1-s2.0-S0143974X0700017X-main

Journal of Constructional Steel Research 63 (2007) 1603–1615www.elsevier.com/locate/jcsr

Analysis of a bolted T-stub strengthened by backing-plates with regardto Eurocode 3

Z. Al-Khataba, A. Bouchaırb,∗

a Aleppo University, Faculty of Mechanical Engineering, Syriab CUST-LGC, Blaise Pascal University, Civil Engineering, Rue des Meuniers, 63174 Aubiere-cedex, France

Received 21 July 2006; accepted 31 January 2007

Abstract

The connection reinforcement by backing-plates, which is of low cost and simple to realize, can give performances which get closer tothat of stiffeners in the tension zone. Currently, in the Eurocode 3, this reinforcement mode is considered only for resistance. The T-stub, amain component in the bolted connections under bending, is analysed at first in the non-reinforced stage as a calibration of the finite elementsmodelling approach. Then, the T-stub is reinforced by backing-plates with various geometrical configurations. The aim is to better understandthe behaviour and the mechanical contribution of backing-plates considering the stiffness and the resistance. The 3D model developed in thefinite elements software Cast3m takes into account elastic plastic behaviour of the materials, the large displacement and the unilateral contactbetween the connected parts. This model shows a good capacity to represent the available experimental results. The sensitivity analysis of T-stubsreinforced by backing-plates to various parameters such as the thickness of the backing-plate and the pre-tension force allows us to show that thecontribution of the backing-plate in term of stiffness is significant in some cases and in particular with preloaded bolts. The T-stub behaviour isanalysed on the basis of the global load–displacement curves, the evolution of the bolt load and the contact pressure due to the prying force effect.The analysis shows that the resistance of the T-stub increases in a significant way with the thickness of backing-plate. This increase has an upperlimit due to the failure mode change which occurs for lower values of loads according to the analytical model, which gives a reserve of resistancefor the reinforced T-stub.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Backing-plates; Finite element model; Non-linear analysis; Prying force; Resistance; Stiffness; Steel connection

1. Introduction

The bolted end-plate beam to column connections (Fig. 1)are widely used in steel structures. They are essentiallyloaded by bending moment and shear force. They show acomplex mechanical behaviour, because the load transfer isdone partially by the contact between the connected parts(bolts, plates and I or H rolled sections). Their mechanicalbehaviour, represented by a non-linear moment–rotation curve,can be described by three main parameters which are stiffness,resistance and rotation capacity. The non-linear behaviour isdue generally to the evolution of the contact areas, to the plasticbehaviour of materials and to the large displacements. Thereare several ways to define this moment–rotation curve but the

∗ Corresponding author. Tel.: +33 473407532; fax: +33 473407494.E-mail address: [email protected] (A. Bouchaır).

0143-974X/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2007.01.012

most common are: the empirical approaches based on tests, thesophisticated numerical methods and the analytical methods.These latter, used by the Eurocode 3 (EC3) [1,2], give a verygood compromise between the simplicity of use, the precisionand the general character.

The component method proposed in the EC3 consists indecomposing the bolted beam to column connection into itselementary components, among which the more important isthe equivalent T-stub in the tension zone (column flange andend-plate in bending). The other components to be consideredin tension zone are the column web and the beam web intransverse tension and the bolts in tension. These componentsare to be assembled with the column web panel in shear, thecolumn web in transverse compression and the beam flangeand web in compression. These components are characterizedin terms of resistance, stiffness and ductility. The stiffnessand the resistance are well defined in the Eurocode 3 for a

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1604 Z. Al-Khatab, A. Bouchaır / Journal of Constructional Steel Research 63 (2007) 1603–1615

Fig. 1. Column flanges reinforced in the tension zone by backing-plates or by welded stiffeners and definition of the T-stub.

large family of connections. They characterize the semi-rigidbehaviour of the connections where the initial stiffness is aparameter to be used for the structural analysis. The jointstiffness is compared to that of the connected beam, to considerwhether the connection is rigid, semi-rigid or simple. However,the ductility is used to define whether a plastic redistributionis possible in the connection. If one of the components in theconnection has a brittle behaviour, the resistance is to be limitedto the elastic one. The ductility is more difficult to characterizein comparison to the stiffness or the resistance.

The basic properties of the components are then assembledaccording to suitable rules to supply the equivalent characteris-tics of the whole connection and to give also useful indicationson the more influential component.

The T-stub can be considered as an elementary connectionin itself. It is one of the main components of a beam to columnbolted joint when the structural elements are I or H hot rolledbeams or welded plate girder [2]. It represents in that casethe tension zone of such a joint (Fig. 1(b)). The behaviourof the elementary T-stub was widely studied by experimental[3], analytical [4,5] or numerical [6] approaches. However,the T-stubs reinforced by backing-plates were studied onlyexperimentally with a limited number of tests [7–9]. This is whythe numerical study is developed and calibrated on the basis ofthese experimental results.

In the beam to column connection, the local reinforcementof the column flanges is sometimes necessary to improve theconnection properties. Traditionally, it is obtained by the useof transverse welded stiffeners on the column web and flanges(Fig. 1(a)). However, tests showed that the reinforcement bybacking-plates bolted on the internal face of the column flangesand covering at least two rows of bolts in the tension zone(Fig. 1(b)) can be an interesting alternative to the weldedstiffeners [7–9] as well in terms of the initial stiffness as of theresistance for the T-stub alone and as a consequence for thewhole beam to column connection.

The EC3 covers the T-stubs reinforced by backing-platesthrough formulae of resistance similar to those of theelementary T-stubs. These formulae admit implicitly that theT-stub flange and its connected backing-plate behave as platessimply superposed. The European standard prescribes theminimal values of the length and the width of backing-plates.However, it does not consider the increase of the T-stub stiffnessdue to the use of backing-plates that tests, although limited,show clear evidence, in particular with preloaded bolts [9].

The present study aims to analyse more precisely thebehaviour of these T-stubs using the finite element numericalmodelling. Such a modelling is calibrated in advance, bycomparing results obtained by means of 2D and 3D models[10,11] to the experimental results for elementary T-stubs(not reinforced) [3,12,13]. It is then used to make aparametrical study and to observe the evolution, during theload until the failure, of various parameters difficult to measureexperimentally, such as, for example, the prying forces and theevolution of the contact area under the T-stub flange. Also, themodel is used to make an evaluation of the Eurocode 3 formulaeapplied to the resistance and the stiffness of elementaryT-stubs with available experimental results to calibrate themodel. Afterwards, the model is applied to T-stubs strengthenedby backing-plates to analyse the effect of the backing-platethickness and the bolts preload.

2. Analytical approach

In practice, according to the EC3, the equivalence betweenthe T-stub and the tension zone in the joint is realized by thedefinition of an effective length for each bolt row which is basedon the plastic mechanisms. This effective length is used for thecalculation of the strength and the stiffness of the T-stub.

In this paper, the EC3’s formulae are used to determine the“design” characteristics of resistance and stiffness of the T-stub.They are then compared to the results given by the numericalmodel described before.

2.1. Strength of the T-stub according to the EC3

In the EC3, the resistance of the T-stub is calculated usingthe simple plastic analysis (yield analysis) of a T-stub in 2D forwhich three possible failure modes are distinguished accordingto the value of the ratio between the flange resistance in bendingand that of the bolt in tension (Fig. 2):

- The first failure mode, associated with the failure of theflange by forming of plastic hinges in the flange (web–flangejunction and bolts row axis).

- The second failure mode, corresponding to the appearanceof one plastic hinge in the flange and to the failure of thebolts in tension.

- The third failure mode, characterized by the failure of thebolts in tension.

Z. Al-Khatab, A. Bouchaır / Journal of Constructional Steel Research 63 (2007) 1603–1615 1605

Fig. 2. T-stub failure modes according to EC3.

The resistance of the T-stub is given by the weakest of theresistance values associated with the three failure modes abovementioned.

The failure mode 1 is of ductile nature because it involvesthe appearance of a large plastic deformation in the flangeunder bending and that the steel used in the construction usuallypresents a large ductility. This allows a large redistribution ofthe loads between the bolts rows represented by the T-stubsdefined in the whole connection.

It will be noticed that the backing-plate contribution to theT-stub resistance concerns only the T-stubs with failure mode 1,because this mode is the only one which mobilizes significantlythe backing-plate bending. If the other failure modes aredeterminant, backing-plate does not give sensitive improvementof the resistance. To find the formulae for the T-stub resistancewithout backing-plate, it is sufficient to eliminate the termsrelated to the backing-plate for the failure mode 1.

The head of the bolt, the nut and the washers have non-negligible diameters in comparison with the width of T-stubflange and the distance of the bolt to the median axis of theT-stub. As a result the loads are not transmitted on aconcentrated zone to the flange but by way of an unknowncontact area between the bolt (or the washer) and the flange.This effect is considered with a formula of resistance called“alternative” which is based on the hypothesis of uniformdistribution of stresses under the head of the bolt and the nut(or the washers). So, the resistance of the T-stub with the failuremode 1 is modified as indicated in Eq. (1). The other failuremodes (2 and 3) remain unchanged (Eqs. (2) and (3)).

Mode 1: FT,Rd,1 =4MP + 2Mbp

mor

FT,Rd,1 =(32n − 2dw) MP + 16·n·Mbp

8mn − dw (m + n)(alternative) (1)

Mode 2: FT,Rd,2 =2MP + n

∑Bt,Rd

m + n(2)

Mode 3: FT,Rd,3 =

∑Bt,Rd (3)

with:

MP and Mbp: plastic moments in the T-stub flange and thebacking-plate;∑

Bt,Rd : resistance of the bolts in tension in the T-stub;n et m: dimensions defining the bolt positions (Fig. 3);dw: diameter of the contact area between the bolt head and

the flange.

Fig. 3a. Initial model for the T-stub stiffness calculation.

2.2. Stiffness of the T-stub according to EC3

The stiffness of a T-stub is defined by likening the behaviourof this T-stub to that of a beam. In a beam–column connection,a first T-stub, representing the column flange in bending,is connected to a second T-stub with the same bolts row,representing the end-plate in bending, connected to the firstT-stub by bolts in tension. The contributions of these twoT-stubs in bending and the bolts in tension are assembledin series with their associated coefficients of stiffness. EveryT-stub is modelled as a continuous beam on four supports. Twoof these supports are elastic supports representing the boltsbearing loads considered as applied in their axis (Fig. 3a). Inthe analytical model used in the Eurocode 3, the interactionbetween the T-stub and the bolts is not considered directly,because the load in the bolt is taken equal to the applied one,while it depends in reality on the ratio between the flange andthe bolt stiffness. To take into account this effect, a factor equalto 1.6 is applied to the stiffness of the bolts. This stiffness isobtained by considering equivalence between the initial model(Fig. 3a) and the simplified one (Fig. 3b) with n = 1.25m. Thismeans taking into account a prying force equal to 14% of theload applied to the T-stub (Q/F = 14%), which is equivalentto an increase of the load in the bolt by 28%.

So, the elastic stiffness (Kini) of two T-stubs connected bybolts is based on the elementary stiffness of the end-plate (kep),that of the column flange (kcf ) and that of the bolt (kb). It isgiven by the following relationship:

Kini =E

1kcf

+1

kep+

1kb

(4)

where E is the modulus of elasticity of steel. According to thesimplified model of the EC3 (Fig. 3b), the stiffnesses of the twoT-stubs, with one row of two bolts in a bolted connection, aregiven by the following expressions:

kcf = 0.852·Leff·t3

cf

m3 .

(3m + n3m + 4n

),


Fig. 3b. Simplified model for the stiffness calculation of the T-stub (Eurocode 3).

Fig. 4. T-stubs studied (two rows of bolts with backing-plates).

kep = 0.852·Leff·t3

ep

m3

(3m + n

3m + 4n

)and kb = 1.6

As

Lb. (5)

In these expressions giving the stiffnesses of the T-stubs (kcfand kep), the formulae of the EC3 given below can be found ifn is replaced by 1.25m:

kcf = 0.85Leff·t3

cf

m3 , kep = 0.85Leff·t3

ep

m3 (6)

where: Leff is the effective length of the flange defined forthe resistance and the 0.85 is a coefficient of reduction forthe stiffness, tcf is the flange thickness, tep is the end-platethickness, m is the distance defined in Fig. 3, As is the tensilestress area of the bolt and Lb its conventional length defined inthe EC3 by:

Lb =

∑t + 2·tw +

12(th + tn) (7)

with: tw, th and tn the thicknesses of the washer, the head ofthe bolt and the nut and

∑t the sum of the thicknesses of the

assembled plates.

3. Numerical modelling

In this paper, the finite element modelling is used torepresent the behaviour of various T-stubs. It allows a goodrepresentation of the two sources of non-linearities: The steelplastification and the evolution of the contact area. In agreementwith the principles of the EC3, two symmetric T-stubs with oneor two rows of bolts are examined. The load is applied by the

controlled displacement so as to better control the evolution ofthe system. For the material characteristic law, the conventionalstress–strain curve is used (σ–ε) defined from the coupontensile tests. No significant modification was found if the realupdated curve is used (true stresses) [12]. The limit and theyield plastic laws are governed by the Von Mises criterionwith isotropic strain hardening. The results obtained by thenumerical modelling are compared to the experimental resultsto check and to calibrate the approach used in the numericalmodelling before continuing the parametrical studies.

2D and 3D finite elements models are applied to fivespecimens with available experimental results, consisting ofelementary T-stubs (not strengthened by backing-plates) ofvarious geometrical configurations. The results consideringthe global load–displacement curves are used to calibrate themodel without backing-plates [10,11]. Then, the FEM (FiniteElement Model) thus calibrated is applied to the same T-stubsbut with the flange strengthened by backing-plates. The globalbehaviour of these reinforced T-stubs is analysed to understandand to quantify their sensitivity to the effects of the backing-plate thickness and the pre-tension load in the bolt.

3.1. Experimental results: Geometrical and material data

The studied T-stubs are symmetrical with two rows of twobolts (T1, P1K, P2K) [3,12], or one row of two bolts (T4, T7)[13] (Fig. 4). This choice is justified by the fact that the T-stubs were designed to present a failure mode 1, the onlyone, as indicated before, able to be influenced by the presenceof the backing-plate. Besides, these T-stubs have different


(a) Whole T-stub model. (b) T-stub model. (c) T-stub model — backing-plate.

Fig. 5. A 2D model of the elementary T-stub or strengthened by backing-plates.

Table 1Characteristics of T-stubs used in the modelling

Test T-stub origin fy (MPa) Bolts Dimensions (mm) Fpret. (kN)P w L b e e1

T1IPE300t f = 10.7

4312*2*M12

40 90 80 150 30 20 –8.8

P1K270

2*2*M16 55 100 105 150 25 25 30P2K 10.9 90 100 210 150 25 60 30

T4HEA260t f = 12.5

3001*2*M24

143.8 118.8 260 58.1 59.4 –8.8

T7HEA300t f = 14

3081*2*M27

156.6 122.8 300 71.7 61.4 –8.8

geometrical configurations, which allows us to check the effectsof the T-stub length and the position of bolts represented bythe parameters (L , P, e) (Fig. 4). Detailed data related to theconcerned specimens are given in Table 1.

3.2. Description and calibration of the numerical model

The finite element type used in the modelling was selectedon the basis of the modelling of simply supported beam underbending by means of various types of linear and quadraticelements in 2D and 3D. On the basis of the comparative study ofthe obtained results, finite elements with quadratic formulationare used in bending considering two elements on the thicknessof each plate (flange and backing-plate). Different densities ofmeshing were tested; those selected for the continuation of thestudy guarantee a good compromise between the size of theelements and the stability of the numerical solution [14].

The contact areas represent an important part of theboundary conditions for the numerical study of the T-stubmodelled in 2D and 3D (Figs. 5 and 6). The calibration ofthe contact for the two models is realized on the basis of aHertz contact example for which the analytical solution is wellknown. Lagrange’s multipliers are used to model the unilateralcontact without friction in all the contact areas situated betweenthe head of the bolt and the backing-plate or the flange surface,between the backing-plate and the flange and between theflange and the symmetrical part (foundation). The friction hasa large effect on the behaviour of the T-stub (particularly with

large displacements or pre-tension). In this study, the frictionis not considered to have a comparison with the EC3 formulawhich is considering two simply superposed plates (backing-plate and column flange). In fact, the case without friction ismore conservative regarding the mechanical characteristics andis less demanding for erection. However, the 3D model requiresa particular adaptation. In fact, even for an uniform appliedload on the finite element face CU20 (quadratic with 20 nodes),the distribution of the nodal forces is not uniform between thecorner and the middle nodes (different signs) [14]. To avoidthis situation, it is easy to introduce a supplementary conditionon the nodal displacements between the middle node and thecorner nodes on all the border of all the elements situated onthe surfaces in contact.

The examined T-stubs have two plans of symmetry (T4 andT7) or three plans of symmetry (T1, P1K and P2K). Thisallows us to model only a quarter of the T-stub, with onerow of two bolts, in 2D or in 3D. However, it is necessaryto introduce the suitable boundary conditions on the segmentsof the circumference corresponding to the axes of symmetry.The 2D model with plan stresses presents a simple solution butleads to a simplification of the geometry for the bolts and theholes. So, different thicknesses are considered for the T-stubweb, the flange, the holes and the bolts [15]. The bolt shank isnot superposed geometrically to the hole and the bolt head is inunilateral contact with the flange (or the backing-plate). Finiteelements with eight nodes (QUA8) are used in the analysis. Itwill be noticed that the case of the whole T-stub (Fig. 5(a))


(a) Whole model for a non-reinforcedT-stub.

(b) Short T-stub with backing-plate. (c) Long T-stub with backing-plate.

Fig. 6. 3D models for the T-stub: A whole model with double T-stub, short and long T-stubs with backing-plates (XOY: U z = 0, YOZ: U x = 0 and ZOX = contact).

was also considered but it shows few differences in comparisonwith the “reduced” case (Fig. 5(b) and (c)). Thus, this simplifiedmodel is chosen for the 2D modelling. The models chosen andcalibrated for the elementary T-stub (not strengthened) are usedto study the T-stub provided with backing-plates.

The backing-plates are of rectangular shape, with the samesteel as the strengthened flange and have minimal dimensions(length and width) given by the EC3. As written above, theparametrical study concerned the effects of the thickness ofbacking-plates and the pre-tension load in the bolts on thestiffness and the resistance of the T-stub. The same approachwas used for all the analysed T-stubs, without or with backing-plates. To avoid a premature plastification of the web in theT-stub, the finite elements of the web are considered with ahigh fictitious elastic limit. This allows the concentration of thestudy on the part of the T-stub considered according to the EC3approach (flange, backing-plate and bolts).

As for the elementary T-stubs, the T-stubs with backing-plates are characterized by two or three plans of symmetry.This allows us to model only a quarter of the T-stub with onerow of two bolts. The unilateral contact is considered under theflange, between the flange and the backing-plate and betweenthe backing-plate and the washer under the bolt head. For moresimplification, the head of the bolt is connected perfectly to thewasher. This reduces the contact management to the interfacebetween the washer and the backing-plate or the flange. Thishypothesis is acceptable regarding the stiffness ratio betweenthe bolt head and the washer. A comparative study showed thatthe effect is very small [14]. The flexibility of the bolt due tothe contact between the threading and the nut can be taken intoaccount by a fictitious length larger than the real one (Eq. (7))(Fig. 5(c)) or by a diameter lower than the real one (Fig. 5(a)and (b)).

The 3D model takes into account the transverse deforma-bility of the T-stub by considering the third dimension. As for

Fig. 7. Definition of the initial stiffness and the plastic resistance.

the 2D model, the case of a quarter of the T-stub (Fig. 6(b))is studied then compared with the result of the whole T-stub(Fig. 6(a)). The simplified models represented on Fig. 6(b) and(c) are used for all the calculations made afterward. For the sim-plified model, the shank of the bolt is considered as a cylinderof constant nominal diameter by taking an equivalent lengthwhich takes into account the low rigidity of the threaded zoneand that due to the flexibility of the nut. So, the contact condi-tions under the flange and the boundary conditions on the boltshank are situated in different plans. So, as for the 2D model,a double T-stub with a complete bolt is considered (Fig. 6(a)).The bolt is modelled with a variable section on its length totake into account the difference between the gross section andthe threaded part and to obtain a stiffness value close to the re-ality. Finite elements with twenty nodes (CU20) are used in the3D analysis because the solid elements with eight nodes (lin-ear formulation) are too stiff and have to be used with a highnumber of elements to obtain acceptable results in bending.

4. Analysis of the results

The results of finite element analyses are based mainlyon the global load–displacement curves (F–δ). From thesecurves, the plastic resistance (Fp) and the initial stiffness(Kini) are defined (Fig. 7). The plastic resistance is usually


Fig. 8. Global (F–δ) curves for a short T-stub (T1).

Fig. 9. Global (F–δ) curves for a long T-stub (P2K).

obtained by the intersection between the two lines tangent tothe load–displacement curve at the initial and the final stages.The various numerical results are compared to the experimentalresults on the basis of the global curves. Also, the evolution ofthe prying forces, which are difficult to measure experimentally,is drawn versus the applied load. They are used to observe theevolution of the load in the bolt, which serves for defining thelimit between the failure modes 1 and 2.

4.1. Comparison of the results between 2D and 3D models

The short T-stubs have rather a two-dimensional behaviour.They can be efficiently represented by a 2D model whereas thelong T-stubs require the use of a 3D model. To illustrate theinfluence of the size of the T-stub and the type of modelling,comparisons are done between two T-stubs where one is short(T1 or T4) and the other is long (P2K).

For a short T-stub, the curves (F–δ) from the 2D modelare in good agreement in term of initial stiffness with thecurve obtained experimentally. However, the resistance is alittle underestimated. The 3D model is also satisfactory forthe initial stiffness and it agrees better with the experimentalcurve at the final ultimate phase. Besides, it can be observed anexcellent agreement between the numerical curves (F–δ) with3D model obtained using the Cast3m software package [16] andthose obtained using another finite element code (Lagamine)[6] (Fig. 8). For a long T-stub, only the 3D model is in goodagreement with experimental results (Fig. 9).

Fig. 10. Evolution of the load in the bolt (T-stub T1): Comparison of the 2Dand 3D models.

4.2. Evolution of the bolt load

The evolution of the real load in the bolt has to be knownbecause it can influence the resistance of the T-stub and,in particular, influence the failure mode really observed. Itincludes the prying effect. The curves representing the bolt loadevolution versus the global load applied to the T-stub showthat, while presenting comparable shapes, it remains alwayshigher with the 2D model than with the 3D model (Fig. 10).As a consequence, the plastification of the bolt arises earlierif the 2D model is used. The Fig. 10 shows the case of ashort T-stub where the prying force increases the load in thebolt, considering the applied load to the T-stub, by more than40%, while the value used implicitly in the simplified analyticalmodel of the EC3 is about 28%. The long T-stubs show similarevolutions [14]. On the curves representing the evolution of theload in the bolt versus the applied load, three phases can bedistinguished: elastic phase, phase with partial plastification ofthe flange and the phase of the bolt plastification.

4.3. Effect of the backing-plate thickness

The effect of backing-plate is observed for various values ofthe ratio (tbp/t f ), and the thickness of the flange is maintainedconstant. This ratio varies between 0.4 and a maximum fromwhich the failure mode of the T-stub changes.

The results are observed, on one hand, on the basis ofthe global load–displacement curve (F–δ) which allows thedetermination of the resistance and the stiffness of the T-stubto be compared to the design values. On the other hand, theevolution of the load in the bolt is observed. It allows theanalysis of the evolution of the failure mode according to thedefinition used by the EC3 (modes 1–3). The prying force istaken into account because its influence is very important onthe bolt loads.

The load–displacement curves (Figs. 11 and 12) show thatthe thickness of the backing-plate has a large influence on theglobal behaviour of the T-stub, and in particular the resistance.This resistance increases in a relatively important way untilthe level from which the failure moves into the bolts. Thecurves show that the thickness of the backing-plates has a moreimportant effect on the resistance than on the stiffness for the


Fig. 11. Global load–displacement curves (F–δ) with different backing-platesthicknesses (long T-stub P2K) — 3D model.

Fig. 12. Global load–displacement curves (F–δ) with different backing-platesthicknesses (T-stub T4) — 2D and 3D models.

considered long and short T-stubs. Furthermore, Fig. 12 showsthat the 2D model, although satisfactory for the elementaryT-stub, is less efficient for the case with backing-plates andparticularly for the limit from which the failure mode changes.The evolution of the global curves (F–δ) shows a maximalratio, of the backing-plate to the flange thickness, smaller in2D model because the bolt has to carry a higher load.

4.4. Contact pressure due to prying action and deformationpattern

In the T-stub, prying forces develop to balance the bendingof the flange. The value of these forces and the position of their

resultant depend on the contact area between the two equivalentT-stubs representing the column flange and the end-plate. Theresultant of the prying forces and its position are not constantsand their evolution depends on the applied load.

As qualitative information, the distribution of the normalpressure of contact under the flange and the deformation of theT-stub, are given for the cases without and with backing-plates.In both cases, the shape of the contact pressure distributionunder the flange shows the same tendency. The contact pressureis uniformly distributed on all the length of the short T-stub(T1) (Fig. 13) and on a shorter zone for the long T-stub (P2K)(Fig. 14). However, for the two T-stubs, maximal pressure ofcontact for the same applied load F , equal to 100 kN for T1 andto 80 kN for P2K, is greatly higher for the cases with backing-plates (tbp/t f = 1).

The deformed shape of the short T-stub flange shows auniform deformation on the length [14] while the long T-stub(P2K), shows a relatively complex shape of deformation. In thislast case, the contact areas are greatly dependent on how theT-stub is or not provided with backing-plates (Fig. 15). It seemsclearly that the backing-plate makes the flange more stiff for thesame level of load (F = 280 kN), which is the ultimate load ofthe T-stub without backing-plates. This ultimate load is takenequal to that relative to an imposed displacement of 10 mm.

The deformed shape of the T-stub flange without and withbacking-plates, under their ultimate load chosen equal to theimposed displacement δ of 10 mm, shows a wider area ofcontact for the case with backing-plates (Fig. 16).

4.5. Comparison of resistance between EC3 and FEM

The analysis of the increase of resistance due to the backing-plates is based on the comparison between the results givenby the finite element modelling and those of the analyticalformulae given by the EC3. The modelling values are takenfrom the global load–displacement curves (F–δ) showed before(Fig. 11).

The values given by the analytical formulae of the EC3 werecalculated on the basis of the nominal values of geometricalcharacteristics and the real values (measured) of the mechanicalcharacteristics. The effective length used in these formulae forone row of bolts was taken equal to the weakest of the values

Fig. 13. Distribution of the contact pressure under the flange (T1): Without and with backing-plates (tbp/t f = 1).


Fig. 14. Distribution of the contact pressure under the flange (P2K): Without and with backing-plates (tbp/t f = 1).

Fig. 15. Deformed shapes of the long T-stub flange (P2K): The same load (F = 280 kN).

Fig. 16. Deformed shapes of the long T-stub flange (P2K): Ultimate loads (δ = 10 mm).

given by the EC3’s formulae for the individual mechanism andthe mechanism of group. Thus, for the T-stubs T1, P1K andP2K, the effective length Leff, governed by a mechanism ofgroup (non-circular patterns), is equal to P (T1 and P2K) orto (e1 + 0.5P). Table 2 gives the resistances for the T-stubswithout backing-plates obtained according to:

- the finite element numerical modelling;- the two formulae proposed by the Eurocode 3 for the failure

mode 1, which should be the only one to take a benefit fromthe presence of backing-plates.

Also, the values of the effective length used in the analyticalformulae and different ratios of resistances are given in Table 2.


Table 2Resistances of the elementary T-stubs (FE model and EC3 formulae)

T-stub T1 P1K P2K T4 T7∑leff 80 105 180 118.8 122.8

Fp (kN)FEM 185 150 230 185 230EC3-simple 134 94 161 114 141EC3-alternative 157 115 196 135 167 Mean SD

FEM/EC3-simple 1.38 1.60 1.43 1.62 1.63 1.53 0.12FEM/EC3-alternative 1.18 1.30 1.17 1.37 1.38 1.28 0.10

Table 3aResistances of the T-stubs: The backing-plate and the flange have the samethickness

T-stub T1 P1K P2K T4 T7

tbp/t f 1 1 1 1 1

Fp (kN)FEM 225 190 310 255 310EC3-simple 201* 141 242 170 212EC3-alternative 239* 176 302 205 254 Mean SD


It can be observed that the formula called “alternative” givesvalues higher by about 20% if compared to those given by the“simplified” formula. The ratio between the resistances givenby the modelling and those given by the formulae of the EC3is on average equal to 1.28 for the “alternative” formula and to1.53 for the “simplified” formula (Table 2).

For the T-stubs with backing-plates, two cases areconsidered for the comparison: (a) the thickness of the backing-plate is equal to the thickness of the flange (tbp/t f = 1), and (b)the thickness for which the failure mode 1 is not dominant. Inview of the results given by the finite element modelling, andin comparison with the elementary T-stubs (without backing-plates), the case (a) seems to give an increase of the plasticresistance (Fp) of 20%–40% (Tables 2 and 3a) and the case(b) gives an increase of 30%–140% (Tables 2 and 3b). Thecomparison between the results given by the modelling andthose given by the analytical formulae of the Eurocode 3 isgiven in Tables 3a and 3b. The comparison of the results ofthe modelling to those of the “alternative” formula shows thatthis one gives satisfactory results for the case (a) and lowerresults for the case (b). These results lead us to think that thereinforcement by backing-plates is interesting from the pointof view of the resistance. However, the analytical formulae,and in particular the “alternative” formula, should be used withcare for large thicknesses of backing-plates. This has to bemoderated by the fact that the analytical formulae “detect”the transition to the failure 2 earlier than the calculation byfinite elements, and in the two cases, the failure mode 3 doesnot occur (Table 4). The thicknesses of backing-plates usedare higher than those given by the analytical formulae of theEC3. The values with (*) (Tables 3a and 3b) correspond tothose for which the failure mode 2 according to the analyticalformulae is already reached. However, the values are kept forthe coherence of the presentation. So, the formulae of the EC3remain conservative in comparison with the finite elementsresults.

Table 3bResistances of the T-stubs: The backing-plate thickness is the maximum for thefailure mode 1 (according to the FEM)


tbp/t f 1.1 2.4 1.8 2.2 2.4

Fp (kN)FEM 235 315 390 430 550EC3-simple 215* 366* 423* 389* 548*EC3-alternative 256* 468* 537* 475* 669* Mean SD


Table 4Resistance of the T-stubs with the failure modes 2 and 3


Fp EC3 (kN) — mode 2 182 301 321 362 495(Fp–FEM/Fp–EC3 — mode 2) 1.29 1.05 1.21 1.19 1.11Fp EC3 (kN) — mode 3 296 651 651 619 805

Table 5Resistances of the T-stubs with a thickness of “transition” for the backing-plate


(tbp/t f ) max. — EC3 0.6 1.7 1.1 1.8 1.9Fp–FEM (kN) 200 270 320 365 500Fp – EC3 — alternative (kN) 182 301 320 362 495Fp EC3/Fp FEM 0.91 1.12 1 0.99 0.99

In the standard approach, the field of application of the“analytical” formula is determined by the maximum thicknessof backing-plate from which the failure mode changes from themode 1 to the mode 2 (Table 5). To obtain the expression of thisthickness called “of transition”, it is possible to use the equalitybetween the expressions giving the resistances for the failuremodes 1 and 2. The resistance for the failure mode 1 is FT,Rd,1(alternative method) (Eq. (1)) and that for the failure mode 2is FT,Rd,2 (Eq. (2)). This transition thickness is given by thefollowing formula (Eq. (8)).

tbp =

√1

m + n·

{[2mn − dw (m + n)] · (0.9·nb·As · fub)

Leff· fy− t2

f (m + 2n)

}.

(8)

For the T-stubs examined in the present paper, the values ofthe thickness “of transition” determined according to Eq. (8)are indicated in the first line of Table 5. It shows that the ratiostbp/t f from Table 3b are situated outside the field of applicationresumed in Table 5. The values of resistances given by the finiteelement model and the analytical values calculated again withthe new backing-plates thicknesses are shown in Table 5. Thereis noticed a good agreement between these two types of results.A difference remains for the short T-stub T1 which seems notable to be reinforced because it changes the failure mode witha low value of the backing-plate thickness.

To better see the evolution of the failure modes accordingto the relative thickness of backing-plates [tbp/t f ], the resultsof the modelling are compared to those of the EC3 in termsof ratio of resistances [Fp bp/Fp] of the T-stubs with and


Fig. 17. Ratios of the resistances associated with failure mode 1 for the T-stubswith and without backing-plates (P2K and T4).

without backing-plates (Fig. 17). This same ratio, calculatedon the basis of the analytical formulae of the EC3, has highervalues because the European standard leads to resistance valuesof the T-stub without backing-plates significantly lower thanthe modelling. This allows the conclusion that the calculationaccording to the EC3 detects the failure mode 2 before thefinite element modelling. As a consequence, the connectionscalculated with a failure mode 1 according to the EC3 still havea reserve of resistance before developing really the failure mode2 which is less ductile.

4.6. Effect of the backing-plates on the stiffness

The formulae of the EC3 do not offer the possibility to takeinto account a possible effect of the backing-plate on the initialstiffness of a T-stub. This lets us understand that this effect is atleast negligible. To examine this possible effect, a comparisonis made between the values of the stiffness given by the finiteelement modelling and those calculated according to the EC3.

The values of the initial stiffness given by the numericalanalysis are determined from the global curves F–δ byconsidering the initial slope for a value of displacement equalto 0.3 mm which corresponds to the elastic phase of the globalbehaviour. The results of the finite element modelling show thatthe addition of backing-plates produce a small modification ofthe initial stiffness although it is associated with an increase ofthe bolt length and by a consequence with a modification ofthe ratio between the stiffness of the flange and the bolt. So,for a backing-plate of a thickness equal to that of the flange,the increase of the stiffness does not exceed 15% (Table 6a).However, for backing-plates with the thickness “of transition”,the increase of the stiffness can reach 60% for the T-stubs T4and T7 (Table 6b). The curves related to the analysed T-stubs(Fig. 18) show the evolution of the ratio of stiffness betweenthe cases with and without backing-plates [Kini bp/Kini nobp]

versus the relative thickness of the backing-plates [tbp/t f ].There can be shown a small evolution of the stiffness fora relative thickness of the backing-plate equal to unity, butthe difference becomes more marked with the increase of therelative thickness. This is true for the long and short T-stubs.

We examine now the effect of the bolts pre-tension on themechanical behaviour of the T-stub with backing-plates. For all

Table 6aInitial stiffness of the T-stubs without and with backing-plates (tbp/t f = 1)

T1 P1K P2K T4 T7

Kini nobp EC3 (kN/mm) 155 156 247 176 198Kini nobp FEM (kN/mm) 164 176 217 121 141tbp/t f 1 1 1 1 1Kini bp FEM (kN/mm) 178 195 236 141 162∆ FEM (bp/nobp) (%) 8.5 10.8 8.7 16.5 14.9

Table 6bInitial stiffness of the T-stubs without and with backing-plates (tbp/t f “oftransition”)

T1 P1K P2K T4 T7

Kini nobp EC3 (kN/mm) 155 156 247 176 198Kini nobp FEM (kN/mm) 164 176 217 121 141tbp/t f 0.6 1.7 1.1 1.8 1.9Kini bp FEM (kN/mm) 166 226 244 191 228∆ FEM (bp/nobp) (%) 1 28.4 12.4 57.8 61.7

Fig. 18. Evolution of the initial stiffness ratio between the T-stubs withand without backing-plates versus the relative thickness of the backing-plate(numerical modelling).

the examined T-stubs, this pre-tension load is taken equal to thevalue specified in the EC3:

Fp.cd = 0.7 · fub · As . (9)

To introduce the pre-tension into the numerical model, weproceed as follows. At first, the T-stub is taken without externalload, and a displacement is applied on the bolt shank to producethe pre-tension and to calibrate the relation between the pre-tension and the bolt elongation in the T-stub configuration.Then, this displacement is applied to the bolt shank beforethe introduction of the external load applied to the T-stubweb. The results of various simulations are compared on thebasis of the global curves (F–δ) (Fig. 19). There is noticedan evident influence of the pre-tension in the elastic phase.The initial stiffness is significantly increased with the pre-tension and it is amplified with the presence of backing-platesbecause the pre-tension enhances more the contact betweenthe plates even without friction. In the phase close to theultimate load, the pre-tension loses gradually its effect, as itwas expected. However, for the plastic resistance (Fp), it keepsa non-negligible influence. The friction has a non-negligible


Fig. 19. Global load–displacement curves F–δ (T4) : (tbp/t f = 1.8).

Table 7Initial stiffness for T-stubs without and with backing-plates (thickness “oftransition”) and pre-tension


Fp.cd (kN) 57.5 126.6 126.6 240.7 312.9Kini nobp & pret. (kN/mm) 224.9 219.1 291.8 165.6 203.3(tbp/t f ) max EC3 0.6 1.7 1.1 1.8 1.9Kini bp & pret. (kN/mm) 237.8 423.5 384.1 328.8 414.9∆ (%) 5.7 93.3 31.6 98.5 104.1

Table 8Initial stiffness for T-stubs with backing-plates (thickness “of transition”) andpre-tension


Fp.cd (kN) 57.5 126.6 126.6 240.7 312.9(tbp/t f )max EC3 0.6 1.7 1.1 1.8 1.9Kini bp (kN/mm) 166 226 244 191 228Kini bp & pret. (kN/mm) 237.8 423.5 384.1 328.8 414.9∆ (%) 43.2 86.4 57.4 72.1 82

effect because it develops a higher equivalent moment ofinertia for the backing-plate and the column flange. However,we considered the case without friction to obtain the mostconservative values with a connection which does not requirea particular preparation of surface. Indeed, the backing-platescan be used as a reinforcement of existing structures. In realityeven without friction, the bolt pre-tension mobilizes a widersurface of contact between the backing-plate and the columnflange. Table 7 is established for the thickness “of transition”of backing-plates, according to the formulae of the Eurocode3, with bolts pre-tension given by Eq. (9). The results ofthe numerical model show that backing-plates produce a highincrease of the initial stiffness of the T-stub, if bolts are withpre-tension. This increase can reach 100% and it is due tothe contact zone which is larger than without pre-tension.This enhances the collaboration between the backing-plate andthe flange even without friction effect. The Table 8 showsthat for a T-stub with backing-plate, the pre-tension increasesthe initial stiffness in an important way for all the studiedT-stubs.

5. Conclusion

Different T-stubs were analysed using a finite element modelto observe their global behaviour as well as the evolution andthe distribution of the contact pressure, their deformed shapesand the loads in the bolts.

A comparison made between 2D and 3D models allowedus to define the limits of the first model, which is certainlyeasier to implement. The 2D model gives satisfactory resultsfor short T-stubs. For some geometries, only the 3D model withconsideration of the contact, the material and the geometricalnon-linearities allows a faithful representation of the T-stubbehaviour. This 3D model allows the description of variousphenomena which occur during the load application on theT-stub and gives information which is difficult to obtainexperimentally.

The 3D models applied to the T-stubs with backing-platesshow that the resistances increase with the thickness of thebacking-plate. However, as care was taken to exclude anyplastification of the web in the T-stub, this increase has anupper limit defined by the bolt failure. This corresponds to thefailure mode 2 which succeeds the failure mode 1 and allowsthe definition of a thickness “of transition” corresponding tothis situation. The analytical model of the EC3 gives a thickness“of transition” of backing-plate lower than that determined bythe finite element model. This shows that a reserve of resistanceand ductility exists if the resistances of T-stubs with backing-plates are calculated according to the EC3.

The analytical formula proposed by the EC3 applied tothe T-stubs with backing-plates gives satisfactory results forthe resistance in comparison with numerical results whichwere previously calibrated on the basis of the experimentalresults. So, numerical models showed that the contribution ofthe backing-plates is interesting regarding the stiffness for themajority of the tested T-stubs. This contribution is more markedif bolts with pre-tension are used. In that case, the backing-plates produce an increase of the plastic resistance and theinitial stiffness for all the tested T-stubs.

The results of the modelling will be useful for studyingmany other cases of strengthened T-stubs. They will be usedin particular in the development of an analytical model whichwould allow the consideration of the evolution of the stiffnessgiven by the backing-plates.

References

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