Local buckling of steel plates in composite members with tie bars under axial compression

Engineering Structures 205 (2020) 110097

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Local buckling of steel plates in composite members with tie bars under axial compression

T

Hong-Song Hua, , Peng-Peng Fangb, Yang Liub, Zi-Xiong Guob, Bahram M. Shahroozc ⁎

a

Intelligence and Automation in Construction Fujian Province Higher-educational Engineering Research Centre, College of Civil Engineering, Huaqiao University, Xiamen 361021, China b College of Civil Engineering, Huaqiao University, Xiamen 361021, China c Dept. of Civil and Architectural Engineering and Construction Management, University of Cincinnati, 765 Baldwin Hall, Cincinnati, OH 45221-0071, USA

ARTICLE INFO

ABSTRACT

Keywords: Local buckling Tie bar Concrete-filled steel tube (CFST) Composite shear wall

Tie bars have been frequently used in square or rectangular concrete-filled steel tubular (CFST) columns and concrete-filled steel plate (CFSP) composite shear walls for delaying local buckling of the steel plates and also improving the concrete confinement. This study investigated the entire response of steel plates in composite members with tie bars. Twelve specimens were designed and tested under axial compression. In these specimens, only the steel plates resisted the axial load, and the concrete was only used to restrain the inward buckling of the steel plates. The effects of the width-to-thickness ratio of the steel plate and the ratio of the vertical spacing of the tie bars to the steel plate width (abbreviated as vertical-spacing-to-width ratio) were investigated. It was found that the increase in the performance of the steel plates due to the tie-bar restraint was quite limited when the vertical-spacing-to-width ratio was 1/2 and became more significant when this ratio decreased to 1/3. The elastic buckling stresses of rectangular plates having boundary conditions similar to those in composite members were computed using the finite element method for further strength evaluation of the steel plates. It was found that the effect of the columns of tie-bars on the elastic buckling stress increased as the vertical-spacing-to-width ratio decreased and was negligible when this ratio was larger than 0.6. Simplified equations for estimating the maximum strength and reserved strength (defined as the average stress corresponding to the average strain of 3%) of the steel plates in composite members with tie bars were developed based on the test results and elastic buckling analyses.

1. Introduction In recent years, concrete-filled steel tubular (CFST) columns have increasingly gained popularity in structural applications, such as highrise building structures, bridges, etc. [1]. In CFST columns with square or rectangular sections, the inward local buckling of the steel plates is effectively prevented due to the existence of the concrete infill; therefore, the steel tube in CFST columns exhibits higher structural performance than the corresponding bare steel tube. Nevertheless, the steel plates in CFST columns can still buckle outward; thus, when the widthto-thickness ratio of the steel tube exceeds a certain value, the yield strength of the section cannot be developed due to the premature buckling of the steel plates [2]. For the past three decades, the structural performance of square CFST columns has been investigated extensively (e.g., [3–6]), and several researchers (e.g., [2,7–10]) have attempted to develop

⁎

empirical formulae to estimate the compressive strengths of the steel plates in square CFST columns. These formulae were mainly developed based on the experimental results of two types of axial compression tests. In the first test method (e.g., [8,9]), only the steel tube of the CFST column was loaded, which was achieved by making the top surface of the concrete infill slightly lower than that of the steel tube. Although this method can directly measure the axial load resisted by the steel tube, it excludes the effect of the transversal stresses in the steel plates induced by the lateral expansion of concrete that exists in real CFST columns. The other method is the conventional axial loading test, in which the steel tube and concrete infill are loaded simultaneously. Since the axial load carried by the steel tube or concrete infill cannot be directly measured, the compressive strength of the steel tube is determined based on an assumed concrete load at the maximum total load (e.g., [7,11]). The correctness of this method depends primarily on the accuracy of the assumed concrete load, which involves many

Corresponding author. E-mail address: [email protected] (H.-S. Hu).

https://doi.org/10.1016/j.engstruct.2019.110097 Received 9 April 2019; Received in revised form 19 November 2019; Accepted 13 December 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Structures 205 (2020) 110097

H.-S. Hu, et al.

Concrete

Butt weld

Steel tube

Concrete

Tie bar

investigate the local buckling and post-buckling behavior of steel plates in composite members with restraints provided by tie bars or other similar elements (e.g., shear studs). Liang et al. [29] studied the local and post-local buckling strength of steel plates in CFSP (called as “double skin” in their paper) composite panels under biaxial compression and in-plane shear using the finite element method. Cai and Long [30] theoretically studied the elastic local buckling of steel plates in rectangular CFST columns with tie bars. Zhang et al. [31] compiled the available test results of CFSP composite walls under axial compression conducted by researchers in Japan and Korea, and further conducted parametric studies using finite element analyses. Based on the experimental and numerical results, they proposed a plate slenderness (ratio of stud spacing to plate thickness) limit to ensure that steel plates yield prior to local buckling. Yang et al. [32] tested ten CFSP composite walls under axial compression and obtained similar results as Zhang et al. [31]. The above studies are all limited to elastic local buckling except the work of Liang et al. [29]. Since slender steel plates can carry additional loads well after elastic buckling and plates satisfying appropriate slenderness limits can still buckle in inelastic range which causes strength degradation [33], it is necessary to investigate the complete load-deformation response of steel plates loaded into large deformations, which is of especial importance for ductile design of composite structures under seismic loading. The study presented herein investigated the entire response of steel plates in composite members, with tie bars, axially loaded to large deformations. The specimens were designed so that the axial load was only resisted by the steel plates and the concrete was only used to restrain the inward buckling of the steel plates. The effects of the widthto-thickness ratio of the steel plate and the ratio of the vertical spacing of the tie bars to the steel plate width were investigated in the tests. Elastic buckling analyses using the finite element method were also conducted to compute the elastic buckling stresses corresponding to various arrangements of tie bars. Simplified equations were developed for estimating the strengths of the steel plates considered herein.

Steel tube

Tie bar Nut Fillet weld

(a) Concrete

(b) Fillet weld

Tie bar

Steel plate

(c) Fig. 1. Typical composite members with tie bars: (a) CFST column with oblique tie bars [13]; (b) CFST column with orthogonal tie bars [15]; and (c) CFSP composite shear walls with tie bar connections [20].

uncertainties. More recently, Hu et al. [12] devised a new test method that can directly measure the axial loads carried by the concrete infill and steel tube. The test results of five specimens with varying width-tothickness ratios indicate that most of the available empirical formulae could reasonably estimate the compressive strength of the steel tube despite of their formulation differences. To further improve the structural performance of square or rectangular CFST columns, some researchers [13–16] have proposed to use tie bars to connect the adjacent [Fig. 1(a)] or opposite [Fig. 1(b)] faces of the steel tube. This additional configuration can delay local buckling of the steel plates and also improve the concrete confinement provided by the steel tube, since the out-of-plane displacements of the steel plates are restrained at the tie points. Tie bars have also been widely used in concrete-filled steel plate (CFSP) composite shear walls for connecting the two surface plates (e.g., [17–20]). A common configuration of CFSP composite shear walls using tie bar connections is shown in Fig. 1(c) [20]. In these composite shear walls, local buckling of steel plates is directly related to the tie bar spacing; therefore, an in-depth understanding of the effect of tie bar constraints on the local buckling behavior of steel plates is essential for determining appropriate spacing of tie bars [21]. When referring to the construction procedure of the tie bar connections, there generally exist two methods. The most common technique is to drill holes into the steel plates first and then insert the tie bars through the holes; the tie bars can then be welded to the steel plates using fillet welds [Fig. 1(c)] or be fixed using nuts [Fig. 1(b)]. The other technique involves a patented welding procedure, which was adopted in a composite wall system called Bi-Steel [22]. Another method commonly used for stiffening the steel plates in composite members is to attach longitudinal stiffeners to the inner surface of the steel plates (e.g., [23–26]). Compared to tie bars, longitudinal stiffeners are more effective in delaying local buckling of the steel plate. A longitudinal stiffener, particularly if its moment of inertia is sufficiently large, provides a continuous line of restraint [26] whereas a tie bar only restrains a discrete point on the plate, but longitudinal stiffeners require a large amount of welding. Hence, tie bars are preferred to be used in CFSP composite shear walls in the U.S. practice [19,20]. Another advantage of tie bars, compared to longitudinal stiffeners, is that they can improve concrete confinement provided by the steel plates since the tie bars connect two plates together, as shown in Fig. 1. In China, tie bars and longitudinal stiffeners are often used together in composite structure practice (e.g., [27,28]). By far, very limited research [29–32] has been conducted to

2. Experimental program In a conventional axial loading test of CFST columns, the steel tube and concrete infill are loaded simultaneously, and only the total load can be directly measured. Since the load-deformation response of the concrete infill in CFST columns with tie bars is unknown, the axial force resisted by the steel plate at a certain axial deformation cannot be accurately determined if the conventional testing method is used. To directly obtain the axial load resisted by the steel plates in CFST columns with tie bars, specimens with the details as shown in Fig. 2 were designed. Each specimen comprised a steel tube with two end plates, concrete infill, and tie bars with their ends welded to the opposite faces of the steel tube. The steel tube was fabricated by welding four steel plates of the same thickness using complete joint penetration groove welds. For each direction of the principal axis, four tie bars were arranged at equal spacing along the length of the specimen. To ensure that the axial force would not be transferred to the concrete infill, a polytetrafluoroethylene (PTFE) layer was added between the steel tube and the concrete infill, and foam boards with a thickness of 10 mm were placed between the adjacent tie bars and on the inner face of the end plates. The foam boards divided the concrete into several separate parts with each part connected to two tie bars (one in each principal direction). Since the elastic modulus of the foam board is very small, the axial force resisted by the steel tube would not be transferred to the concrete until the foam boards were completely squashed. It should be noted that the plate segments beside the foam boards were not laterally restrained by the concrete infill; however, since the length of these plate segments (i.e., the thickness of the foam boards, 10 mm) was very small compared to the steel tube width, local buckling would not solely occur in these regions. Therefore, the presence of foam boards would not alter the behavior of the steel plates when they did not exist. There was a 2

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0.5sv

End plate Steel tube

B

sv

L

sv

Tie bar

A

B

A

Foam board sv

Concrete 0.5sv

PTFE layer B-B section B A-A section Fig. 2. Details of the specimens with tie bars.

square hole in the middle of each end plate in order to spread the PTFE layer and place concrete. The foam board was first positioned before placing concrete on it; this procedure was repeated until the top foam board was installed. For comparison, specimens without tie bars were also designed as shown in Fig. 3. Since tie bars were not included in these specimens, there is no need to separate the concrete into different parts; for simplicity, a 40-mm gap was left between the top end plate and the top surface of concrete so that the top end plate would not touch the concrete during axial loading; this configuration is equivalent to using 4 foam boards with a thickness of 10 mm equally spaced along the length of the specimen, as adopted in the specimens with tie bars. To prevent local buckling in the gap region, the steel tube was strengthened by welding an additional steel plate on each face of it. It should be noted that transverse stresses in the steel plates induced by lateral expansion of concrete cannot be simulated in this test. However, previous studies on square CFST columns without tie bars [12] indicate that the empirical formulae developed on the basis of this kind of testing procedure could still reasonably estimate the compressive strength of the steel tube in real CFST columns. Hence, the testing method is deemed acceptable. The specimens with and without tie bars are denoted as WB and WOB specimens, respectively, hereinafter. In total, twelve specimens were fabricated. The outer width of the

steel tube, B, was 300 mm for all the specimens. The specimens were divided into four groups according to the wall thickness of the steel tube. Each group included two WB specimens and one WOB specimen. The tie bar spacing, sv, of the two WB specimens was 100 mm and 150 mm with the resulting values of sv/B equal to 1/3 and 1/2, respectively. Since the tie bar spacing rather than the steel tube length is the primary parameter that influences the local buckling behavior of steel plates, the length of the steel tube, L, was taken as 4 times the corresponding tie bar spacing to make the load-displacement curves of different specimens comparable. Consequently, L was to 400 and 600 mm for the specimens with sv = 100 and 150 mm, respectively. The length of the unstrengthened region of the WOB specimen, L, was 600 mm. The measured wall thicknesses (t) of the steel tubes ranged from 3.94 mm to 9.44 mm, resulting in the width-to-thickness (B/t) ratios ranging from 31.8 to 76.1. The measured diameter of the tie bars was 9.85 mm. Table 1 shows the geometric properties of all the specimens. In the designation of each specimen, the first number represents the nominal wall thickness of the steel tube, and the second number represents the tie bar spacing; if the second number is replaced by the letter “N”, it means that this is a WOB specimen. For example, specimen SP4-100 is the specimen with a nominal wall thickness of 4 mm and tie bar spacing of 100 mm. The measured cylinder (150 × 300 mm) strength, fc′, of the concrete filled in the specimens was 33 MPa. The measured stress-strain relationships of the steel plates and the tie bar from tensile tests are shown in Fig. 4. Although Q235 steel was used for all the steel plates, the ultimate strain of the 8 mm plate was considerably lower than those of the other plates. Nevertheless, this difference would not affect the local buckling behavior of the specimens since the ultimate strain would not be possibly reached under compression. The yield strength, fy, and the tensile strength, fu, of the steel plate corresponding to each specimen are given in Table 1. The stress-strain relationship of the tie bar did not have a yield plateau; hence, the stress corresponding to 0.2% offset strain was taken as the yield strength of the tie bar. The yield strength and the tensile strength of the tie bar were 445 MPa and 506 MPa, respectively. After the specimens were fabricated, the out-of-plane geometric imperfections of each face of the steel tube were measured using a linear variable displacement transducer (LVDT) installed on a motordriven linear guide rail, as shown in Fig. 5(a). The measuring paths of the LVDT are illustrated in Fig. 5(b). The plane passing through three

50

End plate Strengthening plate Gap Steel tube

L

Concrete

B

B

A

A

PTFE layer

B-B section B A-A section

Fig. 3. Details of the specimens without tie bars. 3

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Table 1 Geometric and material properties and primary test results. Specimen designation

L (mm)

t (mm)

B/t

sv (mm)

sv/B

fy (MPa)

fu (MPa)

wap (mm)

wap/B

Nm (kN)

Nlb (kN)

δm/L (%)

δlb/L (%)

SP4-100 SP4-150 SP4-N SP6-100 SP6-150 SP6-N SP8-100 SP8-150 SP8-N SP10-100 SP10-150 SP10-N

400 600 600 400 600 600 400 600 600 400 600 600

3.94 3.94 3.94 5.90 5.90 5.90 7.78 7.78 7.78 9.44 9.44 9.44

76.1 76.1 76.1 50.8 50.8 50.8 38.6 38.6 38.6 31.8 31.8 31.8

100 150 – 100 150 – 100 150 – 100 150 –

1/3 1/2 – 1/3 1/2 – 1/3 1/2 – 1/3 1/2 –

376 376 376 343 343 343 302 302 302 324 324 324

511 511 511 473 473 473 425 425 425 468 468 468

0.67–1.07 0.46–0.93 0.78–1.94 0.47–0.77 0.55–1.54 0.48–1.15 0.44–0.78 0.67–1.85 0.51–1.44 0.36–0.90 0.61–1.24 0.69–1.26

1/448–1/280 1/652–1/323 1/385–1/155 1/638–1/390 1/545–1/195 1/625–1/261 1/682–1/385 1/448–1/162 1/588–1/208 1/833–1/333 1/491–1/242 1/435–1/238

1356 1221 1114 2182 2071 1914 3044 2716 2660 3943 3602 3634

1332 1081 1109 2182 2055 1679 2970 2715 2660 3943 3601 3610

0.30 0.20 0.19 0.45 0.24 0.23 1.52 0.49 0.41 2.07 0.48 0.54

0.32 0.14 0.20 0.48 0.26 0.15 1.17 0.48 0.41 2.03 0.50 0.52

Fig. 6 shows some representative results of the measured imperfections, in which the outward imperfection is taken as positive and the inward imperfection is taken as negative. The four corners of one face were seldom close to the same plane, except in few cases such as that shown in Fig. 6(d). The vertical distance of the fourth corner relative to the plane determined by the other three corners was mostly between 0.5 and 2 mm, and the maximum measured value reached about 3 mm. The imperfection contours of some faces such as those shown in Fig. 6(a), (b) and (d) are to some extent similar to the corresponding first buckling mode of the steel tube; for other faces such as that shown in Fig. 6(c), no distinct waves can be observed. The level of imperfection of a certain face can be represented by wap = (wmax-wmin)/ 2, where wmax and wmin are the measured maximum and minimum imperfections of this face. The minimum and maximum values of wap and wap/B among the four faces of each specimen are given in Table 1. The minimum wap/B among the four faces of each specimen ranges from 1/833 to 1/385 with an average of 1/531; and the maximum wap/ B ranges from 1/390 to 1/155 with an average of 1/242, which is close to the value of 1/200 recommended by Eurocode 3 [34] to be included in finite element models. All the specimens were tested under monotonic axial compression using a 10,000-kN universal testing machine. Four vertical LVDTs were installed symmetrically to measure the axial shortening of the steel tube; strain gauges were attached along the center lines of the back and left faces (the front face is the one that faces the fixed camera as shown

600

Stress (MPa)

450 300

4 mm plate 6 mm plate 8 mm plate 10 mm plate Tie bar

150 0 0.00

0.06

0.12 0.18 Strain

0.24

0.30

Fig. 4. Stress-strain relationships of the steel plates and the tie bar.

selected corners was used as the reference plane to determine the imperfections. The data measured along the two diagonal paths were used to determine the out-of-plane coordinate of the fourth corner. The data measured along the two longitudinal edges were then used to determine the imperfections along them, which were further used with the data measured on the transversal paths to determine the imperfections along them. Eleven and nine transversal paths were used for the specimens with L = 600 mm and 400 mm, respectively, to ensure that the imperfection distribution could be adequately determined.

2 3

Linear guide rail

LVDT

Motor (a)

4 1

5 6

5 6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14 15

14 15 (b)

Fig. 5. Measurement of geometric imperfections: (a) Measuring apparatus; and (b) Measuring paths. 4

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Imperfection (mm)

Imperfection (mm)

H.-S. Hu, et al.

Dis

tanc

e al

(a)

ong

Dis

g lon e a m) c n (m sta Di idth w

leng

th ( mm

alon g

leng

th (

g lon e a m) c n (m sta Di idth w

mm

)

Imperfection (mm)

(b)

Imperfection (mm)

)

tanc e

Dis

tanc

(c)

e al

ong

leng

th (

Dis

tanc e al

g lon e a m) c n (m sta Di idth w

mm

)

ong

(d)

leng

th (

g lon e a m) c n (m sta Di idth w

mm

)

Fig. 6. Measured geometric imperfections of some faces: (a) SP4-100 (left face); (b) SP4-150 (left face); (c) SP8-150 (left face); and (d) SP8-N (left face).

2400

N (kN)

1800 1200 600 0

(a)

(b)

(c)

0.0

Local buckling occurred N=Nm N=0.9Νm N=0.75Nm 0.5

1.0

δ/L (%)

1.5

2.0

(d)

Fig. 7. Development of local buckling deformation in specimen SP6-100 (red dots denote the locations of tie bars): (a) N = Nm; (b) N decreased to 0.9Nm; (c) N decreased to 0.75Nm; and (d) N-δ/L curve. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

in Fig. 7) to monitor the strain distributions. The axial load was applied under displacement control. The rate of the axial shortening, δ, was controlled as 2 × 10−5L per second during the entire loading process [35]. The test was terminated when the concrete infill started to carry axial load or the measured axial shortening, δ, reached 0.03L.

this result will be presented later. Specimens with the same configuration exhibited similar local buckling behavior; thus, the three specimens with the nominal plate thickness of 6 mm are taken as representatives to illustrate the development of local buckling of the specimens herein. To demonstrate the progression of the local buckling deformation, three limit states corresponding to the peak axial load, Nm, 0.9Nm (representing slight decrease in the load-carrying capacity of the steel tube), and 0.75Nm (when local buckling deformation had become evident) on the descending branch are selected. Figs. 7–9 show the conditions of the steel plates at these limit states for specimens SP6100, SP6-150 and SP6-N, respectively, along with the corresponding axial load versus the normalized axial shortening, δ/L, curves with the points for the limit states. In Figs. 7 and 8, the red dots denote the locations of tie bars. For specimens SP6-100 and SP6-150, local buckling deformation was not observed until the axial load, N, reached the peak load, Nm; whereas for specimen SP6-N, local buckling deformation

3. Experimental results 3.1. Development of local buckling The axial load, Nlb, and normalized axial shortening, δlb/L, corresponding to the initiation of local buckling are given in Table 1. In general, the value of δlb/L of the specimen with sv = 150 mm was close to that of the WOB specimen in the same group; however, when the tie bar spacing decreased to 100 mm, the value of δlb/L increased noticeably in comparison to the corresponding WOB specimen. The reason for 5

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2400

N (kN)

1800 1200 600 0 0.0 (a)

(b)

Local buckling occurred N=Nm N=0.9Νm N=0.75Nm 0.5

(c)

1.0

δ/L (%)

1.5

2.0

(d)

Fig. 8. Development of local buckling deformation in specimen SP6-150 (red dots denote the locations of tie bars): (a) N = Nm; (b) N decreased to 0.9Nm; (c) N decreased to 0.75Nm; and (d) N-δ/L curve. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

was observed at approximately 0.88Nm prior to the peak load. These trends indicate that the steel plates of specimens SP6-100 and SP6-150 were mainly in the inelastic stage when local buckling occurred, while those of specimen SP6-N buckled elastically and exhibited post-buckling reserve strength. The buckling modes of specimens SP6-150 and SP6-N were similar, i.e., both of them exhibited a single bulge between the longitudinal edges. The buckling regions of specimen SP6-150 were restrained between two adjacent tie bars. As shown in Fig. 7(c), the tie bars restrained the height of the buckled region to nearly 100 mm, i.e., the space of the tie bars, but the regions away from the tie bars buckled over approximately 130 mm. Therefore, specimen SP6-100 experienced a “barbell-shaped” bulge after buckling. Local buckling deformation accumulated in the initial buckling region as the axial shortening increased. When the axial load decreased to 0.75Nm, the out-of-plane displacements at the bulge peaks were approximately 11, 8, and 17 mm for specimens SP6-100, SP6-150, and SP6-N, respectively. Fig. 10 presents the buckling modes at δ/L = 1.5% for all the specimens except for specimens SP8-100 and SP10-100. For these two specimens, no evident buckling deformation can be observed at δ/L = 1.5%; thus, for specimens SP8-100 and SP10-100, the buckling modes at δ/L = 3.0% are presented instead. It is evident that as the steel plate thickness decreases, the buckling deformation at a given δ/L becomes more evident. Since the inward displacements of the steel plates are restrained by the concrete infill, the buckling region of the steel plates in the WOB specimens can be considered as a rectangular steel plate that can only

buckle in one direction with four clamped edges. The elastic buckling stress, σcr, of this type of plate was derived by Timoshenko and Gere [36] and is given by cr

=k

2E

12(1

(1)

v 2)(b t ) 2

where the plate buckling coefficient, k, is given by (2)

k = 4(b2 a2 + a2 b2 + 2 3)

In Eqs. (1) and (2), E is the elastic modulus; ν is the Poisson’s ratio; and a, b, and t are the plate length, width, and thickness, respectively. The minimum value of k is obtained when a/b = 1; thus, in theory, the length of the buckled region should be equal to the width of the steel tube, B. However, for the WOB specimens tested in this study, the lengths of the buckling regions were all smaller than B, varying from approximately 180 mm (0.6B) to 240 mm (0.8B). One reason for this result is that the value of k does not vary much when the aspect ratio, a/ b, is in the vicinity of 1.0, as shown in Fig. 11; the ratios of the values of k at a/b = 0.6, 0.7, and 0.8 to the minimum value of k are 1.43, 1.20, and 1.08, respectively. On the other hand, the region where local buckling occurs is significantly affected by the initial geometric imperfections, i.e., local buckling tends to occur in the regions where initial geometric imperfections are relatively large. The combined effects of the above two factors led to the observed variations in the lengths of the buckled regions. The measured lengths of the buckling regions also indicate that the tie bars would have negligible effect on the local

2400

N (KN)

1800 1200 600 0 0.0 (a)

(b)

(c)

Local buckling occurred N=Nm N=0.9Νm N=0.75Nm 0.5

1.0

δ/L (%)

1.5

2.0

(d)

Fig. 9. Development of local buckling deformation in specimen SP6-N: (a) N = Nm; (b) N decreased to 0.9Nm; (c) N decreased to 0.75Nm; and (d) N-δ/L curve. 6

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δ/L=1.5%

δ/L=1.5%

δ/L=1.5%

δ/L=1.5%

δ/L=1.5%

(a)

δ/L=3.0%

δ/L=1.5%

(b)

δ/L=1.5%

δ/L=1.5%

δ/L=3.0%

δ/L=1.5%

(c)

δ/L=1.5%

(d)

Fig. 10. Buckling modes of all the specimens (red dots denote the locations of tie bars): (a) 4-mm plate specimens; (b) 6-mm plate specimens; (c) 8-mm plate specimens; and (d) 10-mm plate specimens. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

section; however, the trend becomes different after local buckling. Therefore, the stress calculated by N/As (As is the cross-sectional area of the steel tube) is an average value over the section, and is referred to as the average stress, σav, herein. In the pre-peak stage, since the axial strains are basically uniform along the length of the steel tube, the axial shortenings measured by the vertical LVDTs can be directly used to determine the strain value, i.e., δ/L. In the post-peak stage, however, the increasing axial shortening is localized in the buckled region while other regions are elastically unloaded. By considering this fact, the average strain of the failure region, εav, which is of interest, can be determined by

120

k

90 60 30

k=15

0 0.00

0.25

0.50

0.75 a/b

1.00

1.25

1.50

av

=

[

L( (

m) m

L

(

av_m

av )

Es )(L

L p)] L p ( >

m)

(3)

where δm and σav_m are the axial shortening and the average stress corresponding to the peak axial load; Es is the elastic modulus of the steel; and Lp is the length of the buckled region. In the second part of Eq. (3), (σav_m-σav)/Es represents the reduction of the average strain of the unbuckled region from the peak point; hence, δm/L-(σav_m-σav)/Es is the average strain of the unbuckled region for any point in the postpeak range, and δ-(δm/L-(σav_m-σav)/Es)(L-Lp) is the corresponding axial shortening of the buckled region. The value of Lp used to compute the average strain value for each specimen was from the experimental results; these values are equal to 150 mm for the specimens with the tie bar spacing, sv = 150 mm, and range from 120 to 150 mm and 180 to

Fig. 11. Value of k with respect to a/b.

buckling behavior of the steel plates when the tie bar spacing is larger than 0.6B to 0.8B. 3.2. Average stress-strain curves The behavior of the steel plates at different loading stages can be presented well by their stress-strain curves. Up to the buckling load, the axial stresses of the steel tube are uniformly distributed over the 7

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1.2

1.2

1.0

σav/fy

0.8

0.6

0.4

0.2

0.2

0.0

0.0

1.5

(a)

3.0 εav (%)

4.5

0.0

6.0

1.2

1.0

1.0

0.8

0.8

0.6

SP8-100 SP8-150 SP8-N

0.0

1.5

3.0

εav (%)

0.6

4.5

6.0

SP10-100 SP10-150 SP10-N

0.4 0.2

0.2 0.0

SP6-100 SP6-150 SP6-N

(b)

1.2

0.4

(c)

0.6

0.4

σav/fy

σav/fy

0.8

σav/fy

1.0

SP4-100 SP4-150 SP4-N

0.0

1.5

3.0 εav (%)

4.5

0.0

6.0

0.0

1.5

(d)

3.0

εav (%)

4.5

6.0

Fig. 12. Normalized average stress-average strain curves: (a) 4 mm-thick plate specimens; (b) 6 mm-thick plate specimens; (c) 8 mm-thick plate specimens; and (d) 10 mm-thick plate specimens.

240 mm for the specimens with sv = 100 mm and the WOB specimens, respectively. Fig. 12 shows the normalized average stress, σav/fy, versus the average strain curves for all the specimens. In general, the WB specimens with sv = 150 mm have higher strength and post-peak performance than the corresponding WOB specimens; however, since the lengths of the buckling regions of the WOB specimens are only slightly larger than 150 mm, the increase in the performance of the steel plates due to the additional tie bar restraint is quite limited when sv = 150 mm. The maximum strength (stress) of specimen SP6-150 is about 8% higher than that of specimen SP6-N, but the average stress of specimen SP6-150 decreases more rapidly in the post-peak stage; after the average strain reaches approximately 2%, the σav/fy-εav curves of the two specimens almost coincide. Specimens SP4-150 and SP4-N with thinner plates exhibit similar trends as specimens SP6-150 and SP6-N. The pre-peak branches of specimens SP8-150 and SP8-N are close to each other, but specimen SP8-150 exhibits better post-peak behavior than specimen SP8-N. The σav/fy-εav curves of specimens SP10-150 and SP10-N are almost the same, indicating that the tie bars have little effect on increasing the performance of the steel plates in specimen SP10150. When a tie bar spacing of 100 mm is used, an increase of more than 8% in the strength of the steel plates is achieved in comparison to the case without tie bars regardless of the steel plate thickness used. The maximum strength of specimen SP4-100 is about 15% higher than that of specimen SP4-N and 4% higher than that of specimen SP4-150; and the maximum strength of specimen SP6-100 is about 14% higher than that of specimen SP6-N and 5% higher than that of specimen SP6-150. For the σav/fy-εav curves of specimens SP8-100 and SP10-100 with larger steel plate thicknesses, there appears a strain-hardening branch after the average stress approximately reaches the yield strength. This behavior is because local buckling occurred at a considerably large average strain for these two specimens, i.e., 1.4% and 2.0% for specimens SP8-100 and SP10-100, respectively.

The stress-strain curves of all the specimens exhibit ductile postpeak behavior. As shown in Fig. 12, when the average strain reaches a value of 5%, the remaining strength is still appreciable when compared to the corresponding maximum strength. Assuming that the neutral axis is located at the center of the cross section of a CFST column, the extreme compression fiber strain is equal to curvature multiplied by one half the cross-sectional height. Furthermore, if the length of the plastic hinge is assumed to be not less than one half the cross-sectional height, the plastic hinge rotation (which is the length of the plastic hinge multiplied by curvature) would not be less than the extreme compression fiber strain. Therefore, for a plastic rotation limit of 0.03 rad for the plastic hinge, it is reasonable to take a strain limit of 3% for the compression flange of a CFST column. Based on this inference, the average stress corresponding to the average strain of 3%, referred to as the reserved strength, is considered herein. Table 2 shows the values of the maximum strength, fm (i.e., the peak value of the average stress σav), and reserved strength, fr, for all the specimens. It can be seen that the values of fm/fy, fr/fy, and fr/fm increase as the plate slenderness decreases. For the specimens having fm/fy > 0.95, the corresponding values of fr/fm are larger than 0.85, indicating that the full development of the yield strength can also assure the steel plate to have a strength higher than 0.8fy at large deformations. 4. Strength evaluation 4.1. General approach For design of composite members with tie bars, it is desirable to have simplified equations to estimate the compressive strengths of the steel plates. As it is commonly known, a slender plate can support additional loads after the plate buckles; the maximum average stress that the plate can resist is generally referred to as the post-buckling strength. On the other hand, even if the elastic buckling stress exceeds the yield 8

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Table 2 Maximum and reserved strengths of the specimens. Ra

Specimen designation

SP4-100 SP4-150 SP4-N SP6-100 SP6-150 SP6-N SP8-100 SP8-150 SP8-N SP10-100 SP10-150 SP10-N

a

R=

Test results

0.76 0.93 1.12 0.48 0.59 0.71 0.34 0.42 0.51 0.29 0.36 0.43

fy

cr

=

B t

2 s ) fy kt 2Es

12(1

Predictions from Eqs. (6) and (7)

fy (MPa)

fm (MPa)

fr (MPa)

fm/fy

fr/fy

fr/fm

fm/fy

fr/fy

376 376 376 343 343 343 302 302 302 324 324 324

290 278 252 313 297 275 334 298 291 360 329 332

190 169 177 237 213 215 311 282 253 348 312 313

0.77 0.74 0.67 0.91 0.87 0.80 1.11 0.99 0.96 1.11 1.02 1.02

0.51 0.45 0.47 0.69 0.62 0.63 1.03 0.93 0.84 1.07 0.96 0.97

0.66 0.61 0.70 0.76 0.72 0.78 0.93 0.95 0.87 0.97 0.95 0.94

0.78 0.72 0.67 0.94 0.86 0.80 1.07 0.99 0.92 1.10 1.05 0.98

0.55 0.47 0.40 0.79 0.67 0.58 1.00 0.88 0.76 1.00 1.00 0.86

where kt is defined in Eq. (4).

strength of the steel to a certain extent, the average stress of the steel plate may still not be able to reach the yield strength due to the influence of residual stresses and geometric imperfections. In most design codes (e.g., [34,37]), the interaction between yielding and local buckling and the influence of residual stresses and geometric imperfections are considered, and the compressive strengths of the steel plates are evaluated based on the values of the elastic buckling stress and the yield strength. To adopt this approach to evaluate the strengths of the steel plates in composite members with tie bars, the elastic buckling stress must be determined first.

were also conducted to determine an appropriate mesh size. Fig. 14 shows the evolution of the computed elastic buckling stress with respect to the mesh size for the case with sv/B = 0.2 (minimum value of sv/B used in the analysis) and B/t = 40. It can be observed that the computed buckling stress converges after the number of elements along the plate width exceeds 80. For consistency, a mesh size of B/100 was selected for all the cases. Nine values of sv/B, namely 1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2 and 5 values of nc, namely 1, 2, 3, 4, 5 were considered to cover the common arrangements of tie bars in practice. The aspect ratio, a/B, was determined for each sv/B using trial and error: (1) in the beginning, select a relatively large a/B so that a complete bulge can be formed in the middle region; (2) gradually reduce a/B until there was only one bulge in the middle region and no additional bulges near the two loaded edges, and the desired value of a/B is, thus, determined. The obtained values of a/B are 1.0 for sv/B ranging from 0.5 to 1.0; and 0.8, 0.65, and 0.5 for sv/B = 0.4, 0.3, and 0.2, respectively. After obtaining the elastic buckling stresses, the buckling coefficients were computed using Eq. (1). The computed buckling coefficients of all 45 cases are plotted against sv/B in Fig. 15, where different marks are used to represent the results corresponding to different nc. Fig. 16 shows the buckling modes of six representative cases. Similar to the effect of the aspect ratio on the buckling coefficient of the rectangular plate with four edges clamped as demonstrated in Section 3.1, the buckling coefficient increase due to reduction of sv/B is very limited for larger sv/B, say sv/B ≥ 0.7. It can also be found that for sv/B ≥ 0.6, the effect of nc on the value of buckling coefficient is negligible. This trend is because for this range of sv/B, increasing nc basically does not change the buckling mode, which can be clearly observed in Fig. 16(a)-(c). For smaller values of sv/B, increasing the value of nc changes the buckling mode as evident from Fig. 16(d)-(f), and, thus, increases the value of the buckling coefficient. Based on the nonlinear regression analyses of the finite element results, the following simplified equation was developed for the buckling coefficient, kt:

4.2. Elastic buckling stresses The elastic buckling stress, σcr, corresponding to different arrangements of tie bars can still be computed using Eq. (1) (in which b is equal to the width of the steel tube, B, for the test specimens in this study), except that the value of the buckling coefficient needs to be determined anew. The arrangement of the tie bars is determined by two parameters, i.e., the ratio of the vertical spacing to the steel tube width, sv/B, and the number of tie bar columns, nc. As shown in Figs. 7 and 8, despite multiple rows of tie bars along the length of the specimen, local buckling occurred only in one region within two adjacent rows of tie bars; therefore, a rectangular plate containing two rows of tie bars is analyzed. To obtain buckling modes in accordance with those in real composite members, the four edges and the longitudinal lines above the upper tie bar and below the lower tie bar of the steel plate are clamped; furthermore, the aspect ratio of the plate, a/B, is adjusted for different sv/B to suppress the additional unwanted buckling of the regions near the two loaded edges. Fig. 13(a) illustrates the boundary conditions of the plate with sv/B = 0.5 and nc = 2. Because of the complexity of the boundary conditions of the steel plates considered, theoretical solutions are basically impossible; instead, the finite element software ABAQUS [38] was used to compute the elastic buckling stresses. Fig. 13(b) shows the details of the finite element model for the case with sv/B = 0.5 and nc = 2. The steel plates were modeled using a square four-node 3D shell element with reduced integration (S4R). The rotational degrees of freedom (DOFs) and the out-of-plane translational DOF at the four edges and the longitudinal lines above the upper tie bar and below the lower tie bar were constrained to achieve the clamped boundary conditions. Symmetric constraints were imposed on the two centerlines of the steel plate to eliminate the in-plane rigid-body motion and achieve a symmetric constraint. Uniform loads with a unit magnitude were equally applied at the two loaded edges, and the elastic buckling stress was then computed using eigenvalue buckling analysis. Mesh sensitivity analyses

k t = 5.0

1 + (s v B )1.6 (s v B )1.6 + 0.2(s h B )1.4

(4)

where sh/B is the ratio of the horizontal spacing of the tie bars (or the spacing between the longitudinal edge and the tie bar) to the steel tube width, and is equal to 1/(1 + nc); and sv/B is taken as 1 if sv/B > 1. As shown in Fig. 15, the values computed using Eq. (4) are in good agreement with those computed from finite element analyses.

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Clamped Clamped

Tie bar location

Clamped

Clamped

sv

a

Clamped

Clamped

Clamped Clamped

B (a)

(b)

σcr (MPa)

Fig. 13. Analyzed plate with sv/B = 0.5 and nc = 2: (a) illustration of the boundary conditions; (b) finite element model.

7000

herein, the slenderness parameter can be written as

6000

R=

4000

0

20

40

60

80

100

cr

=

B t

120

fm

Number of elements along the plate width

fy

Fig. 14. Relationship between the computed elastic buckling stress and the mesh size (sv/B = 0.2 and B/t = 40).

fr fy

=

0.70 R0.4

1.1

=

0.44 R0.8

1.0

80 nc

40

0.2

0.4

sv/B

0.6

0.8

s

(5)

(6) (7)

In this study, the local buckling behavior of the steel plates in the composite members with tie bars was investigated. Eight specimens with tie bars (WB specimens) and four specimens without tie bars (WOB specimens) were tested under axial compression. The specimens were carefully designed so that the axial load was resisted only by the steel plates and the concrete was only used to restrain the inward buckling of the steel plates. The test parameters were the width-to-thickness (B/t) ratio of the steel plate and the ratio of the vertical spacing of the tie bars to the steel plate width, sv/B. The key findings from the test results are as follows:

20

0 0.0

kt

2 s ) fy 2E

5. Conclusions

Eq. (4)

1 2 3 4 5

60

kt

FEA

12(1

where Es and νs are the elastic modulus and the Poisson’s ratio of the steel, respectively. The equation for computing the buckling coefficient, kt, is given in Eq. (4), and the steel plates in the members without tie bars are equivalent to the case of sv/B = 1 and nc = 1. As shown in Fig. 17, the slenderness parameter, R, can basically capture the trends of fm/fy and fr/fy. Based on the test data, the following equations were determined to estimate the maximum and reserved strengths:

5000

3000

fy

1.0

(1) For the specimens with sv/B = 1/2, the buckling mode was a single bulge restrained between two adjacent tie bars. The specimens with sv/B = 1/3 exhibited a “barbell-shaped” bulge, in which the neck was formed between two adjacent tie bars. The lengths of the buckling regions of the WOB specimens varied from approximately 0.6B to 0.8B due to the fact that the elastic buckling stress does not vary much when the length of the buckling region is in the range of 0.6B to B. (2) The increase in the performance of the steel plates due to the

Fig. 15. Buckling coefficients of the analyzed plates.

4.3. Design equations Most design codes (e.g., [34,37]) and researchers (e.g., [7,8,10,39]) adopt the effective-width concept to estimate the compressive strengths of steel plates, in which the normalized strength, fm/fy, is a function of the slenderness parameter, R = fy cr . For the steel plates investigated 10

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H.-S. Hu, et al.

Fig. 16. Buckling modes of representative cases: (a) sv/B = 0.7, nc = 1; (b) sv/B = 0.7, nc = 3; (c) sv/B = 0.7, nc = 5; (d) sv/B = 0.3, nc = 1; (e) sv/B = 0.3, nc = 3; and (f) sv/B = 0.3, nc = 5.

1.2

1.2

1.1

1.0 0.8

0.9

fr/fy

fm/fy

1.0

0.6

0.8

Test data Eq. (6)

0.7 0.6 (a)

0.0

0.2

0.4

Test data Eq. (7)

0.4 0.6 R

0.8

1.0

0.2

1.2

(b)

0.0

0.2

0.4

0.6 R

0.8

1.0

1.2

Fig. 17. Characteristic strengths versus the slenderness parameter: (a) maximum strength; (b) reserved strength.

additional tie-bar restraint was quite limited when sv/B = 1/2, since the lengths of the buckling regions of the WOB specimens were only slightly larger than 0.5B. When sv/B = 1/3 was used, a significant increase in the performance of the steel plates was achieved in comparison to the case without tie bars.

estimating these strengths were developed based on the test results and the results from elastic buckling analyses.

To estimate the compressive strength of steel plates considered, the elastic buckling stresses of rectangular plates having boundary conditions similar to those in the composite members were computed using the finite element method. The value of sv/B, and the number of tie bar columns, nc, were varied in the models. The effect of sv/B was found to be consistent with the test results. The elastic buckling stress is expected to increase if sv/B is decreased, which corresponds to increasing the value of nc. However, the elastic buckling stress increased marginally when sv/B ≥ 0.6. The slenderness parameter, defined as the square root of the ratio of the yield stress to the elastic buckling stress, basically captures the trends of the maximum strength and reserved strength (defined as the average stress corresponding to the average strain of 3%) normalized by the yield strength. Simplified equations for

Hong-Song Hu: Conceptualization, Methodology, Writing - original draft, Project administration. Peng-Peng Fang: Software, Validation, Formal analysis, Investigation. Yang Liu: Writing - review & editing. ZiXiong Guo: Writing - review & editing. Bahram M. Shahrooz: Writing - review & editing.

CRediT authorship contribution statement

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Acknowledgements

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