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Numerical study of hybrid tubular K-joints with circular braces and square chord in stainless steel
Thin-Walled Structures 145 (2019) 106390
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Numerical study of hybrid tubular K-joints with circular braces and square chord in stainless steel
T
Ran Fenga,b,∗, Junwu Lina a b
School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen, 518055, China State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou, 510640, China
ARTICLE INFO
ABSTRACT
Keywords: Circular brace Finite-element analysis (FEA) Hybrid tubular K-joint Parametric study Square chord Stainless steel
This paper introduces a finite-element analysis (FEA) on hybrid tubular K-joints with circular braces and square chord in stainless steel. The finite-element models (FEMs) were established by using the shell element S4R for circular braces, square chord and welds. The FEMs were validated by comparing the failure modes, failure strengths and joint deformation curves obtained from the experiments and FEA. A parametric study was carried out on 162 FEMs to evaluate the influences of brace diameter/chord width ratio (β = d1/b0), brace/chord thickness ratio (τ = t1/t0), chord width/thickness ratio (2γ = b0/t0), overlap ratio (Ov=(q/p) × 100%) of overlapped tubular K-joints and eccentricity (e) of gapped tubular K-joints. The numerical results show that the joint strengths of overlapped tubular K-joints increased with the increment of the β, τ and Ov values, as well as the decrement of the 2γ value, while the joint strengths of gapped tubular K-joints increased with the increment of the β and τ values, as well as the decrement of the 2γ and e values. The comparison of the failure strengths with design strengths demonstrates that the design formulae of CIDECT and Australian/New Zealand Standard (AS/NZS) are generally unconservative, whereas the design strengths determined by Eurocode (EC3) and Chinese Code are relatively close to the failure strengths for overlapped tubular K-joints. The design formulae of AS/NZS are generally unconservative, whereas the design formulae of CIDECT, EC3 and Chinese Code are generally conservative for gapped tubular K-joints. The design equations are proposed by introducing the correction factors that considered the influences of geometrical parameters of β, τ, 2γ and Ov, which were validated to be accurate for hybrid tubular K-joints in stainless steel.
1. Introduction A tubular joint is one of the efficient joint forms commonly used in steel tubular structures. It has the advantages of less steel consumption, good mechanical behaviour, clear path of force transmission and large bearing capacity. Hybrid tubular K-joints with circular braces and square chord meet the requirements of structural form and mechanical properties and are easy to design and construct, which are widely used in practical engineering. On the other hand, stainless steel structures have the advantages of good durability, easy processing, high-temperature resistance, excellent mechanical properties and beautiful appearance. They have been paid more and more attention in architectural and structural designs. Therefore, hybrid tubular K-joints with circular braces and square chord in stainless steel have good engineering application prospects. Many types of research were conducted by finite-element analysis (FEA) on carbon steel tubular joints. The crack modelling of circular
∗
tubular joints was investigated by Cao et al. [1] using the FEA. Various kinds of crack tip models in connection with the crack elements and the neighbourhoods were discussed. A process to transform the crack elements of a plane curve into those of a doubly curved semi-elliptical surface around a joint intersection was also introduced. Lee [2] reviewed the modelling techniques applied in the FEA of tubular joints for the strengths, stress fields and stress intensity factors. The guidance on material curve input, model discretization, choice of elements, weld modelling, results interpretation and FEA limitations were also detailed. An eight-node thick shell element was applied by Fung et al. [3,4] to model the completely overlapped tubular joints with lap brace in axial compression. The finite-element model (FEM) was verified and employed in the parametric study to investigate the strengths and behaviour of tubular joints. Shao et al. [5] studied the influences of geometrical parameters on the stress distributions of tubular T-joints under axial loads. A FEM for welded tubular T-joints was developed, which was used to analyze 140 tubular T-joints with different
Corresponding author. School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen, 518055, China. E-mail address: [email protected] (R. Feng).
https://doi.org/10.1016/j.tws.2019.106390 Received 12 April 2019; Received in revised form 4 September 2019; Accepted 5 September 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
Table 1 Comparison of failure strengths between tests and FEA. Specimen
K–C150 × 3-B108 × 3-O1.8 K–C150 × 3-B108 × 3-O50.9 K–C150 × 3-B108 × 3-O50.9-R K–C150 × 3-B108 × 3-G72.26 K–C150 × 3-B133 × 3-O20.3 K–C150 × 3-B133 × 3-O60.1 K–C150 × 3-B133 × 3-G36.91 K–C150 × 3-B133 × 3-G36.91-R K–C200 × 4-B108 × 3-O34.5 K–C200 × 4-B108 × 3-G47.26 K–C200 × 4-B108 × 3-G147.26 K–C200 × 4-B133 × 3-O46.8 K–C200 × 4-B133 × 3-G11.91 K–C200 × 4-B133 × 3-G111.91 Mean COV
Ov
1.8% 50.9% 50.9% – 20.3% 60.1% – – 34.5% – – 46.8% – –
g (mm)
– – – 72.26 – – 36.91 37.91 – 47.26 147.26 – 11.91 111.91
e (mm)
0.00 −37.50 −37.50 37.50 0.00 −37.50 37.50 37.50 −50.00 0.00 50.00 −50.00 0.00 50.00
β
0.72 0.72 0.72 0.72 0.88 0.88 0.88 0.88 0.54 0.54 0.54 0.66 0.66 0.66
2γ
τ
1.00 1.00 1.00 0.96 1.04 1.03 1.02 1.02 0.74 0.72 0.71 0.76 0.76 0.77
50.48 50.96 50.62 51.04 51.03 50.56 50.64 50.80 50.19 50.70 50.52 50.34 50.55 50.70
Test results
FEA results
Comparison
Failure mode
NTest (kN)
Failure mode
NFEA (kN)
NTest/NFEA
A A A A A A A A A A A A A A
200.38 253.96 243.23 102.56 252.39 295.77 194.31 201.91 260.19 159.72 161.36 289.88 202.37 186.60
A A A A A A A A A A A A+C A A
206.79 240.53 240.53 100.43 238.34 278.11 178.43 178.43 246.74 160.35 149.92 277.94 192.26 193.03
0.97 1.06 1.01 1.02 1.06 1.06 1.09 1.13 1.05 1.00 1.08 1.04 1.05 0.97 1.04 0.044
+B
+B +B+C +B
Note: A = Chord face plastification; B=Weld fracture; C = Local buckling failure of brace.
geometrical parameters. Shao [6] also used a developed FEM to analyze the stress distributions along the weld toe on the chord surface for tubular T- and K-joints under axial loads. A total of 140 tubular T-joints and 420 tubular K-joints were included in the parametric study to investigate the geometrical effects. Wang et al. [7] studied the seismic behaviour of the thick-walled circular hollow section (CHS) tubular Xjoints subjected to out-of-plane bending. The experimental behaviour was simulated by the FEA to further reveal the significant experimental observations. Lesani et al. [8] carried out the FEA using finite-element program ABAQUS to present the failure patterns, ultimate static strengths and detailed behaviour of un-stiffened tubular T- and Y-joints under axial compression. The FEMs were also developed by Ozyurt et al. [9] using finite-element program ABAQUS to investigate the resistances of welded tubular joints with CHS or SHS brace members under axial loads at various elevated temperatures. The numerical studies on stainless steel tubular joints are relatively few. A total of 172 FEMs were developed by Feng and Young [10] in the parametric study on the square and rectangular hollow section (SHS and RHS) tubular T-joints, X-joints and X-joints with chord preload in stainless steel. The design equations were proposed and verified to be accurate. The numerical investigation was also conducted by Feng and Young [11] on SHS and RHS tubular T-joints, X-joints and X-joints with chord preload in stainless steel at elevated temperatures. The FEA results were used to assess the existing design guidelines and the accurate design rules were developed. Both experimental and numerical analyses were carried out by Feng et al. [12] on empty and grouted tubular Xjoints with chord preload in stainless steel. The design formulae were derived to determine the joint strengths. Mussa and Mutalib [13] applied finite-element software LUSAS to study the influences of brace/ chord width ratio (β) and brace/chord thickness ratio (τ) on the strengths and behaviour of SHS and RHS tubular X-joints in stainless steel. The previous literatures exhibit that the researches on carbon steel tubular joints are relatively mature. However, the numerical analyses on stainless steel tubular joints, especially stainless steel hybrid tubular joints are relatively rare, although stainless steel is nowadays increasingly used in tubular structures owing to its excellent mechanical and technical properties. It is worth noting that the constitutive relation of stainless steel is notably different from that of carbon steel, which exhibits the obvious non-linear material behaviour without distinct yield plateau. Hence, the current design rules for carbon steel tubular joints may not be suitable for the design of stainless steel tubular joints. In
order to promote the applications of stainless steel in practical engineering of tubular structures, more studies need to be performed by FEA on stainless steel tubular joints to derive the corresponding design guidelines, in particular for stainless steel hybrid tubular joints. Therefore, the FEMs of hybrid tubular K-joints with circular braces and square chord in stainless steel were established in this study using finite-element software ABAQUS 6.14, which were calibrated against the experimental results. A total of 162 hybrid tubular K-joints in stainless steel were simulated in the parametric study using the verified FEMs. The design equations are proposed for hybrid tubular K-joints in stainless steel, whose accuracy and reliability were evaluated by the experimental and numerical results. 2. A brief summary of experimental work A total of 14 hybrid tubular K-joints including 7 overlapped and 7 gapped tubular K-joints were tested to investigate the load-carrying capacities of hybrid tubular K-joints in stainless steel. All specimens were fabricated from austenitic stainless steel (AISI 304) with circular braces fully welded to the continuous square chord. The hidden seam of the overlapped brace for overlapped tubular K-joint was unwelded. The inclined angles (θ) between the axes of circular braces and square chord of all specimens kept in consistence as 45°. The overall length (L0) of the square chord was chosen as 6b0, while the length (L1) of the axis from the top of circular brace to the top flange of square chord was chosen as 3d1. The effects of the key geometrical parameters including the brace diameter/chord width ratio (β = d1/b0), eccentricity (e), overlap ratio (Ov=(q/p) × 100%) of overlapped tubular K-joints and gap distance (g) of gapped tubular K-joints on the failure modes and joint strengths were studied. The key geometrical parameters of all specimens are presented in Table 1, with the symbols shown in Fig. 1. The test setup of hybrid tubular K-joints in stainless steel is displayed in Fig. 2. The square chord of the specimens was subjected to axial compression force, while the circular braces were pin-connected to the reaction frame that resulted in the compression and tension forces applied to the circular braces separately. The monotonic multilevel load control was used for loading until the peak load of the specimens was reached. Then the displacement control was applied for replacement in the post-ultimate stage. The experimental results including the failure modes, failure strengths and joint deformation curves are detailed in Ref. [14].
2
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
Fig. 1. Schematic sketch of hybrid tubular K-joints in stainless steel.
developed by some researchers and introduced in some design specifications as below. Ramberg-Osgood model (R–O model) was usually used to describe the stress-strain relationships of non-linear metal materials, which is a classical theoretical model representing the constitutive relations of elasto-plastic materials in solid mechanics. This model was first developed by Osgood and Ramberg [15] in 1943, whose basic idea laid in the assumption that the total strains of materials consisted of elastic strain (ϵe) and plastic strain (ϵp) as follows:
=
e
+
p;
e
=
E0
;
p
n
=
K
,
(1)
where E0 is the initial modulus of elasticity, K is the correlation coefficient of strain hardening, and n is the strain hardening index of materials reflecting the non-linear degree of stress-strain curves of materials. By carrying out experimental and numerical investigations, Mirambell and Real [16] proposed a constitutive relation model for stainless steel materials based on the R–O model as follows:
Fig. 2. Test setup of hybrid tubular K-joints in stainless steel.
3. Development and validation of FEMs
=
3.1. Constitutive relation models of stainless steel materials
E0 0.2
+ 0.002 +
( ),
0.2
E0.2
n
<
0.2
+
pu
(
0.2 0.2
u
0.2
)
n 0.2, u
,
0.2
(2)
where σ0.2 is the static 0.2% tensile proof stress, σu is the ultimate stress, E0.2 is the tangent modulus at σ0.2, εpu is the ultimate plastic strain, and n’0.2,u is the strain hardening index for the stress of materials between
A large number of researches proved that stainless steel is a typical non-linear metal material. The stress-strain relationships of stainless steel materials were reported and the constitutive relation models were 3
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
Fig. 3. Comparison of stress-strain curves of stainless steel CHS and SHS tubes.
σ0.2 and σu. Rasmussen [17] proposed a full-range constitutive relation model in 2003 for stainless steel materials based on the R–O model as follows:
=
+ 0.002
E0
0.2
E0.2
in which,
+
u
n
0.2
=
m
0.2
u
m = 1 + 3.5 u
( ), < ( ) +
in which,
u
0.2
b= 0.2,
1 , for austenitic and duplex stainless 0.2 + 185e 0.2 1 0.0375(n 5) 0.2, for ferritic stainless steel 0.2 + 185e E0 0.2 ; e = 0.2 ; E0.2 = ; 0.2 = 0.2 E0 1 + 0.002n/ e E0 u
steels
+ 0.002;
E0
=
E0
( ),
+ 0.002
0.2 E0.2
0.2 a
n
0.2
+ 0.008 + (
< 1.0
(
1 E0
1 E0.2
) ×(
1.0
0.2 0.2
)
n0.2,1.0
+
>
(5)
2.0
n
+ 0.002
in which, n =
fy E
fy
, fy
+
Ey
ln 20 ln
fy
fy RP0.01
+
u
fy fu
m
fy
fy
, fy <
, m = 1 + 3.5 f , E y = u
< fu E 1 + 0.002n
(6) E , fy
u
=1
fy fu
A.
where E, fy and fu are the eigenvalues of stainless steel materials, RP0.01 is the stress at the residual strain of 0.01%, Ey is the tangent modulus at the yield stress, ϵu is the ultimate strain, and A is the elongation after fracture. In Chinese Code (CECS 410: 2015) [20], the linear elastic model was employed for the stress-strain relationships of materials. The constitutive relation model of stainless steel materials was developed for
0.2,
2.0
,
E
0.002 +
0.2 0.2 )
b 2.0 2.0 ) 2.0 (1 ± 2.0)
where the operator of plus sign in the equations represents the stainless steel in tension, while the operator of minus sign in the equations represents the stainless steel in compression. In Annex C of Part 1–4 of Eurocode (EC3) [19], the constitutive relation model of stainless steel materials employed in the numerical simulation was presented as follows:
(3)
0.2
± u
0.2
A full-range three-stage constitutive relation model was developed by Quach et al. [18] in 2008 for stainless steel materials, which was represented by three basic parameters of the R–O model as follows:
b
0.2 (1
u (1 ± u)
0.2
=1
=
a=
2.0
(4) 4
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
analysis to consider the material non-linearity as follows:
=
E0
+ 0.002
0.002 +
( ),
f0.2 E0
n
f0.2
f0.2
+
f0.2 E0.2
in which, m = 1 + 3.5
reference points by means of displacement and rotation in the FEMs, which can rotate in the plane around the x-axis of the local coordinate system only. On the other hand, a reference point was also assigned at the centre of the loading end of square chord to couple the degrees of freedom of all nodes at the loading end of square chord by using the (*COUPLING) procedure in the ABAQUS library. Another local coordinate system was established at the reference point with the connected face and the axis of square chord defined as the x-axis and z-axis directions, respectively. The axial compression force was applied at the reference point by means of displacement loading along the axis of square chord, which was identical to the loading applications in the experimental work.
f0.2 fu
+
u
(
f0.2 fu
, E0.2 =
f0.2
) ,f m
0.2
E0
E 1 + 0.002 0
f0.2
,
< u
< fu
=1
(7)
f0.2 fu
.
where f0.2 is the standard value of yield strength, and fu is the standard value of ultimate strength. The stress-strain curves of stainless steel CHS and SHS tubes determined by the constitutive relation models developed by Mirambell and Real [16], Rasmussen [17], Quach et al. [18], as well as adopted in Eurocode [19] and Chinese Code [20] are compared with the testing curves obtained from the tensile coupon tests reported in Ref. [14], as illustrated in Fig. 3. The comparison exhibits that the stress-strain curves of stainless steel CHS and SHS tubes determined by the constitutive relation models proposed by Rasmussen [17] and Quach et al. [18], respectively, agreed well with the testing curves. Therefore, the constitutive relation models of stainless steel materials proposed by Rasmussen [17] and Quach et al. [18] were employed in the FEA to simulate the material properties of circular braces and square chord of hybrid tubular K-joints in stainless steel, respectively.
3.3. Validation of FEMs The developed FEMs for hybrid tubular K-joints in stainless steel were validated against the corresponding experimental results summarized in Ref. [14] before its application for parametric study. The test failure strengths are compared with those predicted by the FEA, as listed in Table 1. The comparison indicates that the deviations between the experimental and FEA results are less than 10% for all specimens, having the mean value of test/FEA strength ratio (NTest/NFEA) of 1.04 and the coefficient of variation (COV) of 0.044. Hence, the FEMs were validated to be accurate. In addition, the test failure modes are also compared with those obtained from the FEA, as illustrated in Fig. 5 and presented in Table 1. The comparison indicates that the failure modes obtained from the FEA include chord face plastification and local buckling failure of brace, which agreed well with the experimental results. Hence, the FEMs were validated to be accurate. However, the weld fracture at the joint intersection area observed in the experimental work was not obtained from the FEA since the weld materials in the FEA are perfect without any defects. The testing curves of axial load versus vertical displacement and axial load versus chord deformation are also compared with those obtained from the FEA, as shown in Figs. 6 and 7, respectively. The comparison indicates that the FEA results generally have a good agreement with the experimental results, except for the minor deviations resulted from the test measurements and the weld fracture at the joint intersection area occurred in the experimental work only. Therefore, the FEMs of hybrid tubular K-joints in stainless steel were validated to be accurate by comparing the failure modes, failure strengths and joint deformation curves, which were used for the succeeding parametric study.
3.2. Establishment of FEMs The general purpose FEA program ABAQUS 6.14 [21] was implemented in this study for the numerical simulation of hybrid tubular K-joints in stainless steel. The quadrilateral shell element namely S4R was used to model the circular braces, square chord and welds. The S4R represents a 4-node quadrilateral shell element with finite film strain and linear reduced integration, in which each node has three translational and three rotational degrees of freedom. The S4R shell element is a general type of shell element with good adaptability, which can be applied to both the thick shell model and the thin shell model. For the modelling of hybrid tubular K-joints in stainless steel, the stresses along the thickness direction can be neglected since the thickness is much smaller than the geometrical dimensions along the length direction. Hence, the thin shell model with the use of S4R shell element was employed in the FEMs, whose performance was stable and characteristics were more suitable for the numerical simulation. On the other hand, both initial geometrical imperfections and residual stresses were not taken into consideration in the FEMs due to their negligible influences on the behaviour of welded tubular joints based on the previous literatures [8,10–12,22]. For the modelling of the welds, a ring shell element was added to simulate the weld seam at the joint intersection area, whose thickness is half of the wall thickness of the square chord. The overall mesh size of the FEMs is 10 × 10 mm with the length-to-width ratio of 1.0, except for the local mesh refinement at the welds and joint intersection area to take the complex stress distributions into account. The finite-element meshes of hybrid tubular K-joints in stainless steel, in particular, the weld seam are presented in Fig. 4. In the experimental work, the square chord of the specimens was subjected to axial compression, which resulted in the circular braces rotated in the plane around the fixed hinge axes. A reference point was assigned at the centre of the end face of circular braces to couple the degrees of freedom of all nodes at the end face of circular braces by using the (*COUPLING) procedure in the ABAQUS library. A local coordinate system was established at the reference points with the fixed hinge axis and the axis of circular brace defined as the x-axis and y-axis directions, respectively. The boundary conditions were applied at the
4. Parametric study 4.1. General The validated FEMs were employed for parametric study to investigate the influences of key geometrical parameters including the brace diameter/chord width ratio (β = d1/b0), brace/chord thickness ratio (τ = t1/t0), chord width/thickness ratio (2γ = b0/t0), eccentricity (e), overlap ratio (Ov) of overlapped tubular K-joints and gap distance (g) of gapped tubular K-joints on the failure modes and joint strengths of hybrid tubular K-joints in stainless steel. All these geometrical parameters were identified as the key influential parameters based on the corresponding experimental work [14], some other previous literatures [3,4] and the current design guidelines. A total of 162 FEMs including 81 overlapped and 81 gapped tubular K-joints with the dimensions of circular braces and square chord selected from the range of practical applications were investigated in the parametric study. The
5
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
Fig. 4. Finite-element meshes of hybrid tubular K-joints in stainless steel.
outer width (b0) of square chord was chosen as a constant value of 300 mm, the overall length (L0) of square chord was chosen as a constant value of 1800 mm, the overall length (L1) of circular brace was chosen as four times of outer diameter (d1), and the inclined angle (θ) between circular brace and square chord was chosen as a constant value of 45°. The validity ranges of key geometrical parameters for hybrid tubular K-joints in stainless steel designed in the experimental work and FEA, as well as defined in the current design guidelines are listed in
Table 2. The comparison exhibits that the validity ranges of key geometrical parameters designed in the FEA are far beyond those defined in the current design guidelines. 4.2. Failure modes and joint strengths The typical failure modes of hybrid tubular K-joints in stainless steel obtained from the parametric study mainly include chord face
Fig. 5. Comparison of failure modes between tests and FEA.
6
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
Fig. 6. Comparison of axial load versus vertical displacement curves between tests and FEA.
plastification, local buckling failure of brace and the combination of these two failure modes, as shown in Fig. 8. It can be concluded from the comparison that most of the specimens failed by chord face plastification, while the overlapped tubular K-joints are normally subjected to local buckling failure of brace, and the gapped tubular K-joints are normally subjected to the combination of two failure modes. It was demonstrated in Ref. [14] that the joint strengths of hybrid tubular K-joints in stainless steel were determined by the strength design criterion and deformation design criterion, depending on which resulted in the lower strength value. For the strength design criterion, the joint strengths were determined by the peak loads of the specimens. For the deformation design criterion, the joint strengths were determined by the axial loads at the chord deformation limit of 0.03b0 of the specimens. The failure strengths of hybrid overlapped and gapped tubular K-joints in stainless steel obtained from the parametric study were determined based on the strength and deformation design criteria. The comparison of failure strengths with design strengths for hybrid tubular K-joints in stainless steel is listed in Table 3.
strengths, in particular, the specimens having large τ value and small 2γ value, which also resulted in the increase of the joint stiffnesses. It is also shown from Fig. 9a, d and 9e that the failure strengths of hybrid overlapped tubular K-joints increased with the decrement of the 2γ value and the increment of the failure strengths by decreasing the 2γ value is greater for the specimens having large β and τ values. This may attribute to the increment of the wall thickness of square chord, which resulted in the increase of the failure strengths, in particular, the specimens having large β and τ values, which also resulted in the increase of the joint stiffnesses. It is shown from Fig. 9b, d and 9f that the failure strengths of hybrid overlapped tubular K-joints increased with the increment of the τ value and the increment of the failure strengths by increasing the τ value is greater for the specimens having large β value and small 2γ value. This may attribute to the increment of the wall thickness of circular brace, which resulted in the increase of the resistance of circular brace to local buckling that enhanced the failure strengths, in particular, the specimens having large β value and small 2γ value, which also resulted in the increase of the joint stiffnesses. It is also shown from Fig. 9c, e and 9f that the failure strengths of hybrid overlapped tubular K-joints increased with the increment of the Ov value. However, the increment is relatively small. The influence of the Ov on the failure strengths of hybrid overlapped tubular K-joints is smaller than other geometrical parameters.
4.3. Failure strengths of hybrid overlapped tubular K-joints The influential curved surfaces of key geometrical parameters including β, τ, 2γ and Ov on the failure strengths of hybrid overlapped tubular K-joints are plotted in Fig. 9. It is shown from Fig. 9a–c that the failure strengths of hybrid overlapped tubular K-joints increased with the increment of the β value and the increment of the failure strengths by increasing the β value is greater for the specimens having large τ value and small 2γ value. This may attribute to the increment of the diameter of circular brace, which resulted in the increase of the failure
4.4. Failure strengths of hybrid gapped tubular K-joints The influential curved surfaces of key geometrical parameters including β, τ, 2γ and e on the failure strengths of hybrid gapped tubular K-joints are also plotted in Fig. 10. It is illustrated from Fig. 10 that the 7
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
Fig. 7. Comparison of axial load versus chord deformation curves between tests and FEA. Table 2 Validity range of geometrical parameters for hybrid tubular K-joints. Parameter
Ov
e (mm)
β
τ
2γ
CIDECT [23] EC3 [24] AS/NZS [25] Chinese Code [26] Test FEA
20%≤Ov 25%≤Ov ≤ 100% 25%≤Ov ≤ 100% 25%≤Ov ≤ 100% 1.8%≤Ov ≤ 60.1% 20%≤Ov ≤ 100%
e ≤ 0.25b0 – – – −50≤e ≤ 50 −150≤e ≤ 150
0.25≤β ≤ 0.8 0.4≤β ≤ 0.8 0.4≤β ≤ 0.8 0.4≤β ≤ 0.8 0.54≤β ≤ 0.88 0.2≤β ≤ 1.0
τ ≤ 1.0 – τ ≤ 1.0 – 0.71≤τ ≤ 1.04 0.5≤τ ≤ 2.0
15≤2γ 15≤2γ ≤ 35 15≤2γ ≤ 35 15≤2γ ≤ 35 50.19≤2γ ≤ 51.04 30≤2γ ≤ 150
failure strengths of hybrid gapped tubular K-joints increased with the increment of the β and τ values, as well as the decrement of the 2γ and e values. However, the influence of the e on the failure strengths of hybrid gapped tubular K-joints is relatively small. Furthermore, the increment of the failure strengths by increasing the β and τ values, as well as decreasing the 2γ and e values is greater for the specimens with large β and τ values, as well as small 2γ value. This may attribute to the identical reasons elaborated above for hybrid overlapped tubular Kjoints, in particular, the specimens with large β and τ values, as well as small 2γ value, which also resulted in the increase of the joint stiffnesses.
and parametric study with the design strengths determined by the current design guidelines for hybrid overlapped and gapped tubular Kjoints in stainless steel is summarized in Table 3 and plotted in Figs. 11 and 12, respectively. The comparison in Table 3 and Fig. 11 exhibits that the design formulae of CIDECT [23] and AS/NZS [25] are generally unconservative, having the mean values of failure/design strength ratios (NF/NCI and NF/NAN) of 0.84 and 0.78, respectively, for hybrid overlapped tubular K-joints. Whereas, the design strengths determined by the design formulae of EC3 [24] and Chinese Code [26] are relatively close to the failure strengths, having the mean values of failure/design strength ratios (NF/NEC and NF/NCC) of 1.07 and 1.07, respectively, for hybrid overlapped tubular K-joints. The COVs of the comparison of the four design guidelines are all quite large, which indicates that the deviations of the predictions between the failure strengths and design strengths of hybrid overlapped tubular K-joints in stainless steel are quite large. This may attribute to the notably different material properties between carbon steel and stainless steel, as well as the large range of key geometrical parameters designed in the FEA, which are far beyond the validity ranges defined in the current design guidelines. The comparison in Table 3 and Fig. 12 also exhibits that the design
5. Proposed design equations 5.1. Assessment of current design guidelines It is worth noting that the joint strengths of hybrid tubular K-joints with circular braces and square chord can be determined by the design formulae of CIDECT [23], Eurocode 3 (EC3) [24], Australian/New Zealand Standard (AS/NZS) [25] and Chinese Code (GB 50017-2017) [26]. The comparison of the failure strengths obtained from the tests 8
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
Fig. 8. Typical failure modes of hybrid tubular K-joints in stainless steel obtained from parametric study.
EC3 [24] and Chinese Code [26] are generally conservative, having the mean values of failure/design strength ratios (NF/NCI, NF/NEC and NF/ NCC) of 1.46, 1.16 and 1.31, respectively, for hybrid gapped tubular Kjoints. The COVs of the comparison of the four design guidelines are also large, which are smaller than those calculated in the comparison for hybrid overlapped tubular K-joints. Hence, the design formulae of EC3 [24] are most accurate among the current design guidelines for the design of hybrid overlapped and gapped tubular K-joints in stainless steel.
Table 3 Comparison of failure strengths with design strengths for hybrid tubular Kjoints in stainless steel. Specimen (A total of 7+81 overlapped K-joints)
Comparison NF/NCI
NF/NEC
NF/NAN
NF/NCC
NF/NP
Mean COV
0.84 0.443
1.07 0.587
0.78 0.615
1.07 0.587
1.03 0.246
Specimen (A total of 7+81 gapped Kjoints) Mean COV
Comparison NF/NCI NF/NEC
NF/NAN
NF/NCC
NF/NP
1.46 0.355
0.83 0.323
1.31 0.339
1.03 0.263
1.16 0.331
5.2. Proposed design equations The design equations are proposed for hybrid tubular K-joints in stainless steel based on the design formulae of EC3 [24] for carbon steel tubular joints. For hybrid overlapped tubular K-joints in stainless steel, a correction factor (ψo = N/Ni) is proposed based on the overlap influence function given in Chinese Code (CECS 280: 2010) [27] by considering the influences of geometrical parameters including β, τ, 2γ
formulae of AS/NZS [25] are generally unconservative, having the mean value of failure/design strength ratio (NF/NAN) of 0.83 for hybrid gapped tubular K-joints. Whereas, the design formulae of CIDECT [23], 9
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R. Feng and J. Lin
Fig. 9. Influential curved surfaces of geometrical parameters on failure strengths of hybrid overlapped tubular K-joints in stainless steel.
and Ov as below: o
= k1
k2 k3 (2
For 50%≤Ov < 80%, (8)
) k 4 Ov k5
Ni =
By taking logarithm on both sides of Eq. (8),
N ln = ln k1 + k2 ln Ni
+ k3 ln + k 4 ln(2 ) + k5 ln Ov
o
0.467
0.398 (2
)
0.371O
v
Ni =
(9)
(10)
0.540
(11)
o Ni
in which. For 25%≤Ov < 50%,
Ni =
4
f yi ti beff + be, ov +
Ov (2di 50
4t i ) /
M5
4ti]/
M5
(13)
4
f yi ti [be, ov + 3di
beff =
10 f y 0 t0 d b0 / t0 f yi ti i
di
be, ov =
10 f yj t j d dj / t j f yi ti i
di
4ti]/
M5
(14)
(15)
where γM5 = 1.0 is the partial safety factor, di and ti are the outer diameter and wall thickness of overlapping brace, respectively, dj and tj are the outer diameter and wall thickness of overlapped brace, respectively, fy0 is the yield stress of chord, fyi and fyj are the yield stresses of overlapping and overlapped braces, respectively. For hybrid gapped tubular K-joints in stainless steel, a correction factor (ψg = N*/Ni*) is also proposed by considering the influences of geometrical parameters including β, τ and 2γ as below:
The joint strengths of hybrid overlapped tubular K-joints with circular braces and square chord in stainless steel can be calculated as below:
N=
f yi ti [beff + be, ov + 2di
For Ov≥80%,
where k1, k2, k3, k4 and k5 are the coefficients to be determined by the multivariate linear regression analysis based on the FEA results in the parametric study. Hence, the correction factor (ψo) was determined as below:
= 1.966
4
g
(12)
= m1
m2 m3 (2
)m4
By taking logarithm on both sides of Eq. (16), 10
(16)
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
Fig. 10. Influential curved surfaces of geometrical parameters on failure strengths of hybrid gapped tubular K-joints in stainless steel.
ln
N Ni
= ln m1 + m2 ln
+ m3 ln + m4 ln(2 )
kn = 1.3
(17)
= 1.428
0.208 0.272 (2
)
0.097
8.9
The comparison of the failure strengths obtained from the tests and parametric study with the design strengths determined by the proposed design equations for hybrid overlapped and gapped tubular K-joints in stainless steel is summarized in Table 3 and plotted in Fig. 13a and b, respectively. A good agreement between the failure strengths and the proposed design strengths was achieved, having the mean values of failure/proposed design strength ratio (NF/NP) of 1.03 and 1.03 and the
(19)
4
0.5k
sin
2 n f y0 t 0 i
d1 + d 2 / 2b0
M5
(22)
5.3. Assessment of proposed design equations
in which,
Ni =
(21)
where θi* is the inclined angle between the axes of ith brace and chord, d1* and d2* are the outer diameters of the 1st and 2nd circular braces, respectively.
(18)
* g Ni
1.0
kn = 1.0
The joint strengths of hybrid gapped tubular K-joints with circular braces and square chord in stainless steel can be calculated as below:
N* =
n
For n ≤ 0 (in tension),
where m1, m2, m3 and m4 are the coefficients to be determined by the multivariate linear regression analysis based on the FEA results in the parametric study. Hence, the correction factor (ψg) was determined as below: g
0.4
(20)
For n > 0 (in compression), 11
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R. Feng and J. Lin
Fig. 11. Comparison of failure strengths with design strengths for hybrid overlapped tubular K-joints in stainless steel.
COV of 0.246 and 0.263 for hybrid overlapped and gapped tubular Kjoints in stainless steel, respectively. Therefore, the proposed design equations were validated to be accurate for hybrid tubular K-joints with circular braces and square chord in stainless steel.
key geometrical parameters including β, τ, 2γ, Ov of overlapped tubular K-joints and e of gapped tubular K-joints on the joint strengths were evaluated. The failure modes mainly include chord face plastification, local buckling failure of brace and the combination of these two failure modes. Most of the specimens failed by chord face plastification. The numerical results in the parametric study exhibit that the joint strengths of hybrid overlapped tubular K-joints increased with the increment of the β, τ and Ov values, as well as the decrement of the 2γ value, while the joint strengths of hybrid gapped tubular K-joints increased with the increment of the β and τ values, as well as the decrement of the 2γ and e values. In addition, the joint stiffnesses increased with the increment of the β and τ values, as well as the decrement of the 2γ value. The comparison of the failure strengths with design strengths indicates that the design formulae of CIDECT and AS/NZS are generally unconservative for hybrid overlapped tubular K-joints. Whereas, the design strengths determined by the design formulae of EC3 and Chinese Code are relatively close to the failure strengths for hybrid overlapped tubular K-joints. In addition, the design formulae of AS/NZS are generally unconservative for hybrid gapped tubular K-joints. Whereas, the design formulae of CIDECT, EC3 and Chinese Code are generally conservative for hybrid gapped tubular K-joints. The design formulae of EC3 are most accurate among the current design guidelines for hybrid overlapped and gapped tubular K-joints in stainless steel. The design
6. Conclusions The FEA was performed in this study on hybrid tubular K-joints with circular braces and square chord in stainless steel. Some constitutive relation models of stainless steel materials are introduced and compared with the results of tensile coupon tests. It is shown that the constitutive relation models proposed by Rasmussen and Quach et al. agreed well with the test results, which were employed in the FEA to simulate the material properties of circular braces and square chord of hybrid tubular K-joints in stainless steel, respectively. The FEMs were established by using the shell element S4R for circular braces, square chord and welds. The axial compression force and boundary conditions were applied at the reference points by means of displacement loading. The FEMs were validated by comparing the failure modes, failure strengths and joint deformation curves obtained from the experiments and FEA. A total of 162 FEMs including 81 overlapped and 81 gapped tubular K-joints were investigated in the parametric study. The influences of 12
Thin-Walled Structures 145 (2019) 106390
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Fig. 12. Comparison of failure strengths with design strengths for hybrid gapped tubular K-joints in stainless steel.
equations are proposed by introducing the correction factors (ψo and ψg) for hybrid overlapped and gapped tubular K-joints in stainless steel, respectively, which considered the influences of geometrical parameters of β, τ, 2γ and Ov. A good agreement between the failure strengths and the proposed design strengths was achieved. The proposed design equations were validated to be accurate for hybrid tubular K-joints with circular braces and square chord in stainless steel.
Acknowledgements The authors are grateful for the financial support from National Natural Science Foundation of China (Grant No. 51528803), Natural Science Foundation of Guangdong Province of China (Grant No. 2018A030313208), State Key Laboratory of Subtropical Building Science (South China University of Technology, Grant No. 2018ZA02), and Guangdong Provincial Key Laboratory of Durability for Marine Civil Engineering, Shenzhen Durability Center for Civil Engineering
Fig. 13. Comparison of failure strengths with proposed design strengths for hybrid tubular K-joints in stainless steel. 13
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
(Shenzhen University, Grant No. GDDCE 18-5).
Structures, AS/NZS 4673: 2001, Sydney, Australia, 2001. [26] Chinese Code, Code for Design of Steel Structures, GB 50017-2017, Beijing, China, 2017 (in Chinese). [27] Chinese Code, Technical Specification for Structures with Steel Hollow Sections vol. 280, CECS, Beijing, China, 2010 2010. (in Chinese).
References [1] J.J. Cao, G.J. Yang, J.A. Packer, F.M. Burdekin, Crack modeling in FE analysis of circular tubular joints, Eng. Fract. Mech. 61 (5) (1998) 537–553. [2] M.M.K. Lee, Strength, stress and fracture analyses of offshore tubular joints using finite elements, J. Constr. Steel Res. 51 (3) (1999) 265–286. [3] T.C. Fung, C.K. Soh, W.M. Gho, F. Qin, Ultimate capacity of completely overlapped tubular joints: I. An experimental investigation, J. Constr. Steel Res. 57 (8) (2001) 855–880. [4] T.C. Fung, C.K. Soh, W.M. Gho, Ultimate capacity of completely overlapped tubular joints. II: behavioural study, J. Constr. Steel Res. 57 (8) (2001) 881–906. [5] Y.B. Shao, S.T. Lie, S.P. Chiew, Evaluation of geometrical effect on stress distribution for tubular T-joints under axial load, Int. J. Space Struct. 21 (2) (2006) 77–92. [6] Y.B. Shao, Geometrical effect on the stress distribution along weld toe for tubular Tand K-joints under axial loading, J. Constr. Steel Res. 63 (10) (2007) 1351–1360. [7] W. Wang, Y.Y. Chen, X.D. Meng, R.T. Leon, Behavior of thick-walled CHS X-joints under cyclic out-of-plane bending, J. Constr. Steel Res. 66 (6) (2010) 826–834. [8] M. Lesani, M.R. Bahaari, M.M. Shokrieh, Detail investigation on un-stiffened T/Y tubular joints behavior under axial compressive loads, Steel Constr. 80 (80) (2013) 91–99. [9] E. Ozyurt, Y.C. Wang, K.H. Tan, Elevated temperature resistance of welded tubular joints under axial load in the brace member, Eng. Struct. 59 (2014) 574–586. [10] R. Feng, B. Young, Design of cold-formed stainless steel tubular T- and X-joints, J. Constr. Steel Res. 67 (3) (2011) 421–436. [11] R. Feng, B. Young, Design of cold-formed stainless steel tubular joints at elevated temperatures, Eng. Struct. 35 (2012) 188–202. [12] R. Feng, Y. Chen, J.G. Wei, K. He, L.Q. Fu, Behaviour of grouted stainless steel tubular X-joints with CHS chord under axial compression, Thin-Walled Struct. 124 (2018) 323–342. [13] M.H. Mussa, A.A. Mutalib, Effect of geometric parameters (β and τ) on behaviour of cold formed stainless steel tubular X-joints, Int. J. Steel Struct. 18 (3) (2018) 821–830. [14] R. Feng, J.W. Lin, X.L. Mou, Experiments on hybrid tubular K-joints with circular braces and square chord in stainless steel, Eng. Struct. 190 (2019) 52–65. [15] W.R. Osgood, W. Ramberg, Description of Stress-Strain Curves by Three Parameters. Technical Note No. 902, National Advisory Committee for Aeronautics, Washington DC, 1943. [16] E. Mirambell, E. Real, On the calculation of deflections in structural stainless steel beams: an experimental and numerical investigation, J. Constr. Steel Res. 54 (1) (2000) 109–133. [17] K.J.R. Rasmussen, Full-range stress-strain curves for stainless steel alloys, J. Constr. Steel Res. 59 (1) (2003) 47–61. [18] W.M. Quach, J.G. Teng, K.F. Chung, Three-stage full-range stress-strain model for stainless steels, J. Struct. Eng., ASCE 134 (9) (2008) 1518–1527. [19] Eurocode 3 (EC3), Design of Steel Structures − Part 1-4: General RulesSupplementary Rules for Stainless Steels, European Committee for Standardization, EN 1993-1-4: 2006, CEN, Brussels, Belgium, 2006. [20] Chinese Code, Technical Specification for Stainless Steel Structures, vol. 410, CECS, Beijing, China, 2015 2015. (in Chinese). [21] ABAQUS Analysis User's Manual, ABAQUS, Inc., 2014Version 6.14. [22] B. Kato, I. Nishiyama, T-joints made of rectangular tubes, Oil Shale Symposium Proceedings. St. Louis, USA, 1980, pp. 663–679. [23] J.A. Packer, J. Wardenier, X.L. Zhao, G.J. van der Vegte, Y. Kurobane, Design Guide for Rectangular Hollow Section (RHS) Joints under Predominantly Static Loading. CIDECT Design Guide No. 3, second ed., Comité International pour le Développement et l’Étude de la Construction Tubulaire (CIDECT), Geneva, Switzerland, 2009. [24] Eurocode 3 (EC3). Design of Steel Structures − Part 1-8: Design of Joints. European Committee for Standardization, EN 1993-1-8: 2005, CEN. Brussels, Belgium, 2005. [25] Australian/New Zealand Standard (AS/NZS), Cold-Formed Stainless Steel
Notation A: Elongation of tensile coupon after fracture b0: Outer width of square chord COV: Coefficient of variation di: Outer diameter of overlapping brace dj: Outer diameter of overlapped brace d1: Outer diameter of circular brace d1*: Outer diameter of 1st circular brace d2*: Outer diameter of 2nd circular brace e: Eccentricity Ey: Tangent modulus at yield stress E0: Initial modulus of elasticity E0.2: Tangent modulus at 0.2% proof stress fu: Standard value of ultimate strength fyi: Yield stress of overlapping brace fyj: Yield stress of overlapped brace fy0: Yield stress of chord f0.2: Standard value of yield strength g: Gap distance between two braces of gapped tubular K-joint K: Correlation coefficient of strain hardening L0: Overall length of square chord L1: Length of axis from top of circular brace to top flange of square chord n: Strain hardening index of materials n’0.2,u: Strain hardening index for stress of materials between σ0.2 and σu NAN: Design strength determined by design formulae of AS/NZS NCC: Design strength determined by design formulae of GB 50017-2017 NCI: Design strength determined by design formulae of CIDECT NEC: Design strength determined by design formulae of EC3 NF: Failure strength NFEA: Joint strength obtained from FEA NP: Design strength determined by proposed design equations NTest: Joint strength obtained from test Ov: Overlap ratio of overlapped tubular K-joint (q/p × 100%) p: Length of major axis of brace and chord intersection ellipse q: Overlap distance between two braces of overlapped tubular K-joint RP0.01: Stress at residual strain of 0.01% ti: Wall thickness of overlapping brace tj: Wall thickness of overlapped brace t0: Wall thickness of square chord t1: Wall thickness of circular brace β: Brace diameter/chord width ratio (d1/b0) γM5: Partial safety factor 2γ: Chord width/thickness ratio (b0/t0) ϵe: Elastic strain ϵp: Plastic strain εpu: Ultimate plastic strain εu: Ultimate strain θ: Inclined angle between circular brace and square chord θi*: Inclined angle between axes of ith brace and chord σu: Ultimate stress σ0.2: Static 0.2% tensile proof stress τ: Brace/chord thickness ratio (t1/t0) ψg: Correction factor for hybrid gapped tubular K-joint in stainless steel ψo: Correction factor for hybrid overlapped tubular K-joint in stainless steel
14
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Full length article
Numerical study of hybrid tubular K-joints with circular braces and square chord in stainless steel
T
Ran Fenga,b,∗, Junwu Lina a b
School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen, 518055, China State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou, 510640, China
ARTICLE INFO
ABSTRACT
Keywords: Circular brace Finite-element analysis (FEA) Hybrid tubular K-joint Parametric study Square chord Stainless steel
This paper introduces a finite-element analysis (FEA) on hybrid tubular K-joints with circular braces and square chord in stainless steel. The finite-element models (FEMs) were established by using the shell element S4R for circular braces, square chord and welds. The FEMs were validated by comparing the failure modes, failure strengths and joint deformation curves obtained from the experiments and FEA. A parametric study was carried out on 162 FEMs to evaluate the influences of brace diameter/chord width ratio (β = d1/b0), brace/chord thickness ratio (τ = t1/t0), chord width/thickness ratio (2γ = b0/t0), overlap ratio (Ov=(q/p) × 100%) of overlapped tubular K-joints and eccentricity (e) of gapped tubular K-joints. The numerical results show that the joint strengths of overlapped tubular K-joints increased with the increment of the β, τ and Ov values, as well as the decrement of the 2γ value, while the joint strengths of gapped tubular K-joints increased with the increment of the β and τ values, as well as the decrement of the 2γ and e values. The comparison of the failure strengths with design strengths demonstrates that the design formulae of CIDECT and Australian/New Zealand Standard (AS/NZS) are generally unconservative, whereas the design strengths determined by Eurocode (EC3) and Chinese Code are relatively close to the failure strengths for overlapped tubular K-joints. The design formulae of AS/NZS are generally unconservative, whereas the design formulae of CIDECT, EC3 and Chinese Code are generally conservative for gapped tubular K-joints. The design equations are proposed by introducing the correction factors that considered the influences of geometrical parameters of β, τ, 2γ and Ov, which were validated to be accurate for hybrid tubular K-joints in stainless steel.
1. Introduction A tubular joint is one of the efficient joint forms commonly used in steel tubular structures. It has the advantages of less steel consumption, good mechanical behaviour, clear path of force transmission and large bearing capacity. Hybrid tubular K-joints with circular braces and square chord meet the requirements of structural form and mechanical properties and are easy to design and construct, which are widely used in practical engineering. On the other hand, stainless steel structures have the advantages of good durability, easy processing, high-temperature resistance, excellent mechanical properties and beautiful appearance. They have been paid more and more attention in architectural and structural designs. Therefore, hybrid tubular K-joints with circular braces and square chord in stainless steel have good engineering application prospects. Many types of research were conducted by finite-element analysis (FEA) on carbon steel tubular joints. The crack modelling of circular
∗
tubular joints was investigated by Cao et al. [1] using the FEA. Various kinds of crack tip models in connection with the crack elements and the neighbourhoods were discussed. A process to transform the crack elements of a plane curve into those of a doubly curved semi-elliptical surface around a joint intersection was also introduced. Lee [2] reviewed the modelling techniques applied in the FEA of tubular joints for the strengths, stress fields and stress intensity factors. The guidance on material curve input, model discretization, choice of elements, weld modelling, results interpretation and FEA limitations were also detailed. An eight-node thick shell element was applied by Fung et al. [3,4] to model the completely overlapped tubular joints with lap brace in axial compression. The finite-element model (FEM) was verified and employed in the parametric study to investigate the strengths and behaviour of tubular joints. Shao et al. [5] studied the influences of geometrical parameters on the stress distributions of tubular T-joints under axial loads. A FEM for welded tubular T-joints was developed, which was used to analyze 140 tubular T-joints with different
Corresponding author. School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen, 518055, China. E-mail address: [email protected] (R. Feng).
https://doi.org/10.1016/j.tws.2019.106390 Received 12 April 2019; Received in revised form 4 September 2019; Accepted 5 September 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 145 (2019) 106390
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Table 1 Comparison of failure strengths between tests and FEA. Specimen
K–C150 × 3-B108 × 3-O1.8 K–C150 × 3-B108 × 3-O50.9 K–C150 × 3-B108 × 3-O50.9-R K–C150 × 3-B108 × 3-G72.26 K–C150 × 3-B133 × 3-O20.3 K–C150 × 3-B133 × 3-O60.1 K–C150 × 3-B133 × 3-G36.91 K–C150 × 3-B133 × 3-G36.91-R K–C200 × 4-B108 × 3-O34.5 K–C200 × 4-B108 × 3-G47.26 K–C200 × 4-B108 × 3-G147.26 K–C200 × 4-B133 × 3-O46.8 K–C200 × 4-B133 × 3-G11.91 K–C200 × 4-B133 × 3-G111.91 Mean COV
Ov
1.8% 50.9% 50.9% – 20.3% 60.1% – – 34.5% – – 46.8% – –
g (mm)
– – – 72.26 – – 36.91 37.91 – 47.26 147.26 – 11.91 111.91
e (mm)
0.00 −37.50 −37.50 37.50 0.00 −37.50 37.50 37.50 −50.00 0.00 50.00 −50.00 0.00 50.00
β
0.72 0.72 0.72 0.72 0.88 0.88 0.88 0.88 0.54 0.54 0.54 0.66 0.66 0.66
2γ
τ
1.00 1.00 1.00 0.96 1.04 1.03 1.02 1.02 0.74 0.72 0.71 0.76 0.76 0.77
50.48 50.96 50.62 51.04 51.03 50.56 50.64 50.80 50.19 50.70 50.52 50.34 50.55 50.70
Test results
FEA results
Comparison
Failure mode
NTest (kN)
Failure mode
NFEA (kN)
NTest/NFEA
A A A A A A A A A A A A A A
200.38 253.96 243.23 102.56 252.39 295.77 194.31 201.91 260.19 159.72 161.36 289.88 202.37 186.60
A A A A A A A A A A A A+C A A
206.79 240.53 240.53 100.43 238.34 278.11 178.43 178.43 246.74 160.35 149.92 277.94 192.26 193.03
0.97 1.06 1.01 1.02 1.06 1.06 1.09 1.13 1.05 1.00 1.08 1.04 1.05 0.97 1.04 0.044
+B
+B +B+C +B
Note: A = Chord face plastification; B=Weld fracture; C = Local buckling failure of brace.
geometrical parameters. Shao [6] also used a developed FEM to analyze the stress distributions along the weld toe on the chord surface for tubular T- and K-joints under axial loads. A total of 140 tubular T-joints and 420 tubular K-joints were included in the parametric study to investigate the geometrical effects. Wang et al. [7] studied the seismic behaviour of the thick-walled circular hollow section (CHS) tubular Xjoints subjected to out-of-plane bending. The experimental behaviour was simulated by the FEA to further reveal the significant experimental observations. Lesani et al. [8] carried out the FEA using finite-element program ABAQUS to present the failure patterns, ultimate static strengths and detailed behaviour of un-stiffened tubular T- and Y-joints under axial compression. The FEMs were also developed by Ozyurt et al. [9] using finite-element program ABAQUS to investigate the resistances of welded tubular joints with CHS or SHS brace members under axial loads at various elevated temperatures. The numerical studies on stainless steel tubular joints are relatively few. A total of 172 FEMs were developed by Feng and Young [10] in the parametric study on the square and rectangular hollow section (SHS and RHS) tubular T-joints, X-joints and X-joints with chord preload in stainless steel. The design equations were proposed and verified to be accurate. The numerical investigation was also conducted by Feng and Young [11] on SHS and RHS tubular T-joints, X-joints and X-joints with chord preload in stainless steel at elevated temperatures. The FEA results were used to assess the existing design guidelines and the accurate design rules were developed. Both experimental and numerical analyses were carried out by Feng et al. [12] on empty and grouted tubular Xjoints with chord preload in stainless steel. The design formulae were derived to determine the joint strengths. Mussa and Mutalib [13] applied finite-element software LUSAS to study the influences of brace/ chord width ratio (β) and brace/chord thickness ratio (τ) on the strengths and behaviour of SHS and RHS tubular X-joints in stainless steel. The previous literatures exhibit that the researches on carbon steel tubular joints are relatively mature. However, the numerical analyses on stainless steel tubular joints, especially stainless steel hybrid tubular joints are relatively rare, although stainless steel is nowadays increasingly used in tubular structures owing to its excellent mechanical and technical properties. It is worth noting that the constitutive relation of stainless steel is notably different from that of carbon steel, which exhibits the obvious non-linear material behaviour without distinct yield plateau. Hence, the current design rules for carbon steel tubular joints may not be suitable for the design of stainless steel tubular joints. In
order to promote the applications of stainless steel in practical engineering of tubular structures, more studies need to be performed by FEA on stainless steel tubular joints to derive the corresponding design guidelines, in particular for stainless steel hybrid tubular joints. Therefore, the FEMs of hybrid tubular K-joints with circular braces and square chord in stainless steel were established in this study using finite-element software ABAQUS 6.14, which were calibrated against the experimental results. A total of 162 hybrid tubular K-joints in stainless steel were simulated in the parametric study using the verified FEMs. The design equations are proposed for hybrid tubular K-joints in stainless steel, whose accuracy and reliability were evaluated by the experimental and numerical results. 2. A brief summary of experimental work A total of 14 hybrid tubular K-joints including 7 overlapped and 7 gapped tubular K-joints were tested to investigate the load-carrying capacities of hybrid tubular K-joints in stainless steel. All specimens were fabricated from austenitic stainless steel (AISI 304) with circular braces fully welded to the continuous square chord. The hidden seam of the overlapped brace for overlapped tubular K-joint was unwelded. The inclined angles (θ) between the axes of circular braces and square chord of all specimens kept in consistence as 45°. The overall length (L0) of the square chord was chosen as 6b0, while the length (L1) of the axis from the top of circular brace to the top flange of square chord was chosen as 3d1. The effects of the key geometrical parameters including the brace diameter/chord width ratio (β = d1/b0), eccentricity (e), overlap ratio (Ov=(q/p) × 100%) of overlapped tubular K-joints and gap distance (g) of gapped tubular K-joints on the failure modes and joint strengths were studied. The key geometrical parameters of all specimens are presented in Table 1, with the symbols shown in Fig. 1. The test setup of hybrid tubular K-joints in stainless steel is displayed in Fig. 2. The square chord of the specimens was subjected to axial compression force, while the circular braces were pin-connected to the reaction frame that resulted in the compression and tension forces applied to the circular braces separately. The monotonic multilevel load control was used for loading until the peak load of the specimens was reached. Then the displacement control was applied for replacement in the post-ultimate stage. The experimental results including the failure modes, failure strengths and joint deformation curves are detailed in Ref. [14].
2
Thin-Walled Structures 145 (2019) 106390
R. Feng and J. Lin
Fig. 1. Schematic sketch of hybrid tubular K-joints in stainless steel.
developed by some researchers and introduced in some design specifications as below. Ramberg-Osgood model (R–O model) was usually used to describe the stress-strain relationships of non-linear metal materials, which is a classical theoretical model representing the constitutive relations of elasto-plastic materials in solid mechanics. This model was first developed by Osgood and Ramberg [15] in 1943, whose basic idea laid in the assumption that the total strains of materials consisted of elastic strain (ϵe) and plastic strain (ϵp) as follows:
=
e
+
p;
e
=
E0
;
p
n
=
K
,
(1)
where E0 is the initial modulus of elasticity, K is the correlation coefficient of strain hardening, and n is the strain hardening index of materials reflecting the non-linear degree of stress-strain curves of materials. By carrying out experimental and numerical investigations, Mirambell and Real [16] proposed a constitutive relation model for stainless steel materials based on the R–O model as follows:
Fig. 2. Test setup of hybrid tubular K-joints in stainless steel.
3. Development and validation of FEMs
=
3.1. Constitutive relation models of stainless steel materials
E0 0.2
+ 0.002 +
( ),
0.2
E0.2
n
<
0.2
+
pu
(
0.2 0.2
u
0.2
)
n 0.2, u
,
0.2
(2)
where σ0.2 is the static 0.2% tensile proof stress, σu is the ultimate stress, E0.2 is the tangent modulus at σ0.2, εpu is the ultimate plastic strain, and n’0.2,u is the strain hardening index for the stress of materials between
A large number of researches proved that stainless steel is a typical non-linear metal material. The stress-strain relationships of stainless steel materials were reported and the constitutive relation models were 3
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Fig. 3. Comparison of stress-strain curves of stainless steel CHS and SHS tubes.
σ0.2 and σu. Rasmussen [17] proposed a full-range constitutive relation model in 2003 for stainless steel materials based on the R–O model as follows:
=
+ 0.002
E0
0.2
E0.2
in which,
+
u
n
0.2
=
m
0.2
u
m = 1 + 3.5 u
( ), < ( ) +
in which,
u
0.2
b= 0.2,
1 , for austenitic and duplex stainless 0.2 + 185e 0.2 1 0.0375(n 5) 0.2, for ferritic stainless steel 0.2 + 185e E0 0.2 ; e = 0.2 ; E0.2 = ; 0.2 = 0.2 E0 1 + 0.002n/ e E0 u
steels
+ 0.002;
E0
=
E0
( ),
+ 0.002
0.2 E0.2
0.2 a
n
0.2
+ 0.008 + (
< 1.0
(
1 E0
1 E0.2
) ×(
1.0
0.2 0.2
)
n0.2,1.0
+
>
(5)
2.0
n
+ 0.002
in which, n =
fy E
fy
, fy
+
Ey
ln 20 ln
fy
fy RP0.01
+
u
fy fu
m
fy
fy
, fy <
, m = 1 + 3.5 f , E y = u
< fu E 1 + 0.002n
(6) E , fy
u
=1
fy fu
A.
where E, fy and fu are the eigenvalues of stainless steel materials, RP0.01 is the stress at the residual strain of 0.01%, Ey is the tangent modulus at the yield stress, ϵu is the ultimate strain, and A is the elongation after fracture. In Chinese Code (CECS 410: 2015) [20], the linear elastic model was employed for the stress-strain relationships of materials. The constitutive relation model of stainless steel materials was developed for
0.2,
2.0
,
E
0.002 +
0.2 0.2 )
b 2.0 2.0 ) 2.0 (1 ± 2.0)
where the operator of plus sign in the equations represents the stainless steel in tension, while the operator of minus sign in the equations represents the stainless steel in compression. In Annex C of Part 1–4 of Eurocode (EC3) [19], the constitutive relation model of stainless steel materials employed in the numerical simulation was presented as follows:
(3)
0.2
± u
0.2
A full-range three-stage constitutive relation model was developed by Quach et al. [18] in 2008 for stainless steel materials, which was represented by three basic parameters of the R–O model as follows:
b
0.2 (1
u (1 ± u)
0.2
=1
=
a=
2.0
(4) 4
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R. Feng and J. Lin
analysis to consider the material non-linearity as follows:
=
E0
+ 0.002
0.002 +
( ),
f0.2 E0
n
f0.2
f0.2
+
f0.2 E0.2
in which, m = 1 + 3.5
reference points by means of displacement and rotation in the FEMs, which can rotate in the plane around the x-axis of the local coordinate system only. On the other hand, a reference point was also assigned at the centre of the loading end of square chord to couple the degrees of freedom of all nodes at the loading end of square chord by using the (*COUPLING) procedure in the ABAQUS library. Another local coordinate system was established at the reference point with the connected face and the axis of square chord defined as the x-axis and z-axis directions, respectively. The axial compression force was applied at the reference point by means of displacement loading along the axis of square chord, which was identical to the loading applications in the experimental work.
f0.2 fu
+
u
(
f0.2 fu
, E0.2 =
f0.2
) ,f m
0.2
E0
E 1 + 0.002 0
f0.2
,
< u
< fu
=1
(7)
f0.2 fu
.
where f0.2 is the standard value of yield strength, and fu is the standard value of ultimate strength. The stress-strain curves of stainless steel CHS and SHS tubes determined by the constitutive relation models developed by Mirambell and Real [16], Rasmussen [17], Quach et al. [18], as well as adopted in Eurocode [19] and Chinese Code [20] are compared with the testing curves obtained from the tensile coupon tests reported in Ref. [14], as illustrated in Fig. 3. The comparison exhibits that the stress-strain curves of stainless steel CHS and SHS tubes determined by the constitutive relation models proposed by Rasmussen [17] and Quach et al. [18], respectively, agreed well with the testing curves. Therefore, the constitutive relation models of stainless steel materials proposed by Rasmussen [17] and Quach et al. [18] were employed in the FEA to simulate the material properties of circular braces and square chord of hybrid tubular K-joints in stainless steel, respectively.
3.3. Validation of FEMs The developed FEMs for hybrid tubular K-joints in stainless steel were validated against the corresponding experimental results summarized in Ref. [14] before its application for parametric study. The test failure strengths are compared with those predicted by the FEA, as listed in Table 1. The comparison indicates that the deviations between the experimental and FEA results are less than 10% for all specimens, having the mean value of test/FEA strength ratio (NTest/NFEA) of 1.04 and the coefficient of variation (COV) of 0.044. Hence, the FEMs were validated to be accurate. In addition, the test failure modes are also compared with those obtained from the FEA, as illustrated in Fig. 5 and presented in Table 1. The comparison indicates that the failure modes obtained from the FEA include chord face plastification and local buckling failure of brace, which agreed well with the experimental results. Hence, the FEMs were validated to be accurate. However, the weld fracture at the joint intersection area observed in the experimental work was not obtained from the FEA since the weld materials in the FEA are perfect without any defects. The testing curves of axial load versus vertical displacement and axial load versus chord deformation are also compared with those obtained from the FEA, as shown in Figs. 6 and 7, respectively. The comparison indicates that the FEA results generally have a good agreement with the experimental results, except for the minor deviations resulted from the test measurements and the weld fracture at the joint intersection area occurred in the experimental work only. Therefore, the FEMs of hybrid tubular K-joints in stainless steel were validated to be accurate by comparing the failure modes, failure strengths and joint deformation curves, which were used for the succeeding parametric study.
3.2. Establishment of FEMs The general purpose FEA program ABAQUS 6.14 [21] was implemented in this study for the numerical simulation of hybrid tubular K-joints in stainless steel. The quadrilateral shell element namely S4R was used to model the circular braces, square chord and welds. The S4R represents a 4-node quadrilateral shell element with finite film strain and linear reduced integration, in which each node has three translational and three rotational degrees of freedom. The S4R shell element is a general type of shell element with good adaptability, which can be applied to both the thick shell model and the thin shell model. For the modelling of hybrid tubular K-joints in stainless steel, the stresses along the thickness direction can be neglected since the thickness is much smaller than the geometrical dimensions along the length direction. Hence, the thin shell model with the use of S4R shell element was employed in the FEMs, whose performance was stable and characteristics were more suitable for the numerical simulation. On the other hand, both initial geometrical imperfections and residual stresses were not taken into consideration in the FEMs due to their negligible influences on the behaviour of welded tubular joints based on the previous literatures [8,10–12,22]. For the modelling of the welds, a ring shell element was added to simulate the weld seam at the joint intersection area, whose thickness is half of the wall thickness of the square chord. The overall mesh size of the FEMs is 10 × 10 mm with the length-to-width ratio of 1.0, except for the local mesh refinement at the welds and joint intersection area to take the complex stress distributions into account. The finite-element meshes of hybrid tubular K-joints in stainless steel, in particular, the weld seam are presented in Fig. 4. In the experimental work, the square chord of the specimens was subjected to axial compression, which resulted in the circular braces rotated in the plane around the fixed hinge axes. A reference point was assigned at the centre of the end face of circular braces to couple the degrees of freedom of all nodes at the end face of circular braces by using the (*COUPLING) procedure in the ABAQUS library. A local coordinate system was established at the reference points with the fixed hinge axis and the axis of circular brace defined as the x-axis and y-axis directions, respectively. The boundary conditions were applied at the
4. Parametric study 4.1. General The validated FEMs were employed for parametric study to investigate the influences of key geometrical parameters including the brace diameter/chord width ratio (β = d1/b0), brace/chord thickness ratio (τ = t1/t0), chord width/thickness ratio (2γ = b0/t0), eccentricity (e), overlap ratio (Ov) of overlapped tubular K-joints and gap distance (g) of gapped tubular K-joints on the failure modes and joint strengths of hybrid tubular K-joints in stainless steel. All these geometrical parameters were identified as the key influential parameters based on the corresponding experimental work [14], some other previous literatures [3,4] and the current design guidelines. A total of 162 FEMs including 81 overlapped and 81 gapped tubular K-joints with the dimensions of circular braces and square chord selected from the range of practical applications were investigated in the parametric study. The
5
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Fig. 4. Finite-element meshes of hybrid tubular K-joints in stainless steel.
outer width (b0) of square chord was chosen as a constant value of 300 mm, the overall length (L0) of square chord was chosen as a constant value of 1800 mm, the overall length (L1) of circular brace was chosen as four times of outer diameter (d1), and the inclined angle (θ) between circular brace and square chord was chosen as a constant value of 45°. The validity ranges of key geometrical parameters for hybrid tubular K-joints in stainless steel designed in the experimental work and FEA, as well as defined in the current design guidelines are listed in
Table 2. The comparison exhibits that the validity ranges of key geometrical parameters designed in the FEA are far beyond those defined in the current design guidelines. 4.2. Failure modes and joint strengths The typical failure modes of hybrid tubular K-joints in stainless steel obtained from the parametric study mainly include chord face
Fig. 5. Comparison of failure modes between tests and FEA.
6
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Fig. 6. Comparison of axial load versus vertical displacement curves between tests and FEA.
plastification, local buckling failure of brace and the combination of these two failure modes, as shown in Fig. 8. It can be concluded from the comparison that most of the specimens failed by chord face plastification, while the overlapped tubular K-joints are normally subjected to local buckling failure of brace, and the gapped tubular K-joints are normally subjected to the combination of two failure modes. It was demonstrated in Ref. [14] that the joint strengths of hybrid tubular K-joints in stainless steel were determined by the strength design criterion and deformation design criterion, depending on which resulted in the lower strength value. For the strength design criterion, the joint strengths were determined by the peak loads of the specimens. For the deformation design criterion, the joint strengths were determined by the axial loads at the chord deformation limit of 0.03b0 of the specimens. The failure strengths of hybrid overlapped and gapped tubular K-joints in stainless steel obtained from the parametric study were determined based on the strength and deformation design criteria. The comparison of failure strengths with design strengths for hybrid tubular K-joints in stainless steel is listed in Table 3.
strengths, in particular, the specimens having large τ value and small 2γ value, which also resulted in the increase of the joint stiffnesses. It is also shown from Fig. 9a, d and 9e that the failure strengths of hybrid overlapped tubular K-joints increased with the decrement of the 2γ value and the increment of the failure strengths by decreasing the 2γ value is greater for the specimens having large β and τ values. This may attribute to the increment of the wall thickness of square chord, which resulted in the increase of the failure strengths, in particular, the specimens having large β and τ values, which also resulted in the increase of the joint stiffnesses. It is shown from Fig. 9b, d and 9f that the failure strengths of hybrid overlapped tubular K-joints increased with the increment of the τ value and the increment of the failure strengths by increasing the τ value is greater for the specimens having large β value and small 2γ value. This may attribute to the increment of the wall thickness of circular brace, which resulted in the increase of the resistance of circular brace to local buckling that enhanced the failure strengths, in particular, the specimens having large β value and small 2γ value, which also resulted in the increase of the joint stiffnesses. It is also shown from Fig. 9c, e and 9f that the failure strengths of hybrid overlapped tubular K-joints increased with the increment of the Ov value. However, the increment is relatively small. The influence of the Ov on the failure strengths of hybrid overlapped tubular K-joints is smaller than other geometrical parameters.
4.3. Failure strengths of hybrid overlapped tubular K-joints The influential curved surfaces of key geometrical parameters including β, τ, 2γ and Ov on the failure strengths of hybrid overlapped tubular K-joints are plotted in Fig. 9. It is shown from Fig. 9a–c that the failure strengths of hybrid overlapped tubular K-joints increased with the increment of the β value and the increment of the failure strengths by increasing the β value is greater for the specimens having large τ value and small 2γ value. This may attribute to the increment of the diameter of circular brace, which resulted in the increase of the failure
4.4. Failure strengths of hybrid gapped tubular K-joints The influential curved surfaces of key geometrical parameters including β, τ, 2γ and e on the failure strengths of hybrid gapped tubular K-joints are also plotted in Fig. 10. It is illustrated from Fig. 10 that the 7
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Fig. 7. Comparison of axial load versus chord deformation curves between tests and FEA. Table 2 Validity range of geometrical parameters for hybrid tubular K-joints. Parameter
Ov
e (mm)
β
τ
2γ
CIDECT [23] EC3 [24] AS/NZS [25] Chinese Code [26] Test FEA
20%≤Ov 25%≤Ov ≤ 100% 25%≤Ov ≤ 100% 25%≤Ov ≤ 100% 1.8%≤Ov ≤ 60.1% 20%≤Ov ≤ 100%
e ≤ 0.25b0 – – – −50≤e ≤ 50 −150≤e ≤ 150
0.25≤β ≤ 0.8 0.4≤β ≤ 0.8 0.4≤β ≤ 0.8 0.4≤β ≤ 0.8 0.54≤β ≤ 0.88 0.2≤β ≤ 1.0
τ ≤ 1.0 – τ ≤ 1.0 – 0.71≤τ ≤ 1.04 0.5≤τ ≤ 2.0
15≤2γ 15≤2γ ≤ 35 15≤2γ ≤ 35 15≤2γ ≤ 35 50.19≤2γ ≤ 51.04 30≤2γ ≤ 150
failure strengths of hybrid gapped tubular K-joints increased with the increment of the β and τ values, as well as the decrement of the 2γ and e values. However, the influence of the e on the failure strengths of hybrid gapped tubular K-joints is relatively small. Furthermore, the increment of the failure strengths by increasing the β and τ values, as well as decreasing the 2γ and e values is greater for the specimens with large β and τ values, as well as small 2γ value. This may attribute to the identical reasons elaborated above for hybrid overlapped tubular Kjoints, in particular, the specimens with large β and τ values, as well as small 2γ value, which also resulted in the increase of the joint stiffnesses.
and parametric study with the design strengths determined by the current design guidelines for hybrid overlapped and gapped tubular Kjoints in stainless steel is summarized in Table 3 and plotted in Figs. 11 and 12, respectively. The comparison in Table 3 and Fig. 11 exhibits that the design formulae of CIDECT [23] and AS/NZS [25] are generally unconservative, having the mean values of failure/design strength ratios (NF/NCI and NF/NAN) of 0.84 and 0.78, respectively, for hybrid overlapped tubular K-joints. Whereas, the design strengths determined by the design formulae of EC3 [24] and Chinese Code [26] are relatively close to the failure strengths, having the mean values of failure/design strength ratios (NF/NEC and NF/NCC) of 1.07 and 1.07, respectively, for hybrid overlapped tubular K-joints. The COVs of the comparison of the four design guidelines are all quite large, which indicates that the deviations of the predictions between the failure strengths and design strengths of hybrid overlapped tubular K-joints in stainless steel are quite large. This may attribute to the notably different material properties between carbon steel and stainless steel, as well as the large range of key geometrical parameters designed in the FEA, which are far beyond the validity ranges defined in the current design guidelines. The comparison in Table 3 and Fig. 12 also exhibits that the design
5. Proposed design equations 5.1. Assessment of current design guidelines It is worth noting that the joint strengths of hybrid tubular K-joints with circular braces and square chord can be determined by the design formulae of CIDECT [23], Eurocode 3 (EC3) [24], Australian/New Zealand Standard (AS/NZS) [25] and Chinese Code (GB 50017-2017) [26]. The comparison of the failure strengths obtained from the tests 8
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Fig. 8. Typical failure modes of hybrid tubular K-joints in stainless steel obtained from parametric study.
EC3 [24] and Chinese Code [26] are generally conservative, having the mean values of failure/design strength ratios (NF/NCI, NF/NEC and NF/ NCC) of 1.46, 1.16 and 1.31, respectively, for hybrid gapped tubular Kjoints. The COVs of the comparison of the four design guidelines are also large, which are smaller than those calculated in the comparison for hybrid overlapped tubular K-joints. Hence, the design formulae of EC3 [24] are most accurate among the current design guidelines for the design of hybrid overlapped and gapped tubular K-joints in stainless steel.
Table 3 Comparison of failure strengths with design strengths for hybrid tubular Kjoints in stainless steel. Specimen (A total of 7+81 overlapped K-joints)
Comparison NF/NCI
NF/NEC
NF/NAN
NF/NCC
NF/NP
Mean COV
0.84 0.443
1.07 0.587
0.78 0.615
1.07 0.587
1.03 0.246
Specimen (A total of 7+81 gapped Kjoints) Mean COV
Comparison NF/NCI NF/NEC
NF/NAN
NF/NCC
NF/NP
1.46 0.355
0.83 0.323
1.31 0.339
1.03 0.263
1.16 0.331
5.2. Proposed design equations The design equations are proposed for hybrid tubular K-joints in stainless steel based on the design formulae of EC3 [24] for carbon steel tubular joints. For hybrid overlapped tubular K-joints in stainless steel, a correction factor (ψo = N/Ni) is proposed based on the overlap influence function given in Chinese Code (CECS 280: 2010) [27] by considering the influences of geometrical parameters including β, τ, 2γ
formulae of AS/NZS [25] are generally unconservative, having the mean value of failure/design strength ratio (NF/NAN) of 0.83 for hybrid gapped tubular K-joints. Whereas, the design formulae of CIDECT [23], 9
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Fig. 9. Influential curved surfaces of geometrical parameters on failure strengths of hybrid overlapped tubular K-joints in stainless steel.
and Ov as below: o
= k1
k2 k3 (2
For 50%≤Ov < 80%, (8)
) k 4 Ov k5
Ni =
By taking logarithm on both sides of Eq. (8),
N ln = ln k1 + k2 ln Ni
+ k3 ln + k 4 ln(2 ) + k5 ln Ov
o
0.467
0.398 (2
)
0.371O
v
Ni =
(9)
(10)
0.540
(11)
o Ni
in which. For 25%≤Ov < 50%,
Ni =
4
f yi ti beff + be, ov +
Ov (2di 50
4t i ) /
M5
4ti]/
M5
(13)
4
f yi ti [be, ov + 3di
beff =
10 f y 0 t0 d b0 / t0 f yi ti i
di
be, ov =
10 f yj t j d dj / t j f yi ti i
di
4ti]/
M5
(14)
(15)
where γM5 = 1.0 is the partial safety factor, di and ti are the outer diameter and wall thickness of overlapping brace, respectively, dj and tj are the outer diameter and wall thickness of overlapped brace, respectively, fy0 is the yield stress of chord, fyi and fyj are the yield stresses of overlapping and overlapped braces, respectively. For hybrid gapped tubular K-joints in stainless steel, a correction factor (ψg = N*/Ni*) is also proposed by considering the influences of geometrical parameters including β, τ and 2γ as below:
The joint strengths of hybrid overlapped tubular K-joints with circular braces and square chord in stainless steel can be calculated as below:
N=
f yi ti [beff + be, ov + 2di
For Ov≥80%,
where k1, k2, k3, k4 and k5 are the coefficients to be determined by the multivariate linear regression analysis based on the FEA results in the parametric study. Hence, the correction factor (ψo) was determined as below:
= 1.966
4
g
(12)
= m1
m2 m3 (2
)m4
By taking logarithm on both sides of Eq. (16), 10
(16)
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Fig. 10. Influential curved surfaces of geometrical parameters on failure strengths of hybrid gapped tubular K-joints in stainless steel.
ln
N Ni
= ln m1 + m2 ln
+ m3 ln + m4 ln(2 )
kn = 1.3
(17)
= 1.428
0.208 0.272 (2
)
0.097
8.9
The comparison of the failure strengths obtained from the tests and parametric study with the design strengths determined by the proposed design equations for hybrid overlapped and gapped tubular K-joints in stainless steel is summarized in Table 3 and plotted in Fig. 13a and b, respectively. A good agreement between the failure strengths and the proposed design strengths was achieved, having the mean values of failure/proposed design strength ratio (NF/NP) of 1.03 and 1.03 and the
(19)
4
0.5k
sin
2 n f y0 t 0 i
d1 + d 2 / 2b0
M5
(22)
5.3. Assessment of proposed design equations
in which,
Ni =
(21)
where θi* is the inclined angle between the axes of ith brace and chord, d1* and d2* are the outer diameters of the 1st and 2nd circular braces, respectively.
(18)
* g Ni
1.0
kn = 1.0
The joint strengths of hybrid gapped tubular K-joints with circular braces and square chord in stainless steel can be calculated as below:
N* =
n
For n ≤ 0 (in tension),
where m1, m2, m3 and m4 are the coefficients to be determined by the multivariate linear regression analysis based on the FEA results in the parametric study. Hence, the correction factor (ψg) was determined as below: g
0.4
(20)
For n > 0 (in compression), 11
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Fig. 11. Comparison of failure strengths with design strengths for hybrid overlapped tubular K-joints in stainless steel.
COV of 0.246 and 0.263 for hybrid overlapped and gapped tubular Kjoints in stainless steel, respectively. Therefore, the proposed design equations were validated to be accurate for hybrid tubular K-joints with circular braces and square chord in stainless steel.
key geometrical parameters including β, τ, 2γ, Ov of overlapped tubular K-joints and e of gapped tubular K-joints on the joint strengths were evaluated. The failure modes mainly include chord face plastification, local buckling failure of brace and the combination of these two failure modes. Most of the specimens failed by chord face plastification. The numerical results in the parametric study exhibit that the joint strengths of hybrid overlapped tubular K-joints increased with the increment of the β, τ and Ov values, as well as the decrement of the 2γ value, while the joint strengths of hybrid gapped tubular K-joints increased with the increment of the β and τ values, as well as the decrement of the 2γ and e values. In addition, the joint stiffnesses increased with the increment of the β and τ values, as well as the decrement of the 2γ value. The comparison of the failure strengths with design strengths indicates that the design formulae of CIDECT and AS/NZS are generally unconservative for hybrid overlapped tubular K-joints. Whereas, the design strengths determined by the design formulae of EC3 and Chinese Code are relatively close to the failure strengths for hybrid overlapped tubular K-joints. In addition, the design formulae of AS/NZS are generally unconservative for hybrid gapped tubular K-joints. Whereas, the design formulae of CIDECT, EC3 and Chinese Code are generally conservative for hybrid gapped tubular K-joints. The design formulae of EC3 are most accurate among the current design guidelines for hybrid overlapped and gapped tubular K-joints in stainless steel. The design
6. Conclusions The FEA was performed in this study on hybrid tubular K-joints with circular braces and square chord in stainless steel. Some constitutive relation models of stainless steel materials are introduced and compared with the results of tensile coupon tests. It is shown that the constitutive relation models proposed by Rasmussen and Quach et al. agreed well with the test results, which were employed in the FEA to simulate the material properties of circular braces and square chord of hybrid tubular K-joints in stainless steel, respectively. The FEMs were established by using the shell element S4R for circular braces, square chord and welds. The axial compression force and boundary conditions were applied at the reference points by means of displacement loading. The FEMs were validated by comparing the failure modes, failure strengths and joint deformation curves obtained from the experiments and FEA. A total of 162 FEMs including 81 overlapped and 81 gapped tubular K-joints were investigated in the parametric study. The influences of 12
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Fig. 12. Comparison of failure strengths with design strengths for hybrid gapped tubular K-joints in stainless steel.
equations are proposed by introducing the correction factors (ψo and ψg) for hybrid overlapped and gapped tubular K-joints in stainless steel, respectively, which considered the influences of geometrical parameters of β, τ, 2γ and Ov. A good agreement between the failure strengths and the proposed design strengths was achieved. The proposed design equations were validated to be accurate for hybrid tubular K-joints with circular braces and square chord in stainless steel.
Acknowledgements The authors are grateful for the financial support from National Natural Science Foundation of China (Grant No. 51528803), Natural Science Foundation of Guangdong Province of China (Grant No. 2018A030313208), State Key Laboratory of Subtropical Building Science (South China University of Technology, Grant No. 2018ZA02), and Guangdong Provincial Key Laboratory of Durability for Marine Civil Engineering, Shenzhen Durability Center for Civil Engineering
Fig. 13. Comparison of failure strengths with proposed design strengths for hybrid tubular K-joints in stainless steel. 13
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(Shenzhen University, Grant No. GDDCE 18-5).
Structures, AS/NZS 4673: 2001, Sydney, Australia, 2001. [26] Chinese Code, Code for Design of Steel Structures, GB 50017-2017, Beijing, China, 2017 (in Chinese). [27] Chinese Code, Technical Specification for Structures with Steel Hollow Sections vol. 280, CECS, Beijing, China, 2010 2010. (in Chinese).
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Notation A: Elongation of tensile coupon after fracture b0: Outer width of square chord COV: Coefficient of variation di: Outer diameter of overlapping brace dj: Outer diameter of overlapped brace d1: Outer diameter of circular brace d1*: Outer diameter of 1st circular brace d2*: Outer diameter of 2nd circular brace e: Eccentricity Ey: Tangent modulus at yield stress E0: Initial modulus of elasticity E0.2: Tangent modulus at 0.2% proof stress fu: Standard value of ultimate strength fyi: Yield stress of overlapping brace fyj: Yield stress of overlapped brace fy0: Yield stress of chord f0.2: Standard value of yield strength g: Gap distance between two braces of gapped tubular K-joint K: Correlation coefficient of strain hardening L0: Overall length of square chord L1: Length of axis from top of circular brace to top flange of square chord n: Strain hardening index of materials n’0.2,u: Strain hardening index for stress of materials between σ0.2 and σu NAN: Design strength determined by design formulae of AS/NZS NCC: Design strength determined by design formulae of GB 50017-2017 NCI: Design strength determined by design formulae of CIDECT NEC: Design strength determined by design formulae of EC3 NF: Failure strength NFEA: Joint strength obtained from FEA NP: Design strength determined by proposed design equations NTest: Joint strength obtained from test Ov: Overlap ratio of overlapped tubular K-joint (q/p × 100%) p: Length of major axis of brace and chord intersection ellipse q: Overlap distance between two braces of overlapped tubular K-joint RP0.01: Stress at residual strain of 0.01% ti: Wall thickness of overlapping brace tj: Wall thickness of overlapped brace t0: Wall thickness of square chord t1: Wall thickness of circular brace β: Brace diameter/chord width ratio (d1/b0) γM5: Partial safety factor 2γ: Chord width/thickness ratio (b0/t0) ϵe: Elastic strain ϵp: Plastic strain εpu: Ultimate plastic strain εu: Ultimate strain θ: Inclined angle between circular brace and square chord θi*: Inclined angle between axes of ith brace and chord σu: Ultimate stress σ0.2: Static 0.2% tensile proof stress τ: Brace/chord thickness ratio (t1/t0) ψg: Correction factor for hybrid gapped tubular K-joint in stainless steel ψo: Correction factor for hybrid overlapped tubular K-joint in stainless steel
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