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Numerical analysis of slender elliptical concrete filled columns under axial compression
Thin-Walled Structures 77 (2014) 26–35
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Numerical analysis of slender elliptical concrete filled columns under axial compression X.H. Dai a, D. Lam a,n, N. Jamaluddin b, J. Ye c a
School of Engineering, Design and Technology, University of Bradford, Bradford, UK Faculty of Civil and Environmental Engineering, Universiti Tun Hussein Onn Malaysia, Malaysia c Department of Engineering, Lancaster University, Lancaster, UK b
art ic l e i nf o
a b s t r a c t
Article history: Received 24 June 2013 Received in revised form 28 November 2013 Accepted 28 November 2013 Available online 15 December 2013
This paper presents a non-linear finite element model (FEM) used to predict the behaviour of slender concrete filled steel tubular (CFST) columns with elliptical hollow sections subjected to axial compression. The accuracy of the FEM was validated by comparing the numerical prediction against experimental observation of eighteen elliptical CFST columns which carefully chosen to represent typical sectional sizes and member slenderness. The adaptability to apply the current design rules provided in Eurocode 4 for circular and rectangular CFST columns to elliptical CFST columns were discussed. A parametric study is carried out with various section sizes, lengths and concrete strength in order to cover a wider range of member cross-sections and slenderness which is currently used in practices to examine the important structural behaviour and design parameters, such as column imperfection, non-dimension slenderness and buckling reduction factor, etc. It is concluded that the design rules given in Eurocode 4 for circular and rectangular CFST columns may be adopted to calculate the axial buckling load of elliptical CFST columns although using the imperfection of length/300 specified in the Eurocode 4 might be overconservative for elliptical CFST columns with lower non-dimensional slenderness. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Numerical modelling Elliptical CFST column Axial compressive load Buckling reduction factor Design buckling curves
1. Introduction Composite CFST columns are increasingly used for high-rises building structures, owing to their excellent structural behaviour such as superior load-bearing capacity, high ductility, good energy dissipation and fire-resistance etc. which arises from the composite action of two different materials in one structural member. Hot rolled elliptical hollow section (EHS) represents a new range of cross-sections recently available in construction market. This new shape has an aesthetically pleasing appearance and improved structural efficiency due to its different major and minor axis. Regarding the concrete filled columns with steel tubular hollow sections, concrete filled composite columns with circular hollow sections (CHS) have the advantages over columns with other section shapes due to the circular sections provide an uniform confinement to the concrete core. The concrete filled columns with square hollow section (SHS) and rectangular hollow section (RHS) have many common virtues to those with circular hollow sections, such as high ductility and large energyabsorption capacity. However, the disadvantage of square and rectangular sections lies with the stress concentration at the corners of the steel sections and may result in cracking [1,2]. Extensive
experimental studies and numerical analyses have been conducted to investigate the structural behaviour of CFST columns with circular, square and rectangular sections, which included the effects of concrete confinement, geometry of hollow section, column slenderness and constituent material properties [3–6]. In particular, the confinement effect of concrete in stub CFST composite columns was well recognised [7–13]. Although the compressive behaviour of the elliptical CFST stub columns have been investigated worldwide [14– 16], there is currently a lack of design rules for the behaviour of elliptical CFST slender columns. The main objective of the research is to understand the axial compressive behaviour of slender CFST columns with elliptical hollow sections. This paper is mainly focuses on the following aspects: (1) develop a reasonable nonlinear FE model through ABAQUS/Standard solver to investigate the overall buckling behaviour of slender elliptical CFST columns; (2) conduct a parametric study to highlight important design parameters; (3) assess the adaptability of the current design rules provided in Eurocode 4 [17] for elliptical CFST columns.
2. Experimental study n
Corresponding author. Tel.: þ 441274234054; fax: þ441274234525. E-mail address: [email protected] (D. Lam).
0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.11.015
To assess the accuracy of the FE model, this section will provide a brief description of the experimental setup and boundary conditions
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27
Table 1 Summary of measured specimen dimensions, material properties and maximum axial compressive loads. Specimen ID
2a 2b t (mm)
Le (mm)
f y (MPa)
f u (MPa)
Es (GPa)
f cu (MPa)
f ck (MPa)
Ec (MPa)
N EXP (kN)
TG1-D150-C30 TG1-D150-C60 TG1-D150-C100 TG1-D200-C30 TG1-D200-C60 TG1-D200-C100 TG2-D150-C30 TG2-D150-C60 TG2-D150-C100 TG2-D200-C30 TG2-D200-C60 TG2-D200-C100 TG3-D150-C30 TG3-D150-C60 TG3-D150-C100 TG3-D200-C30 TG3-D200-C60 TG3-D200-C100
150.9 75.4 4.0 150.4 75.2 4.1 150.3 75.2 4.1 197.5 100.2 5.2 197.4 100.1 5.1 197.7 100.1 5.1 150.5 75.4 4.1 150.7 75.2 4.2 150.7 75.4 4.1 197.6 100.2 5.1 197.7 100.1 5.1 197.3 100.0 5.2 150.0 75.1 4.2 150.2 75.0 4.0 150.1 75.0 4.1 197.5 100.3 5.2 197.8 100.1 5.1 197.7 100.1 5.1
1677 1678 1676 1679 1678 1678 1965 1965 1966 1965 1966 1966 2680 2682 2681 2678 2679 2678
410 410 410 350 350 350 410 410 410 350 350 350 410 410 410 350 350 350
545 545 545 500 500 500 545 545 545 500 500 500 545 545 545 500 500 500
201 201 201 209 209 209 201 201 201 209 209 209 201 201 201 209 209 209
26 58 91 29 64 91 20 65 100 42 63 98 23 63 100 29 59 101
20 52 78 27 42 71 13 50 72 29 41 76 18 54 72 22 50 74
31,649 35,040 39,889 31,719 34,149 38,698 32,122 36,250 36,390 33,689 37,001 43,699 25,422 34,480 44,027 27,018 35,901 45,685
651 743 923 938 1064 1480 484 663 871 968 1237 1411 327 427 547 839 947 1072
The setup enabled the joint to have a full rotational capacity about the balls centre, therefore the effective length, Le of composite columns is calculated from the centres of the two steel balls. For each test, horizontal and vertical strain gauges were installed on the steel hollow section surface along the longitudinal axis, to record the hoop and axial strains respectively. Linear variable displacement transducers (LVDTs) were placed alongside the length of the column to measure the horizontal displacements of the column along the length. The vertical axial displacement of the column was recorded via the LVDT in the actuator. NEXP in Table 1 gives the maximum axial compressive loads obtained from the tests. 3. Development of the finite element model Fig. 1. Measured stress–strain curves of steel materials.
3.1. General of slender elliptical CFST columns under compressive loads. Detailed information about the setup and test results is given in other publications [18,19]. In total, eighteen slender elliptical CFST columns, with two hollow section sizes (150 75 4 mm, 200 100 5 mm), three column lengths (1.5 m, 1.8 m and 2.5 m) and three nominal concrete strength (30, 60 and 100 MPa) were tested to investigate the axial compressive behaviour and failure modes. Measured specimen dimensions are summarised in Table 1. In which a, b and t are major radius, minor radius and wall thickness of the elliptical steel hollow section respectively; Le is the effective length of column, which is the buckling length between the supports. Before each specimen was tested, steel coupons, cut from the long elliptical hollow section were tested in tension to acquire the mechanical properties of steel materials. The average stress–strain curves for two different steel hollow sections are shown in Fig. 1, with the main characteristic parameters given in Table 1 where fy, fu and Es are the yield strength, ultimate strength and elastic modulus of the steel sections. In addition to the steel properties, concrete cubes and cylinders, cast using the same batch of concrete used for the column specimen were also tested to determine the concrete properties. The measured unconfined concrete cube and cylinder compressive strengths as well as the elastic modules of the concrete; tested on the test date are presented in Table 1, in which, fcu, fck and Ec are the characteristic cube, cylinder compressive strengths and the elastic modulus of the unconfined concrete. The experimental setup is as shown in Fig. 2. The pin-ended joint at the column ends was provided by the combination of a high strength steel ball (80 mm in dia.) with two 40 mm thick steel plates.
It is undoubtedly that full-scale testing provides the most effective method in understanding the behaviour of a structural member, however experimental studies with high numbers of specimens are costly and time consuming. This leads to the application of numerical method to carry out parametric study. Numerical model has been successfully utilised in prediction the axial compressive behaviour of stub CFST columns [16]. This section describes the numerical finite element method (FEM) adopted to predict the structural behaviour of slender elliptical CFST columns under axial compression. As observed from the experimental studies, the failure mechanism of a slender CFST column under axial compression was characterised by overall buckling although local buckling might present at the compressive region of the steel section in some cases. This differs from the failure mode of the stub CFST columns which was characterised mainly by local buckling, yielding of the steel section and concrete crushing. Therefore for the modelling of the slender composite column, apart from the effect of the constituent of member materials, cross-section shapes and boundary conditions, the slenderness and imperfection of column must be taken into consideration in the FE simulations. 3.2. Finite element model For the research presented in this paper, ABAQUS/standard software package [20] was used to develop a nonlinear FEM to capture the structural behaviour of the slender elliptical CFST columns under axial compression. As described in the previous section, under compressive load the dominant deformation of a
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Fig. 2. Experimental setup and failure mode. (a) Experimental setup, (b) Aftertesting.
Elliptical CFST column and end-pinned arrangement
Cross section
Deformation of elliptical CFST column under compression
Fig. 3. Deformation of a slender elliptical CFST column under compression.(a) Experiment, (b) FE modelling.
slender composite column was characterised by global buckling of the first buckling mode shape as shown in Fig. 3. To model the pinended joint as in the experiment, the steel balls and associated plates were modelled. Fig. 3 shows the structure and FE mesh of the composite column member together with the supports.
Three-dimensional 8-node solid element (C3D8) was adopted for the FE model. Some researchers believe that only shell elements can be used for buckling analysis and this is certainly untrue. Element C3D8 is a general solid element and can be used for displacement, stress/strain analysis as well as for buckling
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
analysis. The composite columns presented in this paper include slender elliptical steel tubes and a concrete core. With the wall thickness of steel tube in the region of 4–5 mm, using either the solid element or the shell element in modelling will give similar and close to identical results. To demonstrate this, a simple example of a thin-walled elliptical steel tube, which is 2.68 m long with major outer diameter of 200 mm, minor outer diameter of 100 mm and wall thickness of 5 mm. The thin-walled steel tube is modelled using one layer of shell elements (SR4) and two layers of three-dimensional 8-node solid elements (C3D8) through the wall thickness of the tube respectively. The material properties with fy ¼350 MPa given in Table 1 are used for the model and the results are shown in Fig. 4. The results show that both models using shell elements and solid elements are able to capture the local and overall buckling effects of the thin-walled tube structure. Fig. 5 shows the comparison of load vs. deformation using the solid and shell elements respectively and the results are seen to be almost indistinguishable. A sensitivity study on mesh sizes to the structural behaviour was performed, in which element sizes ranged from 5 mm to 50 mm were used to select the most effective element size for the FE model. Using large element sizes caused difficulty to ensure simulation accuracy, while using too small element size would consume high amount of computer time and cause convergence difficulty. From the mesh sensitivity study, it was concluded that the appropriate mesh sizes would range from 7.5 to 25 mm depending on the member cross-section dimensions. For instance, for the 150 75 4 mm and 200 100 5 mm EHS sections, the element sizes of 7.5 mm and 10 mm are used respectively. For EHS section 500 250 10 mm, the element size of 25 mm is used. Two layers of element were adopted in the wall thickness to avoid premature failure due to convergence difficulty.
29
The boundary conditions were applied to the rigid plate as reference points. At both ends, the rigid plate was tied to the bearing plate. Contact pairs were defined as surface-to-surface contacts to account for the interaction between steel tube and concrete core, steel ball and the plates. “Hard contact” was assumed for the normal contact behaviour and reopening was allowed. In the tangential direction, the general bond action between concrete core and steel hollow section was simulated through the contact friction. Different friction coefficients ranged from 0.1 to 0.5 in the tangential direction were investigated, however from the sensitivity study, it was found that friction coefficients gave little effect to axial resistance but small friction coefficient is likely to cause convergent problem at large deformation. Therefore a friction coefficient of 0.3 was adopted in this study for the interaction between concrete core and steel hollow section. Theoretically, the friction between steel ball and the plates is expected to be zero without damping; however friction between the ball and plate existed in practice. Based on the sensitivity analysis, a friction coefficient between 0.04 and 0.1 may be suitable to account for the friction between the steel ball and the steel plates. In test, grease was used to reduce the friction between the steel ball and the plates. Different to the stub column tests, the imperfection might give significant effect to the compressive behaviour of a slender member. Due to the difficulty in measuring all the imperfection accurately and to apply all the local imperfections to a structural member in the FE modelling, only the global imperfection (the measured value of initial out of straightness) was applied. Chan and Gardner investigated the effect of imperfection to the buckling behaviour of elliptical steel hollow sections in compression and an imperfection of Le/1000 was suggested in their numerical modelling to obtain consistent results to the current design codes [21]. However, to validate the FE Model, the measured imperfection was used. From the test specimens listed in Table 1, the measured global imperfection values (out of straightness) were in the range from 0.5 to 1.5 mm. A comparative analysis demonstrated such imperfection value was close to the value of Le/2000. Therefore an imperfection value of Le/2000 was assumed in the research presented in this paper. 3.3. Stress–stain model of confined concrete
(b) Shell element for steel tube
(a) Solid element for steel tube
Fig. 4. Both shell and solid element types used in slender steel tube column models.
As described in the previous section, when a slender elliptical CFST column subjected to axial compression, its deformation shape follows column's first buckling mode in the minor axis direction. In such cases, the stress and strain states of the steel tube and concrete core are different from those of the stub elliptical CFST columns in which the section and concrete core were subjected to pure compression during the loading course, however when a slender CFST column under compressive load, both steel tube section and concrete core might experience different stress and strain states
Fig. 5. Effect of element types of steel tube to load-deformation relationships.
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in loading process, and might be partly in compression and partly in tension after the flexural deflection of the column reach certain level. The differences in deformation and stress–strain behaviour of the slender to the stub elliptical CFST columns will result in different confinement effect to the concrete. It has been recognised that the mechanical properties of confined concrete changes depending on the confinement boundary conditions. For CFST columns, the confinement is uniform throughout the contact interaction between the concrete core and steel tube, therefore the hollow section shape and dimensions, material properties might affect the total confinement effect and the confining force distribution along the perimeter of the tube, which leads to different confinement effect to the infill concrete. Han et al. proposed special models for stress–strain relationships of concrete filled in circular and square hollow sections based on experiments and numerical simulations. [9,11] The proposed models were further verified in prediction of the axial compressive behaviour of CFST columns with circular, square and rectangular sections. [22–24] So far there is no special consideration for the confinement of concrete in slender elliptical CFST columns although Dai and Lam developed a confined concrete stress–strain model for elliptical stub CFST columns but it is not suitable for elliptical slender CFST columns [16]. In this paper, the confined concrete stress–strain model developed by Han et al. [9,11] for circular sections is modified and extended its application to slender elliptical CFST columns. Detailed stress–strain model of the confined concrete in circular steel hollow sections were given in these publications [9,11]. In which the basic form is described as following: 8 < 2x x2 ; ðx r 1Þ; y¼ ð1Þ x : β0 ðx 1Þ2 þ x; ðx 4 1Þ; where: x ¼ ε=ε0 ; y ¼ s=s0 ; s0 ¼ f ck ðN=mm2 Þ; ε0 ¼ εc þ 800ξ0:2 10 6 ; εc ¼ ð1300 þ 12:5f ck Þ 10 6 ; 7
β0 ¼ ð2:36 10 5 Þ½0:25 þ ðξ 0:5Þ ðf ck Þ0:5 0:5 Z 0:12 In the above formulae, f ck is the cylinder compressive strength of the concrete, ξ is the so called confinement factor whose expression is described as following: ξ¼
As f y Ac f ck
ð2Þ
where As and Ac are the cross-sectional areas of the steel tube and concrete core. f y is the yield strength of steel and f ck is the characteristic strength of concrete and f ck ¼0:8f cu , f cu is the cube compressive strength of unconfined concrete. The Modulus of pffiffiffiffiffiffi Elasticity of the confined concrete is described as Ec ¼ 4700 f ck MPa and the Poisson ratio of the confined concrete was taken as 0.2. As mentioned previously, the concrete core might be partly in tension during loading therefore the tensile stress–strain model for concrete must be introduced in the numerical simulations. In the absence of special test data, the concrete tensile stress and corresponding strain might be taken as 10% of the concrete corresponding compressive stress and strain. This method was adopted to form the tensile stress–strain model of concrete for the numerical simulations presented in this paper.
Table 2 Comparison of axial compressive loads by experiments and numerical predictions. Specimen ID
N EXP (kN)
N FE (kN)
NFE N EXP
TG1-D150-C30 TG1-D150-C60 TG1-D150-C100 TG1-D200-C30 TG1-D200-C60 TG1-D200-C100 TG2-D150-C30 TG2-D150-C60 TG2-D150-C100 TG2-D200-C30 TG2-D200-C60 TG2-D200-C100 TG3-D150-C30 TG3-D150-C60 TG3-D150-C100 TG3-D200-C30 TG3-D200-C60 TG3-D200-C100
651 743 923 938 1065 1480 484 663 871 968 1237 1411 327 427 547 839 947 1073
595 712 999 1041 1213 1455 487 731 843 1015 1256 1409 301 432 498 814 949 1083
0.914 0.958 1.082 1.110 1.139 0.983 1.006 1.103 0.968 1.049 1.015 0.999 0.920 1.012 0.910 0.970 1.002 1.009
Average COV
1.008 0.065
3.4. Validation of the FE model The accuracy of the FE model was verified through comparing the relationship and failure modes predicted by the numerical method described in the previous sections against the experimental results. In numerical modelling, the measured dimensions of tested specimens and measured material properties were adopted. The confined concrete compressive stress–strain model is in accordance to Eqs. (1) and (2) and using the measured unconfined concrete parameters given in Table 1. The confined concrete tensile stress–strain model was taken as 10% of its compressive stress and strain values. Due to the measured initial out of straightness for some specimens was not available; a monotonic global imperfection value of Le/2000 was adopted. Table 2 compares the maximum axial compressive load of slender composite columns predicted by FE model (N FE ) with that test results (N EXP ) and the good agreement are demonstrated by the ratio of NFE =N EXP being close to unity. The difference between the prediction and measured value was possibly resulted from the difference between the material property and imperfection adopted in the FE model. Fig. 6 shows the typical axial compressive load vs. mid-height lateral deflection curves observed by the experiments and the FEM predictions. Both the experiment and numerical results showed that the failure modes of slender composite columns were characterised by global buckling in the minor axis direction although in some cases, local buckling was also observed on the compressive side of the column at midheight. For the very slender elliptical CFST columns, the mode of failure is dominated by global buckling.
4. Parametric study 4.1. Axial compressive behaviour There are many factors that might affect the axial compressive behaviour of a slender CFST column. These factors include the members' non-dimensional slenderness, material properties, the sectional shape, dimensions and column lengths. After the verification of the FE model, parametric study was performed using the measured material properties and geometries, a detailed parametric study was conducted to assess the axial compressive
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
31
Fig. 6. Comparison of axial compressive load to mid-span deflection curves predicted by FE modelling against experimental recordings.
behaviour of slender elliptical CFST columns with typical the hollow section dimensions from 150 75 4 EHS up to 500 250 10 EHS and member non-dimensional slenderness range from 0.25 up to 2.93. The following section presented the axial buckling loads of concrete filled composite columns with different elliptical steel hollow sections and slenderness. In total; 64 elliptical CFST columns with five section sizes range from 150 75 4 to 500 250 10; six column lengths from 1.5 to 4.5 m and three concrete grades as shown in Table 3 were adopted for the parametric study. The measured steel properties taken from section 200 100 5 EHS were used. For infill concrete, the unconfined cube strengths of 30 MPa, 60 MPa and 100 MPa were assumed for concrete with nominal grades C30, C60 and C100 respectively. Modulus of Elasticity of 23.7, 34.9 and 43.3 GPa was used for the respective model. Due to absence of imperfection value, a global imperfection of Le/2000 of the first buckling mode shape was assumed and applied for the modelling. In addition the confined concrete stress–strain model was derived in accordance to Eqs. (1) and (2) where the cylinder strength of unconfined concrete was taken as 85% of the cube strength of the unconfined concrete. Table 3 gives the maximum axial
compressive load N FE of for all selected elliptical CFST columns. Using the design rules provided in Eurocode 4 for circular and rectangular CFST columns with an equivalent diameter D ¼2a2/b for elliptical section, important buckling load parameters, N cr , Npl;Rd , λ and χ, were calculated and showed in Table 3, where buckling load reduction factor, χ is taken asN FE /N pl;Rd . Fig. 7 compares the predicted buckling reduction factors, χ vs. the non-dimensional slenderness, λ taken from Table 3. Eurocode 4 proposed that the buckling curve (a) to be used for circular and rectangular CFST columns. It can be found from Fig. 7, the predicted reduction factors χ for most elliptical CFST columns lie above or close to the curve (a) except for the selected columns with non-dimensional slenderness λ is less than 0.4, therefore the buckling curve (a) might be safely used for the design of elliptical CFST columns with global imperfection not exceeding Le/2000. 4.2. Effect of column imperfection to axial compressive behaviour The member imperfection may be different for individual members due to the fabrication tolerance, transport method and design allowance. It has been recognised that column imperfection
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Table 3 Summary of selected specimen dimensions and axial compressive behaviour. Specimen ID
2a 2b t (mm)
Le (mm)
f y (MPa)
f cu (MPa)
N FE (kN)
N cr (kN)
N pl;Rd (kN)
λ
χ
PG1-D150-C30 PG1-D150-C60 PG1-D150-C100 PG1-D200-C30 PG1-D200-C60 PG1-D200-C100 PG1-D300-C30 PG1-D300-C60 PG1-D300-C100 PG1-D400-C30 PG1-D400-C60 PG1-D500-C30 PG1-D500-C60 PG2-D150-C30 PG2-D150-C60 PG2-D150-C100 PG2-D200-C30 PG2-D200-C60 PG2-D200-C100 PG3-D150-C30 PG3-D150-C60 PG3-D150-C100 PG3-D200-C30 PG3-D200-C60 PG3-D200-C100 PG3-D300-C30 PG3-D300-C60 PG3-D300-C100 PG3-D400-C30 PG3-D400-C60 PG3-D500-C30 PG3-D500-C60 PG4-D150-C30 PG4-D150-C60 PG4-D150-C100 PG4-D200-C30 PG4-D200-C60 PG4-D200-C100 PG5-D150-C30 PG5-D150-C60 PG5-D150-C100 PG5-D200-C30 PG5-D200-C60 PG5-D200-C100 PG5-D300-C30 PG5-D300-C60 PG5-D300-C100 PG5-D400-C30 PG5-D400-C60 PG5-D500-C30 PG5-D500-C60 PG6-D150-C30 PG6-D150-C60 PG6-D150-C100 PG6-D200-C30 PG6-D200-C60 PG6-D200-C100 PG6-D300-C30 PG6-D300-C60 PG6-D300-C100 PG6-D400-C30 PG6-D400-C60 PG6-D500-C30 PG6-D500-C60
150 75 4 150 75 4 150 75 4 200 100 5 200 100 5 200 100 5 300 150 8 300 150 8 300 150 8 400 200 8 400 200 8 500 250 10 500 250 10 150 75 4 150 75 4 150 75 4 200 100 5 200 100 5 200 100 5 150 75 4 150 75 4 150 75 4 200 100 5 200 100 5 200 100 5 300 150 8 300 150 8 300 150 8 400 200 8 400 200 8 500 250 10 500 250 10 150 75 4 150x75 4 150x75 4 200x100 5 200x100 5 200x100 5 150 75 4 150 75 4 150 75 4 200 100 5 200 100 5 200 100 5 300 150 8 300 150 8 300 150 8 400 200 8 400 200 8 500 250 10 500 250 10 150 75 4 150 75 4 150 75 4 200 100 5 200 100 5 200 100 5 300 150 8 300 150 8 300 150 8 400 200 8 400 200 8 500 250 10 500 250 10
1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1800 1800 1800 1800 1800 1800 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 3000 3000 3000 3000 3000 3000 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500
410 410 410 360 360 360 360 360 360 360 360 360 360 410 410 410 360 360 360 410 410 410 360 360 360 360 360 360 360 360 360 360 410 410 410 360 360 360 410 410 410 360 360 360 360 360 360 360 360 360 360 410 410 410 360 360 360 360 360 360 360 360 360 360
30 65 100 30 65 100 30 65 100 30 65 30 65 30 65 100 30 65 100 30 65 100 30 65 100 30 65 100 30 65 30 65 30 65 100 30 65 100 30 65 100 30 65 100 30 65 100 30 65 30 65 30 65 100 30 65 100 30 65 100 30 65 30 65
665 793 901 1066 1355 1608 2584 3304 3997 4035 5389 6311 8641 570 657 742 1038 1301 1530 343 387 428 880 1039 1191 2471 3091 3605 3894 5066 6070 8176 243 276 301 696 802 907 180 205 220 534 620 688 2200 2609 2989 3738 4763 5885 7791 111 125 133 335 385 416 1625 1852 2078 3373 4120 5618 7335
1041 1107 1145 3197 3411 3537 16,932 17,991 18,609 49,645 49,646 106,868 115,922 723 769 795 2220 2369 2456 375 398 412 1151 1228 1273 6096 6477 6699 17,872 17,872 38,472 41,732 260 277 286 799 853 884 191 203 210 587 627 650 3110 3305 3418 8040 8721 19,629 21,292 118 125 129 355 379 393 1881 1999 2068 5516 5516 11,874 12,880
750 972 1194 1140 1539 1939 2660 3520 4439 4235 5734 6859 9238 750 972 1194 1140 1539 1939 750 972 1194 1140 1539 1939 2671 3560 4449 3948 5635 6198 8722 750 972 1194 1140 1539 1939 750 972 1194 1140 1539 1939 2671 3560 4449 3984 5635 6224 8804 668 890 1112 1140 1539 1939 2671 3560 4449 3984 5635 6224 8804
0.849 0.937 1.021 0.597 0.672 0.740 0.396 0.442 0.488 0.292 0.340 0.253 0.282 1.018 1.125 1.225 0.717 0.806 0.888 1.415 1.562 1.702 0.995 1.120 1.234 0.662 0.741 0.815 0.470 0.561 0.401 0.457 1.697 1.874 2.042 1.194 1.343 1.481 1.980 2.187 2.383 1.393 1.567 1.728 0.927 1.038 1.141 0.704 0.804 0.563 0.643 2.383 2.669 2.934 1.791 2.015 2.221 1.192 1.335 1.467 0.850 1.011 0.724 0.827
0.887 0.816 0.755 0.935 0.880 0.829 0.971 0.939 0.900 0.953 0.940 0.920 0.935 0.760 0.676 0.622 0.911 0.845 0.789 0.457 0.399 0.359 0.772 0.675 0.614 0.925 0.868 0.810 0.986 0.899 0.979 0.937 0.324 0.284 0.252 0.611 0.521 0.468 0.240 0.211 0.184 0.469 0.403 0.355 0.824 0.733 0.672 0.938 0.845 0.946 0.885 0.166 0.140 0.120 0.294 0.250 0.215 0.608 0.520 0.467 0.847 0.731 0.903 0.833
might give significant influence to the axial compressive behaviour. Therefore a sensitivity study using numerical modelling was conducted to investigate the effect of global imperfection to the maximum axial compressive load and buckling behaviour of elliptical CFST columns. Various initial global imperfections range from Le/8000 to Le/250 mm was used. The range represents the possible initial global imperfection of composite columns,
including initial out of straightness in fabrication due to fabrication. Fig. 8 compares the predicted buckling reduction factors of selected elliptical CFST columns with 5 different global imperfections against to the design buckling curves provided in Eurocode 3 for hollow steel members and CFST members with circular and rectangular sections [25]. It can be seen from Fig. 8, if the imperfection of Le/1000 is used, only the predicted buckling
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
Fig. 7. Comparison of predicted buckling reduction factors of elliptical CFST columns with initial global imperfection of Le/2000.
Reduction factor: χ
1.2 1 0.8
a0
a
b
c
d
IMP: Le/1000
0.6 0.4 0.2 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Non-dimensional slenderness: λ
Reduction factor: χ
1.2 1 0.8
a0 b d IMP: Le/4000
a c IMP: Le/2000
0.6 0.4 0.2 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Non-dimensional slenderness: λ
Reduction factor: χ
1.2 1 0.8
a0 b d IMP: Le/250
a c IMP: Le/500
0.6 0.4 0.2 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Non-dimensional slenderness: λ
Fig. 8. Comparison of buckling behaviour of elliptical CFST columns with various initial global imperfections to design buckling curves provided in EC3.
to Le/2000, then most selected elliptical CFST columns will lie above the design curve (a). From the tests, an imperfection of Le/2000 was normally observed in most specimens as discussed earlier, the predicted reduction factors of selected columns with various imperfections was compared against the predicted reduction factor of column with an imperfection of Le/2000 as given in Table 4, the difference can clearly be observed. Obviously, with the imperfection increasing from Le/2000 to Le/250, the average axial compressive capacity is reduced by up to 19% on average. If the imperfection is increased from Le/1000 to Le/250; the average axial compressive capacity is reduced by up to 15% on average. Similar conclusion was observed by Chan and Gardner in their research on elliptical steel hollow section members [20]. However, if the imperfection is reduced from Le/2000 to Le/8000, the average compressive capacity is only increased by 4%. Base on the parametric study and sensitivity analysis, it is concluded that the imperfection of Le/2000 should be used for the FE modelling.
5. Design to Eurocode 4 Currently the Eurocode 4 does not covered CFST columns with elliptical hollow sections although simplified design rules for CFST columns with circular, square and rectangular sections are available. In accordance to Eurocode 4, the design axial compressive load of a circular CFST composite column is determined from the plastic resistance and buckling behaviour by introducing partial factor and imperfection characteristics according to the structural features of the member itself. According to the comparison and analysis presented in the previous sections, the design rules provided in Eurocode 4 for CFST columns with circular sections appear to be unsuitable for the elliptical CFST columns with low slenderness. This section gives further review to the application of the design method for the elliptical CFST columns. Unlike the stub CFST columns whose compressive capacity and failure mode are governed by member yielding and local buckling, the design of a slender composite column is generally dominated by its global buckling behaviour. Therefore relevant design buckling curves of non-dimensional slenderness λ to buckling load reduction factor χ relationships are adopted for the design. The non-dimensional slenderness λ is expressed as: sffiffiffiffiffiffiffiffiffiffiffiffi N pl;Rd λ¼ ð3Þ Ncr Npl;Rd is the plastic resistance of a composite column from crosssection strength, it may be calculated by adding the plastic resistances of its components. For CFST columns without reinforcement, N pl;Rd is described as: Npl;Rd ¼ Ac f ck þ As f y
ð4Þ
where Ac is the cross-sectional area of the concrete core, f ck is the unconfined compressive concrete cylinder strength, As is the crosssectional area of the steel tube, f y is the yield strength of the steel. Ncr is the elastic buckling resistance of the composite column which is determined by: Ncr ¼
π 2 ðEIÞef f L2e
ð5Þ
In which Le is the effective length and ðEIÞef f is the effective sectional stiffness of the composite column which is calculated by: ðEIÞef f ¼ Es I s þ ke Ecm I c
reduction factors; χ with λ higher than 0.8 lie above the design curve (a). If the imperfection is increased to Le/500, the predicted buckling reduction factors χ of most of the selected columns lie below the design curve (a). However, if the imperfection is reduced
33
ð6Þ
where Es and I s are the elastic modulus and the second area moment of the steel tube section, Ecm andI c are the elastic modulus and the second area moment of the uncrushed concrete core, ke is a coefficient which is taken as 0.6.
34
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
Table 4 Comparison of buckling reduction factors for elliptical CFST columns with different global imperfections.
Specimen ID
PG1-D150-C30 PG1-D150-C60 PG1-D150-C100 PG1-D200-C30 PG1-D200-C60 PG1-D200-C100 PG1-D300-C30 PG1-D300-C60 PG1-D300-C100 PG1-D400-C30 PG1-D400-C60 PG1-D500-C30 PG1-D500-C60 PG2-D150-C30 PG2-D150-C60 PG2-D150-C100 PG2-D200-C30 PG2-D200-C60 PG2-D200-C100 PG3-D150-C30 PG3-D150-C60 PG3-D150-C100 PG3-D200-C30 PG3-D200-C60 PG3-D200-C100 PG3-D300-C30 PG3-D300-C60 PG3-D300-C100 PG3-D400-C30 PG3-D400-C60 PG3-D500-C30 PG3-D500-C60 PG4-D150-C30 PG4-D150-C60 PG4-D150-C100 PG4-D200-C30 PG4-D200-C60 PG4-D200-C100 PG5-D150-C30 PG5-D150-C60 PG5-D150-C100 PG5-D200-C30 PG5-D200-C60 PG5-D200-C100 PG5-D300-C30 PG5-D300-C60 PG5-D300-C100 PG5-D400-C30 PG5-D400-C60 PG5-D500-C30 PG5-D500-C60 PG6-D150-C30 PG6-D150-C60 PG6-D150-C100 PG6-D200-C30 PG6-D200-C60 PG6-D200-C100 PG6-D300-C30 PG6-D300-C60 PG6-D300-C100 PG6-D400-C30 PG6-D400-C60 PG6-D500-C30 PG6-D500-C60 Average COV
Geometrical and material properties λ
As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As
in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in
Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
0.849 0.937 1.021 0.597 0.672 0.740 0.396 0.442 0.488 0.292 0.340 0.253 0.282 1.018 1.125 1.225 0.717 0.806 0.888 1.415 1.562 1.702 0.995 1.120 1.234 0.662 0.741 0.815 0.470 0.561 0.401 0.457 1.697 1.874 2.042 1.194 1.343 1.481 1.980 2.187 2.383 1.393 1.567 1.728 0.927 1.038 1.141 0.704 0.804 0.563 0.643 2.383 2.669 2.934 1.791 2.015 2.221 1.192 1.335 1.467 0.850 1.011 0.724 0.827
Ratio of reduction factor χ of columns with different imperfections to χ of column with imperfection Le/2000 Le/8000
Le/4000
Le/2000
Le/1000
Le/500
Le/250
1.041 1.040 1.040 1.041 1.041 1.041 1.041 1.040 1.041 1.040 1.040 1.040 1.041 1.041 1.041 1.040 1.041 1.041 1.041 1.042 1.040 1.039 1.041 1.040 1.041 1.041 1.040 1.041 1.041 1.041 1.041 1.041 1.040 1.039 1.040 1.041 1.040 1.041 1.038 1.043 1.043 1.041 1.040 1.039 1.040 1.040 1.040 1.041 1.041 1.040 1.041 1.042 1.043 1.033 1.041 1.040 1.037 1.041 1.040 1.041 1.040 1.041 1.040 1.060
1.028 1.028 1.028 1.029 1.028 1.028 1.029 1.028 1.029 1.027 1.028 1.028 1.028 1.028 1.028 1.027 1.027 1.028 1.028 1.028 1.028 1.028 1.028 1.028 1.029 1.028 1.028 1.028 1.028 1.028 1.029 1.029 1.028 1.028 1.028 1.028 1.027 1.028 1.025 1.028 1.033 1.028 1.027 1.028 1.028 1.027 1.027 1.029 1.028 1.027 1.028 1.030 1.029 1.025 1.027 1.028 1.028 1.028 1.029 1.028 1.027 1.029 1.025 1.044
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.957 0.958 0.958 0.958 0.958 0.958 0.958 0.957 0.958 0.957 0.957 0.958 0.958 0.958 0.959 0.957 0.957 0.959 0.958 0.958 0.957 0.958 0.959 0.957 0.958 0.958 0.957 0.958 0.958 0.958 0.958 0.958 0.957 0.958 0.956 0.957 0.958 0.957 0.958 0.957 0.962 0.957 0.958 0.958 0.958 0.958 0.957 0.958 0.959 0.958 0.958 0.958 0.964 0.958 0.959 0.960 0.958 0.959 0.958 0.957 0.957 0.958 0.958 0.956
0.896 0.896 0.895 0.896 0.897 0.896 0.897 0.896 0.897 0.896 0.896 0.897 0.896 0.896 0.896 0.895 0.896 0.897 0.896 0.897 0.895 0.894 0.896 0.896 0.897 0.896 0.896 0.896 0.897 0.897 0.897 0.896 0.895 0.894 0.893 0.895 0.896 0.895 0.896 0.896 0.897 0.896 0.896 0.896 0.896 0.896 0.896 0.897 0.896 0.895 0.896 0.898 0.900 0.892 0.895 0.896 0.893 0.896 0.896 0.897 0.896 0.896 0.891 0.885
0.804 0.804 0.804 0.804 0.805 0.805 0.804 0.804 0.804 0.804 0.804 0.804 0.804 0.804 0.805 0.804 0.805 0.805 0.805 0.805 0.805 0.805 0.804 0.804 0.805 0.804 0.804 0.805 0.804 0.804 0.805 0.805 0.802 0.803 0.802 0.804 0.804 0.803 0.804 0.806 0.804 0.804 0.804 0.806 0.805 0.804 0.804 0.805 0.805 0.803 0.805 0.807 0.807 0.800 0.803 0.804 0.805 0.804 0.804 0.805 0.804 0.804 0.795 0.780
1.041 0.003
1.028 0.002
1.000 0.000
0.958 0.001
0.896 0.002
0.804 0.003
For CFST column with circular sections, if λ from Eq. (3) does not exceed 0.5 and e=D o 0:1, where e is the eccentricity of the loading given by section moment divided by the axial load, D is the
outer diameter of the circular steel tube, an expression given in Eq. (7) may be used to consider the effect of concrete confinement. For elliptical CFST columns, an equivalent diameter, De ¼2a2/b is
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
suggested to replace the outer diameter of the circular column which gave a reasonable estimation of the axial capacity of the elliptical CFST columns. fy t N pl;Rd ¼ As ηs f y þ Ac f ck 1 þ ηc ð7Þ D f ck whereAc , f ck , As and f y areas specified as in Eq. (4), t is the thickness of the hollow section respectively, ηs and ηc are functions of the non-dimensional slenderness λ of the composite column and are determined by: ηs ¼ 0:25ð3 þ2λÞ but with ηs r1:0 2
ηc ¼ 4:9 18:5λ þ 17λ
but with ηc Z 0
35
Eurocode 4 may be used for axial compressive behaviour design of elliptical CFST columns. (3) The initial imperfection has a significant effect to the axial compressive behaviour of slender elliptical CFST columns. Initial imperfection of Le/2000 should be used for the FE modelling. (4) From the experimental observation and subsequent parametric study presented in the paper, it would appear that the design buckling curve (a) provided in the Eurocode 3 for circular and rectangular CFST columns is suitable for the design of slender elliptical CFST columns.
ð8aÞ ð8bÞ
Acknowledgement
When calculate the design load NEd of a composite member in compression, the member imperfection and buckling effects should be taken into consideration if λ Z 0:2. The design load N Ed should satisfy:
The research reported in this paper is partly funded by the research grant from the Engineering and Physical Science Research Council (EP/G002126/1) in the UK who are gratefully acknowledged.
N Ed r 1:0 χN pl;Rd
References
ð9Þ
N pl;Rd is the plastic resistance given by Eqs. (4) and (7) but in which f y should be replaced by design strength f yd if the partial factor γ M1 is not taken as unity. χ is the reduction factor for the relevant buckling mode given in Eurocode 3 in terms of the relevant nondimensional slenderness λ and the equivalent buckling curve. In Eurocode for members in axial compression the design value of χ for the appropriate λ is determined from the relevant buckling curves which can be expressed by: χ¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi but with χ r1:0 Φ þ Φ2 λ2
ð10Þ
2
where Φ ¼ 0:5½1 þ αðλ 0:2Þ þ λ , λ given by Eq. (3). α is the imperfection factor whose value changes according to the member characteristics. For the design of the composite members with λ r 0:2, the global buckling effect may be ignored and therefore χ may be taken as 1.0. The above design method currently given in Eurocode 4 for CFST columns for circular and rectangular sections suggested that an initial imperfection of Le/300 is used. However based on the comparison and analysis given in the previous sections, it would appear that the imperfection of Le/300 is not suitable for modelling the elliptical CFST columns. If the imperfection of Le/300 were used for FE modelling, the results would be over-conservative.
6. Conclusions This paper provided a non-linear FE model developed through ABAQUS/Standard solver to investigate the axial compressive behaviour of slender elliptical CFST columns. After the numerical method being verified by comparing the predicted results against the experimental observations, an extensive parametric study was presented. Based on the comparison and analysis, the following conclusions can be made: (1) The FE numerical method developed via ABAQUS/Standard solver, combined with the special stress–strain model for confined concrete in slender elliptical hollow sections could be used to predict the axial compressive behaviour of slender elliptical CFST columns. (2) Both experimental and numerical modelling results demonstrated that the failure mode of a slender CFST column under compressive loads was global buckling with the first buckling mode shape. The simplified design method provided in
[1] Zhang S, Guo L, Ye Z, Wang Y. Behavior of steel tube and confined high strength concrete for concrete-filled rhs tubes. Adv Struct Eng 2005;8(2):101–16. [2] Zhang S, Guo L. Behaviour of high strength concrete-filled slender rhs steel tubes. Adv Struct Eng 2007;10(4):337–51. [3] Schneider SP. Axially loaded concrete filled steel tubes. ASCE J Struct Eng 1998;124(10):1125–38. [4] O’Shea MD, Bridge RQ. Design of circular thin-walled concrete filled steel tubes. ASCE J Struct Eng 2000;126(11):1295–303. [5] Giakoumelis G, Lam D. Axial capacity of circular concrete filled tube columns. J Constr Steel Res 2004;60(7):1049–68. [6] Han LH, Liu W, Yang YF. Behavior of thin walled steel tube confined concrete stub columns subjected to axial local compression. Thin-Walled Struct 2008;46(2):155–64. [7] Saw HS, Liew JY. Assessment of current methods for the design of composite columns in buildings. J Constr Steel Res 2000;53(2):121–47. [8] Hu HT, Huang CS, Wu MH, Wu YM. Nonlinear analysis of axially loaded concrete-filled tube columns with confinement effect. ASCE J Struct Eng 2003;129(10):pp1322–9. [9] Han LH, Yao GH, Zhao XL. Tests and calculations of hollow structural steel (HSS) stub columns filled with self-consolidating concrete (SCC). J Constr Steel Res 2005;61(9):1241–69. [10] Ellobody E, Young B, Lam D. Behaviour of normal and high strength concrete-filled compact steel tube circular stub columns. J Constr Steel Res 2006;62(7):706–15. [11] Han LH, Yao GH, Tao Z. Performance of concrete-filled thin-walled steel tubes under pure torsion. Thin-Walled Struct 2007;45(1):24–36. [12] Dabaon M, El-Khoriby S, El-Boghdadi M, Hassanein MF. Confinement effect of stiffened and unstiffened concrete-filled stainless steel tubular stub columns. J Constr Steel Res 2009;65(8-9):1846–54. [13] Lam D, Gardner L. Structural design of stainless steel concrete filled columns. J Constr Steel Res 2008;64(11):1275–82. [14] Yang H, Lam D, Gardner L. Testing and analysis of concrete-filled elliptical hollow sections. Eng Struct 2008;30(12):3771–81. [15] Zhao XL, Packer JA. Tests and design of concrete-filled elliptical hollow section stub columns. Thin-Walled Struct 2009;47(6-7):617–28. [16] Dai X, Lam D. Numerical modelling of axial compressive behaviour of short concrete-filled elliptical steel tube columns. J Steel Constr Res 2010;66(7): pp931–pp942. [17] BS EN 1994-1-1: Eurocode 4. Design of composite steel and concrete structures Part 1-1: General rules and rules for buildings. London: British Standards Institution; 2004. [18] Jamaluddin N. Behaviour of concrete filled elliptical hollow sections [Ph.D. thesis]. Leeds, UK: University of Leeds; 2011. [19] Jamaluddin N, Lam D, Dai XH, Ye J. An experimental study on elliptical concrete filled columns under axial compression. J Constr Steel Res 2013;87:6–16. [20] ABAQUS Standard User's Manual. Hibbitt, Karlsson and Sorensen, Inc. Version 6.12, USA; 2011. [21] Chan TM, Gardner L. Flexural buckling of elliptical hollow section columns. ASCE J Struct Eng 2009;135(5):546–57. [22] Lu H, Han LH, Zhao XL. Analytical behavior of circular concrete-filled thin-walled steel tubes subjected to bending. Thin-Walled Struct 2009;47(3):346–58. [23] Tao Z, Uy B, Han LH, Wang ZB. Analysis and design of concrete-filled stiffened thin-walled steel tubular columns under axial compression. Thin-Walled Struct 2009;47(12):1544–56. [24] Huang H, Han LH, Tao Z, Zhao XL. Analytical behaviour of concrete-filled double skin steel tubular (CFDST) stub columns. J Constr Steel Res 2010;66(4): 542–55. [25] BS EN 1993-1-1: Eurocode 3. Design of steel structures Part 1-1: General rules and rules for buildings. London: British Standards Institution; 2005.
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Numerical analysis of slender elliptical concrete filled columns under axial compression X.H. Dai a, D. Lam a,n, N. Jamaluddin b, J. Ye c a
School of Engineering, Design and Technology, University of Bradford, Bradford, UK Faculty of Civil and Environmental Engineering, Universiti Tun Hussein Onn Malaysia, Malaysia c Department of Engineering, Lancaster University, Lancaster, UK b
art ic l e i nf o
a b s t r a c t
Article history: Received 24 June 2013 Received in revised form 28 November 2013 Accepted 28 November 2013 Available online 15 December 2013
This paper presents a non-linear finite element model (FEM) used to predict the behaviour of slender concrete filled steel tubular (CFST) columns with elliptical hollow sections subjected to axial compression. The accuracy of the FEM was validated by comparing the numerical prediction against experimental observation of eighteen elliptical CFST columns which carefully chosen to represent typical sectional sizes and member slenderness. The adaptability to apply the current design rules provided in Eurocode 4 for circular and rectangular CFST columns to elliptical CFST columns were discussed. A parametric study is carried out with various section sizes, lengths and concrete strength in order to cover a wider range of member cross-sections and slenderness which is currently used in practices to examine the important structural behaviour and design parameters, such as column imperfection, non-dimension slenderness and buckling reduction factor, etc. It is concluded that the design rules given in Eurocode 4 for circular and rectangular CFST columns may be adopted to calculate the axial buckling load of elliptical CFST columns although using the imperfection of length/300 specified in the Eurocode 4 might be overconservative for elliptical CFST columns with lower non-dimensional slenderness. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Numerical modelling Elliptical CFST column Axial compressive load Buckling reduction factor Design buckling curves
1. Introduction Composite CFST columns are increasingly used for high-rises building structures, owing to their excellent structural behaviour such as superior load-bearing capacity, high ductility, good energy dissipation and fire-resistance etc. which arises from the composite action of two different materials in one structural member. Hot rolled elliptical hollow section (EHS) represents a new range of cross-sections recently available in construction market. This new shape has an aesthetically pleasing appearance and improved structural efficiency due to its different major and minor axis. Regarding the concrete filled columns with steel tubular hollow sections, concrete filled composite columns with circular hollow sections (CHS) have the advantages over columns with other section shapes due to the circular sections provide an uniform confinement to the concrete core. The concrete filled columns with square hollow section (SHS) and rectangular hollow section (RHS) have many common virtues to those with circular hollow sections, such as high ductility and large energyabsorption capacity. However, the disadvantage of square and rectangular sections lies with the stress concentration at the corners of the steel sections and may result in cracking [1,2]. Extensive
experimental studies and numerical analyses have been conducted to investigate the structural behaviour of CFST columns with circular, square and rectangular sections, which included the effects of concrete confinement, geometry of hollow section, column slenderness and constituent material properties [3–6]. In particular, the confinement effect of concrete in stub CFST composite columns was well recognised [7–13]. Although the compressive behaviour of the elliptical CFST stub columns have been investigated worldwide [14– 16], there is currently a lack of design rules for the behaviour of elliptical CFST slender columns. The main objective of the research is to understand the axial compressive behaviour of slender CFST columns with elliptical hollow sections. This paper is mainly focuses on the following aspects: (1) develop a reasonable nonlinear FE model through ABAQUS/Standard solver to investigate the overall buckling behaviour of slender elliptical CFST columns; (2) conduct a parametric study to highlight important design parameters; (3) assess the adaptability of the current design rules provided in Eurocode 4 [17] for elliptical CFST columns.
2. Experimental study n
Corresponding author. Tel.: þ 441274234054; fax: þ441274234525. E-mail address: [email protected] (D. Lam).
0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.11.015
To assess the accuracy of the FE model, this section will provide a brief description of the experimental setup and boundary conditions
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
27
Table 1 Summary of measured specimen dimensions, material properties and maximum axial compressive loads. Specimen ID
2a 2b t (mm)
Le (mm)
f y (MPa)
f u (MPa)
Es (GPa)
f cu (MPa)
f ck (MPa)
Ec (MPa)
N EXP (kN)
TG1-D150-C30 TG1-D150-C60 TG1-D150-C100 TG1-D200-C30 TG1-D200-C60 TG1-D200-C100 TG2-D150-C30 TG2-D150-C60 TG2-D150-C100 TG2-D200-C30 TG2-D200-C60 TG2-D200-C100 TG3-D150-C30 TG3-D150-C60 TG3-D150-C100 TG3-D200-C30 TG3-D200-C60 TG3-D200-C100
150.9 75.4 4.0 150.4 75.2 4.1 150.3 75.2 4.1 197.5 100.2 5.2 197.4 100.1 5.1 197.7 100.1 5.1 150.5 75.4 4.1 150.7 75.2 4.2 150.7 75.4 4.1 197.6 100.2 5.1 197.7 100.1 5.1 197.3 100.0 5.2 150.0 75.1 4.2 150.2 75.0 4.0 150.1 75.0 4.1 197.5 100.3 5.2 197.8 100.1 5.1 197.7 100.1 5.1
1677 1678 1676 1679 1678 1678 1965 1965 1966 1965 1966 1966 2680 2682 2681 2678 2679 2678
410 410 410 350 350 350 410 410 410 350 350 350 410 410 410 350 350 350
545 545 545 500 500 500 545 545 545 500 500 500 545 545 545 500 500 500
201 201 201 209 209 209 201 201 201 209 209 209 201 201 201 209 209 209
26 58 91 29 64 91 20 65 100 42 63 98 23 63 100 29 59 101
20 52 78 27 42 71 13 50 72 29 41 76 18 54 72 22 50 74
31,649 35,040 39,889 31,719 34,149 38,698 32,122 36,250 36,390 33,689 37,001 43,699 25,422 34,480 44,027 27,018 35,901 45,685
651 743 923 938 1064 1480 484 663 871 968 1237 1411 327 427 547 839 947 1072
The setup enabled the joint to have a full rotational capacity about the balls centre, therefore the effective length, Le of composite columns is calculated from the centres of the two steel balls. For each test, horizontal and vertical strain gauges were installed on the steel hollow section surface along the longitudinal axis, to record the hoop and axial strains respectively. Linear variable displacement transducers (LVDTs) were placed alongside the length of the column to measure the horizontal displacements of the column along the length. The vertical axial displacement of the column was recorded via the LVDT in the actuator. NEXP in Table 1 gives the maximum axial compressive loads obtained from the tests. 3. Development of the finite element model Fig. 1. Measured stress–strain curves of steel materials.
3.1. General of slender elliptical CFST columns under compressive loads. Detailed information about the setup and test results is given in other publications [18,19]. In total, eighteen slender elliptical CFST columns, with two hollow section sizes (150 75 4 mm, 200 100 5 mm), three column lengths (1.5 m, 1.8 m and 2.5 m) and three nominal concrete strength (30, 60 and 100 MPa) were tested to investigate the axial compressive behaviour and failure modes. Measured specimen dimensions are summarised in Table 1. In which a, b and t are major radius, minor radius and wall thickness of the elliptical steel hollow section respectively; Le is the effective length of column, which is the buckling length between the supports. Before each specimen was tested, steel coupons, cut from the long elliptical hollow section were tested in tension to acquire the mechanical properties of steel materials. The average stress–strain curves for two different steel hollow sections are shown in Fig. 1, with the main characteristic parameters given in Table 1 where fy, fu and Es are the yield strength, ultimate strength and elastic modulus of the steel sections. In addition to the steel properties, concrete cubes and cylinders, cast using the same batch of concrete used for the column specimen were also tested to determine the concrete properties. The measured unconfined concrete cube and cylinder compressive strengths as well as the elastic modules of the concrete; tested on the test date are presented in Table 1, in which, fcu, fck and Ec are the characteristic cube, cylinder compressive strengths and the elastic modulus of the unconfined concrete. The experimental setup is as shown in Fig. 2. The pin-ended joint at the column ends was provided by the combination of a high strength steel ball (80 mm in dia.) with two 40 mm thick steel plates.
It is undoubtedly that full-scale testing provides the most effective method in understanding the behaviour of a structural member, however experimental studies with high numbers of specimens are costly and time consuming. This leads to the application of numerical method to carry out parametric study. Numerical model has been successfully utilised in prediction the axial compressive behaviour of stub CFST columns [16]. This section describes the numerical finite element method (FEM) adopted to predict the structural behaviour of slender elliptical CFST columns under axial compression. As observed from the experimental studies, the failure mechanism of a slender CFST column under axial compression was characterised by overall buckling although local buckling might present at the compressive region of the steel section in some cases. This differs from the failure mode of the stub CFST columns which was characterised mainly by local buckling, yielding of the steel section and concrete crushing. Therefore for the modelling of the slender composite column, apart from the effect of the constituent of member materials, cross-section shapes and boundary conditions, the slenderness and imperfection of column must be taken into consideration in the FE simulations. 3.2. Finite element model For the research presented in this paper, ABAQUS/standard software package [20] was used to develop a nonlinear FEM to capture the structural behaviour of the slender elliptical CFST columns under axial compression. As described in the previous section, under compressive load the dominant deformation of a
28
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
Fig. 2. Experimental setup and failure mode. (a) Experimental setup, (b) Aftertesting.
Elliptical CFST column and end-pinned arrangement
Cross section
Deformation of elliptical CFST column under compression
Fig. 3. Deformation of a slender elliptical CFST column under compression.(a) Experiment, (b) FE modelling.
slender composite column was characterised by global buckling of the first buckling mode shape as shown in Fig. 3. To model the pinended joint as in the experiment, the steel balls and associated plates were modelled. Fig. 3 shows the structure and FE mesh of the composite column member together with the supports.
Three-dimensional 8-node solid element (C3D8) was adopted for the FE model. Some researchers believe that only shell elements can be used for buckling analysis and this is certainly untrue. Element C3D8 is a general solid element and can be used for displacement, stress/strain analysis as well as for buckling
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
analysis. The composite columns presented in this paper include slender elliptical steel tubes and a concrete core. With the wall thickness of steel tube in the region of 4–5 mm, using either the solid element or the shell element in modelling will give similar and close to identical results. To demonstrate this, a simple example of a thin-walled elliptical steel tube, which is 2.68 m long with major outer diameter of 200 mm, minor outer diameter of 100 mm and wall thickness of 5 mm. The thin-walled steel tube is modelled using one layer of shell elements (SR4) and two layers of three-dimensional 8-node solid elements (C3D8) through the wall thickness of the tube respectively. The material properties with fy ¼350 MPa given in Table 1 are used for the model and the results are shown in Fig. 4. The results show that both models using shell elements and solid elements are able to capture the local and overall buckling effects of the thin-walled tube structure. Fig. 5 shows the comparison of load vs. deformation using the solid and shell elements respectively and the results are seen to be almost indistinguishable. A sensitivity study on mesh sizes to the structural behaviour was performed, in which element sizes ranged from 5 mm to 50 mm were used to select the most effective element size for the FE model. Using large element sizes caused difficulty to ensure simulation accuracy, while using too small element size would consume high amount of computer time and cause convergence difficulty. From the mesh sensitivity study, it was concluded that the appropriate mesh sizes would range from 7.5 to 25 mm depending on the member cross-section dimensions. For instance, for the 150 75 4 mm and 200 100 5 mm EHS sections, the element sizes of 7.5 mm and 10 mm are used respectively. For EHS section 500 250 10 mm, the element size of 25 mm is used. Two layers of element were adopted in the wall thickness to avoid premature failure due to convergence difficulty.
29
The boundary conditions were applied to the rigid plate as reference points. At both ends, the rigid plate was tied to the bearing plate. Contact pairs were defined as surface-to-surface contacts to account for the interaction between steel tube and concrete core, steel ball and the plates. “Hard contact” was assumed for the normal contact behaviour and reopening was allowed. In the tangential direction, the general bond action between concrete core and steel hollow section was simulated through the contact friction. Different friction coefficients ranged from 0.1 to 0.5 in the tangential direction were investigated, however from the sensitivity study, it was found that friction coefficients gave little effect to axial resistance but small friction coefficient is likely to cause convergent problem at large deformation. Therefore a friction coefficient of 0.3 was adopted in this study for the interaction between concrete core and steel hollow section. Theoretically, the friction between steel ball and the plates is expected to be zero without damping; however friction between the ball and plate existed in practice. Based on the sensitivity analysis, a friction coefficient between 0.04 and 0.1 may be suitable to account for the friction between the steel ball and the steel plates. In test, grease was used to reduce the friction between the steel ball and the plates. Different to the stub column tests, the imperfection might give significant effect to the compressive behaviour of a slender member. Due to the difficulty in measuring all the imperfection accurately and to apply all the local imperfections to a structural member in the FE modelling, only the global imperfection (the measured value of initial out of straightness) was applied. Chan and Gardner investigated the effect of imperfection to the buckling behaviour of elliptical steel hollow sections in compression and an imperfection of Le/1000 was suggested in their numerical modelling to obtain consistent results to the current design codes [21]. However, to validate the FE Model, the measured imperfection was used. From the test specimens listed in Table 1, the measured global imperfection values (out of straightness) were in the range from 0.5 to 1.5 mm. A comparative analysis demonstrated such imperfection value was close to the value of Le/2000. Therefore an imperfection value of Le/2000 was assumed in the research presented in this paper. 3.3. Stress–stain model of confined concrete
(b) Shell element for steel tube
(a) Solid element for steel tube
Fig. 4. Both shell and solid element types used in slender steel tube column models.
As described in the previous section, when a slender elliptical CFST column subjected to axial compression, its deformation shape follows column's first buckling mode in the minor axis direction. In such cases, the stress and strain states of the steel tube and concrete core are different from those of the stub elliptical CFST columns in which the section and concrete core were subjected to pure compression during the loading course, however when a slender CFST column under compressive load, both steel tube section and concrete core might experience different stress and strain states
Fig. 5. Effect of element types of steel tube to load-deformation relationships.
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in loading process, and might be partly in compression and partly in tension after the flexural deflection of the column reach certain level. The differences in deformation and stress–strain behaviour of the slender to the stub elliptical CFST columns will result in different confinement effect to the concrete. It has been recognised that the mechanical properties of confined concrete changes depending on the confinement boundary conditions. For CFST columns, the confinement is uniform throughout the contact interaction between the concrete core and steel tube, therefore the hollow section shape and dimensions, material properties might affect the total confinement effect and the confining force distribution along the perimeter of the tube, which leads to different confinement effect to the infill concrete. Han et al. proposed special models for stress–strain relationships of concrete filled in circular and square hollow sections based on experiments and numerical simulations. [9,11] The proposed models were further verified in prediction of the axial compressive behaviour of CFST columns with circular, square and rectangular sections. [22–24] So far there is no special consideration for the confinement of concrete in slender elliptical CFST columns although Dai and Lam developed a confined concrete stress–strain model for elliptical stub CFST columns but it is not suitable for elliptical slender CFST columns [16]. In this paper, the confined concrete stress–strain model developed by Han et al. [9,11] for circular sections is modified and extended its application to slender elliptical CFST columns. Detailed stress–strain model of the confined concrete in circular steel hollow sections were given in these publications [9,11]. In which the basic form is described as following: 8 < 2x x2 ; ðx r 1Þ; y¼ ð1Þ x : β0 ðx 1Þ2 þ x; ðx 4 1Þ; where: x ¼ ε=ε0 ; y ¼ s=s0 ; s0 ¼ f ck ðN=mm2 Þ; ε0 ¼ εc þ 800ξ0:2 10 6 ; εc ¼ ð1300 þ 12:5f ck Þ 10 6 ; 7
β0 ¼ ð2:36 10 5 Þ½0:25 þ ðξ 0:5Þ ðf ck Þ0:5 0:5 Z 0:12 In the above formulae, f ck is the cylinder compressive strength of the concrete, ξ is the so called confinement factor whose expression is described as following: ξ¼
As f y Ac f ck
ð2Þ
where As and Ac are the cross-sectional areas of the steel tube and concrete core. f y is the yield strength of steel and f ck is the characteristic strength of concrete and f ck ¼0:8f cu , f cu is the cube compressive strength of unconfined concrete. The Modulus of pffiffiffiffiffiffi Elasticity of the confined concrete is described as Ec ¼ 4700 f ck MPa and the Poisson ratio of the confined concrete was taken as 0.2. As mentioned previously, the concrete core might be partly in tension during loading therefore the tensile stress–strain model for concrete must be introduced in the numerical simulations. In the absence of special test data, the concrete tensile stress and corresponding strain might be taken as 10% of the concrete corresponding compressive stress and strain. This method was adopted to form the tensile stress–strain model of concrete for the numerical simulations presented in this paper.
Table 2 Comparison of axial compressive loads by experiments and numerical predictions. Specimen ID
N EXP (kN)
N FE (kN)
NFE N EXP
TG1-D150-C30 TG1-D150-C60 TG1-D150-C100 TG1-D200-C30 TG1-D200-C60 TG1-D200-C100 TG2-D150-C30 TG2-D150-C60 TG2-D150-C100 TG2-D200-C30 TG2-D200-C60 TG2-D200-C100 TG3-D150-C30 TG3-D150-C60 TG3-D150-C100 TG3-D200-C30 TG3-D200-C60 TG3-D200-C100
651 743 923 938 1065 1480 484 663 871 968 1237 1411 327 427 547 839 947 1073
595 712 999 1041 1213 1455 487 731 843 1015 1256 1409 301 432 498 814 949 1083
0.914 0.958 1.082 1.110 1.139 0.983 1.006 1.103 0.968 1.049 1.015 0.999 0.920 1.012 0.910 0.970 1.002 1.009
Average COV
1.008 0.065
3.4. Validation of the FE model The accuracy of the FE model was verified through comparing the relationship and failure modes predicted by the numerical method described in the previous sections against the experimental results. In numerical modelling, the measured dimensions of tested specimens and measured material properties were adopted. The confined concrete compressive stress–strain model is in accordance to Eqs. (1) and (2) and using the measured unconfined concrete parameters given in Table 1. The confined concrete tensile stress–strain model was taken as 10% of its compressive stress and strain values. Due to the measured initial out of straightness for some specimens was not available; a monotonic global imperfection value of Le/2000 was adopted. Table 2 compares the maximum axial compressive load of slender composite columns predicted by FE model (N FE ) with that test results (N EXP ) and the good agreement are demonstrated by the ratio of NFE =N EXP being close to unity. The difference between the prediction and measured value was possibly resulted from the difference between the material property and imperfection adopted in the FE model. Fig. 6 shows the typical axial compressive load vs. mid-height lateral deflection curves observed by the experiments and the FEM predictions. Both the experiment and numerical results showed that the failure modes of slender composite columns were characterised by global buckling in the minor axis direction although in some cases, local buckling was also observed on the compressive side of the column at midheight. For the very slender elliptical CFST columns, the mode of failure is dominated by global buckling.
4. Parametric study 4.1. Axial compressive behaviour There are many factors that might affect the axial compressive behaviour of a slender CFST column. These factors include the members' non-dimensional slenderness, material properties, the sectional shape, dimensions and column lengths. After the verification of the FE model, parametric study was performed using the measured material properties and geometries, a detailed parametric study was conducted to assess the axial compressive
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
31
Fig. 6. Comparison of axial compressive load to mid-span deflection curves predicted by FE modelling against experimental recordings.
behaviour of slender elliptical CFST columns with typical the hollow section dimensions from 150 75 4 EHS up to 500 250 10 EHS and member non-dimensional slenderness range from 0.25 up to 2.93. The following section presented the axial buckling loads of concrete filled composite columns with different elliptical steel hollow sections and slenderness. In total; 64 elliptical CFST columns with five section sizes range from 150 75 4 to 500 250 10; six column lengths from 1.5 to 4.5 m and three concrete grades as shown in Table 3 were adopted for the parametric study. The measured steel properties taken from section 200 100 5 EHS were used. For infill concrete, the unconfined cube strengths of 30 MPa, 60 MPa and 100 MPa were assumed for concrete with nominal grades C30, C60 and C100 respectively. Modulus of Elasticity of 23.7, 34.9 and 43.3 GPa was used for the respective model. Due to absence of imperfection value, a global imperfection of Le/2000 of the first buckling mode shape was assumed and applied for the modelling. In addition the confined concrete stress–strain model was derived in accordance to Eqs. (1) and (2) where the cylinder strength of unconfined concrete was taken as 85% of the cube strength of the unconfined concrete. Table 3 gives the maximum axial
compressive load N FE of for all selected elliptical CFST columns. Using the design rules provided in Eurocode 4 for circular and rectangular CFST columns with an equivalent diameter D ¼2a2/b for elliptical section, important buckling load parameters, N cr , Npl;Rd , λ and χ, were calculated and showed in Table 3, where buckling load reduction factor, χ is taken asN FE /N pl;Rd . Fig. 7 compares the predicted buckling reduction factors, χ vs. the non-dimensional slenderness, λ taken from Table 3. Eurocode 4 proposed that the buckling curve (a) to be used for circular and rectangular CFST columns. It can be found from Fig. 7, the predicted reduction factors χ for most elliptical CFST columns lie above or close to the curve (a) except for the selected columns with non-dimensional slenderness λ is less than 0.4, therefore the buckling curve (a) might be safely used for the design of elliptical CFST columns with global imperfection not exceeding Le/2000. 4.2. Effect of column imperfection to axial compressive behaviour The member imperfection may be different for individual members due to the fabrication tolerance, transport method and design allowance. It has been recognised that column imperfection
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Table 3 Summary of selected specimen dimensions and axial compressive behaviour. Specimen ID
2a 2b t (mm)
Le (mm)
f y (MPa)
f cu (MPa)
N FE (kN)
N cr (kN)
N pl;Rd (kN)
λ
χ
PG1-D150-C30 PG1-D150-C60 PG1-D150-C100 PG1-D200-C30 PG1-D200-C60 PG1-D200-C100 PG1-D300-C30 PG1-D300-C60 PG1-D300-C100 PG1-D400-C30 PG1-D400-C60 PG1-D500-C30 PG1-D500-C60 PG2-D150-C30 PG2-D150-C60 PG2-D150-C100 PG2-D200-C30 PG2-D200-C60 PG2-D200-C100 PG3-D150-C30 PG3-D150-C60 PG3-D150-C100 PG3-D200-C30 PG3-D200-C60 PG3-D200-C100 PG3-D300-C30 PG3-D300-C60 PG3-D300-C100 PG3-D400-C30 PG3-D400-C60 PG3-D500-C30 PG3-D500-C60 PG4-D150-C30 PG4-D150-C60 PG4-D150-C100 PG4-D200-C30 PG4-D200-C60 PG4-D200-C100 PG5-D150-C30 PG5-D150-C60 PG5-D150-C100 PG5-D200-C30 PG5-D200-C60 PG5-D200-C100 PG5-D300-C30 PG5-D300-C60 PG5-D300-C100 PG5-D400-C30 PG5-D400-C60 PG5-D500-C30 PG5-D500-C60 PG6-D150-C30 PG6-D150-C60 PG6-D150-C100 PG6-D200-C30 PG6-D200-C60 PG6-D200-C100 PG6-D300-C30 PG6-D300-C60 PG6-D300-C100 PG6-D400-C30 PG6-D400-C60 PG6-D500-C30 PG6-D500-C60
150 75 4 150 75 4 150 75 4 200 100 5 200 100 5 200 100 5 300 150 8 300 150 8 300 150 8 400 200 8 400 200 8 500 250 10 500 250 10 150 75 4 150 75 4 150 75 4 200 100 5 200 100 5 200 100 5 150 75 4 150 75 4 150 75 4 200 100 5 200 100 5 200 100 5 300 150 8 300 150 8 300 150 8 400 200 8 400 200 8 500 250 10 500 250 10 150 75 4 150x75 4 150x75 4 200x100 5 200x100 5 200x100 5 150 75 4 150 75 4 150 75 4 200 100 5 200 100 5 200 100 5 300 150 8 300 150 8 300 150 8 400 200 8 400 200 8 500 250 10 500 250 10 150 75 4 150 75 4 150 75 4 200 100 5 200 100 5 200 100 5 300 150 8 300 150 8 300 150 8 400 200 8 400 200 8 500 250 10 500 250 10
1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1800 1800 1800 1800 1800 1800 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 3000 3000 3000 3000 3000 3000 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500
410 410 410 360 360 360 360 360 360 360 360 360 360 410 410 410 360 360 360 410 410 410 360 360 360 360 360 360 360 360 360 360 410 410 410 360 360 360 410 410 410 360 360 360 360 360 360 360 360 360 360 410 410 410 360 360 360 360 360 360 360 360 360 360
30 65 100 30 65 100 30 65 100 30 65 30 65 30 65 100 30 65 100 30 65 100 30 65 100 30 65 100 30 65 30 65 30 65 100 30 65 100 30 65 100 30 65 100 30 65 100 30 65 30 65 30 65 100 30 65 100 30 65 100 30 65 30 65
665 793 901 1066 1355 1608 2584 3304 3997 4035 5389 6311 8641 570 657 742 1038 1301 1530 343 387 428 880 1039 1191 2471 3091 3605 3894 5066 6070 8176 243 276 301 696 802 907 180 205 220 534 620 688 2200 2609 2989 3738 4763 5885 7791 111 125 133 335 385 416 1625 1852 2078 3373 4120 5618 7335
1041 1107 1145 3197 3411 3537 16,932 17,991 18,609 49,645 49,646 106,868 115,922 723 769 795 2220 2369 2456 375 398 412 1151 1228 1273 6096 6477 6699 17,872 17,872 38,472 41,732 260 277 286 799 853 884 191 203 210 587 627 650 3110 3305 3418 8040 8721 19,629 21,292 118 125 129 355 379 393 1881 1999 2068 5516 5516 11,874 12,880
750 972 1194 1140 1539 1939 2660 3520 4439 4235 5734 6859 9238 750 972 1194 1140 1539 1939 750 972 1194 1140 1539 1939 2671 3560 4449 3948 5635 6198 8722 750 972 1194 1140 1539 1939 750 972 1194 1140 1539 1939 2671 3560 4449 3984 5635 6224 8804 668 890 1112 1140 1539 1939 2671 3560 4449 3984 5635 6224 8804
0.849 0.937 1.021 0.597 0.672 0.740 0.396 0.442 0.488 0.292 0.340 0.253 0.282 1.018 1.125 1.225 0.717 0.806 0.888 1.415 1.562 1.702 0.995 1.120 1.234 0.662 0.741 0.815 0.470 0.561 0.401 0.457 1.697 1.874 2.042 1.194 1.343 1.481 1.980 2.187 2.383 1.393 1.567 1.728 0.927 1.038 1.141 0.704 0.804 0.563 0.643 2.383 2.669 2.934 1.791 2.015 2.221 1.192 1.335 1.467 0.850 1.011 0.724 0.827
0.887 0.816 0.755 0.935 0.880 0.829 0.971 0.939 0.900 0.953 0.940 0.920 0.935 0.760 0.676 0.622 0.911 0.845 0.789 0.457 0.399 0.359 0.772 0.675 0.614 0.925 0.868 0.810 0.986 0.899 0.979 0.937 0.324 0.284 0.252 0.611 0.521 0.468 0.240 0.211 0.184 0.469 0.403 0.355 0.824 0.733 0.672 0.938 0.845 0.946 0.885 0.166 0.140 0.120 0.294 0.250 0.215 0.608 0.520 0.467 0.847 0.731 0.903 0.833
might give significant influence to the axial compressive behaviour. Therefore a sensitivity study using numerical modelling was conducted to investigate the effect of global imperfection to the maximum axial compressive load and buckling behaviour of elliptical CFST columns. Various initial global imperfections range from Le/8000 to Le/250 mm was used. The range represents the possible initial global imperfection of composite columns,
including initial out of straightness in fabrication due to fabrication. Fig. 8 compares the predicted buckling reduction factors of selected elliptical CFST columns with 5 different global imperfections against to the design buckling curves provided in Eurocode 3 for hollow steel members and CFST members with circular and rectangular sections [25]. It can be seen from Fig. 8, if the imperfection of Le/1000 is used, only the predicted buckling
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
Fig. 7. Comparison of predicted buckling reduction factors of elliptical CFST columns with initial global imperfection of Le/2000.
Reduction factor: χ
1.2 1 0.8
a0
a
b
c
d
IMP: Le/1000
0.6 0.4 0.2 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Non-dimensional slenderness: λ
Reduction factor: χ
1.2 1 0.8
a0 b d IMP: Le/4000
a c IMP: Le/2000
0.6 0.4 0.2 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Non-dimensional slenderness: λ
Reduction factor: χ
1.2 1 0.8
a0 b d IMP: Le/250
a c IMP: Le/500
0.6 0.4 0.2 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Non-dimensional slenderness: λ
Fig. 8. Comparison of buckling behaviour of elliptical CFST columns with various initial global imperfections to design buckling curves provided in EC3.
to Le/2000, then most selected elliptical CFST columns will lie above the design curve (a). From the tests, an imperfection of Le/2000 was normally observed in most specimens as discussed earlier, the predicted reduction factors of selected columns with various imperfections was compared against the predicted reduction factor of column with an imperfection of Le/2000 as given in Table 4, the difference can clearly be observed. Obviously, with the imperfection increasing from Le/2000 to Le/250, the average axial compressive capacity is reduced by up to 19% on average. If the imperfection is increased from Le/1000 to Le/250; the average axial compressive capacity is reduced by up to 15% on average. Similar conclusion was observed by Chan and Gardner in their research on elliptical steel hollow section members [20]. However, if the imperfection is reduced from Le/2000 to Le/8000, the average compressive capacity is only increased by 4%. Base on the parametric study and sensitivity analysis, it is concluded that the imperfection of Le/2000 should be used for the FE modelling.
5. Design to Eurocode 4 Currently the Eurocode 4 does not covered CFST columns with elliptical hollow sections although simplified design rules for CFST columns with circular, square and rectangular sections are available. In accordance to Eurocode 4, the design axial compressive load of a circular CFST composite column is determined from the plastic resistance and buckling behaviour by introducing partial factor and imperfection characteristics according to the structural features of the member itself. According to the comparison and analysis presented in the previous sections, the design rules provided in Eurocode 4 for CFST columns with circular sections appear to be unsuitable for the elliptical CFST columns with low slenderness. This section gives further review to the application of the design method for the elliptical CFST columns. Unlike the stub CFST columns whose compressive capacity and failure mode are governed by member yielding and local buckling, the design of a slender composite column is generally dominated by its global buckling behaviour. Therefore relevant design buckling curves of non-dimensional slenderness λ to buckling load reduction factor χ relationships are adopted for the design. The non-dimensional slenderness λ is expressed as: sffiffiffiffiffiffiffiffiffiffiffiffi N pl;Rd λ¼ ð3Þ Ncr Npl;Rd is the plastic resistance of a composite column from crosssection strength, it may be calculated by adding the plastic resistances of its components. For CFST columns without reinforcement, N pl;Rd is described as: Npl;Rd ¼ Ac f ck þ As f y
ð4Þ
where Ac is the cross-sectional area of the concrete core, f ck is the unconfined compressive concrete cylinder strength, As is the crosssectional area of the steel tube, f y is the yield strength of the steel. Ncr is the elastic buckling resistance of the composite column which is determined by: Ncr ¼
π 2 ðEIÞef f L2e
ð5Þ
In which Le is the effective length and ðEIÞef f is the effective sectional stiffness of the composite column which is calculated by: ðEIÞef f ¼ Es I s þ ke Ecm I c
reduction factors; χ with λ higher than 0.8 lie above the design curve (a). If the imperfection is increased to Le/500, the predicted buckling reduction factors χ of most of the selected columns lie below the design curve (a). However, if the imperfection is reduced
33
ð6Þ
where Es and I s are the elastic modulus and the second area moment of the steel tube section, Ecm andI c are the elastic modulus and the second area moment of the uncrushed concrete core, ke is a coefficient which is taken as 0.6.
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Table 4 Comparison of buckling reduction factors for elliptical CFST columns with different global imperfections.
Specimen ID
PG1-D150-C30 PG1-D150-C60 PG1-D150-C100 PG1-D200-C30 PG1-D200-C60 PG1-D200-C100 PG1-D300-C30 PG1-D300-C60 PG1-D300-C100 PG1-D400-C30 PG1-D400-C60 PG1-D500-C30 PG1-D500-C60 PG2-D150-C30 PG2-D150-C60 PG2-D150-C100 PG2-D200-C30 PG2-D200-C60 PG2-D200-C100 PG3-D150-C30 PG3-D150-C60 PG3-D150-C100 PG3-D200-C30 PG3-D200-C60 PG3-D200-C100 PG3-D300-C30 PG3-D300-C60 PG3-D300-C100 PG3-D400-C30 PG3-D400-C60 PG3-D500-C30 PG3-D500-C60 PG4-D150-C30 PG4-D150-C60 PG4-D150-C100 PG4-D200-C30 PG4-D200-C60 PG4-D200-C100 PG5-D150-C30 PG5-D150-C60 PG5-D150-C100 PG5-D200-C30 PG5-D200-C60 PG5-D200-C100 PG5-D300-C30 PG5-D300-C60 PG5-D300-C100 PG5-D400-C30 PG5-D400-C60 PG5-D500-C30 PG5-D500-C60 PG6-D150-C30 PG6-D150-C60 PG6-D150-C100 PG6-D200-C30 PG6-D200-C60 PG6-D200-C100 PG6-D300-C30 PG6-D300-C60 PG6-D300-C100 PG6-D400-C30 PG6-D400-C60 PG6-D500-C30 PG6-D500-C60 Average COV
Geometrical and material properties λ
As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As As
in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in
Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
0.849 0.937 1.021 0.597 0.672 0.740 0.396 0.442 0.488 0.292 0.340 0.253 0.282 1.018 1.125 1.225 0.717 0.806 0.888 1.415 1.562 1.702 0.995 1.120 1.234 0.662 0.741 0.815 0.470 0.561 0.401 0.457 1.697 1.874 2.042 1.194 1.343 1.481 1.980 2.187 2.383 1.393 1.567 1.728 0.927 1.038 1.141 0.704 0.804 0.563 0.643 2.383 2.669 2.934 1.791 2.015 2.221 1.192 1.335 1.467 0.850 1.011 0.724 0.827
Ratio of reduction factor χ of columns with different imperfections to χ of column with imperfection Le/2000 Le/8000
Le/4000
Le/2000
Le/1000
Le/500
Le/250
1.041 1.040 1.040 1.041 1.041 1.041 1.041 1.040 1.041 1.040 1.040 1.040 1.041 1.041 1.041 1.040 1.041 1.041 1.041 1.042 1.040 1.039 1.041 1.040 1.041 1.041 1.040 1.041 1.041 1.041 1.041 1.041 1.040 1.039 1.040 1.041 1.040 1.041 1.038 1.043 1.043 1.041 1.040 1.039 1.040 1.040 1.040 1.041 1.041 1.040 1.041 1.042 1.043 1.033 1.041 1.040 1.037 1.041 1.040 1.041 1.040 1.041 1.040 1.060
1.028 1.028 1.028 1.029 1.028 1.028 1.029 1.028 1.029 1.027 1.028 1.028 1.028 1.028 1.028 1.027 1.027 1.028 1.028 1.028 1.028 1.028 1.028 1.028 1.029 1.028 1.028 1.028 1.028 1.028 1.029 1.029 1.028 1.028 1.028 1.028 1.027 1.028 1.025 1.028 1.033 1.028 1.027 1.028 1.028 1.027 1.027 1.029 1.028 1.027 1.028 1.030 1.029 1.025 1.027 1.028 1.028 1.028 1.029 1.028 1.027 1.029 1.025 1.044
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.957 0.958 0.958 0.958 0.958 0.958 0.958 0.957 0.958 0.957 0.957 0.958 0.958 0.958 0.959 0.957 0.957 0.959 0.958 0.958 0.957 0.958 0.959 0.957 0.958 0.958 0.957 0.958 0.958 0.958 0.958 0.958 0.957 0.958 0.956 0.957 0.958 0.957 0.958 0.957 0.962 0.957 0.958 0.958 0.958 0.958 0.957 0.958 0.959 0.958 0.958 0.958 0.964 0.958 0.959 0.960 0.958 0.959 0.958 0.957 0.957 0.958 0.958 0.956
0.896 0.896 0.895 0.896 0.897 0.896 0.897 0.896 0.897 0.896 0.896 0.897 0.896 0.896 0.896 0.895 0.896 0.897 0.896 0.897 0.895 0.894 0.896 0.896 0.897 0.896 0.896 0.896 0.897 0.897 0.897 0.896 0.895 0.894 0.893 0.895 0.896 0.895 0.896 0.896 0.897 0.896 0.896 0.896 0.896 0.896 0.896 0.897 0.896 0.895 0.896 0.898 0.900 0.892 0.895 0.896 0.893 0.896 0.896 0.897 0.896 0.896 0.891 0.885
0.804 0.804 0.804 0.804 0.805 0.805 0.804 0.804 0.804 0.804 0.804 0.804 0.804 0.804 0.805 0.804 0.805 0.805 0.805 0.805 0.805 0.805 0.804 0.804 0.805 0.804 0.804 0.805 0.804 0.804 0.805 0.805 0.802 0.803 0.802 0.804 0.804 0.803 0.804 0.806 0.804 0.804 0.804 0.806 0.805 0.804 0.804 0.805 0.805 0.803 0.805 0.807 0.807 0.800 0.803 0.804 0.805 0.804 0.804 0.805 0.804 0.804 0.795 0.780
1.041 0.003
1.028 0.002
1.000 0.000
0.958 0.001
0.896 0.002
0.804 0.003
For CFST column with circular sections, if λ from Eq. (3) does not exceed 0.5 and e=D o 0:1, where e is the eccentricity of the loading given by section moment divided by the axial load, D is the
outer diameter of the circular steel tube, an expression given in Eq. (7) may be used to consider the effect of concrete confinement. For elliptical CFST columns, an equivalent diameter, De ¼2a2/b is
X.H. Dai et al. / Thin-Walled Structures 77 (2014) 26–35
suggested to replace the outer diameter of the circular column which gave a reasonable estimation of the axial capacity of the elliptical CFST columns. fy t N pl;Rd ¼ As ηs f y þ Ac f ck 1 þ ηc ð7Þ D f ck whereAc , f ck , As and f y areas specified as in Eq. (4), t is the thickness of the hollow section respectively, ηs and ηc are functions of the non-dimensional slenderness λ of the composite column and are determined by: ηs ¼ 0:25ð3 þ2λÞ but with ηs r1:0 2
ηc ¼ 4:9 18:5λ þ 17λ
but with ηc Z 0
35
Eurocode 4 may be used for axial compressive behaviour design of elliptical CFST columns. (3) The initial imperfection has a significant effect to the axial compressive behaviour of slender elliptical CFST columns. Initial imperfection of Le/2000 should be used for the FE modelling. (4) From the experimental observation and subsequent parametric study presented in the paper, it would appear that the design buckling curve (a) provided in the Eurocode 3 for circular and rectangular CFST columns is suitable for the design of slender elliptical CFST columns.
ð8aÞ ð8bÞ
Acknowledgement
When calculate the design load NEd of a composite member in compression, the member imperfection and buckling effects should be taken into consideration if λ Z 0:2. The design load N Ed should satisfy:
The research reported in this paper is partly funded by the research grant from the Engineering and Physical Science Research Council (EP/G002126/1) in the UK who are gratefully acknowledged.
N Ed r 1:0 χN pl;Rd
References
ð9Þ
N pl;Rd is the plastic resistance given by Eqs. (4) and (7) but in which f y should be replaced by design strength f yd if the partial factor γ M1 is not taken as unity. χ is the reduction factor for the relevant buckling mode given in Eurocode 3 in terms of the relevant nondimensional slenderness λ and the equivalent buckling curve. In Eurocode for members in axial compression the design value of χ for the appropriate λ is determined from the relevant buckling curves which can be expressed by: χ¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi but with χ r1:0 Φ þ Φ2 λ2
ð10Þ
2
where Φ ¼ 0:5½1 þ αðλ 0:2Þ þ λ , λ given by Eq. (3). α is the imperfection factor whose value changes according to the member characteristics. For the design of the composite members with λ r 0:2, the global buckling effect may be ignored and therefore χ may be taken as 1.0. The above design method currently given in Eurocode 4 for CFST columns for circular and rectangular sections suggested that an initial imperfection of Le/300 is used. However based on the comparison and analysis given in the previous sections, it would appear that the imperfection of Le/300 is not suitable for modelling the elliptical CFST columns. If the imperfection of Le/300 were used for FE modelling, the results would be over-conservative.
6. Conclusions This paper provided a non-linear FE model developed through ABAQUS/Standard solver to investigate the axial compressive behaviour of slender elliptical CFST columns. After the numerical method being verified by comparing the predicted results against the experimental observations, an extensive parametric study was presented. Based on the comparison and analysis, the following conclusions can be made: (1) The FE numerical method developed via ABAQUS/Standard solver, combined with the special stress–strain model for confined concrete in slender elliptical hollow sections could be used to predict the axial compressive behaviour of slender elliptical CFST columns. (2) Both experimental and numerical modelling results demonstrated that the failure mode of a slender CFST column under compressive loads was global buckling with the first buckling mode shape. The simplified design method provided in
[1] Zhang S, Guo L, Ye Z, Wang Y. Behavior of steel tube and confined high strength concrete for concrete-filled rhs tubes. Adv Struct Eng 2005;8(2):101–16. [2] Zhang S, Guo L. Behaviour of high strength concrete-filled slender rhs steel tubes. Adv Struct Eng 2007;10(4):337–51. [3] Schneider SP. Axially loaded concrete filled steel tubes. ASCE J Struct Eng 1998;124(10):1125–38. [4] O’Shea MD, Bridge RQ. Design of circular thin-walled concrete filled steel tubes. ASCE J Struct Eng 2000;126(11):1295–303. [5] Giakoumelis G, Lam D. Axial capacity of circular concrete filled tube columns. J Constr Steel Res 2004;60(7):1049–68. [6] Han LH, Liu W, Yang YF. Behavior of thin walled steel tube confined concrete stub columns subjected to axial local compression. Thin-Walled Struct 2008;46(2):155–64. [7] Saw HS, Liew JY. Assessment of current methods for the design of composite columns in buildings. J Constr Steel Res 2000;53(2):121–47. [8] Hu HT, Huang CS, Wu MH, Wu YM. Nonlinear analysis of axially loaded concrete-filled tube columns with confinement effect. ASCE J Struct Eng 2003;129(10):pp1322–9. [9] Han LH, Yao GH, Zhao XL. Tests and calculations of hollow structural steel (HSS) stub columns filled with self-consolidating concrete (SCC). J Constr Steel Res 2005;61(9):1241–69. [10] Ellobody E, Young B, Lam D. Behaviour of normal and high strength concrete-filled compact steel tube circular stub columns. J Constr Steel Res 2006;62(7):706–15. [11] Han LH, Yao GH, Tao Z. Performance of concrete-filled thin-walled steel tubes under pure torsion. Thin-Walled Struct 2007;45(1):24–36. [12] Dabaon M, El-Khoriby S, El-Boghdadi M, Hassanein MF. Confinement effect of stiffened and unstiffened concrete-filled stainless steel tubular stub columns. J Constr Steel Res 2009;65(8-9):1846–54. [13] Lam D, Gardner L. Structural design of stainless steel concrete filled columns. J Constr Steel Res 2008;64(11):1275–82. [14] Yang H, Lam D, Gardner L. Testing and analysis of concrete-filled elliptical hollow sections. Eng Struct 2008;30(12):3771–81. [15] Zhao XL, Packer JA. Tests and design of concrete-filled elliptical hollow section stub columns. Thin-Walled Struct 2009;47(6-7):617–28. [16] Dai X, Lam D. Numerical modelling of axial compressive behaviour of short concrete-filled elliptical steel tube columns. J Steel Constr Res 2010;66(7): pp931–pp942. [17] BS EN 1994-1-1: Eurocode 4. Design of composite steel and concrete structures Part 1-1: General rules and rules for buildings. London: British Standards Institution; 2004. [18] Jamaluddin N. Behaviour of concrete filled elliptical hollow sections [Ph.D. thesis]. Leeds, UK: University of Leeds; 2011. [19] Jamaluddin N, Lam D, Dai XH, Ye J. An experimental study on elliptical concrete filled columns under axial compression. J Constr Steel Res 2013;87:6–16. [20] ABAQUS Standard User's Manual. Hibbitt, Karlsson and Sorensen, Inc. Version 6.12, USA; 2011. [21] Chan TM, Gardner L. Flexural buckling of elliptical hollow section columns. ASCE J Struct Eng 2009;135(5):546–57. [22] Lu H, Han LH, Zhao XL. Analytical behavior of circular concrete-filled thin-walled steel tubes subjected to bending. Thin-Walled Struct 2009;47(3):346–58. [23] Tao Z, Uy B, Han LH, Wang ZB. Analysis and design of concrete-filled stiffened thin-walled steel tubular columns under axial compression. Thin-Walled Struct 2009;47(12):1544–56. [24] Huang H, Han LH, Tao Z, Zhao XL. Analytical behaviour of concrete-filled double skin steel tubular (CFDST) stub columns. J Constr Steel Res 2010;66(4): 542–55. [25] BS EN 1993-1-1: Eurocode 3. Design of steel structures Part 1-1: General rules and rules for buildings. London: British Standards Institution; 2005.