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Design of cold-formed steel unequal angle compression members
ARTICLE IN PRESS
Thin-Walled Structures 45 (2007) 330–338 www.elsevier.com/locate/tws
Design of cold-formed steel unequal angle compression members Ben Younga,, Ehab Ellobodyb a
Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong Department of Structural Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt
b
Received 20 July 2006; received in revised form 2 February 2007; accepted 22 February 2007 Available online 9 May 2007
Abstract Cold-formed steel unequal angles are non-symmetric sections. The design procedure of non-symmetric sections subjected to axial compression load could be quite difficult. The unequal angle columns may fail by different buckling modes, such as local, flexural and flexural–torsional buckling as well as interaction of these buckling modes. The purpose of this study is to investigate the behaviour and design of cold-formed steel unequal angle columns. A nonlinear finite element analysis was conducted to investigate the strength and behaviour of unequal angle columns. The measured initial local and overall geometric imperfections as well as the material properties of the angle specimens were included in the finite element model. The finite element analysis was performed on fixed-ended columns for different lengths ranged from stub to long columns. It is demonstrated that the finite element model closely predicted the experimental ultimate loads and the behaviour of cold-formed steel unequal angle columns. Hence, the model was used for an extensive parametric study of cross-section geometries. The column strengths obtained from the parametric study were compared with the design strengths calculated using the North American Specification for cold-formed steel structural members. It is shown that the current design rules are generally unconservative for short and intermediate column lengths for the unequal angles. Therefore, design rules of cold-formed steel unequal angle columns are proposed. r 2007 Elsevier Ltd. All rights reserved. Keywords: Buckling; Cold-formed steel; Column; Finite element; Modelling; Steel structures; Structural design; Thin-walled structures; Unequal angle
1. Introduction In recent years, cold-formed steel sections are become popular because of its high strength and light weight. Although, cold-formed steel angle columns are simple structural members, the design rules are quite complicated. So far, limited test data has been found in the literature on cold-formed steel equal and unequal angle columns. Experimental investigations have been found in the literature on the behaviour of cold-formed and hot-rolled steel plain angle columns as summarized in Yu [1] and Popovic et al. [2]. Cold-formed steel plain angle columns have been tested by Popovic et al. [2] and Young [3]. Investigation on the column behaviour of lipped angle sections with equal flange widths, known as equal lipped angles in this paper, have been presented by Zavelani and Faggiano [4], Gaylord and Wilhoite [5], Raghunathan et al. Corresponding author Tel.: +852 2859 2674; fax: +852 2559 5337.
E-mail address: [email protected] (B. Young). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.02.015
[6], Marsh [7], Young [8], and Young and Ellobody [9]. However, very limited column test data were found in the literature on lipped angle sections with unequal flange widths, known as unequal lipped angles in this paper. Young [10] conducted a series of tests on cold-formed steel unequal lipped angle columns. The column strengths and failure modes were presented. Finite element analysis of cold-formed steel unequal lipped angle columns could provide better understanding to the complex buckling behaviour of non-symmetric angles. Numerical investigation on cold-formed steel columns has been presented by Young and Yan [11], Bakker and Peko¨z [12], Ellobody and Young [13] and other researchers. The primary objective of this study is to investigate the strength and behaviour of cold-formed steel unequal lipped angle columns. Thus, design rules of cold-formed steel unequal angle columns are proposed. The investigation was carried out in two stages. The first stage is to measure the initial local geometric imperfections and corner
ARTICLE IN PRESS B. Young, E. Ellobody / Thin-Walled Structures 45 (2007) 330–338
Nomenclature
Pn
Ae Bf 1 Bf 2 Bl b E eFE
Pp
eTest Fe Fn Fy L le l ex , l ey lt P PFE
effective area overall width of long flange (long leg) overall width of short flange (short leg) overall width of lip flat flange width Young’s modulus axial shortening obtained from finite element analysis at failure axial shortening obtained from tests at failure elastic buckling stress critical buckling stress yield stress is taken as 0.2% proof stress (s0:2 Þ actual column length column effective length column effective length for buckling about the principal x-axis and y-axis column effective length for torsional buckling axial compressive load ultimate load obtained from finite element analysis
PTest ro1 ri rx , ry t x, y x o , yo lc s0:2 sex , sey st su
nominal axial strength calculated using North American Specification (unfactored design strength) proposed design strengths (unfactored design strength) test ultimate load (test strength) polar radius of gyration of cross section about the shear centre inside corner radius of specimen radii of gyration of cross section about the principal x-axis and y-axis plate thickness of specimen principal coordinates distances from the shear centre to centroid along the principal x-axis and y-axis elongation (tensile strain) after fracture based on gauge length of 50 mm non-dimensional slenderness static 0.2% tensile proof stress elastic buckling stresses for flexural buckling about the principal x-axis and y-axis elastic buckling stress for torsional buckling static tensile strength
material properties of cold-formed steel unequal lipped angle columns. The second stage is to develop an accurate finite element model to simulate the behaviour of these columns. The results obtained from the finite element analysis were verified against the test results obtained by Young [10]. Parametric study was performed to investigate the effect of cross-section geometries on the strength and behaviour of unequal lipped angle columns. The results obtained from the finite element analysis were compared with design strengths calculated using the North American Specification (NAS) [14] for cold-formed steel structural members. 2. Experimental investigation
Bl Corner A
ri
t
Bf1
2.1. General A series of column tests on cold-formed steel unequal lipped angle sections has been conducted by Young [10]. The angle specimens were tested between fixed-ended conditions at different column lengths ranged from stub to long columns. Fig. 1 shows the definition of symbols for a cold-formed steel unequal lipped angle section. Three series of unequal lipped angles were tested, having a nominal long flange (long leg) width of 80 mm, a nominal short flange (short leg) width of 50 mm, and a nominal lip width of 16 mm. The nominal plate thicknesses were 1.0, 1.5 and 1.9 mm. The three series are labelled U1.0, U1.5 and U1.9 according to their nominal thickness. The test specimens are labelled such that the test series and specimen length could be identified from the label. For
331
Corner C
Corner B Bl
Bf2 Fig. 1. Definition of symbols for cold-formed steel unequal lipped angle section.
example, the label ‘‘U1.0L1500’’ defines the specimen belonged to test Series U1.0, and the fourth letter ‘‘L’’ indicates the length of the specimen. The last three or four
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digits are the nominal column length of the specimen in mm (1500 mm). The measured inside corner radius (ri Þ was 3.0 mm for Series U1.0, and 3.5 mm for Series U1.5 and U1.9 specimens. The initial overall geometric imperfections of the column specimens were measured. The maximum overall geometric imperfections at mid-length were 1=1410, 1=1690 and 1=1300 of the specimen length for Series U1.0, U1.5 and U1.9, respectively. The measured cross-section dimensions and initial overall geometric imperfections of the test specimens are detailed in Young [10]. The lipped angle test specimens were brake-pressed from high strength zinc-coated structural steel sheets having nominal yield stresses of 450 and 550 MPa. Tensile coupon tests were conducted to obtain the material properties in the flat portion of the test specimens. The measured material properties in the flat portion of the test specimens obtained from the coupon tests are summarized in Table 1. The measured Young’s modulus (EÞ, static 0.2% proof stress (s0:2 Þ, static tensile strength (su Þ, and elongation after fracture (Þ based on a gauge length of 50 mm are shown in Table 1. The tensile coupon tests of the flat portion of the specimens are detailed in Young [10]. The initial local geometric imperfections and corner material properties of the test specimens have not been reported, and the values of these measurements are also important for finite element analysis. Therefore, the initial local imperfections and corner material properties of the unequal lipped angle specimens belonged to the same batch of specimens as the column tests are measured in this study and reported in this paper. A servo-controlled hydraulic testing machine was used to apply compressive axial force to the column specimen. The experimental ultimate loads and the failure modes of the unequal lipped angle column tests are detailed in Young [10]. The failure modes of the columns involved local buckling (L), flexural–torsional buckling (FT) and the interaction of these two modes. 2.2. Material and imperfections measurements The material properties in the corners of the unequal lipped angle section of Series U1.9 were determined. The material properties of the corners of Series U1.0 and U1.5 were extrapolated from the material properties of the Series Table 1 Measured and predicted material properties Test series
Portion
E (GPa)
s0:2 (MPa)
su (MPa)
(%)
U1.0
Flat Corners A, C Corner B Flat Corners A, C Corner B Flat Corners A, C Corner B
212 206 206 216 206 206 196 206 206
605 710 695 530 620 610 505 590 580
615 750 745 560 655 650 535 625 620
7.8 2.5 2.9 11.5 2.5 2.9 10.3 2.5 2.9
U1.5
U1.9
Note: 1 ksi ¼ 6.89 MPa.
U1.9 according to the measured 0.2% tensile proof stress (s0:2 Þ in the flat portions of each series. The measured and predicted material properties of the corner and flat coupons are summarized in Table 1. Initial local geometric imperfections have a considerable effect on the buckling behaviour of cold-formed steel columns due to the thinner wall thickness of cold-formed steel sections compared with hot-rolled steel sections. An unequal lipped angle test specimen of 300 mm in length for Series U1.9 was used for the measurement of local imperfections. The measurements were taken at the middle and quarter lengths of the specimen. Readings were taken at regular intervals and maximum magnitude of local plate imperfection was 0.01 mm, which is equal to 0.53% of the angle thickness. The same factor was used to predict the initial local geometric imperfections of the angle specimens for Series U1.0 and U1.5. The measurement procedures for the corner material properties and initial local geometric imperfections of the unequal lipped angle specimens are identical to those for the equal lipped angle specimens as detailed in Young and Ellobody [9]. 3. Finite element model 3.1. General The finite element program ABAQUS [15] was used to simulate the behaviour of the unequal lipped angle columns tested by Young [10]. The measured geometry, initial local and overall geometric imperfections, and material properties at the flat and corner portions of the unequal lipped angle columns were included in the finite element model. Two types of analyses were carried out. The first is Eigenvalue analysis, and the second is loaddisplacement nonlinear analysis. The four-noded doubly curved shell elements with reduced integration S4R were used in the model. The S4R element has six degrees of freedom per node and proved to give accurate solutions from previous research as shown in Ellobody and Young [13] and Young and Ellobody [9]. The finite element mesh size of 10 mm 10 mm (length by width) for the flat portions and finer mesh size was used at the corners. Fig. 2 shows the deformed shape of the finite element mesh for the unequal lipped angle column specimen U1.5L1500. In the finite element model, the ends of the columns were fixed against all degrees of freedom except for the displacement at the loaded end in the direction of the applied load, which is identical to the testing procedures for the unequal lipped angle columns. The nodes other than the two ends were free to translate and rotate in any directions. The load was applied in increments using the modified RIKS method available in the ABAQUS library. The load was applied as static uniform loads at each node of the loaded end, which is identical to the experimental investigation. The nonlinear geometry parameter (NLGEOM) was included to deal with the large displacement analysis. The stress–strain curves for the flat and
ARTICLE IN PRESS B. Young, E. Ellobody / Thin-Walled Structures 45 (2007) 330–338
Short leg
Long leg Fig. 2. Deformed shape of unequal lipped angle column U1.5L1500.
corner portions were used in the analysis. The initial local and overall geometric imperfections were incorporated in the model. Superposition of local buckling mode as well as overall buckling mode with measured magnitudes was carried out. These buckling modes were obtained by eigenvalue analysis of the column with very large b=t ratio and very small b=t ratio to ensure local and overall buckling occurs, respectively, where b is the flat flange width and t is the plate thickness of the angle sections. The shape of a local buckling mode as well as overall buckling mode is taken as the lowest buckling mode (eigenmode 1) in the analysis. Short columns having lengths of 250 and 625 mm were modelled for local imperfection only. All buckling modes predicted by ABAQUS Eigenvalue analysis are normalized to 1.0, therefore, the buckling modes are factored by the measured magnitudes of the initial local and overall geometric imperfections. Previous studies by Ellobody and Young [13] and Young and Ellobody [9] on cold-formed steel equal plain and lipped angle columns, have shown that the effect of residual stresses on the column capacity is negligible. In this study, it is assumed that the effect of residual stresses on the column strength of cold-formed steel unequal lipped angle columns is also negligible. Hence, residual stresses were not included in the finite element model.
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obtained from the test strengths. A maximum difference of 8% was observed between experimental and numerical column strengths for specimen U1.9L3000. The mean values of PTest =PFE ratio are 1.04, 1.03 and 1.06 with the corresponding coefficients of variation (COV) of 0.025, 0.029 and 0.019 for series U1.0, U1.5 and U1.9, respectively, as shown in Table 2. The mean values of eTest =eFE ratio are 1.00, 0.96 and 1.03 with the corresponding COV of 0.069, 0.067 and 0.063 for Series U1.0, U1.5 and U1.9, respectively. Generally, good agreement has been achieved for most of the columns. Three buckling modes were predicted by the finite element analysis for the unequal lipped angle columns. The buckling modes are the local buckling (L), flexural buckling (F) and flexural–torsional buckling (FT). The local buckling was observed in the stub column tests of 250 mm length for Series U1.0, U1.5 and U1.9, and for most of the columns in Series U1.0, which was also predicted by the finite element analysis. For all the three columns series, flexural–torsional buckling was observed experimentally and confirmed numerically for columns lengths ranged from 625 to 3000 mm. Flexural buckling was predicted by the finite element analysis for the long columns, but was not observed in the tests. Fig. 4 shows the applied load against the axial shortening curves of column U1.5L1500 obtained from the test and finite element analysis. It is shown that the column initial stiffness and behaviour reflects good agreement between experimental and finite element results. The experimental ultimate load was 50.1 kN with an axial shortening at the ultimate load of 4.15 mm compared with 48.3 kN and 4.33 mm, respectively, predicted by the finite element analysis. Figs. 5 and 6 show the deformed shapes of the unequal lipped angle column observed experimentally and confirmed by the finite element analysis in different views. It can be seen that the angle cross section is twisted and flexural buckled about the major axis from its original position resulting in a flexural–torsional buckling mode. The finite element model closely predicted the failure mode of the columns. 4. Parametric study
3.2. Verification of finite element model The finite element results are verified against the test results conducted by Young [10]. A total of 21 cold-formed steel unequal lipped angle columns were analyzed using the finite element model. The comparison of the ultimate loads (PTest and PFE Þ, axial shortening (eTest and eFE Þ at the ultimate loads and failure modes obtained experimentally and numerically are shown in Table 2. Fig. 3 plotted the relationship between the ultimate load and the column effective length ðl e ¼ L=2Þ for Series U1.0, U1.5 and U1.9, where L is the actual column length. It can be seen that good agreement has been achieved between both the experimental and numerical results for most of the columns. Generally, the column strengths obtained from the finite element analysis are slightly less than those
A parametric study was performed using the verified finite element model to investigate the effects of crosssection geometries on the strength and behaviour of coldformed steel unequal lipped angle columns. A total of 35 cold-formed steel unequal lipped angle columns was analyzed in the parametric study. Five series of columns U0.55, U0.7, U0.85, U3.0 and U5.0 having plate thicknesses of 0.55, 0.7, 0.85, 3.0 and 5.0 mm, respectively, were studied. All angle sections had the overall flange widths Bf 1 and Bf 2 of 84 and 54 mm, respectively, which is having approximately the same flange widths as the test specimens. The inner corner radius (ri Þ was taken as 3.0 mm for Series U0.55, U0.7 and U0.85, while the inner corner radius for Series U3.0 and U5.0 was taken as 3.5 and 5.0 mm, respectively. The five series had the flat long flange
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Table 2 Comparison between test and finite element results Specimen
Test
Analysis
PTest (kN) U1.0L250 U1.0L625 U1.0L1000 U1.0L1500 U1.0L2000 U1.0L2500 U1.0L3000 Mean COV
42.5 35.5 26.6 25.9 23.4 21.5 16.4 – –
U1.5L250 U1.5L625 U1.5L1000 U1.5L1500 U1.5L2000 U1.5L2500 U1.5L3000 Mean COV
90.0 69.9 51.6 50.1 42.9 36.1 29.1 – –
U1.9L250 U1.9L625 U1.9L1000 U1.9L1500 U1.9L2000 U1.9L2500 U1.9L3000 Mean COV
121.6 86.9 68.7 64.3 55.8 50.5 36.3 – –
eTest (mm)
Failure mode
0.92 1.07 1.48 3.05 3.91 4.05 4.40
L L þ FT L þ FT L þ FT L þ FT L þ FT FT
– –
– –
0.93 1.15 1.77 4.15 4.20 4.25 4.45
L FT FT FT FT FT FT
– –
– –
1.35 1.22 2.80 4.20 4.40 5.30 –
L FT FT FT FT FT FT
– –
– –
PFE (kN) 42.4 34.4 26.7 24.6 22.0 20.5 15.4 – – 90.2 70.2 51.0 48.3 40.3 34.1 27.2 – – 114.7 84.4 64.1 62.6 52.9 47.1 33.6 – –
Test/Analysis eTest eFE
eFE (mm)
Failure mode
PTest PFE
1.00 1.03 1.56 3.31 3.71 3.92 4.02
L L þ FT L þ FT L þ FT L + F þ FT L + F þ FT F þ FT
1.00 1.03 1.00 1.05 1.06 1.05 1.06
0.92 1.04 0.95 0.92 1.05 1.03 1.09
– –
– –
1.04 0.025
1.00 0.069
1.01 1.05 1.97 4.33 4.35 4.47 4.67
L FT FT FT F þ FT F þ FT F þ FT
1.00 1.00 1.01 1.04 1.06 1.06 1.07
0.92 1.10 0.90 0.96 0.97 0.95 0.95
– –
– –
1.03 0.029
0.96 0.067
1.28 1.16 2.52 4.41 4.62 5.00 –
L FT FT FT F þ FT F þ FT F þ FT
1.06 1.03 1.07 1.03 1.05 1.07 1.08
– –
– –
1.06 0.019
1.05 1.05 1.11 0.95 0.95 1.06 – 1.03 0.063
Note: 1 kip ¼ 4.45 kN; 1 in ¼ 25.4 mm.
150 Test U1.9 FE U1.9
Axial load, P (kN)
120
Test U1.5 FE U1.5
90
Test U1.0 FE U1.0
60
30
0 0
500
1000 1500 Effective length, le (mm)
2000
Fig. 3. Ultimate loads of unequal lipped angle columns for Series U1.0, U1.5 and U1.9.
width-to-thickness ratio (b=tÞ of 139.8, 109.4, 89.8, 23.7 and 12.8 for Series U0.55, U0.7, U0.85, U3.0 and U5.0, respectively. The length of the lips (Bl Þ was kept at 17.0 mm
for all series. Table 3 summarizes the dimensions of the five series analyzed in the parametric study. Each series of columns consists of seven column lengths of 250, 625, 1000, 1500, 2000, 2500 and 3000 mm. The maximum initial local geometric imperfection magnitude was taken as the measured value of 0.53% of the plate thickness. The maximum initial overall geometric imperfection magnitude was taken as the average of the measured maximum overall imperfections of the tested series, which is equal to L=1450, where L is the column length. The stress–strain curves of the flat and corner portions of Series U1.9 were used in the parametric study. A summary of the parametric study results is presented in Table 4. The ultimate loads (PFE Þ, axial shortening at ultimate load (eFE Þ and failure modes are given in Table 4. 5. Current design rules The NAS [14] for the design of cold-formed steel structural members provides design rules for non-symmetric angle columns. In this study, the lipped angle sections with unequal flange widths are non-symmetric
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60
Axial load (kN)
50 40 30 20 Test 10
FE
0 0
2
4
6
8
10
Shortening (mm) Fig. 4. Load-axial shortening curves for column U1.5L1500.
Fig. 6. Comparison of experimental and finite element analysis deformed shapes of column U1.5L1500 (view into short leg). (a) Experimental; (b) FE analysis.
Fig. 5. Comparison of experimental and finite element analysis deformed shapes of column U1.5L1500 (view into long leg). (a) Experimental; (b) FE analysis.
sections. The columns were compressed between fixed ends. The fixed-ended bearings were restrained against the major and minor axis rotations as well as twist rotations and warping. The effective lengths (l e Þ for principal x-axis (l ex Þ and principal y-axis (l ey Þ flexural buckling as well as torsional buckling (l t Þ are assumed equal to one-half of the
column length (LÞ for the fixed-ended columns. The shift in the line of action of the internal force is balanced by a shift in the line of action of the external force for fixed-ended column [16]. Hence, the unequal lipped angles were designed as concentrically loaded compression members in accordance with Section C4 of the NAS Specification [14]. It should be noted that angle sections shall be designed for an additional bending moment, for which the additional moment is calculated as the axial load multiplied by an eccentricity of 1=1000 of the column length as recommended in the NAS Specification. The required additional bending moment was based on the investigation of plain angle sections with equal flange widths. Hence, the additional moment may not be needed for the unequal
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Specification [14] are as follows:
Table 3 Dimensions of unequal lipped angle sections in parametric study Series
U0.55 U0.7 U0.85 U3.0 U5.0
Flanges
Pn ¼ Ae F n ,
Lips
Thickness
Radius
Bf 1 (mm)
Bf 2 (mm)
Bl (mm)
t (mm)
ri (mm)
84.0 84.0 84.0 84.0 84.0
54.0 54.0 54.0 54.0 54.0
17.0 17.0 17.0 17.0 17.0
0.55 0.7 0.85 3.0 5.0
3.0 3.0 3.0 3.5 5.0
Note: 1 in ¼ 25.4 mm.
Table 4 Summary of parametric study results Specimen
where Ae is the effective area calculated at the critical buckling stress (F n Þ. The effective area that accounts for local buckling is calculated in accordance with Section B of the NAS Specification. The critical buckling stress (F n Þ that accounts for overall buckling is determined as 2
for lc p1:5 ) F n ¼ ð0:658lc ÞF y , " # 0:877 for lc 41:5 ) F n ¼ F y, l2c where rffiffiffiffiffiffi Fy lc ¼ , Fe
FE analysis PFE (kN)
(1)
eFE (mm)
Failure mode
U0.55L250 U0.55L625 U0.55L1000 U0.55L1500 U0.55L2000 U0.55L2500 U0.55L3000
11.5 9.9 6.8 4.4 3.4 3.0 2.7
0.39 0.56 0.75 1.10 1.70 2.01 2.59
L L L þ FT L þ FT L þ FT L + F þ FT F þ FT
U0.7L250 U0.7L625 U0.7L1000 U0.7L1500 U0.7L2000 U0.7L2500 U0.7L3000
18.8 15.9 9.9 6.5 5.6 5.3 4.9
0.33 0.60 0.88 1.12 1.79 2.16 2.63
L L L þ FT L þ FT L þ FT L + F þ FT F þ FT
U0.85L250 U0.85L625 U0.85L1000 U0.85L1500 U0.85L2000 U0.85L2500 U0.85L3000
26.6 22.7 15.3 12.0 10.7 9.8 7.9
0.58 0.65 1.27 1.80 1.97 2.39 2.89
L L L þ FT L þ FT L þ FT L þ F þ FT F þ FT
U3.0L250 U3.0L625 U3.0L1000 U3.0L1500 U3.0L2000 U3.0L2500 U3.0L3000
182.2 121.4 90.7 76.7 63.5 54.7 41.0
1.64 1.03 1.53 1.87 2.06 2.50 2.98
L FT FT FT F þ FT F þ FT F þ FT
U5.0L250 U5.0L625 U5.0L1000 U5.0L1500 U5.0L2000 U5.0L2500 U5.0L3000
326.1 195.8 164.0 136.2 109.1 84.8 66.3
2.58 1.74 1.79 1.99 2.22 2.63 2.99
L FT FT FT F þ FT F þ FT F þ FT
Note: 1 kip ¼ 4.45 kN; 1 in ¼ 25.4 mm.
lipped angle columns, and the required additional moment was not included in calculating the design strengths. The design rules of nominal axial strength (Pn Þ for concentrically loaded compression members in the NAS
(2) (3)
(4)
where F y is the yield stress which is taken as the 0.2% proof stress (s0:2 Þ, and F e is the elastic buckling stress that was obtained by taking the smallest value of the following cubic equation: F 3e ðr2o1 x2o y2o Þ F 2e ½r2o1 ðsex þ sey þ st Þ ðsey x2o þ sex y2o Þ þ F e r2o1 ðsex sey þ sey st þ sex st Þ ðsex sey st r2o1 Þ ¼ 0, ð5Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ro1 ¼ r2x þ r2y þ x2o þ y2o is the polar radius of gyration of cross section about the shear centre, xo and yo are the distances from the shear centre to centroid along the principal x-axis and y-axis, respectively, rx and ry are the radii of gyration of cross-section about the principal x-axis and y-axis, respectively, sex and sey are the elastic buckling stresses for flexural buckling about the principal x-axis and y-axis, respectively, and st is the elastic buckling stress for torsional buckling. Alternatively, F e may be determined by rational elastic buckling analysis. 6. Proposed design rules The column design rules proposed by Young [3] and Young and Ellobody [9] for singly-symmetric cold-formed steel plain and lipped angle sections respectively, are used to predict the design strengths of the non-symmetric lipped angle sections. The proposed equations required simple modification to the critical buckling stress (F n Þ in the NAS Specification [14]. The proposed design equations are as follows: 2
for lc p1:4 ) F n ¼ ð0:5lc ÞF y , " # 0:5 for lc 41:4 ) F n ¼ F y, l2c
(6) (7)
where the non-dimensional slenderness ðlc Þ is identical to that in Eq. (4), except that the elastic buckling stress ðF e Þ is the least of the elastic flexural, torsional and flexural– torsional buckling stress determined in accordance with Sections C4.1–C4.4 of the NAS Specification [14] rather
ARTICLE IN PRESS B. Young, E. Ellobody / Thin-Walled Structures 45 (2007) 330–338
The column strengths (PFE Þ obtained from finite element analysis in the parametric study of Series U0.55, U0.7, U0.85, U3.0 and U5.0 are compared with the nominal (unfactored) design strengths (Pn Þ calculated using the current NAS Specification [14] and the proposed design strengths (Pp Þ, as shown in Figs. 7–11. The column strengths PFE , Pn and Pp are plotted against the effective length l e in Figs. 7–11, where the effective length is assumed equal to one-half of the column length. For the unequal lipped angle sections of Series U0.55 and U0.7, the current design strengths (Pn Þ predicted by the NAS Specification are generally unconservative compared to the column strengths (PFE Þ obtained from the finite element analysis, except for angle sections with the effective lengths greater than or equal to 1000 mm, as shown in Figs. 7 and 8. Similarly, the current design strengths Pn are unconservative compared to the column strengths PFE , except for angle sections with the effective lengths greater than or equal to 750 mm for Series U0.85 and U3.0, as shown in Figs. 9 and 10. For Series U5.0, the current design strengths Pn are unconservative compared to the column strengths PFE for all column lengths, as shown in Fig. 11. The proposed design strengths Pp are conservative compared to the column strengths PFE , except for angle sections with the effective lengths of 125 and 312.5 mm for Series U0.55, U0.7 and U0.85, as shown in Figs. 7–9. For Series U3.0, the proposed design strengths Pp are conservative compared to the column strengths PFE for all column lengths, as shown in Fig. 10. For Series U5.0,
PFE
25 Axial load, P (kN)
7. Comparison of column strengths
30
Pn Pp
20 15 10 5 0 0
500
1000
1500
Fig. 8. Comparison of column strengths for Series U0.7.
40 35
PFE
30
Pn Pp
25 20 15 10 5 0 0
500
1000
1500
2000
Effective length, le (mm) Fig. 9. Comparison of column strengths for Series U0.85.
250
20
PFE
PFE 16
200
Pn Pp
Axial load, P (kN)
Axial load, P (kN)
2000
Effective length, le (mm)
Axial load, P (kN)
than solving the cubic equation. The proposed column design strengths (Pp Þ were than computed as Pp ¼ Ae F n , where Ae is the effective area calculated at critical buckling stress (F n Þ. The measured material properties of the flat portion of Series U1.9, tabulated in Table 1, were used to calculate the current and proposed design strengths.
337
12
8
Pn Pp
150
100
50
4
0
0 0
500
1000 1500 Effective length, le (mm)
Fig. 7. Comparison of column strengths for Series U0.55.
2000
0
500
1000
1500
Effective length, le (mm) Fig. 10. Comparison of column strengths for Series U3.0.
2000
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angle sections of Series U5.0, and conservatively predicted the column strengths for angle sections of Series U0.55, U0.7, U0.85 and U3.0, except for some of the angle sections with short effective lengths.
Axial load, P (kN)
400 350
PFE
300
Pn Pp
250 200
References
150 100 50 0 0
500
1000 1500 Effective length, le (mm)
2000
Fig. 11. Comparison of column strengths for Series U5.0.
the proposed design strengths Pp are accurately predicted compared to the column strengths PFE for all column lengths, as shown in Fig. 11. 8. Conclusions The behaviour and design of cold-formed steel unequal angle compression members have been investigated. A finite element model for the analysis of cold-formed steel lipped angle sections with unequal flange widths has been presented. Geometric and material nonlinearities were included in the finite element model. The model has been verified against column test results. It is shown that the strength and behaviour of unequal angle columns predicted using the finite element analysis are generally in good agreement with the experimental results. Therefore, a parametric study has been performed using the finite element model. The column strengths obtained from the finite element analysis were compared with the current design strengths calculated using the North American Specification for cold-formed steel structural members. It is shown that the current design strengths calculated using the North American Specification are generally unconservative for the cold-formed steel unequal angle columns, except for some of the angle sections with long column lengths. Hence, design rules of cold-formed steel unequal angle columns have been proposed. The proposed design strengths accurately predicted the column strengths for
[1] Yu WW. Cold-formed steel design. 3rd ed. New York: Wiley; 2000. [2] Popovic D, Hancock GJ, Rasmussen KJR. Compression tests on cold-formed angles loaded parallel with a leg. Journal of Structural Engineering ASCE 2001;127(6):600–7. [3] Young B. Tests and design of fixed-ended cold-formed steel plain angle columns. Journal of Structural Engineering ASCE 2004; 130(12):1931–40. [4] Zavelani A, Faggiano P. Design of cold-formed latticed transmission towers. Journal of Structural Engineering ASCE 1985;111(11): 2427–45. [5] Gaylord EH, Wilhoite GM. Transmission towers: design of coldformed angles. Journal of Structural Engineering ASCE 1985; 111(8):1810–25. [6] Raghunathan MD, Natarajan PR, Muralidharan K. Prototype test on X-bracing panels using cold-formed lipped angles. Thin-walled Structures 1996;25(1):31–46. [7] Marsh C. Design of single and multiple angle columns and beams. Journal of Structural Engineering ASCE 1997;123(7):847–56. [8] Young B. Experimental investigation of cold-formed steel lipped angle concentrically loaded compression members. Journal of Structural Engineering ASCE 2005;131(9):1390–6. [9] Young B, Ellobody E. Buckling analysis of cold-formed steel lipped angle columns. Journal of Structural Engineering ASCE 2005;131(10):1570–9. [10] Young B. Column tests of cold-formed steel non-symmetric lipped angle sections. In: Proceedings of the 10th East Asia-Pacific conference on structural engineering and construction analytical and computational methods. Bangkok, Thailand: Asian Institute of Technology; 2006. p. 65–70. [11] Young B, Yan J. Finite element analysis and design of fixed-ended plain channel columns. Finite Elements in Analysis and Design 2002; 38(6):549–66. [12] Bakker MCM, Peko¨z T. The finite element method for thin-walled members—basic principals. Thin-walled Structures 2003;41(2–3): 179–89. [13] Ellobody E, Young B. Behavior of cold-formed steel plain angle columns. Journal of Structural Engineering ASCE 2005;131(3): 457–66. [14] NAS. North American specification for the design of cold-formed steel structural members, North American cold-formed steel specification, Washington, D.C: American Iron and Steel Institute; 2001. [15] ABAQUS standard user’s manual, vols. 1, 2 and 3, Version 6.3. Hibbitt, Karlsson and Sorensen, Inc., 2003. [16] Young B, Rasmussen KJR. Behaviour of cold-formed singly symmetric columns. Thin-walled Structures 1999;33(2):83–102.
Thin-Walled Structures 45 (2007) 330–338 www.elsevier.com/locate/tws
Design of cold-formed steel unequal angle compression members Ben Younga,, Ehab Ellobodyb a
Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong Department of Structural Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt
b
Received 20 July 2006; received in revised form 2 February 2007; accepted 22 February 2007 Available online 9 May 2007
Abstract Cold-formed steel unequal angles are non-symmetric sections. The design procedure of non-symmetric sections subjected to axial compression load could be quite difficult. The unequal angle columns may fail by different buckling modes, such as local, flexural and flexural–torsional buckling as well as interaction of these buckling modes. The purpose of this study is to investigate the behaviour and design of cold-formed steel unequal angle columns. A nonlinear finite element analysis was conducted to investigate the strength and behaviour of unequal angle columns. The measured initial local and overall geometric imperfections as well as the material properties of the angle specimens were included in the finite element model. The finite element analysis was performed on fixed-ended columns for different lengths ranged from stub to long columns. It is demonstrated that the finite element model closely predicted the experimental ultimate loads and the behaviour of cold-formed steel unequal angle columns. Hence, the model was used for an extensive parametric study of cross-section geometries. The column strengths obtained from the parametric study were compared with the design strengths calculated using the North American Specification for cold-formed steel structural members. It is shown that the current design rules are generally unconservative for short and intermediate column lengths for the unequal angles. Therefore, design rules of cold-formed steel unequal angle columns are proposed. r 2007 Elsevier Ltd. All rights reserved. Keywords: Buckling; Cold-formed steel; Column; Finite element; Modelling; Steel structures; Structural design; Thin-walled structures; Unequal angle
1. Introduction In recent years, cold-formed steel sections are become popular because of its high strength and light weight. Although, cold-formed steel angle columns are simple structural members, the design rules are quite complicated. So far, limited test data has been found in the literature on cold-formed steel equal and unequal angle columns. Experimental investigations have been found in the literature on the behaviour of cold-formed and hot-rolled steel plain angle columns as summarized in Yu [1] and Popovic et al. [2]. Cold-formed steel plain angle columns have been tested by Popovic et al. [2] and Young [3]. Investigation on the column behaviour of lipped angle sections with equal flange widths, known as equal lipped angles in this paper, have been presented by Zavelani and Faggiano [4], Gaylord and Wilhoite [5], Raghunathan et al. Corresponding author Tel.: +852 2859 2674; fax: +852 2559 5337.
E-mail address: [email protected] (B. Young). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.02.015
[6], Marsh [7], Young [8], and Young and Ellobody [9]. However, very limited column test data were found in the literature on lipped angle sections with unequal flange widths, known as unequal lipped angles in this paper. Young [10] conducted a series of tests on cold-formed steel unequal lipped angle columns. The column strengths and failure modes were presented. Finite element analysis of cold-formed steel unequal lipped angle columns could provide better understanding to the complex buckling behaviour of non-symmetric angles. Numerical investigation on cold-formed steel columns has been presented by Young and Yan [11], Bakker and Peko¨z [12], Ellobody and Young [13] and other researchers. The primary objective of this study is to investigate the strength and behaviour of cold-formed steel unequal lipped angle columns. Thus, design rules of cold-formed steel unequal angle columns are proposed. The investigation was carried out in two stages. The first stage is to measure the initial local geometric imperfections and corner
ARTICLE IN PRESS B. Young, E. Ellobody / Thin-Walled Structures 45 (2007) 330–338
Nomenclature
Pn
Ae Bf 1 Bf 2 Bl b E eFE
Pp
eTest Fe Fn Fy L le l ex , l ey lt P PFE
effective area overall width of long flange (long leg) overall width of short flange (short leg) overall width of lip flat flange width Young’s modulus axial shortening obtained from finite element analysis at failure axial shortening obtained from tests at failure elastic buckling stress critical buckling stress yield stress is taken as 0.2% proof stress (s0:2 Þ actual column length column effective length column effective length for buckling about the principal x-axis and y-axis column effective length for torsional buckling axial compressive load ultimate load obtained from finite element analysis
PTest ro1 ri rx , ry t x, y x o , yo lc s0:2 sex , sey st su
nominal axial strength calculated using North American Specification (unfactored design strength) proposed design strengths (unfactored design strength) test ultimate load (test strength) polar radius of gyration of cross section about the shear centre inside corner radius of specimen radii of gyration of cross section about the principal x-axis and y-axis plate thickness of specimen principal coordinates distances from the shear centre to centroid along the principal x-axis and y-axis elongation (tensile strain) after fracture based on gauge length of 50 mm non-dimensional slenderness static 0.2% tensile proof stress elastic buckling stresses for flexural buckling about the principal x-axis and y-axis elastic buckling stress for torsional buckling static tensile strength
material properties of cold-formed steel unequal lipped angle columns. The second stage is to develop an accurate finite element model to simulate the behaviour of these columns. The results obtained from the finite element analysis were verified against the test results obtained by Young [10]. Parametric study was performed to investigate the effect of cross-section geometries on the strength and behaviour of unequal lipped angle columns. The results obtained from the finite element analysis were compared with design strengths calculated using the North American Specification (NAS) [14] for cold-formed steel structural members. 2. Experimental investigation
Bl Corner A
ri
t
Bf1
2.1. General A series of column tests on cold-formed steel unequal lipped angle sections has been conducted by Young [10]. The angle specimens were tested between fixed-ended conditions at different column lengths ranged from stub to long columns. Fig. 1 shows the definition of symbols for a cold-formed steel unequal lipped angle section. Three series of unequal lipped angles were tested, having a nominal long flange (long leg) width of 80 mm, a nominal short flange (short leg) width of 50 mm, and a nominal lip width of 16 mm. The nominal plate thicknesses were 1.0, 1.5 and 1.9 mm. The three series are labelled U1.0, U1.5 and U1.9 according to their nominal thickness. The test specimens are labelled such that the test series and specimen length could be identified from the label. For
331
Corner C
Corner B Bl
Bf2 Fig. 1. Definition of symbols for cold-formed steel unequal lipped angle section.
example, the label ‘‘U1.0L1500’’ defines the specimen belonged to test Series U1.0, and the fourth letter ‘‘L’’ indicates the length of the specimen. The last three or four
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digits are the nominal column length of the specimen in mm (1500 mm). The measured inside corner radius (ri Þ was 3.0 mm for Series U1.0, and 3.5 mm for Series U1.5 and U1.9 specimens. The initial overall geometric imperfections of the column specimens were measured. The maximum overall geometric imperfections at mid-length were 1=1410, 1=1690 and 1=1300 of the specimen length for Series U1.0, U1.5 and U1.9, respectively. The measured cross-section dimensions and initial overall geometric imperfections of the test specimens are detailed in Young [10]. The lipped angle test specimens were brake-pressed from high strength zinc-coated structural steel sheets having nominal yield stresses of 450 and 550 MPa. Tensile coupon tests were conducted to obtain the material properties in the flat portion of the test specimens. The measured material properties in the flat portion of the test specimens obtained from the coupon tests are summarized in Table 1. The measured Young’s modulus (EÞ, static 0.2% proof stress (s0:2 Þ, static tensile strength (su Þ, and elongation after fracture (Þ based on a gauge length of 50 mm are shown in Table 1. The tensile coupon tests of the flat portion of the specimens are detailed in Young [10]. The initial local geometric imperfections and corner material properties of the test specimens have not been reported, and the values of these measurements are also important for finite element analysis. Therefore, the initial local imperfections and corner material properties of the unequal lipped angle specimens belonged to the same batch of specimens as the column tests are measured in this study and reported in this paper. A servo-controlled hydraulic testing machine was used to apply compressive axial force to the column specimen. The experimental ultimate loads and the failure modes of the unequal lipped angle column tests are detailed in Young [10]. The failure modes of the columns involved local buckling (L), flexural–torsional buckling (FT) and the interaction of these two modes. 2.2. Material and imperfections measurements The material properties in the corners of the unequal lipped angle section of Series U1.9 were determined. The material properties of the corners of Series U1.0 and U1.5 were extrapolated from the material properties of the Series Table 1 Measured and predicted material properties Test series
Portion
E (GPa)
s0:2 (MPa)
su (MPa)
(%)
U1.0
Flat Corners A, C Corner B Flat Corners A, C Corner B Flat Corners A, C Corner B
212 206 206 216 206 206 196 206 206
605 710 695 530 620 610 505 590 580
615 750 745 560 655 650 535 625 620
7.8 2.5 2.9 11.5 2.5 2.9 10.3 2.5 2.9
U1.5
U1.9
Note: 1 ksi ¼ 6.89 MPa.
U1.9 according to the measured 0.2% tensile proof stress (s0:2 Þ in the flat portions of each series. The measured and predicted material properties of the corner and flat coupons are summarized in Table 1. Initial local geometric imperfections have a considerable effect on the buckling behaviour of cold-formed steel columns due to the thinner wall thickness of cold-formed steel sections compared with hot-rolled steel sections. An unequal lipped angle test specimen of 300 mm in length for Series U1.9 was used for the measurement of local imperfections. The measurements were taken at the middle and quarter lengths of the specimen. Readings were taken at regular intervals and maximum magnitude of local plate imperfection was 0.01 mm, which is equal to 0.53% of the angle thickness. The same factor was used to predict the initial local geometric imperfections of the angle specimens for Series U1.0 and U1.5. The measurement procedures for the corner material properties and initial local geometric imperfections of the unequal lipped angle specimens are identical to those for the equal lipped angle specimens as detailed in Young and Ellobody [9]. 3. Finite element model 3.1. General The finite element program ABAQUS [15] was used to simulate the behaviour of the unequal lipped angle columns tested by Young [10]. The measured geometry, initial local and overall geometric imperfections, and material properties at the flat and corner portions of the unequal lipped angle columns were included in the finite element model. Two types of analyses were carried out. The first is Eigenvalue analysis, and the second is loaddisplacement nonlinear analysis. The four-noded doubly curved shell elements with reduced integration S4R were used in the model. The S4R element has six degrees of freedom per node and proved to give accurate solutions from previous research as shown in Ellobody and Young [13] and Young and Ellobody [9]. The finite element mesh size of 10 mm 10 mm (length by width) for the flat portions and finer mesh size was used at the corners. Fig. 2 shows the deformed shape of the finite element mesh for the unequal lipped angle column specimen U1.5L1500. In the finite element model, the ends of the columns were fixed against all degrees of freedom except for the displacement at the loaded end in the direction of the applied load, which is identical to the testing procedures for the unequal lipped angle columns. The nodes other than the two ends were free to translate and rotate in any directions. The load was applied in increments using the modified RIKS method available in the ABAQUS library. The load was applied as static uniform loads at each node of the loaded end, which is identical to the experimental investigation. The nonlinear geometry parameter (NLGEOM) was included to deal with the large displacement analysis. The stress–strain curves for the flat and
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Short leg
Long leg Fig. 2. Deformed shape of unequal lipped angle column U1.5L1500.
corner portions were used in the analysis. The initial local and overall geometric imperfections were incorporated in the model. Superposition of local buckling mode as well as overall buckling mode with measured magnitudes was carried out. These buckling modes were obtained by eigenvalue analysis of the column with very large b=t ratio and very small b=t ratio to ensure local and overall buckling occurs, respectively, where b is the flat flange width and t is the plate thickness of the angle sections. The shape of a local buckling mode as well as overall buckling mode is taken as the lowest buckling mode (eigenmode 1) in the analysis. Short columns having lengths of 250 and 625 mm were modelled for local imperfection only. All buckling modes predicted by ABAQUS Eigenvalue analysis are normalized to 1.0, therefore, the buckling modes are factored by the measured magnitudes of the initial local and overall geometric imperfections. Previous studies by Ellobody and Young [13] and Young and Ellobody [9] on cold-formed steel equal plain and lipped angle columns, have shown that the effect of residual stresses on the column capacity is negligible. In this study, it is assumed that the effect of residual stresses on the column strength of cold-formed steel unequal lipped angle columns is also negligible. Hence, residual stresses were not included in the finite element model.
333
obtained from the test strengths. A maximum difference of 8% was observed between experimental and numerical column strengths for specimen U1.9L3000. The mean values of PTest =PFE ratio are 1.04, 1.03 and 1.06 with the corresponding coefficients of variation (COV) of 0.025, 0.029 and 0.019 for series U1.0, U1.5 and U1.9, respectively, as shown in Table 2. The mean values of eTest =eFE ratio are 1.00, 0.96 and 1.03 with the corresponding COV of 0.069, 0.067 and 0.063 for Series U1.0, U1.5 and U1.9, respectively. Generally, good agreement has been achieved for most of the columns. Three buckling modes were predicted by the finite element analysis for the unequal lipped angle columns. The buckling modes are the local buckling (L), flexural buckling (F) and flexural–torsional buckling (FT). The local buckling was observed in the stub column tests of 250 mm length for Series U1.0, U1.5 and U1.9, and for most of the columns in Series U1.0, which was also predicted by the finite element analysis. For all the three columns series, flexural–torsional buckling was observed experimentally and confirmed numerically for columns lengths ranged from 625 to 3000 mm. Flexural buckling was predicted by the finite element analysis for the long columns, but was not observed in the tests. Fig. 4 shows the applied load against the axial shortening curves of column U1.5L1500 obtained from the test and finite element analysis. It is shown that the column initial stiffness and behaviour reflects good agreement between experimental and finite element results. The experimental ultimate load was 50.1 kN with an axial shortening at the ultimate load of 4.15 mm compared with 48.3 kN and 4.33 mm, respectively, predicted by the finite element analysis. Figs. 5 and 6 show the deformed shapes of the unequal lipped angle column observed experimentally and confirmed by the finite element analysis in different views. It can be seen that the angle cross section is twisted and flexural buckled about the major axis from its original position resulting in a flexural–torsional buckling mode. The finite element model closely predicted the failure mode of the columns. 4. Parametric study
3.2. Verification of finite element model The finite element results are verified against the test results conducted by Young [10]. A total of 21 cold-formed steel unequal lipped angle columns were analyzed using the finite element model. The comparison of the ultimate loads (PTest and PFE Þ, axial shortening (eTest and eFE Þ at the ultimate loads and failure modes obtained experimentally and numerically are shown in Table 2. Fig. 3 plotted the relationship between the ultimate load and the column effective length ðl e ¼ L=2Þ for Series U1.0, U1.5 and U1.9, where L is the actual column length. It can be seen that good agreement has been achieved between both the experimental and numerical results for most of the columns. Generally, the column strengths obtained from the finite element analysis are slightly less than those
A parametric study was performed using the verified finite element model to investigate the effects of crosssection geometries on the strength and behaviour of coldformed steel unequal lipped angle columns. A total of 35 cold-formed steel unequal lipped angle columns was analyzed in the parametric study. Five series of columns U0.55, U0.7, U0.85, U3.0 and U5.0 having plate thicknesses of 0.55, 0.7, 0.85, 3.0 and 5.0 mm, respectively, were studied. All angle sections had the overall flange widths Bf 1 and Bf 2 of 84 and 54 mm, respectively, which is having approximately the same flange widths as the test specimens. The inner corner radius (ri Þ was taken as 3.0 mm for Series U0.55, U0.7 and U0.85, while the inner corner radius for Series U3.0 and U5.0 was taken as 3.5 and 5.0 mm, respectively. The five series had the flat long flange
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Table 2 Comparison between test and finite element results Specimen
Test
Analysis
PTest (kN) U1.0L250 U1.0L625 U1.0L1000 U1.0L1500 U1.0L2000 U1.0L2500 U1.0L3000 Mean COV
42.5 35.5 26.6 25.9 23.4 21.5 16.4 – –
U1.5L250 U1.5L625 U1.5L1000 U1.5L1500 U1.5L2000 U1.5L2500 U1.5L3000 Mean COV
90.0 69.9 51.6 50.1 42.9 36.1 29.1 – –
U1.9L250 U1.9L625 U1.9L1000 U1.9L1500 U1.9L2000 U1.9L2500 U1.9L3000 Mean COV
121.6 86.9 68.7 64.3 55.8 50.5 36.3 – –
eTest (mm)
Failure mode
0.92 1.07 1.48 3.05 3.91 4.05 4.40
L L þ FT L þ FT L þ FT L þ FT L þ FT FT
– –
– –
0.93 1.15 1.77 4.15 4.20 4.25 4.45
L FT FT FT FT FT FT
– –
– –
1.35 1.22 2.80 4.20 4.40 5.30 –
L FT FT FT FT FT FT
– –
– –
PFE (kN) 42.4 34.4 26.7 24.6 22.0 20.5 15.4 – – 90.2 70.2 51.0 48.3 40.3 34.1 27.2 – – 114.7 84.4 64.1 62.6 52.9 47.1 33.6 – –
Test/Analysis eTest eFE
eFE (mm)
Failure mode
PTest PFE
1.00 1.03 1.56 3.31 3.71 3.92 4.02
L L þ FT L þ FT L þ FT L + F þ FT L + F þ FT F þ FT
1.00 1.03 1.00 1.05 1.06 1.05 1.06
0.92 1.04 0.95 0.92 1.05 1.03 1.09
– –
– –
1.04 0.025
1.00 0.069
1.01 1.05 1.97 4.33 4.35 4.47 4.67
L FT FT FT F þ FT F þ FT F þ FT
1.00 1.00 1.01 1.04 1.06 1.06 1.07
0.92 1.10 0.90 0.96 0.97 0.95 0.95
– –
– –
1.03 0.029
0.96 0.067
1.28 1.16 2.52 4.41 4.62 5.00 –
L FT FT FT F þ FT F þ FT F þ FT
1.06 1.03 1.07 1.03 1.05 1.07 1.08
– –
– –
1.06 0.019
1.05 1.05 1.11 0.95 0.95 1.06 – 1.03 0.063
Note: 1 kip ¼ 4.45 kN; 1 in ¼ 25.4 mm.
150 Test U1.9 FE U1.9
Axial load, P (kN)
120
Test U1.5 FE U1.5
90
Test U1.0 FE U1.0
60
30
0 0
500
1000 1500 Effective length, le (mm)
2000
Fig. 3. Ultimate loads of unequal lipped angle columns for Series U1.0, U1.5 and U1.9.
width-to-thickness ratio (b=tÞ of 139.8, 109.4, 89.8, 23.7 and 12.8 for Series U0.55, U0.7, U0.85, U3.0 and U5.0, respectively. The length of the lips (Bl Þ was kept at 17.0 mm
for all series. Table 3 summarizes the dimensions of the five series analyzed in the parametric study. Each series of columns consists of seven column lengths of 250, 625, 1000, 1500, 2000, 2500 and 3000 mm. The maximum initial local geometric imperfection magnitude was taken as the measured value of 0.53% of the plate thickness. The maximum initial overall geometric imperfection magnitude was taken as the average of the measured maximum overall imperfections of the tested series, which is equal to L=1450, where L is the column length. The stress–strain curves of the flat and corner portions of Series U1.9 were used in the parametric study. A summary of the parametric study results is presented in Table 4. The ultimate loads (PFE Þ, axial shortening at ultimate load (eFE Þ and failure modes are given in Table 4. 5. Current design rules The NAS [14] for the design of cold-formed steel structural members provides design rules for non-symmetric angle columns. In this study, the lipped angle sections with unequal flange widths are non-symmetric
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60
Axial load (kN)
50 40 30 20 Test 10
FE
0 0
2
4
6
8
10
Shortening (mm) Fig. 4. Load-axial shortening curves for column U1.5L1500.
Fig. 6. Comparison of experimental and finite element analysis deformed shapes of column U1.5L1500 (view into short leg). (a) Experimental; (b) FE analysis.
Fig. 5. Comparison of experimental and finite element analysis deformed shapes of column U1.5L1500 (view into long leg). (a) Experimental; (b) FE analysis.
sections. The columns were compressed between fixed ends. The fixed-ended bearings were restrained against the major and minor axis rotations as well as twist rotations and warping. The effective lengths (l e Þ for principal x-axis (l ex Þ and principal y-axis (l ey Þ flexural buckling as well as torsional buckling (l t Þ are assumed equal to one-half of the
column length (LÞ for the fixed-ended columns. The shift in the line of action of the internal force is balanced by a shift in the line of action of the external force for fixed-ended column [16]. Hence, the unequal lipped angles were designed as concentrically loaded compression members in accordance with Section C4 of the NAS Specification [14]. It should be noted that angle sections shall be designed for an additional bending moment, for which the additional moment is calculated as the axial load multiplied by an eccentricity of 1=1000 of the column length as recommended in the NAS Specification. The required additional bending moment was based on the investigation of plain angle sections with equal flange widths. Hence, the additional moment may not be needed for the unequal
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Specification [14] are as follows:
Table 3 Dimensions of unequal lipped angle sections in parametric study Series
U0.55 U0.7 U0.85 U3.0 U5.0
Flanges
Pn ¼ Ae F n ,
Lips
Thickness
Radius
Bf 1 (mm)
Bf 2 (mm)
Bl (mm)
t (mm)
ri (mm)
84.0 84.0 84.0 84.0 84.0
54.0 54.0 54.0 54.0 54.0
17.0 17.0 17.0 17.0 17.0
0.55 0.7 0.85 3.0 5.0
3.0 3.0 3.0 3.5 5.0
Note: 1 in ¼ 25.4 mm.
Table 4 Summary of parametric study results Specimen
where Ae is the effective area calculated at the critical buckling stress (F n Þ. The effective area that accounts for local buckling is calculated in accordance with Section B of the NAS Specification. The critical buckling stress (F n Þ that accounts for overall buckling is determined as 2
for lc p1:5 ) F n ¼ ð0:658lc ÞF y , " # 0:877 for lc 41:5 ) F n ¼ F y, l2c where rffiffiffiffiffiffi Fy lc ¼ , Fe
FE analysis PFE (kN)
(1)
eFE (mm)
Failure mode
U0.55L250 U0.55L625 U0.55L1000 U0.55L1500 U0.55L2000 U0.55L2500 U0.55L3000
11.5 9.9 6.8 4.4 3.4 3.0 2.7
0.39 0.56 0.75 1.10 1.70 2.01 2.59
L L L þ FT L þ FT L þ FT L + F þ FT F þ FT
U0.7L250 U0.7L625 U0.7L1000 U0.7L1500 U0.7L2000 U0.7L2500 U0.7L3000
18.8 15.9 9.9 6.5 5.6 5.3 4.9
0.33 0.60 0.88 1.12 1.79 2.16 2.63
L L L þ FT L þ FT L þ FT L + F þ FT F þ FT
U0.85L250 U0.85L625 U0.85L1000 U0.85L1500 U0.85L2000 U0.85L2500 U0.85L3000
26.6 22.7 15.3 12.0 10.7 9.8 7.9
0.58 0.65 1.27 1.80 1.97 2.39 2.89
L L L þ FT L þ FT L þ FT L þ F þ FT F þ FT
U3.0L250 U3.0L625 U3.0L1000 U3.0L1500 U3.0L2000 U3.0L2500 U3.0L3000
182.2 121.4 90.7 76.7 63.5 54.7 41.0
1.64 1.03 1.53 1.87 2.06 2.50 2.98
L FT FT FT F þ FT F þ FT F þ FT
U5.0L250 U5.0L625 U5.0L1000 U5.0L1500 U5.0L2000 U5.0L2500 U5.0L3000
326.1 195.8 164.0 136.2 109.1 84.8 66.3
2.58 1.74 1.79 1.99 2.22 2.63 2.99
L FT FT FT F þ FT F þ FT F þ FT
Note: 1 kip ¼ 4.45 kN; 1 in ¼ 25.4 mm.
lipped angle columns, and the required additional moment was not included in calculating the design strengths. The design rules of nominal axial strength (Pn Þ for concentrically loaded compression members in the NAS
(2) (3)
(4)
where F y is the yield stress which is taken as the 0.2% proof stress (s0:2 Þ, and F e is the elastic buckling stress that was obtained by taking the smallest value of the following cubic equation: F 3e ðr2o1 x2o y2o Þ F 2e ½r2o1 ðsex þ sey þ st Þ ðsey x2o þ sex y2o Þ þ F e r2o1 ðsex sey þ sey st þ sex st Þ ðsex sey st r2o1 Þ ¼ 0, ð5Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ro1 ¼ r2x þ r2y þ x2o þ y2o is the polar radius of gyration of cross section about the shear centre, xo and yo are the distances from the shear centre to centroid along the principal x-axis and y-axis, respectively, rx and ry are the radii of gyration of cross-section about the principal x-axis and y-axis, respectively, sex and sey are the elastic buckling stresses for flexural buckling about the principal x-axis and y-axis, respectively, and st is the elastic buckling stress for torsional buckling. Alternatively, F e may be determined by rational elastic buckling analysis. 6. Proposed design rules The column design rules proposed by Young [3] and Young and Ellobody [9] for singly-symmetric cold-formed steel plain and lipped angle sections respectively, are used to predict the design strengths of the non-symmetric lipped angle sections. The proposed equations required simple modification to the critical buckling stress (F n Þ in the NAS Specification [14]. The proposed design equations are as follows: 2
for lc p1:4 ) F n ¼ ð0:5lc ÞF y , " # 0:5 for lc 41:4 ) F n ¼ F y, l2c
(6) (7)
where the non-dimensional slenderness ðlc Þ is identical to that in Eq. (4), except that the elastic buckling stress ðF e Þ is the least of the elastic flexural, torsional and flexural– torsional buckling stress determined in accordance with Sections C4.1–C4.4 of the NAS Specification [14] rather
ARTICLE IN PRESS B. Young, E. Ellobody / Thin-Walled Structures 45 (2007) 330–338
The column strengths (PFE Þ obtained from finite element analysis in the parametric study of Series U0.55, U0.7, U0.85, U3.0 and U5.0 are compared with the nominal (unfactored) design strengths (Pn Þ calculated using the current NAS Specification [14] and the proposed design strengths (Pp Þ, as shown in Figs. 7–11. The column strengths PFE , Pn and Pp are plotted against the effective length l e in Figs. 7–11, where the effective length is assumed equal to one-half of the column length. For the unequal lipped angle sections of Series U0.55 and U0.7, the current design strengths (Pn Þ predicted by the NAS Specification are generally unconservative compared to the column strengths (PFE Þ obtained from the finite element analysis, except for angle sections with the effective lengths greater than or equal to 1000 mm, as shown in Figs. 7 and 8. Similarly, the current design strengths Pn are unconservative compared to the column strengths PFE , except for angle sections with the effective lengths greater than or equal to 750 mm for Series U0.85 and U3.0, as shown in Figs. 9 and 10. For Series U5.0, the current design strengths Pn are unconservative compared to the column strengths PFE for all column lengths, as shown in Fig. 11. The proposed design strengths Pp are conservative compared to the column strengths PFE , except for angle sections with the effective lengths of 125 and 312.5 mm for Series U0.55, U0.7 and U0.85, as shown in Figs. 7–9. For Series U3.0, the proposed design strengths Pp are conservative compared to the column strengths PFE for all column lengths, as shown in Fig. 10. For Series U5.0,
PFE
25 Axial load, P (kN)
7. Comparison of column strengths
30
Pn Pp
20 15 10 5 0 0
500
1000
1500
Fig. 8. Comparison of column strengths for Series U0.7.
40 35
PFE
30
Pn Pp
25 20 15 10 5 0 0
500
1000
1500
2000
Effective length, le (mm) Fig. 9. Comparison of column strengths for Series U0.85.
250
20
PFE
PFE 16
200
Pn Pp
Axial load, P (kN)
Axial load, P (kN)
2000
Effective length, le (mm)
Axial load, P (kN)
than solving the cubic equation. The proposed column design strengths (Pp Þ were than computed as Pp ¼ Ae F n , where Ae is the effective area calculated at critical buckling stress (F n Þ. The measured material properties of the flat portion of Series U1.9, tabulated in Table 1, were used to calculate the current and proposed design strengths.
337
12
8
Pn Pp
150
100
50
4
0
0 0
500
1000 1500 Effective length, le (mm)
Fig. 7. Comparison of column strengths for Series U0.55.
2000
0
500
1000
1500
Effective length, le (mm) Fig. 10. Comparison of column strengths for Series U3.0.
2000
ARTICLE IN PRESS B. Young, E. Ellobody / Thin-Walled Structures 45 (2007) 330–338
338
angle sections of Series U5.0, and conservatively predicted the column strengths for angle sections of Series U0.55, U0.7, U0.85 and U3.0, except for some of the angle sections with short effective lengths.
Axial load, P (kN)
400 350
PFE
300
Pn Pp
250 200
References
150 100 50 0 0
500
1000 1500 Effective length, le (mm)
2000
Fig. 11. Comparison of column strengths for Series U5.0.
the proposed design strengths Pp are accurately predicted compared to the column strengths PFE for all column lengths, as shown in Fig. 11. 8. Conclusions The behaviour and design of cold-formed steel unequal angle compression members have been investigated. A finite element model for the analysis of cold-formed steel lipped angle sections with unequal flange widths has been presented. Geometric and material nonlinearities were included in the finite element model. The model has been verified against column test results. It is shown that the strength and behaviour of unequal angle columns predicted using the finite element analysis are generally in good agreement with the experimental results. Therefore, a parametric study has been performed using the finite element model. The column strengths obtained from the finite element analysis were compared with the current design strengths calculated using the North American Specification for cold-formed steel structural members. It is shown that the current design strengths calculated using the North American Specification are generally unconservative for the cold-formed steel unequal angle columns, except for some of the angle sections with long column lengths. Hence, design rules of cold-formed steel unequal angle columns have been proposed. The proposed design strengths accurately predicted the column strengths for
[1] Yu WW. Cold-formed steel design. 3rd ed. New York: Wiley; 2000. [2] Popovic D, Hancock GJ, Rasmussen KJR. Compression tests on cold-formed angles loaded parallel with a leg. Journal of Structural Engineering ASCE 2001;127(6):600–7. [3] Young B. Tests and design of fixed-ended cold-formed steel plain angle columns. Journal of Structural Engineering ASCE 2004; 130(12):1931–40. [4] Zavelani A, Faggiano P. Design of cold-formed latticed transmission towers. Journal of Structural Engineering ASCE 1985;111(11): 2427–45. [5] Gaylord EH, Wilhoite GM. Transmission towers: design of coldformed angles. Journal of Structural Engineering ASCE 1985; 111(8):1810–25. [6] Raghunathan MD, Natarajan PR, Muralidharan K. Prototype test on X-bracing panels using cold-formed lipped angles. Thin-walled Structures 1996;25(1):31–46. [7] Marsh C. Design of single and multiple angle columns and beams. Journal of Structural Engineering ASCE 1997;123(7):847–56. [8] Young B. Experimental investigation of cold-formed steel lipped angle concentrically loaded compression members. Journal of Structural Engineering ASCE 2005;131(9):1390–6. [9] Young B, Ellobody E. Buckling analysis of cold-formed steel lipped angle columns. Journal of Structural Engineering ASCE 2005;131(10):1570–9. [10] Young B. Column tests of cold-formed steel non-symmetric lipped angle sections. In: Proceedings of the 10th East Asia-Pacific conference on structural engineering and construction analytical and computational methods. Bangkok, Thailand: Asian Institute of Technology; 2006. p. 65–70. [11] Young B, Yan J. Finite element analysis and design of fixed-ended plain channel columns. Finite Elements in Analysis and Design 2002; 38(6):549–66. [12] Bakker MCM, Peko¨z T. The finite element method for thin-walled members—basic principals. Thin-walled Structures 2003;41(2–3): 179–89. [13] Ellobody E, Young B. Behavior of cold-formed steel plain angle columns. Journal of Structural Engineering ASCE 2005;131(3): 457–66. [14] NAS. North American specification for the design of cold-formed steel structural members, North American cold-formed steel specification, Washington, D.C: American Iron and Steel Institute; 2001. [15] ABAQUS standard user’s manual, vols. 1, 2 and 3, Version 6.3. Hibbitt, Karlsson and Sorensen, Inc., 2003. [16] Young B, Rasmussen KJR. Behaviour of cold-formed singly symmetric columns. Thin-walled Structures 1999;33(2):83–102.