Abstract
Connection between multicell shaped concrete-filled steel tube (MCFST) columns and steel beams gives full play of strength and properties of steel tube and core concrete and has many advantages in mechanical and seismic performance. Shear behavior of the connection is crucial to damage performance of special-shaped CFST structure under seismic load. However, few studies focused on shear behavior of the MCFST connection. This study investigates elastoplastic shear behavior of the MCFST connection. Nine specimens were tested under constant axial load and lateral low cyclic loading. The test results showed that as height to thickness ratio of column section increased, the plastic status of the stresses of the flange limb steel webs was obviously different from those of the web limb steel webs, and initial shear stiffness of the panel zone increased. In order to give analytical methods for shear behavior of the MCFST connection under compressive and shear force at the material level to engineers, according to unified theory the analytical model of MCFST composite material in the panel zone is equivalent to that of concrete-filled circular steel tube of web and flange limbs considering the difference of the stresses of the steel webs and interaction between the web and flange limbs. Based on the previous research and analysis of test results, migration and evolution are carried out from analytical models for shear behavior of the MCFST connection at the material level to that at the member level. The analytical expression of initial shear stiffness of the MCFST connection is derived, and the theoretical results correlate well with the experimental results. In addition, the analytical expression of shear stiffness of the MCFST connection in plastic hardening stage is obtained. The calculation formulas are proposed to predict the shear stiffness of the MCFST connection in the elastic and plastic hardening stages by geometrical parameters, material property parameters, and stress state of the steel webs.
1. Introduction
In recent years, reinforced concrete frame structures with special-shaped columns have been widely used in China. Because thickness of the special-shaped column is equal to that of infilled walls, room effective area and design freedom of personalization can be increased significantly, and comfortability of residence and perception of space can be enhanced effectively and design freedom of personalization increases [1, 2]. Several disadvantages of the special-shaped reinforced concrete columns are discovered as the application scope of the columns expands. The loading capacity of the columns is lower than that of rectangular columns due to limitations in height to thickness ratio of the columns, concrete strength, and steel ratio. In addition, the special-shaped cross section weakens overall performance of the columns due to performance differences between web and flange column limbs. As a result, shear performance and ductility of special-shaped reinforced concrete connection are worse than those of the rectangular column beam connection, limiting application scope of reinforced concrete frames with special-shaped columns in areas prone to earthquakes up to magnitude 8. In response, it has become essential to increase shear capacity and performance of special-shaped connection and results in comprehensive investigation of mechanical behavior of special-shaped concrete-filled steel tubular structural components in areas prone to seismic activity [3, 4]. Studies have shown that as tightening force the core concrete of concrete-filled steel tube (CFST) components is in triaxial compression stress state, compressive strength, elastic and plastic modulus of the core concrete are improved. Additionally, steel tubes can be used as forms during concrete pouring, which reduces construction costs and increases construction speed. Based on the test and numerical results, simplified formulas for the fire resistance and fire protection layer of L-shaped concrete-filled steel tubular columns were proposed by Yang et al. [5], followed by the parametric analysis. The overall flexural stability performance of the T-shaped irregularly concrete-filled steel tube column (T-ICFSTCs) under combined axial loads and moments was studied by Liu et al. [6]. Based on the simplified FE model verified by the experimental results, the parametric analysis of the overall flexural stability capacity of the T-ICFSTCs was carried out, and a preliminary design formula of the overall flexural stability capacity of the T-ICFSTCs was proposed. Experimental and numerical studies to understand the performance of T-shaped multicell CFST columns to cold-formed U-shaped steel-concrete composite beams (CUCB) joint were conducted by Cheng et al. [7]. The calculation formula for such joint flexural capacity as well as detailing for joint stiffness was proposed. Researchers have been investigating the reliable, accurate, and consistent models for solving structural engineering problems [8]. In order to estimate the compressive strength of geopolymer concrete, a three-step machine learning (ML) approach was employed by Mansouri et al. [8]. Research results showed that both the individual and the hybrid models could predict the compressive strength of geopolymer concrete with high accuracy.
In order to predict the shear stiffness of CFST structural components for engineers by design parameters, a mechanical model was established by Wu et al. [9, 10] for calculating the shear stiffness of bidirectional bolted connections for CFST columns and H-beams, and the elastoplastic behavior of connections consisting of CFST columns and steel beams was studied by Fukumoto and Morita [11] and Ou et al. [12]. The panel zone of the connections is shown in Figure 1. In Figure 1, Hbw and Lc are the height and depth of the panel zone, respectively, and Hbw is the distance between the sectional centroids of the upper and the lower steel beam flanges. The shear stiffness of the panel zone was superposition of shear stiffnesses of the steel tube and the core concrete, but the confinement effect of the steel tube on the core concrete was not taken into account. Analytical models to predict shear stiffness of HSS beam and panel zone were established by Qin et al. [13]. A softening parameter β in the model was proposed to account for the fact that tensile strain of the steel tube could soften confinement on the core concrete. Previous research has shown that the confinement effect of the steel tube on the core concrete is a key factor affecting mechanical behaviors of the CFST components under axial compressive force, pure torsion, and bending moment [14]. So, it has become essential to develop analytical models to evaluate mechanical performance of the CFST components considering the confinement effect of the steel tube on the core concrete.
According to unified theory [10, 15, 16], concrete-filled steel tube was regarded as an unified body composed of a composite material, whose behaviour was changed with change of physics parameters of materials, geometrical parameters of members, types of cross-sections, and stresses states. Unified theory has been used to analyse mechanical behavior of special-shaped CFST components and CFST components composed of multiple materials. The variation of confinement effect of special-shaped CFST columns was investigated when cross-sectional shapes change from triangle, square, and regular polygon to circular [17]. Based on analysis of effective and ineffective confined regions of the core concrete, a cross-sectional formula was obtained to calculate axial compressive strength, and the relationship between cross-sectional interior angle and confinement effect was deduced by regression analysis. Based on a constitutive model of confined concrete, a method for calculating maximum strength of L-shaped CFST columns with and without binding bars was proposed [18]. In order to analyse the confinement effect on the core concrete of unstiffened and stiffened L-and T-shaped steel tube [19], L-and T-shaped CFST was equivalent to square CFST with the same peripheral dimensions. In addition, the confinement factor of each cavity of the column was obtained by considering each cavity as an independent rectangular CFST. By defining the equivalent square section and the angle of boundary between effectively and ineffectively confined areas of the core concrete [20], the constraining force of the multicell T-shaped concrete-filled steel tubular on the core concrete was characterized. Based on the unified theory, the variation of confinement effect of multicell T-shaped steel tubular on the core concrete was investigated when steel plate thickness changed [17], and by splitting a special-shaped cross section with multiple cavities into separate polygons and considering the influence of shape efficiency and regularity. In addition, the strength of each cavity CFST was calculated and summated to obtain the total axial compressive strength. For CFST composed of multiple materials, the equivalent confinement coefficient of unified material could be obtained by combining the confinements from every confining material [21], and one equivalent confinement coefficient coming from circular steel tube was proposed with consideration of different sections of confining materials.
Existing research results [18, 22–24] showed that interaction between column limbs, the stress status of special-shaped steel tube, and the confinement of the steel tube on the core concrete affected the mechanical behavior of special-shaped CFST components. The mono-columns of L-shaped CFST stub columns fabricated worked more cooperatively [25]. The filling of concrete inside vertical steel plates could enhance the overall performance of the connection between the mono-columns. In addition, two side mono-columns and the vertical steel plates yielded first, and the yield region developed gradually to central mono-column steel tube.
However, the existing research on mechanical behavior of special-shaped CFST components has not consider the influence of interaction between column limbs and stress status of special-shaped steel tube. This study investigates elastoplastic shear behavior of the MCFST connection. An experimental study on MCFST connection under constant axial load and lateral cyclic loading was conducted to investigate elastoplastic shear behavior of MCFST connection. The test results showed that the web and flange limbs had different plasticity status, and the plasticity status was influenced by height to thickness ratio of column section, and the initial shear stiffness of the panel zone increased as height to thickness ratio of column section increased. In order to predict the shear behavior of the MCFST connection, analytical methods for shear behavior of the MCFST connection under compressive and shear force at the material level are conducted based on the unified theory by being equivalent to that of concrete-filled circular steel tube of web and flange limbs. To consider the influence of interaction between column limbs, the core concrete is divided into effective and ineffective confined regions, and the force equilibrium of the integral multicell shaped steel tube is analyzed. The influence of stress status of special-shaped steel tube is considered by calculation of constraining force on the core concrete. Based on the previous research and analysis of test results, migration and evolution are carried out from the analytical models for shear behavior of MCFST connection at the material level to that at the member level. The analytical expressions of shear stiffnesses of the MCFST connection in elastic and plastic hardening stages are obtained, and it involve geometrical parameters, material property parameters, and stress state of the MCFST connection.
2. Experimental Program
2.1. Specimen Design and Fabrication
Nine specimens were subjected to axial compression force and lateral low cyclic loading to investigate shear failure mode and seismic behavior of MCFST connection. The specimens were manufactured at a 1/2 scale, and the distance H between upper and lower side load position of the specimens was 1500 mm, and the distance from antibending points of beam to the panel zone was 1200 mm. The thickness of column limb is 120 mm shown as in Figure 2. The specimens were designed to ensure that flexural failure in the panel zone did not occur before shear failure. Thus, thickness of column flange plates was increased to ensure flexural capacity of the specimens before the shear failure in the panel zone. Details of the specimens are shown in Figure 2, and geometry of the column section is depicted in Figure 3, where Hbw is the distance between the sectional centroids of the upper and the lower beam flanges, H is the distance between the upper and the lower column ends, Hc1 is the distance between the upper column end and the upper beam flange, Hc2 is the distance between the lower column end and the lower beam flange, Hb is the height of the beam cross section, Lb is the distance between the lateral surface of the column and the beam end in the vertical direction, Lc is the depth of the steel tube of the web limb, is the width of the flange of the beam end, N is the axial force on the column top, P is the horizontal force on the column top, and Vb is the constrained force at the beam end.
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The specimens were divided into three groups according to cross-sectional shapes (+, T, and L). The design parameters of +-shaped specimens were axial compression ratio n of column and the height to thickness ratio β of column section. The ratio of height-to-thickness of column sections β is defined as β = Lc/Hw. The axial compression ratio of the columns n is defined as n = N/[σcB(Ac + αEAs)], where N is load applied to the column, σcB is the axial compressive strength of the concrete core, Ac and As are respectively the sectional areas of the core concrete and the steel tube, and αE = Es/Ec, where Ec and Es are the elastic moduli of the concrete core and the steel tube, respectively. The design parameters of T- and L-shaped specimens were the height to thickness ratio of column section. The design parameters of the specimens are listed in Table 1.
The specimens consisted of all-welded steel beams, special-shaped CFST columns, and connections with internal diaphragm. The +and T-shaped column consisted of 3 rectangular steel tubes and core concrete, and L-shaped column consisted of 2 rectangular steel tubes and core concrete. All the steel tubes were connected by fillet welds. The column flange plates were connected to the internal diaphragms and the beam flanges by groove welding.
2.2. Material Properties
Q235B and Q345 steel were used in the test. Standard tensile steel samples were fabricated, and tensile tests were conducted according to Chinese National Standard (GB/T228.1-2010) [26]. The measured results of stress-strain relationship of the samples conformed to trilinear model of stress-strain relationship [11], as shown in Figure 4. In Figure 4, σsy, σsr, and σsB are the yield stress, the stress at plastic stiffness reduction point, and the ultimate stress, respectively, αs1 and αs2 are the stiffness reduction ratios at yield point and the plastic stiffness reduction point, respectively. Material strength and thickness of the steel plates of webs and flanges of the beams and columns are listed in Table 2.
Nine 150 mm × 150 mm × 150 mm concrete cubes were fabricated, sampled, and tested according to Chinese National Standard (GB/T50081-2002) [27]. The average cubic strength and elastic modulus of concrete for all specimens on the test day were equal to 43 MPa and 33,256 MPa, respectively.
2.3. Test Set-Up and Loading Scheme
An electro-hydraulic servo actuator with a capacity of 1000 kN and a hydraulic Jack with a capacity of 1500 kN were used in the test. The Jack and column end could be moved in real-time using roller device between antiforce beam and Jack as shown in Figure 5. Mixed force-displacement loading was applied in the test.
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As shown in Figure 6, the increment of each loading level was 40 kN in force loading stage, and loops cycled once. Testing was conducted until lateral load decreased below 85% of ultimate lateral load. Strain gauges were used to measure the strain of the steel webs and flanges in the panel zone, and column and beam ends to investigate shear behavior of the panel zone, as shown in Figure 7. Dial gauges S1∼S4 were used to measure shear deformation of the panel zone, and column end angle θc and beam end angle θb were obtained from displacement gauges W5, W6, and S5∼S8, as shown in Figure 8.
Based on the strain gauge measurements, the principal stresses of the steel webs are calculated as follows [28]:where , , and .
According to 4th strength theory, equivalent stress σ is obtained using the following formula [29]:
The calculation model of shear deformation γ of the panel zone of specimens +J4, +J5, TJ1, TJ2, LJ1, and LJ2 is shown in Figure 9, and the shear deformation γ is obtained as follows [30]:where , γ1 is the rotation angle in the vertical direction, γ2 is the rotation angle in the horizontal direction, and are the tensile lengths in the direction of diagonal when the shear deformations of the panel zone are produced, and and are the compressive lengths in the direction of diagonal when the shear deformations of the panel zone are produced.
The shear deformation γ of the panel zone of specimens +J1, +J2, and +J3 was obtained by the relative rotation angle θj of the panel zone as follows [31]:where θb is the rotation angle of the beam end and θc is the rotation angle of the column end.
3. Test Results and Discussion
3.1. Failure Process and Lateral Shear Force and Deformation Curves
The shear force V of the panel zone can be calculated as follows [32]:
The shear force V-shear deformation γ curves are shown in Figure 10. The characteristic points are marked in the curves.
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First, an axial force was applied at upper end of column and increased gradually to design value. Subsequently, lateral monotonic load was applied at the column end. The curves show a linear relationship for event O-A, which indicates that the steel webs of web and flange limbs were in linear-elastic stage. At event B, the stiffness of the curves decreases significantly, which demonstrates that the specimens reached yielding state as shown in Figure 11. At event C, the specimens reached positive and negative limit state. At event D, the welding connecting web and flange of the web limb cracked in the region which was approximately the same height as internal diaphragm as shown in Figure 11. After event D, the cracks widened and extended in the direction of the specimen length. After the test, steel tubes were stripped and concrete core was exposed as shown in Figure 12. Cracks formed in the core concrete of the web and flange limbs and extended in diagonal direction of the panel zone and near intersection of the internal diaphragm and the column flange.
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3.2. Performance-Based Analysis of Elastoplastic Stress of the Steel Webs
The equivalent stress σ-shear deformation curves are shown in Figure 13. At positive and negative events A, the web limb steel webs of specimens TJ1 and LJ2 exceed yielding point, and the web limb steel webs of specimen +J3 are close to yielding point, and none of the flange limb steel webs of the specimens reaches the yielding point, except for corner region of the flange limb steel webs of specimen +J2. At positive event A, the web limb steel webs of specimen +J1 exceed yielding point, and the corner regions of the web limb steel webs of specimens +J5 and TJ2 exceed yielding point, and the central regions of the web limb steel webs of specimen LJ1 are close to yielding point. At negative event A, the web limb steel webs of specimens +J2 and +J5 exceed yielding point, and the web limb steel webs of specimen +J4 are close to yielding point. At positive and negative events A, the curves of the corner regions of the web limb steel webs of specimens +J1, +J2, +J3, +J4, TJ1, and TJ2 are identical to those of the central regions, and the curves of the corner regions of the flange limbs steel webs of specimens +J3, +J4, +J5, TJ1, LJ1, and LJ2 are identical to those of the central regions.
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At positive and negative events B, the equivalent stresses of the web limb steel webs of all specimens exceed the yield strength, and the equivalent stresses of the web limb steel webs of specimens +J2, +J3, TJ2, and LJ1 exceed ultimate strength, and the central regions of the web limb steel webs of specimen LJ2 exceed the ultimate point. At positive event B, the central regions of the web limb steel webs of specimens +J1 and +J5 exceed the ultimate point. At negative event B, the web limb steel webs of specimens +J1 and +J5 exceed the ultimate point and the corner regions of the web limb steel webs of specimens +J4 and TJ1 exceed the ultimate point. These results indicate a fully developed plastic zone in the web limb steel webs. At positive and negative events B, except for the corner regions of the flange limb steel webs of specimen +J2, the equivalent stresses of the flange limb steel webs of all specimens do not exceed the yield strength.
After positive event B, the equivalent stresses of the web limb steel webs of specimens +J1, +J2, +J3, and LJ1 exceed the ultimate strength and decrease gradually, and the equivalent stresses of the flange limb steel webs of specimens +J4 and TJ1 increase gradually. After negative event B, the equivalent stresses of corner regions of the web limb steel webs of specimens +J2 decrease gradually, and the equivalent stresses of centeral regions of the web limb steel webs of specimens +J1 and +J2 and corner regions of the web limb steel webs of specimen +J3 increase, the equivalent stresses of centeral regions of the web limb steel webs of specimen +J3 were almost unchanged, and the equivalent stresses of the web limb steel webs of specimen +J5 decrease and then increase. After positive and negative events B, the equivalent stresses of the flange limb steel webs of specimens +J1 and +J3 and the centeral regions of the flange limb steel webs of specimen +J2 increase. The growth rates of the stresses of the flange limb steel webs of specimens +J4, +J5, LJ1, and LJ2 are significantly lower than those of the web limb, and the curves of the equivalent stresses of the flange limb steel webs are close to horizontal line. These results demonstrate that plasticity status of the flange limb steel webs are obviously different from those of the web limb steel webs as height to thickness ratio of column section increases and the cross section shape changes.
At positive event C, the equivalent stresses of centeral regions of the flange limb steel webs of specimens +J1 exceed ultimate strength, the equivalent stresses of the flange limb steel webs of specimens +J2 and +J3 exceed ultimate strength. At negative event C, the central regions of the flange limb steel webs of specimens +J1 exceed ultimate point, and the equivalent stresses of the flange limb steel webs of specimen +J2 do not exceed yield strength. At event C, the central regions of the flange limb steel webs of specimens TJ1 and TJ2 exceed ultimate point, whereas the corner regions of the flange limb steel webs of specimens TJ1 and TJ2 does not exceed ultimate point. At event C, the flange limbs steel webs of specimens +J4, +J5, LJ1, and LJ2 do not exceed yield point, whereas the web limb steel webs of specimens +J1,+J3, +J4, +J5, TJ1, TJ2, LJ1, and LJ2 exceed ultimate point.
After event C, the equivalent stresses of the flange limb steel webs of specimens +J3 and TJ1 exceed ultimate strength, and the equivalent stresses of central regions of the flange limb steel webs of specimens +J1 and TJ2 exceed ultimate strength. The plastic zones of the flange limb steel webs of specimens +J1, TJ1, +J3, and TJ2 develop sufficiently. The maximum stress of the flange limb steel webs of specimens +J4, +J5, and LJ1 are 167 N/mm2, 99 N/mm2, and 102 N/mm2, respectively. In all loading process, the equivalent stresses of the flange limb steel webs of specimens +J4, +J5, LJ1, and LJ2 and the corner regions of the flange limb steel webs of specimen TJ2 do not exceed the yield point. As the height to thickness ratio of the column section increases, the maximum stress of the corner regions of the flange limb steel webs decreases.
3.3. Effecting Factors on Initial Shear Stiffness of the Panel Zone
Shear stiffness Ksc of the panel zone can be determined by the following formula [31]:
The shear stiffness Ksc of the curves for event O-A is initial shear stiffness of the panel zone. Figures 14–16 show shear force-deformation envelope curves of the specimens.
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Figures 14(a)–14(c) show that with other condition remaining unchanged, the height to thickness ratio of column section has a significant influence on the initial shear stiffness Ksc. From Figure 14(a), it can be seen that Ksc of the panel zone of specimen +J4 with the sectional height to thickness ratio 3 is higher than that of specimen +J1 with the sectional height to thickness ratio 2, and Ksc of the panel zone of specimen +J5 with the sectional height to thickness ratio 4 is obviously higher than that of specimen +J4. As seen in Figure 14(b), Ksc of the panel zone of specimen TJ2 with the sectional height to thickness ratio 3 under the positive loading is slightly higher than that of specimen TJ1 with the sectional height to thickness ratio 2, and Ksc of the panel zone of specimen TJ2 under the negative loading is obviously higher than that of specimen TJ1. As shown in Figure 14(c), Ksc of the panel zone of specimen LJ2 with the sectional height to thickness ratio 3 under the positive loading is obviously higher than that of specimen LJ1 with the sectional height to thickness ratio 2, and Ksc of the panel zone of specimen LJ2 under the negative loading is slightly higher than that of specimen LJ1. In summary, as the increase of the height to thickness ratio of the column section Ksc increases.
As shown in Figure 15, Ksc of the panel zone of specimens +J1, TJ1, and LJ1 with the height to thickness ratio β of 2 under positive loading is almost the same. The elastic stages of the curves of specimens +J4, TJ2, and LJ2 under negative loading almost coincide, and the differences of Ksc of the panel zone between specimens TJ1 and LJ1 under negative loading is negligible. In summary, the effect of column section shape on Ksc of the panel zone is not significant.
In Figure 16 for event O-A, the curves of specimens +J1, +J2, and +J3 almost coincide, which indicates that the effect of the axial compression ratio on Ksc of the panel zone is not significant.
4. Elastoplastic Analysis Model of Unified Material in Panel Zone Based on Unified Theory
Based on the above analysis results, at event B under combined action of axial force, bending moment, and shear force, the web limb steel webs yield first, which indicates that the specimens are in elastoplastic phase. For events BC, the yielding of the web limb steel webs and the damage caused by crack development of the core concrete lead to shear stiffness degradation of the specimens. As the shear stiffness of the steel tubes and the core concrete decreases, the stress distribution of the steel tubes and the core concrete changes which results in ring-tightening force [14]. The core concrete and special-shaped steel tube affected by the ring-tightening force can be considered as a unified material. The performance of the unified material reflects combined performance of the core concrete and special-shaped steel tube and is evaluated by composite performance index [16].
Existing research results show that lateral expansion deformation of the core concrete causes outward bending of the steel tube walls. Bending rigidity of the central region of the steel tube is low, so constraining force of the central region of the steel tube on the core concrete is low [33]. In contrast, bending rigidity of the corner region of the steel tube is high, so constraining force on the core concrete is high. The constraining force F on the core concrete is in diagonal direction, and the force F is the summations of forces F1 and F2 along the web and flange at the corner region of the steel web as shown in Figure 17(a).
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The regional boundary of effective restraint area of the core concrete is a parabola and initial angle of the parabola is approximately φ1 and φ2. The effective restraint area of the core concrete is shaded area, and the weak restraint area is nonshaded area as shown in Figure 17, where tw1 is the thickness of the web limb steel webs, tw2 is the thickness of the flange limb steel webs, aw is the length of the core concrete of the web limb in web direction, af is the length of the core concrete of the flange limb in web direction, af is the length of the core concrete of the web limb protruding the flange limb, is the width of the core concrete of the web limb in web direction, and bf is the width of the core concrete of the flange limb in web direction.
The local coordinate system for external contour line of the shaded area for specimens +J1∼+J3 is shown in Figure 18.
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The intersection point between starting point of the external contour lines of region S2 and X-axis is Q(−awm/2, 0). The slope of the parabola at the intersection point is defined as follows:
The expression of the external contour lines is as follows:
The area between the external contour lines of region S1 and Y-axis is given as follows:
The area surrounded by the external contour lines of region S2 is defined as follows:
The ineffective restraint area Swie of the core concrete of the web limb is given as follows:
The previous results show that the smaller the thickness of steel plate of the steel tube is, the worse confinement effect of the steel tube on the core concrete is [17]. In order to reflect the influence of the steel plate whose thickness is smaller than average thickness of all the steel plates of the steel tube in a limb on the confinement effect, a correction factor Ψi is introduced, and it is determined by the following formula:where is the average thickness of all the steel plates of the steel tube of a limb, i is the length of the steel plate on a side, and ti is the thickness of the steel plate on a side. The modified ineffective restraint area of the core concrete of the web limb is given as follows:
Thus, the effective restraint area Swe of the core concrete of the web limb is given as follows:
The area surrounded by the external contour lines of S3 is defined as follows:
The area surrounded by the external contour lines of S4 is given as follows:
Thus, the effective restraint area Sfe of the core concrete of the flange limb is given as follows:
The analysis model of constraining force in intersection area between the web and flange limbs is shown in Figure 19. The constraining stress pwI of the web limb is acted on lower side of boundary I-I, and it is derived from the constraining force of the web limb steel tubes and the lateral confinement of the flange limb core concrete. The constraining stress pfI of the flange limb is acted on upper side of boundary I-I, and it is derived from the constraining force of the flange limb steel tubes and the lateral confinement of the web limb core concrete.
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As a result of the lateral confinement of the core concrete, the steel web I-I is in equilibrium of forces, and pwI and pfI satisfy the equilibrium condition of the forces as follows:
According to the previous research [33], the constraining force of square steel tube on the core concrete is equivalent to even distribution of uniform lateral stress. Similarly, the constraining force of the steel tubes of the web and flange limbs on the core concrete is equivalent to even distribution of uniform lateral stress prf and prw, respectively. The analysis model of the constraining stress of the steel tubes of the web and flange limbs is shown in Figure 20. The model includes the right side flanges, and the webs of the web and flange limbs.
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According to characteristics of constraining stress of square steel tube, the following equation can be obtained as follows:
Because the core concrete of the web limb is a continuous integrity, the following equation can be given as follows:
According to equation (18), the following equation is deduced as follows:where pr is the constraining stress of the steel tubes of the web and flange limbs. According to the equilibrium condition in web direction, the following equation of the +-shaped specimens can be obtained:where tf1 is the thickness of the web limb flange, tf2 is the thickness of the flange limb flange, is the length of the web limb steel tube in web direction, lf is the length of the flange limb steel tube in web direction, σf is the transverse tensile stress of flange limb steel webs, is the transverse tensile stress of the web limb steel webs, and σf and are obtained by the measurement results of the strain gauges.
Confinement coefficient is defined based on the concrete-filled circular steel tube analysis model [14, 16]. Existing research results show that considering the influence of ineffective restraint concrete area on the weakness of the confinement coefficient of square and regular octagon steel tube [17], based on unified theory the effective restraint concrete area of concrete-filled square and regular octagon steel tube is equivalent to that of concrete-filled circular steel tube, and mechanical indices of concrete-filled square and regular octagon steel tube correlate well with those of concrete-filled circular steel tube. So, the effective restraint concrete area of the web and flange limbs of the multicell shaped concrete-filled steel tubular (MCFST) is equivalent to the core concrete area of the web and flange limbs of concrete-filled circular steel tubular, respectively, and the steel tube area of the web and flange limbs is equivalent to circular steel tube area of the web and flange limbs, respectively. The analysis model of the constraining force of MCFST in the panel zone is transferred equivalently to that of concrete-filled circular steel tube as shown in Figure 21.
Where pcw and pcf are the constraining stress of the equivalent circular steel tubes of the web and flange limbs, respectively, p is the constraining stress of the equivalent circular steel tubes, r1 is the equivalent radius of the effective restraint core concrete regions of the web limb, tew1 is the equivalent thickness of the web limb steel tube, α1 is the ratio of the web limb steel content, fc is the axial compressive strength of the core concrete, σ1 is equal to the yield strength fy of the steel webs, r2 is the equivalent radius of the effective restraint core concrete regions of the flange limbs, tew2 is the equivalent thickness of the flange limb steel tube, α2 is the ratio of the flange limb steel content, and σ2 is the average principal tensile stress of the flange limb at event B.
According to the characteristics of the constraining stress of square steel tube, the following equation can be obtained:
According to equation (18), the following equation is deduced as follows:
Because the sum of the constraining stress of the steel tube on the core concrete of multicell shaped concrete-filled steel tube is equivalent to that of the concrete-filled circular steel tube, the following formula can be derived:
Substituting equations (22)–(26), the mechanical equilibrium equation can be obtained as follows:
Therefore, the constraining stress p of +-shaped specimens is given as follows:
Based on the experimental results, at event B the stresses of the web limb steel webs of all specimens exceed the yield strength, whereas the stresses of the flange limb steel webs of all specimens do not exceed the yield strength except for the corner regions of the flange limb steel webs of specimen +J2. Therefore, the damage of the concrete-filled circular steel tubular was controlled by stress state of steel webs. The following equation can be obtained by the equilibrium conditions of the forces in the web limb:
The confinement coefficient of the unified material reflects the influence of the constraining force on combined working between the steel tubes and the core concrete. The confinement coefficient ξ+w of the web limb in the panel zone of +-shaped specimens can be obtained as follows:
Substituting equation (30) to (29), the mechanical equilibrium equations can be obtained as follows:
Similarly, the following equation can be obtained by the equilibrium condition of the forces in the flange limbs:
Because the average principal tensile stress of the flange limb steel webs of all specimens do not exceed the yield strength, k is the ratio of the average principal tensile stress to yield strength and determined as follows:
Confinement coefficient ξ+f of the flange limbs in the panel zone of +-shaped specimens can be obtained by the following formula:
Substituting equation (35) to (34), the mechanical equilibrium equations can be obtained as follows:
Similarly, the confinement coefficient ξ of the unified material in the whole panel zone consisting of the web and flange limbs of +- and T-shaped specimens can be obtained by the following formulas:where As1 and As2 are the areas of the steel tubes of the web and flange limbs, respectively and Ac1 and Ac2 are the areas of the effective restraint core concrete of the web and flange limbs, respectively. Similarly, the confinement coefficient ξ of L-shaped specimens can be obtained by the following formula:
The confinement coefficient ξ+w of the web limb of +shaped specimens can be obtained by the following formula:
In addition, the confinement coefficient ξ+f of the flange limbs of +shaped specimens can be deduced by the following formula:
Similarly, the confinement coefficient ξTw of the web limb of T-shaped specimens can be obtained by the following formula:
In addition, the confinement coefficient ξTf of the flange limbs of T-shaped specimens can be deduced by the following formula:
In addition, the confinement coefficient ξLw of the web limb of L-shaped specimens can be obtained by the following formula:
In addition, the confinement coefficient ξLf of the flange limbs of L-shaped specimens can be deduced by the following formula:
5. Analysis and Calculation for Elastoplastic Shear Modulus in Panel Zone Based on Unified Theory
5.1. Analysis for Elastoplastic Shear Modulus in Panel Zone Based on Unified Theory
The shear force-deformation relation for the panel zone based on the experimental results is shown in Figure 22. According to the unified theory, the average shear stress of the unified material of the panel zone under the shear force V of the panel zone can be obtained by the following formula:where Asc is the area of the equivalent analysis model of constraining force of the panel zone, for the +- and T-shaped specimens, , for the L-shaped specimens, . The average shear stress-deformation relation for the panel zone is shown in Figure 23. The shear modulus , , and of the unified material of the panel zone can be obtained by the following formulas:
Based on the unified theory [14] and above results, A denotes the starting point of elastoplastic stage of the web limb steel webs. In the elastic stage OA, the stresses of the steel tubes of the web and flange limbs and the core concrete are small, and the steel tubes have no confinement effect on the core concrete. A is the state point which indicates that the steel tube goes into the elastoplastic phase. In the elastoplastic stage AB, with the increase of loading more areas of the web limb steel webs yield. At event a0, because microcracks form in the core concrete and extend, and the steel tubes provide an increasing constraining force to the core concrete. At event B, almost all areas of the web limb steel webs of all the specimens yield, which indicates that the unified material of the panel zone reaches the yield point. In the plastic hardening stage BC, the core concrete can support the steel tubes and prevent local buckling of the steel tubes. After event B, the loading capacity of the specimens continues to increase, and the specimens exhibit good elastoplastic performance. In Figure 22, VA, VB, and VC are the shear forces of the panel zone at state point A, B, and C, respectively, γA, γB, and γC are the shear deformations of the panel zone at state point A, B, and C, respectively, , , and are the average shear stresses of the unified material of the panel zone at state point A, B, and C, respectively, , , and are the shear strains of the unified material of the panel zone at state point A, B, and C, respectively, Ksc is the initial shear modulus of the panel zone, and are the shear moduli of the panel zone in the elastoplastic and the plastic hardening stages, respectively, is the initial shear modulus of the unified material of the panel zone, and and are the shear moduli of the unified material of the panel zone in the elastoplastic and the plastic hardening stages, respectively.
5.2. Calculation for Initial Shear Modulus in Panel Zone Based on Unified Theory
According to the unified theory of CFST, the composite shear strength of the unified material at event a0 is obtained as follows [14, 30]:where and . Based on , the composite shear strength of the unified material at event A is derived as follows [23]:
The shear strain γA of the unified material at event A is obtained as follows:
The shear modulus Gsct of the unified material at event A is obtained by the following formula:
In the stage OA the steel tubes of the web and flange limbs and the core concrete in the panel zone are linear-elastic, and as a whole can be taken as the unified material. According to the previous research [11], the initial shear modulus of the panel zone is equal to the shear modulus of the unified material in the stage OA. According to experimental results, the ratio of height to thickness of column section has obviously influenced on initial shear modulus Ksc of the panel zone. It is assumed that f (β) is influence coefficient of the ratio of height to thickness of column section on the initial shear modulus , and is obtained as follows:
Mathematical expression f (β) is obtained by the following regression analysis:
Figure 24 shows the comparison of theoretical shear modulus Gsc and experimental results of shear modulus . From Figure 24, it is seen that the theoretical results correlate well with the experimental results.
5.3. Calculation for Elastoplastic Shear Modulus in Panel Zone in the Plastic Harding Stage
As a result of the combined effect of axial compression and shear force, the stress conditions of the unified material of the panel zone include compressive stress σsN and shear stress τsc, as shown in Figure 25.
σsN is given as follows:where Ac and As are cross-sectional areas of the core concrete and steel tube in the panel zone, respectively.
The principal stress of the unified material can be expressed as follows:where σsc1 and σsc3 is the principal tensile and compressive stress of the unified material unit, respectively. According to Von Mises yield criterion, the principal stresses satisfy the following equation:where is yield stress of the unified material in simple shear test. can be expressed as follows [16]:
Equation (53) is substituted into equation (54) to obtain the following equation:
Thus, the composite shear strength of the unified material of the panel zone at event B under axial compression and shear force can be obtained as follows:
Based on the unified theory and test results, it is assumed that is the function which expresses the variety rule of the shear modulus of the unified material of the panel zone in elastoplastic stage with the change of average shear stress of the unified material of the panel zone as shown in Figure 23. In elastoplastic stage AB as a result of crack development of the core concrete, the shear modulus is obtained as follows [14]:where , and is shear modulus of the panel zone in plastic hardening stage BC. By formula (47), is given by
From Figure 23, the midpoint of AB is M. According to formula (58), shear modulus at M is and is equal to the slope kAB of line AB. Based on the previous study results [11], kAB can be expressed as follows:where is the combined axial compressive strength of the unified material, is the combined axial compressive proportional limit of the unified material, and is the combined axial compressive modulus of the unified material, , , , and .
The equation can be concluded as follows:
is solved by formula (61).
Substituting into formula (58), can be obtained. According to the unified theory, in the stage AB microcracks of the core concrete appear and develop, and in the stage BC microcracks of the steel tubes appear and develop. The experiments showed that macrocracks of the steel tubes appeared after the ultimate bearing capacity of the panel zone. The calculation method for composite strengths and shear modulus of the unified material based on unified theory in the stage AC has considered the influence of the occurrence and development of the microcracks of the core concrete and the steel tubes. So, the analytical expression of the shear stiffness of the panel zone in the plastic hardening stage BC deduced by the shear modulus and composite strength has considered the influence of the occurrence and development of the microcracks of the core concrete and the steel tubes.
Based on the analysis results in Figure 10, the range of the absolute values for the shear deformations of the panel zone at event B is 0.0005–0.005, and the range of the absolute values for the shear deformations in the panel zone at event C is 0.0008–0.011. So, the shear deformation of the panel zone at elastoplastic stage is large deformation situation. According to the unified theory of CFST, the shear deformation of the unified material at event B is obtained as follows [14]:
Based on the confinement coefficient ξ of the specimens obtained by formulas (36) and (37), the maximum value of the shear deformation of the unified material is 0.011, which indicates that the shear deformation in the panel zone at elastoplastic stage is in the range of the shear deformation of the unified material at event B. So, it is reasonable that formula (63) is used to control the shear deformation of the panel zone at elastoplastic stage.
6. Conclusions
Elastoplastic analysis of shear behavior of MCFST connection based on unified theory was conducted. The following conclusions can be drawn based on the results:(1)Nine specimens were tested under constant axial load and lateral cyclic loading. When the specimens reached yield point, the stiffness of the curves decreased significantly, and the stresses of the web limb steel webs of all specimens exceeded yield strength, and the stresses of the flange limb steel webs of all specimens did not exceed the yield strength except for corner regions of the flange limb steel webs of specimen +J2. These results demonstrate that in all loading process plasticity status of the stresses of the flange limb steel webs are obviously different from those of the web limb steel webs as height to thickness ratio of column section increases and cross section shape changes. As height to thickness ratio increases, initial shear stiffness of the panel zone increases. The effect of column section shape and axial compression ratio on the initial shear stiffness of the panel zone is not significant.(2)In order to give the analytical model for the shear behavior of the MCFST connection under compressive and shear force at the material level to engineers, based on unified theory the analytical model of multicell shaped concrete-filled steel tube in the panel zone is transferred equivalently to that of concrete-filled circular steel tube of the web and flange limbs considering difference of stress state and interaction between the web and flange limbs. First, the core concrete is divided into effective and ineffective restraint regions based on the analysis of lateral confinement of neighboring limb and the influence of steel plate thickness on the confinement effect. Second, the constraining force of steel tubes of the web and flange limbs on the core concrete is equivalent to even distribution of the uniform lateral stress. Third, according to the equilibrium condition of forces the constraining stress in the effective restraint concrete area is obtained, and confinement coefficient of the web and flange limbs of unified material in the panel zone is deduced. Eventually, the confinement coefficient of unified material in the whole panel zone, and the analytical expressions of composite shear strengths and shear modulus of the unified material in elastic and elastoplastic stages of the specimens are obtained.(3)Based on the previous research and analysis of test results, migration and evolution are carried out from analytical models for the shear behavior of the MCFST connection at the material level to that at the member level. By regression analysis, the analytical expression of initial shear modulus of the unified material in the panel zone is proposed, and the theoretical results correlate well with the experimental results. In addition, the analytical expression of shear stiffness of the MCFST connection in plastic hardening stages is obtained. The calculation formulas are proposed to predict the shear stiffness of the MCFST connection in the elastic and plastic hardening stages by geometrical parameters, material property parameters, and stress state of the MCFST connection.
Notations
| Hbw (mm): | Distance between sectional centroids of upper and lower beam flanges |
| H (mm): | Distance between upper and lower column ends |
| Hc1 (mm): | Distance between the upper column end and upper beam flange |
| Hc2 (mm): | Distance between the lower column end and lower beam flange |
| Hb (mm): | Height of beam cross section |
| N (kN): | Axial force on column top |
| P (kN): | Horizontal force on column top |
| Vb (kN): | Constrained force at beam end |
| Lb (mm): | Distance between the lateral surface of column and beam end in vertical direction |
| Lc (mm): | Depth of steel tube of web limb |
| Hw (mm): | Width of flange of beam end |
| n: | Axial compression ratio of column |
| β: | Ratio of height-to-thickness of column section |
| σcB (N/mm2): | Axial compressive strengths of core concrete |
| Ac and As (mm2): | Sectional areas of concrete core and steel tube |
| Ec and Es (MPa): | Elastic moduli of concrete core and steel tube |
| σsy, σsr, and σsB (N/mm2): | Yield stress, stress at plastic stiffness reduction point, and ultimate stress |
| αs1 and αs2: | Stiffness reduction ratios at yield point and plastic stiffness reduction point |
| θc and θb: | Column and beam end angles |
| and (N/mm2): | The principal stresses of the steel webs |
| σ (N/mm2): | Equivalent stress according to 4th strength theory |
| γ: | Shear deformation of panel zone |
| γ1: | Rotation angle in vertical direction |
| γ2: | Rotation angle in horizontal direction |
| and (mm): | Tensile lengths in diagonal direction when shear deformations of panel zone are produced |
| and (mm): | Compressive lengths in diagonal direction when shear deformations of panel zone are produced |
| θj: | Relative rotation angle of panel zone |
| θb: | Rotation angle of beam end |
| θc: | Rotation angle of column end |
| V (kN): | Shear force of panel zone |
| Ksc (MPa): | Shear stiffness of panel zone |
| F (kN): | Constraining force on core concrete in diagonal direction |
| F1 and F2 (kN): | Components of constraining forces on core concrete along web and flange |
| φ1 and φ2: | Initial angles of regional boundary of effective restraint area of core concrete |
| tw1 and tw2 (mm): | Thicknesses of web and flange limbs steel webs |
| aw and af (mm): | Lengths of core concrete of web and flange limbs in web direction |
| bw (mm): | Width of core concrete of web limb in web direction |
| bf (mm): | Width of core concrete of flange limb in web direction |
| Swie (mm2): | Ineffective restraint area of web limb core concrete |
| Ψi: | Correction factor reflecting influence of steel plate whose thickness is smaller than average thickness of all steel tube plates in a limb on confinement effect |
| (mm): | Average thickness of all steel plates of steel tube of a limb |
| i (mm): | Length of steel plate on a side |
| ti (mm): | Thickness of steel plate on a side |
| (mm2): | Modified ineffective restraint area of web limb core concrete |
| Swe (mm2): | Effective restraint area of web limb core concrete |
| Sfe (mm2): | Effective restraint area of flange limb core concrete |
| pwI (N/mm2): | Constraining stress of web limb acting on the lower side of common boundary of web and flange limbs |
| pfI (N/mm2): | Constraining stress of flange limb acting on the upper side of common boundary of web and flange limbs |
| prw (N/mm2): | Uniformly distributed stress equivalent to pwI |
| prf (N/mm2): | Uniformly distributed stress equivalent to pfI |
| pr (N/mm2): | Constraining stress of steel tubes of web and flange limbs |
| tf1, tf2 (mm): | Thicknesses of web and flange limbs flanges |
| and lf (mm): | Lengths of web and flange limbs steel tube in web direction |
| σf and σw (N/mm2): | Transverse tensile stresses of flange and web limbs steel webs |
| pcw and pcf (N/mm2): | Constraining stresses of equivalent circular steel tubes of web and flange limbs |
| p (N/mm2): | Constraining stress of equivalent circular steel tubes |
| r1 and r2 (mm): | Equivalent radii of effective restraint core concrete of web and flange limbs |
| tew1 and tew2 (mm): | Equivalent thicknesses of the web and flange limbs steel tubes |
| α1 and α2: | Ratios of web and flange limbs steel contents |
| fc (N/mm2): | Axial compressive strength of core concrete |
| σ1 and σ2 (N/mm2): | Average principal tensile stresses of web and flange limbs at event B |
| ξ: | Confinement coefficient of unified material |
| ξ+w and ξ+f: | Confinement coefficients of unified material of web and flange limbs in panel zone of +-shaped specimens |
| ξTw and ξTf: | Confinement coefficients of unified material of web and flange limbs in panel zone of T-shaped specimens |
| ξLw and ξLf: | Confinement coefficients of unified material of web and flange limbs in panel zone of L-shaped specimens |
| k: | Ratio of average principal tensile stress to yield strength |
| Asc (mm2): | Area of equivalent analysis model of constraining force of panel zone |
| As1 and As2 (mm2): | Areas of steel tubes of web and flange limbs in equivalent analysis model of constraining force of panel zone |
| Ac1 and Ac2 (mm2): | Areas of effective restraint core concrete of web and flange limbs |
| (N/mm2): | Average shear stress of unified material of panel zone under shear force at column end |
| VA, VB, and VC (kN): | Shear forces of panel zone at state point A, B, and C |
| γA, γB, and γC: | Shear deformations of panel zone at state point A, B, and C |
| , , and (N/mm2): | Average shear stresses of unified material of panel zone at state point A, B, and C |
| , , and : | Shear strains of unified material of panel zone at state point A, B, and C |
| Ksc (MPa): | Initial shear modulus of panel zone |
| and (MPa): | Shear moduli of panel zone in elastoplastic and plastic hardening stages |
| (MPa): | initial shear modulus of unified material of panel zone |
| and (MPa): | Shear moduli of unified material of panel zone in elastoplastic and plastic hardening stages |
| (N/mm2): | Composite shear strength of unified material at event a0 |
| (N/mm2): | Composite shear strength of unified material at event A |
| Gsct (MPa): | Shear modulus of unified material at event A |
| γA: | Shear strain of unified material at event A |
| E and F: | Calculation parameters for composite shear strength of unified material at event a0 |
| f (β): | Influence coefficient of ratio of height to thickness of column section on initial shear modulus |
| σsN and τsc (N/mm2): | Compressive and shear stresses of unified material of panel zone |
| Ac and As (mm2): | Cross-sectional areas of core concrete and steel tube in panel zone |
| σsc1 and σsc3 (N/mm2): | Principal tensile and compressive stresses of unified material unit |
| (N/mm2): | Yield stress of unified material in simple shear test |
| (N/mm2): | Composite shear strength of unified material of panel zone at event B under axial compression and shear force |
| F (β): | Function which express variety rule of shear modulus of unified material of panel zone in elastoplastic stage with change of average shear stress of unified material of panel zone |
| A1 and B1: | Calculation parameters for function f(β) |
| kAB: | Slope of line AB |
| (N/mm2): | Combined axial compressive strength of unified material |
| (N/mm2): | Combined axial compressive proportional limit of unified material |
| (MPa): | Combined axial compressive modulus of unified material. |
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no.: 51808195), and National Natural Science Foundation of China (Grant no.: 50978217). The financial support is highly appreciated.