Numerical analysis and formulae for SCF reduction coefficients of CFRP-strengthened CHS gap K-joints

Engineering Structures 210 (2020) 110369

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Numerical analysis and formulae for SCF reduction coefficients of CFRPstrengthened CHS gap K-joints

T

Guowen Xua,b, Lewei Tonga,b, , Xiao-Ling Zhaoc, Haiming Zhoud, Fei Xud ⁎

a

State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China Department of Structural Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China c Department of Civil and Environmental Engineering, UNSW Sydney, NSW 2052, Australia d Shanghai Yichang Carbon Fibre Materials Co., Ltd., Shanghai 200081, China b

ARTICLE INFO

ABSTRACT

Keywords: Carbon fibre reinforced polymer CFRP strengthening Circular hollow section gap K-joint Finite element analysis SCF reduction coefficient Parametric formulae

The focus of this paper is on estimating stress concentration factor (SCF) reduction coefficients (ψ) of circular hollow section (CHS) gap K-joints strengthened with carbon fibre reinforced polymer (CFRP) sheets (CFRP-CHS K-joints), so that the SCFs can be calculated on the basis of the existing formulae for unstrengthened joints. Three-dimensional finite element models are developed for both CFRP-strengthened and unstrengthened CHS gap K-joints to calculate the SCFs, which are verified using previous test data. A parametric study is then carried out to explore the effects of various parameters (11 independent CFRP strengthening parameters and 5 basic geometric parameters describing the steel K-joints) on the efficiency of CFRP strengthening in terms of SCF reduction coefficient ψ. It is found that ψ is primarily affected by the CFRP relative reinforcement rates, the CFRP-to-steel tensile modulus ratios, the adhesive modulus, and non-dimensional geometric parameters. Through regression analyses, parametric formulae are finally developed for estimating ψ at six critical hot spots in CFRP-CHS K-joints under balanced axial loading. The proposed parametric formulae agree well with the experimental data.

1. Introduction Since the last century, welded tubular structures have been increasingly used in civil engineering, particularly after the 1950s, by which time the problems in the manufacturing, end preparation, and welding had been solved [1]. However, due to severe stress concentration and inevitable welding defects, fatigue is often experienced by welded tubular joints, especially those subjected to long-term cyclic loading. Due to aging and increasing service loads of these early structures, the proper strengthening of welded tubular joints is urgently needed worldwide [2]. Attributed to the excellent mechanical properties, strengthening techniques using carbon fibre reinforced polymer (CFRP) have been receiving increasing attention, not only for concrete structures [3,4] but also for steel structures [5,6]. It has been proved that bonding CFRP sheets or plates to a steel plate, beam, or welded plate-to-plate joint is an effective method of improving its fatigue performance [7–16,2]. Regarding welded tubular joints, however, very limited research on CFRP fatigue strengthening is available in the literature. Accordingly, this paper focuses on CFRP-strengthened tubular K-joints made of

⁎

circular hollow sections (CHS) (CFRP-CHS K-joints). In 2003 and 2006, Pantelides et al. [17] and Fam et al. [18] successively retrofitted cracked aluminium CHS K-joints using unidirectional CFRP and glass-fibre-reinforced polymer (GFRP) sheets. They found that the static ultimate capacity of the cracked K-joints could be effectively restored by wrapping FRP (CFRP or GFRP) sheets around the joints. In addition, Fam et al. [18] demonstrated that CFRP is more effective than GFRP. In 2007, the fatigue behaviour of GFRP-strengthened aluminium CHS K-joints was tested by Nadauld and Pantelides [19]. More recently, Fu et al. [20] proposed an improved method to strengthen steel CHS gap K-joints using CFRP sheets. However, the study by Fu et al. only dealt with the static ultimate capacity of such joints. In the hot spot stress method for fatigue design, stress concentration factors (SCF) are essential. Therefore, it is worthy to investigate the SCFs of CFRP-strengthened tubular joints. Sadat et al. [21,22] discussed SCFs of FRP-strengthened CHS T-joints under various brace loading via pure finite element (FE) analysis. They concluded that the SCFs could be reduced by up to 50% by bonding FRP to the joints, and the influences of several FRP parameters, such as fibre orientation, FRP

Corresponding author at: State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China. E-mail address: [email protected] (L. Tong).

https://doi.org/10.1016/j.engstruct.2020.110369 Received 3 December 2019; Received in revised form 9 February 2020; Accepted 10 February 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.

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thickness and wrapping length, were investigated Using the strengthening method by Fu et al. [20], Tong et al. [23] experimentally investigate the SCFs of CFRP-CHS K-joints. It was found that the maximum SCFs were reduced by about 15%–20%. However, due to the limited number of samples, a thorough understanding of the influences of various parameters on the CFRP strengthening effect is still required. In fatigue design of welded tubular joints, SCFs are generally calculated using existing SCF formulae. SCF formulae for unstrengthened CHS gap K-joints are readily available, such as those recommended by CIDECT [24] and Morgan and Lee [25]. However, no direct SCF formulae for CFRP-CHS K-joints are available at present. A feasible way to calculate the SCFs of a CFRP-CHS K-joint is to multiply the corresponding SCFs of an unstrengthened joint by the corresponding SCF reduction coefficients (ψ) caused by CFRP strengthening. Therefore, an understanding of how to calculate ψ is fundamental to solving the problem. A theoretical method based on equivalent section analysis and the local degree of bending (DoB) was proposed by Tong et al. [23] for estimating ψ. However, this theoretical method only considers local sectional changes, not changes in overall stiffness along the intersection line. Besides, it relies on the empirical assumption of strain distribution in different CFRP layers and calculation of the DoB values. This paper aims to develop new parametric formulae for ψ of CFRP-CHS K-joints based on extensive FE analyses, which consider both local and overall factors. The new formulae are expected to be more accurate and easier to use. In this paper, a three-dimensional (3D) FE model for a CFRP-CHS Kjoint is developed and verified using previous experimental results. Next, the verified FE modelling strategy is employed to study the influences of several parameters on the CFRP strengthening efficiency in terms of the SCF reduction coefficient (ψ). The parameters considered are the wrapping length, number of CFRP layers, tensile moduli and thicknesses of the CFRP sheets, thickness and modulus of the adhesive, and five basic geometric parameters describing the steel K-joints. Finally, parametric formulae for ψ of the CFRP-CHS K-joints subjected to balanced axial loading are obtained through regression analyses and verified using the test data.

been successfully applied in the analysis of hot spot stress on concretefilled tubular joints [26]. To balance the computational cost and accuracy, the meshes near the weld intersection line were refined (see Fig. 4(d): ‘Refined region’). The refined regions were set according to the extrapolation region for CHS tubular joints in CIDECT [24]. There was a transition region between the refined region and coarse region. The meshes in the transition region became progressively coarser away from the brace-chord junction. The mesh sizes were assigned according to a meshing sensitivity analysis. Overall, six layers of elements were constructed in the wall thickness direction. In the refined region, the mesh size along the intersection line was one third the brace wall thickness (t1/3); the mesh size along the extrapolation direction was min{D 6, t 6} , where D refers to the width of the extrapolation region (see Fig. 4(e)), and t refers to the corresponding wall thickness (t0 or t1, see Fig. 2). For the specimens, full penetration welds were adopted to connect the braces and the chord. Following the AWS specification D1.1 [27] and Chinese standard JGJ 81-2002 [28], the weld shape and dimensions of the CHS gap K-joints were simplified, as illustrated in Fig. 5. Standard linear elastic properties of low carbon steel were assumed for the steel joint, i.e. Young’s modulus Es = 206 GPa, and Poisson’s ratio νs = 0.3. The weld metal was assumed to have the same elastic properties as steel. 3.2. CFRP sheets The CFRP sheets were modelled using a shell element S4R. The shell elements were generated from the surface elements of the steel joint so that the meshes of the CFRP sheets were identical to those of the steel surfaces (see Fig. 4(e)). A perfect fit between the steel and CFRP meshes is beneficial to improve the efficiency and accuracy of the FE analysis. Following the CFRP strengthening scheme shown in Fig. 1, the shell elements for the CFRP sheets were divided into 11 different regions, as shown in Fig. 6. Different composite layup sections were assigned to different regions to consider the variations in the CFRP fibre orientation. Specially, the small part of connecting CFRP in Region ① (see Fig. 1) was ignored for simplification, which is conservative. Table 3 lists the typical CFRP layup of the FE model of CK1-23 joint. The fibre of each CFRP layer is oriented anticlockwise from the corresponding reference orientation (see Fig. 6). More specifically, the rotation angle of the connecting CFRP on the brace was 90°, and that on the chord was estimated by the following equation [29]:

2. Brief summary of the CFRP strengthening scheme and experimental study The current study is based on an experimental investigation conducted by Tong et al. [23], where eight CFRP-CHS K-joints were tested. The CFRP strengthening scheme is shown in Fig. 1, in which three types of CFRP sheets were employed: chord CFRP (bidirectional CFRP sheet), connecting CFRP (unidirectional CFRP sheet), and anchoring CFRP (unidirectional CFRP sheet). Table 1 lists the basic information of the tested specimens. The main parameter symbols of a CFRP-CHS K-joint are defined in Fig. 2. Under axial loading, the strain concentration factors (SNCFs) at critical hot spots (see Fig. 3) were measured as given in Table 2.

cf

= arctan

4 ·sin + (6 + ) (4 2 arcsin ) sin

(1)

where ζ, β, and θ are defined in Fig. 2. The CFRP and adhesive were assumed to be linear elastic. The tensile moduli of the bidirectional and unidirectional CFRP sheets were 130 and 250 GPa, respectively, which were based on nominal thicknesses of 0.211 and 0.167 mm, respectively [23]. The in-plane shear modulus (G12) and Poisson’s ratio (ν12) of the CFRP were extracted from manufacturer’s data sheet: G12 = 7.17 GPa and ν12 = 0.3 [20]. The tensile modulus (Ea) and Poisson’s ratio (νa) of the adhesive were 2.2 GPa [23] and 0.3, respectively. Measured from a series of coupon samples [29], the average thickness of the adhesive between two layers of the CFRP (ta1) was approximately 0.5 mm. The thickness of the adhesive between the CFRP and the steel (ta0) was assumed to be half the average value (0.25 mm), as shown in Fig. 6(b).

3. Numerical modelling of CFRP-CHS K-joints 3D FE models of CFRP-CHS K-joints were developed using the ABAQUS software package to simulate the steel joints, the applied CFRP sheets and the interaction between the two parts. Fig. 4 shows an example of an FE model of a CFRP-CHS K-joint. Only half of the model was constructed due to symmetry. 3.1. Steel joint The steel tubular joint was modelled using a linear solid element C3D8I, which is better at solving the convergence problem that occurs in contact analysis than a quadratic element (such as C3D20R). With sufficiently fine meshes, C3D8I can approximate the hot spot strain and stress as successfully as a quadratic element. This type of element has

3.3. Interaction of the steel and the CFRP Surface-based cohesive contact was applied in the FE models to simulate the bond-slip behaviour of the CFRP-steel interface. Based on Fernando’s research [30], the normal stiffness (Knn) and shear stiffness 2

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STEP 1: Chord CFRP (× ncf0 layers) Bidirectional CFRP bonded around the chord near the conjunction region to improve the rigidity of the chord.

Patch 1

Patch 1 Patch 2

Patch 2

Lcf0 (800mm)

Overlapping area

STEP 2: Connecting CFRP (× ncf1 layers) 1. Connecting CFRP in the gap region: unidirectional CFRP sheets bonded along the brace axes through the gap region. 2. Connecting CFRP in the side region: unidirectional CFRP sheets bonded along the axis of one brace and wrapped around the chord.

STEP 3: Anchoring CFRP (× 2 layers) Unidirectional CFRP around the braces or chord between the brace-chord junction and the end of the CFRP strengthening region, mainly for anchorage and suppression of peeling effect in crown heel and crown toe.

Fig. 1. Strengthening scheme of a CHS gap K-joint using CFRP sheets.

(Kss and Ktt) of the bond-slip behaviour were estimated using the following equations:

where Ea and Ga are the tensile and shear moduli of the adhesive, respectively; and ta0 is the thickness of the bottom adhesive layer. The adhesive shear modulus Ga equals Ea 2(1 + va) with va = 0.3. In the FE analyses for SNCFs and SCFs, the bond-slip behaviour was assumed

(2)

Knn = Ea ta0, Kss = Ktt = 3(Ga ta0 )0.65 Table 1 Basic information of the test specimens. Specimen no.

Chord

Brace

Angle

Number of CFRP layers

Before strengthening

After strengthening

d0 (mm)

t0 (mm)

d1 (mm)

t1 (mm)

θ (°)

Chord CFRP (ncf0)

Connecting CFRP (ncf1)

Anchoring CFRP (ncf2)

K1-13 K1-43 K1-23 K1-21 K1-25 K2-23 K3-23 K4-23

CK1-13 CK1-43 CK1-23 CK1-21 CK1-25 CK2-23 CK3-23 CK4-23

219 219 219 219 219 219 219 219

8.22 8.15 8.27 8.04 8.09 6.18 8.00 6.00

127 127 127 127 127 127 127 89

6.21 6.18 6.22 6.20 6.18 4.47 8.10 4.52

45 45 45 45 45 45 45 45

1 4 2 2 2 2 2 2

3 3 3 1 5 3 3 3

2 2 2 2 2 2 2 2

3

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2γ = d0/t0 τ = t1/t0 β = d1/d0 ζ = g/d0 Connecting CFRP ncf2 g

t0

Brace

Chord

d0

ncf1

Lcf0

θ ncf2

ncf0 Chord CFRP

stress and deformation in the joint are reduced by CFRP strengthening. The SNCFs determined from FE analyses (SNCFFE) were compared with the experimental results (SNCFTest) for the unstrengthened and CFRP-strengthened joints, respectively. The comparisons are shown in Figs. 8 and 9. The mean value and corresponding coefficient of variation (CV) of the ratio SNCFFE/SNCFTest are given for each joint. For the unstrengthened joints, the mean value of the ratio SNCFFE/SNCFTest varies within 0.93–1.14. For the CFRP-strengthened joints, the mean value of the ratio SNCFFE/SNCFTest varies within 0.99–1.17. The comparisons indicate that the SNCFs of both the unstrengthened and CFRPstrengthened joints could be captured using the FE modelling with acceptable accuracy.

Anchoring CFRP

5. Parametric study

Fig. 2. Schematic diagram of a CFRP-CHS K-joint.

5.1. General Brace

Brace B0 C0 Crown heel

B180 B90

Crown toe

C180

C0 Crown heel

C90 Saddle

Tensile

Chord

A total of 16 independent parameters are needed to determine a CFRP-CHS K-joint without eccentricity that has been strengthened using the current method (see Table 4). To investigate the effects of these parameters on the CFRP strengthening efficiency, a parametric study was conducted using the verified FE modelling method. To identify the effects of CFRP strengthening on SCFs experienced by the CFRP-CHS K-joints, a non-dimensional SCF reduction coefficient ψ was introduced, which is defined as follows:

B0

B90

C180

C90 Saddle

Axial force

B180

Compressive

=

SCFCFRP SCF0

(3)

where SCFCFRP and SCF0 refer to the SCFs of the CFRP-strengthened and unstrengthened joints, respectively. Each CFRP-strengthened joint as well as its unstrengthened counterpart was analysed, allowing ψ to be calculated using Eq. (3). As in the method described in CIDECT [24], only the SCFs on the tensile side, i.e., along the measuring lines at the crown heel (C0 and B0), saddle (C90 and B90), and crown toe (C180 and B180) on the tensile brace and chord were computed (see Fig. 3). Typical balanced axial loading was applied.

Fig. 3. Loading condition and hot spot stress measuring lines in a CHS gap Kjoint.

elastic. 4. Verification of the FE modelling Assuming the same loading conditions as in the experiments [23], FE analyses of the tested samples were performed. An extrapolation method was adopted to obtain the SNCFs and SCFs at the critical hot spots (C0, C90, C180 and B0, B90 and B180, see Fig. 3). Fig. 7 shows two examples of von Mises stress distributions and deformation of the CFRP-strengthened and unstrengthnened joints. Obviously, both the

5.2. Effects of CFRP strengthening parameters One unstrengthened joint was used as the reference, and the effects of the various CFRP strengthening parameters were analysed. The

Table 2 Experimental SNCFs of the test specimens. Specimen no.

SNCFs in chord

SNCFs in brace

Tensile side C0 Unstrengthened joints K1-13 0.37 K1-43 0.32 K1-23 0.48 K1-21 0.45 K1-25 0.59 K2-23 0.80 K3-23 0.91 K4-23 1.32 CFRP-strengthened joints CK1-13 0.42 CK1-43 0.08 CK1-23 0.27 CK1-21 0.45 CK1-25 0.43 CK2-23 0.32 CK3-23 0.66 CK4-23 0.68

Compressive side

Tensile side

Compressive side

C90

C180

C0

C90

C180

B0

B90

B180

B0

B90

B180

2.70 2.52 2.27 2.34 2.63 3.41 3.67 3.85

3.10 3.20 2.81 2.56 2.45 2.99 3.34 3.45

1.32 1.47 1.49 1.09 1.37 2.16 2.05 2.47

2.44 2.20 2.60 2.18 2.18 3.45 4.01 3.81

4.13 3.51 2.66 2.78 2.50 2.85 4.86 3.44

1.59 1.28 1.31 1.39 1.60 1.34 1.34 1.75

1.38 1.69 1.39 1.22 1.54 1.71 1.81 2.96

1.91 2.32 2.06 1.85 2.25 2.49 1.71 2.23

1.37 1.35 1.46 1.18 2.03 1.97 1.21 1.15

1.54 1.93 1.76 1.55 1.74 2.41 2.05 2.56

2.80 1.76 2.41 2.77 2.43 2.55 2.12 2.43

2.31 2.05 1.89 2.10 2.07 2.45 3.10 2.73

2.52 2.59 2.47 2.34 2.16 2.32 3.08 3.14

1.29 1.04 1.35 1.17 1.22 1.33 1.97 1.53

2.00 1.71 1.97 1.98 1.56 2.57 3.22 2.86

3.07 3.02 2.09 2.15 1.75 2.61 3.62 3.25

1.51 0.84 0.99 1.31 1.20 1.02 1.00 1.06

1.24 1.30 1.11 1.19 1.27 1.27 1.61 2.29

1.63 2.17 1.86 1.79 1.70 1.92 1.60 2.16

1.32 1.11 1.39 1.33 1.89 1.31 1.41 0.65

1.36 1.55 1.43 1.44 1.28 1.87 1.82 1.83

2.19 1.54 2.05 2.36 1.74 2.46 1.81 2.30

4

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Fig. 4. FE model of a CFRP-CHS K-joint.

Fig. 5. Weld shape and dimensions in FE modelling of CHS gap K-joints: (a) crown heel; (b) saddle; (c) crown toe.

reference model was assumed to have the following geometric parameters: t0 = 8.22 mm, τ = 0.75, 2γ = 26.6, β = 0.58 and θ = 45°.

Lcf 0,min = max{220 + 2d1 sin

+ g , 9.5 d 0 t 0 + d1 sin

+ g}

(4)

In the following analyses, Lcf0 was determined by Eq. (4), and Lcf1 was 200 mm (> 110 mm).

5.2.1. CFRP strengthening ranges Figs. 10 and 11 show the effects of the two CFRP strengthening range parameters (Lcf0 and Lcf1) on ψ. Neither Lcf0 nor Lcf1 has a significant effect on ψ. This is particularly true for large Lcf0 and Lcf1, i.e., Lcf0 ≥ 800 mm and Lcf1 ≥ 200 mm, under the conditions of the experiment. Following the method developed by Fu et al. in their research on the static ultimate capacity of CFRP-CHS K-joints [20], the minimum value of Lcf0 is suggested to be determined by Eq. (4). A minimum Lcf1 value of 110 mm is suggested as per Fu et al. [20]. The result is based on the concept of effective bond length.

5.2.2. Number of CFRP layers The effects of the number of chord CFRP layers (ncf0), connecting CFRP layers (ncf1), and anchoring CFRP layers (ncf2) are discussed in this section.

• Number of anchoring CFRP layers (n

cf2)

With constant ncf0 and ncf1, ncf2 was varied from 1 to 4. As can be seen from Fig. 12, the effect of ncf2 on ψ is small, because the fibres of the unidirectional anchoring CFRP are perpendicular to the tube axes. 5

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(a) T

Z

Brace-Com-Cys Z Reference orientation R T 0°

Brace-Ten-Cys

R

Z

9

8

0° Reference orientation 0°

7

10

11

6

5

T 1

R

2

4

3

1

Chord-Cys

(b)

(c) tcf1=0.167mm

Simplified

tcf + ta

ta tcf tcf ta/2

CFRP

Orientation of CFRP fibre

Adhesive

tcf + ta

ta

Adhesive

CFRP sheet

tcf + ta

ncf (ta+tcf )

ta/2 tcf

9

Reference orientation θcf = 0° Steel surface

Steel

Axis direction

Fig. 6. Diagram of a CFRP composite shell in FE modelling: (a) subregions of the CFRP meshes; (b) simplified ‘CFRP-adhesive’ composite section; (c) layup of the ‘CFRP-adhesive’ composite section.

In particular, the ψ at B0 slightly increases with an increase in ncf2, which may be due to the change in the stiffness distribution along the brace-chord junction. Because the anchoring CFRP is essential to protect and restrain early debonding of the other CFRP sheets, ncf2 = 2 was chosen and is suggested for practical applications.

• Number of chord CFRP layers (n (ncf1)

cf0)

location. However, with an increase in ncf1, this effect gradually weakens, particularly at the crown toe (C180 and B180). An increase in ncf0 also yields a decrease in ψ (except at B0). However, the effects at the crown toe (C180 and B180) are small because the chord CFRP does not cover the gap region (see Fig. 1). Furthermore, an increase in ncf1 usually leads to a larger reduction in the SCFs than an increase in ncf0 does. At B0, the ψ is positively correlated with ncf0, which is similar to ncf2. This is primarily attributed to the change in the stiffness distribution along the brace-chord junction. The chord CFRP fails to cover the entire

and connecting CFRP layers

As shown in Fig. 13, ψ is negatively correlated with ncf1 at each Table 3 Typical CFRP layup in the FE modelling (specimen CK1-23). CFRP region

CFRP type

Local reference coordinate system*

Rotation Angle θcf (°)**

① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ⑪

Chord CFRP & anchoring CFRP Chord CFRP & connecting CFRP Chord CFRP & connecting CFRP Chord CFRP & connecting CFRP & anchoring CFRP Connecting CFRP Connecting CFRP Connecting CFRP & anchoring CFRP Connecting CFRP & anchoring CFRP Connecting CFRP & anchoring CFRP Connecting CFRP Connecting CFRP

Chord-Cys Chord-Cys Chord-Cys Chord-Cys Chord-Cys Chord-Cys Chord-Cys Brace-Ten-Cys Brace-Com-Cys Brace-Ten-Cys Brace-Com-Cys

[0, 0]ch/[0, 0]an [0, 0]ch/[36, 36, 36]co [0, 0]ch/[−36, −36, −36]co [0, 0]ch/[−36, 36, −36, 36, −36, 36]co/[0, 0]an [36, 36, 36]co [−36, −36, −36]co [90, 90, 90]co/[0, 0]an [90, 90, 90]co/[0, 0]an [90, 90, 90]co/[0, 0]an [90, 90, 90]co [90, 90, 90]co

Notes: (i) ‘*’ See Fig. 6. (ii) ‘**’ The subscripts “ch”, “co,” and “an” refer to the chord CFRP, connecting CFRP and anchoring CFRP, respectively. 6

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Fig. 7. Examples of von Mises stress distributions and deformation of the CFRP-strengthened and unstrengthnened joints (deformation scale factor: 200): (a) unstrengthened joint; (b) CFRP-strengthened joint (CFRP is not displayed).

connecting region (see Fig. 1), hence the deformation of the chord outside the connecting region is restrained more effectively than that of the connecting region. Consequently, if the hot spot B0 experiences the maximum hot spot stress, ncf0 and ncf2 should be minimized.

thickness parameters are discussed separately as follows.

• Thickness of the adhesive between the CFRP and the steel t

a0

The bond between the CFRP and the steel is key to ensuring the strengthening effectiveness. According to Eq. (2), the stiffness of the bond-slip behaviour is directly impacted by the thickness of the bottom adhesive layer (ta0). Generally, a smaller ta0 leads to stiffer bond-slip behaviour, more efficient force transfer from the steel to the CFRP, and more effective strengthening. As shown in Fig. 16(a), an increase in ta0 leads to a gradual increase in ψ.

5.2.3. Tensile moduli and thicknesses of the CFRP sheets The chord CFRP and connecting CFRP are discussed separately. Fig. 14 focuses on the effect of the tensile modulus (Ecf0) and thickness (tcf0) of the chord CFRP on ψ. In Fig. 14, the horizontal axis represents the non-dimensional CFRP modulus (NEcf0), or thickness (Ntcf0), which has been normalised by the benchmark value (i.e., 130 GPa for Ecf0 and 0.211 mm for tcf0). For example, NEcf0 = Ecf0/130. It can be seen from Fig. 14 that changing Ecf0 has almost the same effect on ψ as changing tcf0 does. Specifically, the ψ at C0 and C90 decreases with an increase in Ecf0 or tcf0, whereas Ecf0 and tcf0 have negligible effect on ψ at the other locations (i.e., C180, B0, B90, and B180) within the ranges considered. Fig. 15 illustrates the effects of the connecting CFRP modulus (Ecf1) and thickness (tcf1). Generally, ψ decreases with an increase in Ecf1 or tcf1–except at B0. In the brace, the effects of Ecf1 and tcf1 on ψ are quite similar, whereas in the chord, tcf1 has more of an effect on ψ than does Ecf1. This is primarily attributed to the increase in the sectional bending stiffness, which is due to the increase in tcf1. According to Morgan and Lee (1998) [25], the chord wall generally suffers from a higher DoB than the brace wall does along the intersection line, leading to a more significant reduction in the SCFs in the chord [23].

• Adhesive thickness between two adjacent CFRP layers t

a1

However, the situation reverses when one considers the effect of ta1 on ψ. As shown in Fig. 16(b), ψ decreases with an increase in ta1 within the parameter ranges considered in this study. This is primarily because of the increase in the sectional bending stiffness due to the increase in ta1. Furthermore, ta0 and ta1 are generally correlated, which means that the above two opposing phenomena work simultaneously and partially offset one another in practice. In Fig. 16(c), ta0 and ta1 increase simultaneously by assuming ta = ta1 = 2 ta0. It can be seen that ψ generally decreases with an increase in ta–except at B0. In addition, the effects observed in Fig. 16(c) are less significant than those observed in Fig. 16(b). Due to the difficulty of controlling the adhesive thickness in the practical application of CFRP to CHS K-joints, a fixed value of 0.5 mm, which was obtained from a series of coupon samples, was adopted for ta.

5.2.4. Thickness of the adhesive There are two types of adhesive thicknesses: the thickness of the adhesive between the CFRP and the steel (ta0), and the thickness of the adhesive between two adjacent CFRP layers (ta1). The effects of the two 7

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4

3

3

2

2

5 4

0°

90°

SNCF

0

Tensile

K1-23

180°

Compressive 180°

90°

0

0°

5

Mean: 1.04 CV: 0.16

4

3

3

2

2

1

5 4

0°

90°

SNCF

0

Tensile

K1-25

180°

Compressive 180°

90°

0

0°

5

Mean: 1.05 CV: 0.12

4 3

2

2

5 4

0°

90°

SNCF

0

Tensile

K3-23

180°

Compressive 180°

90°

0

0°

5

Mean: 1.01 CV: 0.13

4

3

3

2

2

1 0

Tensile 0°

90°

180°

Compressive 180°

90°

0°

90°

Tensile 0°

90° K2-23

0°

90° K4-23

0

0°

180°

180°

90°

180°

K1-21

90°

0°

Mean: 1.14 CV: 0.13

Compressive 180°

Location

90°

Location 0°

Compressive 180°

90°

Location 0°

Mean: 1.05 CV: 0.18

Tensile 0°

Compressive

Mean: 0.93 CV: 0.14

Tensile

1

Location

180°

Chord,SNCFTest Chord,SNCFFE Brace,SNCFTest Brace,SNCFFE

1

Location

Mean: 1.01 CV: 0.14

Tensile

1

Location

3

1

K1-43

1

Location

SNCF

1

SNCF

5

Mean: 1.00 CV: 0.11

SNCF

4

K1-13

SNCF

5

SNCF

G. Xu, et al.

180°

Compressive 180°

90°

Location 0°

Fig. 8. Comparison of the SNCFs of experimental and FE results for unstrengthened joints.

5.2.5. Modulus of the adhesive The stiffness of the bond-slip behaviour is related to the tensile modulus as per Eq. (2). Thus when the tensile modulus of the adhesive (Ea) is increased, the cooperation between the CFRP and the steel as well as that between the adjacent CFRP layers is enhanced. Additionally, the stiffness of the ‘CFRP-adhesive’ composite section is increased. Therefore, the CFRP strengthening efficiency increases significantly as Ea increases (see Fig. 17). However, with the increase in Ea, the increase in the CFRP strengthening efficiency becomes less significant.

5.3.1. Effect of tube wall thickness The tube wall thicknesses (t0 and t1) are important for evaluating the strengthening system, because they are closely related to the relative reinforcement rate of the CFRP. Fig. 18 contains plots showing the relationships between the chord wall thickness t0 and ψ for constant τ (=0.75), 2γ (=24.0) and β (=0.6). With simultaneous increases in t0 and t1 (t1 = 0.75 t0), the relative reinforcement rate of the CFRP decreases, hence ψ gradually increases–except at B0. The trend of ψ at B0 is different because it is negatively correlated with the reinforcement rate of the connecting CFRP, but positively correlated with the reinforcement rate of the chord CFRP (refer to Section 5.2.2 and Fig. 13). Once t0 is determined, t1 is linearly related to the non-dimensional parameter τ (t1 = τ·t0). In Fig. 19, the effect of the parameter τ on ψ when θ = 45° is shown. Generally, ψ increases with an increase in τ at the crown heel (C0 and B0) and crown toe (C180 and B180). However, the increase in ψ slows down as τ increases. In some cases, ψ even decreases as τ increases, e.g., when 2γ = 24 at C0 and B0. At the brace saddle (B90), as τ increases, ψ decreases when τ < 0.5 but slightly increases when τ > 0.5. The ψ at C90 is minimally affected by τ.

5.3. Effects of the geometric parameters of the steel joint For a CHS K-joint without eccentricity, the five geometric parameters listed in Table 4 can be divided into three categories: i) thickness or thickness-related parameters: t0 and τ; ii) shape-related parameters: β and θ; and iii) scale-related parameter: 2γ (when t0 is determined). With the CFRP strengthening parameters kept constant (ncf0 = 2, ncf1 = 3, tcf0 = 0.211 mm, tcf1 = 0.167 mm, Ecf0 = 130 GPa, and Ecf1 = 250 GPa), the geometric parameters were varied separately to investigate their effects on ψ. The ranges of the geometric parameters were chosen according to CIDECT [24] as listed in Table 4. Some typical graphs are provided below (see Figs. 18–20).

5.3.2. Effects of 2γ, β, and θ The effects of 2γ, β and θ are much complicated. Theoretically, they can affect ψ at least through the following possible ways: 8

Engineering Structures 210 (2020) 110369

3

5

Mean: 0.99 CV: 0.10

4

0°

90°

180°

Compressive 180°

CK1-23

1

Location

90°

0

0°

5

Mean: 1.06 CV: 0.17

4

3

3

2

2

1

5 4

0°

90°

SNCF

0

Tensile 180°

Compressive 180°

CK1-25

0

0°

5

Mean: 1.05 CV: 0.10

4

3

3

2

2

1

5 4

0°

90°

SNCF

0

Tensile 180°

Compressive 180°

CK3-23

0

0°

5

Mean: 1.02 CV: 0.16

4

3

3

2

2

1 0

Tensile 0°

90°

180°

Compressive 180°

0

0°

Compressive

180°

180°

90°

Compressive

180°

180°

90°

Compressive

180°

180°

Location

90°

CK4-23

90°

0°

Mean: 0.99 CV: 0.15

0°

Mean: 1.08 CV: 0.18

Tensile 0°

Location

90°

CK2-23

Tensile 0°

0°

Mean: 1.17 CV: 0.13

Tensile 0°

Location

90°

CK1-21

1

Location

90°

90°

1

Location

90°

0°

1

Location

90°

Tensile

SNCF

Tensile

SNCF

4

Mean: 1.05 CV: 0.15

2

1

5

CK1-43

3

2

0

SNCF

CK1-13

SNCF

4

Chord,SNCFTest Chord,SNCFFE Brace,SNCFTest Brace,SNCFFE

SNCF

5

SNCF

G. Xu, et al.

Compressive

180°

180°

Location

90°

0°

Fig. 9. Comparison of SNCFs between experimental and FE results for CFRP-strengthened joints. Table 4 Geometric and CFRP strengthening parameters of a CFRP-strengthened CHS gap K-joint. Component

Chord CFRP Connecting CFRP Anchoring CFRP Adhesive Steel joint

CFRP strengthening parameters and ranges

Geometric parameters and ranges

Strengthening range (mm)

Number of layers

Elastic modulus (GPa)

Thickness (mm)

t0 (mm)

τ

2γ

β

θ (°)

Lcf0: [500, 1200]* Lcf1: [50, 400] – – –

ncf0: [0, 4] ncf1: [1,8] ncf2: [1,4] – –

Ecf0: [130, 520] Ecf1: [250, 1000] Ecf2 = Ecf1 Ea: [1.0, 10.0] –

tcf0: [0.106, 0.633] tcf1: [0.084, 0.501] tcf2 = tcf1 ta: [0.25, 1.5] –

– – – – [5.4, 16]

– – – – [0.25, 1.0]

– – – – [24, 60]

– – – – [0.3, 0.6]

– – – – [30, 60]

* “[ ]” denotes mathematical interval, which refers to the parameter range considered in the parametric study.

• ψ versus 2γ

(i) Degree of bending (DoB) [25]. According to Tong et al. [23], ψ will decrease as DoB increase. (ii) The stiffness distribution along the brace-chord junction. (iii) The rotation angle of the connecting CFRP (θcf) on the chord (refer to Eq. (1)).

ψ and 2γ are positively correlated at C0 but negatively correlated at B0. The effects of 2γ on ψ are relatively small at the crown toe and saddle locations. Within the considered parameter ranges, when 2γ increases from 24 to 60, the maximum change in ψ is less than 5%. At B180 and C180, ψ slightly decreases with an increase in 2γ when β = 0.3, whereas no significant correlation is found between ψ and 2γ when β ≥ 0.45.

Fig. 20 presents plots of ψ against 2γ for different values of β and θ with the tube thicknesses kept constant. From Fig. 20, the following effects of 2γ, β, and θ on ψ can be observed:

9

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G. Xu, et al.

1.0

ψ

circles, triangles, and squares. The ψ at C0 is positively correlated with θ, and the effect is more prominent as 2γ gets smaller. At B0, ψ is negatively correlated with θ when β = 0.3, but no significant correlation was found when β ≥ 0.45. At C90 and C180, the effects of θ on ψ are small. Within the considered parameter ranges, the maximum changes in ψ are less than 3%. At B90, ψ is negatively correlated with θ. At B180, ψ increases with an increase in θ when β ≥ 0.45 (hollow labels), whereas the effect of θ is smaller when β = 0.3 (solid labels).

ψ vs. Lcf0 , Lcf1 = 200 mm

0.9 0.8 C0 C90 C180

0.7 0.6

250

500

B0 B90 B180 750

6. Study of the formulae for ψ It is noteworthy that the ratio of the ‘CFRP-adhesive’ composite section thickness to the steel thickness (i.e., the relative reinforcement rate) has a significant effect on ψ. In the current study, three types of relative reinforcement rates were therefore defined, namely, s0, s1, and s01 (see Eq. (5)).

Lcf0 (mm) 1000

1250

Fig. 10. Lcf0 versus ψ.

1.0

ψ

s0 =

ψ vs. Lcf1 , Lcf0 = 800 mm

0.9

C0 C90 C180

0.6

0

100

B0 B90 B180 200

Lcf1 (mm)

300

400

500

Fig. 11. Lcf1 versus ψ.

1.0

ψ

ψ vs. ncf2 , ncf0 = 2, ncf1 = 3

T0 = ncf 0 (tcf 0 + ta )

(6)

T1 = ncf 1 (tcf 1 + ta)

(7)

0

0.9 0.8

=

E0 , Es

1

=

E1 Es

(8)

E0 = (Ecf 0 tcf 0 + Ea ta ) (tcf 0 + ta )

(9)

E1 = (Ecf 1 tcf 1 + Ea ta ) (tcf 1 + ta)

(10)

where ta = 0.5 mm.

C0 C90 C180

0.7 0.6

(5)

where ta equals 0.5 mm. In Fig. 21, ψ is plotted against t0 with constant values of s0, s1, and s01. A slight scale effect is found, i.e., ψ decreases slightly with an increase in t0, even though the values of s0, s1, and s01 are kept constant. However, the scale effect is small. Generally, the changes in ψ are within 3% as t0 increases from 8 mm to 16 mm. Therefore, for simplicity, a constant chord wall thickness of 8 mm was adopted for development of the formulae for ψ. The non-dimensional parameters were defined to consider the effects of the tensile moduli of the ‘CFRP-adhesive’ composite, as shown in Eq. (8).

0.8 0.7

T t T T0 T , s1 = 1 , s01 = 1 = 1 · 1 = ·s1 t0 t1 t0 t0 t1

0

1

6.1. Formulae for ψ

B0 B90 B180 2

Based on the previous analysis and multiple FE modelling results, the mathematical regression model for ψ was constructed as follows:

ncf2 3

4

5

(11)

= [(f1 · f2 + A0 ) ]

Fig. 12. ncf2 versus ψ.

f1 = A1 (1 +

• ψ versus β

f2 =

By comparing the solid and hollow labels in Fig. 20, the influence of β on ψ can be seen. In general, an increase in β would result in a decrease in the ψ at C0 and B0. However, the trend reverses when 2γ and θ become large (e.g., 2γ = 60 and θ = 60°). At the saddle locations (C90 and B90) and chord crown toe (C180), the influence of β on ψ is small. Within the considered parameter ranges, when β increases from 0.3 to 0.6, the changes in ψ are less than 3% at C90 and B90 and less than 4% at C180. At B180, ψ is negatively correlated with β when θ is small (e.g., θ = 30°). However, for θ ≥ 45°, ψ is positively correlated with β when 2γ is larger than 35.

=

=

0

) A2 (1

+

1

) A3 (1

+ s0 B4

B0 + B1· + B2· + B3·

12 Ea 2.2

) A 4 (1

B5

0.45 C0 + C1· 1 s01+ C2·( 1 s01 )2

D0 · s1 D1+ D2·

+ D3· + D4·

\;or

+ s1

) A5 (1

45 =

Ea 2.2

B6

0.5

+ s01

) A6

+ B7

C0+ C1· 1 s1+ C2·( 1 s1 )2

+ D5 for B90 and 1 for the other cases

+ A7

< 1.25

(12) (13)

\;

60

(14) (15)

where f1 is related to the relative reinforcement rates of CFRP, f2 is written in terms of the non-dimensional geometric parameters, ξ is a correction term for the tensile modulus of the adhesive, and λ is an additional correction term for the ψ at B90. A0–A7, B0–B7, C0–C2, and D0–D5 are constants that were determined by multiple regression analysis. Through multiple nonlinear regression analyses based on a database contains 1011 models, the parametric formulae for ψ were obtained and

• ψ versus θ The influence of θ on ψ can be seen in Fig. 20 by comparing the 10

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1.0

ψ

1.0

C0

ψ

1.0

C90

0.9

0.9

0.9

0.8

0.8

0.8

0.7

0.7

0.7

nNcf0=0 cf0 = 0 nNcf0=2 cf0 = 2 nNcf0=4 cf0 = 4

0.6 0.5

0

1.0

2

ncf0 increases 4

ψ

ncf1

6

B0

8

0.6

0.5

ncf1 0

1.0

ncf0 increases

0.8

2

0.7

4

ψ

6

8

0.9

0.9

0.8

0.8 ncf0 increases

2

4

6

8

0.5

ncf1 0

2

4

ψ

6

8

B180

ncf0 increases

0.7 0.6

0.6

ncf1

0.5 1.0

B90

0.7

0

ncf0 increases

ncf0 increases 0.6

0.9

0.6

C180

ψ

ncf1 0

2

4

6

8

0.5

ncf1 0

2

4

6

8

Fig. 13. ncf1 versus ψ for different ncf0.

are summarised in Table 5. In particular, the formula for C0 was derived from the database when θ ≥ 45°. When θ < 45°, no SCF reduction was performed at C0, because the SCF was observed to be smaller than those at the other locations in either CFRP-strengthened or unstrengthened CHS gap K-joints. As can be seen from Fig. 22, the SCFs at C0 derived from the considered models were less than 0.6 times the maximum of the SCF values in the chord. Furthermore, the chord crown heel even suffered from compressive stress when θ < 35° (see Fig. 22). The proposed formulae are valid when used to calculate the SCF reduction coefficients ψ of CFRP-CHS K-joints subjected to balanced axial loading. The validity ranges are noted in Table 5. Moreover, ψ was assumed to have a maximum value of 1.0 (except at B0). At B0, the SCF increased due to the change in the stiffness distribution caused by CFRP strengthening (see Figs. 12 and 13); thus, no maximum limit was set.

( FE ). The comparison shows that the ratio of FE to FOR is almost 1.0 with a very low CV. This indicates that FOR agree well with FE statistically and the fitting of the regression analysis is highly accurate. 6.2.2. Comparison between the formulae and the experiments Additionally, ψ of the eight tested CFRP-CHS K-joints at the critical locations were calculated using the proposed parametric formulae ( FOR ) and compared with the test results ( Test ) in Tong et al. [23]. The ratios of FOR to Test are listed in Table 6. One sees that the calculated FOR are in good agreement with the test results. The SCFs of the CFRP-CHS K-joints were obtained from the following equation, which is based on the definition of ψ:

where SCF0 can be determined through simple FE analysis or from the existing SCF formulae for unstrengthened CHS gap K-joints, such as those recommended by Morgan and Lee (ML formulae) [25]. In Fig. 24, the calculated maximum SCFCFRP of the experimental joints is compared with the corresponding experimental results (SCFTest). In Fig. 24(a), the SCF0 were extracted from FE analysis. The resulting SCFCFRP (SCFψ,FE)

6.2. Verification of the proposed formulae 6.2.1. Comparison of ψ determined from the formulae and FE analysis In Fig. 23, the SCF reduction coefficients determined from the proposed parametric formulae ( FOR ) are compared with the FE results

1.0

ψ

1.0

ψ vs. NEcf0 or Ntcf0, Chord Ecf0 = 130 GPa, tcf0 = 0.211mm

0.9

0.8

Ecf0 changes C0 C90 C180

0.6 0.5

ψ

0

1

2

Ecf0 = 130 GPa, tcf0 = 0.211mm

0.7

tcf0 changes C0 C90 C180

Ecf0 changes B0 B90 B180

0.6

NEcf0 or Ntcf0 3

ψ vs. NEcf0 or Ntcf0, Brace

0.9

0.8 0.7

(16)

SCFCFRP = ·SCF0

0.5

4

0

1

tcf0 changes B0 B90 B180 2

NEcf0 or Ntcf0 3

Fig. 14. Effects of the chord CFRP modulus and thickness on SCF reduction coefficients. 11

4

Engineering Structures 210 (2020) 110369

G. Xu, et al.

0.9

ψ

1.0

ψ vs. NEcf1 or Ntcf1, Chord

ψ vs. NEcf1 or Ntcf1, Brace

0.9

0.8 0.7

0.8

Ecf1 = 250 GPa tcf1 = 0.167mm

0.6

Ecf1 changes C0 C90 C180

0.5 0.4

ψ

0

0.7

tcf1 changes C0 C90 C180

1

2

Ecf0 = 250 GPa tcf0 = 0.167mm Ecf1 changes B0 B90 B180

0.6

NEcf1 or Ntcf1 3

0.5

4

0

tcf1 changes B0 B90 B180

1

2

NEcf1 or Ntcf1 3

4

Fig. 15. Effects of the connecting CFRP modulus and thickness on SCF reduction coefficients.

(a)

(b)

1.1

ψ vs. ta0, ta1 = 0.5mm

ψ

(c)

1.1

1.1

ψ vs. ta1, ta0 = 0.25mm

ψ

1

0.9

0.9

0.9

0.8

0.8

0.8

0.7

0.7

C0 C90 C180

0.7 0.6 0.5

B0 B90 B180

0.6

ta0 (mm) 0

0.25

0.5

0.75

1

0.5

ψ vs. ta, ta = ta1 = 2ta0

ψ

1

1

0.6 ta1 (mm) 0

0.25

0.5

0.75

1

1.25

1.5

0.5

ta (mm) 0

0.25

0.5

0.75

1

1.25

1.5

Fig. 16. Effects of the adhesive thickness on SCF reduction coefficients: (a) ta0 versus ψ; (b) ta1 versus ψ; (c) ta versus ψ.

ψ

1.0

1.0

ψ vs. Ea, ncf0 = 2, ncf1 = 3

ψ vs. t0

ψ

0.9

0.9

0.8

0.8

C0 C90 C180

B0 B90 B180

τ = 0.75, 2γ = 24.0, β = 0.6 ncf0 = 2, ncf1 = 3

t0 (mm)

0.7

C0 C90 C180

0.7 0.6

0

2.5

5

B0 B90 B180 7.5

10

0.6

Ea (GPa) 12.5

0.5

15

Fig. 17. Effect of the adhesive modulus on SCF reduction coefficients.

4

5.4

8

12

16

Fig. 18. Effect of the wall thickness on SCF reduction coefficients (with constant CFRP parameters).

are in good agreement with SCFTest. In Fig. 24(b), the SCF0 were determined using the ML formulae [25]. The resulting SCFCFRP (SCFψ,ML) are generally found to be larger than SCFTest. This is primarily because the ML formulae conservatively estimate the SCF0, which produces results that can safely and reasonably be used for engineering applications.

developed to estimate ψ. The main conclusions are the following: (1) CFRP strengthening on the chord and brace (Lcf0 and Lcf1) has a negligible effect on ψ if the bond length requirement is met. Minimum values of Lcf0 and Lcf1 are suggested. (2) Increasing the number of layers (ncf0, ncf1, and ncf2), tensile moduli (Ecf0 and Ecf1), and thicknesses (tcf0 and tcf1) of the CFRP sheets improves the strengthening effectiveness to varying degrees. Unidirectional connecting CFRP (ncf1, Ecf1, and tcf1) is more effective than bidirectional chord CFRP (ncf0, Ecf0, and tcf0). Furthermore, increasing the thickness of the connecting CFRP (tcf1) is more effective than increasing its tensile modulus (Ecf1)–particularly along the chord–due to the increased sectional bending stiffness and large DoB at these locations. However, the effects of the thickness and

7. Conclusions In this study, 3D FE models were developed to analyse the SCFs of CFRP-CHS K-joints. After verifying the model using previously reported experimental results, a parametric study was conducted to explore the effects of CFRP strengthening and geometric parameters on the SCF reduction coefficients ψ of CFRP-CHS K-joints subjected to balanced axial loading. Through regression analyses, parametric formulae were 12

Engineering Structures 210 (2020) 110369

G. Xu, et al.

Fig. 19. Effect of the parameter τ (brace wall thickness t1) on ψ.

Fig. 20. Effect of the parameter 2γ on ψ for different θ and β.

modulus of the chord CFRP (tcf0 and Ecf0) are smaller. (3) At B0, the ψ slightly increases with an increase in ncf0 or ncf2 due to changes in the stiffness distribution. Therefore, the number of chord CFRPs (ncf0) should be minimized if B0 experiences the maximum hot spot stress on a CFRP-CHS K-joint. An ncf2 of 2 is suggested to restrain early debonding of the other CFRP.

(4) Increasing the thickness of the bottom adhesive layer (ta0) weakens the strengthening effect. Conversely, properly increasing the thickness of the interlayer adhesive (ta1) improves the strengthening effect. A fixed value of 0.5 mm is suggested for ta1 in practical applications. An adhesive with a high modulus significantly increases the efficiency of the CFRP strengthening. 13

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G. Xu, et al.

(a)

(b)

1.0

ψ

1.0

ψ vs. t0, τ = 0.5, s0 = 0.18, s1 = 0.67

0.9

0.9

0.8

0.8

0.7

0.7 C0 C90 C180

0.6 0.5

4

B0 B90 B180

8

ψ

ψ vs. t0, τ = 1.0, s0 = 0.18, s1 = 0.33

C0 C90 C180

0.6 t0 (mm)

12

0.5

16

4

B0 B90 B180

8

12

t0 (mm) 16

Fig. 21. Effect of the wall thickness with a constant relative reinforcement rate (s0 and s1): (a) τ = 0.5; (b) τ = 1.0. Table 5 Parametric formulae for the SCF reductions coefficient ψ in CFRP-strengthened CHS gap K-joints. Location

Formulae:

C0 (θ ≥ 45°):

f1 = 5.000(1 +

( ) =( )

f2 =

= [(f1 · f2 + A 0 ) ] 0.044 (1 + 0) 2.261 0.603 + 0.042

12

( ) ( ) 0.45

0.239 (1 + 0) 0.335 0.21 + 0.015

f1 = 0.763(1 +

( ) =( )

f2 =

1) 1.068 1.56

( ) =( )

0.285 (1

1)

0.469 (1

0.006

12

ψ ≤ 1.0

f1 =

+ s0 )

0.139 (1

12

( ) =( )

B180:

0.138 (1 1) 0.191 + 0.0086

+ s0 )

f1 =

0.265(1 +

0.12 (1

+ s01 )0.089 + 1.235

1.298

0.5

+ s1)

0.0437 (1

+ s01)

0.185

1.459

0.243

45

) 2.996 + 1.785

12

5.824

+ 0.949 60

1.25

0.038 (1 + 0) 14.962 + 13.847 + 0.16

2 Ea 1.952 1 s1 2.357( 1 s1 ) , 2.2

A0 = 1.007

0.039

1

( ) =( )

( )

0.363 (1

0.176

A0 = 0.95, λ = 1

0.093

0.45

0.081(1 + s1

0.564

45

( ) ( )

12

+ s0 )0.168 (1 + s1)

( )

0.45

=

f2 =

0.504

A0 = 0.885, λ = 1

( )

2 Ea 0.228 + 0.926 1 s1 1.372( 1 s1 ) , 2.2

ψ ≤ 1.0

0.692

+ s01)

0.119

45

0.146 (1 1) 0.673

+

0.242 (1

+ s1)

( ) ( )

0.063 (1 0) 1.347 0.004 + 0.835

1.189(1 +

f1 = 1.515(1 + f2 =

0.411 (1

+ s0)

0.267

2 Ea 0.37 + 3.98 1 s01 6.923( 1 s01 ) , 2.2

B90:

0.555

4.795

A0 = 0.852, λ = 1

0.45

( ) =( )

f2 =

0.731

0.061

0.654

2 Ea 0.12 + 2.733 1 s01 5.467( 1 s01 ) , 2.2

B0:

+ s01)

+ s01)

45

f1 = 0.682(1 +

f2 =

0.067 (1

45

2 Ea 0.188 + 2.096 1 s01 4.634( 1 s01 ) , 2.2

ψ ≤ 1.0 C180:

+ s1)

2.254

A0 = 0.970, λ = 1

( )

12

0.138 (1

+ s0 )

0.102

2 Ea 2.579 1 s01 3.789( 1 s01 ) , 2.2

ψ ≤ 1.0 C90:

0.062 (1

1)

1)

0.452 (1

( )

1.049

0.45

+ s0)

( )

< 1.25

0.096 (1

0.618

45

+ s1)

60 0.361 (1

+ s01)

0.316

+ 0.209

4.549

A0 = 0.921, λ = 1

ψ ≤ 1.0 Range of validity: (1) no eccentricity; (2) equal braces; (3) balanced axial loading 0 s0 0.46 30° 60° 0 < s1 2.66 0.3 0.6 0 0.757 0 24 2 60 0 < 1 1.223 0.25 1.0

0

n cf 0

4; 1

ncf 1

8

130GPa

Ecf 0

520GPa

250GPa

Ecf 1

1000GPa

1.0GPa

14

Ea

10.0GPa

Note: (1) ta = 0.5 mm.(2) When ncf0 = 0, η0 = 0.

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Fig. 22. Ratio of the SCFs at C0 (SCFC0) to the maximum SCF in the chord (SCFC,max): (a) CFRP-strengthened joints; (b) unstrengthened joints.

Fig. 23. Comparison of ψ determined from formulae and FE analysis.

(5) The CFRP strengthening efficiency is closely related to the thickness parameters t0 and τ. In most cases, ψ increases as the tube thickness increases. The parameters 2γ, β, and θ have smaller effects on the ψ at the crown toe and saddle locations. Although the ψ at C0 is significantly influenced by 2γ and θ, it does not dominate the fatigue failure of CFRP-CHS K-joints due to the relatively small SCF, especially when θ < 45°. (6) Three types of relative CFRP reinforcement rates (s0, s1, and s01), the CFRP-to-steel tensile modulus ratios (η0, η1), the adhesive modulus (Ea), and the non-dimensional geometric parameters (τ, 2γ, θ, and β) are considered in the proposed parametric formulae for ψ. A comparison with the test results confirms the validity of the proposed formulae and their usefulness in estimating ψ of CFRPCHS K-joints under balanced axial loading.

Through a combination use of existing SCF formulae for unstrengthened CHS gap K-joints (i.e. CIDECT [24] and ML formulae [25]) and the proposed ψ formulae, SCF and hot spot stress of CFRP-CHS Kjoints can be determined. Then fatigue life of CFRP-CHS K-joints can be estimated using an Sr,hs-N curve similar to that for un-strengthened joints. Further discussion on Sr,hs-N curve of CFRP-CHS K-joints is being carried out. CRediT authorship contribution statement Guowen Xu: Methodology, curation, Writing - original Conceptualization, Validation, Funding acquisition, Writing 15

Formal analysis, Investigation, Data draft, Visualization. Lewei Tong: Supervision, Project administration, review & editing. Xiao-Ling Zhao:

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Table 6 Comparison of ψ determined from formulae and experiments. Specimen no.

Calculated ψ (ψFOR)

CK1-13 CK1-43 CK1-23 CK1-21 CK1-25 CK2-23 CK3-23 CK4-23 Mean CV

Ratio of ψFOR to ψTest

C90

C180

B0

B90

B180

0.85 0.81 0.84 0.90 0.79 0.80 0.83 0.79

0.86 0.84 0.85 0.93 0.79 0.82 0.86 0.82

0.92 0.94 0.93 0.96 0.90 0.90 0.94 0.91

0.91 0.88 0.90 0.95 0.86 0.86 0.90 0.87

0.87 0.85 0.86 0.95 0.80 0.82 0.88 0.81

(a)

C90

B0

B90

B180

0.99 1.01 1.03 1.02 1.02 1.11 1.01 1.13 1.04 0.05

1.01 1.11 1.06 1.06 0.98 1.14 0.99 1.07 1.05 0.05

1.06 1.24 1.05 0.98 1.08 0.98 1.15 0.98 1.06 0.09

1.05 1.12 1.12 1.01 1.04 1.09 1.04 1.05 1.06 0.04

0.98 0.96 1.01 1.00 1.10 1.14 0.97 0.97 1.02 0.07

(b)

6.0

SCFψ,FE

6.0

SCF0: by FE analysis

4.5

4.5

3.0

3.0

Brace,max

1.5

0.0

C180

SCFTest 1.5

3.0

4.5

SCF0 : by ML formulae

Brace,max

1.5

Chord,max

0.0

SCFψ,ML

0.0

6.0

Chord,max SCFTest 0.0

1.5

3.0

4.5

6.0

SCFψ,FE/SCFTest

Chord, max

Brace, max

SCFψ,ML/SCFTest

Chord, max

Brace, max

Mean CV

1.00 0.07

1.02 0.07

Mean CV

1.05 0.11

1.28 0.12

Fig. 24. Comparison of the maximum SCFs in CFRP-CHS K-joints between formulae and experimental results.

Funding acquisition. Haiming Zhou: Resources. Fei Xu: Resources.

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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement The authors are grateful for the financial support from Shanghai Civil Engineering Peak Discipline Program of Shanghai Municipal Education Commission (No. 2019010301). References [1] Wardenier J, Packer JA, Zhao XL, van der Veget GJ. Hollow sections in structural applications. 2nd ed. Delft, Netherlands: Bouwen met Staal; 2010. [2] Zhao XL. FRP-strengthened metallic structures. Boca Raton, FL, USA: Taylor & Francis; 2013. [3] Teng JG, Chen JF, Smith ST, Lam L. FRP: strengthened RC structures. Frontiers Phys 2002:266. [4] Teng JG, Chen JF, Smith ST, Lam L. Behaviour and strength of FRP-strengthened RC structures: a state-of-the-art review. Proc Inst Civil Engineers - Struct Build 2003;156:51–62. [5] Zhao XL, Motavalli M. A brief review of fatigue strengthening of metallic structures in civil engineering using fibre reinforced polymers. Adv Mater Res

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