Calculation method for the shear bearing capacity of diaphragm-through connections

Abstract

This article presents the results of experimental, numerical, and theoretical studies into the shear behavior of panel zone of diaphragm-through connections between concrete-filled square steel tubular columns and steel beams. The current design code does not cover the yield and ultimate shear strengths of such connections in China. Five tee-shaped diaphragm-through connections were fabricated and tested that they comply with the principle of “strong members and weak panel zone.” Three-dimensional nonlinear finite element models were developed to study the shear behavior of the panel zone under low-frequency cyclic loading. Based on the test results, the force transfer mechanism and the effect of different factors on the panel zone shear capacity were analyzed. The width and deformation of compression strut of core concrete were taken into account. A simple manual calculation method was also proposed for evaluating the shear strength of panel zone of diaphragm-through connections. Good agreement was found between theoretical and test results for both yield and ultimate shear strengths for connections.

Keywords

calculation method concrete-filled square steel tubular column diaphragm-through connection experimental investigation panel zone shear capacity

Introduction

Buildings with composite steel–concrete framing are increasingly popular around the world, as they combine the advantages of high erection speed and the ductility of the steel structures, together with the high compressive strength of the concrete. In these frame systems, the panel zone of the beam-to-column moment connection is subjected to large forces under lateral seismic loading. Compared with the shear force in the column tip or beam tip, the shear force in the panel zone is nearly four to six times higher. However, current design codes in China do not have relevant provisions to calculate the shear strength of panel zone for diaphragm-through connections.

Over the past 30 years, some work has been done to study the shear behavior of panel zone of the diaphragm-through connections. Kawaguchi (1987), Kawaguchi et al. (1996), and Morino et al. (1993) proposed a model composed of a multi-linear shear–deformation relationship model for the steel tube and a trilinear shear–deformation relationship model for the concrete core. However, neither the influence of axial load nor the confinement to the concrete core provided by steel tubes was considered in their model. An analytical model for the restoring force characteristics of the shear panel in the subassemblies was also proposed by Nishiyama et al. (2004). Fukumoto and Morita (2005) proposed a nonlinear shear force–deformation model for predicting the elastoplastic behavior of the panel zone. The proposed model includes a superposed model based on a trilinear shear–deformation relationship for the steel tube and bilinear for the concrete core, and a simple model provided as a trilinear model having a yield strength point and an ultimate strength point for this panel zone. Yin Qin et al. (2014a, 2014b, 2014c) proposed an analytical model for shear strength according to the simplified trilinear shear–deformation relationship, and a theoretical method that the contribution of steel frame mechanism in the panel zone was considered was proposed to evaluate the shear strength of the concrete compression strut at the yielding point of the steel tube. Rong et al. (2013a, 2013b) used the finite element methods (FEMs) to analyze the shear behavior of the panel zone of diaphragm-through connections, and the numerical data have a good agreement with the test results.

In this research, five tee-shaped diaphragm-through connections were tested under cyclic loading to investigate the panel zone behavior. A new model and hand calculation method for the width of the compression strut of core concrete were proposed. This article also proposes a method for evaluating shear strength of the panel zone of diaphragm-through connections and compares predicted and experimental results.

Experimental investigation

Test specimens introduction

The test program consisted of tests on five subassemblages to investigate the shear behavior of the panel zone in the concrete-filled steel tube (CFST) column to steel beam moment connections. All specimens were designed according to Code for Design of Steel Structures (GB50017-2003, 2003) and Technical Specification for Structures with Concrete-Filled Rectangular Steel Tubular Members (CECS159, 2004). In order to study the shear behavior of panel zone of diaphragm-through connection, all connections, panel zones, columns, and beams were designed in such a way that all yielding would start to occur from the panel zone, satisfying the “strong-member and weak-joint” concept. The parameters investigated in the study include panel zone steel tube thickness, concrete strength, diaphragm plate thickness, and axial compression ratio. Table 1 summarizes the test matrix containing these specimens.

Table 1.

Test specimens.

Specimen	Column section (mm)	Panel zone (mm)	Steel beam (mm)	Diaphragm (mm)	Concrete	Axial ratio
SJ-1	250 × 250 × 12	250 × 250 × 6	250 × 250 × 8 × 14	14	None	0.2
SJ-2	250 × 250 × 12	250 × 250 × 6	250 × 250 × 8 × 14	14	C30	0
SJ-3	250 × 250 × 12	250 × 250 × 6	250 × 250 × 8 × 14	14	C40	0.2
SJ-4	250 × 250 × 12	250 × 250 × 8	250 × 250 × 8 × 14	14	C40	0.2
SJ-5	250 × 250 × 12	250 × 250 × 8	250 × 250 × 8 × 14	16	C40	0

Steel used for all specimens is Q235B. Cold-formed square steel tube and welded H-shaped steel beam were used in the specimen construction. Butt welds are adopted to connect the diaphragm plate and beam flange. The welding was carried out by shield metal arc welding process using electrode E43. The details of the test specimens are shown in Figure 1.

Figure 1.

Details of the diaphragm-through connection: (a) tee-shaped specimen, (b) overhead view, (c) section view, and (d) axonometric view.

Materials for the tensile test coupons were cut from the square tube, flanges and webs of the steel beams, and diaphragm plates. Values of the yield strength and ultimate strength of the steel members are obtained by testing one strip coupon of each member and have been listed in Table 2. The values of the concrete strength at the age of 28 days, and on the day of testing, are summarized in Table 3.

Table 2.

Steel material properties.

Element	No.	$f'_{y}$ (MPa)	$f'_{u}$ (MPa)	f_y (MPa)	f_u (MPa)
Panel zone (6 mm)	1	312.6385	437.6938	307.9	443.9
	2	299.1558	426.0704
	3	311.9793	467.9689
Panel zone (8 mm)	1	338.9831	527.307	335.2	507.9
	2	348.8813	523.322
	3	317.705	473.1776
Beam web (8 mm)	1	316.8106	458.3643	336.5	471.2
	2	345.506	480.9985
	3	347.2463	474.1248
Column (12 mm)	1	347.2222	454.6958	372.9	485.3
	2	397.9272	504.5881
	3	373.5626	496.6685
Diaphragm (14 mm)	1	253.1765	424.338	262.1	437.3
	2	255.382	434.1494
	3	277.6286	453.3429
Beam flange (14 mm)	1	292.6874	446.1698	263.3	429.9
	2	260.5667	434.2778
	3	236.7424	409.2262
Diaphragm (16 mm)	1	258.1129	432.1215	259.8	425.7
	2	257.6007	421.253
	3	263.8006	423.8594

Table 3.

Concrete material properties.

Grade	No.	$f'_{c}$ (MPa)	f_c (MPa)
C30	1	37.4	38.4
	2	40.9
	3	36.7
C40	1	48.7	49.0
	2	49.3
	3	48.9

The test configuration is depicted in Figure 2. The hydraulic actuator had a maximum force capacity of 1000 kN and a stroke of ±350 mm. It was positioned horizontal and applied to the beam tip. The specimens were first force-controlled and then displacement-controlled. The horizontal force was loaded once in the positive direction and once in the reverse direction in the beam ends at each load step before yield. After yield, the specimen would be controlled by the displacement. The increment is chosen as Δy (the yield displacement of the specimen), and each step is repeated for three cycles.

Figure 2.

Test setup.

Test results and discussion

The main test results and the key parameters characterizing the behavior of the test specimens are presented in Table 4. The yield moment was computed by multiplying the force at the beam tip by the distance between the beam tip and the face of the column. The yield shear load of the panel zone for all the specimens was calculated based on the condition of static equilibrium.

Table 4.

Test results.

Specimen	Measured response			Inelastic deformation mode
Specimen	Yield moment, M_y (kN m)	Yield shear, V_y (kN)	Rotation angle (%)	Inelastic deformation mode
SJ-1	131.7	506.6	1.22	Panel zone yield
SJ-2	260.3	1001.4	2.84	Panel zone yield
SJ-3	268.0	1031.1	1.4	Panel zone yield
SJ-4	270.4	1040.1	1.49	Panel zone yield
SJ-5	286.8	1113.6	2.47	Panel zone yield

The failure mode of all specimens is typically shearing destruction of panel zone. This behavior is due to the “strong-member weak-joint” design. Bending deformation and local buckling were found at the end of the diaphragm and the beam flange near the column face. The final failure mode of each specimen is tension failure of welds’ case by overturning moment. Cracks appeared on the weld tip and extended along the weld connecting the diaphragm and beam flange, which led to the performance degradation and failure of the connection. Visible crossing cracks were found in the core concrete of panel zone, as shown in Figure 3.

Figure 3.

Failure mode: (a) shearing destruction, (b) beam flange buckling, (c) weld crack, and (d) concrete crash.

Finite element analysis

Finite element model

A comprehensive three-dimensional finite element model was generated for the test specimen using the ANSYS finite element program. Elements SOLID95 and SOLID65 were used to model the steel and concrete core, respectively. The interface between the steel and concrete was modeled using contact elements (TARGE170 and CONTAC174 of ANSYS). Element PRETS179 was used to simulate pre-tension force of high-strength bolt.

Similar to the loading of the test specimen, horizontal loads were applied at the beam ends, and constant axial load was applied on the column. The value applied to the finite element model was the same as that during the test. The detailed loading procedure realized in the ANSYS can be summarized as follows:

Step 1. The pre-stress on the high-strength bolts was applied.

Step 2. The pre-stress was locked until the failure of the specimen.

Step 3. The vertical axial force was applied to the top of the column.

Steps 4 and 5. The cyclic horizontal force was applied to the beam tip. The load was displacement-controlled, and the corresponding displacements for steps 4 and 5 are (+) 6 mm and (−) 6 mm, respectively, where (+) denotes the push of the specimen from the left to the right, and (−) denotes the pull from the right to the left.

The similar loading procedures of steps 4 and 5 were repeated in steps 6–9. The corresponding displacements for each step were (+) 12 mm, (−) 12 mm, (+) 18 mm, and (−) 18 mm.

Actual material properties were used in the analysis. Figure 4 shows the typical finite element mesh used in this study to model the geometry of the test specimen.

Figure 4.

Finite element model.

The von Mises yield criterion with kinematic hardening rule was used to define the material yield surface, and an associated flow rule was used to determine the plastic deformation. The trilinear stress–strain relation was used to model the steel material. The bilinear kinematic hardening model was used to model the bolt. The Willam–Warnke concrete model was adopted as the failure criterion for the infilled concrete where multi parameters are required as inputs to describe the model.

Model verification

In order to verify the analytical model, the results from the finite element analysis are compared to the experimental data. Figure 5 shows the comparison between the experimental and analytical load–displacement relationships at the beam end for SJ-1. Good agreement is observed between the experimental and analytical load–displacement relationships. This illustrates that the analytical model is capable in predicting the behavior of the joint in both the linear and the nonlinear stages. In the sections to follow, selected results from the finite element analysis results are discussed to develop mechanical behaviors of panel zone.

Figure 5.

Comparison of skeleton curve and envelop for SJ-1: (a) skeleton curve and (b) envelop.

Numerical results

Stress distribution of whole model

Specimen SJ-2 is analyzed to determine the structural behavior and shear failure mechanism of the panel zone of diaphragm-through connection. The ultimate stress and deformation distribution are shown in Figure 6. The shear deformation of the connections and high stress mainly concentrate at the panel zone when connections’ damage occurs. The outer columns and beams remain elastic while the steel tube web and core concrete of panel zone yield.

Figure 6.

The von Mises stress of SJ-2.

Stress distribution of steel tube of panel zone

Figure 7 shows the stress distribution of steel tube in the panel zone at the different load stages. The stress level is higher in the web part of the steel tube with a large distribution area, while the flange part the stress level is lower with a smaller distribution area. In the web part of steel tube, the stress distribution of high level appears mainly in the diagonal direction, as shown in Figure 7(b). Until panel zone yield, the flange part of steel tube still suffered a low stress state.

Figure 7.

The von Mises stress distribution of steel tube: (a) step 5, (b) step 7, and (c) step 9.

Stress distribution of core concrete

The stress distribution of the concrete core in the panel zone at the different load stages is shown in Figure 8. It can be found that the core concrete mainly suffered from oblique force. With the increase in the load, stress of the core concrete increased gradually; however, the width of the compression diagonal struts tends to remain the same. The core concrete formed a strut to resist the shear force of the panel zone. Therefore, the compression strut model can be used in the shear capacity analysis of the connections.

Figure 8.

The von Mises stress distribution of core concrete: (a) step 5, (b) step 7, and (c) step 9.

Stress distribution of diaphragm plate

As shown in Figure 9, it can be found that the inner part of diaphragm plate (inside the red line in Figure 9(a)) is always subjected a low-level stress, even at the ultimate load stage. Instead, high stress concentrates in the external part of diaphragm plate at the ultimate load stage. The reason of this behavior is that the external part of diaphragm plate and steel beam flange exhibited large plastic deformation when the panel zone yielded. These results indicate that the diaphragm plate does not contribute too much to resist the shear force but has a positive role for developing the plastic deformation.

Figure 9.

The von Mises stress distribution of diaphragm plate: (a) step 5, (b) step 7, and (c) step 9.

Superposed model of shear force for panel zone

Modeling of panel zone

In steel frame structure systems, the panel zone of the diaphragm-through connection is subjected to large forces under lateral seismic loading, as shown in Figure 10. In the evaluation of shear strength of the diaphragm-through connections, CECS159 (2004) superposed a shear capacity calculation method for the panel zone of steel beam to the CFST column moment connections that are resisted by the steel tube web, the diagonal concrete core strut, and the steel frame which composite with flange part of steel tube and diaphragm plate. AIJ-SRC (1998) characterized a distinguished failure mode, which resulted from the insufficient shear capacity in the panel zone that is resisted only by the steel tube web and the diagonal concrete core strut.

Figure 10.

Panel zone subjected to seismic loading.

The experimental and nonlinear finite element analysis results indicate that the shear capacity in the panel zone of diaphragm-through connections is resisted only by the steel tube web and the diagonal concrete core strut in this article. A new calculating method for the width of compression strut of core concrete is proposed. The total shear strength is equal to the sum of their strengths at identical deformation. This can be expressed as follows

V = V_{s} + V_{c} = V_{web} + V_{c}

(1)

where V_s, V_c, and V_web are shear strengths of the steel, the concrete strut, and the shear force resisted by the steel tube web, respectively.

The shear–deformation relationship of the diaphragm-through connections is assumed to be a trilinear curve (Kanatani et al., 1987), as shown in Figure 11, in which the first critical point, corresponding to yield strength, is defined as the point where the steel tube reaches its yield shear strength, and the second critical point, corresponding to ultimate strength, is defined as the point where the diagonal infilled concrete strut crushes.

Figure 11.

Assumed trilinear shear–deformation skeleton curves of diaphragm-through connections.

Contribution of the web of steel tubes

During the past three decades, considerable research has been undertaken to study the shear bearing capacity of the web part of steel tube. Some simplifying calculation formulas and kinds of accurate computing methods have been obtained. For the reason that the web part of steel tube’s thickness is far smaller than the width and height, the stress problem could be considered as a plain strain and plain stress. The stress state of the web part of steel tube which subjected to the axial load and horizontal load is illustrated in Figure 12.

Figure 12.

Principal strain of steel tube in panel zone.

The plane of maximum and minimum principal stress is given by the following equation

σ_{1, 2} = \frac{σ_{x} + σ_{y}}{2} \pm \sqrt{{(\frac{σ_{x} + σ_{y}}{2})}^{2} + τ_{xy}^{2}}

(2)

where σ_1,2 is the plane of maximum and minimum principal stress, σ_x,y is actual stress in x and y direction, and τ_xy is the actual shear stress, respectively.

The von Mises stress σ_m in the steel web panel being designed equals the yield stress f_y while the web part of steel tube is led to the yielded state. Based on the fourth strength theory of material and the von Mises yield criterion, this can be expressed as follows

\sqrt{(\frac{[{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ_{3})}^{2} + {(σ_{3} - σ_{1})}^{2}]}{2}) ⩽ [f_{y}]}

(3)

where σ_1,2,3 are the plane principal stress in different directions, and f_y is the yield stress of steel tube of panel zone, respectively.

According to equations 2 and 3, the yield shear stress of web part of steel tube can be computed by

τ_{xy} = \frac{\sqrt{f_{y}^{2} - σ_{s}^{2}}}{\sqrt{3}}

(4)

σ_{s} = \frac{N_{s}}{A_{s}} = \frac{N E_{s}}{E_{s} A_{s} + E_{c} A_{c}}

(5)

Thus, the shear capacity of the steel tube of panel zone can be obtained as follows

V_{sy} = A_{w} τ_{xy} = \frac{A_{s}}{2} τ_{xy} = \frac{2 t_{c} (B - t_{c}) \sqrt{f_{y}^{2} - σ_{s}^{2}}}{\sqrt{3}}

(6)

where V_sy is the yield shear strength of the web part of the steel tube of panel zone, B is the width of the panel zone steel tube, t_c is the thickness of the web part of steel tube, f_y is the yield strength of the web part of steel tube, σ_s is the axial stress applied on the column tubes, E_s is the modular of the steel tube, E_c is the modular of the concrete, A_s is the sectional area of the steel tube, A_s is the sectional area of the web part of steel tube, A_s is the sectional area of the core concrete, and N is the axial load applied the column.

Contribution of core concrete

As illustrated in Figure 10, the bending moments M forced on the beam tip and column tip near the panel zone are equivalent to the horizontal tensile force and press force V_M acted on the beam flanges, respectively. The corner of the core concrete resisted the vertical press and horizontal press. Which means that the core concrete is subjected to diagonal press force and orthogonal tensile force, as shown in Figure 13.

Figure 13.

Principal strain of steel tube in panel zone.

The shear strength of the core concrete can be decided by the horizontal component of the compressive stress of the compression strut, this can be expressed as follows

V_{c} = F_{s} \cos θ = f_{c} \cdot a \cdot (D - 2 t_{c}) \cdot \cos θ

(7)

where V_c is shear strength of the core concrete, α is the width of the compression strut, and D is the length of the core concrete, $\cos θ = B - 2 t_{c} / \sqrt{{(B - 2 t_{c})}^{2} + (H - 2 t_{s})}$ .

Currently, the width of the compression strut of core concrete is usually determined by experimental phenomena. Because of the test conditions’ limitations, empirical formulas have different features and applicable ranges for different tests. The theoretical compression strut mechanism was widely used for calculating the shear strength of the core concrete, but different calculated width of the compression strut was adopted in different computing methods, such as the existed research results for Nishiyama et al. (2004), Fukumoto (2005), and Yin Qin et al. (2014a, b, c).

According to the results of concrete slitting test and finite element analysis, a new computing method for the width of compression strut of core concrete was proposed in this article. The vertical width and horizontal width of the compression strut are similar to 1/4 scale of the core concrete height and width. The shear strength calculation methods of panel zone of diaphragm-through connections can be expressed as follows

\begin{matrix} V_{cy} = 0.8 V_{c} = 0.8 \cdot f_{c} \cdot \\ \sqrt{{(\frac{B - 2 t_{c}}{4})}^{2} + {(\frac{H_{b} - 2 t_{s}}{4})}^{2} \cdot \frac{(D - 2 t_{c}) (B - 2 t_{s})}{\sqrt{{(B - 2 t_{c})}^{2} + {(H - 2 t_{s})}^{2}}}} \\ = 0.8 \cdot f_{c} \cdot \frac{(D - 2 t_{c}) \cdot (B - 2 t_{s})}{4} \end{matrix}

(8)

where V_cy is the yield shear strength of the core concrete, D is the length of the core concrete, t_s is the thickness of the diaphragm plate, 0.8 is the reduction factor of concrete compression strength, and b_o = (B−2t_c)/4 and h₀ = (H_b−2t_s)/4 are the vertical width and horizontal width of the compression strut, respectively.

The yield shear strength V_y of the diaphragm-through connections is based on superposition of the yield strength of the steel tube V_sy and the strength contribution of core concrete V_cy, given by

V_{y} = \frac{2 t_{c} (B - t_{c}) \sqrt{f_{y}^{2} - σ_{s}^{2}}}{\sqrt{3}} = 0.8 \cdot f_{c} \cdot \frac{(D - 2 t_{c}) \cdot (B - 2 t_{s})}{4}

(9)

The ultimate shear strength point in this model corresponds to where the concrete core reaches its ultimate strength. Therefore, the ultimate shear strength V_u is based on the superposition of the ultimate shear strength of the steel tube and the through-diaphragm V_su on the ultimate strength of the concrete core V_cu, as obtained in equation (10)

V_{u} = \frac{2 t_{c} (B - t_{c}) \sqrt{f_{y}^{2} - σ_{s}^{2}}}{\sqrt{3}} = 0.8 \cdot f_{c} \cdot \frac{(D - 2 t_{c}) \cdot (B - 2 t_{s})}{4}

(10)

Comparison of predicted and experimental results

The estimated yield and ultimate shear strengths using proposed theoretical formulas for through-diaphragm connections are shown in Table 5 in comparison with values obtained from the previous experimental study.

Table 5.

Comparison of test and theoretical results (in kN).

Specimen	Axial ratio	Test result		Calculated value		V_py/V_y	V_pu/V_u
Specimen	Axial ratio	V_py	V_pu	V_y	V_u	V_py/V_y	V_pu/V_u
SJ-1	0.2	506.6	734.1	497.4	724.9	1.02	1.01
SJ-2	0	1001.4	1177.2	942.7	1166.9	0.97	1.01
SJ-3	0.2	1031.1	1265.3	1062.3	1287.1	0.97	0.98
SJ-4	0.2	1040.1	1105.97	1261.5	1553.7	0.82	0.71
SJ-5	0	1113.6	1359.6	1261.5	1553.7	0.88	0.88

It can be seen from Table 5 that the theoretical results are in good agreement with the experimental data. The actual steel tube thickness of panel zone is thinner than the designed value, which caused the observed yield, and the ultimate shear strength of specimen SJ-4 is significantly smaller than the specimen SJ-5 and the calculated value. It can be found that the changed diaphragm-through thickness had no effect on shear strength of panel zone.

Conclusion

In this article, the yield and ultimate shear strengths of panel zone of diaphragm-through connections between concrete-filled square steel tube columns and steel beams were studied based on the experimental phenomena and finite element analysis results of five specimens with different steel tube thickness, concrete strength, diaphragm-through thickness, and axial load ratio.

The main findings can be summarized as follows:

All specimens yielded in the panel zone and presented significant deformation. Yielding of the panel zone was gradual, and no rapid loss of strength was observed. Local buckling occurred on the steel beam flanges, and then, the fracture was found on the weld. Crossing cracks were observed in the infilled concrete of panel zone.

Based on the experimental phenomena and finite element analysis results, the modification has been made to the principal width of the compression strut of core concrete, and a new calculating method for the width of compression strut of core concrete is proposed.

A simple manual calculation method for shear strength of panel zone of diaphragm-through connections was proposed. The shear capacity in the panel zone is resisted by the steel tube web and the diagonal concrete core strut. The computed results for both yield and ultimate shear strengths of panel zone using the proposed formulas are in good agreement with the experimental results.

The yield and ultimate shear strengths were not affected by diaphragm-through thickness. The concrete strength grade has little effect on the shear capacity but contributes in preventing local buckling.

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: The funding for this investigation was provided by the National Natural Science Foundation of China (nos 51268054, 51468061) and Natural Science Foundation of Tianjin City, China (no. 13JCQNJC07300).

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