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4 annular plates in transmission towers
Thin-Walled Structures 144 (2019) 106271
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Full length article
Design resistance of longitudinal gusset-tube K-joints with 1/4 annular plates in transmission towers
T
Fang Li, Hong-zhou Deng*, Xiao-yi Hu Department of Structural Engineering, Tongji University, Shanghai, 200092, China
ARTICLE INFO
ABSTRACT
Keywords: Gusset-tube connections K-joints Annular plates Design resistance Full-scale experiments Transmission towers
In this paper, the design resistance of longitudinal gusset-tube K-joints stiffened with 1/4 annular plates was studied. Full-scale experiments of the K-joints were carried out. The branch loads of the K-joints were equivalent to a vertical shear force (Pv ) and a bending moment (Mw ). Substantial parametric studies revealed that annular plates had little impact on the ultimate shear force capacity (Pv,u ). Nevertheless, they could significantly improve the ultimate moment capacity (Mw,u ) and restrain the chord wall deformation. Moreover, the ultimate moment capacity of K-joints reinforced with annular plates was about 1.3–2.3 times that of K-joints without annular plates. The minimum construction dimensions for annular plates were set to R / d0 = 0.25 and tr / t0 = 0.75.
1. Introduction In China, transmission towers are being widely built to meet the increasing need for electricity. Tubular steel towers are quite appropriate for long-span transmission towers due to the characteristics of the circular hollow section (CHS), of which the most significant are not having a weak axis and having the same performance in all directions [1]. In tubular steel transmission towers, longitudinal plate-to-CHS connections are famous for their easy fabrication, simple installation and efficient construction. Fig. 1 shows some examples of the application of gusset-tube connections in the tubular steel transmission towers. For plate-to-CHS connections, excessive transverse deformations are quite large and usually govern the nominal strength of the connections, resulting in connection failure before the material is fully utilized. Because of the negative influence of excessive transverse deformations on the ultimate strength, local reinforcements are usually adopted. Although fabricators do not favour these reinforcements, they can effectively improve the ultimate strength of plate-to-CHS connections [2,3]. A common method for reinforcement is to arrange annular plates on both ends of the gusset plate (shown in Fig. 1). In this paper, planar Kjoints strengthened by 1/4 annular plates were studied. Much research has been done on gusset-tube connections. Voth and Packer [4–7] did a battery of investigations into the ultimate strength of plate-to-CHS connections, including longitudinal T-type, transverse Ttype, longitudinal X-type, transverse X-type, and skewed X-type. Kim [8,9] investigated the capacity of longitudinal plate-to-tube X-joints when the plate was subjected to in-plane bending. Zapata et al. [10]
*
used the upper bound theorem of plastic collapse to estimate the capacity of T-type transverse plate-tube connections. Yang et al. [11] studied the behaviour of X-type plate-to-CHS connections stiffened by external annular plates when the plate was subjected to an in-plane bending moment. Wardenier et al. [12] did a short review of equations for plate-to-CHS connections in the existing codes and literature. In Fig. 1, it is evident that the connections applied in the transmission towers are usually spatial and have more than one gusset plate. Due to restrictions on experimental finance and conditions, there is almost no research on spatial gusset-tube connections. In practice, the planar K-joints that are shown in Fig. 1(d) are usually employed to analyse the spatial connections because planar behaviours are the basis of spatial behaviours. However, in the relevant standards [13–16], there are no methods to calculate the ultimate strength of the K-joints directly. Only methods to calculate the ultimate capacity of the gussettube connections when they are subjected to an in-plane bending moment (Mw ) are found. In the existing literature, there are only a few recordings of K-joints. Kim [17] studied the ultimate capacity of Kjoints without annular plates by simplifying the axial brace forces into a vertical component force (Pv ) and a chord wall moment (Mw ). Then interaction equations of the component force (Pv ), the chord wall moment (Mw ), and the axial chord force (N ) were proposed. Qu [18,19] predicted the ultimate strength of K-joints with and without annular plates using the virtual work principle and energy theory. His research all related to Q690 high strength steel. As the behaviours are complicated in K-joints stiffened by annular plates and there is no mature theory for these connections, the
Corresponding author. E-mail address: [email protected] (H.-z. Deng).
https://doi.org/10.1016/j.tws.2019.106271 Received 3 January 2019; Received in revised form 16 May 2019; Accepted 25 June 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Gusset-tube connections in transmission towers.
equivalent model employed by Kim [17] was adopted to analyse the behaviours of K-joints in this paper. The equivalent chord wall moment (Mw ) can also be simplified to forces on the annular plates (PR ), as seen in Li's research [20] on tube-gusset KT-joints with 1/4 annular plates. The relationship between Mw and PR is Mw = PR B , where B is the height of the gusset plate. JSTA [13] has given a set of formulas to calculate the capacity of 1/4 annular plates when they are subject to tensile or compressive forces. However, these formulas are too complicated and conservative. Formulas that have a more concise expression and can predict the capacity more accurately need to be proposed. In this paper, K-joints with 1/4 annular plates were analysed and compared with K-joints without annular plates. Full-scale experiments, ultimate strength and failure modes of K-joints with 1/4 annular plates were introduced and investigated. Then, numerical parametric analysis was conducted to obtain empirical formulas to predict the ultimate strength of K-joints. Finally, the proposed equations were compared with current guidelines.
The K-1 joints have two additional 1/4 annular plates (width R , thickness t r ) on both ends of the gusset plate compared with the K-0 joints. 7-mm fillet welds were applied in the connection between the annular plate and the gusset plate, between the gusset plate and the chord, as well as between the annular plate and the chord. In the test, both ends of the chord were arranged with endplates and rib plates for loading convenience (Fig. 2(b)). The K-1 specimens were equipped with eight evenly arranged strain gauges on the chord surface, positioned 50 mm below the bottom surface of the annular plate (Fig. 2(c)). The material properties were obtained from coupon tests. Each material type had three identical coupons. The average material properties are shown in Table 2. 2.2. Experimental installation of K-1 specimens In the experiment, static in-plane loading was conducted on the K-1 specimens (Fig. 3). At the loading end of the chord, one transitional device was inserted between the 10,000 kN hydraulic-servo actuator and the chord. Lateral bracing was used to ensure the stability of the experimental setup during the loading procedure. One end of the lateral bracing was linked to the transitional device and the other end was hinged to the lateral support. Thus, only the vertical displacement of the chord loading end is free. The support end of the chord was a spherical hinge, which was introduced in detail in Ref. [2]. The upper branch tube was compression loaded using a 2000 kN hydraulic jack and the lower branch tube was tensile loaded using a 1000 kN hydraulic jack.
2. Experimental design 2.1. Test connections Fig. 2(b) illustrates the full-scale test specimens of K-joints with 1/4 annular plates, abbreviated as “K-1 joints”. Detailed dimensions of the two identical K-1 joints in the test are listed in Table 1. To investigate the influence of annular plates on the ultimate strength, K-joints without annular plates (abbreviated as “K-0 joints”), shown in Fig. 2(a), were adopted. In this paper, K-0 and K-1 joints are both called “Kjoints”. From the top view of the K-joints, it is clear that both the K-0 and K-1 joints are symmetrical with the S-S cross-section. K-joints have one longitudinal plate (length B , thickness t1) that is coplanar with the chord axis. Two branch tubes are connected to the gusset plate via an inserted plate and bolts. These two branch tubes are on the same plane as the gusset plate and they have angles of 1 and 2 with the chord axis respectively. The intersection of two branch tube axes is on the chord axis.
2.3. Test loads Fig. 4(a) and Table 3 show the loads and boundary conditions of the K-1 specimens in the test. The static loading had two stages. First, the loads were proportionally applied to the chord and the two branch tubes at a specific value. Then, the chord load remained constant and both branches continued loading until connection failure. In the test, due to the influence of the lateral bracing, the boundary conditions of 2
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Fig. 2. Dimension diagrams of K-0 and K-1 joints.
the K-1 joint only had vertical freedom on the loading end. As for the support end, it can be seen as a hinge. In the numerical analysis, the support was a hinge and only the lateral displacement was restrained at the loading end. The branch forces in the K-joints can be simplified into the equivalent models in Fig. 4(b). The branch forces are resolved into a vertical load (Pv ), a horizontal load (Ph ) and a chord wall moment (Mw ) at the intersection of the main tube face and the gusset plate. The chord wall moment can also be simplified to loads on the annular plates (PR,1, PR,2 ). The value of Ph is usually zero. The equivalent force calculations are shown in Fig. 4(b).
Table 2 Average material properties of the K-1 specimens. Members
Steel grade
E (GPa)
Chord Gusset plate Annular plate
Q345 Q345 Q345
191 210 204
0.26 0.28 0.3
f y (MPa)
fu (MPa)
Elongation (%)
470 350 480
600 602 608
26.7 27.5 25.0
Note: the properties of the weld matched those of the chord.
lower of N1 and N3% . N1 usually corresponds to chord buckling and is determined by the first yield point in the load-displacement curve. N3% is generally related to chord plastification and is the load that corresponds to the 3%d 0 deformation limit [15,21,22]. As shown in Fig. 5, the transverse deformation ( ) of the chord can be calculated as and BB = B + B (for a detailed introduction please refer to Ref. [2]). The corresponding load for which the transverse deformation first reaches 3%d 0 is defined as the ultimate load. In the parametric analysis of K-1 joints, the annular plates may present two conditions (Fig. 6), no apparent deformation (Mode A) or visible deformation (Mode B). The capacity of the connection will be different when the condition of the annular plates changes.
3. Resistance definition of K-joints When the strengths of braces, gusset plates and weld are guaranteed, the main failure modes for K-joints are usually chord plastification, chord buckling, and the combination of these. Punching shear failure was not considered in this paper because the chord in transmission towers always has small diameter-to-thickness ratios and large axial compression ratios. A detailed explanation can be found in Ref. [2]. The ultimate strength of the plate-to-CHS connection is set to be the Table 1 Dimensions of the K-1 specimens in the test. Specimens
K-1-1 K-1-2
Chord (mm)
Gusset plate (mm)
Annular plate (mm)
Angle (° )
d0
t0
L0
L1
L2
L1
L2
B
t1
R
tr
1
2
273 273
6 6
1801 1800
850 850
951 950
268 266
436 435
614 615
12.0 12.2
60 60
6 6
44 44
55 55
3
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Fig. 3. Experimental installation of K-1 specimens.
4. FE modelling and experimental results
computation, the universal software ABAQUS [23,24] and Python programming language were used. ABAQUS is powerful in nonlinear analysis, and Python cuts the labour of modelling. Brick element C3D8I was applied to the connection and material modelling. Nonlinear material properties were applied, and large deformation was switched on. The ABAQUS/Standard module was employed. The Static/general procedure and full Newton solver were adopted for the material
Experimental and theoretical analysis is vital in the studies of Kjoint behaviour. However, due to financial restrictions on the experiments and ambiguous hypothesis in theoretical studies, empirical analysis serves as a bridge between the experimental research and theoretical analysis. As the empirical analysis requires a great deal of
Fig. 4. Loads of the K-joints. 4
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Table 3 K-1 specimen loading. Specimens
Stages
N (kN)
F1 (kN)
F2 (kN)
K-1-1 and K-1-2
Stage 1 Stage 2
1092 1092
276 720
234 610
Notes: In Stage 2, N is constant. F1 and F2 are accumulated values. The load directions are shown in Fig. 4(a).
analysis. Two kinds of algorithms (Static/general and Static/riks) were used in the connection calculation. 4.1. FE modelling of K-1 joints Fig. 6. Annular plate Conditions.
As the K-1 joints are symmetrical, only half of the joints were established. Symmetric loading and boundary conditions were applied on the joint, shown in Fig. 7. The gusset plate had six elements in thickness while the other members had three. The meshes were refined on both ends of the gusset plate. The 7-mm fillet welds were duplicated in the FE models. A 1-mm gap was built between the gusset plate and the chord, between the gusset plate and the annular plate, and between the annular plate and the chord. In this way, forces will be transferred through the welds. The corner of the gusset plate was cut off for the K-1 joints. To validate the test results more precisely, the branches, the endplates and the rib plates were also built in the FE models of the K-1 joints. Reference points (RP1 to RP4) were established at each position of loading or support. These reference points were coupled with nodes at the corresponding position. Loads and boundary conditions were applied to the reference points, as is illustrated in Fig. 7. In the parametric analysis, the equivalent FE model in Fig. 7(b) was adopted.
load-strain curves of the strain gauges were compared. From the loadstrain curves shown in Fig. 8(c), it is obvious that the initial stiffness is the same between the two test specimens, and between the test and FE results. Fig. 8(b) shows that the axial strain distribution along the chord circumference also presents good agreement between the two test specimens, as well as between the FE model and the test specimens. From Fig. 8(a), the load-displacement curve of the equivalent model (shown in Fig. 7(b)) agrees quite well with that of the FE model with braces (shown in Fig. 7(a)). The failure modes of the two K-1 specimens are both chord buckling (Fig. 9), which appears at the tensile end of the gusset plate. Thus, the ultimate capacities of the two K-1 specimens are not controlled by the 3%d 0 deformation limit. The numerical model has the same failure modes. A “push-pull” deformation mechanism can be found in the K-1 joints. At the 2-2 cross section in Fig. 9(b), the chord is close to an elliptical shape. At the 1-1 cross section, the chord is almost squashed flat by the annular plate, but the shape of the chord is still close to circular. Kim's [17] test specimens were adopted to further validate the FE model correction, which was done in Ref. [2]. The comparison proved that the FE models can predict the ultimate capacities of Kim's specimens well. Meanwhile, the failure modes of the FE models match well with those of the test specimens (Fig. 10(a and b)). The “push-pull” deformation mechanism of the K-0 joints is even more obvious than that of the K-1 joints. At the compressive branch end, the chord wall has a
4.2. Validation of FE models with experimental results The two K-1 test specimens have the same ultimate capacity (Nsum )—1960 kN, while the ultimate capacity of the FE model with braces is 2160 kN—10% larger than the test results because the test loading stopped when buckling appeared in the specimen. Then the loads remained unchanged until the joint failed. Another phenomenon is that the initial stiffness of the FE curve differs from that of the test results. This difference might have been caused by the precision of the displacement meter. (Fig. 8(a)). To further validate the initial stiffness,
Fig. 5. Measurement of the chord wall deformation.
5
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Fig. 7. FE models of K-joints.
“heart” shape; at the other end, it has a “water drop” shape (Fig. 10(c)). It is obvious that annular plates can prevent transverse deformation effectively. Thus, the validation in this section proves that the FE models are reliable.
load ratio ( = N /Ny ), chord thickness (t 0 ), gusset plate length (B ), chord diameter (d 0 ), annular plate width (R ), annular plate thickness (t r ), chord yield strength (f y ), and annular plate yield strength (fry ). In the parametric analysis, only one of the parameters changed in values, while the other parameters remained unchanged. Then, the ultimate strength equations for Mw, u and Pv, u were derived. A chord length of L1 = L 2 = 3d 0 was applied in the parametric analysis below. In the analysis, the gusset plate and the weld were guaranteed to be safe. Some parametric ranges applied to the transmission towers are
5. Parametric studies A great deal of parametric studies were conducted, including axial
Fig. 8. Validation between the test and FE results (K-1 joints).
6
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Fig. 9. Failure modes of the K-1 specimens.
stated below:
In Figs. 11–15, the graphs of the K-1 joints and K-0 joints were put together corresponding to capacity. Therefore, it is easy to find out the relationship between these two kinds of joints. First, the ultimate strength of the vertical component force (Pv, u ) was analysed under combined load N and Pv :
◆ The diameter-to-thickness ratio is limited to 20 d 0/ t0 50 [2]. ◆ In the guidelines [15,22], the gusset plate length to chord diameter ratio is 1 B /d 0 4 . While in transmission towers, it is usually 1 B /d 0 2.5. 1. ◆ The axial load ratio of the chord is restricted to 0 ◆ In transmission towers, constructional design is usually adopted for the annular plate dimensions. Nevertheless, parametric analysis was still carried out to determine the influence of the annular plate dimensions on the ultimate strength of the K-joints.
• The failure modes of the K-1 joints are mainly chord buckling. • From Figs. 11(b)–Fig. 15(b), the K-1 and K-0 joints have the same trend in all parameters concerning P . • The P capacity of K-1 joints is always a bit larger than that of K-0 joints. It can be concluded that annular plates barely influence P . • The chord axial load ratio ( ) has a decreasing linear influence on v, u
v, u
v, u
Pv, u as
increases.
Fig. 10. Failure modes of the K-0 joints.
7
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Fig. 11. Impact of
• All the other parameters (t , d , B and f ) affect P in an increasing linear way as the parametric value increases. • The annular plate width (R ) and thickness (t ) have no impact on the 0
0
v, u
y
r
value of Pv, u for K-1 joints (Fig. 16(b) and Fig. 17(b)).
•
Secondly, the ultimate moment capacity (Mw, u ) is analysed under combined load N and Mw :
• The M • • •
•
w, u capacity of K-1 joints is more complicated compared with that of K-0 joints. It is because the capacity of K-1 joint is different when the deformation condition of annular plates changes (shown in Fig. 6). The moment capacity of the K-1 joints is much larger than that of the K-0 joints with the same dimensions (Figs. 11 ~ 15). The chord axial load ( ) has a non-linear decreasing impact on Mw, u when increases. The slope of the curves changes at the point of = 0.7 . When the condition of the annular plates is Mode A, the moment capacity grows linearly with increases of the following parametric
values — t 0 , d 0 , B and fy . The moment capacity stays unchanged with increasing d 0 when Mode B appears. For the other three parameters (t 0 , B and fy ), the moment capacity increases linearly with increasing parameter values when Mode B dominates. With the increase of the annular plate width (R ) and thickness (t r ), Mode B appears first and the moment capacity increases. When R and t r reach a specific value, Mode A emerges and the moment capacity will remain constant. When R/ d 0 0.25 and tr / t0 0.75, deformation is mainly dominated by Mode A and this phenomenon is unaffected by the chord diameter-to-thickness ratio. This means the moment capacity of K-1 joints no longer changes when the dimension of the annular plates increases to a certain value. In this condition, blindly increasing the size of annular plates would be wasteful and futile.
6. Proposed equations for K-1 joints From the above studies, the ultimate strength formulas of the vertical component force (Pv, u ) and wall moment (Mw, u ) for the K-1 joints
Fig. 12. Impact of t 0
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Fig. 13. Impact of d 0
are derived as Eqs. (1)–(6), which were obtained using multiple nonlinear regression analysis and SPSS software. The yield strength (Ny ) is the ultimate axial load strength of the chord (Nu in Eq. (1)). Only the compressive axial load was investigated (Eq. (2)–(5)). The ultimate strength equations for the K-0 joints have been analysed in Ref. [2] and the equations are listed in Table 4.
e
(1)
Pv,u =
3.6(B /d 0) 0.35 (B /t 0) 0.2d 0 t 0 fy (1
Mw, u =
Mw, u (B) , 0
),
(2)
0.5
min(Mw, u (A) , Mw, u (B) ), 0. 5 <
Mw, u (A) = 5.6(B /d 0)
= N / Ny
0.28 (d
1.0
0.28Bt 2 (1 0 /t 0) 0 e
Mw, u (B) = 6.45B (t02 f y + 0.16Rtr fry ) 1
(0.7 2
(3)
1.7
) 2)
(6)
where all the symbols are explained in the symbols list and fy is the yield strength of the chord. The application of Eqs. (2)–(6) is limited to 20 d 0/ t0 50 , 1 B /d 0 2.5, 0 1. The proposed equations for K-1 joints were compared with the existing guidelines, as shown in Table 4. JSTA [13] offers equations for plate-to-CHS connections with 1/4 annular plates when the plate is subjected to bending moments or the annular plate subjected to axial forces. The equations in Q/GDW [14] are almost the same as those in JSTA [13]. JSTA [13] also considers punching shear failures based on ring plates. K-1 joints can also be seen as I-section to CHS chord T-type joints. Fortunately, EN 1993-1-8 [25] and ISO 14346 [26] have formulas for these kinds of joints. These two standards do not consider the condition that the annular plates have distinct out-of-plane deformation. The regression coefficients were achieved using multiple regression analyses between the equations and numerical data using SPSS software. From the curves in Fig. 18, it is obvious that the proposed equations have good agreement with the FE results considering both Mw, u and Pv, u .
◆ Proposed equations for K-1 joints (chord in compression):
Nu = Ny
= 345(fy /345) 0.8
(4) (5)
Fig. 14. Impact of B
9
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Fig. 15. Impact of fy
Fig. 16. Impact of R
Fig. 17. Impact of t r
10
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The predictions of JSTA [13] (Q/GDW [14]) and EN 1993-1-8 [25] are either much smaller or larger than the FE results and the calculated values from ISO 14346 [26] are larger than the FE results. The regression statistics shown in Table 4 prove that the proposed equations have a more precise prediction for the ultimate strengths of K-1 joints than those of the other guidelines. Moreover, the proposed equations have a more concise expression than JSTA [13] (Q/GDW [14]). In engineering practice, these three loads (N , Mw and Pv ) act on the K-joints simultaneously. In Eqs. (1)–(6), the axial load of the chord was considered in Mw, u and Pv, u . Thus, only an interaction equation of Mw and Pv needed to be derived (Eq. (7)). Because the vertical component load and the moment cancel each other out (Fig. 4(b)), the interaction curves of K-joints is larger than 1.0 in the coordinates. Comparing the interaction figures of K-0 and K-1 joints (Fig. 19), the scattered points of K-1 joints are more concentrated in the middle of the interaction curve than those of K-0 joints. This means the ultimate capacity of K-1 joints improves a lot compared with K-0 joints. As there are no more experimental data in the existing literature, the proposed equations were only compared with the test results in this paper (Table 5). This indicates that the proposed equations can predict the design resistance of K-1 joints well. The calculation procedures were introduced in Ref. [2]. Interaction equation for K-1 joints (Mw versus Pv ):
Not available
m4
1.65m3n + 3.03m2n2
2.68mn3 + n4 = 1,
(m=Mw / Mw,u, n= Pv / Pv,u)
(7)
7. Effect of annular plates in K-1 joints
) 0.25Mw, u (P) = 0.82B t 0d 0 fy
The effect of the annular plates can be regarded as the distinguishing features between K-0 and K-1 joints. Based on transmission towers, these features can be summarized as follows:
• From Table 4, the ultimate vertical component capacity (P
0.2
(1
•
d0 2t 0
( )
• •
(Punching shear failure)
Mw, u = min(Mw, u (A), Mw, u (P) ) Mw, u (A) = 9. 68Bt02 f y
shear failure)
•
• ISO 14346 [26]
> 0.4
0.3 (1 + ))Mw, u (P) = 0.82B t 0d 0 fy (Punching Mw, u = min(Mw, u (A), Mw, u (P) ) Mw, u (A) = 14Bt 02 f y (1 EN 1993-1-8 [25]
0.67)2,
0.4 = N /Ny
(1.67 × 1
1.0,
t0)/2 + tr KN = Be = 1.52 t0 (d0 Be t 0 , Rtr
K=
Mw, u (P) = 2Rtr fry KN / 3 (Punching shear failure)
Mw, u (B) = 4 2 BR2tr fry KN /d 0 (K > 1.0 )
1.0 ) 2K2) KN / d0 (K Mw, u (B) = 2 2 BR2tr fry (1 + 2K
Mw, u = min(Mw, u (A), Mw, u (B) , Mw, u (P) )Mw, u (A) = 21Bt 02 fy KN / 2
1
= 345(fy e
Mw, u = 6.35(B /d 0)0.24Bt 02 fy KN KN =
Proposed for K-0 joints [2]
JSTA [13] (Q/GDW [14])
1 2,
0
(0.7 0.28Bt 2 (1 0 e 0 /t 0) 0.28 (d
/345)0.8
Mw, u (A) = 5.6(B /d0)
0.5
min(Mw,u (A) , Mw, u (B) ), 0. 5 <
Mw,u (B) , 0
Mw, u = Proposed for K-1 joints
Mw, u (kN.m) Equations
Table 4 Proposed equations for K-joints and formulas in the guidelines.
1.0
1.7 )2) Mw, u (B) = 6.45B (t02 f y + 0.16Rtr fry ) 1
2
Not available
Not available
0 t 0 f y KN K N
0 t 0 f y KN K N
=
=1
,0
0.92 0.8 , 0 1 , 0.4 <
1
1
= N /Ny
0.4
Mw, u (A) : 0.943
0.2d
Pv,u = 3.5(B/d0)0.35 (B/t 0)
Pv,u : 0.998
Mw, u (A) : 0.866 Mw, u (B) : 0.637
Mw, u (A) : 0.999 Mw, u (B) : 0.994 0.2d
Pv,u = 3.6(B/d0)0.35 (B/t 0)
Mw, u : 0.998 Pv,u : 0.998
Statistics R2
Pv,u (kN)
F. Li, et al.
v, u ) of K-1 joints is about 3% larger than that of K-0 joints. Therefore, the annular plates do not have an influence on Pv, u . From Table 4, the ultimate moment capacity (Mw, u ) of K-1 joints is higher than that of K-0 joints. However, it is difficult to measure this difference using a specific value because of the influence of annular plates. Nevertheless, a general range can be derived—the ultimate moment capacity of K-1 joints is about 1.3–2.3 times that of K-0 joints. The ultimate strength capacity (under the combined action of N , Mw and Pv ) of K-1 joints is about 1.3–2.3 times that of K-0 joints, which can also be observed from the interaction curves in Fig. 19. Annular plates can also significantly restrain the transverse deformation of the chord. First, annular plates can increase the contact area of the gusset plate with the chord. Thus, stress concentration can be effectively prevented. Second, annular plates play the role of a “hoop”, which can strengthen the integrity of the connections. In the parametric analysis, the following conclusions have been made: ① Under combined N and Mw , the Mw,u capacity will remain unchanged when R/ d 0 0.25 and tr / t0 0.75. ② Under combined action of N and Pv or combined actions of N , Mw and Pv , annular plates do not impact Pv,u . Moreover, in transmission towers, the chord axial load ratio is usually higher than 50%, which means the value of Mw is relatively small. Thus, the dimensions of the annular plates do not need to be very large and will meet the requirements if they satisfy the minimum constructional dimensions defined below. The minimum constructional dimensions for annular plates are: R/ d 0 = 0.25 and tr / t0 = 0.75.
8. Conclusions In this paper, the failure modes and ultimate strength capacities of gusset-tube K-joints strengthened by 1/4 annular plates (K-1 joints) 11
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Fig. 18. Comparison of the equations for K-1 joints.
Fig. 19. Pv versus Mw interaction. Table 5 Validation of proposed equations for K-1 joints with test results. Members
Mw, u (kN·m)
Pv, u (kN)
d= d 0/2
K-1 specimens
126.17
785.31
0.1365
e (m)
Pv (kN)
N (kN)
Nu,sum(P) (kN)
Nu,sum(T) (kN)
Nu,sum(P) Nu,sum(T)
975.89
1092.00
2067.89
1960.00
1.05
Note: Nu,sum(P) , Nu,sum(T) : P-Proposed equation results, T-Test results. 12
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were studied. Full-scale experiments were carried out on K-1 joints. Kjoints without annular plates (K-0 joints) were used to compare with the K-1 joints. Equivalent models were adopted in the parametric analysis. Then ultimate capacity equations of Pv,u and Mw,u were proposed. Annular plates do not have much impact on Pv,u . However, they can appreciably improve the moment capacity (Mw,u ). The ultimate moment capacity (Mw,u ) of K-1 joints is about 1.3–2.3 times that of K-0 joints. Annular plates can also restrict the transverse deformation of the chord. In the end, the structural design of the annular plates was given.
transverse deformation of the chord wall axial load ratio of the chord (= N / Ny ) References [1] J. Wardenier, J.A. Packer, X.L. Zhao, et al., Hollow Sections in Structural Applications, Bouwen met Staal, The Netherlands, 2010. [2] F. Li, H.Z. Deng, X.Y. Hu, Experimental and numerical investigation into ultimate capacity of longitudinal plate-to-circular hollow section K- and DK-joints in transmission towers, Thin-Walled Struct. 143 (2019), https://doi.org/10.1016/j.tws. 2019.106240. [3] B. Lv, Z. Chen, H. Li, et al., Research on the ultimate bearing capacity for steel tubular transmission tower's joints with annular plate, Appl. Mech. Mater. 166 (2012) 379–384. [4] A.P. Voth, J.A. Packer, Branch plate-to-circular hollow structural section connections. I: experimental investigation and finite-element modeling, J. Struct. Eng. 138 (8) (2012) 995–1006. [5] A.P. Voth, J.A. Packer, Branch plate-to-circular hollow structural section connections. II: X-type parametric numerical study and design, J. Struct. Eng. 138 (8) (2012) 1007–1018. [6] A.P. Voth, J.A. Packer, Numerical study and design of skewed X-type branch plateto-circular hollow section connections, J. Constr. Steel Res. 68 (1) (2012) 1–10. [7] A.P. Voth, J.A. Packer, Numerical study and design of T-type branch plate-to-circular hollow section connections, Eng. Struct. 41 (2012) 477–489. [8] W.-B. Kim, K.-J. Shin, H.-H. Choi, Study on the ultimate strength of gusset platecircular hollow section (CHS) joint, J. Korean Soc. Steel Const. 23 (5) (2011) 523–533. [9] W.-B. Kim, K.-J. Shin, H.-D. Lee, et al., Strength equations of longitudinal plate-tocircular hollow section (CHS) joints, Int. J. Steel Struct. 15 (2) (2015) 499–505. [10] L.M. Zapata, C. Graciano, D.G. Zapata-Medina, Ultimate strength of transversal Tbranch plate-to-CHS connections under compression, Thin-Walled Struct. 112 (2017) 92–97. [11] Z. Yang, H. Deng, X. Hu, Strength of longitudinal X-type plate-to-circular hollow section (CHS) connections reinforced by external ring stiffeners, Thin-Walled Struct. 131 (2018) 500–518. [12] J. Wardenier, J. Packer, R. Puthli, Simplified design equations for Plate‐to‐CHS T and X joints for use in codes, Steel Const. 11 (2) (2018) 146–161. [13] The Japanese Steel Tubular Association, Electricity Transmitting Steel Tubular Tower Manufacture Norm, (2015) [in Japanese]. Japan. [14] State Grid Corporation of China, Technical Regulation for Conformation Design of Steel Tubular Towers of Transmission Lines, (2015) [in Chinese], Q/GDW 13912015. China. [15] Design Guide for Circular Hollow Section (CHS) Joints under Predominantly Static Loading, second ed., CIDECT, Geneva (Switzerland), 2008. [16] J. Packer, D. Sherman, Hollow Structural Section Connections, American Institute of Steel Construction, USA, 2012. [17] W.-B. Kim, Ultimate strength of tube–gusset plate connections considering eccentricity, Eng. Struct. 23 (11) (2001) 1418–1426. [18] S. Qu, X. Wu, Q. Sun, Experimental study and theoretical analysis on the ultimate strength of high-strength-steel tubular K-Joints, Thin-Walled Struct. 123 (2018) 244–254. [19] S.-Z. Qu, X.-H. Wu, Q. Sun, Experimental and numerical study on ultimate behaviour of high-strength steel tubular K-joints with external annular steel plates on chord circumference, Eng. Struct. 165 (2018) 457–470. [20] X.L. Li, L. Zhang, X.M. Xue, et al., Prediction on ultimate strength of tube-gusset KTjoints stiffened by 1/4 ring plates through experimental and numerical study, ThinWalled Struct. 123 (2018) 409–419. [21] L. Lu, G. De Winkel, Y. Yu, et al., Deformation Limit for the Ultimate Strength of Hollow Section Joints, 6th International Symposium on Tubular Structures, (1994), pp. 341–347. [22] International Institute of Welding (IIW), Static Design Procedure for Welded Hollow Section Joints—Recommendations, (2012) IIW Doc. XV-1402-12. Paris. [23] Abaqus 6.14 [Computer Software], Dassault Systèmes, USA, 2014. [24] Dassault Systèmes Simulia Corp, ABAQUS 6.14 Online Documentation, (2014) USA. [25] Eurocode 3, Design of Steel Structures - Part 1-8: Design of Joints, BS EN 1993-1-8, (2010) Brussels, Belgium. [26] ISO 14346, 2013: Static Design Procedure for Welded Hollow-Section Joints Recommendations, (2013) Geneva, Switzerland.
Conflicts of interest The authors declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgements The authors gratefully acknowledge the National Natural Science Foundation of China (Grant No. 51578421). Symbols
d0 t0 B t1 R tr L0 1
E fy
fu fry N Nu Ny Nsum Nu,sum
F1 Pv Pv,u
Ph PR1 Mw Mw, u
external diameter of the chord chord wall thickness gusset plate length gusset plate thickness annular plate width annular plate thickness L1, L 2 , L1, L 2 chord length shown in Fig. 2(c) 2 angle between the branch tubes and the chord modulus of elasticity poisson ratio yield strength in Table 2 or yield strength of the chord in the equations ultimate strength yield strength of the annular plate axial compressive load of the chord ultimate axial load capacity of the chord yield axial load capacity of the chord sum of all the vertical loads (=N +Pv ) ultimate vertical capacity when K- or DK-joints are subjected to combined loads (N and Pv ; or N and Mw ; or N , Pv and Mw ) F2 axial load of the branch tubes vertical component load of the brace loads (= F1 cos 1 + F2 cos 2 ) ultimate vertical load applied on the gusset plate when K- or DK-joints are subjected to combined load N and Pv transverse component load of the brace loads (=F1 sin 1 F2 sin 2 ) PR2 equivalent component load on the annular plate (=Mw /B ) equivalent bending moment on the gusset plate (= Pv d 0/2 ) ultimate bending moment applied on the gusset plate when K- or DK-joints are subjected to combined load N and Mw
13
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Full length article
Design resistance of longitudinal gusset-tube K-joints with 1/4 annular plates in transmission towers
T
Fang Li, Hong-zhou Deng*, Xiao-yi Hu Department of Structural Engineering, Tongji University, Shanghai, 200092, China
ARTICLE INFO
ABSTRACT
Keywords: Gusset-tube connections K-joints Annular plates Design resistance Full-scale experiments Transmission towers
In this paper, the design resistance of longitudinal gusset-tube K-joints stiffened with 1/4 annular plates was studied. Full-scale experiments of the K-joints were carried out. The branch loads of the K-joints were equivalent to a vertical shear force (Pv ) and a bending moment (Mw ). Substantial parametric studies revealed that annular plates had little impact on the ultimate shear force capacity (Pv,u ). Nevertheless, they could significantly improve the ultimate moment capacity (Mw,u ) and restrain the chord wall deformation. Moreover, the ultimate moment capacity of K-joints reinforced with annular plates was about 1.3–2.3 times that of K-joints without annular plates. The minimum construction dimensions for annular plates were set to R / d0 = 0.25 and tr / t0 = 0.75.
1. Introduction In China, transmission towers are being widely built to meet the increasing need for electricity. Tubular steel towers are quite appropriate for long-span transmission towers due to the characteristics of the circular hollow section (CHS), of which the most significant are not having a weak axis and having the same performance in all directions [1]. In tubular steel transmission towers, longitudinal plate-to-CHS connections are famous for their easy fabrication, simple installation and efficient construction. Fig. 1 shows some examples of the application of gusset-tube connections in the tubular steel transmission towers. For plate-to-CHS connections, excessive transverse deformations are quite large and usually govern the nominal strength of the connections, resulting in connection failure before the material is fully utilized. Because of the negative influence of excessive transverse deformations on the ultimate strength, local reinforcements are usually adopted. Although fabricators do not favour these reinforcements, they can effectively improve the ultimate strength of plate-to-CHS connections [2,3]. A common method for reinforcement is to arrange annular plates on both ends of the gusset plate (shown in Fig. 1). In this paper, planar Kjoints strengthened by 1/4 annular plates were studied. Much research has been done on gusset-tube connections. Voth and Packer [4–7] did a battery of investigations into the ultimate strength of plate-to-CHS connections, including longitudinal T-type, transverse Ttype, longitudinal X-type, transverse X-type, and skewed X-type. Kim [8,9] investigated the capacity of longitudinal plate-to-tube X-joints when the plate was subjected to in-plane bending. Zapata et al. [10]
*
used the upper bound theorem of plastic collapse to estimate the capacity of T-type transverse plate-tube connections. Yang et al. [11] studied the behaviour of X-type plate-to-CHS connections stiffened by external annular plates when the plate was subjected to an in-plane bending moment. Wardenier et al. [12] did a short review of equations for plate-to-CHS connections in the existing codes and literature. In Fig. 1, it is evident that the connections applied in the transmission towers are usually spatial and have more than one gusset plate. Due to restrictions on experimental finance and conditions, there is almost no research on spatial gusset-tube connections. In practice, the planar K-joints that are shown in Fig. 1(d) are usually employed to analyse the spatial connections because planar behaviours are the basis of spatial behaviours. However, in the relevant standards [13–16], there are no methods to calculate the ultimate strength of the K-joints directly. Only methods to calculate the ultimate capacity of the gussettube connections when they are subjected to an in-plane bending moment (Mw ) are found. In the existing literature, there are only a few recordings of K-joints. Kim [17] studied the ultimate capacity of Kjoints without annular plates by simplifying the axial brace forces into a vertical component force (Pv ) and a chord wall moment (Mw ). Then interaction equations of the component force (Pv ), the chord wall moment (Mw ), and the axial chord force (N ) were proposed. Qu [18,19] predicted the ultimate strength of K-joints with and without annular plates using the virtual work principle and energy theory. His research all related to Q690 high strength steel. As the behaviours are complicated in K-joints stiffened by annular plates and there is no mature theory for these connections, the
Corresponding author. E-mail address: [email protected] (H.-z. Deng).
https://doi.org/10.1016/j.tws.2019.106271 Received 3 January 2019; Received in revised form 16 May 2019; Accepted 25 June 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Gusset-tube connections in transmission towers.
equivalent model employed by Kim [17] was adopted to analyse the behaviours of K-joints in this paper. The equivalent chord wall moment (Mw ) can also be simplified to forces on the annular plates (PR ), as seen in Li's research [20] on tube-gusset KT-joints with 1/4 annular plates. The relationship between Mw and PR is Mw = PR B , where B is the height of the gusset plate. JSTA [13] has given a set of formulas to calculate the capacity of 1/4 annular plates when they are subject to tensile or compressive forces. However, these formulas are too complicated and conservative. Formulas that have a more concise expression and can predict the capacity more accurately need to be proposed. In this paper, K-joints with 1/4 annular plates were analysed and compared with K-joints without annular plates. Full-scale experiments, ultimate strength and failure modes of K-joints with 1/4 annular plates were introduced and investigated. Then, numerical parametric analysis was conducted to obtain empirical formulas to predict the ultimate strength of K-joints. Finally, the proposed equations were compared with current guidelines.
The K-1 joints have two additional 1/4 annular plates (width R , thickness t r ) on both ends of the gusset plate compared with the K-0 joints. 7-mm fillet welds were applied in the connection between the annular plate and the gusset plate, between the gusset plate and the chord, as well as between the annular plate and the chord. In the test, both ends of the chord were arranged with endplates and rib plates for loading convenience (Fig. 2(b)). The K-1 specimens were equipped with eight evenly arranged strain gauges on the chord surface, positioned 50 mm below the bottom surface of the annular plate (Fig. 2(c)). The material properties were obtained from coupon tests. Each material type had three identical coupons. The average material properties are shown in Table 2. 2.2. Experimental installation of K-1 specimens In the experiment, static in-plane loading was conducted on the K-1 specimens (Fig. 3). At the loading end of the chord, one transitional device was inserted between the 10,000 kN hydraulic-servo actuator and the chord. Lateral bracing was used to ensure the stability of the experimental setup during the loading procedure. One end of the lateral bracing was linked to the transitional device and the other end was hinged to the lateral support. Thus, only the vertical displacement of the chord loading end is free. The support end of the chord was a spherical hinge, which was introduced in detail in Ref. [2]. The upper branch tube was compression loaded using a 2000 kN hydraulic jack and the lower branch tube was tensile loaded using a 1000 kN hydraulic jack.
2. Experimental design 2.1. Test connections Fig. 2(b) illustrates the full-scale test specimens of K-joints with 1/4 annular plates, abbreviated as “K-1 joints”. Detailed dimensions of the two identical K-1 joints in the test are listed in Table 1. To investigate the influence of annular plates on the ultimate strength, K-joints without annular plates (abbreviated as “K-0 joints”), shown in Fig. 2(a), were adopted. In this paper, K-0 and K-1 joints are both called “Kjoints”. From the top view of the K-joints, it is clear that both the K-0 and K-1 joints are symmetrical with the S-S cross-section. K-joints have one longitudinal plate (length B , thickness t1) that is coplanar with the chord axis. Two branch tubes are connected to the gusset plate via an inserted plate and bolts. These two branch tubes are on the same plane as the gusset plate and they have angles of 1 and 2 with the chord axis respectively. The intersection of two branch tube axes is on the chord axis.
2.3. Test loads Fig. 4(a) and Table 3 show the loads and boundary conditions of the K-1 specimens in the test. The static loading had two stages. First, the loads were proportionally applied to the chord and the two branch tubes at a specific value. Then, the chord load remained constant and both branches continued loading until connection failure. In the test, due to the influence of the lateral bracing, the boundary conditions of 2
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Fig. 2. Dimension diagrams of K-0 and K-1 joints.
the K-1 joint only had vertical freedom on the loading end. As for the support end, it can be seen as a hinge. In the numerical analysis, the support was a hinge and only the lateral displacement was restrained at the loading end. The branch forces in the K-joints can be simplified into the equivalent models in Fig. 4(b). The branch forces are resolved into a vertical load (Pv ), a horizontal load (Ph ) and a chord wall moment (Mw ) at the intersection of the main tube face and the gusset plate. The chord wall moment can also be simplified to loads on the annular plates (PR,1, PR,2 ). The value of Ph is usually zero. The equivalent force calculations are shown in Fig. 4(b).
Table 2 Average material properties of the K-1 specimens. Members
Steel grade
E (GPa)
Chord Gusset plate Annular plate
Q345 Q345 Q345
191 210 204
0.26 0.28 0.3
f y (MPa)
fu (MPa)
Elongation (%)
470 350 480
600 602 608
26.7 27.5 25.0
Note: the properties of the weld matched those of the chord.
lower of N1 and N3% . N1 usually corresponds to chord buckling and is determined by the first yield point in the load-displacement curve. N3% is generally related to chord plastification and is the load that corresponds to the 3%d 0 deformation limit [15,21,22]. As shown in Fig. 5, the transverse deformation ( ) of the chord can be calculated as and BB = B + B (for a detailed introduction please refer to Ref. [2]). The corresponding load for which the transverse deformation first reaches 3%d 0 is defined as the ultimate load. In the parametric analysis of K-1 joints, the annular plates may present two conditions (Fig. 6), no apparent deformation (Mode A) or visible deformation (Mode B). The capacity of the connection will be different when the condition of the annular plates changes.
3. Resistance definition of K-joints When the strengths of braces, gusset plates and weld are guaranteed, the main failure modes for K-joints are usually chord plastification, chord buckling, and the combination of these. Punching shear failure was not considered in this paper because the chord in transmission towers always has small diameter-to-thickness ratios and large axial compression ratios. A detailed explanation can be found in Ref. [2]. The ultimate strength of the plate-to-CHS connection is set to be the Table 1 Dimensions of the K-1 specimens in the test. Specimens
K-1-1 K-1-2
Chord (mm)
Gusset plate (mm)
Annular plate (mm)
Angle (° )
d0
t0
L0
L1
L2
L1
L2
B
t1
R
tr
1
2
273 273
6 6
1801 1800
850 850
951 950
268 266
436 435
614 615
12.0 12.2
60 60
6 6
44 44
55 55
3
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Fig. 3. Experimental installation of K-1 specimens.
4. FE modelling and experimental results
computation, the universal software ABAQUS [23,24] and Python programming language were used. ABAQUS is powerful in nonlinear analysis, and Python cuts the labour of modelling. Brick element C3D8I was applied to the connection and material modelling. Nonlinear material properties were applied, and large deformation was switched on. The ABAQUS/Standard module was employed. The Static/general procedure and full Newton solver were adopted for the material
Experimental and theoretical analysis is vital in the studies of Kjoint behaviour. However, due to financial restrictions on the experiments and ambiguous hypothesis in theoretical studies, empirical analysis serves as a bridge between the experimental research and theoretical analysis. As the empirical analysis requires a great deal of
Fig. 4. Loads of the K-joints. 4
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Table 3 K-1 specimen loading. Specimens
Stages
N (kN)
F1 (kN)
F2 (kN)
K-1-1 and K-1-2
Stage 1 Stage 2
1092 1092
276 720
234 610
Notes: In Stage 2, N is constant. F1 and F2 are accumulated values. The load directions are shown in Fig. 4(a).
analysis. Two kinds of algorithms (Static/general and Static/riks) were used in the connection calculation. 4.1. FE modelling of K-1 joints Fig. 6. Annular plate Conditions.
As the K-1 joints are symmetrical, only half of the joints were established. Symmetric loading and boundary conditions were applied on the joint, shown in Fig. 7. The gusset plate had six elements in thickness while the other members had three. The meshes were refined on both ends of the gusset plate. The 7-mm fillet welds were duplicated in the FE models. A 1-mm gap was built between the gusset plate and the chord, between the gusset plate and the annular plate, and between the annular plate and the chord. In this way, forces will be transferred through the welds. The corner of the gusset plate was cut off for the K-1 joints. To validate the test results more precisely, the branches, the endplates and the rib plates were also built in the FE models of the K-1 joints. Reference points (RP1 to RP4) were established at each position of loading or support. These reference points were coupled with nodes at the corresponding position. Loads and boundary conditions were applied to the reference points, as is illustrated in Fig. 7. In the parametric analysis, the equivalent FE model in Fig. 7(b) was adopted.
load-strain curves of the strain gauges were compared. From the loadstrain curves shown in Fig. 8(c), it is obvious that the initial stiffness is the same between the two test specimens, and between the test and FE results. Fig. 8(b) shows that the axial strain distribution along the chord circumference also presents good agreement between the two test specimens, as well as between the FE model and the test specimens. From Fig. 8(a), the load-displacement curve of the equivalent model (shown in Fig. 7(b)) agrees quite well with that of the FE model with braces (shown in Fig. 7(a)). The failure modes of the two K-1 specimens are both chord buckling (Fig. 9), which appears at the tensile end of the gusset plate. Thus, the ultimate capacities of the two K-1 specimens are not controlled by the 3%d 0 deformation limit. The numerical model has the same failure modes. A “push-pull” deformation mechanism can be found in the K-1 joints. At the 2-2 cross section in Fig. 9(b), the chord is close to an elliptical shape. At the 1-1 cross section, the chord is almost squashed flat by the annular plate, but the shape of the chord is still close to circular. Kim's [17] test specimens were adopted to further validate the FE model correction, which was done in Ref. [2]. The comparison proved that the FE models can predict the ultimate capacities of Kim's specimens well. Meanwhile, the failure modes of the FE models match well with those of the test specimens (Fig. 10(a and b)). The “push-pull” deformation mechanism of the K-0 joints is even more obvious than that of the K-1 joints. At the compressive branch end, the chord wall has a
4.2. Validation of FE models with experimental results The two K-1 test specimens have the same ultimate capacity (Nsum )—1960 kN, while the ultimate capacity of the FE model with braces is 2160 kN—10% larger than the test results because the test loading stopped when buckling appeared in the specimen. Then the loads remained unchanged until the joint failed. Another phenomenon is that the initial stiffness of the FE curve differs from that of the test results. This difference might have been caused by the precision of the displacement meter. (Fig. 8(a)). To further validate the initial stiffness,
Fig. 5. Measurement of the chord wall deformation.
5
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Fig. 7. FE models of K-joints.
“heart” shape; at the other end, it has a “water drop” shape (Fig. 10(c)). It is obvious that annular plates can prevent transverse deformation effectively. Thus, the validation in this section proves that the FE models are reliable.
load ratio ( = N /Ny ), chord thickness (t 0 ), gusset plate length (B ), chord diameter (d 0 ), annular plate width (R ), annular plate thickness (t r ), chord yield strength (f y ), and annular plate yield strength (fry ). In the parametric analysis, only one of the parameters changed in values, while the other parameters remained unchanged. Then, the ultimate strength equations for Mw, u and Pv, u were derived. A chord length of L1 = L 2 = 3d 0 was applied in the parametric analysis below. In the analysis, the gusset plate and the weld were guaranteed to be safe. Some parametric ranges applied to the transmission towers are
5. Parametric studies A great deal of parametric studies were conducted, including axial
Fig. 8. Validation between the test and FE results (K-1 joints).
6
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Fig. 9. Failure modes of the K-1 specimens.
stated below:
In Figs. 11–15, the graphs of the K-1 joints and K-0 joints were put together corresponding to capacity. Therefore, it is easy to find out the relationship between these two kinds of joints. First, the ultimate strength of the vertical component force (Pv, u ) was analysed under combined load N and Pv :
◆ The diameter-to-thickness ratio is limited to 20 d 0/ t0 50 [2]. ◆ In the guidelines [15,22], the gusset plate length to chord diameter ratio is 1 B /d 0 4 . While in transmission towers, it is usually 1 B /d 0 2.5. 1. ◆ The axial load ratio of the chord is restricted to 0 ◆ In transmission towers, constructional design is usually adopted for the annular plate dimensions. Nevertheless, parametric analysis was still carried out to determine the influence of the annular plate dimensions on the ultimate strength of the K-joints.
• The failure modes of the K-1 joints are mainly chord buckling. • From Figs. 11(b)–Fig. 15(b), the K-1 and K-0 joints have the same trend in all parameters concerning P . • The P capacity of K-1 joints is always a bit larger than that of K-0 joints. It can be concluded that annular plates barely influence P . • The chord axial load ratio ( ) has a decreasing linear influence on v, u
v, u
v, u
Pv, u as
increases.
Fig. 10. Failure modes of the K-0 joints.
7
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Fig. 11. Impact of
• All the other parameters (t , d , B and f ) affect P in an increasing linear way as the parametric value increases. • The annular plate width (R ) and thickness (t ) have no impact on the 0
0
v, u
y
r
value of Pv, u for K-1 joints (Fig. 16(b) and Fig. 17(b)).
•
Secondly, the ultimate moment capacity (Mw, u ) is analysed under combined load N and Mw :
• The M • • •
•
w, u capacity of K-1 joints is more complicated compared with that of K-0 joints. It is because the capacity of K-1 joint is different when the deformation condition of annular plates changes (shown in Fig. 6). The moment capacity of the K-1 joints is much larger than that of the K-0 joints with the same dimensions (Figs. 11 ~ 15). The chord axial load ( ) has a non-linear decreasing impact on Mw, u when increases. The slope of the curves changes at the point of = 0.7 . When the condition of the annular plates is Mode A, the moment capacity grows linearly with increases of the following parametric
values — t 0 , d 0 , B and fy . The moment capacity stays unchanged with increasing d 0 when Mode B appears. For the other three parameters (t 0 , B and fy ), the moment capacity increases linearly with increasing parameter values when Mode B dominates. With the increase of the annular plate width (R ) and thickness (t r ), Mode B appears first and the moment capacity increases. When R and t r reach a specific value, Mode A emerges and the moment capacity will remain constant. When R/ d 0 0.25 and tr / t0 0.75, deformation is mainly dominated by Mode A and this phenomenon is unaffected by the chord diameter-to-thickness ratio. This means the moment capacity of K-1 joints no longer changes when the dimension of the annular plates increases to a certain value. In this condition, blindly increasing the size of annular plates would be wasteful and futile.
6. Proposed equations for K-1 joints From the above studies, the ultimate strength formulas of the vertical component force (Pv, u ) and wall moment (Mw, u ) for the K-1 joints
Fig. 12. Impact of t 0
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Fig. 13. Impact of d 0
are derived as Eqs. (1)–(6), which were obtained using multiple nonlinear regression analysis and SPSS software. The yield strength (Ny ) is the ultimate axial load strength of the chord (Nu in Eq. (1)). Only the compressive axial load was investigated (Eq. (2)–(5)). The ultimate strength equations for the K-0 joints have been analysed in Ref. [2] and the equations are listed in Table 4.
e
(1)
Pv,u =
3.6(B /d 0) 0.35 (B /t 0) 0.2d 0 t 0 fy (1
Mw, u =
Mw, u (B) , 0
),
(2)
0.5
min(Mw, u (A) , Mw, u (B) ), 0. 5 <
Mw, u (A) = 5.6(B /d 0)
= N / Ny
0.28 (d
1.0
0.28Bt 2 (1 0 /t 0) 0 e
Mw, u (B) = 6.45B (t02 f y + 0.16Rtr fry ) 1
(0.7 2
(3)
1.7
) 2)
(6)
where all the symbols are explained in the symbols list and fy is the yield strength of the chord. The application of Eqs. (2)–(6) is limited to 20 d 0/ t0 50 , 1 B /d 0 2.5, 0 1. The proposed equations for K-1 joints were compared with the existing guidelines, as shown in Table 4. JSTA [13] offers equations for plate-to-CHS connections with 1/4 annular plates when the plate is subjected to bending moments or the annular plate subjected to axial forces. The equations in Q/GDW [14] are almost the same as those in JSTA [13]. JSTA [13] also considers punching shear failures based on ring plates. K-1 joints can also be seen as I-section to CHS chord T-type joints. Fortunately, EN 1993-1-8 [25] and ISO 14346 [26] have formulas for these kinds of joints. These two standards do not consider the condition that the annular plates have distinct out-of-plane deformation. The regression coefficients were achieved using multiple regression analyses between the equations and numerical data using SPSS software. From the curves in Fig. 18, it is obvious that the proposed equations have good agreement with the FE results considering both Mw, u and Pv, u .
◆ Proposed equations for K-1 joints (chord in compression):
Nu = Ny
= 345(fy /345) 0.8
(4) (5)
Fig. 14. Impact of B
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Fig. 15. Impact of fy
Fig. 16. Impact of R
Fig. 17. Impact of t r
10
Thin-Walled Structures 144 (2019) 106271 Mw, u (A) : 0.466
The predictions of JSTA [13] (Q/GDW [14]) and EN 1993-1-8 [25] are either much smaller or larger than the FE results and the calculated values from ISO 14346 [26] are larger than the FE results. The regression statistics shown in Table 4 prove that the proposed equations have a more precise prediction for the ultimate strengths of K-1 joints than those of the other guidelines. Moreover, the proposed equations have a more concise expression than JSTA [13] (Q/GDW [14]). In engineering practice, these three loads (N , Mw and Pv ) act on the K-joints simultaneously. In Eqs. (1)–(6), the axial load of the chord was considered in Mw, u and Pv, u . Thus, only an interaction equation of Mw and Pv needed to be derived (Eq. (7)). Because the vertical component load and the moment cancel each other out (Fig. 4(b)), the interaction curves of K-joints is larger than 1.0 in the coordinates. Comparing the interaction figures of K-0 and K-1 joints (Fig. 19), the scattered points of K-1 joints are more concentrated in the middle of the interaction curve than those of K-0 joints. This means the ultimate capacity of K-1 joints improves a lot compared with K-0 joints. As there are no more experimental data in the existing literature, the proposed equations were only compared with the test results in this paper (Table 5). This indicates that the proposed equations can predict the design resistance of K-1 joints well. The calculation procedures were introduced in Ref. [2]. Interaction equation for K-1 joints (Mw versus Pv ):
Not available
m4
1.65m3n + 3.03m2n2
2.68mn3 + n4 = 1,
(m=Mw / Mw,u, n= Pv / Pv,u)
(7)
7. Effect of annular plates in K-1 joints
) 0.25Mw, u (P) = 0.82B t 0d 0 fy
The effect of the annular plates can be regarded as the distinguishing features between K-0 and K-1 joints. Based on transmission towers, these features can be summarized as follows:
• From Table 4, the ultimate vertical component capacity (P
0.2
(1
•
d0 2t 0
( )
• •
(Punching shear failure)
Mw, u = min(Mw, u (A), Mw, u (P) ) Mw, u (A) = 9. 68Bt02 f y
shear failure)
•
• ISO 14346 [26]
> 0.4
0.3 (1 + ))Mw, u (P) = 0.82B t 0d 0 fy (Punching Mw, u = min(Mw, u (A), Mw, u (P) ) Mw, u (A) = 14Bt 02 f y (1 EN 1993-1-8 [25]
0.67)2,
0.4 = N /Ny
(1.67 × 1
1.0,
t0)/2 + tr KN = Be = 1.52 t0 (d0 Be t 0 , Rtr
K=
Mw, u (P) = 2Rtr fry KN / 3 (Punching shear failure)
Mw, u (B) = 4 2 BR2tr fry KN /d 0 (K > 1.0 )
1.0 ) 2K2) KN / d0 (K Mw, u (B) = 2 2 BR2tr fry (1 + 2K
Mw, u = min(Mw, u (A), Mw, u (B) , Mw, u (P) )Mw, u (A) = 21Bt 02 fy KN / 2
1
= 345(fy e
Mw, u = 6.35(B /d 0)0.24Bt 02 fy KN KN =
Proposed for K-0 joints [2]
JSTA [13] (Q/GDW [14])
1 2,
0
(0.7 0.28Bt 2 (1 0 e 0 /t 0) 0.28 (d
/345)0.8
Mw, u (A) = 5.6(B /d0)
0.5
min(Mw,u (A) , Mw, u (B) ), 0. 5 <
Mw,u (B) , 0
Mw, u = Proposed for K-1 joints
Mw, u (kN.m) Equations
Table 4 Proposed equations for K-joints and formulas in the guidelines.
1.0
1.7 )2) Mw, u (B) = 6.45B (t02 f y + 0.16Rtr fry ) 1
2
Not available
Not available
0 t 0 f y KN K N
0 t 0 f y KN K N
=
=1
,0
0.92 0.8 , 0 1 , 0.4 <
1
1
= N /Ny
0.4
Mw, u (A) : 0.943
0.2d
Pv,u = 3.5(B/d0)0.35 (B/t 0)
Pv,u : 0.998
Mw, u (A) : 0.866 Mw, u (B) : 0.637
Mw, u (A) : 0.999 Mw, u (B) : 0.994 0.2d
Pv,u = 3.6(B/d0)0.35 (B/t 0)
Mw, u : 0.998 Pv,u : 0.998
Statistics R2
Pv,u (kN)
F. Li, et al.
v, u ) of K-1 joints is about 3% larger than that of K-0 joints. Therefore, the annular plates do not have an influence on Pv, u . From Table 4, the ultimate moment capacity (Mw, u ) of K-1 joints is higher than that of K-0 joints. However, it is difficult to measure this difference using a specific value because of the influence of annular plates. Nevertheless, a general range can be derived—the ultimate moment capacity of K-1 joints is about 1.3–2.3 times that of K-0 joints. The ultimate strength capacity (under the combined action of N , Mw and Pv ) of K-1 joints is about 1.3–2.3 times that of K-0 joints, which can also be observed from the interaction curves in Fig. 19. Annular plates can also significantly restrain the transverse deformation of the chord. First, annular plates can increase the contact area of the gusset plate with the chord. Thus, stress concentration can be effectively prevented. Second, annular plates play the role of a “hoop”, which can strengthen the integrity of the connections. In the parametric analysis, the following conclusions have been made: ① Under combined N and Mw , the Mw,u capacity will remain unchanged when R/ d 0 0.25 and tr / t0 0.75. ② Under combined action of N and Pv or combined actions of N , Mw and Pv , annular plates do not impact Pv,u . Moreover, in transmission towers, the chord axial load ratio is usually higher than 50%, which means the value of Mw is relatively small. Thus, the dimensions of the annular plates do not need to be very large and will meet the requirements if they satisfy the minimum constructional dimensions defined below. The minimum constructional dimensions for annular plates are: R/ d 0 = 0.25 and tr / t0 = 0.75.
8. Conclusions In this paper, the failure modes and ultimate strength capacities of gusset-tube K-joints strengthened by 1/4 annular plates (K-1 joints) 11
Thin-Walled Structures 144 (2019) 106271
F. Li, et al.
Fig. 18. Comparison of the equations for K-1 joints.
Fig. 19. Pv versus Mw interaction. Table 5 Validation of proposed equations for K-1 joints with test results. Members
Mw, u (kN·m)
Pv, u (kN)
d= d 0/2
K-1 specimens
126.17
785.31
0.1365
e (m)
Pv (kN)
N (kN)
Nu,sum(P) (kN)
Nu,sum(T) (kN)
Nu,sum(P) Nu,sum(T)
975.89
1092.00
2067.89
1960.00
1.05
Note: Nu,sum(P) , Nu,sum(T) : P-Proposed equation results, T-Test results. 12
Thin-Walled Structures 144 (2019) 106271
F. Li, et al.
were studied. Full-scale experiments were carried out on K-1 joints. Kjoints without annular plates (K-0 joints) were used to compare with the K-1 joints. Equivalent models were adopted in the parametric analysis. Then ultimate capacity equations of Pv,u and Mw,u were proposed. Annular plates do not have much impact on Pv,u . However, they can appreciably improve the moment capacity (Mw,u ). The ultimate moment capacity (Mw,u ) of K-1 joints is about 1.3–2.3 times that of K-0 joints. Annular plates can also restrict the transverse deformation of the chord. In the end, the structural design of the annular plates was given.
transverse deformation of the chord wall axial load ratio of the chord (= N / Ny ) References [1] J. Wardenier, J.A. Packer, X.L. Zhao, et al., Hollow Sections in Structural Applications, Bouwen met Staal, The Netherlands, 2010. [2] F. Li, H.Z. Deng, X.Y. Hu, Experimental and numerical investigation into ultimate capacity of longitudinal plate-to-circular hollow section K- and DK-joints in transmission towers, Thin-Walled Struct. 143 (2019), https://doi.org/10.1016/j.tws. 2019.106240. [3] B. Lv, Z. Chen, H. Li, et al., Research on the ultimate bearing capacity for steel tubular transmission tower's joints with annular plate, Appl. Mech. Mater. 166 (2012) 379–384. [4] A.P. Voth, J.A. Packer, Branch plate-to-circular hollow structural section connections. I: experimental investigation and finite-element modeling, J. Struct. Eng. 138 (8) (2012) 995–1006. [5] A.P. Voth, J.A. Packer, Branch plate-to-circular hollow structural section connections. II: X-type parametric numerical study and design, J. Struct. Eng. 138 (8) (2012) 1007–1018. [6] A.P. Voth, J.A. Packer, Numerical study and design of skewed X-type branch plateto-circular hollow section connections, J. Constr. Steel Res. 68 (1) (2012) 1–10. [7] A.P. Voth, J.A. Packer, Numerical study and design of T-type branch plate-to-circular hollow section connections, Eng. Struct. 41 (2012) 477–489. [8] W.-B. Kim, K.-J. Shin, H.-H. Choi, Study on the ultimate strength of gusset platecircular hollow section (CHS) joint, J. Korean Soc. Steel Const. 23 (5) (2011) 523–533. [9] W.-B. Kim, K.-J. Shin, H.-D. Lee, et al., Strength equations of longitudinal plate-tocircular hollow section (CHS) joints, Int. J. Steel Struct. 15 (2) (2015) 499–505. [10] L.M. Zapata, C. Graciano, D.G. Zapata-Medina, Ultimate strength of transversal Tbranch plate-to-CHS connections under compression, Thin-Walled Struct. 112 (2017) 92–97. [11] Z. Yang, H. Deng, X. Hu, Strength of longitudinal X-type plate-to-circular hollow section (CHS) connections reinforced by external ring stiffeners, Thin-Walled Struct. 131 (2018) 500–518. [12] J. Wardenier, J. Packer, R. Puthli, Simplified design equations for Plate‐to‐CHS T and X joints for use in codes, Steel Const. 11 (2) (2018) 146–161. [13] The Japanese Steel Tubular Association, Electricity Transmitting Steel Tubular Tower Manufacture Norm, (2015) [in Japanese]. Japan. [14] State Grid Corporation of China, Technical Regulation for Conformation Design of Steel Tubular Towers of Transmission Lines, (2015) [in Chinese], Q/GDW 13912015. China. [15] Design Guide for Circular Hollow Section (CHS) Joints under Predominantly Static Loading, second ed., CIDECT, Geneva (Switzerland), 2008. [16] J. Packer, D. Sherman, Hollow Structural Section Connections, American Institute of Steel Construction, USA, 2012. [17] W.-B. Kim, Ultimate strength of tube–gusset plate connections considering eccentricity, Eng. Struct. 23 (11) (2001) 1418–1426. [18] S. Qu, X. Wu, Q. Sun, Experimental study and theoretical analysis on the ultimate strength of high-strength-steel tubular K-Joints, Thin-Walled Struct. 123 (2018) 244–254. [19] S.-Z. Qu, X.-H. Wu, Q. Sun, Experimental and numerical study on ultimate behaviour of high-strength steel tubular K-joints with external annular steel plates on chord circumference, Eng. Struct. 165 (2018) 457–470. [20] X.L. Li, L. Zhang, X.M. Xue, et al., Prediction on ultimate strength of tube-gusset KTjoints stiffened by 1/4 ring plates through experimental and numerical study, ThinWalled Struct. 123 (2018) 409–419. [21] L. Lu, G. De Winkel, Y. Yu, et al., Deformation Limit for the Ultimate Strength of Hollow Section Joints, 6th International Symposium on Tubular Structures, (1994), pp. 341–347. [22] International Institute of Welding (IIW), Static Design Procedure for Welded Hollow Section Joints—Recommendations, (2012) IIW Doc. XV-1402-12. Paris. [23] Abaqus 6.14 [Computer Software], Dassault Systèmes, USA, 2014. [24] Dassault Systèmes Simulia Corp, ABAQUS 6.14 Online Documentation, (2014) USA. [25] Eurocode 3, Design of Steel Structures - Part 1-8: Design of Joints, BS EN 1993-1-8, (2010) Brussels, Belgium. [26] ISO 14346, 2013: Static Design Procedure for Welded Hollow-Section Joints Recommendations, (2013) Geneva, Switzerland.
Conflicts of interest The authors declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgements The authors gratefully acknowledge the National Natural Science Foundation of China (Grant No. 51578421). Symbols
d0 t0 B t1 R tr L0 1
E fy
fu fry N Nu Ny Nsum Nu,sum
F1 Pv Pv,u
Ph PR1 Mw Mw, u
external diameter of the chord chord wall thickness gusset plate length gusset plate thickness annular plate width annular plate thickness L1, L 2 , L1, L 2 chord length shown in Fig. 2(c) 2 angle between the branch tubes and the chord modulus of elasticity poisson ratio yield strength in Table 2 or yield strength of the chord in the equations ultimate strength yield strength of the annular plate axial compressive load of the chord ultimate axial load capacity of the chord yield axial load capacity of the chord sum of all the vertical loads (=N +Pv ) ultimate vertical capacity when K- or DK-joints are subjected to combined loads (N and Pv ; or N and Mw ; or N , Pv and Mw ) F2 axial load of the branch tubes vertical component load of the brace loads (= F1 cos 1 + F2 cos 2 ) ultimate vertical load applied on the gusset plate when K- or DK-joints are subjected to combined load N and Pv transverse component load of the brace loads (=F1 sin 1 F2 sin 2 ) PR2 equivalent component load on the annular plate (=Mw /B ) equivalent bending moment on the gusset plate (= Pv d 0/2 ) ultimate bending moment applied on the gusset plate when K- or DK-joints are subjected to combined load N and Mw
13