Yield and ultimate strengths determination of a blind bolted endplate connection to square hollow section column

Engineering Structures 111 (2016) 345–369

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Yield and ultimate strengths determination of a blind bolted endplate connection to square hollow section column Zhi-Yu Wang ⇑, Qing-Yuan Wang Department of Civil Engineering & Mechanics, Institute of Architecture and Environment, Sichuan University, Chengdu, Chengdu 610065, PR China Sichuan Provincial Key Laboratory of Failure Mechanics and Engineering Disaster Prevention & Mitigation, Sichuan University, Chengdu 610065, PR China

a r t i c l e

i n f o

Article history: Received 17 February 2015 Revised 23 November 2015 Accepted 24 November 2015 Available online 7 January 2016 Keywords: Blind bolts Endplate joint Yield/ultimate strengths Hollow section column

a b s t r a c t This paper evaluates the yield and ultimate strengths of a blind bolted endplate to square tubular column connection in joint tensile region. It outlines a research work of eleven connection tests with focus on the strengths of the connection components under tensile loads. The influences of the bolt gauge width, slenderness of tubular columns and bolt size details on the strengths of the connections in three failure modes are reported. Based on the experimental test results, yield strength determination methods are examined and discussed on the basis of deformations of the tube connecting face and the overall connection. Finite element models explicitly allowing for the configuration details of the Hollo-Bolt are then presented and compared with the test results. Subsequently, an analytical model is proposed for the prediction of yield strength with due consideration given to the plastification of SHS column and combined action of the blind bolt related to three failure modes. Based on an augmented consideration of membrane effect and circumferential locking action of the Hollo-Bolt, the analytical model is also extended for the prediction of ultimate strength. Additionally, two geometric parameters in characterizing the failure modes of the connection are discussed from a numerical parametric study. The analytical results presented in this paper provide some essential information for the strength design recommendations of such a blind bolted endplate connection to square tubular column. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Square hollow section (SHS) members without asymmetrical properties and weaknesses of minor axis have superior compression strength over steel open section members. It is also evident that SHS members have relatively low surface area and high strength to weight ratio which have resulted in their good application as column members in single-storey and multi-storey buildings. The conventional structural frame system is often involved with the welded connections between open section beams and SHS columns. However, the application of this connecting method is affected by on-site welding errors and labour costs. Recently, as an alternative, great attention has been drawn to the blind bolted connections to SHS columns in preference of their shop preparation and on-site bolting. The development of newly blind bolting technology, also named as one-side bolting, nowadays has made it possible to overcome the difficulties in the installation of bolted ⇑ Corresponding author at: Department of Civil Engineering & Mechanics, Institute of Architecture and Environment, Sichuan University, Chengdu, Chengdu 610065, PR China. E-mail address: [email protected] (Z.-Y. Wang). http://dx.doi.org/10.1016/j.engstruct.2015.11.058 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

connections to SHS columns with rarely sufficient free access. The AJAX Oneside bolt system [1] is such an application and can be analogous to the configuration of standard bolt when the collapsible washer is guided through the bolt hole and located against the joint, as shown in Table 1(a). The use of Lindapter Hollo-Bolt [2] is another application focused in this study. As shown in Table 1(b), while the threaded cone is drawn towards the head of the bolt as it is installed, the legs or the flaring part of the sleeves (flaring sleeves) are splayed out to provide a secure mechanical interlock of the connection. In this sense, the tension of the Hollo-Bolt is obtained as the sloping surface of the flaring sleeve is locked against circumferential edge of the bolt clearance hole (circumferential locking action). It is noted that this action is distinguished from that of the nut and the washer in the standard bolts which are installed underneath flat face of gripped plate. Fundamental and comprehensive testing [3–7] has been conducted on the tensile and shear capacities of the Hollo-Bolt and its related blind bolted connections to SHS columns. Based on these test results, this type of blind bolted connection has been recognized in the design guide books [8,9] for simple connection design. An exemplified application is the use of Hollo-Bolts in SHS steel splices and bracket connections in the project of Huntington

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Nomenclature Special symbols adopted are not summarized below, but their explanations or illustrations are included in the texts and graphs CD, FD annotations related to overall connection and its tube connecting face deformation respectively FE, EX finite element modelling and experimental analysis respectively ESA stress area around the bottom edge of bolt clearance hole HSS hollow structural section SG strain gauge SHS square hollow section Asl effective shearing area of Hollo-Bolt sleeve Bus, Buh ultimate strengths of standard bolts in tension and Hollo-Bolt sleeve in shear respectively b0 outside width of square tubular member bs gap distance (slot width) between adjacent sleeve pieces b0 , c0 length and width of an equivalent rectangular area respectively in the evaluation of the failure load in tension zone as suggested in Gomes equations b bolt gauge width-to-tube width ratio (g/b0) b1, c1 coefficients related to yield load in SCI equation C01 , C02 , C03 , C1, C2, C3, C4 curve fitting constant values diameter of bolt clearance hole dbh dba effective diameter of bolt clamping area dsh, Ash nominal diameter and stress area of bolt shank respectively dsl outer diameter of Hollo-Bolt sleeve dtcm diameter of intermediate circle between upper and lower parts of threaded cone dm mean diameter of bolt head energy dissipation due to the yield line deformation Dd E expenditure of energy for the external work done by the external load, P Fb bolt tensile load Fp bolt preload Fpo additional pullout load of Hollo-Bolt Fg connection tensile load Fy,CD, Fy,FD yield loads determined from F–D curves related to the deformations of overall connection and tube connecting face deformation respectively F 3%b0 connection load determined by a 3%b0 deformation limit Fy,G-loc, Fy,G-glo yield loads calculated from local and global failure mechanisms of Gomes formula respectively Fy,Yeo, Fy,SCI yield load calculated from Yeoman formula and SCI formula respectively

Botanical Gardens [10]. In contrast, there is currently still a significant gap of knowledge on semi-rigid and partial strength blind bolted connections to SHS columns which limit their application in the moment-resisting steel frame. To fill this gap, several experimental research works have been performed on the blind bolted open section beam-to-SHS column joints [11–14] and their equivalent blind bolted T-stub assemblies [15–19]. In the latter studies, the performance of the tension zone of the joint can be idealized as shown in Fig. 1. This equivalence has also gained its significance within the design philosophy of component method [20] in which the joint behaviour is determined by assembling the contributions of individual components of known mechanical properties. Prelim-

Fy,mode i, Fy,mode ii, Fy, mode iii yield loads calculated from proposed equations corresponding to flexural yielding modes i, ii and iii respectively Fy, Fu yield strength and ultimate (maximum carried) strength of the connection respectively g bolt gauge width htc distance between centres of joint compression and tension zones K dimensionless nut factor k correction factor in Gomes equations for flexural and combined mechanisms li, hi length and rotation of ith yield lines respectively Mp plastic moment per unit length Nib,x, Nib,y contact force per unit length around bolt clearance hole edge in the x and y directions respectively nv number of bolts in connection nr number of rows of bolts in connection P external load assumed in the yield line analysis p bolt pitch width tc wall thickness of square tubular member ts element thickness of Hollo-Bolt sleeve in contact with threaded cone DCD, DFD deformation of overall connection and that of tube connecting face respectively Dy,CD yield deformation of overall connection Hollo-Bolt sleeve flaring angle, taken as 10° in this ha study R radius of yield fan Tin input torque value d unit deflection at bolt position dt deflection between the bottom surface of flange plate and the centre of deformed bolt clearance hole Ec, ry modulus of elasticity and yield stress of tube steel material respectively rus, ruh ultimate tensile stresses of standard bolt shank and Hollo-Bolt sleeve materials respectively rn, rr, rc normal, radial and circumferential stresses on the ESA respectively rm bolt tensile load induced meridional stress on the bolt clearance hole t Poisson’s ratio k ratio of R to b0 g geometric correction factor related to local deformation of bolt clearance hole W0 tube wall thickness-to-half wall width ratio (tc/0.5b0) er , ec radial and circumferential strains around bolt clearance hole respectively jFC, Τs tangential curvature and stress on the shear area of flaring sleeves respectively

inary work [21] has indicated the feasibility of the component method for the analysis of bolted connection to SHS column connections. However, the inherent flexibility of the SHS column connecting face and the distinct configuration & clamping mechanism of the blind bolt differ significantly from the deflection of the flange plate and deformation of the standard bolt in the classical T-stub assembly model codified in current design guidance. The design strength of the bolted connection is determined by the summation of the contribution of an individual bolt-row in the tension zone of the joint [22]. Some recent studies have been conducted to obtain the experimental observation of the connections with different type of blind bolts to tubular sections for

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Table 1 Schematic of typical blind bolting systems. (a) AJAX Oneside bolt system [1] Connecting plate

Solid washer Nut

Bolt shank Bolt head

Collapsible washer

(b) Lindapter Hollo-Bolt system [2]

Collar Bolt head

Sleeve

Leg or flaring sleeve Bolt shank

Threaded cone

Joint tension zone

Blind bolt

H beam

SHS column

Fig. 1. Illustration of tension zone of an extended endplate connection to a SHS column.

design [23]. With regard to the contribution of the flexibility of the SHS column connecting face, several relevant experimental studies have been reported [24–27] for the tensile behaviour of transverse branch plate-to-rectangular hollow structural section (HSS) member connections. Packer and Morris [28] studied the strength of the tension branch members welded orthogonally to I-section webs using combined flexural and punching shear yield line analyses. Afterwards, Gomes et al. [29] appears to be the first one extending the combined yield line mechanism of aforementioned welded connection [28] into the strength study of beam (I-section)-tocolumn (I-section) minor-axis bolted connection. Although this model was also suggested [21] and referred [30] in an analogical design strength prediction of the bolted connections to HSS columns, no further consideration was given for the interaction between the bolt in tension and the column connecting face in flexural yielding. Yeomans [13,31] suggested a straight yield line formula for the hollow section face plastification capacity which appears in the current CIDECT design guide 9 [9]. Likewise, this formula was derived on the basis of the welded joint, i.e. a rectangular to rectangular hollow section T-joint, and gave inexplicit consideration of the size effect of the blind bolt. As a result, the group tension test results of the Hollo-Bolted connections in his report showed that this formula may give much lower prediction of

failure loads, i.e. approximately 30–50% error. Therefore, it is of great necessity to better understand plastification of the SHS column face and its combined effect of the blind bolt so that more accurate evaluation of the connection strength can be performed from an efficient and economic design standpoint. For the purpose of improving the strength evaluation model previously reported in the literature, an attention was drawn in gaining evidence on the characterization of the blind bolts and SHS columns in the tension zone of the joint. This paper presents an experimental study consisting eleven connection specimens with three typical failure modes in analogy with that in Eurocode 3 [22]. The connection behaviour patterns related to the plastic deformation on the SHS column connecting face and sidewall are compared and discussed. Afterwards, a finite element model (FE), explicitly allowing for the local geometry of three part assembled unit of the Hollo-Bolt, is presented to trace the failure mechanism of the blind bolt as well as the connection. Based on the results of experimental test and finite element modelling, the analytical models are proposed for the yield strength calculation allowing for the flexural strength of the SHS column connecting face and the combined action of the blind bolt. Following the previous work conducted by the authors [32], in particular, the mechanical interlock mechanism related circumferential locking action of the

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Hollo-Bolt and its effect on the bolt pullout are taken in the development of the analytical model for the ultimate strength calculation. Finally, the accuracy and applicability of the proposed analytical model with the range of related geometric parameters in the strength evaluation are discussed. 2. Experimental programme As stated above, the component method codified in Eurocode 3 [22] is a general approach in characterizing the bolted joints, provided that the elementary components governing the joint behaviour are identified. With regard to the blind bolted endplate to square tubular column connections studied herein, three joint components can be readily identified as: endplate in bending, SHS column wall in flexural deformation and blind bolt in tension. The strength of the first component can be evaluated by means of an equivalent T-stub assembly as well documented in Eurocode 3 [22,33]. Hence, this part of contribution is not considered in details in this study. Rather, the strength of the connection is determined by examining the capacity of the other unrecognized connection components under two failure sources: plastification of the SHS column connecting face and tension of the blind bolt, given that the endplate is adequately designed and is noncritical. The yield and ultimate strengths are determined from test load–deformation relationships. 2.1. Geometric and material properties of test specimens Eleven test specimens were designed with the purpose of attaining the development of three failure modes related to the SHS column and the blind bolt. To this end, the geometric parameters of the bolted endplate connection, such as the bolt gauge width (g), the tube wall thickness (tc) and the bolt shank diameter of the blind bolt (dsh) have been determined for each specimen. The endplate with the thickness of 25 mm was designed so as to behave as a rigid plate and isolate the plastification of the SHS column and the tension of the blind bolt. The general configuration of the test specimens is illustrated in Fig. 2. All test specimens have been assembled by T-stubs and SHS columns with outside width of 150 mm and length of 550 mm. Apart from the bolt gauge width (g), the bolt pitch width (p) was kept as 100 mm for all test specimens. According to the force transfer mechanism of the Hollo-Bolt (HB) previously studied by the authors [32], the geometric parameters of the Hollo-Bolt are defined as the diameter of bolt clearance hole (dbh), the outer diameter of Hollo-Bolt sleeve (dsl), the diameter of intermediate circle between upper and lower parts of threaded cone (dtcm), the element thickness of flaring sleeve (ts), the width of the slot between the adjacent flaring sleeves (bs) and sleeve flaring angle (ha) in characterizing the effective contact area between the threaded cone and the flaring sleeve, as shown in Fig. 3(a). Since there are still some other types of blind bolts, e.g. the AJAX Oneside

Loading direction T stub Bolt

T stub flange

SHS column 0 c

0

Fig. 2. Geometric parameters of blind bolted T-stub-to-SHS connection.

bolt system, have washer & nut clamping similar to the standard bolt, the M16 standard bolt (ST) of 4.8 class with similar strength of the flaring sleeves of the Hollo-Bolt was also adopted in fastening the components of the specimens. It was considered that the bolt strengths are not too strong so that the combined failure mode of the bolt and the tube wall could take place in an assumed mechanism. The effective diameter of the clamping area (dba) of the standard bolt is taken as the mean value of longitudinal and transverse lengths of the bolt head or nut, as shown in Fig. 3(b). The geometric details of the test specimens are listed in Table 2 and represented as ‘BC-(ST/HB)dsh–tc–g’, which can be referred from foregoing annotations. Of test specimens, BC-HB16-5-60, BC-HB165-90, and BC-HB16-8-60 are used in the representation of the tensile region of a full-sized blind bolted endplate joint with the same geometry reported by Yeomans [13,31]. Additionally, the previous test results [15] of similar connections with 200  200 mm section SHS columns of 8 mm, 10 mm and 12.5 mm tube wall thicknesses and S355 steel materials, labelled as RF-HB16-8-90, RF-HB16-1090 and RF-HB16-12.5-90 respectively, are added in Table 2 for a comparison. The Hollo-Bolt size, bolt gauge width and bolt pitch width are kept as HB16, 90 mm and 100 mm respectively for these referred connections. The steel materials used for hot-rolled SHS columns and standard bolts conform to the standards GB/T1591-2008 [34] and ISO 898-1:2009 [35,36] respectively. Tensile tests were performed using three identical coupon specimens extracted from the tubes for each kind of section and the bolts of the same batch to obtain average results. Related mechanical property and chemical composition are described in Table 3. Also, the mean value of three hardness test was used to determined strength of the Hollo-Bolt sleeve which often becomes the weakest part carrying the load in such an assembled unit. 2.2. Test set-up and instrumentation In the test connections, the T-stub and the SHS column were clamped together with the standard bolts and the Hollo-Bolts. All bolts were tightened with a special torque wrench in accordance with required torque values as: (1) 101 Nm for the M16 standard bolt; (2) 45 Nm and 190 Nm for the HB10 Hollo-Bolt and the HB16 Hollo-Bolt respectively [2]. The general torque equation [38] relating input torque value (Tin) to the preload (Fp) can be referred as:

T in ¼ F p Kdsh

ð1Þ

where K is the dimensionless nut factor, which can be taken as 0.2 for steel fasteners used in steel joints. Therefore, the bolt preload can be recast into:

Fp ¼

5T in dsh

ð2Þ

Therefore, the preloads for the test bolts can be calculated as 31.56 kN, 22.5 kN and 59.38 kN for the M16 standard bolt, HB10 Hollo-Bolt and HB16 Hollo-Bolt respectively. Besides, the preload can also be determined experimentally by converting the measured strains on the bolt shank below the head. Former reported tests [39,40] have shown the difference of the preloads for the HolloBolts between theoretical calculation and experimental measurement is small; thus the preload of the connection in this study was evaluated based on the theoretical calculation above. All connection specimens have been tested by means of a test rig developed in the Civil Engineering Laboratory at Sichuan University. The rig was composed of the assemblage of a series of steel profiles and connecting plates in which the test specimen was kept inside the reaction buttress as shown in Fig. 4. Two hydraulic jacks with each capacity of 500 kN and stroke of 150 mm were adopted to apply simultaneous jacking loads to the

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sl

s

Bolt shank core Flaring sleeve

0.5

sl

sh

0.5

bh

bh

T0

T0

Threaded cone contact area

sl tcm

s

T1

T1

a

i

s

ba

sh

i

ii

ii

T0-T0

tcm

T1-T1

(a) Details with a Hollo-Bolt

(b) Details with a standard bolt

Fig. 3. Geometric parameters for bolted connections.

Table 2 Connection geometric parameters. Connection identification

SHS member properties

BC-ST16-5-60 BC-HB10-5-60 BC-HB16-5-60 BC-ST16-5-90 BC-HB10-5-90 BC-HB16-5-90 BC-ST16-8-60 BC-HB10-8-60 BC-HB16-8-60 BC-ST16-8-90 BC-HB10-8-90 RF-HB16-8-90 [15] RF-HB16-10-90 [15] RF-HB16-12.5-90 [15]

Bolt properties (mm)

b0 (mm)

tc (mm)

g (mm)

b

W0

dsh

dbh

dba

dsl

dtcm

ts

bs

150 150 150 150 150 150 150 150 150 150 150 200 200 200

5 5 5 5 5 5 8 8 8 8 8 8 10 12.5

60 60 60 90 90 90 60 60 60 90 90 90 90 90

0.4 0.4 0.4 0.6 0.6 0.6 0.4 0.4 0.4 0.6 0.6 0.45 0.45 0.45

0.067 0.067 0.067 0.067 0.067 0.067 0.107 0.107 0.107 0.107 0.107 0.08 0.1 0.125

16 10 16 16 10 16 16 10 16 16 10 16 16 16

18 19 28 18 19 28 18 19 28 18 19 28 28 28

25 19 28 25 19 28 25 19 28 25 19 28 28 28

– 17.5 26 – 17.5 26 – 17.5 26 – 17.5 26 26 26

– 17 25 – 17 25 – 17 25 – 17 25 25 25

– 3 4 – 3 4 – 3 4 – 3 4 4 4

– 1.2 1.4 – 1.2 1.4 – 1.2 1.4 – 1.2 1.4 1.4 1.4

Table 3 Chemical compositions and mechanical properties of connection materials. Components

SHS column Bolt Hollo-Bolt sleeve

Chemical compositions (%)

Mechanical properties

C

Si

Mn

P

S

Yield stress (MPa)

Ultimate stress (MPa)

Elongation (%)

0.15 0.09 –

0.22 0.26 –

1.24 0.46 –

0.016 0.021 –

0.01 0.009 –

348 389 396

483 495 502

27 23 14

Rigid cross beam

Load cell

Auxiliary fixing assembly

T junction plate

T stub

Test connection

Jack movement Hydraulic jack Reaction buttress

Fig. 4. Experimental test set-up.

two ends of rigid cross beam with stiffeners. The jacking load sustained by an overhead rigid beam was then transferred as a tensile action to the web of the T-stub through double bolted T junction

plates. On the other hand, the restraint of the test specimen was provided via a very stiffer reaction buttress consisting of 300  200  8  10 mm H-section beams with end and mid-span

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2.3. Experimental results and observations 2.3.1. Failure mode and local plasticity pattern As already mentioned, the test connections were designed so that their strengths were determined by the plastification of the SHS column connecting face and the tension of the blind bolt.

Load, F (kN)

240 180 120 60 0

Deformation, Δ (mm)

stiffeners. Whilst the hydraulic jack was in incrementally jacking, it had a downward reaction which counteracted the pushing load of the specimen on the flange of the buttress. Using two sets of auxiliary fixing assembly, two load cells were installed between the hydraulic jack and the end of the rigid cross beam to monitor the incremental loads applied by two hydraulic jacks. The defining connection load (F) was taken by summating the simultaneous forces measured by these two load cells. All the test specimens were instrumented with four linear variable differential transformers (LVDTs) to measure the connection deformations. As shown in Fig. 5, LVDT1 and LVDT2 were positioned at the bilateral sides of the T-stub flange surfaces for the measurement of overall connection deformation. LVDT3 positioned at the middle of the tube sidewall face and LVDT4 positioned at the bottom of SHS mid-face were used to measure the deformation of the sidewall and the deformation of the tube connecting face from underneath side respectively. The connection deformation (DCD) was defined as the change in the distance between the mean value measured from LVDT1 and LVDT2 and that measured from LVDT3. Similarly, the deformation of the tube connecting face (DFD) was taken as the difference between the measured values of LVDT4 and LVDT3. The output and instant information of load and displacement were recorded by automatic data acquisition software. The initial loading rate of movement for the hydraulic jack of 0.004 mm per second was applied. Subsequent to the load–displacement curve becoming nonlinear, the loading rate was incrementally increased to 0.02 mm per second until the end of the test. During the test, the simultaneous working mode was applied to ensure that the applied load ratio was controlled within ±2% between two jacks (i.e. F1/F2). The effectiveness of this control has been confirmed from test monitoring data, as illustrated in Fig. 6 for the specimen BC-HB16-5-60. To allow for the strain distribution on the SHS column in the directions parallel and perpendicular to the bolt row, twelve strain gauges were located across the reverse side of the tube connecting face around the bolt clearance hole & SHS mid-face and on sidewall face. In particular, two strain gauges were installed at a given distance perpendicular to the bolt row to capture the strain following an assumption of 45° stress dispersion from the centre of the bolt, i.e. 0.5(b0  g) + 10 mm, as shown in Fig. 7. Since the T-stub was designed as rigid and remained elastic during loading progress, the majority of the connection deformation can be attributed to plastification of the SHS column connecting face and tension of the bolt.

40 30 20 10

0 110%

F1 /F 2 (kN)

350

105% 100% 95% 90%

Normalized loading time Fig. 6. Schematic of loading data of two hydraulic jacks (BC-HB16-5-60).

During the test, the load was applied until the test was stopped as the bolts were completely failed or pulled out of the bolt clearance holes, where upon a sudden drop in the strength occurred. The deformation pattern of the connection was recorded at each load step. As expected, the failure modes of test connections are generally attributed to the SHS column face and the bolt, for which an analogy can be drawn to the three possible failure modes of the T-stub model documented in Eurocode 3 [22]. As indicated from eleven test specimens, the following three failure modes can be identified as:  Failure mode i: complete flexural yielding of SHS column face and bolt pullout. This case occurs for the connections with the SHS column wall of considerably lower strength and stiffness than the bolt. All test specimens with SHS column wall thickness of 5 mm exhibit this failure mode. As shown in Table 4 (a), this failure mode is characterized by significant plastification around the bolt clearance hole on the SHS column connecting face. Also, there is an obvious inward deformation at the corner of the SHS column connecting face. In the loading progress, a significant localized bulging effect can be observed on the tube face around the bolt holes as a result of bearing action of the bolt head & washer for the standard bolt and the flaring sleeve for the Hollo-Bolt against the gripped area and the edge of the bolt clearance hole on the SHS column respectively. This effect finally leads to the bolt pullout from the bolt clearance hole on the SHS column connecting face. The flaring sleeve in contact with the edge of the bolt clearance hole is suffered from local depression as a result of clamping. Upon the bolt pullout, the flaring sleeve is locally squeezed and folded inward as a result of the hoop constraint of the bolt clearance hole.

Loading direction Bolt T stub flange

2

1

SHS connecting face 3 SHS 150X150

Fig. 5. Schematic of test instrument arrangement.

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SG-4 SG-12

SG-3

75

SG-2

SG-9 SG-5

75

SG-1

150

SG-10 SG-11

SG-6 100

150

SG-7

55

20

Tube connecting face

75

SG-8

Tube sidewall Fig. 7. Labels and locations of strain gauge on SHS column.

 Failure mode ii: bolt failure with local plastification of SHS column face. This case occurs for the connections with SHS column wall and the bolt of equivalent lower strength and stiffness. All the test specimens with SHS column wall thickness of 8 mm show this failure mode. As shown in Table 4(b), this failure mode is characterized by bolt failure and much less deformation with smaller plastification area than failure mode i. The corner of the tube connecting face represents limited inward deformation as well as local yielding. The typical failure of the bolt is the bolt shank failure for the standard bolt and the shear of the flaring sleeve of the Hollo-Bolt, resulting in the eventual pullout of reminder part of the bolt after failure.  Failure mode iii: bolt failure only. This case occurs when the SHS column wall has much greater strength and stiffness in contrast to the bolt. Only the specimen ‘BC-HB10-8-90’ in the test exhibits this failure mode as the shear failure of the flaring sleeve of the Hollo-Bolt without noticeable flexural deformation on the SHS column connecting face. This failure mode is clearly distinguishable from mode i especially when the bolt fails with limited plastification on the SHS column face. Since the connection tensile action is transferred to the SHS column face through the bolt component, aforementioned strain gauges were mainly used to trace the plastification development in the vicinity of the bolt clearance hole on the SHS column connecting face. The results of strain measurement are discussed below in contrast to the connection failure modes as well as the influential geometric parameters, i.e. tube wall slenderness and bolt size details.  Plasticity development in the longitudinal direction of the SHS column: as shown in Fig. 8(a) and (b), the specimens of the failure mode i (e.g. BC-HB16-5-60) exhibits first yielding and greater plastification at the corner of the tube connecting face

351

(SG-10). Afterwards, simultaneous plasticity development can be observed at the locations within the bolt gauge width range (SG-1 & SG-9) and at assumed stress dispersion point (SG-2 & SG-3) in the vicinity of the bolt clearance hole. SG-2 at the SHS mid-face with the distance of ‘0.5(b0  g)  tc + 10’ away from the bolt centre shows a greater plastic strain development than SG-3 which deviates 0.5g from the face centreline in transverse distance. In contrast, a significant difference can be observed for the specimens of failure mode ii (e.g. BC-HB16-860), as shown in Fig. 8(c) and (d). In this case, the corner of the tube connecting face (SG-10) shows limited and less plastification with respect to the other locations. Also, a close strain measurement of SG-2 and SG-3 indicates a nearly identical plasticity progress around the bolt clearance hole which can be attributed to the reduction of the plastic deformation at the connecting face and the sidewall with the increase of the SHS column wall thickness. In the case of failure mode iii, a similar trend but reduced magnitude of strains can be observed in the longitudinal direction of the SHS column for the specimen BCHB10-8-90, as shown in Fig. 8(e). In addition, the increase of the size of the Hollo-Bolt appears to notably increase the local plasticity around the bolt clearance hole especially for the failure mode i, as shown from the strain results of SG-2, SG-3 and SG-10 in Fig. 8(a) and (b).  Plasticity development in the transverse direction of the SHS column: similar to the plasticity development discussed above, significant plastification near the bolt clearance hole on the SHS column face for the failure mode i can be observed from the transverse strain measurement of SG-4 & SG-6 at the corner of the tube connecting face in Fig. 8(a) and (b). In contrast, such a difference is greatly reduced for the failure mode ii, although the surface close to the sidewall (SG-4 & SG-6) still exhibits relatively higher strain than the inner side of the SHS mid-face (SG-5). For failure mode iii, all these measurements are below yield strain which indicates further decreased deformation at the transverse direction of SHS column face. On the other hand, the strain measurement of SG-5 at the SHS mid-face indicates different progress of plastic deformation for the connections with varying failure modes. At the initial stage of loading, the inner side of the SHS mid-face is subjected to compression due to the clamping effect of the bolt. The resultant compressive strain increases constantly on the condition that the SHS column face has less flexibility which provides sufficient restraints for the connections in failure modes ii and iii as shown in Fig. 8 (c–e). Conversely, for the connections with flexible tube face in failure mode i, the bolt clearance hole is prone to suffer local expanding and bulging outside producing tensile stress on the inner side of the SHS mid-face. Therefore, the results in such case exhibit a transition of the strain from compression to tension when the connections reveal progressive yielding as shown in Fig. 8(a and b). Additionally, similar observation can be made for higher plasticity outside than inside of the bolt gauge range as the connection is in post-yielding. This can be evidenced from the greater value of SG-11 in contrast to that of SG-12, especially for the failure modes ii and iii, as shown in Fig. 8(c–e).  Plasticity development on the sidewall of the SHS column: as mentioned previously, as the SHS column sidewall deforms inward at the corner of the connecting face under loading progress, greater strain can be expected at the location of SG-7 for the failure mode i than that for the failure modes ii and iii. Besides, an obvious increase of the extent of plasticity can be observed for the connection with the increase of the size of the Hollo-Bolt by comparing the measurement of SG-7 in Fig. 8(c) and (d). In contrast, the SHS column sidewall midface almost remains elastic for all the connection cases plotted herein.

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Depression of sleeve after pullout

Flexural yielding Tube corner inward deformation (b) Failure mode ii (BC-HB10-8-60)

Shear of most of flaring sleeves

Local yielding Local plastification (c) Failure mode iii (BC-HB10-8-90)

Shear of all flaring sleeves

No visible deformation

2.3.2. Connection load–deformation response The mechanical response of the connection is evaluated through an examination of the relations of measured load (F) versus connection deformation (DCD) and tube connecting face deformation (DFD) As shown in Fig. 9, the test F–D plots are given in groups related to the bolt gauge width (g) varying from 60 mm to 90 mm and tube wall thickness (tc) varying from 5 mm to 8 mm. In each group, the test results are compared for the connections with different size of the bolt. The yield strength is defined as significant yielding in the critical region of the connection representing a clear nonlinearity in the force–deformation relation. Given that the plastification of the SHS column connecting face governs the strength of the connection when the tube wall is relatively thin, an imposed deformation limit of 3% of the connecting face width for rectangular hollow section [37,41] is annotated with its corresponding limit states design strength ðF 3%b0 Þ as illustrated in Fig. 10. Besides, an alternative

method of classical bilinear approximation of the yield loads is also given in the determination of the yield load of the connection. In this method, the first line represents the initial stiffness while the second line is the stiffness tangent to the inelastic branch of the curve. The yield load is then defined as the intersection of two straight lines approximating the F–D curve. This method has been widely used in the study of tensile behaviour of bare steel HSS members (e.g. [42]) and welded joints (e.g. [43]). As such, this yield load determination method was adopted in this study under the consideration that the test SHS columns have notable strain hardening response and formation of yield line mechanism comparable to aforementioned models reported in the literature. Using this determination method, the yield strengths corresponding to the connection deformation (DCD) and tube connecting face deformation (DFD) are given as Fy,CD and Fy,FD respectively. Additionally, the ultimate strength, Fu, is determined as the maximum carried load of the connection shown in the F–D plots.

Z.-Y. Wang, Q.-Y. Wang / Engineering Structures 111 (2016) 345–369

Fig. 8. Measurement of strains on the tube connecting faces and sidewall surfaces.

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250

250

3%b0 limit

150 100 50 0

3%b0 limit

200

BC-ST16-5-60-CD BC-HB10-5-60-CD BC-HB16-5-60-CD

0

10

20

BC-ST16-5-60-FD BC-HB10-5-60-FD BC-HB16-5-60-FD

30

Load, F (kN)

Load, F (kN)

200

150 100 BC-ST16-5-90-CD BC-HB10-5-90-CD BC-HB16-5-90-CD

50 0

40

0

10

(a) tc=5mm & g =60mm

200

0

0

10

20

BC-ST16-8-60-FD BC-HB10-8-60-FD BC-HB16-8-60-FD

30

Load, F (kN)

Load, F (kN)

3%b0 limit

300

100

30

40

(b) tc=5mm & g =90mm 400

BC-ST16-8-60-CD BC-HB10-8-60-CD BC-HB16-8-60-CD

20

Deformation, Δ (mm)

Deformation, Δ (mm) 400

BC-ST16-5-90-FD BC-HB10-5-90-FD BC-HB16-5-90-FD

3%b0 limit

300 200

BC-ST16-8-90-CD BC-ST16-8-90-FD BC-HB10-8-90-CD BC-HB10-8-90-FD

100 0

40

0

10

20

30

Deformation, Δ (mm)

Deformation, Δ (mm)

(c) tc=8mm & g =60mm

(d) tc=8mm & g =90mm

40

Fig. 9. Load (F)–deformation (D) relations for test connections.

u

FD

CD

0

y,FD y,CD

Load,

(kN)

3%b

Δ=3%

Deformation, Δ (mm)

0

Fig. 10. Schematic of yield load and ultimate load determination methods from F–D curves.

The test results of yield and ultimate strengths of the test connections are listed in Table 5. Previous experimental results [15] of similar connections to 200  200 mm section SHS columns with the tube wall thickness of 8 mm, 10 mm and 12.5 mm are also plotted in Fig. 11 for the sake of comparison. The strength of test connections will be discussed in details with respect to failure modes, geometric parameters and yield strength determination methods in the following sections. In the case of failure mode i (complete flexural yielding of SHS column face and bolt pullout), since the flexural deformation of the SHS column face governs the connection behaviour, the increase of the gauge distance from 60 mm (Fig. 9a) to 90 mm (Fig. 9b) leads to an obvious enhancement of the strength and stiffness. From a theoretical point of view, this can be explained as greater force is required with less lever arm distance between location of the bolt in tension as a loading point and the tube sidewall acting as a boundary restraint. On the other hand, it is noted that the size of the bolt has significant influence on the strength of the connection. In both graphs, the connection with HB16 Hollo-Bolt (dbh = 28 mm,

Table 5 Summary of test results of connection strength. Connection identification

BC-ST16-5-60 BC-HB10-5-60 BC-HB16-5-60 BC-ST16-5-90 BC-HB10-5-90 BC-HB16-5-90 BC-ST16-8-60 BC-HB10-8-60 BC-HB16-8-60 BC-ST16-8-90 BC-HB10-8-90 RF-HB16-8-90 [15] RF-HB16-10-90 [15] RF-HB16-12.5-90 [15]

Fy,CD (kN)

58.57 53.79 78.46 97.79 79.80 129.25 174.60 169.47 192.19 242.15 212.80 168.42 268.14 351.72

Fy,FD (kN)

60.52 55.16 81.62 101.89 81.34 132.24 181.87 176.40 196.39 245.99 219.30 – – –

F 3%b0 (kN)

76.52 65.52 87.01 110.59 115.09 157.24 183.14 181.87 200.10 – – 178.34 279.91 417.63

Fu (kN)

192.59 122.58 208.97 197.14 142.98 244.53 214.81 205.68 314.90 292.00 252.46 346.14 352.41 444.95

Failure modes

i i i i i i ii ii ii ii iii i ii ii

Ratios Fy,CD/Fy,FD

F 3%b0 =F y;CD

F 3%b0 =F y;FD

0.97 0.98 0.96 0.96 0.98 0.98 0.96 0.96 0.98 0.98 0.97 – – –

1.17 1.22 1.11 1.13 1.44 1.22 1.05 1.07 1.04 – – 1.06 1.04 1.19

1.13 1.19 1.07 1.09 1.41 1.19 1.01 1.03 1.02 – – – – –

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500

3% b0 limit

Load, F (kN)

400 300 200

RF-HB16-8-90 RF-HB16-10-90 RF-HB16-12.5-90

100 0

0

5

10

15

20

25

30

Deformation, Δ (mm) Fig. 11. Referred F–D curves for the connections with 200  200 mm SHS columns [15].

dsh = 16 mm) has nearly 70% higher ultimate strength in contrast to that with HB10 Hollo-Bolt (dbh = 19 mm, dsh = 10 mm) which can be roughly proportional to the enlargement ratio of the bolt clearance hole diameter. Also, the connection with ST16 standard bolt (dbh = 18 mm, dsh = 16 mm) exhibits nearly 40–50% higher ultimate strength than that with HB10 Hollo-Bolt (dbh = 19 mm, dsh = 10 mm) even if the bolt clearance hole is almost the same. This can be attributed to a much larger clamping area of the bolt head and washer in the standard bolt with respect to that of the flaring sleeve of the Hollo-Bolt. Hence, the flexural strength and plastification of the SHS column face is greatly influenced by the clamping area induced by the size of the bolt and its related component. In the case of failure mode ii (bolt failure with local plastification of SHS column face), the tube wall slenderness of these specimens is reduced by increasing the tube wall thickness. As a result of less plasticity development on the SHS column face, the strength and stiffness of the connections are significantly increased when comparing Fig. 9(a) and (c). However, such an increase is compensated by an apparently reduction of the deformity of the connection especially for the case when the bolt becomes the weakest component in the connection as shown in Fig. 9(d). Besides, it seems that the ultimate strength of the connection in this failure mode is less sensitive to the size of the bolt with respect to the case in the failure mode i. Regarding the failure mode iii, as expected, the specimen BC-HB10-8-90 shows much higher strength and stiffness but less deformability when comparing the other specimens with HB10 Hollo-Bolts.

2.3.3. Application of yield strength determination method A notional ultimate deformation limit (3%b0) based criterion and a classical bilinear approximation were described in the previous section to determine the strength of the connection. Although the proposal of former method is well based on the study of rigid welded joints, it was employed by Yeomans [31] to find the ultimate capacity of the flowdrill joints involving relatively small drilled bolt holes and tapped threads. To discuss the applicability of this method in this study, the imposed deformation limit of 3% of the width of the SHS column connecting face is also plotted as an indicator in Fig. 9 for comparing Fy,CD and Fy,FD in the load–deformation responses with related failure modes. For the specimens in failure mode i, although the shape of F– DCD curves is similar to that of F–DFD curves, there is an apparent lag of the tube connecting face deformation (DFD) behind that the connection deformation (DCD) since the initial stage of loading, as shown in Fig. 9(a) and (b). For the specimens in failure mode ii, in contrast, this lag is slightly reduced and its resultant difference can be observed after the yielding of the connection, as shown in

355

Fig. 9(c). This behaviour can be attributed to the pullout of the bolt body from the bolt clearance holes suffered from local deformation and enlargement, thereby indicating that, in addition to the tube connecting face deformation, the bolt elongation and its pullout displacement are also contributed to the overall connection deformation. Besides, the ultimate capacity of the connection in failure mode iii may even be achieved prior to the notional 3%b0 deformation limit, as shown in Fig. 9(d). Table 5 lists the strength results of the test connections using aforementioned notional ultimate deformation limit (3%b0) based criterion and classical bilinear approximation in the determination of strength. It can be seen that the former method results in even 44% higher prediction of strength than the latter related to the connection deformation and the tube connecting face deformation in the case of the failure mode i. Thus, unlike welded joints, the use of the notional ultimate deformation limit (3%b0) based criterion in the strength determination of the bolted connections herein is plagued by non-equivalent representation of the connection deformation to the tube connecting face deformation, thereby underestimating the yielding or failure of the connections induced by redistribution of internal forces. Alternatively, the use of the classical bilinear approximation in the idealization of the force versus connection deformation curve and force versus tube connecting face deformation curve resulted in almost identical (within 5%) comparative values of the yield strength for all the test specimens. This suggests that similar to the welded joints, the strength corresponding to the intersection of two straight lines approximating the F–D curve can be adopted to indicate the resistance of the plastic flexibility of the tube face together with plastic deformation of the bolt. Thus, the strength derived from the classical bilinear approximation was adopted in this study in the representation of the yield strength of the connection. 3. Finite element modelling The foregoing experimental tests provided general observations about the failure mode; however, they are limited by the number of the specimens and available locations of measurement. In this regard, the finite-element analysis being an efficient and costeffective tool to supplement experimental investigations was adopted in this study to further the understanding of the failure behaviour and to expand the range of geometric parameters studied. The finite element numerical models were developed using the ANSYS Academic Research, v.12.1 software package [44]. 3.1. Description of finite element models The three-dimensional finite element models were constructed to replicate all relevant properties of test specimens. In particular, a geometric spatial surface modelling technology has been adopted in replicating the flaring sleeve and threaded cone of the Hollo-Bolt so that all the related parameters can be remained as consistent with the test specimens. To accurately capture the stress behaviour in the region around the bolt clearance holes where likely failures would initiate, a refined mesh was made on the SHS column connecting face and the sleeves of the Hollo-Bolt. The coarse meshes were used for low stress areas such as the column end supports and the T-stub, as shown in Fig. 12. The rationality of the mesh refinement was examined by a sensitivity study which allows the optimum representation of model. 8-node solid brick elements (SOLID185) with three degrees of freedom per node were used in the definition of the solid component parts in the model. 3D interface contact and target elements, TARGE 170 and CONTA 174, with friction coefficient of 0.3 were

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Reaction force

Square tubular column

Applied force

Hollo-Bolt Symmetric boundary condition

T-stub Fig. 12. Representation of connection model with mesh layout and boundary condition.

used to simulate interacting surface condition related to: T-stub flange bottom surface and tube connecting face, bottom surface of the bolt head and top surface of the straight part of the sleeves, flaring sleeve and its restraint surface. Using 2-node PRETS179 elements with one translation degree of freedom, the bolt preload was applied as the bolt shank is cut along the existing element boundaries at control nodes followed by defining a control node on the pretension section normal to the preload. The input preload magnitude was based on the aforementioned calculated values. The restraints were applied in the finite element models in accordance with those for the test specimens. Given the connection has symmetry about longitudinal and transverse planes, i.e. planes passing the midline of bolt gauge width and the midline of bolt pitch width, only one quarter of the model was simulated in the modelling. Symmetric displacement boundary conditions were defined for the nodes along such two planes of symmetry. To avoid the error of nodal force caused by target mesh density, SURF154 element allowing for the structural surface effect was imported and the surface load was applied incrementally on the nodes of the web end surface of the T-stub. A nonlinear time step analysis with standard Newton–Raphson algorithm was employed for each model taken into account nonlinear material properties and large deformation conditions. Model validation and failure mode modelling are highlighted in the following sections.

3.2. Validation of the finite element models The finite element (FE) models developed in this study were validated through a comparison with the connection responses measured in the experiment (EX). As shown in Fig. 13, it can be seen that the FE model prediction agrees well with the experimental measurement related to the initial stiffness, yield deformation and strength. A minor discrepancy in this comparison can be observed for the post yield stiffness and ultimate tensile capacity. For example, the FE modelling predictions of the specimens BCST16-5-90 and BC-HB10-5-90 in failure mode i demonstrate slightly higher post yield stiffness and ultimate tensile strength than experimental measurements. This observation can be owing to the difference in the replication of the local plastification on the tube connecting face induced by the Hollo-Bolt pullout action as well as circumferential locking action. For the specimens BCHB16-5-90 and BC-HB10-8-90 in failure modes ii and iii, the dis-

crepancy mainly exists after the yield point which produces 4% and 7% lower estimation of ultimate tensile capacity respectively. These differences can be attributed to the development of local plastification not only on the tube connecting faces but also on the flaring sleeves as mentioned earlier. Therefore, despite not being able completely account for the connection plastic force–deformation response, the prediction by FE modelling is of good accuracy in capturing the yielding of the connections.

3.3. FE model based failure mode analysis To further understand the failure mode of the connection in this study, the stress distributions on the connecting face of SHS column and the Hollo-Bolt were analyzed as a supplement to the experimental strain measurement. The numerical stress contours of tube connecting face and bolted regions in the groups of the specimens with HB10 Hollo-Bolts are plotted in Table 6. For the ease of visualization, the original FE model was expanded in dihedral symmetry view to display a panorama of an overall connection. The stress contours by other words can also be regarded as an indication of the stress flow on the connecting components. From the stress contours in the ‘‘Bolted regions” portion of Table 6, apparently the specimens in failure mode i (BC-HB10-560 and BC-HB10-5-90) exhibit wider range of highly stressed tube connecting face around the bolt hole and flaring sleeve than these in failure mode ii (i.e. BC-HB10-8-60) and failure mode iii (i.e. BCHB10-8-90). This observation agrees with the assumptions of the connection failure mode outlined in Section 2.3.1. Moreover, it can be inferred from plotted stress contours that the bolt shank of the Hollo-Bolt can be regarded as the primary part to carry the tensile load of the connection as long as the circumferential locking action between the flaring sleeve and the bolt clearance hole of the SHS column face remains effective. Greater stressed bolt shank section can be identified for the connections in failure modes ii and iii referring to specimens BC-HB10-8-60 and BC-HB10-8-90 respectively. A special attention was paid to the development of plastification on the SHS connecting face which has great influence on the flexural strength of the connection. This observation was also taken as a referred indication for the yield line pattern outlined in the subsequent section. Comparing the numerical stress contours on the tube connecting face, it can be observed from the

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BC-ST16-5-60-EX BC-ST16-5-90-EX BC-ST16-8-60-EX BC-ST16-8-90-EX

300

400

BC-ST16-5-60-FE BC-ST16-5-90-FE BC-ST16-8-60-FE BC-ST16-8-90-FE

Load, F (kN)

Load, F (kN)

400

200 100 0

0

10

20

30

BC-HB10-5-60-EX BC-HB10-5-90-EX BC-HB10-8-60-EX BC-HB10-8-90-EX

300

BC-HB10-5-60-FE BC-HB10-5-90-FE BC-HB10-8-60-FE BC-HB10-8-90-FE

200 100 0

40

0

10

Deformation, Δ (mm)

20

30

40

Deformation, Δ (mm)

(a) Group with M16 standard bolt 400

(b) Group with HB10 Hollo-Bolt

BC-HB16-5-60-EX BC-HB16-5-90-EX BC-HB16-8-60-EX

BC-HB16-5-60-FE BC-HB16-5-90-FE BC-HB16-8-60-FE

Load, F (kN)

300 200 100 0

0

10

20

30

40

Deformation, Δ (mm)

(c) Group with HB16 Hollo-Bolt Fig. 13. Comparison of Load (F)–deformation (D) relations obtained by experiment and finite element analysis.

Table 6 Typical stress contours on the tube connecting faces and bolted regions.

‘‘Tube connecting face” portion of Table 6 that the specimens BC-HB10-5-60 and BC-HB10-5-90 in the failure mode i, are characterized by noticeable plastification in the regions of two bolt rows with complete yielding across the bolt gauge width and spreading outward with the yield boundary in curved pattern around the bolt clearance hole. With the increase of the bolt gauge width, the pronounced yielding expands outside the two bolt row areas and gathers at the centreline of the SHS column connecting face. For the case of the failure mode ii, the specimen, e.g. BCHB10-8-60, exhibits significant yielding locally around the bolt clearance hole and at the corner of the tube connecting face. By contrast, as for the specimen BC-HB10-8-90 in the failure mode iii, the plastification area is greatly reduced to the edge area of

the bolt clearance hole on the side facing the corner of the tube connecting face. As the tube connecting face deflects under tensile loads, the sidewall may suffer certain inward deformation at the corner of the SHS column. To account for this factor, the connection deformation (DCD) is compared against the sidewall inward deformation (Din) for the specimens with HB10 Hollo-Bolt, as shown in Fig. 14. It can be seen that the overall deformation is approximately three times greater than sidewall inward deformation. The connections in failure mode i display similar trend with ultimate sidewall deformation at 5.2 mm and 7.5 mm for BC-HB10-5-90 and BC-HB10-5-60 respectively. In contrast, these in failure mode ii and iii show much lower deformation of 2.5 mm and 1.3 mm for

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which are taken as b0 = g + 0.9dba and c0 = p + 0.9dba respectively; g and p are bolt gauge width and bolt pitch width respectively. In the punching shear mechanism, the plastic load is calculated as the minimum of the loads related to the punching perimeter around the rectangular (b0  c0 ) and that around each bolt head as:

1

Δcd /Δy,cd

24

0.5 0

18

0

0.15

0.3



Δcd, (mm)

Δin /Δy,cd

F y;GlocP ¼ min 12

BC-HB10-5-60 (Mode i) BC-HB10-5-90 (Mode i) BC-HB10-8-60 (Mode ii)

0

BC-HB10-8-90 (Mode iii) 0

2

4

6

 ð6Þ

where nv is the number of bolts. dba is effective diameter of bolt clamping area as shown in Fig. 3. As far as the compression and tension zones of the joint are accounted, the global behaviour can be represented by assuming equal contribution of both zones to the tensile load. If the distance between centres of joint compression and tension zones, htc, is known, the corresponding tensile capacity (Fy,G-glo) was further suggested as:

Measured point for Δin

6

0

2ry tc ðb þ c0 Þ nv pdba ry t c pffiffiffi pffiffiffi ; 3 3

8

Δin (mm) Fig. 14. Comparison of overall deformation and sidewall inward deformation of the connection with HB10 Hollo-Bolt.

BC-HB10-8-60 and BC-HB10-8-90 respectively. By normalizing the connection deformation (DCD) with respect to the yield deformation (Dy,CD), it can be seen that when the connection becomes yield (Dcd/Dy,cd = 1), the sidewall inward deformations (Din) account for nearly 15%, 25% and 29% in contrast to the connection deformation (DCD) or the connections in failure modes i, ii and iii respectively. 4. Analytical model Based on the characterization of the plastification on the SHS connecting face from finite element modelling, it is the purpose of this section to evaluate the flexural yield strength related to the connection failure mode i & ii and the ultimate strength incorporating the local clamping action of the Hollo-Bolt and the membrane action of the SHS column connecting face. The newly proposed analytical model is presented in this section to permit a comparison with existing design formula and experimental results, thereby facilitating the understanding of its applicability. 4.1. Present design formulae for bolted connections to SHS columns

F y;G-glo

8   kF r t2 > 0 tc < y;G-loc þ y4 c 2b þ p þ b2hb 0 2 htc 0   ¼ ry t2c 2b0 > : kF y;G-loc þ þ p þ 2 2 4 htc

htc b0 b0

P1

htc b0 b0

<1

ð7Þ

As an alternative model, Yeomans [12,31] reported a formula developed for flowdrill connections which was also extended for the calculation of Hollo-Bolt insert as:

"

ry t2c f ðkÞi 2ðp  dsl Þ

F y;Yeo ¼ h

sl 1  bgd 0 t c

b0  t c


0:5 # g  dsl þ4 1 b0  tc

ð8Þ

where dsl is the diameter of Hollo-Bolt insert which is taken as the outer diameter of Hollo-Bolt sleeve. f(k) = 1 + k 6 1 in which k is the column stress divided by the column yield stress. In view of test condition, f(k) was taken as 1.0 for the case studied herein. The formula (8) has also been incorporated in the design formulae for bolt connections to RHS columns in current CIDECT design guide 9 [9]. Another similar calculation was provided by the Steel Construction Institute & the British Constructional Steelwork Association for simple connections [8], which is labelled as ‘‘SCI” herein for short. The corresponding formula for the tying capacity of SHS column wall, Fy,SCI, was suggested as:

F y;SCI ¼

  2ry t2c ðnr  1Þp  0:5nr dbh þ 1:5ð1  b1 Þ0:5 ð1  c1 Þ0:5 1  b1 b0  3t c

ð9Þ As mentioned previously, the formula given by Gomes et al. [29] appears to be a good starting point in the calculation of tensile strength of the SHS column connecting face. In this model, the local failure mechanism is referred herein considering the yield line pattern localized only in the tension zone. Its corresponding tensile capacity (Fy,G-loc) was given as the lowest value of the plastic loads related to the flexural mechanism (Fy,G-loc-F) and the punching shear mechanism (Fy,G-loc-P), i.e.

F y;G-loc ¼ minðF y;G-loc-F ; F y;G-loc-P Þ

ð3Þ

In the flexural mechanism, the force was assumed to act on a given rectangular (b0  c0 ) and the plastic load of its related optimized mechanism is given as:

F y;G-loc-F ¼ k ( k¼

pry t2c

"


0 0:5

b 1 b0

0

1  bb0

2c0 þ pb0

# ð4Þ

0

if ðb þ c0 Þ=b0 > 0:5

1; 0

0

0

0:7 þ 0:6ðbb0þc Þ ; if ðb þ c0 Þ=b0 < 0:5

ð5Þ

where b0 and tc are the outside width and wall thickness of square tubular member respectively, ry is the yield stress of tube steel material. b0 and c0 are the length and width of a given rectangular

where b1 = g/(b0  3tc) and c1 = dbh/(b0  3tc). nr is the number of rows of bolts under tensile loading. This formula has also been applied in a similar study of yield line analysis by Barnett [15]. The design formulae given by Gomes, Yeomans and SCI were adopted to predict the strength of the connection and compared with the experimental results. htc was assumed to be 340 mm as reported by Yeomans [12] for a typical joint with such connection details. These results are listed in Table 7. A further evaluation of the comparison is made in Fig. 15, where the experimental data are plotted versus the predicted strength divided by the maximum value of Fy,CD, i.e. 351.72 kN. Here, it can be observed that the Gomes formula exhibits a consistent overestimated yield strength with average value of 27.80% and its error becomes greater for the connections in failure mode ii in contrast to failure mode i. This can be due to the fact that the Gomes formula was originally developed for the minor axis connections of open sections where the rigidity at the edge of the root radius determined the boundary limits of the yield line pattern. It is noted that, however, under a tension load on the connecting face, the sidewall face and root radius of the SHS column provide less stiff restraint when comparing with the flange plate restraint in the open sections. In this case, the plastification is not only well developed on the tube connecting face but also extended to the root radius and adjacent sidewall as

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Z.-Y. Wang, Q.-Y. Wang / Engineering Structures 111 (2016) 345–369 Table 7 Summary of test failure modes and strength results with referred theoretical prediction. Connection identification

BC-ST16-5-60 BC-HB10-5-60 BC-HB16-5-60 BC-ST16-5-90 BC-HB10-5-90 BC-HB16-5-90 BC-ST16-8-60 BC-HB10-8-60 BC-HB16-8-60 BC-ST16-8-90 BC-HB10-8-90 RF-HB16-8-90 [15] RF-HB16-8-90 [15] RF-HB16-8-90 [15]

Test results

Theoretical prediction using referred formulae

Fy,CD (kN)

Fy,FD (kN)

Fu (kN)

Failure modes

58.57 53.79 78.46 97.79 79.80 129.25 174.60 169.47 192.19 242.15 212.80 168.42 268.14 351.72

60.52 55.16 81.62 101.89 81.34 132.24 181.87 176.40 196.39 245.99 219.30 – – –

192.59 122.58 208.97 197.14 142.98 244.53 214.81 205.68 314.90 292.00 252.46 346.14 352.41 444.95

i i i i i i ii ii ii ii iii i ii ii

shown in Table 6. On the other hand, from the comparative results, it can be seen that the Yeomans formula in most cases underestimates the yield strength of the connection with average value of 17.44%. Likewise, greater errors can be seen for the connections in failure mode ii than these in failure mode i. Moreover, it is noteworthy that the Yeomans formula is unable to give reasonable allowance for the influence of the size of the bolt on the strength of the connection. As an example shown in Table 7, it can be seen that within each group of identical bolt gauge width and tube wall thickness, Yeomans formula even gives smaller value of predicted strength for the connections with larger size of the Hollo-Bolt. This minors the shortcoming in applying the Yeomans formula for the connections with Hollo-Bolts and oversized bolt clearance holes which are different from that for the flowdrill joints involving relatively small drilled bolt holes as a likely unchanged parameter. Likewise, SCI formula represents slightly larger underestimation of the yield strength with average value of 19.78%. Consequently, the limitations of present design formulae can be identified in their theoretical consideration for the geometric details of the bolt and the plastification of the SHS column with the combination of bolt behaviour in the failure mode ii. Regarding this, a new yield line model is needed as an improvement of present design formulae and will be presented in following section.

Gomes formula [29]

Yeoman formula [12,31]

SCI formula [8]

Fy,G-loc (kN)

Error (%)

Fy,Yeo (kN)

Error (%)

Fy,SCI (kN)

Error (%)

74.75 70.22 78.11 141.31 120.17 158.47 201.17 187.69 211.27 444.19 354.33 173.94 281.48 420.89

27.6% 30.6% 0.4% 44.5% 50.6% 22.6% 15.2% 10.7% 9.9% 83.4% 66.5% 3.3% 5.0% 19.7%

58.06 56.99 54.05 73.01 71.05 65.91 149.06 146.21 138.48 189.00 183.71 142.60 222.93 321.18

0.87% 5.94% 31.11% 25.34% 10.96% 49.00% 14.63% 13.73% 27.95% 21.95% 13.67% 15.33% 16.86% 8.68%

51.10 51.25 47.88 73.29 73.22 68.09 139.35 139.74 130.01 215.85 215.42 127.46 205.79 307.38

12.76% 4.72% 38.98% 25.06% 8.24% 47.32% 20.19% 17.54% 32.35% 10.86% 1.23% 24.32% 23.25% 12.61%

Average: Max/Min: St. Dev.:

27.80% 83.44% 0.24

17.44% 49% 0.13

19.78% 47.32% 0.13

by Packer & Morris [46] and Eurocode 3 [22], additional consideration was given to a fan or curve collapse pattern for the localized action of the bolt. Following the procedure and application previously reported, therefore, the realistic factors of safety against collapse can be ensured provided that the possible failure patterns are sufficiently recognized. In this study, the method used to calculate the yield strength related to each failure mode is that of rigid plastic collapse. The following assumptions are made in the derivation of formulae:  The membrane action is ignored in the yield line analysis which means only the moderate deformation of the SHS column connecting face is concerned in this model.  The bolt is located at the centre of the bolt clearance hole and only transmits axial tension force as a point load inducing the uplift of the SHS column connecting face.  The steel yield stress and the bolt ultimate stress are the mean values from the material of the same batch with that applied in the test specimens.  The geometry of the SHS column wall and the cross-sectional area of the bolt shank are assumed to be of nominal values of thickness (tc) & outside width (b0) and stress area (Ash) respectively.

4.2. Derivation of a new yield strength analytical model 1.5

Gomes model Yeoman model

1.2

Fy,CD / Fy,CD-max

The yield line approach is an upper bound method of determining the load carrying capacity of a plate based on an assumed collapse mechanism [45]. Within this approach, the yield line pattern is effectively a series of plastic hinges indicating the collapse manner of the plate and should be considered with regard to the loading conditions, support conditions and plate geometries. The yield model analysis can be executed using the virtual working principle, in which the collapse load can be calculated by equating the external work done by the applied loads to the internal energy dissipation along the yield lines. So far, the yield line approach has been successfully applied in the determination of the strength of welded connections made up of hollow sections [43] and bolted T-stub connections [22,33]. Regarding the former applications, the chord face yielding of the connections are represented by straight yield lines (beam pattern) going through the intersection of exes of rotation and terminating at the plate boundary. Referring the latter applications suggested

SCI model

0.9

0.6

0.3

0

0

0.3

0.6

0.9

1.2

1.5

Fy,prediction / Fy,CD-max Fig. 15. Comparison of yield strengths from experiment and predicted by referred models.

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 The factors affecting the discontinuity of the F–D curve at the very beginning stage of loading, such as the loss of the bolt pretension and the settlement of the connection, are not accounted in the modelling of the linear elastic response of the connection.  The effective bolt clamping areas are estimated from the contact condition of the bolt head or the nut for the standard bolt and of the flaring sleeves (i.e. ring contact) for the Hollo-Bolt. The finite element models previously reported have also been employed in characterizing the yield line patterns related to the failure modes i and ii. As already discussed in Section 3.3, the test connections have two typical development of plastification on the SHS column connecting face which corresponds to two failure modes, as shown in Fig. 16. The characterizing parameters are: b0, p, R (radius of yield fan) and b (bolt gauge width-to-nominal tube width ratio) and related angle parameters are: / = arccos [0.5(1  b)b0/R] and W = arccos(0.5bb0/R). For the purpose of determining the internal work done by the plastic hinges associated with each of the yield line pattern, the length of the yield lines and its corresponding rotation related to the energy dissipation are described below. The plastic moment per unit length [45] of the yield line is given by:

mp ¼

ry t2c

ð10Þ

4

and the general form of energy dissipation due to the yield line deformation is expressed as:

D ¼ ni mp

X

li hi

ð11Þ

where ni, li and hi are the number, length and rotation of the yield lines in ith yield line category. Given the size of the bolt has significant influence on the strength of the connection, a special attention is paid to the local deformation at the bolt clearance hole. It was considered that the pullout load of the bolt enforces the round edge of the bolt clearance hole deforming outwards and then the flaring sleeve of the Hollo-Bolt slipping through the bolt clearance hole. In such circumstances, the deformation of the connection (governed by bolt pullout), i.e. the displacement (d) between the T-stub flange (rigid connecting plate) bottom surface and undeformed tube connecting face, differs from the deformation at the edge of the bolt clearance hole with a gap distance of ‘0.5dbad/R’. The deformation at the bolt clearance hole and also, the maximum deformation of the connecting face of the SHS column, can then be given in a general form as ‘ (1–0.5dba/R)d’. 4.2.1. Mechanism type i In failure mode i, plastic hinges form at both the bolt line and the corner of the SHS column connecting face. Besides, the edge of the bolt clearance hole undergoes certain flexural deformation with the uplift action of the bolt. Therefore, the yield line pattern for the failure mode i, i.e. mechanism type i, is plotted in Fig. 16 (a). This pattern combines the yield fans and straight yield lines. In the yield fans, the hogging yield lines radiate from the centre of bolt clearance hole while the sagging yield lines centre around the bolt clearance hole with a curved boundary of radius R and intersect with each other at the SHS mid-face. Besides, in the straight yield lines, the hogging parts connect the bolt clearance hole centre while the sagging parts join the points where the yield fan curved boundaries and the tube sidewall boundaries intersect. Assuming the external work done by the external load, P, the corresponding expenditure of energy can be expressed in the form:


 dba E ¼ P 1  0:5g d R

ð12Þ

where g is the coefficient allowing for non-uniform local deformation of the bolt clearance hole. dba is effective diameter of bolt clamping area which can be obtained from the size of the bolt head or the nut for the standard bolt as shown in Fig. 3 and from the bolt hole diameter for the Hollo-Bolt, i.e. dba = dbh. It is assumed that the non-uniform local deformation of the bolt clearance hole is contributed not only by the flexibility provided by the SHS column, including the restraints of the sidewall face and root radius, but also by the interaction of the bolt. Based on the results of finite element parametric study with geometric details of the connection, the coefficient, g, was determined as plotted in Fig. 17, in relation with dba, W0 and b as:

g ¼ ð0:01b þ 0:003Þð15400w30 þ 3880w20  321w0 þ 9:52Þe0:196dba ð13Þ As such, the gap distance and maximum deformation of the connecting face of the SHS column can be recast as 0.5gdbad/R and (1–0.5gdba/R)d respectively, as shown in Fig. 16(a). Based on the yield line pattern of the mechanism type i, the internal work calculation for each part of pattern can be given below: 1. Yield fans (j), a1o1b1, a2o2b1, a3o3b2, a4o4b2: In the determination of the energy dissipation in the yield fans, the yield lines in meridional direction (e.g. o1c1) are subjected to no torsional moments and transverse forces. Thus, the energy dissipation in this manner is contributed by the bending moments in tangential direction and the restrained moments in circumferential direction. For a fan sector with the angle of dh, the tangential curvature (jFC) is given by:

jFC ¼

d rR

ð14Þ

Given the yield fan is represented by a number of fan sectors with angle range of p–/–W and radius range of R and recalling the general form of energy dissipation in Eq. (11), the energy dissipation contributed by the bending moments in tangential direction can be expressed by:

Z p/ Z

R

Z p/ Z

R

d rdrdh rR 

 ð1  bÞb0 bb0  arccos d ¼ 4mp p  arccos 2R 2R

Dd-j-1 ¼ 4mp

w

0

jFC rdrdh ¼ 4mp

w

0

ð15Þ

For the restrained edge of the curved boundaries of the yield fans, a constant yield line rotation, i.e. d/R, and length, Rdh, can be expected; therefore, the corresponding energy dissipation can be given as:

Z p/ d Rdh R w 

 ð1  bÞb0 bb0  arccos d ¼ 4mp p  arccos 2R 2R

Dd-j-2 ¼ 4mp

ð16Þ

Then, the energy dissipation of the yield fans can be obtained as the sum of Dd-j-1 and Dd-j-2:

Dd-j ¼ Dd-j-1 þ Dd-j-2 

 ð1  bÞb0 bb0  arccos d ¼ 8mp p  arccos 2R 2R

ð17Þ

2. Yield lines (k), a1o0 1, a2o0 2, a3o0 3, a4o0 4, a1o1, a2o2, a3o3, a4o4: The sagging yield lines along the tube sidewall are divided as the one outside the range of bolt pitch width (yield lines, a1o0 1, a2o0 2, a3o0 3, a4o0 4) and the others inside the range of bolt pitch

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g bh

0.5 (1-0.5

b

/ / )

ba ba

/ 0.5

c a

Hogging yield lines

T3

1

1

ba

a

b

2

0

1

o'

o'

1

2

o

o

1

2

T3

+2 o

o

3

o'

0

4

o'

4

3

a

a

4

3

b

2

Sagging yield lines

0.5 (1- ) 0

=

0.5

0

(1- )

=

0.5

0

0

0

(1- ) 0.5 (1- ) 0

0

g

(a)

i bh

b

/ -0.5 (1- ) 0.5 0

t

0

(1- )

0.5 0

c

a

a

11

12

11

o'

o

11

o

11

0

o'

12

12

T4

T4 Hogging yield lines

o

o'

o

13

14

o'

13

14 0

a

a

13

0

Sagging yield lines

= -2

14

0

-2 0

ii Fig. 16. Proposed yield line patterns on the SHS column.

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Z

4

η

3

hm lm ¼ 2

28

Dd-m ¼ ni mp

ψ0

0.09

0.12

Fig. 17. Non-uniform local deformation coefficient g with varying dba, W0 and b.

width (yield lines, l). The former one is also coincided with the vector component of the yield lines a1o1, a2o2, a3o3, a4o4 along the tube sidewall direction, thus having identical yield line rotation and length. Such rotations or component rotation about the tube sidewall can be easily calculated using the distance of the line normal to these yield lines and passing through the centre of the bolt as:

d 0:5b0 ð1  bÞ

ð18Þ

Also, the original length of each of yield lines a1o0 1, a2o0 2, a3o0 3, a4o0 4 is equal to the projected length of each of yield lines a1o1, a2o2, a3o3, a4o4 as: 2 0:5

lk ¼ fR2  ½0:5b0 ð1  bÞ g

ð19Þ

Thus, the energy dissipation of yield lines k can be written as:

X

2 0:5

d 0:5b0 ð1  bÞ

2 0:5

8mp ½4R2  ð1  bÞ2 b0  d ð1  bÞb0

ð20Þ

3. Yield lines (l), o0 1o0 3, o0 2o0 4, o1o3, o2o4: Similar to the calculation of the rotation of yield lines k with sagging yield lines, o0 1o0 3, o0 2o0 4, and hogging yield lines, o1o3, o2o4, the rotation and length of each of yield lines l, can be obtained as:


 1 dba 1  0:5g d; 0:5b0 ð1  bÞ R

ll ¼ p  dba

ð21Þ

Then, the corresponding energy dissipation of yield lines l can be given by:

X

li hi ¼

8mp ðp  dba Þ dba ð1  0:5g Þd ð1  bÞb0 R

Dd ¼ Ddj þ Ddk þ Ddl þ Ddm

 ð1  bÞb0 bb0  arccos ¼ 8mp d p  arccos 2R 2R )
 2 2 0:5 2 ½4R  ð1  bÞ b0  dba 2ðp  dba Þ þ þ 4mp d 1  0:5g ð1  bÞb0 ð1  bÞb0 R

0:5bb0 0:5dba ð25Þ þ arcsin  arcsin R R Recalling Eq. (10) and equating the expenditure of energy by the external loads, Eq. (12), and internal dissipation of energy in the yield lines, Eq. (25) gives:

F y;mode i ) P ¼

2ry t2c



p  arccos


 ð1  bÞb0 2R

1  0:5g dRba )
 2 0:5 bb0 ½4R2  ð1  bÞ2 b0  þ  arccos 2R ð1  bÞb0

2ðp  d Þ 0:5bb0 0:5dba ba þ ry t2c þ arcsin  arcsin ð1  bÞb0 R R ð26Þ

li h i

¼ 8mp fR2  ½0:5b0 ð1  bÞ g

Dd-l ¼ ni mp

ð24Þ

And the overall energy dissipation of yield lines becomes: 0.06

Ddk ¼ ni mp

X

li h i

 dba 0:5bb0 0:5dba ¼ 4mp d 1  0:5g arcsin  arcsin R R R

16

0 0.03

hl ¼ hk ¼

ð23Þ

The energy dissipation of yield lines l can then be given as:

2

1

¼

ð1  0:5g dRba Þd

0:5dba

24

hk ¼

0:5bb0

dx 0:5 ðR2  x2 Þ

 dba 0:5bb0 0:5dba arcsin ¼ 2d 1  0:5g  arcsin R R R

β=0.3 β=0.5 β=0.7

dba (mm)

ð22Þ

4. Yield lines (m), o1o2, o3o4: The hogging yield lines o1o2 and o3o4 are expected to rotate along the restrained edge of the yield fan. It is noted that such rotation varies with the distance of the line normal to o1o2 or o3o4 and passing through the intersection with the restrained edge of the yield fan. Furthermore, the uplift deformation of both yield lines is represented by (1–0.5gdbh/R)d/R instead of d. Given the symmetry of the restrained edge about the SHS mid-face, the overall rotation of each yield line can be calculated by doubling the half length in the following integral formula:

Regarding R as the only variable, Eq. (26) represents an upper bound solution for the forces which induce yielding. The minimum value of load capacity for this yield line mechanism can then be calculated by setting to zero the derivative of P with respect to R, which yields:

h i 0:5 0:5 R ¼ 0:5b0 b2  b þ 0:5 þ 0:5½9b4  18b3 þ 13b2  4b þ 1 ð27Þ In this equation, therefore, the radius of yield fan can be determined exclusively from b which facilitates the evaluation of the scope of plastification on the SHS column face for the design. Substituting Eqs. (27) into (26), the corresponding yield strength for the failure mode i can then be calculated. 4.2.2. Mechanism type ii and iii In failure mode ii, noticeable yielding appears locally at the corner of the tube connecting face and around the bolt clearance hole preceded by bolt failure. Unlike the foregoing failure mode i which is governed by the flexure of the SHS column connecting face under the bolt tension load, the failure mode ii is featured by combining both the bolt in tension and the SHS column connecting face in flexible yielding, both of which should be taken into account in the internal energy dissipation. Mechanism type ii, as plotted in Fig. 16(b), was adopted to represent yield line pattern in the failure mode ii. In this yield line pattern, a quarter circle curved boundary of radius R is present at each corner of this pattern, which is connected by sagging yield lines in the longitudinal and transverse directions of the SHS column faces; meanwhile, the hogging yield lines connect the circle centres of the curved boundary, rather than these of the bolt clearance hole, thus permitting elastic extension of the bolts as well as the deformation of the connecting face of

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Z.-Y. Wang, Q.-Y. Wang / Engineering Structures 111 (2016) 345–369

the SHS column. Since the circle centre is within bolt gauge width, the range R can be identified as:

0:5b0 ð1  bÞ 6 R < 0:5b0

ð28Þ

Given the SHS column wall is relatively stiffer in this mechanism with respect to that in the foregoing mechanism type i, the local deformation at the bolt clearance hole is ignored; on the other hand, the bolt is assumed to suffer plastic deformation (dt) when the overall connection deformation (d) takes place. According to the geometric relations in Fig. 16(b), it is obvious that dt can be given by:

  0:5b0 ð1  bÞ d dt ¼ 1  R

ð29Þ

The corresponding expenditure of energy for the external work done by the external load, P, can be expressed as:

E ¼ Pd

ð30Þ

Likewise, the internal work calculation for each part of pattern can be given as follows: 1. Yield fans (n), a11o11o0 11, a12o12o0 12, a13o13o0 13, a14o14o0 14: Similar to the calculation of the energy dissipation in the yield fans as outlined in Section 4.2.1, the yield lines in meridional direction (e.g. o11a11) can be accounted with consideration of the tangential curvature (jFC). Then the energy dissipation contributed by the bending moments in tangential direction for the summation of four quarters of the circle can be written as:

Z Dd-n-1 ¼ 4mp

p 2

0

Z

Z

R

jFC rdrdh ¼ 4mp 0

0

p 2

Z

R 0

d rdrdh ¼ 2mp pd rR ð31Þ

and also, the energy dissipation contributed by the restrained edge of the curved boundaries of the yield fans is:

Z Dd-n-2 ¼ 4mp

p 2

0

d Rdh ¼ 2mp pd R

sleeve, thus determining its ultimate strength. This sort of shear strength instead of tensile strength of bolt shank is highlighted herein for the Hollo-Bolt. Given the flaring sleeves are the part splayed out from the original insert, the periphery length of one piece of the flaring sleeve can be taken as a quarter of its perimeter minus corresponding slot width (bs) between adjacent sleeve pieces, i.e. pdsl/4  bs. Based on the experimental and numerical observation, the projected shear area is assumed between the bottom edge of the bolt clearance hole and the intermediate circle between upper and lower parts of threaded cone. This area is also inclined with the angle of 45° to the surface of the sleeve, as illustrated in Fig. 18; therefore, the shearing of flaring sleeve governed tensile strength of a Hollo-Bolt can be estimated as:

pffiffiffi pffiffiffi Buh-1 ¼ ruh ts ðpdsl  4bs Þ 2= 3

ð37Þ

where ruh is the ultimate stress of the Hollo-Bolt sleeve. Considering four bolts in the connection studied herein, the overall bolt ultimate strengths, Bu, can be obtained from the strength of the standard bolt (Bus) or that of the Hollo-Bolt (Buh) as:

Bus ¼ 4Bus-1 ¼ 2:88rus Ash

ð38Þ

Buh ¼ 4Buh-1 ¼ 3:27ruh ts ðpdsl  4bs Þ

ð39Þ

The energy dissipation of the bolt plastic deformation can then be calculated as:

  0:5b0 ð1  bÞ Dd-q ¼ Bu dt ¼ Bu d 1  R

ð40Þ

Summing the energy dissipation of yield lines in this mechanism, so that

Dd ¼ Dd-n þ Dd-o þ Ddp þ Dd-q     p þ b0 0:5b0 ð1  bÞ ¼ 4mp d p  2 þ þ Bu d 1  R R

ð32Þ

so the energy dissipation of the yield fans (n) becomes

Dd-j ¼ Dd-j-1 þ Dd-j-2 ¼ 4mp pd

bh sl

ð33Þ

2. Yield lines (o), o11o13, o12o14, o0 11o0 13, o0 12o0 14: With the identical length of p and rotation of d/R for the yield lines (o), the counterpart energy dissipation can be given by:

Ddo ¼

4mp pd R

45

4mp ðb0  2RÞd R

τs

o

c

Shear area

ð34Þ

s

a

3. Yield lines (p), a11a12, a13a14, o11o12, o13o14: Similar to Eq. (34), the corresponding energy dissipation is:

Dd-p ¼

ð41Þ

tcm

ð35Þ

s

bh

sl

sl

Bus-1 ¼

cM2

Shear area s

bh

ð36Þ

where rus and Ash are the ultimate stress and the stress area of the standard bolt respectively. cM2 is the partial safety factor for the resistance of bolts which can be taken as 1.25. As mentioned in Section 2.3, with distinct configuration details, the Hollo-Bolt in tension is featured by shearing of the flaring

bh

sl

s

0:9rus Ash

bh

s

4. Bolt plastic deformation work (q) According to the Eurocode 3 [22], the ultimate strength of a standard bolt without the consideration of partial safety factor can be computed as:

Threaded cone

sl

s

bh

sl

Fig. 18. Schematic of shear area in the calculation of shear strength.

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Z.-Y. Wang, Q.-Y. Wang / Engineering Structures 111 (2016) 345–369

Similar to Eq. (26), equating Eqs. (30)–(41) yields:

 P ¼ 4mp

p2þ

p þ b0 R



  0:5b0 ð1  bÞ þ Bu 1  R

also, a simplified form can also be written for Fy,mode

F y;mode ii



 p þ b0 1b þ Bu 1  ) P ¼ ry t2c 1:14 þ 2k kb0

ð42Þ ii

meridional stress of the flaring sleeve (rm) against the edge of the bolt clearance hole is produced from the bolt tensile load (Fb) acting on the threaded cone (Nib,x) and can be expressed in the form:

rm ¼

as:

ð43Þ

where k = R/b0. Recalling Eq. (28) yields the limit of k as 0.5(1  b) < k < 0.5. From the results of finite element parametric study, the range of k was further identified as between 0.55(1  b) and 0.65 (1  b), in which k = 0.6(1  b) was taken in this study.

F b tan ha ðpdsl  4bs Þtc

ð47Þ

Since the flaring sleeve is meridionally and circumferentially restrained, the corresponding stresses rr and rc can be assumed to be equal and negative (in compression). Similarly, the radial strain and circumferential strain induced by the radial deformation of the bolt clearance hole can be written as:

er ¼ en ¼

dtcm þ costsha  dbh dbh

4.3. Derivation of a new ultimate strength analytical model

ð48Þ

According to the generalized Hooke’s law, the relations between rr, Following the flexural yielding of the connecting face of the SHS column face, the load–deformation response of test connections exhibits a pronounced and modest load enhancement for the connections in failure mode i and ii respectively due to the membrane action. This action, therefore, plays an important role in the determination of the ultimate strength of the connection. Assuming that transverse tube face tension with little bending contributes to the most part of the membrane strength, this action is related to the geometric parameters of the bolt gauge width (g) and SHS column wall thickness (tc) and outside width (b0). Since the out-of-plane deformation induced by the transverse membrane force normal to the tube connecting face is closely relevant to the transverse slope of the yielding tube face [47] and thus approximately inversely proportional to 0.5W0 (i.e. tc/b0) and b (i.e. g/b0). Therefore, the following formula incorporating the above factors was assumed for the ultimate strength contributed by the SHS column face as:



 C 02 C0 1þ 3 F 0u ¼ C 01 F y 1 þ 0:5w0 b

ð44Þ

where Fy is the yield strength of the connection related to aforementioned mechanism type. Parameters C01 , C02 and C03 are curvefitting constants. As observed in the failure mode of the connections mentioned earlier, the ultimate state of the Hollo-Bolt is featured by localized plastification and bulging around the bolt clearance hole. This may also contribute to the ultimate strength of the connection of such type. Unlike the standard bolt with the nut and the washer underneath flat face of gripped plate, the clamping of the Hollo-Bolt is formed with aforementioned circumferential locking action. As illustrated in Fig. 19, the original clamping induced bearing stress is assumed to disperse in 45° through the edge of the bolt clearance hole into the gripped plates between the collar and the flaring sleeve of the Hollo-Bolt. Then the stressed area around the bottom edge of bolt clearance hole (ESA) was taken as within the diameter range between ‘dbh’ and ‘dbh + 0.2tctan(45°–ha)’, and thus, the stress area of ESA around the bolt clearance hole can be given as: 2

2

AESA ¼ 0:25pf½dbh þ 0:4t c tanð45  ha Þ  dbh g

ð45Þ

where the angle of the Hollo-Bolt flaring sleeve, ha, is taken as 10° in this study. And the vertical contact force induced stress (rn) normal to the tube connecting face can be written as:

rn ¼

Fb Fb n o ¼ AESA 0:25p ½d þ 0:4t c tanð45  ha Þ2  d2 bh bh

ð46Þ

Prior to the bolt pullout, the sleeve is locally squeezed in which the critical horizontal thickness can be given as ‘0.5ts/cos(ha)’. The load transfer mechanism and its resultant stress distribution of the Hollo-Bolt on the bolt clearance hole are shown in Fig. 20. The

rc, rn and er can be given by:

er ¼

1 ½rr  lðrn þ rc Þ Ec

ð49Þ

where Ec is the modulus of elasticity. With rr = rc, Eq. (49) can be transformed into the form as:

rr ¼

Ec er þ lrn 1l

ð50Þ

Given the zero resultant force at the meridional direction of the bolt clearance hole, the summation of the stresses of rm and rr should be equal and in opposite reaction, i.e.

rm þ rr ¼ 0

ð51Þ

Substituting Eq. (47) and Eq. (50) into Eq. (51), the tensile load (Fb) for of a single bolt can be obtained and then, as the negative demand of Fb, the pullout force (Fpo) can be expressed as:

F po ¼ F b ¼

AESA Ec er ðpdsl  4bs Þt c AESA ð1  lÞ tan ha þ lðpdsl  4bs Þt c

ð52Þ

Therefore, by combining the contributed of the SHS column face, the ultimate strength of the connection can be expressed by:



 C2 C3 1þ Fu ¼ C1Fy 1 þ þ C 4 nv F po w0 b

ð53Þ

where nv is the number of bolts, which is taken as four for the connection studied herein. C1, C2 and C3 are constant values for the extent of contribution of the flexibility of SHS column face, while C4 is a constant value for the extent of contribution of the bolt pullout and localized plastification. 4.4. Validation of proposed analytical model For the sake of validating the proposed analytical model, a comparison of strength results obtained from analytical model prediction and experiment measurement is listed in Table 8. A further comparison is plotted in Fig. 21, with the experimental data versus predicted strength divided by the maximum value of Fy,CD, i.e. 351.72 kN, for yield strength and that by the maximum value of Fu, i.e. 444.95 kN, for ultimate strength. The predictions of yield strength are based on the formula calculation in Eq. (26) for the mechanism type i and Eq. (43) for the mechanism type ii and iii. Also, the predictions from both formulae are rearranged by fitting to each failure mode (Fy,ff). Initially, it can be seen that the mechanism type i almost in all cases shows relatively lower yield strength prediction for the connections in failure mode ii and can be taken as the minimum strength as normally considered for design. On the other hand, the rearrangement of the prediction values (Fy,ff) for each failure mode better matches

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Z.-Y. Wang, Q.-Y. Wang / Engineering Structures 111 (2016) 345–369

ch

ch

Undeformed (initial) prestressed area 45

Deformed prestressed area

o

bh

bh

a

a

o

45 -

T5

a

c

T5

s a

s

ESA o

c

tan(45 - a)

Deformed flaring sleeve

Undeformed flaring sleeve tcm

s

/cos( a)

Geometry of undeformed and deformed prestressed areas

o

0.2 c tan(45 - a)

Edge stressed area (ESA)

bh

bh

o

0.2 c tan(45 - a)

Photo of a part of deformed flaring sleeve

T5-T5

Fig. 19. Stress condition through gripped plates and flaring sleeves.

the experimental results of the yield strengths with an average and maximum errors of 2.3% and 10.2% respectively and a standard deviation of 0.06 which suggest a significant improvement from previous predictions given by referred design formulae as mentioned in Table 7 of Section 4.1. These slight discrepancies can be attributed to the local effects, such as the tube sidewall deformation, deflection of bolt clearance hole and variation of the shear area of the Hollo-Bolt, which are ignored in the idealization of the yield line model. The predictions of ultimate strength are based on the formula calculations in Eq. (53) for the connections with the standard bolts and the Hollo-Bolts. The values of C1, C2, C3 and C4 were determined by minimizing the coefficient of variation of the experimental ultimate strength fitted in the formulae above. The resultant expressions can be generally given as: For the connections with four standard bolts:

Mechanism type i :



0:12 0:22 þ 0:8F po 1þ F u ¼ 0:1F y 1 þ w0 b

ð54Þ



0:06 0:25 þ 0:14F po Mechanism type ii : F u ¼ 0:6F y 1 þ 1þ w0 b ð55Þ in which, the standard deviations from regression analysis are 1.52% and 1.60% for Eqs. (54) and (55) respectively. For the connections with four Hollo-Bolts:

Mechanism type i :



 0:06 0:12 1þ þ 1:24F po F u ¼ 0:5F y 1 þ w0 b ð56Þ

Mechanism type ii :



 0:06 0:28 1þ þ 1:24F po F u ¼ 0:3F y 1 þ w0 b ð57Þ

in which, the standard deviations from regression analysis are 7.6% and 5.1% for Eqs. (56) and (57) respectively. The calculated ultimate strengths are also summarized and listed in Table 8. The greatest error of 12.9% can be observed for

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bh

sl

s

bh

sl

b sl s

i b,y

bh

T6

c i b,x

T6 s

bh

sl

Bolt hole bh s

s sl

Bolt hole a s

bh

sl

tcm

T6-T6 sl s

bh

Flaring sleeve

sl

s

Threaded cone

s

Fig. 20. Bolt axial force induced circumferential stress around bolt clearance hole and flaring sleeve.

Table 8 Comparison of test strength results and calculation from proposed equations.

the connections with HB16 Hollo-Bolts (specimen BC-HB16-5-60) which indicates additional consideration should be made for the development of post yield strength on the local regions of the tube connecting face influenced by large size of the Hollo-Bolt. Notwith-

standing that, the general agreement trend over the comparison in Fig. 21 with an average error of 1.2% and standard deviation of 0.07 suggests a satisfied representation of the ultimate strength by proposed analytical model.

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1.5

0.8 Fu,CD / Fu,CD-max

1.2

Fy,CD / Fy,CD-max

1

Proposed model Gomes model Yeomans model SCI model

0.9 0.6 0.3 0

0.6 0.4 0.2

0

0.3

0.6

0.9

1.2

1.5

0

0

0.2

0.4

0.6

0.8

Fy,prediction / Fy,CD-max

Fu,prediction / Fu,CD-max

(a) Yield strength ratio

(b) Ultimate strength ratio

1

Fig. 21. Comparison of strengths of connections from experiment and predicted by proposed models.

b 0 =150 b 0 =400 SHS 200x5 1 β Upper bound

b0 =200 SHS 150x5 SHS 200x15

b0 =300 SHS 150x12 SHS 300x6

0.8 0.6

Selected range Lower bound

0.4 0.2 0 8

10

12

14

16

Hollo-Bolt size Fig. 22. Limits of b with the size of Hollo-Bolt.

4.5. Failure mode related geometric parameters Based on the validated finite element models and proposed analytical models, a numerical parametric study was performed to study the failure mode of the connection and its related geometric parameters. The bolt gauge width-to-tube width ratio (b) and tube wall thickness-to-half wall width ratio (W0) were chosen in line with counterpart representations in the aforementioned formulae. In accordance with the design guidance of the connection with the Hollo-Bolt [2,8], the minimum bolt gauge width and the minimum edge distance from the centre of the bolt to the adjacent

edge of inner and outer SHS column sidewall are required with varying bolt size. Following this requirement, the upper bound and lower bound of b can be obtained through a conversion for the connections with the SHS column width ranging between 150 and 300 and the Hollo-Bolt ranging between HB08 and HB16, as illustrated in Fig. 22. The range of b between 0.25 and 0.75 was chosen to cover the primary part of the configuration details in this parametric study. Also, the SHS columns with the tube wall thicknesses ranging between 4 mm and 12 mm and the tube breadth ranging between 150 mm and 400 mm were used to represent the variation of W0. Using repeated analyses from closely matched data obtained from numerical analysis and analytical model, the ratio between applied load (F) and bolt ultimate strength (Bu) can be correlated with the each failure mode. Following the consideration given above, a three dimensional diagram can be drawn for the relation between the load ratios and the geometric ratios of b and W0 with respect to each failure mode as shown in Fig. 23. A fitting boundary surface is also plotted to differentiate the test data in each failure mode. In this diagram, the following conditions can be obtained for each mechanism typology as:  Mechanism type i (failure mode i) occurs for F/RBu 6 0.16e2b and W0 6 0.084;  Mechanism type ii (failure mode ii) occurs for 0.16e2b < F/RBu 6 0.9 and 0.084 < W0 < 0.125;  Mechanism type iii (failure mode iii) occurs for 0.9 < F/RBu 6 1 Additionally, from the range of variation, b and W0 are recommended as the control parameters in the determination of the

Fig. 23. Effects of b and W0 on the failure mode of the connections.

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strength of connection in the mechanism type i and the mechanism type ii respectively. 5. Conclusion remarks The yield and ultimate strengths of a blind bolted endplate to square tubular column connection have been examined for the evaluation of the strength of such connection in joint tensile region. An experimental programme consisting of eleven test connections assembled by T-stubs and SHS columns with the standard bolts and the Hollo-Bolts has been described. The local behaviour manner and its corresponding strength have been identified for the connections in failure modes i, ii and iii. The load–deformation responses were analyzed for the determination of justified yield strength of the test connections. The finite element model explicitly allowing for the component details of the Hollo-Bolt was developed to further the understanding of the local response of the connections. Afterwards, an analytical model was proposed for the prediction of yield strength and ultimate strength of the connections. In this model, the yield line analysis has been adopted for the plastification of SHS column connecting face with combined action of the blind bolt in tension whereby the theoretical formulae have been derived for three failure modes. Moreover, based upon augments relating to membrane forces and circumferential locking action of the bolt, an expression for the prediction of ultimate strength has also been proposed. Finally, the accuracy and applicability of the proposed analytical model have been examined and discussed. Based on the results of experimental test, finite element modelling and analytical model study, the main conclusions can be summarized as follows: 1. The experimental inspection of the connection behaviour has indicated three failure modes which can be analogy with their counterparts in Eurocode 3. The failure mode i is featured by complete flexural yielding of SHS column face and bolt pullout while the failure mode ii is characterized by bolt failure with local plastification of SHS column face. The failure mode iii can be regarded as an extreme case in which the flaring sleeves of the Hollo-Bolt becomes the weakest part of the components and dominant in the failure of the connection. It was shown that the connection strength is sensitive to the tube wall slenderness and bolt gauge width with respect to outside breadth of SHS column. Moreover, the size of the blind bolt also has significant influence on the local plastification of SHS column and thus the strength of the connection especially for the case of failure mode i. 2. It was shown that the deformational behaviour of the SHS column connecting face differs from that of the overall connection due to localized deformation of bolt clearance hole and the bolt pullout, thereby perplexing the use of notional ultimate deformation limit of tube breadth in the representation of the yield strength of the connection. Alternatively, the classical bilinear approximation method appears to be promising by providing almost identical results of yield strengths based on the load–deformation curves of the tube connecting face and the overall connection. 3. The finite element models achieved acceptable correlation with limited experimental results in terms of force–deformation response of the connection. The numerical stress distribution exhibited significant plastification around the bolt clearance hole on the tube connecting face which can be represented by curved yielding manner. It appears that the bolt shank of the Hollo-Bolt can be regarded as the primary part to carry the tensile load as long as the circumferential locking action between the flaring sleeve and the bolt clearance hole of the SHS column face remains effective.

4. The predicted yield strength and ultimate strength results from the proposed analytical model were found to agree well with the experimental results. These good agreements confirm that in contrast to referred design formulae, the proposed analytical model is able to give better allowance for the plastification of SHS column connecting face with combined action of the blind bolt. Moreover, despite some differences due to the assumption of the bolt size details, the consideration of the membrane forces and circumferential locking action of the Hollo-Bolt are justified in the prediction of the ultimate strength of the connection. 5. The results of the parametric study indicated that the flexural yield strengths of the connections for failure mode i and ii are mostly influenced by b and W0 respectively, which can be regarded as key parameters in the design accordingly. Although the difference may exist when comparing the geometry of the blind bolts and the SHS columns considered in this study with those in engineering practice, the presented analytical models provide a basis for further elaboration of design parameters of the full-scale joints with similar connection details. Acknowledgements The research work described in this paper was supported by the National Natural Science Foundation of PR China (Nos. 51308363 & 11327801), the Scientific Research Foundation for the Returned Overseas Chinese Scholars (No. 2013-1792-9-4) and the Program for Changjiang Scholars & Innovative Research Team in University (No. IRT14R37). The experiments were carried out in the Civil Engineering Laboratory at Sichuan University. The authors are grateful for the contributions of the students, X.K. Liu, H. Xue, and Y.L. Chen, who have assisted in the experimental tests. References [1] One side structural fastener. AJAX Engineered Fasteners, Australia. . [2] Type HB Hollo-Bolt, Lindapter. United Kingdom. . [3] Tabsh SW, Mourad S. Resistance factors for blind bolts in direct tension. Eng Struct 1997;12(19):995–1000. [4] Klippel S. Recent design developments with blind mechanically operated bolt systems for use with hollow section steelwork. J Constr Steel Res 1998;46(1– 3):267–8. [5] Occhi F. Hollow section connections using (Hollofast) HolloBolt expansion bolting. CIDECT Report 6G-14E/96; 1996. [6] Korol RM, Ghobarah A, Mourad S. Blind bolting W-shape beams to HSS columns. J Struct Eng ASCE 1993;119(12):3463–81. [7] Liu Y, Málaga-Chuquitaype, Elghazouli AY. Behaviour of beam-to-tubular column angle connections under shear loads. Eng Struct 2012;42:434–56. [8] SCI/BCSA. Joints in Steel Construction: Simple Connections. The Steel Construction Institute (SCI) & The British Constructional Steelwork Association (BCSA); 2005. [9] Kurobane Y, Packer JA, Wardenier J, Yeomans N. Design guide for structural hollow section column connections. Koln, Germany: CIDECT and Verlag TüV Rheinland GmbH; 2004. [10] Melnick SL. Product case study-Huntington botanical gardens. Modern Steel Constr 2003;10:69. [11] France JE. Bolted connections between open section beams and box columns. University of Sheffield, United Kingdom, in fulfilment of the requirements for Ph.D. degree; 1997. [12] Yeomans N. Rectangular hollow section column connections using the Lindapter HolloBolt. In: Proc. 8th int. symposium on tubular structures, Singapore; 1998. p. 559–66. [13] Yeomans N. I-beam to rectangular hollow section column T-connections. In: Proc. 9th int. symposium on tubular structures, Düsseldorf, Germany; 2001. p. 119–26. [14] Elghazouli AY, Málaga-Chuquitaype C, Castro JM, Orton AH. Experimental monotonic and cyclic behaviour of blind-bolted angle connections. Eng Struct 2009;31:2540–53. [15] Barnett TC. The behaviour of a blind bolt for moment resisting connections in hollow steel sections. University of Nottingham, United Kingdom, in fulfilment of the requirements for Ph.D. degree; 2001. [16] Goldsworthy H, Gardner AP. Feasibility study for blind bolted connections to concrete-filled circular steel tubular columns. Struct Eng Mech 2006;24 (4):463–78.

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