Mechanics of Optimal Structural Design: Minimum Weight Structures

565 Pages·2009·5.866 MB·English

by  David W. A. Rees

MECHANICS OF
OPTIMAL STRUCTURAL
DESIGN
Mechanics of Optimal Structural Design: Minimum Weight Structures David W. A. Rees
© 2009 John Wiley & Sons, Ltd. ISBN: 978-0-470-74623-3
MECHANICS OF
OPTIMAL STRUCTURAL
DESIGN
Minimum Weight Structures
David W. A. Rees
SchoolofEngineeringandDesign,BrunelUniversity,Uxbridge,UK
A John Wiley and Sons, Ltd., Publication
Thiseditionfirstpublished2009
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Rees,D.W.A.(DavidW.A.),1947-
Mechanicsofoptimalstructuraldesign:minimumweightstructures/DavidRees.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-74623-3 (cloth)
1. Lightweightconstruction. I. Title.
TA663.R432009
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2009025913
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Contents
Preface xi
Glossary of Terms xv
Key Symbols xix
Chapter 1 Compression of Slender Struts 1
1.1 Introduction 1
1.2 Failure Criteria 1
1.3 Solid Cross-Sections 3
1.4 Thin-Walled, Tubular Sections 6
1.5 Thin-Walled, Open Sections 13
1.6 Summary of Results 24
References 25
Exercises 25
Chapter 2 Compression of Wide Struts 29
2.1 Introduction 29
2.2 Failure Criteria 29
2.3 Cellular Sections 31
2.4 Open Sections 37
2.5 Corrugated Sandwich Panel 57
2.6 Summary of Results 60
References 61
Exercise 61
Chapter 3 Bending of Slender Beams 65
3.1 Introduction 65
3.2 Solid Cross-Sections 66
3.3 Thin-Walled, Tubular Sections 69
3.4 Open Sections 76
3.5 Summary of Results 88
References 89
Exercises 89
vi Contents
Chapter 4 Torsion of Bars and Tubes 91
4.1 Introduction 91
4.2 Solid Cross-Sections 92
4.3 Thin-Walled, Open Sections 99
4.4 Thin-Walled, Closed Tubes 109
4.5 Multi-Cell Tubes 121
References 130
Exercises 130
Chapter 5 Shear of Solid Bars, Tubes and Thin Sections 135
5.1 Introduction 135
5.2 Bars of Solid Section 136
5.3 Thin-Walled Open Sections 143
5.4 Thin-Walled, Closed Tubes 159
5.5 Concluding Remarks 170
References 171
Exercise 171
Chapter 6 Combined Shear and Torsion in Thin-Walled Sections 173
6.1 Introduction 173
6.2 Thin-Walled, Open Sections 173
6.3 Thin-Walled, Closed Tubes 177
6.4 Concluding Remarks 189
References 190
Exercises 190
Chapter 7 Combined Shear and Bending in Idealised Sections 193
7.1 Introduction 193
7.2 Idealised Beam Sections 193
7.3 Idealised Open Sections 201
7.4 Idealised Closed Tubes 210
References 221
Exercises 221
Chapter 8 Shear in Stiffened Webs 223
8.1 Introduction 223
8.2 Castellations in Shear 223
8.3 Corrugated Web 226
8.4 Flat Web with Stiffeners 231
References 237
Exercises 237
Chapter 9 Frame Assemblies 239
9.1 Introduction 239
9.2 Double-Strut Assembly 239
9.3 Multiple-Strut Assembly 244
Contents vii
9.4 Cantilevered Framework 247
9.5 Tetrahedron Framework 253
9.6 Cantilever Frame with Two Struts 256
9.7 Cantilever Frame with One Strut 259
References 264
Exercises 264
Chapter 10 Simply Supported Beams and Cantilevers 265
10.1 Introduction 265
10.2 Variable Bending Moments 265
10.3 Cantilever with End-Load 271
10.4 Cantilever with Distributed Loading 281
10.5 Simply Supported Beam with Central Load 292
10.6 Simply Supported Beam with Uniformly Distributed Load 303
10.7 Additional Failure Criteria 316
References 322
Exercises 323
Chapter 11 Optimum Cross-Sections for Beams 325
11.1 Introduction 325
11.2 Approaching Optimum Sections 326
11.3 Generalised Optimum Sections 328
11.4 Optimum Section, Combined Bending and Shear 330
11.5 Solid, Axisymmetric Sections 331
11.6 Fully Optimised Section 341
11.7 Fully Optimised Weight 345
11.8 Summary 355
References 356
Exercises 356
Chapter 12 Structures under Combined Loading 357
12.1 Introduction 357
12.2 Combined Bending and Torsion 357
12.3 Cranked Cantilever 359
12.4 Cranked Strut with End-Load 362
12.5 Cranked Bracket with End-Load 365
12.6 Portal Frame with Central Load 368
12.7 Cantilever with End and Distributed Loading 371
12.8 Centrally Propped Cantilever with End-Load 377
12.9 End-Propped Cantilever with Distributed Load 385
12.10 Simply Supported Beam with Central-Concentrated and Distributed
Loadings 390
12.11 Centrally Propped, Simply Supported Beam with Distributed Load 395
References 400
Exercises 400
viii Contents
Chapter 13 Encastre´ Beams 403
13.1 Introduction 403
13.2 Central-Concentrated Load 403
13.3 Uniformly Distributed Load 418
13.4 Combined Loads 437
References 463
Exercises 463
Chapter 14 Plastic Collapse of Beams and Frames 465
14.1 Introduction 465
14.2 Plane Frames 466
14.3 Beam Plasticity 468
14.4 Collapse of Simple Beams 474
14.5 Encastre´ Beams 478
14.6 Continuous Beams 481
14.7 Portal Frames 486
14.8 Effect of Axial Loading upon Collapse 497
14.9 Effect of Shear Force upon Collapse 500
14.10 Effect of Hardening upon Collapse 505
References 507
Exercises 507
Chapter 15 Dynamic Programming 511
15.1 Introduction 511
15.2 Single-Span Beam 511
15.3 Two-Span Beam 513
15.4 Three-Span Beam 515
15.5 Design Space 517
Reference 520
Exercises 520
Appendix A Mechanical Properties 521
A.1 Non-Metals 521
A.2 Metals and Alloys 522
References 524
Appendix B Plate Buckling Under Uniaxial Compression 525
B.1 Wide and Slender Struts 525
B.2 Plates with Supported Sides 527
B.3 Inelastic Buckling 530
B.4 Post-Buckling 533
References 534
Appendix C Plate Buckling Under Biaxial Compression and Shear 537
C.1 Biaxial Compression 537
C.2 Pure Shear 539
Contents ix
C.3 Inelastic Shear Buckling 541
References 541
Appendix D Secondary Buckling 543
D.1 Buckling Modes 543
D.2 Local Compressive Buckling 544
D.3 Global Buckling 545
D.4 Local Shear Buckling 547
References 547
Bibliography 549
Index 553
Preface
Design is the process of finding a solution to a problem, with all the constraints and
requirementsthatitpresents.Accordingly,therewouldnormallybemorethanonedesign
solution, depending upon the requirements perceived from the application. Specifically,
an optimum design will narrow the solutions to one or two depending upon the criteria
adopted. For example, the criteria may relate to structural form, the material character-
istics, ease of manufacture and assembly, a demand for standard parts, or a specified
mechanical performance involving aerodynamics, heat transfer, wear resistance, etc. In
principle, the designer would wish to optimise some merit factor common to all of these.
In practice, an optimisation can be achieved when related items are considered together
but more usually specific requirements are optimised in isolation, as with attaining min-
imal waste in machining and maximal heat transfer in material selection. In this book
we are concerned with minimising the weight of load bearing structures through making
informed choices upon their structural forms. In fact, we find that this problem can be
treated in terms of its building blocks. This amounts to optimising all the key elements
of a structure – struts, ties, beams, shear panels, plates, etc. – in terms of their strength
and weight contributions.
An almost infinite number of structural forms can bear load. Whilst a strut, a beam, a
braced frame, a tripod would usually be adequate, the choice rests with which is best for
the particular application. In the fields of aerospace, transport and structural engineering
thereductioninweighthasbecomeakeyfactorwiththeirassociatedenvironmentalissues.
Quite apart from the obvious reduction in material cost, the improvement in performance
from the reduction in weight becomes an overriding factor when designing to reduce
emissions in road vehicles and aircraft. In this context the usual reference is made to a
strength-to-weight ratio, which implies a maximising of strength for a given weight. In
fact,it is the weight that isto be minimised with respectto a requiredstrengthandso the
term weight-to-strength is a more precise description of the approach to be adopted here.
Though the structures considered in this book are many and varied, they share the
common theme of optimising their weight to strength. The question as to whether an
optimised design exists for thin-walled tubes, Ts, Is and channel sections is answered by
applying additional design criteria involving both global and local buckling. Generally,
the solution is based upon similar loading stresses from all sources being reached simul-
taneously within each design criterion adopted. For example, in a thin-walled open strut,
with an I or T cross-section, Euler’s global stress for flexural buckling in the length and
the local buckling stress for the sections’ flanges are made to coincide with the limiting
design stress. Struts as compression members, both wide and narrow, appear typically
as columns, pillars and walls (Chapters 1 and 2). Beams and bar sections under uniform
xii Preface
bending moments and torques are treated in Chapters 3 and 4. Various types of strut and
beam sections are covered. Solid sections in standard shapes are usually more amenable
to an optimum design, but tubular and thin-walled sections obviously present less weight
at the outset. Thin sections present wall thickness as a further variable to an optimum
design and an additional, local buckling failure mode within their walls.
Transverse shear and torsion in thin-walled tubes are given special consideration
(Chapters 5 and 6) because of their ability to sustain a constant shear flow irrespective
of the wall thickness, unlike an open section. However, a pure flexure is only possible
when shear forces act at the shear centre of a tube. With shear forces displaced from the
shear centre, an example of combined loading upon a structure arises through bending,
torsion and shear. The effect is similar to when a torque is not applied at the centre
of twist, this giving rise to a similar loading combination (Chapter 6). Here we might
simplify the problem by idealising a section in which webs carry shear force and booms
carry bending moments. The interested reader can find the detail of these more subtle
aspects of optimum design within Chapter 6 and 7. Methods of reducing weight while
preserving the shear resistance of webs are considered in Chapter 8. Corrugations, Z and
top-hat stiffeners are effective in offsetting the risk of buckling in thin webs over long
lengths anddepths. A method of optimising the design of pin-jointed frame assemblies is
given in Chapter 9. Whilst the structures considered consist of relatively few bars, they
are sufficient to show how the approach may be extended to plane and three-dimensional
frames with many bars.
Beams which feature as lateral carriers for vertical loading are allowed in Chapters 10
and 11 to vary in cross-section. In particular, the beam supports are altered from resting
uponsimpleknifeedgesorrollerstohavingoneorbothendsbuiltin.Theinfluenceofboth
shearforceandbendingmomentuponanoptimumbeamdesignhasasignificantinfluence
uponweightreductionscomparedtostandarddesignsinuniformsection.Savingsofover
50% are possible by allowing the section to vary whilst maintaining the load bearing
integrity. In the case of combined bending and shear in beams, their effects can be
separated because at the surface, where the bending stress is at its maximum, the shear
stress is zero. Conversely, at the neutral axis, where the bending stress is zero, the shear
stress is at its maximum. So in a given cross-section we need only alternate from their
respective maximum stress positions as either bending or shear governs the design of the
section depth. However, we might seek an ideal optimum design in which all points in
the depth of a section are stressed equally, say at a given equivalent (von Mises) stress.
Chapter 11 shows that all the common solid beam sections are inefficient, even with
the aforementioned weight savings. The ideal section is found numerically, so that when
its depth is allowed to change with the varying force and moment, the greatest possible
weight saving is achieved.
InChapter12furtherexamplesofcombinedloadinginstructurespresentthemselvesin
various ways: tension-torsion in shafts, bending and torsion of beams, compression and
bending of struts and combinations of concentrated and distributed loadings in beams.
Limbs are separated within free-body diagrams within which the design stress is limited
byayieldcriterionbeforetheappropriateloadingcriteriaareapplied.HerethevonMises
yield criterion is used to account for combined loading and also in converting material
yield stress from the shear to the tension mode.