# Experimental Study on Hysteretic Behavior of Double-Plate Reinforced Overlapped K-Joints

Authors: Wenwei Yang; Shuntao Li; Ruhao Yan; Yaqi Suo
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

The lifetime of hollow section tubular joints frequently can be shortened owing to the occurrence of the welded cracks and the plastic deformation of chords under the cyclic loading, because of the deficient radial bearing capacity of the steel tube. To avoid such failures, this paper proposes a novel method to strengthen the chord with double plates at the intersection of the chord and braces. To further investigate the efficiency of this strengthening method on hysteretic performance and energy depletion ability of the overlapped K-joints with hollow sections, two unreinforced K-joints and two reinforced K-joints were fabricated. By loading on the braces with collaborative cyclic loading, the joints failure modes, hysteresis curve, and skeleton curve were obtained. The bearing capacity, ductility, and energy depletion of the joints were assessed and the restoring force model of joints was proposed. The results show that the failure mode of the unreinforced joint is the plastic failure of the surface of the chord. For the K-RC1 (double-plate reinforced square hollow section tubular K-joints), cracks appeared at the junction weld between the through brace and the overlapped brace. However, cracks extended along the weld at the intersection of the chord and the through brace for K-CC1 (double-plate reinforced circular hollow section tubular K-joints). There is no obvious deformation on the chord surface of reinforced joints. Experimental results reveal that the mechanical properties of the joints can be improved effectively by such reinforcement measures and that the plastic deformation of the chord can also be restrained. Meanwhile, the reinforcement measures demonstrate the ability to avoid the risk of large stress concentration of the chord in the area where the braces and chords are intersected. The bearing capacity of the joint was increased; however, the ductility of the joint was weakened.


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## Body

## 1. Introduction

Hollow structural sections are characterized by light weight and high strength, excellent axial compression resistance and torque resistance, and beautiful artistic modeling, among others [1, 2]. Hence, hollow structural sections have been widely used in many popular structures such as terminal buildings, bridges, and large sports centers, in which circle, square, and ellipse are the primary shapes [3]. The lifetime of hollow section tubular joints frequently is shortened by the occurrence of the corrosion [4, 5], welded cracks [6], and the plastic deformation of chords under the cyclic loading because of the deficient radial bearing capacity of the steel tube [1, 7]. Relevant design specifications and guidelines for reinforced steel joints have also been proposed accordingly [8, 9]. As one of the most widely used steel tubular truss joints in the field of the building structure, overlapped K-joints are generated in the actual construction process of the steel tubular truss when two braces (overlap brace and through brace) intersect on the surface of a chord. There are many additional factors that affect the mechanical behavior of the overlapped joints, including the geometric parameter of the joints, the shape of brace and chord (circular, square, or rectangular), different reinforcement measures, and the hidden weld [10]. In overlapped K-joints, the overlap of the intersections of the brace (hidden invisible) can be welded to the chord or not welded [11]. Compared to the joints that do not weld the hidden welds, the joints of the welded hidden welds have the better bearing capacity, but the ductility and hysteresis performance are reduced [12]. Nevertheless, Chinese Code of Steel Structures Design does not specify whether such welds should be welded or not [13].If the hidden weld of overlapped K-joint is welded, it must be welded according to the welding sequence before the overlapped brace is assembled, which is difficult to achieve in the actual construction process. Based on this situation, the hidden welds of overlapped K-joints are usually not welded. However due to the lack of stiffness of the chord members, it is easy to cause plastic deformation of the circular hollow steel tubular truss (CHS). To prevent this unfavorable situation from happening on the unreinforced tubular joints, scholars have adopted different approaches such as pouring concrete into the steel tube [14], adding a double-plate outer chord [15] and adopting internal ring stiffeners [16]. Although the strengthening methods mentioned above have improved the bearing capacity of joints in varying degrees, there are also some new problems to be introduced. In-filled concrete in steel pipes can improve the bearing capacity and rigidity of the joints [17, 18] and also can better resist impact [19]; however, pouring concrete into the chords increases self-weight of joints and introduces additional construction problems such as debonding and voids [20]. Additionally, many studies [21–23] have shown that adopting internal ring stiffeners can effectively increase the static strength of the joints, which made the process of specimens’ preparation cumbersome. In summary, the reinforced joints with a double plate are very convenient and fast. It is to weld the cross brace to the double plate and then to weld the double plate on the steel tubular truss [24]. For example, some scholars [25, 26] have proposed to use the same thickness for the reinforcement plate and the chord for more economical purposes.At present, there are many static studies for the CHS K-joints. Kurobane et al. summarized a large amount of data on the results of tubular joint experiments [27]. Zhao et al.’s research on overlapped CHS K-joints [28] shows that whether the hidden weld has no significant influence on the bearing capacity of the K-joints, but it will affect the failure mode. Kim [29] proposed a new bearing capacity formula for the K-joints. In addition, many researchers [30–37] have carried out experimental studies on different types of steel tube joints and obtained many valuable conclusions.It can be seen from the above statements that it is not difficult to find that the static behavior of reinforced or unreinforced joints has been studied in detail. At the same time, the hysteretic performance of the joints also represents the energy dissipation performance [38–41], when the structure is subjected to earthquake or other vibrations. Compared to the static performance of joints, the research on joints hysteresis performance is rare, but it is very important to evaluate the joints. According to references [1, 42–44], the authors carried out the experimental tests on the hysteretic behavior of unreinforced joints and the results showed that the weld toe cracking is the main failure mode of unreinforced joints. In reference [24], through the hysteretic behavior test of tubular joints strengthened by double steel plate and filled concrete, it is found that although such a reinforcement pattern can greatly increase the bearing capacity of the joints, the ductility is actually reduced. In this paper, to further investigate the efficiency of this strengthening method on hysteretic performance and energy depletion ability of the overlapped K-joints, four overlapped K-joints under coordinated cyclic loading are experimentally studied. The failure modes, hysteresis curves, and skeleton curves of the joints were obtained. The bearing capacity, ductility, and energy depletion of the joints were estimated quantitatively, and the restoring force model of the joints is proposed based on this.


## 2. Experimental Design

### 2.1. Specimens Design

Two unreinforced overlapped K-joints and two double plates reinforced overlapped K-joints were designed in this test, according to the design guidelines [45]. First, weld the through brace to the chord, then weld the overlapped brace to the through brace and the chord. For reinforced joints, the plate was welded to the chord by surrounding welding, and then two braces were welded to the plate. Design of the overlapped K-joints is shown in Figure 1. In addition to the double plate, Q345 steel was used in the chords and braces. As shown in Table 1, “K-RC” means that the chord of the overlapped K-joint is square and the brace is circular, and “K-CC” means that the chord and brace are both circular. Geometrical dimensions of the chord and brace cross section were defined as D(B) × T and d × t, respectively, as well as the double plate D0(B0) × T0. Lc and L were the length of chord and double plate, respectively. Lb was the vertical length of the brace. θ is the angle between the axis of brace and chord. e is eccentricity, which is the distance between the axis of a chord and the intersection point of double braces’ axes(α = 2Lc/D(B), β = d/D(B), γ = D(B)/(2T), τ = t/T).Figure 1
Design of the overlapped K-joints. (a) K-RC or K-CC. (b) K-RC1or K-CC1.
(a)
(b)Table 1
Geometrical parameters of the overlapped K-joints.
Specimen
L
c (mm)
L
b (mm)
D(B) × T (mm)
d × t (mm)
L (mm)
D
0(B0) × T0 (mm)
θ
e (mm)
α
β
Γ
τ
K-RC
1200
450
200 × 8
95 × 5
—
—
45°
55
12
0.48
12.5
0.63
K-RC1
1200
450
200 × 8
95 × 5
450
212 × 6
45°
55
12
0.48
12.5
063
K-CC
1200
450
159 × 6
89 × 5
—
—
45°
35
15.1
0.56
13.3
0.83
K-CC1
1200
450
159 × 6
89 × 5
450
171×6
45°
35
15.1
0.56
13.3
0.83


### 2.2. Material Properties

Steels of different specifications are made into three standard samples of each group and standard tensile tests are carried out to determine the material properties of the steel members [46]. Finally, the results, including the yield strength (fy), tensile strength (fu), yield strain (εy), elongation (δ), and elasticity modulus (E), are shown in Table 2. In addition, the yielding strength and ultimate strength of the double plate is 368.7 MPa and 526.8 MPa.Table 2
Test result of steel tubular material properties.
Specification
f
y (MPa)
f
u (MPa)
E (GPa)
ε
y = fy/E
f
u/fy
δ (%)
200 × 200 × 8
391.1
533.0
226.2
0.0017
1.36
35.3
95 × 5
377.6
515.6
214.6
0.0017
1.37
25.9
159 × 6
386.5
531.5
222.4
0.0017
1.38
34.2
89 × 5
373.1
510.1
210.2
0.0018
1.37
26.6


### 2.3. Test Program

The loading devices used in the test include reaction frame, upright column, cross beam, transfer beam, test system, and loading system, as shown in Figure2. The specimens were placed in a common plane with the reaction frame and the left end plate of chord was fixed with the transfer beam and the right end plate was a sliding support. Two braces were articulated along their axis with two hydraulic jacks, which can provide tension and compression for 500 kN, respectively. The reciprocating load was exerted on the end plate of the braces by hydraulic jacks. That is, the magnitude of the load applied to the through brace and the overlapped brace is synchronous.Figure 2
Loading device.
(a)
(b)To eliminate the installation gap and check whether the instrument is normal, a 25%-yield load was applied to the specimen before formal loading. The force-displacement hybrid loading mode was adopted, and the loading scheme is shown in Figure3. The formal loading is divided into two stages: the first stage was force-controlled with three levels and the increment of each stage was 25% of the yield load with two circulations. The second stage was displacement-controlled. When the first stage is completed, the displacement Δy corresponding to the yield load of the brace is taken as the increment. Each level load circulated three times, until the specimen failure.Figure 3
Loading scheme.The measurement contents of the test include reaction force, displacement, local deformation of the specimen, stress distribution of brace, and strain distribution of the specimen. Each joint test specimen was equipped with five displacement transducers, 16 strain gauges, and 16 strain rosettes for simultaneous data acquisition. The layout of strain gauge and displacement transducer is shown in Figure4. Each specimen is equipped with strain gauge and T1–T16 strain rosettes to monitor the strain in the complex area, the distance of which from the weld was 20 mm. Strain rosettes were arranged parallel to the weld lines. The S1–S8 strain gauges were arranged in the circumferential direction of the brace and, respectively, located on the line 60 mm from line 1 and line 2. The S9–S16 strain gauges were arranged in the circumferential direction of the chord and located on the line 150 mm from the parallel line 3. The displacement transducers D1–D4 were used to monitor deformation of the braces relative to chord, which were installed on the top surface of the square chord, which were located on the line 50 mm from the parallel line 3. The displacement transducer D5 monitored the deformation of the chord’s bottom, which were located on the bottom surface of the square chord or circular chord.Figure 4
Measurement points of specimen.


## 2.1. Specimens Design

Two unreinforced overlapped K-joints and two double plates reinforced overlapped K-joints were designed in this test, according to the design guidelines [45]. First, weld the through brace to the chord, then weld the overlapped brace to the through brace and the chord. For reinforced joints, the plate was welded to the chord by surrounding welding, and then two braces were welded to the plate. Design of the overlapped K-joints is shown in Figure 1. In addition to the double plate, Q345 steel was used in the chords and braces. As shown in Table 1, “K-RC” means that the chord of the overlapped K-joint is square and the brace is circular, and “K-CC” means that the chord and brace are both circular. Geometrical dimensions of the chord and brace cross section were defined as D(B) × T and d × t, respectively, as well as the double plate D0(B0) × T0. Lc and L were the length of chord and double plate, respectively. Lb was the vertical length of the brace. θ is the angle between the axis of brace and chord. e is eccentricity, which is the distance between the axis of a chord and the intersection point of double braces’ axes(α = 2Lc/D(B), β = d/D(B), γ = D(B)/(2T), τ = t/T).Figure 1
Design of the overlapped K-joints. (a) K-RC or K-CC. (b) K-RC1or K-CC1.
(a)
(b)Table 1
Geometrical parameters of the overlapped K-joints.
Specimen
L
c (mm)
L
b (mm)
D(B) × T (mm)
d × t (mm)
L (mm)
D
0(B0) × T0 (mm)
θ
e (mm)
α
β
Γ
τ
K-RC
1200
450
200 × 8
95 × 5
—
—
45°
55
12
0.48
12.5
0.63
K-RC1
1200
450
200 × 8
95 × 5
450
212 × 6
45°
55
12
0.48
12.5
063
K-CC
1200
450
159 × 6
89 × 5
—
—
45°
35
15.1
0.56
13.3
0.83
K-CC1
1200
450
159 × 6
89 × 5
450
171×6
45°
35
15.1
0.56
13.3
0.83


## 2.2. Material Properties

Steels of different specifications are made into three standard samples of each group and standard tensile tests are carried out to determine the material properties of the steel members [46]. Finally, the results, including the yield strength (fy), tensile strength (fu), yield strain (εy), elongation (δ), and elasticity modulus (E), are shown in Table 2. In addition, the yielding strength and ultimate strength of the double plate is 368.7 MPa and 526.8 MPa.Table 2
Test result of steel tubular material properties.
Specification
f
y (MPa)
f
u (MPa)
E (GPa)
ε
y = fy/E
f
u/fy
δ (%)
200 × 200 × 8
391.1
533.0
226.2
0.0017
1.36
35.3
95 × 5
377.6
515.6
214.6
0.0017
1.37
25.9
159 × 6
386.5
531.5
222.4
0.0017
1.38
34.2
89 × 5
373.1
510.1
210.2
0.0018
1.37
26.6


## 2.3. Test Program

The loading devices used in the test include reaction frame, upright column, cross beam, transfer beam, test system, and loading system, as shown in Figure2. The specimens were placed in a common plane with the reaction frame and the left end plate of chord was fixed with the transfer beam and the right end plate was a sliding support. Two braces were articulated along their axis with two hydraulic jacks, which can provide tension and compression for 500 kN, respectively. The reciprocating load was exerted on the end plate of the braces by hydraulic jacks. That is, the magnitude of the load applied to the through brace and the overlapped brace is synchronous.Figure 2
Loading device.
(a)
(b)To eliminate the installation gap and check whether the instrument is normal, a 25%-yield load was applied to the specimen before formal loading. The force-displacement hybrid loading mode was adopted, and the loading scheme is shown in Figure3. The formal loading is divided into two stages: the first stage was force-controlled with three levels and the increment of each stage was 25% of the yield load with two circulations. The second stage was displacement-controlled. When the first stage is completed, the displacement Δy corresponding to the yield load of the brace is taken as the increment. Each level load circulated three times, until the specimen failure.Figure 3
Loading scheme.The measurement contents of the test include reaction force, displacement, local deformation of the specimen, stress distribution of brace, and strain distribution of the specimen. Each joint test specimen was equipped with five displacement transducers, 16 strain gauges, and 16 strain rosettes for simultaneous data acquisition. The layout of strain gauge and displacement transducer is shown in Figure4. Each specimen is equipped with strain gauge and T1–T16 strain rosettes to monitor the strain in the complex area, the distance of which from the weld was 20 mm. Strain rosettes were arranged parallel to the weld lines. The S1–S8 strain gauges were arranged in the circumferential direction of the brace and, respectively, located on the line 60 mm from line 1 and line 2. The S9–S16 strain gauges were arranged in the circumferential direction of the chord and located on the line 150 mm from the parallel line 3. The displacement transducers D1–D4 were used to monitor deformation of the braces relative to chord, which were installed on the top surface of the square chord, which were located on the line 50 mm from the parallel line 3. The displacement transducer D5 monitored the deformation of the chord’s bottom, which were located on the bottom surface of the square chord or circular chord.Figure 4
Measurement points of specimen.


## 3. Analysis of Experimental Results

### 3.1. Failure Modes

The failure modes of the specimens are shown in Figure5. The specimens are in the elastic status before yielding and there is no obvious deformation of the chords and braces. With the increase of load, the failure modes of the specimens were different after yielding.Figure 5
The failure modes of the specimens. (a) K-RC, (b) K-RC1, (c) K-CC, and (d) K-CC1.
(a)
(b)
(c)
(d)In the process of collaborative cyclic loading, the failure mode of K-RC and K-CC specimens without reinforcement can be divided into two stages namely: chord yielding and local cracking. With the increase of cyclic loading, the concave and convex deformation on the surface of the chords appears gradually as yield strength of the unreinforced joints being reached, due to different stress states imposed by the brace, as shown in Figures5(a) and 5(c). The failure modes of the reinforced joints and the unreinforced joints were very different. In terms of the K-RC1 joint, first of all, the cracks appeared along the weld at the intersection of the two braces and then the crack propagated along the weld to the saddle point (A point), with a faster-spreading speed than before, as shown in Figure 5(b). As for K-CC1 joint, the failure mode can be divided into two stages, namely, initial cracking and crack propagation. Cracks firstly appeared at point B under collaborative cyclic loading and then extended along the weld at the intersection of the chord and the through brace to point C, which is shown in Figure 5(d). In general, great changes have taken place in the failure modes of the joints reinforced by the double plate. The reinforcement measure of the double plate can significantly increase the mechanical properties of the K-joints, restrain the plastic deformation of the chord, and avoid the large stress concentration of the chord in the area where the braces and chords intersect.It is well known that repeated loads tend to cause local buckling of the structure leading to brittle failure. Furthermore, the stiffness of steel tube at weld connection is not continuous, which may become stress raisers leading to cracking in the weld toe. Such high stress raisers are also the main cause of brittle failure in this area and the welding quality also causes brittle failure, because in the process of welding the chord and the brace, a large welding residual tensile stress is generated along the direction of the weld toe, which causes such a welded joint to cause such brittle failure at the weld region. The experimental tests on the four specimens have verified the above discussion. After a period of loading, cracks could be observed at the welding seam at the intersection of a chord and brace, extending along the weld, which is coincident with the failure modes shown in Figures5(a)–5(d).


### 3.2. Hysteretic Curve

In order to evaluate the seismic capacity of overlapped K-joints under collaborative cyclic loading, it is essential to evaluate the hysteretic performance of the K-joints by the load-displacement curves. The hysteretic curves of the test specimens are shown in Figure6. The longitudinal axis is the load P exerted by the hydraulic jack, as shown in Figure 6, and the load of through brace is a positive number in tension and negative number in compression. The transverse axis is the axial displacement δ of the through brace, whose direction is the same as that of P.Figure 6
The hysteretic curves of the joints. (a) K-RC, (b) K-RC1, (c) K-CC, and (d) K-CC1.
(a)
(b)
(c)
(d)By analyzing the hysteretic curves of the K-joints under the reciprocating load, the following characteristics were observed: (1) the stiffness of joints hardly decreases until the weld cracks occur, which can be seen in Figure6. The area enveloped by the hysteretic curves is very small, and the hysteretic curves are narrow and slender in the elastic stage; the stiffness of the joints changes little, there is hardly residual deformation and the energy dissipating capacity of joints is poor. After yielding, the joints entered the inelastic stage, the slope of hysteretic curve decreased, the stiffness of the joints degenerated, and the area enveloped by the hysteretic loop increased gradually. (2) Compared to the reinforced joints with double plate, the hysteretic curves of the unreinforced joints have better plumpness and better energy dissipating capacity. Furthermore, there is a “pinch” phenomenon in the hysteretic curves of double-plate reinforced overlapped K-joints. (3) For reinforcement joints K-RC1 and K-CC1, the area of the hysteretic loop under tension is smaller than that under compression with the increase of cyclic displacement, which indicates that the braces under compression enter the energy dissipating stage earlier than the braces under tension. On the contrary, the areas of hysteretic loops with unreinforced joints are approximately equal under tension and compression.The bearing capacity increased continuously before the specimen yields. The comparison results of the bearing capacity of each joint are shown in Table3. The ratios of tensile ultimate load and compression ultimate load of reinforced and unreinforced joints are defined as rt and rc, respectively. Compared to the unreinforced joints, the ultimate bearing capacity of the reinforced joints is higher. For instance, the maximum tensile load and compressive load of the K-RC1 are 16.8% and 63.5% higher than that of the joints K-RC, and the maximum tensile load and compressive load of K-RC1 are 13.9% and 95.0% higher than that of the joints K-CC, respectively. In terms of the unreinforced joints, the bearing capacity of both K-RC is slightly higher than that of the K-CC. Conversely, the bearing capacity of the K-RC1 both is smaller than that of the K-CC1. The above discussion shows that the hysteretic behavior and load-bearing capacity of overlapped K-joints with different cross section forms, strengthened by double plate, are different.Table 3
The control load and displacement ductility coefficient of joints.
Specimen
Tension
Compression
r
t (%)
r
c (%)
δ
u (mm)
δ
y (mm)
μ
P
u (kN)
P
y (kN)
P
u (kN)
P
y (kN)
K-RC
245.1
198.2
−230.5
−190.8
—
—
13.5
6.1
2.21
K-RC1
286.2
260.5
−376.8
−328.7
16.8
63.5
16.4
9.7
1.69
K-CC
240.2
168.3
−221.9
−172.5
—
—
12.5
5.6
2.23
K-CC1
334.5
312.8
−432.7
−389.2
13.9
95.0
17.1
10.2
1.68


### 3.3. Skeleton Curve and Ductility Coefficient

The skeleton curves of the joints are plotted in the same coordinate system as shown in Figure7. As can be seen from the figure, the load-displacement curve is approximated to a straight line when the joint is in the elastic status. With the increase of cyclic displacement, the unreinforced joints enter the inelastic stage earlier than the double-plate reinforced joints. In the experiment, the plate was welded to the chord only by surrounding welding, and the intermediate bonding surface was not welded. Therefore, under cyclic loading, the deformation of the plate causes a slight separation between the plate and the chord, which is why the initial stiffness of the reinforced joint is smaller than that of the unreinforced joint. Of course, this also makes the deformation capacity of the reinforced joints better.Figure 7
The skeleton curve.The double-plate reinforcement has improved the strength of the overlapped K-joints and the improvement of compressive capacity is more obvious. The ultimate capacityPu in tension of the K-RC1 and K-CC1 are, respectively, 16.8% and 13.9% higher than the K-RC and K-CC. However, the ultimate capacity Pu in compression of the K-RC1 and K-CC1 are, respectively, 63.5% and 95.0% higher than the K-RC and K-CC. The above phenomenon is caused by the closure of cracks in the braces of the joints under compression. Hence, the compressive strength of joints increases more than the tensile strength. As far as the section form of the chord is concerned, the difference between the ultimate capacity in tension and compression of K-RC and K-CC is not more than 5%. The tensile and compressed yield load (Py) of K-RC are 17.8% and 10.5% higher than that of the joints K-CC, while the tensile and compressive yield load(Py) of K-RC1 are 20.1% and 18.6% smaller than that of the joints K-CC1.The ductility of the structure determines its seismic capacity and the displacement ductility ratioμ is used to evaluate the deformability of the K-joints. The ductility coefficient μ is defined as δu/δy, δu, and δy are the ultimate displacement and the yield displacement of a joint, respectively, [1] and the calculation results of μ are shown in Table 3. As shown in the calculation results, the displacement ductility coefficient of unreinforced joints are almost equal, as well as the reinforced joints. Furthermore, the displacement ductility coefficients of unreinforced joints are about 1.3 times of those of K-RC1 and K-CC1. The yield displacement and ultimate displacement of reinforced joints are obviously larger than those of unreinforced joints. In terms of seismic resistance, the ultimate bearing capacity of the double-plate reinforced joints are significantly increased, while the deformation capacity of the joints are slightly reduced. Therefore, for the seismic resistance of double-plate reinforced K-joints, this reinforcement is still an effective and worthwhile reinforcement.


### 3.4. Energy Dissipation

Energy dissipation capacity is an important index in evaluation of mechanical properties of the K-joints. The energy dissipating capacity of joints is measured by the area enveloped byP-δ hysteretic curves. In this paper, the energy dissipation factor ηa is used as the evaluation index of earthquake resistance of the K-joints, which is defined as the following equation (1). As shown in Figure 8, where SABC and SCDA are the total area surrounded by the hysteresis curve of the outermost circle, that is, the sum of tensile energy and compression energy. SOBE and SODF are the dissipated linear energy in tensile and compressive stages, respectively. Furthermore, the equivalent viscous damping ratio (he) is introduced, which is shown in equation (2).(1)ηa=SABC+SCDASΔOBE+SΔODF,(2)he=ηa2π.Figure 8
Typical hysteretic curve.In addition, the cumulative energy dissipation coefficientET and the cumulative energy dissipation ratio η are introduced to measure the cumulative energy dissipation during the whole loading process. The relevant formulas are shown below:(3)ET=∑i=1nEi++Ei−,(4)η=ETEy,(5)Ey=Py×δy2.In equation (3), n is the number of times of cyclic loading, Ei+ and Ei− are the energy dissipations under tension and compression in the ith loading, respectively. Ey is the nominal elastic potential energy of overlapped K-joints [1], which can be calculated by the yield load Py and yield displacement δy in Figure 7.The results of the energy dissipation of the overlapped K-joints are shown in Table4. ηa and he of reinforced K-joints are smaller than those of unreinforced K-joints. In addition, the he of the unreinforced K-joints under collaborative cyclic loading is close to 0.3, which indicates that the unreinforced K-joints have good energy dissipation capability. The cumulative energy dissipation coefficients (ET) of the reinforced K-joints are higher than those of the unreinforced joints. On the contrary, the accumulative energy dissipation ratio (η) of the unreinforced K-joints is greater than that of the reinforced joints. For instance, the accumulative energy dissipation ratio of the K-RC is 92.2% higher than that of the joint K-RC1 and the K-CC is 100.5% higher than for the K-CC1. As far as the section form of the chord is concerned, the accumulative energy dissipation ratio (η) of K-RC is 6.5% smaller than that of K-CC, while the accumulative energy dissipation ratio (η) of K-RC1 is 8.5% higher than that of K-CC1, even if the strengthening measures of adding double plate effectively has improved the ultimate bearing capacity of overlapped K-joints. However, the effect of this strengthening method on the seismic performance of overlapped K-joints is different. The energy dissipation capacity of reinforced joints is smaller than the unreinforced joints.Table 4
Energy dissipation details of the joints.
Specimen
S
(ABC+CDA) (kN·mm)
S
Δ(OBE+ODF) (kN·mm)
η
a
h
e
E
T (kN·mm)
E
y (kN·mm)
η
K-RC
4862.8
2796.5
1.74
0.278
22369.7
605.6
36.9
K-RC1
3720.6
5168.6
0.72
0.115
27319.5
1428.9
19.2
K-CC
4510.4
2684.7
1.68
0.268
19176.5
486.8
39.34
K-CC1
5848.7
6410.8
0.91
0.145
31554.6
1787.8
17.7Based on the cumulative energy dissipating coefficientET in Table 4, the relationship curve between the cumulative energy dissipating coefficient and displacement is obtained, as shown in Figure 9. The longitudinal axis is the cumulative energy dissipating coefficient ET, which corresponds to the cumulative energy consumption under this displacement and the transverse axis is the average of the absolute values of compression and tension displacement. Figure 9 shows that the energy dissipating coefficient ET of the joints is small before yielding and increases as the displacement increases, which indicates that the energy dissipating capacity of the joint is poor. After the yielding occurred, the energy dissipating coefficient of the joint became bigger gradually with the increase of control displacement, which indicates that the joint has excellent plastic energy dissipating capacity.Figure 9
The relationship curve ofE and displacement.


### 3.5. Load Transfer and Failure Mechanism

As stated above, failure modes, hysteretic performance, bearing capacity, and ductility of joints vary with the presence of double plates. To clarify the influence of double plates on the mechanical properties of K-joints in detail, K-RC and K-RC1 are selected to analyze and compare their load transfer schematics in Figure10. Because all the failures of specimens were caused by cracking under tension, the load transfer path of the through brace under tension load is discussed. The force transfer schematics of the joints with circular chord are similar to that of the joints with the square chord. As shown in Figure 10, only the transfer schematics of K-joints with square chord were given.Figure 10
The load transfer mechanism of the through brace under tension. (a) K-RC and (b) K-RC1.
(a)
(b)In terms of the unreinforced K-joint, the tensile load on the through brace is decomposed into horizontal and vertical directions, both of which are 0.707P in magnitude. Then, the vertical force acts directly on the upper surface of the chord. Meanwhile, the overlap brace is in state of compression. Ultimately, because the radial stiffness of the chord is weak, concave, and convex deformations occur on the surface of a square steel tube. At this time, the damage of the weld between the through and the overlap brace is not obvious, only small cracks appear. For the reinforced joint K-RC1, it is different from the load transfer mechanism of K-RC. Because there is the double plate between the chord and the brace of K-RC1, so the load is transferred by the double plate. Since the plate is connected to the chord by the surrounding weld, the load is transmitted to the chord through the weld. When the through brace is subjected to an upward load, the weld is subjected to an eccentric load in the vertical direction, which is also the cause of the additional bending moment. Therefore, hysteretic performance and failure modes of joints are inevitably affected by the double plate. On the one hand, because of the presence of the double plate, the additional bending moments M are generated at the connection between the through brace and the double plate when the through brace is stretched, as shown in Figure 10(b). On the other hand, compressed overlapped brace and tensioned through brace are subjected to a force of 0.707P equal in size and opposite in direction, which means that the weld between the through and the overlap braces is subjected to vertical upward and downward forces with 0.354P in size, respectively. In summary, the two factors stated above both contribute to the cracking and expansion of weld between the through and the overlapped brace, which is in good agreement with the failure mode of the strengthened joints shown in Figure 5.


## 3.1. Failure Modes

The failure modes of the specimens are shown in Figure5. The specimens are in the elastic status before yielding and there is no obvious deformation of the chords and braces. With the increase of load, the failure modes of the specimens were different after yielding.Figure 5
The failure modes of the specimens. (a) K-RC, (b) K-RC1, (c) K-CC, and (d) K-CC1.
(a)
(b)
(c)
(d)In the process of collaborative cyclic loading, the failure mode of K-RC and K-CC specimens without reinforcement can be divided into two stages namely: chord yielding and local cracking. With the increase of cyclic loading, the concave and convex deformation on the surface of the chords appears gradually as yield strength of the unreinforced joints being reached, due to different stress states imposed by the brace, as shown in Figures5(a) and 5(c). The failure modes of the reinforced joints and the unreinforced joints were very different. In terms of the K-RC1 joint, first of all, the cracks appeared along the weld at the intersection of the two braces and then the crack propagated along the weld to the saddle point (A point), with a faster-spreading speed than before, as shown in Figure 5(b). As for K-CC1 joint, the failure mode can be divided into two stages, namely, initial cracking and crack propagation. Cracks firstly appeared at point B under collaborative cyclic loading and then extended along the weld at the intersection of the chord and the through brace to point C, which is shown in Figure 5(d). In general, great changes have taken place in the failure modes of the joints reinforced by the double plate. The reinforcement measure of the double plate can significantly increase the mechanical properties of the K-joints, restrain the plastic deformation of the chord, and avoid the large stress concentration of the chord in the area where the braces and chords intersect.It is well known that repeated loads tend to cause local buckling of the structure leading to brittle failure. Furthermore, the stiffness of steel tube at weld connection is not continuous, which may become stress raisers leading to cracking in the weld toe. Such high stress raisers are also the main cause of brittle failure in this area and the welding quality also causes brittle failure, because in the process of welding the chord and the brace, a large welding residual tensile stress is generated along the direction of the weld toe, which causes such a welded joint to cause such brittle failure at the weld region. The experimental tests on the four specimens have verified the above discussion. After a period of loading, cracks could be observed at the welding seam at the intersection of a chord and brace, extending along the weld, which is coincident with the failure modes shown in Figures5(a)–5(d).


## 3.2. Hysteretic Curve

In order to evaluate the seismic capacity of overlapped K-joints under collaborative cyclic loading, it is essential to evaluate the hysteretic performance of the K-joints by the load-displacement curves. The hysteretic curves of the test specimens are shown in Figure6. The longitudinal axis is the load P exerted by the hydraulic jack, as shown in Figure 6, and the load of through brace is a positive number in tension and negative number in compression. The transverse axis is the axial displacement δ of the through brace, whose direction is the same as that of P.Figure 6
The hysteretic curves of the joints. (a) K-RC, (b) K-RC1, (c) K-CC, and (d) K-CC1.
(a)
(b)
(c)
(d)By analyzing the hysteretic curves of the K-joints under the reciprocating load, the following characteristics were observed: (1) the stiffness of joints hardly decreases until the weld cracks occur, which can be seen in Figure6. The area enveloped by the hysteretic curves is very small, and the hysteretic curves are narrow and slender in the elastic stage; the stiffness of the joints changes little, there is hardly residual deformation and the energy dissipating capacity of joints is poor. After yielding, the joints entered the inelastic stage, the slope of hysteretic curve decreased, the stiffness of the joints degenerated, and the area enveloped by the hysteretic loop increased gradually. (2) Compared to the reinforced joints with double plate, the hysteretic curves of the unreinforced joints have better plumpness and better energy dissipating capacity. Furthermore, there is a “pinch” phenomenon in the hysteretic curves of double-plate reinforced overlapped K-joints. (3) For reinforcement joints K-RC1 and K-CC1, the area of the hysteretic loop under tension is smaller than that under compression with the increase of cyclic displacement, which indicates that the braces under compression enter the energy dissipating stage earlier than the braces under tension. On the contrary, the areas of hysteretic loops with unreinforced joints are approximately equal under tension and compression.The bearing capacity increased continuously before the specimen yields. The comparison results of the bearing capacity of each joint are shown in Table3. The ratios of tensile ultimate load and compression ultimate load of reinforced and unreinforced joints are defined as rt and rc, respectively. Compared to the unreinforced joints, the ultimate bearing capacity of the reinforced joints is higher. For instance, the maximum tensile load and compressive load of the K-RC1 are 16.8% and 63.5% higher than that of the joints K-RC, and the maximum tensile load and compressive load of K-RC1 are 13.9% and 95.0% higher than that of the joints K-CC, respectively. In terms of the unreinforced joints, the bearing capacity of both K-RC is slightly higher than that of the K-CC. Conversely, the bearing capacity of the K-RC1 both is smaller than that of the K-CC1. The above discussion shows that the hysteretic behavior and load-bearing capacity of overlapped K-joints with different cross section forms, strengthened by double plate, are different.Table 3
The control load and displacement ductility coefficient of joints.
Specimen
Tension
Compression
r
t (%)
r
c (%)
δ
u (mm)
δ
y (mm)
μ
P
u (kN)
P
y (kN)
P
u (kN)
P
y (kN)
K-RC
245.1
198.2
−230.5
−190.8
—
—
13.5
6.1
2.21
K-RC1
286.2
260.5
−376.8
−328.7
16.8
63.5
16.4
9.7
1.69
K-CC
240.2
168.3
−221.9
−172.5
—
—
12.5
5.6
2.23
K-CC1
334.5
312.8
−432.7
−389.2
13.9
95.0
17.1
10.2
1.68


## 3.3. Skeleton Curve and Ductility Coefficient

The skeleton curves of the joints are plotted in the same coordinate system as shown in Figure7. As can be seen from the figure, the load-displacement curve is approximated to a straight line when the joint is in the elastic status. With the increase of cyclic displacement, the unreinforced joints enter the inelastic stage earlier than the double-plate reinforced joints. In the experiment, the plate was welded to the chord only by surrounding welding, and the intermediate bonding surface was not welded. Therefore, under cyclic loading, the deformation of the plate causes a slight separation between the plate and the chord, which is why the initial stiffness of the reinforced joint is smaller than that of the unreinforced joint. Of course, this also makes the deformation capacity of the reinforced joints better.Figure 7
The skeleton curve.The double-plate reinforcement has improved the strength of the overlapped K-joints and the improvement of compressive capacity is more obvious. The ultimate capacityPu in tension of the K-RC1 and K-CC1 are, respectively, 16.8% and 13.9% higher than the K-RC and K-CC. However, the ultimate capacity Pu in compression of the K-RC1 and K-CC1 are, respectively, 63.5% and 95.0% higher than the K-RC and K-CC. The above phenomenon is caused by the closure of cracks in the braces of the joints under compression. Hence, the compressive strength of joints increases more than the tensile strength. As far as the section form of the chord is concerned, the difference between the ultimate capacity in tension and compression of K-RC and K-CC is not more than 5%. The tensile and compressed yield load (Py) of K-RC are 17.8% and 10.5% higher than that of the joints K-CC, while the tensile and compressive yield load(Py) of K-RC1 are 20.1% and 18.6% smaller than that of the joints K-CC1.The ductility of the structure determines its seismic capacity and the displacement ductility ratioμ is used to evaluate the deformability of the K-joints. The ductility coefficient μ is defined as δu/δy, δu, and δy are the ultimate displacement and the yield displacement of a joint, respectively, [1] and the calculation results of μ are shown in Table 3. As shown in the calculation results, the displacement ductility coefficient of unreinforced joints are almost equal, as well as the reinforced joints. Furthermore, the displacement ductility coefficients of unreinforced joints are about 1.3 times of those of K-RC1 and K-CC1. The yield displacement and ultimate displacement of reinforced joints are obviously larger than those of unreinforced joints. In terms of seismic resistance, the ultimate bearing capacity of the double-plate reinforced joints are significantly increased, while the deformation capacity of the joints are slightly reduced. Therefore, for the seismic resistance of double-plate reinforced K-joints, this reinforcement is still an effective and worthwhile reinforcement.


## 3.4. Energy Dissipation

Energy dissipation capacity is an important index in evaluation of mechanical properties of the K-joints. The energy dissipating capacity of joints is measured by the area enveloped byP-δ hysteretic curves. In this paper, the energy dissipation factor ηa is used as the evaluation index of earthquake resistance of the K-joints, which is defined as the following equation (1). As shown in Figure 8, where SABC and SCDA are the total area surrounded by the hysteresis curve of the outermost circle, that is, the sum of tensile energy and compression energy. SOBE and SODF are the dissipated linear energy in tensile and compressive stages, respectively. Furthermore, the equivalent viscous damping ratio (he) is introduced, which is shown in equation (2).(1)ηa=SABC+SCDASΔOBE+SΔODF,(2)he=ηa2π.Figure 8
Typical hysteretic curve.In addition, the cumulative energy dissipation coefficientET and the cumulative energy dissipation ratio η are introduced to measure the cumulative energy dissipation during the whole loading process. The relevant formulas are shown below:(3)ET=∑i=1nEi++Ei−,(4)η=ETEy,(5)Ey=Py×δy2.In equation (3), n is the number of times of cyclic loading, Ei+ and Ei− are the energy dissipations under tension and compression in the ith loading, respectively. Ey is the nominal elastic potential energy of overlapped K-joints [1], which can be calculated by the yield load Py and yield displacement δy in Figure 7.The results of the energy dissipation of the overlapped K-joints are shown in Table4. ηa and he of reinforced K-joints are smaller than those of unreinforced K-joints. In addition, the he of the unreinforced K-joints under collaborative cyclic loading is close to 0.3, which indicates that the unreinforced K-joints have good energy dissipation capability. The cumulative energy dissipation coefficients (ET) of the reinforced K-joints are higher than those of the unreinforced joints. On the contrary, the accumulative energy dissipation ratio (η) of the unreinforced K-joints is greater than that of the reinforced joints. For instance, the accumulative energy dissipation ratio of the K-RC is 92.2% higher than that of the joint K-RC1 and the K-CC is 100.5% higher than for the K-CC1. As far as the section form of the chord is concerned, the accumulative energy dissipation ratio (η) of K-RC is 6.5% smaller than that of K-CC, while the accumulative energy dissipation ratio (η) of K-RC1 is 8.5% higher than that of K-CC1, even if the strengthening measures of adding double plate effectively has improved the ultimate bearing capacity of overlapped K-joints. However, the effect of this strengthening method on the seismic performance of overlapped K-joints is different. The energy dissipation capacity of reinforced joints is smaller than the unreinforced joints.Table 4
Energy dissipation details of the joints.
Specimen
S
(ABC+CDA) (kN·mm)
S
Δ(OBE+ODF) (kN·mm)
η
a
h
e
E
T (kN·mm)
E
y (kN·mm)
η
K-RC
4862.8
2796.5
1.74
0.278
22369.7
605.6
36.9
K-RC1
3720.6
5168.6
0.72
0.115
27319.5
1428.9
19.2
K-CC
4510.4
2684.7
1.68
0.268
19176.5
486.8
39.34
K-CC1
5848.7
6410.8
0.91
0.145
31554.6
1787.8
17.7Based on the cumulative energy dissipating coefficientET in Table 4, the relationship curve between the cumulative energy dissipating coefficient and displacement is obtained, as shown in Figure 9. The longitudinal axis is the cumulative energy dissipating coefficient ET, which corresponds to the cumulative energy consumption under this displacement and the transverse axis is the average of the absolute values of compression and tension displacement. Figure 9 shows that the energy dissipating coefficient ET of the joints is small before yielding and increases as the displacement increases, which indicates that the energy dissipating capacity of the joint is poor. After the yielding occurred, the energy dissipating coefficient of the joint became bigger gradually with the increase of control displacement, which indicates that the joint has excellent plastic energy dissipating capacity.Figure 9
The relationship curve ofE and displacement.


## 3.5. Load Transfer and Failure Mechanism

As stated above, failure modes, hysteretic performance, bearing capacity, and ductility of joints vary with the presence of double plates. To clarify the influence of double plates on the mechanical properties of K-joints in detail, K-RC and K-RC1 are selected to analyze and compare their load transfer schematics in Figure10. Because all the failures of specimens were caused by cracking under tension, the load transfer path of the through brace under tension load is discussed. The force transfer schematics of the joints with circular chord are similar to that of the joints with the square chord. As shown in Figure 10, only the transfer schematics of K-joints with square chord were given.Figure 10
The load transfer mechanism of the through brace under tension. (a) K-RC and (b) K-RC1.
(a)
(b)In terms of the unreinforced K-joint, the tensile load on the through brace is decomposed into horizontal and vertical directions, both of which are 0.707P in magnitude. Then, the vertical force acts directly on the upper surface of the chord. Meanwhile, the overlap brace is in state of compression. Ultimately, because the radial stiffness of the chord is weak, concave, and convex deformations occur on the surface of a square steel tube. At this time, the damage of the weld between the through and the overlap brace is not obvious, only small cracks appear. For the reinforced joint K-RC1, it is different from the load transfer mechanism of K-RC. Because there is the double plate between the chord and the brace of K-RC1, so the load is transferred by the double plate. Since the plate is connected to the chord by the surrounding weld, the load is transmitted to the chord through the weld. When the through brace is subjected to an upward load, the weld is subjected to an eccentric load in the vertical direction, which is also the cause of the additional bending moment. Therefore, hysteretic performance and failure modes of joints are inevitably affected by the double plate. On the one hand, because of the presence of the double plate, the additional bending moments M are generated at the connection between the through brace and the double plate when the through brace is stretched, as shown in Figure 10(b). On the other hand, compressed overlapped brace and tensioned through brace are subjected to a force of 0.707P equal in size and opposite in direction, which means that the weld between the through and the overlap braces is subjected to vertical upward and downward forces with 0.354P in size, respectively. In summary, the two factors stated above both contribute to the cracking and expansion of weld between the through and the overlapped brace, which is in good agreement with the failure mode of the strengthened joints shown in Figure 5.


## 4. Restoring Force Model of K-Joints

The Clough [47] model and the Takeda [48] model are the classical restoring force models in the field of seismic engineering. Based on the existing restoring force model, the hysteretic curve of the joint is fitted by an envelope curve. Firstly, the fitted skeleton curves are shown in Figure 11. The dimensionless parameter δ/δu is defined as the transverse axis and P/Pu is defined as the longitudinal axis. Pu represents the ultimate bearing capacity of joints and δu is the ultimate displacement of joints under the cyclic loading. As shown in Figure 11, the skeleton curve of the joints is a double-folding line model and the joints undergo elastic and strengthening stages under the cyclic loading. The restoring force model of joints is shown in Figure 12 and the polyline function expression of the restoring force model is as follows:(6)Line segment 0A:PPu=keδδu,P≤Py,(7)Line segment AB:PPu=kpδδu+α,Pu≥P≥Py,(8)Line segment BF:PPu=kdδδu+β,Pu≥P≥Py,where ke denotes the elastic stiffness of joints; kp denotes the elastoplasticity stiffness of joints; and kd denotes the residual stiffness of joints after degenerating and is equal to the elastic stiffness ke. According to the double-folding line model, the loading and unloading stiffness of the joints can be calculated by fitting the hysteretic curve obtained from the test. Finally, the restoring force model of joints is proposed, which can be seen in Figure 12.Figure 11
Loading and unloading stiffness of the joints.
(a)
(b)Figure 12
Restoring force model of the joints. (a) K-RC, (b) K-CC, (c) K-RC1, and (d) K-CC1.
(a)
(b)
(c)
(d)As shown in Figure12, the coincidence degree between the restoring force models of unreinforced joints and the hysteretic curves obtained from tests is higher than that of reinforced joints. In terms of the double-plate reinforced K-joints, although the bearing capacity of joints has been improved obviously due to the existence of the double plate, the opening of welds is not conducive to the improvement of tensile strength. Based on this situation, the restoring force models of the reinforced joints is modified and the ultimate tensile strength of reinforced joints is reduced by 30%, which caused that the compressed capacity of reinforced joints is higher than the tensile capacity of reinforced joints. Therefore, the restoring force models of reinforced joints will not assume a symmetrical shape; as a whole, which is consistent with the results of the test above. As shown in Table 5, the average values of the absolute values of the tension and compression bearing capacity are taken as the bearing capacity of the joints. The difference between the model results and the test results is not more than 13.0%. Compared to the unreinforced joints, the bearing capacity of the reinforced joints in test is smaller than that of the restoring force models. But the error between them is less than 2%. It shows that the restoring force model is reliable for the double-plate reinforced overlapped K-joints.Table 5
Comparison between test results and model results.
Specimen
Test results (kN)
Model results (kN)
Error (%)
Tension
Compression
Average
Tension
Compression
Average
K-RC
245.1
−230.5
237.8
270.2
−254.7
262.5
10.3
K-RC1
286.2
−376.8
331.5
251.6
−404.2
327.9
1.1
K-CC
240.2
−221.9
231.1
268.1
−252.1
260.1
12.5
K-CC1
334.5
−432.7
383.6
313.6
−443.8
378.7
1.3


## 5. Conclusions

The quasistatic loading scheme was adopted to carry out the reciprocating loading tests for the four overlapped K-joints along the axis of the brace. The experimental results and the restoring force model are compared and analyzed. Influence of the double-plate reinforced stress on the failure mode and hysteretic behavior of the joints was explored. The conclusions are as follows:(1)
Failure of unreinforced joints occurs at the weld position on the chord surface and the tiny cracks appeared along the weld. For the K-RC1 joint, cracks appeared at the junction weld between the through brace and the overlap brace. However, cracks extended along with the weld at the intersection of the chord and the through brace for K-CC1. There is no obvious deformation on the chord surface of reinforced joints.(2)
The double-plate reinforced measures effectively restrain the plastic deformation of the chord and avoid the large stress concentration of the chord in the intersecting area of the brace and chord. The ultimate capacity (Pu) in tension of K-RC1 and K-CC1 are, respectively, 16.8% and 13.9% higher than the corresponding unreinforced joints. The ultimate capacity (Pu) in compression of K-RC1 and K-CC1 are 63.5% and 95.0% higher than the corresponding unreinforced joints. But the double-plate reinforced measures reduce the ductility of the joint by 30%. In addition, compared to the unreinforced joints, energy dissipation capacity of reinforced joints is smaller.(3)
According to the double-folding line model, the loading and unloading stiffness of the joints can be calculated by fitting the hysteretic curve obtained from the test. And, based on the test results, the restoring force model of joints is proposed. The error between the bearing capacity of the reinforced joints in the test and the restoring force model is less than 2%. It shows that the restoring force model is reliable for the double-plate reinforced overlapped K-joints



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*Source: 3183206-2020-01-03.xml*