Study on shear capacity of connections with external stiffening rings between square steel tubular columns and steel beams

Abstract

This paper studied the shear behavior of the connections with external stiffening rings between square steel tubular columns and steel beams by experimental, numerical and analytical methods. Two connections with external stiffening rings were tested under low cyclic loading to investigate the effect of axial compression ratio on the shear behavior and capacity of the connection. The test result showed that the change of the axial compression ratio had little effect on the shear capacity of the connection while the ductility of the connection was decreasing with the increase of the axial compression ratio. Seven nonlinear finite element models were designed to investigate the seismic behavior of the connection under cyclic test. Parametric studies are carried out to study the influence of the following parameters on the shearing capacity and deformation in panel zone: the width and the height of the steel tube in panel zone and the thickness of the external stiffening rings. Finally, based on the model considering the post-buckling strength of the web of the steel tube in panel zone, a calculation formula was fitted by the results of the finite element simulation.

Keywords

connection external stiffening ring square steel tubular column shear capacity calculation method

Introduction

There are various forms of connections between square steel tubular column and steel beam, such as square stiffener connection, T-stiffener connection, strengthening bolt connection and so on, among which, connection with external stiffening rings, is one of the most popular connections for keeping the integrity of the steel tubular column and constructing convenience. After the Hyogoken-Nanbu Earthquake in 1995, the researchers (Nakashima, 1998) found that the shear failure in the panel zone of the connection with external stiffening rings was one of the main damage forms. In view of the shear failure modes, a lot of researches have been carried out and a series of design methods have been developed.

According to the results of low cyclic loading test of four interior diaphragm connections, Fukumoto (2001) proposed that the shear capacity of interior diaphragm connection was composed of three parts: the web of the steel tube in panel zone, the flange of the steel tube in panel zone and concrete compression strut. On this basis, the formula of the shear capacity of the connection was derived using the principle of virtual work, and it was also considered to be applicable to the connection with external stiffening rings. Morino (1992) carried out low cyclic loading tests on 10 connections with external stiffening rings and interior diaphragm connections between rectangular concrete-filled steel tubular columns and H-type steel beams separately. The test results showed that the connection which occurred bending failure at the end of the column has better energy dissipation ability than those occurring shear failure in panel zone. Fukumoto and Morita (2005) studied the connections with external stiffening rings and interior diaphragm connections made of high strength materials. The research parameters were section form, weak component type, the diameter thickness ratio of steel tubular column and material strength. According to the experimental results, the shear force- displacement model and the calculation method of shear capacity in panel zone of the connections were put forward. Nie et al. (2006, 2008), Nie and Qin (2007, 2008) studied the seismic performance and shear capacity of the connection with external stiffening rings and the interior diaphragm connection. Based on the principles of deformation compatibility in the elastic, yield, strengthening and damage stages of the steel and concrete in panel zone, the principle of virtual work was used to derive the calculation method of the shear capacity of the panel zone in connection. Wang et al. (2018) developed a finite element (FE) model to evaluate the behavior of steel beam-to-column joints subjected to bending moments composed of hollow structural section beams and columns. Then, a parametric analysis was carried out to investigate the effects of steel yield strength, end-plate thickness, beam thickness, column wall thickness, bolt diameter, number of bolts and location. The result shows the end-plate thickness and column wall thickness have a significant influence on the joint behavior, and the layout of double bolt-rows in tension is recommended for joints with extended end-plates. Finally, an analytical model was derived based on the component method to predict the moment-rotation relationships for the sub-assemblies with extended end-plates. Du et al. (2018) adopted six finite element analysis models for the panel zone of connections between concrete-filled square steel tubular columns and steel–concrete composite beams with external diaphragms under cyclic loading to study the influence of the ratio of width to thickness of column, the concrete strength of panel zone, the axial load ratio of the column and the ratio of depth to width of panel zone on the shearing capacity and deformation in panel zone. Finally, a constitutive model of shear performance is then proposed for the panel zone of the composite connections. Cao et al. (2018) presented a new type of connection called bottom-through-diaphragm and top-ring connection to square steel tubular (SST) columns. Three different configurations were test to investigate the seismic behavior of this new connection. The failure modes, hysteretic curves, rotation capacity, stiffness degradation, skeleton curves, ductility, and energy dissipation of these connections were analyzed. The experimental results indicated that the new bottom-through-diaphragm and top-ring connection to square steel tubular columns exhibited stable and plump hysteretic curves, favorable ductility, and excellent energy dissipation. However, the above calculation methods and models did not take into account the post-buckling strength of the web of the steel tube in panel zone, and some even did not consider the contribution of the flange of the steel tube in panel zone to the shear capacity of connection, which did not match the experimental phenomenon and will underestimate the shear capacity of the connection with external stiffening rings. Thus, an analytical model which can be used to properly predict the shear capacity of connection with external stiffening rings is still necessary.

In this paper, two connections with external stiffening rings between square steel tubular columns and steel beams were tested under low cyclic loading to investigate the effect of axial compression ratio on the shear behavior and the shear capacity of the connection. Nonlinear finite element models were developed to service the calculation method. Based on experiments and finite element simulation analysis, an analytical model was developed to evaluate the shear capacity of connection with external stiffening rings.

Experimental programs

Specimens design

In order to investigate the shear capacity of connections with external stiffening rings between square steel tubular columns and steel beams under seismic cyclic loading, two full size cruciform specimens have been designed. Assume that the inflection points of columns and beams occurred at the mid-point and these inflection points are the boundaries of specimens. In this cruciform substructure, the upper inﬂection point was restrained from moving in the horizontal direction while the lower was restrained both in horizontal and vertical direction, the left and right inﬂection points were free in loading direction, all inﬂection points could cause rotation but no displacement outside the plane. To ensure the shear failure occurs at connection, the thickness of the steel tube wall of the panel zone was designed to be thinner than other parts in specimen according to the principle of “strong member – weak joint failure mechanism”. The two specimens are same in geometry size but different in axis pressure acting on the upper inﬂection point. The details of these two specimens are shown in Figure 1 and Table 1.

Figure 1.

The details of the specimens.

Table 1.

Dimensions of specimens.

Specimen	Column (mm)	Beam (mm)	Panel zone (mm)	Thickness of ring plate (mm)	Axial compression ratio	Axis pressure (kN)
TS1	200×200×12	250×200×8×14	200×200×6	14	0.4	679.78
TS2	200×200×12	250×200×8×14	200×200×6	14	0.6	1019.67

Material properties

The steel tubes of the specimens were made of cold-formed square steel tube. The beams of the specimens were made of H-shaped steel. All the steel components of the connection were connected to each other by full penetration butt welds and fillet welds with backing bars. They were processed and welded together in the factory in advance. The strength grade of steel of all the specimens was Q235B, defined in Notations for designations of iron and steel (2008). According to the “metallic materials at room temperature tensile test method” (2002), standard tensile pieces of steel from steel tubes and beams of the specimens were cut. The dimension of the tension bar was 180 mm×40 mm. The material properties of the steel are listed in Table 2.

Table 2.

Material properties of steel.

Thickness of the steel(mm)	$f_{y}$ (N/mm²)	$f_{u}$ (N/mm²)	$E_{s}$ (GPa)
5.85	365	471.67	200
7.66	345	468.33	196
10	335	426.67	199
11.9	280	423.47	199
14.08	280	441.28	193

Test setup and loading procedure

Loading and measuring devices are shown in Figure 2. Before the actual experiment, specimens were preloaded to ensure good contact with the test device and all the test equipment worked properly. Then, the axial compressive load was applied to the column. In this experiment, both load control and displacement control were used in the test: firstly, reversed cyclic load was applied before yielding, the level difference in force-control loading is 20 kN, when the specimen approaches yield, the level difference is 10 kN; Then after yielding, a reversed cyclic displacement history with monotonically increasing amplitude was applied as illustrated in Figure 3 where $Δ y$ is the yielding displacement. Every different displacement amplitude was reversed for three times. When the slope of the force-displacement curve dropped significantly, specimens were considered to be yielding. The loading was continued until failure occurred or the load carrying capacity drops to the 85% of peak value. During the loading process, in order to get the stable deformation data of the connection, the experimental data was collected after keeping the load for 5 minutes per level. The controlled displacement in the loading process was measured by the LVDT (Linear Variable Differential Transformer) placed at the end of the beam corresponding to the location of a vertical actuator. The displacement data collected by LVDT and the force data collected by force sensor were processed by the computer to form a real-time force-displacement curve.

Figure 2.

Loading and measuring devices.

Figure 3.

Loading system.

Observations

For the specimen TS1, before yielding, the load-displacement curve was linear. At the load of 58.92 $kN$ , a concavity appeared on the force-displacement curve, which indicated that the connection had yielded, with a yield displacement ( $Δ y$ ) of 10.685 mm. However, the deformation of the panel zone was so insignificant that no major phenomenon was observed at this time. Beyond this point, the loading continued to increase with the displacement control method. The applied displacements were equal to multiples of yield displacement ( $Δ y$ ) at the beam end, and the slope of the load-displacement curve became smaller. Local buckling occurred in the steel tube webs of the panel zone during the second cycle in three times of yield displacement ( $3 Δ y$ ). The load reached its peak value of 86.4 $kN$ , corresponding to 38.425 mm during the second cycle in four times of yield displacement (4 $Δ y$ ). When the loading displacement was five times of yield displacement ( $5 Δ y$ ), the webs in panel zone bulged inward significantly. As the experiment went on, the peak value of each cycle decreased, and the load carrying capacity dropped to the 85% of the peak value during the first cycle in eight times of yield displacement ( $8 Δ y$ ), the external stiffening ring slightly bent, but the steel tube of the panel zone was seriously deformed and couldn’t bear load, thus the test was terminated. Figure 4 shows the state of specimen TS1 during different loading processes.

Figure 4.

The state of specimen TS1 during different loading processes.

For the specimen TS2, at the load of 56.175 kN, the corresponding displacement was 11.245 mm, the panel zone yielded. There was no local buckling in the panel zone until two times of yield displacement ( $2 Δ y$ ). Meanwhile, the rectangular panel zone started to become rhombus. As the load went on, the deformation of the panel zone was becoming more and more serious and the internal force of the specimen was redistributed, the beam flanges began to bear more load transferred from the beam end. The maximum load was $84.0 kN$ , and the corresponding displacement was $31.325 mm$ . After that, the load decreased rapidly to 85% of the peak value during the first cycle in five times of yield displacement ( $5 Δ y$ ), and the test was terminated. Same as specimen TS1, the external stiffening ring of specimen TS2 appeared slightly bent. Figure 5 shows the state of specimen TS2 during different loading processes.

Figure 5.

The state of specimen TS2 during different loading processes.

Discussion of the results of tests

From Figures 4 and 5, the rectangular panel zone became rhombus when loading, which indicated that shear failure occurred in both two specimens. In the process of loading, the web of the panel zone firstly buckled and then formed a tensile field, whose edge was partly located at the flange of the steel tube in the panel zone, and partly on the external stiffening rings plate. When the stress in the tensile field reached the ultimate stress, the specimen could not bear more load and reached its maximum shear capacity. Load-displacement hysteresis loops for two specimens and the envelope curves of load-displacement hysteresis loops, which are called skeleton curves, are showed in Figure 6. The shapes of two hysteresis loops were in a plump shuttle type. This shape indicated that both connections had excellent energy dissipation capacity. The strength decreased gradually after it reached the maximum point. The slopes of the loading curves of the specimens decreased with the increase of the cyclic load. However, the slopes of the unloading curves were almost unchanged, which indicated bigger loading stiffness degradation and smaller unloading stiffness degradation. Comparing the hysteresis curves of TS1 and TS2, it can be found that the hysteresis curve of TS1 was more plentiful than TS2, which indicated that the ductility of the connection with external stiffening ring was decreasing with the increase of the axial compression ratio. For skeleton curves, they were all S-shaped, which indicated that all the specimens had elastic, plastic and damage stages during the tests. The peak point of the skeleton curve of the TS1 was almost equal to that of the TS2, which indicated that the change of the axial compression ratio had little effect on the shear capacity of the connection.

Figure 6.

Curves of load-displacement hysteresis loops and skeleton curves.

Finite element analysis

Preparing for the parameter fitting in analytical investigation below, a three-dimensional (3D) nonlinear finite element model was developed using the general finite element program ANSYS to investigate the influence of three parameters on shear capacity.

Experimental studies on the connections with external stiffening rings between square steel tubular column and H-shaped steel beam were introduced in the former content. Based on the experimental specimens TS1 and TS2, finite element models TF1 and TF2 were established to validate the accuracy of finite element models.

Materials

The steel tube, steel beam and external stiffening ring were assumed to be the elastic-plastic material in the finite element modeling, which was expressed specifically by trilinear kinematic hardening model as shown in Figure 7 (Yu, 2015). The Poisson’s ratio of steel was assumed as 0.3. The strength of steel in different thickness is shown in Table 2 in accordance with the data tested by material experiment. Using Von Mises yield criterion in steel constitutive relation.

Figure 7.

Typical strain-stress relationship of steel used in the analysis.

Element type, mesh and boundary condition

SOLID186 element with 20 nodes and three degrees of translational freedom at each node was used to simulate the steel tube, external stiffening ring and H-beam.

To simulate the inflection points of the columns and the beams, the displacement of three directions was constrained at the bottom of the column, the top of the column was restricted by the displacement in two directions, the displacement in the direction of the column axis was relaxed. The degrees of freedom of all the nodes at the bottom of the column were coupled to one of nodes, and then constrain the three translational degrees of freedom of this point. After treatment with this method, the boundary condition of column bottom is close to hinge. The axis pressure applied to the steel tubular columns was firstly applied to the column, and the horizontal cyclic displacement loads were applied to the end of the beam in the following step.

Verification

Comparisons of the load-displacement skeleton curves between the numerical analysis and tests are shown in Figure 8. Failure modes are compared in Figure 9. The ultimate loads of the tested connections are compared with the numerical analysis as listed in Table 3. It is found that failure modes consist with the experimental phenomena well. Though finite element analysis overestimates the ultimate load, considering that material defects are not considered in the finite element model, so the difference of the ultimate load between the finite element model and test within the allowable range.

Figure 8.

Load-displacement curves comparison between FEA and test results.

Figure 9.

Failure modes comparison between FEA and test results.

Table 3.

The comparison between experiment and FEM in ultimate load.

Test specimen	Ultimate load ( $kN$ )	FEM specimen	Ultimate load( $kN$ )	Ratio
TS1	86.4	TF1	93.5	0.924
TS2	84.0	TF2	93.0	0.903

Parameter Studies

Taking into account the calculation formula of shear capacity of connections with external stiffening rings between square steel tubular columns and steel beams in the fourth part of the article, factors that were taken into discussion include: width of the steel tube in panel zone ( $a$ ), height of the steel tube in panel zone ( $h$ ) and thickness of the external stiffening ring ( $t$ ). Key factors of different models are marked in Table 4.

Table 4.

Key factors of different finite element models.

Specimen	Steel tube( $mm$ )	Panel Zone( $mm$ )	Steel beam( $mm$ )	External stiffening ring thickness( $mm$ )	Axial compression ratio
SJ1	200×200×12	200×200×6	250×200×8×14	14	0.4
SJ2	180×180×12	180×180×6	250×200×8×14	14	0.4
SJ3	220×220×12	220×220×12	250×200×8×14	14	0.4
SJ4	200×200×12	200×200×6	225×200×8×14	14	0.4
SJ5	200×200×12	200×200×6	275×200×8×14	14	0.4
SJ6	200×200×12	200×200×6	200×250×8×12	12	0.4
SJ7	200×200×12	200×200×6	200×250×8×16	16	0.4

In order to investigate the influence of the width of the steel tube in panel zone ( $a$ ), three cases with different values of 180 $mm$ , 200 $mm$ and 220 $mm$ for SJ2, SJ1, and SJ3 were designed in this parametric study. The load-displacement hysteretic curves and skeleton curves of SJ1, SJ2, and SJ3 are shown in Figure 10. From these two pictures, it can be found that the hysteretic curves of SJ2 is the fullest while the fullness of the hysteresis curves of SJ1 and SJ3 is close. The peaks of the skeleton curves are 90.7 $kN$ , 95.6 $kN$ and 104.1 $kN$ . The width of the steel tube in panel zone increases by 22.2% while the shear capacity of the connection increases by 14.8%. The above phenomena shows that the seismic performance and the ultimate loads of connections with external stiffening rings between square steel tubular columns and steel beams are improved with the increase of the width of the steel tube in panel zone, however, when the width of the steel tube in panel zone has increased to a certain value, the seismic performance of the specimen will not be improved with the increase of the width of the steel tube in panel zone.

Figure 10.

Load-displacement curves comparison between SJ1, SJ2 and SJ3.

To investigate the influence of the height of the steel tube in panel zone ( $h$ ), three cases with different values of 225 mm, 250 mm and 275 mm for SJ4, SJ1 and SJ5 were designed in this parametric study. The load-displacement hysteretic curves and skeleton curves of SJ1, SJ4, and SJ5 are shown in Figure 11. The hysteresis curves of SJ1, SJ4, and SJ5 are in similar shape and the fullness of the hysteresis curves is also close. The peaks of skeleton curves of SJ4, SJ1, and SJ5 are 85.6 $kN$ , 95.6 $kN$ and 106.0 $kN$ . The height of the steel tube in panel zone increases by 22.2% while the shear capacity of the connection increases by 23.8%. The above phenomena show that the seismic performance of connections with external stiffening rings between square steel tubular columns and steel beams has nothing to do with the height of the steel tube in panel zone while the ultimate load is positive correlation with the height of the steel tube in panel zone.

Figure 11.

Load-displacement curves comparison between SJ1, SJ4 and SJ5.

In order to investigate the influence of the thickness of the external stiffening ring ( $t$ ), the three models SJ1, SJ6 and SJ7 were established. The thickness of SJ1, SJ6 and SJ7 is 14 $mm$ , 12 $mm$ , 16 $mm$ , respectively. The hysteresis curves of SJ1, SJ6, and SJ7 are similar in shape, size and fullness, shown in Figure 12. In the picture of skeleton curves, as the thickness of the external stiffening ring ( $t$ ) increases from 12 $mm$ to 16 $mm$ , which increases by 33.3%, the shear capacity of the specimens increase by only 6.2% from 92.2 $kN$ to 97.9 $kN$ . The hysteretic curves and skeleton curves indicate that the seismic performance of connections with external stiffening rings between square steel tubular columns and steel beams has nothing to do with the thickness of the external stiffening ring and the ultimate load increases with the increase of the thickness of the external stiffening ring, but is less than that of the width of the steel tube in panel zone.

Figure 12.

Load-displacement curves comparison between SJ1, SJ6 and SJ7.

These three parameters whose influence on this connection performance could be already pretty much known, therefore, the article did not provide other details, only showed the maximum load of each specimen, shown in Table 5.

Table 5.

The maximum load of the specimens in finite element analysis.

Specimen	maximum load ( $kN$ )	considering parameter
SJ1	95.6	basic specimen
SJ2	90.7	width of the steel tube in panel zone ( $a$ )
SJ3	104.1	width of the steel tube in panel zone ( $a$ )
SJ4	85.6	height of the steel tube in panel zone ( $h$ )
SJ5	106.0	height of the steel tube in panel zone ( $h$ )
SJ6	92.2	thickness of the external stiffening ring ( $t$ )
SJ7	97.9	thickness of the external stiffening ring ( $t$ )

Analytical investigation

It can be seen from the test results that the following failure characteristics were observed when shear failure occurred at the connection with external stiffening rings between square steel tubular columns and steel beams. The webs of the steel tubes in panel zone concaved obviously, there were plastic hinges at the flanges of the steel tubes in panel zone. Based on this, it is considered that the shear capacity of the connection is provided by three parts, which are the webs of the steel tubes in panel zone, the flanges of the steel tubes in panel zone and the external stiffening ring.

The shear strength of webs of the steel tubes in panel zone

From the loading process of specimens in test, buckling load is not the failure load of the web of the steel tube in panel zone, and there is a large carrying capacity after buckling (post-buckling strength). The tension field comes into being in web to prevent lateral displacement when the web is buckled. So, the shear strength of webs of the steel tubes in panel zone can be divided into two parts: shear buckling strength ( $V_{cr}$ ) and post-buckling strength ( $V_{tf}$ ). According to post-buckling strength theory of thin-web beam, the shear strength of webs of the steel tubes in panel zone is deduced referring to AISC specifications (2010) and EC3-ENV-1993 specifications (1992), show as equation (1).

V_{w} = V_{cr} + V_{tf}

(1)

where $V_{w}$ is the shear strength of webs of the steel tubes in panel zone, $V_{cr}$ is the shear strength of webs of the steel tubes in panel zone in the point where the web of steel tube in panel zone is buckling and $V_{tf}$ is the shear strength providing by tension field.

The shear buckling strength ( $V_{cr}$ )

The buckling shear strength of the webs of steel tube in panel zone can be calculated by equation (2).

V_{cr} = 2 a t_{w} τ_{cr}

(2)

where a is the width of steel tube in panel zone, $τ_{cr}$ is shear buckling stress in the webs of steel tube in panel zone and $t_{w}$ is the thickness of steel tube in panel zone.

For a specific boundary condition, the formula for calculating the buckling stress $τ_{cr}$ of a plate can be derived by elastic-plastic mechanics, and the change of the boundary condition can be expressed by the buckling coefficient $k$ . However, for the steel tube web in the panel zone, its boundary conditions are complex, so it is impossible to deduce the formula for calculating the buckling stress $τ_{cr}$ by using elastic-plastic mechanics. Therefore, a coefficient m similar to the buckling coefficient $k$ is introduced, assuming that:

τ_{cr} = f_{wy} / m

(3)

where $f_{wy}$ is the uniaxial tension yield stress of steel tube in panel zone, m is a scale factor that is currently unknown.

The shear strength provided by tension field ( $V_{tf}$ )

Considering the influence of the steel beam webs on the stiffness of the flanges of the steel tubes in panel zone and the results of finite element analysis, the boundary of the tension field is assumed to be partly on external stiffening rings while another part is on the flanges of the steel tubes in panel zone as shown in Figure 13. The shear strength provided by tension field is equal to the horizontal component of tension by tension field which can be calculated by equation (4).

V_{tf} = 2 s t_{w} σ_{f} \cos γ

(4)

where $s$ is the width of tension field, $σ_{f}$ is the tensile stress in tension field and $γ$ is the angle between tension field and horizontal direction.

Figure 13.

The tension field in panel zone.

In equation (4), s, $σ_{f}$ and $γ$ are three unknowns in order to calculate $V_{tf}$ .

The calculation of the tensile stress in tension field ( $σ_{f}$ )

The stress distribution of the web unit under shear and tension field is shown in Figure 14. Projecting $τ_{cr}$ to the direction of tension field as well as the direction perpendicular to the direction of tension field. However, the result is very complex. It is noted the direction of tension field is close to the direction of $τ_{cr}$ , so adding them directly is one of effective ways to simplify. The value of $σ_{1}$ and $σ_{2}$ in Figure 15 can be obtained via equation (5).

σ_{1} = τ_{cr} + σ_{f}

(5a)

σ_{2} = - τ_{cr}

(5b)

where $σ_{1}$ and $σ_{2}$ are the principal stresses in web.

Figure 14.

The stress distribution of the web unit.

Figure 15.

The stress distribution of the web unit after projecting.

The tensile stress in tension field ( $σ_{f}$ ) can be obtained by substituting $σ_{1}$ and $σ_{2}$ into the fourth strength theorem. However, the result is too complex to application in practice. Referring to the treatment in AISC specifications (2010), when calculating the shear capacity of thin web beams with transverse stiffeners, AISC uses the tension field method to consider the effect of post-buckling strength. The fourth strength theory under plane stress is used in the tension field method when calculating $σ_{1}$ (C-point ordinates).

σ_{1}^{2} + σ_{2}^{2} + σ_{1} σ_{2} \leq f_{y}^{2}

(6)

Drawing elliptic curve by using the formula above, shown in Figure 16.

Figure 16.

The elliptic curve of equation (6).

Because the AB is a curve, the calculated $σ_{1}$ (C-point ordinates) is complex. If AB is approximated to a straight line, the slope of the straight line AB is $(\sqrt{3} - 1)$ for $τ_{y} = f_{y} / \sqrt{3}$ , and $σ_{1} = f_{y} - τ_{cr} (\sqrt{3} - 1)$ .

so, the approximate value of $σ_{f}$ can be calculated by equation (7).

σ_{f} = f_{wu} - τ_{cr} \sqrt{3}

(7)

where $f_{wu}$ is the uniaxial tension ultimate stress of steel tube in panel zone.

The calculation of the width of tension field ( $s$ )

The calculation model of the width of tension field is shown in Figure 14. According to the phenomenon observed in tests and the result obtained by finite element analysis, assume there are plastic hinges existing at point 1 and point 2.

In the right picture of Figure 17, the value of $d$ can be confirmed based on the equilibrium of moment shown as in equation (8).

2 M_{p} = σ_{f} t_{w} d^{2} \cos γ

(8)

where $M_{p}$ is the ultimate bending moment of the flange of the steel tube in panel zone, d is the anchorage length of the tension field along the height of the web of the steel tube in panel zone.

Figure 17.

The calculation model of the width of tension field.

$M_{p}$ can be classified by equation (9) referring to material mechanics.

M_{p} = 0.25 {at}_{w}^{2} f_{wu}

(9)

According to geometric relation, the width of tension field ( $s$ ) can be expressed by the anchorage length of the tension field along the diaphragm ( $d$ ) as shown in equation (10).

s = a \sin γ - (h_{0} - 2 d) \cos γ

(10)

where $h_{0}$ is the height of steel tube in panel zone.

Substituting equations (8) and (9) into equation (10), it gives equation (11), which is the calculating formula of the width of tension field ( $s$ ).

s = a \sin γ - (h_{0} - 2 \sqrt{a t_{w} f_{wu} / (2 σ_{f} \cos γ)}) \cos γ

(11)

The calculation of the inclination of tension field ( $γ$ )

From the introduction of Narayanan (1983), the inclination of tension field can’t be calculated directly. The best value of the inclination of tension field can be obtained by iterative method. From the result of iterative method, all values of $γ$ between $\frac{θ}{2}$ and $θ$ are conservative and the change rate of the shear strength providing by tension field is very slow with the change of $γ$ , where $θ$ is the diagonal inclination of the web of steel tube in panel zone. Therefore, the value of $γ$ is determined by equation (12).

γ = 2 θ / 3

(12)

In summary, substituting equations (3), (7), (11) and (12) into equation (4), it gives equation (13), which is the calculating formula of the shear strength providing by tension field.

\begin{matrix} V_{tf} = 2 t_{w} (f_{wu} - \sqrt{3} f_{wy} / m) \cos 2 θ / 3 \\ (a \sin 2 θ / 3 - h_{0} \cos 2 θ / 3 + \sqrt{2 a t_{w} f_{wu} / (f_{wu} - \sqrt{3} f_{wy} / m})) \end{matrix}

(13)

According to the contents above in this study, the calculating formula of the shear strength of webs of the steel tubes in panel zone is shown as in equation (14).

\begin{matrix} V_{w} = V_{cr} + V_{tf} = 2 a t_{w} f_{wy} / m + 2 t_{w} (f_{wu} - \sqrt{3} f_{wy} / m) \\ \cos \frac{2}{3} θ (a \sin \frac{2}{3} θ - h_{0} \cos \frac{2}{3} θ + \sqrt{2 a t_{w} f_{wu} / (f_{wu} - \sqrt{3} f_{wy} / m})) \end{matrix}

(14)

The shear strength of the flanges of the steel tubes in panel zone

It can be assumed that there are four plastic hinges formed at the corners of the flanges of the steel tubes in panel zone referring the results of tests and finite element simulation, shown as Figure 18.

Figure 18.

The calculation model of the width of the shear strength of the flanges.

Equation (15) is obtained from the moment equilibrium conditions of the flanges of the steel tubes in panel zone.

V_{f} = 4 M_{fp} / (h - t_{d})

(15)

where $M_{fp}$ is the ultimate bending moment of the web of the steel tube in panel zone, h is the sum of the heights of steel tube in panel zone and external stiffening rings, $t_{d}$ is the thickness of the external stiffening ring.

$M_{fp}$ can be classified by equation (16) referring to material mechanics.

M_{fp} = 0.25 {at}_{w}^{2} f_{wu}

(16)

The shear strength of the external stiffening ring

According to the tests and finite element analysis results, under the shear deformation mechanism of the panel zone, the external stiffening ring produces bending deformation with the increase of load and the yield lines are formed, which start from the corner of the square steel tubular column and perpendicular to the hypotenuse of the external stiffening ring, as shown in Figure 16.

According to the principle of virtual work and combined with Figure 19, the shear strength of the external stiffening rings is deduced by equation (17).

V_{r} = 4 M_{rp} / (h - t_{d})

(17)

where $M_{rp}$ is the ultimate bending moment of the external stiffening ring,

Figure 19.

The yield lines in the external stiffening ring.

which is calculated by equation (18)

M_{rp} = 0.25 (a + 2 L_{d} \sin α) t_{d}^{2} f_{ru}

(18)

where $L_{d}$ is the vertical distance from the corner of the square steel tubular column to the hypotenuse of the external stiffening ring, $f_{ru}$ is the uniaxial tension ultimate stress of the external stiffening ring, $α$ is the angel between yield line and horizontal direction.

In summary, the shear capacity of the connection with external stiffening rings between square steel tubular columns and steel beams can be expressed by equation (19).

V_{u} = V_{w} + V_{f} + V_{r}

(19)

where $V_{u}$ is the shear capacity of the connection

$V_{f}$ and $V_{r}$ are two calculated parameters, but there is still an unknown scale factor m. The relationship between the force applied to the beam ends and the shear capacity of the connection is confirmed by Rong (Rong et al., 2018). The tests and finite element method results are listed in Table 6. According to the results of FEM to fit the value of scale factor m, it can be discovered that the results of analytical calculation is in good accordance with that of FEM when $m = 1.35$ .

Table 6.

The comparison between analytical calculation and FEM.

Specimen	FEM ( $kN$ )	Analytical calculation ( $kN$ )	Ratio
SJ1	824.6	824.5	1.000
SJ2	790.0	756.4	0.957
SJ3	889.1	891.7	1.003
SJ4	832.5	844.4	1.014
SJ5	820.2	808.2	0.985
SJ6	788.1	780.5	0.990
SJ7	852.3	871.4	1.022
Average			0.996
Variance			0.00046

The difference among the results of the analytical calculation and FEM is within 5%, which proves that the method to calculate the shear capacity of the connection with external stiffening rings between square steel tubular columns and steel beams in this paper is effective.

Conclusions

Two connections with external stiffening rings were tested and seven finite element models were developed under low cyclic loading to investigate the shear capacity of the connection with external stiffening rings between square steel tubular columns and steel beams. Based on the test results, the following main conclusions are obtained:

The change of the axial compression ratio has little effect on the shear capacity of the connection while the ductility of the connection is decreasing with the increase of the axial compression ratio.

In a certain range, seismic performance and the ultimate loads of connections with external stiffening rings are improved with the increase of the width of the steel tube in panel zone.

Seismic performance of connections with external stiffening rings has nothing to do with the height of the steel tube in panel zone while the ultimate load is positive correlation with the height of the steel tube in panel zone.

The seismic performance of connections with external stiffening rings has nothing to do with the thickness of the external stiffening ring and the ultimate load increases with the increase of the thickness of the external stiffening ring, but is less than that of the width of the steel tube in panel zone.

A model which considering the post-buckling strength of the web of the steel tube in panel zone for the shear capacity of the connection with external stiffening rings is proposed. The model is developed by superposition of models for steel tube and external stiffening rings. Finally, a calculation formula based on this model is fitted.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundations of China [No.51468061].

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