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An experimental investigation into perforated and non-perforated steel storage rack uprights
Thin-Walled Structures 112 (2017) 159–172
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An experimental investigation into perforated and non-perforated steel storage rack uprights
MARK
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Xianzhong Zhaoa, , Chong Rena,b, Ru Qina a b
Department of Structural Engineering, Tongji University, Shanghai 200092, China Department of Civil Engineering, Shanghai University, Shanghai 200444, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Storage rack uprights Perforation Axial compression tests Distortional-global buckling interaction Direct Strength Method (DSM)
This paper presents an experimental investigation into the behaviour of steel storage rack uprights subjected to axial compression. Material tensile tests were carried out to determine the material properties of cold-formed steel uprights. Geometrical imperfection measurements were recorded for the specimens prior to testing. A total of 67 specimens were tested under axial compression, including four different cross-sections, various lengths and specimens with and without perforations. The focus of the study is to investigate the influence of perforations on the performance and failure mode of steel storage rack uprights, and comparisons of performance and failure modes between perforated and non-perforated members are provided. The interaction of distortional-global buckling is also discussed, and the results of this study explicitly show that the strengths obtained from the tests highlight the underestimation of the existing distortional strength curve of the Direct Strength Method (DSM) on perforated steel storage rack uprights. Hence, based on the test results, a modified DSM for perforated uprights is proposed.
1. Introduction Cold-formed steel sections are widely used as steel uprights, beams and bracings in steel storage rack structures. Upright members used in storage racking generally have many perforations, and the perforated uprights have arrays of holes along the length, which allow the beam to be connected at variable heights and the bracing to be bolted to form the frames. The stability behaviour is of prime importance for coldformed steel members, but under the influence of continuous perforations the buckling behaviour and load capacity of an upright may vary. The rising demand of cold-formed steel in industry necessitates simple and reliable design methods, and attracts researchers to investigate its structural behaviour. Over recent decades, researchers have published a number of journal articles reviewing and summarising the development of cold-formed steel members. The perforated sections and non-standard restraint conditions make a numerical analysis too complicated to be used in the design of storage rack structures. Therefore, the current design codes for steel storage racks (North America (RMI) [1], Australia and New Zealand (AS/NZS4084) [2] and Europe (EN 15512) [3]) provide test procedures to obtain the strength requirements for storage rack design. Several researchers have also investigated the behaviour of cold-formed steel uprights by experimentation. Casafont et al. [4] presented an experimental inves-
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tigation into the structural behaviour of steel storage rack uprights subjected to compression. Koen [5] performed an experiment on stub uprights and complete upright frames to determine a series of reduction factors for the effective length of the uprights in compression. A series of compression tests examining perforation patterns were carried out by Rhodes and Schneider [6]. These concentrated on plain channel crosssections with perforation layouts systematically varied to examine the effects of perforation position, dimensions and quantity on the section performance. Roure et al. [7] gathered a comprehensive set of experimental results from upright cross-sections subject to compression. These papers only focus on short or intermediate length uprights. However, intermediate and high uprights are widely used in high-rise steel storage rack structures which have had a dramatic increase in use in recent years, and it is therefore essential to study their stability behaviour. Another important characteristic of cold-formed steel uprights is the presence of continuous perforations on the web and flanges. The perforations have been demonstrated by researchers to have a significant influence on the stability behaviour of uprights. Compression tests on 24 short and intermediate length cold-formed steel uprights with and without slotted web holes were conducted by Moen and Schafer [8]. More recently, Moen and Schafer [9] reported the tentative use of the direct strength method (DSM) for perforated thin-walled sections, and concluded that practical testing is necessary in
Corresponding author. E-mail address: [email protected] (X. Zhao).
http://dx.doi.org/10.1016/j.tws.2016.11.016 Received 11 April 2016; Received in revised form 23 October 2016; Accepted 21 November 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 112 (2017) 159–172
X. Zhao et al.
In this paper, an experimental investigation into the behaviour of cold-formed steel storage rack uprights subjected to axial compression is presented. Material tensile tests were carried out to determine material properties of cold-formed steel uprights. Geometrical imperfection measurements were recorded for the specimens prior to testing. Compression tests were conducted on 67 steel rack uprights, with and without perforations, in various sections and lengths. On the basis of the test investigations, the influence of parameters including the crosssections, the length of specimens, and the perforations on the ultimate load bearing capacity of rack uprights are studied. Comparisons of the performance and failure modes of perforated and non-perforated uprights are examined in detail. The Direct Strength Method (DSM) is employed to compare with the test results. A modified DSM for perforated uprights obtained from test results is also proposed.
Table 1 Properties of test specimens. Specimen
b (mm) h (mm) t (mm) Anet (mm2) L (mm)
75-55-1.8 75-55-1.8-H
75 75
55 55
1.8 1.8
376.7 279.1
90-66-1.8
90
66
1.8
500.8
90-66-1.8-H
90
66
1.8
405.0
90-66-2.0-H
90
66
2.0
450.0
100-80-1.8 100 100-80-1.8-H 100
80 80
1.8 1.8
547.0 449.2
100-80-2.0-H 100
80
2.0
499.1
100-80-2.3-H 100
80
2.3
574.0
120-80-2.0 120 120-80-2.0-H 120
80 80
2.0 2.0
653.8 554.9
250, 650, 1200, 1800 250, 450, 650, 900, 1200, 1500, 1800, 2200 300, 700, 1200, 2000, 2700 300, 500, 700, 900, 1200, 1500, 2000, 2400, 2700 300, 700, 1200, 2000, 2700 300,700,1200,1800,2800 300, 500, 700, 900, 1200, 1500, 1800, 2300, 2800 300, 700, 1200, 1800, 2800 300, 700, 1200, 1800, 2800 375, 1125, 1875, 2650 375, 750, 1125, 1500, 1875, 2250, 2650, 3000
2. Test program 2.1. Specimens A total of 67 tests on short, intermediate and long specimens with and without holes were categorised into four types of cross-section (see Table 1). The specimens were labelled to specify section type: for example, 90-66-1.8-H-900 represents a web length (b) of 90 mm, a flange height (h) of 66 mm, a thickness (t) of 1.8 mm and an overall length of 900 mm. The ‘H’ indicates that the specimen is perforated, while for a specimen without perforations, the ‘H’ is absent. Fig. 1 shows the cross-sectional geometry and illustrates the typical perforation locations and dimensions at the web and flange. It can be seen from the figure that triangular and circular holes are located at the web and flanges respectively; these perforations along the length of the upright allow the beam to be connected at variable heights, and steel bracings are usually bolted to the upright and act to stiffen the framework. It should be mentioned that for reasons of commercial confidentiality some geometrical details of the complex cross-sections are not permitted to be published.
Note: H – Perforated member.
the design of storage rack structures. The type of perforations described in the above papers is different from the perforations in the uprights of storage racks. There is a significant difference between the influence of large web holes on the performance of an upright and that of the perforations systematically located in the web and flanges. Local, distortional and global buckling are three typical types of buckling which a cold-formed steel member may experience [10]. Since the 1970 s there has been substantial research activity in the field of cold-formed structures which has led to numerous published works on the local, distortional and global buckling of cold-formed steel sections, (for example see [11–14]). However, single buckling modes were the focus of the above studies, while research on interactions involving distortional buckling has been conducted very recently. Kwon et al. [15] experimentally investigated the interaction between distortional buckling and material yield. Crisan et al. [16] carried out an experimental study on the interaction between distortional and overall buckling of perforated uprights. Pedro et al. [17] reported experimental and numerical results concerning local-distortional buckling interaction in fixed-end cold-formed steel web-stiffened lipped channel uprights. Dinis and Camotim investigated the interaction between local-distortional buckling ([18–20]) and distortional-global buckling [21] using the finite element method. Nandini and Kalyanaraman [22] investigated the interaction between distortional buckling and lateral-torsional buckling, also using the finite element method. Ren et al. [23] studied the distortional buckling interactions of cold-formed steel purlins under uplift loading in purlin-sheeting systems. Due to the complex nature and the limitations of research into buckling interactions on perforated rack members, and the effects of systematic perforations and strong stiffeners which eliminate the occurrence of local buckling, the interaction of distortional-global buckling on intermediate and long perforated members is thus considered to be sufficiently important to warrant further investigation.
2.2. Material properties A series of tensile coupon tests were conducted to determine the basic stress–strain response of the cold-formed steel specimens. Flat coupon samples were cut longitudinally from the rear flange of the cold-formed uprights. Three coupon samples selected from a batch of the specimen were tested using a computer controlled material tensile machine. The tension test followed the standard test procedure for tension testing of metallic materials in GB/T228-2002 [24]. The coupon sample was fully clamped at both ends, leaving a specified clear distance in the middle section to measure the elongation under tension. The goal of the coupon test was to obtain the precise tensile yield and ultimate strength for each steel rack upright specimen. A summary of the average ultimate and average 0.2% proof strengths obtained from three coupon samples are given in Table 2. It can be seen from the table that the yield strengths are found to be great in the perforated specimens than in the non-perforated specimens, indicating that the perforations have significant impact on the yield strength. This can be explained by the perforating procedure, which influences the cold forming process. The bending residual stresses produced from the coldforming process are minor [25,26] and their influence on the performance of the cold-formed steel uprights are thus ignored.
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k
Fig. 1. (a) Cross-sectional geometries. (b) Perforation positions. (c) Regular perforations in web and flanges.
Table 2 Tensile test results. Specimen
fy (N/mm2) (perforated)
fy (N/mm2) (non-perforated)
fu (N/mm2) (perforated)
fu (N/mm2) (non-perforated)
75-55-1.8 90-66-1.8 90-66-2.0 100-80-1.8 100-80-2.0 100-80-2.3 120-80-2.0
371.3 373.7 341.7 349.9 359.1 368.8 299.3
314.3 339.2 – 342.2 – – 297.0
545.0 556.3 496.5 531.1 539.1 553.5 558.3
535.3 561.8 – 530.0 – – 549.7
Fig. 2. Imperfection measurement set-up.
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75-55-1.8-250
Fig. 3. Imperfection measurement positions for each cross-section.
2.3. Initial geometrical imperfection measurements
90-66-1.8-H-900
100-80-2.0-H-1200 Fig. 5. Imperfection measurements. Fig. 4. Definition of geometric imperfections. Table 3 Imperfection magnitudes of perforated specimens (L < 1500 mm).
The selections of distribution and magnitude of imperfections are crucial due to the imperfection sensitive characteristic of cold-formed steel. A custom built test rig was employed to physically measure the geometrical imperfections of the cold-formed steel uprights, as shown in Fig. 2. Fig. 3 shows the imperfection measurement locations for each cross-section. Five linear variable differential transformers (LVDTs) were placed at the middle of the web, and at the front and rear flanges, to record the imperfection magnitudes of each segment. The LVDTs were travelled smoothly along the longitudinal direction of the specimen and the measurements from these five LVDTs provide comprehensive details about the geometrical imperfections of the specimen. Fig. 4 illustrates the definition of the geometrical imperfection magnitudes, which are Lweb, D1 and D2, respectively. The “local” imperfection magnitude at the web, Lweb is approximated by:
L web = d p5
(1)
where dp5 is the y-component of displacement of the cross-section, recorded by LVDT(5) at the middle of the web. The “distortional” imperfection magnitude for both flanges, D1 and D2 are approximated by:
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Specimen
Lweb (mm)
D1 (mm)
D2 (mm)
75-55-1.8-H-250 75-55-1.8-H-450 75-55-1.8-H-650 75-55-1.8-H-900 75-55-1.8-H-1200 90-66-1.8-H-300 90-66-1.8-H-500 90-66-1.8-H-700 90-66-1.8-H-900 90-66-1.8-H-1200 90-66-2.0-H-300 90-66-2.0-H-700 90-66-2.0-H-1200 100-80-1.8-H-300 100-80-1.8-H-500 100-80-1.8-H-700 100-80-1.8-H-900 100-80-1.8-H-1200 100-80-2.0-H-300 100-80-2.0-H-700 100-80-2.0-H-1200 100-80-2.3-H-300 100-80-2.3-H-700 100-80-2.3-H-1200 120-80-2.0-H-375 120-80-2.0-H-750 120-80-2.0-H-1125 120-80-2.0-H-1500
0.32 0.22 0.32 1.79 0.27 1.59 1.13 1.28 1.03 3.58 0.16 0.98 1.56 0.68 0.67 0.64 0.61 1.80 0.43 0.21 0.43 0.62 1.31 1.81 0.32 0.63 1.62 0.80
0.61 0.30 0.72 2.19 2.44 1.43 4.01 3.12 4.67 5.19 2.58 5.86 3.50 0.76 0.88 1.29 1.65 2.58 1.23 1.46 2.94 2.30 1.70 1.01 2.10 1.90 1.24 1.66
0.43 0.91 0.56 1.63 2.08 0.96 2.67 4.23 4.89 4.39 0.19 3.83 4.29 0.62 0.77 1.09 0.99 0.87 0.89 0.86 2.30 1.09 0.47 0.95 2.16 0.90 0.98 0.33
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Table 4 Imperfection magnitudes of non-perforated specimens (L < 1500 mm). Specimen
Lweb (mm)
D1 (mm)
D2 (mm)
75-55-1.8-250 75-55-1.8-650 75-55-1.8-1200 90-66-1.8-300 90-66-1.8-700 90-66-1.8-1200 100-80-1.8-300 100-80-1.8-700 100-80-1.8-1200 120-80-2.0-375 120-80-2.0-1125
0.32 0.07 2.10 0.73 1.86 0.03 0.36 0.51 0.50 0.47 0.43
1.06 2.45 1.40 2.10 5.02 3.11 0.59 0.66 0.71 0.90 2.86
0.65 1.92 1.70 2.62 5.79 4.75 1.37 2.68 2.51 2.01 2.84
Fig. 8. LVDT positions.
Fig. 6. Test set-up.
Fig. 9. Initial imperfection obtained from the Southwell plot (100-80-1.8-2800).
Fig. 7. Two-directional hinged joint base.
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[27], and detailed examples are presented in the next section.
Table 5 Southwell plot estimations of perforated and non-perforated specimens (L≥1500 mm). Specimen
δ0 (mm)
75-55-1.8-H-1500 75-55-1.8-H-1800 75-55-1.8-H-2200 90-66-1.8-H-1500 90-66-1.8-H-2000 90-66-1.8-H-2400 90-66-1.8-H-2700 90-66-2.0-H-2000 90-66-2.0-H-2700 100-80-1.8-H-1500 100-80-1.8-H-1800 100-80-1.8-H-2300 100-80-1.8-H-2800 100-80-2.0-H-1800 100-80-2.0-H-2800 100-80-2.3-H-1800 100-80-2.3-H-2800 120-80-2.0-H-1875 120-80-2.0-H-2250 120-80-2.0-H-2625 120-80-2.0-H-3000 75-55-1.8–1800 90-66-1.8–2000 90-66-1.8–2700 100-80-1.8–1800 100-80-1.8–2800 120-80-2.0–1875 120-80-2.0–2625
2.12 0.65 0.15 2.20 0.75 2.11 0.74 6.21 1.83 0.55 0.38 0.52 0.92 0.24 0.80 0.11 1.43 1.75 0.04 1.81 0.44 0.20 0.15 0.32 0.35 1.88 0.05 0.42
D1 =
D2 =
2.4. Test setup and instrumentations Following the aforementioned preparations, an axial compression test was carried out. As shown schematically in Fig. 6, the test specimens were placed vertically, using a 50 t hydraulic jack and a vertical reaction frame system to apply an axial load. The upper crosshead and lower actuator were both fitted with a two-directional hinged joint base (see Fig. 7). The two-directional hinged joint base is a special device to simulate the simply supported boundary condition. In this device, the top plate has a vertical slot underneath to connect with the knife on the upper surface of the adjacent middle plate, and the middle plate also has horizontal slots underneath to match the knife on the upper surface of the bottom plate. In this way the device applies sufficient restraint on torsion in the web of the specimen in the longitudinal direction, but not on rotation in the normal and tangential directions. Compared with a fixed boundary condition, the role of this device is to maximise the slenderness ratio for a fixed length of upright as well as to control the range of the slenderness ratio when the specimen is under the axial compression load in the frame system. All specimens under axial compression were monitored by a number of strain gauges and LVDTs. The strain gauges were set up for measuring the longitudinal strain at the mid-span of the cross-section. To demonstrate the deformations of members in load-displacement curves, several LVDTs were positioned at the middle span of the cross-section to determine the displacements of the flanges and the web, and also at the upper and lower support base plates to measure the longitudinal shortening displacement of specimens, as shown in Fig. 8. After all the specimens and equipment were in place, 10% of the ultimate load was initially applied to the specimen for centring. An incrementally increasing load was then added to the specimen until it reached the failure point. Deflections recorded by the LVDTs, and strains monitored by the strain gauges were collected at each incremental load step. This process was repeated for each specimen. The Southwell plot method [28] is employed herein to determine the imperfection for specimens longer than 1500 mm in length. Fig. 9 illustrates the linear relationship between deflection (δ) and ratio of deflection and axial load (δ/P), which were obtained from the axial compression test. It can be seen in this figure that the magnitude of the initial deflection, δ0, is intercepted on the abscissa. The initial deflection (δ0) can be assumed as an initial imperfection of a long specimen, which is determined from experimental measurements of deflection and axial load. A straight line is then plotted which best fits the measurement points and the slope (1/Pcr), where Pcr is the elastic buckling load of the long specimen. Table 5 shows the initial imperfections obtained from the Southwell plot method. Note that some extremely large imperfections (i.e. 90-66-2.0-H-2000) or tiny imperfections (i.e. 12080-2.0–1875) occur in some specimens, even though other specimens do not have a consistent ratio of initial imperfection to length. Because the buckling interactions, such as distortional-global buckling interaction, may affect the deflection of the tested members, the Southwell plot method is not accurate enough for estimating the geometrical imperfections of the specimens, which are dominated by the buckling interactions. As a consequence of this, a conservative estimate (L/1000 [29]) is recommended for the magnitude of the imperfection of long specimens experiencing global buckling (L≥1500 mm).
h ( d p1 − d p 3 ) dh
(2)
h (d p 2 − d p 4 ) dh
(3)
where h is the height of the flange, dp1, dp2, dp3 and dp4 are the xcomponent of displacement of the cross-section, recorded by LVDT(1)(4) at each middle flange, and dh is the distance between the imperfection measurements between the front flange and the rear flange. Fig. 5 shows the imperfection measurements of non-perforated and perforated uprights of various lengths (where the length is less than 1500 mm). It can be seen in this figure that the distortional imperfection has a significant influence on the short and intermediate specimens. It also can be seen that the lines of imperfection measurements at the perforation zones fluctuate, indicating that the perforations have an impact on the magnitude of the imperfections. Detailed imperfection measurements including for perforated and non-perforated uprights are given in Tables 3 and 4 respectively, for specimens less than 1500 mm in length. It should be noted that this test rig only records web local imperfections and flange distortional imperfections for specimens less than 1500 mm in length, because for specimens longer than 1500 mm the extra deflection generated from the self-weight of a horizontally placed specimen may affect the imperfection measurements. Alternatively, the Southwell plot method can be adopted to determine the initial imperfection of long specimens experiencing global buckling
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Table 6 Ultimate loads and failure modes. Specimen (perforated)
Pcr,1 (kN)
Buckling Mode of Pcr,1
Pu,test (kN)
Failure mode
Specimen (nonperforated)
Pcr,1 (kN)
Buckling mode of Pcr,1
Pu,test (kN)
Failure mode
75-55-1.8-H-250 75-55-1.8-H-450 75-55-1.8-H-650 75-55-1.8-H-900 75-55-1.8-H-1200 75-55-1.8-H-1500 75-55-1.8-H-1800 75-55-1.8-H-2200 90-66-1.8-H-300 90-66-1.8-H-500 90-66-1.8-H-700 90-66-1.8-H-900 90-66-1.8-H-1200 90-66-1.8-H-1500 90-66-1.8-H-2000 90-66-1.8-H-2400 90-66-1.8-H-2700 90-66-2.0-H-300 90-66-2.0-H-700 90-66-2.0-H-1200 90-66-2.0-H-2000 90-66-2.0-H-2700 100-80-1.8-H-300 100-80-1.8-H-500 100-80-1.8-H-700 100-80-1.8-H-900 100-80-1.8-H-1200 100-80-1.8-H-1500 100-80-1.8-H-1800 100-80-1.8-H-2300 100-80-1.8-H-2800 100-80-2.0-H-300 100-80-2.0-H-700 100-80-2.0-H-1200 100-80-2.0-H-1800 100-80-2.0-H-2800 100-80-2.3-H-300 100-80-2.3-H-700 100-80-2.3-H-1200 100-80-2.3-H-1800 100-80-2.3-H-2800 120-80-2.0-H-375 120-80-2.0-H-750 120-80-2.0-H-1125 120-80-2.0-H-1500 120-80-2.0-H-1875 120-80-2.0-H-2250 120-80-2.0-H-2625 120-80-2.0-H-3000
469.00 243.74 198.26 170.84 133.40 88.90 72.76 42.22 417.37 375.15 362.53 316.62 218.84 159.05 99.25 72.04 58.38 513.13 451.76 252.38 112.43 66.19 515.13 387.82 248.55 197.87 182.73 154.74 146.37 125.88 87.19 630.10 291.28 219.90 182.20 97.23 826.61 363.41 277.53 239.05 112.24 485.83 254.77 205.50 173.56 164.35 154.45 124.24 96.60
DB DB DB FTB FB FTB FB FB LB LB LB LB FTB FTB FTB FTB FTB LB LB FTB FTB FTB LB DB DB DB DB DB DB FB FB LB DB DB DB FB LB DB DB DB FB LB DB DB DB DB DB FB FB
91.47 82.77 75.17 79.20 58.70 59.79 44.57 29.66 129.68 127.97 122.53 114.92 134.49 91.63 73.92 58.70 48.14 154.06 143.19 125.02 81.07 54.67 134.96 114.77 125.02 120.67 112.75 104.05 97.68 88.37 59.79 158.25 148.62 127.97 119.74 83.55 185.58 142.10 142.25 119.12 86.97 171.92 142.25 128.90 119.43 104.21 90.07 77.49 64.29
DB DB DB FB DB+FB FTB DB+FTB DB+FTB DB DB DB DB DB+FTB FTB FTB FTB FTB DB DB DB FTB FTB DB DB DB DB DB+FB DB+FB DB+FB DB+FB FTB DB DB DB DB+FB FTB DB DB DB+FB DB+FB DB+FB DB DB DB DB+FB DB+FB DB+FB DB+FB DB+FB
75-55-1.8-250 – 75-55-1.8-650 – 75-55-1.8-1200 – 75-55-1.8-1800 – 90-66-1.8-300 – 90-66-1.8-700 – 90-66-1.8-1200 – 90-66-1.8-2000 – 90-66-1.8-2700 – – – – – – – 100-80-1.8-700 – 100-80-1.8-1200 – 100-80-1.8-1800 – 100-80-1.8-2800 – – – – – – – – – – 120-80-2.0-375 – 120-80-2.0-1125 – 120-80-2.0-1875 – 120-80-2.0-2625 –
549.53 – 253.67 – 157.43 – 62.588 – 538.99 – 462.09 – 255.54 – 114.79 – 67.13 – – – – – – – 275.06 – 206.91 – 175.75 – 98.94 – – – – – – – – – – 560.93 – 243.46 – 190.94 – 139.79 –
DB – DB – FTB – FTB – LB – LB – DB – FTB – FTB – – – – – – – DB – DB – DB – FB – – – – – – – – – – LB – DB – DB – FB –
112.13 – 95.04 – 84.02 – 53.58 – 177.66 – 162.29 – 134.49 – 100.63 – 67.40 – – – – – – – 140.24 – 139.21 – 104.67 – 79.20 – – – – – – – – – – 190.40 – 153.90 – 129.52 – 87.43 –
DB – DB – DB+FB – DB+FB – DB – DB – DB – FTB – FTB – – – – – – – DB – DB – DB+FB – FTB – – – – – – – – – – DB – DB – DB – DB+FTB –
3. Test results and analysis
in Table 6 for perforated and non-perforated specimens, respectively. Fig. 10 depicts the typical load-displacement curves of specimens with different cross-sections and lengths. The ordinate axis represents compressive load, and in the main chart the abscissa axis represents the displacement of flanges and webs recorded from LVDTs at the middle span cross-section, and in the inset chart the abscissa axis represents the longitudinal shortening displacement. It should be noted that for the middle span cross-sections of the main charts, the maximum displacements correspond to the ultimate failure loads. It can be seen in
3.1. Load-displacement curves Compression tests were conducted on 67 specimens, including four different cross-sections, various lengths, and specimens with and without perforations. Ultimate loads and failure modes recorded from the compression tests and critical buckling loads and buckling modes (first buckling modes) obtained from finite element analyses (FEA) are shown
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75-55-1.8-H-250 (DB)
75-55-1.8-H-1500 (FTB)
75-55-1.8-H-1200 (DB+FB)
100-80-1.8-H-2800 (FTB) Fig. 10. The load-displacement curves.
Fig. 10(a) that the obvious displacements of flanges, together with almost no movement at the web-flange junctions, reveals that distortional buckling (DB) is the main failure mode for the short specimen (75-55-1.8-H-250). In Fig. 10(b) the buckling modes of specimen 75-551.8-H-1200 is dominated by the distortional-flexural buckling interaction (DB+FB). The large displacements of the web-flange junctions and the equivalent displacements of the flanges indicate that not only flexural buckling exists, but also that distortional buckling has an influence on the failure mode of the specimen. Nevertheless, In Fig. 10(c) and (d), for 75-55-1.8-H-1500 and 100-80-1.8-H-2800, the buckling modes are mainly controlled by flexural-torsional buckling (FTB), which the shape of cross-section keeps non-deformable when the buckling is occurring. It can also be seen in the inset charts of Fig. 10, that for the load-longitudinal shortening displacement curve of perforated and non-perforated specimens which failed due to distortional bucking, distortional-flexural buckling interaction and flexural-torsional buckling, the ultimate loads of perforated specimens are lower than the ultimate loads obtained from non-perforated specimens. This indicates that, in terms of axial compression, the perforations have a significant influence on the performance of the cold-formed steel uprights due to the change of cross-section area. The specimens were classified into three categories according to the buckling failure mode: distortional buckling (DB) failure specimens, global buckling failure specimens (flexural-torsional buckling (FTB) or flexural buckling (FB))
and interaction of distortional-global (DB+E) buckling failure specimens, typical examples of which are shown in Fig. 11.
3.2. Influences of cross-sections Fig. 12 shows the parametric distribution of different buckling modes for specimens with perforations and without perforations, respectively. It can be seen that distortional buckling and the interaction of distortional-global (distortional-flexural-torsional buckling interaction or distortional-flexural buckling interaction) are the main buckling failure modes for both perforated and non-perforated specimens. The buckling mode is only dominated by distortional buckling where the length of the perforated specimen is shorter than 1250 mm, and the length of the non-perforated specimen is shorter than 2000 mm. The buckling mode is mainly controlled by the interaction of distortional-flexural or distortional-flexural-torsional buckling where the flange-thickness ratio (h/t) of the perforated specimen is greater than 40. Flexural-torsional buckling failure of perforation specimens has a lower flange-thickness ratio (h/t). Fig. 13 shows a comparison of the normalised compression load capacity obtained from tests on specimens with and without perforations, plotted against length for four different cross-sections. This comparison takes the same thickness and length for non-perforated and perforated specimens. It is seen from this figure that load decreases
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75-55-1.8-H-250 (DB)
75-55-1.8-250 (DB)
90-66-1.8-H-900 (DB)
120-80-2.0-H-1125 (DB)
Fig. 11. Typical buckling failure modes (front and side views).
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75-55-1.8-H-1200 (DB+FB)
75-55-1.8-1200 (DB+FB)
90-66-1.8-H-1200 (DB+FTB)
100-80-2.0-H-1800 (DB+FB) Fig. 11. (continued)
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75-55-1.8-H-1500 (FTB)
90-66-2.0-H-2000 (FTB)
100-80-1.8-H-2800 (FTB)
100-80-1.8-2800 (FTB) Fig. 11. (continued)
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study, the upright fails when a global buckling mode develops and interacts/couples with the distortional buckling. However, in the crosssection of 90-66, the buckling interactions are rarely observed (see Table 6 for details). This is because when an upright buckles, it usually changes from a state of compression to a combined state of compression and bending. However, compared with the others cross-sections, the shape of 90-66 cross-section is special and is close to a lipped channel section, and the cross-section has extreme strong flange stiffeners, which provide high flexural rigidity to avoid flexural buckling. Consequently, the failure modes are mainly controlled by distortional buckling of the compression flange-lip systems where the upright is short, and the uprights have low torsional rigidity about their minor axes, rendering the members prone to flexural-torsional buckling at mid-span where the upright is long. 4. Comparisons between DSM and test results The complex cross-sections of the perforated steel storage rack uprights subjected to axial compression studied in the present paper are not stipulated in the Direct Strength Method (DSM) included in the North American Specification (AISI S100) [30]. This section addresses the applicability of the Direct Strength Method (DSM) for estimating the ultimate strength of uprights failing in different buckling modes, and describes the modified DSMs obtained from the test results. To consider the influence of the perforations in the design of steel storage rack uprights subjected to axial compression, comparisons between the DSM predictions and the ultimate strength of perforated uprights and nonperforated uprights obtained from testing are illustrated in Fig. 14. DB and DB+E test results compared with the DSM (DB) curve and DB+E and E test results compared with the DSM (E) curve are correspondingly presented in Fig. 14(a) and (b). The slenderness of the upright is represented by λ=(Py/Pcr,1)0.5, where Pcr,1 is the critical distortional buckling load for Fig. 14(a) and the critical global buckling load for Fig. 14(b), respectively. Note that the Pcr,1 is calculated using the first eigenvalue of buckling from a finite element analysis (FEA). It should be mentioned that the critical buckling load obtained from the theoretical method or the finite strip method (FSM) does not consider the influence of perforations, and the FEA is thus used to determine the critical buckling load of perforated uprights. Specimens of identical crosssectional dimensions, thickness and length are compared, with the only difference being whether the specimen has perforations. It can be observed from Fig. 14 that the ultimate loads of non-perforated specimens are reduced by such perforations, indicating that the perforations have a significant influence on the performance. It also can be seen in this figure that the predictions of distortional buckling failure obtained from the test are generally underestimated by the distortional buckling curve of the DSM, but the predictions of global buckling failure determined from the experiment are in good agreement with those of the DSM. Additionally, for distortional-global buckling interaction failures the predictions obtained from the test are below the distortional buckling strength curve of the DSM, and for some specimens, the predictions are even lower than the predicted global buckling curve of the DSM. This implies that the ultimate strength may be eroded significantly when the member experiences buckling interaction phenomena, and thus the current DSM distortional buckling strength curve does not offer a reliable design for perforated steel storage rack uprights subjected to axial compression. As a consequence of this, the modifications to the current DSM distortional strength curve are given as follows:
Fig. 12. Parametric distribution of different buckling modes.
with increasing specimen length, and the perforations have an influence on the load capacity for small cross-sections. The load capacity is reduced for sections 75-55-1.8 and 90-66-1.8 when perforations are present. However, the load capacity is almost identical whether perforations are present or absent for sections 100-80-1.8 and 12080-2.0. The former observation reveals that the perforations have an influence on the ultimate load bearing capacity of rack uprights in small sections. The latter observation implies that the influence of the perforations is insignificant in large section specimens. The explanations for above observations are that in terms of perforation ratio, the effective area of the small cross-sections is sensitive to the perforations, and the ultimate load is thus significantly reduced in perforated members. Whereas the effective area of the large cross-sections is impacted insignificantly by the perforations, therefore the differences of ultimate load between perforated members and non-perforated members are negligible. It is worth noting here that the effect of perforations on the performance of an upright is considerable, and this illustrates the significance of ensuring that storage uprights have an appropriate cross-section. One of the concerns in the buckling interaction of cold-formed steel members is the post-buckling strength of distortional buckling, which allows it to combine with local buckling and global buckling. In the
For λ ≤ 0. 568
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Pnd =1 Py
(4)
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(a) Cross-section 75-55-1.8.
(c) Cross-section 100-80-1.8.
(b) Cross-section 90-66-1.8.
(d) Cross-section 120-80-2.0.
Fig. 13. Ratio of compression load versus length for different cross section members with and without perforations.
For λ
> 0. 568
where λ =
⎛ ⎛ P ⎞0.35⎞⎛ P ⎞0.8 Pnd cr ,1 ⎟⎟ ⎟⎜⎜ cr ,1 ⎟⎟ = ⎜⎜1 − 0. 4⎜⎜ Py Py ⎠ ⎟⎝ Py ⎠ ⎝ ⎝ ⎠
5. Conclusions This paper has presented an experimental investigation into the behaviour of both perforated and non-perforated steel storage rack uprights subjected to axial compression. Comprehensive experimental studies including material tensile testing, geometrical imperfection measurement and axial compression testing have been presented and described, and the corresponding results obtained from the tests have been listed and discussed. Comparisons of performance and failure mode between perforated and non-perforated members have been provided. The results show that the perforations have a significant influence not only on the performance but also on the buckling failure mode of cold-formed steel storage rack uprights. The interaction of distortional-global buckling has also been studied. The results of this study explicitly show that the Direct Strength Method (DSM) has been demonstrated to be unreliable for predicting the ultimate strength of uprights failing in buckling interactions for perforated uprights. Hence, based on the test results, a modified DSM distortional buckling strength curve for perforated uprights has been proposed.
(5)
Py Pcr,1
Pcr,1 =Critical elastic buckling load (first buckling mode) Py = Anet Fy where Anet is the net cross-sectional area, and Pcr,1 is the critical elastic buckling load (first buckling mode). Fig. 15 shows a comparison of the DSM predictions with test results for the perforated specimens, which are dominated by distortional buckling (in this case, Pcr,1 is the critical elastic distortional buckling load). A modified DSM for the distortional strength curve obtained from the test results is also provided in this figure. It should be mentioned that the distortional-global buckling interaction failures are treated as distortional buckling failures for the purposes of investigating buckling interactions.
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References [1] RMI, Specification for the design, testing and utilization of industrial steel storage racks, ANSI MH16.1-2012', Rack Manufacturers Institute, Technical report, 2012. [2] AS/NZS 4084, Steel storage racking, Standards Australia/Standards New Zealand, Sydney, Australia, 2012. [3] EN 15512, Steel static storage systems, European Technical Committee CEN/TC 344, European Specifications, 2009. [4] M. Casafont, M. Pastor, F. Roure, T. Pekoz, An experimental investigation of distortional buckling of steel storage rack uprights, Thin-Walled Struct. 49 (2011) 933–946. [5] D. Koen, Structural capacity of light gauge steel storage rack uprights. School of Civil Engineering, University of Sydney, 2008. [6] J. Rhodes, F. Schneider, The compressional behaviour of perforated elements, in: Proceedings of the Twelfth International Specialty Conference on Cold-formed Steel Structures, St. Louis, Missouri, USA, 1994. [7] F. Roure, M. Pastor, M. Casafont, M. Somalo, Stub upright tests for racking design: experimental testing, FE analysis and EC3, Thin-Walled Struct. 49 (2011) 167–184. [8] C. Moen, B. Schafer, Experiments on cold-formed steel uprights with holes, ThinWalled Struct. 46 (2008) 1164–1182. [9] C. Moen, B. Schafer, Elastic buckling of cold-formed steel columns and beams with holes, Eng. Struct. 31 (2009) 2812–2824. [10] G.J. Hancock, Local, distortional, and lateral buckling of I-beams, J. Struct. Div. (ASCE) 104 (11) (1978) 1787–1798. [11] R.A. La Boube, Literature survey of cold-formed sections, in: Proceedings of the Eighth International Specialty Conference on Cold Formed Steel Structures. St. Louis, USA. University of Missouri-Rolla, Missouri, 1986. [12] W. Yu, Cold-Formed Steel Design, John Wiley and Sons, New-York, 2000. [13] G.J. Hancock, Cold-formed steel structures, J. Constr. Steel Res. 59 (2003) 473–487. [14] M. Macdonald, M.A. Heiyantuduwa, J. Rhodes, Recent developments in the design of cold-formed steel members and structures, Thin-Walled Struct. 46 (2008) 1047–1053. [15] Y.B. Kwon, B.S. Kim, G.J. Hancock, Compression tests of high strength cold-formed steel channels with buckling interaction, J. Constr. Steel Res. 65 (2) (2009) 278–289. [16] A. Crisan, V. Ungureanu, D. Dubina, Behaviour of cold-formed steel perforated sections in compression: part 1—Experimental investigations, Thin-Walled Struct. 61 (2012) 86–96. [17] B. Pedro, B. Young, D. Camotim, Strength, interactive failure and design of webstiffened lipped channel uprights exhibiting distortional buckling, Thin-Walled Struct. 81 (2014) 195–209. [18] P.B. Dinis, D. Camotim, N. Silvestre, FEM-based analysis of the local plate/ distortional mode interaction in cold-formed steel lipped channel uprights, Comput. Struct. 85 (2007) 1461–1474. [19] P.B. Dinis, D. Camotim, Local/distortional mode interaction in cold-formed steel lipped channel beams, Thin-Walled Struct. 48 (2010) 771–785. [20] D. Camotim, P. Dinis, Coupled instabilities with distortional buckling in coldformed steel lipped channel uprights, Thin-Walled Struct. 49 (2011) 562–575. [21] P.B. Dinis, D. Camotim, Post-buckling behaviour and strength of cold-formed steel lipped channel uprights experiencing distortional/global interaction, Comput. Struct. 89 (2011) 422–434. [22] P. Nandini, V. Kalyanaraman, Interaction of local and distortional buckling with lateral torsional buckling of cold-formed steel beams. in: Proceedings of the Fifth International Conference on Coupled Instabilities in Metal Structures, Sydney, Australia, 23–25 June 2008. [23] C. Ren, X. Zhao, Y. Chen, Buckling behaviour of partially restrained cold-formed steel zed purlins subjected to transverse distributed uplift loading, Eng. Struct. 114 (2016) 14–24. [24] GB/T228-2002, Metallic materials tensile test procedure at ambient temperature. (In Chinese) Beijing, 2002. [25] R. Cruise, L. Gardner, Residual stress analysis of structural stainless steel sections, J. Constr. Steel Res. 64 (3) (2008) 352–366. [26] L. Gardner, R. Cruise, Modeling of residual stresses in structural stainless steel sections, J. Struct. Eng. 135 (1) (2009) 42–53. [27] P. Mandala, C.R. Calladineb, Lateral-torsional buckling of beams and the Southwell plot, Int. J. Mech. Sci. 44 (2002) 2557–2571. [28] N.S. Trahair, M.A. Bradford, D.A. Nethercot, L. Gardner, The Behaviour and Design of Steel Structures to EC3, fourth edition, Taylor & Francis, London and New York, 2008. [29] ECCS, European recommendations for the design of light gauge steel members. ECCS. Brussels, 1987. [30] AISI, North American specification for the design of cold-formed steel structural members, AISI S100-12. Washington, D.C., American Iron and Steel Institute, 2012.
Fig. 14. Comparison between the DSM and test results.
Fig. 15. Comparison of the current DSM distortional strength curve and modified strength curve of perforated members.
Acknowledgement The authors gratefully acknowledge the support provided by WAP logistics equipment (Shanghai) Ltd. The authors would like to acknowledge the financial support received from the China Postdoctoral Science Foundation (2014M561517).
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An experimental investigation into perforated and non-perforated steel storage rack uprights
MARK
⁎
Xianzhong Zhaoa, , Chong Rena,b, Ru Qina a b
Department of Structural Engineering, Tongji University, Shanghai 200092, China Department of Civil Engineering, Shanghai University, Shanghai 200444, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Storage rack uprights Perforation Axial compression tests Distortional-global buckling interaction Direct Strength Method (DSM)
This paper presents an experimental investigation into the behaviour of steel storage rack uprights subjected to axial compression. Material tensile tests were carried out to determine the material properties of cold-formed steel uprights. Geometrical imperfection measurements were recorded for the specimens prior to testing. A total of 67 specimens were tested under axial compression, including four different cross-sections, various lengths and specimens with and without perforations. The focus of the study is to investigate the influence of perforations on the performance and failure mode of steel storage rack uprights, and comparisons of performance and failure modes between perforated and non-perforated members are provided. The interaction of distortional-global buckling is also discussed, and the results of this study explicitly show that the strengths obtained from the tests highlight the underestimation of the existing distortional strength curve of the Direct Strength Method (DSM) on perforated steel storage rack uprights. Hence, based on the test results, a modified DSM for perforated uprights is proposed.
1. Introduction Cold-formed steel sections are widely used as steel uprights, beams and bracings in steel storage rack structures. Upright members used in storage racking generally have many perforations, and the perforated uprights have arrays of holes along the length, which allow the beam to be connected at variable heights and the bracing to be bolted to form the frames. The stability behaviour is of prime importance for coldformed steel members, but under the influence of continuous perforations the buckling behaviour and load capacity of an upright may vary. The rising demand of cold-formed steel in industry necessitates simple and reliable design methods, and attracts researchers to investigate its structural behaviour. Over recent decades, researchers have published a number of journal articles reviewing and summarising the development of cold-formed steel members. The perforated sections and non-standard restraint conditions make a numerical analysis too complicated to be used in the design of storage rack structures. Therefore, the current design codes for steel storage racks (North America (RMI) [1], Australia and New Zealand (AS/NZS4084) [2] and Europe (EN 15512) [3]) provide test procedures to obtain the strength requirements for storage rack design. Several researchers have also investigated the behaviour of cold-formed steel uprights by experimentation. Casafont et al. [4] presented an experimental inves-
⁎
tigation into the structural behaviour of steel storage rack uprights subjected to compression. Koen [5] performed an experiment on stub uprights and complete upright frames to determine a series of reduction factors for the effective length of the uprights in compression. A series of compression tests examining perforation patterns were carried out by Rhodes and Schneider [6]. These concentrated on plain channel crosssections with perforation layouts systematically varied to examine the effects of perforation position, dimensions and quantity on the section performance. Roure et al. [7] gathered a comprehensive set of experimental results from upright cross-sections subject to compression. These papers only focus on short or intermediate length uprights. However, intermediate and high uprights are widely used in high-rise steel storage rack structures which have had a dramatic increase in use in recent years, and it is therefore essential to study their stability behaviour. Another important characteristic of cold-formed steel uprights is the presence of continuous perforations on the web and flanges. The perforations have been demonstrated by researchers to have a significant influence on the stability behaviour of uprights. Compression tests on 24 short and intermediate length cold-formed steel uprights with and without slotted web holes were conducted by Moen and Schafer [8]. More recently, Moen and Schafer [9] reported the tentative use of the direct strength method (DSM) for perforated thin-walled sections, and concluded that practical testing is necessary in
Corresponding author. E-mail address: [email protected] (X. Zhao).
http://dx.doi.org/10.1016/j.tws.2016.11.016 Received 11 April 2016; Received in revised form 23 October 2016; Accepted 21 November 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.
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In this paper, an experimental investigation into the behaviour of cold-formed steel storage rack uprights subjected to axial compression is presented. Material tensile tests were carried out to determine material properties of cold-formed steel uprights. Geometrical imperfection measurements were recorded for the specimens prior to testing. Compression tests were conducted on 67 steel rack uprights, with and without perforations, in various sections and lengths. On the basis of the test investigations, the influence of parameters including the crosssections, the length of specimens, and the perforations on the ultimate load bearing capacity of rack uprights are studied. Comparisons of the performance and failure modes of perforated and non-perforated uprights are examined in detail. The Direct Strength Method (DSM) is employed to compare with the test results. A modified DSM for perforated uprights obtained from test results is also proposed.
Table 1 Properties of test specimens. Specimen
b (mm) h (mm) t (mm) Anet (mm2) L (mm)
75-55-1.8 75-55-1.8-H
75 75
55 55
1.8 1.8
376.7 279.1
90-66-1.8
90
66
1.8
500.8
90-66-1.8-H
90
66
1.8
405.0
90-66-2.0-H
90
66
2.0
450.0
100-80-1.8 100 100-80-1.8-H 100
80 80
1.8 1.8
547.0 449.2
100-80-2.0-H 100
80
2.0
499.1
100-80-2.3-H 100
80
2.3
574.0
120-80-2.0 120 120-80-2.0-H 120
80 80
2.0 2.0
653.8 554.9
250, 650, 1200, 1800 250, 450, 650, 900, 1200, 1500, 1800, 2200 300, 700, 1200, 2000, 2700 300, 500, 700, 900, 1200, 1500, 2000, 2400, 2700 300, 700, 1200, 2000, 2700 300,700,1200,1800,2800 300, 500, 700, 900, 1200, 1500, 1800, 2300, 2800 300, 700, 1200, 1800, 2800 300, 700, 1200, 1800, 2800 375, 1125, 1875, 2650 375, 750, 1125, 1500, 1875, 2250, 2650, 3000
2. Test program 2.1. Specimens A total of 67 tests on short, intermediate and long specimens with and without holes were categorised into four types of cross-section (see Table 1). The specimens were labelled to specify section type: for example, 90-66-1.8-H-900 represents a web length (b) of 90 mm, a flange height (h) of 66 mm, a thickness (t) of 1.8 mm and an overall length of 900 mm. The ‘H’ indicates that the specimen is perforated, while for a specimen without perforations, the ‘H’ is absent. Fig. 1 shows the cross-sectional geometry and illustrates the typical perforation locations and dimensions at the web and flange. It can be seen from the figure that triangular and circular holes are located at the web and flanges respectively; these perforations along the length of the upright allow the beam to be connected at variable heights, and steel bracings are usually bolted to the upright and act to stiffen the framework. It should be mentioned that for reasons of commercial confidentiality some geometrical details of the complex cross-sections are not permitted to be published.
Note: H – Perforated member.
the design of storage rack structures. The type of perforations described in the above papers is different from the perforations in the uprights of storage racks. There is a significant difference between the influence of large web holes on the performance of an upright and that of the perforations systematically located in the web and flanges. Local, distortional and global buckling are three typical types of buckling which a cold-formed steel member may experience [10]. Since the 1970 s there has been substantial research activity in the field of cold-formed structures which has led to numerous published works on the local, distortional and global buckling of cold-formed steel sections, (for example see [11–14]). However, single buckling modes were the focus of the above studies, while research on interactions involving distortional buckling has been conducted very recently. Kwon et al. [15] experimentally investigated the interaction between distortional buckling and material yield. Crisan et al. [16] carried out an experimental study on the interaction between distortional and overall buckling of perforated uprights. Pedro et al. [17] reported experimental and numerical results concerning local-distortional buckling interaction in fixed-end cold-formed steel web-stiffened lipped channel uprights. Dinis and Camotim investigated the interaction between local-distortional buckling ([18–20]) and distortional-global buckling [21] using the finite element method. Nandini and Kalyanaraman [22] investigated the interaction between distortional buckling and lateral-torsional buckling, also using the finite element method. Ren et al. [23] studied the distortional buckling interactions of cold-formed steel purlins under uplift loading in purlin-sheeting systems. Due to the complex nature and the limitations of research into buckling interactions on perforated rack members, and the effects of systematic perforations and strong stiffeners which eliminate the occurrence of local buckling, the interaction of distortional-global buckling on intermediate and long perforated members is thus considered to be sufficiently important to warrant further investigation.
2.2. Material properties A series of tensile coupon tests were conducted to determine the basic stress–strain response of the cold-formed steel specimens. Flat coupon samples were cut longitudinally from the rear flange of the cold-formed uprights. Three coupon samples selected from a batch of the specimen were tested using a computer controlled material tensile machine. The tension test followed the standard test procedure for tension testing of metallic materials in GB/T228-2002 [24]. The coupon sample was fully clamped at both ends, leaving a specified clear distance in the middle section to measure the elongation under tension. The goal of the coupon test was to obtain the precise tensile yield and ultimate strength for each steel rack upright specimen. A summary of the average ultimate and average 0.2% proof strengths obtained from three coupon samples are given in Table 2. It can be seen from the table that the yield strengths are found to be great in the perforated specimens than in the non-perforated specimens, indicating that the perforations have significant impact on the yield strength. This can be explained by the perforating procedure, which influences the cold forming process. The bending residual stresses produced from the coldforming process are minor [25,26] and their influence on the performance of the cold-formed steel uprights are thus ignored.
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k
Fig. 1. (a) Cross-sectional geometries. (b) Perforation positions. (c) Regular perforations in web and flanges.
Table 2 Tensile test results. Specimen
fy (N/mm2) (perforated)
fy (N/mm2) (non-perforated)
fu (N/mm2) (perforated)
fu (N/mm2) (non-perforated)
75-55-1.8 90-66-1.8 90-66-2.0 100-80-1.8 100-80-2.0 100-80-2.3 120-80-2.0
371.3 373.7 341.7 349.9 359.1 368.8 299.3
314.3 339.2 – 342.2 – – 297.0
545.0 556.3 496.5 531.1 539.1 553.5 558.3
535.3 561.8 – 530.0 – – 549.7
Fig. 2. Imperfection measurement set-up.
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75-55-1.8-250
Fig. 3. Imperfection measurement positions for each cross-section.
2.3. Initial geometrical imperfection measurements
90-66-1.8-H-900
100-80-2.0-H-1200 Fig. 5. Imperfection measurements. Fig. 4. Definition of geometric imperfections. Table 3 Imperfection magnitudes of perforated specimens (L < 1500 mm).
The selections of distribution and magnitude of imperfections are crucial due to the imperfection sensitive characteristic of cold-formed steel. A custom built test rig was employed to physically measure the geometrical imperfections of the cold-formed steel uprights, as shown in Fig. 2. Fig. 3 shows the imperfection measurement locations for each cross-section. Five linear variable differential transformers (LVDTs) were placed at the middle of the web, and at the front and rear flanges, to record the imperfection magnitudes of each segment. The LVDTs were travelled smoothly along the longitudinal direction of the specimen and the measurements from these five LVDTs provide comprehensive details about the geometrical imperfections of the specimen. Fig. 4 illustrates the definition of the geometrical imperfection magnitudes, which are Lweb, D1 and D2, respectively. The “local” imperfection magnitude at the web, Lweb is approximated by:
L web = d p5
(1)
where dp5 is the y-component of displacement of the cross-section, recorded by LVDT(5) at the middle of the web. The “distortional” imperfection magnitude for both flanges, D1 and D2 are approximated by:
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Specimen
Lweb (mm)
D1 (mm)
D2 (mm)
75-55-1.8-H-250 75-55-1.8-H-450 75-55-1.8-H-650 75-55-1.8-H-900 75-55-1.8-H-1200 90-66-1.8-H-300 90-66-1.8-H-500 90-66-1.8-H-700 90-66-1.8-H-900 90-66-1.8-H-1200 90-66-2.0-H-300 90-66-2.0-H-700 90-66-2.0-H-1200 100-80-1.8-H-300 100-80-1.8-H-500 100-80-1.8-H-700 100-80-1.8-H-900 100-80-1.8-H-1200 100-80-2.0-H-300 100-80-2.0-H-700 100-80-2.0-H-1200 100-80-2.3-H-300 100-80-2.3-H-700 100-80-2.3-H-1200 120-80-2.0-H-375 120-80-2.0-H-750 120-80-2.0-H-1125 120-80-2.0-H-1500
0.32 0.22 0.32 1.79 0.27 1.59 1.13 1.28 1.03 3.58 0.16 0.98 1.56 0.68 0.67 0.64 0.61 1.80 0.43 0.21 0.43 0.62 1.31 1.81 0.32 0.63 1.62 0.80
0.61 0.30 0.72 2.19 2.44 1.43 4.01 3.12 4.67 5.19 2.58 5.86 3.50 0.76 0.88 1.29 1.65 2.58 1.23 1.46 2.94 2.30 1.70 1.01 2.10 1.90 1.24 1.66
0.43 0.91 0.56 1.63 2.08 0.96 2.67 4.23 4.89 4.39 0.19 3.83 4.29 0.62 0.77 1.09 0.99 0.87 0.89 0.86 2.30 1.09 0.47 0.95 2.16 0.90 0.98 0.33
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Table 4 Imperfection magnitudes of non-perforated specimens (L < 1500 mm). Specimen
Lweb (mm)
D1 (mm)
D2 (mm)
75-55-1.8-250 75-55-1.8-650 75-55-1.8-1200 90-66-1.8-300 90-66-1.8-700 90-66-1.8-1200 100-80-1.8-300 100-80-1.8-700 100-80-1.8-1200 120-80-2.0-375 120-80-2.0-1125
0.32 0.07 2.10 0.73 1.86 0.03 0.36 0.51 0.50 0.47 0.43
1.06 2.45 1.40 2.10 5.02 3.11 0.59 0.66 0.71 0.90 2.86
0.65 1.92 1.70 2.62 5.79 4.75 1.37 2.68 2.51 2.01 2.84
Fig. 8. LVDT positions.
Fig. 6. Test set-up.
Fig. 9. Initial imperfection obtained from the Southwell plot (100-80-1.8-2800).
Fig. 7. Two-directional hinged joint base.
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[27], and detailed examples are presented in the next section.
Table 5 Southwell plot estimations of perforated and non-perforated specimens (L≥1500 mm). Specimen
δ0 (mm)
75-55-1.8-H-1500 75-55-1.8-H-1800 75-55-1.8-H-2200 90-66-1.8-H-1500 90-66-1.8-H-2000 90-66-1.8-H-2400 90-66-1.8-H-2700 90-66-2.0-H-2000 90-66-2.0-H-2700 100-80-1.8-H-1500 100-80-1.8-H-1800 100-80-1.8-H-2300 100-80-1.8-H-2800 100-80-2.0-H-1800 100-80-2.0-H-2800 100-80-2.3-H-1800 100-80-2.3-H-2800 120-80-2.0-H-1875 120-80-2.0-H-2250 120-80-2.0-H-2625 120-80-2.0-H-3000 75-55-1.8–1800 90-66-1.8–2000 90-66-1.8–2700 100-80-1.8–1800 100-80-1.8–2800 120-80-2.0–1875 120-80-2.0–2625
2.12 0.65 0.15 2.20 0.75 2.11 0.74 6.21 1.83 0.55 0.38 0.52 0.92 0.24 0.80 0.11 1.43 1.75 0.04 1.81 0.44 0.20 0.15 0.32 0.35 1.88 0.05 0.42
D1 =
D2 =
2.4. Test setup and instrumentations Following the aforementioned preparations, an axial compression test was carried out. As shown schematically in Fig. 6, the test specimens were placed vertically, using a 50 t hydraulic jack and a vertical reaction frame system to apply an axial load. The upper crosshead and lower actuator were both fitted with a two-directional hinged joint base (see Fig. 7). The two-directional hinged joint base is a special device to simulate the simply supported boundary condition. In this device, the top plate has a vertical slot underneath to connect with the knife on the upper surface of the adjacent middle plate, and the middle plate also has horizontal slots underneath to match the knife on the upper surface of the bottom plate. In this way the device applies sufficient restraint on torsion in the web of the specimen in the longitudinal direction, but not on rotation in the normal and tangential directions. Compared with a fixed boundary condition, the role of this device is to maximise the slenderness ratio for a fixed length of upright as well as to control the range of the slenderness ratio when the specimen is under the axial compression load in the frame system. All specimens under axial compression were monitored by a number of strain gauges and LVDTs. The strain gauges were set up for measuring the longitudinal strain at the mid-span of the cross-section. To demonstrate the deformations of members in load-displacement curves, several LVDTs were positioned at the middle span of the cross-section to determine the displacements of the flanges and the web, and also at the upper and lower support base plates to measure the longitudinal shortening displacement of specimens, as shown in Fig. 8. After all the specimens and equipment were in place, 10% of the ultimate load was initially applied to the specimen for centring. An incrementally increasing load was then added to the specimen until it reached the failure point. Deflections recorded by the LVDTs, and strains monitored by the strain gauges were collected at each incremental load step. This process was repeated for each specimen. The Southwell plot method [28] is employed herein to determine the imperfection for specimens longer than 1500 mm in length. Fig. 9 illustrates the linear relationship between deflection (δ) and ratio of deflection and axial load (δ/P), which were obtained from the axial compression test. It can be seen in this figure that the magnitude of the initial deflection, δ0, is intercepted on the abscissa. The initial deflection (δ0) can be assumed as an initial imperfection of a long specimen, which is determined from experimental measurements of deflection and axial load. A straight line is then plotted which best fits the measurement points and the slope (1/Pcr), where Pcr is the elastic buckling load of the long specimen. Table 5 shows the initial imperfections obtained from the Southwell plot method. Note that some extremely large imperfections (i.e. 90-66-2.0-H-2000) or tiny imperfections (i.e. 12080-2.0–1875) occur in some specimens, even though other specimens do not have a consistent ratio of initial imperfection to length. Because the buckling interactions, such as distortional-global buckling interaction, may affect the deflection of the tested members, the Southwell plot method is not accurate enough for estimating the geometrical imperfections of the specimens, which are dominated by the buckling interactions. As a consequence of this, a conservative estimate (L/1000 [29]) is recommended for the magnitude of the imperfection of long specimens experiencing global buckling (L≥1500 mm).
h ( d p1 − d p 3 ) dh
(2)
h (d p 2 − d p 4 ) dh
(3)
where h is the height of the flange, dp1, dp2, dp3 and dp4 are the xcomponent of displacement of the cross-section, recorded by LVDT(1)(4) at each middle flange, and dh is the distance between the imperfection measurements between the front flange and the rear flange. Fig. 5 shows the imperfection measurements of non-perforated and perforated uprights of various lengths (where the length is less than 1500 mm). It can be seen in this figure that the distortional imperfection has a significant influence on the short and intermediate specimens. It also can be seen that the lines of imperfection measurements at the perforation zones fluctuate, indicating that the perforations have an impact on the magnitude of the imperfections. Detailed imperfection measurements including for perforated and non-perforated uprights are given in Tables 3 and 4 respectively, for specimens less than 1500 mm in length. It should be noted that this test rig only records web local imperfections and flange distortional imperfections for specimens less than 1500 mm in length, because for specimens longer than 1500 mm the extra deflection generated from the self-weight of a horizontally placed specimen may affect the imperfection measurements. Alternatively, the Southwell plot method can be adopted to determine the initial imperfection of long specimens experiencing global buckling
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Table 6 Ultimate loads and failure modes. Specimen (perforated)
Pcr,1 (kN)
Buckling Mode of Pcr,1
Pu,test (kN)
Failure mode
Specimen (nonperforated)
Pcr,1 (kN)
Buckling mode of Pcr,1
Pu,test (kN)
Failure mode
75-55-1.8-H-250 75-55-1.8-H-450 75-55-1.8-H-650 75-55-1.8-H-900 75-55-1.8-H-1200 75-55-1.8-H-1500 75-55-1.8-H-1800 75-55-1.8-H-2200 90-66-1.8-H-300 90-66-1.8-H-500 90-66-1.8-H-700 90-66-1.8-H-900 90-66-1.8-H-1200 90-66-1.8-H-1500 90-66-1.8-H-2000 90-66-1.8-H-2400 90-66-1.8-H-2700 90-66-2.0-H-300 90-66-2.0-H-700 90-66-2.0-H-1200 90-66-2.0-H-2000 90-66-2.0-H-2700 100-80-1.8-H-300 100-80-1.8-H-500 100-80-1.8-H-700 100-80-1.8-H-900 100-80-1.8-H-1200 100-80-1.8-H-1500 100-80-1.8-H-1800 100-80-1.8-H-2300 100-80-1.8-H-2800 100-80-2.0-H-300 100-80-2.0-H-700 100-80-2.0-H-1200 100-80-2.0-H-1800 100-80-2.0-H-2800 100-80-2.3-H-300 100-80-2.3-H-700 100-80-2.3-H-1200 100-80-2.3-H-1800 100-80-2.3-H-2800 120-80-2.0-H-375 120-80-2.0-H-750 120-80-2.0-H-1125 120-80-2.0-H-1500 120-80-2.0-H-1875 120-80-2.0-H-2250 120-80-2.0-H-2625 120-80-2.0-H-3000
469.00 243.74 198.26 170.84 133.40 88.90 72.76 42.22 417.37 375.15 362.53 316.62 218.84 159.05 99.25 72.04 58.38 513.13 451.76 252.38 112.43 66.19 515.13 387.82 248.55 197.87 182.73 154.74 146.37 125.88 87.19 630.10 291.28 219.90 182.20 97.23 826.61 363.41 277.53 239.05 112.24 485.83 254.77 205.50 173.56 164.35 154.45 124.24 96.60
DB DB DB FTB FB FTB FB FB LB LB LB LB FTB FTB FTB FTB FTB LB LB FTB FTB FTB LB DB DB DB DB DB DB FB FB LB DB DB DB FB LB DB DB DB FB LB DB DB DB DB DB FB FB
91.47 82.77 75.17 79.20 58.70 59.79 44.57 29.66 129.68 127.97 122.53 114.92 134.49 91.63 73.92 58.70 48.14 154.06 143.19 125.02 81.07 54.67 134.96 114.77 125.02 120.67 112.75 104.05 97.68 88.37 59.79 158.25 148.62 127.97 119.74 83.55 185.58 142.10 142.25 119.12 86.97 171.92 142.25 128.90 119.43 104.21 90.07 77.49 64.29
DB DB DB FB DB+FB FTB DB+FTB DB+FTB DB DB DB DB DB+FTB FTB FTB FTB FTB DB DB DB FTB FTB DB DB DB DB DB+FB DB+FB DB+FB DB+FB FTB DB DB DB DB+FB FTB DB DB DB+FB DB+FB DB+FB DB DB DB DB+FB DB+FB DB+FB DB+FB DB+FB
75-55-1.8-250 – 75-55-1.8-650 – 75-55-1.8-1200 – 75-55-1.8-1800 – 90-66-1.8-300 – 90-66-1.8-700 – 90-66-1.8-1200 – 90-66-1.8-2000 – 90-66-1.8-2700 – – – – – – – 100-80-1.8-700 – 100-80-1.8-1200 – 100-80-1.8-1800 – 100-80-1.8-2800 – – – – – – – – – – 120-80-2.0-375 – 120-80-2.0-1125 – 120-80-2.0-1875 – 120-80-2.0-2625 –
549.53 – 253.67 – 157.43 – 62.588 – 538.99 – 462.09 – 255.54 – 114.79 – 67.13 – – – – – – – 275.06 – 206.91 – 175.75 – 98.94 – – – – – – – – – – 560.93 – 243.46 – 190.94 – 139.79 –
DB – DB – FTB – FTB – LB – LB – DB – FTB – FTB – – – – – – – DB – DB – DB – FB – – – – – – – – – – LB – DB – DB – FB –
112.13 – 95.04 – 84.02 – 53.58 – 177.66 – 162.29 – 134.49 – 100.63 – 67.40 – – – – – – – 140.24 – 139.21 – 104.67 – 79.20 – – – – – – – – – – 190.40 – 153.90 – 129.52 – 87.43 –
DB – DB – DB+FB – DB+FB – DB – DB – DB – FTB – FTB – – – – – – – DB – DB – DB+FB – FTB – – – – – – – – – – DB – DB – DB – DB+FTB –
3. Test results and analysis
in Table 6 for perforated and non-perforated specimens, respectively. Fig. 10 depicts the typical load-displacement curves of specimens with different cross-sections and lengths. The ordinate axis represents compressive load, and in the main chart the abscissa axis represents the displacement of flanges and webs recorded from LVDTs at the middle span cross-section, and in the inset chart the abscissa axis represents the longitudinal shortening displacement. It should be noted that for the middle span cross-sections of the main charts, the maximum displacements correspond to the ultimate failure loads. It can be seen in
3.1. Load-displacement curves Compression tests were conducted on 67 specimens, including four different cross-sections, various lengths, and specimens with and without perforations. Ultimate loads and failure modes recorded from the compression tests and critical buckling loads and buckling modes (first buckling modes) obtained from finite element analyses (FEA) are shown
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75-55-1.8-H-250 (DB)
75-55-1.8-H-1500 (FTB)
75-55-1.8-H-1200 (DB+FB)
100-80-1.8-H-2800 (FTB) Fig. 10. The load-displacement curves.
Fig. 10(a) that the obvious displacements of flanges, together with almost no movement at the web-flange junctions, reveals that distortional buckling (DB) is the main failure mode for the short specimen (75-55-1.8-H-250). In Fig. 10(b) the buckling modes of specimen 75-551.8-H-1200 is dominated by the distortional-flexural buckling interaction (DB+FB). The large displacements of the web-flange junctions and the equivalent displacements of the flanges indicate that not only flexural buckling exists, but also that distortional buckling has an influence on the failure mode of the specimen. Nevertheless, In Fig. 10(c) and (d), for 75-55-1.8-H-1500 and 100-80-1.8-H-2800, the buckling modes are mainly controlled by flexural-torsional buckling (FTB), which the shape of cross-section keeps non-deformable when the buckling is occurring. It can also be seen in the inset charts of Fig. 10, that for the load-longitudinal shortening displacement curve of perforated and non-perforated specimens which failed due to distortional bucking, distortional-flexural buckling interaction and flexural-torsional buckling, the ultimate loads of perforated specimens are lower than the ultimate loads obtained from non-perforated specimens. This indicates that, in terms of axial compression, the perforations have a significant influence on the performance of the cold-formed steel uprights due to the change of cross-section area. The specimens were classified into three categories according to the buckling failure mode: distortional buckling (DB) failure specimens, global buckling failure specimens (flexural-torsional buckling (FTB) or flexural buckling (FB))
and interaction of distortional-global (DB+E) buckling failure specimens, typical examples of which are shown in Fig. 11.
3.2. Influences of cross-sections Fig. 12 shows the parametric distribution of different buckling modes for specimens with perforations and without perforations, respectively. It can be seen that distortional buckling and the interaction of distortional-global (distortional-flexural-torsional buckling interaction or distortional-flexural buckling interaction) are the main buckling failure modes for both perforated and non-perforated specimens. The buckling mode is only dominated by distortional buckling where the length of the perforated specimen is shorter than 1250 mm, and the length of the non-perforated specimen is shorter than 2000 mm. The buckling mode is mainly controlled by the interaction of distortional-flexural or distortional-flexural-torsional buckling where the flange-thickness ratio (h/t) of the perforated specimen is greater than 40. Flexural-torsional buckling failure of perforation specimens has a lower flange-thickness ratio (h/t). Fig. 13 shows a comparison of the normalised compression load capacity obtained from tests on specimens with and without perforations, plotted against length for four different cross-sections. This comparison takes the same thickness and length for non-perforated and perforated specimens. It is seen from this figure that load decreases
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75-55-1.8-H-250 (DB)
75-55-1.8-250 (DB)
90-66-1.8-H-900 (DB)
120-80-2.0-H-1125 (DB)
Fig. 11. Typical buckling failure modes (front and side views).
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75-55-1.8-H-1200 (DB+FB)
75-55-1.8-1200 (DB+FB)
90-66-1.8-H-1200 (DB+FTB)
100-80-2.0-H-1800 (DB+FB) Fig. 11. (continued)
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75-55-1.8-H-1500 (FTB)
90-66-2.0-H-2000 (FTB)
100-80-1.8-H-2800 (FTB)
100-80-1.8-2800 (FTB) Fig. 11. (continued)
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study, the upright fails when a global buckling mode develops and interacts/couples with the distortional buckling. However, in the crosssection of 90-66, the buckling interactions are rarely observed (see Table 6 for details). This is because when an upright buckles, it usually changes from a state of compression to a combined state of compression and bending. However, compared with the others cross-sections, the shape of 90-66 cross-section is special and is close to a lipped channel section, and the cross-section has extreme strong flange stiffeners, which provide high flexural rigidity to avoid flexural buckling. Consequently, the failure modes are mainly controlled by distortional buckling of the compression flange-lip systems where the upright is short, and the uprights have low torsional rigidity about their minor axes, rendering the members prone to flexural-torsional buckling at mid-span where the upright is long. 4. Comparisons between DSM and test results The complex cross-sections of the perforated steel storage rack uprights subjected to axial compression studied in the present paper are not stipulated in the Direct Strength Method (DSM) included in the North American Specification (AISI S100) [30]. This section addresses the applicability of the Direct Strength Method (DSM) for estimating the ultimate strength of uprights failing in different buckling modes, and describes the modified DSMs obtained from the test results. To consider the influence of the perforations in the design of steel storage rack uprights subjected to axial compression, comparisons between the DSM predictions and the ultimate strength of perforated uprights and nonperforated uprights obtained from testing are illustrated in Fig. 14. DB and DB+E test results compared with the DSM (DB) curve and DB+E and E test results compared with the DSM (E) curve are correspondingly presented in Fig. 14(a) and (b). The slenderness of the upright is represented by λ=(Py/Pcr,1)0.5, where Pcr,1 is the critical distortional buckling load for Fig. 14(a) and the critical global buckling load for Fig. 14(b), respectively. Note that the Pcr,1 is calculated using the first eigenvalue of buckling from a finite element analysis (FEA). It should be mentioned that the critical buckling load obtained from the theoretical method or the finite strip method (FSM) does not consider the influence of perforations, and the FEA is thus used to determine the critical buckling load of perforated uprights. Specimens of identical crosssectional dimensions, thickness and length are compared, with the only difference being whether the specimen has perforations. It can be observed from Fig. 14 that the ultimate loads of non-perforated specimens are reduced by such perforations, indicating that the perforations have a significant influence on the performance. It also can be seen in this figure that the predictions of distortional buckling failure obtained from the test are generally underestimated by the distortional buckling curve of the DSM, but the predictions of global buckling failure determined from the experiment are in good agreement with those of the DSM. Additionally, for distortional-global buckling interaction failures the predictions obtained from the test are below the distortional buckling strength curve of the DSM, and for some specimens, the predictions are even lower than the predicted global buckling curve of the DSM. This implies that the ultimate strength may be eroded significantly when the member experiences buckling interaction phenomena, and thus the current DSM distortional buckling strength curve does not offer a reliable design for perforated steel storage rack uprights subjected to axial compression. As a consequence of this, the modifications to the current DSM distortional strength curve are given as follows:
Fig. 12. Parametric distribution of different buckling modes.
with increasing specimen length, and the perforations have an influence on the load capacity for small cross-sections. The load capacity is reduced for sections 75-55-1.8 and 90-66-1.8 when perforations are present. However, the load capacity is almost identical whether perforations are present or absent for sections 100-80-1.8 and 12080-2.0. The former observation reveals that the perforations have an influence on the ultimate load bearing capacity of rack uprights in small sections. The latter observation implies that the influence of the perforations is insignificant in large section specimens. The explanations for above observations are that in terms of perforation ratio, the effective area of the small cross-sections is sensitive to the perforations, and the ultimate load is thus significantly reduced in perforated members. Whereas the effective area of the large cross-sections is impacted insignificantly by the perforations, therefore the differences of ultimate load between perforated members and non-perforated members are negligible. It is worth noting here that the effect of perforations on the performance of an upright is considerable, and this illustrates the significance of ensuring that storage uprights have an appropriate cross-section. One of the concerns in the buckling interaction of cold-formed steel members is the post-buckling strength of distortional buckling, which allows it to combine with local buckling and global buckling. In the
For λ ≤ 0. 568
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Pnd =1 Py
(4)
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(a) Cross-section 75-55-1.8.
(c) Cross-section 100-80-1.8.
(b) Cross-section 90-66-1.8.
(d) Cross-section 120-80-2.0.
Fig. 13. Ratio of compression load versus length for different cross section members with and without perforations.
For λ
> 0. 568
where λ =
⎛ ⎛ P ⎞0.35⎞⎛ P ⎞0.8 Pnd cr ,1 ⎟⎟ ⎟⎜⎜ cr ,1 ⎟⎟ = ⎜⎜1 − 0. 4⎜⎜ Py Py ⎠ ⎟⎝ Py ⎠ ⎝ ⎝ ⎠
5. Conclusions This paper has presented an experimental investigation into the behaviour of both perforated and non-perforated steel storage rack uprights subjected to axial compression. Comprehensive experimental studies including material tensile testing, geometrical imperfection measurement and axial compression testing have been presented and described, and the corresponding results obtained from the tests have been listed and discussed. Comparisons of performance and failure mode between perforated and non-perforated members have been provided. The results show that the perforations have a significant influence not only on the performance but also on the buckling failure mode of cold-formed steel storage rack uprights. The interaction of distortional-global buckling has also been studied. The results of this study explicitly show that the Direct Strength Method (DSM) has been demonstrated to be unreliable for predicting the ultimate strength of uprights failing in buckling interactions for perforated uprights. Hence, based on the test results, a modified DSM distortional buckling strength curve for perforated uprights has been proposed.
(5)
Py Pcr,1
Pcr,1 =Critical elastic buckling load (first buckling mode) Py = Anet Fy where Anet is the net cross-sectional area, and Pcr,1 is the critical elastic buckling load (first buckling mode). Fig. 15 shows a comparison of the DSM predictions with test results for the perforated specimens, which are dominated by distortional buckling (in this case, Pcr,1 is the critical elastic distortional buckling load). A modified DSM for the distortional strength curve obtained from the test results is also provided in this figure. It should be mentioned that the distortional-global buckling interaction failures are treated as distortional buckling failures for the purposes of investigating buckling interactions.
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Fig. 14. Comparison between the DSM and test results.
Fig. 15. Comparison of the current DSM distortional strength curve and modified strength curve of perforated members.
Acknowledgement The authors gratefully acknowledge the support provided by WAP logistics equipment (Shanghai) Ltd. The authors would like to acknowledge the financial support received from the China Postdoctoral Science Foundation (2014M561517).
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