Steel Connection Analysis

Steel Connection Analysis

Paolo RugarliCastalia S.r.l.Milan, Italy

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Library of Congress Cataloging-in-Publication Data

Names: Rugarli, Paolo, 1963– author.Title: Steel connection analysis / Paolo Rugarli.Description: Hoboken, NJ : John Wiley & Sons, 2018. | Includes bibliographicalreferences and index. |

Identifiers: LCCN 2017053858 (print) | LCCN 2018000726 (ebook) |ISBN 9781119303480 (pdf) | ISBN 9781119303534 (epub) |ISBN 9781119303466 (cloth)

Subjects: LCSH: Steel, Structural – Testing. | Finite element method. |Building, Iron and steel–Joints.

Classification: LCC TA475 (ebook) | LCC TA475 .R84 2018 (print) | DDC624.1/821–dc23

LC record available at https://lccn.loc.gov/2017053858

Cover Design: WileyCover Image: © tonymax/iStockphoto

Set in 10/12pt Warnock by SPi Global, Pondicherry, India

10 9 8 7 6 5 4 3 2 1

To all my dears

Fiorenza dentro da la cerchia antica,ond’ella toglie ancora e terza e nona,si stava in pace, sobria e pudica.

Non avea catenella, non corona,non gonne contigiate, non cinturache fosse a veder più che la persona.

Dante, Paradiso, XV, 97-102

Contents

Preface xv

1 Introduction 11.1 An Unsolved Problem 11.2 Limits of Traditional Approaches 2

1.2.1 Generality 21.2.2 Member Stress State Oversimplification 31.2.3 Single Constituent Internal Combined Effects Linearization 41.2.4 Single Constituent External Combined-Effects Neglect 71.2.5 Neglecting Eccentricities 81.2.6 Use of Envelopes 91.2.7 Oversimplification of Plastic Mechanisms Evaluation 111.2.8 Evaluation of Buckling Phenomena 13

1.3 Some Limits of the Codes of Practice 141.3.1 Problem of Coded Standards 141.3.2 T-Stub in Eurocode 3 151.3.3 Eurocode 3 Component Model 171.3.4 Distribution of Internal Forces 201.3.5 Prying Forces 201.3.6 Block Tearing 21

1.4 Scope of This Book 211.5 Automatic Modeling and Analysis of 3D Connections 231.6 Acknowledgments 24

References 24

2 Jnodes 272.1 BFEM 272.2 From the BFEM to the Member Model 29

2.2.1 Physical Model and the Analytical Model 292.2.2 Member Detection: Connection Codes 312.2.3 An Automatic Algorithm for Straight Prismatic Member Detection 342.2.4 Member Data Structure 362.2.5 Member Classification at a Node 362.2.6 Member Mutual Alignment Coding 37

vii

2.3 Jnodes 402.3.1 Need for the Jnode Concept 402.3.2 Jnode Definition 41

2.4 Jnode Analytics 422.4.1 Classification of Jnodes 422.4.2 Simple Jnodes 422.4.3 Hierarchical Jnodes 422.4.4 Central Jnodes 432.4.5 Cuspidal Jnodes 432.4.6 Tangent Jnodes 442.4.7 Constraints 452.4.8 Summary of Jnode Classification 462.4.9 Setting Connection Codes: Examples 46

2.5 Equal Jnodes Detection 492.5.1 Toponode 492.5.2 Jnode Data Structure 492.5.3 Superimposable Member Couples 502.5.4 Criteria to Assess Jnodes Equality 512.5.5 Algorithm to Find Equal Jnodes 522.5.6 Examples 55

2.6 Structural Connectivity Indices 562.7 Particular Issues 59

2.7.1 Symmetries 592.7.2 Splitting of Jnodes 602.7.3 Mutual Interaction of Different Jnodes, Jnode Clusters 612.7.4 Tolerances 63

2.8 Jclasses 63References 64

3 A Model for Connection 653.1 Terminology 653.2 Graphs of Connections 663.3 Subconstituents vs Layouts 693.4 Classification of Connections 70

Reference 72

4 Renodes 734.1 From Jnode to Renode Concept 734.2 BREP Geometrical Description of 3D Objects 734.3 The Scene 75

4.3.1 Generality 754.3.2 Members 774.3.3 Typical Fittings 784.3.4 Connectors 79

4.4 Dual Geometry 834.5 Automatic Connection Detection 85

Contentsviii

4.5.1 Faces in Contact 854.5.2 Bolt Layouts 864.5.3 Weld Layouts 89

4.6 Elementary Operations 914.7 Renode Logic and the Chains 93

4.7.1 Minimum Compliance Criteria for Renode Good Design 934.7.2 Chains 944.7.3 Finding Chains 96

4.8 Prenodes 1024.9 After Scene Creation 103

5 Pillars of Connection Analysis 1055.1 Equilibrium 105

5.1.1 Generality 1055.1.2 Statics of Free Rigid Bodies 108

5.2 Action Reaction Principle 1115.3 Statics of Connections 115

5.3.1 Equilibrium of Members in Renodes: Proper and Dual Models 1155.3.2 Force Packets for Compound Members 1195.3.3 Primary Unknowns: Iso-, Hypo-, and Hyperconnectivity 124

5.4 Static Theorem of Limit Analysis 1275.5 The Unsaid of the Engineering Simplified Methods 1305.6 Missing Pillars of Connection Analysis 130

5.6.1 Buckling 1315.6.2 Fracture 1475.6.3 Slip 1505.6.4 Fatigue 152

5.7 Analysis of Connections: General Path 153References 154

6 Connectors: Weld Layouts 1556.1 Introduction 1556.2 Considerations of Stiffness Matrix of Connectors 1566.3 Introduction to Weld Layouts 1606.4 Reference Systems and Stresses for Welds 1626.5 Geometrical Limitations 165

6.5.1 Penetration Weld Layouts 1656.5.2 Fillet Weld Layouts 166

6.6 Penetration-Weld Layouts (Groove Welds) 1676.6.1 Generality 1676.6.2 Simple Methods to Evaluate the Stresses 1686.6.3 Weld Layout Cross-Section Data 1706.6.4 Stiffness Matrix 1726.6.5 Special Models 1856.6.6 Example 188

6.7 Fillet-Welds Weld Layouts 196

Contents ix

6.7.1 The Behavior of Fillet Welds 1966.7.2 Numerical Tests of Fillet Welds in the Linear Range 2076.7.3 The Stiffness Matrix of a Single Fillet Weld 2126.7.4 Instantaneous Center of Rotation Method in 3D 2146.7.5 Computing the Stresses in Fillet Welds from the Forces Applied to the

Layout 2316.7.6 Fillet Welds Using Contact and Friction 233

6.8 Mixed Penetration and Fillet Weld Layouts 235References 235

7 Connectors: Bolt Layouts and Contact 2377.1 Introduction to Bolt Layouts 2377.2 Bolt Sizes and Classes 2387.3 Reference System and Stresses for Bolt Layouts 2407.4 Geometrical Limitations 243

7.4.1 Eurocode 3 2447.4.2 AISC 360-10 244

7.5 Not Preloaded Bolt Layouts (Bearing Bolt Layouts) 2447.5.1 Shear and Torque 2447.5.2 Axial Force and Bending 249

7.6 Preloaded Bolt Layouts (Slip Resistant Bolt Layouts) 2667.6.1 Preloading Effects 2667.6.2 Shear and Torque 2747.6.3 Axial Force and Bending 275

7.7 Anchors 2777.8 Stiffness Matrix of Bolt Layouts and of Single Bolts 282

7.8.1 Generality 2827.8.2 Not Preloaded Bolts 2837.8.3 Preloaded Bolts 2927.8.4 Non-Linear Analysis of Bolts 293

7.9 Internal Force Distribution 2967.9.1 General Method 2967.9.2 Bearing Surface Method to Compute Forces in Bolts 3027.9.3 Instantaneous Center of Rotation Method 3067.9.4 Examples 307

7.10 Contact 316References 317

8 Failure Modes 3198.1 Introduction 3198.2 Utilization Factor Concept 3208.3 About the Specifications 3268.4 Weld Layouts 328

8.4.1 Generality 3288.4.2 Penetration Weld Layouts 3288.4.3 Fillet Weld Layouts 332

Contentsx

8.5 Bolt Layouts 3378.5.1 Resistance of Bolt Shaft 3378.5.2 Sliding and Resistance of No-Slip Connections 3428.5.3 Pull-Out of Anchors, or Failure of the Anchor Block 345

8.6 Pins 3468.6.1 Eurocode 3 3468.6.2 AISC 360-10 347

8.7 Members and Force Transferrers 3478.7.1 Generality 3478.7.2 Local Failure Modes 3508.7.3 Fracture Failure Modes 3588.7.4 Global Failure Modes 373References 382

9 Analysis: Hybrid Approach 3859.1 Introduction 3859.2 Some Basic Reminders About FEM Analysis of Plated-Structures 386

9.2.1 FEM Analysis as an Engineering Tool 3869.2.2 Linear Models 3879.2.3 Linear Buckling Analysis 3889.2.4 Material Non-Linearity 3909.2.5 Geometrical Non-Linearity 3929.2.6 Contact Non-Linearity 3949.2.7 Non-Linear Analysis Control 396

9.3 IRFEM 4009.3.1 Goal 4009.3.2 Hypotheses 4019.3.3 Construction 4029.3.4 Examples 4089.3.5 Results 4119.3.6 Remarks on the Use of IRFEM 413

9.4 Connector Checks 4189.4.1 Weld Checks 4189.4.2 Bolt Resistance Checks 4199.4.3 Pull-Out Checks 4199.4.4 Slip Checks 4199.4.5 Prying Forces 419

9.5 Cleats and Members Non-FEM Checks 4269.5.1 Action Reaction Principle 4269.5.2 Bolt Bearing 4289.5.3 Punching Shear 4289.5.4 Block Tearing 4289.5.5 Simplified Resistance Checks 429

9.6 Single Constituent Finite Element Models 4309.6.1 Remarks on the Finite Element Models of Single Constituents

(SCOFEM) 430

Contents xi

9.6.2 Stiffeners 4329.6.3 Meshing 4339.6.4 Constraints 4379.6.5 Loading 4399.6.6 Members: Deciding Member-Stump-Length 4439.6.7 Compatibility Issues 444

9.7 Multiple Constituents Finite Element Models (MCOFEM) 4459.7.1 Goal and Use 4459.7.2 Mesh Compatibility Between Constituents and Connector

Elements 4469.7.3 Saturated Internal Bolt Layouts and Contact Non-Linearity 4479.7.4 Constraints 4489.7.5 Stabilizing Springs and Buckling of Members 4489.7.6 Need for Rechecks 449

9.8 A Path for Hybrid Approach 449References 450

10 Analysis: Pure FEM Approach 45110.1 Losing the Subconnector Organization 45110.2 Finite Elements for Welds 455

10.2.1 Introduction 45510.2.2 Penetration Welds 45710.2.3 Fillet Welds 460

10.3 Finite Elements for Bolts 46310.3.1 Introduction 46310.3.2 Bolts in Bearing: No Explicit Bolt-Hole Modeling 46410.3.3 Bolts in Bearing: Explicit Bolt-Hole Modeling 46510.3.4 Preloaded Bolts: No Explicit Bolt-Hole Modeling 46810.3.5 Preloaded Bolts: Explicit Bolt-Hole Modeling 46810.3.6 Effect of the Bending Moments in Bolt Shafts 46910.3.7 Example: A Bolted Splice Joint Using PFEM 469

10.4 Loads 47810.4.1 PFEM 47810.4.2 MCOFEM 479

10.5 Constraints 48010.5.1 PFEM 48010.5.2 MCOFEM 480

10.6 Checking of Welds and Bolts 48010.7 Checking of Components 48110.8 Stiffness Evaluation 48210.9 Analysis Strategies 484

Reference 484

11 Conclusions and Future Developments 48511.1 Conclusions 48511.2 Final Acknowledgments 486

11.2.1 Reasons of This Project 486

Contentsxii

11.3 Future Developments 487References 488

Appendix 1: Conventions and Recalls 489A1.1 Recalls of Matrix Algebra, Notation 489A1.2 Cross-Sections 490A1.3 Orientation Matrix 492A1.4 Change of Reference System 493A1.5 Pseudocode Symbol Meaning 493

Appendix 2: Tangent Stiffness Matrix of Fillet-Welds 495A2.1 Tangent Stiffness Matrix of a Weld Segment 495A2.2 Modifications for Weld Segments Using Contact 499A2.3 Tangent Stiffness Matrix of a Weld Layout for the Instantaneous Center

of Rotation Method 500

Appendix 3: Tangent Stiffness Matrix of Bolts in Shear 503A3.1 Tangent Stiffness Matrix of a Bolt 503A3.2 Tangent Stiffness Matrix of a Bolt Layout for the Instantaneous Center

of Rotation Method 505

Symbols and Abbreviations 507

Index 513

Contents xiii

Preface

Around 17 years ago, at the end of the 1990s, when I started my research on steel con-nections with the aim of developing some reliable and general software, able to tackle,hopefully, every connection, I often felt like giving up. The problem was tremendouslycomplex, and the general rules of mechanics difficult to relate to the problem to be faced;there was a huge gap to be bridged.Initially I thought that only a system able to learn from the analyst could deal with such

a complex problem, learning ad hoc rules to be later applied, case by case. However, I wasable to move some steps forward, finding what in the second chapter of this book isnamed the jnode, its analytics, and all the related concepts. My first useful result wasdetecting equal jnodes. Several years were then necessary to develop the tools neededto create the scene, that is, to place the constituents in their proper position, freelyplacing them interactively in 3D space, in the specific context of steel connection study.The mechanical problem of connection analysis, to be tackled with a general approach,was however still unsolved. I was prepared to develop an expert system able to learn fromthe user how to recognize specific subproblems to be faced, by simple ad hoc rules.This was the tentative generalization of the methods widely used by engineers, butwas not the solution I was searching for.Adopting the concept of the force packet, and recognizing that the connections

could be classified as isoconnected or hyperconnected, I finally understood, in 2008, thata simplified finite element model that in this work is named IRFEM, could be used tocompute the force packets flowing into the connectors for a generic set of connections.Then, by the action and reaction principle, a cornerstone for connection analysis, theforces loading the constituents could be known, and by finite element models of singleconstituents using plate–shells, coherent and well rooted Von Mises stress maps couldbe obtained. This is what I call the hybrid approach and is described in Chapter 9 ofthis book.The door was then opened for the automatic creation of finite element models of

constituents (2008), and from there, in 2012, to the complete automatic modeling ofthe whole node, using what I call here the pure fem approach (PFEM). This is seen asa special case of the hybrid approach and is discussed in Chapter 10.What initially seemed an inextricable tangle could indeed be solved in strict observ-

ance of the main principles of mechanics and of plasticity theory.Several issues are still to be better solved, but a general well rooted method is now

available, that can be applied to every connection configuration, from the simplest to

xv

the most complex. Indeed, I think this is a useful result, because a part of the methodcan be implemented with relative ease.I am well aware that several issues are pending and must still be better tackled.

However, after many years of solitary work, I think the time has come to explain whatI have researched and to propose my work for the attention of my colleagues.Anything can be improved, but the structural analysis problem of analyzing steel

connections having a generic geometrical configuration, regardless of the number ofloading combinations, is now solvable with automatic tools.

Paolo RugarliMilan, 17 May 2017

Prefacexvi

1

Introduction

1.1 An Unsolved Problem

Steel connection analysis and checking is one of themost complex problems in structuralengineering, and even though we use very powerful computing tools, it is still generallydone using very simplistic approaches.From the point of view of a typical structural engineer, the problem to solve is to design

and check nodes,1 not single connections, i.e. a number of connections between a num-ber of different members –maybe tens or even hundreds of load combinations, inclinedmember axes, and generic stress states. In a typical 3D structure there may be several tensof such nodes (Figure 1.1), or maybe even hundreds, which may be similar, or may bedifferent from one another; identifying nodes that are equal is one of the problems thatthe designer has to face in order to reduce the number of different possible solutions, andin order to get a rational design. However, this problem of detecting equal nodes has notbeen sufficiently researched, and there are currently no tools that are able to properlysolve this issue.If posed with the due generality, the problem of checking 3D nodes of real structures

has not been solved by automatic computing tools. Also, because a general method oftackling all these problems is apparently still lacking, usually a few “cooking recipes” havebeen used to solve a limited number of typical, recurring (2D assimilated) nodes. Indeed,it often happens that true, real world nodes have to be analyzed by such recipes, despitethe fact that the basic hypotheses needed to apply these recipes do not always hold true.This poses a serious problem because although these “cooking recipes” have been widelyused, in the past few years they have been applied to 3D structures designed usingcomputer tools, in the non-linear range, perhaps in seismic areas, and with the aim ofreducing the weight of steel.The effects of such oversimplification have already been seen inmany structures where

steel connections have failed, especially in seismic areas (e.g. Booth 2014), but even innon-seismic areas (e.g. White et al. 2013, Bruneau et al. 2011). Generally speaking,it is well known that connections are one of the most likely points of weakness of steelstructures, one of the most cumbersome to design – indeed one of the least designed –and one of the least software-covered in structural engineering.

1 It will be seen that the term node is too generic for the aims of steel connection analysis. In this introductorychapter, however, it will be used due to its widespread diffusion.

1

Steel Connection Analysis, First Edition. Paolo Rugarli.© 2018 John Wiley & Sons Ltd. Published 2018 by John Wiley & Sons Ltd.

This book describes the research efforts made by the author since 1999 to tacklethese issues, and it proposes a general set of methods to deal with these problems(see Section 1.6 for more details).

1.2 Limits of Traditional Approaches

1.2.1 Generality

Traditional approaches to connection design have been extensively used for many years,and are still widely used. Usually they imply several simplifying hypotheses, which areneeded in order to apply them in by hand computation. The equivalent of by hand com-putation is today a “simple spreadsheet” often written very quickly for each given job. Aswith every other form of calculus, they are prone to serious errors (slips and lapses – seeReason 1990 for a general study of human error, and Rugarli 2014 for a discussion onvalidation of structural models; for spreadsheets programming errors, see the EuropeanSpreadsheet Risk Interest Group web site).There are several possible design situations where the use of traditional approaches is

completely justified. These approaches are rooted in the traditional 1D or 2D design. Theuse of 2D design needed intense by hand computation or the use of graphic tools up tothe 1970s; at that time there was no need and no specific legal requirements for checkingtens or may be hundreds load combinations, and safety factors were much higher thanthose used nowadays. When dealing with such situations, today – for example simpledetermined structures under elementary actions – the use of traditional approaches isstill useful. So, it would not be sensible to exclude them completely. Indeed, they willnever lose their utility, especially as one of several possible cross-checking tools thatcan be used to detect possibly unsound designs.

Figure 1.1 A possible node of a 3D structure.

Steel Connection Analysis2

However, in current design practice, we almost always use 3D methods of analysisapplied to highly redundant structures, sometimes in the plastic range, automatic com-puterized checks, with minimum weight often being a must, and safety factors have beenreduced to their minimum. (Currently the material safety factor for limit state design is1.0 in Eurocode 3. The load safety factor for dead loads is lower than that valid for liveloads. The maximum loads are applied with a reduction factor ψ to take into account thereduced probability of contemporary occurrence. All these practices were not, as such, intraditional designs, which means that they used higher safety factors.)In summary, while traditional design of structures was often simple, 2D, and was

designed bymaking extensive use of safe-side envelopes both for loads and for resistance,today things are not so easy; indeed, they are much more complex. While virtually alldesign steps have been semi-automated (modeling, checking members, drawing them,and even cutting them into true 3D pieces by means of computer numerical control,CNC), the checking of connections has remained at the traditional level, more or lessupgraded to the modern era by the use of spreadsheets and dedicated, ad hoc software.As mentioned, several simplifications are widely used in traditional approaches. The

following sections will briefly summarize them.

1.2.2 Member Stress State Oversimplification

Members in highly redundant 3D structures are often nonsymmetrical (such as in indus-trial plants or architects’ innovative designs), and under the effect of combined loadcases, they are always loaded in the most general way. If they are not: (a) fully hingedat both extremities, (b) straight, and (c) with no transverse load applied, they will in gen-eral exhibit all six internal forces components: an axial force, two shears, one torque andtwo bending moments, referred to the principal axes of the member cross-section.Idealizing the connection in such a way that somemember internal forces components

are considered zero at the connection is still a widely used practice. While this is justifiedwhen the connections are specifically conceived with that aim, this is unjustified for con-nections that are not so designed. In a typical moment resisting frame (MRF) ideallydesigned to work in a plane, beam-to-column connections that must transfer bendingand shear in one plane (and axial force) will always transfer the bending and the shearalso in the other plane – and of course torque. So the internal forces to deal with are notthree, but six. Sometimes it is said that the torque and out-of-plane bending are avoidedby “the concrete slab”, or by something equivalent, but often the concrete slab does notexist or cannot be considered a true restraint, or its true effect is questionable.A simple beam hinged at an extremity (e.g. Figure 1.2), will transfer the shear, and will

not transfer bending moment if the connection is light and does not use flanges, but itwill also transfer the axial force and, if any, the shear in the other direction. However,textbooks usually refer to “shear connections” and only recently, under the flag of “robustdesign” (a replacement for correct design) has this axial force finally – sometimes, in sometextbooks, – been considered (e.g. the Green Books by SCI).This systematic neglect of some internal forces which have, however, been computed

introduces a clear mismatch in the design process. Simply, load paths are interrupted(Figure 1.3) and the corresponding forces are thrown away: recalling the variationalcrimes of the finite element literature, this can be called a connection-design crime,morespecifically an equilibrium crime. Usually no one cares, and no one mention it.

Introduction 3

1.2.3 Single Constituent Internal Combined Effects Linearization

Not only are some components of member internal stress states thrown away, but theremaining components are tackled one at a time, as if the connection were loaded onlyby a single member internal force component. The typical example is the axial force plus(one) bending moment loading condition, for beam-to-column connection or for a baseplate. As already pointed out, this loading condition is itself usually the result of a con-nection-design crime. However, several possible combinations of N and M can beapplied to the connection (two infinities), leading to an infinite number of possible stressstates. This is usually tackled by computing the limit for the axial force, Nlim, and for the

Figure 1.2 Flexible end plate connection (“shear” connection).

Figure 1.3 Traditional design applied to computerized analysis: no way for the load path.

Steel Connection Analysis4

bending moment, Mlim, as if they were acting alone, and then the mutual interaction iscomputed by simply drawing a straight line in the (N, M) plane. So the design safetycondition becomes

U =NNlim

+MMlim

≤ 1 0 1 1

where the utilization ratio,U, can be considered as the reciprocal of the “limit”multiplierλ = 1/U. It must be underlined that this limit condition is not applied to the membercross-section, but to the member connections, implicitly considering all the possiblefailure modes: bolt bearing, block tear, generic resistance of constituents, buckling ofplates, punching shear, weld-resistance, and so on.As there are quite a number of plastic failures implicitly included in the typical design

formulae (e.g. bolt bearing), this must not be considered as a superposition of effects,which would only be valid in linear range.It must instead be considered a simplification of the limit domain, assuming that it can

safely be considered convex, so that a straight line would be a safe simplification; ascan be seen, the previous equation is the equation of a straight line in the (N, M) plane.There are several issues to be discussed here.The first is that this choice clearly lays aside every possible “realistic” computation of a

safety measure. In particular, the utilization ratio U, a pure number and a much usedsafety index, which must be lower than 1 in the safety region, is usually computed asU = PO/AO (Figure 1.4), that is, the ratio of the distance of the applied stress state P(N, M) from the origin, to the distance of point A from the origin, A being the pointwhere the straight line joining P and O meets the limit (linearized) domain. If the truedomain were convex the correct utilization ratio would have been U = PO/QO, whichcan be much lower. So this method is not very realistic, and can be too much on thesafe side.By posing

λN =Nlim N

λM =Mlim M

O N/NIim

convex

Q

A

PC

M/MIim1

linear

concave

Figure 1.4 Limit domain for aconnection; P(N, M) is the stress state fora single member assumed.

Introduction 5

Equation 1.1. becomes

1λlinear

=1λN

+1λM

≤ 1

A similar result can be found in Fraldi et al. 2010, a paper dealing with the problemof finding some bound of the limit multiplier under combined loadings, once the limitmultipliers of single loadings are known; there it is formally proved, in the frameworkof classic plasticity, that the “true” combined-loading multiplier λ, is surely such as tosatisfy the following inequality:

λ≥1λN

+1λM

−1

= λlinear

which, considering Figure 1.4, simply means

QO ≥AO

The second issue is that in order to be confident that the limit domain, consideringall the possible failure modes, is convex, no buckling effect must be possible at loadlevels lower than those leading to the first failure mode, that is, the failure mode whichis met first, linearly increasing the stress state from (0, 0) to (N,M). If this is the case, thenU = PO/CO which is much higher than PO/AO. A good design of connection shouldalways ensure that the first failure mode is plastic, i.e. ductile, and avoid brittle failuremodes. However, this cannot be considered an implicit condition but must be assuredby correct sizing and proper numerical checks, which presents a serious problem.The third issue is related to signs. If the connection is not doubly symmetrical, then it

can be expected that reversing signs can lead to different limit values, perhaps due tothe buckling effects which must, however, always be kept in consideration, if only to beproven irrelevant. So, to be applicable when signs are reversed, two points in plane(N, M) are not enough and four must instead be evaluated, doubling the effort.It can be concluded that the practice of drawing a linear domain, considering only the

equilibrium-crime survivors, has several limits and can also be: (a) too much on the safeside because the “true” limit multiplier can be much higher than that obtained by thelinearized limit domain, (b) not on the safe side, if some buckling mode (possibly asso-ciated with sign reversal) has not properly been accounted for and checked.This is not an academic discussion. The point Q which would be obtained by

increasing the couple (N, M) and searching for the first failure mode, in classic plas-ticity, may well be quite different from point A. This can be understood when con-sidering for instance the plastic yield lines related to the possible failure modes: theplastic-limit resistance failure of parts of the member itself or of its connected con-stituents. These yield lines are tightly related to the load configuration applied. Withonly load N (load increasing along the horizontal axis of Figure 1.4), a yield lines setwould be found, related to some mechanism. With only load M applied (load alongthe vertical axis of Figure 1.4), another yield lines set would be found. With a possiblecombination of N and M, a third, possibly completely different yield lines set wouldbe found (load along the inclined straight line in Figure 1.4). So, linearizing meansforgetting the true load state and doing a purely numerical simplification, with nophysical meaning.

Steel Connection Analysis6

Finally, premature buckling of connections subconstituents is one of themost frequentfailure modes, especially in seismic areas where internal forces sign reversal is a normalcondition. So the problem of its correct evaluation is a real problem.

1.2.4 Single-Constituent External Combined-Effects Neglect

One typical simplification of traditional connection design is that when dealing with anode where n members are joined, the connections are evaluated one by one, consid-ering two or at most three members at a time, and no more. So, if the node is likeA-B-C, where B is the main member and A and C are secondary, connection designis often carried on by considering A-B, and so checking A, B, and their connectors,and then B-C, and so once more B, then C, and then their connectors. However, thisworking method is not correct, as the effects on B of A and C are contemporary, andnot separate.Consider for instance the node of Figure 1.5: all the five “slave” members act together

on the column, which is the “master”member used as a reference for all other members.Considering the effects of each slave member on the column separately would only bepossible if the superposition principle were to hold true (and it would not in plasticanalysis), and only if the – very different – effects of each member on the column werecorrectly summed. But this is not what is usually done in designing connections. Usually,the effects will at most be grouped considering typical member configurations, like twoopposite beams joined to a web, or two opposite beams joined to flanges, and traditionalmethods are definitely not able to take all these members into account.What is clearly dangerous here is the possibility that combined effects could

drive the common member to failure, or, more generally, the common constituents.C

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Introduction 7

Without a clear and coherent computational method to consider the sum of theeffects of all the connections, the evaluation of the combined effects is often leftto improbable envelopes or the sum of physically meaningless quantities. And thisis a very strong weakness.Besides, it is important to note that the problem is not only related tomaster members,

but in general to all constituents. As all the members in a node are directly or indirectlytied together, it is not unusual for the internal forces acting in a member can flow to theconnections of another member, a possibility that is implicitly excluded by traditionaldesign methods and that can instead be easily observed when a complete finite elementmodel of a node is set up.Connections are definitely more complex than a one-to-one, or a one-to-two relation-

ship between members. Connections have inside themselves the same complexity as thewhole structure.

1.2.5 Neglecting Eccentricities

True structures and true connections very often have relevant eccentricities that shouldbe ideally considered, but that are very often neglected.The first type of eccentricity is that of members’ axis lines, which in the actual con-

struction are often not as in the finite element model. This leads to possibly severe addi-tional moments that can induce stresses comparable to, or even higher than, thosecomputed considering members to be fully aligned with their computational schemata.As it is very lengthy to properly take into account all these eccentricities by hand, in a 3Dcontext, it is still dangerously considered normal practice to neglect them. This choice isstrengthened by the equilibrium crime, which, neglecting some internal forces compo-nents, also neglects the additional moments that theymight drive in some constituents ofthe connections. However, if a force F is offset by e, the additional moment is Fe, which,assuming a resisting lever b, leads to an additional force equal to Fe/b. If e is much lowerthan b, then the additional force is negligible, but if it is not, the additional force mayconsiderably increase the nominal one.To compute the additional moment, in a 3D context, a more precise rule would be (P is

the true point of application, O the point where the force is moved to, F is the forceapplied)

M= P−O F

and there are three moment components.The second type of eccentricity is in the connection area where, due to constructional

needs, the true layout of bolts, welds, stiffeners, and cleats may not be that effectivelyassumed in the connection simplified – often 2D – computational model, especiallywhen considering forces and moments flowing into connectors. Indeed, this computa-tion requires ideally simple, but in true practice quite boring and error-prone vector pro-ducts. For instance, considering the angle connected to the node in Figure 1.6, while theeccentricity in an horizontal plane will probably be considered, the eccentricity in thevertical plane (the center of the diagonal is not at the elevation of plate mid-thickness)will probably be neglected, which is not correct.

Steel Connection Analysis8

1.2.6 Use of Envelopes

One of the key features of traditional approaches is that they had to deal with a limitednumber of loading conditions, usually computed by considering envelopes of notionalload cases that were themselves envelopes (maximum wind, plus maximum snow,and so on).Nowadays the number of loading conditions, expressed in the form of load combina-

tions, is quite high. Referring to Eurocodes and applying the combination rules there pro-vided, it is not unusual to get hundreds or even thousands of combinations (Rugarli2004). This means that the traditional way of computing connections, two or more load-ing conditions, can be obtained by only assuming a special kind of envelope, which con-siders maximum or minimum values of different internal forces components actingtogether. Owing to the equilibrium crime the number of different components of inter-nal forces is often limited to two, so the notional combinations are usually quite a few.2 Isthat approach on the safe side?Well, provided that, when increasing the absolute value ofthe internal forces all the load effects do always increase, then this might be the case.Unfortunately, this is not always true, and it is not true when a decrease of some internalforce, for some failure modes, is more dangerous than an increase, such as in slip checksunder compression, or when a specific mix of internal forces leads to worst results.

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2 The set of load combinations has to be really complete, otherwise the “maximum” is not really the worst valuethat the structure might experience, but this is another problem, well rooted in using “realistic” combinationsets and not unrealistic, but enveloping.

Introduction 9

Consider a very simplified example (Figure 1.7), a slip resistant simple support connec-tion relying on the compression N acting over a friction plane having friction constant k.Let Nlim be the maximum compression allowable without failure (vertical line bound).A shear V is also applied, and it must be that V < kN to avoid slip. Several couples (N,V) have been computed in a number of load combinations. To check the connection,the maximum N is taken with maximum V. Is that safe? Figure 1.7 clearly shows thatit is not. The limit domain is here a triangle.Point A, neglected by this “envelope” rule, is outside the limit domain. Point D is

obtained by mixing maximum N (point C) with maximum V (point B). So consideringthe maximum absolute values as acting together is not always a safe approach, at leastwhen slip can be a failure mode. It’s interesting to note also that adding to checks pointE beside point D, i.e. the point with minimum N, the problem would not have beensolved, and only mixing the minimum N with the maximum V would have been onthe safe side, but possibly, too much.Indeed, mixing maxima and/or minima can be a quite over-safe approach. Assuming

that the governing failure mode has a linear limit domain (Figure 1.8), this way of com-puting can sometimes really lead to an overestimate of utilization.

V

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B: max { V }C: max { N }

Figure 1.7 Friction connection. Point A is outsidethe limit domain, point D is inside.

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Figure 1.8 Overevaluation of utilization factorusing contemporary maxima.

Steel Connection Analysis10

1.2.7 Oversimplification of PlasticMechanisms Evaluation

Traditional by hand or by spreadsheet approachesall try to evaluate the plastic limit load of complex3D assembly of steel plates. As this is in general avery complex task, it is not surprising that thenumber of such computations is reduced to theminimum possible (as it has been shown, by lin-earizing the limit domain), and that quite oftenthe problem is tackled by using simplified andquite regular geometries.One problem here is that real world connec-

tions do not always comply with such geometries,so the analyst is pushed to force his/her probleminto one that is solvable. According to somecomputational tools, the problem to be solved isalways a T-stub, pulled, compressed, or bent, withregular bolt “rows”,. But this is not always realistic– for example see Figure 1.9.Not only can the geometry of the steel plates be

quite different, but the bolt layout (not tomentionthe loading condition see Section 1.2.2), can bedifferent. So, if for some good reason a bolt or astiffener has to be shifted, or if a plate is not rec-tangular, or if the footprint of the cross-section and stiffeners is not as regular as in thetextbooks, the computational model is simply not able to deal with the problem. Muchserious effort has been spent in order to categorize the local failure modes related to typ-ical connections, so that the limit multiplier under simple loads (axial force, bendingmoment) of typical assemblies could be evaluated. For instance, theGreen Book referringto moment resisting connections (MRC), published by Steel Construction Institute (SCI1995), is an excellent book which lists all the possible failure modes and partial yield linesrelated to typical connections, basically considering the T-stub idealization.Evaluating the limit load is then a matter of summing up different contributions,

analyzing different possible failure lines and modes (Figure 1.10), and finally gettingthe minimum value. Table 2.4 of the Green Book for moment connections lists 11 pat-terns for elementary yield lines,3 used in order to evaluate a final effective length, Leff,to be introduced in the formula for the problem at hand (see also Eurocode 3, Part1.8, §6.2 and subsections):

M =Leff t2fy

4

Figure 1.9 Real world momentconnection (courtesy CE-N CivilEngineering Network, Bochum,Germany).

3 They are: (i) circular yielding, (ii) side yielding, (iii) side yielding near beam flange or stiffener, (iv) sideyielding between two stiffeners, (v) corner yielding, (vi) corner yielding near a stiffener, (vii) double curvature,(viii) group end yielding, (ix) corner yielding, (x) individual end yielding, (xi) circular yielding. For each of thesepatterns an Leff formula is provided.

Introduction 11

where t is the thickness of the plate where yield lines willappear (e.g. a base plate, an end plate, or a cap plate of acolumn), fy is its yield strength, andM is a limit bendingmoment. The formula is clearly notional and is exactwhen considering the plastic moment of a plate oflength Leff and of thickness t. The use of the typical pat-terns is not easy, nor particularly intuitive, and it maywell lead to errors that are hardly detectable; it is upto the user of such tables to properly mix and composethe typical patterns in a reliable way (see Figure 1.10).Single effects are evaluated and then summed. Forinstance, to get the Leff of “a bolt row below the beamflange of a flush end plate”, we have to evaluate the finalLeff related to that bolt row as a function of the individualpatterns Leff,i as follows:

Min Maxii+ iii2

, ii , i

but only if some specific geometrical limitations aremet,otherwise we have to use

Min Max ii, iii , i

where “i”means “pattern number i”. Of course this takes into account only that bolt row.It is not necessary to get further into that here, but it has to be realized that the eval-

uation of the limit load multiplier is out of reach of these methods when applied togeneric 3D models loading conditions. So, the problem should in general be tackledfor what it is: there are six internal forces flowing at the end of each member, many loadcombinations, many failure modes, a geometry that may well not be forcible into aT-stub, bolt rows that could well be moved (or perhaps no row may be available), andso on.Classical simplified resistance checks continued to use some kind of simplified geom-

etry for the flow-lines of stresses, widely used for instance in the strut and tie method(STM), and these are simplified ways of computing plastic mechanisms. For instance,assuming a 30 line of stress flow (e.g. see AISC Steel Construction Manual, 14th ed.§9, “Whitmore section”), and computing some “effective” resistance cross-section toevaluate its limit or ultimate load, is one of these simplified and not always sufficient waysof computing a plastic mechanism. Often, approximations of this type are applied to gus-set plates under complex membrane stress (Figure 1.11). Cutting a complex solid with anideal plane and computing a net-cross-section to be checked in a beam-like way is anotherfrequently used simplification. Using simple structural schemata, such as cantilevers,simply supported beams, or struts and ties, extracted from more complex 3D scenes,to be checked for their limit plastic loads is another way to try to assess plastic mechan-isms of complex structural configurations. So, when considering what could be called“generic resistance checks”, the traditional approaches try to evaluate a plastic mechan-ism by means of simplified tools. Basically, a generic resistance check is a check that theconstituent – usually having an irregular shape – is able to carry the loads applied with no

Figure 1.10 An example ofevaluation of Leff by summingeffects. Red lines are the yieldlines whose total length has tobe evaluated.

Steel Connection Analysis12