7 This work deals with the predict ion of the bearing resistance of elementary shear connections 8 constituted by gusset plate s and square hollow sections fastened with long bolts . The research is 9 motivated by the lack of specific rules in international codes for this particular configuration. In fact, 10 while in lap shear connections, single or double -sided, the holes are always confined out of plane , in 11 case of connections with long bolts and gusset plates (typical of tubular structures) , the holes are 12 constrained out of plane outwards , but they are free to buckle inward s, providing a reduction of the 13 bearing strength . Within this framework, t he paper is aimed at the development of an analytical 14 formulation for the prediction of the bearing resistance of connections with long bolts and square 15 hollow sections ( SHS ). The proposal is based on the calculation of the post -buckling load of the tube 16 plate subjected to be aring at the bolt hole , combining the Winter’s formula (originally developed to 17 predict the post -buckling strength of steel plates accounting for geometrical and mechanical 18 imperfections ) with a simplified FE model developed i n SAP 2000. The procedure deve loped is 19 presented in detail proposing a design equation for connections with long bolts , based on a formula 20 suggested recently by Može (2014 , 2016) suitable for the inclusion in code provisions such as EC3 . The 21 accuracy of the design equation is check ed on the experimental data reported in a previous work of 22 the same authors properly extended with parametric FE simulations to cope for variation of 23 geometrical parameters . 24 25 Keywords: Bearing resistance, Buckling , SHS, Shear, Model, Eurocode 3 26 -2- 1. Introduction 27 Bolte d connections are widely used in steelwork because they are easy to manufacture while 28 provid ing significant benefits in terms of constructability . In fact , they allow to split complex 29 structures in simpler parts which can be welded in the shop (assuring a high quality of welds) and 30 bolted on the construction site. Almost all bolted connections, in order to provide an appropriate 31 response , need to activate the typical mechanisms for the transfer of shear force s, thus requir ing to 32 check resistance accounting for different possible failure modes . In the most elementary type of 33 connections , plates are simply overlapped according to single -sided or double -sided lap shear 34 configurations , providing the load transfer through the activation of bearing stresses and shear forces 35 in plate and bolt. The bearing stress es develop in the area of the plate ahead of the bolt shank and, 36 usually, due to a combination of material strain hardening and stress triaxi ality, they achieve level s 37 much beyond the yield limit of steel [1-3]. In order to account for this triaxial stress state , promoted by 38 the lateral confinement of the bolt head or nut , currently, all the national and international codes 39 suggest to adop t an average value of the bearing stress which can achieve , depend ing on the level of 40 confinement , up to four times the ultimate resistance (fu) of the base material [4-6]. 41 In case of classical double -sided and single -sided lap shear connections, the bearing strength has been 42 extensively investigated over the last twenty years both for standard and Cold -Formed Steelwork 43 (CFS) . However , if reference is made, as an example, to the European (Eurocode 3 part 1.8 [4] and 44 Eurocode 3 part 1.3 [5]) and North -American design codes (AISC 360 -10 [7] and AISI S100 [6]), many 45 differences , mainly related to the background documentation followed to develop the standardized 46 design procedures can be still recognized . Generally , it is common practice to merge in the same 47 formula the checks for the so-called bearing and tear -out failure modes, expressing the bearing 48 resistance at the bolt hole as the product of an average bearing stress multiplied by the bearing area. 49 However , even thoug h the equation is expressed in this way in all the most widespread standards , the 50 definition of the design bearing factor s does not find a common agreement . As an example, in AISI 51 S100 the bearing stress may range , in case of standard holes , from 1.35 fu (for very thin plates and 52 minimum confinement ) to 4 fu (for thick plates and maximum confinement ). Differently , but w ith a 53 similar equation, AISC 360 -10 limits the average bearing stress to an upper bound value of 3fu,. 54 Conversely , in European practice, Eurocode 3 part 1.8 limits the bearing factor to a maximum value 55 equal to 2.5fu being, therefore, much more conservative than all the other design provisions . This lack 56 of consistence between the various design provisions has already been evidenced by several authors 57 (e.g. [2, 8, 9]) . Though, the definition of a unique and commonly accepted criterion is still far from 58 being achieved. In the European framework, t he most recent proposals are those provided by the 59 research group of the University of Ljubljana that , in a series of works published since 2010 [1-3], has 60 -3- suggest ed a more accurate procedure for the European practice , developing a new equation validated 61 on a large set of experiments including mild and high strength steels. This new formula considers the 62 possibility to achieve an average bearing stress equal to 3fu, limited , eventually , by the development of 63 tear -out failure mode s, which are accounted for with an approach similar to the one reported in the 64 earlie r versions of the AISC specification (1993) [9,10] . Examples of o ther recent works, which 65 underline the continuous need for more accurate design procedures , as well as the complexity of the 66 topi c, are those of Salih et al. [11] , Kim & Kuwamura [12] and Kymaz [13] regarding the behavior of 67 bearing in stainless steel connections , Draganic et al . [8] or Teh & Uz [14] dealing with single sided 68 connections , the series of works by Teh and coauthors [14-17] or other recent experimental works 69 [18-24]. 70 71 Fig. 1 – Example of a t ypical configuration of a joint with tubular elements , gusset plates and long bolts 72 As herein described , the technical literature regarding the bearing resistance at bolt holes is very wide . 73 However , it is practically exclusively devoted to sheet -to-sheet connections in double -sided or single - 74 sided configurations, neglecting other possible configur ations, such as the case of connections with 75 long bolts fastening structural elements with a gap (e.g. hollow sections , channels or simply spaced 76 gusset plates ). Nevertheless , even in absence of regulations and specific experimental works , such a 77 kind of connection s are widely used in constructions, in CFS assemblies and ordinary steelwork, for 78 trusses, racking systems, bracings, scaffoldings, etc. (Fig.1) . Indeed, at the author s best knowledge, 79 currently th e only explicit specification in design standards for this connection typology is that 80 reported in AISC 360 -10 [7] (Equation J7 -1), while there are only very few theoretical and 81 experimental works, such as that of Yu & Panyanouvong [25] . 82 The main difference between a traditional sheet -to-sheet connec tion and a connection with long bolts 83 passing completely through the member is that due to the lack of installation space , the bolt head or 84 nut can confine the plates subjected to bearing only outwards, while inwards they are free to buckle 85 (Fig.2 ). Clearly, compared to the classical overlapped configurations, this lead s to a strong decrease of 86 -4- the co nfinement , thus reduc ing the value of the maximum achievable bearing stress. In fact, as already 87 evidence d in [26] , in many cases, in this type of connection , due to the development of a particular out - 88 of-plane failure mode of the plate, the bearing stress es cannot attain values as high as those reported 89 neither in the recent proposal of Može [3] (i.e. 3fu) nor in the AISI S100 or EC3 provisions (i.e. 2.25 fu 90 [26] or 2. 5fu [4,5]) . 91 a) Sheet -to-sheet connection (full lateral confinement) b) Connection with a gap (confinement only on one side) Fig. 2 – Different confinement effect in sheet -to-sheet and connections with a gap 92 Within this framework, aiming to provide a contribut ion toward s the development of specific design 93 guidelines for connections composed by gusset plates, squared hollow sections and long bolts, the 94 main g oal of this paper is to investigate the behaviour of this joint typology, suggesting an analytical 95 formula able to predict the bearing strength at bolt holes . This is made introducing in the recent 96 proposal for revision of Eurocode 3 of Može et al. [2 -3] a further check to be considered only in case of 97 connections with long bolts and Squared Hollow Sections (SHS) . The new analytical formulation is 98 based on the calculation of the post -buckling load of the tube plate subjected to the bearing action of 99 the bolt, combining the Winter’s equation [27] (originally developed to predict the post -buckling 100 strength of steel plates under compression, accounting for geometrical and mechanical imperfections) 101 with a simplified F E model developed in SAP 2000 [28] . This model is used to determine the elastic 102 buckling load for the particular loading condition (patch load wit h partial rigidity constraints) which, 103 otherwise, would be quite complex in a closed form. The procedure developed is presented in detail 104 proposing a design equation whose accuracy is checked on the experimental data reported in a 105 previous work of the same authors properly extended with parametric FE simulations [26] . The 106 additional FE analyses are carried out in order to consider the variation of the main geometrical 107 parameters influencing the bearing resistance. 108 109 F F/2 F/2 F F/2 F/2 gap SHS plate plate plate plate plate tube’s wall tube’s wall -5- 2. Current design practice for bearing strength at bolt holes 110 In order to provide a state of the art o n the current practice for the bearing strength at bolt holes, in 111 the following the main suggestions available in design standards are summarized . 112 The AISI S100 code is essentially based on the works of Yu, Zadanfarrokh & Bryan, LaBoube & Yu , 113 Wallace, Schuster and LaBoube and Rogers & Hancock [29-41]. In particular, Rogers & Hancock first 114 and Wallace, Schuster and LaBoube after , conducted tests on CFS bolted connections defining , as it is 115 made in all the codes, the bearing strength as dependent on the tensile strength of the plate ( fu), 116 thickness of the sheet ( t) and bolt diameter ( d). Additionally, the authors after verification with a 117 database including various research reports , expressed the stress bearing co efficient as the product of 118 a modification factor mf , multiplied by a bearing factor C dependent, in turn, on the t/d ratio. The 119 product of these two coefficients may lead , for connections with standard holes, to values of the 120 bearing stress ranging from a minimum of 1.35 fu (d/t>22 , single -sided or outside sheet of a double 121 shear connection without washers or only one washer) to a maximum of 4 fu (d/t<10 , inside sheet of 122 double shear connection with or without washers): 123 AISI S100 𝐹𝑏= 𝐶𝑚 𝑓𝑓𝑢𝑡𝑑 * (1) 𝐶=3 𝑓𝑜𝑟𝑑 𝑡≤10− 𝐶=4−0.1 𝑑 𝑡 𝑓𝑜𝑟𝑑 𝑡≤22 / 𝐶=1.8 𝑓𝑜𝑟𝑑 𝑡>22** (2) *modification factors reported in Table J3.3.1 -2 of AISI S100 (0.75<m f<1.33 for standard holes) ** values for connections with standard holes Furthermore, with a very advanced approach , AISI S100 propose s also a reduced value of the bearing 124 strength to be used only in the lo ad combinations where the bolt hole s deformation is a design 125 consideration expressing, eve ntually, the need to limit deformations at bolt holes under service load 126 conditions . Practically, the same design procedures given in AISI S100 are reported also in the 127 Australian and New Zealand Standard AS/NZS 4600:2005 [42] for CFS, whose background documents 128 are mainly the works of Rogers & Hancock [34-41]. 129 AISC 360 -10 neglects the dependence on the t/d ratio and the influence of the connection typology, but 130 provides a similar design equation expressing the bearing strength as equal to 1.2 or 1.5 times the 131 clear distance in the bolt force direction (lc) multiplied by the plate thickness and ultimate stress of the 132 material ( Fu), with a limitation to the maximum achievable bearing stress equal to 2.4Fu or 3Fu. Also in 133 this case, t he alternative choice of the factors 1.2 or 1.5 and 2.4 or 3, depends on whether the 134 deformation at bolt holes is or it is not a design consideration: 135 AISC 360 -10 𝐹𝑏= 1.5𝑙𝑐𝑡𝐹𝑢<3dt𝐹𝑢 * (3) -6- (normal connections in standard holes) 𝐹𝑏= 1.2𝑙𝑐𝑡𝐹𝑢<2.4dt𝐹𝑢 ** (4) * if the bolt hole deformation is not a design consideration ** if the bolt hole deformation is a design consideration at service loads Additionally, for connections made using bolts that pass completely through an unstiffened box 136 member or HSS, AISC 360 -10 suggests to make reference to the equation normally used for pins in 137 bearing , expressing the bearing strength as the area in bearing ( Apb) multiplied by 1.8 times the 138 nominal yield stress ( Fy): 139 AISC 360 -10 (connections in HSS members ) 𝐹𝑏= 1.8𝐹𝑦𝐴𝑝𝑏<1.5𝑙𝑐𝑡𝐹𝑢<3dt𝐹𝑢 (5) In European practice the current rules for connections in normal steelwork and CFS are those 140 contained in Eurocode 3 part 1.8 and part 1.3 respectively [4,5], whose background is based on the 141 work done at the end of the 80s by one of the working group s for the Eurocodes [43 -45] which, at the 142 time, provided a synthesis of the most recent experiences of those ages. On the base of these analyses a 143 model covering the basic failure modes given in Fig. 3 was developed. The equations reported in the 144 Eurocode s are very similar to the design formulas of AISC 360 but more conservative and complicated 145 owing to the introduction of a dependence between the bearing factor and the distance of the hole 146 center from the lateral edge (e2). 147 (a) (b) (c) Fig. 3 –Splitting failure (a), Bearing Failure (b), Tear -out failure(c). 148 The dependence of the bearing factor on e2, expressed by the k coefficient, as already evidenced by 149 Može et al [2] was included in the formula due to the lack of experimental evidence and is relevant 150 only when e2 is lower than 1.5 d0. Practically it adds a check for the splitting failure mode (Fig.3 a) and a 151 check of the holed section of the plate in tension , which is in any case covered in the Eurocode by a 152 specific check of the net cross -section resistance : 153 Eurocode 3 part 1.8 (connections joining plates starting from a thickness of 4 mm) 𝐹𝑏= 𝑘α𝑓𝑢𝑡𝑑 * 𝛼=𝑚𝑖𝑛 (𝑒1 3𝑑0; 𝑓𝑢𝑏 𝑓𝑢;1) for end bolts 𝛼=𝑚𝑖𝑛 (𝑝1 3𝑑0−3 4; 𝑓𝑢𝑏 𝑓𝑢;1) for inner bolts (6) -7- 𝑘=𝑚𝑖𝑛 (2.8𝑒2 𝑑0−1.7;2.5) for edge bolts 𝑘=𝑚𝑖𝑛 (1.4𝑝2 𝑑0−1.7;2.5) for inner bolts * with: e1= distance from the edge in the direction of the force (end distance) e2= distance from the edge orthogonally to the force (edge distance) p1= spacing in the direction of the force p2= spacing orthogonally to the force d0= diameter of the bolt hole In Eurocode 3 part 1.3 , an equation with the same structure of part 1.8 is suggested , but the 154 dependence of the bearing resistance on the plate thickness is introduced with a coefficient kt, 155 correcting the resistance given by Eq.(6) while assuming a value of k equal to 2.5 : 156 Eurocode 3 part 1.3 (connections joining plates with 1.25 mm<t<4 mm) 𝐹𝑏= 2.5𝑘𝑡α𝑏𝑓𝑢𝑡𝑑 𝑘𝑡=(0.8𝑡+1.5) 2.5 𝑓𝑜𝑟 0.75 𝑚𝑚 ≤𝑡≤1.25 𝑚𝑚 ; 𝑘𝑡=1 𝑓𝑜𝑟 𝑡>1.25 𝑚𝑚 𝛼𝑏=𝑚𝑖𝑛 (𝑒1 3𝑑;1) (7) From the review of the code provisions previously reported , it is clear that the only design equation 157 specifically devoted to the case of long bolts passing through a section with a gap is the one proposed 158 by AISC 360 -10, namely Eq.(5). Conversely , all the other formulas herein reported are not specifically 159 referre d to the case of long bolts and , therefore , they should not be blindly extended due to potential 160 unsafe predictions . In fact, in a previous work of the same authors a preliminary investigation dealing 161 with the accuracy of current code formulations was performed , demonstrating that the equation s 162 formerly given are not able to provide a satisfactory prediction of the actual connection behaviour 163 [26] . Indeed , for connections made with Squared Hollow Sections (SHS), gusset plates and long bolts , it 164 was already shown in [26] that Eurocode 3 provides conservative results in the range of high plate 165 thicknesses (t>4mm ) while providing unsafe predictions for low plate thicknesses (t<4mm ). 166 Conversely, AISI S100, accounting more accurately for the dependence on the t/d ratio, provides a 167 better approximation for lower thicknesses assuming a modification factor equal to 0.75 but in any 168 case, as it is not specifically calibrated for the case of long bolts, still provides a signi ficant 169 overestimation of the resistance in many cases. Similarly, also the provisions of AISC 360 -10 are not 170 accurate and in fact, Eq s.(3-4) lead in general to an overestimation of the resistance, while, Eq.(5) 171 provide s a significant underestimation. 172 This preliminary review justifies the work hereinafter reported , considering the need for the definition 173 of design rules more specifically devoted to the case of long bolts. As above said, the model presented 174 afterwards follow s the proposal recently made in view of a new Eurocode standard by Može [2,3] , 175 -8- introducing a further check for the plate buckling to be considered only in case of connections with 176 long bolts and a gap. In particular, the analysis reported in the next sectio ns is specifically devoted to 177 connections made up of gusset plates and SHS members . Anyway, the same methodology hereinafter 178 presented could be applied to extend the approach also to other joint configurations with long bolts. 179 As it will be used for the ca libration of the equation reported afterwards, t he recent proposal of Može 180 [2,3] is herein reported . It regards an update of the EC3 part 1.8 formula in which the bearing strength , 181 consistently with other international codes, can assume a maximum value equ al to 3. Furthermore, in 182 view of a simplification, the coefficient k is removed, while a correction factor (kb) accounting for the 183 steel grade is introduce d: 184 Može [2,3] 𝐹𝑏= 𝑘𝑏α𝑓𝑢𝑡𝑑 𝛼=𝑚𝑖𝑛 (𝑒1 𝑑0; 3) for end bolts 𝛼=𝑚𝑖𝑛 (𝑝1 𝑑0−1 4; 3) for inner bolts 𝑘𝑏=1 for steel grade s lower than S460 𝑘𝑏=0.9 for steel grade s higher or equal to S460 (8) However, i t is worth noting that in the original work s of Može [2,3] the bearing factor for a single 185 bolt ed connection can achieve a maximum stress equal to 5 fu. Nevertheless, a reduction of the bearing 186 coefficient to 3 is suggested by the same author to introduce, indirectly, a control on the development 187 of brittle failure modes. 188 3. Main outcomes of the previous activity [26] 189 In order to evaluate the behavior of connections with long bolts passing through tubular members 190 subjected to shear forces , an experimental programme regarding 24 specimens was performed at the 191 Laboratory for Materials and St ructures of the University of Liège (Fig.4) [26] . 192 a) SHS 70x70mm with M12 bolts, thickness equal to 2, 2.5 or 4 mm and steel S235 or HX420LAD b) SHS 70x70 mm with M16 bolts, thickness equal to 2, 2.5 or 4 mm and steel S235 or HX420LAD Fig. 4 – Specimens typologies examined in [26] 193 -9- The results of this experimental work , which was followed also by the development of a FE model in 194 ABAQUS and further parametric analyses to cope for variation of geometrical properties , are reported 195 in detail in [2 6] and are hereinafter reported to summarize the main outcomes of this activity , which is 196 the main reference for the further developments described in this pap er. The specimens consisted in 197 elementary connections made up of tubular SHS members fastened with external gusset plates and 198 with a single long bolt , subjected to tensile axial forces in order to generate a shear in the connection 199 (Fig.4) . The specimens w ere realized both with S235 and HX420LAD steel , considering three different 200 tube s thickness , namely 2, 2.5 and 4 mm and two bolt s diameters (M12 and M16). For every 201 considered co mbination of steel grade, tube thickness and bolt diameter, four nominally equal 202 specimens were tested. 203 a) HX-2-M16 b) HX-2-M12 c) HX-2.5-M12 d) HX-2.5-M16 e) S235 -4-M12 f) S235 -4-M16 Fig. 5 – Experimental Force -Displacement curves [26] 0102030405060708090100 0 2 4 6 8 10 12 14 16 18 20F[kN] s[mm]HX-2-M16 Test 1 Test 2 Test 3 Test 4 01020304050607080 0 2 4 6 8 10 12 14 16 18 20F[kN] s[mm]HX-2-M12 Test 5 Test 6 Test 7 Test 8 0102030405060708090100 0 2 4 6 8 10 12 14 16 18F[kN] s[mm]HX-2,5-M12 Test 9 Test 10 Test 11 Test 12 020406080100120140 0 2 4 6 8 10 12 14 16 18 20F[kN] s[mm]HX-2,5-M16 Test 13 Test 14 Test 15 Test 16 020406080100120 0 1 2 3 4 5 6 7 8 9 10F[kN] s[mm]S235 -4-M12 Test 1 Test 2 Test 3 Test 4 020406080100120140160180 0 2 4 6 8 10 12 14 16 18 20F[kN] s[mm]S235 -4-M16 Test 5 Test 6 Test 7 Test 8 -10- The tests, whose results are summarized in Fig s.5, showed a response similar for all the specimens in 204 terms of force -displacement , distinguishing only the cases where the bolt failure was prominent from 205 the cases where the hole failure was observed. In the first case the response was characterized by the 206 achievement of the maximum resistance and then by a sudden drop (e.g. Fig.5e) ; in the second case, 207 the response was more ductile and characterized by the development of a smooth softening branch 208 after the attainment of the peak force (e.g. Fig.5f) (the labels reported in the graphs titles of Figs.5 209 identify the steel grade (HX which stands f or HX420LAD or S235), the tube thickness (2, 2.5 or 4 mm) 210 and the bolt diameter (M12 or M16)). 211 a) Loca l buckling failure at the bolt hole (Test HX -2-M16) b) Loca l buckling failure at the bolt hole , particular of the bump (Test HX -2-M16) c) Tear -out failure (Test S235 -4-M16) d) Bolt failure (Test S235 -4-M12) Fig. 6 – Typical failure modes observed in the tests reported in [26] 212 -11- Even though the shape of the force -displacement curves for all the specimens was in line with the 213 expectations, the observed failure modes were very peculiar, identifying the development of a new 214 failure mechanism , which is normally not accounted for in traditional design equations . Indeed , in 215 cases with high strength steel (HX420LAD grade) and low thickness of the SHS tube (equal to 2 and 2.5 216 mm ) a failure mode characterized by the development of local bumps in the bearing area ahead the 217 bolt shank was observed (Figs.6a,b) . Conversely, in all the other tests with higher tube thickness , due 218 to the lower local slenderness of the tube hole under beari ng (higher thickness over bolt diameter 219 ratio) the traditional failure modes w ere identified , namely tear -out, bearing or shear failure of the 220 bolt shank (Figs.6c,d ). It is useful to observe that, since the bumps appear into an area sufficiently far 221 from the possible confinement, similar phenomena could b e possible observed in the outer plates of 222 lap connections. 223 The experimental results were used to calibrate a FE model in ABAQUS extending the experimental 224 sample with further analyses devoted to consider other geometries. Subsequently, all the data 225 collected, experimental and simulated, were used to verify the accuracy of the currently available 226 equations (see section 2) evidencing the need for a more accurate formulation able to predict the 227 bearing resistance in case of connections with long bolts and members with a gap , accounting also for 228 the possible development of local buckling phenomena at the bolt holes , such as those observed in the 229 experiments . Indeed , the comparison of the experimental and FE results with the EC3 and AISI S100 230 formulation s reported in [26] evidenced that the codified models are not able to pre dict accurately the 231 connection resistance, even though the application of AISI S100 equations with the minimum value of 232 the modification factor allowed for standard holes ( mf=0.75) provides a safe prediction and a better 233 approximation at least in the low thickness over bolt diame ter ratio range (Fig.7). In all the other cases 234 both EC3 part 1.3 or 1.8 and AISI S100 specification do not seem accurate. 235 236 Fig. 7 – Accuracy of EC3 and AISI S100 equations 237 050100150200250300 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Resistance [kN] t/dFEM simulations Bolt failure (EC3) Bearing failure EC3 -1.8FE Simulations vs EC3 Bearing failure EC3 -1.3 050100150200250300 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Resistance [kN] t/dFEM simulationsAISI S100 (2007) -mf=1 AISI S100 (2007) -mf=0.75Bolt failure (EC3)FE Simulations vs AISI S100AISI S100 (2016) - mf=1 AISI S100 (2016) - mf=0.75 -12- 4 Prediction of the critical buckling load for unconfined bolt holes under bearing 238 This section is devoted to the definition of a procedure to evaluat e the post -elastic buckling load of 239 connections of a SHS tube with through -all long bolts. The knowledge of s uch a buckling load value is 240 necessary in order to account in design for the particular failure mode that was observed in the 241 experiments reported in [26] , previously briefly described (Figs.6a,b) . The prediction of the post - 242 elastic failure load , with the approa ch described afterwards in this section, is intended to introduce in 243 the recent equation proposed by Može et al. [2 -3] (validated for stand ard lap shear connections ) a 244 further check for local buckling to be considered only in case of connections with long bolts and 245 Squared Hollow Sections (SHS), where this particular failure mode can actually develop . The Može et 246 al. model [2-3] is selected because it was already demonstrated to be more simple and more accurate 247 than EC3 model and, additionally, it has alrea dy been calibrated in view of the definition of a new 248 standard for the European practice. As aforesaid, t he new analytical formulation is developed 249 combining Winter’s equation [27] with a FE model developed in SAP 2000 [28] . The purpose of the FE 250 model is to provide a prediction of the elastic buckling load (necessary to apply Winter’ s equation) 251 which otherwise, for the particular loading and restraining conditions would be very complex in a 252 closed form. Therefore, in the following the procedure adopted to define the elastic critical buckling 253 load is described in detail and, afterwards the Winter ’s approach is applied in order to account for 254 geometrical and mechanical non -linearities. Subsequently, by means of a parametric analysis a simple 255 design equation expressed as a function of the main geometrical parameters of the connection under 256 investiga tion is calibrate d through regression analysis of simulated data. The details of the procedure 257 are hereinafter reported. 258 259 Fig. 8 – Scheme of the analysed connection 260 L B Hd0 H Top/Bottom PlateLateral Plate z yyzTop Plate Bottom PlateLateral Plate Lateral PlatetCross Section -13- 4.1 Evaluation of the elastic and post -elastic buckling load 261 The procedure applied in this work to assess the value of the elastic critical load leading to the local 262 buckling of the tube hole is carried out starting from the analysis of a very simple sub-structure 263 extracted from the whole connection representative of the zone where the local buckling effects are 264 more significant. In particular, in view of simplification, the sub -structure is idealized extracting from 265 the tube only the end part of the plate directly loaded from the bolt force , whose dimensions are the 266 width of the tube (H) and the distance between the centre of the hole and the free edge of the tube ( e1) 267 (Fig.8 ). This is the zone that, as also demonstrated by the experimental analysis, is more significantly 268 affected by the local buckling phenomena. 269 Dealing with the modelling of buckling phenomena, it is well known that the definition of the 270 restraining conditio ns applied at the edges of the plate is of paramount importance, due to the 271 influence that the constraints have on the value of the critical buckling load. This is the reason why, 272 even though in the present work only a small portion of the tube is studied, the continuity of the 273 analysed plate with the remaining parts of the tube is recovered by considering appropriate boundary 274 conditions applied on the edges of the element (Fig. 9). 275 276 Fig. 9 - Boundary conditions 277 In particular, in the following, the considerations made to define the stiffness of appropriate springs 278 able to account for the rotational, shear and axial behaviour of t he remaining parts of the tube plates 279 FF/2d 0A B CDEF z yVertical supports + Rotational springsFree edge Vertical supports + Rotational springsSpring in Direction z + Support in Direction x Springs in Direction x/z Springs in Direction x/z Spring in Direction z + Support in Direction x H e1 -14- are described. In particular, in order to account for the remaining portions of the tube the following 280 constraints are applied at the edges of the analysed plate (Fig. 9): 281 • at the intersection of the edges of the plate with the lateral walls of the tube rotational springs 282 and vertical supports are applied (edges A -B and D -C); 283 • on the ed ge loaded under compression by the bolt, in -plane and out -of-plane springs are 284 introduced both on the portions of the edge next to the hole (edges A -F and D -E) and in the 285 corners of the plate (points A and D). Furthermore, in the zone of the hole, a unifor m load 286 modelling the shear action transferred by the bolt is introduced; 287 • at the free edge of the tube no boundary conditions are introduced (edge B -C); 288 Fig. 10 – Scheme used to define the rotational stiffness due to the lateral plates of the tube 289 Ther efore, o n the sides of the plate parallel to the load (from A to B and from D to C, fig. 9), out-of- 290 plane supports and rotational springs are applied in order to model the axial and rotational stiffness of 291 the lateral plates of the tube. In particular, in order to model the axial stiffness of the lateral plates of 292 the tube, on these edges, the out -of-plane supports are defined as rigid elements. Conversely, the 293 stiffness of the rotational springs is determined by defining the value of the bending moment needed 294 to generate a unitary rotation on the lateral plates of the tube (Fig. 10). By adopting th is approach, the 295 following expression is obtained: 296 𝑘,𝑙𝑎𝑡𝑒𝑟𝑎𝑙 =2𝐸𝐼 𝐻 (9) 297 where E is the modulus of elasticity of steel, I is the moment of inertia of the lateral tube plate per unit 298 of length and H is the height of the tube plate. 299 The side that goes from B to C is considered free; a similar assumption is made for the portion of the 300 edge on which the load is applied that goes from E to F. In addition, to take into account for the 301 yx SymmetrySymmetry yx Symmetry Symmetry M =1H H yx SymmetrySymmetry yx Symmetry Symmetry M =1H H -15- stiffness of the remaining part of the tube behind the analysed plate, various consi derations are made. 302 In particular, a t points A and D of the model (the corners of the plate ) extensional springs in the z 303 direction accounting for the stiffness of the portion of the lateral plate behind the bolt are applied . The 304 stiffness of such springs is easily calculated as: 305 306 𝑘𝑧,𝑐𝑜𝑟𝑛𝑒𝑟𝑠 =𝐸(𝐴𝑙𝑝2⁄) 𝐿 (10) 307 where L is the length of the tube and Alp is the area of the lateral wall of tube calculated as the product 308 of the tube thickness t and H/2 . In the same way , in the zone going from A to F and from E to D 309 extensional springs are spread along the edges both in the x and in the z direction . As already made for 310 the stiffness 𝑘𝑧,𝑐𝑜𝑟𝑛𝑒𝑟𝑠 , the value of the stiffness of the springs spread on the loaded edge is determined 311 in order to acco unt for the axial deformability of the remaining part of the top plate located behind the 312 bolt: 313 𝑘𝑧,𝑒𝑑𝑔𝑒 =𝐸(𝐴𝑡𝑝2⁄) 𝐿 (11) where Atp is the area of the top plate of the tube calculated as the product of the tube thickness t and H. 314 Conversely, the stiffness of the springs in the z direction is determined in order to account for the out- 315 of-plane deformability of the portion of the tube behind the bolt hole. To this scope, the s imple scheme 316 of plate fully restrained on three edges loaded with a unitary force applied on the free edge 317 represented in Fig. 11 has been studied. Therefore, in order to determine the value of the stiffness of 318 the springs to be applied on the holed side of the plate several schemes with the force applied in 319 different points of the free edge have been solved. 320 321 Fig. 11 – Scheme used to evaluate the out -of-plane stiffness of the springs applied on the holed edge 322 The result of this analysis is delivered in Fig. 1 2 where the value of the generic spring stiffness is 323 represented versus the distance of the point of application of the force from the mid -point of the edge . 324 z yyxFully restrained Fully restrainedFully restrainedF =1HL -16- From such a figure it is possible to observe tha t the out -of-plane stiffness of the plate is almost 325 constant in the central area, while in the end regions it asymptotically tends to infinity due to the 326 presence of the full restraints . In order to maintain a simple structure of the model and to avoid to 327 introduce a variable value of the springs stiffness, the out -of-plane stiffness of the springs has been 328 assumed constant and equal to the value of the force needed to obtain a unitary displacement at the 329 mid -point of the plate edge. Conversely, at the corn ers of the plate, as already said, a fixed support has 330 been applied. This choice is justified by the fact that, from some preliminary analyses, no appreciable 331 differences were found in the results considering a variable value of the springs stiffness or a constant 332 value. 333 334 Fig. 12 – Stiffness for out -of-plane springs 335 Under the reported hypotheses, the value of the elastic buckling load of the plate with the above 336 described boundary conditions has been determined by developing an appropriate FE model in 337 SAP2000 v.16 software package. T he model has been developed by means of the following steps : 338 geometrical characterization of the plate , definition of the material properties, definition of the 339 boundary conditions and choice of the elements and size of the mesh. The geometry of the plate has 340 been defined b y using the automatic tools provided by the software, which are able to generate and to 341 partition in a very simple way the plate, after defining the dimensions and the number of divisions of 342 the elements . The plate section has been provided by introducing in the model the thickness of the 343 plate and by choosing the shell formulation. In particular, in the proposed FE model the so -called 344 “thin -shell” formulation has been used, i.e. a shell formulation following the Kirchhoff model [46] , 345 where the transverse s hear deformations are neglected. Due to the typical aspect ratio and thickness 346 of the analysed plates, this is considered a reasonable hypothesis. The material properties of the steel 347 plate have been simply described by means of an elastic isotropic model with elastic modulus equal to 348 210.000 MPa and Poisson’s ratio equal to 0.3. The optimal mesh subdivision has been defined after 349 y- -17- performi ng several preliminary analyses devoted to investigate the dependence of the result from the 350 mesh size. After these prel iminary analyses a minimum dimension of the area element equal to 2.5 mm 351 x 2.5 mm has been chosen (Fig.1 3). 352 Fig. 13 – SAP2000 model of the plate 353 An example of the results of one of the analyses presented in the next section is delivered in Fig.1 4, 354 where the shapes of the first four buckling modes of one on the analysed cases are represented. As it is 355 possible to verify, the 1st buckling shape, i.e the buckling shape characterized by the lowest value of the 356 load multiplier, is the one typically observed also in the experimental analysis. 357 e1=48 mm, H=60 mm, t = 0.50 mm, d=16 mm 1st mode – cr=105.3 2nd mode – cr=306.8 d e1 H=2e 2 -18- 3rd mode – cr=441.1 Fig. 14 – Buckling modes for a plate with thickness equal to 0.50 mm 358 After determining the elastic buckling load, in order to obtain the actual value of the ultimate buckling 359 load it is necessary to account for two additional parameters that normally provide a significant 360 influence: the imperfections and the non -linear behaviour of steel. In fact, both these factors, as widely 361 recognized in technical literature, may provide a significant variation to the value of the buckling 362 resistance calcul ated by means of an elastic analysis and, therefore, they need to be properly 363 accounted for. In this work, in order to account for these factors, the empirical approach of Winter 364 [27], already proposed by EC3 and by the AISI specification, is followed. Suc h an approach , on which 365 the effective width concept is based, provides to correct the value of the ultimate load by means of the 366 following expression: 367 𝑁𝑐𝑟,𝑝𝑒=𝑁𝑝𝑙𝜌=𝑁𝑝𝑙[(1−0,22 𝜆𝑝̅̅̅)1 𝜆𝑝̅̅̅]≤𝑁𝑝𝑙 (12) where 𝜆𝑝̅̅̅ is the non-dimensional slenderness parameter expressed as: 368 𝜆𝑝̅̅̅=(𝑁𝑝𝑙 𝐹𝑐𝑟,𝑒𝑙)1/2 (13) where Fcr,el is the value of the elastic buckling load, evaluated in this work by means of the buckling 369 analys es described previously, and Npl is the value of the plastic load. It is worth noting that, according 370 to the Winter’s methodology, the two coefficients 1 𝜆𝑝̅̅̅̅ and (1−0,22 𝜆𝑝̅̅̅̅) reported in Eq.12 account for the 371 influence of the mechanical non -linearity of the material and for the imperfections respectively 372 [27,47]. The value of the plastic load Npl, in the current case, can be determined following one of the 373 approaches mentioned in section 2. In this work, the recent expression propose d by Može , namely 374 Eq.(8), is selected. This formulation , as demonstrated by the recent validation studies, is able to 375 accurately predict the plastic resistance of a hole in bearing. 376 -19- As far as the p rocedure to determine the crit ical buckling load at the bolt hole is defined 377 (determination of the elastic buckling load by means of the SAP 2000 simple model previously 378 described and of the post -elastic buckling load by means of the Winter’s formula) , in the next section, a 379 parametric analysis is carried out in order to calibrate a simple design equation through regression 380 analysis of a set of simulated data. 381 4.2 Calibration of a design equation 382 Basing on the procedure previously described , the aim of this section is to set up a simple design 383 equation able to account for the buckling check in connections with SHS and trough -all long bolts. This 384 can be made introducing in Eq.(8) a correction coefficient lower or equal to one accounting for the 385 particular buckling failure mechanism arising in such a type of connections. In particular, with the aim 386 to keep the same structure of the formulation propose d by Može , it is proposed to include in Eq.(8) an 387 additional coefficient 𝛼𝑠𝑡, covering the buckling failure mode, calibrated by means of a parametric 388 analysis performed applying the procedure previously described . Equating the ratio between the 389 ultimate bearing resistance calculated according to Eq.(8) and the post -elastic bucklin g strength 390 provided by Eq.(12), a new factor 𝛼𝑠𝑡, to be added in the bearing resistance formula of Može , can be 391 easily defined: 392 𝐹𝑏 𝑁𝑐𝑟,𝑝𝑒=𝐹𝑏 𝑁𝑝𝑙𝜌=𝛼𝑠𝑡(𝑘𝛼𝑑𝑡 𝑓𝑢) (𝑘𝛼𝑑𝑡 𝑓𝑢)𝜌=1 (14) 𝛼𝑠𝑡=𝜌 (15) where 𝜌=(1−0,22 𝜆𝑝̅̅̅̅)1 𝜆𝑝̅̅̅̅ ≤1 is the buckling factor evaluated according to the Winter’s formula 393 (Eqs.12, 13). 394 Following this app roach, in order to define a simple formula to be used in design, the main issue is to 395 determine a correlation between  and the main geometrical parameters of the analysed connection. 396 According to the results reported in [26 ], the factors that influence the value of the ultimate load in 397 case of buckling failure mode are t/d, e1/d and e2/d ratios. Therefore, the parametric study proposed in 398 this section has been mainly devoted to investigate the influence of these three parame ters on the 399 value of . To this scope, FE simulations have been performed by carrying out three groups of analyses 400 according to the procedure previously defined , based on the determination of the elastic and post - 401 elastic buckling load by means of the simpl ified SAP 2000 model and the Winter’s formula . A first 402 group of analy ses devoted to check the influence of the t/d ratio. In this case the analyses have been 403 carried out fixing the values of e1/d equal to 3 and of e2/d equal to 2.5 varying the bolt diameter d and 404 plate thickness t case by case. In particular, in this first group of analyses the values of d have been 405 assumed equal to 12 and 16 mm and the t/d ratio has been varied in the range 0.05/0.15. The second 406 -20- group of analyses has been carried out in order to investigate the influence of the e1/d ratio. In this 407 case the analyses have been carried out by fixing the value of e2/d equal to 2.5 and by varying t/d in 408 between 0.1 and 0.3 and e1/d in between 3 and 5. The third group of analyses has been carried out 409 with the aim to investigate the dependence of the buckling resistance on the parameter e2/d. In this 410 case e2/d has been varied in between 1.88 and 3.5, t/d has been varied in the range 0.03 and 0.15 and 411 e1/d has b een fixed as equal to 3. For each of the considered combination of geometrical properties , the 412 procedure previously reported to determine the value of the elastic and post -elastic buckling load has 413 been applied and, subsequently, the values of the critical elastic buckling load and of the reduction 414 factor calculated by means of the Winter’s approach have been calculated. The results of these 415 analyses are summarized in Tab.1. 416 Table 1 – Results of the parametric analysis. 417 1st group of analyses – influence of t/d 418 t/d e1/d e2/d d [mm] t [mm] e1 [mm] e2 [mm] Ncrit,elastic* [kN] fu** [MPa] Npl= Fb [Eq.8] [kN] λp [Eq.13]  [Eq.13] Ncr,pe [Eq.12] [kN] 0.05 3 2.5 16 0.8 48 40 3.51 499 18.58 2.30 0.39 7.30 0.075 3 2.5 16 1.2 48 40 11.84 499 27.87 1.53 0.56 15.56 0.1 3 2.5 16 1.6 48 40 28.08 499 37.16 1.15 0.70 26.13 0.125 3 2.5 16 2 48 40 54.84 499 46.45 0.92 0.83 38.41 0.15 3 2.5 16 2.4 48 40 94.77 499 55.74 0.77 0.93 51.83 0.05 3 2.5 12 0.6 36 30 1.86 499 10.35 2.36 0.38 3.98 0.075 3 2.5 12 0.9 36 30 6.30 499 15.52 1.57 0.55 8.50 0.1 3 2.5 12 1.2 36 30 14.95 499 20.69 1.18 0.69 14.30 0.125 3 2.5 12 1.5 36 30 29.20 499 25.87 0.94 0.81 21.06 0.15 3 2.5 12 1.8 36 30 50.46 499 31.04 0.78 0.92 28.48 2nd group of analyses – influence of e 1/d 0.1 3 2.5 16 1.6 48 40 26.976 499 37.16 1.17 0.69 25.73 0.1 3.5 2.5 16 1.6 56 40 27.365 499 43.36 1.18 0.69 26.36 0.1 4 2.5 16 1.6 64 40 27.558 499 49.55 1.18 0.69 26.44 0.1 4.5 2.5 16 1.6 72 40 27.684 499 55.74 1.18 0.69 26.48 0.2 3 2.5 16 3.2 48 40 212.934 499 74.32 0.59 1.06 78.96 0.2 3.5 2.5 16 3.2 56 40 218.957 499 86.71 0.59 1.06 81.38 0.2 4 2.5 16 3.2 64 40 220.512 499 99.10 0.59 1.06 81.49 0.2 4.5 2.5 16 3.2 72 40 221.516 499 111.49 0.59 1.06 81.57 0.3 3 2.5 16 4.8 48 40 718.672 499 111.49 0.39 1.12 124.95 0.3 3.5 2.5 16 4.8 56 40 739.028 499 130.07 0.39 1.12 128.90 0.3 4 2.5 16 4.8 64 40 744.221 499 148.65 0.39 1.12 128.78 0.3 4.5 2.5 16 4.8 72 40 747.638 499 167.23 0.39 1.12 128.70 3rd group of analyses – influence of e 2/d 0.031 3 1.9 16 0.5 48 30 0.948 499 11.61 3.50 0.27 3.11 0.063 3 1.9 16 1 48 30 7.59 499 23.23 1.75 0.50 11.61 0.094 3 1.9 16 1.5 48 30 25.59 499 34.84 1.17 0.70 24.23 0.125 3 1.9 16 2 48 30 60.69 499 46.45 0.87 0.86 39.74 0.16 3 1.9 16 2.5 48 30 118.53 499 58.07 0.70 0.98 56.88 0.0313 3 2.5 16 0.5 48 40 0.698 499 11.61 4.08 0.23 2.69 0.05 3 2.5 16 0.8 48 40 3.51 499 18.58 2.30 0.39 7.30 0.075 3 2.5 16 1.2 48 40 11.84 499 27.87 1.53 0.56 15.56 0.1 3 2.5 16 1.6 48 40 28.08 499 37.16 1.15 0.70 26.13 0.125 3 2.5 16 2 48 40 54.84 499 46.45 0.92 0.83 38.41 -21- t/d e1/d e2/d d [mm] t [mm] e1 [mm] e2 [mm] Ncrit,elastic* [kN] fu** [MPa] Npl= Fb [Eq.8] [kN] λp [Eq.13]  [Eq.13] Ncr,pe [Eq.12] [kN] 0.15 3 2.5 16 2.4 48 40 94.77 499 55.74 0.77 0.93 51.83 0.0313 3 3.5 16 0.5 48 56 0.51 499 11.61 4.77 0.20 2.32 0.05 3 3.5 16 0.8 48 56 2.402 499 18.58 2.78 0.33 6.15 0.075 3 3.5 16 1.2 48 56 7.115 499 27.87 1.98 0.45 12.52 0.1 3 3.5 16 1.6 48 56 16.774 499 37.16 1.49 0.57 21.28 0.125 3 3.5 16 2 48 56 32.77 499 46.45 1.19 0.68 31.81 0.15 3 3.5 16 2.4 48 56 56.62 499 55.74 0.99 0.78 43.72 0.05 3 4.5 16 0.8 48 72 1.651 499 18.58 3.35 0.28 5.18 0.075 3 4.5 16 1.2 48 72 5.908 499 27.87 2.17 0.41 11.53 0.1 3 4.5 16 1.6 48 72 13.209 499 37.16 1.68 0.52 19.25 0.125 3 4.5 16 2 48 72 25.820 499 46.45 1.34 0.62 28.95 0.15 3 4.5 16 2.4 48 72 44.617 499 55.74 1.12 0.72 40.05 0.2 3 4.5 16 3.2 48 72 105.766 499 74.32 0.84 0.88 65.39 0.225 3 4.5 16 3.6 48 72 150.584 499 83.61 0.75 0.95 79.08 0.25 3 4.5 16 4 50 72 206.493 499 96.78 0.68 0.99 95.23 *Calculated with the procedure reported in Section 3.1, namely by means of the FE model in SAP 2000 419 **Experimental data reported in [2 6] 420 As it is possible to verify from Table 1, the value of the elastic critical load shows a significant 421 dependence on parameters t/d and e2/d, while its dependence on e1/d seems to be less significant . 422 Furthermore, as demonstrated by the first group of analyses , the value of the reduction factor  423 calculated by means of the Winter’s approach shows practically the same variation trend with respect 424 to the t/d ratio even though a different value of the bolt diameter is considered . This confirm s that  425 depend s on the t/d ratio and not on the bolt diameter d itself. Additionally, f rom the simulated set of 426 data it is possible also to observe that, as expected, t/d is directly correlated to the value of the load 427 with a correlation that is practically linear (Fig.15), while e2/d and e1/d are inversely correlated to 428  (Fig s.16, 17). 429 430 Fig. 15 – Regression curve s of the analysed data – dependence of  on t/d 431 -22- From the physical point of view, t he influence of the parameter e2/d on the reduction coefficient  is 432 related to the influence of the lateral t ube walls on the value of the elastic buckling load. In fact, as 433 expected, the more the tube walls are close to the hole (low er values of e2/d), the higher is the buckling 434 critical load. 435 In order to provide a coefficient to be used in practical design , based on the simulated set of data, a 436 linear multi -parametric regression analys is has been carried out with the aim to find a law providing 437 the value of  as a function of t/d, and e2/d. As verified with former tentative regression models, 438 parameter e1/d provides a negligible influence on the overall value of  and, consequently, can be 439 excluded . Similarly, as reported in Fig. 17, also e2/d is weakly correlated to  Therefore, considering 440 these dependencies, two alternative equation s have been defined , with correlation coefficient s equal 441 to 0.85 and 0.83 respectively : 442 𝜌=3.36𝑡 𝑑−0.043𝑒2 𝑑+0.39 – model 2 (16.1) 𝜌=3.24𝑡 𝑑+0.292 – model 1 (16.2) As a result , combining Eq s.(16) and Eq. (15), the Eq.(8) proposed by Može , can be adapted to the case 443 of connections with SHS and long bolts with the two possible alternative models : 444 𝐹𝑏= αα𝑠𝑡𝑓𝑢𝑡𝑑 𝛼=𝑚𝑖𝑛 (𝑒1 𝑑0; 3) 𝛼𝑠𝑡=min (3.36𝑡 𝑑−0.043𝑒2 𝑑+0.39;1) – model 1 (17.1) 𝑜𝑟 𝛼𝑠𝑡=min (3.24𝑡 𝑑+0.29;1) – model 2 (17.2) 445 Fig. 1 6 – Regression curve of the analysed data – dependence of  on e1/d 446 y = -0.06x + 0.88 R² = 1.00y = -0.06x + 1.24 R² = 1.00y = 0.01x + 1.11 R² = 0.39 0.000.200.400.600.801.001.20 2.5 3 3.5 4 4.5 5[Eq.13] e1/dvs e 1/d =0.1 =0.2 =0.3t/d t/d t/d t/d -23- Finally, by equating Eq s.16 to one it is also possible to find two different expression of the limit value 447 of t/d separating the buckling and bearing /tear -out failure modes. If t/d of the connection is higher 448 than this limit value, it means that the check for the buckling failure mode can be neglected in practical 449 design (obviously, the second equation herein reported is simpler, yet more approximated) : 450 (𝑡 𝑑) 𝑙𝑖𝑚𝑖𝑡 ,1=0.0128𝑒2 𝑑+0.181 - model 1 (18.1) (𝑡 𝑑) 𝑙𝑖𝑚𝑖𝑡 ,2=0.22 - model 2 (18.2) 451 Fig. 1 7 – Regression curve of the analysed data – dependence of  on e2/d 452 5. Model validation 453 The accuracy of the capacity model previously described has been verified by comparison with 454 experimental data and FE results currently available in technical literature. In particular, the reference 455 dataset is composed by the 41 experimental data and FE simulation s reported in a recent paper of the 456 same authors [26 ], which ha s been extended in this work with further 12 FE simulations , performed 457 running again the same FE model already validated in [26] in order to consider also other cases with a 458 different bolt diameter. Both the experimental and FE results are referred to the same layout, namely 459 that described in Fig.4 , which is an elementary connection composed by a SHS tube, fastened with a 460 single long bolt. 461 The experimental cases of the reference datas et are 24 [26] and regard connections made with SHS 462 tubes of nominal thickness varying from 2 to 4 mm, two different steel grades (S235 and HXLAD420 ), 463 two different bolt diameters (M12 and M16) and the following range of variation of the main 464 geometrical p arameters: 0.1< t/d<0.3; 2.4< e1/d<3.5, 1.9< e2/d<3.5. Conversely, the FE simulations, both 465 -24- those carried out in [26] and the additional ones performed in this work, are referred to further 29 466 cases considering a wider range of variability of the geometries . In particular, the thickness of the 467 tubes considered in the FE simulations varies from 0.50 to 6 mm, while the geometrical parameters 468 vary in the following range: 0.1< t/d<0.4; 1< e1/d<5.6, 0.9< e2/d<2.5 . 469 The experimental or simula ted values of the ultimate load s, reported in Table 2 (together with a 470 summary of the geometrical and mechanical properties o f each one of the analysed cases) , have been 471 compared to the values of the load predicted by means of the two models previously set up , expressed 472 by means of Eqs.(16), (17). Additionally , in order to u nderline the improved accuracy obtained with the 473 proposed models with respect to the codified procedures , the bearing strengths have also been 474 calculated by means of the formulations proposed by : i) AISI S100 (Eqs.1 -2), assuming modifica tion 475 factors alternatively equal to 0.75 or 1; ii) with the equations proposed by AISC 360 -10 for normal lap 476 shear connection s (Eq.(3)) or for connecti ons of with long bolts (Eq.(5)); iii) with the Eurocode 3 part 477 1.3 and 1.8 equations (Eqs.(6) and (7)) ; iv) with the model of Mo že (Eq.(8)). Obviously in all the 478 analysed cases , the strength of the connection (to be compared with the experimental or simulated 479 data) has been determined as minimum between the bearing strength at the bolt hole and the bolt 480 resistance. The bolt resistance has been calculated according to the EC3 approach as: 481 𝐹𝑏𝑜𝑙𝑡 ,𝐸𝐶3=0.60𝐴𝑏𝑓𝑢𝑏 (19) where Ab is the bolt area which, in the experimental cases, is on one side of the tube the gross area, 482 while on the other side is the net area, and in the cases simulated by means of the FE models it is on 483 both sides the net area, and 𝑓𝑢𝑏 is the ultimate stress of the bolt material which is considered in all the 484 cases to be 800 MPa. 485 Table 2 – Set of Reference Data 486 t/d e1/d e2/d e1/d0 D d0 t e1 e2 fy fu FFEM-Fexp [mm] [mm] [mm] [mm] [mm] [MPa] [MPa] [kN] Experimental Data [26] 1 0.1 3.1 1.9 3.0 16.0 16.5 2.0 50.3 29.7 398.0 496.0 69.3 2 0.1 3.0 1.9 2.9 16.0 16.5 2.1 48.7 29.7 398.0 496.0 80.8 3 0.1 3.1 1.9 3.0 16.0 16.5 2.1 49.0 29.7 398.0 496.0 82.0 4 0.1 3.1 1.9 3.0 16.0 16.5 2.1 49.1 29.8 398.0 496.0 91.0 5 0.2 3.3 2.5 3.1 12.0 12.5 2.0 39.3 29.9 398.0 496.0 67.6 6 0.2 3.3 2.5 3.2 12.0 12.5 2.1 40.0 30.2 398.0 496.0 70.8 7 0.2 3.3 2.5 3.2 12.0 12.5 2.1 39.9 30.2 398.0 496.0 76.6 8 0.2 3.3 2.5 3.2 12.0 12.5 2.0 39.9 30.2 398.0 496.0 69.8 9 0.2 3.3 2.5 3.2 12.0 12.5 2.6 39.6 29.8 434.0 509.0 89.6 10 0.2 3.3 2.5 3.2 12.0 12.5 2.6 40.0 30.0 434.0 509.0 90.3 11 0.2 3.5 2.5 3.4 12.0 12.5 2.5 41.9 29.7 434.0 509.0 87.4 12 0.2 3.4 2.5 3.3 12.0 12.5 2.6 40.8 29.7 434.0 509.0 90.8 13 0.2 3.1 1.9 3.0 16.0 16.5 2.6 49.6 29.9 434.0 509.0 121.6 14 0.2 3.2 1.9 3.1 16.0 16.5 2.5 51.7 30.0 434.0 509.0 120.0 15 0.2 3.2 1.9 3.1 16.0 16.5 2.6 50.9 30.0 434.0 509.0 117.8 -25- t/d e1/d e2/d e1/d0 D d0 t e1 e2 fy fu FFEM-Fexp 16 0.2 3.1 1.9 3.0 16.0 16.5 2.6 49.5 30.0 434.0 509.0 104.4 17 0.3 2.8 2.5 2.7 12.0 12.5 3.9 33.2 30.3 401.0 427.0 103.1 18 0.3 2.8 2.5 2.7 12.0 12.5 3.9 33.4 30.4 401.0 427.0 100.4 19 0.3 2.8 2.5 2.6 12.0 12.5 3.9 33.1 30.5 401.0 427.0 97.2 20 0.3 2.8 2.5 2.7 12.0 12.5 3.8 34.1 30.4 401.0 427.0 100.2 21 0.2 2.6 1.9 2.5 16.0 16.5 3.8 41.5 30.4 401.0 427.0 160.4 22 0.2 2.4 1.9 2.3 16.0 16.5 3.8 37.6 30.4 401.0 427.0 164.1 23 0.2 2.5 1.9 2.5 16.0 16.5 3.8 40.6 30.5 401.0 427.0 161.7 24 0.2 2.6 1.9 2.5 16.0 16.5 3.8 41.3 30.3 401.0 427.0 163.0 FE simulation reported in [26] 25 0.0 3.1 1.9 3.0 16.0 16.5 0.5 50.0 30.0 443.0 499.0 9.7 26 0.1 3.1 1.9 3.0 16.0 16.5 1.0 50.0 30.0 443.0 499.0 23.8 27 0.1 3.1 1.9 3.0 16.0 16.5 1.5 50.0 30.0 443.0 499.0 42.8 28 0.1 3.1 1.9 3.0 16.0 16.5 2.0 50.0 30.0 443.0 499.0 72.9 29 0.2 3.1 1.9 3.0 16.0 16.5 2.5 50.0 30.0 443.0 499.0 103.9 30 0.2 3.1 1.9 3.0 16.0 16.5 3.0 50.0 30.0 443.0 499.0 125.6 31 0.3 3.1 1.9 3.0 16.0 16.5 4.0 50.0 30.0 443.0 499.0 134.4 32 0.4 3.1 1.9 3.0 16.0 16.5 6.0 50.0 30.0 443.0 499.0 134.1 33 0.2 5.6 1.9 5.5 16.0 16.5 2.5 90.0 30.0 443.0 499.0 141.9 34 0.2 5.0 1.9 4.8 16.0 16.5 2.5 80.0 30.0 443.0 499.0 141.9 35 0.2 4.4 1.9 4.2 16.0 16.5 2.5 70.0 30.0 443.0 499.0 141.8 36 0.2 2.5 1.9 2.4 16.0 16.5 2.5 40.0 30.0 443.0 499.0 96.5 37 0.2 1.9 1.9 1.8 16.0 16.5 2.5 30.0 30.0 443.0 499.0 71.2 38 0.2 1.6 1.9 1.5 16.0 16.5 2.5 25.0 30.0 443.0 499.0 54.0 39 0.2 1.0 1.9 1.0 16.0 16.5 2.5 16.5 30.0 443.0 499.0 39.8 40 0.2 3.1 0.9 3.0 16.0 16.5 2.5 50.0 15.0 443.0 499.0 109.0 41 0.2 3.1 1.3 3.0 16.0 16.5 2.5 50.0 20.0 443.0 499.0 107.0 New FE Simulations 42 0.0 3.3 2.5 3.2 12.0 12.5 0.5 40.0 30.0 443.0 499.0 8.2 43 0.1 3.3 2.5 3.2 12.0 12.5 1.0 40.0 30.0 443.0 499.0 20.3 44 0.1 3.3 2.5 3.2 12.0 12.5 1.5 40.0 30.0 443.0 499.0 35.4 45 0.2 3.3 2.5 3.2 12.0 12.5 2.0 40.0 30.0 443.0 499.0 67.5 46 0.2 3.3 2.5 3.2 12.0 12.5 2.5 40.0 30.0 443.0 499.0 80.7 47 0.3 3.3 2.5 3.2 12.0 12.5 3.0 40.0 30.0 443.0 499.0 84.7 48 0.3 3.3 2.5 3.2 12.0 12.5 4.0 40.0 30.0 443.0 499.0 84.3 49 0.2 3.3 1.3 3.2 12.0 12.5 2.5 40.0 15.0 443.0 499.0 80.9 50 0.2 3.3 1.7 3.2 12.0 12.5 2.5 40.0 20.0 443.0 499.0 82.3 51 0.1 3.3 1.3 3.2 12.0 12.5 1.5 40.0 15.0 443.0 499.0 40.2 52 0.1 3.3 1.7 3.2 12.0 12.5 1.5 40.0 20.0 443.0 499.0 38.8 53 0.1 3.3 2.5 3.2 12.0 12.5 1.5 40.0 30.0 443.0 499.0 37.8 For the sake of simplicity , the results of the validation study are summarized in Table 3 reporting for 487 the 8 considered models , the professional factors calculated for each one of the 53 cases , determined 488 as the ratio between the predicted over experimental/simulated value of the maximum load. 489 Additionally, in the same table the min/max and mean values, standard deviation and coefficient of 490 variation of the professional factors obtained with the considered models are reported . Due to the size, 491 the complete table containing the detailed results of the application of the various models is delivered 492 only in a specific a nnex. The results of the application of the mode ls are also rep orted in Fig. 18, where 493 the experimental or simulated values are graphically compared to the calculated ones . 494 495 -26- Table 3 – Accuracy of the models 496 AISI S100 (Eqs.1,2) AISC (Eqs.3,5) EC3 Moze Proposal (1) Proposal (2) mf=0.75 mf=1 Eq.3 Eq.5 Eqs.6,7 Eq.8 Eq.16. 1,17.1 Eq.16. 2,17.2 PF PF PF PF PF PF PF PF Experimental Data [26] 1 1.05 1.40 1.40 0.67 1.17 1.40 1.04 0.99 2 0.93 1.24 1.24 0.60 1.02 1.22 0.92 0.88 3 0.90 1.20 1.20 0.58 0.99 1.19 0.88 0.84 4 0.81 1.08 1.08 0.52 0.90 1.08 0.80 0.76 5 0.80 1.07 1.07 0.52 0.89 1.07 0.91 0.90 6 0.78 1.04 1.04 0.50 0.87 1.04 0.89 0.88 7 0.73 0.97 0.97 0.47 0.81 0.97 0.84 0.83 8 0.78 1.04 1.04 0.50 0.87 1.04 0.89 0.88 9 0.79 1.05 1.05 0.54 0.88 1.05 1.05 1.04 10 0.78 1.04 1.04 0.53 0.86 1.04 1.03 1.02 11 0.79 1.05 1.05 0.54 0.88 1.05 1.04 1.02 12 0.77 1.03 1.03 0.53 0.86 1.03 1.03 1.01 13 0.77 1.03 1.03 0.53 0.86 1.03 0.88 0.84 14 0.78 1.03 1.03 0.53 0.86 1.03 0.87 0.83 15 0.81 1.09 1.09 0.56 0.91 1.09 0.93 0.89 16 0.90 1.19 1.19 0.61 0.91 1.19 1.01 0.96 17 0.86 0.92 0.92 0.65 0.85 0.92 0.92 0.92 18 0.89 0.94 0.94 0.67 0.88 0.94 0.94 0.94 19 0.91 0.97 0.97 0.69 0.89 0.97 0.97 0.97 20 0.87 0.95 0.95 0.66 0.88 0.95 0.95 0.95 21 0.73 0.97 0.97 0.55 0.68 0.81 0.81 0.81 22 0.71 0.95 0.87 0.53 0.60 0.72 0.72 0.72 23 0.72 0.96 0.96 0.54 0.66 0.79 0.79 0.79 24 0.72 0.96 0.96 0.54 0.66 0.80 0.80 0.80 FE simulation reported in [26] 25 1.11 1.48 2.47 1.31 1.56 2.47 1.02 0.97 26 1.21 1.61 2.01 1.07 1.54 2.01 1.04 0.99 27 1.23 1.64 1.68 0.89 1.40 1.68 1.05 1.00 28 0.99 1.31 1.31 0.70 1.09 1.31 0.96 0.92 29 0.86 1.15 1.15 0.61 0.96 1.15 0.96 0.92 30 0.86 1.14 1.14 0.61 0.95 1.14 1.07 1.03 31 1.07 1.12 1.12 0.76 1.12 1.12 1.12 1.12 32 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 33 0.63 0.84 0.84 0.45 0.70 0.84 0.70 0.67 34 0.63 0.84 0.84 0.45 0.70 0.84 0.70 0.67 35 0.63 0.84 0.84 0.45 0.70 0.84 0.70 0.67 36 0.93 1.24 1.23 0.66 0.84 1.00 0.84 0.80 37 1.26 1.68 1.14 0.90 0.85 1.02 0.85 0.81 38 1.66 2.22 1.16 1.16 0.93 1.12 0.93 0.89 39 Not applicable to this case 1.00 0.94 0.80 40 0.82 1.10 1.10 0.59 0.92 1.10 0.96 0.88 41 0.84 1.12 1.12 0.60 0.76 1.12 0.96 0.89 New FE Simulation s 42 0.99 1.31 2.19 1.17 1.39 2.19 0.93 0.94 43 1.24 1.65 1.77 0.94 1.36 1.77 1.00 0.99 44 1.14 1.52 1.52 0.81 1.27 1.52 1.07 1.06 45 0.80 1.06 1.06 0.57 0.89 1.06 0.90 0.89 46 0.83 1.00 1.00 0.59 0.93 1.00 1.00 1.00 -27- AISI S100 (Eqs.1,2) AISC (Eqs.3,5) EC3 Moze Proposal (1) Proposal (2) 47 0.95 0.96 0.96 0.68 0.96 0.96 0.96 0.96 48 0.96 0.96 0.96 0.91 0.96 0.96 0.96 0.96 49 0.83 1.00 1.00 0.59 0.93 1.00 1.00 1.00 50 0.82 0.98 0.98 0.58 0.91 0.98 0.98 0.98 51 1.01 1.34 1.34 0.71 1.12 1.34 1.01 0.94 52 1.04 1.39 1.39 0.74 1.16 1.39 1.03 0.97 53 1.07 1.43 1.4 0.76 1.19 1.43 1.00 0.99 Min 0.63 0.84 0.84 0.45 0.60 0.72 0.70 0.67 Max 1.66 2.22 2.47 1.31 1.56 2.47 1.12 1.12 Mean Value 0.92 1.16 1.17 0.67 0.96 1.15 0.94 0.91 st. dev. 0.19 0.26 0.33 0.20 0.21 0.34 0.10 0.10 CV 0.21 0.23 0.28 0.30 0.22 0.29 0.11 0.11 The results summarized in Fig.1 8 and Table 3 show that the proposed model s, due to the inclusion of a 497 corrective coefficient able to properly account for the buckling failure mode detected in [26] , are able 498 to accurately predict the resistance of the connection. In fact, with the proposed regression s (Eqs.16) , 499 the model s provide an accurate estimate of the load , with a professional factor that is on average equal 500 to 0.9 4 or 0.91 , with a very low coefficient of variation, equal to the 11%. As expected, due to the 501 assumption related to the maximum achievable bearing strength which, following the suggestion made 502 by Može [3] has been limited to 3, the model results to be slightly con servative. Conversely, as already 503 preliminarily pointed o ut in [26], the codified models do not seem to provide a sufficient accuracy and, 504 in fact, even though some of them (e.g. EC3 and AISI S100 with modification factor equal to 0.75) can 505 achieve mean va lues of the professional factors close to one, they are not able to accurately predict the 506 resistance in many cases. This is demonstrated by the high dispersion of the professional factors 507 (ranging from 0.6 to 1.6 for EC3 and AISI S100) and high coefficien ts of variation (21% and 22% for 508 AISI S100 and EC3). As expected, t he model of Može , (Eq.8) and the model suggested by AISC 360 -10 509 (Eq.(3) ), being developed for overlapped connections with a full out -of-plane confinement of the bolt 510 hole , overestimate sign ificantly the value of the ultimate capacity and provide high values of the 511 coefficient of variation, (28% and 29% for AISC and Može models). Additionally, as also expected, the 512 model suggested by AISC 360 -10 for connections with long bolts (Eq.5) signific antly underes timates 513 the resistance because it represents mainly a conservative extension of the formula normally used for 514 pins in bearing. Indeed in case of application of Eq. (5) the mean v alue of the professional factor results 515 to be equal to 0.6 7 with a coefficient of variation equal to the 30%. 516 -28- 517 518 519 520 Fig. 18 – Accuracy of the models (literature and proposed) 521 522 020406080100120140160180200 0 20 40 60 80 100 120 140 160 180 200Experimental/Simulated data Predicted [Eq.1]Accuracy of AISI S100 mf=0.75 020406080100120140160180200 0 20 40 60 80 100 120 140 160 180 200Experimental/Simulated data Predicted [Eq.1]Accuracy of AISI S100 mf=1 020406080100120140160180200 0 20 40 60 80 100 120 140 160 180 200Experimental/Simulated data Predicted [Eq.3]Accuracy of AISC specification -1 020406080100120140160180200 0 20 40 60 80 100 120 140 160 180 200Experimental/Simulated data Predicted [Eq.5]Accuracy of AISC specification -2 020406080100120140160180200 0 20 40 60 80 100 120 140 160 180 200Experimental/Simulated data Predicted [Eq.6,7]Accuracy of EC3 part 1.8 and 1.3 model 020406080100120140160180200 0 20 40 60 80 100 120 140 160 180 200Experimental/Simulated data Predicted [Eq.8]Accuracy of Može model 020406080100120140160180200 0 20 40 60 80 100 120 140 160 180 200Experimental/Simulated data Predicted [Eqs.16.1,17.1]Accuracy of the Proposed Model -1 020406080100120140160180200 0 20 40 60 80 100 120 140 160 180 200Experimental/Simulated data Predicted [Eqs.16.2,17.2]Accuracy of Proposed model -2 -29- 6. Conclusions 523 In this work the behaviour of connections made with SHS and long bolts has been investigated. The 524 work has been mainly motivated by the preliminary results obtained in a previous experimental 525 analysis developed at t he L aboratory on Materials and Structures of the U niversity of Liège. The 526 previous work has evidenc ed a particular buckling failure mode which is not explicitly considered 527 neither in the current EC3 formulas nor in other codified design procedures or literature models, such 528 as AISI S100, AISC 360 -10 or the model recently proposed by Može in view of the definition of a new 529 Eurocode standard . In fact, such experimental analysis pointed out that in case of connections made 530 with tubular members and long bolts, the lack of confinement effect in the internal part of the tube 531 may lead during the load process to the development of a particular type of out-of-plane local buckling 532 of the bolt hole (Fig.6). Therefore, due to this experimental observation, the analyse s performed in this 533 paper have been devo ted to the develop ment of a design equation for this particular connection 534 typology, based on the recent proposal of Može , able to account for the possible development of local 535 buckling at the bolt hole . 536 In order to define a design formula, a procedure able to predict the ultimate load of the tube plate 537 subjected to the action applied from the bolt in bearing has been set up . This procedure has been 538 based on the application of the Winter’s approach for the determination of the post -elastic buckling 539 load and on the development of a simplified FE model in SAP 2000 to determine the elastic buckling 540 load. Afterwards , exploiting the developed approach , a parametric analysis has been carried out in 541 order to define a correlation between the reduction of resistance due to the buckling failure mode and 542 the geo metrical properties of the tube plate. The analysis has shown that the local buckling effects, in 543 the analysed connection, may be significant depending on the values of t/d, e1/d and of e2/d 544 demonstrating that the buckling check is necessary if t/d is lower than one of the two limit value s 545 given by Eq s. (18). Finally, in order to define a simple coefficient to be used in design to account for 546 this particular failure mode, two different reg ression analyse s of the data generated by mea ns of 547 parametric simulations have been carried out, proposing to calculate the bearing resistance at bolt 548 hole in case of connections made with SHS and long bolts adopting the model s described by Eqs.(17). 549 The proposed design equation s have been validated against a reference set of data composed by 53 550 tests or FE simulations , in part collected from [26] and in part developed within this work . The 551 comparison between the data and the predictions obtained with the proposed equation s and with 552 other six literature models has evidenced the poor accuracy of the classical bearing models (calibrated 553 mainly on lap shear connections) for this specific type of connection, as well as the improved accuracy 554 obtained with the proposed m odel s which are able to achieve mean professional factor equal to 555 0.94 /0.91 , with a coefficient of variation lower than the 1 1%. 556 -30-