Section 3 - Notes on the modelling of bolted end-plate joints through the component method

(1)

Bolted connections are widely used in design practice because they allow to split complex structures in smaller elements providing the possibility to implement easily a design philosophy based on shop welding and site bolting. The main advantage deriving from the use of the bolted joints is the removability and the easy buildability. Nevertheless, the use of bolts and plates which can deform in bending may increase significantly the flexural deformability of a joint which, possibly, may suffer from a lower stiffness which can result in a semi-rigid behaviour. Bolted connections, nevertheless, may be stiffened both in the panel zone (similarly to what can be done for welded joints) and in the end-plates with additional ribs, allowing to control the resistance and stiffness of the bolted parts.

Fig. 1– Different types of two-sides extended end-plate joint configurations

The connection side, in case of bolted joints plays, indeed, a fundamental role because the typology of stiffeners and the extension of the end-plate, may drastically change the capacity of the connection to transfer bending moments. End-plate connections are characterized by the presence of a plate welded at the end of the beam, which is then bolted to the column.

Fig. 2– One-side vs two-sides extended configurations

(3)

They can be divided in different categories according to the extension of the end-plate and position of the bolt rows, in particular we distinguish:

• Extended end-plate joints: when the end-plate extends over the beam flanges with one or more bolt rows. They can be, in turn, subdivided in:

- one-side extended end-plate joints when the extended plate is located only in the upper or lower part, providing to the connection a high resistance only for hogging or sagging bending moment respectively. This connection typology is typically suitable for elements where the bending moment is always only positive or always only negative. This may be the case, for instance, of structures bearing only gravity loads;

- two-sides extended end-plate joints when the extended part is located both at the upper and lower sides of the joint. This provides to the connection a symmetric behaviour which may be suitable for application in frames subjected to seismic or wind loading conditions;

• Flush end-plate joints: when the end-plate extends only up to the beam flanges and the bolt rows are all contained within the size of the beam;

Fig. 3– Flush end-plate connections with/without stiffeners

The main difference between a flush and an extended end-plate configuration is due to the size of the lever arm which is case of a flush end-plate joint is surely lower than the beam height, while in case of an extended end-plate joint is typically equal or larger than the beam size.

Fig. 4– Flush vs extended end-plate joints

(4)

While the response of a bolted extended end-plate joint may range more easily from the full-strength- rigid to the partial-strength/semi-rigid behaviour, on the contrary, in case of flush end-plate joints, the reduction of the connection lever arm has at least the following consequences: i) the connection is in most of the cases semi-rigid or pinned, depending on the stiffness of the end-plate; ii) the reduced lever arm, for a given bending moment, provides an increase of the actions both in the tension and compression side of the joint. Additionally, since the connection lever arm is lower than the beam size the component beam flange in compression is prone to govern the joint failure mode; iii) the reduction of the lever arm reduces drastically the stiffness, owing to the dependence of the flexural stiffness on the square of the lever arm.

Fig. 5– Approximate values of the lever arms for extended end-plate joints with two active bolt rows and flush end-plate joints according to EC3 part 1.8

The prediction of stiffness and resistance of bolted end-plate beam-to-column joints may be carried out through the component method. Within this framework, the elementary components are in part similar to those already introduced for welded joints and, in some other part, have to be added in order to consider the behaviour of bolted plates which, in these connections, represent one of the main sources of resistance and deformability. The approach to model the connection through the component method is always the same and, therefore, the joint components are first identified, then modelled individually and finally assembled into a mechanical model representative of the connection physical response.

(5)

Also in case of a bolted joint, some of the components contribute in terms of both resistance and stiffness (elastic-plastic components) and some others only in terms of resistance (rigid-plastic components).

Fig. 6 – Active components of a bolted end-plate joint The active components in case of a bolted beam-to-column joint are:

• The column web panel in shear (cws);

• The column web in transverse compression (cwc);

• The column web in tension (cwt);

• The beam flange under local compression (bfc);

• The column flange in bending (cfb);

• The end-plate in bending (epb);

• The bolts in tension (bt);

• The beam web in tension (bwt);

where four of the active components (highlighted in bold) have already been characterized with reference to welded joints, while the remaining four are those typical of bolted joints and they are mainly introduced to consider the bending of the end-plate (epb) and column flange (cfb) which are fastened one to another through the bolts (bt). In the following, it is assumed that the bolts are checked in shear according to the traditional methodologies, while, in the subsequent part of this document, the attention is focused on the bending response of bolted beam-to-column joints, which is characterized through the component method as currently codified in EC3 part 1.8. All the components listed before govern both the strength and stiffness of the joint, made exception for the beam flange in compression (bfc) and the beam web in tension (bwt) that, at most, can provide a limitation to the connection resistance. In the

(6)

response of a bolted joint the geometry of the plates and position of the bolt rows plays a primary role, because it governs more than the other components the stiffness and resistance of the connection.

Column web panel in shear

Idealization FEM

Column web panels in tension/compression

Column flange in bending

(7)

End-plate in bending

Idealization FEM

Beam flange in compression

The evaluation of the flexural strength and of the rotational stiffness of the joint requires the definition of strength and stiffness of each component and of the lever arm. In case of bolted end-plate joints, according to EC3 part 1.8, the lever arm could be fixed equal to the distance between the beam flanges centre provided that there are only two active bolt rows in tension equidistant from the beam flange.

More in general, the lever arm should be calculated based on the stiffness of the single bolt rows which is governed by the response of several components working in series. In fact, at any bolt row the force is transferred from the beam to the column through the end-plate which is fastened with bolts to the column flange. Therefore, in every bolt row, there is the contemporary participation of four components which are loaded by the same force which passes through the bolts, bends the connected plates and arrives to the column web.

(8)

Fig. 7 – Extended end-plate joint modelling

Therefore, in order to get the overall connection stiffness, it is necessary, first of all to estimate the stiffness of each bolt row, which can be seen as a macro-component whose stiffness is given by the inverse of the sum in series of the deformability of the single components aligned on the same bolt row, namely:

, = 1

+ 1

And subsequently, the stiffness of the connection side (excluding the column web in compression and the column web in shear contributions) has to be calculated, accounting for the rotational stiffness of the single bolt rows.

Fig. 8 – Equivalent translational spring modelling a bolt row

Since the lever arm of each bolt row (yi) is calculated fixing centre of compression in correspondence of the mid-thickness of the beam flange, the rotational stiffness of the connection side, through simple equilibrium equations can be written as:

(9)

, = ,

or, equivalently as:

, = ,

where:

= ∑ ,

∑ _,

In which zeq represents the position of the barycentre of the tensile forces of the connection in elastic range, namely the equivalent lever arm. Given the usual practice of EC3 part 1.8 of representing the rotational springs, as equivalent translational springs placed at a distance equal to the lever arm from the centre of compression, the connection stiffness can be also written as:

, = ^, =∑ _,

Fig. 9 – Equivalent translational spring modelling the connection side

With the scheme given in figure, which is analogous to that adopted for welded joints, the stiffness of the connection can be calculated as the sum in series of three components: i) the column web in shear, ii) the column web in compression and iii) the macro-component modelling the connection, namely:

= = 1/ "#"+ 1/²_"#$+ 1/ _",

The connection resistance, instead, can be calculated as:

%,&'= ()* +, _{,&' -}, , _{.,&' -}, , -, , /

(10)

whereas ,000 are the resistances of the joint components, _- is the plastic value of the joint lever arm and , -, , is the plastic resistance of the i-th bolt row determined as the minimum resistance between the five components contributing to the resistance of the i-th row, namely:

, -, , = ()*1, ,&', , ,&', , ,&', , ,&', , ,&'2

The plastic value of the joint lever arm depends only on the resistance of the active bolt rows and on their distance from the centre of compression (yi), namely:

-=∑ , ,

∑ , ,

The procedure suggested by EC3 part 1.8 to calculate the resistance of the connection provides to assess the plastic resistance of the rows starting from the upper line of bolts and subsequently to calculate the resistance of the other bolt rows, one by one, up to the closest to the centre of compression. The procedure to determine , , , from the top to the bottom row, stops when the sum of the resistances of the bolt rows ∑ , , equals the resistance of the column web in compression, of the column web panel in shear or of the beam flange in compression. This condition assures that the translational equilibrium is always satisfied.

The connection resistance in case of extended end-plate connections is more complicated to determine and, in some cases it can be tedious because of the long calculations. This is the reason why EC3 part 1.8, in some specific cases accepts simplifications, namely:

• If the bolt rows in tension are only two, the lever arm may be approximated with the beam depth, minus the beam flanges thickness;

• It is possible to consider the web disconnected from the end-plate if the bolts distance of beam-to-column joints with two active rows only from the beam flange in the tension side is the same.

(11)

These two assumptions, in most of the practical cases, allow to strongly simplify the calculation as it will be shown afterwards.

The notes hereinafter reported have the following objectives:

1. Introduce the design equations to define stiffness and resistance of the basic joint components of a bolted joint (they will be useful also for the modelling of column bases);

2. Presenting the sub-model of the T-stub;

3. Reporting examples of characterization of a bolted beam-to-column joint according to the component method;

4. Proposing simple design rules for designing the resistance of bolted joints;

5. Comparing the results of the component method with an FE model realized with IDEA-STATICA CBFEM;

6. Presenting the main features of a simple excel spreadsheet based on the component method.

After studying these notes the student should be able to: i) check a bolted beam-to-column joint according to the component method with a full understanding of the component behaviour; ii) design a bolted joint for a given value of the required bending strength; iii) use/construct a spreadsheet for modelling bolted extended end-plate joints.

1.2 Technological aspects and historical development of bolted joints

Structural steel frames have been used since the 20th century, when steel effectively replaced cast iron and wrought iron as a construction material. Before the 1920s, connections and steel frame elements were constructed as very complex built-up members with riveted plates. In ordinary practice, before the 20’s, steel structures were built exclusively assembling girders and columns by means of riveted joints.

Fig. 11 – Rivets and High Strength Bolts

The rivet is a mechanical fastener that consists, before the installation, of a shaft with a head only on one end. Typically, rivets are positioned in pre-drilled holes and the termination of the shaft without the

(12)

head is mechanically deformed to about 1.5 times the original diameter. The installation of rivets, in past was made manually by means of a hammer and more recently by means of pneumatic machines.

Noteworthy examples of totally riveted structures are built worldwide as, for instance, the “Eiffel Tower” in Paris or the bridge “Dom Luis I” in Oporto. At the beginning of development of steel structures, the entire frame structure was then encased in concrete for fire protection. Very few steel structures were designed for seismic loadings, because wind loads were considered predominant before 1930, and for some structures even this request was not considered. During this period, in 1928, the first manual AISC (American Institute of Steel Construction) appeared which included specifications for steel frame structures. Standard hot rolled shapes appeared after 1920. These were connected with riveted angles, became standard connections and were used until 1960. Connection stiffness and strength was enhanced by non-structural masonry walls and concrete for fire protection. In structural design of these structures the seismic forces were considered, but these forces were simplified and often much smaller than those currently used. Between the 1920s and 1950s the introduction of High Strength Bolts represented a significant innovation. High strength bolts allowed to fasten plates through high contact pressures, leading to the development of the so-called slip resistant joints.

Fig. 12 – Examples of famous riveted structures

In this type of connections, the force transfer is achieved by means of the friction exploited between two clamped surfaces. The adoption of High Strength bolts allowed significant time-savings associated with the ease of installation due to threads and washers. Besides, the adoption of this system favorites the realization of smaller connections, reducing simultaneously deformability of frames under lateral loads

(13)

materials. Meanwhile seismic design procedures have evolved toward similar methods to those used in modern seismic design. It was recognized that seismic forces can be extremely large, and the structures could be calculated for seismic design forces much smaller if they would consider inelastic capacity of the building and its connections. This has led to increased interest in the inelastic behaviour of structures (FEMA, 2000). During the 1960s welding became more economical and practical and welding procedures were developed. Frame structures with welded-flange-welded-web connections were built and later these were replaced with welded-flange bolted-web connections, due to numerous researches at the time and due to the economy of these connections over the welded web connections. This type of connection became the standard connection used for seismic design (FEMA, 2000).

During 1970-1994, many steel structures have been built in the USA using welded beam-column joints.

Indeed, this type of connection provides economical execution and it was believed that possess a high capacity of deformability under bending moment action. But the Northridge earthquake of 17 January 1994 damaged these connections and cracks of various sizes appeared in welds (James, 1997; Mahin, 1997). There was no collapse of structures with welded beam-column connections but came out the brittle nature of these joints. In the next period new design codes were required for providing truly ductile zones at the end beams of these structures. These new design specifications permit beam-column connection deflection (rotation), without exceeding their loadbearing capacity. A feasible solution of this requirement is the semi-rigid (flexible) beam-column connection. This concept of semi-rigid connection was considered an erection imperfection (Alexa and Moldovan, 2005), in the early 20th century and later modern design codes (ASRO, 2006), classified it as recommended connection in seismic regions. The experimental results and researches (Alexa and Moldovan, 2005), showed that semi-rigid beam-column connection is obtained by bolting, not welding (Mathe, 2009). As a natural consequence of the brittle behaviour of welded beam-column connections during the 1994 Northridge earthquake, the bolted connections have been gaining poles (Mahin, 1997).

(14)

2 Component modelling of bolted joints

The modelling of the joint components briefly described in the introduction is hereinafter properly addressed, giving, wherever necessary, all the needed insights on the historical developments and component behavior. The components are characterized both in terms of stiffness and resistance and, subsequently, the model assembly procedures provided by EC3 part 1.8 are examined in detail. For what concerns the column web panel in shear and the column web in transverse compression, the methodology of calculation of stiffness and resistance is the same explained for welded joints. Similarly, the column web in tension follows the same approach, but the calculation of the effective length has to be based on the analysis of the bolted T-stubs representing the column flange in bending presented in the next sections. Similarly, the modelling of the beam web in tension component will follow an approach similar to the column web in tension, but with a value of the effective width properly calculated based on the effective length concept. In the next sections, the modelling of bolted joints through the equivalent T-stub model is first introduced and, afterwards, the single joint components are characterized one by one.

2.1 Component identification and modelling

2.1.1 Modelling of bolted components through equivalent T-stubs

In case of bolted joints it is of primary importance to account for the influence of the plate deformability and for the development of contact forces which act at the beam-column interface. The traditional hypotheses adopted in the elastic design of beam-to-column joints, which are based on the assumption of rigid plates and neglect the development of contact forces, are very limiting. In fact, an equal stiffness for all the bolt lines and the assumption that the plate rotates rigidly around a centre of compression arbitrarily selected, typically imply a distribution of forces in the bolt rows which provides a safe estimate of the reality only when plates are actually rigid, such as for joints with haunches or stiffening plates.

Fig. 13 – Elastic distribution of forces in an end-plate connection under the assumption of rigid plate

(15)

In all the other cases, the application of such a simplified modelling approach provides strong approximations which may be removed only through the proper modelling of the plates’ deformability.

Fig. 14 – Elastic distribution of forces in an end-plate connection under the assumption of rigid plate and centre of compression at the lower flange mid-thickness

It is common practice to describe the components of bolted connections through the sub-model of the so-called bolted T-stub. A T-stub is a tee element, connected through the flanges with a couple of bolts.

It is intended to represent the behaviour of the steel plates of a bolted connection which are connected with one or more bolt lines. Such plates, which in the specific case of a bolted end-plate connection can be either the end-plate or the column flange, with the T-stub sub-model are imagined extracted from the connection and separately modelled.

(16)

Fig. 15 – Bolted T-stubs extracted from the column flange in bending and from the end-plate A noteworthy analogy which may help the reader to understand better the objectives of modelling bolted connections through equivalent T-stubs can be related to a beam-to-column connection similar to the extended end-plate joint, which is the joint with upper and lower split tees. In this typology the beam is fastened to the column through the flanges by means of a couple of bolted Tee elements. In both the double split tee joint and in the extended end-plate joint, as given in figure, the upper flange is in tension and, under the applied actions the forces give rise to bending in the upper bolted elements of the connection. In case of a double split tee joint, the upper tee must carry a force equal to the bending moment divided by the distance between the tee elements. Similarly, in case of a bolted end-plate joint

(17)

the upper side of the connection has to bear tensile forces. In both cases at the beam-column interface the deformations of the plates cause the development of contact forces.

Fig. 16 – T-stubs and end-plate connections in undeformed and deformed configuration

The result of the tensile forces applied at the level of the upper beam flange is that external loads bend the T-stub flange/end-plate overloading the bolts which will have to transfer not only the force provided by the beam flange, but also the contact force which develop due to the deflection of the plates. This is clear from next figure, where the response of a bolted T-stub in bending is represented. Under the action 2T, due to the flange plate bending, it happens that the plates push one on the other giving rise to the contact force Q, while due to translational equilibrium the bolt will have to carry a force equal to T+Q. It is very difficult to assess the contact force distribution in complex joint configurations with simple calculations and it is usually preferred to model the single bolt rows of end-plates as equivalent T-stubs extracted from the joint and treated separately characterizing into an easier way their elastic stiffness and plastic resistance.

(18)

Fig. 17 – Forces in a T-stub from Zoetemejier (1974)

The design method at the base of the theory of bolted T-stubs is based on the assumption that the plasticization is large enough to allow the adoption of the most favorable static equilibrium. Simple plastic hinges are thought to form eventually at the bolt and the web line with bending moments equal to:

34 = 5

6

4 8

where b is the width of the T-stub and t is the thickness of the t-stub flange plate. The collapse mechanisms which can form are three depending on the relative resistance of bolt and plate, namely:

• Mode-1 (Strong bolt-weak plate): in which the collapse is achieved with the formation of four plastic hinges in correspondence of the bolt lines and at the flange-web plates attachments. In this case contact forces develop, and they are assumed applied at the tip of the flange plate;

• Mode-2 (Intermediate): in which the collapse is achieved due to the contemporary failure of bolt and plate. In this case only two plastic hinges are forming at the flange-web intersection.

• Mode-3 (Strong plate-weak bolt): in which the collapse is achieved due to the detachment of the T-stub (no contact forces), the plate is particularly strong, and the resistance is achieved due to the failure of the bolts;

(19)

Fig. 18 – Failure modes of a bolted T-stub

The collapse loads of these three plastic mechanisms can be easily calculated through equilibria or through the virtual work principle. Both strategies are shown hereinafter. The characteristic geometric parameters of a bolted T-stub, which will be used in the subsequent calculations, are the distance of the bolt line from the plastic hinge forming in correspondence of the web plate (m) and the distance of the bolt line from the tip of the plate (n) which in practical design cases cannot exceed 1.25 m. The distance m is conventionally defined in EC3 part 1.8 as:

• ( = 9 − 0.8> with r equal to the root radius, for T-stubs modelling the column flange in bending (hot-rolled profiles)

or

• ( = 9 − 0.8$ with $ equal to the weld thickness, for T-stubs modelling the end-plates (welded elements)

Fig. 19 – Geometric parameters of a bolted T-stub

(20)

2.1.2 Resistance of bolted T-stubs Mechanism type-1

Resistance assessed with equilibria

Fig. 20 – Bending moments and forces in Mode-1

In case of mechanism type-1, under the application of a tensile force F to the T-stub, a double-curvature bending moment develops in the plate, with the formation of four plastic hinges, two at the bolt lines and two at the flange-web attachment. Contact forces develop in the plate (Q) and the bolt has to carry, because of translational equilibria, a force B = F/2+Q. The contact force is conventionally considered applied at the tip of the plate. In reality a non-linear distribution of contact forces exists, and its actual shape depends on the relative stiffness between the connected plates.

Fig. 21 – Single side of the flange plate of a bolted T-stub

In order to define the collapse load, in this case, three equilibria can be written, namely a translational equilibrium, a rotational equilibrium around point C) and a rotational equilibrium around point B):

(21)

⎩ ⎪

⎨

⎪ ⎧ ,

2 + C = D

34+ D* =

,

2 (( + *)

C* = ₃₄ → C = *³⁴

The third equation simply provides that the contact force, in this case is equal to the plastic moment divided by n. Conversely, from the first two equations the collapse load and the bolt force can be obtained:

H

,

2 +

*³⁴

= D

34 + D* =

,

2 (( + *)

Substituting the first equation in the second, yields:

-+ I

,

2 +

* J * =^-

,

2 (( + *) →

² ^- ⁼

, 2 (

Thus:

, =4 _- ( Resistance assessed with virtual work principle

Fig. 22 – Assigned kinematic mechanism External work:

K = ,

L ≈

,

N(

Internal work:

(22)

K =

4 ₃₄N External work = Internal work:

,

N( = 4 ₃₄N → ,₁ = 4 ₃₄ (

Mechanism type-2

Fig. 23 – Bending moments and forces in Mode-2

In case of mechanism type-2, under the application of a tensile force F to the T-stub, bending moment develops in the plate, with the formation of two plastic hinges at the flange-web attachment. Contact forces develop in the plate (Q) and the bolt fails due to the contemporary action of the external and contact force. The contact force also in this case is conventionally considered applied at the tip of the plate. In reality, a non-linear distribution of contact forces exists, even though in this case the assumption is closer to the reality because of the higher stiffness of the plate.

Fig. 24 – Single side of the flange plate of a bolted T-stub

(23)

In order to define the collapse load and the magnitude of the contact forces, in this case, two equilibria are sufficient, namely a translational equilibrium and a rotational equilibrium around point C):

H ,

2 + C = D

^'

→ C = D

_'

− , 2

34+

D

_'* =

,

2 (( + *)

The second equation yields:

, =2 _-+ 2

D

_>9*

(( + *)

Resistance assessed with virtual work principle

K = ,

L ≈

,

N

(( + *)

Internal work:

K =

2 ₃₄N +

D

_'N*

External work = Internal work:

,

N

(( + *)

= 2 ₃₄N +

D

_'N* → ,₂ = 2 ₃₄+ 2

D

_'*

(( + *)

Mechanism type-3

(24)

Fig. 26 – Force distribution in mode-3 (Q=0)

In this case the translational equilibrium is sufficient to determine the collape load, which is equal to:

,_O = 2D _'

In which Brd is the bolt resistance.

Resistance assessed with virtual work principle

K = ,

L Internal work:

K =

2D_>9L External work = Internal work:

,

L = 2D_>9L → ,₃ = 2D_>9

(25)

2.1.3 Resistance of equivalent T-stubs according to EC3 part 1.8

From all the previous considerations it is easy to observe that the resistance of a T-stub (with two bolts) can be calculated as the minimum collapse load of the three characteristic failure modes, namely:

,_{Q . R} = ()*S, ; , ; ,_OU Where:

, =4 _- (

, =2 _-+ 2

D

_>9*

(( + *)

,_O = 2D _' and:

34 = 5 ₈₈

6

4 V ₀8 , D_>9= 0.9X₅8_Y V ₂

Whereas 5 is the effective width hereinafter defined, A^b is the bolt area and fu is the bolt material ultimate stress. It is possible to demonstrate that the collapse load of a T-stub depends only on a parameter, commonly called Z (not to be confused with the shear panel transformation parameter):

Z =,

,_O = 4 _- 2D _'(

Based on this parameter, by equating the previous collapse loads it is possible to verify that the failure mode depends only on [. In particular, when [ is lower than \ /(1+2 )^{, with} =]/^, the failure mode is type-1, conversely when [ is larger than 2 the failure mode is type-3. In the intermediate cases the failure mode is type-2. This is described in figure.

Fig. 28 – Influence of β on the failure mode

(26)

The classical theory here described is based on an equivalent beam model. Nevertheless, in some specific cases, if the flange of the T-stub is particularly wide, other three-dimensional failure modes may arise. The collapse loads of these alternative failure modes are based on the yield-line theory and were developed by Zoetemejier in 1974. They are mostly devoted to model the case of the column flange in bending which is, typically, very wide and for which a beam failure mode is usually not possible. In these cases, an effective width value is defined, while the value of the collapse load is calculated always through the previous equations. The yield line theory assumes the formation of a yield line pattern in the flange. These patterns can eventually turn the flange into a mechanism since a yield line a continuous formation of a plastic hinge along a strait or curved line is formed. The flange becomes a collection of rigid parts which can turn relatively to each other in the yield lines. Once the family of yield lines is selected the internal work stored in the lines and the external work are equated leading to the definition of the collapse load. For design purposed this load is equated to the resistance determined through the beam model, deriving an equivalent value of the width which provides in a beam model the same load of the actual failure mechanism. The analysis of the yield line patterns is complex and more details can be found in the original work of Zoetemejier, in case of an isolated bolted T.stub three yield-line patterns are individuated:

Non-circular pattern:

5 = 4( + 1.25*

Circular pattern:

5 = 2a(

Beam pattern:

5 _O = 5 The effective width to be used in calculation is:

5 = ()*15 ; 5 ; 5 2

(27)

Fig. 29 – 3D failure modes of a bolted T-stub

Other cases are possible in reality because of the vicinity of bolts. In particular, mixed yield line patterns are possible leading to other failure mechanisms which are translated in EC3 part 1.8 in effective width values accounting for group behaviour of more bolt lines.

Fig. 30 – Examples of bolt group behaviour

Another important case arises when bolts are close to stiffeners, such as for example in case of the first internal bolt row of an extended end-plate connection. In this circumstance, tables of the effective widths are given in EC3 part 1.8 accounting for the change of the yield line pattern and the influence of the stiffener on the load bearing capacity of the bolt line. An example is provided in figure, but more details are given in the worked examples.

(28)

Fig. 31 – Bolt rows with stiffeners 2.1.4 Stiffness of equivalent T-stubs

The resistance of an equivalent T-stub can be calculated by means of a simple beam model where the force is supposed to spread from the bolt with a 45° angle.

Fig. 32 – Scheme for the stiffness calculation Under this assumption it results that the plate deformability is equal to:

L =D(^O

3bc =,(^O

6bc = 12,(^O

6b5 _88,$66^O → Q . R =, L = 0.5

b5 _88,$66^O (^O Similarly, in EC3 part 1.8 the stiffness of a bolted T-stub is calculated as:

(29)

Q . R = 0.9b5 6^O (^O

In which the increase of the factor from 0.5 to 0.9 accounts essentially for the effect of the bolt preloading which, even in case of low tightening values, usually provides a significant contribution. The stiffness of the T-stub has to be combined usually with the stiffness of the bolts in tension, which is defined as:

= 1.6bX e

Where Lb is the bolt conventional length defined as the sum of the connected plates thicknesses plus half-thickness of head and nut plus, eventually, the thickness of the washers. The factor 1.6 instead of 2 is usually justified by the detrimental contribution of the parasite bending moments acting in the bolt due to plate deformability. Based on the T-stub theory herein presented, in the next sections the design equations for each joint component are summarized.

2.1.5 End-plate in bending

All the procedures for the characterization of an extended end-plate connection are based on the characterization of the single bolt rows. Every bolt row (composed by two bolts) is seen in EC3 part 1.8 as an equivalent T-stub. Therefore, the design formula for the calculation of bolt-rows stiffness and resistance are those previously reported, namely:

Resistance

fghi= ^j]Sfk; f\; flU with:

fk =mn_ho,ghi

^

f\=\nho,ghi+ \pqr] (^ + ]) fl= \pqr

and:

nho,ghi=igss,ghitgh\

m unv sw , pqr=v. xyisz

un\

(30)

Stiffness

= 0.9b5 , 6^O (^O

The design resistance and failure mode of an end-plate in bending, together with the associated bolts in tension, should be taken as similar to those of an equivalent T-stub flange, for both: each individual bolt- row required to resist tension; each group of bolt-rows required to resist tension. The groups of bolt- rows either side of any stiffener connected to the end-plate should be treated as separate equivalent T- stubs. In extended end-plates, the bolt-row in the extended part should also be treated as a separate equivalent T-stub. The design resistance and failure mode should be determined separately for each equivalent T-stub.

In case of the external bolt row, EC3 part 1.8 calls ex the parameter previously called n and e the distance of the bolt from the lateral edge. Instead w is the bolt pitch. In case of an end-plate several cases arise in practice: the external bolt row is an unstiffened T-stub, similar to those previously presented; the first internal bolt row below the tension flange is stiffened because the bolt is close both to the beam web and flange. The other bolt rows, if present, are treated as provided in Table 6.6 of EC3 part 1.8. In the assembly procedure each bolt row, if not separated by a stiffener from the other bolt rows, has to be considered both as an isolated element and as part of a bolt group.

Fig. 33 – Representative equivalent T-stubs defined in EC3

Since the assembly procedure provides to calculate the resistance of the bolt rows, one by one, from the top part of the connection to the lower, at any bolt row the following conditions must be verified to determine the resistance of the end-plate in bending:

(31)

Resistance of the spring modelling the extended end plate at the generic bolt row 1^st bolt row (type-1 – only isolated behaviour): , 351,{9 = , 351,{9

2^nd bolt row (type-2 – only isolated behaviour because it is separated from the 1^st row by the beam flange): , _352,{9 = , _352,{9

3^rd bolt row (type-3 if isolated – type-3 + type 2 when in group with 2^nd row): , 353,{9= min1, 353,{9;, 353+2,{9− , _,&'2

4^th bolt row (type-3 if isolated – type-3 + type-3 when in group with 3^rd row or type-3 + type-3 + type 2 when in group with 2^nd and 3^rd row2): , 354,{9= min1, 354,{9;, 354+3+2,{9− ,_O,&'−

,_,&';, _353+2,{9− , _,&'2

…… and so on

In previous equations with , ,&' it is indicated the resistance of the bolt row considered isolated from the others, with , _•( _)•( )•⋯( •),&' it is indicated the resistance of the bolt row considered as a part of a group and with ,,&' it is indicated the resistance of the bolt row determined at the previous step of the analysis as the minimum of the resistance of the component in series at the bolt line (epb, cfb, bwt, cwt, bt). EC3 part 1.8 table is reported hereinafter and a graphic explanation of the effective widths is provided.

(32)

Fig. 34 – Effective length for an end-plate according to EC3 part 1.8

Fig. 35 – Geometrical explanation of the effective lengths for the external bolt row

For the case of the first bolt row below the tension flange, which is stiffened, for the calculation of the effective width reference has to be made to the scheme given in figure proposed by EC3 part 1.8.

Type-1

Type-2

Type-3 Type-4

(33)

Fig. 36 – Effective width calculation for the first bolt row below the beam tension flange (stiffened)

2.1.6 Column flange in bending

The design formula for the calculation of stiffness and resistance of a bolt row on the side of the column flange are the same as before:

Resistance

, = ()*S, ; , ; ,_OU with:

, =4 -,

(

, =2 -, + 2D '* (( + *) ,O= 2D'

(34)

and:

-, =5 , 6

4V‚ƒ 8^„ , D _'=0.9X 8R

V‚

Stiffness

= 0.9b5 , 6^O (^O

The design resistance and failure mode of an unstiffened column flange in transverse bending, together with the associated bolts in tension, should be taken as similar to that of an equivalent T-stub flange, for both: i) each individual bolt-row required to resist tension; ii) each group of bolt-rows required to resist tension. The dimensions emin and m should be determined from Figure 6.8 of EC3 part 1.8.

Fig. 37– Geometrical parameters of the t-stubs on the column flange side

Transverse stiffeners and/or appropriate arrangements of diagonal stiffeners may be also used to increase the design resistance of the column flange in bending. The different cases are individuated in Fig. 6.9 of EC3 part 1.8 and tables 6.4 and 6.5. As for the end-plate in bending, the resistance has to be calculated according to the following scheme:

Resistance of the springs modelling a column flange in bending without stiffeners 1^st bolt row (type-2 – only isolated behaviour): ,"851,{9 = ,"851,{9

2^nd bolt row (type-3 if isolated – type-3 + type-2 if in group with the 1^st bolt row): ,"852,{9 =

min1, ;, − , 2

(35)

3^rd bolt row (type-3 if isolated – type-3 + type-3 if in group with the 2^nd bolt row or type-3 + type- 3+type-2 if in group with the 1^st and 2^nd bolt row): ,"853,{9 = min1,"853,{9;,"853+2,{9−

,_,&';,"853+2+1,{9− , _,&'− ,_,&'2

…… and so on

Resistance of the springs modelling the column flange in bending in presence of stiffeners 1^st bolt row (type-1 – only isolated behaviour): ,"851,{9 = ,"851,{9

2^nd bolt row (type-4 – only isolated behaviour because it is separated from the first row by the stiffener):

,"852,{9 = ,"852,{9

3^rd bolt row (type-3 if isolated – type-3 + type-4 if in group with the 2^nd bolt row): ,"853,{9= min1,"853,{9;,"853+2,{9− , _,&'2

4^th bolt row (type-3 if isolated – type-3 + type-3 if in group with the the 3^rd row or type-3+type-3+type- 4 if in group with the 2^nd and 3^rd bolt rows): ,"854,{9 = min1,"854,{9;,"854+3+2,{9− ,_O,&'− ,_,&';,"853+2,{9− , _,&'2

…… and so on

In previous equations with , ,&' it is indicated the resistance of the bolt row considered isolated from the others, with , •( )•( )•⋯( •),&' it is indicated the resistance of the bolt row considered as a part of a group and with ,,&' it is indicated the resistance of the bolt row determined at the previous step of the analysis as the minimum of the resistance of the component in series at the bolt line (epb, cfb, bwt, cwt, bt). EC3 part 1.8 tables are reported hereinafter with a graphic explanation of the effective widths.

Fig. 38 – Representative equivalent T-stubs defined in EC3

(36)

Fig. 39 – Effective lengths of bolt rows adjacent to a stiffener and geometric explanation

Fig. 40 – Effective lengths of bolt rows of an unstiffened column flange in bending and geometric It must be added here,

missing in EC3 part 1.8

Type-1 Type-4

Type-3

Type-2

(37)

2.1.7 Column web in tension

Stiffness and resistance of the column web in panel tension is calculated with the same methodology already pointed out for welded joints Nevertheless, in this case, the effective length of the column flange in bending is adopted for the resistance and stiffness calculation. The design equations are herein summarized. In this case, the possibility of bolt group behaviour has also to be accounted for as done for the other components aligned on the same bolt line.

, _,&' = ω5 _, 6 8„

V‚ƒ

"#6 = 0.7b5 _88,"#66_"#

9_"#

Similarly, to what has been reported before with reference to the column flange in bending it is necessary to calculate the resistance according to the following logical scheme:

Resistance of the springs modelling the column web in tension without stiffeners 1^st bolt row (type-2 – only isolated behaviour): ,"#61,{9 = ,"#61,{9

2^nd bolt row (type-3 if isolated – type-3 + type-2 if in group with the 1^st bolt row): ,"#62,{9 = min1,"#62,{9;,"#62+1,{9− , _,&'2

3^rd bolt row (type-3 if isolated – type-3 + type-3 if in group with the 2^nd bolt row or type-3 + type- 3+type-2 if in group with the 1^st and 2^nd bolt row): ,"#63,{9= min1,"#63,{9;,"#63+2,{9−

,_,&';,"#63+2+1,{9− , _,&'− , _,&'2

…… and so on

Resistance of the springs modelling column web in tension in presence of stiffeners 1^st bolt row (type-1 – only isolated behaviour): ,"#61,{9 = ,"#61,{9

2^nd bolt row (type-4 – only isolated behaviour): ,"#62,{9 = ,"#62,{9

3^rd bolt row (type-3 if isolated – type-3 + type-4 if in group with the 2^nd bolt row): ,"#63,{9= min1,"#63,{9;,"#63+2,{9− , _,&'2

(38)

4^th bolt row (type-3 if isolated – type-3 + type-3 if in group with the the 3^rd row or type-3+type-3+type- 4 if in group with the 2^nd and 3^rd bolt rows): ,"854,{9 = min1,"#64,{9;,"#64+3+2,{9− ,_O,&'−

,_,&';,"#63+2,{9− , _,&'2

…… and so on

2.1.8 Beam web in tension

The resistance of the beam web in tension is calculated with the same method already pointed out for the column web in tension, while its stiffness is assumed infinite. Nevertheless, in this case, ω factor is equal to 1 and the effective length of the end-plate in bending is adopted for the calculation, considering, wherever needed the effective lengths of the bolt group. Obviously, in this case the bolt rows outside the column flange have not a beam web in tension component. The design equation for the resistance is the following:

, ,&'= 5 , 6 8„

V‚ƒ

1^st bolt row (type-1 – only isolated behaviour): not relevant 2^nd bolt row (type-2 – only isolated behaviour): ,5#62,{9= ,5#62,{9

3^rd bolt row (type-3 if isolated – type-3 + type 2 when in group with 2^nd row): ,5#63,{9= min1,_5#63,{9;,_5#653+2,{9− , _,&'2

4^th bolt row (type-3 if isolated – type-3 + type-3 when in group with 3^rd row or type-3 + type-3 + type 2 when in group with 2^nd and 3^rd row2): ,5#64,{9= min1,5#64,{9;,5#64+3+2,{9− ,_O,&'−

,_,&';,5#63+2,{9− ,_,&'2

…… and so on

2.1.9 Beam flange in compression

The possibility that the local resistance of the beam flange in compression is exceeded should be taken into account including in the joint mechanical model an additional rigid-plastic spring whose resistance is equal to:

, ,&' = ^,&' =(ℎ − 6 ) ^,&'

(39)

This component is characterized through the bending resistance of the profile, which is translated into an equivalent translational spring located in correspondence of the compressed beam flange. This practice is reasonable only if the lever arm of the connection is lower than (hb-tbf), which is not the case of extended end-plate beam-to-column joints. Furthermore, as already done for welded joints, it has to be noted that, when seismic design of structures with full-strength joints is considered, it is necessary to omit this component. In fact, by definition, a seismic-resistant full-strength joint is characterized by a flexural strength equal to 1.1Vˆ‰ ,&'=1.375 _,&', which accounts for the occurrence of yielding in the beam, considering randomness and strain-hardening. The inclusion of the beam flange in bending in the joint model leads, in case of seismic design, to an evident paradox, which is that the connection resistance with the beam flange in bending included in the joint components would always be lower than the required strength. A general rule, valid also for the other connection typologies under pure bending, is that the beam flange in compression could be omitted when the connection lever arm (z) is higher than or equal to (hb-tbf).

2.1.10 Column web in shear and column web in compression

The resistance of these two components is calculated with the same procedures pointed out for welded beam-to-column joints.

(40)

2.2 Model assembly and summary

Based on all the previous considerations, hereinafter, the design equations are summarized, reporting the equations to determine the design resistance and the stiffness of each joint component and the equations to define stiffness and resistance of the whole extended end-plate bolted joint. A similar procedure can be individuated for bolted connections with double split tee or top and seat angles. Some examples are given directly in the worked examples.

Column web panel in shear Resistance:

unstiffened - , .,&' =^ƒ.Š‹_•√O‘^{Œ• Ž}

’“

with continuity plates - , .,&' =^ƒ.Š‹_•√O‘^{Œ• Ž}

’“+^”‚_•˜^{•–,•—}

with doubler plates - , .,&' =^ƒ.Š(‹_•√O‘^Œ•^•^{™ ™}⁾^Ž

’“

with doubler plates and continuity plates - , .,&'=^ƒ.Š(‹_•√O‘^Œ•^•^{™ ™}⁾^Ž

’“ +^”‚_•˜^{•–,•—}

Stiffness:

unstiffened - _.= 0.38^š‹_{˜ •}^Œ•

with doubler plates - _.= 0.38^š(‹^Œ•_{˜ •}^•^{™ ™}⁾

Column web panel in transverse compression Resistance + Buckling

› = œ1 −^ƒ. • ≤ 1 with: 0.932

Ÿ

^{——,•¡• Ž}_š _•¡¢ ^'^•¡ or 0.932

Ÿ

^{——,•¡• Ž}_š( _•¡_• _™^'₎¢^•¡ if doubler plates are present

= 1 )8 £‰ < 0.78„

= 1.7 − ^¥^Œ

Ž )8 £‰ > 0.78„

unstiffened - , ,&' = › ω5 , 6 _‘^Ž

’¢ #)6ℎ 5 , = 26 + 6 + 2$ + 5§6 + > ¨

(41)

with continuity plates - , ,&'= ω(5 , 6 )_‘^Ž

’“+ X _‘^Ž

’“

with doubler plates - , ,&'= › ω5 , (6 + 6.)_‘^Ž

’¢

’“ + X _‘^Ž

’“

Stiffness

unstiffened - "#" = ^0.7b588,"#"6"#

9"#

with doubler plates - "#" = ^0.7b588,"#"(6"#+6$) 9"#

with continuity plates - = ∞ Beam flange in compression Resistance

, ,&' = ^,&' =(ℎ − 6 ) ^,&'

End-plate in bending Resistance

, = ()*S, ; , ; ,OU with:

, =4 -,

(

, =2 -, + 2D '* (( + *) ,O= 2D '

and:

-, =5 , 6

4 V‚ƒ 8„ , D '=0.9X 8R

V‚

(42)

With the effective length of the single bolt row calculated according to the scheme described in the previous section

Stiffness

= 0.9b5 , t-®O

(^O

Column flange in bending Resistance

, = ()*S, ; , ; ,OU with:

, =4 -,

(

, =2 _-, + 2D _'* (( + *) ,O= 2D '

and:

-, =5 , 6

4 V‚ƒ 8„ , D '=0.9X 8_R V‚

With the effective length of the single bolt row calculated according to the scheme described in the previous section

Stiffness

= 0.9b5 , 6^O (^O

Column web panel in transverse tension Resistance

, ,&'= ω5 , 6 8„

V‚ƒ

with doubler plates - , ,&'= ω5 , (6 + 6.)_‘^Ž

’“

(43)

With the effective length of the single bolt row calculated according to the scheme described in the previous section.

Stiffness

"#6 = 0.7b5 _88,"#66_"#

9_"#

with doubler plates - _"#6 = ^0.7b5^88,"#6₉ ⁽⁶^"#⁺⁶^$⁾

"#

Beam web in tension Resistance

, ,&'= 5 , 6 8_„

V‚ƒ

With the effective length of the single bolt row calculated according to the scheme described in the previous section.

Joint stiffness

%= 1/ + 1/ .+ 1/ ,

with:

, =∑ _,

(44)

, = 1 + 1 + 1 + 1

and:

= ∑ _,

∑ _,

Joint resistance

The connection resistance can be calculated as:

%,&'= ()* +, _{,&' -}, , _{.,&' -}, , _{,&' -}, , -, , /

whereas ,000 are the resistances of the joint components, _- is the plastic value of the joint lever arm and , -, , is the plastic resistance of the i-th bolt row determined as the minimum resistance between the five components contributing to the resistance of the i-th row, namely:

, -, , = ()*1, ,&', , ,&', , ,&', , ,&', , ,&'2

- =^{∑ ¯}_{∑ ¯}^°±,²^„^²

°±,² – considering only the active bolt lines

(45)

3 Design applications

Worked Example 1

Calculate strength and stiffness of the isolated bolted T-stub described in figure.

Fig. 41 – T-stub to be checked

The design parameters of the T-stub are the following:

( = 9 − 0.8$ = 60 − 0.8 × 10 = 52 ((

* = 45 (( (45 ≤ 1.25( = 65 ((, ) X = 245 ((

8R = 800 ´µ 8„'= 275 ´µ

T-stub resistance

The possible yield-line patterns are equal to those typical of an isolated T-stub with two bolts only, namely circular pattern, non-circular pattern and beam pattern. The effective length, making reference to a single row of two bolts is equal to:

Circular pattern:

5 = 2a( = 326.6 ((

5 _• = 4( + 1.25* = 264 ((

(46)

Beam pattern:

5 = 5 = 77 mm

The effective width to be used in calculation is:

5 = ()*15 ; 5 _• ; 5 2 = 77 ((

- =5 6

4 V_‚ƒ 8_„ =77 ∙ 20

4 ∙ 1 275 = 2117500 ·((

D _' = 0.9 ∙ 245 ∙ 800

1.25 = 141120 ·

The resistance associated with the three failure modes provide:

, =4 _-

( =4 ∙ 2117500

52 ∙ 10^O = 163 ·

, =2 _-+ 2

D

_>9*

(( + *)

⁼ 2 ∙ 2117500 + 2 ∙ 141120 ∙ 45

(52 + 45)

∙ 10^O = 207 · ,_O = 2D _' = 282 ·

Therefore

,Q . R , ˆ- . = ()*S, ; , ; ,_OU = 163 ·

Leading to a failure mode Type-1. The overall resistance of the T-stub is therefore equal to 163 kN.

The stiffness of the T-stub can be simply calculated as:

Q . R = 0.9b5 6^O

(^O = 0.9210000 ∙ 77 ∙ 20^O

52^O∙ 10^O = 828 ·/((

The stiffness of the bolts instead is calculated as:

= 1.6bX e

If the bolt grip length is equal to 53 mm (20+15+0.9x20):

= 1.6210000 ∙ 245

53 ∙ 10^O = 1553 ·/((

Therefore, the overall stiffness of T-stub and bolts can be calculated as:

(47)

Q = I 1

Q . R + 1

J = 540 ·/((

Comparison with the results of CBFEM

A FE model of the T-stub has been prepared with IDEA-STATICA CBFE software. The results in terms of force-diplacement curve are the following:

Fig. 42 – Force-displacement curve with IDEASTATICA

The comparison between IDEA-STATICA and the component method are in good agreement especially in terms of resistance, in fact:

,%,¯š‚

,%,¸‚ =176

163 = 1.08

and

%,¯š‚

%,¸‚ =284

540 = 0.53

Even though resistance is accurately predicted, we notice that the scatter on the stiffness prediction is much larger. This could be due to a limitation of the CBFEM which, probably, does not account for the beneficial effect provided by the bolts preloading. The failure mode, instead, is well predicted and in both methodologies results to be a mode-1, namely plate failure due to the formation of a couple of plastic hinges.

(48)

Fig. 43 – FE results in terms of plastic strains

Fig. 44 – FE results in terms of equivalent stress

(49)

Worked Example 2

Calculate strength and stiffness of the isolated bolted T-stub described in figure.

Fig. 45 – T-stub to be checked

The design parameters of the T-stub are the following:

( = 9 − 0.8$ = 60 − 0.8 × 10 = 52((

* = 45 (( (45 ≤ 1.25( = 65) X = 245 ((

8R = 800 ´µ 8„'= 275 ´µ

T-stub resistance

The possible yield-line patterns are equal to those typical of an isolated T-stub with two bolts only, namely circular pattern, non-circular pattern and beam pattern. The effective length, making reference to a single row of two bolts is equal to:

Circular pattern:

5 = 2a( = 326.6 ((

Section 3 - Notes on the modelling of bolted end-plate joints through the component method

Contents