal loads in the case of uniformly distributed loads and out-of-joint point loads. Finite element interpolations were also used to established expressions for stresses and strains in members. The results (stresses, displacements, support reactions, natural frequencies and mode shapes) were obtained for both pin and rigid jointed plane truss structures through the aid of computer program using MATLAB codes written in this work. The results were cross checked using commercial structural engineering software called ROBOT structural engineering software. 1.5 SIGNIFICANCE OF STUDY This study will hopefully be relevant in the following respect: a. The analysis and design of plane truss structures irrespective of nodal condition, loading and support condition.

b. Finite element analysis modeling of plane truss structures with rigid and pin joints.

c. Software development. 1.6 SCOPE OF WORK This work used the finite element analysis technique to formulate stiffness and mass matrices for plane truss elements, it derives stresses and strain equations, equations for equivalent nodal loads at the joints for both uniformly distributed loads and point loads at any section of the truss element other than the joints for rigid jointed plane truss structures. It also reviewed the analysis of pin jointed plane truss structures using finite element analysis technique. In static analysis, nodal displacements, support reactions, member axial and bending stresses are sort. Whereas, in dynamic analysis, natural frequencies and mode shapes of the plane truss structure are sort. For the determination of natural frequency, cases of negative and complex eigen value problem is outside the scope of this work. Only real and positive or zero eigen values were encountered as usually the case with vibration problems [36]. The computer program to be developed will be based on MATLAB codes and the commercial program that will be used to cross check the outputs of the developed program will be ROBOT structural analysis software. 8

1.7 LIMITATION OF STUDY This work is limited to static and modal (natural frequency and mode shapes) of plane truss structures. The joint condition of the plane truss is either rigid or pin; it does not in any way consider partial fixity. 9

CHAPTER TWO LITERATURE REVIEW 2.1 HISTORY OF FINITE ELEMENT METHOD The Finite Element Method (FEM), sometimes referred to as Finite Element Analysis (FEA), is a computational technique used to obtain approximate solution of boundary value problem in engineering [2]. The Finite Element Method is a numerical procedure that can be applied to obtain solution to a variety of problem in engineering. The origin of the modern Finite Element Method may be traced back to the early 1900s, when Edne Investigators approximated and modeled elastic continuum using discrete equivalent elastic bars. However, Coarant has been credited with being the first person to develop the Finite Element Method [15]. In a paper published in the early 1940s, Coarant used piece wise polynomial interpolation over triangular sub-regions to investigate torsion problem. The next significant step in the utilization of Finite Element Methods was taken by Boeng in the 1950s when Boeng, followed by others, used triangular stress elements to model airplane signs [7]. Yet, it was not until 1960s that they made the term ―Finite Element Method� However, the work of Rayleigh and Rite cannot be left out when mentioning the history of Finite Element Method as both used trial functions (in our context, interpolation functions) to approximate solutions of differential equations which was also used by GalarKin [7]. In the late 1940, aircraft engineers dealt with inventions of the jet engine and the needs for more sophisticated analysis of air frame structures to withstand larger loads associated with higher speeds. These engineers, without the benefit of modern computers, developed matrix method of force analysis collectively known as the flexibility method in which unknown are the force and the known are displacements [8]. The Finite Element Method, in its most often – used form, corresponds to the displacement in response to applied force systems. Displacement as a term is quite general in the Finite Element Method and can represent physical displacement, temperature, or fluid velocity. During the decade of the 1960s and 1970s the Finite Element Method was extended to application in plate bending, shell bending, pressure vessels and general three dimensional problems in elastic structural analysis as well as to fluid flow and heat transfer [24]. Further extension of the method to large 10

deflections and dynamic analysis also occurred during this period. [17] also gave excellent history of Finite Element Method. The Finite Element Method is computationally intensive owing to the required operations on very large matrices. In the early years, applications were performed using mainframe computers, which, at the time, were considered to be very powerful, high-speed tools for use in engineering analysis. As outlined by [13], during 1960s, the Finite Element Software‘s Code NASTRAN was developed in conjunction with the space exploration program of the United States. NASTRAN was the first major Finite Element Software Code. It was and still is capable of hundreds of thousands of degree of freedom (nodal field variable computations). In the years since the development of NASTRAN, many commercial packages have been introduced for Finite Element Analysis. Among this are ANSYS, ALGOR and COSMOS/M. 2.2 BASIC STEPS ON THE FINITE ELEMENT METHOD. The basic steps involved in any Finite Element Analysis consist of the following as stated in [9]. Preprocessing Phase; 1. Create and discretize the solution domain into Finite Elements, that is subdivide the problem into nodes and elements.

2. Assume a shape function to represent the physical behaviour of an element that is, an approximate continuous function is assumed to represent to solution of an element.

3. Develop equations for an element.

4. Assemble the elements to present the entire problem, construct the global stiffness matrix.

5. Apply boundary conditions, initial conditions and loading solution phase.

6. Solve a set of linear or non-linear algebraic equations simultaneously to obtain nodal results such as displacement values at different nodes. Post process phase 1. Obtain other important information. At this point you may be interested in values of principal stresses. 11

2.3 DEFINITION OF TRUSS A truss is an engineering structure consisting of straight members connected at their ends by means of bolts, rivets, pins, or weld. The members found in trusses may consist of steel or aluminum tubes, wooden struts, metal bars, angles and channels. Trusses offer practical solutions to many structural problems in engineering, such as power transmission towers, bridges, and roofs of buildings. A plane truss according to [30] is defined as a truss whose members lie in a single plane. The force acting on such a truss must also lie in this plane. Members of a truss are generally considered to be two-force members. This means that internal forces act in equal and opposite directions, along the members as shown [10]; load 3 compression 3 two- 3 1 2 1 force member Tension 2 Figure 2.1 A simple truss subjected to a load. In the analysis that follows, it is assumed that the member are connected together by smooth pins and by bolt and socket joint in three – dimensional trusses. Moreover, it can be shown that as long as the center lines of the joining member intersect at a common point, trusses with bolted or welded joint may be treated as having smooth pins (no bending). Another important assumption deals with the way loads are applied. All loads must be applied at the joints-[10]. Usually the weights of member are negligible compared to those of the applied loads. However, if the weights of the member are to be considered, then half of the weight of each member is applied to the connecting joints. Statically determinate truss problem are analyzed by the methods of joints or sections. These methods do not provide information on deflection of the joints because the truss members are assumed to be rigid bodies. Statically indeterminate problems are difficult to analyze by using both methods of joints and sections. From [3], the Finite Element Method allows us to remove the rigid body restriction and solve statically indeterminate problem. Figure [2.2] depicts example of statically determinate and statically indeterminate problems. 12

load y load 3 x 3 1 2 1 2 Figure 2.2 Statically Indeterminate and Determinate Structure 2.4 FINITE ELEMENT INTERPOLATION FUNCTIONS FOR GENERAL ELEMENT

FORMULATION Interpolation functions are used to formulate finite element equations for various types of physical problems, for example trusses [8]. Formulation of finite element characteristics requires differentiation and integration of the interpolation function in various forms. Owing to the simplicity with which polynomial functions can be differentiated and integrated, polynomials are the most commonly used interpolation functions [13]. Displacement field for one dimension truss element is expressed via first degree polynomials: . . . . . . . . . 2.1 In terms of nodal displacement, Equation 2.1 is determined to be equivalent to . . . . . . . 2.2 The coefficients a0 and a1 are obtained by applying the nodal conditions u(x = 0) = u1 and u(x = L) = u2. Then, collecting coefficients of the nodal displacements, the interpolation functions are obtained as . . . . . . . 2.3 For plane trusses which has 4 degrees of freedom and the displacement field is represented as the cubic polynomial 13

. . . . . . . 2.4 which is ultimately to be expressed in terms of interpolation functions and nodal variables as

. . . . . 2.5 Rewriting Equation 2.4 as the matrix product, . . . . . . . . . . . . . . . 2.6 the nodal conditions . . . . . . . . . . . 2.7 are applied to obtain . . . . . . . . . 2.8 . . . . . . . . . 2.9 . . . . . . . . . 2.10 . . . . . . . . . 2.11 The last four equations are combined into the equivalent matrix form 14

. . . . . . 2.12 The system represented by equation 2.12 can be solved for the polynomial coefficients by inverting the coefficient matrix to obtain . . . . . 2.13 The interpolation functions can now be obtained by substituting the coefficients given by equation 2.3 into equation 2.13 and collecting coefficients of the nodal variables. However, the following approach is more direct and algebraically simpler. Substitute equation 2.13 into Equation 2.6 and equate to Equation 2.5 to obtain . . . . . 2.14 The interpolation functions are 15

. . . 2.15 2.6 MATRIX METHODS Behaviour of all types of structure is described by differential equation – [10]. In practice, it is common to represent trusses as planar structures for which the approximate or exact solution of the individual member are available. By using those solutions, relationship between force and displacement in the end of the member in combination with equilibrium and compatibility equations at the joint and support can be written which yields a system of algebraic equation that describe the behaviour of the structure. Thus, these simultaneous equation can be using matrix methods once the equation are written in matrix form taking advantage of the power computing capabilities of the computers-[30], the large system of equations can be solved quickly. [30] also, outlined the two general approaches of Matrix Method as flexibility and stiffness approaches and the former approach is used in this dissertation work to achieve Finite Element Formulations. 2.7 DIRECT STIFFNESS APPROACH As mentioned earlier that in Matrix Method there are generally two approaches the flexibility and the stiffness approach and the latter is used in this work for Finite Element Formulation. [8] showed that every element in the plane truss structure is idealized as a line element with sections properties (A) area, (I) moment of inertia, and (J) torsion constant and material properties as (E) modulus of elasticity (G) shear rigidity, (NU) poison ratio. [13], considered the whole planar structure as mesh, formed by individual Finite Elements connected together at joints called Nodal Points. Each element consists of 2 nodes and each is associated with corresponding degree of freedom (DOF). This is defined by [34] as ―the number of independent coordinates necessary to define the configuration of the system�. The frame of reference to identify entire structure is called Global Axis with X-axis along the member longitudinally applied to the nodes as point load and the corresponding displacement is measured at nodal point. Thus relationship between nodal force and corresponding nodal displacement is written in terms of stiffness equation or flexibility 16 Fi(Ui) Fj (Uj) i k j

equation. Also, [19] explains the procedure involved in the formation of member stiffness matrix and summarized in three steps; 1. Equilibrium

2. Compatibility

3. Hooke‘s law For illustration, he considered one dimensional spring element which has 2 DOF, the element force – displacement relationship related 2 nodal forces (fi, fj) to 2 nodal displacements through 2 X 2 stiffness matrix. Fe = K Ue . . . . . . . . . . . . 2.16 Expanding the above equation, we get: Fi = K11 Vi + K12 Uj . . . . . 2.17 Fj = K21 Vi + K22 Uj . . . . . . 2.18 In matrix form, we get: F1 = K11 K12 U1 . . . . . . . .. . . . . . . . 2.19 F2 K21 K22 U2 Thus K11, K12, K21, K22 are to be defined Applying springs law, force (f) vs displacement (�) is a linear curve, it can be written as; {F} = [K] {�} . . . . . . . . . . . . 2.20 Figure 2.3 Nodal forces We need to find (F) in terms of end displacements Ui and Uj using compatibility conditions. Fi = K(Ui – Uj) . . . . . . . . 2.21 Fi = K(Uj – Ui) . . . . . . . . . . 2.22 Now applying equilibrium equation; 17

�fx = 0 . . . . . . . . . 2.23 Fi + Fj = 0 i.e, Fi = -Fj . . . . . . . . . 2.24 Thus, Fi = -Fj = -(-KUi + KUj) . . . . . . . 2.25 In matrix form we can be written as; Fi = K -K U1 . . . . . . . . . . . 2.26 Fj -K K U2 Thus, this is the member stiffness matrix, the fundamental stiffness equation is of the form; {F} = [K] {�} . . . . . . . . . . 2.27 Where, (K) is the element stiffness matrix. In case of an axially loaded member, i unit displacement is applied keeping all the other degrees of freedom fixed, then the force applied will be equal to the stiffness of the member or in other words, nodal forces and nodal displacements are connected with each other by element stiffness matrix. This is formulated depending upon the material and sectional properties of the element. 2.8 FINITE ELEMENT FORMULATION OF PLANE TRUSS USING THE CONCEPT OF DIRECT

STIFFNESS METHOD In general,[18] used two frames of reference to described plane truss problem in Finite Element Analysis; global coordinate system and local frame of reference. Choosing a fixed coordinate system, XY: (1) to represent the location joint (node) and to keep track of the orientation of each member (element) using angles such as �; (2) to apply the constraints and the applied loads in terms of their respective global components; and (3) to represent the solution that is, the displacement of each joint in global direction. We also need a local, or an element coordinate system to describe the two force member behaviour of individual member (elements). The relationship between the local (element) description and the global descriptions is shown in fig. [2.4]. [18] gives the relationship between global displacement and local displacement according to the following equations; Uix = Uixcos� - Uiysin� . . . . . . . . . 2.28 Uiy = Uixsin� - Uiycos� . . . . . . . . . 2.29 18 Uiy uiy i Uix uix j Ujx ujx Ujy ujy Fiy fiy fix Fix Fjx Fjy fjx fjy -- 2.86 y x Uix Uiy Ujx Ujy uix uiy ujx ujy

Uix = Uixcos� - Ujysin� . . . . . . . . . 2.30 Uiy = Ujxsin� - Ujycos� . . . . . . . . . . 2.31 By putting the above equations in matrix from, we have : {U} = [T] {u} . . . . . . . . . . . . . . 2.32 Figure 2.4: Relationship between local and global coordinates. Where {U} = , , .. . . . . 2.32a and {u} = . . . . . . 2.32b cos � -sin� 0 0 sin � cos� 0 0 [T] = 0 0 cos � -sin� . . . 2.33 0 0 sin � cos� {U} and {u} represent the displacements of nodes i and j with respect to the global XY and the local xy frame of references, respectively. [T] is the transformation matrix that allows for the transfer of local 19 Fix Fiy Fjx

Fjy fix fiy fjx fjy AE L fix fiy fjx fjy uix uiy ujx ujy K 0 -K 0 0 0 0 0 -K 0 K 0 0 0 0 0

deformation to their respective global values. In a similar way, the local and global forces may be related according to the equations as stated in [26]. Fix = fix cos� - fiy sin� Fiy = fix sin� + fiy cos� Fjx = fjx cos� - fjy sin� . . . . . . . . . 2.34 Fjy = fjx sin� + fjy cos� Or in matrix form; {F} = [T] {f} . . . . . . 2.35 Where; {F} = . . . . . . . . 2.36 Are components of forces acting at nodes i and j with respect to global coordinates and {f} = - - . . . . . . . . . 2.37 Represent the local components of the forces at nodes i and j. Also, [7] relates internal forces and displacements through the stiffness matrix; = . . . . 2.38 Where K = Keq = , . . . . 2.38a and using the matrix form, this can be written as; {F} = [K]{U} . . . . . . . . . . . 2.39 After substituting for {f} and {u} in terms of {F} and {U}, we have; [T]-1 {F} = [K] [T]-1 {U} . . . . . . . . . . 2.40 Where [T]-1 is the inverse of the transformation matrix [T] and is 20 cos � sin� 0 0 -sin � cos� 0 0 0 0 cos � sin� 0 0 -sin � cos� Fix Fiy Fjx Fjy cos2� sin�cos� -cos2� -sin�cos� sin�cos� sin2� -sin�cos� -sin2� -cos2� -sin�cos� cos2� sin�cos� sin�cos� -sin2� sin�cos� sin2� Uix Uiy Ujx Ujy cos2� sin�cos� -cos2� -sin�cos� sin�cos� sin2� -sin�cos� -sin2� -cos2� -sin�cos� cos2� sin�cos� sin�cos� -sin2� sin�cos� sin2�

[T]-1 = . . . . . . . . . 2.41 Multiplying both sides of eqn. 2.40 by 2.41 and simplifying, he obtained; {F} = [T]- [K] [T]-1 {U} . . . . . . . . . . . 2.42 By substituting for value of the [T], [K], [T]-1 and {U} matrices in eqn. 2.42 and multiplying, He obtained; = K . . . 2.43 Equations (2.43) express the relationship between the applied forces, the element stiffness matrix [K](e), and the global deflection of the nodes of an arbitrary element the stiffness matrix [K](e) for any member (element) of the truss is K(e) = K . . . 2.44 This can be written in short form as; [K](e) = K c2 cs -c2 -cs cs s2 -cs -s2 . . . . . . . . . 2.45 -c2 -cs c2 cs -cs -s2 cs s2 Where K is the stiffness coefficient of the element as in equation 2.38a 21

2.9 DIRECTION COSINE In practice, a finite element model is constructed by defining nodes at specified coordinate locations followed by definition of elements by specification of the nodes connected by each element [19]. For the case at hand, nodes i and j are defined in global coordinates by (Xi, Yi) and (Xj, Yj). Using the nodal coordinates, element length is readily computed as . . . . . . . . . 2.46 and the unit vector directed from node i to node j is . . . . . 2.47 Where I and J are unit vectors in global coordinate directions X and Y respectively. Recalling the definition of the scalar product of two vectors and referring again to Figure 2.6, the trigonometric values required to construct the element transformation matrix are also readily determined from the nodal coordinates as the direction cosines in Equation 2.47 . . . . . . . . . . 2.48 . . . . . . . . . 2.49 Thus, the element stiffness matrix of a bar element in global coordinates can be completely determined by specification of the nodal coordinates, the cross sectional area of the element, and the modulus of elasticity of the element material. 2.10 ASSEMBLY OF GLOBAL STIFFNESS MATRIX According to [19], the technique of directly assembling the global stiffness matrix for a finite element model of a truss is discussed in terms of the simple two-element system depicted in Figure 2.6. Assuming the geometry and material properties to be completely specified, the element stiffness matrix in the global frame can be formulated for each element using Equation 2.47 to obtain 22

. . . . . . 2.50

. . . . . 2.51 In this depiction of the stiffness matrices for the two individual elements, the numbers to the right of each row and above each column indicate the global displacement associated with the corresponding row and column of the element stiffness matrix. Thus, we combine the nodal displacement correspondence table with the individual element stiffness matrices. For the element matrices, each individual component is now labeled as associated with a specific row-column position of the global stiffness matrix and can be added directly to that location. For example, Equation 2.51 shows that the k(2), K24 component of element 2 is to be added to global stiffness component K46 (and via symmetry K64). Thus, we can take each element in turn and add the individual components of the element stiffness matrix to the proper locations in the global stiffness matrix. 2.11 STATIC ANALYSIS OF PLANE TRUSS STRUCTURES In static analysis, stiffness matrix is used in formulating the equilibrium equation of a truss system given as [18]; [K]{U} = {F} . . . . . . . . . 2.52 Where [K] is the Global Stiffness Matrix, {U} is the Vector of Nodal Displacements, and {F} is the Vector of Applied Nodal Forces. In solving equation 2.52 boundary conditions and constraint forces must be taken into account in order to have unique solution to the displacements. 2.12 BOUNDARY CONDITION, CONSTRAINT FORCES 23

Having obtained the global stiffness matrix via either the equilibrium equations or direct assembly, the system displacement equations for the example truss of Figure 2.3 are of the form . . . . . . . . 2.52a As noted, the global stiffness matrix is a singular matrix; therefore, a unique solution to Equation 2.52a cannot be obtained directly. However, in developing these equations, we have not yet taken into account the constraints imposed on system displacements by the support conditions that must exist to preclude rigid body motion. In this example, we observe the displacement boundary conditions . . . . . . . . . . . . . . 2.53 leaving only U5 and U6 to be determined. Substituting the boundary condition values and expanding Equation 2.52a we have, formally, . . . . . . 2.54 as the reduced system equations (this is the partitioned set of matrix equations, written explicitly for the active displacements). In this example, F1, F2, F3, and F4 are the components of the reaction forces at constrained nodes 1 and 2, while F5 and F6 are global components of applied external force at node 3. Given the external force components, the last two of Equations 2.54 can be explicitly solved for displacements U5 and U6. The values obtained for these two displacements are then substituted into the constraint equations (the first four of Equations 2.54) and the reaction force components computed. 24

A more general approach to application of boundary conditions and computation of reactions is as follows. Letting the subscript c denote constrained displacements and subscript a denote unconstrained (active) displacements, the system equations can be partitioned to obtain . . . . . . . . 2.55 . . . . . . . . 2.56 where we have assumed that the specified displacements {Uc} are not necessarily zero, although that is usually the case in a truss structure. 2.13 ELEMENT STRESS AND STRAIN The final computational step in finite element analysis of a truss structure is to utilize the global displacements obtained in the solution step to determine the strain and stress in each element of the truss [26]. For an element connecting nodes I and j, the element nodal displacements in the element coordinate system are given by Equations 2.55 and 2.56 as . . . . 2.57 . . . . 2.58 and the element axial strain (utilizing Equation 2.57 and the discretization and interpolation functions of Equation 2.58) is then . . . . . . 2.58a Where, L (e) is element length. The element axial stress is then obtained via application of Hooke‘s law as . . . . . . 2.59 Note, however, that the global solution does not give the element axial displacement directly. Rather, the element displacements are obtained from the global displacements via Equations 2.58a. Recalling Equations 2.57 and 2.58, the element strain in terms of global system displacements is 25

. . . . 2.60 Where, [R] is the element transformation matrix defined by Equation 2.60. The element stresses for the bar element in terms of global displacements are those given by . . . . 2.61 As the bar element is formulated here, a positive axial stress value indicates that the element is in tension and a negative value indicates compression per the usual convention. Note that the stress calculation indicated in Equation 2.61 must be performed on an element-by-element basis. 2.14 MODAL ANALYSIS OF TRUSS STRUCTURES [14] studied ‗ Interval natural frequency and mode shape analysis for truss structures with interval parameters‘. In this paper, euler-bernoulli beam theory was applied for modeling of stepped piezoelectric beams to analyze different numerical methods for obtaining modal analysis of the beam. Results from standard numerical approaches that relied on the discretization of the stepped beam (assumed modes and finite element methods), were compared with the exact solution of the exact transcendental eigenvalue problem for the infinite dimensional system. An accurate and manageable novel method was obtained that improved the assumed modes basis functions with special jump functions. Numerical results were compared with experimental data and the adopted beam model was checked for accuracy and then validated. 2.15 MODAL ANALYSIS OF PLANE FRAMES [36] studied ‗Modal analysis and seismic response of steel frames with connection dampers‘. In between the connection of end plate and column flange or between the angle and member flange, dissipation materials may be placed in order to minimize dynamic response of steel frame of bolted 26

connections. He derived the mass matrix, stiffness matrix and damping matrix for the frame of bolted connections. He derived the mass , stiffness matrix and damping matrix for frame using a combination of the finite element method and the direct stiffness method to idealize bolted connections and energy dissipation material as rotational spring and damper respectively. Using the complex modal analysis, dynamic characteristics of the frame was analyzed and the effects of connection stiffness and rotational damper on natural frequency and modal damping ratio were required. In order to observe the effect of connection stiffness and rotational damper on the seismic performance of the frame, a generalized pseudo- excitation method was introduced for the vibration analysis of the frame subject to earthquake excitation. The parametric studies on the example frame with and without the connection dampers showed that there was an optional damper damping coefficient by which the modal damping ratio of the frame can be considered increased and the seismic responses, including both lateral displacements and internal forces, can be significantly reduced. [5], studied ‗vibration and modal analysis of steel frame with semi- rigid connections‘. In practical design, frames are assumed to be either rigid or pinned joints. Strictly speaking, all joints are semi-rigid, so this assumption does not normally represent the actual behaviour of a realistic steel frame. The response of a steel frame will be of interest to the designer for the purpose of ensuring safety and serviceability of the structure under the action of dynamic loads. This paper was addressed to the extension of the well-known stiffness matrix method of analysis of framed structures to cover a more general case of vibration and stability analysis of flexibly connected steel frames. 2.16 CONSISTENT MASS MATRIX IN PLANE TRUSS STRUCTURES ASSUMING PIN

CONNECTION [1], admonishes that in general case of solid structures, the mass is distributed geometrically throughout the structure and the inertia properties of the structure depend directly on the mass distribution. In modeling plane structures, additional considerations are required, and the mass matrix modified, accordingly. [26] work, he says that when a truss undergoes deflection, either statically or dynamically, individual elements experience both axial and transverse displacements resulting from overall structural displacement and element interconnections at nodes. In static analysis, transverse 27 �U �t = �U �t U1 U2 V1 V2 1 0 U1 V1 U2 V2 0 N12 O N2N2 O O Ni2 O N1N2 N1N2 O N2 O O N1N2 O N2

displacement of elements is ignored owing to the assumption of pin connections, hence free rotation. However, in dynamic case, transverse motion introduces additional kinetic energy, which must be taken into account. [27], considered the differential volume of a bar element undergoing both axial and transverse displacement as shown in fig.2.91. He assumed a dynamic situation such that both . . . . . . . . . 2.62 displacement components vary with position and time. The Kinetic Energy of the differential volume is; dT = ½eAdx [(�u/�t )2 + (�v/�t)2] = ½eAdx (U2+V2) . . . . . . 2.63 and the Kinetic Energy of the bar becomes: T = ½eA � U-2dx + ½eA � V-2dx . . . . . . . 2.64 By observation, the transverse displacement can be expressed in terms of the transverse displacement of the element nodes, using the same interpolation functions as for axial displacement, thus; U(x,t) = N1 (x) U1 (t) + N2 (x) U2 (t) . . . . . . . . 2.65 V(x,t) = N1 (x) V1 (t) + N2 (x) V2 (t) . . . . . . . . . . 2.66 Using matrix notation, the velocities are written as; U(x1t) = [N1 N2] . . . . . . . . . 2.67 V(x1t) = [N1 N2] . . . . . . . . . 2.68 And element Kinetic Energy becomes; T = ½ eA {U}T � [N]T [N] dx {U} + ½ eA {V}T � [N]T [N] dx {V} . . . . . . 2.69 Expressing nodal velocities ag {�} = . . . . . . . . . . . . . 2.70 The Kinetic Energy expression can be rewritten in the form; T = ½ {�}T [M2(e)] {�}= ½{�}T eA � dx {�} . . . . 2.71 28 2 0 1 0 0 2 0 1 1 0 2 0 . . . . 2.73 . . . 2.74 N12 O NiN2 O O N12 O N1N2 N1N2 O N2 O O N1N2 O N2 U2 cos� sin� U3 V2 -sin� cos� U4 U3 U2 U4 V2

From equation 2.72, the Mass Matrix of the bar element in two dimension is identified [M2(e)] = eA � dx = eAL/6 2.72 The Mass Matrix defined by eqn. 1.10 is described in the element (local) coordinate system, since the axial and transverse directions are defined in terms of the axis of the element. How, then, is this Mass Matrix transformed to the global coordinate system of a structure? Recalling that when stiffness matrix for a bar element was derived, the axial displacements were expressed in terms of global displacements via rotation transformation of the element x axis. To reiterate, the transverse displacements were not considered, as no stiffness is associated with the transverse motion. Now, however, the transverse displacements must be included in the transformation to global coordinates because of the associated mass and kinetic energy – [13 ]. [12], illustrated this principle by using fig. 2.4 which depicts a single node of a bar element oriented at angle � relative to the x-axis of a global coordinate system. Nodal displacements in the element frame are U2, V2 and corresponding global displacements are U3, U4, respectively. As the displacement in the two coordinate systems must be the same, he had; U2 = U3cos� + U4 sin� V2 = U3cos� + U4 sin� or = Figure 2.5: The relation of element and global displacements at a single node. 29 U1 V1 U2 V2 cos� sin� 0 0 -sin� cos� 0 0 0 0 cos� sin� 0 0 -sin� cos� U1 U2 U3 U4 2C2 2CS C2 CS 2CS 2S2 CS S2 C2 CS 2C2 2CS CS S2 2CS 2S2 eAL 6 2 0 1 0 0 2 0 1 1 0 2 0 0 1 0 2 C2 CS 0 0 CS S2 0 0 0 0 C2 CS 0 0 CS S2

As the same relation holds at the other element node, the complete transformation is; = = = [R]{U} . . . . 2.75 Since the nodal velocities are related by the same transformation, substitution into the kinetic energy expression shows that the Mass Matrix in the global coordinate system is; [M2(e)] = [R]T[M2(e)][R] . . . . . . . . . . . . . . 2.76 If the matrix multiplication in eqn. 2.914 are performed for an arbitrary angle, the resulting global Mass Matrix for a bar element is found to be; [Me] = . . . . . . . 2.77 Which is the same as: [Me] = eAl/6 . . . . . . . 2.78 Also, the lumped Mass Matrix can obtain similarly as shown below; [Me] = eAL/6 . . . . . . 2.79 And, the result is exactly the same as the Mass Matrix in the element coordinate system regardless of element orientation in the global system. This phenomenon should come as no surprise since mass is an absolute scalar property and therefore independent on coordinate system. 30

2.17 DEGREES OF FREEDOM The term ―degrees of freedom� is equal to the number of independent co-ordinates or measurements required to define completely the configuration of the structure at any instant of time; in other words, it is equal to the number of independent types of motion possible in the structure [15] Using the lumped mass procedure, the number of degrees of freedom is equal to the number of masses. Consequently we have framed and truss structures with single with Single Degree of Freedom (SDOF) and others with Many Degrees of Freedom (MDOF). 2.18 NATURAL FREQUENCY The frequency of free vibration is termed the natural frequency [27]. For a given structure there are as many natural frequencies as there are degrees of freedom. Natural frequencies depend on mass and stiffness of the structure. Calculations of stiffness, where necessary are to be based on a proper evaluation, shearing, axial and torsional deformations of the members [24]. In most practical structures, the effects of shearing deformation do not substantially affect the natural frequencies [6]. Consequently, shear deformation effect is often neglected in dynamic analysis of structures. The unit of natural frequency, w, is radians/seconds or cycles/seconds. 2.19 RESONANCE

Resonance occur when the natural frequency, w, coincides with the frequency of forcing, ө, i.e., w = 0. If w= 0 is substituted in equation of motion for free vibration of an SDOF system given by:

X(t) = Asin (wt + ө) . . . . . . . . 2.80 Gives: X(t) = � . . . . . . . . 2.81 which means that in such a case the displacement becomes infinitely large leading to the destruction of the structure. The resonance phenomenon described here is known as theoretical resonance [20]. This is so because it is not possible for displacement to be infinity in real life. However it must be noted that large deflections occur during resonance and this situation is not a good one for any structure. Avoidance of resonance is a major concern in dynamic analysis. 31

2.20 EQUATION OF MOTION FOR FREE UNDAMPED VIBRATIONS OF A PLANE TRUSS

STRUCTURE SYSTEM USING FINITE ELEMENT METHOD Free vibration occurs when, for example, a truss structure is displaced and then suddenly released. Then it vibrates freely with its natural frequency in the absence of external load or applied disturbance. Therefore, according to [14], free-vibration response of a multiple degrees-of-freedom system is described by Equation below: . . . . . . . . . 2.82 Assuming that we have solved for the natural circular frequencies and the modal amplitude vectors via the assumed solution form qi (t ) = Ai sin(�t + �), substitution of a particular frequency �i into equation 2.82 gives . . . . . . 2.83 And it is repeated for any other frequency �j. Examination of Equation 2.83 in light of known facts about the stiffness and mass matrices reveals that the differential equations are coupled, at least through the stiffness matrix, which is known to be symmetric but not diagonal. The phenomenon embodied here is referred to as elastic coupling, as the coupling terms arise from the elastic stiffness matrix. In consistent mass matrices, the equations are also coupled by the non-diagonal nature of the mass matrix; therefore, the term inertia coupling is applied when the mass matrix is not diagonal-[13]. [36] show, however, that the modal characteristics embodied in the equations of motion can be used to advantage in examining system response to harmonic (sinusoidal) forcing functions. The so-called harmonic response is a capability of essentially any finite element software package. In the absence of externally applied nodal forces or in the case of free vibration of the structure Equation 2.83 is a system of P homogeneous, linear second-order differential equations in the independent variable time. Hence, we have an eigenvalue problem in which the eigenvalues are the natural circular frequencies of oscillation of the structural system, and the eigenvectors are the amplitude vectors (mode shapes) corresponding to the natural frequencies. The frequency equation is represented by the determinant 32

|−�2[M] + [K ]| = 0 . . . . . . . . . . 2.84 If formally expanded, this determinant yields a polynomial of order P in the variable �2. Solution of the frequency polynomial results in computation of P natural circular frequencies and P modal amplitude vectors. 2.21 DETERMINATION OF NATURAL FREQUENCY OF A PLANE TRUSS The equation of motion for free vibration (eqn. 2.84) can be solved as an eigenvalue problem in order to determine the natural frequency of vibration. Thus a further discussion on the determination of natural frequency of vibration of an MDOF system the understanding of the application of eigenvalue and eigenvector to dynamic (or vibration) requires However, it must be pointed out that the non-trivial solution of (Eqn. 2.84) is the eigen value analysis. 2.22 EIGENVALUE AND EIGENVECTOR IN DYNAMIC ANALYSIS Eigenvalue Eigenvectors have a particular application to dynamic analysis in the determination of natural frequencies and mode shapes of vibrating systems. In vibration (or dynamic) analysis, the eigenvalues correspond to natural frequencies of vibration, and the eigenvectors to the relative vibration amplitudes or vibratory displacements of individual masses [24]. All masses vibrate in phase with the same natural frequency [24]. The eigenvector associated with any particular eigenvalue represents the amplitudes of the masses when they are vibrating at that natural frequency. This vector is termed the mode shape vector of the system since its elements represent the displacements of the masses [27]. It must emphasized that eigenvector is concerned with only relative values (i.e ratio of the displacements) and not absolute values. 2.23 DEFINITION OF EIGENVALUE AND EIGENVECTOR An eigenvalue is a scalar quantity. An eigenvector is a column vector. For a square matrix of order N (an N x N matrix ) there will be N eigenvalues, each with its associated eigenvector. The relationship which exists between a matrix and its eigenvalues and eigenvectors is given by the equation.

[A]{x} = ʎ{x} . . . . . . . . . 2.85 33

Where, [A] is the square matrix {x} is the eigenvector

ʎ is the eigenvalue, in other words the eigenvector is a vector which, when pre-multiplied by the matrix [A],

yields the same eigenvector multiplied by a scalar being the eigenvalue [22] 2.24 POSITIVE, NEGATIVE COMPLEX EIGENVALUES Eigenvalues are the roots of a polynomial. For any arbitrarily chosen matrix the eigenvalues can be positive, negative, zero, imaginary or complex. This can lead to difficulties in general manipulation of matrices, but in the case of matrices associated with vibrating systems all the eigenvalues are real and positive (or zero) [35]. Cases of complex and negative eigenvalues would not arise in this work. 2.25 SOLUTION OF EIGENVALUE PROBLEM In finite element analysis of plane truss structures, the governing equation for free vibration of the structure is given by Eqn 2.83 as shown below −�2[M] + [K ] = 0 . . . . . . . . . 2.86 This is an eigenvalue problem. For an example consider the structure in Fig 2.5, Stiffness matrix denoted by K which columns and rows equals the number of displacements of the structure and M is the mass matrix which has the dimension as the stiffness matrix. Considering plane truss of fig. 2.6 below: Figure 2.6 Plane truss 34

Therefore, Eq. 2.86 is as follows: K11 K12 K13 K14 K15 K16 M11 M12 M13 M14 M15 M16 ` K21 K22 K23 K24 K25 K26 M21 M22 M23 M24 M25 M26 K31 K32 K33 K34 K35 K36 -�2 M31 M32 M33 M34 M35 M36 = 0 . . 2.87 K41 K42 K43 K44 K45 K46 M41 M42 M43 M44 M45 M46 K51 K52 K53 K54 K55 K56 M51 M52 M53 M54 M55 M56 K61 K62 K63 K64 K65 K66 M61 M62 M63 M64 M65 M66 If supports 1 and 2 are pinned, then applying constraints, equation above reduces to: K33 K34 -�2 M33 M34 = 0 . . . . . . . 2.88 K43 K44 M43 M44 expanding gives K33-�2M33 K34-�2M34 = 0. . . . . . . 2.89 K43-�2M43 K44-�2M44 the solution of the above eigenvalue problem is the determinant of the left hand side equals zero as follows: K33-�2M33 K34-�2M34 = 0. . . . . . . . 2.90 K43-�2M43 K44-�2M44 Using Cramer‘s rule [22 ] to solve above equation gives: (K33-�2M33 )( K44-�2M44) - (K43-�2M43 )( K44-�2M44) = 0 . . . . . 2.91 Eqn 2.91 is a characteristic of degree n whose solution gives natural frequencies of the structure. However, for complex plane truss structure which large number of degrees of freedom, it is difficult to solve the eigenvalue problem manually, so computer program should be develop to solve this problem to achieve more accuracy. In chapter four of this work MATLAB program is developed to solve this problem. 35

2.26 DETERMINATION OF EIGENVECTORS Having determined the eigenvalues by finding the solutions of the characteristic polynomial equation, the eigenvectors can be determined by substituting these values back into equation 2.83 [18]. For each value of �n (for n= 1, 2, 3, ….. n.) there corresponds an eigenvector An whose elements are A1, A2, A3 . . . . An that are yet unknown. That is, [−�2[M] + [K ] ][A]= 0 . . . . . . . . . . 2.92 K11 K12 K13 K14 K15 K16 M11 M12 M13 M14 M15 M16 A1 ` K21 K22 K23 K24 K25 K26 M21 M22 M23 M24 M25 M26 A2 K31 K32 K33 K34 K35 K36 -�2 M31 M32 M33 M34 M35 M36 A3 = 0 . . . 2.93 K41 K42 K43 K44 K45 K46 M41 M42 M43 M44 M45 M46 A4 K51 K52 K53 K54 K55 K56 M51 M52 M53 M54 M55 M56 A5 K61 K62 K63 K64 K65 K66 M61 M62 M63 M64 M65 M66 A6 Unless a further process is employed the solution for these unknown A values will be zero (or non-trivial) for every eigenvalue �n. For this reason the eigenvector is a set of A values in a particular ratio. To do this, the above equation is normalized by introducing one into the forcing matrix that was initially zero due to free vibration [18] as follows: K11 K12 K13 K14 K15 K16 M11 M12 M13 M14 M15 M16 A1 1 ` K21 K22 K23 K24 K25 K26 M21 M22 M23 M24 M25 M26 A2 0 K31 K32 K33 K34 K35 K36 -�2 M31 M32 M33 M34 M35 M36 A3 = 0 . . 2.94 K41 K42 K43 K44 K45 K46 M41 M42 M43 M44 M45 M46 A4 0 K51 K52 K53 K54 K55 K56 M51 M52 M53 M54 M55 M56 A5 0 K61 K62 K63 K64 K65 K66 M61 M62 M63 M64 M65 M66 A6 The above equation can then be solved to yield a set of eigenvectors which is the relative displacements for each natural frequency of vibration. 36

2.27 MODE SHAPES A structural system has exactly the same number of normal modes as degrees of freedom [27]. This means that there will, altogether, be n mode shapes. Each of the amplitude vectors Ak is called a normal mode shape, A1 is the first mode shape, A2 is the second mode shape, An is the nth mode shape [36]. Absolute values of amplitudes are not necessary in defining the characteristic mode shapes, only relative values of amplitudes are required for this purpose. 2.28 ELEMENTARY BEAM THEORY Figure 3.1a depicts a simply supported beam subjected to a general, distributed, transverse load q(x ) assumed to be expressed in terms of force per unit length. The coordinate system is as shown with x representing the axial coordinate and y the transverse coordinate. The usual assumptions of elementary beam theory are applicable here: 1. The beam is loaded only in the y direction. 2. Deflections of the beam are small in comparison to the characteristic dimensions of the beam. 3. The material of the beam is linearly elastic, isotropic, and homogeneous. 4. The beam is prismatic and the cross section has an axis of symmetry in the plane of bending. Figure 2.7 (a) Simply supported beam subjected to arbitrary (negative) distributed load. (b) Deflected

beam element. (c) Sign convention for shear force and bending moment. 37

Figure 2.8 Beam cross sections: (a) and (b) satisfy symmetry conditions for the simple bending theory,

(c) does not satisfy the symmetry requirement. Considering a differential length dx of a beam after bending as in Fig 3.1b (with the curvature greatly exaggerated), it is intuitive that the top surface has decreased in length while the bottom surface has increased in length. Hence, there is a ―layer� that must be under formed during bending. Assuming that this layer is located distance _ from the center of curvature O and choosing this layer (which, recall, is known as the neutral

surface) to correspond to y = 0, the length after bending at any position y is expressed as . . . . . . . . . . 2.95 and the bending strain is then . . 2.96 From basic calculus, the radius of curvature of a planar curve is given by . . 2.97 where v = v(x) represents the deflection curve of the neutral surface. In keeping with small deflection theory, slopes are also small, so Equation 3.3 is approximated by . . . 2.98 such that the normal strain in the direction of the longitudinal axis as a result of bending is 38

and the corresponding normal stress is . . 2.100 where, E is the modulus of elasticity of the beam material. Equation 3.6 shows that, at a given cross section, the normal stress varies linearly with distance from the neutral surface. As no net axial force is acting on the beam cross section, the resultant force of the stress distribution given by Equation 4.6 must be zero. Therefore, at any axial position x along the length, we have . . . 2.101 Noting that at an arbitrary cross section the curvature is constant, Equation 3.7 implies . . . . . 2.102 which is satisfied if the xz plane (y = 0) passes through the centroid of the area. Thus, we obtain the well-known result that the neutral surface is perpendicular to the plane of bending and passes through the centroid of the cross-sectional area. Similarly, the internal bending moment at a cross section must be equivalent to the resultant moment of the normal stress distribution, so . 2.103 39

The integral term in Equation 4.9 represents the moment of inertia of the cross sectional area about the z axis, so the bending moment expression becomes . . 2.104 Combining Equations 3.6 and 3.10, we obtain the normal stress equation for beam bending: . . 2.105 Note that the negative sign in Equation 4.11 ensures that, when the beam is subjected to positive bending moment per the convention depicted in Figure 3.1c, compressive (negative) and tensile (positive) stress values are obtained correctly depending on the sign of the y location value. 2.28 COMMERCIAL SOFTWARE In this work, the commercial software used is Autodesk Robot Structural analysis. It is a Finite Element Application that can perform static and dynamic of plane trusses with any type of complexity. It is an integrated graphic program for modeling, analyzing and designing various types of structures. It lets you carry out calculations and verify results and also create documentation for the design and calculated structure. During static analysis the problem is simplified to [K] {U} = {F} and after that, boundary conditions are applied to formed matrices. The system is solved by means of cholesky (LU) factorization which is a fast arimethic procedure. Finally, after geometric transformation, the program computes the values of the axial forces in the local coordinate system of each member as well as the deformed shape of each element. The dynamic problem is described by the following differential equation. [M]{U} + [U]{U} + [K]{U} = {F} . . . . . . . . . 2.106 Where, M is the mass; C the Damping and K the Stiffness Matrix of the Structure to be analyzed. F is the loading (time-dependent) for all degrees of freedom. For free vibration, the dynamic equation is 40

expressed as follows; (K – W2M) � = O. the program now solves the eigene value problem to obtain its natural frequencies. 2.29 MATLAB COMPUTING LANGUAGE According to [21],� MATLAB is a high-level computer language for scientific computing and data visualization built around an interactive programming environment�. It is becoming the premiere platform for scientific computing at educational institutions and research establishments. The great advantage of an interactive system is that programs can be tested and debugged quickly, allowing the user to concentrate more on the principles behind the program and less on programming itself. Since there is no need to compile, link and execute after each correction, MATLAB programs can be developed in much shorter time than equivalent FORTRAN or C programs. On the negative side, MATLAB does not produce stand-alone applications—the programs can be run only on computers that have MATLAB installed. MATLAB has other advantages over mainstream languages that contribute to rapid program development: MATLAB contains a large number of functions that access proven numerical libraries, such as LINPACK and EISPACK. This means that many common tasks (e.g., solution of simultaneous equations) can be accomplished with a single function call. -There is extensive graphics support that allows the results of computations to be plotted with a few statements. -All numerical objects are treated as double-precision arrays. Thus there is no need to declare data types and carry out type conversions. 2.30 WRITING FUNCTIONS SUBROUTNE IN MATLAB MATLAB provides a convenient tool by which we can a program write using collections of MATLAB commands. This approach is similar to other common programming languages. It is quite useful especially when we write a series of MATLAB commands compatible files in a text file. This text file is edited and saved for later use. According to [25], the text file should have filename format, normally called m-files. That is all MATLAB subroutines should end with .m extension so that MATLAB recognizes them as MATLAB 41

compatible files. [16] illustrates the procedure as follows: make a text file using any text editor. For example, generate a file called func1.m, the file func1.m should start with the following fie header. Function[ov1, ov2,-- ----] =func1(iv, iv2, ------) . . . . . . 2.96 Where iv, iv2 ---- are input variables, while ov1, ov2, ---- are output variables. The input variables are specific variables and the output variables are dummy variables, for any variable can be used. 42

CHAPTER THREE METHODOLOGY 3.1 ANALYSIS OF FLEXURE BEAM ELEMENT Using the elementary beam theory, the 2-D beam or flexure element is now developed with the aid of the first theorem of Castigliano. The assumptions and restrictions underlying the development are the same as those of elementary beam theory with the addition of 1. The element is of length L and has two nodes, one at each end. 2. The element is connected to other elements only at the nodes. 3. Element loading occurs only at the nodes. Recalling that the basic premise of finite element formulation is to express the continuously varying field variable in terms of a finite number of values evaluated at element nodes, we note that, for the flexure element, the field variable of interest is the transverse displacement v(x) of the neutral surface away from its straight, undeflected position. As depicted in Figure 4.3a and 4.3b, transverse deflection of a beam is such that the variation of deflection along the length is not adequately described by displacement of the end points only. The end deflections can be identical, as illustrated, while the deflected shape of the two cases is quite different. Therefore, the flexure element formulation must take into account the slope (rotation) of the beam as well as end-point displacement. In addition to avoiding the potential ambiguity of displacements, inclusion of bea elements, thus precluding the physically unacceptable discontinuity depicted in Figure 3.1c. Figure 3.1 (a) and (b) Beam elements with identical end deflections but quite different deflection

characteristics. (c) Physically unacceptable discontinuity at the connecting node. In light of these observations regarding rotations, the nodal variables to be associated with a flexure element are as depicted in Figure 3.2. Element nodes 1 and 2 are located at the ends of the element, and the nodal variables are the transverse displacements v1 and v2 at the nodes and the slopes 43

(rotations) _1 and _2. The nodal variables as shown are in the positive direction, and it is to be noted that the slopes are to be specified in radians. For convenience, the superscript (e) indicating element properties is not used at this point, as it is understood in context that the current discussion applies to a single element. When multiple elements are involved in examples to follow, the superscript notation is restored. The displacement function v(x) is to be discretized such that nodal displacement . . . . . . . . . . . 3.1 subject to the boundary conditions . . . . . . . . . . . . 3.2 . . . . . . . . . . . . . 3.3 . . . . . . . . . . . . . 3.4 . . . . . . . . . . . . . . 3.5 Considering the four boundary conditions and the one-dimensional nature of the problem in terms of the independent variable, we assume the displacement function in the form . . . . . . . 3.6 Application of the boundary conditions 3.2–3.5 in succession yields . . . . . . . . . . . . 3.7 44

............3.8 ………3.9 ……….3.10 Figure 3.2 Bending moment diagram for a flexure element. Sign convention per the strength

of materials theory. Equations 3.7–3.10 are solved simultaneously to obtain the coefficients in terms of the nodal variables as . . . . . . 3.11 . . . . . . 3.12 . . . . . . 3.13 ………………………………3.14 Substituting Equations 3.11–3.14 into Equation 4.17 and collecting the coefficients of the nodal variables results in the expression: 45

. . . . . . 3.15 which is of the form . . . .. . . . 3.16 or, in matrix notation, . . . . . . . . . . 3.17 where N1, N2, N3, and N4 are the interpolation functions that describe the distribution of displacement in terms of nodal values in the nodal displacement vector . For the flexure element, it is convenient to introduce the dimensionless length coordinate . . . . . . . . . . 3.18 so that Equation 3.26 becomes . . . 3.19 Using Equation 2.11 in conjunction with Equation 3.17, the normal stress distribution on a cross section located at axial position x is given by …………………………………………………….3.20 Since the normal stress varies linearly on a cross section, the maximum and minimum values on any cross section occur at the outer surfaces of the element, where distance y from the neutral surface is 46

largest. As is customary, we take the maximum stress to be the largest tensile (positive) value and the minimum to be the largest compressive (negative) value. Hence, we rewrite Equation 3.30 as . . . . . . . . . . . . . . 3.21 . . .3.22 Observing that Equation 3.21 indicates a linear variation of normal stress along the length of the element and since, once the displacement solution is obtained, the nodal values are known constants, we need calculate only the stress values at the cross sections corresponding to the nodes; that is, at x = 0 and x = L. The stress values at the nodal sections are given by . . . . . . . . . . . 3.23 . . . . . . . . . .. 3.24 3.2 FLEXURE ELEMENT STIFFNESS MATRIX The total strain energy is expressed as . … . . .. . . .. . .. . . .. . . . . . . . .. . . . . . . .. . . . 3.25 where V is total volume of the element. Substituting for the stress and strain per Equations 3.5 and 3.6, into equation 3.25 . . . . . .. . . . . . .. . . . . . .. . . ..3.26 47

which can be written as: . . . .. . .. . .. . . . … .. . . .. .3.27 Again recognizing the area integral as the moment of inertia Iz about the centroidal axis perpendicular to the plane of bending, we have . . .. . . .. . ….. .. . .. . .. .. . . .. . . . . 3.28 Equation 3.38 represents the strain energy of bending for any constant cross-section beam that obeys the assumptions of elementary beam theory. For the strain energy of the finite element being developed, we substitute the discretized displacement relation of Equation 3.27 to obtain . . .. .3.29 as the approximation to the strain energy. Applying the first theorem of Castigliano to the strain energy function with respect to nodal displacement v1 gives the transverse force at node 1 as ..3.30 while application of the theorem with respect to the rotational displacement gives the moment as 48

.3.31 For node 2, the results are ..3.32 ..3.33 Equations 3.29–3.32 algebraically relate the four nodal displacement values to the four applied nodal forces (here we use force

in the general sense to include applied moments) and are of the form . . . . . . . . . . . 3.34 where kmn, m, n = 1, 4 are the coefficients of the element stiffness matrix. By

comparison of Equations 3.40–3.43 with the algebraic equations represented by matrix Eqution 4.44, it is seen that . . . . . . . . 3.35 and the element stiffness matrix is symmetric, as expected for a linearly elastic element. Prior to computing the stiffness coefficients, it is convenient to convert the integration to the dimensionless length variable = x/L

by noting . . . . . . . . . . . . 3.36 49

. . … ..

. . . . . 3.37 so the integrations of Equation 3.45 become ..3.38

The stiffness coefficients are then evaluated as follows: . . 3.39 Continuing the direct integration gives the remaining stiffness coefficients as . . . . . 3.40 50

The complete stiffness matrix for the flexure element is then written as . . . . . . . . 3.41 3.3 STIFFNESS MATRIX FOR RIGID JOINTED PLANE TRUSS MEMBER Due to fixity of the joints, this class of element is modeled as a flexure element with axial loading as shown on fig.3.4. It is seen that in addition to the nodal transverse deflections and rotations, there are displacements in the nodes. This means that, the total degrees of freedom for each element are six with each node having three global degrees of freedom, two displacements in global axis and one rotation. Figure 3. 3: Nodal displacements of rigid jointed plane truss member This being the case, we can simply add the spartial element stiffness matrix to the flexure element stiffness matrix to obtain the 6 × 6 element stiffness matrix for a rigid jointed plane truss element as follows: 51

3.42 which is seen to be simply 3.43 and is a non-coupled superposition of axial and bending stiffnesses. For plane truss structures, orientation of the element in the global coordinate system must be considered. Fig 3.5a depicts an element oriented at an arbitrary angle from the X axis of a global reference frame and shows the element nodal displacements. Before proceeding, note that it is convenient here to reorder the element stiffness matrix given by Equation 3.43 so that the element displacement vector in the element reference frame is given as . . . . . . 3.44 52

Figure 3.4: (a) Nodal displacements in the element coordinate system. (b) Nodal

displacements in the global coordinate system. And the element stiffness matrix becomes . . 3.45 Using Fig 3.4 the element displacements are written in terms of the global displacements as Equations 3.46 can be written in matrix form as 53

. . 3.47 where [R] is the transformation matrix that relates element displacements to global displacements. Therefore, the 6 × 6 element stiffness matrix in the global system is given by . . . . . . . . . . . . 3.48 Carrying out matrix multiplication gives: a3 a4 a5 -a3 -a4 a5 a4 a6 a7 -a4 -a6 a7 K = a5 a7 a1 -a5 -a7 a2 . . . . . . 3.49 -a3 -a4 -a5 a3 a4 -a5 -a4 -a6 -a7 a4 a6 -a7 a5 a7 a2 -a5 -a7 a1 Where, a1= 4EI/L a2 = 2EI/L a3 = (AE/L)c2 + (12EI/L3)s2 a4 = (AE/L)cs - (12EI/L3)cs . . . . . . 3.50 a5 = -(6EI/L2)s a6 = (AE/L)s2 + (12EI/L3)c2 a7 = (6EI/L2)c and, A=cross sectional area of element in m2, E is modulus of elasticity in kn/m2, L is the length of element in metre(m) c = s = 54

3.4 EQUIVALENT NODAL LOAD VECTOR OF RIGID JOINTED PLANE TRUSS

SUBJECTED TO OUT OF JOINT TRANVERSE LOADING In the derivation of bending stress of a flexure element loads were restricted to be applied at the nodes. But in some real cases loads may not only act at the nodes since we can have point loads at any section of the element or distributed loads in the case of dead loads. In this section, we derive work equivalent nodal load vectors that will actually give the same impact of the applied load at the nodes. Having in mind that the rigid jointed plane truss elements are modeled as a flexure element, the following derivation holds: 1. A point load on the rigid jointed plane truss bar. This is modeled as a flexure element with fixed ends. The truss element is divided into two elements with a node at the point of application of the force as shown below. Figure 3.6: Flexure element with node at the point of application of a point load along the element Recalling stiffness matrix of a flexure element as: . . . . . . 3.51 Considering element 1, stiffness matrix for element 1 is as follows: 12 6x -12 6x K1= EI/x3 6x 4x2 -6x 2x2 . . . . . . . . 3.52 -12 -6x 12 -6x 6x 2x2 -6x 4x2 Considering element 2, stiffness matrix for element 2 is as follows: 55

12 6(L-x) -12 6(L-x) 6(L-x) 4(L-x)2 -6(L-x) 2(L-x)2 . . . . . . 3.53 K2= EI/(L-x)3 -12 -6(L-x) 12 -6(L-x) 6(L-x) 2(L-x)2 -6(L-x) 4(L-x)2 Assembling the global stiffness matrix, the resultant components of the global stiffness matrix are. K11 = K11(1) = 12EI/x3 K12 = K12(1) = 6EIx/x3 K13 = K13(1) = -12EI/x3 K14 = K14(1) = 6EIx/x3 K15 = 0 K16 = 0 K21 = K21(1) = 6EIx/x3 K22 = K22(1) = 6EIx/x3 K23 = K23(1) = -6EIx/x3 K24 = K24(1) = 2EIx2/x3 K25 = K25(1) = 0 K26 = K26(1) = 0 . . . . . . 3.54 K12 = K12(1) = 6EIx/x3 K31 = K31(1) = -12EI/x3 K32 = K32(1) = -6EIx/x3 K33 = K33(1) + K33(2) = 12EI/x3 + 12EI/(L-x)3 K34 = K34(1) + K34(2) = -6EIx/x3 + 12EI(L-x)/(L-x)3 K35 = K35(2) = -12EI/(L-x)3 K36 = K36(2) = 6EI(L-x)/(L-x)3 K41 = K41(1) = 6EIx/x3 K42 = K42(1) = 2EIx2/x3 K43 = K43(1) + K43(2) = -6EIx/x3 + 6EI(L-x)/(L-x)3 K44 = K44(1) + K44(2) = 4EIx2/x3 + 4EI(L-x)2/(L-x)3 56

K45 = K45(2) = -6EI(L-x)/(L-x)3 K46 = K46(2) = 2EI(L-x)2/(L-x)3 K51 = 0 K52 = 0 K53 = K53(2) = -12EI/(L-x)3 K54 = K54(2) = -6EI(L-x)/(L-x)3 K55= K55(2) = 12EI/(L-x)3 . . . . . . 3.54 K56 = K33(2) = -6EI(L-x)/(L-x)3 K61 = 0 K62 = 0 K63 = K63(2) = 6EI(L-x)/(L-x)3 K64 = K64(2) = 2EI(L-x)2/(L-x)3 K65 = K33(2) = -6EI(L-x)/(L-x)3 K66 = K66(2) = 4EI(L-x)2/(L-x)3 Therefore the global stiffness matrix is as follows: K= K11 K12 K13 K14 K15 K16 K21 K22 K23 K24 K25 K26 K31 K32 K33 K34 K35 K36 . . . . . . 3.55 K41 K42 K43 K44 K45 K46 K51 K52 K53 K54 K55 K56 K61 K62 K63 K64 K65 K66 From the general form, KU = F . . . . . . . . . . . 3.56 v1

Ө1 Where, U = v2

Ө2 . . . . . . . . . . 3.57 v3

Ө3 57

F1 M1 F = F2 . . . . . . . . . . . 3.58 M2 F3 M3 K11 K12 K13 K14 K15 K16 v1 F1

K21 K22 K23 K24 K25 K26 Ө1 M1 K31 K32 K33 K34 K35 K36 v2 F2 . . . 3.59

K41 K42 K43 K44 K45 K46 Ө2 = M2 K51 K52 K53 K54 K55 K56 v3 F3

K61 K62 K63 K64 K65 K66 Ө3 M3 Consider a fixed ended truss member with the following support conditions Applying boundary conditions:

v1= v3 = Ө1 = Ө3 = 0 . . . . . . . . . . . 3.60 Therefore, K33 K34 v2 = F2. . . . . . . 3.61

K43 K44 Ө2 M2 But, F2 = -P and M2 = 0; Substituting the expressions for K33, K34, K43 and K44 in the above equation we obtain: 12EI/x3 + 12EI/(L-x)3 -6EIx/x3 + 12EI(L-x)/(L-x)3 v2 = -P . . . 3.62

-6EIx/x3 + 6EI(L-x)/(L-x)3 4EIx2/x3 + 4EI(L-x)2/(L-x)3 Ө2 0 At x = L/2, V2 = -PL3/ (192EI) . . . . . . . . . . 3.63

Ө2 = 0 Substituting the nodal displacement values into the constraints equation gives: F1 = P/2 M1 = PL/8 M2 = 0 . . . . . . . . . . . . . 3.64 F2 = -P M3 = -PL/8 F3 = P/2 58

Therefore, the equivalent force vector, for a rigid truss member subjected to point load at the centre is as follows: F1 -P/2 F = M1 = -PL/8 . . . . . . . . 3.65 F3 -P/2 M3 PL/8 2. Uniformly distributed load on a fixed ended plane truss member. The restriction that loads be applied only at element nodes for the flexure element must be dealt with if a distributed load is present. The usual approach is to replace the distributed load with nodal forces and moments such that the mechanical work done by the nodal load system is equivalent to that done by the distributed load. Referring to Figure 3.7, the mechanical work performed by the distributed load can be expressed as: Figure 3.7: work equivalent nodal forces and moment for a uniformly distributed load . . . . . . . . . 3.66 The objective here is to determine the equivalent nodal loads so that the work expressed in Equation 3.66 is the same as: . . . 3.67 59

where F1q , F2q are the equivalent forces at nodes 1 and 2, respectively, and M1q and M2q are the equivalent nodal moments. Substituting the discretized displacement function given by Equation 3.16, the work integral becomes: . . . 3.68 Comparison of Equations 3.67 and 3.68 shows that . . . . . . . . . 3.69 Hence, the nodal force vector representing a distributed load on the basis of work equivalence is given by Equations 3.69. For example, for a uniform load q(x ) = q = constant, and also substituting the interpolation functions N1 = 1 – 3x2 /L2 +2x3 /L3 N2 = x – 2x2 /L + x3 /L2 . . . . . . 3.70 N3 = 3x2 /L2 – 2x3 /L3 N4 = x3/L2 – x2 /L and integrating them within specified limits yields: 60

F1q qL/2 F = M1q = -qL2/12 . . . . . . 3.71 F2q qL/2 M2q qL2/12 Which is the nodal equivalent loads on the element. 3.5 BENDING STRESSES IN RIGID JOINTED PLANE TRUSS MEMBER For plane truss structures, orientation of the element in the global coordinate system must be considered. Fig 3.4a and b depicts an element oriented at an arbitrary angle from the X axis of a global reference frame and shows the element nodal displacements in both local and global coordinates. This orientation of the element was not considered in deriving bending stresses of equations 3.22 and 3.23, so re-arranging the equations considering the orientation of the element in global reference frame yields as follows: From eq. 3.23, . . . . 3.72

σx (x=0) = ymaxE[ 6v2/L2 – 6v1/L2 - 4Ө1/L - 2Ө2/L] . . . . . . . . 3.73 re-arranging in an ascending order gives:

σx (x=0) = ymaxE[ 6v1/L2 - 4Ө1/L + 6v2/L2 - 2Ө2/L] . . . . . . . . . 3.74

substituting eq. 3.46 for v1, Ө1, v2, Ө2 in eq. 3.74 gives: σx (x=0) = ymaxE[ 6sU1/L2 – 6cU2/L2 - 4U3/L – 6s U4/L2 + 6c U5/L2 + 2 U6/L] . . . . 3.75 in matrix form gives: σx (x=0) = ymaxE[ 6s/L2 -6c/L2 -4/L -6s /L2 6c /L2 2 /L] U1 U2 U3 . . . . 3.76 U4 U5 U6 Similarly, eq. 3.24 is re-arranged as follows: 61

σx (x=L) = ymaxE[ -6s/L2 6c/L2 2/L 6s /L2 -6c /L2 4 /L] U1 U2 U3 . . . . 3.77 U4 U5 U6 3.6 AXIAL STRESSES IN RIGID JOINTED PLANE TRUSS MEMBER From, σx = Eε . . . . . . . . . . 3.78 where, ε = (u2 – u1)/L . . . . . . . . 3.79 σx = E(u2 – u1)/L . . . . . . . . . . 3.80 substituting, expressions for u1, u2 in eq. 3.46 Into eq. 3.80 Gives: σx = E[ -cU1 – sU2 +cU4 +sU5] . . . . . . . . 3.81 converting the above equation into matrix form gives σx = E[ -c -s 0 c s 0 ] U1 U2 U3. . . . . . . . 3.82 U4 U5 U6 3.7 MODAL ANALYSIS (NATURAL FREQUENCY) 3.7.1 FORMATION OF LUMP MASS MATRIX By taking the flexural effects of the members into consideration and by using the lump mass method, the mass influence coefficient for axial effects of rigid jointed truss element is found out. Combining the mass matrix for flexural effects with the matrix for axial effects we obtain the lump mass matrix for a uniform element of a plane truss with rigid joint in reference to the modal coordinates. [ 34 ] Lump mass matrix is given the following equation: 62

F1 = �AL/2 1 0 0 0 0 0 U1 F2 0 1 0 0 0 0 U2 F3 0 0 1 0 0 0 U3 . . . . .3.83 F4 0 0 0 1 0 0 U4 F5 0 0 0 0 1 0 U5 F6 0 0 0 0 0 1 U6 or in condensed notation, {F} = [Me] {U} . . . . . . . . . . . 3.84 In which [M] is the lump mass matrix for the element of a rigidly jointed plane truss. 3.7.2 TRANSFORMATION FROM LOCAL TO GLOBAL CO-ORDINATE SYSTEM Repeating the procedure of transformation as applied to stiffness matrix , for the lump mass matrix we obtain in the similar manner {F} = [M] {U} ……………………………………. 3.85 In which, {M} =[R]T[Me][R] …………………………… 3.86 3.7.3 CALCULATION OF EIGEN VALUE AND EIGEN VECTOR The structure is not excited externally in free vibration mode that is no force or support motion acts on it. So, under condition of free motion, dynamic analysis can be carried out and the important properties like natural frequencies mode shapes corresponding to the natural frequency can be obtained. 3.7.4 NATURAL FREQUENCIES Since free vibration mode is considered, the structure is not under influence of any external force. Hence, the force vector in stiffness equation or flexible equations is taken as zero. By taking the above condition into consideration, the stiffness equation can be represented as: ……………………………………… 3.87 The solution of the above equation for undamped structure is in the form, 63

Ui = ai + sin(�t-a) i = 1,2,3…….n …………… 3.88 When represented as vector, {U} = {a}sin(�t-a) …………………………………………. 3.89 Where ai is the amplitude of motion of nth coordinate and is the and n is the number of degree of freedom. Substituting equation 3.89 in equation 3.87, we get -�2[M]{a}sin(�t-a) + [K]{a}sin(�t-a) = 0 …………………… 3.90 Or [[K] - �2[M]] {a} = {0} ……………………………………. 3.91 The mathematical problem for the formulation of the above equation is called eigen problem. As the right hand side of the equation is equal to zero. It can be considered as a set of n number of homogeneous linear equations with n unknown displacements ai and �2 as the unknown parameter, amplitude ‗a‘ cannot be zero. Its solution is non-trivial. So, [[K] - �2[M]] = 0 ………………………………. 3.92 The polynomial equation of degree n in �2 obtained is the characteristic equation and the values of � are the natural frequency of the structure. The values that satisfy equation 3.92can be substituted in equation 3.91 to obtain the amplitudes in terms of arbitrary constant. 3.8 ORTHOGONALITY OF PRINCIPAL MODE The principal modes of vibration of systems with multiple degrees of freedom share a fundamental mathematical property known as orthogonality. The free-vibration response of a multiple degrees-of-freedom system is described by Equation below: ……………………………. 3.93 Assuming that we have solved for the natural circular frequencies and the modal amplitude vectors via the assumed solution form qi (t ) = Ai sin(�t + �), substitution of a particular frequency �i into Equation 3.93 gives ………………………….. 3.94 64

and for any other frequency �j Multiplying Equation 3.93 by {A( j )}T and Equation 3.94by {A(i )}T gives . . . . . 3.95 …………….. 3.96 Subtracting Equation 3.96 from Equation 3.95, we have ………………… 3.97 In arriving at the result represented by Equation 3.97, we utilize the fact from matrix algebra that [A]T [B][C] = [C]T [B][A] , where [A], [B], [C] are any three matrices for which the triple product is defined. As the two circular frequencies in Equation 10.104 are distinct, we conclude that ………………………….. 3.98 Equation 3.98 is the mathematical statement of orthogonality of the principal modes of vibration. For a system exhibiting P degrees of freedom, we define the modal matrix as a P × P matrix in which the columns are the amplitude vectors for each natural mode of vibration; that is, ……………………… 3.99 and consider the matrix triple product [S] = [A]T [M][A] . Per the orthogonality condition, Equation 3.99, each off-diagonal term of the matrix represented by the triple product is zero; hence, the matrix [S] = [A]T [M][A] is a diagonal matrix. The diagonal (nonzero) terms of the matrix have magnitude …………………. 3.100 As each modal amplitude vector is known only within a constant multiple the modal amplitude vectors can be manipulated such that the diagonal terms described by Equation 3.100 can be made to assume any desired numerical value. In particular, if the value is selected as unity, so that …………………… 3.101 where [I ] is the P × P identity matrix. 65

Amplitude vectors are then normalized as follows: The corresponding diagonal term of the modal matrix is ……………….. 3.102 If we redefine the terms of the modal amplitude vector so that …………………. 3.103 the matrix described by Equation 3.103 is indeed the identity matrix. Having established the orthogonality concept and normalized the modal matrix, we return to the general problem described by Equation 3.93, in which the force vector is no longer assumed to be zero. For reasons that will become apparent, we introduce the change of variables …………………………. 3.104 Where, { p} is the column matrix of generalized displacements, which are linear combinations of the actual nodal displacements {q}, and [A] is the normalized modal matrix. Equation 3.93 then becomes ……………………. 3.105 Premultiplying by [A]T , we obtain …………………. 3.106 Now we must examine the stiffness effects as represented by [A]T [K ][A] . Given that [K] is a symmetric matrix, the triple product [A]T [K ][A] is also a symmetric matrix. Following the previous development of orthogonality of the principal modes, the triple product [A]T [K ][A] is also easily shown to be a diagonal matrix. The values of the diagonal terms are found by multiplying Equation 3.106 by A(i )T to obtain 66

………………… 3.107 If the modal amplitude vectors have been normalized as described previously, Equation 3.107 is …………………………. 3.108 hence, the matrix triple product [A]T [K ][A] produces a diagonal matrix having diagonal terms equal to the squares of the natural circular frequencies of the principal modes of vibration; that is, ………………. 3.109 Finally, Equation 10.115 becomes ……………………… 3.110 with matrix [�2] representing the diagonal matrix defined in Equation 3.111. where A is the mode shape(eigen vector). 67

CHAPTER FOUR ANALYSIS AND RESULTS 4.1 ANALYSIS In this section, static and dynamic analyses of rigid jointed and pin jointed plane truss structures are considered. Other forms of loadings other than the conventional loading (load must act at the joint) such as uniformly distributed loads and point loads are also considered. Equations and derivations emphasized in the previous sections using finite element analysis techniques are used in the analysis. Parameters such as stresses in members, joints displacements, and reactions at the supports are sorted in the case of static analysis. While, natural frequencies and mode shapes are sort in the case of dynamic (modal) analysis. Both rigid and pin jointed plane truss structures are analyzed for these parameters and their results are compared. Robot structural analysis software, a commercial structural analysis software is used to cross check the results obtained from the computer program written in this project using MATLAB codes to analyze the plane truss structures. The program is based on the finite element method of structural analysis and is developed using MATLAB. Care was taken during input process as it has a great impact on the results. The input data in this program are: unit area of material, density of material, modulus of elasticity of material, nodal coordinates, prescribed degree of freedom of the system (PrescribedDof), and element nodes connections. Numerical example 1 ( pin jointed plane truss structures) Consider a modified warren truss for Railway Bridge shown in/ figure 4.1. The dimensions, nodal numberings, supports and loadings are as also shown on fig. 4.1. The elements are composed of 50 x 50 x 6 equal angle iron with the foll0wing properties: modulus of elasticity(E)= 205EKN/m2, density(�)=7850Kg/m3, area(A) =0.000569m2 . 68

Figure 4.1 Warren truss for Railway Bridge. The nodal coordinates and elements numbers are as follows: Figure 4.2 Nodal coordinates and element numbering. Element stiffness in local coordinates are calculated using the following equations: Ki=AE/Li. where Ki is the stiffness for each element, Li is the length of each element, A is the area of element and E is the modulus of elasticity for the element. i.e, K1=0.000569*205e+09/3=38881666.667KN/m2. Transforming element stiffnesses in to global coordinate system by utilizing equation 2.45 as follows: Ke=Ke c2 sc - c2 -sc Sc s2 -sc -s2 - c2 -sc c2 sc -sc -s2 sc s2 69

0 0 0 0 K1 = 1.0e+007 3.8881666667 0 -3.88816666667 0 0 0 0 0 -3.8881666667 0 3.8881666667 0 K1 = K2 =K3 . . . . . . K18 1.374674508 1.374674508 -1.374674508 -1.374674508 K19 = 1.0e+007 1.374674508 1.374674508 -1.374674508 -1.374674508 -1.374674508 -1.374674508 1.374674508 1.374674508 -1.374674508 -1.374674508 1.374674508 1.374674508 1.374674508 -1.374674508 -1.374674508 1.374674508 K20 = 1.0e+007 -1.374674508 1.374674508 1.374674508 -1.374674508 -1.374674508 1.374674508 1.374674508 -1.374674508 1.374674508 -1.374674508 -1.374674508 1.374674508 K20 = K21 . . . . K24 1.374674508 1.374674508 -1.374674508 -1.374674508 K25 = 1.0e+007 1.374674508 1.374674508 -1.374674508 -1.374674508 -1.374674508 -1.374674508 1.374674508 1.374674508 -1.374674508 -1.374674508 1.374674508 1.374674508 K25 = K26 . . . . K28 K29 = 1.0e+007 0 0 0 0 0 3.888166667 0 -3.888166667 0 0 0 0 0 -3.888166667 0 3.888166667 70

K29 = K30 . . . . K37 Assembling the element global stiffness matrices into structure global stiffness matrix, K as follows: K= Columns 1 through 8 1.0e+008 * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.5553 0 -0.77763 0 0 0 0 0 0 0.77763 0 0 0 0 0 0 -0.77763 0 2.10513 0. -0.77763 0 0 0 0 0 0 1.327503 0 0 0 0 0 0 -0.77763 0 1.5553 0 0 0 0 0 0 0 0 0.77763 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.77763 0 0 0 0 71

0 0 0 0 -0.27493 0.27493 0 0 0 0 0 0 0.27493 -0.27493 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 -0.27493 -0.27493 0 0 0 0 0 0 -0.27493 -0.27493 0 -0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 9 through 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.105136 0 -0.77763 0 0 0 0 0 0 1.3275 0 0 0 0 0 0 72

-0.77763 0 1.555267 0 -0.77763 0 0 0 0 0 0 0.77763 0 0 0 0 0 0 -0.77763 0 2.10513 0 -0.77763 0 0 0 0 0 0 1.327503 0 0 0 0 0 0 -0.77763 0 1.555267 0 0 0 0 0 0 0 0 0.77763 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.27493 0.27493 0 0 0 0 0 0 0.27493 -0.27493 0 0 0 0 0 0 0 0 0 0 -0.274934 0.274934 0 0 0 -0.77763 0 0 0.274934 -0.274934 0 0 -0.27493 -0.27493 0 0 0 0 0 0 -0.27493 -0.27493 0 -0.77763 -0.77763 0 0 0 0 0 0 0 -0.274934 -0.274934 0 -0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 73

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 17 through 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.77763 0 0 0 0 0 0 -0.27493 0.27493 0 0 0 0 0 0 0.27493 -0.27493 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.105136 0 -0.777633 0 0 0 0 0 0 1.327503 0 0 0 0 0 0 -0.77763 0 1.555267 0 0 0 0 0 0 0 0 0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.32750 0 0 0 0 0 0 0 0 1.3275 74

0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.2749349 0.2749349 0 0 0 0 0 0 0.2749349 -0.2749349 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 -0.2749349 -0.2749349 0 0 0 0 0 0 -0.2749349 -0.2749349 0 -0.77763 0 0 0 0 Columns 25 through 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.27493 -0.27493 0 0 0 0 0 -0.77763 -0.27493 -0.27493 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 -0.27493 0.27493 0 -0.77763 -0.27493 -0.27493 0 0 0.27493 -0.27493 0 0 0 0 75

0 0 0 0 0 0 0 0.77763 0 0 0 0 0 0 -0.27493 0.27493 0 0 0 0 0 0 0.27493 -0.27493 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.555267 0 -0.77763 0 0 0 0 0 0 0.77763 0 0 -0.77763 0 0 0 -0.77763 0 2.10513 0 0 0 0 0 0 0 0 1.32750 1.555267 0 -0.77763 0 0 0 -0.77763 0 0 0.77763 0 0 0 0 0 0 -0.77763 0 2.1051 0 0 0 0 0 0 0 0 1.3275 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 76

0 0 0 0 0 0 0 0 Columns 33 through 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.27493 -0.27493 0 0 0 0 0 -0.77763 -0.27493 -0.27493 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 -0.27493 0.27493 0 0 -0.27493 -0.27493 0 0 0.27493 -0.27493 0 -0.77763 -0.27493 -0.27493 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 77

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.77763 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.555267 0 -0.77763 0 0 0 0 0 0 0.77763 0 0 0 0 0 0 -0.77763 0 2.1051 0 -0.77763 0 0 0 0 0 0 1.3275 0 0 0 0 0 0 -0.77763 0 1.55526 0 -0.777633 0 0 0 0 0 0 0.77763 0 0 0 0 0 0 -0.77763 0 1.3275 0 0 0 0 0 0 0 0 1.3275 From, KU=F … 4.1 where K is the structure global stiffness matrix which is a 40x40 matrix, U is the global displacement matrix which ranges from U1 to U40 and it is a 40x1 column matrix; F is the force vector that contains all the forces applied to the structure and ranges from F1 to F40 and also a 40 x 1column matrix. 1.0e+008 * 0 0 0 - - - 0 0 0 U1 F1 0 0 0 - - - 0 0 0 U2 F2 0 0 1.5553 - - - 0 0 0 U3 F3 - - - - - - - - - - - - - - - - - - - - - = - - - - - - - - - - - - 0 0 0 - - - 0.77763 0 0 U38 F38 0 0 0 - - - 0 1.3275 0 U39 F39 0 0 0 - - - 0 0 1.3275 U40 F40 78

Nodes 1 and 11 are constrained by their support conditions. Therefore, U1 = U2 = U21 = U22 = 0. The above equation is then reduced to 36 x 36 matrix equation. The unknowns Us which are global displacements are then sort by solving the equation using MaTlab as follows: U3 = -17.4890mm U4 = -247.4879mm U5 = -34.9779mm U6 = -522.6793mm U7 = -23.1472mm U8 = -709.2508mm U9 = -11.3164mm U10 = -857.6120mm U11 = 9.2589mm U12 = -879.2891mm U13 = 29.8341mm U14 = -845.2668mm U15 = 38.5786mm U16 = -687.6468mm U17 = 47.3231mm U18 = -497.9890mm U19 = 23.6615mm U20 = -228.9702mm U23 = 191.4746mm U24 = -247.4879mm U25 = 154.4392mm 79

U26 = -527.8231mm U27 = 117.4037mm U28 = -709.2508mm U29 = 61.3362mm U30 = -862.7558mm U31 = 5.2686mm U32 = -879.2891mm U33 = -49.2558mm U34 = -850.4106mm U35 = -103.7802mm U36 = -687.6468mm U37 = -136.1862mm U38 = -503.1328mm U39 = -168.5922mm U40 = -228.9702mm Back substituting the values of Us into eq. 4.1 And then solving for only constrained Fs yields reaction forces as follows: F1 = 1450KN F2 = 700KN F21 = -1750KN F22 = 830KN Computing stress in each element: This is obtained as a function of displacement as follows: Stress, σ = E/Le[-c –s c s]U Element 1: Cx = (x2 – x1) /Le = 3/3 =1 Cy = (y2 – y1) /Le = 0/3 =0 80

Le = √((y2 – y1)2 + (x2 – x1)2) =3 U=[0 0 -0.017489 -0.2474879]1 σ1 = 205e+09/3[-1 0 1 0]*U = 1195081667KN/m2 = -1195.081667Mpa Element 2: Cx = (x2 – x1) /Le = 3/3 =1 Cy = (y2 – y1) /Le = 0/3 =0 Le = √((y2 – y1)2 + (x2 – x1)2) =3 U=[-0.017489 -0.2474879 -0.034977 -0.5226793]1 σ2= 205e+09/3[-1 0 1 0]*U = 1195013333KN/m2 = -1195.013337Mpa element 3 Cx = (x2 – x1) /Le = 3/3 =1 Cy = (y2 – y1) /Le = 0/3 =0 Le = √((y2 – y1)2 + (x2 – x1)2) =3 U=[ -0.034977 -0.5226793 -0.0231472 -0.7092508]1 σ3= 205e+09/3[-1 0 1 0]*U = 808369667KN/m2 = 808.369667Mpa repeating the same process for other elements yield the following stresses: σ4 = 808.435852372579Mpa σ5 = 1405.975395430570 Mpa σ6 = 1405.975395430570 Mpa σ7 = 0.597539543057997 Mpa σ8 = 597.539543057999 Mpa σ9 = -1616.871704745153 Mpa σ10 = -1616.871704745153 Mpa σ11 = -2.530755711775044 Mpa σ12 = -2530.755711775046 Mpa σ13 = -3831.282952548325 Mpa σ14 = -3831.282952548325 Mpa 81

σ15 -3725.834797891031 Mpa σ16 = -3725.834797891030 Mpa σ17 = -2214.411247803168 Mpa σ18 = -2214.411247803163 Mpa σ19 = -1913.786367359019 Mpa σ20 = -2062.912577802581 Mpa σ21 = 1665.242683286424 Mpa σ22 = 671.067946996019 Mpa σ23 = -323.106789294374 Mpa σ24 -1317.281525584774 Mpa σ25 = 1814.368893729979 Mpa σ26 = 820.194157439570 Mpa σ27 = -173.980578850811 Mpa σ28 = -1168.155315141210 Mpa σ29 = 0 Mpa σ30 = -351.49384885 Mpa 7649 σ31 = 0 Mpa σ32 = -351.4 Mpa 93848857641 σ33 = 0 Mpa σ34 = -351.4938488 Mpa 57641 σ35 = 0 Mpa σ36 = -351.49384885 Mpa 7649 σ37 = 0 Mpa 82

DYNAMIC ANALYSIS ( NATURAL FREQUENCY – FREE VIBRATION) From, equation 2.72 Consistent mass matrix Me = �AL/6 2 0 1 0 0 2 0 1 1 0 2 0 0 1 0 2 Therefore, the consistent mass matrix for all the elements in the structure are as follows: M1 = 4.46665 0 2.233325 0 0 4.46665 0 2.233325 2.233325 0 4.46665 0 0 2.233325 0 4.46665 M1 = M2 =M3 . . . . . . M18 6.316797 0 3.1583985 0 M19 = 0 6.316797 0 3.1583985 3.1583985 0 6.316797 0 0 3.1583985 0 6.316797 M19 = M20 =M21 . . . . . . M28 83

4.46665 0 2.233325 0 M29 0 4.46665 0 2.233325 2.233325 0 4.46665 0 0 2.233325 0 4.46665 M29 = M30 =M31 . . . . . . M37 Assembling the element by element mass matrix into the structure global mass matrix, M as follows: M= Columns 1 through 8 10.7834 0 2.2333 0 0 0 0 0 0 10.7834 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 26.0335 0 2.2333 0 0 0 0 2.2333 0 26.0335 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 0 0 0 0 0 2.2333 0 13.3999 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 84

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.1584 0 2.2333 0 3.1584 0 0 0 0 3.1584 0 2.2333 0 3.1584 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 3.1584 0 2.2333 0 0 0 0 0 0 3.1584 0 2.2333 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 9 through 16 0 0 0 0 0 0 0 0 85

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 26.0335 0 2.2333 0 0 0 0 0 0 26.0335 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 26.0335 0 2.2333 0 0 0 0 2.23332 0 26.0335 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 0 0 0 0 0 2.2333 0 13.3999 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.15834 0 0 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 86

3.1584 0 2.2333 0 3.1584 0 0 0 0 3.1584 0 2.2333 0 3.1584 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 3.1584 0 2.2333 0 0 0 0 0 0 3.1584 0 2.2333 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 17 through 24 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 26.0335 0 2.2333 0 0 0 0 0 87

0 26.0335 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 10.7834 0 0 0 0 0 0 2.2333 0 10.7834 0 0 0 0 0 0 0 0 21.5669 0 0 0 0 0 0 0 0 21.5669 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 3.1584 0 2.2333 0 3.1584 0 0 0 0 3.1584 0 2.2333 0 3.1584 0 0 Columns 25 through 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.2333 0 3.1584 0 0 0 0 0 88

0 2.2333 0 3.1584 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 3.1584 0 2.2333 0 3.1584 0 0 0 0 3.15834 0 2.2333 0 3.1584 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 13.3999 0 2.2333 0 0 0 0 0 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 26.0335 0 2.2333 0 0 0 0 2.2333 0 26.0335 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 26.0335 0 0 0 0 0 0 2.2333 0 26.0335 89

0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 33 through 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.2333 0 3.1584 0 0 0 0 0 0 2.2333 0 3.1584 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 3.1584 0 2.2333 0 3.1584 0 90

0 0 0 3.1584 0 2.2333 0 3.1584 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 0 0 3.1584 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 13.3999 0 2.2333 0 0 0 0 0 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 26.0335 0 0 0 0 0 0 2.233325 0 26.0335 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 13.3999 0 2.2333 0 0 0 0 2.2333 0 21.5669 0 0 0 0 0 0 2.233325 0 21.5669 From, K - �2M =0; 91

Substituting the values of K and M into the above equation, providing constraints through the supports and solving the above equation as an eigen value problem using MaTlab yields � which is the modal natural frequencies of free vibration of the structure as follows: Modes Natural frequency(Hetz) 1 10.53 2 27.05 3 49.30 4 53.91 5 81.29 6 94.34 7 110.16 8 123.34 9 157.41 10 158.95 11 189.64 12 189.64 13 197.22 14 218.08 15 245.23 16 261.83 17 300.37 18 305.41 19 373.05 20 374.07 21 377.43 22 377.60 23 379.11 92

24 381.67 25 385.50 26 395.32 27 396.69 28 396.83 29 407.40 30 438.82 31 465.14 32 482.51 33 517.50 34 519.48 35 539.95 36 556.86 Mode shapes using orthogonality principle as discussed in chapter three, each column represent mode shape for each natural frequency. Column 1 to 3 0.0027 -0.0126 0.0359 0.0335 0.0515 0.0220 0.0020 -0.0200 0.0402 0.0463 0.0524 0.0257 0.0014 -0.0271 0.0425 0.0568 0.0312 0.0235 -0.0000 -0.0273 0.0436 0.0589 -0.0000 0.0000 -0.0014 -0.0271 0.0425 -0.0568 -0.0312 -0.0235 -0.0020 -0.0200 0.0402 93

0.0463 -0.0524 -0.0257 -0.0027 -0.0126 0.0359 0.0335 -0.0515 -0.0220 -0.0013 -0.0064 0.0184 0.0156 -0.0344 0.0075 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0122 -0.0250 0.0243 0.0156 0.0339 -0.0074 -0.0099 -0.0195 0.0348 0.0336 0.0522 0.0230 -0.0076 -0.0137 0.0435 0.0462 0.0517 0.0245 -0.0038 -0.0102 0.0516 0.0569 0.0317 0.0245 -0.0000 -0.0066 0.0573 0.0587 -0.0000 0.0000 0.0038 -0.0102 0.0516 0.0569 -0.0317 -0.0245 0.0076 -0.0137 0.0435 0.0462 -0.0517 -0.0245 0.0099 -0.0195 0.0348 0.0336 -0.0522 -0.0230 0.0122 -0.0250 0.0243 0.0156 -0.0339 0.0074 Columns 4 through 10 -0.0143 0.0102 -0.0511 -0.0106 0.0287 0.0055 0.0124 94

0.0586 -0.0458 -0.0142 -0.0237 0.0184 -0.0512 -0.0447 -0.0153 -0.0037 -0.0349 -0.0260 0.0029 -0.0025 -0.0103 0.0195 0.0376 -0.0514 0.0576 0.0588 0.0224 0.0296 -0.0156 -0.0168 -0.0122 -0.0349 -0.0242 -0.0096 -0.0293 -0.0317 0.0553 0.0166 0.0175 -0.0607 0.0262 -0.0023 0.0000 -0.0179 0.0000 0.0000 -0.0288 0.0000 -0.0380 -0.0639 -0.0000 0.0233 -0.0857 -0.0000 -0.0588 -0.0000 0.0156 -0.0168 0.0122 0.0349 -0.0242 0.0096 -0.0293 -0.0317 -0.0553 0.0166 0.0175 0.0607 0.0262 0.0023 0.0153 -0.0037 0.0349 0.0260 0.0029 0.0025 -0.0103 0.0195 -0.0376 -0.0514 0.0576 -0.0588 0.0224 -0.0296 0.0143 0.0102 0.0511 0.0106 0.0287 -0.0055 0.0124 0.0586 0.0458 -0.0142 -0.0237 -0.0184 -0.0512 0.0447 0.0074 0.0055 0.0282 0.0062 0.0169 -0.0051 0.0102 0.0487 0.0585 0.0420 -0.0504 0.0573 0.0968 -0.0890 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0200 0.0073 -0.0415 -0.0199 0.0184 0.0492 0.0668 0.0461 -0.0518 0.0365 -0.0403 -0.0446 0.0635 0.0577 -0.0070 -0.0085 -0.0468 -0.0318 0.0190 0.0346 0.0523 0.0618 -0.0514 -0.0167 -0.0289 0.0241 -0.0749 -0.0652 0.0066 -0.0235 -0.0436 -0.0366 0.0136 0.0050 0.0145 0.0184 0.0336 -0.0443 0.0470 0.0465 0.0154 0.0209 0.0033 -0.0124 -0.0238 -0.0210 -0.0176 0.0028 -0.0027 -0.0335 0.0623 0.0187 0.0208 -0.0772 0.0397 -0.0019 0.0000 0.0005 0.0000 0.0000 -0.0433 0.0000 -0.0171 -0.0606 -0.0000 0.0195 -0.0706 -0.0000 -0.0397 -0.0000 95

-0.0033 -0.0124 0.0238 0.0210 -0.0176 -0.0028 -0.0027 -0.0335 -0.0623 0.0187 0.0208 0.0772 0.0397 0.0019 -0.0066 -0.0235 0.0436 0.0366 0.0136 -0.0050 0.0145 0.0184 -0.0336 -0.0443 0.0470 -0.0465 0.0154 -0.0209 0.0070 -0.0085 0.0468 0.0318 0.0190 -0.0346 0.0523 0.0618 0.0514 -0.0167 -0.0289 -0.0241 -0.0749 0.0652 0.0200 0.0073 0.0415 0.0199 0.0184 -0.0492 0.0668 0.0461 0.0518 0.0365 -0.0403 0.0446 0.0635 -0.0577 Columns 11 through 17 0.0078 0.0196 -0.0506 -0.0573 -0.0467 -0.0700 0.0104 0.0308 -0.0305 -0.0238 0.0095 -0.0079 0.0013 -0.0094 0.0101 0.0521 -0.0487 -0.0861 -0.0034 -0.0271 -0.0145 -0.0834 0.0657 0.0574 -0.0060 0.0308 0.0309 -0.0041 0.0057 0.0559 -0.0173 -0.0507 0.0349 0.0458 -0.0056 0.0578 -0.0209 -0.0234 -0.0003 0.0035 0.0001 0.0106 -0.0000 0.0788 -0.0209 -0.0000 0.0000 0.0930 0.0068 -0.1133 -0.0000 -0.0000 0.0371 -0.0310 0.0000 0.0000 -0.0057 0.0559 -0.0173 0.0507 -0.0349 0.0458 -0.0056 0.0578 0.0209 0.0234 -0.0003 0.0035 -0.0001 -0.0106 -0.0101 0.0521 -0.0487 0.0861 0.0034 -0.0271 -0.0145 -0.0834 -0.0657 -0.0574 -0.0060 0.0308 -0.0309 0.0041 -0.0078 0.0196 -0.0506 0.0573 0.0467 -0.0700 0.0104 0.0308 0.0305 0.0238 0.0095 -0.0079 -0.0013 0.0094 -0.0058 0.0120 -0.0389 0.0416 0.0513 -0.0660 0.0329 -0.0283 -0.0431 0.0007 -0.0318 -0.0053 0.0252 0.0004 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 96

0.0045 -0.0387 -0.0246 0.0657 -0.0554 0.0671 0.0633 -0.0157 0.0227 0.0016 -0.0111 0.0001 -0.0016 0.0017 0.0101 -0.0523 -0.0016 0.0727 0.0071 0.0560 -0.0280 0.0553 -0.0554 -0.0493 0.0257 -0.0451 0.0215 0.0478 0.0090 -0.0358 0.0265 0.0287 0.0684 -0.0060 -0.0830 -0.0461 0.0349 0.0287 -0.0028 0.0055 0.0016 0.0006 0.0067 -0.0395 0.0764 0.0195 0.0698 -0.0453 0.0115 0.1052 -0.0388 -0.0487 0.0019 0.0218 0.0061 -0.0568 0.0000 -0.0235 0.0782 -0.0000 -0.0000 -0.0420 0.0914 -0.0622 -0.0000 -0.0000 0.0119 -0.0058 0.0000 -0.0000 -0.0067 -0.0395 0.0764 -0.0195 -0.0698 -0.0453 0.0115 0.1052 0.0388 0.0487 0.0019 0.0218 -0.0061 0.0568 -0.0090 -0.0358 0.0265 -0.0287 -0.0684 -0.0060 -0.0830 -0.0461 -0.0349 -0.0287 -0.0028 0.0055 -0.0016 -0.0006 -0.0101 -0.0523 -0.0016 -0.0727 -0.0071 0.0560 -0.0280 0.0553 0.0554 0.0493 0.0257 -0.0451 -0.0215 -0.0478 -0.0045 -0.0387 -0.0246 -0.0657 0.0554 0.0671 0.0633 -0.0157 -0.0227 -0.0016 -0.0111 0.0001 0.0016 -0.0017 Columns 18 through 24 -0.0482 0.0029 0.0010 0.0028 -0.0005 -0.0023 -0.0003 -0.0067 0.0572 0.0322 -0.0001 -0.0667 0.0779 0.0650 0.0429 -0.0280 0.0205 -0.0161 0.0576 0.0313 -0.0192 0.0422 0.0646 0.0859 0.0836 -0.0437 -0.0165 -0.1046 0.0818 -0.0051 -0.0007 -0.0042 0.0003 0.0040 0.0002 0.0027 -0.0387 0.0698 0.0779 0.0492 0.0152 0.0461 0.0000 0.0000 -0.0392 -0.0000 -0.0476 -0.0000 0.0366 -0.0525 -0.1061 0.0000 0.0032 -0.0000 0.1124 0.0000 97

-0.0818 0.0051 -0.0007 0.0042 0.0003 -0.0040 0.0002 0.0027 -0.0387 -0.0698 0.0779 -0.0492 0.0152 -0.0461 -0.0429 0.0280 0.0205 0.0161 0.0576 -0.0313 -0.0192 0.0422 0.0646 -0.0859 0.0836 0.0437 -0.0165 0.1046 0.0482 -0.0029 0.0010 -0.0028 -0.0005 0.0023 -0.0003 -0.0067 0.0572 -0.0322 -0.0001 0.0667 0.0779 -0.0650 0.0618 -0.0368 -0.0003 -0.0410 -0.0683 0.0366 -0.0046 -0.0176 -0.0138 0.0221 -0.0192 -0.0156 -0.0130 -0.0103 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0628 0.0002 -0.0044 -0.0036 -0.0108 0.0041 0.0010 0.0051 -0.0043 0.0074 0.0137 0.0052 -0.0098 -0.0237 0.0174 0.0045 0.0126 -0.0079 0.0107 0.0267 0.0175 0.0330 -0.0656 -0.0203 0.0224 0.0800 -0.1011 -0.0967 -0.0439 -0.0006 0.0051 0.0025 0.0114 -0.0027 -0.0007 -0.0087 -0.0483 -0.0835 -0.0783 0.0353 -0.0110 0.0496 -0.0490 -0.0219 -0.0114 0.0315 -0.0029 -0.0500 -0.0171 -0.0138 0.0420 -0.0725 -0.0691 -0.0814 0.0129 -0.0792 0.0000 0.0000 -0.0057 -0.0000 -0.0116 -0.0000 0.0004 0.0107 0.0906 -0.0000 -0.0403 0.0000 -0.0884 -0.0000 0.0490 0.0219 -0.0114 -0.0315 -0.0029 0.0500 -0.0171 -0.0138 0.0420 0.0725 -0.0691 0.0814 0.0129 0.0792 0.0439 0.0006 0.0051 -0.0025 0.0114 0.0027 -0.0007 -0.0087 -0.0483 0.0835 -0.0783 -0.0353 -0.0110 -0.0496 -0.0174 -0.0045 0.0126 0.0079 0.0107 -0.0267 0.0175 0.0330 -0.0656 0.0203 0.0224 -0.0800 -0.1011 0.0967 -0.0628 -0.0002 -0.0044 0.0036 -0.0108 -0.0041 0.0010 98

0.0051 -0.0043 -0.0074 0.0137 -0.0052 -0.0098 0.0237 Columns 25 through 31 0.0003 0.0004 0.0044 0.0017 -0.0269 0.0320 0.0613 0.0324 -0.0395 -0.0019 0.0059 0.0287 0.0146 -0.0051 -0.0055 -0.1043 0.0012 -0.0025 -0.0480 0.0663 -0.0253 -0.0828 0.0039 0.0319 0.0256 -0.0262 0.0110 0.0125 -0.0002 -0.0002 -0.0052 -0.0004 0.0439 -0.0523 -0.0381 0.0647 0.0310 0.0020 -0.0027 -0.0078 -0.0049 -0.0108 0.0000 0.1047 -0.0000 -0.0028 -0.0000 -0.0000 0.0845 -0.1244 0.0000 0.0032 -0.0000 0.0346 -0.0141 0.0000 0.0002 -0.0002 0.0052 -0.0004 -0.0439 0.0523 -0.0381 0.0647 -0.0310 0.0020 0.0027 -0.0078 -0.0049 0.0108 0.0055 -0.1043 -0.0012 -0.0025 0.0480 -0.0663 -0.0253 -0.0828 -0.0039 0.0319 -0.0256 -0.0262 0.0110 -0.0125 -0.0003 0.0004 -0.0044 0.0017 0.0269 -0.0320 0.0613 0.0324 0.0395 -0.0019 -0.0059 0.0287 0.0146 0.0051 0.0130 0.1011 0.0299 -0.0242 -0.0733 0.1068 -0.0685 0.0147 -0.0170 0.1354 -0.1372 0.0243 -0.0092 0.0082 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0004 -0.0373 -0.0050 0.0032 0.0482 0.0461 -0.0546 -0.0194 -0.0038 -0.1133 -0.1170 -0.0300 0.0060 0.0124 0.0121 0.0102 0.0109 0.0027 -0.0628 -0.0662 0.0893 -0.0488 0.0490 0.0461 0.0366 -0.0278 -0.0104 0.0035 0.0001 0.0363 0.0027 -0.0039 -0.0291 -0.0273 -0.0221 99

0.0388 -0.0010 -0.0268 -0.0223 0.0182 -0.0154 -0.0111 -0.0022 -0.0028 -0.0126 -0.0009 0.0912 0.1003 -0.0541 -0.1181 -0.0369 0.0072 0.0102 0.0112 0.0034 0.0110 -0.0000 -0.0360 0.0000 0.0041 -0.0000 0.0000 0.0688 0.0624 -0.0000 -0.0038 0.0000 -0.0274 0.0189 0.0000 0.0022 -0.0028 0.0126 -0.0009 -0.0912 -0.1003 -0.0541 -0.1181 0.0369 0.0072 -0.0102 0.0112 0.0034 -0.0110 -0.0001 0.0363 -0.0027 -0.0039 0.0291 0.0273 -0.0221 0.0388 0.0010 -0.0268 0.0223 0.0182 -0.0154 0.0111 -0.0121 0.0102 -0.0109 0.0027 0.0628 0.0662 0.0893 -0.0488 -0.0490 0.0461 -0.0366 -0.0278 -0.0104 -0.0035 0.0004 -0.0373 0.0050 0.0032 -0.0482 -0.0461 -0.0546 -0.0194 0.0038 -0.1133 0.1170 -0.0300 0.0060 -0.0124 Columns 32 through 36 -0.0542 -0.0676 0.0578 -0.0352 -0.0455 -0.0039 0.0017 0.0016 0.0028 -0.0019 0.0349 0.1218 -0.0609 0.0937 0.0709 -0.0047 -0.0009 0.0018 0.0004 0.0015 0.0335 -0.0417 0.0356 -0.0570 -0.0731 -0.0093 -0.0048 -0.0062 0.0011 -0.0008 -0.1129 0.0000 -0.0000 0.1157 0.0876 0.0000 -0.0029 0.0057 0.0000 0.0000 0.0335 0.0417 -0.0356 -0.0570 -0.0731 0.0093 -0.0048 -0.0062 -0.0011 0.0008 0.0349 -0.1218 0.0609 0.0937 0.0709 0.0047 -0.0009 0.0018 -0.0004 -0.0015 100

-0.0542 0.0676 -0.0578 -0.0352 -0.0455 0.0039 0.0017 0.0016 -0.0028 0.0019 0.0912 -0.0757 0.0367 0.0362 0.0265 -0.0062 0.0045 -0.0042 -0.0016 -0.0024 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0447 -0.0259 -0.0518 -0.0193 0.0268 -0.0075 -0.0080 0.0063 -0.0040 -0.0038 0.1058 0.0898 0.1057 0.0679 -0.0551 0.0019 0.0002 -0.0021 -0.0002 0.0022 -0.0174 -0.0403 -0.0797 -0.0470 0.0639 0.0112 0.0026 -0.0013 -0.0022 -0.0017 -0.0650 0.0548 0.0639 0.1074 -0.0858 0.0039 0.0004 0.0051 -0.0001 0.0008 0.0554 -0.0000 -0.0000 -0.0575 0.0780 0.0000 0.0085 -0.0047 -0.0000 0.0000 -0.0650 -0.0548 -0.0639 0.1074 -0.0858 -0.0039 0.0004 0.0051 0.0001 -0.0008 -0.0174 0.0403 0.0797 -0.0470 0.0639 -0.0112 0.0026 -0.0013 0.0022 0.0017 0.1058 -0.0898 -0.1057 0.0679 -0.0551 -0.0019 0.0002 -0.0021 0.0002 -0.0022 -0.0447 0.0259 0.0518 -0.0193 0.0268 0.0075 -0.0080 0.0063 0.0040 0.0038 101

NUMERICAL EXAMPLE 2: RIGID JOINTED PLANE TRUSS STRUCTURES Considering figure 4.1 as a plane truss structure whose joints have replaced with rigid joints and the loadings and support conditions remains the same. SOLUTION 2 The free body diagram of the structure is as follows: k1 = K1 = K2 =K3 . . . . . . K18 1.0e+007 * 3.8882 -3.8882 0 0 0 0 -3.8882 3.8882 0 0 0 0 0 0 0.0766 -0.0766 0.1149 0.1149 0 0 -0.0766 0.0766 -0.1149 -0.1149 0 0 0.1149 -0.1149 0.2299 0.1149 0 0 0.1149 -0.1149 0.1149 0.2299 K19 = K20 . . . K27 1.0e+007 * 2.7493 -2.7493 0 0 0 0 -2.7493 2.7493 0 0 0 0 0 0 0.0271 -0.0271 0.0575 0.0575 0 0 -0.0271 0.0271 -0.0575 -0.0575 0 0 0.0575 -0.0575 0.1625 0.0813 0 0 0.0575 -0.0575 0.0813 0.1625 K28 = K29 …. K36 102

1.0e+007 * 3.8882 -3.8882 0 0 0 0 -3.8882 3.8882 0 0 0 0 0 0 0.0766 -0.0766 0.1149 0.1149 0 0 -0.0766 0.0766 -0.1149 -0.1149 0 0 0.1149 -0.1149 0.2299 0.1149 0 0 0.1149 -0.1149 0.1149 0.2299 Assembling the element global stiffness matrices into structure global stiffness matrix, K as follows: K= 1.0e+008 * Columns 1 through 7 0.5276 -0.3888 0 0 0 0 0 -0.3888 0.7853 -0.3888 0 0 0 0 0 -0.3888 1.0629 -0.3888 0 0 0 0 0 -0.3888 0.7853 -0.3888 0 0 0 0 0 -0.3888 1.0629 -0.3888 0 0 0 0 0 -0.3888 0.7776 -0.3888 0 0 0 0 0 -0.3888 1.0553 0 0 0 0 0 0 -0.3888 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 -0.0077 -0.1388 0 0 0 0 0 0 -0.0077 0 0 0 0 103

0 0 -0.1388 -0.0077 -0.1388 0 0 0 0 0 0 -0.0077 0 0 0 0 0 0 -0.1388 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 104

0 0 -0.0196 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 -0.0196 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 -0.0115 -0.0041 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 -0.0041 -0.0115 -0.0041 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 -0.0041 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 8 through 14 0 0 0 0 -0.1388 0 0 0 0 0 0 -0.0077 0 0 0 0 0 0 -0.1388 -0.0077 -0.1388 0 0 0 0 0 0 -0.0077 0 0 0 0 0 0 -0.1388 0 0 0 0 0 0 0 -0.3888 0 0 0 0 0 0 0.7776 -0.3888 0 0 0 0 0 -0.3888 1.0553 -0.3888 0 0 0 0 105

0 -0.3888 0.7776 -0.3888 0 0 0 0 0 -0.3888 0.5276 0 0 0 0 0 0 0 0.6741 -0.3888 0 0 0 0 0 -0.3888 0.7853 -0.3888 0 0 0 0 0 -0.3888 1.0629 0 0 0 0 0 0 -0.3888 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 0 -0.1388 0 0 0 0 0 0 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 0 0 0 0 106

0 0 0 0 0 0 0 0 -0.1361 0 0.1361 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0.0041 0.0115 0.0041 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0.0196 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0.0196 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 -0.0041 0 0 0 Columns 15 through 21 0 0 0 0 0 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0077 -0.1388 0 0 0 0 0 107

0 0 0 0 0 0 0 0 -0.1388 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 0 0 0 0 0 0 0 -0.1361 0 0 0 0 0 0 0 -0.3888 0 0 0 0 0 0 0.7853 -0.3888 0 0 0 0 0 -0.3888 1.0553 -0.3888 0 0 0 0 0 -0.3888 0.7776 -0.3888 0 0 0 0 0 -0.3888 1.0553 -0.3888 0 0 0 0 0 -0.3888 0.7776 -0.3888 0 0 0 0 0 -0.3888 0.6665 0 0 0 0 0 0 0 0.1465 0 0 0 0 0 0 -0.0077 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 0 0 0 0 0 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 0 108

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0156 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0115 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0081 0 Columns 22 through 28 109

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0077 0 0 0 0 0 0 0.4041 -0.0077 0 0 0 0 0 -0.0077 0.6818 -0.0077 0 0 0 0 0 -0.0077 0.4041 -0.0077 0 0 0 0 0 -0.0077 0.6818 -0.0077 0 0 0 0 0 -0.0077 0.0153 -0.0077 0 0 0 0 0 -0.0077 0.2930 -0.0077 0 0 0 0 0 -0.0077 0.0153 0 0 0 0 0 0 -0.0077 110

0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.3888 -0.1388 0 0 0 0 0 0 -0.3888 0 0 0 0 0 0 -0.1388 -0.3888 -0.1388 0 0 0 0 0 0 -0.3888 0 0 0 0 0 0 -0.1388 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 111

0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 29 through 35 0 0 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0.1361 0 0 0 0 0 0 0 -0.1388 0 0 0 0 0 0 -0.3888 0 0 0 0 0 0 -0.1388 -0.3888 -0.1388 0 0 0 0 0 0 -0.3888 0 0 0 0 0 0 -0.1388 -0.3888 112

0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0077 0 0 0 0 0 0 0.2930 -0.0077 0 0 0 0 0 -0.0077 0.0153 -0.0077 0 0 0 0 0 -0.0077 0.1465 0 0 0 0 0 0 0 0.6741 -0.0077 0 0 0 0 0 -0.0077 0.4041 -0.0077 0 0 0 0 0 -0.0077 0.6818 -0.0077 0 0 0 0 0 -0.0077 0.4041 0 0 0 0 0 0 -0.0077 0 0 0 0 0 0 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 0 -0.1388 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 0 0 0 0 0 0.0115 -0.0156 0 0 0 0 0 0 0 0.0115 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 113

0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 -0.0041 0 0 0 0 Columns 36 through 42 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 -0.1361 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1361 0 0 0 0 0 0 0 0.0041 0.0115 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0156 0.0115 114

0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 0 -0.1388 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.1388 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0077 0 0 0 0 0 0 0.2930 -0.0077 0 0 0 0 0 -0.0077 0.0153 -0.0077 0 0 0 0 0 -0.0077 0.2930 -0.0077 0 0 0 0 0 -0.0077 0.0153 -0.0077 0 0 0 0 0 -0.0077 0.2853 0 0 0 0 0 0 0 0.0392 0.0115 0 0 0 0 0 0.0115 0.0690 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 -0.0041 0 0 0 0 0 0 0 0 0 115

0 0 0 0 0.0041 0 0 0 0 0 0 0 0.0081 0.0115 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 0 0 0 0 0 0.0115 0 -0.0115 0 0 0 0 0 0.0115 -0.0115 0 0 Columns 43 through 49 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0196 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 -0.0196 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0.0041 0.0115 0.0041 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0.0041 0 0.0041 0 0 0 0 0 0 0 0 0 116

0 0 0 0 0.0041 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 -0.0115 0 0.0115 0 0 0 0 0 -0.0115 0 0.0115 0 0 0 0 0 -0.0115 0 0.0115 0 0 0 0 0 -0.0115 0 0.0115 0 0 0 0 0 -0.0115 0 0.0115 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0.1015 0.0115 0 0 0 0 0 0.0115 0.0690 0.0115 0 0 0 0 0 0.0115 0.1015 0.0115 0 0 0 0 0 0.0115 0.0460 0.0115 0 0 117

0 0 0 0.0115 0.0785 0.0115 0 0 0 0 0 0.0115 0.0460 0.0115 0 0 0 0 0 0.0115 0.0785 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0.0081 0.0115 0.0081 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0.0081 0 0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0081 0 0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0081 Columns 50 through 56 0 0 -0.0041 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 -0.0041 -0.0115 -0.0041 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 -0.0041 -0.0115 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0.0196 0 0 0 0 0 0 0 0.0115 0 0 0 118

0 0 0 0 0.0196 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 -0.0115 -0.0156 0 0 0 0 0 0 0 0.0115 0.0115 0 0 0 0 0 -0.0115 0 0.0115 0 0 0 0 0 -0.0115 0 0.0115 0 0 0 0 0 -0.0115 0 0.0115 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 -0.0115 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 0 0.0115 0 0 0 0 119

0 0 0.0081 0.0115 0.0081 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0.0081 0.0115 0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0.0460 0.0115 0 0 0 0 0 0.0115 0.0392 0 0 0 0 0 0 0 0.0785 0.0115 0 0 0 0 0 0.0115 0.0690 0.0115 0 0 0 0 0 0.0115 0.1015 0.0115 0 0 0 0 0 0.0115 0.0690 0.0115 0 0 0 0 0 0.0115 0.0785 0 0 0 0 0 0 0.0115 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 Columns 57 through 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 -0.0041 0 -0.0041 120

0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0041 0 0 0 0 0 0 0 -0.0041 0 0.0041 0 0 0 0 0 0 0 -0.0041 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0115 0 0 0 0 0.0115 0 0 -0.0115 0 0.0115 0 121

0 - 0.0115 0 0.0115 0 0 -0.0115 -0.0115 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 0 0 0.0081 0 0.0081 0 0 0 0 0 0 0 0.0081 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0115 0 0 0 0.0460 0.0115 0 0 0.0115 0.0785 0.0115 0 0 0.0115 0.0460 0.0115 0 0 0.0115 0.0555 From, equation 2.39, re-arranged as: KU=F …………. .. 4.1 where K is the structure global stiffness matrix which is a 60x60 matrix, U is the global displacement matrix which ranges from U1 to U60 and it is a 60x 1 column matrix; F is the force vector that contains all the forces applied to the structure and ranges from F1 to F60 and also a 60x1 column matrix. 122

1.0e+007 * 0.528 -0.388 0 - - - 0 0 0 U1 F1 -0.389 0.785 -0.389 - - - 0 0 0 U2 F2 0 -0.389 1.0629 - - - 0 0 0 U3 F3 - - - - - - - - - - - - - - - - - - - - - = - - - - - - - - - - - - 0 0 0 - - 0.079 0.0115 0 U58 F58 0 0 0 - - - 0.0115 0.046 0.0115 U59 F59 0 0 0 - - - 0 0.0115 0.0555 U60 F60 Nodes 1 and 11 are constrained by their support conditions. Therefore, U1 = U2 = U21 = U22 = 0. The above equation is then reduced to 36 x 36 matrix equation. The unknowns Us which are global displacements are then sort by solving the equation using MaTlab as follows: Displacements ans = Nodal no. Displacements(mm) 1 0 2 -17.6528 3 -34.5937 4 -23.0798 5 -11.1990 6 9.1795 7 29.5282 8 38.3798 9 46.8052 123

10 23.7829 11 0 12 188.1022 13 152.0888 14 115.5010 15 60.4113 16 5.1525 17 -48.6066 18 -102.1369 19 -134.1649 20 -165.5578 21 0 22 -244.5623 23 -510.6602 24 -696.1420 25 -838.7415 26 -862.7453 27 -826.5600 28 -674.8394 29 -486.3495 30 -226.4994 31 0 32 -244.0003 33 -514.8480 34 -695.6633 35 -842.9540 36 -862.2697 124

37 -830.7725 38 -674.3600 39 -490.5351 40 -225.8891 41 -75.9824 42 -81.5706 43 -71.3280 44 -52.7979 45 -26.5478 46 1.8791 47 29.6675 48 54.5128 49 70.3972 50 76.9059 51 68.8070 52 -79.6811 53 -72.5563 54 -51.9986 55 -26.9141 56 1.8144 57 30.1628 58 53.5860 59 71.7345 60 75.0509 Back substituting the values of Us into eq. 4.1 and then solving for only constrained Fs yields reaction forces as follows: 1.0e+003 * 125

F1 = 1459.5Kn F11 = -1759.5Kn F21 = 770Kn F31 = 830Kn Computing stress in each element: This is obtained as a function of displacement as follows: Stress, σ = E/Le[-c –s c s]U Element 1: Cx = (x2 – x1) /Le = 3/3 =1 Cy = (y2 – y1) /Le = 0/3 =0 Le = √((y2 – y1)2 + (x2 – x1)2) =3 U=[0 0 -0.0176258 -0.2445623]1 σ1 = 205e+09/3[-1 0 1 0]*U = -1206.3Mpa Element 2: Cx = (x2 – x1) /Le = 3/3 =1 Cy = (y2 – y1) /Le = 0/3 =0 Le = √((y2 – y1)2 + (x2 – x1)2) =3 U=[-0.0176528 -0.2445623 -0.0345937 -0.5106602]1 σ2= 205e+09/3[-1 0 1 0]*U = 1157.6Mpa element 3 Cx = (x2 – x1) /Le = 3/3 =1 Cy = (y2 – y1) /Le = 0/3 =0 Le = √((y2 – y1)2 + (x2 – x1)2) =3 U=[ -0.0345937 -0.5106602 -0.0230798 -0.6961420]1 σ3= 205e+09/3[-1 0 1 0]*U = -786.8Mpa repeating the same process for other elements yielded the following stresses: 126

axialstresses = Element No. Stresses(10^ 3Mpa) 1 1.2063 2 1.1576 3 -0.7868 4 -0.8119 5 -1.3925 6 -1.3905 7 -0.6049 8 -0.5757 9 1.5732 10 1.6252 11 2.4609 12 2.5002 13 3.7645 14 3.7760 15 3.6735 16 3.6579 17 2.1886 18 2.1452 19 1.9099 20 2.0613 21 -1.5021 22 -0.5596 23 0.3872 24 1.3348 127

25 -1.6433 26 -0.7016 27 0.2452 28 1.1927 29 -0.0384 30 0.2862 31 -0.0327 32 0.2879 33 -0.0325 34 0.2879 35 -0.0328 36 0.2860 37 -0.0417 Bending Stresses from equations 3.23 and 3.24 as shown below: which are stresses at first and second element jointe respectively and local coordinate system of the element. Converting the above equations to universal coordinate system, the following parameters are utilized; 128

σx (x=0) = ymaxE[ 6s/L2 -6c/L2 -4/L -6s /L2 6c /L2 2 /L] U1 U2 U3 U4 U5 U6 Similarly, equation 3.24 is re-arranged as follows: σx (x=L) = ymaxE[ -6s/L2 6c/L2 2/L 6s /L2 -6c /L2 4 /L] U1 U2 U3 U4 U5 U6 For element 1, substituting the following parameters into the above equations: ymax = 0.0205m E = 205e9N/mm2 L= 3m c = 1 s = 0 U1 = 0 U2 = 0 U3 = -0.0759824m U4 = -0.0176528m U5 = -0.2445623 U5 = -0.0815706 And multiplying the matrices gives the following bending stresses at the nodes: σ1 (x=0) = -15.2375N/mm2 σ1 (x=L) = 487.9613N/mm2 Repeating the above process yields bending stresses for the other elements within the structure as follows: 129

Bendingstresses(Kn/mm2) = Element σ1 (x=L) σ1 (x=0) 2 488.2874 -117.3092 3 267.9059 -23.9764 4 178.0499 -102.8376 5 -86.7702 -3.4016 6 -173.9684 -70.1221 7 -411.5606 36.4989 8 -419.8619 -19.0995 9 -549.0194 99.8550 10 -396.4210 33.5625 11 515.6240 -129.0280 12. 245.7098 -11.9403 13. 196.6983 -116.1679 14 -101.7761 11.1213 15 -162.5837 -85.8509 16. -419.3337 53.4494 17 -415.7322 -37.0662 18. -549.7652 119.9382 19 284.8148 38.2213 20. -263.8010 -45.9321 21 291.4347 -27.7108 22 242.8663 -8.6059 23 112.9332 -6.4457 24. -68.2735 -18.0135 25 -245.3825 31.5243 130

26 -211.2927 10.0899 27 -91.4649 5.9018 28 83.7952 15.4053 29 243.7148 98.5563 30 132.7860 83.3723 31 108.7718 51.0305 32 64.5246 24.5588 33 45.5991 -4.1490 34 28.2792 -33.2226 35 -23.3029 -59.3056 36 -51.2703 -92.1636 37 -176.5447 -105.5304 Natural Frequency Analysis Consistent mass matrix is given by the following equations: M1 = M1 = M2 =M3 . . . . . . M18 1.0e+008 * 4.4666 0 0 2.2333 0 0 0 4.9771 2.1057 0 1.7229 -1.2443 0 2.1057 1.1486 0 1.2443 -0.8614 2.2333 0 0 4.4666 0 0 0 1.7229 1.2443 0 4.9771 -2.1057 0 -1.2443 -0.8614 0 -2.1057 1.1486 M19 = M20 . . M27 6.3168 0 0 3.1584 0 0 0 7.0387 4.2114 0 2.4365 -2.4886 0 4.2114 3.2486 0 2.4886 -2.4365 3.1584 0 0 6.3168 0 0 0 2.4365 2.4886 0 7.0387 -4.2114 131

0 -2.4886 -2.4365 0 -4.2114 3.2486 M28 = M29 …. M36 4.4666 0 0 2.2333 0 0 0 4.9771 2.1057 0 1.7229 -1.2443 0 2.1057 1.1486 0 1.2443 -0.8614 2.2333 0 0 4.4666 0 0 0 1.7229 1.2443 0 4.9771 -2.1057 0 -1.2443 -0.8614 0 -2.1057 1.1486 Assembling the element by element mass matrix into the structure global mass matrix, M as follows: M= 1.0e+008 * Columns 1 through 10 9.2494 0 0 0 0 0 0 0 0 0 0 10.5923 0 0 0 0 0 0 0 0 0 0 20.1578 0 0 0 0 0 0 0 0 0 0 10.5923 0 0 0 0 0 0 0 0 0 0 20.1578 0 0 0 0 0 0 0 0 0 0 10.5923 0 0 0 0 0 0 0 0 0 0 20.1578 0 0 0 0 0 0 0 0 0 0 10.5923 0 0 0 0 0 0 0 0 0 0 20.1578 0 0 0 0 0 0 0 0 0 0 10.5923 0 0 0 0 0 0 0 0 0 0 -3.6849 0 3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3.6849 0 3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 132

0 0 0 0 -3.6849 0 3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3.6849 0 3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3.6849 0 1.5341 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 0 0 0 0 0 0 0 0 0 2.2333 -0.526 -2.1057 -0.5265 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -0.5265 -2.1057 -0.5265 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -0.5265 -2.1057 -0.5265 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -0.5265 -2.1057 -0.5265 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -0.5265 -2.1057 -1.7597 1.7229 0 0 0 0 0 0 0 0 0 -2.4886 1.7229 0 0 0 0 0 0 0 133

0 0 -6.0079 1.7229 0 0 0 0 0 0 0 0 0 -2.4886 1.7229 0 0 0 0 0 0 0 0 0 -6.0079 1.7229 0 0 0 0 0 0 0 0 0 -2.4886 1.7229 0 0 0 0 0 0 0 0 0 -6.0079 1.7229 0 0 0 0 0 0 0 0 0 -2.4886 1.7229 0 0 0 0 0 0 0 0 0 -6.0079 1.7229 0 0 0 0 0 0 0 0 0 -2.4886 0 0 0 0 0 0 0 0 0 0 1.7229 0.8614 1.7229 0 0 0 0 0 0 0 0 0 0.8614 0 0 0 0 0 0 0 0 0 1.7229 0.8614 1.7229 0 0 0 0 0 0 0 0 0 0.8614 0 0 0 0 0 0 0 0 0 1.7229 0.8614 1.7229 0 0 0 0 0 0 0 0 0 0.8614 0 0 0 0 0 0 0 0 0 1.7229 0.8614 1.7229 0 0 0 0 0 0 0 0 0 0.8614 0 0 0 0 0 0 0 0 0 1.7229 0.8614 Columns 11 through 20 0 -3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.6849 0 -3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.6849 0 -3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 134

0 0 0 0 0 3.6849 0 -3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.6849 0 -3.6849 0 0 0 0 0 0 0 0 0 0 9.7598 0 0 0 0 0 0 0 0 3.6849 0 22.2888 0 0 0 0 0 0 0 0 0 0 13.9104 0 0 0 0 0 0 0 0 0 0 27.2659 0 0 0 0 0 0 0 0 0 0 13.9104 0 0 0 0 0 0 0 0 0 0 27.2659 0 0 0 0 0 0 0 0 0 0 13.9104 0 0 0 0 0 0 0 0 0 0 27.2659 0 0 0 0 0 0 0 0 0 0 13.9104 0 3.6849 0 0 0 0 0 0 0 0 22.7993 0 0.5265 0 0 0 0 0 0 0 0 0 -2.2333 0 0 0 0 0 0 0 0 0 0.5265 -2.2333 0.5265 0 0 0 0 0 0 0 0 0 -2.2333 0 0 0 0 0 0 0 0 0 0.5265 -2.2333 0.5265 0 0 0 0 0 0 0 0 0 -2.2333 0 0 0 0 0 0 0 0 0 0.5265 -2.2333 0.5265 0 0 0 0 0 0 0 0 0 -2.2333 0 0 0 0 0 0 0 0 0 0.5265 -2.2333 0.5265 2.1057 0 0 0 0 0 0 0 0 -2.2333 -1.5341 0 0 0 0 0 0 0 0 0.5265 0 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 135

0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 0 0 0 0 0 0 0 0 2.2333 0 2.1057 -0.5265 0 0 0 0 0 0 0 2.2333 0 0 1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.7229 0 1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.7229 0 1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.7229 0 1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.7229 0 1.7229 1.7229 0 0 0 0 0 0 0 0 0 -3.0040 0 0 0 0 0 0 0 0 -1.7229 0 0 1.7229 0 0 0 0 0 0 0 0 0 -1.2443 1.7229 0 0 0 0 0 0 0 0 0 -1.2443 1.7229 0 0 0 0 0 0 0 0 0 -1.2443 1.7229 0 0 0 0 0 0 0 0 0 -1.2443 1.7229 0 0 0 0 0 0 0 0 0 -1.2443 1.7229 0 0 0 0 0 0 0 0 0 -1.2443 1.7229 0 0 0 0 0 0 0 0 0 -1.2443 1.7229 1.7229 0 0 0 0 0 0 0 0 -1.2443 136

Columns 21 through 30 1.5341 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 0 0 0 0 0 0 0 0 0 2.1057 0.5265 -2.2333 0.5265 0 0 0 0 0 0 0 0 0 -2.2333 0 0 0 0 0 0 0 0 0 0.5265 -2.2333 0.5265 0 0 0 0 0 0 0 0 0 -2.2333 0 0 0 0 0 0 0 0 0 0.5265 -2.2333 0.5265 0 0 0 0 0 0 0 0 0 -2.2333 0 0 0 0 0 0 0 0 0 0.5265 -2.2333 0.5265 0 0 0 0 0 0 0 0 0 -2.2333 0 0 0 0 0 0 0 0 0 0.5265 -2.2333 5.9313 0 0 0 0 0 0 0 0 0 0 10.0819 0 0 0 0 0 0 0 0 0 0 19.6473 0 0 0 0 0 0 0 0 0 0 10.0819 0 0 0 0 0 0 137

0 0 0 0 19.6473 0 0 0 0 0 0 0 0 0 0 10.0819 0 0 0 0 0 0 0 0 0 0 19.6473 0 0 0 0 0 0 0 0 0 0 10.0819 0 0 0 0 0 0 0 0 0 0 19.6473 0 0 0 0 0 0 0 0 0 0 10.0819 0 0 0 0 0 0 0 0 0 0 3.6849 0 -3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.6849 0 -3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.6849 0 -3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.6849 0 -3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.6849 0 3.0040 0 0 0 0 0 0 0 0 0 -0.8614 1.2443 0 0 0 0 0 0 0 0 0 -0.8614 1.2443 0 0 0 0 0 0 0 0 0 -0.8614 1.2443 0 0 0 0 0 0 0 0 0 -0.8614 1.2443 0 0 0 0 0 0 0 0 0 -0.8614 1.2443 0 0 0 0 0 0 0 0 0 -0.8614 1.2443 0 0 0 0 0 0 0 0 0 -0.8614 1.2443 0 0 0 0 0 0 0 0 0 -0.8614 1.2443 0 0 0 0 0 0 0 0 0 -0.8614 1.2443 0 0 0 0 0 0 0 0 0 -0.8614 138

-1.7229 0 1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.7229 0 1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.7229 0 1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.7229 0 1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.7229 0 Columns 31 through 40 0 -0.5265 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -0.5265 -2.105 -0.5265 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -0.5265 -2.1057 -0.5265 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -0.5265 -2.1057 -0.5265 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -0.5265 -2.1057 -0.5265 2.2333 0 0 0 0 0 0 0 0 -2.1057 -1.5341 0 0 0 0 0 0 0 0 -0.5265 0 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 139

0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0 0 0 0 0 0 0 0 2.1057 0 2.2333 0.5265 0 0 0 0 0 0 0 2.1057 0 0 3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3.6849 0 3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3.6849 0 3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3.6849 0 3.6849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3.6849 0 3.6849 0 0 0 0 0 0 0 0 0 0 9.2494 0 0 0 0 0 0 0 0 -3.6849 0 19.4812 0 0 0 0 0 0 0 0 0 0 10.5923 0 0 0 0 0 0 0 0 0 0 23.9479 0 0 0 0 0 0 0 0 0 0 10.5923 0 0 0 0 0 0 0 0 0 0 23.9479 0 0 0 0 0 0 0 0 0 0 10.5923 0 0 0 0 0 0 0 0 0 0 23.9479 0 0 0 0 0 0 0 0 0 0 10.5923 0 -3.6849 0 0 0 0 0 0 0 0 22.7993 0 1.7229 0 0 0 0 0 0 0 0 0 1.7229 0 0 0 0 0 0 0 0 140

0 1.7229 1.7229 1.7229 0 0 0 0 0 0 0 0 0 1.7229 0 0 0 0 0 0 0 0 0 1.7229 1.7229 1.7229 0 0 0 0 0 0 0 0 0 1.7229 0 0 0 0 0 0 0 0 0 1.7229 1.7229 1.7229 0 0 0 0 0 0 0 0 0 1.7229 0 0 0 0 0 0 0 0 0 1.7229 1.7229 1.7229 0 0 0 0 0 0 0 0 0 1.7229 -1.7597 0 0 0 0 0 0 0 0 1.7229 0 -3.5194 0 0 0 0 0 0 0 0 0 -0.8614 0 0 0 0 0 0 0 0 0 0 -0.8614 -3.5194 0 0 0 0 0 0 0 0 0 -0.8614 0 0 0 0 0 0 0 0 0 0 -0.8614 -3.5194 0 0 0 0 0 0 0 0 0 -0.8614 0 0 0 0 0 0 0 0 0 0 -0.8614 -3.5194 0 0 0 0 0 0 0 0 0 -0.8614 0 0 1.7229 0 0 0 0 0 0 0 -0.8614 -4.7636 Columns 41 through 50 -1.7597 0 0 0 0 0 0 0 0 0 1.7229 -2.4886 0 0 0 0 0 0 0 0 0 1.7229 -6.0079 0 0 0 0 0 0 0 0 0 1.7229 -2.4886 0 0 0 0 0 0 0 0 0 1.7229 -6.0079 0 0 0 0 0 0 0 0 0 1.7229 -2.4886 0 0 0 0 141

0 0 0 0 0 1.7229 -6.0079 0 0 0 0 0 0 0 0 0 1.7229 -2.4886 0 0 0 0 0 0 0 0 0 1.7229 -6.0079 0 0 0 0 0 0 0 0 0 1.7229 -2.4886 0 0 0 0 0 0 0 0 0 1.7229 1.7229 0 -1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.7229 0 -1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.7229 0 -1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.7229 0 -1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.7229 0 3.0040 -0.8614 0 0 0 0 0 0 0 0 0 1.2443 -0.8614 0 0 0 0 0 0 0 0 0 1.2443 -0.8614 0 0 0 0 0 0 0 0 0 1.2443 -0.8614 0 0 0 0 0 0 0 0 0 1.2443 -0.8614 0 0 0 0 0 0 0 0 0 1.2443 -0.8614 0 0 0 0 0 0 0 0 0 1.2443 -0.8614 0 0 0 0 0 0 0 0 0 1.2443 -0.8614 0 0 0 0 0 0 0 0 0 1.2443 -0.8614 0 0 0 0 0 0 0 0 0 1.2443 0 0 0 0 0 0 0 0 0 0 1.7229 1.7229 1.7229 0 0 0 0 0 0 0 0 0 1.7229 0 0 0 0 0 0 0 142

0 0 1.7229 1.7229 1.7229 0 0 0 0 0 0 0 0 0 1.7229 0 0 0 0 0 0 0 0 0 1.7229 1.7229 1.7229 0 0 0 0 0 0 0 0 0 1.7229 0 0 0 0 0 0 0 0 0 1.7229 1.7229 1.7229 0 0 0 0 0 0 0 0 0 1.7229 0 0 0 0 0 0 0 0 0 1.7229 1.7229 12.0158 -2.1057 0 0 0 0 0 0 0 0 -2.1057 11.1028 -2.1057 0 0 0 0 0 0 0 0 -2.1057 25.1802 -2.1057 0 0 0 0 0 0 0 0 -2.1057 11.1028 -2.1057 0 0 0 0 0 0 0 0 -2.1057 25.1802 -2.1057 0 0 0 0 0 0 0 0 -2.1057 11.1028 -2.1057 0 0 0 0 0 0 0 0 -2.1057 25.1802 -2.1057 0 0 0 0 0 0 0 0 -2.1057 11.1028 -2.1057 0 0 0 0 0 0 0 0 -2.1057 25.1802 -2.1057 0 0 0 0 0 0 0 0 -2.1057 11.1028 0 0 0 0 0 0 0 0 0 -2.1057 -4.2114 -2.1057 -4.2114 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -4.2114 -2.1057 -4.2114 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -4.2114 -2.1057 -4.2114 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -4.2114 -2.1057 -4.2114 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -4.2114 -2.1057 143

Columns 51 through 60 0 1.7229 0 0 0 0 0 0 0 0 0 0.8614 0 0 0 0 0 0 0 0 0 1.7229 0.8614 1.7229 0 0 0 0 0 0 0 0 0 0.8614 0 0 0 0 0 0 0 0 0 1.7229 0.8614 1.7229 0 0 0 0 0 0 0 0 0 0.8614 0 0 0 0 0 0 0 0 0 1.7229 0.8614 1.7229 0 0 0 0 0 0 0 0 0 0.8614 0 0 0 0 0 0 0 0 0 1.7229 0.8614 1.7229 0 0 0 0 0 0 0 0 0 0.8614 -3.0040 0 0 0 0 0 0 0 0 1.7229 0 0 0 0 0 0 0 0 0 0 0 1.7229 -1.2443 0 0 0 0 0 0 0 0 0 1.7229 -1.2443 0 0 0 0 0 0 0 0 0 1.7229 -1.2443 0 0 0 0 0 0 0 0 0 1.7229 -1.2443 0 0 0 0 0 0 0 0 0 1.7229 -1.2443 0 0 0 0 0 0 0 0 0 1.7229 -1.2443 0 0 0 0 0 0 0 0 0 1.7229 -1.2443 0 -1.7229 0 0 0 0 0 0 0 1.7229 -1.2443 0 -1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.7229 0 -1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 144

0 0 0 1.7229 0 -1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.7229 0 -1.7229 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.7229 0 -1.7229 -0.8614 0 0 0 0 0 0 0 0 0 -1.7597 0 0 0 0 0 0 0 0 1.7229 0 -3.5194 -0.8614 0 0 0 0 0 0 0 0 0 0 -0.8614 0 0 0 0 0 0 0 0 0 -3.5194 -0.8614 0 0 0 0 0 0 0 0 0 0 -0.8614 0 0 0 0 0 0 0 0 0 -3.5194 -0.8614 0 0 0 0 0 0 0 0 0 0 -0.8614 0 0 0 0 0 0 0 0 0 -3.5194 -0.8614 0 0 0 0 0 0 0 0 0 0 -0.8614 1.7229 0 0 0 0 0 0 0 0 -4.7636 0 -4.2114 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -4.2114 -2.1057 -4.2114 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -4.2114 -2.1057 -4.2114 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -4.2114 -2.1057 -4.2114 0 0 0 0 0 0 0 0 0 -2.1057 0 0 0 0 0 0 0 0 0 -4.2114 -2.1057 -4.2114 -2.1057 0 0 0 0 0 0 0 0 -2.1057 8.1873 0 0 0 0 0 0 0 0 -4.2114 145

0 12.6230 -2.1057 0 0 0 0 0 0 0 0 -2.1057 7.2743 -2.1057 0 0 0 0 0 0 0 0 -2.1057 13.7715 -2.1057 0 0 0 0 0 0 0 0 -2.1057 7.2743 -2.1057 0 0 0 0 0 0 0 0 -2.1057 13.7715 -2.1057 0 0 0 0 0 0 0 0 -2.1057 7.2743 -2.1057 0 0 0 0 0 0 0 0 -2.1057 13.7715 -2.1057 0 0 0 0 0 0 0 0 -2.1057 7.2743 -2.1057 -4.2114 0 0 0 0 0 0 0 -2.1057 8.7944 Calculating for natural frequency due to free vibration: Substituting the values of K and M into the equation 2.86, providing constraints through the supports and solving the above equation as an eigen value problem using MATLAB yields � which is the modal natural frequencies of free vibration of the structure as follows: Modes natural Frequency 1 14.1997 2 35.2392 3 62.4290 4 66.1365 5 74.8869 6 80.9680 7 83.2866 8 86.7196 9 88.9510 10 92.4987 11 97.7490 12 98.9537 13 101.6521 146

14 104.3928 15 107.2557 16 111.3204 17 118.0483 18 127.8771 19 130.3102 20 140.9521 21 144.5286 22 160.2040 23 162.8571 24 172.0447 25 185.4750 26 195.3959 27 202.2813 28 209.6131 29 214.5448 30 227.8287 31 236.0370 32 265.1959 33 274.1926 34 301.8753 35 306.2569 36 324.8325 37 345.6500 38 379.0416 39 385.4925 40 387.2919 147

41 400.8484 42 412.1143 43 423.5731 44 428.4148 45 438.1661 46 456.0091 47 458.6797 48 464.0935 49 472.5337 50 477.4471 51 491.4296 52 502.1988 53 504.2689 54 552.8848 55 594.0167 56 620.5068 Mode shapes Columns 1 through 6 0.0020 0.0078 -0.0232 -0.0071 0.0013 0.0069 0.0040 0.0156 -0.0446 -0.0139 0.0042 0.0132 0.0033 0.0253 -0.0487 -0.0165 0.0099 0.0133 0.0026 0.0347 -0.0504 -0.0177 0.0150 0.0126 0.0007 0.0357 -0.0505 0.0001 0.0111 0.0091 -0.0012 0.0359 -0.0486 0.0178 0.0064 0.0058 -0.0022 0.0270 -0.0464 0.0192 0.0045 0.0060 -0.0032 0.0175 -0.0422 0.0191 0.0024 0.0067 -0.0016 0.0088 -0.0217 0.0096 0.0014 0.0026 148

0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0159 0.0330 -0.0326 -0.0209 0.0147 0.0134 -0.0128 0.0258 -0.0449 -0.0057 0.0098 0.0131 -0.0097 0.0181 -0.0549 0.0105 0.0063 0.0122 -0.0046 0.0134 -0.0644 0.0065 0.0080 0.0146 0.0005 0.0086 -0.0702 0.0018 0.0097 0.0168 0.0056 0.0132 -0.0620 -0.0026 0.0121 0.0116 0.0107 0.0176 -0.0500 -0.0066 0.0135 0.0069 0.0139 0.0252 -0.0391 0.0091 0.0069 0.0039 0.0169 0.0322 -0.0259 0.0246 -0.0007 0.0038 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0205 -0.0444 0.0141 0.0550 0.0056 -0.0148 0.0441 -0.0681 -0.0222 0.0670 -0.0221 -0.0077 0.0614 -0.0707 -0.0280 0.0295 -0.0174 -0.0004 0.0754 -0.0448 -0.0230 -0.0348 0.0032 0.0092 0.0786 -0.0063 0.0038 -0.0706 0.0188 0.0004 0.0759 0.0349 0.0360 -0.0374 0.0119 -0.0159 0.0625 0.0626 0.0387 0.0137 -0.0021 -0.0247 0.0451 0.0636 0.0298 0.0638 -0.0223 -0.0076 0.0215 0.0440 -0.0058 0.0579 -0.0159 -0.0046 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0204 -0.0438 0.0138 0.0521 0.0020 -0.0137 0.0441 -0.0689 -0.0232 0.0698 -0.0245 -0.0078 0.0612 -0.0700 -0.0263 0.0279 -0.0174 0.0003 0.0754 -0.0453 -0.0243 -0.0369 0.0035 0.0110 0.0784 -0.0063 0.0042 -0.0678 0.0174 0.0014 0.0760 0.0353 0.0370 -0.0394 0.0128 -0.0159 149

0.0623 0.0619 0.0373 0.0130 -0.0023 -0.0207 0.0452 0.0645 0.0303 0.0672 -0.0238 -0.0056 0.0214 0.0435 -0.0057 0.0551 -0.0151 -0.0023 0.0063 -0.0182 0.0255 0.0611 0.2008 0.0084 0.0071 -0.0113 -0.0042 0.0038 -0.0685 -0.0167 0.0067 -0.0073 -0.0026 0.0173 0.0415 0.0180 0.0051 0.0029 -0.0014 -0.0177 -0.0150 -0.0181 0.0031 0.0079 0.0113 -0.0232 0.0175 0.0384 0.0002 0.0127 0.0087 -0.0040 0.0001 -0.0382 -0.0022 0.0112 0.0218 0.0076 -0.0043 0.0486 -0.0049 0.0052 -0.0017 0.0169 -0.0021 -0.0594 -0.0063 -0.0017 0.0069 0.0187 -0.0190 0.1081 -0.0072 -0.0096 -0.0100 -0.0078 0.0096 -0.0764 -0.0067 -0.0168 0.0218 -0.0213 -0.0093 0.1296 0.0069 -0.0075 -0.0156 -0.0121 -0.0775 -0.0019 0.0065 -0.0044 -0.0051 -0.0011 0.0061 -0.0001 0.0047 0.0065 -0.0022 -0.0292 -0.0015 -0.0019 0.0028 0.0101 0.0036 -0.0141 0.0061 -0.0043 -0.0003 0.0141 0.0053 0.0090 -0.0016 -0.0211 -0.0026 0.0108 0.0039 0.0125 -0.0023 -0.0092 -0.0052 0.0036 -0.0119 0.0212 -0.0048 -0.0191 -0.0066 -0.0030 -0.0065 0.0068 -0.0013 -0.0058 -0.0073 -0.0108 -0.0122 -0.0215 0.0152 -0.0535 Columns 11 through 20 0 0 0 0 0 0 0 0 0 0 150

0.0007 -0.0016 -0.0045 -0.0023 -0.0057 0.0009 0.0032 0.0083 -0.0095 -0.0056 0.0012 -0.0026 -0.0085 -0.0044 -0.0092 0.0044 0.0016 0.0106 -0.0185 -0.0134 0.0013 -0.0040 -0.0081 0.0005 -0.0006 0.0051 0.0017 0.0020 -0.0074 -0.0042 0.0003 -0.0052 -0.0070 0.0066 0.0088 0.0085 0.0001 -0.0026 0.0012 0.0076 -0.0029 -0.0097 -0.0086 0.0115 0.0114 0.0061 0.0009 -0.0017 0.0031 0.0037 -0.0063 -0.0138 -0.0086 0.0144 0.0101 0.0067 0.0016 -0.0000 0.0096 -0.0010 -0.0072 -0.0066 0.0031 0.0062 0.0039 0.0044 -0.0045 -0.0060 0.0115 0.0026 -0.0072 0.0006 0.0136 -0.0023 -0.0017 0.0045 -0.0052 -0.0126 0.0096 0.0035 -0.0043 -0.0002 0.0074 -0.0024 0.0011 0.0023 -0.0027 -0.0051 0.0060 -0.0005 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0022 -0.0020 -0.0035 -0.0020 -0.0069 0.0094 -0.0008 0.0090 -0.0164 -0.0105 0.0009 -0.0100 -0.0102 0.0061 0.0014 0.0092 0.0084 0.0108 -0.0074 0.0009 -0.0015 -0.0161 -0.0176 0.0132 0.0107 0.0124 0.0112 0.0096 -0.0010 0.0103 -0.0010 -0.0122 -0.0078 0.0104 0.0047 0.0084 0.0081 -0.0003 0.0033 -0.0009 -0.0033 -0.0057 0.0039 0.0077 -0.0021 0.0088 0.0020 -0.0064 0.0080 -0.0095 -0.0024 -0.0101 -0.0021 0.0092 0.0086 0.0067 0.0001 -0.0114 0.0055 0.0050 -0.0032 -0.0162 -0.0079 0.0097 0.0160 0.0086 0.0016 -0.0147 0.0061 0.0171 -0.0045 -0.0072 0.0000 0.0073 0.0083 0.0038 -0.0056 -0.0094 0.0111 0.0104 -0.0043 0.0028 0.0085 0.0035 -0.0023 0.0014 -0.0075 -0.0070 0.0109 0.0025 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0113 -0.0271 0.0156 0.0360 0.0346 -0.0092 0.0217 -0.0184 0.0447 0.0344 -0.0013 -0.0249 -0.0164 0.0238 0.0226 -0.0069 0.0268 0.0072 0.0209 0.0277 0.0109 -0.0066 -0.0067 -0.0160 -0.0247 -0.0120 -0.0000 0.0101 -0.0271 -0.0362 0.0080 0.0187 0.0335 -0.0194 -0.0300 -0.0146 -0.0118 -0.0164 -0.0043 -0.0372 0.0135 0.0291 0.0132 -0.0129 0.0091 -0.0024 -0.0134 -0.0120 -0.0048 0.0271 0.0101 -0.0176 -0.0313 0.0058 0.0361 -0.0041 0.0017 -0.0017 -0.0111 0.0392 -0.0042 -0.0201 -0.0336 0.0265 0.0108 -0.0040 0.0030 0.0293 0.0072 -0.0163 151

-0.0028 0.0276 0.0215 -0.0032 -0.0245 -0.0122 -0.0162 0.0177 0.0125 -0.0279 -0.0148 0.0212 0.0457 -0.0339 -0.0193 0.0023 -0.0011 -0.0122 0.0003 -0.0075 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0074 -0.0224 0.0091 0.0301 0.0313 -0.0074 0.0174 -0.0185 0.0399 0.0297 -0.0030 -0.0259 -0.0207 0.0247 0.0238 -0.0052 0.0296 0.0139 0.0211 0.0349 0.0075 -0.0055 -0.0023 -0.0103 -0.0201 -0.0119 0.0026 0.0108 -0.0239 -0.0304 0.0072 0.0209 0.0379 -0.0204 -0.0343 -0.0148 -0.0099 -0.0188 0.0017 -0.0433 0.0130 0.0217 0.0118 -0.0142 0.0043 -0.0032 -0.0140 -0.0121 -0.0042 0.0215 0.0106 -0.0219 -0.0354 0.0069 0.0395 -0.0043 0.0055 -0.0050 -0.0171 0.0418 -0.0010 -0.0162 -0.0307 0.0217 0.0152 -0.0030 0.0046 0.0267 0.0048 -0.0121 -0.0017 0.0310 0.0238 -0.0033 -0.0276 -0.0145 -0.0199 0.0176 0.0128 -0.0251 -0.0115 0.0212 0.0386 -0.0235 -0.0219 0.0004 -0.0047 -0.0100 0.0031 -0.0075 0.0539 -0.0293 0.0594 0.0168 -0.0205 -0.0390 0.0545 0.0460 0.0108 0.0222 0.0071 -0.0287 0.0387 0.0620 0.0764 0.0575 -0.0667 -0.1083 0.0152 -0.0340 -0.0538 0.0553 -0.1119 -0.0704 -0.0145 0.0140 -0.0400 -0.0365 -0.0095 -0.0274 0.0665 -0.0378 0.0433 -0.0158 -0.0471 0.0507 -0.0538 0.0770 -0.0837 -0.0093 -0.1080 0.0581 0.0455 0.0697 0.0440 0.0014 0.0209 0.0308 -0.0135 0.0273 0.0699 0.0312 -0.0279 -0.0341 0.0115 0.0766 0.0164 0.0064 0.1060 0.0705 -0.0670 -0.1244 -0.0028 -0.0076 -0.0616 -0.0041 -0.0232 -0.0055 0.0296 -0.0200 0.0098 0.0561 -0.0148 0.0592 -0.0427 0.0354 0.0723 -0.0005 -0.0335 -0.1228 0.0324 0.0056 0.0334 -0.0809 0.0733 0.0202 0.0414 -0.0078 -0.0283 0.0013 -0.0443 -0.0124 0.0302 -0.0328 -0.0523 0.0202 0.0357 -0.0804 -0.0716 0.1270 0.0936 0.0374 -0.0914 0.1570 -0.0696 -0.0283 -0.0575 0.0033 0.0290 0.0209 -0.0135 0.0042 -0.0266 -0.0195 -0.0389 -0.0641 0.0922 0.0939 0.0256 0.0644 0.0144 -0.0163 0.0402 0.0300 0.0131 0.0700 -0.0593 0.0288 -0.0762 -0.0153 0.0253 0.0002 0.0238 -0.0158 -0.0150 -0.0969 0.0504 -0.0880 0.0638 -0.0574 0.0214 -0.0082 -0.0083 -0.0270 -0.0301 0.0958 -0.0364 0.0344 0.0481 0.0345 152

0.0248 -0.0134 -0.0364 0.0084 0.0552 -0.0943 -0.0160 -0.0247 -0.1161 0.0132 0.0097 0.0248 -0.0102 0.0172 0.0065 0.0835 0.0862 0.0068 0.0292 -0.0356 -0.0002 0.0370 0.0292 -0.0110 0.0041 -0.0722 -0.1146 0.0457 0.0472 0.0414 -0.0045 0.0084 -0.0008 0.0374 -0.0880 0.0310 0.0393 -0.0756 -0.0395 0.0333 -0.0293 -0.0310 0.0013 -0.0356 0.0772 -0.0094 -0.0116 0.0712 0.0487 -0.0939 Columns 21 through 30 0 0 0 0 0 0 0 0 0 0 -0.0277 0.0042 0.0079 0.0024 -0.0115 0.0089 0.0159 0.0007 0.0075 0.0080 -0.0512 0.0088 0.0131 0.0065 -0.0204 0.0143 0.0247 0.0000 -0.0000 0.0134 -0.0371 -0.0137 0.0014 0.0080 -0.0198 0.0043 -0.0049 0.0012 0.0234 0.0254 -0.0172 -0.0331 -0.0089 0.0108 -0.0153 -0.0049 -0.0290 0.0016 0.0341 0.0283 0.0031 -0.0066 -0.0181 0.0055 0.0009 -0.0121 -0.0231 0.0288 0.0308 0.0275 0.0203 0.0200 -0.0234 0.0010 0.0180 -0.0169 -0.0133 0.0431 0.0210 0.0174 0.0377 0.0158 0.0024 -0.0014 0.0189 0.0090 -0.0037 0.0257 0.0147 0.0247 0.0504 0.0079 0.0265 -0.0031 0.0182 0.0315 0.0056 0.0069 0.0038 0.0231 0.0275 0.0038 0.0155 -0.0017 0.0092 0.0149 0.0042 0.0022 0.0028 0.0134 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0516 -0.0021 0.0078 0.0314 -0.0331 0.0171 0.0267 -0.0169 -0.0759 -0.0239 -0.0532 -0.0166 0.0046 0.0369 -0.0348 0.0124 0.0111 -0.0217 -0.0674 -0.0222 -0.0483 -0.0252 0.0047 0.0242 -0.0223 0.0002 -0.0206 -0.0226 -0.0169 -0.0004 -0.0241 -0.0136 -0.0089 0.0209 -0.0010 -0.0160 -0.0292 -0.0050 -0.0086 -0.0094 0.0077 -0.0054 -0.0251 0.0048 0.0204 -0.0290 -0.0199 0.0261 -0.0004 -0.0232 0.0349 0.0050 -0.0103 0.0001 0.0381 -0.0146 0.0042 0.0099 -0.0030 -0.0222 0.0574 0.0222 0.0041 -0.0076 0.0392 0.0139 0.0260 -0.0213 -0.0055 -0.0049 0.0545 0.0211 0.0071 -0.0086 0.0415 0.0022 0.0435 -0.0590 -0.0014 -0.0124 153

0.0440 0.0136 0.0154 -0.0070 0.0248 -0.0251 0.0489 -0.0753 0.0042 -0.0162 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0340 -0.0516 -0.0270 0.0228 -0.0060 -0.0184 -0.0580 -0.0194 -0.1183 -0.0162 -0.0115 -0.0212 0.0072 -0.0148 -0.0003 -0.0023 -0.0252 -0.0131 0.0530 0.0398 -0.0296 0.0652 0.0160 -0.0064 0.0019 0.0179 0.0594 0.0145 -0.0507 0.0278 0.0207 0.0072 -0.0250 -0.0220 0.0090 -0.0207 -0.0084 0.0368 -0.0079 -0.0615 -0.0056 -0.0665 0.0295 0.0045 -0.0253 0.0206 -0.0293 -0.0503 0.0385 0.0064 0.0154 0.0149 0.0166 -0.0100 -0.0133 0.0363 0.0243 -0.0147 -0.0308 0.0471 -0.0391 0.0249 -0.0602 -0.0007 0.0018 -0.0547 -0.0131 0.0581 0.0060 0.0141 -0.0231 -0.0110 0.0072 0.0013 -0.0289 -0.0297 -0.0023 -0.0329 0.0093 -0.0466 0.0185 -0.0146 0.0457 0.0005 -0.0225 0.1039 -0.0492 0.0763 -0.0189 0.0164 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0318 -0.0442 -0.0242 0.0208 -0.0045 -0.0161 -0.0506 -0.0182 -0.0960 -0.0101 -0.0123 -0.0292 0.0084 -0.0141 -0.0033 -0.0019 -0.0322 -0.0198 0.0691 0.0434 -0.0280 0.0572 0.0166 -0.0070 0.0033 0.0137 0.0472 0.0108 -0.0418 0.0134 0.0199 0.0118 -0.0313 -0.0231 0.0115 -0.0286 -0.0114 0.0483 -0.0196 -0.0851 -0.0053 -0.0603 0.0241 0.0010 -0.0203 0.0168 -0.0223 -0.0422 0.0240 -0.0008 0.0204 0.0145 0.0232 -0.0111 -0.0136 0.0482 0.0302 -0.0205 -0.0447 0.0530 -0.0347 0.0233 -0.0540 -0.0020 -0.0006 -0.0444 -0.0084 0.0413 0.0014 0.0058 -0.0272 -0.0111 0.0034 0.0010 -0.0290 -0.0459 0.0057 -0.0564 0.0127 -0.0666 0.0181 -0.0113 0.0404 -0.0001 -0.0192 0.0826 -0.0374 0.0528 -0.0134 0.0051 -0.0042 -0.0111 0.0016 0.0034 -0.0165 0.0257 0.0826 0.0376 0.1134 -0.0506 -0.0420 -0.0143 -0.0143 0.0491 -0.0487 0.0436 0.1256 0.0432 0.1282 -0.0583 0.0070 0.0047 -0.0174 0.0354 -0.0138 0.0034 -0.0014 -0.0049 0.0194 0.0012 0.0929 0.0118 -0.0709 0.1673 -0.0328 -0.0076 -0.0480 -0.0254 0.0404 0.0729 -0.0010 0.0123 0.0095 0.0540 0.0007 -0.0046 0.0075 0.0205 0.0013 0.0326 -0.0432 0.0741 0.1105 0.1078 0.0897 -0.0349 -0.0098 0.0687 0.0102 -0.0612 154

-0.0138 -0.0083 0.0200 0.0271 0.0417 0.0052 -0.0158 -0.0072 0.0247 -0.0214 -0.0043 -0.1154 0.0106 0.0352 0.1476 0.0398 0.0356 -0.0415 -0.0012 0.0634 -0.0002 -0.0181 -0.0162 0.0061 0.0428 -0.0080 0.0152 0.0095 -0.0018 0.0467 -0.0673 0.0249 -0.1090 0.0012 0.0727 0.0410 -0.0052 -0.0155 0.0123 -0.0329 0.0031 0.0220 -0.0318 -0.0029 0.0343 0.1089 -0.0374 0.0199 0.0045 -0.0512 0.0362 -0.0368 -0.0029 -0.0056 -0.0055 0.0102 0.0347 0.0230 0.0556 -0.0763 -0.0589 0.0554 0.0869 -0.1529 0.0789 -0.0412 -0.1348 -0.0421 -0.0156 0.0085 -0.0192 0.0475 -0.0170 0.0068 0.0011 -0.0050 -0.0191 -0.0194 -0.0052 0.1152 0.0194 -0.1267 -0.0590 -0.1412 -0.0295 0.0092 0.1225 0.0796 -0.0599 0.0492 0.0231 -0.0295 0.0432 -0.0022 -0.0018 0.0017 -0.0058 0.0334 0.0161 -0.1007 0.0252 0.1156 -0.0834 -0.0632 -0.1304 0.0791 -0.0642 -0.1082 0.0501 -0.0602 -0.0443 0.0076 -0.0439 -0.0026 0.0158 -0.0137 0.0213 -0.0293 -0.0059 0.0899 0.0506 -0.0061 0.1152 -0.0030 -0.1333 -0.1686 0.0225 0.0689 -0.0138 0.0945 0.0247 -0.0103 0.0431 0.0002 -0.0318 0.0582 -0.0413 0.0472 -0.0016 -0.0376 Columns 31 through 40 0 0 0 0 0 0 0 0 0 0 -0.0172 -0.0202 0.0031 0.0130 0.0529 0.0313 0.0269 -0.0286 -0.0761 0.0102 -0.0327 -0.0309 0.0023 0.0202 0.0727 0.0356 0.0356 -0.0318 -0.0789 0.0181 -0.0446 -0.0516 -0.0066 0.0256 0.0849 0.0005 0.0296 -0.0409 -0.0247 0.0558 -0.0466 -0.0420 -0.0177 0.0091 0.0433 -0.0423 0.0007 -0.0164 0.0505 0.0457 -0.0559 -0.0611 -0.0094 0.0062 -0.0015 -0.0457 -0.0071 0.0478 0.0822 0.0402 -0.0492 -0.0400 0.0047 -0.0042 -0.0467 -0.0090 -0.0074 0.0693 0.0321 0.0074 -0.0470 -0.0489 0.0200 -0.0064 -0.0808 0.0397 -0.0435 0.0222 -0.0241 -0.0023 -0.0280 -0.0236 0.0306 0.0025 -0.0629 0.0637 -0.0525 -0.0450 -0.0500 0.0001 -0.0181 -0.0182 0.0172 -0.0002 -0.0476 0.0567 -0.0434 -0.0477 -0.0487 0.0039 155

-0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0148 -0.0011 0.0396 0.0595 -0.0422 0.0470 0.0416 0.0677 0.0125 -0.0076 0.0277 0.0086 0.0291 0.0222 -0.0418 -0.0019 -0.0240 -0.0062 0.0351 -0.0241 0.0383 0.0109 -0.0007 -0.0168 -0.0085 -0.0416 -0.0596 -0.0735 0.0032 -0.0103 0.0409 0.0344 -0.0334 -0.0658 -0.0058 -0.0370 -0.0265 -0.0119 -0.0589 -0.0075 0.0269 0.0241 -0.0444 -0.0542 0.0027 -0.0081 0.0403 0.0534 -0.0482 0.0141 0.0182 0.0492 -0.0377 -0.0277 0.0128 0.0390 0.0777 -0.0076 0.0257 0.0016 0.0064 0.0279 -0.0026 0.0054 0.0123 0.0400 0.0406 -0.0525 0.0709 -0.0016 -0.0039 0.0415 0.0244 0.0487 0.0418 -0.0210 -0.0306 0.0057 0.0130 0.0119 -0.0122 0.0207 0.0227 0.0549 0.0324 -0.0434 -0.0823 0.0568 -0.0610 0.0197 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0703 -0.0313 0.0214 -0.0150 0.0652 0.0387 0.0112 0.0025 -0.0363 -0.0182 0.0412 -0.0066 -0.0129 0.0569 -0.0085 0.0075 0.0197 -0.0088 -0.0200 0.0080 -0.0757 0.0205 -0.0454 -0.0314 -0.0058 -0.0329 -0.0179 0.0057 0.0541 -0.0074 0.0248 -0.0503 0.0424 0.0259 -0.0020 -0.0170 0.0061 -0.0041 0.0112 0.0182 -0.0721 0.0644 -0.0010 -0.0131 -0.0553 0.0015 0.0037 0.0345 0.0019 -0.0362 0.0496 -0.0541 0.0009 -0.0361 -0.0021 -0.0044 -0.0108 0.0124 0.0065 0.0082 -0.0240 0.0763 0.0589 0.0194 -0.0119 0.0318 -0.0117 -0.0459 -0.0463 -0.0104 -0.0002 -0.0332 -0.0657 -0.0070 -0.0174 0.0351 -0.0141 -0.0039 -0.0106 0.0055 0.0240 0.0517 0.0229 0.0110 0.0446 -0.0304 0.0101 0.0088 0.0153 0.0114 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0415 -0.0165 0.0137 0.0019 0.0173 0.0156 0.0071 0.0194 0.0116 -0.0059 0.0749 -0.0087 -0.0001 0.1026 -0.0375 0.0338 0.0457 0.0526 0.0106 -0.0227 -0.0405 0.0217 -0.0281 -0.0215 -0.0224 -0.0245 -0.0203 -0.0084 0.0137 -0.0136 0.0604 -0.0732 0.0628 0.0367 -0.0060 -0.0418 -0.0262 -0.0650 -0.0022 -0.0159 -0.0362 0.0522 -0.0001 -0.0188 -0.0294 0.0085 0.0120 0.0082 -0.0284 -0.0032 0.0849 -0.0792 -0.0194 -0.0721 0.0060 -0.0136 0.0156 0.0404 -0.0437 0.0236 156

-0.0074 0.0596 0.0301 0.0134 0.0108 0.0161 0.0068 -0.0248 0.0141 0.0046 0.0029 -0.0472 -0.1010 -0.0137 -0.0178 0.0660 0.0109 -0.0379 0.0628 0.0210 0.0169 0.0379 0.0108 0.0149 0.0367 -0.0286 -0.0060 0.0262 -0.0004 0.0074 0.0085 -0.0020 0.0416 -0.0874 0.0474 0.0282 -0.0506 0.0310 0.0437 -0.0748 0.0174 -0.0041 0.0526 -0.0802 0.0523 0.0445 -0.0435 0.0359 0.0312 -0.0820 -0.0024 -0.0192 0.0245 -0.0263 0.0717 0.0662 -0.0262 0.0348 0.0051 -0.1127 0.0363 -0.0224 -0.0286 0.0489 0.0579 0.0429 -0.0087 -0.0029 -0.0232 -0.1062 -0.0162 -0.0217 -0.0583 0.0519 0.0216 -0.0270 -0.0020 -0.0286 -0.0063 -0.1129 -0.0222 -0.0650 -0.0218 0.0489 0.0067 -0.0706 0.0380 -0.0301 -0.0086 -0.0719 -0.0413 -0.0358 0.0227 -0.0083 -0.0360 -0.0690 0.0681 -0.0050 -0.0183 -0.0643 -0.0025 -0.0702 0.0514 -0.0515 -0.0392 -0.0401 0.0554 0.0024 0.0040 -0.0225 -0.0076 -0.0161 0.0403 -0.0236 -0.0356 0.0296 0.0020 -0.0294 -0.0059 0.0141 -0.0236 -0.0312 -0.0438 0.0142 -0.0422 0.0860 -0.0821 -0.0177 -0.0255 0.0522 -0.0137 0.0013 -0.0582 0.0430 -0.0077 0.0816 -0.1220 0.0112 -0.0238 0.0879 -0.0228 -0.0209 0.0797 -0.1217 0.0963 0.0798 -0.0674 0.0784 0.0682 -0.1461 -0.0024 -0.0299 0.0524 -0.0502 0.0992 0.1118 -0.0649 0.0931 0.0691 -0.2141 0.0216 0.0027 -0.0708 0.0538 0.0354 0.0455 -0.0496 0.0182 0.0190 -0.2284 -0.0220 -0.0100 -0.1230 0.1001 -0.0111 -0.0478 -0.0209 -0.0484 -0.0367 -0.2352 -0.0533 -0.0242 -0.0266 0.0628 -0.0400 -0.0983 0.0676 -0.0537 -0.0772 -0.1706 -0.0618 -0.0264 0.0572 -0.0376 -0.0635 -0.1166 0.1453 -0.0461 -0.0725 -0.1187 0.0107 -0.0246 0.1103 -0.0768 -0.0315 -0.0589 0.1213 -0.0491 -0.0018 -0.0418 0.0139 0.0154 0.0688 -0.0486 -0.0190 0.0306 0.0397 -0.0442 0.0321 0.0317 -0.0003 0.0067 -0.0629 0.0290 -0.0012 0.0912 -0.1119 0.0113 -0.0034 0.1079 Columns 41 through 50 0 0 0 0 0 0 0 0 0 0 157

0.0788 -0.0038 -0.0052 -0.0038 0.0130 0.0633 -0.0006 0.0296 0.0389 -0.0781 0.0556 0.0012 -0.0073 0.0102 0.0106 0.0134 0.0046 0.0191 0.0127 -0.0431 -0.0390 -0.0072 0.0034 -0.0065 -0.0143 -0.0572 0.0236 0.0480 0.0305 -0.0300 -0.0758 0.0002 -0.0076 -0.0031 -0.0179 -0.0269 -0.0047 -0.0344 -0.0323 0.0151 0.0098 -0.0062 0.0206 -0.0208 -0.0079 -0.0029 -0.0255 -0.0918 -0.0521 -0.0301 0.0796 0.0022 -0.0005 -0.0128 0.0214 0.0297 -0.0004 0.0287 -0.0013 0.0368 0.0410 -0.0389 0.0061 0.0096 0.0518 0.0323 0.0446 0.0821 -0.0302 0.0706 -0.0435 -0.0082 0.0017 0.0115 -0.0045 -0.0173 0.0106 -0.0145 0.0233 0.0077 -0.0628 0.0024 0.0170 0.0364 -0.0894 -0.0182 -0.0107 -0.0714 0.0862 0.0346 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0113 0.0085 0.0058 0.0087 -0.0077 -0.0505 -0.0044 -0.0420 -0.0546 0.0629 -0.0014 -0.0244 0.0135 -0.0604 -0.0150 0.0829 -0.0009 0.0501 0.0881 -0.0270 0.0189 -0.0093 -0.0118 -0.0041 0.0140 0.0467 0.0061 0.0220 0.0016 0.0273 0.0091 0.0255 0.0413 0.0758 0.0478 -0.0922 -0.0156 -0.0475 -0.0017 -0.0578 -0.0250 0.0162 0.0038 0.0095 -0.0190 -0.0136 -0.0029 -0.0011 0.0556 -0.0026 -0.0198 -0.0486 -0.0411 -0.0245 -0.0695 0.0707 0.0334 0.0187 -0.1007 -0.0471 0.0281 -0.0129 -0.0157 -0.0071 0.0358 -0.0188 0.0063 -0.0167 -0.0048 -0.0308 0.0179 0.0345 0.0149 0.0039 0.0279 -0.0298 -0.0839 0.0211 0.0545 0.0716 -0.0256 0.0200 0.0080 -0.0037 -0.0431 0.0338 0.0442 -0.0053 -0.0424 -0.0400 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0406 0.0079 -0.0091 0.0214 0.0285 0.0865 -0.0527 -0.1401 -0.0354 0.0866 0.0235 -0.0158 0.0193 -0.0488 -0.0339 -0.0462 0.0229 0.0482 -0.0113 0.0691 -0.1016 0.0369 -0.0747 0.1214 0.0628 0.0612 0.0252 0.0723 0.0225 0.0496 -0.0283 -0.0219 0.0627 -0.0339 0.0162 0.0157 -0.0219 -0.0291 0.0611 -0.0310 0.0940 0.0602 -0.1324 0.0027 -0.0915 -0.0387 0.0039 -0.0194 0.0960 -0.0530 0.0320 -0.0282 0.0468 0.0565 0.0097 -0.0030 0.0274 0.0131 -0.0313 -0.0637 -0.0679 0.0554 -0.0688 -0.1197 0.1204 -0.0359 0.0205 -0.0345 -0.0384 -0.0744 158

-0.0196 -0.0641 -0.0255 0.0174 -0.0439 0.0027 -0.0258 -0.0108 -0.0055 -0.0041 0.0079 0.1482 0.0805 -0.0016 -0.0667 0.0640 0.1293 -0.0226 -0.0249 -0.0029 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0219 -0.0006 0.0087 -0.0105 -0.0235 -0.0844 0.0375 0.0904 0.0015 -0.0420 -0.0448 0.0305 -0.0349 0.0825 0.0506 0.0760 -0.0257 -0.0313 0.0407 -0.1066 0.0215 -0.0171 0.0344 -0.0584 -0.0277 -0.0157 -0.0206 -0.0502 -0.0143 -0.0277 0.0482 0.0468 -0.1073 0.0618 -0.0283 -0.0498 0.0158 0.0072 -0.0915 0.0176 -0.0157 -0.0176 0.0612 0.0091 0.0447 0.0230 -0.0026 0.0266 -0.0495 0.0341 -0.0511 0.0560 -0.0826 -0.1058 -0.0158 0.0340 -0.0379 0.0084 -0.0008 0.0731 0.0041 -0.0213 0.0232 0.0528 -0.0663 0.0097 -0.0265 0.0128 0.0235 0.0412 0.0353 0.1356 0.0526 -0.0200 0.0535 -0.0116 -0.0123 0.0259 0.0212 0.0259 0.0057 -0.0446 -0.0328 -0.0064 0.0252 -0.0196 -0.0870 0.0278 -0.0148 -0.0246 -0.0292 0.0047 0.0046 -0.0010 -0.0049 -0.0111 -0.0042 -0.0044 -0.0085 0.0062 -0.0115 0.0039 0.0056 -0.0053 -0.0048 -0.0001 -0.0035 0.0059 -0.0013 -0.0102 0.0052 0.0060 0.0099 -0.0099 -0.0090 -0.0051 -0.0078 -0.0002 -0.0153 0.0111 0.0091 0.0004 0.0187 -0.0246 -0.0169 -0.0173 -0.0001 0.0303 0.0007 0.0078 -0.0379 0.0115 0.0184 -0.0059 -0.0066 0.0004 -0.0172 0.0166 -0.0059 0.0143 -0.0466 0.0118 0.0408 -0.0090 -0.0095 -0.0018 -0.0365 -0.0165 -0.0106 -0.0145 0.0003 0.0335 0.0242 0.0041 -0.0267 0.0084 -0.0485 0.0213 -0.0082 -0.0197 0.0236 0.0215 0.0302 0.0172 -0.0163 0.0222 -0.0499 0.0546 -0.0432 -0.0193 -0.0127 0.0517 0.0160 0.0004 -0.0294 0.0117 -0.1152 0.0548 -0.0628 -0.0546 -0.0314 0.0613 0.0202 0.0130 -0.0786 0.0183 -0.1498 0.0490 -0.0582 -0.0536 -0.0236 0.1384 0.0478 0.0049 -0.1105 0.0547 -0.1902 0.0938 -0.1286 -0.0997 -0.0484 0.0121 0.0123 -0.0015 -0.0166 -0.0547 0.0021 0.0130 -0.0282 0.0006 -0.0400 0.0058 0.0263 -0.0358 -0.0322 -0.0294 -0.0043 0.0297 -0.0055 0.0004 -0.0024 0.0048 0.0259 -0.0446 -0.0284 0.0062 -0.0175 0.0202 -0.0212 0.0248 -0.0064 0.0199 0.0540 -0.0129 -0.0016 -0.0140 -0.0390 0.0098 -0.0206 -0.0035 159

-0.0470 0.0385 0.0617 0.0020 -0.0052 -0.0013 -0.0591 0.0282 0.0011 0.0056 -0.0565 0.0402 0.0533 0.0081 -0.0564 0.0308 -0.0760 0.0466 -0.0543 -0.0296 -0.0148 0.0616 0.0430 0.0199 -0.0662 0.0229 -0.1205 0.0583 -0.0640 -0.0616 0.0049 0.0999 0.0421 0.0174 -0.0680 0.0196 -0.2416 0.1071 -0.0852 -0.0586 -0.0122 0.1444 0.0366 0.0012 -0.1079 0.0496 -0.3394 0.1584 -0.1804 -0.1429 Columns 51 through 60 0 0 0 0 0 0 0 0 0 0 0.1684 0.0143 0.0427 -0.0117 0.0135 0.0266 0.1336 0.1173 -0.0854 0.0440 0.0085 -0.0528 -0.0028 -0.0154 0.0314 0.0115 -0.0654 -0.0973 0.0962 -0.0578 -0.1386 -0.0861 -0.0147 0.0304 -0.0297 -0.0290 -0.0574 0.0573 -0.1469 0.1185 -0.0094 -0.0232 0.0464 0.0372 -0.0002 0.0465 0.0880 0.0558 0.0633 -0.0939 0.1163 -0.0106 0.0420 0.0225 -0.0470 -0.0490 -0.0116 -0.1499 -0.0111 0.1495 0.0299 -0.0381 -0.0167 -0.0200 -0.0444 0.0017 -0.0921 0.0637 -0.0542 -0.0969 -0.0649 -0.0730 -0.0209 -0.0518 0.0241 -0.0356 0.1108 0.0306 0.1366 0.1259 -0.0212 0.0096 0.0743 -0.0503 -0.0042 0.0042 0.0473 -0.0926 -0.0978 -0.0647 0.0709 0.0041 0.0842 0.0331 0.0420 -0.0727 -0.1306 0.1204 0.0896 0.0497 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0032 0.0465 -0.0118 -0.0173 0.0140 0.0102 0.0198 0.0378 -0.0371 0.0223 -0.0317 0.0315 0.0274 0.0878 -0.1249 -0.0734 0.0268 -0.0245 0.0397 -0.0284 0.0138 0.0238 -0.0277 -0.0470 0.0683 0.0361 -0.0567 -0.0240 -0.0221 0.0336 0.0100 -0.0001 -0.0683 -0.0678 -0.0577 -0.1233 0.0374 0.0276 0.0190 -0.0434 -0.0293 0.0404 0.0293 0.0452 0.0231 0.0782 0.0283 -0.0253 0.0226 0.0337 -0.0169 0.0160 0.0113 0.0117 0.0955 -0.1255 -0.0015 0.0090 -0.0277 -0.0429 0.0102 0.0113 -0.0487 -0.0011 -0.0628 0.0615 -0.0340 0.0405 0.0377 0.0221 0.0359 -0.0506 -0.0773 0.1143 0.0864 -0.0474 0.0428 -0.0370 -0.0379 -0.0268 160

-0.0150 0.0219 0.0114 -0.0378 -0.0342 0.0219 -0.0111 0.0049 0.0023 0.0008 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0748 0.0595 -0.0083 0.0191 -0.0283 -0.0132 0.0223 0.0352 -0.0334 0.0195 0.0150 0.0965 -0.0094 0.0196 -0.0505 -0.0351 -0.0026 -0.0346 0.0420 -0.0272 0.0395 0.1038 -0.0381 -0.0239 -0.0133 -0.0303 -0.0232 -0.0030 -0.0401 0.0425 -0.0092 0.0391 -0.0659 -0.0858 0.0423 -0.0214 0.0207 0.0225 0.0266 -0.0440 -0.0548 0.0662 -0.0342 -0.0251 0.0461 -0.0087 0.0260 -0.0319 0.0062 0.0507 -0.0348 0.0656 0.0172 0.0773 0.0611 -0.0030 -0.0236 0.0211 -0.0233 -0.0447 0.0048 0.0491 -0.0459 0.0767 0.0244 0.0072 -0.0127 0.0222 0.0444 0.0410 0.0317 -0.0214 -0.1202 0.0545 -0.0196 0.0374 0.0201 -0.0341 -0.0395 -0.0278 0.0145 -0.0232 -0.0906 0.0317 -0.0121 0.0165 0.0044 0.0140 0.0183 0.0125 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 0.0748 -0.0233 0.0036 -0.0263 0.0326 0.0193 0.0005 -0.0014 0.0021 -0.0011 -0.0189 -0.1200 0.0205 0.0052 0.0143 0.0106 -0.0089 -0.0148 0.0221 -0.0168 -0.0320 -0.0723 0.0151 -0.0064 0.0447 0.0433 0.0050 0.0276 -0.0248 0.0142 0.0108 -0.0620 0.0643 0.0749 -0.0621 -0.0097 0.0244 0.0156 0.0077 -0.0230 0.0332 -0.0297 0.0421 0.0496 -0.0321 0.0356 -0.0357 -0.0173 -0.0164 0.0235 0.0435 -0.0833 -0.0134 -0.0793 -0.0428 -0.0437 -0.0104 0.0064 -0.0172 -0.0214 -0.0027 -0.0349 0.0058 -0.0673 -0.0502 0.0272 0.0288 -0.0158 0.0148 0.0250 -0.0320 0.0111 0.1220 -0.0401 0.0429 -0.0482 0.0132 -0.0191 -0.0183 -0.0112 -0.0180 0.0346 0.0758 -0.0384 -0.0037 -0.0101 -0.0356 0.0355 0.0325 0.0212 -0.0249 -0.0125 -0.0029 0.0016 0.0011 0.0015 -0.0172 -0.0108 0.0054 -0.0013 0.0331 0.0026 0.0107 -0.0019 0.0026 0.0093 0.0336 0.0368 -0.0313 0.0184 0.0099 0.0058 0.0030 -0.0010 0.0014 0.0064 -0.0053 -0.0147 0.0164 -0.0084 -0.0263 -0.0021 -0.0079 0.0054 -0.0150 -0.0111 -0.0292 -0.0041 -0.0308 0.0346 -0.0063 0.0124 -0.0059 -0.0114 0.0027 0.0127 0.0092 0.0161 0.0069 -0.0117 0.0189 0.0165 -0.0006 -0.0122 -0.0029 -0.0053 0.0150 -0.0338 0.0119 0.0392 161

-0.0043 0.0259 -0.0054 -0.0031 -0.0019 0.0127 -0.0223 0.0075 -0.0094 -0.0093 -0.0265 0.0124 -0.0217 0.0010 0.0081 0.0032 0.0057 0.0282 0.0419 0.0319 -0.0221 0.0352 -0.0359 0.0016 -0.0167 0.0338 0.0090 -0.0098 -0.0033 0.0022 0.0018 0.0338 -0.0424 0.0193 -0.0126 0.0209 -0.0283 0.0269 0.0239 0.0166 -0.0208 0.0521 -0.0914 0.0127 -0.0383 0.0545 -0.0142 0.0221 0.0272 0.0220 0.0131 -0.0187 0.0005 -0.0165 0.0252 0.0214 -0.0019 0.0138 -0.0181 0.0136 0.0096 0.0036 0.0185 0.0250 -0.0338 -0.0063 0.0225 0.0133 -0.0064 0.0044 0.0032 0.0048 0.0100 0.0047 0.0053 0.0252 -0.0151 -0.0029 -0.0200 0.0282 0.0038 -0.0058 -0.0287 -0.0420 -0.0129 -0.0237 -0.0135 0.0088 -0.0161 0.0139 -0.0027 0.0171 -0.0066 -0.0171 -0.0143 0.0314 -0.0003 -0.0115 0.0071 0.0343 -0.0173 0.0344 0.0157 0.0015 0.0260 -0.0108 0.0043 -0.0150 -0.0001 0.0209 -0.0259 0.0493 -0.0125 -0.0181 -0.0224 0.0347 -0.0135 0.0191 0.0328 0.0329 -0.0165 0.0348 -0.0813 0.0287 -0.0092 0.0352 0.0009 0.0166 0.0317 0.0300 -0.0424 0.0856 -0.0966 0.0062 -0.0474 0.0713 -0.0215 0.0357 0.0461 0.0378 Tutorial 3 ( Uniformly distributed loads and out – of – joints point load) Consider the truss structure shown below, all the joints in the structure are rigid jointed, its members have the same sectional properties as the structure in tutorial 2, the loadings are also as shown and all dimensions are in metres. Solution: Work equivalent nodal loads are calculated using equations 3. And 3. As follows: Consider the top cord which is loaded as follows: 162

+ But, Mq1 = ql2/12 = 8 x 32 / 12 = 6Knm = - Mq2 Mp1= PL/8 = 10 x 3/8 = 3.75Knm = - Mp2 Fq1 =- qL/2 = -8 x3/2 =- 12Kn = -Fq2 Fp1 = -P/2 = -10/2 = -5Kn = -Fp2 Moment at the first external node = Mq1 + Mp1= 9.75Knm Moment at the second external node = Mq2 + Mp2 = - 9.75Knm Moment at the internal nodes = Mq1 + Mq2 + Mp1 + Mp2 = 0Knm Force at the external nodes = Fq1 + Fp1 = -17Kn Force at the internal nodes = Fq1 + Fp1 + Fq2 + Fp2 = 34Kn Therefore the work equivalent nodal loads on the structure are shown below: The structure universal coordinate system is the same as the one for tutorial 2 as shown below: 163

Since the structure geometry and member properties are the same with that of tutorial 2, the members‘ stiffness matrices are the same as and the global structure stiffness matrix is the same too. From , KU=F where K is the structure global stiffness matrix which is a 60x60 matrix, U is the global displacement matrix which ranges from U1 to U60 and it is a 1x60 column matrix; F is the force vector that contains all the forces applied to the structure and ranges from F1 to F60 and also a 1x60 column matrix. 1.0e+007 * 0.528 -0.388 0 - - - 0 0 0 U1 F1 -0.389 0.785 -0.389 - - - 0 0 0 U2 F2 0 -0.389 1.0629 - - - 0 0 0 U3 F3 - - - - - - - - - - - - - - - - - - - - - = - - - - - - - - - - - - 0 0 0 - - 0.079 0.0115 0 U58 F58 0 0 0 - - - 0.0115 0.046 0.0115 U59 F59 0 0 0 - - - 0 0.0115 0.0555 U60 F60 Where, F1 - - - - F31 = 0, F32 = F40 = -17Kn, F33 = F34 - - - - F39 = -34Kn, F41 = F42 ---- F51 = 0, F52 = 9.75Knm, F53 = F54 - ---- F59 = 0 and F60 = -9.75knm. Nodes 1 and 11 are constrained by their support conditions. Therefore, U1 = U2 = U21 = U22 = 0. The above equation is then reduced to 36 x 36 matrix equation. The unknowns Us which are global displacements are then sort by solving the equation using MaTlab as follows: Displacements 164

ans = Cordinate no. displacement (mm) 1. 0 2. -3.4751 3. -6.8583 4. -5.1830 5. -3.4410 6. -0.0000 7. 3.4410 8. 5.1830 9. 6.8583 10. 3.4751 11. 0 12. 32.2968 13. 26.0159 14. 19.6250 15. 9.8295 16. 0.0000 17. -9.8295 18. -19.6250 19. -26.0159 20. -32.2968 21. 0 22. -42.2835 23. -89.0292 24. -121.9413 25. -148.0177 165

26. -153.1563 27. -148.0177 28. -121.9413 29. -89.0292 30. -42.2835 31. 0 32. -42.1766 33. -89.6969 34. -121.8490 35. -148.7235 36. -153.0656 37. -148.7235 38. -121.8490 39. -89.6969 40. -42.1766 41. -13.2797 42. -14.3858 43. -12.6366 44. -9.4806 45. -4.9602 46. -0.0000 47. 4.9602 48. 9.4806 49. 12.6366 50. 14.3858 51. 13.2797 52. -12.5699 166

53. -13.0040 54. -9.3160 55. -5.0351 56. -0.0000 57. 5.0351 58. 9.3160 59. 13.0040 60. 12.5699 Back substituting the values of Us into eq. 4.1 And then solving for only constrained Fs yields reaction forces as follows: reactions ans = 1. 271.3490 11. -271.3490 21. 136.0000 31. 136.0000 Computing stress in each element: This is obtained as a function of displacement as follows: Stress, σ = E/Le[-c –s c s]U Element 1: C = (x2 – x1) /Le = 3/3 =1 C = (y2 – y1) /Le = 0/3 =0 Le = √((y2 – y1)2 + (x2 – x1)2) =3 U=[0 0 -0.0176258 -0.2445623]1 167

σ1 = 205e+09/3[-1 0 1 0]*U = -1206.3Mpa Element 2: C = (x2 – x1) /Le = 3/3 =1 C = (y2 – y1) /Le = 0/3 =0 Le = √((y2 – y1)2 + (x2 – x1)2) =3 U=[-0.0176528 -0.2445623 -0.0345937 -0.5106602]1 σ2= 205e+09/3[-1 0 1 0]*U = 1157.6Mpa element 3 C = (x2 – x1) /Le = 3/3 =1 S = (y2 – y1) /Le = 0/3 =0 Le = √((y2 – y1)2 + (x2 – x1)2) =3 U=[ -0.0345937 -0.5106602 -0.0230798 -0.6961420]1 σ3= 205e+09/3[-1 0 1 0]*U = -786.8Mpa Repeating the same process for other elements yield the following stresses: axialstresses = Member Stresses(Mpa) 1. 237.4649 2. 231.1873 3. -114.4778 4. -119.0389 5. -235.1355 6. -235.1355 7. -119.0389 8. -114.4778 9. 231.1873 10. 237.4649 11. 429.1930 168

12. 436.7162 13. 669.3601 14. 671.6797 15. 671.6797 16. 669.3601 17. 436.7162 18. 429.1930 19. 337.5593 20. 337.5593 21. -262.9953 22. -106.0077 23. 54.9021 24. 216.5001 25. -262.9953 26. -106.0077 27. 54.9021 28. 216.5001 29. -7.3080 30. 45.6310 31. -6.3008 32. 48.2289 33. -6.1991 34. 48.2289 35. -6.3008 36. 45.6310 37. -7.3080 Bending Stresses: 169

which are stresses at first and second element joints respectively and local coordinate system of the element. Converting the above equations to universal coordinate system, the following parameters are utilized; σx (x=0) = ymaxE[ 6s/L2 -6c/L2 -4/L -6s /L2 6c /L2 2 /L] U1 U2 U3 U4 U5 U6 Similarly, eq. 3.34 is re-arranged as follows: σx (x=L) = ymaxE[ -6s/L2 6c/L2 2/L 6s /L2 -6c /L2 4 /L] U1 U2 U3 U4 U5 U6 For element 1, substituting the following parameters into the above equations: ymax = 0.0205m E = 205e9N/mm2 L= 3m c = 1 s = 0 U1 = 0 170

U2 = 0 U3 = -0.0759824m U4 = -0.0176528m U5 = -0.2445623 U5 = -0.0815706 And multiplying the matrices gives the following bending stresses at the nodes: σ1 (x=0) = -15.2375N/mm2 σ1 (x=L) = 487.9613N/mm2 Repeating the above process yields bending stresses for the other elements within the structure as follows: Bending stresses(Kn/mm2) = Bending stresses = Member x=0 x=L 1. 84.3580 -0.6505 2. 85.7605 -19.8545 3. 47.9634 -3.6822 4. 33.8312 -18.7023 5. -13.3969 -0.4999 6. -28.2935 -13.3969 7. -71.8254 6.0376 8. -74.4890 -5.1597 9. -100.4630 14.9537 10. -75.0612 3.7494 11. 99.1359 -25.0535 12. 43.3139 -1.4457 13. 37.1991 -20.9793 14. -16.0484 1.9416 171

15. -26.2719 -16.0484 16. -73.1801 8.9856 17. -74.3117 -8.8869 18. -95.4868 26.2699 19. 46.0544 2.3432 20. -46.0544 -2.3432 21. 46.4410 -10.3561 22. 40.7546 -2.0307 23. 18.0614 -1.4179 24. -13.5430 -3.2035 25. -46.4410 10.3561 26. -40.7546 2.0307 27. -18.0614 1.4179 28. 13.5430 3.2035 29. 35.3569 10.5165 30. 19.2994 16.1667 31. 13.4388 9.2586 32. 4.2115 4.9309 33. -0.0000 0.0000 34. -4.2115 -4.9309 35. -13.4388 -9.2586 36. -19.2994 -16.1667 37. -35.3569 -10.5165 172

4.2 RESULTS Some of the results obtained are tabulated below: Table 4.1

Reaction

Forces(KN) S/N

Support reactions (KN) For numerical example 1

Program ROBOT %Diff 1450 1454.89 0.336 -1750 -1754.89 0.279 770 772.72 0.352 830 832.72 0.327