Numerical modelling of concrete-filled lean duplex slender stainless steel tubular stub columns

Journal of Constructional Steel Research 66 (2010) 1057–1068

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Numerical modelling of concrete-filled lean duplex slender stainless steel tubular stub columns M.F. Hassanein ∗ Department of Structural Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt

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Article history: Received 14 September 2009 Accepted 9 March 2010 Keywords: Cold-formed Finite element investigation Lean duplex stainless steel tubes Slender plates Square and rectangular hollow-sections Confinement Compressive strength Structural design

abstract Major technological advances in materials processing have led to the development of duplex stainless steels with exceptional mechanical properties. Duplexes have great potential for expanding future structural design possibilities, enabling a reduction in section sizes leading to lighter structures. The duplex grades offer combination of higher strength than austenitics as well as a great majority of carbon steels with similar or superior corrosion resistance. However, high nickel prices have more recently led to a demand for lean duplexes with low nickel content, such as grade EN 1.4162. Extensive work is needed to include the lean duplex grade EN 1.4162, into design standards such as EN 1993-1-4 and ENV 1994-11. Accordingly, finite element modelling for concrete-filled lean duplex slender stainless steel tubular stub columns of Grade EN 1.4162 is presented in this paper. The paper is predominantly concerned with two parameters: cross-section shape and concrete compressive strength, which have not yet been investigated. The non-linear displacement analysis of the columns was constructed herein based on the confined concrete model provided by Hu et al. (2003) [15]. The behaviour of the columns was investigated using a range of concrete cylinder strengths (25–100 MPa). The overall depth-to-width ratios (aspect ratio) varied from 1.0 to 1.8. The depth-to-plate thickness ratio of the tube sections varied from 60 to 90. The concrete-filled lean duplex slender stainless steel tubular columns were subjected to uniform axial compression over the concrete and stainless steel tube to force the entire section to undergo the same deformations by blocking action. The ABAQUS 6.6 program, as a finite element package, is used in the current work. The results showed that the design rules specified in the ASCE are highly conservative for square and rectangular concrete-filled lean duplex slender stainless steel stub columns while they are conservative in the case of European specifications. A new design strength is, therefore, proposed that is accurately found to represent the behaviour of concrete-filled lean duplex stainless steel tubular stub columns. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Over the last 20 years, significant developments have occurred in materials processing, providing a range of stainless steel material characteristics to suit the demands of various engineering applications. This is due to the aesthetic appearance, high corrosion resistance, ease of maintenance, smooth and uniform surface, high fire resistance, high ductility and impact resistance, reuse and recycling capability, as well as ease of construction of stainless steel structural members. Generally, the austenitic grades feature most prominently within the constructional industry. The most commonly employed grades of austenitic stainless steel are EN 1.4301/1.4307 and EN 1.4401/1.4404, which contain around 8%–11% nickel. Nickel stabilises the austenitic microstructure and

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therefore contributes to the associated favourable characteristics such as formability, weldability, toughness and high temperature properties. However, nickel also represents a significant portion of the cost of austenitic stainless steel. Therefore, high nickel prices have more recently led to a demand for lean duplexes with low nickel content, such as grade EN 1.4162 [1–3]. Despite early applications of lean duplex stainless steel, its structural properties remain to some extent unverified as limited test data on structural components have been reported. Recently, a research project is underway at Imperial College, London, to address these shortcomings, focusing initially on cold-formed hollow sections [4], where the compressive behaviour of lean duplex stainless steel square and rectangular hollow sections was examined through experimental tests. The test results were used to validate finite element (FE) models, which were thereafter employed in parametric studies to expand the range of available structural performance data, studying the influence, in particular, of the cross-section and member slenderness. It is important to

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note that the compound Ramberg–Osgood material model [5–7], which is a two-stage version of the basic Ramberg–Osgood model [8,9], was used during the numerical part of this research. The results of the study showed that a good agreement between experimental and FE results for cold-rolled stainless steel hollow sections was obtained. Hence, the generated stress–strain curve based on the available models, as used in [4], may be used to represent the actual behaviour of the lean duplex stainless steel material. At the same time, extensive work is still needed to include the lean duplex 1.4162 into design standards such as EN 1993-1-4 [10] and EN 1994-1-1 [11]. One of these works is to investigate the behaviour of concrete-filled lean duplex slender stainless steel tubular stub columns, which have not yet been investigated, as presented in the current research. However, the following paragraphs provide a short survey on the composite columns, stainless steel material properties and previous investigations dealing with concrete-filled columns. Composite columns refer to compression members in which a steel element acts compositely with a concrete element, so that both elements resist the compressive force. There is a wide variety of composite column types of varying cross-section, but the most commonly used and studied are encased I-section and concrete-filled steel tubes. In contrast to the encased composite column, the concrete-filled column has the advantage that it does not need any formwork or reinforcement. The concretefilled column offers several advantages related to the structural behaviour over pure steel, reinforced concrete or encased I-section columns. It can be said that a concrete-filled column delivers the economies of a concrete column with the speed of construction and the constructability of a steel column, resulting in significant economies in the overall structure of a building project [12]. On the other hand, the use of stainless steels in structural engineering is limited in conventional constructions due to the high initial material costs, limited structural design guidance, restricted section availability and lack of knowledge about their physical, mechanical and resistance properties among designers, practice engineers and architects. In recent years, the introduction and the revision of the design codes have assisted in the spreading use of the material in conventional structures. The better awareness of the additional benefits of the stainless steel made the material suited to use in construction [2,13,14]. The behaviour of stainless steel sections is different from that of carbon steel sections. Stainless steel sections have a rounded stress–strain curve with no yield plateau and a low proportional limit stress compared to carbon steel sections. Stainless steel design rules have been based on assumed analogies with carbon steel behaviour. Consequently, stainless steel design based on these assumptions leads to overly conservative cross-sections. In addition to the differences in the basic material properties, other differences cannot be ignored, such as the nature of the stress–strain curve, the material’s response for cold-working and high elevated temperature. Because of the non-linear stress–strain behaviour, the design of stainless steel compression members is based on the tangent modulus theory for column buckling. However, there are no test data on concrete-filled lean duplex stainless steel tubular columns. Experimental tests are usually used to investigate the behaviour of concrete-filled steel tubular columns. However, they are often costly and time consuming. Finite element modelling of concrete-filled steel tubular columns can provide an efficient alternative to full-scale concretefilled steel tubular columns. Many researches investigated the behaviour of concrete-filled steel tubular columns using a finite element model. Hu et al. [15] developed a nonlinear finite element model using the ABAQUS program [16] to simulate the behaviour of concrete-filled carbon steel tube columns. The concrete confinement was achieved by matching the numerical results

with experimental results via parametric study. Three years later, Ellobody and Young [17] developed a nonlinear finite element model to investigate the behaviour and design of axially loaded concrete-filled cold-formed high strength stainless steel tubular columns. The concrete confinement, as established in [15], was considered in the model. In a comparison between stiffened and unstiffened normal-strength stainless steel columns [18], finite element models to investigate the behaviour and strength of concrete-filled unstiffened austenitic stainless steel columns were conducted. In this paper, a series of theoretical models to investigate the behaviour and strength of concrete-filled lean duplex slender stainless steel tubular stub columns are reported. The principle target was to build a knowledge basis for the mechanical behaviour of this type of column, which has not yet been investigated. Thirty finite element models were conducted on square and rectangular hollow-sections using different in-filled concrete core strengths with no use of discrete mechanical shear connectors to improve the bond in the stainless steel interface or additional reinforcement besides the stainless steel tubes. The dimensions of the stainless steel tubes were chosen to simulate highly slender cross-sections, as they are 20% upper and lower than the maximum width-tothickness ratio specified by the EN 1993-1-4 [10] in Section 5.2.1, i.e. the maximum value for the width-to-thickness ratio for a flat element allowed by [10] for a stainless steel cross section is 75, if the visual distortion of the cross-section is unacceptable under serviceability loading. Accordingly, the dimensions of the stainless steel tubes herein were chosen in between 60 and 90. In general, the author tried to make this paper self-contained by providing as many details as possible. 2. Current finite element model 2.1. General The approach used to develop the finite element models for the concrete-filled slender lean duplex stainless steel tubular stub columns is similar to that used previously by the current author [18] and other researchers to develop reliable nonlinear finite element (i.e. nonlinear shell element) models for both carbon and stainless steel tubes and solid elements for the concrete core using the ABAQUS [16] computer package. The details of the finite element modelling are given in the following paragraphs. The models used the geometries of the concrete-filled slender stainless steel stub columns which were examined in [18]. The analysis conducted herein is so called a load–displacement nonlinear analysis. From this analysis, the finite element strengths (PFE ), failure modes and load-end shortening relationships are determined. The concrete-filled lean duplex stainless steel tube columns were labelled such that the shape of stainless steel tube and concrete strength could be identified from the label. For example, the label ‘‘SHS1C55’’ defines the specimen with a square hollowsection that belonged to test series SHS1, and the letter ‘‘C’’ indicates the concrete strength followed by its value in MPa (55 MPa). In case of rectangular hollow-section columns, the labels start with RHS. 2.2. Finite element type and mesh Owing to the thin-walled nature of the lean duplex stainless steel tubes Grade EN 1.4162, and in line with similar previous investigations [17,18], shell elements were employed to discretise the models. The four-noded doubly curved shell element with reduced integration S4R [16] has been utilised in this study. The

M.F. Hassanein / Journal of Constructional Steel Research 66 (2010) 1057–1068

S4R element has six degrees of freedom per node and provides an accurate solution to most applications. A convergent study of the mesh had been done by Wu [19] using a range of element sizes for both circular and square concrete-filled tubular (CFT) columns. It was shown that the results of the circular CFT column with 30 (5 × 6) elements were almost identical to those with 192 (16 × 12) elements. Similarly, the results of square CFT columns with 48 (8 × 6) elements were almost identical to those with 135 (15 × 9) elements. Since mesh refinement has very little influence on the numerical results, coarse meshes could be used through the finite element analyses of concrete-filled columns. Accordingly, a mesh of an approximate global size of 25 mm is used in the current modelling for the stainless steel tubes and concrete cores. For concrete core, three dimensional eight-node solid elements, so called C3D8R, was used. To simulate the bond between the stainless steel tube and the concrete core, a surface-based interaction with a contact pressureoverclosure model in the normal direction, and a Coulomb Friction Model in the directions tangential to the surface, are used. In order to construct contact between two surfaces, the slave and master surfaces must be chosen successfully. Generally, if a smaller surface contacts a larger surface, the best is to choose the smaller surface as the slave surface. If the distinction cannot be made, the master surface should be chosen as the surface of the stiffer body or as the surface with the coarser mesh if the two surfaces are on structures with comparable stiffness. The stiffness of the structure and not just the material should be considered when choosing the master and slave surface. Herein, a thin sheet of stainless steel is less stiff than a larger block of concrete core even though the stainless steel material has a higher stiffness than the concrete material. Therefore, the stainless steel surface is chosen as the slave surface whereas the concrete core surface is chosen as the master surface. 2.3. Boundary conditions and load application The concrete-filled lean duplex stainless steel columns considered herein had fixed ends. Merely the displacement at the loaded end in the direction of the applied load was allowed. The other nodes were free to translate and rotate in any direction. A uniform distributed load was applied statically at the top of the upper cover plate using the displacement control. The load was applied in increments using the modified RIKS method available in the ABAQUS library. In the RIKS method, the load is applied proportionally in several load increments. In each load increment the equilibrium iteration is performed and the equilibrium path is tracked in the load–displacement space. This method is often used in static analysis and shows to be a strong method for nonlinear analysis. The non-linear geometry parameter (*NLGEOM) was included to deal with the large displacement analysis. The load application for SHS2C25 is presented in Fig. 1. 2.4. Material modelling 2.4.1. Stainless steel material According to EN 10088-4 [3], the cold formed lean duplex stainless steel Grade EN 1.4162 has a minimum 0.2% proof stress (σ0.2 ) of 530 MPa and an ultimate tensile strength ranging from 700–900 MPa, taken herein as 700 MPa. In a study by Rasmussen et al. [20] describing the development of numerical models for analysing stainless steel plates in compression, it was indicated that anisotropy may not be important for numerical analyses involving monotonic loading. Consequently, the anisotropic nature of the stainless steel material was not included in the current modelling.

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Fig. 1. Boundary conditions and load application on SHS2C25.

The stainless steel material relationship has been modelled as a Von Mises material with isotropic hardening. The nonlinear relationship between stress and strain for stainless steel is generally represented by the Ramberg–Osgood [8] equation as given below;

ε=

σ E0

+ 0.002



σ σ0.2

n

.

(1)

In Eq. (1), (n) is called the nonlinearity index and it is a measure of the non-linearity of the stress–strain behaviour, lower (n) values implying a greater degree of non-linearity. The degree of nonlinearity varies among different grades of stainless steel. As the value of (n) increases the material behaviour tends to converge to the elasto-plastic behaviour of carbon steel (elastic-perfectly plastic behaviour for n = ∞). Grades of low (n) values exhibit higher hardening behaviour and, for a given stress level, benefits of strain hardening comparatively becomes more apparent. Eq. (1) is known to give excellent agreement with experimental stress–strain data up to the 0.2% proof stress (σ0.2 ), however, for higher strains the formulation generally overestimates the corresponding stresses. Therefore, a two-stage version expressing the full-range stress–strain material behaviour of stainless steel was developed in [6]. In this respect, Rasmussen [6] proposed the use of an expression for the complete stress–strain curve for stainless steel alloys. The expression; Eq. (2), involves the conventional Ramberg–Osgood parameters (n, E0 , σ0.2 ) as well as the ultimate tensile strength (σu ) and strain (εu ). The expressions have been shown to produce stress–strain curves which are in good agreement with tests over the full range of strains up to the ultimate tensile strain. Therefore, Eq. (2) is used in the current investigation to generate the stress–strain curve of the lean duplex stainless steel material Grade EN 1.4162, as presented in Fig. 2.

   σ n σ    + 0.002 E0 σ0.2 ε=    σ − σ0.2 σ − σ0.2 m   + εu + ε0.2 E0.2 σu − σ0.2

for σ ≤ σ0.2 (2) for σ > σ0.2 .

In the equations, E0 is the initial modulus of elasticity (e.g. 200 GPa), E0.2 is the tangent modulus of the stress–strain curve at the 0.2% proof stress and is given as; E0.2 =

E0 1 + 0.002n/e

(3)

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confined concrete. The value of (εc ) is usually around the range of 0.002–0.003. A representative value suggested by ACI Committee 318 [23] and used in the analysis is 0.003. The confined concrete strength (fcc ) and the corresponding strain (εcc ) may be calculated from Eqs. (6) and (7): fcc = fco + kσlat



εcc = εc 1 + 5k

Fig. 2. Stress–strain curve of the lean duplex stainless steel material Grade EN 1.4162.

(6)

σlat



fco

.

(7)

Because the concrete in the concrete-filled tube columns is usually subjected to triaxial compressive stresses, the failure of concrete is dominated by the compressive failure surface, expanding with increasing hydrostatic pressure. Hence, a linear Drucker–Prager yield criterion is used to model the yield surface of concrete. The first part of the curve is assumed to be an elastic part up to the proportional limit, which is taken as (0.5f cc ). The initial modulus of elasticity (Ecc ) is highly correlated to its compressive strength and can be calculated with reasonable accuracy from the empirical equation ACI Committee 318 [23] as follows: Ecc = 4700 fcc .

p

(8)

The second part of the curve is the nonlinear portion, starting from the proportional limit stress (0.5f cc ) to the confined concrete strength (fcc ). The stress–strain relationship proposed by Saenz [24] has been widely adopted as the uniaxial stress–strain curve for concrete and it has the following form: Ecc ε

fc = 1 + (R + RE − 2)

Fig. 3. Equivalent uniaxial stress–strain curve for confined concrete.

where (e) is the non-dimensional proof stress given as e = σ0.2 /E0 . The material behaviour provided by ABAQUS allows for a multi linear stress–strain curve to be used. The first part of the multilinear curve represents the elastic part up to the proportional limit stress, with measured Young’s modulus E0 = 200 GPa, and where Poisson’s ratio was taken as 0.3. The proportional limit was found to be σ0.01 = 300 MPa. Since the analysis of post-buckling involves large inelastic strains, the nominal static stress–strain curve was converted to a true stress and logarithmic plastic true strain curve. pl The true stress σtrue and plastic true strain εtrue true, as required by ABAQUS, were calculated using Eqs. (4) and (5):

σtrue = σ (1 + ε) pl εtrue = ln(1 + ε) −

(4)

σtrue E◦

.

(5)

2.4.2. Concrete material The concrete-filled stainless steel tubular columns of small depth to thickness ratios provide high considerable confinement for the concrete. In this case, an equivalent uniaxial stress–strain relationship for confined concrete should be used. On the other hand, high depth to thickness ratios of concrete-filled stainless steel tubular columns provide inadequate confinement for the concrete, therefore the uniaxial stress–strain relationship for unconfined concrete should be used. Mander et al. [21] defined the limiting depth-to-thickness (D/t) ratio between confined and unconfined concrete to be equal to 29.2. Poisson’s ratio (υc ) in the elastic part of concrete under uniaxial compression stress ranges from 0.15 to 0.22, with a representative value of 0.19 to 0.2 according to ASCE [22]. In this numerical modeling, Poisson’s ratio (υc ) of concrete is taken as 0.2. Fig. 3 presents the equivalent uniaxial stress–strain curve for confined concrete, as well as the unconfined stress–strain concrete curve. Three parts of the curve have to be identified in the case of



ε εcc



− (2R − 1)



ε εcc

2

+R



ε εcc

3

(9)

where: R=

RE (Rσ − 1)

(Rε − 1) Ecc εcc RE = . fcc

2

−

1 Rε

(10) (11)

Rσ = Rε = 4 may be used, as recommended by Hu and Schnobrich [25]. In the analysis, Eq. (9) is taken as the equivalent uniaxial stress–strain curve for concrete when the concrete strain (ε ) is less than (εcc ), as can be seen in Fig. 3. When (ε > εcc ), a linear descending line (the third part of the curve) is used to model the softening behaviour of concrete. If (k3 ) is defined as the material degradation parameter, the descending line is assumed to be terminated at the point where fc = rk3 fcc and ε = 11εcc . To account for the effect of different concrete strengths, the degradation parameter (k3 ) should be multiplied by an additional reduction factor (r), as given in [17]. The value of (r) is taken as 1.0 for concrete with cubic strength (fcu ) equal to 30 MPa. The value of (r) is taken as 0.5 for concrete with (fcu ) greater than or equal to 100 MPa. Linear interpolation, as given in [17], is used to determine the value of (r) for concrete cubic strength between 30 and 100 MPa. Generally, the parameters (σlat ) and (k3 ) should be provided in order to completely define the equivalent uniaxial stress–strain relation. These two parameters apparently depend on the widthto-thickness ratio (D/t or B/t), cross-sectional shape, and stiffening means. Consequently, their appropriate values were determined by matching the numerical results with the experimental results via parametric study, as given in [15]. 2.5. Verification of previous lean duplex stainless steel bare columns In order to check the validity of the stainless steel material stress–strain curve as proposed in [6], it was necessary to model

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Fig. 4. Stress–strain curve of the lean duplex stainless steel material Grade EN 1.4162 using the experimental results provided in [4]. Fig. 6. Deformed shape of 80 × 80 × 4-SC2.

Fig. 5. Load-end shortening relationship of the column 80 × 80 × 4-SC2 [4].

the available lean duplex stainless steel bare columns given in [4]. Accordingly, the experimentally tested lean duplex square hollowsection column (80 × 80 × 4-SC2) is modelled herein. The measured dimensions of the columns depth, width and thickness are 80 mm, 80 mm and 3.81 mm, respectively. The length of the column is 332.2 mm and its corner inner radius is 3.6 mm. The column suffered from an initial geometrical imperfection of 0.08 mm. The lean duplex stainless steel material was tested in tension and the following properties were measured: E0 = 199 900 MPa, σ0.2 = 679 MPa, σu = 773 MPa, n = 6.5. All degrees of freedom were restrained at the end cross-sections of the stub column model, apart from vertical translation at the loaded end, which was constrained via kinematic coupling to follow the same vertical displacement. Herein, the stress–strain curve of the material is built using the equations previously presented in Section 2.4.1. It should be noted that the effect of the corner increased strength is neglected in the current verification. The generated material curve is given in Fig. 4. Fig. 5 presents the load-end shortening curve for the lean duplex column (80 × 80 × 4-SC2), while the deformed shape can be seen in Fig. 6, which confirms that the failure mode is achieved by local buckling. The buckles on the adjacent sides occurred in different directions and the boundary conditions for longitudinal boundaries become like those for simple support. Hence, a good agreement is accomplished by using the stress–strain curve of the lean duplex material as given in [6]. 2.6. Verification of previous concrete-filled steel columns Before the comparative study of [18] could be carried out, it was necessary to prove that the established finite element model is capable of simulating the structural behaviour of unstiffened concrete-filled stainless steel tubular stub columns. However, the verification was made for concrete-filled steel tubular

stub columns tested by Schneider [26] and good agreement was achieved between experimental and numerical results. This step was followed by establishing a finite element program to investigate concrete-filled austenitic stainless steel stub columns having the same cross-sections as used in the current paper. As a result, the verification of the material stress–strain curve using [4], as well as the verification of the concrete-filled carbon and austenitic stainless steel columns using [18,26], ensure the accuracy of simulating the concrete-filled lean duplex stainless steel stub columns with a high degree of confidence. However, the current study continues in the following, by using again a minimum 0.2% proof stress (σ0.2 ) of 530 MPa and an ultimate tensile strength of 700 MPa, as specified by the EN 10088-4 [3] and as can be seen in the stress–strain curve of Fig. 2. 2.7. Current concrete-filled stainless steel columns The finite element programme consisted of five test series, including two series of concrete-filled SHS tubes (SHS1 and SHS2) and three series of concrete-filled RHS tubes (RHS1, RHS2 and RHS3), as shown in Fig. 7. The thickness of the tubes is fixed to 2 mm throughout the investigation. The length of the stub columns (L) was chosen to be three times the depth (D) of the SHS or RHS columns to avoid the effects of flexural buckling and end conditions. In line with [15], the corners of the concrete-filled tubes with square and rectangular sections are assumed herein to be exact 90° and corner radii are not considered. Tables 1 and 2 summarize the dimensions of the columns. The cross-section dimensions D and B are outside measurement. In addition, the tables provide the cross-sectional areas for both the stainless steel tubes and the concrete cores of the whole program. Parameters such as r [17] and k3 [15], required for the concrete model, are also presented. The classification of cross-sections and the effective cross-sectional area of the finite element series are calculated according the EN 1993-1-4 [10] and the ASCE [27]. These finite element models were based on the confined concrete model recently available in literature [15,17]. The lateral confining pressure (σlat ) imposed by the current stainless steel tubes was found to be zero for the whole models, as the depth-tothickness ratios for the whole columns are greater than 29.2 [15]. The model neglected the effect of residual stresses because its effect on the column capacity was found by Ellobody [28] to be small. The model also neglected the effect of initial imperfections because the strength reduction is not significant compared to thin-walled hollow tubes owing to the delaying effect of core concrete on the tube local buckling; refer to Tao et al. [29]. Hence, the phenomenon of buckling would not emerge in the deformed

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Table 1 Details of square concrete-filled stainless steel columns. Group

Specimen

Depth D (mm)

Width B (mm)

Thickness t (mm)

D/t

Length L (mm)

L/D

SHS1

SHS1C25 SHS1C40 SHS1C55 SHS1C70 SHS1C85 SHS1C100

120 120 120 120 120 120

120 120 120 120 120 120

2 2 2 2 2 2

60 60 60 60 60 60

360 360 360 360 360 360

3 3 3 3 3 3

SHS2

SHS2C25 SHS2C40 SHS2C55 SHS2C70 SHS2C85 SHS2C100

160 160 160 160 160 160

160 160 160 160 160 160

2 2 2 2 2 2

80 80 80 80 80 80

480 480 480 480 480 480

Thickness t (mm)

D/t

Area of stainless steel (mm2 )

Area of concrete (mm2 )

r[17] k3 [15]

930 930 930 930 930 930

13 448 13 448 13 448 13 448 13 448 13 448

0.99 0.86 0.72 0.59 0.50 0.50

0.43 0.43 0.43 0.43 0.43 0.43

3 3 3 3 3 3

1250 1250 1250 1250 1250 1250

24 328 24 328 24 328 24 328 24 328 24 328

0.99 0.86 0.72 0.59 0.50 0.50

0.4 0.4 0.4 0.4 0.4 0.4

Length L (mm)

L/D

Area of stainless steel (mm2 )

Area of concrete (mm2 )

r[17] k3 [15]

Table 2 Details of rectangular concrete-filled stainless steel columns. Group

Specimen

Depth D (mm)

Width B (mm)

RHS1

RHS1C25 RHS1C40 RHS1C55 RHS1C70 RHS1C85 RHS1100

140 140 140 140 140 140

80 80 80 80 80 80

2 2 2 2 2 2

70 70 70 70 70 70

420 420 420 420 420 420

3 3 3 3 3 3

850 850 850 850 850 850

10 328 10 328 10 328 10 328 10 328 10 328

0.99 0.86 0.72 0.59 0.50 0.50

0.4 0.4 0.4 0.4 0.4 0.4

RHS2

RHS2C25 RHS2C40 RHS2C55 RHS2C70 RHS2C85 RHS2C100

170 170 170 170 170 170

120 120 120 120 120 120

2 2 2 2 2 2

85 85 85 85 85 85

510 510 510 510 510 510

3 3 3 3 3 3

1130 1130 1130 1130 1130 1130

19 248 19 248 19 248 19 248 19 248 19 248

0.99 0.86 0.72 0.59 0.50 0.50

0.4 0.4 0.4 0.4 0.4 0.4

RHS3

RHS3C25 RHS3C40 RHS3C55 RHS3C70 RHS3C85 RHS3C100

180 180 180 180 180 180

100 100 100 100 100 100

2 2 2 2 2 2

90 90 90 90 90 90

540 540 540 540 540 540

3 3 3 3 3 3

1090 1090 1090 1090 1090 1090

16 888 16 888 16 888 16 888 16 888 16 888

0.99 0.86 0.72 0.59 0.50 0.50

0.4 0.4 0.4 0.4 0.4 0.4

strength of the concrete-filled columns are discussed. The finite element strengths (PFE ) of the concrete-filled lean duplex slender stainless steel tubular stub columns of the SHS and RHS are shown in Tables 3 and 4. The comparison was made with the design specifications [10,27]. A comparison is then continued to take place with three other design strengths in order to accomplish a new design method accurately representing the behaviour of concretefilled lean duplex slender stainless steel tubular stub columns. 3.1. Failure modes of current concrete-filled models

Fig. 7. Definition of symbols for concrete-filled stainless steel tubular stub columns.

shape as the geometric imperfections have not been added to the initial geometry. The full details of the concrete-filled stainless steel columns are presented in Tables 1 and 2. The generated stress–strain curve, as explained in Section 2.4.1, was used in this numerical modelling. The concrete compressive strength varied from 25 to 100 MPa. For more details, refer to Tables A.1 and A.2. 3. Results and evaluation The main objective of this research was to investigate the effect of the cross-section shape and the concrete compressive strength in concrete-filled lean duplex slender stainless steel tubular stub columns. The failure modes, effect of concrete cylinder strength and the influence of the stainless steel material on the

It is well known that concrete-filled slender steel stub columns fail by outward local buckling of the steel tubes accompanied by a crushing of the concrete core. In order to ensure this, it was necessary to conduct at the beginning of the research finite element models suffering from initial imperfection. Figs. 8 and 9 represent the first Eigen mode, as well as the deformed shape and the stress contour, of the column RHS2C55, respectively. It is worth pointing out that Fig. 9(a) represents one half of the column by applying a vertical cutting plane. It can be observed that, due to local buckling, several parts of the tube are separated from the concrete core. Again the rest of the parametric study neglects the initial imperfection as its effect on the behaviour of the concretefilled columns is not significant, as shown from the parametric study conducted in [29]. 3.2. Effect of concrete cylinder strength In order to examine the effect of concrete cylinder strength, the efficiency of the cross-sections (PFE /PFE,C25 ) was calculated,

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Table 3 Comparison of finite element strengths with unfactored design strengths according to the ENV 1994-1-1 [10] and the ASCE Standard [27] for square concrete-filled columns. Group

Specimen

PFE (kN)

Aeff,EC3 (mm2 )

PEC4 (kN)

PFE /PEC4

Aeff,ASCE (mm2 )

PASCE (kN)

PFE /PASCE

SHS1

SHS1C25 SHS1C40 SHS1C55 SHS1C70 SHS1C85 SHS1C100

711 882 1111 1287 1442 1598

457 457 457 457 457 457

578 780 982 1183 1385 1587

1.23 1.13 1.13 1.09 1.04 1.01

554 554 554 554 554 554

579 751 922 1094 1265 1437

1.23 1.17 1.20 1.18 1.14 1.11

SHS2

SHS2C25 SHS2C40 SHS2C55 SHS2C70 SHS2C85 SHS2C100

1009 1287 1771 2096 2442 2798

470 470 470 470 470 470

857 1222 1587 1952 2317 2682

1.18 1.05 1.12 1.07 1.05 1.04

577 577 577 577 577 577

823 1133 1443 1753 2064 2374

1.23 1.14 1.23 1.20 1.18 1.18

Ave

1.10

Ave

1.18

COV

0.065

COV

0.039

Table 4 Comparison of finite element strengths with unfactored design strengths according to the ENV 1994-1-1 [10] and the ASCE Standard [27] for rectangular concrete-filled columns. Group

Specimen

PFE (kN)

Aeff,EC3 (mm2 )

PEC4 (kN)

PFE /PEC4

Aeff,ASCE (mm2 )

PASCE (kN)

PFE /PASCE

RHS1

RHS1C25 RHS1C40 RHS1C55 RHS1C70 RHS1C85 RHS1100

711 740 850 1045 1177 1343

446 446 446 446 446 446

495 650 805 959 1114 1269

1.44 1.14 1.06 1.09 1.06 1.06

536 536 536 536 536 536

503 635 767 898 1030 1162

1.41 1.17 1.11 1.16 1.14 1.16

RHS2

RHS2C25 RHS2C40 RHS2C55 RHS2C70 RHS2C85 RHS2C100

805 1152 1537 1822 1891 2243

464 464 464 464 464 464

727 1016 1305 1593 1882 2171

1.11 1.13 1.18 1.14 1.00 1.03

568 568 568 568 568 568

710 955 1201 1446 1691 1937

1.13 1.21 1.28 1.26 1.12 1.16

RHS3

RHS3C25 RHS3C40 RHS3C55 RHS3C70 RHS3C85 RHS3C100

821 964 1253 1555 1872 2158

460 460 460 460 460 460

666 919 1173 1426 1679 1933

1.23 1.05 1.07 1.09 1.11 1.12

560 560 560 560 560 560

655 871 1086 1301 1517 1732

1.25 1.11 1.15 1.19 1.23 1.25

Ave

1.12

Ave

1.19

COV

0.097

COV

0.076

the strength of the concrete-filled column, at least for the range of concrete strengths investigated herein (25–100 MPa). The loadend shortening of the finite element models are also monitored. Fig. 11 represents the load-end shortening relationships for the models for the same Group (RHS3). It can be primarily noticed that the higher the strength of in-filled concrete, the sharper the loadaxial shortening behaviour. This sharper behaviour enforces the column to undergo smaller deformations at relatively higher loads, extending to much higher ultimate loads. 3.3. Influence of stainless steel material on the strength of concretefilled columns

Fig. 8. First eigen buckling mode for RHS2C55.

where PFE,C25 is the finite element strength for the specimen of 25 MPa concrete strength of each group. As a sample of results, Fig. 10 provides the efficiency of Group RHS3, as described above. The figure also includes a trend line that identifies the type of relationship between the strengths of the columns. Simply, it can be noticed that increasing the strength of the concrete core, for the same stainless steel tube, leads, almost, to a linear increase in

A comparative parametric study is conducted under this title between the concrete-filled carbon and lean duplex stainless steel tubular stub columns. The concrete-filled carbon steel column modelling can be checked in several papers with paper [18] of the current author, with his colleagues being one of them. The main target is to examine the influence of the lean duplex stainless steel material on the strength of the concrete-filled columns. The concrete-filled carbon steel columns modelled herein are typical of the concrete-filled lean duplex stainless steel columns except in the definition of the tube material. Therefore, any change in the behaviour of the concrete-filled columns will be directly attributed to the influence of the stainless steel material. Carbon steel material properties, with an elastic modulus E0 = 210 GPa,

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a

Fig. 12. Comparison between load-end shortening relationships for concrete-filled carbon (Car) and lean duplex stainless steel (Sta) stub columns.

b

Fig. 9. (a) Deformed shape and (b) stress contour of RHS2C55.

Fig. 10. Efficiency of Group RHS3.

The results of the current comparative study showed that the lean duplex stainless steel material outfits the concretefilled columns of high compressive strengths; fco > 55 MPa. It should be remembered that in the high-strength concrete case, the confinement becomes less apparent compared to normalstrength concrete mainly because of its lower lateral expansion; for additional information refer to [12]. Hence, the stainless steel material will suffer from less biaxial stress than the case of normal strength concrete. Consequently, the concrete-filled columns benefit from the strain hardening behaviour of the lean duplex stainless steel material more than the case of carbon steel tubes. This explains why the concrete-filled lean duplex stainless steel stub columns have higher capacities than their equivalent concrete-filled carbon steel stub ones, as can be observed in Fig. 12. However, Fig. 12 presents the load-end shortening relationships for columns RHS3C25 and RHS3C70 with carbon (Car) and stainless steel (Sta) tubes. On the contrary, for low concrete strengths, such as fco = 25 MPa, the concrete crushing becomes obvious as a result of the early local buckling for both tubes. Thus, the behaviour of the columns will depend mainly on the initial stiffness, which is higher in the case of carbon steel than the stainless steel. This higher stiffness extends to higher loads for concrete-filled carbon steel columns compared to the concrete-filled columns of stainless steel tubes. In the intermediate range of concrete core strengths, 25 MPa < fco ≤ 55 MPa, the columns of both types of steel become near to each other. 3.4. Comparison with design specifications The finite element strengths (PFE ) for concrete-filled models were compared to the design rules specified by ENV 1994-1-1 [10] using the effective area approach. The comparisons of strengths with design strengths (PEC4 ) for finite element series of concretefilled stainless steel stub columns are given in Tables 3 and 4. Generally, the predicted strength was calculated using Eq. (6.30) of the ENV 1994-1-1 [10]. In this equation, the plastic resistance of the concrete-filled stainless steel tubular columns should be calculated by adding the plastic resistance of the tube material and the concrete core, in the case where no reinforcement was used, as in the current test and finite element specimens, as follows: PEC4 = Aa fyd + 0.85Ac fcd

(12)

where; Fig. 11. Load-end shortening relationship of the Group RHS3.

normal yield stress Fy = 530 MPa and Poisson’s ratio υ = 0.3 are used. The material is assumed to be elastic-perfectly plastic with no strain hardening. Note that the yield stress Fy is chosen identical to that of the 0.2% proof strength (σ0.2 ) of the current lean duplex stainless steel material. However, Group RHS3 is chosen for this parametric study.

PEC4 is the design value of the plastic resistance of the composite section to compressive normal force Aa is the cross-sectional area of the structural steel section Ac is the cross-sectional area of concrete fyd is the design value of the yield strength of structural steel fcd is the design value of the cylinder compressive strength of concrete.

M.F. Hassanein / Journal of Constructional Steel Research 66 (2010) 1057–1068

For concrete filled sections the coefficient 0.85 may be replaced by 1.0, and this is the case which used herein. However, Table 5.2 in the EN 1993-1-4 [10] was used to classify the cross-section type of the test specimens. By applying the limitations in Table 5.2, the finite element models were all of Class 4. As well known, the effective widths may be used in order to reduce the resistance of the cross-section due to the effect of local buckling. For this reason, clause 5.2.3 was used during the calculations of the effective widths in Class 4 cross-sections forming this research. A reduction factor ρ should be taken as follows: For cold formed or welded internal elements, which are used with flat portions, as shown in Fig. 4:

ρ=

0.772

λp

0.125

−

λ2p

but ≤ 1

(13)

where λp is the element slenderness defined as:

λp =

b/t 28.4ε

1065

E◦ is the initial elastic modulus, n is the Ramberg–Osgood parameter which is the strain hardening exponent that defines the degree of roundness of the curve (n = 5). Ac is the cross-sectional area of concrete. fc = 0.8fcu is the design value of the cylinder compressive strength of concrete. The determination of flexural buckling stress and then the design of axial strength of concentrically loaded cold-formed stainless steel compression members, according to the ASCE Standard [27], requires iterative process. From Tables 3 and 4, it can be seen that the predicted values for the concrete-filled lean duplex stainless steel stub columns according to the ASCE [27] are highly conservative, while they are conservative according to the ENV 1994-1-1 [10]. Accordingly, this remark encourages the proposal of suitable design strengths, as given in the following section.

(14)

√ kσ

3.5. Comparison with proposed design strengths

where; t is the relevant thickness kσ is the buckling factor corresponding to the stress ratio ψ and boundary conditions from Table 4.1 in EN 1993-1-5 [30] as appropriate: it is 4.0 for flat portions. This value means that the case of simply support is considered. The assumptions about simple support when determining kσ are normally unfavourable. The assumption is therefore normally a simplification intended for manual calculation. b is the flat element width ε is the material factor defined in Table 5.2 in EN 1993-1-4 [10]. The comparison of the test and finite element strengths (PTest and PFE ) was also made by using the predicted value according to the ASCE Standard [27]. The comparisons of strengths with design strengths (PASCE ) according to the ASCE Standard [27] for finite element series for concrete-filled lean duplex stainless steel stub columns are also given in Tables 3 and 4. In general, clause 3.4 in the ASCE Standard [27] was used to classify the cross-section type of the test specimens. The resistance of a stainless steel crosssection subject to compression with a resultant acting through the centroid of the effective section shall be calculated as follows: PASCE = Ae Fn + 0.85Ac fc

(15)

where: Ae is the effective area calculated at stress Fn . A reduction factor ρ should be taken as follows: For uniformly compressed flat portions:

ρ=

1 − 0.22/λ

λ     1.052  w  f if λ = ≥ 0.673. √ t

k

E◦

(16) (17)

Fn is the flexural buckling stress for doubly symmetric sections, closed cross sections, and any other sections which are not subjected to torsional or torsional-flexural buckling, the flexural buckling stress. Fn , is determined as follows: Fn = Et =

π 2 Et ≤ Fy (KL/r )2 E◦ Fy Fy + 0.002nE◦ Fn /Fy

(18)

In this section, three design strengths are compared to the finite element strengths. In the first one, the design strength is calculated identical to the PEC4 , as explained in Section 3.1, but by using the revised effective width equation which was proposed in [31], as given herein in Eq. (20). This design strength is labelled as PEC4,Mod . The second design strength (PSq ) represents the squash load of the cross-section. The squash load is calculated using the whole crosssectional area of the lean duplex stainless steel tubes. In the third strength (PSq,red ), the contribution of the concrete core is reduced by the factor (0.85) and the whole cross-sectional area of the lean duplex stainless steel tubes is still used. Tables 5 and 6, provides the comparison with these three design strengths.

ρ=

0.772

λp

−

0.079

λ2p

but ≤ 1.

(20)

In the case of concrete-filled lean duplex slender stainless steel stub columns, the revised effective width equation proposed in [31] is found not to make a significant change in the design strengths compared to the original one. The reduced squash strength (PSq,red ) seems to give the best strength among the whole proposed and codes specifications results. However, based on the design approach used in both codes [10,27], a new design strength may be proposed, with a certain reduction factor added so that the formulas would fit in the finite element results of the concrete-filled lean duplex slender stainless steel stub columns. This factor is calibrated with the finite element strengths obtained from the finite element analysis (PFE ) and is found to be 0.96. Therefore, the design equation, as given in Eq. (21), can be used for the design of concrete-filled lean duplex slender stainless steel tubular stub columns. Good agreement between the finite element strength and the proposed design strength is observed, as can be seen in Fig. 13. The average finite element strength of the models over the average proposed strength is 1.01. Finally, this proposed formula should be compared in the future with experimental tests that examine it elsewhere because of a lack of the lean duplex material at present in the manuscript region of origin. PProp = 0.96(Aa fyd + 0.85Ac fcd )

(21)

where;


n−1 .

(19)

PProp is the proposed design value for the concrete-filled lean duplex stainless steel tubular stub columns

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Table 5 Comparison of finite element strengths with unfactored proposed design strengths for square concrete-filled columns. Group

Specimen

PFE (kN)

Aeff,Mod [31] (mm2 )

PEC4,Mod (kN)

PFE /PEC4,Mod

PSq (kN)

PFE /PSq

PSq,red (kN)

PFE /PSq,red

SHS1

SHS1C25 SHS1C40 SHS1C55 SHS1C70 SHS1C85 SHS1C100

711 882 1111 1287 1442 1598

475 475 475 475 475 475

588 790 991 1193 1395 1596

1.21 1.12 1.12 1.08 1.03 1.00

829 1031 1233 1434 1636 1838

0.86 0.86 0.90 0.90 0.88 0.87

779 950 1122 1293 1465 1636

0.91 0.93 0.99 1.00 0.98 0.98

SHS2

SHS2C25 SHS2C40 SHS2C55 SHS2C70 SHS2C85 SHS2C100

1009 1287 1771 2096 2442 2798

483 483 483 483 483 483

864 1229 1594 1959 2324 2689

1.17 1.05 1.11 1.07 1.05 1.04

1271 1636 2001 2365 2730 3095

0.79 0.79 0.89 0.89 0.89 0.90

1179 1490 1800 2110 2420 2730

0.86 0.86 0.98 0.99 1.01 1.02

Ave

1.09

0.87

0.96

COV

0.060

0.039

0.056

Table 6 Comparison of finite element strengths with unfactored proposed design strengths for rectangular concrete-filled columns. Group

Specimen

PFE (kN)

Aeff,Mod [31] (mm2 )

PEC4,Mod (kN)

PFE /PEC4,Mod

PSq (kN)

PFE /PSq

PSq,red (kN)

PFE /PSq,red

RHS1

RHS1C25 RHS1C40 RHS1C55 RHS1C70 RHS1C85 RHS1100

711 740 850 1045 1177 1343

468 468 468 468 468 468

506 661 816 971 1126 1281

1.40 1.12 1.04 1.08 1.05 1.05

709 864 1019 1173 1328 1483

1.00 0.86 0.83 0.89 0.89 0.91

670 802 933 1065 1197 1328

1.06 0.92 0.91 0.98 0.98 1.01

RHS2

RHS2C25 RHS2C40 RHS2C55 RHS2C70 RHS2C85 RHS2C100

805 1152 1537 1822 1891 2243

480 480 480 480 480 480

735 1024 1313 1602 1890 2179

1.09 1.12 1.17 1.14 1.00 1.03

1080 1369 1658 1946 2235 2524

0.75 0.84 0.93 0.94 0.85 0.89

1008 1253 1499 1744 1990 2235

0.80 0.92 1.03 1.04 0.95 1.00

RHS3

RHS3C25 RHS3C40 RHS3C55 RHS3C70 RHS3C85 RHS3C100

821 964 1253 1555 1872 2158

477 477 477 477 477 477

675 928 1182 1435 1688 1942

1.22 1.04 1.06 1.08 1.11 1.11

1000 1253 1507 1760 2013 2267

0.82 0.77 0.83 0.88 0.93 0.95

937 1152 1367 1583 1798 2013

0.88 0.84 0.92 0.98 1.04 1.07

Ave

1.11

0.87

0.96

COV

0.092

0.064

0.078

4. Conclusions

Fig. 13. PFE vs PProp for the concrete-filled lean duplex stainless steel material Grade EN 1.4162.

Aa is the whole cross-sectional area of the structural steel section Ac is the cross-sectional area of concrete fyd is the design value of the yield strength of structural steel fcd is the design value of the cylinder compressive strength of concrete.

This paper is primarily concerned with the behaviour and design of the concrete-filled lean duplex slender stainless steel hollow tubular stub columns of Grade EN 1.4162. These columns have not yet been investigated. A generated stress–strain curve, based on available models, is used to represent the actual behaviour of the lean duplex stainless steel material. The stress–strain curve uses a minimum 0.2% proof stress (σ0.2 ) of 530 MPa and an ultimate tensile strength of 700 MPa, as specified by the EN 100884. First, verifications for the stress–strain curve of the lean duplex material, as well as previous concrete-filled carbon and austenitic stainless steel columns, are made herein to ensure the accuracy of simulating the concrete-filled lean duplex stainless steel stub columns with a high degree of confidence. As a result, the finite element models constructed herein reflect the behaviour of the columns. The paper is predominantly concerned with two parameters: the cross-section shape and the concrete compressive strength. The non-linear displacement analyses of the columns are constructed herein based on the confined concrete model provided by Hu et al. [15]. The behaviour of the columns are investigated using a range of concrete cylinder strengths (25–100 MPa). The overall depth-to-width ratios (aspect ratio) varied from 1.0 to 1.8. The

M.F. Hassanein / Journal of Constructional Steel Research 66 (2010) 1057–1068

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Table A.1 Required key parameters in the finite element models for square columns. Group

Specimen

fcc (MPa)

rk3 fcc (MPa)

Ecc (MPa)

εcc

11εcc

SHS1

SHS1C25 SHS1C40 SHS1C55 SHS1C70 SHS1C85 SHS1C100

25 40 55 70 85 100

10.64 14.79 17.03 17.76 18.28 21.50

23 500 29 725 34 856 39 323 43 332 47 000

0.003 0.003 0.003 0.003 0.003 0.003

0.033 0.033 0.033 0.033 0.033 0.033

SHS2

SHS2C25 SHS2C40 SHS2C55 SHS2C70 SHS2C85 SHS2C100

25 40 55 70 85 100

9.90 13.76 15.84 16.52 17.00 20.00

23 500 29 725 34 856 39 323 43 332 47 000

0.003 0.003 0.003 0.003 0.003 0.003

0.033 0.033 0.033 0.033 0.033 0.033

Table A.2 Required key parameters in the finite element models for rectangular columns. Group

Specimen

fcc (MPa)

rk3 fcc (MPa)

Ecc (MPa)

εcc

11εcc

RHS1

RHS1C25 RHS1C40 RHS1C55 RHS1C70 RHS1C85 RHS1100

25 40 55 70 85 100

9.90 13.76 15.84 16.52 17.00 20.00

23 500 29 725 34 856 39 323 43 332 47 000

0.003 0.003 0.003 0.003 0.003 0.003

0.033 0.033 0.033 0.033 0.033 0.033

RHS2

RHS2C25 RHS2C40 RHS2C55 RHS2C70 RHS2C85 RHS2C100

25 40 55 70 85 100

9.90 13.76 15.84 16.52 17.00 20.00

23 500 29 725 34 856 39 323 43 332 47 000

0.003 0.003 0.003 0.003 0.003 0.003

0.033 0.033 0.033 0.033 0.033 0.033

RHS3

RHS3C25 RHS3C40 RHS3C55 RHS3C70 RHS3C85 RHS3C100

25 40 55 70 85 100

9.90 13.76 15.84 16.52 17.00 20.00

23 500 29 725 34 856 39 323 43 332 47 000

0.003 0.003 0.003 0.003 0.003 0.003

0.033 0.033 0.033 0.033 0.033 0.033

depth-to-plate thickness ratio of the tube sections varied from 60 to 90. The concrete-filled lean duplex stainless steel tube specimens are subjected to uniform axial compression over the concrete and stainless steel tube to force the entire section to undergo the same deformations by blocking action. The ABAQUS 6.6 program, as a finite element package, is used in the current work. The results demonstrated that increasing the strength of the concrete core, for the same stainless steel tube, leads, generally, to a linear increase in the strength of the concrete-filled column, at least for the range of concrete strengths investigated herein (25–100 MPa). Moreover, it is noticed that the higher the strength of in-filled concrete, the sharper the load-axial shortening behaviour. This sharper behaviour enforces the column to undergo smaller deformations at relatively higher loads, extending to much higher ultimate loads. In addition, the comparison between concrete-filled carbon and lean duplex stainless steel stub columns illustrated that the lean duplex stainless steel material outfits the concrete-filled columns of high compressive strengths; fco > 55 MPa. Finally, the outcomes showed that the design rules specified in the ASCE are highly conservative for square and rectangular concrete-filled lean duplex slender stainless steel stub columns while they are conservative in the case of European specifications. Finally, a new design strength is proposed that is found to accord well with the finite element results, nevertheless, it is recommended to be examined through future experimental tests. On the other hand, the current work may be extended to study the effect of different lean duplex stainless cross-section classes (1–3), which would also be considered as introductory for this new material to be included into design standards.

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[23] ACI. ACI 318-95. Building code requirements for structural concrete and commentary. Detroit (USA): American Concrete Institute; 1999. [24] Saenz LP. Discussion of ‘equation for the stress–strain curve of concrete’ by P. Desayi and S. Krishnan. Journal of American Concrete Institute 1964;61: 1229–35. [25] Hu HT, Schnobrich WC. Constitutive modeling of concrete by using nonassociated plasticity. Journal of Materials in Civil Engineering 1989;1(4): 199–216. [26] Schneider SP. Axially loaded concrete-filled steel tubes. Journal of Structural Engineering, ASCE 1998;124(10):1125–38. [27] ASCE. Specification for the design of cold-formed stainless steel structural members (SEI/ASCE 8-02). American Society of Civil Engineers; 2002. [28] Ellobody E. Buckling analysis of high strength stainless steel stiffened and unstiffened slender hollow section columns. Journal of Constructional Steel Research 2007;63:145–55. [29] 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 Structures 2009;47(12):1544–56. [available online]. [30] Eurocode 3. EN 1993-1-5. Design of steel structures—part 1.5: plated structural elements. CEN; 2007. [31] Gardner L, Theofanous M. Discrete and continuous treatment of local buckling in stainless steel elements. Journal of Constructional Steel Research 2008; 64(11):1207–16.