Matrix and Finite Element Analyses of Structures

482 Pages·2022·10.234 MB·English

by  Madhujit Mukhopadhyay, Abdul Hamid Sheikh

Madhujit Mukhopadhyay
Abdul Hamid Sheikh
Matrix and
Finite Element
Analyses
of Structures
Matrix and Finite Element Analyses of Structures
·
Madhujit Mukhopadhyay Abdul Hamid Sheikh
Matrix and Finite Element
Analyses of Structures
Madhujit Mukhopadhyay Abdul Hamid Sheikh
Indian Institute of Technology Indian Institute of Technology
Kharagpur, India Kharagpur, India
ISBN 978-3-031-08723-3 ISBN 978-3-031-08724-0 (eBook)
https://doi.org/10.1007/978-3-031-08724-0
Jointly published with ANE Books Pvt. Ltd.
In addition to this printed edition, there is a local printed edition of this work available via Ane Books in
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African subcontinent).
ISBN of the Co-Publisher’s edition: 978-81-80520-75-4
1st edition: © Authors 2015
© The Author(s) 2022
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To
Our Families
Ratna
Daisy
Kuttus
Aniq
Chuchu
Shaon
Arko
Preface
A revolutionary change has been brought about in the outlook of many fields of
engineering and science with the advent of the electronic digital computer. Amongst
them structural engineering forms an important discipline. With the innovations,
improvements and rapid progress of the development of sophisticated computing
systems, both in terms of hardware and software and with the fading away of the
boundaries of different disciplines and the enlargement of the interdisciplinary work,
never before has so much emphasis been laid in the computer aided analysis and
design as it is today. The main objective of the book is to acquaint the engineers
about the modern computer based techniques used in structural analysis.
This textbook is primarily written as a basic learning tool for senior undergraduate
and postgraduate students in the departments dealing with the mechanics of solid and
structural systems. The material of the book has been developed over a number of
years of teaching senior undergraduate and postgraduate students of different disci-
plines of IIT, Kharagpur. The materials of the book have been presented in sufficiently
simple manner. Basing on the knowledge of classical methods of structural analysis,
the reader is introduced to ‘Matrix and the Finite Element Methods’ and then exposed
to advanced topics in a more gradual manner. The book is self-contained in that most
of the background materials required for the analysis of structures have been dealt
with.
The presentation of the matter of the book is made as simple as possible so that
engineers who are interested may use this for his own purpose. A large number
of diagrams and illustrations make the text easy to understand. The book presents
‘matrix’, ‘finite element’, ‘computer’ and ‘structural analysis’ in a unified and inte-
grated manner. The role of computers is of paramount importance to the present
day structural analysis and design. Computer software and structural analysis have
progressed side by side throughout the text as the matter presented in the text are
primarily meant for large-sized structures. All the computer codes presented are in
two languages—‘FORTRAN’ and ‘C’. The computer programs in ‘C’ are becoming
quite popular now-a-days due to their versatility.
The senior author is grateful to his students—Dr. Manoranjan Barik of National
Institute of Technology, Rourkela and Dr. K. S. R. K. Murthy of Indian Institute of
vii
viii Preface
Technology, Guwahati for the assistance rendered by them. Most of the computer
programs in ‘C’ language have been developed by Mr. Dhirendra Kumar Singh, an
undergraduate student of the Department for which the authors are grateful to him.
The authors are very much thankful to Continuing Education Cell, IIT Kharagpur
for providing financial support for the preparation of the manuscript.
Sincere thanks are due to Mr. Parimal Kumar Roy for typing the manuscript and
preparing a large number drawings. The authors are thankful to Chinmoy Mukherjee
for preparing the necessary of drawings given to him. They also sincerely thank their
wives Ratna and Daisy for the patience and forbearance shown by them during the
preparation of the manuscript.
Kharagpur, India Madhujit Mukhopadhyay
Abdul Hamid Sheikh
About This Book
The main objective of the book is to acquaint the engineers about the computer-based
techniques used in structural analysis. This textbook is primarily written as a basic
learning tool for undergraduate and postgraduate students in the departments dealing
with the mechanics of solids and structural systems. The materials of the book have
been presented in sufficiently simple manner. Based on the knowledge of classical
methods of structural analysis, the reader is introduced to ‘Matrix and the Finite
Element Methods’ and then exposed to advanced topics in a more gradual manner.
The book is self-contained in that most of the background materials required for the
analysis of structures have been dealt with.
The presentation of the matter of the book is made as simple as possible so that
engineers who are interested may use this for its own purpose. A large number of
diagrams and illustrations may help the text easy to understand. The book presents
‘matrix’, ‘finite element’, ‘structural analysis’ and ‘computer implementation’ in
a unified and integrated manner. Computer programs and structural analysis have
progressed side by side throughout the text as the matter presented in the text are
primarily meant for large-sized structures. All the computer codes presented are in
two languages—‘FORTRAN’ and ‘C’.
ix
Contents
1 Basic Concepts of Structural Analysis ........................... 1
1.1 Types of Structures ...................................... 1
1.2 Objective of Structural Analysis ........................... 1
1.3 Materials and Basic Assumptions .......................... 3
1.4 Loads ................................................. 3
1.5 General Methods of Analysis ............................. 4
1.5.1 Equilibrium Conditions .......................... 4
1.5.2 Compatibility Conditions ........................ 5
1.6 Force–Displacement Relationship ......................... 5
1.7 Statical Indeterminacy ................................... 6
1.7.1 Plane Structure ................................. 7
1.7.2 Space Structures ................................ 9
1.8 Kinematic Indeterminacy ................................. 10
1.9 Two Approaches of Structural Analysis ..................... 11
2 Energy Principles ............................................. 13
2.1 Introduction ............................................ 13
2.2 Principle of Virtual Work ................................. 13
2.3 Principle of Complementary Virtual Work .................. 15
2.4 Principle of Minimum Potential Energy .................... 15
2.5 Principle of Minimum Complementary Energy .............. 16
2.6 Castigliano’s Theorems .................................. 16
2.7 Determination of Displacements ........................... 18
References and Suggested Readings .............................. 19
3 Introduction to the Flexibility and Stiffness Matrix Methods ...... 21
3.1 Introduction ............................................ 21
3.2 The Flexibility Matrix Method ............................ 21
3.3 The Stiffness Method .................................... 33
3.4 Incorporation of Different Loading Conditions .............. 50
3.5 Other Types of Loadings ................................. 51
3.5.1 Treatment by the Flexibility Matrix Method ........ 52
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xii Contents
3.5.2 Treatment by the Stiffness Method ................ 54
3.6 Incorporation of Shear Deformation ........................ 58
3.7 Relation Between Flexibility and Stiffness Matrices .......... 59
3.8 Equivalent Joint Loads ................................... 61
3.9 Choice of the Method of Analysis ......................... 62
References and Suggested Readings .............................. 73
4 Direct Stiffness Method ........................................ 75
4.1 Introduction ............................................ 75
4.2 Local and Global Coordinate System ....................... 75
4.3 Transformation of Variables .............................. 76
4.3.1 Transformation of Member Coordinate Axes ....... 76
4.3.2 Transformation of Member Displacement
Matrix ........................................ 78
4.3.3 Transformation of the Member Force Matrix ....... 79
4.3.4 Transformation of the Member Stiffness Matrix ..... 80
4.4 Transformation of the Stiffness Matrix of the Member
of a Truss .............................................. 81
4.5 Transformation of the Stiffness Matrix of the Member
of a Rigid Frame ........................................ 82
4.6 Transformation of the Stiffness Matrix of the Member
of a Grillage ............................................ 85
4.7 Transformation of the Stiffness Matrix of the Member
of a Space Frame ........................................ 87
4.8 Horizontally Circular Curved Beam Element ................ 89
4.9 Overall Stiffness Matrix .................................. 91
4.10 Boundary Conditions .................................... 95
4.10.1 Boundary Conditions Corresponding to Skewed
Supports ...................................... 95
4.11 Computation of Internal Forces ............................ 97
4.12 Computer Program for the Truss Analysis by the Direct
Stiffness Method ........................................ 97
4.13 Computer Program for the Frame Analysis by Direct
Stiffness Method ........................................ 103
4.14 Computer Program for the Grillage Analysis by the Direct
Stiffness Method ........................................ 106
References and Suggested Readings .............................. 111
5 Substructure Technique for the Analysis of Structural Systems .... 113
5.1 Introduction ............................................ 113
5.2 Basic Concepts ......................................... 114
5.3 Direct Stiffness Method Restated .......................... 115
5.4 The Substructure Technique .............................. 116
5.5 An Illustrative Example .................................. 119
5.6 Computer Program for the Truss Analysis
by the Substructure Technique ............................ 125