Basic Structural Analysis

396 Pages·2005·14.38 MB·English

Basic Structural Analysis
(81 Units)
Second Edition
C S flEDDY
Principal
K S R M College of Engineering
Cuddapah
in memory of
myparents
© 1981, 1996, Tata McGraw-HiU Publishing Company Limited
Sixth reprint 2002
RXDYYRQARALZR
No part of this publication can be reproduced in any form or by any
means without the prior written permission of the publishers
This edition can be exported from India only by the publishers,
Tata McGraw-HiU Publishing Company Limited
ISBN 0-07-461366-4
Published by Tata McGraw-Hili Pub1ishing Company Limited,
7 West Patel Nagar, New Delhi 110008, and printed at
Ram BookBinding House, New Delhi 110020.
Preface to the Second Edition
Encouraged by the tremendous response to the first edition, this book has been
revised keeping in mind the valuable suggestions received from the reviewers,
publishers, readers and colleagues.
Keeping the basic approach of the first edition intact, the second edition has
been written to make the book broad-based and gain wider acceptance amongst
teachers and students.
Chapter 3on 'Theory of vectors and matrices' has now been removed from
the main text and placed in Appendix A.
Chapter 7, 'Rolling loads-Influence lines' has been completely revised and
a number of illustrative examples have been added for better conceptual
understanding.
This edition incorporates new chapters on 'Cables and suspension bridges'
(Chapter 8), 'Column analogy' (Chapter 14), and 'Plastic analysis of steel
structures' (Chapter 18).
Chapter 12on 'Moment distribution methods' has been expanded by including
topics like 'No sheer moment distribution, and adding concept building illustrative
examples.
Chapters 17and 18on 'Flexibility and stiffness matrix methods of analysis'
have been rewritten to include a large number of worked-out examples. In
these chapters, the emphasis has been laid on computer applications for which
flow charts for flexibility and stiffness have been provided.
I hope that the above changes in the second edition will widen the scope of
the book and meet the approval of the students, teachers and practising engineers.
Further suggestions for the improvement of the book are welcome.
In the end, I wish to express my sincere thanks to the publishers for their
expert guidance in bringing out this revised edition.
I also appreciate the ardous effort of Shri K Subba Reddy in typing the
manuscript.
C S REDDY
Preface to the First Edition
The use of computers for structural analysis has completely altered the method of
presentation of structural theory. While the student isexpected tobe familiar with
this presentation, itisfarmore important that heunderstands thebasic principles of
~tructural analysis.
This book endeavours to present in one volume, the classical as well as matrix
methods ifstructural analysis. Itisexpected that for sometime tocome, the student
win berequired tostudy both these approaches, forthematrix methods are notvery
different from classical methods-the only difference is in the emphasis laid in
formulating them soastobesuitable forcomputer programming. Anunderstand ing
of the basic principles in both these methods necessarily requires the solving of
simple problems using hand computations.
This book is intended for a course in structural analysis following the usual
course inmechanics ofsolid or,asitismore commonly called, strength ofmaterials.
Itaims to provide asmooth transition from the classical approaches that are based
on physical behaviour of structures in terms of their deflected shapes to aformal
treatment of ageneral class of structures by means of matrix formulation.
Chapters Iand 2 deal with basic principles of structural analysis of simple
structures using only equilibrium equations. Chapter 3isdevoted to the theory of
vectors andmatrices. This review isintended toprovide thebackground material for
the analysis ofspace trusses inChapter 5and matrix methods ofstructural analysis
later in Chapters 14 to 17. Chapter 4 deals with the analysis of plane trusses.
Chapters 6 and 7 deal with displacement calculations by geometric and energy
methods respectively. Chapter 8isdevoted totheapproximate analysis ofstatically
indeterminate structures, while Chapter 9discusses the analysis for moving loads
by influence lines.
Chapters 10 to 13 are devoted to the analysis of statically indeterminate
structures using classical methods, such as consistent displacement, slope-
deflection and moment distribution. Kani's method ispresented in some detail in
Chapter I~.
x Preface to the [<,rstt.dltlOn
Chapters 14and 15 discuss the preliminaries required for the formulation of
matrix methods of structural analysis. The flexibility and stiffness methods of
analysis are presented inChapters 16and 17.Simple examples needing only hand
computations have been illustrated in these chapters. However, the matrix formu-
Contents
lation of the problems and computation techniques employed are suitable for
computer programmes.
Abook such asthis, devoted tobasic aspects ofstructural analysis cannot claim
tocontain anyoriginal work, and only material collected overtheyears ispresented.
The author gratefully acknowledges the sources he has consulted.
The author sincerely thanks allhiscolleagues and students whohelped inwriting
this text. The author isgrateful to his wife for her understanding and forebearance
during the long hours he spent working onthe manuscript. Aword ofappreciation
isdue to his children who refrained from disturbing him. The author also thanks
Usharanjan Bhattacherjee for typing the manuscript and S.P. Hazra formakiQg the vii
Preface to the Second Edition
final diagrams. ix
Prefa(:e to the First Edition
xviii
Sf Unitsfor Structural Engineers
C SREDDY
1
Chapter 1: INTRODUCflON TO STRUCTURAL ANALYSIS
1
1.1 Forms of Structures
4
1.2 Analysis and Design
4
1.3 Loads and Forces
'5
1.3.1 Dead Load
5
1.3.2 Imposed Loads and Forces
7
1.3.3 Load Combinations
8
1.4 Idealizatiod of Structures
9
1.5 Supports and Connections-Conventional Representation
9
1.6 Elastic and Linear Behaviour of Structures
11·
1.7 Principle of Superposition
13
Chapter 2: STATICS OF STRUCTURES
13
2.1 Equations of Equilibrium
14
2.2 Free-body Diagrams
19
2.3 Sign Convention
22
2.4 Simple Cable and Arch Structures
22
2.4.1 Cables
26
2.5 Arches
26
2.5.1 Theoretical Arch or Line of Thrust
27
2.5.2 Actual Arch
27
2.5.3 Types of Arches
34
2.6 Graphic Statics
34
2.6.1 General
2.6.2 Resultant of 1\vo Concurrent Forces 34 6.3 Strain Energy in Members 120
2.6.3 Resultant of Sewnl Forces in a Plane 35 6.3.1 Axially Loaded Members 120
2.6.4 Equilibriant 36 6.3.2 Members Under Bending 121
2.6.5 Funicular Polygon 37 6.3.3 Members Under Shearing 121
2.6.6 Funicular Polygon through 1\vo Points 39 6.3.4 Circular Members in Torsion 122
6.4 Energy Relations in Structural Theory 123
6.4.1 Law of Conservation of Energy 123
Chapter 3: PLANE TRUSSES 47 6.5 Virtual Work 126
6.5.1 Virtual Work on a Rigid Body 127
3.1 Introduction 47 6.5.2 Virtual Work on an Elastic Body 128
3.2 Plane Truss 47 6.6 Betti's and Maxwell's Laws of Reciprocal Deflections 130
3.3 Geometric Stability and Static Determinancy of Trusses 48 6.7 Applications of Virtual Work 132
3.4 Analysis of Trusses 51 6.8 Deflection of Trusses and Frames 140
3.4.1 Assumptions 51 6.9 Castigliano's Theorems 148
3.4.2 Methods of Analysis 51
3.4.3 Subdivided Truss 57..
3.5 Compound and Complex Trusses 59 Chapter 7: ROLLING LOADS AND INFLUENCE LINES 162
3.6 Graphical Analysis of Trusses 61
3.6.1 Analysis of a Simple Truss 61 7.1 Introduction 162
3.6.2 Analysis of a Fink Roof.Truss 63 7.2 A Single Concentrated Load 162
I 7.3 Unifomlly Distributed Load Longer Than the Span 165
Chapter 4: SPACE TRUSSES 69 7.4 Uniformly Distributed Load Shorter Than Span 167
7.5 1\vo Concentrated Loads 170
4.1 Introduction 69 7.6 Series of Concentrated Loads 173
4.2 Simple Space Truss 69 7.6.1 Maximum S.F. at a Section 174
7.6.2 Maximum Bending Moment Under a Given Load 174
4.3 Types of Supports 70 7.6.3 Maximum Bending Moment at a Given Section 174
4.4 Equilibrium and Stability Conditions 70 7.6.4 A9solute Maximum Shear and Moment in Beams 175
4.5 Analysis of Space Trusses • 72 7.7 Equivalent U.D.L 181
7.8 Influence Lines 183
Chapter 5: DISPLACEMENTS-GEOMETRIC METHODS 83 7.8.1 Introduction 183
7.8.2 Influence lines 183
5.1 Deflected Shapes 83 7.8.3 Uses of Influence Lines 185
5.2 Moment-area Method 86 7.8.4 Distributed Loads 193
5.3 Conjugate Beam Method 101 7.8.5 Influence Lines for Statically Determinate Frames
504 Deflection of Trusses-Graphical Method 107 and Beams with Hinges 201
5.4.1 Williot-Mohr Diagram 107 7.9 Influence Lines for Panelled Beams 204
7.10 Influence Lines for Truss Members 207
7.11 Influence Lines for Three-hinged Arches 213
Chapter 6: DISPLACEMENTS-ENERGY METHODS 116 7.11.1 Influence Line for Horizontal Reaction H 213
7.11.2 Influence Line Diagram for Moment 215
6.1 Introduction 116 7.11.3 Influence Line Diagrams for Radial Shear and
6.2 Forms of Elastic Strain Energy 117 Normal Thrust 215
6.2.1 Axial Stress 117 7.11.4 Absolute Maximum Moment in a Three-Hinged
6.2.2 Shearing Stress 118 Parabolic Arch 221
6.2.3 Multi-Axial State of Stress 119 7.12 Influence Lines from Deflected Shapes 226
xiv Contents
330
10.7 1\vo-hinged Arches
349
Chapter 8: CABLES AND SUSPENSION BRIDGES 236 10.8 Influence Lines for Continuous Members
8.1 Introduction 236
364
Chapter 11: SLOPE-DEFLECTION METHOD
8.2 Cables 237
8.2.1 Equation of the Cable 237 364
11.1 Introduction
8.2.2 Horizontal Tension in the Cable 238 365
11.2 Sign Convention
8.2.3 Tension in Cable Supported at Different Levels 239 Development of Slope-deflection Equations 366
11.3
8.2.4 Length of the Cable 242 11.4 Analysis of Continuous Be.ams 368
8.2.5 Effect on Cable Due to Change of Temperature 244 Analysis of Frames With No Lateral Translation of Joints 373
11.5
8.3 Stiffening Girders 249 Analysis of Frames With Lateral Translation of Joints 377
11.6
8.4 Three-hinged Stiffening Girder 249
8.4.1 Single Concentrated Load 250 386
8.4.2 Influence Line for H 252 Chapter 12: MOMENT DISTRmUTION METHOD
8.4.3 I.L. for B.M. at Section X 252
386
8.4.4 Maximum B.M. Under U.D.L. Longer than Span 255 12.1 Introduction
387
8.5 Influence Lines for Stiffening Girder 256 12.1.1 Absolute and Relative Stiffness of Members 387
8.5.1 Influence Line for Shear Force 256 12.1.2 Carry Over Factor (C.O.F.) 388
8.5.2 Uniformly Distributed Load Longer than Span 258 12.1.3 Distribution Factor (D.F.) 389
8.6 1\vo-hinged Stiffening Girder 266 12.2 Devewpmem of Method 400
Analysis of Frames With No Lateral Translation of Joints
8.6.1 Influence Lines for a Single Concentrated Load 12.3 404
Rolling Over the Girder 266 12.4 Analysis of Frames With Lateral Translation of Joints
419
8.6.2 Uniformly Distributed Load Longer than Span 270 12.5 Symmetrical Frames 426
12.6 Multistorey Frames 429'
12.7 No-shear Moment Distribution
Chapter 9: APPROXIMATE ANALYSIS OF STATICALLY
INDETERMINATE STRUCTURES , 276 • 439
Chapter 13: KANI'S METHOD
9.1 Introduction 276 439
9.2 Methods of Analysis 277 13.1 Introduction 439
9.2.1 General 277 13.2 Basic Concept
453
9.2.2 Indeterminate Trusses 277 13.3 Frames Without Lateral Translation of Joints 460
9.2.3 Mill Bents 279 13.4 Frames With Lateral Translation of Joints
9.2.4 Portal Frames 282 13.5 General Case-Storey Columns Unequal in Height and 482
9.2.5 Continuous Beams and Building Frames 283 Bases Fixed or Hinged
9.3 Portal Method 285
9.4 Cantilever Method 287 Chapter 14: COLUMN ANALOGY 493
493
Chapter 10: INDETERMINATE STRUCTURES- 14.1 Introduction
493
COMPATIBILITY METHODS 296 14.2 Development of the Method
495
14.2.1 Sign Convention
500
10.1 Introduction 296 14.2.2 Stiffness and Carry-over Factors 507
10.2 Degree of Indeterminancy and Stability of Structures 297 14.3 Analysis of Frames by the Column Analogy Method 511
10.3 Analysis of Indeterminate Structures 302 14.3.1 Closed Frames 514
10.4 Flexibility Coefficients 305 14.4 Gable Frames
10.5 Theorem of Three Moments 318
10.6 The Method of Least Work 326
14.5 Analysis of Unsymmetrical Frames 517 Chapter 18: STIFFNESS OR DISPLACEMENT METHOD OF
ANALYSIS 622
Chapter 15: MATRIX METIIODS OF STRUCTURAL ANALYSIS 525
18.1 Introduction 622
18.1.1 Stiffness Method-Steps to be Followed 622
15.1 Introduction
525
18.1.2 Effect of Support Displacements, Temperature
15.2 Stiffness and Flexibility Coefficients 529
Changes, etc. 626
15.3 Member Stiffness and Flexibility Matrices 545,
15.4 Energy Concepts in Structures 552 18.2 Development of Stiffness Matrix for a Pin-jointed
15.5 Maxwell's and Betti's Reciprocal Deflections 557 Structure 636
15.6 Strain Energy in Elements and Systems 560 18.2.1 Member Forces 638
18.3 Development of Method for a Structure Having Forces at
all Degrees of Freedom 644
Chapter 16: TRANSFORMATION OF INFORMATION 'IN 18.3.1 Computer Programme for the Stiffness Analysis
STRUCTURES TIIROUGH MATRICES 567 of Kinematically Determinate System 654
18.4 Development of Method for a General Case 654
16.1 Transformation of System Forces to Element Forces 567 18.4.1 Computer Programme for the Stiffness Analysis of
16.2 Transformation of System Displacements to Element Kinematically Indeterminate Structures 665
Displacements 568 18.4.2 Temperature Stresses, Lack of Fit, Support
16.3 Transformation of Element "Flexibility Matrices to System Settlements, etc. 665
Flexibility Matrix 570 18.5 Direct Stiffness Method 667
16.4 Transformation of Element Stiffness Matrices to System 18.6 Analysis by Tridiagonalization of Stiffness Matrix 679
Stiffness Matrix 572 18.7 Comparison of Flexibility and Stiffness Methods 691
16.5 Transformation of Forces and Displacements in General 574
16.6 Tranformation of Information from Member Coordinates
Chapter 19: PLASTIC ANALYSIS OF STEEL STRUCTURES 696
to Structure Coordinates and Vice versa 576
19.1 Introducti9n 696
Chapter 17: FLEXIBILITY OR FORCE METIIOD OF • 19.2 Stress-strain Curve 697
ANALYSIS 583 19.3 Plastic Moment 697
19.3.1 Plastic Modulus, Shape Factor 699
17.1 Introduction 583 19.3.2 Load Factor 700
17.1.1 Flexibility Method-Steps to be Followed 583 19.3.3 Mechanism of Failure 700
17.1.2 Sign Convention 584 19.4 Methods of Analysis 709
17.1.3 Effect of Displacements at Releases 591 19.4.1 Statical Method of Analysis 710
17.2 Generalised Method of Analysis 594 19.4.2 Mechanism Method of Analysis 710
17.3 Statically Determinate Structures 594 19.5 Gable Frames or Frames with Inclined Members 715
17.3.1 Computer Programme for Statically Determinate 19.6 1\\'0 Bay Portal Frame 719
Structure
601
17.3.2 Flow Chart 601 Appendix A 724
17.4 Statically Indeterminate Structures 603 Appendix B 754
17.4.1 Computer Programme for Statically Indeterminate Appendix Cl 755
Structures 614 Appendix C2 756
17.4.2 Flow Chart
614 Select Bibliography 757
17.5 Temperature Stressses, Lack of Fit, Support Settlements,
,etc. Answers to Problems for Practice 759
614
Index 776
Sf Units xix
X 103 mm3 and moment of inertia x 106 mm4. Very small sections, such
3
as light gauge steel sha~es may be listed as section modulus x mm and
moment of inertia x 10 mm4
•
Mass and Density
SI Units for Structural
Mass is a basic quantity in the system. The base unit of mass is the
kilogram (kg). The use of kg should not be confused with the old metric
Engineers
force called kgf.
Material quantities are measured in mass units rather than in weight
or force· units. Thus, the mass per length of a steel beam is expressed in
kg/m, gravity floor loading in kglm2 and the mass of an object in kg.
Mass density is given in kg/m3. In contrast to weight units, these quan-
tities do not depend upon the acceleration due to gravity. Weight is not
used directly in the SI system, but force is obviously caused by gravity
acting on mass.
Force, Moment and Stress
The unit of force is the newton (N), which is the force required to give
1 kg mass 1 m/s2 acceleration. Thus 1 N is 1 kg.m/s2. The newton is a
derived unit that is independent of the acceleration due to gravity. A kilo-
newton (1000 newtons) or kN, which is about 100 kgf, is a convenient
The international system of units (System Internationale d'Unites), com-
quantity in structural analysis and design. Approximating the acceleration
monly called SI, is being adopted allqver the world as a uniform meas-
due to gravity as 9,.81 m/s2, a kg of mass exerts a force of 9.81 N on
urement system. While the complete transition from customary units to
• its support point.
the SI system may take years, the use of SI units in the fields of en- The stress unit is newton per square metre (N/m2) called pascal (Pa).
gineering and science is proceeding rather ~apidly, and it will soon be- This is a very small unit (1 kglcm2 approximates to 98100 Pa) and be-
come necessary for the modern civil engineer to gain experience in using
comes practical only when used with a prefix (k or M). The most con-
the SI system. Fortunately, the chanje.over from the now common MKS
venient SI stress unit for structures is 1,000,000 Pa, the mega pascal or
units to SI units is quite simple, unni<~ the changeover from FPS to MK5 MPa, which is identical to MN/m2 or N/mm2• The modulus of steel is
units. In this book, SI units have been used throughout, with only mi,nor
about 200,000 MPa in SI units.
modifications, to suit the requirements of the engineering world.
Surface loadings and allowable soil pressures have the units of pressure
The basic and derived units for various categories of measurement are
or stress and thus may be expressed in Pascals, but common usage will
discussed in the following sections. dictate their expression in kN/m2 or similar units. Surface loads in par-
ticular are well expressed in kN/m2 because their effects must be con-
verted into kN during structural analysis.
TYPICAL BASIC UNITS Moment is expressed in N.m or kN.m. These units are convenient since
1 N.m is close to 10 kg.cm and I kN.m is close to 1/10 t.m.
Geometry
Angle, Temperature, Energy and Power
The basic unit of length is the metre (m), which together with the mil-
limetre (mm) is used exclusively for geometrical quantities. Although the Plane angles are measured in radians (rad), but degrees are also used.
centimetre (cm) is a convenient quantity, its use is generally avoided in Temperature in the 51 system should be expressed in Kelvin (K) but the
the SI system. The use of mm for section modulus and moment of inertia use of degrees Celsius ("C), formerly called centigrade, is also permis-
involves large numbers for the majority of common flexural members. sible. Kelvin and Celsius are equal for temperature changes since an in-
This problem is met by listing steel sections properties as section modulus crement of 1°Cequals an increment of I K. Energy is expressed in joules
(J), where I J is I N.m. The unit of power is the watt (W) which is
equal to one joule per second (I/s). 1
Some Simple Rules to be Observed in Using SI Units
Prefixes are to be selected from the following table, in which each prefix
is a multiple of 1000. Introduction to Structural
Prefix Symbol
MultIplying factor
Analysis
glga
G 109
mega M 6
10
kilo k 103
milli m 10-3
micro
~ 10-6
nano n 10-9
Compound units, such as for moments, are written with a dot to indi-
cate multiplication, such as kN.m (kilonewton-metre).
CONVERSION FACTORS FOR SI UNITS
=
(Standard Gravitational Acceleration 9.80665 m/s2) 1.1 FORMS OF STRUCTURES
MKS To Sf Units
I. ForcelLoad/Weight ] kgf (kg) = 9.80665 N Any civil engineering structure is conceived keeping in mind its intended
] tonne (t) = 9.80665 kN • use, the materials available, cost and aesthetic considerations. The struc-
2. ForcelLoad/Weight I kgf/m = 9.80665 N/m tural analyst encounters a great variety of structures and these are briefly
per Unit Length I tf/m = 9.8Q665 kN/m reviewed here.
3. Unit Weight I kgf/m3 = 9.80665 N/m3 One of the simplest structures is a simply supported beam, supported
4. Stress/Pressure! on a pin at one end and a roller at the other (Fig. I.la). Such a beam,
Modulus of Elasticity 1 kgf/m2 = 9.80665 N/m2 it may be recalled from the fundamentals of strength of materials, is quite
=
9.80665 Pa
1 kgf/cm2 = 98066.5 N/m2
=
98066.5 Pa
= 98.0665 kN/m2
5. Moment of Force/ I kgf.m = 9.80665 N.m
Bending moment! 1 kgf.cm = 98.0665 x 10-3 N.m
Torque I tf.m = 9.80665 kN.m