Contents
1 Joint typologies in steel structures ... 2 2 Frames and joints classification... 13 3 Effect of semi-rigidity on the response of frames ... 23
Section 1 – Introduction to the structural modelling of beam-to-column joints and classification according to EC3 part 1.8
1 Joint typologies in steel structures
Steel buildings may be realized with different technologies, depending on both technical and architectural constraints. Steel structures may be divided in braced and unbraced depending on the on the presence or absence of a bracing system which is used to reduce lateral drifts and absorb lateral forces such as those generated by wind and seismic loads. Braced frames can be subdivided in two macro-categories namely, concentrically braced or eccentrically braced. Concentrically braced frames (CBFs) are a class of structures which resist to the horizontal loads through vertical concentric truss systems. Concentric means that the axes of the members all converge concentrically in the joints. CBFs provide high stiffness and strength which, especially under seismic actions, tends to attract higher accelerations and seismic forces.
Fig. 1– Typologies of bracing systems
Due to the high stiffness, CBFs usually result in low drift capacity and limited ductility supply. When engaged in plastic range, the primary source of drift capacity in CBFs is constituted by yielding of diagonal elements which assures energy dissipation capacity through the development of plastic hinges located at the ends of the brace and into an intermediate section. The success of this system under
seismic loading conditions is mainly due to the response of the bracings which represent the critical element which must be designed with accuracy.
Fig. 2– Examples of CBFs
In fact, the buckling of braces in the post-yielding phase requires the development of a significant ductility demand both in the members and in gusset plates which have to be properly detailed. In particular, the gusset plates should be designed to allow the formation of a yield line assuring the possibility of rotation at the brace ends in the gusset plates and the formation of a plastic hinge into an intermediate section of the bracing member. For this reason, typically the cross-section of bracings should belong at least to class-1 or class-2.
Fig. 3– Brace buckling and gusset plate out-of-plane deformation
The main feature of CBFs is economic because they allow to obtain solutions for buildings in high seismicity areas with low cost. CBFs have been quite popular since the 1960s mainly because of their economic advantages over MRFs, particularly in cases where the drift requirements govern the design.
Fig. 4– Differences between an EBF and CBF configuration
The CBF system is currently one of the most widely used seismic load resisting systems in steel structures. It is easy to design and the most efficient especially in controlling lateral drifts of buildings.
Nevertheless, it provides significant architectural constraints which in many cases cannot be overcome leading to the selection of different systems such as Eccentrically Braced Frames (EBFs) or Moment Resisting Frames (MRFs). As an alternative to CBFs, especially when the architectural constraints promote their use, EBFs are sometimes adopted. EBF is a frame system in which one end of the braces is connected to the beam instead of a frame node.
Fig. 5– Some configurations of EBFs
They have been introduced after extensive research studies during the ‘70s-‘90s becoming one of the most common systems for providing resistance against seismic actions. Into an EBF system the element which is providing lateral drift capacity under horizontal loads is the part of the beam between the brace and the frame nodes, which is known as a link. Brace forces act in the link as shear and bending forces, so that the link acts as a seismic fuse. Basically, EBFs combine in some part the features of CBFs, with the features of MRFs which are presented afterwards. Basically, they are not as rigid as CBFs nor as deformable as MRFs providing intermediate levels of stiffness and resistance. Different configurations are possible for EBFs, since the eccentricity between the bracing and the beam may be created in different zones and with different arrangements of the bracings.
Clearly, in the EBF frame configuration the crucial element under seismic loading conditions is the link which experiences three forces: shear, axial and flexural. Usually, axial forces have a negligible effect, but the length of the link plays a crucial role because it governs the ratio between shear and bending forces changing the failure mode of the link and its energy dissipation capacity. In practice, based on the ratio between bending and shear forces short, long or intermediate links are distinguished.
Fig. 6– Actions on a link element
Moment Resisting Frames (MRFs) have become very popular in last century and many buildings have been built applying this technique. Architects and building owners usually appreciate layouts obtained using this structural typology, in fact such a system provides large open spaces without the obstruction usually due to the presence of braces or walls. Steel MRFs have been used worldwide especially for applications in low-rise industrial buildings and multi-storey low, medium and high-rise buildings.
MRFs are structures that withstand seismic actions by the bending of girders, columns and connections.
Their main source of stiffness and lateral strength is given by the flexural resistance of members and connections, and the seismic energy dissipation capacity and ductility is provided by the formation of a high number of dissipative zones which can be located in beams, columns or joints depending on the applied design philosophy. Classically, framed structures are designed to possess strong columns, weak beams and full-strength rigid connections, so that the earthquake input energy is dissipated through the plastic engagement of the end of beams and of the end of columns of the first storey.
For the above-mentioned architectural characteristics, often MRFs are preferred to other structural systems, such as concentrically (CBF) and eccentrically (EBF) braced frames, where pendular frames are combined with stiff and strong braced frames.
Fig. 7– Examples of welded steel MRFs
Notwithstanding the undoubted advantages which are possible to obtain by using Moment Resisting Frames, this structural typology possesses some weak points. First of all, the low lateral stiffness can significantly affect the response of the structure both at the ULS and at the SLS. In fact, the susceptibility to second order effects and the fulfillment of the serviceability limit states in terms of maximum lateral drifts, can become governing parameters of the design process, leading to member size greater than the minimum needed for the satisfaction of the strength requirements. So that, especially for high-rise buildings, MRFs can become uneconomical with respect to other systems.
Other problems can be individuated in the difficulties which are possible to encounter in providing adequate stiffness and resistance to joints when these have to be rigid and designed to be over-resistant with respect to the connected beam. In fact, providing adequate stiffness and strength to joints can, in some cases, become onerous. In addition, when welded joints are adopted it is of fundamental importance to avoid as much as possible the use of field welds, which have been demonstrated, by evaluating the damages on steel structures caused by Northridge and Kobe earthquakes, to be unreliable due to the limited deformation capacity and to the presence of welding defects. Therefore, when welding details are adopted it would be better to rely on shop welds.
Fig. 8– Buildings erected applying the column-tree technique
A constructional procedure, which proposes to combine shop welded and field bolted connections is the so-called “column-tree” technique. In this practice, columns are welded with short girder stubs in the shop, successively erected in the field and finally connected by bolting the middle segment of the beam.
Usually, the bolted part of the connection is designed to be full strength.
Fig. 9 – Classification of Moment Resisting Frames
As a consequence, its contribution to the frame stiffness and resistance is negligible and therefore, it does not play a major role on the overall structural behavior. Furthermore, the “column tree” technique assures high reliability and economy. In fact, in general, shop welds are less expensive and guarantee higher performances than field welds, which require inspections and good weather conditions to be executed.
A classification of MRFs, according to (Seismic Design of Bolted Steel Moment-Resisting Frames, 1995), can be arranged with reference to the following:
• the spatial distribution of the frames within the whole building (e.g. perimeter frames, few rigid bays, etc.);
• the type of connections provided to connect girders and columns (riveted, bolted or welded);
• ductility class of the frame system (Low, Medium, High);
• stiffness of the connections (Pinned, Semi-Rigid, Rigid);
• relative resistance of the structural members i.e. beams and columns, panel zone and elements composing the connection.
Fig. 10 – Spatial Distribution of Space Frames
Moment Resisting Frames Pendular Frames
Space Frames
MRFs can be categorized according to their spatial distribution in: Space Frames, Perimeter Frames, Perimeter Frames with only Few Rigid Bays, Planar Frames and Hybrid Systems. This classification can be easily extended also to braced frames, with the exception of space frames which is highly impractical in braced configurations.
Fig. 11 – Spatial Distribution of Perimeter Frames
In case of Space Frames, MRFs are uniformly spread in the structure and all the columns, girders and connections are required to carry both the vertical and lateral loads. Such a kind of typology traditionally necessitates the adoption of expensive rigid full-strength joints and as a consequence can result in structures which are not cost/effective. It is probably for this reason, that the adoption of Perimeter Moment Resisting Frames has gained increasing attention in the last decade. In fact, in this approach, only the exterior frames are part of the earthquake resistant system, while the interior beams and columns compose a pendular organism which has to carry only the gravity loads. Thus, interior beams and columns can be designed to have pinned connections and only the perimeter frames, which resist the horizontal loads, have to be designed with rigid joints. In this manner, the number of expensive connections is reduced and a more economical design is achieved.
Perimeter Frames
Moment Resisting Frames Pendular Frames
Fig. 12 – Spatial Distribution of Perimeter Frames with only few rigid bays
The extension of the concept of perimeter moment resisting frame is the Perimeter Moment Resisting Frame with only few Rigid Bays. In this structural typology only few bays of the exterior frames are called to withstand the seismic action, while the other parts of the structure have to carry only the vertical loads. Even though reducing the number of moment resisting connections provides indisputable advantages, the variation of the structural scheme from the space frame to the perimeter frame with only few rigid bays reduces not only the cost, but also the structural redundancy.
This aspect has been object of discussion in the past, and in particular in the aftermath of Northridge earthquake. In fact, a large percentage of structure that was affected by structural damages, especially by the fracture of welds in correspondence of girder-to-column joints, was realized with perimeter frames and only few rigid bays. The reasons of such an unsatisfactory behavior have to be searched in the reduction of the number of dissipative zones which consequently increases the ductility demand in the dissipative zones, i.e. the plastic hinges.
Perimeter Frames with few Rigid Bays
Moment Resisting Frames Pendular Frames
Fig. 13 – Spatial Distribution of Planar Frames
In case of Planar frames, moment resisting structures are used to resist to lateral forces only in one of the main directions of the building whilst, out of the plane of MRFs other seismic resistant organisms are used to adsorb the seismic action. These structural typologies are frequently used when, for architectural reasons, it is not possible to dispose braced frames on all the facades of the building.
Hybrid systems are defined by the coupling of MRFs with other structural elements, such as reinforced concrete core or walls, infill panels or steel walls. In the first case the support to vertical loads is mainly provided by MRFs and the resistance to the lateral loads is given in part by the core or the walls and in part by the frames. The response of dual systems depends on the relative stiffness of the seismic resisting elements. If the shear resistance provided by the frame is greater than the 50% of the total shear resistance the system is called “frame-equivalent dual system” conversely, if the greater contribute to the resistance for lateral loads is given by the reinforced concrete elements it is called
“wall/core-equivalent dual system”.
Planar Frames
Other Structural System Moment Resisting Frames
Pendular Frames
Fig. 14 – Spatial Distribution of Hybrid Systems
When MRFs are coupled with infill panels, the infills which usually are designed as secondary elements and thus separated from the steel frames, are rigidly connected to the beams and the columns, stiffening the structure with respect to the lateral loads. It is obvious that in this case the designer has to take particular care of the connection between the infill and the structure, avoiding the slippage of the two parts. Finally, MRFs can be coupled to Steel shear walls. These are usually designed to dissipate the seismic input energy by means of cyclic inelastic deformations. Compared to RC walls, steel walls are definitely much lighter, providing advantages to the load-carrying system and to the foundations, quick to erect, if field bolted/shop welded connections are adopted, with low architectural impact and more cost/effective since are faster to construct.
Hybrid Systems
Other Structural System Moment Resisting Frames
Pendular Frames
2 Frames and joints classification
Structural joints may be sorted according to the configuration or according to their stiffness and resistance. With reference to the geometric characteristics EC3 part 1.8 individuates the following categories of beam-to-column joints in steel structures:
1. single sided Beam-to-Column Joints;
2. double sided Beam-to-Column Joints;
3. beam splices;
4. column splices;
5. base plate joints.
Fig. 15 – Joints in MRFs
Often the words “joint” and “connection” are used likewise to individuate the region of intersection between the beam and the column. In general, with the term “connection” it is intended the zone where the mechanical devices (bolts, plates, welds etc.), used to join the beam to the column, are located.
Conversely, with the term “panel zone” it is individuated the region of the column web contained within the flanges of the connected beam. The addition of the connection and of the panel zone it is usually referred as a “joint”.
The structural response of steel frames strongly depends on the behaviour of connections. In fact, stiffness and strength of joints deeply affect the static and dynamic properties of frames and their post- elastic behaviour. Furthermore, also the internal actions arising in the structure, both due to Serviceability and Ultimate limit state loads, depend on the elastic and post-elastic stiffness of the connecting elements. It is well known in technical literature that the actual flexural behaviour of a joint
can be considered as intermediate between the two extreme conditions of infinitely rigid or infinitely deformable.
Fig. 16 – Distinction between panel zone and connection
When the elements composing the connection, i.e. the plates, the bolts and the panel zone, are sufficiently stiff and no relative rotations between the beam and the column occur, in design practice, the joint is usually modelled as a full restraint. Conversely, when the beam is free to rotate with respect to the connected elements, the joint is usually considered as a pin. In all the other cases, i.e. when the connecting elements are neither adequately rigid to allow to model the connection as a full restraint nor so deformable to consider the beam free to rotate, the beam-to-column joint is defined semi-rigid. In general, all beam-to-column connections are semi-rigid and should be modelled considering their actual relationship between the bending moment and the rotation. Notwithstanding simplifications, for design purposes, are usually allowed.
Moreover, a joint can be classified as full-strength, pinned or partial strength on the base of the relative resistance of the joint and of the connected beam. If the flexural strength of the joint is greater than the bending resistance of the beam, the connection is defined “full-strength”. Conversely, if the joint is completely incapable to withstand bending, connection is usually called “pinned”, meaning that it is only capable to transfer shear actions. In all the other cases, i.e. when the joint flexural resistance is lower than the bending strength of beam, the connection is defined “partial strength”. From the standpoint of the location of dissipative zones, different behaviours are obviously expected if full-strength or partial- strength connections are adopted.
Fig. 17 – Different levels of rigidity of a beam-to-column joint
Basically, three cases can be individuated depending on the relative resistance of beams and joints:
• plastic resistance of the beam lower than the ultimate bending resistance of the connection;
• plastic resistance of the beam greater than the ultimate bending resistance of the connection;
• plastic resistance of the beam and ultimate bending resistance of the connection balanced.
In the first case, the beam-connection overall behaviour can be effectively represented with an elastic- plastic model. A plastic hinge will develop in the beam, while the connection will contribute only by means of its elastic stiffness. In the second case, the opposite situation will develop. The dissipative zone is located in the connection and the overall structural behaviour at ULS is influenced by joints inelastic behaviour. In this case, an accurate representation of the connection moment-rotation curve is needed.
In the third case, joint and beam bending resistance is similar and, as a consequence, both the elements can be engaged in plastic range.
Within this framework, it is easy to understand that the design of connections plays a role of fundamental importance for the design of steel frames. In fact, design of joints can be carried out aiming to obtain different values of stiffness, strength and rotational capacity on the base of the desired overall structural behaviour. As an example, if the design is aimed to obtain a weak beam-strong column – strong joint structural behaviour, joints will be designed to be full strength and rigid. So that, thick plates, reinforcing ribs, doubler and continuity plates will be probably adopted to exclude from the
dissipative mechanisms panel zones and connecting elements. Conversely, if the design is aimed to obtain a weak connection-strong beam-strong column behaviour, joints will be designed to be partial strength and the rotational capacity of the connecting elements will have to be carefully evaluated.
Fig. 18 – Structural modelling of pinned, rigid or semi-rigid joints
It is clear from the above considerations that detailing of joint plays a role of fundamental importance within MRFs overall structural behaviour. In fact, joints ultimate behaviour can be completely modified by strengthening some elements rather than others, governing the failure mode by means of simple modifications of the joint detail.
In conclusion, as it will be shown later, on the base of the classification of joints according to strength and stiffness, the following typologies of MRFs can be individuated:
• frames with full-strength rigid connections;
• frames with full-strength semi-rigid connections;
• frames with partial-strength rigid connections;
• frames with partial-strength semi-rigid connections.
Classification of joints and frames are strictly related. As discussed earlier, the overall structural response of MRFs is strongly influenced by joints strength, stiffness and rotational capacity. In fact, the distribution of the internal actions, the structural ductility, the susceptibility to second order effects and the location of the plastic zones are all parameters which are influenced by the existing relationship between the bending moment and the joint rotation.
Last version of Eurocode 3 classifies MRFs according to the following two characteristics:
• the susceptibility to second order effects;
On the base of the first feature MRFs are divided in sway and non-sway. A frame is defined non-sway if its lateral displacements are small enough to assume that the internal actions due to the lateral deformability of the frame are negligible. Conversely, a frame is called sway if the deformed geometry leads to a substantial increase of the internal actions or modify significantly the structural behavior.
According to EC3 a frame can be considered to be non-sway if the following relationship are satisfied:
•
= > 10
for elastic analysis;•
= > 15
for plastic analysis;where is the factor by which the design loading would have to be increased to cause elastic instability in a global mode, is the elastic critical buckling load for global instability mode based on initial elastic stiffnesses and is the design loading on the structure. In all the other cases frames have to be classified as sway.
Moreover, frames are divided in braced or unbraced. In the first case MRFs are stiffened by specific elements which reduce the lateral displacement of at least the 80%, in the second case frames are defined unbraced. A further classification of MRFs, depending on the joints characteristics, is provided by EC3 part 1.8, where frames are categorized in:
• simple: joints do not transmit bending moment to the column;
• continuous: the behavior of the joint may be assumed rigid;
• semi-continuous: the behavior of the joint has to be taken into account by adopting proper models.
In the first case, joints can transfer to the columns only shear and beams are free to rotate, as a consequence the obtained structural system is pendular and joints can be faithfully modeled by means of hinges. In the second case, joints behavior is rigid, and their resistance is greater than the flexural strength of the connected beam, so that connections can be structurally represented by means of full internal restraints. In the case of semi-continuous frames, joints are intermediate between the extreme situation of pinned and rigid-full strength, so that their structural behavior must be properly accounted for by means of accurate models representing the actual moment-rotation curve.
Aiming to individuate quantitatively the boundaries between rigid, semi-rigid and pinned behavior, EC3 part 1.8 introduces the following two parameters:
• =
• =
where is the bending stiffness of the beam, is the bending stiffness of the column, is the steel elastic modulus, and are the beam and column second moments of area, and are the beam and column lengths. Eurocode boundaries are determined by defining “rigid” a joint whose stiffness do not reduce by more than the 5% the Euler buckling load of the structure with full rigid attachments. On the base of the so-called “5% criterion” the following classifications derive:
BEAM-TO-COLUMN JOINTS
φ
Zone 1: rigid, if Sj,ini ≥ kb EIb / Lb
where
(BRACED FRAMES)
kb = 8 for frames where the bracing system reduces the horizontal displacement by at least 80 % (UNBRACED FRAMES)
kb = 25 for other frames, provided that in every storey, Kb/Kc ≥ 0,1 *)
Zone 2: semi-rigid
All joints in zone 2 should be classified as semi-rigid. Joints in zones 1 or 3 may optionally also be treated as semi- rigid.
Zone 3: nominally pinned, if Sj,ini ≤ 0,5EIb / Lb
*) For frames where Kb/Kc < 0,1 all the joints should be classified as semi- rigid.
Key:
Kb is the mean value of Ib/Lb for all the beams at the top of that storey;
Kc is the mean value of Ic/Lc for all the columns in that storey;
Ib is the second moment of area of a beam;
Ic is the second moment of area of a column;
Lb is the span of a beam (centre-to-centre of columns);
Lc is the storey height of a column.
BASE-PLATE JOINTS
φ
Zone 1: rigid, if (BRACED FRAMES) λ0<0,5
0,5<λ0<3,93 and Sj,ini > 7(2 l0 -1)EIc / Lc
λ0>3,93 and Sj,ini > 48EIc / Lc
(UNBRACED FRAMES) Sj,ini > 30EIc / Lc
Zone 2 and 3: semi - rigid
Key:
Ic is the second moment of area of a column;
Lc is the storey height of a column;
λ0 is the slenderness of the column considering both ends pinned
Furthermore, as previously underlined, EC3 part 1.8 provides a classification of joints on the base of the resistance, individuating full strength, partial strength and nominally pinned joints. Such a classification is obtained comparing joint and connected beam/column flexural resistance:
• full strength joints: , , < , ;
• partial strength joints: 0,25 , , < , < , , ;
• nominally pinned joints: , < 0,25 , , .
where , and , , are the joint and beam plastic moment resistance respectively. Dealing with rotational capacity, a further classification of connections has been proposed by (Design of Structural Joints in Building Frames, 2002). The author suggested categorizing connections in:
• ductile;
• semi-ductile;
• brittle;
• pinned.
According to this proposal joints can be defined ductile if their plastic rotation supply is adequate to allow a sufficient redistribution of the internal actions. As a result, ductile connections are suitable for plastic frame analysis. Conversely, connections are defined brittle if their post-elastic capacity is so limited to consider that no redistribution of the internal actions may occur. Semi-ductile joints behave in an intermediate way between ductile and brittle. Moreover, as mentioned before, nominally pinned
connections are designed to transfer only shear to the columns, but a check of the connection rotational capacity must be provided. In fact, pinned joints should be designed to possess enough rotation capacity to admit the imposed rotations.
The quantification of the boundaries between the above defined categories can derive only by a preliminary comparison between the required and available ductility. In addition, up to now, few studies have been devoted to the valuation of the ductility supply of joints. Indeed, the lack of experimental studies on this topic is negatively reflected on EC3 part 1.8 where few indications for the assessment of the rotation capacity of welded and bolted joints are given.
As far as the method of analysis is elastic, elastic-plastic or rigid-plastic, joints modeling and classifications are different. If an elastic analysis of the frame is performed, internal actions and deflections are only influenced by connections rotational stiffness and, as a consequence, the joints have to be classified as pinned, semi-rigid or rigid (Table 2.1).
Table 2.1. Joint Classification
Method of global analysis
Classification of joint
Elastic Nominally pinned Rigid Semi-rigid
Rigid-Plastic Nominally pinned Full-strength Partial-strength Elastic-Plastic Nominally pinned Rigid and full-strength Semi-rigid and
partial-strength Semi-rigid and full-strength
Rigid and partial-strength
Type of joint model Simple Continuous Semi-continuous
If a rigid-plastic analysis is lead, the only parameter which plays a role on the overall response is the connection bending resistance and, as a result, joints can be classified as full-strength, partial strength or pinned. When an elastic-plastic analysis is considered both initial stiffness and flexural resistance have to be accounted for and a classification according to both parameters is needed. In conclusion, on the base of stiffness and strength the following categories can be individuated:
• full-strength rigid connections;
• full-strength semi-rigid connections;
• partial-strength rigid connections;
• partial-strength semi-rigid connections;
In order to model beam-to-column connections, since 1994 the Commission of the European Communities has introduced in Eurocode 3, within its Annex J, the so-called component method. Such a method provides the general rules to obtain a mechanical model able to predict initial stiffness and plastic resistance of joints. Its validity has been verified in last fifteen years with particular reference to steel joints, but in recent times it has been indicated as a general tool for the prediction of the characteristic of any kind of joint, even made of materials different from steel. Indeed, the application of the component method has been recently extended to joints of composite structures and base plate joints.
The component method is mainly based on the following steps:
• identification of the source of strength and stiffness of the joint;
• mechanical modeling of the components;
• assembly of the components;
• classification.
In the first step a set of basic components, characterizing the strength and stiffness of the connection is individuated and the joint, intended as a whole, is decomposed in different components characterizing its moment-rotation behavior. Then, each component is modelled, and its mechanical properties are determined, i.e. stiffness, strength and deformation capacity. This procedure of characterization is made essentially by means of experimental tests, finite element modelling, numerical simulations and analytical modelling. In the third step, all the components are combined to provide a prediction of the whole joint moment-rotation relationship. Finally, as above shown, joints are properly classified in terms of resistance, stiffness and rotational capacity. The main goal of the classification procedure is to simplify, if possible, the joint modelling within the structural analysis, for example by representing in case of fully “rigid” or “deformable” connections the joints with a clamp or a hinge (Joints in steel structures based on eurocode 3, 2004).
Within the framework of EC3 part 1.8, up to now, the application of the component method to structural connections is still limited to a narrow range of cases. In fact, only connections between H- or I- sections hot-rolled or shop-welded profiles are available. In particular, the following joint typologies are considered:
• welded joints;
• bolted end-plate connections;
• bolted flange cleated joints;
• base plate connections;
• minor axis joints where the beam is connected to the web of a H- or I- section;
• steel-concrete composite joints;
• joints with beam haunches.
The prediction of stiffness and strength of such a type of connections is obtained by the assembly of different joint components properly modelled. For each joint component EC3 part 1.8 provides the rules to calculate the plastic resistance and the initial stiffness. The following components are considered:
• column web panel in shear;
• column web in transverse compression;
• column web in transverse tension;
• column flange in bending;
• endplate in bending;
• flange cleat in bending;
• beam or column flange and web in compression;
• beam web in tension;
• plate in tension or compression;
• bolts in tension;
• bolts in shear;
• bolts in bearing;
• concrete in compression including grout;
• base plate in bending under compression;
• base plate in bending under tension;
• anchor bolts in tension;
• anchor bolts in shear;
• anchor bolts in bearing;
• welds;
• haunched beam.
These components are typically sufficient to model all the possible types of major axis beam-to-column joints and exposed base plate joints. Examples of application of the component method to beam-to- column joints is reported in the next section of the notes. In the next section the importance of correctly modelling the beam-to-column joint stiffness is highlighted with few simple examples.
3 Effect of semi-rigidity on the response of frames
The semi-rigidity of beam-to-column joints provides at least the following detrimental effects to the response of braced/unbraced structures:
• can modify the distribution of internal actions in members;
• increases the lateral deformability of unbraced frames with a consequent increase of the susceptibility to 2nd order effects;
• can reduce significantly the buckling strength of members due to the reduction of the stiffness of the end-constraints;
These aspects are treated hereinafter with the aim to provide simple examples which show the effect of semi-rigidity. These examples highlight some critical aspects of the joint classification suggested by EC3 part 1.8. However, the objective is not to criticize the code provisions but, rather, to highlight what are the main parameters affecting the response of semi-rigid frames.
Distribution of the internal actions in members (Beams/Columns)
A simple beam with flexible end-restraints and uniform distributed load is considered in this example.
This elementary system can be easily analysed with the flexibility method (force method) providing a first insight into the response of structures with semi-rigid joints. The stiffness of the connections is called Sj and the corresponding rotation is referred as θj. With the flexibility method, the system can be regarded as an equivalent isostatic beam loaded with a uniform load and, owing to the symmetry, equal and opposite bending moments acting at the two ends M=Sj θj.
Fig. 19– Simple beam with semi-rigid joints With these actions the rotations at the ends are equal to:
" = #$%
24 − $
3 − $
6 = #$%
24 − $
2 Obviously:
" = = #$%
24 − $
2 Which can be rewritten as:
*1 + $
2 , = #$%
24 → = #$%
24 * 2
2 + $ , =#$.
12 *2 ⁄ +$ ,
By defining ⁄ = 0 it follows: $ =#$.
12 *20 + , =#$.
12 * 1
20 / + 1, Obviously, when = ∞, the bending moment is equal to:
3=#$. 12
This equation shows that the bending moment reduces with the reduction of the joint stiffness. This means that the ratio between the bending moment of the beam with flexible ends and the corresponding bending moment of the beam with rigid elements is equal to:
3= * 1
20 / + 1,
Therefore, in case of a braced frame, for nominally rigid connections (according to EC3 part 1.8 a connection is rigid if ≥ 80 ), the error done in the estimate of the bending moment can be up to the 20%, in fact:
6 37
89: = 6 1
2/8 + 17 = 0.80
The same result could be obtained with the stiffness method, considering that the bending moment can be expressed as a function of the stiffness factors as follows (positive bending moments clockwise):
9 =4
$ <"9− " =9:>,9? +2
$ <" − " =9:>, ? −#$. 12
=2
$ <"9− " =9:>,9? +4
$ <" − " =9:>, ? +#$. 12
Considering that in such a simple scheme the end rotation is only due to the joint flexibility ("9 = " = 0) and that due to the symmetry of the scheme, the joint rotations will be equal and opposed (" =9:>, =
−" =9:>,9), it results:
9 = −4
$ " =9:>,9+2
$ " =9:>,9−#$. 12 = −2
$ " =9:>,9−#$. 12
= −2
$ " =9:>,9+4
$ " =9:>,9+#$. 12 =2
$ " =9:>,9+#$.
12 = − 9 Considering that " =9:>,9= 9/ it is easy to verify that the previous expression is obtained:
9*1 + 20 , = −#$.
12 → 9 = −#$.
12 *20 + ,
Where in this case, the negative sign, just indicates that the bending moment is counterclockwise, as previously assumed. Similar considerations could be done for the deflection, verifying the increased flexibility of the beam. The beam deflection at the mid-span, in this case could be calculated as follows:
@ = 5#$A
384 − 2 $.
16 = 5#$A
384 − 4#$A
384 * 1
20 / + 1, Which, obviously, in case of rigid connections provides:
@3= #$A 384 The ratio in this case is:
@
@3= 5 − 4 * 1 20 / + 1,
This equation shows a large error on the estimate of the vertical deflections under the assumption of nominally rigid connections (with ≥ 80 it is equal up to the 80%). The magnitude of these errors highlights an insufficient accuracy in the definition of the bound values suggested by EC3 part 1.8 for joints in braced frames especially for what regards the assessment of deflections.
Frame deformability and sensitivity to 2nd order effects
To understand the influence of the joint deformability over the global stiffness of unbraced structures, reference can be made to a simple cruciform scheme, corresponding to an internal joint extracted from a moment resisting frame.
Fig. 21– Substructure extracted from an unbraced frame
This scheme can be immediately solved considering that the internal actions are available through simple global equilibria (assuming positive the clockwise bending moments):
− = = − ℎ
2
Fig. 22– Effect of rigid, pinned and semi-rigid joints in a frame node
In this scheme, considering that the constraints far from the internal joint are pins, the maximum bending moment in the beam within the framework of the stiffness method can be written as:
= 3
($ /2) <" − " =9:>? =6
$ <" − " =9:>? =6
$ *" − , Therefore:
= 6 /$
61 + 6 /$ 7" = 60
61 + 60 7" → " =61 + 60 7 60
Considering that, as shown before, = .E , it follows:
" =61 + 60 7
60 ℎ
2
Where the factor 61 +FGH
I7 can be regarded as modification factor of the beam stiffness accounting for the joint deformability. Similarly, the column bending moment can be written as:
= 3
(ℎ /2) " − 3 6ℎ2 7.
@2 =6
ℎ " −6
ℎ. @ = ℎ
2 → 60 " −60
ℎ @ = − ℎ 2
Fig. 23– Notation and lateral drift Substituting the previous expression of " into this equation it yields:
2 = −60ℎ
61 + 60 7
60 ℎ
2 +60 ℎ @ Which results in:
ℎ 2 +0
0 *1 +60 , ℎ
2 = 60 ℎ @ Thus:
= ℎ
2 +0
0 *1 +60 , ℎ 2 =
120ℎ @
1 + 00 61 +60 7=12
ℎ% 1
1 + 00 61 +60 7@ The lateral stiffness of the considered cruciform joint is therefore:
J = @ =12 ℎ%
1
1 + 00 61 +60 7 For rigid joints it provides:
J3=12 ℎ%
1 1 + 00
As a consequence, the ratio between the stiffness of the joint with flexible ends and the corresponding scheme with rigid joints is:
JJ3= 1 + 00
1 + 00 61 +60 7= 1 + 00 1 + 00 +60
Being the denominator obviously bigger than the numerator, the effect of the joint deformability results into an increase of the frame flexibility (or reduction of stiffness). It is worth observing that, even though this equation comes from a very simple example, the reduction of stiffness depends on two factors, namely 0 /0 and 60 / . In typical cases, 0 /0 is bigger than the minimum value suggested by EC3 part 1.8 equal to 0.1. Nevertheless, for MRFs in seismic zone, due to the fulfilment of beam-column hierarchy criteria it is usually likely to be lower than 1. Therefore, assuming the bound value for rigid joints for unbraced frames = 250 and fixing alternatively 0 /0 = 0.1 ÷ 1 it is possible to verify that the frame stiffness reduces of J/J3= 0.82 ÷ 0.89. This obviously corresponds to an increase of the lateral displacements of about @/@3 = 1.22 ÷ 1.12. Therefore, owing to the joint flexibility, in practical cases the lateral drift of a frame with nominally rigid joints as defined in EC3 part 1.8 is larger of about 10 and 20%.
A frame undergoing larger lateral displacements is obviously more sensitive also to second-order effects. This typically results into an amplification of the bending moments. In EC3 this is done through the amplified sway moment method. based on the assessment of the critical load multiplier αcr. The critical load can be estimated, for instance, through the Horne method, as also done by EC8 with certain limitations:
= ℎ
M@ = ℎ M@ =Jℎ
M Thus:
,3= J
J3= 1 + 00 1 + 00 +60
With rigid columns, an upper bound of this effect can be obtained (0 /0 = 0):
,3= 1
1 + 60
with = 250 this leads to:
* ,3,
89:= 0.806
which may seem large from the point of view of the critical load multiplier. Nevertheless, from the point of view of the amplification of the first order bending moments, the approximation deriving from the assumption of rigid joints if = 250 is:
08 = 1
1 − 1/ NOP 08,3= 1 1 − 1/ ,3
Considering that for usual cases ,3 is typically higher than 5, which corresponds under the assumption of = 250 to = 0.806 × 5 = 4.03. With this bound value, it is possible to assess the ratio:
08
08,3=
1 − 1/1 1 − 1/1 ,3
= 1.33
1.25 = 0.94
which shows that the maximum error on the estimate of the first order bending moments magnification factor is only about the 6%, which seems an acceptable level of accuracy. The error is even lower for higher values of the critical load.
Detrimental effect on the buckling strength of structural members in braced frames
The buckling resistance of a column into an unbraced frame depends also on the capacity of end restraints to avoid rotation. Obviously, the more the joints are flexible the lower will be the buckling resistance of the column, because of the detrimental effect on the beam stiffness. The buckling length of columns in a braced frame may be determined using sub-frames through the adoption of equivalent bucking length as suggested by EC3 part 1.1. This implies the knowledge of the beam stiffness which depends over the typology of the beam end-restraints far from the joint. The lower is the flexural stiffness of these restraints, the higher is the effect of joint flexibility. To apply the method of the isolated columns it is necessary to define the distribution factors accounting for the stiffness of the columns upper and lower end-restraints:
where, for the scheme reported in next figure are equal to: Kc= K2 =K1=Ic/hc and K12=K11 are the beam distribution factors. To account for the joint flexibility the terms K12 and K11, normally equal to /$ (for rigid joints) must be modified properly.
Fig. 24– Sub-frame considered and distribution factors
Considering that for a beam, with semi-rigid joints on one-side and fixed-end constraints on the other side, the bending moments can be written as:
=4
$ <" − / ? → " = 4$ 61 + 4 $ 7
Therefore, it results that the beam stiffness modification factor is equal to 61 +AHI 7, leading to an equivalent distribution factor equal to:
JRR= JR.= /$
61 + 4 $ 7
Fixing an equal flexural stiffness for beams and columns the distribution factors are equal to:
R= .= 2 /ℎ
2 /ℎ + 2 /$
61 + 4 $ 7
Assuming for simplicity that /$ = /ℎ , yields:
R= . = 1
1 + 1
61 + 4 $ 7
Fig. 25– Buckling length ratio for a column in a non-sway mode
This equation leads for rigid joints to R= .= 1/2 = 0.5 and for semirigid joints to larger values. The critical length, based on this equation may be consequently calculated as:
= ℎ 1 + 0.145< R+ .? − 0.265 R . 2 − 0.364< R+ .? − 0.247 R . For rigid joints, it leads to:
= 0.685ℎ
Providing a critical buckling load, with the assumptions made (braced frame with rigid joints and flexible beams, with /$ = /ℎ ) is equal to:
M ,3= 2.12T. ℎ
Conversely, the critical load (M ) with semi-rigid joints of stiffness can be easily calculated from the previous equations. The reduction of the critical load M /M ,3 is represented in next chart as a function of /( /$ ). It is easy to notice from the figure that assuming the bound value for nominally rigid joints suggested by EC3 part 1.8 equal to /( /$ ) = 8 results into an error on the assessment of the critical buckling load of about the 12% which seems an acceptable level of approximation.
Fig. 26– Reduction of the critical load on the buckling resistance of a column
