Experimental and numerical investigation of SHS truss T-joints reinforced with sidewall plates

Thin-Walled Structures 145 (2019) 106404

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Experimental and numerical investigation of SHS truss T-joints reinforced with sidewall plates

T

N.V. Gomesb, L.R.O. de Limaa,∗, P.C.G. da S. Vellascoa, A.T. da Silvaa, M.C. Rodriguesa, L.F. Costa-Nevesc a

PGECIV - Civil Engineering Post-Graduate Program, UERJ – State University of Rio de Janeiro, Brazil Structural Engineering Department, UERJ – State University of Rio de Janeiro, Brazil c INESCC, Civil Engineering Department - University of Coimbra, Portugal b

ABSTRACT

This paper presents a study on the behaviour of cold-formed SHS truss T-joints reinforced with a pair of sidewall plates. This type of reinforcement may be used in tubular joints where the chord and the brace have the same width and is very efficient particularly for failure modes involving the chord sidewall. An experimental programme and a numerical study are presented and discussed, while the research covered reinforced joints with a pair of sidewall plates and their unreinforced counterparts, the braces being subjected to axial compressive forces. The geometry of the reinforcement plates varied in thickness and in length. The results from the experimental tests were thoroughly discussed, and in a second step were used to calibrate a numerical study that was the basis to a parametric analysis involving the most relevant parameters affecting the joint behaviour. The joint resistances obtained were compared with those forecasted by Eurocode 3 - part 1.8, ABNT NBR 16239, and analytical solutions available in the literature. These comparisons indicated that the available methods for the design of these joints lead to scattered predictions: for the unreinforced joints, the results were very conservative, mainly for increasing chord sidewall slenderness values. On the other hand, the analytical and codes proposals for reinforced joints with sidewall plates with similar thickness of the chord overestimate the observed values in the majority of cases. Finally, this paper proposes two alternative design equations for the studied joints that are more accurate in predicting their resistance when compared to the presented experimental and numerical results.

1. Introduction The structural advantages and aesthetical features of tubular structural elements boosted their use around the world in recent years [1–3]. Many examples in nature depict the excellent properties of the tubular shape when loaded in compression, tension, torsion or bending in any direction. Furthermore, the section's closed shape, without sharp corners, also reduces the surface area to be painted and protected, extending their corrosion protection life [4]. The last decades showed an extensive work being carried out addressing the technological and design issues of these structures [5,6]. The vast majority of international design codes and specific recommendations explicitly cover the design of tubular joints [6–8]. Wardenier et al. [9] and Zhao et al. [11] presented the latest design recommendations for tubular joint incorporating the IIW improvements [8]. More recently, two codes focusing on tubular joints design were also updated with the recent advances in this field [12,13]. Despite the advantages of tubular sections, the joints in tubular structures may still represent a critical issue with many studies indicating that further research is needed, particularly for some geometries. Examples include the study authored by Sadeghi et al. [10] ∗

investigating an innovative I-beam to hybrid fabricated column connection. In many design situations, changing the cross-section thickness without changing the cross-section dimensions may optimise the stiffness and resistance properties of the tubular elements. In other cases, the tubular joint resistance can be enhanced using a doubler plate as reinforcement, avoiding the replacement of the chord or braces with stronger sections, as depicted in Fig. 1. If at first glance this structural solution can be considered expensive due to the additional requirements of welds, when long spans are considered, the use of reinforced joints may lead to more economical designs [14]. Bearing this scenario in mind, the Eurocode 3 [7] and NBR16239 [13] cover the design of these type of reinforced joints, an alternative requiring the reinforcement plate thickness to be at least twice the thickness of the brace member. Traditionally, design rules for hollow sections joints are based on plastic analysis or deformation limit criteria. The use of plastic analysis for determining the joint ultimate limit state implies the establishment of a plastic mechanism with an assumed yield line pattern. Packer et al. [15], Cao et al. [16], Packer [17] and Kosteski et al. [18] presented typical examples of these approaches. Other studies focusing on the structural behaviour of reinforced

Corresponding author. Structural Engineering Department, Faculty of Engineering, FEN, State University of Rio de Janeiro, UERJ, Brazil. E-mail addresses: [email protected], [email protected] (L.R.O. de Lima).

https://doi.org/10.1016/j.tws.2019.106404 Received 11 June 2019; Received in revised form 26 August 2019; Accepted 14 September 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Tubular joints reinforced with flange and sidewall plates.

hollow section joints were performed by Choo et al. [19,20], Van der Vegte et al. [21], Fung et al. [22] and Nassiraei et al. [23–26], and focused mainly on the assessment of the CHS joints structural behaviour. For reinforced joints of SHS or RHS sections under bending, some results relating to their ultimate capacity may be found in Chen & Chen [27], while Chang et al. [28] investigated the behaviour of doublerplate reinforced square hollow section (DPR-SHS) T-joints with a brace under compression. Aguilera & Fam [29] evaluated the efficiency of FRP plates adhesively bonded to the sidewalls of rectangular HSS chords in the vicinity of T-joints with brace members under compression. Young-Bo et al. [30] carried out experiments and numerical simulations of reinforced and unreinforced square tubular T-joints subjected to quasi-static cyclic loads. Feng et al. [31] presented experimental and numerical investigations on collar plates and doubler plates as reinforcement for SHS T-joints under axial compression. On the other hand, when reinforced joints with sidewall plates are considered, there is still a lack of information in the literature. With this scenario in mind, for a better understanding of the effect of a reinforcement plate on the ultimate joint capacity, an experimental study has been carried out comprising the static behaviour of T-joints reinforced by a sidewall plate with the same thickness as the chord. Fig. 2 depicts this geometry that was shown to provide a more economical design when the brace is axially loaded in compression. The main variables of the study were the reinforcing plate length and thickness. Subsequently, a numerical model based on the finite element method was developed to extend the database of the studied sidewall plate reinforcement in SHS T-joints. Furthermore, the reinforced joints’

response was compared to their unreinforced counterparts to evaluate the behavioural differences, namely the resistance enhancement. The experimental and numerical results were finally compared to Eurocode 3 [7], NBR 16239 [13] and other design recommendations. 2. Tubular joints with sidewall failure mode Packer [32] experimentally investigated the behaviour of rectangular hollow sections with the sidewall subjected to transverse compressive loads on isolated, equal-width, cross joints. The results of 31 tests covering a wall slenderness range of 15.3 ≤ h0/t0 ≤ 42.2 were discussed together with further 40 tests from other authors. Based on this discussion, this author proposed an equation to predict the ultimate resistance of full-width tee or cross joint that will be presented in Table 1. The application of this equation resulted in mean value for the ratio of the predicted and experimental values of 1.36 with a small scatter (COV = 0.16) and only 3% of the values out of the range 1.0–2.0. However, neither the chord depth (h0) nor the axial chord preload was explicitly included in the equation, as they were assumed to have little effect on the joints ultimate strength. Afterwards, Davies and Packer [33] rejected this conclusion and instead postulated that the joint strength depends on the chord slenderness (h0/t0) and on the nondimensional bearing length (h1/h0). Zhang et al. [34] performed 12 experimental tests on RHS joints (in which 6 were T-joints) followed by a numerical evaluation to investigate joints with β = 1.0. Based on the obtained results, the authors proposed an equivalent frame tube model to estimate the ultimate

Fig. 2. Reinforced tubular joints with sidewall plates – geometrical properties [7,13]. 2

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Table 1 Design equations for sidewall failure mode of T tubular joints with β = 1.0 Wardenier et al. [4]

N1, W = 2f y0 (or fb ) t0 Packer [32]

N1u,P =

f y b00.3 t01.7 sin

1

h1 1 + 5t 0 sin 1 sin

. 3.8 + 10.75

(9)

1

2

b1 + h1 2b 0

(10)

Eurocode 3, part 1.8 [7] N1, Rd, EC 3 =

kn f y t 0

2h1 + 10t0 with kn = 1 for tension chord stress and kn M 5 sen 1 sen 1

= 1.3 chord with

0.4n/ for compression on the

= 3,46

h0 t0

2

1 sin 1 E f y0

also used in ABNT NBR 16239 [9]

(11)

ABNT NBR16239 [13] N1, Rd, NBR =

kn f y t0 2,2h1 + 11t 0 with a1 sen 1 sen 1

1

=

Zhang et al. [34]

N1u = 2f y . t0. h1e . k1 with k1 = 1.75 25 or k1 = 1.40 k2 =

(

h 0.7 0 h1

)

0,7

if

h1 h0

0.016

h0 t0

if

0,03 h0 t0

(12)

1/2,24

(1 + 04,48) h0 t0

if

h0 t0

> 25 and h1e = h1. k2 with

( )

0.7 or k2 = 0.7

h0 h1

0,2

if

h1 h0

> 0.7

(13)

Geometrical parameters, according to Fig. 2.

resistance of this joint type where the web crippling governs the joint design. The author's formula was applied to 85 tests associated to a mean value of 1.22, and a small data scatter corresponding to a COV = 0.11. Cheng and Becque [35] developed a design methodology for equalwidth RHS X-joints failing by sidewall buckling with a particular focus on the compressive chord preload effects. The proposed design equation was verified against finite elements results calibrated with experiments, covering a wide range of slenderness (h0/t0) up to 50, while the results, according to the authors, led to excellent predictions. The deformation limit criterion proposed by Lu et al. [36] may be used for joints subjected to bending and axial forces. The assessment of the joint resistance is based on the comparison of the deformation at the chord-brace intersection for two load levels: the ultimate load limit, Nu, that corresponds to a chord out-of-plane displacement of 3% b0, and the serviceability limit, Ns, that corresponds to out-of-plane displacement of 1% b0. For joints presenting a peak load (Npeak) with an associated deformation less than 3% b0, this is assumed as the joint resistance. It is relevant to mention that Zhao et al. [11] performed an investigation of the deformation limit criterion and concluded that a better agreement with experiments was achieved considering the 3% b0 as deformation limit solely and discarding the out-of-plane displacement of 1% b0. This change was adopted by IIW [8–11].

t0 ≥ 20–25), the yield stress fy0 should be changed to the fb based on Eurocode 3 [37] buckling curve a. equation (9) is shown in Table 1. Furthermore, as previously mentioned, Packer [32] developed equation (10) for sidewall failure mode of SHS/RHS unreinforced T-joints considering the h0/t0 ratio, see also Table 1. According to Eurocode 3, part 1.8 [7], the sidewall failure mode for unreinforced joints is verified by equation (11) in Table 1, where χ is also evaluated using the Eurocode 3 [37] buckling curve a. The Brazilian code ABNT NBR 16239 [13] is based on Eurocode 3, part 1.8 [7] and considers a similar equation to deal with joints with sidewall failure mode - Table 1. On the other hand, the buckling reduction factor is evaluated using a different equation, see Table 1. Moreover, equation (13) proposed by Zhang et al. [34] considers the influence of the two main parameters, h0/t0 and h0/h1 as described in Table 1 for unreinforced joints. For reinforced joints, the Eurocode 3 part 1.8 [7] and the ABNT NBR 16239 [13] consider the same procedure where the joint resistance is evaluated using equations (11) and (12) replacing the chord thickness t0 by (t0+tp) since tp ≥ 2t1 as presented in equation (6). For the cases where the same section is used for the chord and the brace members, this condition conducts to a reinforcement plate thickness of twice the reference value. As mentioned before, to achieve a more economical solution, this work considers the thickness of the reinforcement plate equal to the chord thickness (tp = t0). The equations proposed by Packer [32] and Zhang et al. [34] were

3. Tubular joints design equations for sidewall failure mode The design codes that cover the resistance of reinforced SHS/RHS Tjoints use currently the equations for unreinforced joints modifying the considered thickness. Initially, the cases of unreinforced SHS/RHS Tjoints with a width ratio β of 1.0 will be considered in this discussion. Wardenier et al. [4] proposed a design equation based on web crippling in the form of either a bearing or buckling failure mode. For bearing, a dispersion force at 22° through the tube wall is considered, being very similar to the case of bearing loads on wide flange sections webs, such as the well-known value of 45° dispersion angle. For the cases where the buckling of the chord sidewall governs the joint failure (h0/

Table 2 Experimental programme – geometrical properties.

3

Test

Chord and brace [mm]

L0 [mm]

L1 [mm]

lp [mm]

tp [mm]

SR1 SR2 CR1 CR2 CR3 CR4

110 × 110 × 6.35 110 × 110 × 6.35 110 × 110 × 6.35 110 × 110 × 6.35 110 × 110 × 6.35 110 × 110 × 6.35

800 800 800 800 800 800

300 300 300 300 300 300

– – 165 165 250 165

– – 6.6 6.6 6.6 9.6

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Fig. 3. Test layout overview.

proposed for joints without the sidewall plate reinforcements. As an initial assessment of the reinforced joints capacity, these equations will be used in this work following the procedure proposed by Eurocode 3 part 1–8 [7] and ABNT NBR 16239 [13], where the chord thickness, t0, is simply replaced in the main equation by the sum of t0 with the used reinforcement plate thickness (t0 + tp). 4. Experimental programme The experimental programme for brace to chord SHS T-joints is summarised in Table 2. All six prototypes adopted ASTM-A36 steel grade: four reinforced tests using a chord wall thickness of 6.35 mm (with three different lp); 9.6 mm sidewall plates; and two unreinforced joints. This sidewall plate thickness does not comply with the tp ≥ 2t1 limit presented in Fig. 2, but leads instead, as discussed, to a more economical design. The chords and the braces were square hollow SHS 110 × 110 × 6.35 sections with 800 mm and 300 mm length, respectively, as depicted in Fig. 3. The reinforcement plates were cut from identical sections of the chords. The fillet welding details of the sidewall plates to the chord can be observed in Figs. 2 and 3(b) where an E70 electrode was used relating to a fw = 485 MPa. The non-dimensional parameters corresponding to the adopted sections were β = 1.0 and 2γ = 17.32. The specimens were instrumented with strain gauges, rosettes and LVDTs to monitor their behaviour according to the layout presented in Fig. 4 [38]. The tests were performed with the specimens fully supported on

Fig. 5. Stub column tests: load - axial displacement.

their entire length, and the brace member applied a pure concentrated force to the chord by to avoid beam mechanisms - see Fig. 4. Moreover, the chord ends were fixed to avoid vertical displacements, as recommended by Lima et al. [14] in their T-joints tests. For all tests, only the braces were axially loaded in compression using a 3000 kN universal Lousenhausen test machine under displacement control with a 0.003 mm/s load rate. The properties of the carbon steel tubes were obtained from tensile coupon tests of the flat part of the cross-section detailed in Gomes [38],

Fig. 4. Tests boundary conditions and LVDT's location. 4

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obtained from the peak load (Npeak) reached before attaining a deformation limit of 3%b0 (N3%b0). For these unreinforced joints, the peak loads were 852.9 kN and 860.4 kN, respectively, where the sidewall failure mode was observed as pointed out in Fig. 8. The two first reinforced joints tests, CR1 and CR2, with a sidewall plate of 6.35 mm (tp = t0) cut from the chord section, increased the joint resistance in 37%. At this point, it is worth observing that this reinforced joint does not meet the design codes requirements in terms of the reinforcement plate thickness (tp > 2t1 but t1 = t0). The third reinforced joint CR3 was selected to investigate the influence of the reinforcement plate length (lp). This length increases from 165 mm (used in tests CR1 and CR2 corresponding to the minimum value according to design codes) to 250 mm. Fig. 6 and Table 3 show that this joint with increased plate length exhibited a resistance of 1191.5 kN, only 0.4% greater than in CR1 and CR2 tests. The failure mode of the test CR3 was characterised by a combination of sidewall failure and brace failure similar to test CR1 presented in Fig. 8. Finally, the reinforced joint CR4 adopting a sidewall reinforcement plate of 9.6 mm (tp = 1.5t0) exhibited the failure mode presented in Fig. 8, involving the brace yielding and followed by local buckling. It should be remarked that this failure sequence is similar to the tests where only the braces in compression were evaluated (Fig. 5). As far as the joint resistance is concerned, the peak load was 1261.7 kN with a sharp decrease of the applied load as observed in the respective loadlateral displacement curve presented in Fig. 6. This value is smaller than 1361.5 kN corresponding to the average of the two stub column tests of the braces subjected to compression load. This phenomenon occurred due to different boundary conditions between the stub column tests and the T-joints tests. To investigate the von Mises stress distribution at the top and sidewalls of the chord, Fig. 7 presents the load - von Mises stress assessed from rosettes distributed according to the schematic figure showed in the graph for the tests SR2, CR1 and CR4. The rosettes R1 and R2 are located at the sidewall at 45 mm from the top face, and R3 and R4 at the top face at 30 mm from the brace face. For the test without reinforcement, SR2, up to the maximum load, similar values of von Mises stress were observed at the top and sidewall as depicted in Fig. 7(a). After the peak load, it may be noted that the von Mises stress at the top face remains constant, whereas at the sidewall a sharp decrease is visualised, which is triggered by local buckling. Fig. 7(b) shows that for the test CR1, the top face is more stressed after the knee of the curve, indicating the influence of the reinforcement over the joint's behaviour. Also the detail of the section cut after failure clearly shows an inwards bending deformation of the vertical brace faces at a higher load stage, resulting from the increase of the bending stiffness and resistance of the chord side wall, that allows this component to bear a stronger load. Finally, Fig. 7(c) indicates that the use of a reinforcement plate with a thickness of 1.5 times the chord thickness moves the failure from the chord to the brace, clearly failing in buckling, with a different pattern than in the previous test. This conclusion is evidenced by the same behaviour observed in the rosettes located at the top and at the sidewall for test CR4. The deformed shapes for the three tests are presented in

Fig. 6. Load-lateral displacement experimental curves.

reporting average results of 321.1 MPa for the proof stress, 439.16 MPa for the ultimate stress, 205044 MPa for Young's modulus (E) and 20.7% for the elongation at fracture based on a 50 mm gauge length. To evaluate the actual response of the brace member compared to the T-joint resistance, two 400 mm length stub column tests were performed instrumented with four strain gauges, one at each face. 5. Experimental results To better analyse the experimental results, the discussion will focus on the load-displacement curves, load - von Mises stress curves, failure modes, and the comparison of the ultimate resistances predicted from design codes and those experimentally observed. At the first stage, two tests were performed with only the brace member subjected to compression load, aiming to evaluate their ultimate loads and to compare these results with the studied T-joint capacities. For these two stub columns tests, the boundary conditions corresponded to both ends fully fixed, being the vertical displacement applied at the top of the specimen. The load-displacement curves of these two tests are presented in Fig. 5. For both tests, the maximum loads were similar, of 1391.8 kN and 1339.4 kN for tests C2 and C3, respectively. These maximum loads are much higher than the values predicted from the codes, that is 876.5 kN calculated using a cross-sectional area of 2720 mm2 and proof stress of 321.7 MPa. This difference can be explained by the fact that the material properties of the corners regions for cold formed sections are increased due to fabrication process [39]. If this modified material were considered in the plastic resistance evaluation, higher values would also be obtained. The shape of the curves characterises the failure mode related to a cross-section yielding (point A) followed by local buckling (point B). Moreover, the load-lateral displacements, with those obtained from LVDT's placed 55 mm from the chord top face for the investigated Tjoints are presented in Fig. 6, while the results are summarised in Table 3. For the unreinforced joints (tests SR1 and SR2), the resistances were Table 3 Comparison of experimental and theoretical results. Test

Failure mode

peak Nexp [kN]

3% b0 Nexp [kN]

NEC3 [kN]

NNBR [kN]

NZhang [kN]

NPacker [kN]

Nexp NEC3

Nexp NNBR

Nexp NZhang

Nexp NPacker

SR1 SR2 CR1 CR2 CR3 CR4

SW SW SW + BYa) SW + BYa) SW + BYa) BY

852.9 860.4 1181.6 1186.3 1191.5 1261.7

850.9 857.1 1176.1 1176.4 1191.5 –

461.8 461.8 1136.7 1136.7 1136.7 –

572.9 572.9 1410.1 1410.1 1410.1 –

542.7 542.7 1085.3 1085.3 1085.3 –

473.3 473.3 1537.8 1537.8 1537.8 –

1.85 1.86 1.04 1.04 1.05 –

1.49 1.50 0.84 0.84 0.84 –

1.57 1.59 1.09 1.09 1.10 –

1.80 1.82 0.77 0.77 0.77 –

a

Small contribution of the brace yielding followed by local buckling. 5

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Fig. 7. Load-von Mises stress (experimental curves).

Fig. 8 where, once again, it is possible to observe that different failure modes were achieved in the tests, i.e., sidewall for test SR1, sidewall combined with brace yielding for test CR1 and finally, brace yielding for test CR4. Table 3 shows a summary of the experimental results and the predicted values obtained from the equations presented in Table 1. As previously mentioned, it is crucial to observe that these equations were developed for T-joints with β = 1 without reinforcement. In the present study, these equations were used changing the chord thickness (t0) by the sum of the chord thickness and the reinforcement plate thickness (t0+tp). The equations from Eurocode 3 [4] and NBR ABNT 16239 [9] consider reinforced joints changing t0 by t0+tp since tp ≥ 2t1. It is essential to highlight that, in principle, the design codes equations cannot be used for the studied reinforced joints since tp ≤ 2t1. It is also worth

mentioning that, as observed in Table 3, for the reinforced joints considered in this work, the equations from Eurocode 3 [4] and Zhang et al. [30] presented a good agreement since the experimental to design formulation ratios were close to 1.0. The equations from ABNT NBR 16239 [9] and Packer [28] are not suitable for these reinforced joints. Moreover, these equations have led to conservative results when unreinforced T-joints, with β = 1, are considered. 6. Numerical model 6.1. Finite element mesh, material and analysis options The numerical model was developed to expand the experimental 6

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Fig. 8. Deformed shapes – experimental tests.

Fig. 9. Reinforced T-joint numerical model.

7

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Fig. 10. Comparison of experimental and numerical curves.

result's dataset. Both unreinforced and reinforced joints numerical models were developed using eight-node solid elements (SOLID185) with three degrees of freedom per node: translations x, y and z available in the ANSYS Element Library [40], therefore considering plasticity, stress stiffening, large deflection and large strain capabilities. A uniformly reduced integration was adopted to prevent mesh locking with one integration point. This option has shown to be more efficient than the B method (selective reduced integration) [40]. The finite element mesh was refined near the welds, where the stress concentration is more likely to occur, and as regular as possible, with well-proportioned elements to avoid numerical problems. The adopted mesh is depicted in Fig. 9(a), where the adopted boundary conditions can also be observed. The weld size was modelled according to the mean measured fillet weld

size (7 mm). Convergence studies were also performed to define the numerical model optimum mesh size [38]. The nodes at the interface between the reinforcement plate and the chord sidewall were considered coupled following the experimental observations. A full nonlinear analysis based on a Newton-Raphson solver was performed considering the material and geometrical nonlinearities (Updated Lagrangian formulation). A multilinear material model (MISO) present in ANSYS software [40] was used considering the stress versus strain curve obtained from tensile coupon tests, i.e., 321.1 MPa for the yield stress, 460.9 MPa for the ultimate stress, 205044 GPa for the Young's modulus (E) and 20.7% for the ultimate strain while the von Mises yield criteria were adopted in the developed finite element model – Fig. 9(b). The tension tests involved solely the flat part of the 8

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Fig. 11. Comparison of experimental and numerical deformed shapes.

SHS, and a resulting mean curve considering the above presented values was adopted to build the finite element model. The nodes associated with lines AB and CD had x and y-direction translations restrained to reproduce the boundary conditions of the previously presented experiments. The nodes depicted in points A, B, C and D had z-direction translations restrained to avoid rigid body displacements. The load was applied in the numerical model at the top brace centre point using a displacement control of a node coupled with all brace top edge nodes.

Regarding the deformed shapes, a good agreement between the tests and the numerical model can also be observed in Fig. 11 for the tests SR1 without reinforcement and for test CR2 with a 165 mm length sidewall reinforcement plate. The presented deformed shapes were obtained by cutting the tested prototypes in the symmetry plane passing through the brace longitudinal axis. In all cases, an expected failure mode associated with a chord sidewall failure was observed, but for reinforced T-joint, large deformations were also noticed at the brace member near to the chord top face.

6.2. Validation of the numerical model

7. Parametric analysis

A comparison of the numerical and experimental-versus displacement curves is depicted in Fig. 10 to validate the developed finite element model. These curves indicate that the numerical model can accurately reproduce the experiments for both unreinforced and reinforced T-joints.

In this section, a parametric analysis is presented based on the developed numerical models previously calibrated with the experiments, aiming to extend the information on unreinforced and reinforced Tjoints structural response with β = 1.0. With these results in hand, the obtained resistances will be compared with the formulations of 9

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Table 4 Unreinforced joints with t0 = t1 = 10.6 mm (β = 1.0). Group

M [mm]

N [mm]

h0 t0

h1 h0

NNUM [kN]

NEC3 [kN]

NNBR [kN]

NZ [kN]

NP [kN]

NNUM NEC3

NNUM NNBR

NNUM NZhang

NNUM NPacker

1A (b0 = h0 = b1 = M and h1 = N)

70 80 90 100 120 130 140 150 175 180 190 200 220 250 260 100 100 100 100 100 120 120 120 120 140 140 150 150 150 200 200 200 100 100 100 100 100 120 120 120 120 140 140 150 150 150 200 200 200

70 80 90 100 120 130 140 150 175 180 190 200 220 250 260 140 150 160 180 200 140 160 180 200 220 260 200 250 300 240 250 320 140 150 160 180 200 140 160 180 200 220 260 200 250 300 240 250 320

6.60 7.55 8.49 9.43 11.32 12.26 13.21 14.15 16.51 16.98 17.92 18.87 20.75 23.58 24.53 9.43 9.43 9.43 9.43 9.43 11.32 11.32 11.32 11.32 13.21 13.21 14.15 14.15 14.15 18.87 18.87 18.87 13.21 14.15 15.09 16.98 18.87 13.21 15.09 16.98 18.87 20.75 24.53 18.87 23.58 28.30 22.64 23.58 30.19

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.40 1.50 1.60 1.80 2.00 1.17 1.33 1.50 1.67 1.47 1.86 1.33 1.67 2.00 1.20 1.25 1.60 0.71 0.67 0.63 0.56 0.50 0.86 0.75 0.67 0.60 0.64 0.54 0.75 0.60 0.50 0.83 0.80 0.63

1072.4 1203.6 1329.3 1448.4 1675.1 1782.6 1886.8 1989.7 2246.9 2296.4 2390.4 2489.6 2672.6 2947.9 3037.1 1697.4 1753.1 1810.5 1919.4 2032.4 1812.7 1940.3 2054.2 2169.0 2395.4 2616.0 2330.8 2616.9 2889.9 2752.4 2819.9 3227.6 1535.8 1546.3 1555.3 1569.2 1580.4 1724.3 1755.6 1777.1 1793.3 2006.1 2036.1 2077.9 2127.7 2162.9 2558.7 2574.6 2661.5

837.0 886.1 932.4 975.8 1052.9 1086.3 1116.0 1141.9 1188.9 1195.3 1204.9 1210.4 1210.3 1186.0 1173.0 1230.9 1294.6 1358.4 1486.0 1613.5 1174.6 1296.3 1418.0 1539.7 1578.6 1809.8 1423.1 1704.3 1985.6 1401.8 1449.6 1784.5 884.7 860.6 835.9 784.9 732.0 1000.3 945.2 887.5 827.7 855.6 723.3 971.2 794.6 635.9 1033.5 990.3 724.0

837.0 904.7 971.9 1038.6 1168.5 1230.9 1290.8 1347.6 1470.6 1491.3 1527.9 1557.8 1596.5 1602.3 1591.9 1310.1 1377.9 1445.8 1581.6 1717.3 1303.6 1438.7 1573.7 1708.8 1825.9 2093.4 1679.5 2011.4 2343.3 1804.1 1865.7 2296.7 917.8 911.1 903.3 884.4 862.1 1062.0 1045.1 1023.3 997.4 1095.1 1018.6 1192.4 1097.8 1034.5 1407.8 1381.9 1277.1

688.6 772.6 853.0 929.9 1072.8 1138.9 1201.4 1260.3 1391.8 1415.4 1460.0 1500.9 1572.1 1651.9 1671.4 1217.1 878.6 895.8 928.0 957.8 1213.6 1350.4 1013.7 1046.2 1151.1 1210.2 1586.4 1229.0 1298.1 1736.6 1794.3 1445.9 1029.3 1029.3 1029.3 1029.3 1029.3 1087.1 1087.1 1087.1 1087.1 1138.6 1138.6 1162.4 1162.4 1162.4 1267.2 1267.2 1267.2

924.8 962.6 997.2 1029.3 1087.1 1113.5 1138.6 1162.4 1217.4 1227.7 1247.8 1267.2 1303.9 1354.9 1370.9 1363.9 1457.0 1554.0 1759.3 1979.8 1226.6 1377.2 1538.9 1711.8 1687.9 2014.1 1472.5 1830.3 2235.9 1463.8 1515.8 1913.2 1023.3 1015.6 1005.9 979.2 942.1 1157.0 1137.4 1107.2 1065.2 1128.7 981.6 1250.0 1073.5 866.6 1387.6 1337.9 982.6

1.28 1.36 1.43 1.48 1.59 1.64 1.69 1.74 1.89 1.92 1.98 2.06 2.21 2.49 2.59 1.38 1.35 1.33 1.29 1.26 1.54 1.50 1.45 1.41 1.52 1.45 1.64 1.54 1.46 1.96 1.95 1.81 1.74 1.80 1.86 2.00 2.16 1.72 1.86 2.00 2.17 2.34 2.82 2.14 2.68 3.40 2.48 2.60 3.68 1.89 0.28

1.28 1.33 1.37 1.39 1.43 1.45 1.46 1.48 1.53 1.54 1.56 1.60 1.67 1.84 1.91 1.30 1.27 1.25 1.21 1.18 1.39 1.35 1.31 1.27 1.31 1.25 1.39 1.30 1.23 1.53 1.51 1.41 1.50 1.52 1.55 1.60 1.68 1.49 1.54 1.60 1.68 1.78 2.07 1.66 1.98 2.50 1.84 1.92 2.71 1.55 0.20

1.56 1.56 1.56 1.56 1.56 1.57 1.57 1.58 1.61 1.62 1.64 1.66 1.70 1.78 1.82 1.39 2.00 2.02 2.07 2.12 1.49 1.44 2.03 2.07 2.08 2.16 1.47 2.13 2.23 1.58 1.57 2.23 1.67 1.70 1.72 1.77 1.83 1.62 1.68 1.74 1.80 1.83 2.00 1.74 1.94 2.09 1.82 1.86 2.08 1.78 0.13

1.16 1.25 1.33 1.41 1.54 1.60 1.66 1.71 1.85 1.87 1.92 1.96 2.05 2.18 2.22 1.24 1.20 1.17 1.09 1.03 1.48 1.41 1.33 1.27 1.42 1.30 1.58 1.43 1.29 1.88 1.86 1.69 1.49 1.50 1.51 1.52 1.54 1.59 1.61 1.63 1.65 1.76 1.79 1.79 1.83 1.86 2.02 2.03 2.10 1.60 0.18

1B (b0 = h0 = b1 = M and h1 = N)

1C (b0 = b1 = h1 = M and h0 = N)

Mean COV

where NEC3 is the joint resistance based on Eurocode 3, part 1.8 [4] – eq. (11); NNBR is based on ABNT NBR 16239 [9] – eq. (12); NZhang is based on Zhang et al. [30] – eq (13) and NPacker is based on Packer [28] – eq (10) and lp, according to eq. (5).

Eurocode 3 - part 1.8 [4], of ABNT NBR 16239 [9] and equations proposed by other authors. The main parameters investigated involved the sidewall slenderness (h0/t0) and bearing width (h1/h0). All sections were classified as class 1 or 2 according to the design codes.

was considered equal to the chord thickness t0, although this structural strategy is not allowed by the design codes, as already discussed. As mentioned, this was used since it can deliver a more economical design. To enlarge the joint dataset, the chord thickness t0 was equal to 10.6 mm for all numerical models used in this section, avoiding premature brace failures. Tables 4 and 5 present a comparison of the numerical results and the corresponding values from equations presented in Table 1 where the chord thickness t0 was replaced by the sum of the chord thickness plus the sidewall reinforcement plate (t0 + tp). The Group 1A presented in Fig. 12(a) considers the same crosssection for the chord and brace members covering values of h0/t0 from 6.60 up to 24.53 keeping constant the bearing width h1/h0 = 1.0. In the second Group 1B, as observed in Fig. 12(b), the chord widths (b0) and heights (h0) were considered equal to width (b1) and the height (h1) of the brace presented values leading h1/h0 to vary from 1.17 up to 2.00,

7.1. Influence of parameters sidewall slenderness (h0/t0) and bearing width (h1/h0) To investigate the influence of sidewall slenderness and bearing width in the T-joints behaviour with β = 1.0, the first dataset was divided into three groups as presented in Table 4 and Table 5 for unreinforced and reinforced joints, respectively. The number of reinforced joints in Table 5 is smaller than the number of unreinforced joints in Table 4 because some joints presented brace failures when the sidewall reinforcement was used. The sidewall reinforcement plate thickness tp 10

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Table 5 Reinforced joints with t0 = t1 = 10.6 mm (β = 1.0). Group

M [mm]

N [mm]

h0 t0

h1 h0

NNUM [kN]

NEC3 [kN]

NNBR [kN]

NZ [kN]

NP [kN]

NNUM NEC3

NNUM NNBR

NNUM NZhang

NNUM NPacker

1A (b0 = h0 = b1 = M and h1 = N)

130 140 150 175 180 190 200 220 250 260 100 100 100 100 100 120 120 120 120 140 140 150 150 150 200 140 150 150 200 200 200

130 140 150 175 180 190 200 220 250 260 140 150 160 180 200 140 160 180 200 220 260 200 250 300 240 220 200 250 240 250 320

12.26 13.21 14.15 16.51 16.98 17.92 18.87 20.75 23.58 24.53 9.43 9.43 9.43 9.43 9.43 11.32 11.32 11.32 11.32 13.21 13.21 14.15 14.15 14.15 18.87 20.75 18.87 23.58 22.64 23.58 30.19

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.40 1.50 1.60 1.80 2.00 1.17 1.33 1.50 1.67 1.47 1.86 1.33 1.67 2.00 1.20 0.64 0.75 0.60 0.83 0.80 0.63

2302.8 2447.0 2586.6 2914.7 2980.5 3108.1 3243.0 3492.2 3849.9 3965.3 2134.1 2211.3 2285.5 2426.8 2561.8 2334.1 2506.4 2667.6 2821.3 3175.2 3460.3 3086.9 3494.6 3838.5 3652.2 2511.3 2631.5 2662.7 3276 3284.2 3333

2801.7 2844.9 2880.0 2930.6 2934.3 2935.3 2928.0 2890.4 2787.0 2743.3 3137.8 3265.3 3392.9 3648.0 3903.1 2994.3 3237.7 3481.1 3724.6 3770.0 4232.6 3442.4 4004.9 4567.4 3310.7 2181.1 2449.6 2004.1 2499.9 2395.6 1751.3

3174.8 3290.6 3398.8 3625.0 3660.9 3722.2 3768.4 3813.0 3765.2 3722.9 3339.7 3475.4 3611.2 3882.7 4154.2 3323.1 3593.3 3863.5 4133.6 4360.7 4895.7 4062.6 4726.4 5390.2 4261.0 2877.3 3152.6 2707.5 3356.5 3236.4 2377.0

2277.7 2402.7 2520.5 2783.6 2830.9 2920.0 3001.9 3144.2 3303.9 3342.7 2434.2 1757.2 1791.6 1856.0 1915.6 2427.2 2700.8 2027.3 2092.4 2302.2 2420.5 3172.8 2458.1 2596.2 3473.3 2190.2 2384.7 2195.5 2815.7 2763.7 2554.1

3617.9 3699.2 3776.6 3955.4 3988.9 4054.2 4117.0 4236.4 4402.1 4454.2 4431.2 4733.8 5048.8 5715.9 6432.4 3985.1 4474.4 5000.0 5561.8 5484.1 6543.9 4784.2 5946.8 7264.5 4755.8 3699.24 3776.61 3776.61 4117.02 4117.02 4117.02

0.82 0.86 0.90 0.99 1.02 1.06 1.11 1.21 1.38 1.45 0.68 0.68 0.67 0.67 0.66 0.78 0.77 0.77 0.76 0.84 0.82 0.90 0.87 0.84 1.10 1.15 1.07 1.33 1.31 1.37 1.90 0.99 0.29

0.73 0.74 0.76 0.80 0.81 0.84 0.86 0.92 1.02 1.07 0.64 0.64 0.63 0.63 0.62 0.70 0.70 0.69 0.68 0.73 0.71 0.76 0.74 0.71 0.86 0.87 0.83 0.98 0.98 1.01 1.40 0.81 0.20

1.01 1.02 1.03 1.05 1.05 1.06 1.08 1.11 1.17 1.19 0.88 1.26 1.28 1.31 1.34 0.96 0.93 1.32 1.35 1.38 1.43 0.97 1.42 1.48 1.05 1.15 1.10 1.21 1.16 1.19 1.30 1.17 0.14

0.64 0.66 0.68 0.74 0.75 0.77 0.79 0.82 0.87 0.89 0.48 0.47 0.45 0.42 0.40 0.59 0.56 0.53 0.51 0.58 0.53 0.65 0.59 0.53 0.77 0.68 0.70 0.71 0.80 0.80 0.81 0.65 0.21

1B (b0 = h0 = b1 = M and h1 = N)

1C (b0 = b1 = h1 = M and h0 = N)

Mean COV

i.e., h1/h0 > 1.0. Finally, in Group 1C - Fig. 12(c), the chord widths (b0) and brace widths (b1) and heights (h1) were the same and the chord height (h0) was varied leading h1/h0 to vary from 0.50 up to 0.86, i.e., h1/h0 < 1.0. The results summarised in Fig. 13 show, for Group 1A, keeping the bearing width constant (h1/h0 = 1.0), for unreinforced (UNR) and reinforced joints (R), that the joint's resistances presented a proportionality with the ratio h0/t0 showing that this parameter influences the Tjoint behaviour. The increasing of the joint is not caused by the increase of the sidewall slenderness but due to other cross-section dimensions as depicted in Table 4. Moreover, according to Fig. 14, it is worth to mention that the application of the equations from Table 1, for unreinforced joints, led to conservative results. For the cases where

reinforced joints are considered, as presented in Table 5, the equation developed by Zhang et al. [34] was accurate, but the dispersion of the results increased as the ratio h0/t0 also increased. For the Group 1B joints, the theoretical resistances tendency to increase was reduced when compared to those obtained from the numerical models. This fact leads to more conservative results as the ratio h0/t0 increases. This fact is minimised when reinforced joints are considered, whose results are presented in Table 5. Moreover, by increasing the ratio h1/h0, the theoretical resistances tendency to increase was boosted, and that does not occur for the resistance obtained from the numerical models. However, the bearing width h1/h0 has an opposite effect when compared to the sidewall slenderness h0/t0 as observed in Fig. 15(a). This trend makes the results less conservative with the

Fig. 12. Geometrical parameters – parametric analysis. 11

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Fig. 13. Sidewall slenderness (h0/t0) influence.

chord sections were used with 120, 150 and 200 mm, to assess the influence of the sidewall slenderness (h0/t0). For each chord section, three RHS were used to evaluate the bearing width (h1/h0). Moreover, for each joint SHS/RHS, three different thickness were considered as 3.0, 7.0 and 10.6 mm. Table 6 presents all geometrical properties of the investigated joints in this second part of the parametric analysis. This table also presents the joint's resistances obtained from the numerical analysis, design codes and author's equations changing t0 by the sum of t0 and tp. Some joints presented brace failure when a sidewall reinforcement plate thickness of 10.6 mm was used, thus being disregarded from the data assessment. The results are summarised in Fig. 16, showing that increasing the sidewall reinforcement plate thickness leads to a small increase of the joint's resistance, independently of the bearing width (h1/h0). In general, the use of the sidewall thickness of 3.0, 7.0 and 10.6 mm led to a joint resistance increase of 19%, 27% and 31%, respectively. In all the studied geometries, adopting a relatively thin reinforcement, of the same thickness as the chord, led to an average gain in the resistance of about 30%, as illustrated in Fig. 17. 8. Proposed alternative formulation Based on the experimental and numerical results presented in the previous sections of this paper, where the main variables that may influence the T-joints behaviour with β = 1.0 were investigated, two new equations are proposed to predict the unreinforced and reinforced joints resistances. These equations are suitable for joints with geometries limited to:

Fig. 14. Comparison of ratios - numerical and theoretical results.

increase of the bearing width, and to significant unsafe results as the ratio h1/h0 increase. For the joints of the Group 1B, the unreinforced joints resistances evaluated from the equations of ABNT NBR 16239 [13] have presented better results when compared to other design codes or authors’ equations. Finally, for the joints of Group 1C, the ratio h0/t0 increase as ratio h1/h0 decrease. According to Fig. 15(b), it can be also be observed that when increasing the chord height h0, a negligible increase of the joint resistance was verified.

• Cross-section class 1 or 2; • b /t , h /t , b /t and h /t ≤ 35; • 0.5 ≤ h /b and h /b ≤ 2.0; • t = t since t = t . 0

p

7.2. Influence of sidewall reinforcement plate thickness (tp)

0

0

0

0

0

0

1

1

1

1

1

1

1

0

These new equations are presented for unreinforced – eq. (14) alternatively, reinforced – eq. (15) T-joints with β = 1.0. The comparison of the obtained results using these new equations and the others presented in Table 1 is summarised in Fig. 18. From this figure, it is

The second part of the parametric analysis investigated the influence of the sidewall reinforcement plate thickness. Three different SHS 12

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Fig. 15. Load-bearing width influence. Table 6 Assessment of the sidewall reinforcement thickness influence with t0 = t1 = 10.6 mm (β = 1.0). Group

M [mm]

N [mm]

h0 t0

h1 h0

tp [mm]

NNUM [kN]

NEC3 [kN]

NNBR [kN]

NZ [kN]

NP [kN]

NNUM NEC3

NNUM NNBR

NNUM NZhang

NNUM NPacker

2A (b0 = h0 = b1 = M and h1 = N)

120 120 120 120 120 120 120 120 120 120 120 120 150 150 150 150 150 150 150 150 150 150 150 150 200 200 200 200 200 200 200 200 200 200 200 200

120 120 120 120 160 160 160 160 200 200 200 200 150 150 150 150 200 200 200 200 250 250 250 250 200 200 200 200 250 250 250 250 320 320 320 320

11.32 11.32 11.32 11.32 11.32 11.32 11.32 11.32 11.32 11.32 11.32 11.32 14.15 14.15 14.15 14.15 14.15 14.15 14.15 14.15 14.15 14.15 14.15 14.15 18.87 18.87 18.87 18.87 18.87 18.87 18.87 18.87 18.87 18.87 18.87 18.87

1.0 1.0 1.0 1.0 1.3 1.3 1.3 1.3 1.7 1.7 1.7 1.7 1.0 1.0 1.0 1.0 1.3 1.3 1.3 1.3 1.7 1.7 1.7 1.7 1.0 1.0 1.0 1.0 1.2 1.2 1.2 1.2 1.5 1.5 1.5 1.5

– 3.0 7.0 10.6 – 3.0 7.0 10.6 – 3.0 7.0 10.6 – 3.0 7.0 10.6 – 3.0 7.0 10.6 – 3.0 7.0 10.6 – 3.0 7.0 10.6 – 3.0 7.0 10.6 – 3.0 7.0 10.6

1675.1 1937.0 2058.5 – 1940.3 2266.3 2428.4 2506.4 2169.0 2534.1 2731.6 2821.3 1989.7 2328.0 2468.0 2586.6 2330.8 2776.7 2974.2 3086.9 2616.9 3125.9 3378.0 3494.6 2489.6 2954.9 3116.0 3243.0 2819.9 3411.6 3615.5 – 3227.6 3952.0 4249.8 –

1052.9 1468.0 2101.8 – 1296.3 1780.3 2506.0 3237.7 1539.7 2092.6 2910.2 3724.6 1141.9 1573.3 2222.8 2880.0 1423.1 1934.1 2689.8 3442.4 1704.3 2295.0 2295.0 4004.9 1210.4 1645.1 2287.8 2928.0 1449.6 1952.0 2685.0 – 1784.5 2381.7 3241.0 –

1168.5 1629.2 2332.7 – 1438.7 1975.8 2781.2 3593.3 1708.8 2322.5 3229.8 4133.6 1347.6 1856.7 2623.2 3398.8 1679.5 2282.5 3174.3 4062.6 2011.4 2708.4 3725.4 4726.4 1557.8 2117.2 2944.4 3768.4 1865.7 2512.3 3455.6 – 2296.7 3065.3 4171.3 –

1072.8 1376.4 1781.2 – 1350.4 1732.6 2242.2 2700.8 1614.3 2071.2 2680.4 3228.7 1260.3 1616.9 2092.5 2520.5 1586.4 2035.4 2634.0 3172.8 1896.4 2433.2 3148.8 3792.9 1500.9 1925.7 2492.1 3001.9 1794.3 2302.1 2979.2 – 2186.0 2804.7 3629.7 –

1087.1 1660.6 2574.1 – 1377.2 2103.7 3260.9 4474.4 1711.8 2614.9 4053.3 5561.8 1162.4 1775.6 2752.3 3776.6 1472.5 2249.3 3486.7 4784.2 1830.3 2796.0 4334.0 5946.8 1267.2 1935.7 3000.4 4117.0 1515.8 2315.5 3589.3 – 1913.2 2922.4 4530.0 –

1.59 1.32 0.98 – 1.50 1.27 0.97 0.77 1.41 1.21 0.94 0.76 1.74 1.48 1.11 0.90 1.64 1.44 1.11 0.90 1.54 1.36 1.47 0.87 2.06 1.80 1.36 1.11 1.95 1.75 1.35 – 1.81 1.66 1.31 – 1.35 0.26

1.43 1.19 0.88

1.56 1.41 1.16 – 1.44 1.31 1.08 0.93 1.34 1.22 1.02 0.87 1.58 1.44 1.18 1.03 1.47 1.36 1.13 0.97 1.38 1.28 1.07 0.92 1.66 1.53 1.25 1.08 1.57 1.48 1.21 – 1.48 1.41 1.17 – 1.27 0.17

1.54 1.17 0.80 – 1.41 1.08 0.74 0.56 1.27 0.97 0.67 0.51 1.71 1.31 0.90 0.68 1.58 1.23 0.85 0.65 1.43 1.12 0.78 0.59 1.96 1.53 1.04 0.79 1.86 1.47 1.01 – 1.69 1.35 0.94 – 1.13 0.35

2B (b0 = h0 = b1 = M and h1 = N)

2C (b0 = b1 = h1 = M and h0 = N)

Mean COV

13

1.35 1.15 0.87 0.70 1.27 1.09 0.85 0.68 1.48 1.25 0.94 0.76 1.39 1.22 0.94 0.76 1.30 1.15 0.91 0.74 1.60 1.40 1.06 0.86 1.51 1.36 1.05 – 1.41 1.29 1.02 – 1.12 0.23

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Fig. 16. Load versus bearing width influence - h1/h0 < 1.0.

with the numerical results is associated with a mean of 1.21 and a COV of 0.09. Values of 1.89 (0.28); 1.55 (0.20); 1.78 (0.13) and 1.60 (0.18) were observed for the equations of Eurocode 3, part 1.8 [4], ABNT NBR 16239 [9], Packer [28] and Zhang et al. [30]. On the other hand, for reinforced joints, a mean of 1.14 with a COV of 0.09 was obtained comparing to 0.99 (0.29); 0.81 (0.20); 1.17 (0.14) and 0.65 (0.21), respectively.

Unreinforced joints NUNR = 0,8 f y t 01,9 2,5

h0 t0

Reinforced joints NR = 0,78 f y (t 02 + tp0,9) 1,5

0,7

+ 10

h0 t0

h1 h0

0,85

+ 13

0,8 1,1

(14)

h1 h0

0,6 1,3

(15) 9. Conclusions This paper presented an experimental and numerical study on the behaviour of cold-formed SHS truss T-joints reinforced with a pair of sidewall plates, with the braces subjected to axial compressive forces. Also, reference tests and numerical models without reinforcement were analysed to assess the influence of the reinforcement introduction and its size. Based on the experimental results, the main conclusions were i) when increasing the reinforcement plate length from 165 to 250 mm, a small increase in the joint resistance was achieved from 1189.3 kN to 1191.5 kN corresponding to a 0.4%; ii) when a 9.6 mm thickness reinforcement plate was used instead of 6.6 mm, once again, a small increase in the joint resistance was observed of around 6% emphasizing that the failure mode was changed from sidewall failure to brace failure. All the results were compared to those delivered by the

Fig. 17. Increase in the T-joints resistances (β = 1.0) by using reinforcement with tp = t0.

possible to conclude that the results of the other equations are more scattered than the proposed alternative formulation. Table 7 summarises the comparison of the application of all equations used in this work. For unreinforced joints, the proposed equation, when compared 14

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Fig. 18. Ratio NNUM/NCODES for T-joints with β = 1.0.

Acknowledgements

Table 7 Summary of results – NNUM/NCODES – Unreinforced and reinforced T-joints with β = 1.0.

Unreinforced Reinforced

Mean COV Mean COV

NNUM NEC3

NNUM NNBR

NNUM NZhang

NNUM NPacker

NNUM NNEW

1.89 0.28 0.99 0.29

1.55 0.20 0.81 0.20

1.78 0.13 1.17 0.14

1.60 0.18 0.65 0.21

1.21 0.09 1.14 0.09

The authors would like to thank CAPES (Finance code 001), CNPq (305143/2015-8; 306042/2013-4; 305026/2017-8) and FAPERJ (E26/203.186/2015; E-26/201.393/2014; E-26/202.789/2017; E-26/ 203.192/2016) for the financial support to this research program. This work has also been partially supported by the Portuguese Foundation for Science and Technology under project grant UID/MULTI/00308/ 2013. Appendix A. Supplementary data

available formulations, based on the Eurocode 3 [7], NBR 16239 [13], ISO 14346 [12] and the design recommendations proposed by Packer [28] and Zhang et al. [30]. These comparisons indicated that the available methods for the joint's design lead to scattered predictions, roughly too conservative for unreinforced joints, but overestimating the resistance for reinforced joints with sidewall plates, namely with a similar thickness of the chord. Afterwards, an extensive parametric analysis based on finite element model calibrated against experimental tests was performed aiming to extrapolate the experimental results and to identify the main parameters that influence the T-joints behaviour with β = 1. With these results in hand, two alternative design equations were proposed for the studied joints that are more accurate in predicting their resistance when compared to the previously presented experimental and numerical results. Also, it was shown that the new formulation might be applied, with reasonable accuracy, to the situation of sidewall reinforcements with the same thickness of the chord, by merely adopting a cut from the same section, simplifying fabrication and leading to a more economical design. This situation is not covered by the above mentioned codes, that require the reinforcement plate to have at least twice the chord face thickness. It was also observed that increasing the sidewall reinforcement plate thickness leads to a small increase of the joint's resistance, independently of other relevant joint parameters.

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Conflicts of interest The authors declare that there is no conflict of interest in the submission of this article: 15

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