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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Discover the theory of structural stability and its applications in crucial areas in engineering Structural Stability Theory and Practice: Buckling of Columns, Beams, Plates, and Shells combines necessary information on structural stability into a single, comprehensive resource suitable for practicing engineers and students alike. Written in both US and SI units, this invaluable guide is perfect for readers within and outside of the US. Structural Stability Theory and Practice: Buckling of Columns, Beams, Plates, and Shell offers: * Detailed and patiently developed mathematical derivations and thorough explanations * Energy methods that are incorporated throughout the chapters * Connections between theory, design specifications and solutions * The latest codes and standards from the American Institute of Steel Construction (AISC), Canadian Standards Association (CSA), Australian Standards (SAA), Structural Stability Research Council (SSRC), and Eurocode 3 * Solved and unsolved practice-oriented problems in every chapter, with a solutions manual for unsolved problems included for instructors Ideal for practicing professionals in civil, mechanical, and aerospace engineering, as well as upper-level undergraduates and graduate students in structural engineering courses, Structural Stability Theory and Practice: Buckling of Columns, Beams, Plates, and Shell provides readers with detailed mathematical derivations along with thorough explanations and practical examples.
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Seitenzahl: 671
Veröffentlichungsjahr: 2020
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Table of Contents
Cover
Title Page
Copyright
Dedication
Foreword
Preface
About the Companion Website
1 Structural Stability
1.1 Introduction
1.2 General Concepts
1.3 Rigid Bar Columns
1.4 Large Displacement Analysis
1.5 Imperfections
References
2 Columns
2.1 General
2.2 The Critical Load According to Classical Column Theory
2.3 Effective Length of a Column
2.4 Special Cases
2.5 Higher-Order Governing Differential Equation
2.6 Continuous Columns
2.7 Columns on Elastic Supports
2.8 Eccentrically Loaded Columns
2.9 Geometrically Imperfect Columns
2.10 Large Deflection Theory of Columns
2.11 Energy Methods
References
3 Inelastic and Metal Columns
3.1 Introduction
3.2 Double Modulus Theory
3.3 Tangent Modulus Theory
3.4 Shanley's Theory for Inelastic Columns
3.5 Columns with Other End Conditions
3.6 Eccentrically Loaded Inelastic Columns
3.7 Aluminum Columns
3.8 Steel Columns
References
4 Beam columns
4.1 Introduction
4.2 Basic Differential Equations of Beam Columns
4.3 Beam Column with a Lateral Concentrated Load
4.4 Beam Columns Subjected to Moments
4.5 Columns with Elastic Restraints
4.6 Beam Columns with Different End Conditions and Loads
4.7 Alternate Method Using Basic Differential Equations
4.8 Continuous Beam Columns
4.9 Slope Deflection Equations for Beam Columns
4.10 Inelastic Beam Columns
4.11 Design of Beam Columns
References
5 Frames
5.1 Introduction
5.2 Critical Loads by the Equilibrium Method
5.3 Critical Loads by Slope Deflection Equations
5.4 Critical Loads by Matrix and Finite Element Methods
5.5 Design of Frame Members
References
6 Torsional Buckling and Lateral Buckling of Beams
6.1 Introduction
6.2 Pure Torsion of Thin-Walled Cross-Sections
6.3 Non-uniform Torsion of Thin-Walled Open Cross-Sections
6.4 Torsional Buckling of Columns
6.5 Torsional Buckling Load
6.6 Torsional Flexural Buckling
6.7 Torsional Flexural Buckling: The Energy Approach
6.8 Lateral Buckling of Beams
6.9 The Energy Method
6.10 Beams with Different Support and Loading Conditions
6.11 Design for Torsional and Lateral Buckling
Problems
References
7 Buckling of Plates
7.1 Introduction
7.2 Theory of Plate Bending
7.3 Buckling of Thin Plates
7.4 Boundary Conditions
7.5 Buckling of Rectangular Plates Uniformly Compressed in One Direction
7.6 The Energy Method
7.7 Buckling of Circular Plates
7.8 The Finite Difference Method
7.9 The Finite Element Method
7.10 Large Deflection Theory of Plates
7.11 Inelastic Buckling of Plates
7.12 Ultimate Strength of Plates in Compression
7.13 Local Buckling of Compression Elements: Design
References
8 Buckling of Shells
8.1 Introduction
8.2 The Large Deflection Theory of Cylindrical Shells
8.3 The Linear Theory of Cylindrical Shells
8.4 Donnell's Linear Equations of Stability of Cylindrical Shells
8.5 The Energy Method
8.6 Application of the Linear Stability Equations
8.7 Failure and Post-buckling Behavior of Cylindrical Shells
8.8 General Shells
8.9 Shells of Revolution
References
Answers to the Problems
Appendix A: Slope Deflection Coefficients for Beam Column Buckling
Appendix B: Torsion Properties of Thin-Walled Open Cross-Sections
Appendix C: Calculus of Variations
C.1 Calculus of Variations
Appendix D: Euler Equations
Reference
Appendix E: Differential Geometry in Curvilinear Coordinates
E.1 Curvature
E.2 Surfaces
E.3 Principal Curvatures
E.4 Derivatives of Unit Vectors along Parametric Lines
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Effective length factor
K
for columns.
Table 2.2 Effective length factor
K
for column in Figure 2.8, with L1 = L2 = L...
Table 2.3 Critical loads for elastically restrained columns.
Table 2.4 Load and mid-length deflection from the large deflection analysis.
Table 2.5 Load and deflection at the top of the cantilever column by the larg...
Chapter 3
Table 3.1 Calculated quantities for column strength curve.
Chapter 4
Table 4.1 Load deflection relation for the beam column.
Table 4.2 Amplification factorsΨ and
C
m
.
Chapter 5
Table 5.1 Moment distribution for gravity load.
Table 5.2 Moment distribution for wind loads.
Chapter 6
Table 6.1 Values of
α
for simply supported I beams with concentrated load...
Table 6.2 Values of
α
for simply supported I beams with uniformly distrib...
Table 6.3 Width thickness ratios for compression elements in flexure.
Chapter 7
Table 7.1 Values of the factor
k
for an axially compressed plate when two load...
Table 7.2 Values of
k
for axially compressed plate when two sides are simply s...
Table 7.3 Factor
k
for axially compressed plate when two loaded sides are simp...
Table 7.4 Factor
k
for axially compressed plate when the two loaded sides are ...
Table 7.5 Effect of the number of terms in the series for
w(x, y
) on
k
.
Table 7.6 Values of
k
for
m
=
n
= 10.
List of Illustrations
Chapter 1
Figure 1.1 Types of equilibrium: (a) stable; (b) unstable; (c) neutral.
Figure 1.2 Bifurcation equilibrium paths: (a) Symmetric stable bifurcation; ...
Figure 1.3 Post-buckling equilibrium paths: (a) Limit load instability; (b) ...
Figure 1.4 Rigid bar under axial force: (a) Rigid bar with axial load; (b) F...
Figure 1.5 Two rigid bars under axial load: (a) Two rigid bars with axial fo...
Figure 1.6 Three-member truss with rigid bars: (a) Three member truss; (b) D...
Figure 1.7 Displacement path of three-member truss.
Figure 1.8 Three rigid bars with two linear springs: (a) Three rigid bars; (...
Figure 1.9 Mode shapes of the three rigid bars with linear springs: (a) Asym...
Figure 1.10 Equilibrium path of rigid bar in Figure 1.4.
Figure 1.11 Rigid bar connected to translational and rotational springs: (a)...
Figure 1.12 Displacement path of rigid bar supported by translational and ro...
Figure 1.13 Displacement path of two rigid bars connected by rotational spri...
Figure 1.14 Imperfect rigid-bar column with rotational spring at the base: (...
Figure 1.15 Equilibrium path of the rigid bar imperfect column with rotation...
Figure 1.16 Imperfect column with two rigid bars and two rotational springs:...
Figure 1.17 Displacement path of two imperfect rigid bars column connected b...
Figure P1.1
Figure P1.3
Figure P1.5
Figure P1.6
Chapter 2
Figure 2.1 Pinned-pinned column under axial load: (a) Pinned-pinned column; ...
Figure 2.2 Load displacement path.
Figure 2.3 Mode shapes of buckling: (a) First mode shape; (b) Second mode sh...
Figure 2.4 Fixed-fixed column under axial load and symmetric mode: (a) Fixed...
Figure 2.5 Fixed-fixed column under axial load and anti-symmetric mode:(a) F...
Figure 2.6 Cantilever column under axial load: (a) Cantilever column; (b) Fr...
Figure 2.7 Fixed-pinned column under axial load: (a) Fixed-pinned column; (b...
Figure 2.8 Pinned-pinned column under axial forces at the ends and at the in...
Figure 2.9 Cantilever column under axial forces at the free end and at the i...
Figure 2.10 Column under axial force shown with internal forces: (a) Column ...
Figure 2.11 Pinned-pinned column under axial load.
Figure 2.12 Cantilever column under axial load.
Figure 2.13 Pinned-guided column under axial load.
Figure 2.14 Continuous column under axial load.
Figure 2.15 Elastically restrained column: hinged base.
Figure 2.16 Elastically restrained column: fixed base.
Figure 2.17 Eccentrically loaded column: (a) Simply supported column; (b) Fr...
Figure 2.18 Load versus maximum total deflection for eccentrically loaded co...
Figure 2.19 Geometrically imperfect column: (a) Initially bent column; (b) F...
Figure 2.20 Load versus maximum total deflection for imperfect column.
Figure 2.21 The Southwell plot.
Figure 2.22 Large deflection under axial load of a pinned-pinned column: (a)...
Figure 2.23 Load deflection graph for pinned-pinned column from large deflec...
Figure 2.24 Large deflection under axial load of cantilever column: (a) Larg...
Figure 2.25 Load deflection graph for a cantilever column from large deflect...
Figure 2.26 Buckling of pinned-pinned column: (a) Pinned-pinned column; (b) ...
Figure 2.27 Deflected shape.
Figure P2.1
Figure P2.2
Figure P2.4
Figure P2.6
Figure P2.7
Figure P2.9
Chapter 3
Figure 3.1 Stress strain diagram for a column under compression.
Figure 3.2 Inelastic pinned-pinned column under axial force: (a) Deflected s...
Figure 3.3 Cross-section of the column.
Figure 3.4 Stresses and strains by double modulus theory: (a) Deformation du...
Figure 3.5 Rectangular cross-section.
Figure 3.6 Inelastic pinned-pinned column by tangent modulus theory: (a) Def...
Figure 3.7 Shanley's rigid bar column with the elastic-plastic link: (a) Sha...
Figure 3.8 Post-buckling behavior of an idealized column.
Figure 3.9 Inelastic column under eccentric load: (a) Eccentrically loaded c...
Figure 3.10 Inelastic eccentrically loaded column: (a) Eccentrically loaded ...
Figure 3.11 Average compressive stress
σ
0
versus mid-height deflection
Figure 3.12 Stress ratio
(
σ
/
σ
0.2
)
versus strain diagram of aluminu...
Figure 3.13 Column strength curve for aluminum alloy 6061-T6.
Figure 3.14 Hot rolled wide flange steel sections: (a) Residual stress in a ...
Figure 3.15 Idealized steel I section and effect of residual stresses.
Figure 3.16 Elastic–plastic idealized I section.
Figure 3.17 Column strength curves for wide flange steel sections.
Figure 3.18 Euler, CRC, and SSRC column strength curves.
Figure 3.19 ECCS (a
0
, a, b, c, and d), AISC (LRFD and ASD), and Euler column...
Figure 3.20 Wide flange steel column.
Chapter 4
Figure 4.1 Beam column with a distributed lateral load and axial force: (a) ...
Figure 4.2 Beam column with a concentrated lateral load and axial force: (a)...
Figure 4.3 Beam column with several concentrated loads.
Figure 4.4 Beam column with a uniformly distributed load and axial force: (a...
Figure 4.5 Beam column with partial distributed load.
Figure 4.6 Beam column with triangular load.
Figure 4.7 Beam column with span moment and axial force: (a) Span moment on ...
Figure 4.8 Beam column with end moment and axial force: (a) End moment on be...
Figure 4.9 Beam column with the end moment on the right support and axial fo...
Figure 4.10 Beam column loaded with two end moments: (a) Beam column with tw...
Figure 4.11 Free body diagram of beam column loaded with two end moments.
Figure 4.12 Beam column in a double curvature due to end moments.
Figure 4.13 Elastically restrained column.
Figure 4.14 Pinned-fixed beam column with concentrated force: (a) Fixed-pinn...
Figure 4.15 Pinned-fixed beam column with uniformly distributed load.
Figure 4.16 Fixed-fixed beam column with a concentrated force.
Figure 4.17 Fixed-fixed beam column with uniformly distributed load.
Figure 4.18 Continuous beam column.
Figure 4.19 Two-span continuous beam column.
Figure 4.20 Two-span beam column with one end fixed.
Figure 4.21 Beam column with ends rotated through
θ
A
and
θ
B
.
Figure 4.22 Beam column with end rotations and relative end displacement.
Figure 4.23 Beam column subjected to end moments and transverse load: (a) Tr...
Figure 4.24 Beam column in a single curvature.
Figure 4.25 Elastic-perfectly plastic beam column subjected to axial force a...
Figure 4.26 Stress distribution in case 1 of an inelastic beam column: (a) S...
Figure 4.27 Stress and strain distributions in Case 2 of an inelastic beam c...
Figure 4.28 Column curves for the inelastic beam column.
Figure 4.29 Load deflection relation for the beam column.
Figure 4.30 Interaction curve for a beam column.
Figure 4.31 Beam column subjected to equivalent moment: (a) Equivalent momen...
Figure 4.32 Beam column in a braced frame.
Figure P4.1
Figure P4.2
Figure P4.3
Figure P4.4
Figure P4.6
Figure P4.7
Chapter 5
Figure 5.1 Symmetrical frame fixed at the base having no sidesway buckling: ...
Figure 5.2 Frames fixed at the base having no sidesway with rigid and extrem...
Figure 5.3 Portal frame fixed at the base with sidesway buckling: (a) Sidesw...
Figure 5.4 Frames fixed at the base having sideway with rigid and flexible b...
Figure 5.5 Symmetrical portal frame hinged at the base with prime bending an...
Figure 5.6 Plot of
k
c
L
versus
k
b
L
.
Figure 5.7 Portal frame fixed at the base without sidesway by slope deflecti...
Figure 5.8 Portal frame fixed at the base with sidesway by slope deflection ...
Figure 5.9 Two-story frame hinged at the base without sidesway by the slope ...
Figure 5.10 Two-bay frame hinged at the supports without sidesway by the slo...
Figure 5.11
θ
B
versus
k
c
L
for a hinged frame with prime bending and no ...
Figure 5.12 Frame fixed at the supports with transverse load and no sidesway...
Figure 5.13
θ
B
versus
k
c
L
for a fixed frame with prime bending and no s...
Figure 5.14 Frame hinged at the supports with transverse load and sidesway....
Figure 5.15
θ
B
versus
k
c
L
for a hinged frame with prime bending and sid...
Figure 5.16 Symmetrical buckling without sidesway in a box frame.
Figure 5.17 Symmetrical buckling of multistory multibay frame without sidesw...
Figure 5.18 Beam element nodal displacements and forces: (a) Frame element d...
Figure 5.19 Global and local coordinates.
Figure 5.20 Distributed forces and equivalent nodal forces.
Figure 5.21 Beam with distributed load and equivalent nodal forces: (a) Beam...
Figure 5.22 Structure and element forces and displacements: (a) Structure co...
Figure 5.23 Second-order frame analysis with the Newton-Raphson iteration me...
Figure 5.24 Large deflection analysis kinematics relations.
Figure 5.25 Internal forces in the element: (a) Axial force and the end mome...
Figure 5.26 Column in a braced frame along with adjoining members.
Figure 5.27 Nomograph for effective length factor
K
of columns in braced fra...
Figure 5.28 Column in unbraced frame along with adjoining members.
Figure 5.29 Member equilibrium for column
BC
.
Figure 5.30 Nomograph for effective length factor
K
of columns in unbraced f...
Figure 5.31 Two-story, two-bay frame's geometry and loading: (a) Gravity loa...
Figure 5.32 End moments and end reactions in columns due to −100 k.ft. (−137...
Figure 5.33 Free body diagrams for the frame under gravity load.
Figure 5.34 Bending moment diagrams for the gravity load from the first-orde...
Figure 5.35 Bending moment diagrams for the wind load from the first-order a...
Figure 5.36 Free body diagrams for the frame under wind load.
Figure 5.37 Two-story, two-bay frame.
Figure P5.1
Figure P5.3
Figure P5.5
Figure P5.6
Chapter 6
Figure 6.1 Twisting of an I-section free to warp: (a) I-section under unifor...
Figure 6.2 Noncircular section under the action of applied torque: (a) Flang...
Figure 6.3 Thin-walled open section bar under non-uniform torsion: (a) Thin ...
Figure 6.4 Channel cross-section.
Figure 6.5 Cross-section having zero warping constant
(
C
w
= 0)
Figure 6.6 Torsional buckling of a doubly symmetric column under axial force...
Figure 6.7 Torsional buckling of an open thin-walled section.
Figure 6.8 Torsional flexural buckling of a thin-walled section: (a) Displac...
Figure 6.9 Singly symmetric section.
Figure 6.10 Equal angle cross-section.
Figure 6.11 Channel cross-section.
Figure 6.12 Torsional flexural buckling of a column: (a) Deformed shape of t...
Figure 6.13 Lateral buckling of beams.
Figure 6.14 Lateral buckling of simply supported, narrow rectangular beam un...
Figure 6.15 Internal moment components: (a) Moments in the
xy
plane; (b) Mom...
Figure 6.16 Lateral buckling of simply supported I beam under pure bending....
Figure 6.17 Lateral buckling of simply supported I beam subjected to a load ...
Figure 6.18 Internal moment components during lateral buckling.
Figure 6.19 Lateral buckling of a simply supported I beam subjected to mid-s...
Figure 6.20 Lateral buckling of I beam acted upon by a concentrated load at ...
Figure 6.21 Internal moment components of a buckled cantilever beam.
Figure 6.22 Lateral buckling of cantilever beam under uniform moment.
Figure 6.23 Lateral buckling of I beam subjected to concentrated load at the...
Figure 6.24 Horizontal deviation of an element due to lateral buckling.
Figure 6.25 Horizontal deviation of element of length
ds
: (a) Element of len...
Figure 6.26 Top plan view of cantilever beam.
Figure 6.27 Lateral buckling of fixed-fixed beams: (a) Warping and lateral b...
Figure 6.28 Unbraced length
L
b
versus nominal moment capacity
M
n
for compact...
Figure 6.29 Simply supported beam laterally braced at third points.
Figure P6.4 (a)
Figure P6.4 (b)
Figure P6.5 Cantilever beam.
Figure P6.6 Simple supported beam.
Chapter 7
Figure 7.1 Plate coordinates and differential element: (a) Coordinates of a ...
Figure 7.2 Displacements in the plate due to bending: (a) Displacements in t...
Figure 7.3 Moments and shears on a plate element.
Figure 7.4 In-plane forces on a plate element.
Figure 7.5 In-plane forces on a deformed plate element.
Figure 7.6 Plate boundary conditions at the free edge.
Figure 7.7 Elastically supported edge with a beam.
Figure 7.8 Simply supported plate under compressive axial force.
Figure 7.9 Variation of buckling load coefficient
k
for uniaxially compresse...
Figure 7.10 Buckling modes of plates with different dimensions.
Figure 7.11 Plates under compressive force with different edges.
Figure 7.12 Plate with loading edges simply supported, side
y = 0
...
Figure 7.13 Plate with loading edges simply supported, sides
y = 0
...
Figure 7.14 Plate with loading edges simply supported, side
y = 0
...
Figure 7.15 Plate with loading edges simply supported, side
y = 0
...
Figure 7.16 Plate with loading edges simply supported, and elastically built...
Figure 7.17 Plate with loading edges simply supported, and elastically restr...
Figure 7.18 Plate with loading edges simply supported, and elastically built...
Figure 7.19 Plate with loading edges simply supported, and supported by beam...
Figure 7.20 Plate subjected to in-plane external force.
Figure 7.21 Shear strain corresponding to deflection
w
.
Figure 7.22 Fixed rectangular plate under compressive forces.
Figure 7.23 Fixed rectangular plate under compressive forces in two perpendi...
Figure 7.24 Simply supported plate subjected to pure shear.
Figure 7.25 Rectangular plate subjected to combined bending and compression....
Figure 7.26 Simply supported rectangular plate with longitudinal stiffener....
Figure 7.27 Simply supported rectangular plate with transverse stiffener.
Figure 7.28 Circular plate subjected to compressive force at the edge: (a) B...
Figure 7.29 Plate element showing shear and compressive force.
Figure 7.30 Grid points for the central finite difference method.
Figure 7.31 Finite difference grid points in two dimensions.
Figure 7.32 Simply supported plate with a biaxial loading.
Figure 7.33 Finite difference mesh for
m
= 4
and
n = 2
...
Figure 7.34 Finite element modeling of a plate: (a) Rectangular plate' (b) R...
Figure 7.35 Axial displacements in plate large deflection analysis.
Figure 7.36 Shear strain in a plate large deflection analysis: (a) Shear str...
Figure 7.37 Simply supported plate under uniaxial compression.
Figure 7.38 Variation of stress in post-buckling range in the middle surface...
Figure 7.39 Simply supported rectangular plate under axial compression.
Figure 7.40 Plate with loading edges simply supported and others clamped.
Figure 7.41 Stress variation across the plate width.
Figure 7.42 Stiffened plate panel: (a) Top view; (b) Front view.
Figure P7.1
Figure P7.2
Figure P7.3
Figure P7.4
Figure P7.5
Figure P7.6
Figure P7.7
Chapter 8
Figure 8.1 Cylindrical shell displacements and forces: (a) Cylindrical shell...
Figure 8.2 Component of
N
y
in the
Z
direction due to initial curvature.
Figure 8.3 Circumferential strain due to radial displacement
w
Figure 8.4 Axially compressed cylinder.
Figure 8.5 Cylinder under the action of axial compression: (a) Buckled shape...
Figure 8.6 Critical axial pressure for cylindrical shells.
Figure 8.7 Cylinder subjected to uniform external pressure: (a) Buckled shap...
Figure 8.8 Critical external lateral pressure for cylindrical shells.
Figure 8.9 Critical shear stress for cylindrical shells subjected to torsion...
Figure 8.10 Cylindrical panel under axial compressive load.
Figure 8.11 Post-buckling behavior of flat plates,
k
= 0.
Figure 8.12 Post-buckling behavior of cylindrical shells,
k
= 30.
Figure 8.13 General shell coordinates.
Figure 8.14 Shell of revolution: (a) Coordinates of shell of revolution; (b)...
Figure 8.15 Circular flat plate.
Figure 8.16 Shallow spherical cap.
Figure 8.17 Conical shell.
Figure 8.18 Toroidal shell meridian.
Figure 8.19 Segments of toroidal shell: (a) Bowed-out segment; (b) Bowed-in ...
Figure 8.20 Critical external lateral pressure for toroidal shells.
Figure P8.7
Appendix C
Figure C.1 Buckling of pinned-pinned column: (a) Pinned-pinned column; (b) F...
Figure C.2 Deflected shape.
Appendix E
Figure E1 Position vectors on a curve.
Figure E2 Shell coordinate system.
Guide
Cover
Table of Contents
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Structural Stability Theory and Practice
Buckling of Columns, Beams, Plates, and Shells
Sukhvarsh Jerath
Professor Emeritus of Civil EngineeringUniversity of North DakotaGrand Forks, ND, USA
This edition first published 2021
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Library of Congress Cataloging-in-Publication Data
Names: Jerath, Sukhvarsh, author.
Title: Structural stability theory and practice : buckling of columns, beams, plates, and shells / Sukhvarsh Jerath.
Description: Hoboken : Wiley, 2021. | Includes index.
Identifiers: LCCN 2020020408 (print) | LCCN 2020020409 (ebook) | ISBN 9781119694526 (paperback) | ISBN 9781119694502 (adobe pdf) | ISBN 9781119694496 (epub)
Subjects: LCSH: Buckling (Mechanics)
Classification: LCC TA656.2 .J47 2021 (print) | LCC TA656.2 (ebook) | DDC 624.1/76—dc23
LC record available at https://lccn.loc.gov/2020020408
LC ebook record available at https://lccn.loc.gov/2020020409
Cover Design: Wiley
Cover Images: Weisse Bruecke 02 © Viaframe/Getty Images, Study of patterns and Lines © Roland Shainidze Photogaphy/Getty Images
Dedicated to my wife Saroj and children Ashley, Rahul, Roger, and Purabi for patiently enduring the years of preparation
Foreword
Structural design is most efficient and usually most economical if the full strength of the material of the member or frame can be achieved under the factored design load. For example, a steel tension strut should have a strength equal to its area times the yield stress, and a beam should be able to achieve the plastic moment at the critical cross-section. This desirable state cannot always be attained: the material may fracture, or the member, the frame, the plate, or the shell may buckle, that is, the structure may become unstable. Most buckling phenomena are sudden, they may not announce their imminent occurrence, and they can result in catastrophic failure. Design codes have extensive clauses that the structural designer must consider in order to safeguard against this undesirable situation. These design specifications provide many formulas and rules that are then routinely applied during the design of a structure. The formulas often are based on approximations and they frequently include shortcuts and numerical coefficients devised by the advisory committees of the code authority. Furthermore, the design codes are not able to account for all possible design situations, especially those occurring during fabrication and erection. It is thus vitally important that the structural engineer be educated in the theory of structural stability and be able to apply this knowledge when using the code formulas or when dealing with stability issues outside the purview of the codes.
The subject of this book is the analysis of the stability of structures and structural elements. Its contents are directed to students and to the practicing structural engineering community. After an introductory chapter that introduces the basic theoretical phenomena and methods of analysis of the structural mechanics of stability design, the following seven chapters lead the student to an understanding of the major practical subdivisions of the structural elements that comprise a designed built system: Columns, beams, frames, plates, and shells. The treatment of each topic follows classical mathematics, emphasizing rigorous analysis in ample detail so the student will have no problem following the derivation and can understand the practical significance of the method and the outcome. Following the derived problems, there is a discussion of advanced methods of analysis, such as finite element analysis or other numerical procedures, and a presentation of the code rules for the topical element of the given chapter. Design rules from the USA, Canada, Europe, and Australia are introduced, and the chapters conclude with several detailed examples of design problems. Several chapters in this book are suitable for a text in an advanced senior course in structural engineering, and it is recommended in its entirety for a course in the first year of a graduate course.
T.V. Galambos
February, 2020
Preface
The material in this book covers the failure of structures due to buckling and structural instability. The book represents an effort to comprehensively describe the principles and theory of structural stability of different types of structures. The failures due to structural instability depend on the structural geometry, size, and the stiffness. It is important to understand structural instability failures because using higher strength materials will not prevent these failures. More structures are failing due to instability because of the increased use of high strength materials and larger structures, such as long bridges, tall buildings, vast sports and public arenas, large storage tanks, bigger airplanes, larger engines, turbines, motors, etc. The enlarged size increases the slenderness of the members of a structure, such that these members reach their stability limit before their material strength. If one looks at the design procedures of different codes, it becomes clear that in many situations the maximum forces members can support is governed by structural instability.
The author felt there is a need for a book on the subject that should be comprehensive, covering theory and practice, and at the same time should be relatively easy to read. To this end, theoretical derivations are given in the most detailed form possible and practical problems are solved to illustrate the use of theory. An attempt is made to tie the theory and the codes of practice.
The author has taught the subject for about 20 years to senior level undergraduate and graduate students, and advised research projects of graduate students in this area. The book should be a valuable text for upper-level undergraduate and graduate students who wish to study the stability of structures courses in aerospace, civil engineering, mechanics, and mechanical engineering departments. It can also be used by practicing engineers as a reference for their designs. It will be a very useful book for engineering libraries. The book uses both FPS and SI units to solve the problems so that the book can be used by students and engineering professionals worldwide. In many cases the nondimensional quantities are used in the formulation of equations.
Chapter 1 presents general concepts of structural stability problem. This chapter introduces the subject by covering the stability of rigid bars using both the equilibrium and energy methods. In this, subject equilibrium of the displaced structure is considered, whereas in the ordinary structural analysis equilibrium equations are formed for the original undeformed geometry. The energy method uses the principle of stationary potential energy. The governing equations for rigid bars are algebraic equations because these bars have infinite stiffness and are easier to solve and formulate. After gaining some fundamental understanding of the subject, Chapter 2 deals with the buckling of columns. Columns with different boundary conditions and supports are covered and critical loads are found by using the classical column theory and energy method. The governing differential equations for different cases and general differential equations applicable to all boundary conditions are derived and solved. Eccentrically loaded columns and imperfect columns are dealt with, in order to know the practical range of failure loads. Large deflection theory is used to know the post-buckling behavior of columns. When the stresses at buckling failure are greater than the elastic limit, the failure is called inelastic buckling. Inelastic behavior is studied in Chapter 3 for metal columns made of aluminum and steel. Different codes of practice used around the world are discussed to solve practical problems in column design.
When structural members are acted on by transverse loads in addition to axial forces, these are called beam columns. In Chapter 4, the concept for solving the instability problems of beam columns is discussed. Basic differential equations of beam columns for different support conditions acted on by different loads are derived and solved. The beam column instability problems are also solved by the slope deflection method of structural analysis except that the equations are derived for the displaced shape of the structure. The continuous and inelastic beam columns are also studied. The design procedures given in different codes are discussed. The critical loads in single-story, multistory, and multibay frames are found in Chapter 5. Frames with and without sidesway and different types of loads and support conditions are studied. Critical loads in frames are found by the equilibrium, slope deflection, matrix and finite element methods. Inelastic buckling of frames is also considered. Design problems for two-story and two-bay frames are solved under the action of gravity and wind loads.
The torsional and lateral buckling of beams is discussed in Chapter 6. The chapter starts with pure and non-uniform torsion in thin-walled open cross-sections. After dealing with general thin-walled open sections, the commonly used I and channel sections are studied. The concepts of St. Venant's and warping torsion are elaborated. Torsional flexural buckling loads of different cross-sections with different boundary conditions are found. The problems are solved by the equilibrium and energy methods. In the second half of Chapter 6, lateral buckling of beams is studied. The lateral buckling phenomenon is studied for beams that are supported differently. The design equations for torsional and lateral buckling are given in order to know the capacity of members under such forces. The concepts of lateral torsional buckling and lateral buckling of beams derived in the book are shown to be related to the equations used by codes.
The buckling of thin rectangular plates with and without stiffeners under different kinds of loading and support conditions is discussed in Chapter 7. Post-buckling behavior and inelastic buckling of plates are also studied. The buckling of circular plates is introduced with solved problems. Linear and nonlinear theories of cylindrical shells are derived, and some problems are solved using linear stability equations. The failure and the post-buckling behavior of cylindrical shells are discussed. Nonlinear and linear equations of stability are derived using curvilinear coordinates for general shells. In Chapter 8, linear equations of general shells are converted to solve problems of the shells of revolution, shallow spherical caps, conical shells, and toroidal shells.
The book was written by keeping in mind those who wish to learn the concepts of buckling and instability of structures. Theoretical derivations are given in detail and are made as simple as possible. The book is also written for those who are in practice and are designing structures so they do not fail due to instability. The author has covered the subject in detail as well as keeping it simple and practical. It is hoped the readers will like and enjoy the book, and find it useful.
Sukhvarsh Jerath
Grand Forks, ND
USA
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Learning Outcomes for all chapters
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1Structural Stability
1.1 Introduction
Structures fail mainly either due to material failure or because of buckling or structural instability. Material failures are governed by the material strength that may be the ultimate strength or the yield point strength of the material. The failure due to structural instability depends on the structural geometry, size, and its stiffness. It does not depend on the strength of the material. It is important to understand the failure due to structural instability, because using a higher strength material will not prevent this type of failure. More and more structures are failing because of stability problems because of the present trend to use high strength materials and large structures. The increase in size increases the slenderness ratio of the members of a structure, and these members reach their stability limit before their material strength. A look at different design codes makes it clear that in many situations the maximum force a system can support is governed by structural instability than by material strength.
An interesting question to ask is, if the material strength is not exceeded, then why does the member fail?. The answer may be that all systems take the path of least resistance when they deform, a basic law of nature. For slender members, it is easier to bend than to shorten under a compressive force resulting in the buckling of the member before it fails by exceeding its material strength. For short members it is easier to shorten than to bend under a compressive force. In practice, there is always a tendency of a slender member to bend sideways even if the intended force is an axial compression. This tendency is due to small accidental eccentricity, unintended lateral disturbing force, imperfections, or other irregularities in the member. For small compressive forces the internal resistance of a member to bending exceeds external action forcing it to bend. As the external forces increase, a limiting load is reached where their overturning effect to bend exceeds the internal resistance to bending of the member. As a result, more and more bending of the system called buckling occurs. The maximum compressive force at which the member can remain in equilibrium in the straight configuration without bending is called the buckling load. A system is called stable if small disturbances cause small deformations of the system configuration. Displaced shape equilibrium and the energy methods are the two most commonly used procedures to solve the buckling loads problem and to study the stability of equilibrium.
Figure 1.1 Types of equilibrium: (a) stable; (b) unstable; (c) neutral.
1.2 General Concepts
Concepts of stability can be explained by considering the equilibrium of a ball resting on three different surfaces [1] shown in Figure 1.1. The ball on the concave surface in Figure 1.1a is in stable equilibrium because any small displacement will increase the potential energy of the ball. The component of the self-weight parallel to the sliding surface will bring the ball back to its original equilibrium position. In Figure 1.1b, the ball rests on a convex surface, a small displacement from its equilibrium position will decrease the potential energy of the ball. The parallel component of the self-weight will slide the ball further from its initial configuration, and the equilibrium is unstable. If the ball is displaced on the flat surface, the potential energy of the ball remains the same, and the ball assumes a new equilibrium position. Thus, potential energy, Π, is a minimum for stable equilibrium, whereas it is a maximum for the unstable equilibrium position, and the potential energy remains the same for the position of neutral equilibrium. Energy methods are based on these concepts for solving the structural stability problems. If ΔΠ > 0, the displaced configuration is stable, whereas for ΔΠ < 0, the displaced shape is in unstable equilibrium, the transition ΔΠ = 0, which is the position of neutral equilibrium gives critical load at which the system becomes unstable by energy method.
Also, since we are studying the state of equilibrium in the slightly displaced position of the body, the equilibrium equations are written based on the displaced shape of the body in the displaced shape equilibrium method. Both methods can be used to formulate the equilibrium equations and calculate the critical loads. However, the displaced equilibrium approach does not give the nature of equilibrium when the critical load is reached. To answer that question, the second variation of potential energy δ2Π is to be considered. The potential energy may be expanded into a Taylor series about the equilibrium state and written as
where
δΠ, δ2Π, and δ3Π are called the first, second and third derivatives respectively of the potential energy Π. The critical load Pcr is obtained from the conditions of equilibrium given by δΠ = 0for any δqi, or for each i [2]. The equilibrium state is stable if ΔΠ > 0. Therefore, the equilibrium state is stable for δ2Π > 0, and is unstable for δ2Π < 0.
Because energy is quadratic, it can also be written as
where
For elastic structures, matrix K represents the stiffness matrix of the structure with regard to its generalized displacements, and Π is the potential energy. The stiffness elements are given by
That shows the stiffness matrix is symmetric. The second variation of the potential energy from Eq. (1.1c) is
For δ2Π > 0, the matrix with elements Kij will be positive definite. A real symmetric matrix is positive definite if and only if all its principal minors are positive, that is,
or
When systems are subjected to compressive forces three types of instabilities can occur: (i) bifurcation of equilibrium; (ii) maximum or limit load instabilities; and (iii) Finite disturbance instability.
1.2.1 Bifurcation of Equilibrium
Equilibrium paths are shown as load displacement plots in Figure 1.2. The equilibrium path starting from the unloaded configuration is called the fundamental or primary path. At a certain load the equilibrium path can continue to be the fundamental path or it could change to an alternate configuration if there is a small lateral perturbation. This alternate path is called the secondary or post-buckling path [3]. The point of intersection between the primary and secondary paths is called the point of bifurcation, and the load corresponding to this point is called the critical load. In Figures 1.2a and 1.2b, the secondary paths are symmetrical. In the symmetric bifurcation the post-buckling load deflection behavior remains the same irrespective of the direction in which the structure bends. It is a stable bifurcation in Figure 1.2a because the load increases with deflection after buckling, axially loaded columns and thin plates subjected to in-plane forces exhibit this behavior. The load decreases below the critical as the deflection increases in the post-buckling stage in Figure 1.2b, and the structure has an unstable bifurcation at the critical load. Guyed towers exhibit this behavior because some of the cables come under compression and are unable to sustain the external forces. If the post-buckling load deflection diagram is affected by the direction of buckling, then the bifurcation is asymmetric as shown in Figure 1.2c. Some framed structures show this kind of behavior.
Figure 1.2 Bifurcation equilibrium paths: (a) Symmetric stable bifurcation; (b) Symmetric unstable bifurcation; (c) Asymmetric bifurcation.
1.2.2 Limit Load Instability
This type of instability is also called snap-through buckling. In this type of buckling, the primary path is nonlinear and once the load reaches a maximum, the point P in Figure 1.3a jumps to Q on another branch of the curve. The load at point P is the critical load in this type of instability. The structure snaps through to a nonadjacent equilibrium position represented by point Q. Spherical caps and shallow arches exhibit this behavior.
1.2.3 Finite Disturbance Instability
This type of instability occurs in cylindrical shells under the action of axial forces shown in Figure 1.3b. The load capacity of the structure drops suddenly at the critical load in Figure 1.3c. The structure takes a non-cylindrical shape after the critical load. The structure continues to take more axial compression in Figure 1.3c after taking another equilibrium configuration. In this type of instability, a finite disturbance of the cylinder or imperfection in the cylinder will lower the critical load considerably and the structure will change equilibrium configuration upon reaching the ideal critical load.
Figure 1.3 Post-buckling equilibrium paths: (a) Limit load instability; (b) Cylindrical shell under axial compression; (c) Finite difference instability.
1.3 Rigid Bar Columns
Columns consisting of rigid bars supported by springs and acted on by axial compression are studied by the displaced shape equilibrium, or by energy methods. At first, the small deflection analysis is considered. The study of rigid bar columns provides a good background on the nature of stability problems and the different methods used to solve them because these systems have limited degrees of freedom.
1.3.1 Rigid Bar Supported by a Translational Spring
1.3.1.1 The Displaced Shape Equilibrium Method
Consider a perfect rigid vertical column supported by a hinge at the bottom and a linear spring of stiffness “k” at the top. The bar is acted on by an axial load shown in Figure 1.4. If there is an accidental lateral disturbance, the spring force, kL sin θ, will bring it back to the vertical position for small axial loads. In this case the restoring moment due to spring force is larger than the overturning moment due to the force P as shown in Eq. (1.2a):
Figure 1.4 Rigid bar under axial force: (a) Rigid bar with axial load; (b) Free-body diagram of displaced shape.
and the vertical position of the bar is stable. The spring force will not be able to bring back the rigid bar to its vertical position for large axial force, because the overturning moment will be larger than the restoring moment shown below
and the vertical position of the bar is unstable. The minimum axial force at which the bar becomes unstable is called the critical load. It is the force at which the equilibrium changes from stable to unstable, and
The critical load, Pcr, can be found by considering the equilibrium of the slightly displaced position of the bar by taking moments of all forces about A in Figure 1.4b as follows:
The same result is obtained from Eqs. (1.2c and 1.2d), hence the critical load can be found by considering the equilibrium of the slightly displaced shape. For small deflections, cosθ ≈ 1, therefore,
1.3.1.2 The Energy Method
The first law of thermodynamics can be used to derive equations used in the energy method. This law, which is a statement of the law of conservation of energy, can be stated as “The work that is performed on a mechanical system by external forces plus the heat that flows into the system from the outside equals the increase of kinetic energy plus the increase of internal energy.”
Here, We, is the work performed on the system by the external forces, Q is the heat that flows into the system, ΔT is the increase of kinetic energy, and ΔU is the increase of internal energy [4]. For an adiabatic change, Q = 0, and for a body in equilibrium, ΔT = 0. This reduces Eq. (1.3a) to
The change in internal energy of an elastic body is determined by the strains, and is called the strain energy. If the system is subjected to conservative forces, We is independent of the path the system takes from the configuration X0 to another configuration X. In this case, the We depends only on the two terminal configurations, and is denoted by −V(X0, X). The function V(X0, X)is called the potential energy of the external forces, and it is always measured as the change in the potential energy, ΔV, from one configuration to another configuration of the system.
Equations (1.3b and 1.3c) can be combined to write
or
V is the potential energy due to external forces, and U is considered the potential energy of the internal forces. Total potential energy of the system is
The total potential energy of a system is a minimum in the position of stable equilibrium, whereas it is a maximum for unstable equilibrium. The critical load can be obtained by equating the first derivative of the total potential energy equal to zero. In Figure 1.4
Substituting , we get
or
giving the same critical load as in Eq. (1.2e). From Eq. (1.3j)
For the initial position,
For , and for in Eq. (1.3n). So the system is in stable equilibrium if P < Pcr, and is in unstable equilibrium for P > Pcr, in the initial position.
1.3.2 Two Rigid Bars Connected by Rotational Springs
1.3.2.1 The Displaced Shape Equilibrium Method
Consider two rigid bars as shown in Figure 1.5. The lower bar is connected to a pin support and a linear rotational spring of stiffness c1 at the bottom. At the top the lower bar is connected to another bar by a linear rotational spring of stiffness c2. The upper bar is free at the top, and the bars are subjected to an axial force of P.
Taking the equilibrium of the lower bar in Figure 1.5c, the sum of the moments of all forces about A is equal to zero,
From Figure 1.5d, sum the moments of all the forces about B and equate it to zero,
sinθ ≈ θ in radians for small values of θ, and Eqs. (1.4a and 1.4b) can be written in the matrix form as
Figure 1.5 Two rigid bars under axial load: (a) Two rigid bars with axial force; (b) Displaced shape; (c) Free body diagram of lower bar; (d) Free body diagram of upper bar; (e) First buckling mode; (f) Second buckling mode.
Equation (1.4d) is an eigenvalue problem. The critical loads P are the eigenvalues, and the angular displacements, θ1 and θ2 are given as eigenvectors. For a nontrivial solution the determinant of the coefficient matrix is zero [5],
or
or
The solution of the quadratic Eq. (1.4f) is given by
If c1 = c2 = c, and L1 = L2 = L
The corresponding eigenvectors are:
1.3.2.2 The Energy Method
The critical load for the two rigid bars shown in Figure 1.5a subjected to an axial force P can be found by using the principle of stationery potential energy. The strain energy of the system in the displaced shape is given by
The potential energy of the external force P is
Total potential energy of the system is
The potential energy of the system must be stationary for equilibrium. The first derivatives of the potential energy function, Π, with respect to θ1 and θ2 are:
For small values of θ, sin θ ≈ θ in radians, and Eqs. (1.5d and 1.5e) can be written in matrix form as:
Equations (1.4c and 1.5f) are the same, giving the same solution for the critical load, Pcr, by the energy method as given before by the displaced shape equilibrium method. For c1 = c2 = c, and L1 = L2 = L, from Eqs. (1.5d and 1.5e)
