reading

Cover
Title Page
Preface
1 Introduction
1. 1.1 An Unsolved Problem
2. 1.2 Limits of Traditional Approaches
3. 1.3 Some Limits of the Codes of Practice
4. 1.4 Scope of This Book
5. 1.5 Automatic Modeling and Analysis of 3D Connections
6. 1.6 Acknowledgments
7. References
2 Jnodes
1. 2.1 BFEM
2. 2.2 From the BFEM to the Member Model
3. 2.3 Jnodes
4. 2.4 Jnode Analytics
5. 2.5 Equal Jnodes Detection
6. 2.6 Structural Connectivity Indices
7. 2.7 Particular Issues
8. 2.8 Jclasses
9. References
3 A Model for Connection
1. 3.1 Terminology
2. 3.2 Graphs of Connections
3. 3.3 Subconstituents vs Layouts
4. 3.4 Classification of Connections
5. Reference
4 Renodes
1. 4.1 From Jnode to Renode Concept
2. 4.2 BREP Geometrical Description of 3D Objects
3. 4.3 The Scene
4. 4.4 Dual Geometry
5. 4.5 Automatic Connection Detection
6. 4.6 Elementary Operations
7. 4.7 Renode Logic and the Chains
8. 4.8 Prenodes
9. 4.9 After Scene Creation
5 Pillars of Connection Analysis
1. 5.1 Equilibrium
2. 5.2 Action Reaction Principle
3. 5.3 Statics of Connections
4. 5.4 Static Theorem of Limit Analysis
5. 5.5 The Unsaid of the Engineering Simplified Methods
6. 5.6 Missing Pillars of Connection Analysis
7. 5.7 Analysis of Connections: General Path
8. References
6 Connectors: Weld Layouts
1. 6.1 Introduction
2. 6.2 Considerations of Stiffness Matrix of Connectors
3. 6.3 Introduction to Weld Layouts
4. 6.4 Reference Systems and Stresses for Welds
5. 6.5 Geometrical Limitations
6. 6.6 Penetration‐Weld Layouts (Groove Welds)
7. 6.7 Fillet‐Welds Weld Layouts
8. 6.8 Mixed Penetration and Fillet Weld Layouts
9. References
7 Connectors: Bolt Layouts and Contact
1. 7.1 Introduction to Bolt Layouts
2. 7.2 Bolt Sizes and Classes
3. 7.3 Reference System and Stresses for Bolt Layouts
4. 7.4 Geometrical Limitations
5. 7.5 Not Preloaded Bolt Layouts (Bearing Bolt Layouts)
6. 7.6 Preloaded Bolt Layouts (Slip Resistant Bolt Layouts)
7. 7.7 Anchors
8. 7.8 Stiffness Matrix of Bolt Layouts and of Single Bolts
9. 7.9 Internal Force Distribution
10. 7.10 Contact
11. References
8 Failure Modes
1. 8.1 Introduction
2. 8.2 Utilization Factor Concept
3. 8.3 About the Specifications
4. 8.4 Weld Layouts
5. 8.5 Bolt Layouts
6. 8.6 Pins
7. 8.7 Members and Force Transferrers
8. References
9 Analysis: Hybrid Approach
1. 9.1 Introduction
2. 9.2 Some Basic Reminders About FEM Analysis of Plated‐Structures
3. 9.3 IRFEM
4. 9.4 Connector Checks
5. 9.5 Cleats and Members Non‐FEM Checks
6. 9.6 Single Constituent Finite Element Models
7. 9.7 Multiple Constituents Finite Element Models (MCOFEM)
8. 9.8 A Path for Hybrid Approach
9. References
10 Analysis: Pure FEM Approach
1. 10.1 Losing the Subconnector Organization
2. 10.2 Finite Elements for Welds
3. 10.3 Finite Elements for Bolts
4. 10.4 Loads
5. 10.5 Constraints
6. 10.6 Checking of Welds and Bolts
7. 10.7 Checking of Components
8. 10.8 Stiffness Evaluation
9. 10.9 Analysis Strategies
10. Reference
11 Conclusions and Future Developments
1. 11.1 Conclusions
2. 11.2 Final Acknowledgments
3. 11.3 Future Developments
4. References
Appendix 1: Conventions and Recalls
1. A1.1 Recalls of Matrix Algebra, Notation
2. A1.2 Cross‐Sections
3. A1.3 Orientation Matrix
4. A1.4 Change of Reference System
5. A1.5 Pseudocode Symbol Meaning
Appendix 2: Tangent Stiffness Matrix of Fillet‐Welds
1. A2.1 Tangent Stiffness Matrix of a Weld Segment
2. A2.2 Modifications for Weld Segments Using Contact
3. A2.3 Tangent Stiffness Matrix of a Weld Layout for the Instantaneous Center of Rotation Method
Appendix 3: Tangent Stiffness Matrix of Bolts in Shear
1. A3.1 Tangent Stiffness Matrix of a Bolt
2. A3.2 Tangent Stiffness Matrix of a Bolt Layout for the Instantaneous Center of Rotation Method
Index
Symbols and Abbreviations
1. Symbols
2. Greek Letters
3. Subscripts
End User License Agreement

List of Tables

Chapter 02
1. Table 2.1 Jnode classification.
2. Table 2.2 Heuristic complexity estimate for hierarchic jnodes of different connectivity.
3. Table 2.3 Hierarchic jnodes relative complexity, assuming a heuristic evaluation formula with m = 1.4 and as reference complexity that of connectivity 2.
4. Table 2.4 Jnodes indices for the structure of Figure 2.20. N is the jnode number. Mark is the jnode mark. NO is the number of occurrences (instances) of each jnode. NM is the number of connected members. Typology is the topological classification of jnode. Constraint is the constraint status. Frequency is the frequency of each jnode. Connectivity is the jnode connectivity value. Complexity is the jnode complexity value, evaluated with m = 1.4.
5. Table 2.5 Permutation of local axes saving physical layout, for cross‐sections having one or two axes of symmetry.
Chapter 05
1. Table 5.1 Imperfection factors for the stability curves according to Eurocode 3.
Chapter 06
1. Table 6.1 Stresses in the welds for six load cases, MPa, N/mm. All welds FR. Stresses n, t_par = t_u, t_per = t_v at extremity “ext”, and force per unit length. Only the worst stressed extremity is listed, for each weld.
2. Table 6.2 Stresses in the welds for six load cases. All welds FR, but the web welds, 2 and 6, that are shear only (rows shaded). Stresses n, t_par = t_u, t_per = t_v at extremity “ext”. Only the worst stressed extremity is listed, for each weld.
3. Table 6.3 Stresses in the welds for six load cases. All welds LO (longitudinally sheared only), flange welds are FR, and web welds, 2 and 6, are shear only (rows shaded). Stresses n, t_par = t_u, t_per = t_v at extremity “ext”. Only the worst stressed extremity is listed, for each weld. As all the welds are LO, t_per is always null.
4. Table 6.4 Stresses in the welds for six load cases. All welds LO (longitudinally sheared only), flange welds are FR, and web welds, 1 and 3, are shear only (rows shaded). Stresses n, t_par = t_u, t_per = t_v at extremity “ext”. Only the worst stressed extremity is listed, for each weld. As all the welds are LO, t_per is always null.
5. Table 6.5 Comparison between the utilization factors computed by ICRM in 3D, with and without the effect due to the angle between the fusion faces (C_w, C_u, and C_v coefficients).
6. Table 6.6 Concentrically loaded weld layout: loading, related utilizations, and difference between utilization factors. Comparison between expected and computed results. Here a precision of about 2.5% on utilization is expected.
7. Table 6.7 A set of generic loading conditions tested.
8. Table 6.8 Utilization and displacement at limit related to the loading conditions of Table 6.7.
Chapter 07
1. Table 7.1 Data for European bolts. D is the diameter of the bolt, D₀ is the normal hole diameter, A is the bolt‐shank area, A_res is the threaded area, s is the distance between two opposite faces of the hexagon, e is the diameter of the circle circumscribed to the hexagon, k is the height of the bolt‐head, and m is the height of the nut.
2. Table 7.2 Material grades for European bolts. f_yb is the yield strength and f_ub is the ultimate strength.
3. Table 7.3 US most frequently used bolt sizes.
4. Table 7.4 Material grades for US bolts. f_yb is the yield strength and f_ub is the ultimate strength.
5. Table 7.5 Estimate of the minimum multiplier to be applied to single plate thickness, to get a bolt diameter such that the bolt bearing resistance is lower than the bolt shear resistance. It is assumed that the bolt resisting shear area is the gross one. The first column lists bolt classes, the first row material grades. The shaded cells are the most frequent couplings.
6. Table 7.6 Values of Q and Q* as a function of L/D.
7. Table 7.7 All bolts fully resistant and with full efficiency. N_B: axial force of bolt. V_uB: shear along axis u. V_vB: shear along axis v. V_B: total shear. M_uB: bending moment about axis u of bolt shaft. M_vB: bending moment about axis v. M_B: total bending moment.
8. Table 7.8 Forces and moments in the bolts assuming that all bolts are shear only. N_B: axial force of bolt. V_uB: shear along axis u. V_vB: shear along axis v. V_B: total shear. M_uB: bending moment about axis u of bolt shaft. M_vB: bending moment about axis v. M_B: total bending moment.
9. Table 7.9 Forces and moments in the bolts assuming mid‐side nodes are shear‐only, while corner nodes are no‐shear. N_B: axial force of bolt. V_uB: shear along axis u. V_vB: shear along axis v. V_B: total shear. M_uB: bending moment about axis u of bolt shaft. M_vB: bending moment about axis v. M_B: total bending moment.
10. Table 7.10 Forces and moments in the bolts assuming mid‐side nodes have an efficiency factor μ_n = 0.5, while corner nodes are no‐shear. N_B: axial force of bolt. V_uB: shear along axis u. V_vB: shear along axis v. V_B: total shear. M_uB: bending moment about axis u of bolt shaft. M_vB: bending moment about axis v. M_B: total bending moment.
Chapter 09
1. Table 9.1 Nodes of the IRFEM model of Figure 9.17 (mm).
2. Table 9.2 The connectivity, type, and meaning of the elements of the IRFEM model in Figure 9.17. N1 is the first node, N2 is the second node. The meaning of the elements is the constituent of the renode the element is modeling.
3. Table 9.3 Possible notional combinations to test connections in a renode where no internal force is available. Combinations from 1 to 12 are obtained by adding the six limit values alone with plus (1–6) and minus sign (7–12). Twenty‐four combinations for each member are so defined.
4. Table 9.4 Prying forces factor in a T‐stub according to Thornton (1985) for several design situations. Bolts are M16 class 8.8.
Chapter 10
1. Table 10.1 Comparison of the main results obtained using IRFEM and the closed formulae discussed in Chapter 6, and the results obtained by rigid MCOFEMs, simulating the IRFEM and the range of applicability of the traditional methods.
2. Table 10.2 Terms of the stiffness matrix for isotropic fillet weld segment and initial stiffness.
3. Table 10.3 Terms of the stiffness matrix for isotropic fillet weld segment and secant stiffness.
4. Table 10.4 Terms of the stiffness matrix for orthotropic fillet weld segment and initial stiffness.
5. Table 10.5 Terms of the stiffness matrix for orthotropic fillet weld segment and modified secant stiffness.
6. Table 10.6 Comparison between (a) the utilizations ratios for several generic loading conditions computed using IRFEM (see also Table 6.8) and the methods of Section 6.7.4.1, and (b) the utilizations computed using a non‐linear rigid MCOFEM.
7. Table 10.7 Single fillet weld segment loading conditions.
8. Table 10.8 Results obtained for the loads applied to a single fillet weld segment (the loads are those of Table 10.7). Displacements d_w, d_u, d_v; angle θ; functions h(p) and q(θ). Forces resulting, to be compared with those of Table 10.7.
9. Table 10.9 Results obtained for the loads applied to a single fillet weld segment (the loads are those of Table 10.7). Displacements d_w, d_u, d_v; angle θ; functions h(p) and q(θ). Forces resulting, to be compared with those of Table 10.7.
10. Table 10.10 HEB 200 cross‐section properties.
11. Table 10.11 Features of the PFEM models. “Y” stands for “Yes”, “N” stands for “No”.
12. Table 10.12 Main results obtained by analyzing the different models whose features are explained in Table 10.11. All models use shear‐only bolts. All models consider the members deformable. The shear V and the axial force N of a single bolt are listed, as well as the total axial displacement d.
13. Table 10.13 Main results obtained by analyzing the different models whose features are explained in Table 10.11. The shear V and the axial force M of a single bolt are listed, as well as the total axial displacement d. All models use fully resistant bolts. All models consider the members deformable.
14. Table 10.14 Results obtained for model E, with bearing and contact non‐linearities activated. Bolts fully reacting, no explicit bolt hole modeling.
15. Table 10.15 Results obtained by with preloaded bolts: model O without holes; model P with holes, linear range; model Q without hole, model R with hole, non‐linear range. Bolts fully resistant. All models consider the members deformable.
Appendix 01
1. Table A1.1 Different conventions for cross‐section principal axes.
2. Table A1.2 Main cross‐section symbols and their meaning.
3. Table A1.3 Symbols used for cross‐section internal forces in this book.

List of Illustrations

Chapter 01
1. Figure 1.1 A possible node of a 3D structure.
2. Figure 1.2 Flexible end plate connection (“shear” connection).
3. Figure 1.3 Traditional design applied to computerized analysis: no way for the load path.
4. Figure 1.4 Limit domain for a connection; P(N, M) is the stress state for a single member assumed.
5. Figure 1.5 All the members connected to the column do act over it at the same time.
6. Figure 1.6 Eccentricity between the point of application of the force N, P, and the weld layout center, W6, O. The vector (P‐O) has the three components (A‐O), (B‐A) and (P‐B), but frequently (P‐B) is neglected.
7. Figure 1.7 Friction connection. Point A is outside the limit domain, point D is inside.
8. Figure 1.8 Overevaluation of utilization factor using contemporary maxima.
9. Figure 1.9 Real world moment connection (courtesy CE‐N Civil Engineering Network, Bochum, Germany).
10. Figure 1.10 An example of evaluation of L_eff by summing effects. Red lines are the yield lines whose total length has to be evaluated.
11. Figure 1.11 Gusset plate under membrane stresses: the three forces in the central area simulate the action transferred by the bolts; weld forces are balanced; fictitious constraints.
12. Figure 1.12 Bridge gusset plate buckling failure.
13. Figure 1.13 A T‐stub.
14. Figure 1.14 Eurocode 3, Part 1.8: column bases only subjected to axial forces.
15. Figure 1.15 Prying forces Q increase tensile forces in bolts T. Simple T‐stub model.
16. Figure 1.16 Block tearing (from Eurocode 3, Part 1.8): forces are aligned with plate sides, and so with the break lines.
Chapter 02
1. Figure 2.1 A beam to column connection.
2. Figure 2.2 Example of two adjacent elements whose nodes are aligned but that, due to the existence of rigid offsets, cannot be considered aligned.
3. Figure 2.3 Sets of finite elements.
4. Figure 2.4 Aligned elements belonging to different members: no end release.
5. Figure 2.5 Member at a node. The member made up of beam elements 1 and 2 is passing at node 8. The member made of beam element 4 is cuspidal at node 8. The member made of element 3 is interrupted and unreleased. The full circle stands for “connection code”. The small full squares mark the node positions.
6. Figure 2.6 Two members alignment: 32‐13‐21.
7. Figure 2.7 Two members alignment: 22‐13‐31.
8. Figure 2.8 Example of a simple jnode in the real world: the column is constrained to a concrete block.
9. Figure 2.9 Example of a hierarchical jnode in the real world: the master is the horizontal (green) beam member, which is passing. All other members are interrupted (courtesy AMSIS Srl, Rovato, Italy).
10. Figure 2.10 Example of a central jnode in the real world: there is no master member, all members are interrupted and attached to the same constituent.
11. Figure 2.11 Hierarchic jnode. There is a vertical passing member, as no connection code has been specified.
12. Figure 2.12 Hierarchic jnode. The upper vertical member is the master. All other members are interrupted, and so are slaves.
13. Figure 2.13 Cuspidal jnode. There are two possible candidates for being a master member: the horizontal passing member, and the vertical cuspidal member. So, there is no clear hierarchy and the jnode is cuspidal.
14. Figure 2.14 Central jnode. All the members are interrupted, so the jnode is central: it has no master.
15. Figure 2.15 Tangent jnode. There are two passing members as no connection code has been specified, and the cross‐section, material and orientation of splice‐aligned members are the same. There is no clear master member.
16. Figure 2.16 Cuspidal jnode. Here there are two cuspidal members and two interrupted members. No master can be clearly set.
17. Figure 2.17 A hierarchical jnode with generic inclinations. This jnode is topologically equal to that of Figure 2.12.
18. Figure 2.18 Simple example of equal jnode detection. Jnode AD (highlighted in yellow) has four instances. Also, jnodes AA, AB and AC have four instances.
19. Figure 2.19 The new Mediateca in Colle Val d’Elsa, Siena (Galluzzi & Associati & Ateliers Jean Nouvel, Galluzzi Associati 2010). The steel structure was very complex and a large number of different jnodes were found (569). A detail of one of the four inclined planes is also sketched, considered as isolated from the surrounding part. Cuspidal and tangent jnodes have been discarded, so some jnodes do not have the jnode symbol. (Courtesy, Galluzzi Associati, Florence.)
20. Figure 2.20 A framed structure with all the jnodes marked. Within the circle is jnode AB.
Chapter 03
1. Figure 3.1 Graph of a column base plate connection.
2. Figure 3.2 A simple bolted splice joint.
3. Figure 3.3 Graph of the splice connection of Figure 3.2. There are two bolt layouts connectors having multiplicity 3 and four bolt layouts connectors having multiplicity 2. The total multiplicity of connectors is 14.
4. Figure 3.4 A two beam to column real node. Connection is by fin plates.
5. Figure 3.5 Graph of the real node of Figure 3.4.
6. Figure 3.6 A more complex node (Fabbrichina, courtesy Studio Galluzzi e Associati, Florence).
7. Figure 3.7 Graph of the real node of Figure 3.6.
8. Figure 3.8 A single six‐bolt layout connector with multiplicity 3 can be transformed into six elementary connectors having multiplicity 3.
Chapter 04
1. Figure 4.1 A jnode and its renode completed.
2. Figure 4.2 A face with its numbered points and outward normal.
3. Figure 4.3 A scene: a constraint block is used (courtesy Amsis srl, Rovato, Italy).
4. Figure 4.4 Initially, members overlap.
5. Figure 4.5 A stiffener for a rolled I or H cross‐section.
6. Figure 4.6 A stiffener is put in place by selecting its insertion point.
7. Figure 4.7 Synthetic representation of a bolt layout having six bolts. Connected parts are not displayed.
8. Figure 4.8 Fillet weld seams definition. The sides of the chosen face are numbered clockwise as the normal of the chosen (blue) face enters into the page.
9. Figure 4.9 Several possible penetration weld seams applied to a thickness.
10. Figure 4.10 Penetration weld layout definition.
11. Figure 4.11 A compound member with a part removed.
12. Figure 4.12 Subplates of a member stump having an I cross‐section.
13. Figure 4.13 Plated subconstituents of a member after two work processes have been applied.
14. Figure 4.14 Coplanar faces. The hexagon face has a positive normal, all other faces have a negative normal. Green surfaces are contact areas. The square face is entirely contained in the hexagon face and is in contact. Leftmost rectangular face has only a part in contact. All other faces are external or tangent.
15. Figure 4.15 Entry face C₁ to add a bolt layout (also displayed). The model, developed using software CSE is by AMIS srl, Rovato, Italy.
16. Figure 4.16 Drilling to air from face C₁.
17. Figure 4.17 The two circular faces of a circular base plate and the two faces of a concrete plinth are the drilled faces stack. Other faces are the footprint of plates and welds and are used to avoid overlaps of the bolts (the bolt heads are drawn). See Figure 4.15.
18. Figure 4.18 An example of addition of a fillet weld layout. The chosen face is usually contained in the other contact face. The plane of the chosen face is common to all weld seams.
19. Figure 4.19 Active faces of weld seams (fillet weld layout).
20. Figure 4.20 Thickness changes. Green ones can be adjusted automatically. The red ones cannot, because it would imply a member cross‐section change and a lack of coherence with the underlying BFEM.
21. Figure 4.21 Structured arrangements imply more chains, here five (two broken and three complete).
22. Figure 4.22 Invalid chains: unconnected (top left), loop (bottom left), broken (top right).
23. Figure 4.23 A three member connection using T1 and P1 as force transferrers. Some of the chains automatically found are listed (the model was prepared using software CSE by AMSIS srl, Rovato, Italy).
24. Figure 4.24 A base node isoconnected (the model was prepared using software CSE by AMSIS srl, Rovato, Italy).
Chapter 05
1. Figure 5.1 In this example an eccentricity has been neglected.
2. Figure 5.2 An example of a base joint with a thick plate, under shear.
3. Figure 5.3 First model to balance the transport moment: extra axial forces in the bolt shafts.
4. Figure 5.4 Second model to balance the moment of transport: additional moments in the bolt shafts.
5. Figure 5.5 A constituent in space under the effect of three force packets.
6. Figure 5.6 Transferring of a force packet from P to Q.
7. Figure 5.7 The 1760 2nd Jesuitical edition of Philosophiae Naturalis Principia Mathematica by Isaac Newton. On the right the Third Law, at page 23 of the first volume (courtesy of the author’s friend Giorgio Nieri, owner of the volumes).
8. Figure 5.8 Nodal zone for a five member real node.
9. Figure 5.9 Member stump extremity N, and the new one F.
10. Figure 5.10 Incorrect member stump definition: too short.
11. Figure 5.11 Forces at far extremity F (pulled bar). Equilibrium is obtained from the forces exerted by the bolts over the member. In the BFEM, those forces are equivalent to s_N applied at extremity N.
12. Figure 5.12 Forces at far extremity F, if external loads are applied. Equilibrium is obtained from the forces exerted by the bolts over the true member. In the BFEM, those forces are equivalent to s_N applied at extremity N.
13. Figure 5.13 Dual model for pulled bar (Figure 5.11). Equilibrium is obtained from the forces exerted by the bolts over the member.
14. Figure 5.14 Dual and true systems under balanced packets (no external load).
15. Figure 5.15 Compound reference system and submember cross‐section reference system.
16. Figure 5.16 Graphs for typical subsystems: isoconnected (top); one time hyperconnected (bottom).
17. Figure 5.17 Apparently isoconnected system, hiding a hypoconnected subsystem and a hyperconnected surrounding.
18. Figure 5.18 Static “safe” theorem of limit analysis does not provide protection from buckling.
19. Figure 5.19 The value of α_R as a function of α_L, for different α_cr, and for buckling curve “b”. If α_L1 < α_L2, then it is always α_R1 < α_R2, for every value of α_cr.
20. Figure 5.20 The value of α_R as a function of α_L, for different α_cr, and for buckling curve “c”. If α_L1 < α_L2, then it is always α_R1 < α_R2, for every value of α_cr.
21. Figure 5.21 The value of α_R as a function of α_L, for different α_cr, and for buckling curve “d”. If α_L1 < α_L2, then it is always α_R1 < α_R2, for every value of α_cr.
22. Figure 5.22 Example of application of the static theorem of limit analysis using the general method. Statically admissible plastic multiplier α_L1 = 2, “true” plastic multiplier α_L2 = 2.5, α_cr = 1.8.
23. Figure 5.23 The general method applied to an eight‐combination model (constant limit load lower bound α_L = 1.15 used). Curve “b” has been used. The minimum value is α_R = 1.003 > 1.0. The first three critical multipliers have been used, so three curves can be distinguished.
24. Figure 5.24 Identical iso‐loaded constituents forming a set.
25. Figure 5.25 Isomorphically loaded constituents.
26. Figure 5.26 Equal force distribution for shear applied to a connector.
27. Figure 5.27 Internal distribution of the forces in a moment connector is not immediately determinable.
28. Figure 5.28 Local stress peak near to a rounded corner (coarse mesh).
29. Figure 5.29 Plate with a circular hole, in tension.
30. Figure 5.30 Stress concentration factor for a pulled plate with a circular hole.
31. Figure 5.31 Iso and hyperconnected no‐slip connectors.
Chapter 06
1. Figure 6.1 Progressive redistribution of elastic forces obtained by lowering the bolt stiffness. Plates are pulled toward the left. Bolt cross‐sections are circles of decreasing diameter (24 mm, 12 mm, 4 mm). This simulates the effect of the bolt bearing plasticization effect.
2. Figure 6.2 A pulled plate fixed by two fillet welds. Total throat is 7.07 mm, plate thickness is 10 mm. The normal stress in the vertical plate is 100 N/mm². The Von Mises stress in the welds is much higher than 100 × 10/7 = 141 N/mm², and reaches the yield stress of 275 N/mm² in extended parts of the fillet welds. Brick and wedge elements are used. Perfect plasticity behavior is assumed.
3. Figure 6.3 Deformed view of the pulled plate (amplification 100×). Maximum Z displacement is 0.0095 mm. Weld and plate length: 50 mm. Plate thickness: 10 mm. Weld seams thickness: 5 mm. Throat size of single weld: 3.53 mm. Force applied: 50 kN.
4. Figure 6.4 Reference systems for a weld layout and single weld seam.
5. Figure 6.5 Throat, leg and thickness definition.
6. Figure 6.6 Ratio between throat thickness a and thickness t, depending on the angle between active faces γ, in degrees. An interpolation formula is also displayed, with the related R².
7. Figure 6.7 The stresses in a weld seam: active face on object O₁.
8. Figure 6.8 Stresses in fillet weld: on the unprojected throat plane (σ, τ) and on a projected throat plane (n, t). Both t_par and τ_par are normal to the paper, positive toward the reader.
9. Figure 6.9 Single‐sided partial penetration weld under the effect of an eccentric force is under bending, with tension at the root of the weld.
10. Figure 6.10 Full penetration weld layout, having seams not extending along all the plate length. Center G and principal axes (u, v) are displayed, as well as insertion point I and reference axes 1 and 2.
11. Figure 6.11 Shear stress due to torsion, elastic method.
12. Figure 6.12 Weld of unstiffened flange.
13. Figure 6.13 Nodal displacements of weld i and of the whole weld layout.
14. Figure 6.14 Example of weld layout (full penetration welds). Eight welds, four having t = 7.5 mm and the two web ones having t = 4.5 mm. Principal axes of weld layout (u, v) are also drawn, as well as u_i direction for all welds.
15. Figure 6.15 Asymmetric weld of a symmetric cross‐section. Welds are now numbered differently. The flange welds are FR, the web welds are shear only. All welds are LO. The loads applied to the weld layout are the same of the other cases.
16. Figure 6.16 Load–displacement curves for fillet welds loaded at different angles θ. The horizontal axis is the ratio of displacement to weld thickness. The vertical axis maps the ratio of the force to the ultimate force of longitudinally loaded fillet welds (R₀, θ = 0). The curves plotted are those of the AISC method. The two straight lines are related to the secant stiffness and modified secant stiffness for θ = 0.
17. Figure 6.17 Normalized secant stiffness as a function of angle θ. K_sn is the normalized secant stiffness curve predicted by the force–displacement curve by AISC. K_sn* is the curve predicted by a Timoshenko beam element using only the stiffnesses valid for θ = 0 and θ = 90°.
18. Figure 6.18 Ratio predicted to exact utilization ratios, using a linearized modified secant stiffness for the welds. Load perpendicular to faying plane.
19. Figure 6.19 Ratio predicted to exact utilization ratios, using a linearized modified secant stiffness for the welds. Load parallel to flange.
20. Figure 6.20 Ratio predicted to exact utilization ratios, using a linearized modified secant stiffness for the welds. Load parallel to web.
21. Figure 6.21 Typical cross‐section of models (γ = 90°).
22. Figure 6.22 Typical model for γ < 90°.
23. Figure 6.23 Typical model for γ > 90°.
24. Figure 6.24 A typical vertical displacement map. Model 2W_7.5_20_10_60.
25. Figure 6.25 Curve q = q(γ): a clear linear law can be detected. Also the points at γ = 80 and γ = 100 have been used.
26. Figure 6.26 Curve q = q(γ) for longitudinally sheared fillet welds. The tendency linear line equation is also plotted, with the R² value.
27. Figure 6.27 The ICRM method.
28. Figure 6.28 unmodified curve h(p).
29. Figure 6.29 The curve q(θ).
30. Figure 6.30 The displacements of a weld segment, j.
31. Figure 6.31 Schematic representation of the incremental‐iterative process. It is implicitly assumed that all the welds are equal and equally loaded. The true curve for generic configurations does not follow h(p) or q(θ).
32. Figure 6.32 Weld layout with fillets having different γ. An inclined element is considered.
33. Figure 6.33 A skewed member welded to a base plate. The fillet welds have γ angle ranging from 63.4° to 116.6°.
34. Figure 6.34 HEB 200 welded to plate by fillet welds.
Chapter 07
1. Figure 7.1 As the bolts can be applied anywhere, they usually imply bending stresses and severe bending of plates.
2. Figure 7.2 Reference system for a generic bolt layout.
3. Figure 7.3 Extremities E_k, check sections Z and points Z_k, force packets s_k and internal forces σ_k at check sections Z_k, for a generic bolt layout.
4. Figure 7.4 The normalized force–displacement curve according to Rex and Easterling (2003). In the abscissa, the normalized displacement Δ = K_iniΔ/R_be. In the ordinate, the ratio is R/R_be.
5. Figure 7.5 The curve q(Δ) defining the shear vs displacement law of a bolt.
6. Figure 7.6 A bolt layout under a slight bending carried only by the axial forces in the shaft.
7. Figure 7.7 A typical example of bolt layout in bending using a bearing surface.
8. Figure 7.8 Model for bearing surface.
9. Figure 7.9 A plate bolted to a flange and one possibly related bearing surface – the whole plate extent. Bearing surface can also be limited to a subpart of the bolted plate. However, no part of the bearing surface may be external to the bolted plate.
10. Figure 7.10 Finite element model of a plate loaded from below. The magenta dots are nodal constraints simulating the welds to the column and to the stiffeners. The dark blue and blue area is where displacements are null or very low: this is where the main part of the pressures are exchanged, as the plate is flexible.
11. Figure 7.11 Cantilever model for bearing surface near to the footprint of loads applied. The span of the cantilever c is a function of the desired load p, the thickness of the plate t, and the yield stress of the plate, f_y.
12. Figure 7.12 A column and four force transferrers welded to a base plate. All the faces are coplanar. The face with red border is the plate face, and the entry face is drilled for bolts.
13. Figure 7.13 First step: the bearing surface is now the face of the column (green) bordered by c.
14. Figure 7.14 Second to fifth step: the terms related to each force transferrer have been added to the bearing surface. The resulting bearing surface becomes outside of the face of the bolted plate, an inadmissible condition.
15. Figure 7.15 Final bearing surface: on the left, c = 47 mm, on the right c = 10 mm.
16. Figure 7.16 Bearing surface examples. With the exception of case A, a cold‐formed “scribble” connected to a plate for test, all cases are taken from real engineering analyses. Case D is a bracket (courtesy B&M, France). Cases B and E use polygon subtraction due to the hole. Case H is a case where the plate bolted to the column flange is wider than the column flange: the red strips are not part of the bearing surface.
17. Figure 7.17 A set of controls designed in such a way as to allow the general definition of the bearing surface (Connection Study Environment, by the author).
18. Figure 7.18 Typical constitutive laws for bearing surface.
19. Figure 7.19 A base plate under pure bending. The candidate bearing surface equals the plate face. Several constitutive laws. Maximum pressure and maximum tensile stress in bolts are highlighted, MPa (results obtained from the CSE software, Connection Study Environment, written by the author).
20. Figure 7.20 A base plate under pure bending. The bearing surface is obtained by considering an edge of 47 mm to the column and stiffeners footprint. Several constitutive laws. Maximum pressure and maximum tensile stress in bolts are highlighted, MPa (results from the CSE).
21. Figure 7.21 The effect of preload of bolts on the plates connected.
22. Figure 7.22 A test model for preload effects. Thickness of plate: 75 mm. Brick elements used. Number of elements 3780. Number of degrees of freedom 13625. The load is applied at the top surface as a constant pressure over a ring. The model rests on a set of frictionless non‐linear no‐tension springs. The green area (bottom right) is in contact. The remaining part lifts up.
23. Figure 7.23 The curve Q = Q(L/D). Q is the factor to be applied to diameter D, to get the external diameter D_eq of the equivalent resisting hollow cylinder, so as to give the same results of Wileman et al., 1991.
24. Figure 7.24 An example of bearing surface of a preloaded bolt layout. This is obtained by considering for each bolt a circular annulus of external diameter 1.75D and internal diameter D₀.
25. Figure 7.25 Variation of internal lever when bearing surface is assumed, bending around vertical axis.
26. Figure 7.26 Ratio (k_ps/k_pc) as a function of (D/t).
27. Figure 7.27 Curve G = G(D/t). D is the anchor diameter, t is the steel plate thickness.
28. Figure 7.28 Displacements d_ui and d_vi of a bolt i, depending or r_w.
29. Figure 7.29 The model for bolt bearing and bolt shear. Degrees of freedom d_u1 and d_u2 are condensed out.
30. Figure 7.30 Generic plastic condition for (N, M) limit domain.
31. Figure 7.31 Bolt layout to be examined.
32. Figure 7.32 Mixed configuration of bolts: mid‐side nodes are shear‐only, corner‐nodes are no‐shear.
33. Figure 7.33 Contact element extremities in simplified models, bearing surface is inside the intersection between the faces in contact.
Chapter 08
1. Figure 8.1 The third law applied to single connectors, bolts, and welds. Once the internal forces flowing into subconnectors are known, by the third law we also know how the constituents are loaded. Bolts also exchange bending moments and axial forces.
2. Figure 8.2 A utilization index is not a measure of the distance from the failure condition. Here, if the applied loads are related to a utilization index U = 0.5, the factor to be applied to get U = 1 is k = 1.41, not k = 2.
3. Figure 8.3 Elliptic limit domain for bolt resistance under tension and shear.
4. Figure 8.4 Modification of the experimental best fit curve by AISC 360‐10. The aim is to make simpler computations and to avoid discontinuity. However, the approximation is made slightly worse. In the green area, the utilization would be assumed >1 instead of <1. In red area it would be assumed < 1, instead of > 1.
5. Figure 8.5 Real‐world welding: a stiffener fillet‐welded to a circular base plate (Milan, photo by the author).
6. Figure 8.6 Partially welded rolled cross‐section using full penetration welds.
7. Figure 8.7 Planes where the stresses to be used for checks must be read, according to AISC 360‐10. Plane 1‐1 must be used for blue plate on the left. Plane 2‐2 for the weld throat. Plane 3‐3 for the green plate, on the right.
8. Figure 8.8 One of the 10.9‐class bolts connecting the engine to the frame of the author’s car, bottom part, sheared and bent to failure due to a clash of the head with an uneven stone (detail). The stone rotated under the wheel and remained blocked between the other stones, acting as a wedge. The bolt fractured at a distance from the force, where the lever was enough to generate sufficient bending. Center Milan, October 2016, photo by the author.
9. Figure 8.9 Limit domain for bolt shaft according to Eurocode 3.
10. Figure 8.10 The structural simplified layout for pins.
11. Figure 8.11 A deliberately generic configuration of the parts and of the bolts. The forces of each bolt have different direction and modulus. In the different thicknesses, all the forces are directed in the same direction, but the boundaries of the parts are different.
12. Figure 8.12 The green faces are the entry faces of a bolt. By using appropriate vector manipulation it is possible to assess if an edge is free or stiffened.
13. Figure 8.13 Distances from edges and pitches.
14. Figure 8.14 Invariance condition for bolt bearing checks.
15. Figure 8.15 In the direction of the force, sometimes no hole can be found. The hole layout is deliberately generic.
16. Figure 8.16 Model for punching shear.
17. Figure 8.17 An example of compressive pressure field over a bearing surface (base plate in bending, neutral axis also drawn). The maximum compressive stress (−3.4 MPa) can be used for crushing checks.
18. Figure 8.18 Typical model for block shear.
19. Figure 8.19 Present model for block shear: (1) totally generic bolt disposition, (2) a subset of bolts may detach, (3) failure paths may be mixed shear and tensile force.
20. Figure 8.20 Longer tensile vs shorter mixed failure paths. Unit thickness assumed.
21. Figure 8.21 A subset Σ_s of the bolt layout (in green), and its (eccentric) resultant F_s. The violet bolts do not belong to the subset Σ_s and when the subset Σ_s is considered, they are discarded.
22. Figure 8.22 Finding the convex hull for a subset of bolts.
23. Figure 8.23 Finding the two farthest bolts A and B. The direction of the resultant F_s of the shears of the bolts belonging to the subset Σ_s is considered. The needed bolts are marked in red.
24. Figure 8.24 Possible failure paths from bolts A and B.
25. Figure 8.25 Nine possible failure paths from a convex hull. It is the geometry that decides which is the worst case, depending on the distances from the plate edges.
26. Figure 8.26 A slight change in the direction of the applied force implies much longer failure paths, if net area is to be used for shear failure path.
27. Figure 8.27 Block shear of a bolt layout under pure torsion. Considering the bolts’ subsets, local failure modes are included.
28. Figure 8.28 Net cross‐sections for a bolted double angle.
29. Figure 8.29 Cross‐sections referring to weakened part.
30. Figure 8.30 Extraction of net cross‐sections from a gusset plate.
31. Figure 8.31 A very limited set of stress peaks due to welding.
32. Figure 8.32 An extended but still limited region where the yield has been crossed.
33. Figure 8.33 A wide region where the yield has been crossed. This constituent has probably undergone excessive loads.
34. Figure 8.34 A clipped cold formed cross‐section and its effective area under pure compression according to Eurocode 3. Note the shift in the centroid of the gross section. Ineffective parts are not filled with color.
35. Figure 8.35 An example of a badly conceived renode: the bolts have all been declared shear only, but if the shear is delivered at the axis of the column, the bolts cannot be shear only. If the analysis is run, the beam will macroscopically rotate due to the lack of bending stiffness of the bolt layout.
Chapter 09
1. Figure 9.1 A pure fem model of a renode. Modification of an existing connection by adding reinforcements.
2. Figure 9.2 A Von Mises stress map of a constituent.
3. Figure 9.3 Buckling in a PFEM model (left) and in two single constituent plus stiffeners models (right).
4. Figure 9.4 Partially yielded model.
5. Figure 9.5 Integration points for surface and thickness.
6. Figure 9.6 Uniaxial constitutive laws.
7. Figure 9.7 Two bolted plates and their plate–shell modeling. The limiting distance between the mid‐surfaces is the sum of half the thicknesses of the two plates. It must be noted that the node positions of the two plates in general do not match.
8. Figure 9.8 Target and contact surfaces.
9. Figure 9.9 An example of the force contact force exchanged as a function of the displacement d (this choice is purely for display purposes).
10. Figure 9.10 An example of PFEM solution considering also contact non‐linearity.
11. Figure 9.11 A renode with compound element and its loaded IRFEM model.
12. Figure 9.12 A shear key applied to a base plate bottom surface.
13. Figure 9.13 A renode and its IRFEM model
14. Figure 9.14 A complex, generic renode and its IRFEM
15. Figure 9.15 A complex generic rendode and its IRFEM model On the left the node built.
16. Figure 9.16 A generic renode and its IRFEM model
17. Figure 9.17 A simple beam‐to‐beam connection and its IRFEM model.
18. Figure 9.18 A base node where all the bolts have been flagged “shear only” (left), under a bending moment equal to 0.1 times the elastic limit for the cross‐section (strong axis bending). Displacement scale 1:1. Clearly the connector is not able to take the loads applied. On the right, the same where all the bolts are “fully resistant”.
19. Figure 9.19 Example of the application of a shear force at the extremity of the physical member. The connection is unloaded, but the column should be considered in axial force plus bending, not under the effect of a simple axial force only.
20. Figure 9.20 False colors representing the utilization factor for single welds of several weld layouts (see also Figure 9.14).
21. Figure 9.21 A pulled T‐stub modeled beam‐like in elastic range. Bolts are hinged to the plate (on the right column of images) or clamped (on the left). Axial forces in bolts are higher if prying is active (see round detail). In plastic range, assuming the plate thin, plastic hinges will form leading to a mechanism (see black stars).
22. Figure 9.22 The three typical failure modes of a pulled T‐stub according to Eurocode 3. Plastic hinges as marked by a red star. Red crosses assigned to bolts mean failure.
23. Figure 9.23 Plasticity distribution at limit for half T‐stub pulled by increasing forces (right long side) and modeled by plate–shell elements. Black dots are in the bolt positions. Bolts modeled by truss elements. The half T‐stub rests over a bed of no‐tension springs. The red parts are plastic.
24. Figure 9.24 Simple Thornton’s model for prying forces.
25. Figure 9.25 The force packets loading a force transferrer at connector extremities are self balanced.
26. Figure 9.26 The forces exerted by the bolts over a constituent.
27. Figure 9.27 A net cross‐section check of a haunch.
28. Figure 9.28 How the third law is applied in hybrid approach.
29. Figure 9.29 Displacement of a base plate SCOFEM under pure strong axis bending of the column (not shown). No bearing surface is used. The deformation is clearly incompatible with that of the base column cross‐section. A compatible displacement would require near‐zero displacement in the whole footprint of the column end.
30. Figure 9.30 A column with its stiffeners as a unique SCOFEM. The “bridge” elements simulating the welds are visible. Plate–shell elements are at middle plane of the plates.
31. Figure 9.31 Segment violation in a mesh is not allowed.
32. Figure 9.32 A sketch model of a member. See also Figure 9.36.
33. Figure 9.33 The polyline referring to a complex bearing surface in the sketch model of a plate. To avoid segment intersection the bearing surface polyline is not closed. The nodes of the bolts and of the welds (hard points) are also displayed. No bolt hole modeling is used in this example.
34. Figure 9.34 Example of stress peaks near to nodes simulating the bolts.
35. Figure 9.35 Dummy constraints application for a plate. Loads due to welds are visible. The maximum reaction modulus is about 0.012 N, each weld layout of the two connected exchange a force P = 612 kN.
36. Figure 9.36 A member constrained at the far extremity F, under the effect of the elementary forces delivered by the (weld) connectors. A double cut has been applied to insert the gusset plate. The elementary nodal forces transferred by the connectors are visible. The mesh is midsize. Finer elements are near to the welds.
37. Figure 9.37 The possible situations of a triangle in a field of pressures.
38. Figure 9.38 Finite element model of a single constituent loaded by the bolt forces, weld forces, and nodal forces related to the pressures exchanged at the bearing surface.
39. Figure 9.39 An example of a perhaps too short member stump.
40. Figure 9.40 Saturated and unsaturated connectors.
41. Figure 9.41 A MCOFEM detail, showing the two sets of equal and opposite forces exchanged at an internal saturated bolt layout using a bearing surface.
Chapter 10
1. Figure 10.1 Beam fillet welded to a column under a tensile force. Both cross‐sections are HEB 200.
2. Figure 10.2 Pulled beam. The beam and the column are rigid. The displacements of the weld segments are all equal. This is a simulation of what traditional approaches do.
3. Figure 10.3 Pulled beam. The beam and the column are modeled with their flexibility resulting from the thickness and the material. The displacements of the weld segments are totally different. A stress concentration at mid beam flange is observed. Top left: deformed view. Top right: false color map of the displacement normal to the column flange. Bottom left: distribution of axial forces in the weld segments. Bottom right: false color map of the normal stresses in weld segments.
4. Figure 10.4 Pulled beam. The beam and the column are modeled with their flexibility resulting from the thickness and the material. Stiffeners added to the column. The displacements of the weld segments are different. A stress concentration at mid beam flange is observed. Top left: deformed view. Top right: false color map of the displacement normal to the column flange. Bottom left: distribution of axial forces in the weld segments. Bottom right: false color map of the normal stresses in weld segments.
5. Figure 10.5 Modeling of weld segments. Left: penetration. Right: fillet.
6. Figure 10.6 Tributary areas for weld segments.
7. Figure 10.7 An example of penetration weld layout modeled in PFEM.
8. Figure 10.8 Flexible and rigid MCOFEMs using penetration welds in pure tension (different scale used).
9. Figure 10.9 The bolt‐plus‐bearing element and the 8 + 8 fictitious beam elements, with end releases applied.
10. Figure 10.10 The bolt hole modeled. Bolt element is normal to the page. The eight bearing elements with their releases are visible, and the local reference system for one of them (axis 3 is normal to page, pointing toward the viewer).
11. Figure 10.11 How to rebuild the overall force R applied to the bearing from the axial force of one radii under generic loading conditions.
12. Figure 10.12 A bolted splice joint.
13. Figure 10.13 Deformation of the models G–N with no bending moments in the bolt shafts. The end‐cones of rigid elements used to apply the loads are visible.
14. Figure 10.14 The bending of the cover plate for tensile forces when the bolt moments are considered (models A–F). No contact pressure assumed.
15. Figure 10.15 If the elements are compressed, and there is an initial gap, bending moment in the cover plates may arise. The contact at the ends acts as an end constraint.
16. Figure 10.16 Non‐dimensional Von Mises stress map for model N of Table 10.13. The disturbed (near and far) and Bernoulli regions of the two members are also marked.
17. Figure 10.17 Stress concentrations in the loaded region at the far extremities of members.
18. Figure 10.18 A Von Mises stress map of a complex PFEM.
19. Figure 10.19 Load–displacement curve for the splice joint of the example. The load is applied using a load parameter λ, ranging from 0 to 1. On the left, is applied a total load equal to 0.5 times the yield axial force N_pl of the member. On the right the load is increased to 0.75 N_pl, and the analysis re‐executed. The non‐linearity increases, but it remains low.