Keywords

1 Introduction

Concrete filled double-skin steel tubular (CFDST) columns with steel inner and outer tubes belong to a category of modern composite columns in which concrete is filled in the space between the two tubes. CFDST columns are optimized to offer benefits and advantages similar to steel–concrete composite construction by taking advantage of strengths of both concrete and steel. The steel tubes act as longitudinal reinforcement and permanent formwork in these columns and also provide potential confinement for enhancing the structural behavior of the concrete infill. The composite action of steel and concrete leads to improved structural characteristics, including potential higher axial-load capacity and ductility, reduced self-weight and cost, better fire resistance and seismic behavior, cost, and space efficiency compared to conventional steel or reinforced concrete columns. CFDST columns find potential applications in onshore and offshore structures to increase buoyancy, medium and high-rise structures, bridge piers, large span buildings for extreme dynamic, and fire loading scenarios [1,2,3,4]. The outer and inner tubes can be provided with multiple cross-sectional shapes such as circular (C) or square (S) tubes. Thus, resulting in multiple configurations of cross sections shown in Fig. 1 such as outer-circle and inner-circle (CC), outer-circle and inner-square (CS), outer-square and inner-circle (SC), and outer square and inner-square (SS).

Fig. 1
figure 1

CFDST column cross sections; CC = circle-circle; CS = circle-square; SC = square-circle; SS = square-square

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Each of these column cross-sectional shapes is expected to result in different axial load compression behavior and ultimate load carrying capacity because of varying degree of concrete confinement provided by outer and inner tubes [5, 6]. Different experimental research studies [7,8,9,10,11,12,13,14,15,16,17] investigated behavior of CFDST columns under axial compression. These research studies focused on individual cross-sectional shapes to study behavior of CFDST columns under axial compressive loading. Other studies [18,19,20,21,22,23,24,25,26,27] validated experimentally observed behavior of CFDST columns using finite element (FE) modeling, and extended studies through parametric studies using validated numerical models for each of the individual cross-sectional shapes. In these research efforts, it is established that increase in material strengths (either of steel tubes or concrete infill) or geometric characteristics (such as thickness of steel tubes or ratio of steel to concrete area in a cross section) enhances column axial load capacity. Whereas, other factors such as slenderness ratios of steel tubes and ratio of inner-to-outer tube diameters have negative influence on CFDST column axial load capacity. The comparative behavior of the CFDST columns with multiple shapes and having equivalent cross-sectional area is not yet available in the present literature.The relative influence of inner and outer tube shapes on the confined concrete behavior is scantly investigated either experimentally or analytically. The outer tube is majorly providing confinement for the sandwiched concrete resulting in enhanced performance than the inner steel tube [1]. The cross-sectional shape of the outer tube can result in varying confinement in the concrete core in the cross section. Such as in SS shape CFDST column, the confinement to concrete is expected to vary from mid-flat sides from a minimum to the maximum at corners compared to more uniform case in a CC shape. Such nonuniform confinements in SS shaped cross section can trigger mid-flat side bucking of the inner steel tube toward the inner void compared to absent or reduced local buckling of CC shapes. Thus, study of comparative behavior of CFDST columns of multiple cross-sectional shapes needs to be investigated to compare their relative structural performances when different shapes are provided with equivalent cross-sectional areas.

The previous research studies, both experimental and numerical, were limited to behavior of CFDST columns with individual cross-sectional shapes. General guidance regarding influence of cross-sectional shape on compressive behavior and nonlinear load carrying capacity of CFDST columns is scant in existing literature and standard design specifications such as International Building Code. To the authors’ knowledge, no major comparative study on the performance of multiple-shapes of CFDST columns has been carried out to investigate and compare their relative performance when all shapes are provided with equivalent cross-sectional areas. The present study builds on the previous research efforts with an objective to perform comparative study on the axial compression behavior of CFDST columns of multiple cross-sectional shapes (recall Fig. 1). The influence on ultimate load carrying capacity was investigated using nonlinear finite element analysis (NLFEA) for four different cross-sectional shapes (i.e., CC, CS, SC, and SS). For this parametric study, all the column shapes were provided with same equivalent cross-sectional areas of inner and outer steel tubes and concrete infill. The accuracy and reliability of nonlinear FE models were first validated against experimental results of 40-documented CFDST column tests (16 specimens for CC and 8 each from CS, SC, and SS shapes) available in existing literature.

2 Finite Element Modeling

The finite element (FE) analysis approach was used in the present study to generate analytical 3D-models of CFDST columns in ANSYS 19.0 [28]. The selection of elements, simulation of boundary conditions and load application, and idealization of contact interfaces between different material interfaces was automated and executed using ANSYS parametric design language (APDL), similar to research studies [29,30,31,32]. Thus, a customized FE modeling framework was developed using APDL for generation of numerical models incorporating structural and geometrical characteristics such as mapped meshing in cross sections, interaction between different structural components, constitutive material modeling, boundary conditions, and load application (Fig. 2). Different cross-sectional shapes (i.e., CC, CS, SC, and SS) considered in this study were provided with mapped meshing across the column components (Fig. 2a). Inner and outer steel tubes and end loading plates were idealized with 4-node shell elements (SHELL181) and concrete infill was simulated using 8-node solid elements (SOLID185). The individual meshed column components are shown in Fig. 2b. The interaction between steel tubes and concrete infill both on inner and outer surfaces was modeled with 2-node rigid beam elements, i.e., kinematic multipoint constraints (MPC184) simulated by zero length elements. Rigid beam MPCs related the shell element rotations for outer and inner tubes to the solid element translations in the concrete core at the concrete-steel interface. The rigid beams thus generated kinematic constraint equations for relating translational DOFs from solid elements to shell element rotational DOFs at the hybrid material interface. The analysis results from MPC interface modeling were compared to using nonlinear contacts, and the analysis results showed insignificant difference. However, the advantage of using rigid beam MPCs stabilized the analysis by avoiding generation of nonlinear contact definition forces at the interface in the numerical models and was thus adopted in this study.

Fig. 2
figure 2

Finite element modeling: a meshing in cross sections; b assembled components of CFDST columns; CC = circle-circle; CS = circle-square; SC = square-circle; SS = square-square

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Surface-to-surface 4-node contact (CONTA173) and target elements (TARGE170) simulated interaction of steel end plates with concrete infill in the cross section. Steel end plates were set as contact surface and concrete infill in the column end cross sections as target for surface-to-surface contacts. The interaction between inner and outer steel tubes with column end plates was idealized by node-to-surface contacts. The edge nodes of steel tubes were set as contact nodes using single node elements (CONTA171) and the column end plates as target surface similar to previously simulated surface-surface contact definition.

FE mesh sizes in the column cross section and in longitudinal direction were seen sensitive to analysis results. Element size of less than 15 mm in cross section and 20 mm in the longitudinal directions was sufficient for obtaining consistently converged results with desired accuracy in terms of axial load–displacement curves. Meshes in the end plates were made slightly coarse with an element size of 15–20 mm for their simulated linear-elastic response.

The boundary conditions have significant influence on column behavior and its load carrying capacities. In this study, all the numerical models were provided with simply supported boundary conditions. The end-plate nodes on both the top and bottom sides of the column were restrained against translations in the plane of cross section, i.e., x- and y-global coordinate directions. Whereas, the translation along the longitudinal direction of column, i.e., z-direction was restrained only for bottom end plate at a single node nearest to the centroid of the column. The top-end plate was free to move in z-coordinate direction. The axial compression was simulated using z-direction prescribed displacement loading of top-end plate. All nodes of the top steel plate were given a displacement along the longitudinal direction of the column equal to 1/6th of the column height. The details of constitutive material modeling are discussed in the following subsections.

2.1 Constitutive Material Modeling

There are two different materials used in CFDST columns, i.e., steel and concrete. Extensive research studies are available in literature for constitutive material models, particularly for simulating behavior of confined concrete in between the two steel tubes. In this study, concrete material plasticity was idealized with multilinear kinematic hardening rule using microplane failure theory [33] in which uniaxial stress–strain curves of concrete are applied on each microplane. Kinematic hardening rule for plastic flow in concrete material was followed considering the expected decreasing trend of the post-peak axial load–deflection behavior, similar to experimentally observed results used for validation of numerical models. At macroscopic level, strain tensors are kinematically related to their microplane counterparts. The steel components used an isotropic-hardening plasticity rule with Von Mises yield criteria. A brief summary of the constitute material models used in this study for concrete and steel based on literature is provided here.

2.1.1 Concrete

In this study, two different nonlinear stress–strain models for confined concrete shown in Fig. 3a, b were used for concrete infill to capture its varying degrees of confinement level depending on DSTC column cross-section shape. Confined stress–strain curve given in study [20] was used in CC and CS numerical models and is referred hereafter as Pagoulatou model. Whereas, stress–strain curve for confined concrete from research study [34] was used in SS shape and is referred as Zhao model in this paper. For columns with SC shape, a modified version of Zhao model incorporating modifications from study [21] was used to simulate concrete confinement behavior. The salient features of each of these concrete stress–strain curves are briefly discussed here.

Fig. 3
figure 3

Material stress–strain curves: a concrete for CC and CS; b concrete for SC and SS; c steel tubes; CC = circle-circle; CS = circle-square; SC = square-circle; SS = square-square

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Figure 3a shows the unconfined and confined stress–strain curves for concrete. Based on Pagoulatou model, peak confined stress (\(f^{\prime}_{{\text{cc}}}\)) and corresponding confined strain (\(\varepsilon_{{\text{cc}}}\)) can be obtained by Eqs. 1 and 2 using lateral confinement factor (\(f_{1}\)) provided elsewhere [20].

$$f^{\prime}_{{\text{cc}}} = f^{\prime}_{c} + k_{1} f_{1} \left( {{\text{MPa}}} \right)$$
(1)
$$\varepsilon_{{\text{cc}}} = \varepsilon_{c} \left( {1 + k_{2} \frac{{f_{1} }}{{f^{\prime}_{c} }}} \right)$$
(2)

where \(k_{1}\) = 4.1 and \(k_{2}\) = 20.5 are fixed constants and \(f^{\prime}_{c}\) is unconfined cylinder strength corresponding to strain \(\varepsilon_{c} = 0.003\). Accordingly, confined stress (\(f^{\prime}\)) at any strain (\(\varepsilon\)) was obtained by Eq. 3 from study [20].

$$f^{\prime} = \frac{{E_{{\text{cc}}} \varepsilon }}{{1 + (R + R_{E} - 2\left( {\frac{\varepsilon }{{\varepsilon_{{\text{cc}}} }}} \right) - \left( {2R - 1} \right)\left( {\frac{\varepsilon }{{\varepsilon_{{\text{cc}}} }}} \right)^{2} + R\left( {\frac{\varepsilon }{{\varepsilon_{{\text{cc}}} }}} \right)^{3} }}{ }\left( {{\text{MPa}}} \right){ }$$
(3)

where \(E_{{\text{cc}}}\) is modulus of elasticity for confined concrete given in Eq. 4 [35]. The parameters \(R_{E}\), \(R\), k3, and r needed for obtaining descending portion of confined stress–strain curve in Fig. 3a were obtained from literature [20].

$$E_{{\text{cc}}} = 4700\sqrt {f^{\prime}_{{\text{cc}}} } { }\left( {{\text{MPa}}} \right)$$
(4)

In Zhao model, the confined stress (\(f^{\prime}\)) at any strain (\(\varepsilon\)) was obtained by Eq. 5, which was originally proposed by Popovics [36] and later modified by Mander et al. [37], to simulate stress–strain behavior of SS columns [34].

$$f^{\prime} = \frac{{f^{\prime}_{{\text{cc}}} { }\left( {\frac{\varepsilon }{{\varepsilon_{{\text{cc}}} }}} \right)r_{m} }}{{r_{m} - 1 + \left( {\frac{\varepsilon }{{\varepsilon_{{\text{cc}}} }}} \right)^{{r_{m} }} }}$$
(5)

where the details of obtaining peak confined stress (\(f^{\prime}_{cc}\)) at strain (\(\varepsilon_{cc}\)) along with all relevant parameters are given elsewhere [34].

The modified version of Zhao model given in study [21] and shown in Fig. 3b was used to simulate confined concrete behavior in SC columns. This stress–strain curve consists of four segments such as OA, AB, BC, and CD. First segment OA of the concrete curve was evaluated as given in research study [37]. The portions AB, BC, and CD of the stress–strain curve were modeled as per recommendations from the literature [38], where \(\beta_{{\text{c}}}\) factor in Fig. 3b reflects confinement effect on concrete ductility.

2.1.2 Steel

Steel stress–strain curve was simulated by typical elasto-plastic curve shown in Fig. 3c, similar to study [39]. The stress (\(f\)) at any strain (\(\varepsilon\)) in the linear-elastic and nonlinear plastic segments of this curve can easily be evaluate by Eqs. (6) and (7), respectively.

$$f = E_{s} \times \varepsilon {\text{ for }}\varepsilon \le \varepsilon_{{{\text{sy}}}}$$
(6)
$$f = f_{{{\text{sy}}}} + 0.01E_{s} \times \left[ {\varepsilon - \varepsilon_{{{\text{sy}}}} } \right]{\text{ for }}\varepsilon > \varepsilon_{{{\text{sy}}}}$$
(7)

where \(f_{{{\text{sy}}}}\) = yield stress at strain (\(\varepsilon_{{{\text{sy}}}}\)). The modulus of elasticity (\(E_{s} )\) and poisson’s ratio (\({\upnu }_{s} )\) of 200 GPa and 0.3, respectively, was used in FE analysis in this study.

3 Results and Discussions

In this study, 80 CFDST column models provided with four different cross-sectional shapes (i.e., CC, CS, SC, and SS) were analyzed using NLFEA. Among these, 40 nonlinear FE analyses (i.e., 16 in CC and 8 each of CS, SC, and SS column shapes) were performed on experimentally tested specimens, having available results, geometry, and material characteristics in documented literature [8, 9, 11,12,13]. Additionally, 40 numerical models with 10 columns for each of CC, CS, SC, and SS shapes were created, such that each of these columns were provided with equivalent cross-sectional areas for each inner tube, outer tube, and concrete infill. The results of these nonlinear FE analyzes are discussed in the subsections below.

3.1 Validation of Numerical Models

Figure 4 shows a comparison of experimental and analytical curves for axial load vs deflection and axial load vs strain results of representative column samples in each considered cross-sectional shape category. Other details, such as location of deflection and strain measurements, material properties and geometric characteristics in analytical study exactly concur with the actual documented experimental studies [8, 9, 11,12,13]. Figure 4a–d, respectively, for CC, CS, SC, and SS shows that adopted NLFEA can capture CFDST column behavior satisfactorily, including nonlinear pre-peak and post-peak curves. The CFDST columns depicted ductile behavior during experiments, which was satisfactorily captured in analytical models for all shapes. This validation study further verified that the degree of concrete confinements was captured correctly by using Pagoulatou material model for CC and CS column shapes and similarly Zhao model in SS column. However, modified Zhao material model used in SC columns resulted in stiffer post-peak behavior compared to experimental results (Fig. 4c). This disparity in post-peak axial load–deflection behavior of SC shaped CFDST columns can be attributed to the fact that the material model used in FE analysis for SC shape in the present study was originally developed by Zhao et al. [34] for SS cross-sectional shape. The modified Zhao material model used in the present paper was used for analytical modeling in SC shapes similar to other studies [21], wherein the difference in the axial load–deflection post-peak behavior was similarly reported.

Fig. 4
figure 4

Axial load–deflection / strain characteristics of CFDST columns, experimental vs analytical results: a circle-circle (CC); b circle-square (CS); c square-circle (SC); d square-square (SS)

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Table 1 summarizes the experimental and analytical ultimate axial load capacities of the columns shown in Fig. 4. It can be seen that NLFEA predicted axial load carrying capacities of CFDST columns in all considered shapes within 1–5% maximum error compared to experimentally evaluated results. The average ratios of axial load capacities evaluated using NLFEA to experimental results (\(P_{{\text{NLFEA}}} /P_{{\text{Exp}.}}\)) on all 40 experimentally tested columns were: 0.98 for CC, 1.05 for CS, 0.99 for SC, and 1.03 for SS. This validation study provided the confidence that adopted NLFEA procedures can be further extended to study influence of column cross-sectional shape on resulting concrete infill confinement and ultimate axial load capacities of CFDST columns.

Table 1 Comparison of experimental and analytical axial load capacities for typical CFDST columns
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3.2 Results of Parametric Study

In this section, the results of NLFEA on additional 40 numerical models depict the influence of column cross-sectional shape on concrete infill confinement and ultimate axial load capacity. Each of 10-sets of numerical models were generated, such that columns across a typical set had same cross-section area of inner tube, outer tube, and concrete infill for four column shapes, i.e., CC, CS, SC, and SS. All hypothetically created columns were provided a nominal length of 650 mm, compressive-cylinder strength of concrete as 44 MPa, and steel yields strengths of 350 MPa and 365 MPa for inner and outer tubes, respectively. Among geometric parameters, ratio of diameter to thickness ratio for inner tube (\(D_{i} /t_{i}\) = 24 to 48) and outer tube (\(D_{0} /t_{0}\) = 45 to 85) was varied in the generated 10-sets of numerical models. However, each generated column set had constant values for \(D_{i} /t_{i}\) and \(D_{0} /t_{0}\) across the considered cross-sectional shapes. Confinement of concrete infill in each of the numerical models was measured by confinement factor. Confinement factor was defined as ratio of analytical ultimate axial load capacity (\(P_{{\text{ult}., \text{NLFEA}}}\)) of CFDST column to its total summed capacity arrived from individual column components, i.e., inner tube (\(f_{yi} .A_{si}\)), outer tube (\(f_{yo} .A_{s0}\)), and concrete infill (\(f_{c}^{,} .A_{c}\)) and is given by Eq. 8.

$$\text{Confinement} \text{factor} \left( {\text{CF}} \right) = \frac{{P_{{\text{ult}., \text{NLFEA}}} }}{{f_{{\text{yi}}} .A_{{\text{si}}} + f_{{\text{yo}}} .A_{s0} + f_{c}^{,} .A_{c} }}$$
(8)

where \(f_{{\text{yi}}} { }\text{and}{ }f_{y0} { }\) = yield stress of inner and outer steel tubes; \(A_{{\text{si}}} { },{ }A_{s0} { }\text{and}{ }A_{c} { }\) = cross-sectional areas of inner tube, outer tube, and concrete infill, respectively.

Figure 5 shows the influence of diameter (or edge length)-to-thickness ratios of inner and outer steel tubes on confinement factor (CF). The inner and outer tube slenderness, i.e., \(D_{0} /t_{0}\) and \(D_{i} /t_{i}\) have negative influence on concrete infill confinement. However, \(D_{0} /t_{0}\) is clearly having more prominent negative effect on CF than \(D_{i} /t_{i}\). Among considered CFDST column shapes, the decrease in CF is seen about 9% in CC, 30% in CS, 22% in SC, and 41% in SS when \(D_{0} /t_{0}\) is varied from 45 to 85 by increasing \(D_{0}\) (Fig. 5a). This indicates that increase in outer tube size (\(D_{0}\)) reduces concrete confinement in all column shapes with the most decrease in SS and the least in CC shape. Figure 5b shows that as \(D_{i} /t_{i}\) is increased from 24 to 48 by increasing \(D_{i}\) at constant \(D_{0} /t_{0}\) ratio, CF decreases by about 23% in CC, 17% in CS and SC, and 11% in SS. Thus, increasing size of inner steel tube (\(D_{i}\)) reduces concrete confinement to greatest degree in CC and lowest in SS. Also Figs. 5a, b depict that among equivalent columns at any particular ratio of \(D_{0} /t_{0}\) or \(D_{i} /t_{i}\), CC provides highest CF, whereas SS shape develops the lowest CF. The other two column shapes, CS and SS have comparable CF values. The relative drop in CF in SS with respect to CC shape is about 11% at \(D_{0} /t_{0}\) = 45, whereas this drop in concrete confinement for SS increase to about 42% with increase in \(D_{0} /t_{0}\) = 85 (Fig. 5a). However, this trend in relative drop for CF values of SS shape is reversed in Fig. 5b for equivalent columns. Such as for \(D_{i} /t_{i}\) = 24, drop in concrete confinement in SS with respect to CC shape is about 32% whereas same reduces to 21% at \(D_{i} /t_{i}\) = 48.

Fig. 5
figure 5

Influence of steel tube slender ratios on confinement factor for multiple-shapes of CFDST columns: a outer steel tube; b inner steel tube; Do = outer tube size; to = outer tube thickness; Di = inner tube size; ti = inner tube thickness

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The influence of cross-sectional shape of CFDST columns with respect to CC on their ultimate axial load capacity (\(P_{{\text{ult}.}}\)) is shown in Fig. 6, and capacity values are summarized in Table 2. The average decrease in \(P_{{\text{ult}.}}\) compared to CC is 12% in CS, 11% in SC, and 29% in SS (Fig. 6). This reduction in column capacities compared to CC can be attributed to the fact that ultimate axial load for each shape has significant contribution from concrete fill confinement. Accordingly, a drop in confinement level is directly correlated to decease in \(P_{{\text{ult}.}}\) for various column shapes. Such as average decrease in CF values compared to CC (recall, Fig. 5a, b) was 12% for CS, 11% SC, and 29% for SS, and these reductions in concrete confinement results in corresponding decrease in axial load capacities.

Fig. 6
figure 6

Ultimate axial load capacity of multiple-shape CFDST columns with respect to circle-circle (CC) shape

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Table 2 Comparison of axial load capacities of equivalent multiple-shape CFDST columns
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4 Summary and Conclusions

In the present study, nonlinear FE modeling methodology was used to reproduce the experimentally observed behavior of CFDST columns of multiple shapes (i.e., CC, CS, SC, and SS) under axial compression. The study utilized documented results of 40 CFDST columns (16 of CC shape and 8 each with CS, SC, and SS shapes) for successful validation of numerical models. The analytical models for all four shapes predicted axial compression behavior of CFDST columns in good agreement with experimental results. The degree of concrete confinement developed in each column shape including pre- and post-peak column behavior under axial compression was successfully captured by the analytical models and was validated against documented experimental results. The analytical values of ultimate axial load capacities of CFDST columns in all considered shapes were within 1–5% maximum errors compared to experimentally evaluated results.

A parametric study was performed on 40 additional CFDST numerical models (10 models each for CC, CS, SC, and SS shapes) to examine influence of cross-section shape on degree of concrete confinement produced and their ultimate axial load capacities. Analytical models were provided with equivalent cross-sectional areas for inner tube, outer tube, and concrete infill. The following conclusions can be drawn from this parametric study:

  • CFDST column cross-sectional shape has a significant influence on degree of confinement produced for concrete infill, thereby directly affecting column behavior from elastic to ultimate states. Compared to CC shape, average decrease in confinement ratios was 12% for CS, 11% for SC, and 29% for SS for inner or outer tube slenderness ratio variations investigated in this study.

  • The ultimate axial load capacity of the CFDST columns showed direct correlation with degree of concrete confinement produced on account of cross-sectional shape of the columns. Average decrease in ultimate column capacity compared to CC was 12% in CS, 11% in SC, and 29% in SS for CFDST columns with equivalent cross-sectional areas.

  • Increase in diameter (or edge length)-to-thickness ratios, i.e., inner and outer tube slenderness has negative influence on concrete confinement in CFDST columns, with outer steel tube having more prominent effect than inner tube. The most decrease in confinement ratio (CF) was observed in SS (41%) and least in CC (9%) shapes for outer tube slenderness ratio variations. However, the maximum decrease in CF is observed for CC (23%) and minimum for SS (11%) shapes when inner tube slenderness ratio is increased.

  • CC shape showed highest confinement ratio and SS the lowest, whereas CS and SS showed comparable confinement for inner or outer tube slenderness ratio variations considered in this study.

Although, the study described herein was performed with rigorous NLFEA, the work was limited to only stub-column behavior. Additional comprehensive future studies are needed for more general understanding on influence of multiple cross-sectional shapes on concrete confinement and ultimate axial load capacities of slender CFDST columns.