Hysteretic behaviors of cold-formed steel beam-columns with hollow rectangular section: Experimental and numerical simulations

Thin-Walled Structures 80 (2014) 217–230

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Hysteretic behaviors of cold-formed steel beam-columns with hollow rectangular section: Experimental and numerical simulations Na Yang a,n, Ya'nan Zhong a, Qing'tong Meng a, Hao Zhang b a b

School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia

art ic l e i nf o

a b s t r a c t

Article history: Received 16 August 2013 Received in revised form 25 February 2014 Accepted 6 March 2014 Available online 17 April 2014

This paper presents a comprehensive experimental and numerical investigation on the cyclic response of cold-formed steel columns with hollow rectangular sections. The present study examined the columns' post-buckling strength and rigidity degradation, deformation and failure modes, ductility, and energy dissipation capacity. The cold-formed steel members exhibited stable hysteretic performance up to the point of local buckling with considerable degradation in strength and ductility. The energy dissipation mechanisms from the in-plane plastic behavior and out-of-plane elastic buckling deformation were identified. The influence of the height-to-width ratio and axial-compression ratio on energy-dissipation and failure mode was also investigated. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Local buckling Cyclic loading Cold-formed steel member with hollow section Ductility Energy dissipation

1. Introduction It is well-known that cold-formed thin-walled steel members with large width-to-thickness ratio are susceptible to local buckling with a post-buckling strength reserve. Over the past decades, extensive analytical, numerical, and experimental investigations on the performance of cold-formed thin-walled steel members have been reported in the literature. A summary of major research developments in cold-formed steel structures was given in Hancock [1,2]. Magnucka reviewed progresses in the field of coldformed beams, focusing particularly on the progresses on buckling and optimal design [3]. Ellobody summarized the research works on the stability of cold-formed steel columns at ambient and fire conditions [4]. These investigations were on the static properties of cold-formed steel members, including the buckling strength, the interaction between different buckling modes, and the postbuckling load-carrying capacity of the member [5–7]. Lightweight steel structures are now increasingly being used in seismically active regions. Cold-formed steel members with hollow rectangular section may be used for critical members and components, and they are expected to undergo inelastic cyclic deformations without suffering from significant loss of strength. For thin-walled steel members, local buckling may significantly

n

Corresponding author. Tel.: þ 86 010 5168 3956; fax: þ86 010 5168 7250. E-mail address: [email protected] (N. Yang).

http://dx.doi.org/10.1016/j.tws.2014.03.004 0263-8231/& 2014 Elsevier Ltd. All rights reserved.

reduce the load-carrying capacity and/or ductility property of the section. On the other hand the post-buckling strength reserve may be utilized to provide the ultimate load resistance. A thorough understanding of the cyclic elastic–plastic behavior of the thinwalled steel members is fundamental to their seismic design. During the last few decades, extensive studies have been undertaken to examine the inelastic cyclic behavior of hot-rolled steel columns, which were on the hysteretic properties under axial tension and compression load [8–10], under a combined constant axial load and cyclic bending load [11–14], or under a combined variable axial load and cyclic bending load [15]. For the hysteretic behavior of cold-formed steel members, the relevant research works focused on the bracing members, which experience the cyclic axial forces. Elchalakani described a series of tests to the failure of fix-ended cold-formed circular tubular braces under cyclic axial loading, and examined the effects of section and member slenderness on the strength, ductility, and energy absorption capacity [16]. Goggins performed quasi-static cyclic tests on bracing specimens made with cold-formed hollow steel, under idealized seismic loading conditions [17]. This paper presents a comprehensive experimental and numerical investigation on the cyclic response of cold-formed steel columns with hollow rectangular sections to assess the energy dissipation mechanism due to the coupling effect of plasticity and local buckling. Moreover, the effects of different geometric parameters on the hysteretic performances and energy-dissipation mechanism are evaluated using finite element (FE) model.

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2. Description of experimental program 2.1. Details of test specimens A total of six cold-formed steel columns with rectangular hollow section (RHS) were tested in the present study. All specimens were made of grade Q235 steel with a nominal yield stress Fy of 235 MPa. Three cross-sectional geometries were considered, namely 160  80  2.49RHS, 120  80  2.49RHS, 120  60  2.49RHS. Two specimen lengths (L) were considered in the cyclic tests – 1800 mm and 2300 mm. Geometric information and load control parameters of the six specimens were listed in Tables 1 and 2. Four groups of specimens were examined. Group I was comprised of R1, R2 and R3, covering different ratio of axial compression. Group II includes R3 and R5, representing different width-to-thickness ratio of web. Group III was comprised of R4 and R5, for different width-to-thickness ratio of flange. Group IV includes R4 and R9, to investigate the effect of different slenderness ratio. The actual material characteristics of the specimens were determined from tensile coupons (dimension: 20  350  3 mm3) taken from the webs of the specimens. Table 2 summarizes the material properties. Note that the cold-forming process caused more significant increase in the yield strength for the members with larger cross-section (e.g., R1, 160  80  3) than those with a smaller section (e.g., R4, 120  60  3).

support toller loading jack

2.2. Test setup and loading scheme

actuator

The test setup and schematic diagram are shown in Fig. 1(a) and (b), respectively. At the ends of the member, stiffening plates were provided with adequate welding to transfer the loads. The dimension of the plates at the top and bottom ends of the specimen were 340  250  10 mm3 and 340  250  20 mm3, respectively. Two actuators placed vertically and horizontally at the top end provide combined compression and bending to the specimen. A special device called “loading head” and a horizontal beam are fixed at the top end to protect the columns from torsional deformation and out-of-plane buckling. The ECCS loading procedure [18] is adopted as shown in Fig. 2. The load step is 0.5δy when δ o δy , in which δy is the estimated yielding displacement evaluated from the monotonic tensile tests. The load step is switched to δy when the cyclic loading displacement reached δy , this procedure was repeated three times and the control displacement was increased until collapse of the specimen.

tested model

R1

R2

R3

R4

R5

R6

Height of section – H (mm) Width of section – b (mm) Thickness- t (mm) Length – L (mm) Slenderness ratio λ Ratio of axial compression n

160 80 2.49 1800 61 0.1

160 80 2.49 1800 61 0.2

160 80 2.49 1800 61 0.3

120 60 2.49 1800 82 0.3

120 80 2.49 1800 82 0.3

120 60 2.49 2300 105 0.3

reaction wall steel plate

anchor bolt

base

Fig. 1. Test setup (a) gernaral view of thin-walled steel members test and (b) schematic diagram of test setup.

Table 1 Design dimensions for six tests. Specimen

support

support

δ/δy

5 4 3 2 1

cycles

-1 -2 -3 -4 -5 Fig. 2. ECCS loading procedures in the quasi-static test [18].

Table 2 Loading control and material parameters for six tests. Specimen

R1

R2

R3

R4

R5

R6

Axial force N (kN) Yielding displacement Δy (cm) Elasticity modulus E (GPa) Yielding stress fy (MPa) Ultimate stress ft (MPa)

33 20 240 330 400

66 20 240 330 400

99 20 240 330 400

73.6 20 210 330 385

82.1 20 189 310 375

73.6 32 210 330 385

Twelve strain gauges ðSGi ; i ¼ 1; 2; ::; 12Þ and four LVDT displacement transducers (DTi, i¼1,…,4) were used to measure the strain and monitor the development of plasticity in the members. The arrangement of the strain gauges and displacement transducers were shown in Fig. 3. The four displacement transducers were placed: (1) on the steel plate welded at the bottom of columns to measure the relative displacement between the plate and the support (DT1), and to observe if the bottom is slipping; (2) on the web at the bottom of the members to monitor the local buckling (DT4); (3) at the center of the columns to record the out-of-plane

N. Yang et al. / Thin-Walled Structures 80 (2014) 217–230

displacement (DT2); and (4) at the top of the columns to monitor the lateral displacement (DT3).

3. Test results 3.1. Deformation and failure process Fig. 4 shows the deformations of all six members at failure. We did not consider the effect of welds on the buckling behavior of component because the welds remain intact during the loading process. It was observed that the failure of the specimens was accompanied by

side 4 side 3

axial load

side 2 side 1

cyclic bending load

20

side 4

SG DT SG DT SG 80 80

SG DT

SG SG SG

3.2. Failure modes and mechanism

160 80

740

SG SG

160

740

SG SG

DT

80

side 3

160 SG

740

significant out-of-plane deformation of the flanges with local buckling in the flange close to the webs. All specimens experienced four stages, including yielding, strain hardening, local buckling, and eventually, the ultimate stage. Take the specimen R1 as a representative case. It exhibited linear behavior in low cyclic load before yielding. As the loads increased, the specimen eventually reached its ultimate load-carrying capacity. The strength and rigidity were degraded because of the defects and spread of plasticity. At 60 mm displacement of the first cycle, audible buckling sound of the buckling wave along the member was heard. Local buckling caused progressive strain localization, which increased with each loading cycle. The magnitude of buckling wave increased with loading and eventually the column failed at a displacement of 80 mm due to multiple local buckling of the webs and flanges. The failure processes of other members were very similar to that of R1. In all tests, local buckling occurred early and had significant influence on the failure. However, the development of plastic deformation appeared to be insignificant.

The strain gauges

160

side 2

20 40

side 1

20 20

40

The displacement transducer

219

900

Fig. 3. Arrangement of the displacement transducers and strain gauges.

The failure modes of all members were local buckling with partial yielding. Although both plastic behavior and buckling deformation were observed at failure, the out-of-plane buckling deformation was the dominant part in the total deformation. The out-of-plane deflection of the column flanges and webs was not observed until the member yielded. Subsequent loading cycles

R1

R2

R3

R4

R5

R6

Fig. 4. Failure patterns of members (column base failure).

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increased the severity of local buckling. The residual out-of plane displacements caused significant strength degradation. Eventually, the buckling of webs and flanges caused the failure. The effect of axial load and local initial deflection in the column led to the unsymmetrical buckling behavior. The serious out-of-plane local buckling deformation is the main factor accounting for the energy-dissipation of the members. The failure of the specimens was controlled by both in-plane plastic deformation and out-of-plane bending deformation (local buckling), and the interactions between these two phenomena. Plastic deformation was not fully developed while the load buckling developed quickly once it occurred. Based on this observation, the out-of-plane bending deformation appears to play a more important role in the plastic energy dissipation.

4. Analysis and discussion of the test results Cyclic responses of the members are discussed based on the following criteria: the hysteretic curve, skeleton curve, energy dissipation, ductility ratio, and the stiffness degradation. 4.1. Hysteretic curve Hysteretic curve is an important index reflecting the seismic behaviors of a member, including the load-carrying capacity, stiffness, energy dissipation capability, and ductility performance. The hysteretic curves of the tested members are plotted in Fig. 5, in which V represents the horizontal load applied to the column, and δ is the horizontal displacement at the column top. Points are marked on the curves at yield displacement, the maximum displacement, the first local buckling, and the ultimate displacement. The local buckling did not occur in the member until the yield strain was reached. After initial buckling, the member compression resistance was reduced with larger axial compression deformations. Post-buckling resistance was also observed after the few cycles loading. The rapid degradation in strength and stiffness observed was due to the severe local buckling. Fig. 6 shows that for R1, R2 and R3 (with applied axial load of 33, 66, 99 kN, respectively), the load-carrying capacity, ductility and energy-dissipation capacity decreased significantly with increasing in the axial load. This suggests that the axial load will weaken the ductility and energy dissipation capacity of the thinwalled column which is highly sensitive to the ratio of axial compression (n). 4.2. Skeleton curve The skeleton curve indicates the load-carrying and bending deformation capacity in the members undergoing a combined cyclic bending and axial load. The skeleton curves for all six tests are plotted in Fig. 6. Under low cyclic loadings, the members experienced three stages, i.e. the yield stage, the strain hardening stage, and the ultimate stage. Based on Fig. 6, the yield load ðP y Þ, the maximum load ðP max Þ and the ultimate load (Pu ¼0.85Pmax), which according to literature [19] were obtained and summarized in Table 3, along with the corresponding displacements Δy , Δmax and Δu . A minus sign in Table 3 indicates that the value corresponds to the negative part of the skeleton curves. It can be seen that the residual deformation is very small at the elastic stage. When the lateral displacement increases, the nonlinear behavior of the member can be found as shown in the hysteretic curves and the strain increases quickly. The curves are not as smooth as in the local buckling stage. After the horizontal load reaches the peak, the curve drops very quickly. The load bearing capacity is reduced although the strain keeps increasing. It was also observed that the

degradation in strength and stiffness seems to be slightly improved in the case of smaller axial-compression ratio (n) (e.g., R1, R2), evaluated as n ¼ N=f y A (N ¼ axial compressive load, fy ¼ yield strength, A ¼area of column specimen). 4.3. Energy dissipation capacity The seismic performance of a structure or a member is highly dependent on its energy dissipation capacity at the plastic stage. The energy dissipation capacity can be characterized by the area of the hysteretic loops. It is assumed that the member failure is initiated when the strength of a member degrades to 0.85 of its maximum strength. The energy dissipation before failure is plotted in Fig. 7. It is found that the energy dissipation mainly occurred in the plastic stage. The energy dissipation curves of R1 and R2 keep on increasing with cycles of loading. However, a peak value of energy dissipation was observed in R3 at the 13th displacement cycle. After that the energy dissipation curve drops because the energy dissipation capacity is weakened by the axial load. Also, the energy-dissipation capacity seems to be slightly improved in the case of smaller axial-compression ratio (n), e.g., R2 versus R3. R1 showed relatively poor performance in energy-dissipation because of its larger initial defects caused by the secondary welding at the bottom. The energy dissipation capacity can also be represented by the energy dissipation coefficient En, defined as En ¼

SðABC þ CDAÞ : SðOBE þ ODFÞ

ð1Þ

where S represents the area (SABC – positive energy dissipation, SCDA – negative energy dissipation, SOBE – positive energy absorbance, SODF – negative energy absorbance), as shown in Fig. 7, the F and X axis can be defined as the reaction force (kN) and the displacement of the member(mm). The energy dissipation coefficient En reflects the ratio of energy dissipation to energy absorbance. The structures can absorb more energy when the stiffness increases. The energy dissipation coefficients of all members in each cycle are shown in Fig. 8. It can be seen that in most cases the coefficients are larger than unity, which suggests the energy dissipation capacity is satisfactory. Fig. 8 shows that the curves are essentially flat at the elastic stage with little energy dissipation. It is also found that at the elastic–plastic stage, all coefficients are increased significantly because the plastic deformation is the main source of energy dissipation. The curves are not monotonous because of the local buckling and the redistribution of stress. The coefficients are decreased when the applied load is 20% less than the ultimate load at the unloading stage. This means that the capacity of energy dissipation declines. Small improvement in energy dissipation coefficients can be seen with an increase in the value of n (R1-0.1, R2-0.2, and R3-0.3). 4.4. Ductility Ductility is a mechanical property that describes the ability of a structure or member to undergo plastic deformation without fracture. Ductility is often characterized by the displacement ductility coefficient, m, evaluated as μ ¼ Δμ =Δy (Δμ ¼ultimate displacement, Δy ¼yield displacement). The ductility ratios for all specimens are listed in Table 3. It is found that the displacement ductility coefficients of all tested members are in the range of 1.65–2.59. According to these results, it is clear that the ratio of axial compression n has a great effect on μ. The ductility ratio of R3 is the worst one which implies the higher axial load n tends to lead to a smaller displacement at failure with a smaller ductility ratio, resulting in a decrease in the total dissipated energy. The plastic deformation only increased by 0.66 times of Δy. Sample R1 did not

N. Yang et al. / Thin-Walled Structures 80 (2014) 217–230

maximum V ( kN ) 14 R1 12 yield local buckling 10 8 6 ultimate 4 2 0 -80 -60 -40 -20 -2 0 20 40 60 80 δ ( mm ) -4 ultimate -6 -8 local buckling -10 yield -12 maximum -14

V ( kN )

R2 maximum V ( kN ) 14 12 yield 10 local buckling 8 6 ultimate 4 2 0 -80 -60 -40 -20 -2 0 20 40 60 80 δ ( mm ) -4 ultimate -6 -8 local buckling yield-10 -12 maximum -14

12 maximum

V ( kN ) 7 6

R3

10 yield

8

221

R4 maximum

5 3

local buckling 4

2 ultimate

2

local buckling

1

ultimate 0

-60

-40

-20

-2

-80 0

20

40

-60

-40

60 δ ( mm )

-20 -1 0 yield

20

40

60

80 ultimate

-2 -3 -4

-6

-5

local buckling

-8 maximum

δ ( mm )

0

-4

ultimate

local buckling

yield

4 6

-6 maximum

yield-10

R5 8 7 maximum 6 5 local buckling yield 4 3 2 ultimate 1 0 -80 -60 -40 -20 -1 0 20 40 60 80 -2 δ ( mm ) yield -3 ultimate local buckling -4 -5 -6 -7 maximum -8 V ( kN )

-7

4 3 2

R6 V ( kN ) maximum yield

local buckling

1

ultimate

ultimate 0 -100 -80 -60 -40 -20 0 -1

20 40 60 80 100 δ ( mm )

-2 yield -3 maximum local buckling

-4 -5

Fig. 5. Hysteretic curves of the specimens.

reflect the effect of n for the reason of larger initial defects caused by the secondary welding at the bottom.

be represented by the secant stiffness: Ki ¼

4.5. Stiffness degradation Stiffness degradation indicates the cumulative damage in the members due to the plastic deformation and local buckling. It can

jþ P i j þj  P i j : jþ Δi j þj  Δi j

ð2Þ

where þPi,  Pi mean the maximum force and þ Δi,  Δi mean the maximum displacement in opposite direction for each cycle. The secant stiffness can be used to describe the pattern of plastic

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N. Yang et al. / Thin-Walled Structures 80 (2014) 217–230

V ( kN )

14

14

Pmax

12 10

-60

-40

8

8

6

6

4

4

2

2

-20 -2 0

20

40

60

80

-80

-60

-40

60

80

δ ( mm )

R1

R2

-10

Pu

-12

Pmax

-14

-14

12

-8

Py

-12

V ( kN )

40

-6

-8 -10

Pmax

20

-4

-6 Pu

Py

0 -20 -20

δ ( mm )

-4 Py

Pu

10

Py

0 -80

Pmax

12

V ( kN )

Pu

6

Pmax

10

V ( kN )

Pu

Pmax

5

Pu

4

8

Py

Py

3

6

2 4

1

2 -80

0 -60

-40

-20

-2

0

20

40

-6

Pu

-8

Pmax

-10

-40

60

0 -20 -1 0

6

R3 Pu

60

80

R4

-4 -5

Pmax

-6

4 Pmax

Pmax

5 4

40

δ ( mm )

-3 Py

7 V ( kN )

20

-2

δ ( mm )

-4 Py

-60

3

Pu

Pu

Py

2

3 2

Py

1

1 -80

-60

-40

0 -20 -1 0 -2 Py

0 20

40

60

80

-100 -80 -60 -40 -20

δ ( mm )

-5 Pmax

40

60

80

100

-3 Py

-4 Pu

0 20 -1

R5

Pu

-2 -3

R6

-6 Pmax

-7

-4

Fig. 6. Skeleton curves of specimens.

deformation and buckling development. Fig. 9 plots rigidity degradation curves of six tested specimens. As shown in Fig. 9, the stiffness degrades quickly after yielding and is close to zero with the displacement increases continuously. The secant stiffness K i ¼ 0 indicates the state at collapse of the member. The yielding points are near δ ¼ 20 mm for R1–R4 and δ ¼ 30 mm for R5 and R6.

The axial load level n has little effect on the rigidity degradation at the elastic stage (δ r20 mm). After that the members with higher compression ratio show more severe rigidity degradation. This is why the stiffness degradation curves for R1–R3 are close with each other when the displacement is small (i.e., smaller than 20 mm). However, the stiffness of R3 degrades more quickly than that

N. Yang et al. / Thin-Walled Structures 80 (2014) 217–230

223

Table 3 Load-carrying strengths and displacement ductility coefficients of members. Results

R1

R2

R3

R4

R5

R6

Py (kN) Δy (mm) Pmax (kN) Δmax (mm) Pu (kN) Δu (mm) μa

9.5/  10.2 20/  20 13.8/  11.8 30/  30 11.7/ 10.0 36.8/  43.1 1.84/2.2

9.5/  9.5 20/  20 12.2/  11.7 40/  40 10.4/  9.9 49.7/  46.2 2.49/2.31

6.3/  7.0 15/  15 11.5/  9.3 30/  30 9.8/  7.9 34.5/  33.0 2.3/2.2

3.0/  3.3 20/  20 5.0/  5.8 40/  40 4.0/  4.9 51.9/  49.4 2.59/2.47

3.1/  3.4 20/  20 5.9/  6.6 40/  40 5.0/  5.6 45.7/  45.4 2.29/2.27

1.9/  2.1 32/  32 3.2/  3.6 64/  80 2.7/  3.1 68.8/  82.7 2.15/2.58

a

μ ¼ Δu =Δy denotes the displacement ductility coefficient.

0.6

K ( kN/m mm ) Test - R1 Test - R2 Test - R3 Test - R4 Test - R5 Test - R6

0.5 0.4 0.3 0.2 0.1

δ ( mm )

0.0 0

10

200

30

40

50

60

70

8 80

90 100

Fig. 9. Rigidity degradation curves.

5. Nonlinear finite element (FE) analysis Fig. 7. Determination for energy dissipation coefficient.

In order to classify the failure modes and predict the energydissipation mechanism between the in-plane plasticity and the out-of-plane buckling deformation effect for members outside the test range, a numerical model was developed using the commercial FE program ANSYS. The numerical results were validated against test results and parametric studies were then conducted.

energy-dissipation coeffecient

18 16 14 12 10 8

R1 R2 R3

5.1. Finite element model

R4 R5 R6

6 4 2 Cycles 0 0

2

4

6

8

10

12

14

16

Fig. 8. Energy dissipation coefficient of specimens.

of R1 and R2 beyond the yielding point because R3 is subjected to a higher axial load. The axial load speeds up the stiffness degradation. According to Fig. 9, the rigidity degradation curves drop sharply, suggesting that local buckling has occurred and the plastic deformation capacity was significantly reduced. It should be noted that the height-to-thickness ratio (H/t) of R3 is larger than that of R5. This led to earlier local buckling in R3 due to the distribution of plastic deformation and damage accumulation.

The physical model and the end restraints are shown in Fig. 10. The tri-linear stress–strain model was adopted because the bilinear stress–strain model cannot include the post-peak portion of the constitution relationship of the material and it cannot represent realistically the large-strain effects. The elastic modulus E is 210 GPa, the plastic modulus E0 is 0.02E, the yield stress s is 310 MPa, the ultimate stress s0 is 400 MPa, the yield strain ε is 0.15%, and the ultimate strain ε0 is 2.3%. Both the web and flange are modeled using SHELL181 shell element in ANSYS. Boundary condition: bottom end is totally fixed that the displacement and rotation in three directions are all restricted. For top end, the displacement in X-direction and rotation in Z-direction are restricted to prevent bending of Z-direction and moving in X-direction. So we can add the vertical load along Y-direction and the horizontal load along Z-direction. Based on the stress–strain relationship obtained from material test of the samples, the tri-linear kinematic hardening material model is adopted with consideration of the Baushinger effect. Baushinger effect is considered in the deformation process of reciprocating loading, unloading and reloading. Plastic strain strengthen caused by positive loading process results to plastic strain soften of the material in the following reverse loading. Initial geometric imperfection was modeled by the following equation [18]: π y π x  L z¼ sin sin : ð3Þ 1000 H b

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N. Yang et al. / Thin-Walled Structures 80 (2014) 217–230

Q

U x = 0, U y = Uz = 1 θ z = 0, θ x = θ y = 1

U x = Uy = Uz = 0 θ x = θ y = θz = 0

Y

Z

X

Fig. 10. Finite element model numerical model (a) physical model, (b) FE model, and (c) Multilinear constitutive model.

where z¼ amplitude of initial geometric imperfection of any location in the plate, x¼the coordinate of x direction of any location in the plate, y¼the coordinate of y direction of any location in the plate, L¼length of the column; H¼length of the plate; and b¼width of the plate.

deformation demand in the bottom web is believed to be responsible for the fracture occurring at the bottom webs of members in test. We can also see from Fig. 12 that the plastic zone development is inadequate. This indicates that the buckling energy dissipation plays a major role in the degradation of the load-carrying capacity, especially for a slender member such as R6.

5.2. Validation of FE models The skeleton curves, deformations at failure and hysteretic curves obtained by FEM and experiments are compared in Figs. 11 and 12, respectively. The comparison shows that the FE models are capable of simulating the hysteretic behavior reasonably well. The drop in compressive resistance after buckling is more pronounced in the FE models than in the tests. In general, the numerical models somewhat overestimate the column stiffness and capacity. This might be because the FE model did not consider the effects of stress concentration and specimen installation errors. The prediction of the failure region is 40 mm  120 mm from the bottom end, which is close to the experimental results. The load-carrying capacity from the finite element model and the tests are compared in Table 4. It can be seen that the FEM results agree reasonably with the test results. For the index of P y and P max , the test results are similar to those from the finite element model with an error less than 20%. 5.3. Plasticity development and out-of-plane deformation Plasticity development and out-of-plane buckling deformation at different stage can be reflected by plastic elements ratio, which is the ratio between the number of plastic elements and the total elements. The plastic ratios for all tests are presented in Table 5. From Table 5, it can be seen that the out-of-plane deformation increases significantly at the ultimate stage because of the development of local buckling. It can be seen that the plasticity development is inadequate as compared to the large out-of-plane buckling deformation when the specimens collapsed. As indicated in Table 5, the plastic deformation caused by material nonlinear behavior plays a less important role in the collapse than the local buckling deformation caused by geometric nonlinear behavior. Fig. 13 shows the plastic strain distribution. The plastic zone extends from bottom to top and the larger plastic zone is within the range of 40 mm 120 mm at the bottom web. Higher

5.4. Parametric studies The test above highlights the coupling effect of plastic development (caused by material nonlinear behavior) and out-of-plane buckling deformation (caused by geometric nonlinear behavior). The combined effects of out-of-plane bending deformation and inplane yielding caused the failure. The coupling mechanism is related to the configurations of the members such as H/b ratio, H¼height of the section, b¼width of the section. In order to give the threshold values as the criteria of failure, twenty FE models were created (RW1, …, RW20), shown in Table 6. Two groups of specimens were included. Group I covers different height of the section, from 60 mm to 200 mm. Group II includes different width of the section, covering 60 mm, 75 mm, 90 mm. Quantitative parametric studies were conducted to understand the factors affecting the nonlinear behavior and the failure mechanisms. Moreover, the plastic element ratio at failure and the maximum out-of-plane deformation were also examined. Various parameters of the members are selected for study from Table 6. The effects of H/b on the collapse modes, and energy dissipation mechanism are studied in the following numerical analysis. 5.4.1. Failure modes and fracture criterion To investigate the cyclic behavior, indices such as the plastic elements ratios at failure and the maximum out-of-plane deformation are evaluated. The analytical member is divided into elements with a dimension of 10  10  40 mm3. The characteristic indices and failure modes are described in Table 7. Three failure modes are identified based on the failure responses. Mode 1: Failure mainly by overall buckling (buckling-dominant failure mode). When the member is failed, most elements of the member are still at elastic stage with insufficient plastic zone developed. The failure is mainly due to severe out-of-plane buckling deformation.

N. Yang et al. / Thin-Walled Structures 80 (2014) 217–230

-80

-60

14 V ( kN ) 12 10 8 6 4 2 0 -40 -20 -2 0 -4 -6 -8 -10 -12 -14

225

V ( kN ) 12 10 8 6 4 2 20

40

60

80

-80

-60

-40

δ ( mm )

0 -20 -2 0

20

-4

FEM - R1 TEST - R1

40 60 80 δ ( mm )

-6

FEM - R2

-8

TEST - R2

-10 -12

6

12 V ( kN ) 9

V ( kN ) 5 4

6

3 2

3

1 0 -60

-40

-20

0

20

40

60

δ ( mm )

-3 -6 -9

-80

-60

-40

0 -20 -1 0 -2

20

40

60

80

δ ( mm )

FEM-R3

-3

FEM-R4

TEST-R3

-4

TEST-R4

-5 -12

-6 4

6 V ( kN ) 5

V ( kN )

4 3

3 2

2 1

1 -80

-60

-40

0 -20 -1 0 -2

20

40 60 80 δ ( mm )

0 -100 -80 -60 -40 -20

0 -1

-3 -4

FEM - R5

-5

TEST - R5

-7

80 100 δ ( mm )

-2 FEM - R6 -3

-6

20 40 60

TEST - R6

-4 Fig. 11. Skeleton curves of finite element model and test results.

Mode 2: Failure mainly by plastic buckling (plastic-dominant failure mode). Significant yielding has occurred and the failure is mainly due to the plastic bending caused by material nonlinear behavior. In contrast, the out-of-plane buckling deformation due to the geometric nonlinear behavior is relatively small. Mode 3: Failure mainly by local buckling (buckling-dominant failure mode). The member collapses with a severe local buckling in the surface with large out-of-plane deformation. The plastic

development zones are not fully developed and the failure is due to excessive buckling deformation. That means the out-of-plane bending energy-dissipation is the main reason of collapse.

5.4.2. Influence of H/b The results in Table 7 and Fig. 13 indicate the important effect of H/b on the hysteretic performance and failure modes. When the

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N. Yang et al. / Thin-Walled Structures 80 (2014) 217–230

sectional size of the member is smaller, the member tends to be more slender. The out-of-plane overall buckling deformation controls the collapse of the member. With increasing H/b ratio, the out-of-plane buckling deformation caused by the geometric nonlinear effect increases notably, and the ratio of the plastic elements decreases. The failure mode changes from the plastic failure (Mode 2) to the local buckling failure (Mode 1 and Mode 3). This indicates that the out-of-plane bending deformation plays an increasingly important role in the collapse, such as Mode 1 and

-80

-60

-40

14 V ( kN ) 12 10 8 6 4 2 0 -20 -2 0 20 40 60 80 δ ( mm ) -4 -6 -8 -10 R1- FEM result -12 R1-Test result -14 Numerical results

R1

-80

Mode 3. Fig. 13 and Table 7 show the relationships between H/b ratio and out-of-plane ultimate deformation and plastic elements ratio (Table 8). For the samples with Mode 1 failure (i.e., RW1, RW2 and RW3 in Table 6), the ultimate deformation is over 110 mm, while the ratio of plastic elements is less than 30% of the total elements. This suggests that the failure is mainly due to the geometric nonlinear effect. For the samples with Mode 3 failure (i.e. RW7, RW8, RW9, RW13,…, RW20), the ultimate out-of-plane deformation ranges

14 V ( kN ) 12 10 8 6 4 2 0 -40 -20 -2 0 -4 -6 -8 -10 -12 -14

-60

20

40

Test results

60 80 δ ( mm )

R2 - FEM result R2 - Test result

Numerical results

R2

Test results

12 V ( kN )10 8 6 4 2 0 -60

-40

-20

-2

0

20

40

60 δ ( mm )

-4 -6 -8 -10 R3

R3 - FEM result R3 - Test result

Numerical results

Test results

Fig. 12. Comparison of failure mode and hysteretic curve between finite element model and test results.

N. Yang et al. / Thin-Walled Structures 80 (2014) 217–230

227

V ( kN ) 6 5 4 3 2 1 -80

-60

-40

0 -20 -1 0

20

40

60

80

δ ( mm)

-2 -3 -4 -5 -6

R4 - FEM result

-7

R4 - Test result

R4 V ( kN )

Numerical results

Test results

Numerical results

Test results

Numerical results

Test results

8 6 4 2 0

-80

-60

-40

-20

0

20

40

-2

60

80

δ (mm )

-4 -6 R5 - FEM result

-8

R5 - Test result

R5

V( kN)

4 3 2 1 0

-100 -80 -60 -40 -20

0 -1

20

40

60

80 100

δ ( mm )

-2 -3 R6 - FEM result

-4

R6 - Test result

R6

Fig. 12. (continued)

from 40 mm to 110 mm and the ratio of plastic elements ranges from 30% to 40% of the total elements of member. For the samples with Mode 2 failure (i.e. RW4, RW5, RW6, RW10, RW11, RW12), the ultimate out-of-plane deformation is less than 40 mm at failure while the plastic elements is more than 40% of the total elements of member.

As shown in Fig. 13, with increasing H/b, the local buckling plays a more important role in the failure mode. The deformation capacity and energy dissipation capacity deteriorate with H/b increasing. To avoid excessive deformation and sudden dynamic instability in practice, the H/b ratio should be equal to or greater than 1.5. On the other hand, the H/b ratio should not exceed 2.0

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Table 4 Load carrying strengths comparison table of finite element model and test results. Results

Specimens R1

R2

R3

R4

R5

R6 1.9/ 1.9

PN y (kN)

9.9/  9.9

9.3/  9.4

6.9/  6.9

3.0/  3.0

3.2/  3.2

P Ty (kN)

9.5/  10.2

9.5/  9.5

6.3/  7.0

3.0/  3.3

3.1/  3.4

1.9/ 2.1

Ratioa PN max (kN)

0.04/0.03 12.6/  12.6

0.02/0.01 11.8/  11.4

0.10/0.01 10.5/  10.8

0/0.09 4.7/  4.9

0.03/0.06 5.3/  5.6

0/0.10 3.0/  3.1

P Tmax (kN) Ratiob

13.8/  11.8

12.2/  11.7

11.5/  9.3

5.0/  5.8

5.9/  6.6

3.2/  3.6

0.09/0.07

0.03/0.03

0.09/0.16

0.06/0.16

0.10/0.15

0.06/0.14

a b

T Ratio ¼ jP Ty  P N y j=P y . T Ratio ¼ jP Tmax  P N max j=P max .

Table 5 Characteristic indices of different stages. Stage

R1

R2

R3

R4

R5

R6

Yield stage Plastic elements ratio Ux (mm) Uz (mm)

1.5% 0.03 0

2.6% 0.04 0

0.2% 0.02 0

5.8% 0.03 0

4.0% 0.04 0

5.1% 0.03 0

Maximum stage Plastic elements ratio Ux (mm) Uz (mm)

16.9% 0.41 0.04

21.1% 0.87 0.32

23.4% 0.81 0.16

23.3% 0.35 0.01

22.7% 0.26 0.02

23.8% 0.21 0.01

Ultimate stage Plastic elements ratio Ux (mm) Uz (mm)

22.8% 23.24 20.99

25.2% 39.43 32.56

27.2 % 46.69 38.44

25.3% 18.60 16.35

28.2% 31.46 27.35

26.1% 24.44 21.75

Ux denotes the maximum out-of-plane displacement of webs. Uz denotes the maximum out-of-plane displacement of flanges.

60 160

b=60

b=60

100 80 60

50

b=90

110

40

b=75

Plastic element %

120

Ultimate deformation ( mm )

b=75

140

b=90

40

30

30 20 40 H/b

40 0.5

1.0

1.5

2.0

2.5

3.0

3.5

H/b 10 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Fig. 13. Ultimate value of characteristic responses for failure.

in order to make full use of stable plastic development energydissipation capacity (the material nonlinear effect), Therefore, in seismic design the optimal H/b ratio should be in the range of 1.5 to 2.0.

6. Conclusions Both experimental and numerical simulations are presented to show the hysteretic performance of cold-formed steel rectangular

hollow columns. The study investigated the buckling strength, post-buckling stiffness, failure mechanism, ductility, and energydissipation capacity. The conclusions of this study can be summarized as follows: 1. The tests showed that the serious out-of-plane local buckling is the main factor accounting for the energy-dissipation of the specimens. The failure of specimens was controlled by both inplane plastic deformation and out-of-plane bending deformation (local buckling) and their interactions. Plastic deformation

N. Yang et al. / Thin-Walled Structures 80 (2014) 217–230

Table 6 Parameters of FE model in numerical analysis. Specimen Dimension

Table 8 List of notations or symbols. Load control

b (mm) H (mm) t (mm) L (mm) n

Material parameters

Δy (cm) fy (MPa) E (GPa)

RW1 RW2 RW3 RW4 RW5 RW6 RW7 RW8 RW9

60

60 75 80 90 100 120 150 180 200

3

1800

0.3 40 32 30 27 24 20 16 13 12

325

206

RW10 RW11 RW12 RW13 RW14 RW15

75

75 90 100 120 150 180

3

1800

0.3 32 27 24 20 16 13

325

206

RW16 RW17 RW18 RW19 RW20

90

90 100 120 150 180

3

1800

0.3 27 24 20 16 13

325

206

Table 7 A summary of characteristic indices. Specimen Ultimate stage

RW1 RW2 RW3 RW4 RW5 RW6 RW7 RW8 RW9 RW10 RW11 RW12 RW13 RW14 RW15 RW16 RW17 RW18 RW19 RW20

229

Plastic elements ratio %

Maximum displacement of Ux (mm)

Maximum displacement of Uz (mm)

Destruction type

23.7 25.8 28.3 51.3 48.5 40.9 38.1 36.7 35.2 49.8 45.4 42.2 39.6 35.4 32.9 38.3 35.4 35.7 35.3 32.1

3.578 16.465 17.005 36.115 32.078 36.346 55.223 42.949 52.941 38.118 36.862 32.492 40.613 44.668 61.92 44.827 41.013 40.264 43.522 48.512

161.242 164.616 167.521 36.137 30.451 30.334 37.990 37.030 46.798 36.391 37.856 39.428 34.851 46.833 46.658 44.968 34.037 47.145 39.064 48.551

1a 1 1 2b 2 2 3c 3 3 2 2 2 3 3 3 3 3 3 3 3

a

1 Overall buckling destruction. 2 Plastic buckling destruction. c 3 Buckling plastic destruction. Ux denotes the maximum out-of-plane displacement of webs. Uz denotes the maximum out-of-plane displacement of flanges. b

has not been fully developed and the development of local buckling becomes faster once it occurs. The out-of-plane bending deformation plays a more dominant role in the plastic energy dissipation. 2. There are three failure modes for thin-walled steel columns subjected to cyclic load actions, i.e. failure by overall buckling, by plastic buckling, and by local buckling. The buckling instability is mainly due to the geometric nonlinear effects of

Nomenclature W H L λ t A n N E Φ fy

Overall width of cross-section (smaller dimension) Overall depth of cross-section (larger dimension) Length of column specimen Slenderness ratio factor Plate thickness of specimen Area of column specimen Axial-compression ratio Axial compressive load Young's modulus Percentage reduction of area Yield strength

ft P max Δmax Pu Δu Py Δy μ ¼ Δu =Δy PN y

Tensile strength Maximum load Maximum displacement Ultimate load Ultimate displacement Yield load Yield displacement Displacement ductility coefficient Yield load from finite element model

P Ty

Yield test load

PN max P Tmax

Maximum load from finite element model Maximum test load

shell-typed structures, while the plastic failure is mainly due to the severe plastic deformations of structural materials and plastic damage in structures. It can also be concluded that the collapse mode is closely related to the height-to-width ratio (H/b) of the plates. This knowledge could be used in seismic design. 3. The columns are highly sensitive to the axial-compression ratio n, with the load-carrying capacity weakens under larger axial load. Larger axial load may induce a premature local buckling and a loss of load-carrying capacity. The recommended axialcompression ratio is n r3 to ensure the ductility and stability capacity of the columns in strong earthquake regions. 4. With increasing H/b, the local buckling plays a more important role in the failure. The deformation capacity and energy dissipation capacity deteriorate with H/b. To avoid excessive deformation and sudden dynamic instability in practice, the H/b ratio should be limited below 1.5. To make full use of stable plastic development energy-dissipation capacity (the material nonlinear effect), the H/b ratio should be ranged from 1.5 to 2.0.

Acknowledgments This study was sponsored by the Fundamental Research Funds for the Central Universities (2012JBM007), 111 Project (B13002), Program for New Century Excellent Talents in University (NCET-11-0571), and National Natural Science Foundation of China (Grant nos. 51078026 and 50778019). These supports are gratefully acknowledged. The writers would also like to thank Professor S.S. Law (Hong Kong Polytechnic university) and Associate Professor Hongjun Xiang Beijing Jiaotong university) for their valuable comments. References [1] Hancock GJ, Rasmussen KJ. Recent research on thin-walled beam-columns. Thin-Walled Struct 1998;32:3–18. [2] Hancock GJ. Cold-formed steel structures. J Constr Steel Res 2003;59:473–87.

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