Numerical modeling of lean duplex stainless steel hollow columns of square, L-, T-, and +-shaped cross sections under pure axial compression

Thin-Walled Structures 53 (2012) 1–8

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Numerical modeling of lean duplex stainless steel hollow columns of square, L-, T-, and þ -shaped cross sections under pure axial compression M. Longshithung Patton, Konjengbam Darunkumar Singh n Department of Civil Engineering, Indian Institute of Technology Guwahati, India

a r t i c l e i n f o

abstract

Article history: Received 5 August 2011 Received in revised form 2 January 2012 Accepted 4 January 2012 Available online 24 January 2012

In this paper, finite element (FE) studies for LDSS (Lean Duplex Stainless Steel) hollow columns with square, L-, T-, and þ -shaped cross sections (i.e., SHC, LHC, THC and þ HC respectively) are presented using ABAQUS, to gain an understanding of the cross sectional shape effects. The LDSS hollow columns having equal material cross-sectional areas with thickness varying from 5 m to 20 mm were subjected to uniform axial compression. Short/Stocky columns with lengths (  1800 mm) of about three times the outer width of square were considered for the analyses. Based on the analyses, it has been found that for all the NRHCs (i.e., LHC, THC and þ HC) considered, a nearly linear variation of Pu with section thicknesses has been observed, although the increase for SHC was relatively slower in thinner sections (t o 12.5 mm). The % increase in Pu with 300% increase in t (from 5 m to 20 mm) has been found to be  1273%, 1252%, 1041% and 679%, respectively, for square, L-, T-, and þ -shaped sections. The gain in Pu for LHC, THC and þ HC sections as compared (expressed as Pu/Pu(sq)) to SHC are in the range 120%–150%, 130%–170%, 140%–230%, respectively. Ductility at ultimate load has been observed to be  0.1–0.6% for SHC, LHC and THC (all thicknesses), and þ HC (tr 10 mm). & 2012 Elsevier Ltd. All rights reserved.

Keywords: Buckling Lean duplex stainless steel section Finite element modeling Square and NRCs sections

1. Introduction Construction industry is generally dominated by carbon steel due to low cost, long experience, applicable design rules and a large variety of strength classes; however it suffers inherently from comparatively low corrosion resistance and higher material cost. As an improvement over carbon steel, various stainless steel types can provide a very wide range of mechanical properties and material characteristics to suit the demands of numerous and diverse engineering applications, along with the advantages of not needing for surface corrosion protection in moderate to highly aggressive environments. Its main advantages include high corrosion resistance, high strength, smooth and uniform surface, esthetic appearance, high ductility, impact resistance and ease of maintenance and construction. These benefits have prompted a moderately upsurge of using stainless steel in construction industry in the recent years. Traditionally, in the constructional industry, austenitic steel grades are used prominently. However, with increasing nickel prices (nickel content of  8%–11% in austenitic stainless steel) there is an escalation in the demand for lean duplexes stainless steel (LDSS) with low nickel content of 1.5%, such as grade EN 1.4162 [1–3]. LDSS grade EN 1.4162 in particular, with twice the mechanical strength of conventional austenitic and ferritic stainless steel, has a potential for use in

n

Corresponding author. Tel.: þ91 361 258 2423; fax: þ 91 361 258 2440. E-mail address: [email protected] (K.D. Singh).

0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2012.01.002

constructions, and its use has seen significant growth and development over the last 20 years. The prime movers for this development have been soaring raw material costs, such as nickel, along with increasing demand for improved corrosion resistance and strength, enabling a reduction in section sizes leading to higher strength to weight ratios. As such, several recent investigations have been attempted on both normal and high strength stainless steel hollow (tube) columns with square, rectangular, circular and elliptical sections [4–11]. In addition to the aforementioned conventional cross-sectional shapes of columns, it is worthwhile to mention that, in the past two decades, the construction industry has shown increasing interest in the use of Non Rectangular Columns (NRCs) (e.g., L-, T-, and þ-shaped cross-sections) especially for reinforced concrete columns [12]. Compared with columns with rectangular/square cross-sections, columns with NRCs have the advantage of providing a flushed wall face, resulting in an enlarged usable indoor floor space area and also in making the interior space more regular. So, NRCs (reinforced concrete) were applied in residential high-rise buildings and were welcomed by architects. These NRCs with L- or T-sections also share unique advantages of having large stiffness and strength in the direction of their longer sections. Whilst several experimental and numerical studies (e.g., [13,14]) on the behavior of reinforced concrete NRCs with L- or T-sections was reported, to the best of authors’ knowledge, there is an apparent lack of systematic studies relating to NRCs hollow columns, particularly for LDSS. Hence, in this paper an attempt has been made to investigate the behavior and strength of LDSS

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hollow columns by employing finite element approach. The main objective of this study is to explore and compare the behavior (e.g., ultimate load and axial deformation at ultimate load) of LDSS square hollow column (SHC) with those of Non-Rectangular Hollow Columns, or NRHCs viz., L-shaped hollow column (LHS), T-shaped hollow column (THC), and þ-shaped hollow column ( þHC) (see Fig. 1) under pure axial compression, but noting that all these column types have equal material cross-sectional areas. First FE analyses on fixed-ended cold formed LDSS for square and NRHC are presented.

2. Finite element modeling 2.1. General In the present work, general purpose commercial finite element software ABAQUS (Version 6.9 EF1 [15] has been used to conduct various parametric studies (e.g., effect of material cross sectional shapes and thicknesses) on load carrying capacity of LDSS NRHCs subjected to pure axial compression. In the initial part of the study, FE analyses have been performed for a square shaped column using various reported material models (e.g., Ramberg–Osgood [16,17] model and its variants [18,19]). By comparing FE results with that of the available experimental data from the literature [20], a suitable material model has been chosen for subsequent analyses for the determination of ultimate

loads and axial deformation/shortening. The FE modeling details are shown in the subsequent sub-sections. 2.2. Geometry and boundary conditions The present modeling approach follows those published in the literature [20–22]. Fig. 2a shows a typical geometry of the LDSS SHC with square cross-section. The bottom part of SHC is fixed while the top loaded part is allowed for free vertical translation (i.e., along the column length direction). The boundary conditions were accomplished using two rigid plates that were tied to the column ends via surface-to-node tie constraints available in ABAQUS [15]. The rigid body reference nodes associated with the rigid plates was then used to restrain all degrees of freedom apart from vertical translation at the loaded end, which was constrained via kinematic coupling to follow the same vertical displacement. A central concentrated normal load was applied statically at the upper rigid plate reference node (RP-2) (Fig. 2(b) and (c)) using displacement control, thus applying uniform pressure at the top edges of the column through the rigid plate. 2.3. Finite element types Shell elements were employed to discretise the models. Fournoded doubly curved shell element with reduced integration S4R [15] with six degrees of freedom per node and known to provides

D

D/ 2

B

t

t

t

D /2

t

D

B/2

D/2

B

B /2

B

B /2

Fig. 1. (a) SHC; (b) LHC, (c) THC, and (d) þ HC (or NRHCs).

Rigid plate

Reference Point Load

Tie constraints

LDSS (SR4 elements)

Square hollow column (SHC)

Fig. 2. Typical Geometry and FE boundary conditions of LDSS SHC.

M.L. Patton, K.D. Singh / Thin-Walled Structures 53 (2012) 1–8

accurate solutions to most applications (for both thin and thick shell problems) has been utilized in this study. The aspect ratio of the element is kept at 1.0 in all the FE models (Fig. 2b). The number of S4R elements used in the analyses for various models ranges from 2400 to 10,000. Linear eigenvalue buckling analysis was then used to achieve mesh convergence on all the FE models under study, by monitoring the first eigen buckling mode with mesh refinement. Typical first eigen modes for SHC and NRHCs are shown in Fig. 8. 2.4. Local geometric imperfection The local geometric properties proposed by Gardner [23] for stub columns have been employed in the FE models. Linear eigenvalue buckling analyses were initially performed to extract the buckling mode shapes. These served as initial geometric imperfection patterns used in the subsequent geometrically and materially non-linear analyses. Modified Riks method [15], (a variation of the classical arc-length method [24–26]), was employed for the non-linear analyses to capture the full (i.e., both pre and post ultimate load) load–deformation response. The first (i.e., lowest) local buckling mode shape was then utilized to perturb the geometry of the columns, by scaling with local imperfection amplitude [23] given by Eq. (1).   s0:2 w0 ¼ 0:023 ð1Þ

scr

t

where, s0:2 is the 0.2% proof stress and scr is the elastic critical buckling stress determined from the buckling analysis given by Eq. (2).   Dl ð2Þ scr ¼ E:E ¼ E:l: L wherel is the eigenvalue obtained from the results of FE analysis, Dl is the initial displacement at the movable end input in the boundary conditions, and L is the length of the column. 2.5. Material modeling The minimum specified material properties of LDSS Grade EN 1.4162 according to EN 10088-4 [3] are 0.2% proof stress (s0.2) of 530 MPa and ultimate stress (su) of 700–900 MPa. The Poisson’s ratio was taken as 0.3. The compressive flat material properties given by Theofanous and Gardner [20] (Table 1) were used in deriving the stress–strain curve of LDSS material through three models viz., Ramberg–Osgood [16,17], Rasmussen [18], and Gardener and Asraf [19] models, as discussed in the following sub-sections. The last two models are modified versions of the original Ramberg–Osgood. These models are tested in the present FE analyses by comparing with an experimental result from Theofanous to Gardener [20] to select a suitable model appropriate for the present work.

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parameters e.g., Young’s modulus (Eo), and material constants K and n. Considering the proof stress as the stress value associated with 0.2% plastic strain the Ramberg–Osgood expression is generally given by Eq. (3).   s s n E ¼ þK ð3Þ s0:2 E0 where K and n are the material nonlinearity indices of the stress– strain behavior. The degree of non-linearity varies with different grades of stainless steel. Using K ¼0.002, Eq. (3) has been reported to give reasonably good predictions of stainless steel material stress–strain behavior up to the 0.2% proof stress (s0.2), but observed to overestimate the corresponding stress beyond that level (Fig. 3).

2.5.2. Rasmussen model Rasmussen [18] adopted Mirambell and Real’s [27] modified Ramberg–Osgood model by using an expression beyond s0.2 for the complete stress–strain curve for stainless steel alloys (See Eq. (4)).   ðss0:2 Þ ss0:2 m E¼ þ Eu  þ Et0:2 ð4Þ E0:2 su s0:2 where, su, eu, m, and E0.2 are the ultimate tensile strength, ultimate strain, additional strain hardening exponent and the tangent stiffness at s0.2 respectively (See Fig. 3). E0.2 is given Eq. (5). E0:2 ¼

s0:2 E0 s0:2 þ 0:002nE0

ð5Þ

The additional strain hardening exponent m proposed by Rasmussen [18] is given Eq. (6); m ¼ 1 þ 3:5

s0:2 su

ð6Þ

The ultimate tensile strength (su), ultimate strain (u) are determined using Eqs. (7) and (8) in terms of s0.2, E0 and n.   0:2 þ 185 sE0:2 s0:2 0 ð7Þ ¼ su 10:0375ðn5Þ

Eu ¼ 1

s0:2 su

ð8Þ

EN 1993-1-4 (Annexure C) [28] adopted the model proposed by Rasmussen [13] as a guideline for material modeling of stainless steel.

2.5.1. Ramberg–Osgood model The Ramberg and Osgood model [16] (Eq. (3)) is a popular material model which provides a smooth stress–strain curve for all values of strain for nonlinear materials with fewer physical Table 1 Compressive flat material properties (Theofanous and Gardner, 2009). Cross-section

80  80  4-SC2

E (Mpa)

197,200

s0.2

s1.0

(MPa)

(MPa)

657

770

Compound R–O coefficients n

n0 0.2,1.0

3.81

3.6

Fig. 3. Illustration of the stress–strain curve of the Ramberg–Osgood model with test results (Ashraf et al., 2006) for an austenitic Grade 1.4301 tensile coupon with s0.2 ¼296 MPa and n¼ 5.8.

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2.5.3. Gardner and Ashraf model In the absence of necking phenomena in compression, Gardner and Nethercot [29] noted that the two stage model given in Eq. (4) was limited to the description of tensile stress–strain behavior because of its dependency on the ultimate stress su and the corresponding strain Eu. Hence, Gardner [23] proposed that 1% proof stress, s1.0 and corresponding strain, Et1.0 be adopted in place of the ultimate stress. The resulting model as recently proposed by Gardner and Ashraf [19] is given by Eq. (9), which applies for stresses greater than s0.2.   ðss0:2 Þ s1:0 s0:2 E¼ þ Et1:0 Et0:2  E0:2 E0:2  n00:2,1:0 ss0:2 þ Et0:2 ð9Þ  s1:0 s0:2 where t0.2 and t1.0 are the total strains at s0.2 and s1.0, respectively 0 and n 0:2,1:0 is a strain hardening exponent. Eq. (9) has been found to give good agreement with measured stress–strain curves in both tension and compression. The validation of the above mentioned three models are shown in Section 2.6. 2.6. Validation of material models In order to validate and compare the three material models mentioned in Section 2.5, the experimental result obtained by Theofanous andGardner [22] for LDSS (Grade EN 1.4162) square hollow-section columns (80  80  4-SC2) has been considered as the benchmark. The measured dimensions and the LDSS material properties (Theofanous and Gardner [20]) for both corner and flat portions are given in the Tables 1 and 2, respectively. It may be noted that this column has been formed using cold-form technique at the corners resulting in different properties between flat and corner portions of the column. Fig. 4 shows the stress–strain plot of the LDSS material used by Theofanous and Gardner [20]), using the materials models viz., Ramberg–Osgood, Rasmussen and Gardner and Ashraf models. It can be seen that beyond the

proof stress (s0.2) the Ramberg–Osgood model predicts higher stress values as compared to both and Gardner and Ashraf and Rasmussen models. The material models shown in Fig. 4 are then used as input parameters to ABAQUS, by converting into true stresss (strue) and true plastic strains (strue ) using the following Eqs. (10) and (11).

strue ¼ snom ð1 þ nom Þ epl true ¼ lnð1 þ nom Þ

ð10Þ

strue E0

where, snom and nom are engineering stress and strain respectively. Imperfection magnitude was seeded in the FE model using Eq. 1. The deformed shape of a typical FE model is shown in Fig. 5. It can be seen from Fig. 5 that the column fails by local buckling. From Fig. 5, it may be noted that the local buckling occurs near the middle length consistent with other FE [2,15] and experimental studies reported in the literature [5,20]. Fig. 6 presents the load-end shortening curve for the LDSS hollow columns (80  80  4-SC2). The variation of load with axial displacement of the top portion of the column is presented in Fig. 6 for various material models discussed above. It is observed all the three models predicts similar trend up to a load of  800 kN. It can be seen from Fig. 6 that compared to Ramber–Osgood and Rasmussen models, Gardner and Ashraf model predicts closer ultimate load (   0.18%) and axial displacement at peak load (   3.85%) than those of the experimental values [20]. Hence for all the subsequent analyses the material model proposed by Gardner and Ashraf [19] is used.

Local buckling

Table 2 Stub column dimensions (Theofanous and Gardner, 2009). Specimen

L (mm)

B (mm)

H (mm)

t (mm)

ri (mm)

w0 (mm)

80  80  4-SC2

332.2

80

80

3.81

3.6

Eq. (1)

L¼ length, B¼ width, H¼ depth, t¼ thickness, ri ¼ internal corner radius, w0 ¼ local geometric imperfection. Fig. 5. Typical FE deformed shape (80  80  4-SC2).

Fig. 4. Stress–strain curves using (a) Ramberg–Osgood, Rasmussen and Gardner & Ashraf material models (LDSS of Grade EN 1.4162 [17]).

ð11Þ

Fig. 6. Variation of load with axial displacement (80  80  4-SC2).

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2.7. Parametric study LDSS hollow columns Various SHC and NRHCs with equal material cross-sectional areas and having same thickness (t) were considered to investigate the effects of cross-sectional shape on the behavior of these stub columns under pure axial compression. Fig. 7 shows the cross-sections considered in the parametric studies for the SHC and NRHCs. The SHC section has a side width of 600 mm. The maximum side (distance between opposite faces) width of all the NRHCs are maintained equal to that of the square section, however the smaller side width is made equal to half the larger width i.e., 300 mm. Altogether, 28 FE models have been studied. The FE modeling method is similar to those discussed in the previous sections. The first eigen buckling modes (Fig. 8) are then used along with local amplitudes proposed by [23] (Eq. (1)) to simulate the local geometric imperfections. The column lengths were set equal to three times the maximum outer dimension (3  600 mm¼ 1800 mm), to avoid the effects of flexural buckling and end conditions, while the thicknesses were varied from 5 mm to 20 mm to encompass a wide range of cross-sectional slenderness ratios. To simplify the modeling process and also due to the lack of experimental material properties for the corner part (in case of cold work), together with the possibility of welding at the corners, corners of the square and NRCs sections are assumed to be at right angles (i.e., corner radii as in the validation (Section 2.6) are not considered). Further, it is assumed that the material properties are same both at the flat and corner regions. Similar material properties as that found out by Theofanus and Gardner [20] are adopted for the FE modeling (Table 1). Results are presented in Fig. 9–13

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followed by hardening prior to buckling. It can be observed from Fig. 10(a) that the ultimate strength of the column increases as it goes from square, L-, T-, to þ-shaped sections. This can be related to the higher stiffening effects of the corner regions and the potential for stress redistribution once local buckling of the wider face plate occurs. As the numbers of corners (right angled joints) are increased the ultimate load also increases. It can be primarily noticed that as it goes from square to þ-shaped section, the sharper is the load–axial deformation behavior (for t¼5 mm). This sharper load-deformation behavior enforces the column to undergo smaller deformations at relatively higher loads. From Fig. 10(a), it can be seen that þHC has the highest ultimate load

3. Results and discussion

Fig. 8. First eigen buckling modes for (a) SHC, (b) LHC, (c) THC, and (d) þ HC.

3.1. Deformed shapes at ultimate load Von-Mises stress (superimposed on deformed shape) at ultimate load are shown in Fig. 9 for SHC and NRHCs (t¼ 5 mm). From Fig. 9, the onset of local buckling modes at ultimate load can be seen along with distribution of von-Mises stress over the surface. 3.2. Variation of ultimate load Variation P/Agfy (where P, Ag, and fy are load, material cross sectional area, and yield stress respectively) with axial displacement is shown for SHC and NRHCs for both 5 mm and 20 mm thick columns is shown in Fig. 10(a). It may be noted that as the perimeter of the cross-sections are same for all the types of sections considered the quantity Agfy remains constant. The term Agfy can be considered as the global yielding load. The ratio of Pu/Agfy o1.0 (Pu is the ultimate load) indicates that the sections are slender and can buckle prior to yield [30]. For Pu/Agfy Z1.0 (i.e., stocky sections), yielding precedes buckling, and may be

Fig. 9. Von-Mises stress (superimposed on deformed shape) at ultimate load of (a) SHC, (b) LHC, (c) THC, and (d) þ HC (t ¼ 5 mm).

300

600

t

t

300

300

300

t

600

600

Fig. 7. Specimen cross-sections. (a) SHC ; (b) LHC, (c) THC, and (d) þ HC.

300

600

t

600

300

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Fig. 10. (a) Variation of P/Agfy with axial displacement for SHC and NRHCs (t ¼ 5 mm and 20 mm). (b) Variation of load with axial displacement for þHC for different thicknesses. (c) von-Mises stress (superimposed on deformed shape) at ultimate and post ultimate loads for t ¼5 mm (a) and (c) and 20 mm (b) and (c) thick þ HC.

Fig. 11. Variation of ultmate load (Pu) with section thicknesses.

(Pu), with (Pu(sq) oPu(L) oPu(T) oPu( þ ) where subscripts sq, L, T andþdenotes the corresponding sections). For 5 mm thick sections an increase in sharpness of the load-displacement curve at ultimate load can also be observed as the cross-section shapes are changed from square -L-T-þ (Fig. 10(a). These kinds of

behavior are found to be typical for 10 mm, 12.5 mm, 15 mm and 17.5 mm sections for SHC, LHC, THC (not shown in Fig. 10(a) for clarity). However, it can be seen that for 20 mm thick sections, it is quite apparent that the sections have become stocky and global yielding (Pu/Agfy Z1.0) takes place prior to buckling (except for SHC where Pu/Agfy  1.0). Again an increase of Pu as the cross-section shapes are changed from square -LT-þ are seen. In the case of þHC, a distinct pattern can be seen in the load-deformation curve, indicative of hardening before buckling i.e., pre-ultimate load and a flatter (plateau) behavior in the post-ultimate load. To gain an insight of the previous observation only for þHC, load-deformation curves are plotted for different thicknesses (t ¼5 mm, 10 mm, 12.5 mm , 15 mm, 17.5 mm and 20 mm) in Fig. 10(b). It can be seen that for thinner sections (tr 10 mm or t/Dr3.3%), the occurrence of sharper peak at Pu is seen, similar to those observed for t ¼5 mm (SHC, LHC, THC and þHC sections) in Fig. 10(a). However, for t Z12.5 mm or t/D Z4.1%), the loaddeformation curve near Pu become flatter for thicknesses greater than 12.5 mm. In the first instance this may be related to the enhanced stiffness due to (a) increasing number of corners (12 corners in þ shaped sections) and (b) higher t/D ratios (thereby increasing stockiness). This led to an increased contributing faces for a possible positive interaction amongst neighboring faces in

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Fig. 12. (a) Variation of ultmate load (Pu) with cross-sectional shapes. (b) Variation of % Pu/Pu(sq) with sectional shapes.

delaying local buckling failure via a hardening process. Hence a competing effect between (a) yielding (identified with Pu/Agfy) and (b) side faces interaction can now be observed which influences in both the ultimate load and deflections at ultimate load. The effect for thickness on the von-Mises stress (superimposed on deformed shape) at ultimate and post ultimate loads are shown in Fig. 10(c) for both 5 mm (Figs. 10(ca) and (cc)) and 20 mm (Figs. 10cb and cc) are shown for þshaped section. At ultimate load (deflection at ultimate load, d ¼ 8.38 mm), for 5 mm thick section, an uneven (wobbly pattern) distribution of vonMises stress (Fig. 10(ca)) can be observed, whilst for 20 mm thick section, almost all the portion has been stressed uniformly (Fig. 10(cb)). Earlier appearance of uneven distribution of stress helps in the achievement of localized buckling (Fig. 10(cc)) for thin (e.g., 5 mm thick section) sections in contrast to thicker sections (e.g., 20 mm thick). In the case of 20 mm (thicker section), a relatively even distribution of von-Mises stress could be seen even at ultimate load due to the hardening effect caused by side face interactions. A mobilization of von-Mises stress in almost the entire column can be seen in Fig. 10(cb). As a result, a longer plateau can be observed at higher thickness ( 412.5 mm), with localized buckling taking place at a much later (hence delayed) post ultimate load, as compared to thinner sections where a very sharp and well defined ultimate load could be seen. The variation of Pu with section thicknesses are shown in Fig. 11 for all the cross-sections tested. It can be seen from Fig. 11 that the variation of Pu with t is nearly linear (R2 ¼  0.99) for NRHCs, in comparison to SHC where the increase in Pu is relatively slower in thinner sections (t o12.5 mm). The % increase in Pu with 300% increase in t (from 5 mm to 20 mm) is order

Fig. 13. (a) Variation of % deflection at Pu for different sectional shapes. (b) Variation of % deflection at Pu with thicknesses.

 1273%, 1252%, 1041%, and 679% for SHC, LHC, THC, and þHC, respectively. For SHC, results from EN 1993-1-4 are also plotted for comparison. It can be seen that although the shape of the Pu vs t from the present FE results is similar, it can be observed that FE results over-predicts the Pu as compared to that of EN 1993-1-4, as also reported for stub columns in the literatures [6,20]. Fig. 12 shows the variation of Pu with sectional shapes (SHCLHC-THC- þHC) for different thicknesses. It can be observed from Fig. 12 that, for all the thicknesses considered, there appears to be a linear increase in Pu as the sections are changed from SHC-LHC-THC- þHC. For changes in section from square toþshape, the increase in Pu are  136%, 114%, 96%, 72%, 48%, 25%, and 30%, respectively for 5 mm, 10 mm, 12.5 mm , 15 mm, 17.5 mm and 20 mm, showing that the increase in Pu is more effective at thinner sections (to12.5 mm) where the increase in more than 70%. Variation of Pu/Pu(sq) with sectional shapes (LHC-THC-þHC) are shown in Fig. 12(b). It can be seen that as compared to square section the gain in Pu for L, T, and þsections are in the range 120%–150%, 130%–170%, 140%–230%, respectively. Thus it can be seen that by switching over to þshape sections, significant gain in ultimate strength can be obtained for all the thicknesses considered, with thinner sections giving more pronounce gain. Thus, it is possible to achieve higher buckling load of LDSS hollow columns with increasing number of corners/sides rather than increasing thickness under pure axial compression scenario, providing a potential application for a much lighter structures and hence economizing cost of construction.

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3.3. Variation of axial deformation at ultimate load Variation of % deflection (d/Ln100%; L¼ 1800 mm and d is the deflection) at Pu are plotted in Figs. 13(a) and (b), to give a measure of ductility at ultimate load. It can be observed that % deflection¼  0.1%–0.6% for SHC, LHC and THC (all thicknesses), and þHC (t r10 mm). Only for thicker sections (t Z12.5 mm) of þHC, a significant % deformation is seen ( Z1.0%). It can be seen that as discussed in the previous section (refer Figs. 10(b) and (c), this behavior is related to the elongated plateau of the loaddeformation plot, as a result of the increasing number of corners along with t/D ratio, leading to significant increase in stiffness.

4. Conclusions Finite element analyses of LDSS (Lean Duplex Stainless Steel) hollow short/stocky columns with square, L-, T-, and þ-shaped cross sections (i.e., SHC, LHC, THC, and þ HC respectively) for various thicknesses are presented using ABAQUS, to understand cross-sectional shape effects on both ultimate load and ductility at ultimate load. Based on the study following conclusions are identified: 1) For all the NRHCs (i.e., LHC, THC, and þHC) considered, a nearly linear variation of Pu with section thicknesses has been observed, although the increase for SHC was relatively slower in thinner sections (to12.5 mm). The % increase in Pu with 300% increase in t (from 5 mm to 20 mm) is order  1273%, 1252%, 1041% and 679% for SHC, LHC, THC, and þHC, respectively. 2) A nearly linear increase in Pu as the sections are changed from square -L-T-þ has also been seen. For changes in section from square toþshape, the increase in Pu are  136%, 114%, 96%, 72%, 48%, 25%, and 30%, respectively, for 5 mm, 10 mm, 12.5 mm , 15 mm, 17.5 mm, and 20 mm, showing that the increase in Pu is more effective at thinner sections (to12.5 mm) where the increase in more than 70%. 3) The gain in Pu for LHC, THC, and þHC sections as compared (expressed as Pu/Pu(sq)) to SHC are in the range 120%–150%, 130%–170%, 140%–230%, respectively, showing that by switching over toþshape sections, significant improvement in ultimate strength can be obtained for all the thicknesses considered, with thinner sections giving more pronounce gain. 4) Ductility at ultimate load (expressed as d/Ln100%; L¼1800 mm and d is the deflection) has been observed to be  0.1%–0.6% for SHC, LHC and THC (all thicknesses), and þHC (t r10 mm). Only for thicker sections (tZ12.5 mm) of þHC, a significant % deformation is seen ( Z1.0%).

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