Ultimate capacity of bulge formed T-joints under brace axial compressive loading

Thin-Walled Structures 129 (2018) 237–250

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Ultimate capacity of bulge formed T-joints under brace axial compressive loading Feilong Nie, Qing Zhang, Xianrong Qin, Yuantao Sun

T

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School of Mechanical and Energy Engineering, Tongji University, Shanghai 201804, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Bulge formed T-joint Unstiffened T-joint Experimental and finite element method Ultimate capacity Brace axial compressive loading

An innovative tubular joint reinforced by a bulge plate was proposed for the first time, named the bulge formed joint. Different from the unstiffened joint, an additional bulge plate is employed to connect the braces and the chord. To investigate the ultimate capacity, firstly full scale experiments and finite element (FE) analyses were conducted for a bulge formed T-joint and its counterpart (an unstiffened joint). Verified by test, the FE method was used to investigate the ultimate capacity of more bugle formed T-joints under brace axial compressive loading. Totally 190 FE models of the bulge formed T-joints and 146 FE models of the unstiffened joints (the counterpart) were analyzed. As a result, four failure modes were found for the bulge formed T-joints, and the ultimate capacity of the bulge formed T-joint was proved to be increased up to 200% compared with that of the unstiffened joint. Finally, the effects of geometrical parameters (τS, η1, η2, α, β, γ and τ) on the ultimate capacity were discussed, among these geometrical parameters, the first three are particularly defined for the bulge plate, while the other four are same as those of the unstiffened joint.

1. Introduction Circular tubular members, due to the low drag coefficient and high strength/weight ratio, are commonly used in offshore structures [1] and many truss structures [2], as shown in Fig. 1a. The connection of the chord and braces results in a tubular joint which is mainly subjected to axial loads. For a tube, the radius strength is far smaller than the axial strength particularly when the thickness is far smaller than the diameter. So the chord surface indentation may occur at the brace/ chord intersection under the ultimate brace axial compressive loading. During the recent decades, various kinds of stiffened joints were investigated to enhance the ultimate capacity. The investigation conducted by Fung et al. [3], Chan et al. [4], Choo et al. [5] and van der Vegte et al. [6] proved the significant increase of the ultimate strength of the joints reinforced with doubler or collar plates. Nassiraei et al. proposed the equations to predict the ultimate bearing capacity of the collar plate reinforced T/Y joints or doubler plate under brace axial compressive loading [7], brace tension loading [8] and in plane bending loading [9]. For the doubler plate reinforced T/Y joints, the equations to predict the ultimate bearing capacity were also proposed under brace axial compressive loading [10], brace tension loading [11] and in plane bending loading [12]. Lee and Llewelyn-Parry [13] investigated the strength of the internal ring stiffened T-joint and discussed the effects of the parameters and the locations of the ring

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stiffeners on the ultimate capacity. Xu et al. [14] investigated the ultimate bearing capacity of the concrete filled tubular joint by experimental method. These stiffened joints were also recommended as alternative methods to enhance the strength of the joint in CIDECT [15] and API [16]. Yang et al. [17] and Shao et al. [18] focused on static strength of the chord reinforced Y and T-joints which increase the chord thickness at the brace/chord intersection. This kind of reinforced joint was also recommended by API [16]. Zhu et al. [19] conducted several experiments to investigate the ultimate capacity of circular hollow section T-joints reinforced by external rings which can be applied to a finished tubular structure. Lesani et al. [20,21] numerically and experimentally investigated the ultimate capacity of the FRP-strength tubular T-joints under brace axial compressive loads. The results showed an evident enhancement in the strength behavior of FRPstrengthened joints, with stresses and deformations moderated up to 50% of the original joint. These reinforced joints can enhance the static strength of the tubular joint, however, some disadvantages still exist. For example, for the internal ring stiffened joint, it is not convenient to weld the ring to the internal surface of the chord with small diameter. All these joints are strengthened without the obvious change of the chord surface, so when more than one brace are welded to the chord, overlap phenomenon (as shown in Fig. 2a) may occur especially for the large brace diameter and large brace/chord inclination angle. As is known, the overlap may lead

Corresponding author. E-mail addresses: nfl[email protected] (F. Nie), [email protected] (Q. Zhang), [email protected] (X. Qin), [email protected] (Y. Sun).

https://doi.org/10.1016/j.tws.2018.02.017 Received 11 September 2017; Received in revised form 5 January 2018; Accepted 19 February 2018 0263-8231/ © 2018 Elsevier Ltd. All rights reserved.

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Fig. 2. (a) Overlap phenomenon of a unstiffened K-joint, (b) Gap between the braces of a bulge formed K-joint.

which increases the spatial area for connecting and effectively avoids the overlap phenomenon, as shown in Fig. 2b. Due to the above advantages, the bulge formed joint has been presented by Chinese patent (patent number 201610008519.3) [23] and has been implemented in a practical engineering by Shanghai Zhenhua Heavy Industries Co., Ltd (ZPMC) (as illustrated in Fig. 1b). As shown in Fig. 1c, the geometric parametric parameters of the bulge formed T-joint includes: TS (thickness of the bulge plate), RS (radius of the bulge plate in XY plane), H (height of the bulge plate), LS (length of the bulge plate), D (chord diameter), d (brace diameter), T (chord thickness), t (brace thickness), LS1 (length of un-bulge part of the bulge plate in XY plane). A constant value of LS1 (i.e. LS1 = 0.38D ) is used in this study due to its limited effect on ultimate capacity, also a too long LS1 may lead to waste of materials. It should be noted that LS which can be expressed as the function of H , RS and LS1 (as shown in Fig. 1c) is not an independent parameter. So totally seven dimensionless geometric parameters are investigated for the bulge formed T-joint: τS , η1, η2 , α , τ , β and γ , among which, the first three parameters are particularly defined for the bulge formed T-joints: τs (Ts / T ),  η1 (2RS / D),  η2 (2H / D) , while the others share the same definitions with those of the unstiffened joint. The geometry of the bulge plate is mainly determined on the strength and stiffness. In this paper, the ultimate capacity of a real scale bulge formed Tjoint and its counterpart (an unstiffened tubular T-joint) was investigated by the experimental method and the finite element (FE) method. The results showed a distinct advantage of the bulge formed T-joint on static strength. The good agreement between the experimental data and the FE results proved that the FE method was able to accurately simulate the static strength of the joints. Then the failure modes, the ultimate capacity and the effects of the geometrical parameters on the ultimate capacity were investigated for the bulge formed T-joints under brace axial compressive loading. In the FE method, the failure modes were obtained based on the elastic-plastic analysis. Other possible failure modes such as chord shear failure and punching failure etc. were not included in this study, which will be investigated in the future report.

Fig. 1. (a) Circular tubular members in the truss beam of a query-side crane; (b) bulge formed joint utilized in a truss structure for the first time; (c) illustration of the bulge formed T-joint.

to complex weld paths and processing technology which are not advocated in the practical engineering. In another aspect, as Sopha et al. [22] said, since the brace wall is thinner than the chord wall in the most offshore structures, the partial overlap joints with hot spot stresses (HSSs) located at the brace side may result in a shorter fatigue life compared with that previous chord failure mode which is common seen in T, Y, N and gapped K joints. Though the overlap can be avoided by enlarging the gap between the braces artificially, however, the change of the gap may lead to the eccentricity between the brace/chord centers and then an additional bending moment occurs. Aiming to the above disadvantages, a new type of stiffened joint called the bulge formed joint was proposed for the first time. In this joint, a bulge plate is welded to the chord surface to connect the braces,

2. Experimental procedures 2.1. Specimen details To investigate the ultimate capacity, two real scale specimens of the bulge formed T-joint and the unstiffened T-joint were designed and fabricated. 238

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The bulge plate, a key member of the bulge formed joint, was fabricated by a 8000-ton Forging Press. The hot pressing method was employed to forge the bulge plate to obtain a favorable final shape with high mechanical performance. For the details of the fabrication, the reader is referred to another report by authors [24]. The fabrication of a bulge formed K-joint in the above report was referred for the bulge formed T-joint in this study. The brace end profiles and grooves were designed based on the recommendation specification for the complete penetration weld of the AWS D 1.1 [25] and manufactured by computerized numerical control machine (CNC). The processed braces were respectively welded to the chord surface of the unstiffened joint and the bulge plate of the bulge formed T-joint. For the bulge formed T-joint, the bulge plate was welded to the chord surface by the fillet weld according to the specification of the AWS D 1.1 [25]. According to AWS D 1.1, the dihedral angle of a tubular joint is the crucial parameter for the determination of the weld size. Based on the dihedral angle, the weld size can be determined according to the prequalified joint details for CJP groove welds in tubular T-, Y-, and K-connections given in AWS D1.1 [25]. The calculation of the dihedral angle and the weld details according to AWS D1.1 [25] were presented in [24]. The weld size of the fillet weld between the bulge plate and the chord surface was equal to the thickness of the bulge plate, which satisfied the requirement of the minimum fillet size given in AWS D1.1 [25]. The weld material was the Supercored 71H with the ultimate strength of 580 MPa. During the assembling, the brace center was perpendicular to the chord center without eccentricity, and experienced workers were employed to ensure the weld quality. It should be noted that the test rig was of high stiffness to simulate the foundation. The main geometrical parameters of the tested unstiffened joint and the tested bulge formed joint are listed in Table 1.

Fig. 3. Details of specimens: (a) the unstiffened tubular T-joint; (b) the bulge formed T-joint.

As shown in Fig. 3a, the unstiffened T-joint consisted of a brace and a chord, with the brace welded to the surface of the chord directly. Two ear plates were welded to the chord ends through the endplates and assembled to the rig by pins. The thickness of the ear plates was 50 mm and the thickness of the endplates was 30 mm. It should be noted that the pin holes in the reaction frame were of the waist-shape to allow the axial displacement of the chord ends, so the effects of the chord axial loads were eliminated. Also, the chord ends were allowed to rotate along the pins. A square plate in length of 250 mm with thickness of 20 mm was welded to the brace end to transfer the axial compressive load from the hydric cylinder. As shown in Fig. 3b, the bulge formed T-joint consisted of a brace, a chord and a bulge plate. Different from the unstiffened joint, the brace and the chord were connected through the bulge plate indirectly. The overall length of the bulge plate was 900 mm (Here the length of the unbulge part was 0.38 times the chord diameter (LS1 = 0.38D ), so the overall length could be calculated by the equation 2(LS1 + RS2 − (RS − H )2 ) ; the height of the bulge plate was 130 mm and the thickness was 12 mm (equal to the chord thickness). The radius of the bulge plate (RS ) in XY plane was 404 mm as shown in Fig. 3b, so the shape of the bulge plate could be uniquely determined. The endplates, ear plates and square plates were respectively welded to the chord ends and the brace end in the same way as that of the unstiffened T-joint. The connection between the ear plates and the test rig was same as that of the unstiffened T-joint, i.e. through the pins and the waistshape holes in the reaction frame. So the supporting condition of the bulge formed T-joint was also same as that of the unstiffened joint, with axial sliding and the rotation along the pins allowed for the chord ends and effect of the chord axial load eliminated.

2.2. Test rig The test rig (as shown in Fig. 4) included the following parts: (1) The reaction frame; (2) The specimens (the bulge formed T-joint and the unstiffened joint); (3) Two pins; (4) The hydraulic loading system; (5) A force sensor; (6) The displacement transducers. Totally four displacement transducers were employed to monitor the vertical displacements at the key positions (the saddle and crown positions of the brace/bulge plate intersection of the bulge formed T-joint or the corresponding positions of the brace/chord intersection of the unstiffened joint). The average displacement of these four transducers was marked as δA . A displacement transducer was employed to monitor the vertical displacement at the bottom of the chord at the mid-span and the corresponding displacement was marked as δB . The ovalisation of the unstiffened joint and the bulge formed joint was calculated by δA − δB , as shown in Fig. 5a and b. 2.3. Test procedure The loading procedure is as follows: Step1: First, the brace axial load was increased from 0 to 20 kN and then unloaded back to 0 kN. It was performed three times to eliminate the friction of during the assembling.

Table 1 Parameters of the unstiffened T-joint and the bulge formed T-joint.

Unstiffened T-joint Bulge formed T-joint Unstiffened T-joint Bulge formed T-joint

D (mm)

d

T

t

L

l

Rs

H

Ts

402 402 αB = 2l/d 6.42 4.78

159 159 α = 2L/D 11.94 11.94

12 12 β = d/D 0.4 0.4

10 10 γ = D/(2T) 16.75 16.75

2400 2400 τ = t/T 0.83 0.83

510 380 τs = Ts/T – 1.00

– 404 η1 = 2Rs/D – 2.01

– 130 η2 = 2H/D – 0.65

– 12 Material Q345 Q345

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Fig. 6. Stress-strain curve of the steel Q345 [26].

Finally the load-ovalisation curves and failure modes of the two specimens were obtained and presented in Section 4 together with the FE results.

Fig. 4. Test rig.

3. FE method 3.1. Material property In the FE simulation, the multi-linear isotropic hardening model of Q345 given by Han and Liu [26] and von Mises yield criterion were adopted. The nominal yield stress and ultimate stress of the Q345 were 345 MPa, 510 MPa respectively; the young's modulus and Poisson's ratio were 203 GPa and 0.3 respectively. As given by the supplier of the specimens, the ranges of the yield stress and the ultimate strength were about ± 10 MPa. An error analysis showed that the above ranges only led to an error within about ± 3% in the ultimate capacity calculation compared with the results obtained by the nominal values. It is an acceptable error in the validation of the FE model by test. So in the study, the nominal material properities of the specimens are presented in Fig. 6 and Table 2, which were then used in the following FE analysis. 3.2. Weld consideration In this study, the welds at the brace/chord intersection of the unstiffened joint and those at the brace/bulge plate intersection of the bulge formed T-joint were complete joint penetration (CJP) welds, and the welds at the bulge plate/chord intersection of the bulge formed Tjoint were fillet welds. As Vegte [27] said, the ignorance of the welds at the brace/chord intersection in the FE model may lead to an under predict of the ultimate capacity as much to 20% below the experimental results. So the welds in the FE simulation were modeled based on the specifications of the AWS D 1.1 [25]. The weld material was Supercored 71H with an ultimate strength of 580 MPa, and the material of the tubular members was Q345 with the ultimate strength of 510 MPa. So the strength of the tubular members is not exceeded that of the weld. In this study, the material of the welds were assumed same as that of the brace, the chord and the bugle plate in the FE analysis since it hardly affects the global behavior [28].

Fig. 5. Definition of the “ovalisation”: (a) unstiffened joint, (b) bulge formed joint.

3.3. Element and mesh

Step2: Increase the brace axial compressive loading gradually from zero to about 480 kN and 880 kN respectively for the unstiffened joint and the bulge formed T-joint. These two values were estimated about 90% the ultimate capacity of the two specimens. The displacement values of the transducers were recorded during the loading process. Step3: At the final stage, the displacement controlled method was adopted to advance the brace end with a rate of 0.2 mm/min before the peak load and with a rate of 1 mm/min after the peak load.

To get more accurate results, 20 node 3D element solid 186 in Table 2 Material parameters of the steel Q345 [26].

240

Steel grade

f y (MPa)

fu (MPa)

ε1(%)

ε2 (%)

ε3 (%)

ε 4 (%)

Q345

345

510

0.170

2

20

25

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(a) original mesh of the unstiffened joint

(b) refined mesh of the unstiffened joint

(c) original mesh of the bulge formed T-joint

(d) refined mesh of the bulge formed T-joint

Fig. 7. Mesh Details: (a) original mesh of the unstiffened joint, (b) refined mesh of the unstiffened joint, (c) original mesh of the bulge formed T-joint, (d) refined mesh of the bulge formed T-joint.

ANSYS was adopted rather than the computationally less-demanding 8 node solid element or shell elements. Solid 186 has the curve profile which is suitable for simulating the complex profiles. The mid-point in the element can increase the accuracy of the calculation results. To capture the accurate failure modes and ultimate capacity under the brace axial compressive loading, two layer meshes were used through the thickness of bulge plate, chord and brace, and convergence test was conducted before the further FE simulation. In the convergence study, different mesh densities were used for the specimens to obtain a suitable mesh size before the following FE simulation. As shown in Fig. 7, for one quarter of the unstiffened joint, totally 2341 elements were used for the original meshes whereas 7974 elements were used for the refined meshes. For one quarter of the bulge formed joint, totally 3910 elements were used for the original meshes whereas 8629 elements were used for the refined meshes. The negligible differences between the load-ovalisation curves obtained from the original meshes and the refined meshes show a good convergence, as shown in Fig. 8.

Fig. 8. Convergence study of load-ovalisation curves for the unstiffened joint and the bulge formed joint.

3.5. Boundary conditions Due to the symmetry of the structure and load, only quarter of the model was adopted in the FE simulation to save the computational cost. For the nodes on the symmetry planes, the displacement perpendicular to the symmetry planes was restrained while the other two transitional degrees of freedom were free. The above constraint conditions could be achieved by applying “Symmetry boundary condition” in ANSYS. For the chord ends, chord axial displacement and the rotation along the pins were allowed to simulate the real constraint in the experiment. Due to the symmetry constraint of the plane of the symmetry, no rigid body motion occurred in the chord axial direction.

3.4. Bulge plate/chord contact of the bulge formed T-joint When the bulge formed T-joint is subjected to the brace axial compressive loading, the internal surface of the bulge plate is tended to deform toward the inner of the chord, while the chord surface contacted to the bulge plate prevent the will-be penetration of the bulge plate. So the contact which allowed the separation but not allowed the penetration was adopted to simulate the interaction between the connected surfaces. ANSYS Contact Technology Guide [29] was referred to conduct the suitable interaction in ANSYS. 241

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failure mode of the tested unstiffened joint is similar to that obtained by Zhu et al. [19], Choo et al. [5] and van der Vegte et al. [6]. The peak brace axial load, which was then taken as the ultimate capacity, was found to be 565.3 kN.

3.6. Nonlinear solver Both the material nonlinearity and the geometrical nonlinearity were considered in the solver, and the arc-length method in ANSYS was adopted to calculate the load-ovalisation curves.

4.2. Failure mode of the bulge formed T-joint

4. Verification of the FE models

In the following passages, we mainly focus on the failure mode and the ultimate capacity of the bulge formed T-joints. The failure mode of the bulge formed T-joint is illustrated in Fig. 10a (the plan view) and b (the cross-section view). Unlike the unstiffened joint whose plastic yield occurred at the chord, the plastic yield of the bulge formed T-joint occurred at the bulge plate, while the plastic deformation at the chord was negligible. It could be observed that the plastic hinge initially occurred at the brace/bulge plate intersection (called “the first plastic hinge”) on the bulge plate, and then developed far away from the intersection (called “the second plastic hinge”) with the continual increase of the brace axial compressive load. The bulge plate between “the first plastic hinge” and “the second plastic hinge” deformed toward the bottom of the chord. However the bulge plate in the brace/bulge plate intersection rotated along the reverse direction compared with that between the first plastic hinge and the second plastic hinge. So the deepest indentation was observed at the brace/bulge intersection and the indentation far away from the intersection was smaller. The bulge plate at the two sides deformed outward slightly. So the “ovalisation” used in the unstiffened joint was also used to describe the deformed cross section of the bulge formed T-joint. It can be observed that the bulge formed T-joint changes the failure from the chord (for the unstiffened joint) to the bulge plate due to the change of structure and load path. For the unstiffened tubular T-joint, the brace axial compressive load is transferred to the chord through the brace/chord intersection (as shown in Fig. 11a), while that of the bulge formed T-joint is transferred to the chord through the bulge plate. Firstly the brace axial compressive load is transferred to the bulge plate through the brace/bulge plate intersection, and then transferred to the chord through two paths: one is the chord/bulge plate intersection (the load path I illustrated in Fig. 11b), the other is the “contact interaction” between the outer chord surface and the internal surface of the bulge

To verify the FE method, the failure modes, load-ovalisation curves and ultimate capacity values obtained from the FE results were compared with the test results. More details about the test were presented in Section 2. 4.1. Failure mode of the unstiffened T-joint For the tested unstiffened T-joint, both the failure modes obtained by the experiment and the FE results are illustrated in Fig. 9a (the plan view) b (the cross-section view), which show a good agreement between the test results and FE results. Fig. 9b (the cross-section view) shows that there are two plastic hinges at the chord wall: one is located at the brace /chord intersection, named as “the first plastic hinge” for the convenience of description; the other is located at the chord wall near the brace/chord intersection (called “the second plastic hinge”). In the test, with the increase of the brace end displacement, the plastic hinge appeared at the brace/chord intersection (i.e. the first plastic hinge) and then moved away from the first plastic hinge in the chord wall and formed the second plastic hinge. As a result, the chord wall between the first plastic hinge and the second plastic hinge moved toward the bottom of the chord. The chord wall in the brace/chord intersection rotated in the reverse direction compared with that between the first plastic hinge and the second plastic hinge and formed a slight outward convex shape. So the deepest indentation was observed at the brace/chord intersection and the smaller indentation was observed at the location far away from the intersection. The two sides of the chord wall deformed outward slightly, so the final shape of the cross section looked like an “oval”. The brace wall remained straight and no yield occurred in it during the test since the axial strength of the brace was higher than the radial strength of the chord. It can be found that the

(a)

the first plastic hinge

the second plastic hinge (b)

Fig. 9. Failure mode of the unstiffened T-joint under brace axial compressive loading. 242

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(a)

the first plastic hinge

the second plastic hinge (b)

Fig. 10. Failure mode of the bulge formed T-joint under brace axial compressive loading.

plate intersection, as a result, the failure occurred at the bulge plate/ brace intersection first and then developed on the bulge plate. The test results shows a great advantage of the bulge formed T-joint, whose ultimate capacity is 990 kN, 1.75 times that of the unstiffened joint. The failure at the ultimate state only occurred at the bulge plate without tremendous deformation in the chord, which is considered to be good for the whole truss structure in the practical engineering. 4.3. Load-ovalisation curves of the bulge formed T-joint and the unstiffened joint As shown in Fig. 12, the load-ovalisation curves of the unstiffened joint and the bulge formed T-joint obtained in test are plotted combined with those obtained by FE analyses. The y-label “load” represents the brace axial compressive loading. The x-label refers to the ovalisation. As described in Section 2.2, the ovalisation of the unstiffened joint refers to the relative vertical deformation between the brace/chord intersection and the chord bottom at the mid-span. For the bulge formed T-joint, the

Fig. 11. (a) Load path of the unstiffened T-joint (b) Load paths of the bulge formed T-joint.

plate (the load path II illustrated in Fig. 11b). The contact interaction is explained as follows: under the axial brace compressive loading, the internal surface of the bulge plate contacted to the chord was tended to deform into the inner of the chord, however the chord surface prevents the will-be penetration of the bulge plate. This interaction transfers partial load from the brace to the chord. On the other hand, the brace/ bulge plate intersection suffers a relative higher stress due to its smaller weld path compared with the bulge plate/chord intersection. So the bulge plate/chord intersection is relative stronger than the brace/bulge

Fig. 12. Load-ovalisation curves of the bulge formed T-joint and the unstiffened joint. 243

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Table 3 Ultimate capacity values of the two specimens.

Table 4 Ranges of the geometrical parameters.

Specimen

Fu, test [kN]

Fu, num [kN]

Fu, num/ Fu, test [kN]

Parameter

Definition

Group 1

Group 2

Unstiffened joint Bulge formed joint

565.3 990

527 978

0.932 0.988

α β γ τ τS η1 η2

2L/ D d/ D D/2T t /T Ts / T 2Rs / D 2H / D

12 0.45 18 0.7 1, 1.25, 1.5, 1.75, 2 2, 3, 4 0.5, 0.7, 0.9

12, 18, 24, 30, 36 0.3, 0.45, 0.6 12, 18, 24 0.4, 0.7, 1.0 1 3 0.7

ovalisation refers to the relative vertical deformation between the brace/bulge plate intersection and the chord bottom at the mid-span. It is observed that the load-ovalisation curve of the bulge formed T-joint locates at an obviously higher position compared with that of the unstiffened joint, which means a remarkably higher strength of the bulge formed T-joint. Exactly, the ultimate strength of the bulge formed Tjoint obtained from the test was 990 kN, which is 1.75 times that of the tested unstiffened joint whose ultimate capacity is 565.3 kN as shown in Table 3. In this table, Fu, test and Fu, num refer to the ultimate capacity obtained in test and obtained by FE analyses respectively. The initial stiffness of the bulge formed T-joint, defined as the slope of the straight line of the load-ovalisation curve before the peak load, was proved to be about 339.4 kN/mm which is 3.11 times that of the unstiffened joint (about 109 kN/mm). From Fig. 9, 10, 12 and Table 3, it can be found that the failure modes, the load-ovalisation curves and the ultimate capacity values of the two specimens obtained by FE analysis show a good agreement with the test results, which indicates that the FE method is feasible to simulate the load-ovalisation behavior of the joints under brace axial compressive loading accurately. The comparison of the two specimens shows a great advantage of the bulge formed T-joint on the ultimate capacity and the initial stiffness compared with the unstiffened joint. In the following passages, the FE method verified by the test was utilized to investigate the more failure modes of the bulge formed T-joints and the effects of the geometrical parameters of the bulge formed T-joint on the ultimate capacity.

(Notes: D = 402 mm and αB = 2l/ d = 5 in this the FE study).

was taken as 5 times the brace diameter (αB = 5). The LS1 of the bulge plate as shown in Fig. 1c was taken as a constant value of 0.38D due to its limited effect on the ultimate capacity. When the LS1 changes from 0.25D to 0.5D, the ultimate capacity only decreases by 0.6% for the case: α = 11.94 , β = 0.4 , γ = 16.75, τ = 0.83, τS = 1, η1 = 2.01, η2 = 0.65. Also, a too large LS1 may lead to the waste of the material since the connected braces are mainly located on the bulge plate. 5.2. Typical failure modes and load-ovalisation curves Based on the FE analyses, four typical failure modes and the corresponding load-ovalisation cures of the bulge formed T-joints under the brace axial load are summarized as follows: (1) Local buckling failure of the bulge plate (LBFBP); (2) Local buckling of the brace (LBFB); (3) Combination of the local buckling failure of the bulge plate and the local buckling of the brace (LBFBP + LBFB); (4) Chord face failure (plastic failure of the chord surface at the bulge plate/chord intersection) (CFF). The four failure modes LBFBP, LBFB, LBFBP + LBFB and CFF are respectively illustrated in Fig. 13a–d, and the corresponding load-ovalisation curves are shown in Fig. 14. The distinct peak loads can be found on the four load-ovalisation curves. As to the ultimate capacity, the corresponding load at the deformation limit (3% D relative to the chord center) was recommended by Lu et al. [30]. Choo et al. [5] modified Lu's deformation limit to a chord ovalisation of 6% D which is relative to the chord bottom to determine the ultimate capacity. In this study, the ovalisations at the peak loads of all the joints were smaller than the 6%D, so the peak loads were taken as the ultimate capacity values.

5. Investigation of the parametric effects on the ultimate capacity 5.1. Investigation scheme Due to the high cost and time consuming, the experimental method was not feasible to investigate the ultimate capacity and failure modes of the bulge formed T-joints with more different parameter configurations. So the FE method verified by test was employed to investigate failure modes and ultimate capacity values of more bulge formed Tjoints with different parameter configurations under the brace axial compressive loading. The bugle formed T-joint, has totally seven geometrical parameters: τS , η1 η2 α , β , γ and τ , among which the first three are particularly defined for the bulge formed T-joint, while the other four parameters share the same definitions as those in the unstiffened joint. To investigate the effects of these parameters on the ultimate capacity, totally two groups of parameters were set for comparison, as shown in group 1 and group 2 listed in Table 4. In group 1, totally 45 FE analyses for bulge formed T-joints were conducted to investigate the effects of the τS , η1 and η2 , with other four parameters α , β , γ and τ kept constant (α = 12, β = 0.45, γ = 18 and τ = 0.7), and an unstiffened joint which shared the same α , β , γ and τ was analyzed as the counterpart. In group 2, totally 145 FE analyzes were conducted for the bulge formed T-joints to investigate the effects of the α , β , γ and τ , with other three parameters τS , η1 and η2 kept constant (τS = 1, η1 = 3 and η2 = 0.7). In the counterpart, 145 FE analyzes for the unstiffened joints were conducted, which shared the same α , β , γ and τ with those of the bulge formed T-joints. Totally 190 FE models of the bulge formed T-joints 146 EF models of the unstiffened joints were analyzed. The chord diameter was taken as the 402 mm and the brace length

5.3. The FE results of the ultimate capacity The FE results of the bulge formed T-joints listed in group1 and group 2 are listed in Tables 5 and 6 respectively. In these tables, the Fu, b refers to the ultimate capacity of the bulge formed T-joint; Fu, u refers to the ultimate capacity of the counterpart (the unstiffened joint which shares the same α , β , γ and τ with the bulge formed T-joint). The data in Table 5 are used to investigate the effects of the parameters τS , η1 and η2 , and the data in Table 6 are used to investigate the effects of α , β , γ and τ . From Tables 5 and 6, it can be found that the ultimate capacity ratio Fu, b/ Fu, u ranges from 1.18 to 3, which means the bulge formed joint increases the ultimate capacity by up to 200% compared with that of the unstiffened joint. 5.4. Effects of τS , η1 and η2 on the ultimate capacity of the bulge formed Tjoint The parameters τS , η1 and η2 are three geometrical parameters particularly defined for the bulge formed T-joint. The parameter τS refers to the ratio of the thickness of the bulge plate to that of the chord. 244

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Fig. 13. Failure modes of the bulge formed T-joints under brace axial compressive loading.

The parameter η1 refers to the ratio of the radius of the bulge plate in XY plane to the radius of the chord as shown in Fig. 1c. The parameter η2 refers to the ratio of the height of the bulge formed T-joint to the radius of the chord. So for a constant chord thickness, the thickness of the bulge plate increases with the increase of τS . For a constant chord diameter, the radius of the bulge plate in XY plane increases with the increase of η1; the height of the bulge plate increases with the increase of η2 . 5.4.1. Effect of η2 on the ultimate capacity From Table 5, it can be observed that the parameter η2 has almost no effect on the ultimate capacity. To give a clear illustration, the data of the case τS = 1 is plotted in Fig. 15. For other cases (τS = 1.25, 1.5, 1.75 and 2) the conclusions are similar, so the corresponding ultimate capacity values are not plotted for the sake of brevity. In the following passages, a constant value of 0.7 was adopted for η2 (η2 = 0.7 ) due to its negligible effect on the ultimate capacity.

Fig. 14. Load-ovalisation curves of the four typical failure modes (LBFBP, LBFB, LBFBP + LBFB and CFF). (Note: In Fig. 14, D = 402 mm, αB = 5, α = 18, τS = 1, η1 = 3, η2 = 0.7).

Table 5 The ultimate capacity values of the bulge formed T-joints with different values of τS , η1 and η2 . η1 = 2 τS

1

1.25

1.5

1.75

2

η1 = 3

η1 = 4

η2

Fu, b (kN)

Fu, b Fu, u

Failure

Fu, b (kN)

Fu, b Fu, u

Failure

Fu, b (kN)

Fu, b Fu, u

Failure

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

1032.9 1042.0 1046.4 1359.2 1376.9 1387.1 1470.7 1497.5 1491.3 1493.7 1503.0 1477.9 1500.9 1499.1 1508.1

2.06 2.08 2.08 2.71 2.74 2.76 2.93 2.98 2.97 2.98 2.99 2.94 2.99 2.99 3.00

LBFBP LBFBP LBFBP LBFBP + LBFB LBFBP + LBFB LBFBP + LBFB CFF LBFB LBFB CFF LBFB LBFB LBFB LBFB LBFB

896.0 898.4 901.0 1198.8 1202.8 1208.1 1474.9 1473.8 1474.3 1476.7 1480.8 1499.5 1477.7 1478.9 1477.8

1.79 1.79 1.80 2.39 2.40 2.41 2.94 2.94 2.94 2.94 2.95 2.99 2.94 2.95 2.94

LBFBP LBFBP LBFBP LBFBP + LBFB LBFBP + LBFB LBFBP + LBFB LBFB LBFB LBFB LBFB LBFB LBFB LBFB LBFB LBFB

815.5 815.0 816.2 1096.6 1096.9 1100.7 1390.9 1390.8 1394.9 1476.4 1477.2 1477.2 1496.4 1480.2 1478.5

1.62 1.62 1.63 2.18 2.19 2.19 2.77 2.77 2.78 2.94 2.94 2.94 2.98 2.95 2.95

LBFBP LBFBP LBFBP LBFBP LBFBP LBFBP LBFBP LBFBP LBFBP LBFB LBFB LBFB LBFB LBFB LBFB

(Notes: In Table 5, D = 402 mm, αB = 5, α = 12, β = 0.45, γ = 18, τ = 0.7). 245

+ + + + + +

LBFB LBFB LBFB LBFB LBFB LBFB

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Table 6 The ultimate capacity values of the bulge formed T-joint with different values of α , β , γ and τ . γ = 12 β = 0.45

β = 0.3 α

12

18

24

30

36

β = 0.6

τ

Fu, b (kN)

Fu, b Fu, u

Failure

Fu, b (kN)

Fu, b Fu, u

Failure

Fu, b (kN)

Fu, b Fu, u

Failure

0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1

902.8 1027.5 1075.2 892.8 1003.4 1049.6 875.1 978.0 977.9 759.6 759.6 759.5 582.9 582.9 582.9

1.28 1.40 1.45 1.41 1.53 1.59 1.55 1.68 1.67 1.51 1.48 1.48 1.32 1.30 1.29

LBFB LBFBP LBFBP LBFB LBFBP LBFBP LBFB CFF CFF CFF CFF CFF CFF CFF CFF

1263.5 1569.9 1628.2 1266.6 1427.4 1427.4 977.9 977.9 977.9 740.4 740.4 740.4 582.9 582.9 582.9

1.32 1.60 1.65 1.55 1.71 1.70 1.44 1.42 1.41 1.31 1.30 1.30 1.22 1.22 1.22

LBFB LBFBP LBFBP LBFB CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF

1699.7 2192.1 2252.6 1427.5 1427.5 1427.6 978.0 978.1 978.1 740.2 740.2 740.2 582.9 582.9 582.9

1.38 1.75 1.79 1.50 1.48 1.48 1.33 1.32 1.32 1.26 1.25 1.25 1.19 1.19 1.18

LBFB LBFBP LBFBP CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF

γ = 18 β = 0.45

β = 0.3 α

12

18

24

30

36

β = 0.6

τ

Fu, b (kN)

Fu, b Fu, u

Failure

Fu, b (kN)

Fu, b Fu, u

Failure

Fu, b (kN)

Fu, b Fu, u

Failure

0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1

533.5 560.6 582.3 525.1 550.1 571.7 516.1 540.4 560.6 485.3 485.3 485.3 389.5 389.4 389.4

1.51 1.54 1.59 1.61 1.64 1.69 1.73 1.77 1.83 1.77 1.74 1.72 1.55 1.52 1.51

LBFBP + LBFB LBFBP LBFBP LBFBP + LBFB LBFBP LBFBP LBFBP + LBFB LBFBP LBFBP CFF CFF CFF CFF CFF CFF

832.4 898.4 930.1 816.9 870.1 898.4 637.8 637.8 637.8 480.4 480.4 480.4 389.4 389.4 389.4

1.68 1.79 1.84 1.85 1.95 2.00 1.63 1.61 1.62 1.39 1.38 1.38 1.29 1.28 1.28

LBFB LBFBP LBFBP LBFB LBFBP LBFBP CFF CFF CFF CFF CFF CFF CFF CFF CFF

1132.8 1278.9 1328.2 938.5 938.6 938.7 637.7 637.7 637.7 480.4 480.4 480.4 389.3 389.3 389.3

1.75 1.95 2.01 1.71 1.68 1.68 1.39 1.37 1.37 1.26 1.25 1.25 1.20 1.20 1.20

LBFB LBFBP LBFBP CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF

γ = 24 β = 0.45

β = 0.3 α

12

18

24

30

36

τ

Fu, b (kN)

Fu, b Fu, u

Failure

0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1 0.4 0.7 1

352.7 366.8 379.3 348.9 362.8 375.3 344.2 357.2 361.0 339.4 352.0 360.2 288.7 288.7 288.8

1.67 1.70 1.74 1.74 1.77 1.82 1.84 1.88 1.89 1.96 2.00 2.03 1.79 1.76 1.75

LBFBP LBFBP LBFBP LBFBP LBFBP LBFBP LBFBP LBFBP LBFBP LBFBP LBFBP CFF CFF CFF CFF

+ LBFB

+ LBFB

+ LBFB

+ LBFB

β = 0.6

Fu, b (kN)

Fu, b Fu, u

Failure

Fu, b (kN)

Fu, b Fu, u

Failure

573.2 601.1 625.0 558.8 587.1 608.9 472.6 472.7 472.7 360.3 360.3 360.3 288.8 288.8 288.8

1.92 1.99 2.05 2.02 2.10 2.16 1.90 1.88 1.87 1.61 1.59 1.58 1.42 1.41 1.40

LBFBP + LBFB LBFBP LBFBP LBFBP + LBFB LBFBP LBFBP CFF CFF CFF CFF CFF CFF CFF CFF CFF

808.7 863.8 907.1 682.4 682.6 682.7 473.0 473.1 473.2 360.0 360.0 359.9 289.0 289.0 289.0

2.06 2.17 2.27 1.95 1.93 1.92 1.56 1.54 1.53 1.36 1.35 1.35 1.25 1.25 1.24

LBFB LBFBP LBFBP CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF CFF

(Notes: In Table 6, D = 402 mm, αB = 5, τS = 1, η1 = 3, η2 = 0.7).

strength of the bulge plate, so the “relative strength” of the brace and chord decreases and the failure is tended to change to the “weaker” component (brace or chord). When τS further increases from 1.5 to 2, the failure mode turns to the local buckling of the brace (LBFB) whose ultimate capacity depends on the brace size and is independent of the

5.4.2. Effects of τS and η1 on the ultimate capacity The effects of τS and η1 are discussed together as shown in Fig. 16. It can be observed that the ultimate capacity increases when τS increases from 1 to 1.5 and then is tended to converge when τS increases from 1.5 to 2. It is due to the fact that the increase of τS enhances the 246

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Fig. 15. Effect of η2 on the ultimate capacity. (D = 402 mm, αB = 5, τS = 1, α = 12, β = 0.45, γ = 18, τ = 0.7).

Fig. 17. Effect of τ on the ultimate capacity of the bulge formed T-joint. (D = 402 mm, αB = 5, τS = 1, η1 = 3, η2 = 0. 7, α = 12, γ = 18).

= 18, τS = 1, η1 = 3 and η2 = 0.7, it can be found that τ has a positive effect on the ultimate capacity, i.e. the ultimate capacity increases with the increase of τ . It is explained as that for these failure modes, the ultimate capacity is related to the brace and brace/bulge plate intersection, which is determined by τ when other parameters remain unchanged. So the strength of the brace and the brace/bulge plate intersection increases with the increase of τ , which thus increases the strength of the joint. For other cases whose failure mode is CFF which usually occurs for the large α (such as most cases in α = 30 and 36), the ultimate capacity is determined by the chord bending strength and independent of brace size, so τ has no effect on the ultimate capacity and no corresponding figures are plotted for the sake of brevity.

Fig. 16. Effects of τS and η1 on the ultimate capacity. (D = 402 mm, αB = 5, η2 = 0.7, α = 12, β = 0.45, γ = 18, τ = 0.7).

bulge plate, so the ultimate capacity at this stage shows a convergence phenomenon. The effects of η1 are discussed for different values of τS . The ultimate capacity decreases with the increase of η1 for the case τS = 1 (failure mode is LBFBP) and 1.25 (failure mode is LBFBP + LBFB). For the case τS = 1.5, the ultimate capacity has no change when η1 increases from 2 to 3 (the failure mode is LBFB) but decreases when η1 increases from 3 (failure mode is LBFB) to 4 (failure mode is LBFBP). For the cases τS = 1.75 and 2, the parameter η1 has no effect on the ultimate capacity. It is due to the fact the failure mode of these two cases is LBFB whose ultimate capacity is dependent on the brace size rather than the bulge plate size. As a summary, if the failure mode depends totally on the bulge plate (for the failure mode LBFBP) or partially on the bulge plate (for the failure mode LBFBP + LBFB), the ultimate capacity decreases with the increase of η1, however, if the failure mode depends on the brace (for the failure mode LBFB), the increase of η1 has no effect on the ultimate capacity.

5.5.2. Effect of α on the ultimate capacity The parmeter α refers to twice the ratio of the chord length to diameter, which is also called the chord slenderness. The effect of α on the ultimate capacity is discussed combined with β and γ with the other parameters kept unchanged: τ = 0.7, τS = 1, η1 = 3, η2 = 0.7. As illustrated in Fig. 18, the increase of α leads to the decrease of the ultimate capacity especially for large β values. The reduction of the ultimate capacity is caused by the effect of the chord bending with the increase of the chord length [4]. 5.5.3. Effect of β on the ultimate capacity The parameter β refers to the ratio of brace diameter to chord diameter. As illustrated in Fig. 18, the effect of β on the ultimate strength was discussed combined with α . Generally speaking, the ultimate capacity increases with the increase of the β for small α values (such as α = 12) and is tended to converge for large α values (α = 30, 36). For small α values, the failure modes are mainly LBFBP, LBFBP + LBFB and LBFB. Obviously the ultimate capacity of theses failure modes is related to the strength of the brace and the brace/bulge plate intersection, so the increase of β enhances the ultimate capacity. For large α values, the failure mode is tended to be the chord face failure (CFF), whose ultimate capacity depends on the chord size and is independent of the brace size, so the increase of β has no effect on the ultimate capacity. For the medium values of α (such as α = 18, 24), the effect of β on the ultimate strength is discussed combined with the failure modes, for the cases whose failure modes are LBFBP, LBFBP + LBFB and LBFB, the ultimate capacity increases with the increase of β , however, for the case whose failure mode is CFF, the increase of β has no effect on the ultimate capacity.

5.5. Effects of α , β , γ and τ on the ultimate capacity of the bulge formed Tjoint The parameters α , β , γ and τ are the other four geometrical parameters of the bulge formed T-joint which share the same definitions as those of the unstiffened joint, which respectively refer to twice the ratio of the chord length to diameter, the brace/chord ratio, half of the ratio of chord diameter to thickness and the brace/chord thickness ratio. So when the chord diameter remains unchanged, the increase of α , β or γ respectively increases the chord length, increases the brace diameter or decreases the chord thickness. The increase of τ increases the brace thickness when the chord thickness remains unchanged.

5.5.4. Effect of γ on the ultimate capacity The parameter γ refers to the ratio of the radius of the chord to the thickness. Comparing the ultimate capacity values at the same locations in the three subplots of Fig. 18, it can be found that the ultimate decreases with the increase of γ . It is due to the factor that increase of γ when other parameters remain unchanged leads to the decrease of the

5.5.1. Effect of τ on the ultimate capacity From Table 6, it can be observed that τ only has a positive effect on the ultimate capacity for small α values (such as α = 12 and 18) whose failure modes are LBFB, LBFBP, and LBFBP + LBFB. Fig. 17 shows the capacity values under different values of τ and β for the case α = 12, γ 247

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Fig. 18. Effects of α , β and γ on the ultimate capacity of the bulge formed T-joint. (D = 402 mm, αB = 5, τS = 1, η1 = 3, η2 = 0. 7, τ = 0.7).

thickness of the chord wall, which thus results in the decrease of the strength of the joint.

small α, but for large α the effect of β on the ultimate capacity tends to converge.

5.6. Effects of α, β, γ and τ on the on the ultimate capacity of the unstiffened T-joint

5.6.4. Effect of γ on the ultimate capacity The increase of γ leads to the decrease of the ultimate capacity. The effects of α, β and γ on the ultimate capacity of the unstiffened T-joint are shown in Fig. 20.

The effects of α, β, γ and τ on the ultimate capacity of the unstiffened T-joint are similar to those on the ultimate capacity of the bulge formed T-joint.

6. Conclusions 1. In this study, an innovative reinforced joint (the bulge formed Tjoint) was proposed for the first time. In the joint, an additional bulge plate is employed to connected the brace and the chord, which results in three new parameters, respectively τS , η1 and η2 , while the other four parameters (α , β , γ and τ ) share the same definitions with those in unstiffened joints. 2. A real scale bulge formed T-joint and an unstiffened joint (the counterpart) were tested to investigate the ultimate capacity under the brace axial compressive loading. The test results show a good agreement with the FE results, which indicates that the FE method is feasible to simulate the failure modes and the load-ovalisation curves accurately. Also the test results prove the great advantage of the bulge formed T-joint on the ultimate capacity and the initial stiffness. Exactly, the ultimate capacity and the initial stiffness of the tested bulge formed T-joint are respectively 1.75 and 3.11 times those of the tested unstiffened T-joint. 3. Verified by test, 190 FE analyzes were conducted for the bulge formed T-joints under the brace axial compressive loading to investigate the failure modes, the ultimate capacity and the effects of the parameters, and 146 FE analyzes were conducted for the unstiffened joints as the counterpart. In these analyzes, the ultimate capacity of the bulge formed T-joint is found to be increased up to 200% compared with the unstiffened joint. Totally four failure modes are found for the bulge formed Tjoints: (1) Local buckling failure of the bulge plate (LBFBP); (2) Local buckling of the brace (LBFB); (3) Combination of the local buckling failure of the bulge plate and the local buckling of the brace (LBFBP + LBFB); (4) Chord face failure (plastic failure of the

5.6.1. Effect of τ on the ultimate capacity For the unstiffened T-joint, with the increase of τ the ultimate capacity increases for small α, however, for large α with the increase of τ the ultimate capacity still increases but tends to converge. Fig. 19 shows the effect of τ on the ultimate capacity of the unstiffened T-joint for the case =12 (a small α) and α=18. For large α, the ultimate capacity tends to converge so no figure was presented for brevity. 5.6.2. Effect of α on the ultimate capacity The increase of α leads to the decrease of the ultimate capacity. 5.6.3. Effect of β on the ultimate capacity The increase of β results in the increase of the ultimate capacity for

Fig. 19. Effect of τ on the ultimate capacity of the unstiffened T-joint (D=402 mm, αB =5, α=12, γ =18). 248

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Fig. 20. Effects of α, β, and γ on the ultimate capacity of the unstiffened T-joint (D =402mm, αB =5, τ =0.7).

L chord length LBFBP Local buckling failure of the bulge plate LBFB Local buckling of the brace LBFBP+LBFB Combination of the local buckling failure of the bulge plate and the local buckling of the brace CFF Chord face failure (plastic failure of the chord surface at the bulge plate/chord intersection) LS length of bulge plate LS1 length of the un-bulge part of the bulge plate RS radius of the bulge plate in the XY plane in Fig. 1c. T chord thickness t brace thickness TS thickness of the bulge plate α chord slenderness ratio (2L/D) αB brace slenderness ratio (2l/d) β brace/chord diameter ratio (d/D) γ chord wall slenderness ratio (D/(2T)) η1 ratio of the radius of the bulge plate in the XY plane to the chord radius (2RS/D) η2 ratio of the height of the bulge plate to the chord radius (2H/ D) θ brace/chord inclination angle μ poisson ratio τ ratio of brace thickness to chord thickness (t/T) τS ratio of bulge plate thickness to chord thickness ratio (TS/T) φ polar angle measured from crown

chord surface at the bulge plate/chord intersection) (CFF). 4. The effects of the three typical parameters τS , η1 and η2 on the ultimate capacity of the bulge formed T-joint are as follows: (1) The effect of η2 on the ultimate capacity is negligible. (2) The ultimate capacity increases when τS increases from 1 to 1.5, and then is tended to converge when τS increases from 1.5 to 2. (3) The ultimate capacity decreases with the increase of the η1 for the case τS = 1 and 1.25. For the case τS = 1.5, the ultimate capacity has no change when η1 increases from 2 to 3, but decreases when η1 increases from 3 to 4. For the cases τS = 1.75 and 2, the parameter η1 has no effect on the ultimate capacity. 5. The effects of τ ,  α, β    and    γ on the ultimate capacity of the bulge formed T-joint are as follows: (1) The ultimate capacity increases with the increase of τ for small α values (such as α = 12 and 18). For other cases whose failure mode is CFF, the parameter τ has no effect on the ultimate capacity. (2) The ultimate capacity generally decreases with the increase of α , especially for large β values. (3) The parameter β only has a positive effect on the ultimate capacity for small α values. For large α values whose failure mode is CFF, β has no effect on the ultimate capacity. (4) The ultimate capacity of the bulge formed joint decreases with the increase of γ . 6. The effects of τ , α, β and γ on the ultimate capacity of the unstiffened T-joint are similar to those on the ultimate capacity of the bulge formed T-joint. Nomeclature D d E F Fu Fu,b Fu,u H

chord diameter brace diameter elastic modulus axial force in the brace ultimate capacity ultimate capacity of the bulge formed T-joint ultimate capacity of the unstiffened joint which shares the same α, β, γ and τ with the bulge formed T-joint. height of the bulge plate

Acknowledgements This work was supported by the National Key Technology Research and Development Program of the Ministry of Science and Technology of China [grant number 2015BAF06B05] and the Science and Technology Commission of Shanghai Municipality, China [grant number 17DZ1204602]. The authors gratefully thank the engineers who helped us in the experiment, and we also gratefully appreciate the helpful comments of reviewers. 249

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