What is a hinge in plastic analysis

Theoretical background


1 Theoretical background May 2017

2 Contents 1 Introduction 2 CBFEM components 2.1 Material model 2.2 Plate model and mesh convergence Plate model Mesh convergence 2.3 Contacts 2.4 Welded connections Direct connections of plates Plastic welded connections 2.5 Bolts 2.6 Pretensioned bolts 2.7 Anchor bolts 2.8 Concrete block construction model Resistance Deformation stiffness 3 Analysis 3.1 Model of the analysis 3.2 Support part and auxiliary elements 3.3 Equilibrium im Node 3.4 Loads Import loads from FEA programs 3.5 Analysis of the strength 3.6 Analysis of the stiffness 3.7 Construction of the part capacity 3.8 Construction resistance of the connection 3.9 Analysis of the stability 3.10 Deformation capacity 4 Component check according to Eurocode 4.1 Plates 4.2 Welded joints Fillet welds Butt welds 4.3 Bolts 4.4 Pretensioned bolts 4.5 Anchors 4.6 Concrete block 4.7 Thrust in the concrete block 5 Testing of the components according to AISC 5.1 plates 5.2 Welded connections Fillet welds CJP butt welds 5.3 Bolts tensile strength and Shear strength of bolts Combined tension and shear in beam connections Beam strength in bolt holes 5.4 Prestressed bolts 5.5 Anchor strength concrete breakout Strength of the concrete breakout of the anchor in shear

3 1 Introduction When designing steel structures, engineers prefer beam parts. However, there are many places in the structure where the theory of parts does not hold, e.g. Welded connections, bolt connections, foundations, bores in walls, taper height at intersections and point loads. Structural analysis in such places is difficult and requires special attention. The behavior is not linear and non-linearities must be respected, e.g. the bending behavior of the plate material, the contact between the end or base plates and the concrete block, one-sided effects of bolts and anchors, welded joints. Construction regulations, e.g. EN and also technical literature offer engineering solution methods. Their general function is the derivation for typical structural forms and simple loads. The component method is used very often. Component method The component method treats the welded joint as a system of interconnected parts. The corresponding model is created for each individual connection type in order to be able to determine the forces and stresses in each component, see the following illustration. 1 pillar net in push, 2 pillar net under pressure, 3 girder flange and net under pressure, 4 pillar flange bent, 5 bolt in pull, 6 end plate bent and 7 pillar net in pull. Components in connection with end plates connected by bolts, described by springs. Each component is tested separately using the appropriate formulas. Since a correct model must be created for each type of connection, the application of the method has limits when solving connections with general shapes and general loads. IDEA StatiCa, together with a project team from the Department of Steel and Timber Structures of Faculty of Civil engineering in Prague and the Institute of Metal and Timber Structures of Faculty of Civil Engineering at the University of Brno University of Technology, developed a new method for advanced construction of structural engineering Steel connections. The name of the method is CBFEM Component Based Finite Element Model and it is: - Generic enough to be used for most types of connections, foundations and details in engineering practice. - Simple and fast enough in daily practice to be able to deliver results in good time that are comparable with current methods and tools. - Understandable enough to provide the designer of structures with clear information regarding the behavior of the connection, its tension, expansion and reserves of the individual components and with regard to the overall safety and reliability. The CBFEM method is based on the idea that the majority of the verified and very useful parts of the CM (component method) should be retained. The weak point of the CM, its general validity in the analysis of stresses in the individual components, has been replaced by modeling and analysis using the finite element method (FEM).

4 2 CBFEM components FEM is a general method that is often used for structural analysis. The application of FEM for modeling connections of any shape seems ideal (Virdi, 1999). An elastic-plastic analysis is required. Steel usually gives in structure. In fact, the results of linear analysis are useless for the construction of connections. FEM models are used for research purposes on the behavior of the joints, which usually apply spatial elements and measured values ​​of material properties. FEM model of the connection for research purposes. It uses 3D spatial elements for plates and bolts. Meshes and flanges of the connected parts are modeled using thin plates in the CBFEM model, for which the known and verified solution is available. The connecting elements, bolts and welded joints, are the most difficult in terms of the analysis model. Modeling such elements in general FEM programs is difficult because the programs do not offer the required properties. Thus, special FEM components had to be developed in order to be able to model the behavior of weld seams and bolts in the connections. CBFEM model of bolt connections on the end plates The connections of parts are modeled as massless points when analyzing the steel frame or the support structure. Equilibrium equations are put together in compounds, and internal forces at the ends of the beams are determined after the entire structure is solved. Indeed, the connection with these forces is strained. The result of the forces of all parts in the connection is zero - the whole connection is in an equilibrium. The actual form of the connection is unknown in the structural model. The designer only defines whether the connection is assumed to be rigid or rotatable.

5 A trustworthy model of the connection that respects the actual state must be created in order to properly construct the connection. Part ends with lengths of 2 3 times the maximum height of the intersection section are used in the CBFEM method. These segments are modeled using shell elements. Theoretical (massless) connection and actual shape of the connection without modified ends of the parts. For a higher precision of the CBFEM model, end forces are applied to 1D parts as loads at the segment ends. Six times the forces of the theoretical connection are transferred to the end of the segment, the values ​​of the forces are retained, but the moments are modified by the effects of the forces on the respective arms. The segment ends at the connection are not connected. The connection must be modeled. So-called manufacturing operations are used in the CBFEM method to model the connection. Manufacturing operations are in particular: cutting, offsets, drilling, stiffening, ribs, end plates and gluing points, angles, wedge plates and others. Fasteners are added to welds and bolts. 2.1 Material model The most common material diagrams used in finite element modeling of structural steel are ideal plastic or elastic models with cold forming and the actual stress-strain diagram. The actual stress-strain diagram is calculated based on the material properties based on tensile tests of structural steel at room temperature. The actual stress and strain can be determined as follows: where the actual stress is the actual strain, the designed stress and the designed strain. The elastic-plastic material with cold forming is modeled according to EN: 2005. The behavior of the material is based on the von Mises strength criterion. It is assumed to be elastic before it reaches the strength value. The ultimate criterion of the limit state for segments that are not prone to jamming is the achievement of a limit value of the essential membrane elongation. A value of 5% is recommended (e.g. EN Appendix C Par. C8 Note 1)

6 Actual stress-strain diagram Constructed stress-strain diagram Ideal plastic material model Plastic limit strain Material diagrams of steel in numerical models The limit value of plastic strain is often discussed. In fact, the ultimate stress has low sensitivity to the plastic strain limit when an ideal plastic model is used. This is shown in the following example of a beam-to-column connection. An open section support IPE 180 is connected to an open section column HEB300 and tensioned with a bending moment. The influence of the limit value of the plastic strain on the resistance of the beam is shown in the following figure. The limit plastic strain varies between 2% and 8%, but the change in moment resistance is less than 4%. Loads Stress Strain Example of the prognosis of the ultimate limit state of a beam-column connection

7 Resistance [knm] / plastic strain [%] Influence of the limit value of the plastic strain on the momentary resistance 2.2 Plate model and mesh convergence Plate model Shell elements are recommended for modeling plates in the FEA construction of structural connections. Square shell elements with 4 nodes at their corners are used. Six degrees of play are allowed for at each node: 3 movements and 3 rotations. Deformations of the elements are divided into membrane and flexural components. The formulation of the behavior of the membrane is based on the work of Ibrahimbegovic (1990). Rotations perpendicular to the face of the element are contemplated. A complete 3D formulation of the element is carried out. The shear deformations existing outside the surface are taken into account in the formulation of the bending behavior of the element, based on the Mindlin hypothesis. MITC4 elements are used, see Dvorkin (1984). The formwork is divided into five integration points along the height of the slab and the plastic behavior is analyzed at each point. This is called the Gauss-Lobatto integration. The non-linear elastic-plastic state of the material is analyzed in each layer on the basis of the known strains. Mesh convergence There are some criteria for mesh generation in the connection model. The control of the connection should be independent of the size of the elements. The mesh generation on a separate disk is problem-free. Particular attention should be paid to complex geometry such as stiffened panels, T-joints, and baseplates. The analysis of the sensitivity with regard to the network discretization should be carried out with complicated geometry. All slabs of intersection sections on beams have a similar division into elements. The size of generated finite elements is limited. The minimum size of an element was set to 10mm, the maximum size of an element is 50mm. Networks on flanges and networks are independent of each other. The standard number of finite elements has been set to 8 elements per intersection section height, as can be seen in the figure below. Mesh on the beam with restrictions between the mesh and the flange plate

8 The network of end plates is separate and independent of the other connecting parts. The default finite element size is set to 16 elements per intersection section height, as shown in the figure. Mesh on the end plate, with 7 elements across the width The following example of a beam-column connection shows the influence of the mesh size on the momentary resistance. An open sectional beam IPE220 is connected to an open HEA200 column and loaded with a bending moment, as can be seen in the following illustration. The critical component is the column panel in the push. The number of finite elements along the crossing section height changes from 4 to 40 and the results are compared. The dashed lines represent 5%, 10% and 15% differences. It is recommended to divide the crossing section height into 8 elements. Beam height Beam-column connection in the model and plastic strain in the ultimate limit state Influence of the number of elements on the moment resistance Number of elements at the edge

9 The study of the mesh sensitivity of a narrow stiffener under pressure from a mesh column panel is presented. The number of elements along the width of the stiffener varies between 4 and 20. The first jamming mode and the influence of the number of elements on the jamming resistance and critical loads are shown in the figure below. A difference of 5% and 10% is shown. It is recommended to use 8 elements along the stiffener width. Width of the reinforcement Number of elements [-] First jamming mode and influence of the number of elements along the reinforcement on the moment resistance The study of the network sensitivity of a T-connection in the train is presented. The geometry of the T-joint is described in section 5.1. Half of the flange width is divided into 8 to 40 elements, and the minimum size of the element is set to 1mm. The influence of the number of elements on the resistance of the T-connection is shown in the figure below. The dashed lines represent 5%, 10% and 15% differences. It is recommended to use 16 elements on half the flange width. Half of the flange width Number of elements [-] Influence of the number of elements on the resistance of the T connection 2.3 Contacts The standard optimization method is recommended for modeling a contact between plates. When the penetration of a nodal point into the opposite contact surface is determined, an optimization stiffness is added between the nodal point and the counterplate. The optimization stiffness is controlled by the heuristic algorithm during non-linear iteration in order to obtain better convergence. The solver automatically finds the penetration point and solves the distribution of the contact force between the penetrated node and the nodes on the counter plate. This allows a contact to be established between the various networks, as shown. The advantage of the optimization method is the automatic assembly of the model. The contact between the plates has an immense influence on the redistribution of the forces in the connection.

10 Example of a division of the plates in the contact between the mesh and the flanges of two overlapping Z-section purlins 2.4 Welded joints There are different options for treating welded joints in numerical models. Large deformations make mechanical analyzes more complex, and it is possible to use different network descriptions, different kinetic and kinematic variables, and constitutive models. The different types of geometric 2D and 3D models and thus finite elements with their possible applications for different levels of accuracy are generally used. The most frequently used material model is the usual rate-independent plasticity model based on the Mises strength criterion. Two approaches used for welded joints are described. Direct joining of panels The first possibility of a welding model between panels is to directly fuse the meshes. The load is transferred to the counter plate by force deformation with restrictions based on the Lagrange formula. The connection is called MPC (multipoint restriction) and relates the nodal points of finite elements of one plate edge to the other. The finite element nodes are not directly connected to each other. The advantage of this approach is the ability to connect networks with different densities. The constraint allows a centerline surface of the bonded plate to be modeled with the offset that takes into account the actual weld configuration and the thickness of the throat piece. The distribution of the load in the welded joint is derived from the MPC, thus the stresses in the section of the neck piece are calculated. This is important for distributing the stress in the plate below the weld joint and for modeling the T-joints. Equal load Multipoint restriction Restriction between the nodes of the networks

11 This model does not respect the stiffness of the welded joint and the stress distribution is conservative. Stress peaks that occur at the ends of the panel edges, in corners and curves, dominate the resistance along the entire length of the welded joint. To eliminate the effect, you can choose between three methods for evaluating the welded joint 1. Maximum stress (conservative) 2. Average stress on the welded joint Evaluation of the stress on welded joints direct connection The program calculates precise values ​​in the welding line. The users can decide how to evaluate the value for control. Method # 1 can be overly conservative in many cases. Method # 2 simulates the situation where the entire weld joint can be plastic. In the majority of cases it is very realistic, but this method is unsuitable for long welded joints, for example. 1.Maximum stress 2. Average stress Plastic welded joints To show the behavior of a welded joint, an improved model of the welded joint is used. A special elastic-plastic element is added between the plates. The element respects the neck piece size of the weld joint, position and orientation. An equivalent welding material with the corresponding dimensions of the weld joint is inserted. The non-linear material analysis is applied and the elastic-plastic behavior in the equivalent weld material is determined. An ideal plastic model is used and the state of plasticity is controlled by stresses in the throat section in the weld joint. The plastic elongation in the welded joint is limited to 5% as in the plate (e.g. EN Annex C Par. C8 Note 1). The stress peaks are redistributed along the longer part of the length of the weld joint.

12 Restriction between the welded connection element and the nodes of the network Evaluation of the tension of the welded connection for plastic welded connections The completely plastic model of welded connections gives real tension values ​​and there is no need to average or interpolate them. The calculated values ​​are used directly for the controls. 2.5 Bolts In the CBFEM (component-based finite element method), the bolt with its behavior in tension, shear and as a carrier is a component that is described by the dependent non-linear springs. The bolt in tension is described by a spring with its initial axial stiffness, construction resistance, bending initialization and deformation capacity. The initial axial stiffness is derived analytically in guideline VDI2230. The model corresponds to the experimental data, see (Gödrich et al 2014). For bending initialization and deformation capacity, it is assumed that plastic deformation only occurs in the threaded part of the bolt shank. The force at the start of bending is where the bending strength is the bolt and the tensile range of the bolt. For materials with a low ratio of ultimate strength to flexural strength, the relation gives higher values ​​than the construction resistance. In order to ensure a positive value of the plastic stiffness, the following should be taken: The deformation capacity of the bolt consists of the elastic deformation of the bolt shank and the plastic deformation of the part with the thread, whereby the initial deformation stiffness of the bolt in tension is according to guideline VDI2230, and

13 where is the limiting plastic elongation given by the value 5% and the length of the threaded part. The tensile force is transmitted to the plates through interpolation connections between the bolt shank and nodes in the plate. The transfer area corresponds to the mean value of the bolt shank and the circle described in the hexagon of the bolt head. The initial stiffness and the design resistance of the bolts in shear are modeled in the CBFEM according to class 3.6 and EN: 2006. The linear behavior up to failure is taken into account. The spring representing the support has a bi-linear deformation force behavior with initial stiffness and construction resistance according to class 3.6 and EN: 2006. According to (Wald et al 2002), the deformation capacity is considered as: The initialization of the bending is expected, see the following figure, at: Force deformation diagram for the carrier of the plate The interaction of axial and shear force in the bolt is according to Table 3.4 of EN: 2006 considered. Only the pressure force is transmitted from the bolt shank to the plate in the bolt hole. It is modeled by interpolation connections between the nodes of the shank and the nodes of the hole edges. The deformation stiffness of the shell element that models the plates distributes the forces between the bolts and simulates the adequate support of the plate. The interaction of axial and thrust force can be incorporated directly into the analysis model. The distribution of forces reflects reality better (see attached diagram). The bolts with a high tensile force take less thrust and vice versa.

14 2.6 Pre-tensioned bolts Pre-tensioned bolts are used when deformation needs to be minimized. The pulling model of the bolt is identical to that for standard bolts. The thrust is not passed on by the carrier, but by friction between the gripping plates. The structural sliding resistance of a preloaded bolt of class 8.8 or 10.9 is subjected to an applied tensile force. Pretensioning force of the bolt with tensile stress range A S, EN (3.7) is to be applied. Construction sliding resistance per bolt EN (3.8) where k S is a coefficient according to Table 3.6, µ is the sliding factor, n is the number of friction surfaces and is a safety factor. IDEA StatiCa Connection checks the functional limit state of preloaded bolts. In the event of a sliding effect, the bolts fail the test. Then the ultimate limit state can be tested as a standard beam test of the bolts. The user can decide which limit state should be checked. Either this is the resistance to the main sliding force or the ultimate state of the bolts in thrust. Both tests on one bolt are not combined in one solution. It is expected that the bolt will have a standard behavior according to the main sliding force and can be tested by the standard beam procedure. The moment load on the connection has a small influence on the thrust capacity. However, we have solved the simple friction test on each bolt using equations (3.8). This test is implemented in the FEM component of the bolt. In general, there is no information as to whether the external tensile load on each individual bolt is based on the moment or the tensile load on the connection. Stress distribution as standard and sliding bolt connection resistant to sliding force

15 2.7 Anchor bolts The anchor bolt is modeled using a similar procedure as the structural bolts. The bolt is fixed to the concrete block on one side. According to EN: 2006, its length L b is taken as the sum of the washer thickness t w, the thickness of the base plate t bp, the mortar thickness t g and the length freely embedded in the concrete, which is assumed with 8d, where d is the bolt diameter. The rigidity in the train is also calculated. The load-deformation diagram of the anchor bolt is shown in the figure below. The values ​​according to ISO 898: 2009 are summarized in the table and the formulas below. Force in the anchor bolt, kn Load-deformation diagram of the anchor bolt

16 Parameters of the anchor bolts, based on ISO 898: 2009 The rigidity of the anchor bolt in shear is assumed to be the rigidity of the structural bolt in shear. The resistance of the anchor bolt is evaluated according to ETAG 001 Annex C or pren. The failure mode of the steel is determined according to the class of EN. 2.8 Concrete block construction model In the CBFEM it is beneficial to simplify the concrete block as 2D contact elements. The connection between the concrete and the base plate only resists pressure. The pressure is transmitted via the Winkler-Pasternak underground model, which shows the deformations of the concrete block. The tensile force between the base plate and the concrete block is borne by the anchor bolts. The shear force between the base plate and the concrete block is transmitted through friction, shear teeth, and by bending the anchor bolts and friction. The resistance of the bolts in thrust is evaluated analytically. Friction and shear teeth are modeled as a total single point restriction on the area of ​​the base area Concrete contact Resistance The resistance of the concrete in 3D printing is determined based on EN: 2006 by calculating the structural beam strength of the concrete in the connection f jd under the effective area A eff of the base plate. The structural support strength of the connection f jd is evaluated according to class of EN: 2006 and class 6.7 of EN: 2005. The quality and thickness of the mortar is brought in with the connection coefficient. For the mortar quality, which is equal to or even better than the quality of the concrete block, 1.0 is expected. It is assumed that the effective area A eff under the base plate is the shape of the cross section of the column, increased by the additional beam width c. where t is the thickness of the base plate, f y is the flexural strength of the base plate, is the partial safety factor for concrete, and the partial safety factor for steel. The calculation of the effective area is repeated until the difference between the additional carrier widths of the current and previous iteration is less than 1mm. The area of ​​the concrete that is in compression is taken from the FEA results. This pressure-loaded area A com allows the position of the neutral axis to be determined. The user can change this area by going to the code setup

17 edited the effective area influence of mesh size. The default value is 0.1, for which the test studies were made. It is not recommended to lower this value. Increasing this value makes the evaluation of the concrete's support resistance more reliable. The value in the code setup determines the limits of the range A com, e.g. the value 0.1 only takes into account areas in which the concrete is higher than 0.1 times the maximum stress of the concrete. The intersection of the area in the pressure A com and the effective area A eff can be used to evaluate the resistance for the generally loaded column foundation for each column shape with each stiffener. The average tension in the effective area A eff is determined as the compressive force divided by the effective area. The component is tested in tension. This process of evaluating the resistance of the pressure-loaded concrete is independent of the network of the base plate, as can be seen in the figure below. Two cases were investigated: loading from pure pressure 1200 kn, and loading from a combination of compressive force 1200 kn and a bending moment 90 kn. Number of elements Number of elements Influence of the number of elements on the prediction of the resistance of the concrete under pressure in the case of pure pressure. Number of elements

18 Number of elements Influence of the number of elements on the prediction of the resistance of compressive concrete in the case of compression and bending Deformation stiffness The stiffness of the concrete block can be predicted as an elastic hemisphere for the construction of the column foundation. The Winkler-Pasternak subsoil model is usually used to simplify the calculation of the foundations. The stiffness of the subsoil is determined by applying the modulus of elasticity of concrete and the effective height of the subsoil, such as: where k is the stiffness in compression, E c is the modulus of elasticity, n is the Poisson coefficient of the concrete foundation, A eff is the effective one Area, A ref is the reference area, d is the base plate width, h is the height of the column foundation, and are coefficients. The following values ​​were used for the coefficients: 3 Analysis 3.1 Model of the analysis The newly developed method (CBFEM Component Based Finite Element Model) enables a quick analysis of the connections of different shapes and configurations. The FEM analysis model is generated automatically. The designer does not create a FEM analysis model, he creates the connection using manufacturing operations, see illustration.

19 Manufacturing operations / objects that can be used to construct the connection Each manufacturing operation brings new details to the connection. Cuts, plates, bolts, welds 3.2 Beam part and auxiliary elements A part of the connection is always defined as a beam. All other parts are connected. The carrier part can be selected by the designer. The carrier part can be continuous or closed in the connection. Completed parts are always specified in the connection. There are several types of connected parts, depending on the load that the parts can accept: - Type N-Vy-Vz-Mx-My-Mz The part can transmit all 6 components of the internal forces. - Type N-Vy-Mz The part can only transfer loads in the XY area internal forces N, Vy, Mz. - Type N-Vz-My The part can only transfer loads in the XZ area internal forces N, Vz, My. - Type X The part can only transfer loads in the X direction normal force N.

20 Plate-plate connection transfers all components of internal forces Ribbed plate connection can only transfer loads in the XZ area Internal forces N, Vz, My Gusset connection Connection of a girder part can only transfer axial forces N Each connection is in the equilibrium status during the analysis of the frame structure. If the final forces of the individual parts are applied in the detailed CBFEM model, the equilibrium status is also fulfilled. It is therefore not necessary to define auxiliary elements in the analysis model. However, for practical reasons

21 the auxiliary element, which withstands all transfers, is fixed at the first end of the carrier part. It does not affect the stress status or internal forces of the connection, but only the representation of the deformations. At the ends of the connected parts, suitable auxiliary elements are specified that take into account all types of the individual parts in order to prevent the occurrence of unstable mechanisms. 3.3 Equilibrium in the node Each node of the 3D FEM model must be in the equilibrium. The equilibrium requirement is correct, but it is not necessary for the construction of simple connections. Part of the connection is always the bearer, and the others are tied up. If only the connection of the connected parts is checked, it is not necessary to maintain an equilibrium. There are therefore two modes of load input: - Simplified for this mode, the carrier part is supported (continuous part on both sides) and the load on the part is not defined - Extended (exactly with equilibrium test). The beam part is supported at one end, the loads are applied to all parts, and the equilibrium must be eroded. The mode can be switched in the Advanced Mode ribbon. The difference between the modes is shown in the following example of a T-connection. The beam has a final bending moment of 41 knm. There is also a normal compression force of 100 kn in the column. In the simplified mode, normal force is not taken into account as the column is supported at both ends. The program only shows the effect of the bending moment of the beam. The effect of normal force is only analyzed in full mode and shown in the results. Simplified entry, the normal force in the column is NOT taken into account.

22 Advanced input, normal forces in the column are taken into account The simplified method is easier for the user, but can only be used if the user wants to investigate the connection parts but not the behavior of the entire connection. If the carrier part is heavily loaded and close to the capacity limit, the extended mode is necessary, which respects all internal forces in the connection. 3.4 Loads The final forces of the parts of the frame analysis model are transferred to the ends of the part segments. When transferring, eccentricities of the parts caused by the connection structure are taken into account. The analysis model created by the CBFEM method corresponds very precisely to the actual connection, with the analysis of the internal forces being carried out on a very idealized 3D FEM 1D model, where the individual beams are modeled using center lines and the connections are modeled using immaterial nodes.

23 Actual form of the connection Theoretical form in the 3D FEM model Connection of a vertical column and a horizontal beam Internal forces are analyzed using 1D parts in the 3D model. An example of the internal forces is shown in the figure below. Bending moment Shear force Internal forces in the horizontal beam. M and V are the final forces on the connection. The effects caused by the parts of the connections are important for the construction of the connection (connection). The effects are shown in the following figure: 1D model of the part CBFEM model dark blue Effects of the parts on the connection. The CBFEM model is drawn in dark blue

24 The moment M and the force V act in the theoretical connection. The point of the theoretical connection does not exist in the CBFEM model, so the load cannot be applied here. The model has to be loaded by the values ​​of M and V, which have to be transferred to the end of the segment at the distance r. In the CBFEM model, the end area of ​​the segment is loaded by the moment Mc and the force Vc. When constructing the connection, its actual position, which is related to the theoretical point of the connection, must be determined and taken into account. The internal forces at the point of the actual connection are mostly different compared to the internal forces at the theoretical point of the connection. Thanks to the precise CBFEM model, the construction is carried out with reduced forces, see moment Mr in the following figure: Bending moment on the CBFEM model. The arrow points to the actual position of the connection. If the connection is loaded, it must be taken into account that the solution of the actual connection must match the theoretical model used to calculate the internal forces. This is true for rigid connections, but it can be completely different for hinges.1D model of the part CBFEM model Hinge position in the theoretical 3D FEM model and in the actual structure

25 The previous figure shows that the hinge position in the theoretical 1D model of the parts deviates from the actual position in the structure. The theoretical model does not match reality. When the calculated internal forces are applied, a significant bending moment is applied in the displaced joint and the joint being constructed is oversized and cannot be constructed. The solution is simply both models must match. Either the hinge must be defined in the correct position in the 1D model of the part, or the shear force must be shifted so that the zero moment is obtained in the hinge position. The shifted distribution of the bending moment on the beam. The zero moment is at the hinge position. The shift of the thrust force can be defined in the table for the definition of the internal forces. The position of the loading effect has a great influence on the correct construction of the connection. To avoid any misunderstandings, the user can choose from the following options: In node / In bolts / In position (node ​​/ bolt / position) node shear force Vz in the theoretical node

26 Bolt shear force Vz in the center of gravity of the bolt Position shear force Vz at the position defined by the user Import loads from FEA programs IDEA StatiCa can import internal forces from third party FEA programs. FEA programs apply an envelope principle of internal forces of combinations. IDEA StatiCa Connection is a program that releases steel connections non-linearly (elastic / plastic material model). This means that envelope combinations cannot be used. IDEA StatiCa searches for extreme values ​​of the internal forces (N, Vy, Vz, Mx, My, Mz) in all combinations at the ends of all parts that are connected in the connection. For each of these extreme values, all other internal forces of this combination are used on all remaining parts. IDEA StatiCa determines the worst combination for each component (plate, weld, stud, etc.) in the joint. The user can modify this list of load cases. He can also work with combinations in the wizard (or BIM), or he can delete some cases directly in the IDEA StatiCa Connection.

27 3.5 Analysis of the strength The analysis of the connection is materially non-linear. The increase in the load is carried out gradually, and the state of tension is sought. There are two optional analysis modes in IDEA Connection: - Reaction of the structure (connection) to the total load. The total defined load (100%) is applied in this mode and the corresponding state of stress and deformation is calculated. - Analysis definition when the ultimate limit status is reached. The box in the code setup Stop at limit strain should be ticked. The status is determined at which the plastic strain reaches the defined limit value. If the defined load is higher than the calculated capacity, the analysis is marked as unsatisfactory and the percentage of the load used is printed out. Note that the analytical resistance of the components, e.g. bolts, can be exceeded. The second mode is more suitable for practical design. The first is preferred for detailed analysis of complex compounds. 3.6 Analysis of the stiffness The CBFEM method enables the analysis of the stiffness of the connection of individual connecting parts. For a proper stiffness analysis, a separate analysis model must be created for each part. Then the stiffness analysis is not influenced by the stiffness of the other parts of the connection, but only by the node itself and the construction of the connection of the analyzed parts. While the beam part is supported for strength analysis (SL part in the figure below), all parts except the one to be analyzed are supported for strength analysis (see two figures below for stiffness analysis for parts B1 and B3).

28 Support elements on parts for the strength analysis Support elements on parts for the stiffness analysis of part B1

29 Supports on parts for stiffness analysis of part B3 Loads can only be applied to analyzed parts. When the bending moment M y is defined, the rotational stiffness around the y-axis is analyzed. When the bending moment M z is defined, the rotational stiffness around the z-axis is analyzed. The axial force N is defined, the axial stiffness of the connection is analyzed. The program generates the complete diagram automatically, it is displayed directly in the GUI and it can be inserted into the output report. The rotational or axial stiffness can be studied for a specific design load. IDEA StatiCa Connection can also examine the interaction of other internal forces. The diagram shows: - level of construction load - limit value of the capacity of the connection for 5% equivalent elongation - limit value of the capacity of the connected parts (also useful for seismic constructions) - 2/3 of the limit capacity for calculating the initial stiffness - value of the initial stiffness S j, ini - value of secant stiffness S js - limits for connection classification rigid and fixed - rotational deformation - rotational capacity stiffness diagram My - y, LE1

30 Rigid welded connection Stiffness diagram My - y, LE1 Semi-rigid bolt connection After reaching a 5% elongation in the column mesh panel in shear, the plastic zones spread rapidly

31 3.7 Part capacity construction IDEA Connection tests the connection against applied construction loads. In many regions at risk of seismic activity, the connection must be tested at the maximum moment that can be transmitted by the connected parts. We calculate this moment in the software and apply it to the specific part. All other parts in the link are supported. The value of the moment is calculated and cannot be edited. The moment is calculated differently for EN and AISC. Capacitance construction EN 1998 R d Resistance of non-dissipative connection R fy Bending strength Capacitance construction AISC M pe The expected moment at the plastic hinge F y Bending strength R y Ratio of the expected bending stress should be the specified minimum bending (Table A3.1) Z x the plastic section module Connections to the Transferring the moment that is equal to the part resistance (connections with full strength) must usually have a much greater stiffening than connections with partial strength. The connected part is not checked. It must be precisely constructed in the global analysis of the structure.

32 3.8 Construction resistance of the connection The designer usually solves the task of constructing a connection / weld seam to transfer the known construction load. However, it is also very useful to know how different the construction is from the limit state, or how large the reserve in the construction is and how safe it is. This can be done quite simply by the type of analysis, construction resistance of the connection. The user enters the construction load as in a standard construction. The software automatically and proportionally increases all loaded components until one of the tests is satisfactory. The user receives the ratio of the maximum load in relation to the construction load. A simple chart is also created. 3.9 Analysis of stability The design guidelines EN and EN provide five categories of finite element analyzes with the following assumptions: 1. Linear material, geometrically linear 2. Non-linear material, geometrically linear 3. Linear material, linear loss of stability Buckling 4. Linear material, geometrical non-linear using imperfections 5. Non-linear material, geometrically non-linear using imperfections A construction process that combines methods 2 and 3 Non-linearity of the material and analysis of the stability is given in Chapter 8 of the EN. The verification of the buckling resistance, based on the FEM results obtained, is described in Annex B of the EN. This process is used for a wide range of structures, except for very thin formwork, for which a geometrically non-linear analysis with initial imperfections is more appropriate (Figures 4 and 5). The process uses stress amplifiers α, which are obtained as the results of the FEM analysis and enable a prediction of the subsequent buckling resistance of the connections. The stress coefficient α ult, k is determined by obtaining the plastic capacity without considering the geometric non-linearity

33 is caused. The testing of the plastic capacity and the general automatic determination of α ult, k is implemented in the software developed. The critical buckling factor α cr is determined, which is obtained by applying the FEM analysis of the linear stability. It is automatically determined in the software using the same FEM model as for the calculation of α ult, k. It should be remembered that the critical point in terms of plastic resistance does not necessarily have to be investigated in the first critical buckling mode. In a complex joint, several buckling modes need to be investigated as they relate to different parts of the joint. The non-dimensional slenderness of the plate is determined: the buckling mode to be investigated The reduction buckling factor is determined according to Annex B of the EN. The reduction factor depends on the slenderness of the plate. The kink curve used shows the influence of the reduction factor on the slenderness of the panel. The assumed buckling factor that is applicable to non-uniform parts is based on the buckling curves of the wearer. The verification is based on the von-mises criterion and the lowered stress method. The buckling resistance is evaluated: Buckling reduction factor p [-] Slenderness of the plate The bending reduction factor p according to EN Annex B Even if the process seems trivial, it is general, robust and easy to automate. The advantage of the process is the advanced FEM analysis of the entire connection, which can be applied to general geometries. It is also included in the valid Eurocode standards. The extended numerical analysis gives a quick overview of the global behavior of the structure and its critical parts and allows quick stiffening in order to prevent instabilities.

34 The slimness limit is shown in Annex B of the EN and defines all cases that have to be evaluated according to the previous process. The resistance to buckling is limited for a slenderness of the plate greater than 0.7. With decreasing thinness, the resistance is regulated by the plastic stretching. The critical limit buckling factor for plate slenderness is 0.7, and the buckling resistance is equal to the plastic resistance and can be calculated as follows: The influence of the plate slenderness on the plastic resistance M ult, k and the buckling resistance M CBFEM is shown in the figure below. The diagram shows the results of the numerical study of a triangular stiffener in a portal frame connection. Panel slenderness Influence of panel slenderness on the resistance of the portal frame connection with thin stiffener It is important to distinguish between global buckling (buckling of all parts) and local buckling (buckling of the individual panels). In the case of global buckling (the plate is an extension of a part, see figure below), it is recommended to check the buckling resistance for the critical buckling factor less than 15. Critical buckling factor for a corner plate as an extension of a structure In most cases of panels in connection local buckling can occur and the maximum value of the critical buckling factor, which a detailed analysis

35 required is usually smaller; it has been confirmed that for stiffeners and column panels in shear it is not necessary to consider buckling if the critical buckling factor is greater than 3. Examples of buckling shapes where buckling can be neglected if the buckling factor is higher than 3 Deformation capacity The deformation capacity / extensibility, together with resistance and stiffness, is one of the three basic parameters that describe the behavior of connections. In resilient moment connections, elasticity is achieved through sufficient rotational capacity. The deformation / rotation capacity is calculated separately for each connection in the weld. The prediction of the deformation capacity of connections is currently being investigated by the component method (CM), but it is not offered as a standardized process. Compared to the very well accepted methods of determining the initial stiffness and resistance of many types of structural connections, there are no generally accepted standard processes for determining rotational capacity. The criteria considered sufficient are chosen to help the engineers in class EN: 2006. In the case of a beam-column connection, in which the constructed moment resistance of the connection M j, rd is regulated by the construction resistance of the mesh column panel in shear, it can be assumed that this has an adequate rotational capacity for plastic global analysis if: where d is the width of the Paneels of the column network, tw is the network thickness and is the strength ratio of the steel bending behavior. In class (2) the plastic distribution between the rows of bolts is limited for connections with an end plate connection with bolts, provided that the constructed moment resistance of the connection depends on the construction resistance of the column flange or the beam end plate in the bent state or the thickness t of the column flange or the Beam end plate is regulated or the tension flange of the clamp meets: where d and f ub are the diameter and strength of the bolt and fy is the bending strength of the relevant plate. The rotational capacity of a welded girder-column connection cannot be assumed to be less than the value given by the following formula, provided that its column network is stiffened in compression but not stiffened in tension, and if its design moment resistance does not depend on the design shear resistance of the Column mesh panel is regulated, see (1):

36 where h b is the depth of the beam and h c is the depth of the column. A rotation capacity of 0.015 radians can be assumed for an unstiffened welded beam-column connection that is in accordance with the regulations of this area. The estimate of the rotational capacity is important for connections exposed to seismic activity, see (Gioncu and Mazzolani, 2002) and (Grecea 2004) and extreme loads, see (Sherbourne AN, Bahaari, 1994 and 1996). The deformation capacity of components has been investigated since the end of the last century (Folay and Vinnakota, Faella et al (2000) carried out tests on T-connections and derived analytical formulas for the deformation capacity. Kuhlmann and Kuhnemund (2000) carried out tests on the column network, the transverse pressures were subjected to different intensities of the axial compressive force in the column. Da Silva et al (2002) predicted a deformation capacity of different intensities of the axial force in the connected beam. Based on the test results in combination with the FE analysis, the basic components are determined using the analytical models Capacities determined by Beg et al (2004) In the work, components are represented by non-linear springs and appropriately combined to determine the rotational capacity of the connection for endplate connections with an elongated or flat endplate and welded connections Components that can significantly contribute to the rotational capacity of the column are recorded: the network in compression, the column network in the pull, the column network in the push, the column flange bent and the end plate bent. The components belonging to the column network are only decisive if there are no stiffeners in the column that can withstand pressure, tension or shear forces. The presence of a stiffener eliminates the corresponding component and its effect on the rotational capacity of the connection can thus be neglected. End plates and pillar flanges are only important for end plate connections in which the components act as a T-connection, which also includes the deformation capacity of the bolts in the train. The questions and limits of the deformation capacity of connections of high-strength steel was examined by Girao at al (2004). 4 Component check according to Eurocode The CBFEM method combines the advantages of the general finite element method and the standard component method. Stresses and internal forces calculated in the precise CBFEM model are used when testing all components. The individual components are tested according to Eurocode EN. 4.1 Plates The resulting equivalent stress (HMH, von Mieses) and the plastic strain are calculated on plates. Since the perfect elastic-plastic material model is used, the test stress does not occupy the plate (the stress never exceeds Fy). An equivalent plastic elongation test is carried out. The limit value of 5% is specified in the Eurocode (EN Para. C Par. C8 Note.1) suggested, this value can be changed by the user in the project settings. The plate element is divided into 7 layers, and the elastic / plastic behavior is examined separately in each layer. The resulting summary lists the most critical test of all 7 layers.

37 The CBFEM method can provide higher stress rather than flexural strength. The reason is the slight inclination of the plastic sector of the stress-strain diagram, which is used in the analysis to improve the stability of the interaction calculation. This is not a problem for the practical construction. The equivalent plastic elongation is exceeded at higher stress, and the connection is not satisfactory anyway. 4.2 Welded connections Fillet welds Construction resistance The tension in the neck area of ​​the fillet welds is determined according to type. Stresses are calculated from the shear forces in the welded joints. The bending moment about the longitudinal welding axis is not taken into account. Welding load where: ß w correlation factor Tab. 4.1