Structural behavior of intermediate length cold-formed steel rack columns with C-stitches

Abstract

This article presents an experimental and numerical investigation on the strength and performance of intermediate length rack column sections with C-stitches under axial compression. The test program consisted of 10 axial concentric compression tests on columns with and without C-stitches under pin end conditions for two different geometric lengths. Finite element (FE) models were developed using commercial FE package ABAQUS considering material and geometric nonlinearities as well as initial geometric imperfections. The elastic buckling properties of the section were calculated using readily available linear elastic buckling analysis tools based on Generalized Beam Theory (GBT) and Finite Strip Method (FSM). Obtained FE results were compared with those obtained experimentally, and once verified the developed FE modeling technique was used to carry out a parametric study to examine changes in structural response due to variations in length, depth and spacing of C-stitches. Observed influences of C-stitches on the behavior and resistance of the considered columns were carefully analyzed, and key design aspects are presented herein.

Keywords

cold-formed steel columns / C-stitches / intermediate length columns / distortional buckling

Introduction

In compression, thin-walled steel open cross-sections primarily demonstrate three modes of instability such as local, distortional, and flexural/flexural-torsional buckling, although the combination of these modes is a very common occurrence [¹–³]. Local buckling occurs at relatively short length, while global buckling takes place at long half wavelengths. Distortional buckling, on the other hand, usually occurs at intermediate half wavelength [⁴]. Distortional buckling, also known as “Stiffener buckling” or “Local torsional buckling” is a mode characterized by a rotation of the flange at the flange-web junction in members with edge-stiffened elements. Distortional post-buckling behavior exhibits considerable asymmetry on the flange-stiffener motion, either inward or outward. Distortional buckling plays a significant role in the use of open cold-formed steel rack columns, and typically governs the load carrying capacity of intermediate length columns [⁵]. The current study proposes a technique to either eliminate or delay the distortional buckling mode for such columns by using appropriate C-stitches.

Distortional buckling for steel storage rack sections was first investigated by Hancock [⁶], and design charts were proposed for computing the relevant critical buckling stress. Lau and Hancock [⁷] proposed simplified formulations to calculate the elastic distortional buckling stress for thin-walled sections with edge-stiffened elements. Lau and Hancock [⁸] also devised design strength curves based on Johnston parabola for distortional buckling using experimental evidences in the inelastic distortional buckling range; the proposed curves, however, did not account for the post-buckling behavior in distortional buckling mode. Kwon and Hancock [⁹] conducted tests on channel sections, with and without intermediate web stiffeners produced from high strength steel demonstrating substantial post-distortional buckling strength; this led them to propose new distortional buckling strength equation for intermediate length columns.

Generalized Beam Theory (GBT) was used by Davises and Jiang [¹⁰] to analyze either individual or selected combinations of individual buckling modes. GBT allows precise expressions to be derived for the critical stress and buckling half wavelength for distortional buckling. Narayanan and Mahendran [¹¹] conducted both experimental and numerical analyses to study the structural behavior of intermediate length columns with innovative and complex cross-section types, whose failure were dominated by distortional buckling. The investigation showed that further research is required for innovative cold-formed steel columns made of thin, high strength steels to improve the accuracy of the design code predictions for distortional buckling strength.

In practice, cold-formed steel columns are manufactured with built-in stiffeners to resist the failure in local buckling to a certain extent leading to failure in distortional buckling mode. By making open sections into partially closed ones, significant improvement in torsional stiffness can be achieved. This technique improves the distortional strength of intermediate length thin-walled open column sections [¹²–¹³].

A series of studies were conducted and reported by Anbarasu et al. [¹⁴–¹⁷] on different innovative open sections for improving the distortional buckling behavior of intermediate open columns using lateral stiffeners/spacers/connectors. In previous studies, simple plate elements were used as lateral stiffeners to connect the lip of the section to improve the distortional buckling strength of columns. But the present study proposes use of C-stitches as an alternative to plate elements to prevent the rotation of flanges to tackle premature distortional buckling. Currently available literature shows that investigations reported on the strength and behavior of rack column sections with C-stitches are very limited, and no previous study focused on the distortional buckling behavior of rack column sections with C-stitches as lateral stiffeners. The present study aims to provide some useful insight into this common failure mode observed in intermediate length columns with specific reference to rack column sections with C-stitches.

The primary objective of this research is to gain an improved understanding of the behavior and resistance of intermediate length rack columns with C-stitches under axial compression through experimental and numerical investigations. Finite element (FE) models were developed using FE package ABAQUS [¹⁸] to simulate the behavior of the considered rack columns. Results obtained from FE models were validated using test results conducted on rack columns. As part of the current study, a total of 10 columns were tested for two different lengths (6 columns with 1250 mm length and 4 columns with 1850 mm length). A comprehensive parametric study was carried out to investigate the influence of C-stitches on the behavior and resistance for a number of chosen intermediate length columns with different depth and spacing for C-stitches. Obtained experimental and numerical results have been used herein to explore the behavior and strength of columns stiffened with C-stitches.

Experimental program

Selection of cross-sectional dimensions

As the current study primarily focuses on distortional buckling issues, geometrical dimensions, particularly the plate slenderness ratio (b/t) of the selected section, were chosen to eliminate the possibility of local buckling based on the limits suggested in AS/NZS 4600-2005 [¹⁹]. The ends of the columns were considered as pinned with warping restraints. The cross-section out-of plane distortion was restrained at the ends by warping restrained end condition. The pin-pin warping fixed boundary condition was reported to have a significant impact on the effective length in torsion (kL₃ = 0.5 kL) [²⁰]. Length and geometric dimensions of the sections were selected using the buckling plot as obtained from GBTUL code [²¹], which is based on GBT, to ensure that distortional buckling occurs at the chosen intermediate length range.

Figure 1 shows the variation of the critical buckling load P_cr with length L (logarithmic scale) for the considered pin-ended rack column section with and without warping restraints. These curves were obtained using finite strip analysis CUFSM [²²] and GBT based GBTUL as detailed in Bebiano et al. [²¹]. Figure 2 shows the geometric dimensions of the column section considered in the current study. Elastic buckling curves obtained from CUFSM and GBTUL for pinned columns with warping free end condition showed that the lowest distortional buckling load was achieved at a length of 1269.3 mm. From Fig. 1, it was observed that the warping fixed end condition improved the elastic distortional buckling when compared against that for warping free end condition. However, the observed increased resistance against distortional buckling achieved for warping fixed ends in pin-ended condition gradually decreased with increase in column length (up to 4000 mm).

The buckling curve for pin-ended column with warping fixed ends exhibited four distinct zones: local buckling mode (L<600 mm), single half-wavelength distortional buckling mode (600<L<2300 mm), multiple half-wavelength distortional buckling mode (2300<L<85000 mm), and global (flexural-torsional) buckling mode (L>8500 mm).

The distortional buckling loads (P_crd) calculated for the chosen column section were 138.7, 102.1, 98.81, and 83.94 kN corresponding to the half-wavelengths of 1250, 1800, 2300, and 2800mm, respectively.

Test specimens

Ten specimens were fabricated by press-braking operation from locally available cold-rolled sheets with 1.56 mm thickness; six specimens were of 1250 mm length, while the remaining four were of 1800 mm length. Length of the specimen was kept parallel to the rolling direction. Both ends of the specimens were milled flat and then welded to 12 mm thick steel end plates for the specimens to achieve warping fixed end sections. C-stitches were made of CFS sheet in C-shape, which were used as transverse stiffeners to connect the edge stiffened element and lips of the rack column sections using self-driving screws. Each of the C-stitches was connected to the lip and the rear flanges of the section by using four 5 mm diameter self-drilling screws as shown in Fig. 3. The fabricated columns were tested with and without C-stitches to investigate the difference in failure characteristics.

Centre line dimensions of specimen cross-section were calculated using the measured overall dimensions. Table 1 shows the average measured cross-sectional dimensions of the test specimens for 1250 and 1800 mm length series.

The specimens are labeled in such a way so that the specimen length, number of C-stitches and the depth of C-stitches could be easily identified. For example, the label “1300-C1-D25” defines the specimen as follows: the first term “1300” indicates the length of the specimen in mm; the second term “C1” indicates the number of C-stitch as one, and the last term “D25” refers to the depth of the C-stitch as 25 mm.

Material properties

Material properties of CFS sheet used in producing the test specimens were determined by tensile coupon tests. Coupons were taken from the longitudinal direction of the CFS sheet used to fabricate column specimens. Tensile coupons were tested in accordance with Australian Standard AS1391 [²³]. Engineering stress-strain curve obtained from coupon test is shown in Fig. 4. Table 2 shows the average material properties obtained from three tension coupon tests carried out as part of the current study.

Test setup and the experimental procedure

Tests were carried out using a loading frame with 1000 kN capacity and hydraulic jacks with a capacity of 400 kN. All necessary details of the test setup have recently been reported by Anbarasu and Murugapandian [²⁴]. The specimens were press-braked from locally available cold-rolled steel sheet of 1.56 mm thickness. The ends of the test specimens were machined to ensure a perfect contact with 12 mm thick steel end plates, which were welded to the specimens to ensure uniform compression of the column.

A load distribution plate was also placed between the load cell and the specimen. Hinged end conditions for the specimens were achieved by placing a spherical ball arrangement between the end plates and the platens as shown in Fig. 5. The effective length of the column was taken as the geometric length plus the depth of two pinned ends, i.e., 1250 mm+ 38 mm (unloaded end) + 24 mm (loaded end) = 1312 mm. Once the specimens were mounted between the platens, their vertical alignments were carefully checked by using a theodolite. Specimens used in the current study had sharp corners, and hence any effect due to cold-work done at the corner regions was considered negligible. The complete experimental test setup for one of the columns is shown in Fig. 5. A pre-load of less than 5 kN was applied to the specimen to ensure that the specimen was in full contact with the end plates; this procedure also helped to hold the test setup in position as well as to eliminate any possible gaps and to retrain any movements between the end plates and the specimen.

The deflection in X and Y directions and the axial deformation in Z direction were measured by using LVDTs. Two LVDTs were placed at L/2 distance, one at the middle of flange, while the other at the middle of the web of the specimen to measure the deflection in X and Y direction. An additional LVDT was placed over the base plate to obtain the axial shortening, i.e., deformation in Z direction. A data acquisition system was used to record the applied load and the readings from transducers at regular intervals during tests. Both lateral and axial deformations of column specimens were recorded for every increment of uniformly applied load using hydraulic jack. Load was applied until the specimens completely failed, and the failure modes were carefully recorded for further investigation.

Test results

The ultimate test load carrying capacities and the corresponding deformed shapes obtained from tests are presented in Table 3. Load versus axial shortening behavior for all specimens were recorded and are shown in Fig. 6(a). It is clearly observed from Fig. 6 that the axial stiffness of specimens increased with increase in number of C-stitches.

As expected, columns without C-stitches for both 1250 and 1800 mm series failed in distortional (D) buckling mode, as presented in Fig. 7; flange lips moved outwards (O-O) with respect to the flange web junction at mid-height in the observed failure mode. 1250-C1-D50 section failed in combined distortional (D) and flexural (F) buckling. Distortional buckling occurred at 1/3 height from the top, while flexural buckling was noticeable at long half wavelength. Introduction of 2 C-stitches in 1250-C2-D50 sections eliminated distortional mode, and the section failed predominantly in flexural buckling. However, some traces of local buckling were visible in flanges near the C-stitches. The deformation of the specimens’ 1250-C0-D0 and 1800-C0-D0 observed at mid-length of the column are shown in Fig. 7.

Specimens with 3, 4, and 5 Nos. of C-stitches in 1250 series showed very localized failure modes. Due to warping-restrained end sections, the failure mechanism also involved the partial yielding of zones close to the column ends as shown Fig. 7(a). The ultimate resistances were almost the same as can be seen from Table 3 demonstrating that changes in cross-section geometry were irrelevant to resistance as the material yielded. There was no evidence of flexural buckling; minor distortional deformations were observed between adjacent C-stitches.

1800-C1-D50 section failed in distortional buckling with flange lips moving outward at mid-height above the C-stitch and moving inward at around mid-height below the C-stitch. 1800-C2-D50 and 1800-C3-D50 sections failed in combined distortional and flexural buckling. No local buckling was observed but both flanges moved in unison at one of the C-stitches triggering failure; this could be categorized as localized flexural buckling. At the early stages of loading (elastic range), all columns exhibited clear distortional buckling deformed configurations. Finally, the distortional deformations were accompanied by local and global buckling modes.

FE analysis

General

FE models were developed for the tested columns by using commercial FE package ABAQUS [¹⁸]. The models were based on the measured centerline dimensions of the tested cross-sections. Two types of residual stresses, i.e., membrane (constant through the thickness) and flexural (varying linearly through the thickness) stresses typically exist in press-braked CFS sections. Flexural residual stresses are generally re-introduced in coupon tests, and are therefore accounted for in FE analysis when material properties are taken from coupons cut from within cross-sections. On the other hand, membrane residual stresses are generally small in cold-formed steel members, and have been reported to have insignificant effect and, hence, were ignored in many numerical studies [¹⁷,²⁵,²⁶]. It is worth noting that the residual stresses originating from the section-forming (press-braking) process were not included in the developed FE models as its effect on the ultimate resistance of such sections was reported to be insignificant [²⁷,²⁸]. All necessary details of the adopted FE modeling technique are discussed in the following sections.

Element type and mesh

Four-node doubly-curved shell (S4R) elements with six degrees of freedom at each node with reduced integration were used to model the rack column sections and C-stitches. These elements are capable of capturing the distortional buckling behavior of members [¹⁶,¹⁷]. An element size of approximately 10 mm ×10 mm for the web and flanges, approximately 10 mm × 7.5 mm (length by width) for the edge stiffener and approximately 12.5 mm × 10 mm (length by width) for C-stitches produced good simulation results; details of this mesh sensitivity analysis was reported by Anbarasu and Sukumar [¹⁴].

Boundary condition and loading

Pin ended conditions with warping-restrained end sections for the tested columns were modeled by restraining rotation about the z-axis at both ends. Translations in x, y, and z directions were restrained at the unloaded end, while the loaded end was free to move in the z direction [¹⁴,¹⁶].

The boundary conditions were applied to the independent master node of the rigid, fixed MPC (Multi Point Constraint) located at the geometric center of the cross-section at the upper and the lower end of the model. Dependent nodes were connected to the master node using rigid beams. MPC acted as a rigid surface that was rigidly connected to the upper and the lower end of the columns as shown in Fig. 8. The reference points for the constraints were considered as the center of the hinged support; reference points were moved away from the end surfaces by a distance of 38 and 24 mm at loaded and unloaded end, respectively.

Screw modeling

The connections between the cross-section and C-stitches were modeled by mesh independent fastener using attachment point techniques available in ABAQUS. CONN3D2 element types, with 2 nodes and 6 degrees of freedom per node, were used to model the self-driving screws [²⁹].

Material properties and geometric imperfections

All specimens were modeled based on isotropic strain-hardening behavior. To account for the elasto-plastic behavior, a bilinear stress-strain curve was incorporated using the material properties obtained from tensile coupon tests. Elastic-perfectly-plastic material was assumed for all parametric analytical models with the value of Young’s Modulus E taken as 200 GPa and yield stress f_y as 250 MPa.

Geometric nonlinearity arising from large deformations as well as initial geometric imperfections were included in the model used for nonlinear analysis. Local and distortional imperfection amplitudes were taken equal to 0.34t and 0.94t (corresponding to 50% cumulative distribution function (CDF) of imperfection amplitude), respectively, where t is the plate thickness following recommendations by Schafer and Pekoz [²⁸]. Along with the local and distortional imperfections, measured global imperfection amplitudes shown in Table 1, were used in FE modeling to simulate the test results. It is worth noting that Kwon and Hancock [⁹] reported that the overall imperfections had little effect on the buckling of intermediate length columns as they typically buckle in local, distortional, or mixed mode of local and distortional buckling. Therefore, FE models used in the parametric study included local and distortional imperfections only; overall imperfections were not included. The adopted FE modeling approach adopted a two stage technique, i.e., linear elastic analysis was carried out to obtain buckling modes, and those modes were appropriately exploited to model initial geometric imperfections in the following nonlinear analysis to obtain reliable load versus axial-shortening behavior.

Validation of the FE modeling technique

The performance of the FE modeling technique was verified by comparing numerically obtained results with those obtained experimentally; Table 3 presents the comparison between test and FE simulation. It is obvious from the presented comparison that very good agreement was achieved between experimental and numerical results with a mean of 0.99 and a standard deviation 0.025; this clearly demonstrates the accuracy and consistency of the numerically simulated results.

Figure 9 compares experimentally obtained deformed shapes of the column specimens with those obtained from FE analysis showing good resemblance. Load vs axial shortening curves obtained by FEA were compared with the corresponding test curve for 1250-C4-D50 in Fig. 10 showing good agreement with the experimental results.

Parametric study

Once validated, the FE modeling technique was used to conduct a parametric study to investigate the influence of the depth and the number of C-stitches on the buckling resistance of rack columns. Following parameters were considered to have a direct influence on the behavior of intermediate length columns with C-stitches:

1 l/l_s ratio, where l is the member slenderness of the column and l_s is the plate slenderness of the C-stitch, which is given by,

{{\displaystyle \lambda }}_{\mathrm{s}}={\displaystyle \frac{d}{t}}\times \sqrt{{\displaystyle \frac{{{\displaystyle \sigma }}_{\mathrm{y}}}{E}}},

\lambda ={\displaystyle \frac{{{\displaystyle l}}_{\mathrm{e}}}{{{\displaystyle r}}_{\min }}}，

where, d and t are the depth and the thickness of C-stitches plate, respectively. l_e is the effective length of the section. r_min is the minimum radius of gyration of the section.

2) Spacing between adjacent C-stiches with respect to the overall column length, a/L, where a is the center-to-center distance between C-stitches and L is the overall length of the column.

3) Aspect ratio of the C-stitch, d/S, where d is the depth of the C-stitch plate and S is the width of the C-stitch plate.

A total of 84 cold-formed steel rack column sections were analyzed as part of the parametric study. The depths of the C-stitches (d) was used as 25, 50, 75, and 100mm, and for each d four specimens were considered by varying the number of C-stitches from 1 to 5. Therefore, a total of 21 analyses were conducted for each length of column including one section without C-stitches. Four lengths of columns were considered for parametric study giving overall slenderness ratio l of 25.06, 34.69, 44.33, and 53.97. The width of the C-stitch (S) was kept constant at 116.8 mm. Hence four groups of sections were formulated based on the d/S ratio of 0.21, 0.43, 0.64, and 0.86. Since the number of C-stitches was varied from 1 to 5, the resulting a/L ratios in the current study were 0.50, 0.33, 0.25, 0.20, and 0.17.

Results of the parametric study

Normalized ratios of ultimate-to-yield stress f_u/f_y for the considered columns were studied to investigate the influence of the considered parameters l/l_s, d/S, and a/L on column resistance. Following sections present the observed variations in f_u/f_y for the considered columns due to changes in key design parameters.

Influence of spacing of C-stitches

The influence of C-stitches spacing on the buckling resistance of rack column sections was investigated for uniform spacing of C-stitches varying from 1 to 5 for four different intermediate lengths (1300, 1800, 2300, and 2800 mm). Figure 11 presents f_u/f_y ratio for the considered columns for variations in C-stitches spacing. FE analysis results clearly showed that the spacing of C-stitches significantly influenced the ultimate buckling resistance of the rack columns.

As the number of C-stitches was increased, the f_u/f_y of rack columns also increased as the buckling resistance increased due to enhanced torsional rigidity. As observed from Figs. 11(a) to 11(d), in terms of increase in f_u/f_y ratio, columns undergoing single half-wave elastic distortional buckling (1300 and 1800 series) outperformed columns showing 3 half-wavelength elastic distortional buckling (2300 and 2800 series). Figure 11 shows that the f_u/f_y ratio decreased with increase in a/L ratio, i.e., for increased spacing between C-stitches. Similar trends for increase in f_u/f_y ratio were observed in Figs. 11(a), 11(b), and 11(c). But in Fig. 11(d), for 2800 series, a sudden drop was observed in the column with 3 stitches, i.e., a/L= 0.25; this may be due to the interference of flexural buckling mode onto predominant distortional mode.

Influence of depth of C-stitches

Figure 12 demonstrates the influence of changing the depth of C-stitches plate on the ultimate buckling resistance of the considered columns. It is obvious from Fig. 12 that the ultimate resistance of columns increases with increase in the depth of C-stitches plate for 1300, 1800, and 2300 series. It is, however, worth noting from Figs. 12(c) and 12(d) that a sudden decrease in strength for RC2800 was observed for a/L= 0.5 corresponding to d/S = 0.21 series due to an apparent influence of local buckling.

Obviously, the center-to-center distance between C-stitches has a considerable effect on the strength of columns. Improved column strengths were obtained as the C-stitches spacing, a was decreased. Increasing the number of C-stitches from 1 to 5 improved the ultimate resistance of columns for all considered cases.

Influence of the plate slenderness ratio of C-stitches

Figure 13 demonstrates the influence of C-stitches plate slenderness ratio on the ultimate buckling resistance. The slenderness ratios for the considered C-stitches were 25.06, 34.69, 44.33, and 53.97. Figure 13 shows that the column resistances expressed as f_u/f_y typically increased with increase in plate slenderness ratio l_s/l for 1300, 1800, and 2300 series. But in 2800 series, there was a sudden drop at l_s/l = 0.125 corresponding to d/S = 0.86; this was primarily due to the interference of flexural failure mode with distortional buckling.

Table 4 presents the observed changes in the ultimate buckling resistance of rack columns due to changes in depth and number of C-stitches considered in the current study.

Discussion

From the numerical parametric study, it is clear that the parameters such as the ratio of the slenderness of C-stitches plate to that of the column l_s/l, the relative spacing between adjacent C-stitches expressed as a/L, and the ratio of depth-to-width of the C-stitch plate d/S have significant influence on the compressive resistance of rack columns. Overall, the addition of C-stitches increased the ultimate resistance of columns by shifting typical failure patters from distortional mode to combinations of local, flexural, and distortional buckling modes depending on the number of C-stitches used along the length. The increase in number and depth of C-stitches considerably enhanced the compressive resistance of columns by improving their torsional rigidity. Addition of C-stitches from 1 to 5 in 1300, 1800, 2300, and 2800 series increased the ultimate buckling resistance from 20%–56%, 16%–46%, 7%–30%, and 10%–38%, respectively. However, it is worth noting that column resistances were somewhat constant or decreased when 2 C-stitches were used for 1300, 1800, and 2300 series as a result of interference of flexural buckling at mid length.

This study clearly shows that significant research is required to accurately understand the complex response of rack columns. Use of C-stitches in an appropriate manner will certainly enhance column resistance. However, the scope of the current study did allow proposing design rules for predicting column resistance. Additional research is currently underway for comprehensive understanding of the structural response and rational design equations for predicting column resistances for rack columns with C-stitch stiffeners will be proposed in the near future.

Conclusions

An experimental and a relevant numerical investigation on the ultimate resistance and behavior of intermediate length cold-formed steel rack column sections with and without C-stitches have been presented in this paper. Compression tests on rack columns with and without C-stitches were performed for two different column lengths. FE models were established and validated using the test results, and consequently, the verified FE models were used to conduct parametric studies to investigate the influence of depth and number of C-stitches on the ultimate compression resistance of rack columns. Overall, use of C-stitches demonstrated some significant improvement in column buckling resistances; as the number of C-stitches was increased, the column compression resistance also increased. However, the presence of a C-stitch at column mid-length is critical in preventing flexural buckling mode to interfere with predominant and desirable distortional buckling mode for intermediate length columns. This study paves the way for further research in this field for in-depth understanding of the behavior of rack columns and to devise reliable design rules for their appropriate use in practical engineering applications.

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