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Probabilistic estimation of the buckling strength of a CFS lipped-channel section with Type 1 imperfection
Thin-Walled Structures 119 (2017) 447–456
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Probabilistic estimation of the buckling strength of a CFS lipped-channel section with Type 1 imperfection
MARK
⁎
Hashmi S.S. Ahmeda, Siddhartha Ghosha, , Mohit Mangalb a b
Structural Safety, Risk & Reliability Lab, Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India Department of Civil and Environmental Engineering, Hong Kong University of Science & Technology, Kowloon, Hong Kong
A R T I C L E I N F O
A B S T R A C T
Keywords: Cold-formed steel Uncertainty Imperfection Local buckling Post-buckling Lipped channel
Local geometric imperfections in a cold-formed steel (CFS) member can significantly alter the force-carrying capacity of the member. These are the dents and undulations which occur during cold-rolling, handling, transportation and erection of CFS members. The buckling strength of a lipped channel section with Type 1 local imperfection is obtained and characterised statistically using finite element analyses and Monte Carlo simulations. The reduction in strength due to these imperfections are found to be significant. The quantification of reduction in strength due to these imperfections is found for different values of non-dimensional slenderness ratio. Based on the statistical analysis, design equations and strength curves are recommended for the buckling strength of geometrically imperfect members. Legitimacy of using a generalised statistics of imperfection, in the case of unavailability of specific data for a particular section, is also verified.
1. Geometric imperfections in cold-formed steel sections Cold-formed steel (CFS) sections are fabricated from thin steel sheets using either press-braking or roll-forming process, by passing the sheet through a number of dies. The characteristic that differentiates CFS sections from hot-rolled ‘structural’ steel sections is that the shape of the cross-section, instead of the thickness of the section, is used for carrying loads [1]. The force carrying capacity of CFS sections depends largely on the shape achieved through a cold-forming process. Deviations from the target shape may affect a CFS section's capacity significantly. However, due to a multiplicity of reasons it is almost impossible to maintain the perfection in the cross-sectional dimensions of a CFS section that is finally used in construction. One major reason is that cold-rolling mills do not adhere to a very strict quality in the coldrolling process, which results in geometrically imperfect sections coming out of these mills. The other significant reason is the loads during handling, transportation, and erection (those loads, for which the section is not typically designed), which cause visible deformations in these thin members. Geometric imperfections (GIs) in CFS sections include global and local deviations. Whereas global behaviours, such as bow, camber and twist, are categorised under global GI, local deviations are characterised by dents and undulations in the member elements (flange, web etc.). As mentioned earlier, these deviations can significantly alter the force-carrying capacity of a section. CFS sections are characterised by the presence of different instability
⁎
Corresponding author. E-mail address: [email protected] (S. Ghosh).
http://dx.doi.org/10.1016/j.tws.2017.07.001 Received 19 October 2015; Received in revised form 7 June 2017; Accepted 1 July 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
modes, such as local, distortional and global buckling, prior to the ultimate ‘failure’ of the member. The design of these sections is generally governed by the ‘post-(local/distortional) buckling’ behaviour. The strength limit states are defined by ‘overall’ buckling, which includes flexural, torsional and torsional-flexural modes. Owing to typical proportions of CFS member elements, these buckling modes (in real structures) are usually elastic [2]. However, some stocky members fail by inelastic (overall) buckling as well. Design standards, such as AISI S100 [3] or AS/NZS 4600 [4], include the effect of inelasticity in the post-(local) buckling behaviour of CFS members. In this article, we focus on the effects of Type 1 local geometric imperfections on the buckling strength of a lipped channel (‘C’) section, in a statistical sense. A lipped C section is considered for this study because these are the most commonly used sections in structures made of coldformed steel, other than ‘Z’ sections. 2. Previous studies on geometric imperfections in CFS sections A local (or, cross-sectional) GI is understood as local unevenness or undulation in the elements of cross-section distributed over the length of the member, which is generally referred as local geometric imperfection or cross-sectional geometric imperfection [5]. Schafer and Peköz [6] categorised the local GIs into two groups: 1. Type 1: Maximum local imperfection in a stiffened element, such as a web
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imperfection magnitude in nonlinear FE analyses of CFS was presented by Bonada et al. [14]. Loughlan et al. [15] examined the failure mechanism of fixed-ended CFS lipped channels with due consideration to the influence of geometric imperfections. Results from the finite element simulations compared favourably with the tests performed, when the local geometric imperfection was properly modelled. They also reported the substantial influence of geometric imperfection on the load-deformation and buckling behaviour from the onset of loading. Dinis et al. [16] analysed the coupled instabilities in CFS lipped channel columns through finite element analyses. They showed the influence of GI on the elastic-plastic post-buckling behaviour of fixed-ended lipped channel columns. They also found the strength estimations based on the direct strength method (DSM) to be inaccurate for these cases. So far, research on the effect of GI on the behaviour of CFS sections primarily looked at aspects such as numerical modelling of GI, selection of GI, impact of model interaction, etc. However, very little work has been reported on the probabilistic performance assessment of the CFS member with geometric imperfections. Considering this fact the main focus of this work is set on the probabilistic estimation of buckling strength of a CFS member with local geometric imperfection.
Fig. 1. Type 1 geometric imperfection of the lipped channel cross-section.
2. Type 2: Maximum deviation from straightness for a lip-stiffened or an unstiffened flange
3. Objective and scope of the work
Fig. 1 illustrates the Type 1 deformation for the case of a lipped channel section. The local deviations are characterised by ‘dents’ and ‘irregular undulations’ in the plate. However, they found that little or no information was available on the specific location of these dents, which made it difficult to find which eigenmode(s) a specific imperfection would trigger. In their detailed statistical treatment of geometric imperfections based on the collected data, Schafer and Peköz [6] considered the amplitude of the imperfection as a random variable, and recommended a direct probabilistic simulation of both the imperfection magnitude and its distribution along the length of the member as the most robust way to assess the effect of geometric imperfections. However, they also pointed to the difficulty with the practical implementation of such an approach, and suggested more affordable but less accurate approaches. Traditionally, the effective width method [3] has been used to evaluate the strength of the CFS member. This method includes the effect of local buckling [2], but not the effect of interaction of different elements of a cross-section [7,8]. In addition, this method is very tedious. The semi-empirical approach adopted from Winter [9], which is a modification of the theoretical equation originally proposed by Karman et al. [10], does not explicitly account for geometric imperfections, either. However, since the modification of Karman et al. [10]'s equation was based on experimental results of real sections, one can expect that the equation(s) in AISI S100 can somewhat represent the imperfection, although not rigorously but in a very limited manner. However, as highlighted before, local geometric imperfections are very random in nature [6,11]. Therefore, a rigorous treatment of this randomness is necessary for a proper assessment of a CFS column's load carrying capacity. In this line, many researchers measured and characterised geometric imperfections in CFS sections [6] and tried to quantify their effect(s) on the capacity of a CFS member. Young and Rasmussen [12] performed compressive tests on two cross-sections of lipped channels, with two different boundary conditions. Prior to the test, detailed measurement of geometric imperfections was performed and reported. Dubina and Ungureanu [13] studied the effect of geometric imperfections on the buckling strength of CFS channels through numerical simulations. They pointed out the necessity of the codification of both size and shape of imperfections for numerical analysis. Also on the basis of their study, they showed that the different shapes of local imperfection have different effects on the buckling strength of the member. The importance of the selection of an initial geometrical
The primary objectives of this work are (i) to find the effect of Type 1 local geometric imperfections on the buckling strength of a CFS section, (ii) to statistically quantify this strength, and (iii) to compare the findings with the recommendations given in design codes (which do not explicitly account for such imperfection effects). Due to the presence of uncertainty in local imperfection magnitudes, a probabilistic framework is preferred to analyse the effect of these imperfections. Additionally, a comparative study between the effects of two different types of imperfection statistics – section-independent ‘generalised’ statistics and section-specific ‘particular’ statistics – is performed. This is to assess the legitimacy of using the generalised statistics of imperfection, in the case that a particular statistic is unavailable. As an initiatory work in this area from a statistical perspective, our study is limited to a single lipped channel section with a Type 1 deformation (Fig. 1). In this work, the yielding of the material is included in the member behaviour, however, torsional and flexural-torsional effects are avoided using suitable constraints.
4. Critical load analysis of the channel section A finite element (FE) approach is adopted for the buckling analysis of the member with some geometric imperfection, in order to properly capture the failure behaviour, including the interaction with local and distortional buckling modes. Second-order inelastic analyses (that is, including both material and geometric nonlinearity) are used to arrive at the ‘critical load’ for the member [17]. Since this load is obtained from the load-deformation curve of the member, the analysis lets us observe the sequence of local/distortional buckling and the overall member buckling. The lipped channel section selected for this study is 362S162-68 with the following dimensions: depth of the web = 92 mm (3.625 in), width of the flange = 41.3 mm (1.625 in), and thickness = 1.7 mm (0.068 in). Six different lengths of the section are considered in order to study the behaviour for different nondimensional slenderness ratios (λc) in the range of 0.5–2.5, where λc is defined as [3]
λc =
Fy Fe
(1)
where Fy is the material yield stress, and Fe is the minimum critical elastic column buckling stress (considering only the flexural mode). 448
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4.1. Incorporation of the imperfection into the FE model Generally, two alternative approaches are adopted in order to incorporate the initial imperfection into the finite element (FE) model for computational simulation of a CFS member's behaviour. One approach would be to measure the imperfections physically and model the very intricate imperfect geometry. This would be accurate but highly impractical as measurements for each and individual section would be prohibitively expensive. Alternatively, buckling shapes can be used to describe the imperfections into the FE model, which is the more common practice followed by researchers in general. In this modal approach of imperfection modelling, a single buckling mode or superposition of various buckling modes, scaled suitably, emulates the imperfection into a perfect member. This technique is quite common and has been adopted by various researchers [5,14,18,19]. The buckling shape is used to describe the distribution of the imperfection over the length of the member with measured statistics of maximum imperfection. In the current paper, the Type 1 local GI is adopted as a perturbation on the lowest eigenmode for the chosen channel cross-section. For this section, this lowest eigenmode corresponds to a local buckling similar to the deviation depicted in Fig. 1, and the deviation at the centre of the web (at the mid-span) is scaled to the maximum imperfection magnitude (d1). Obviously, this modelling also allows overall flexural buckling effects, with which the local buckling may interact before failure, depending on a member's λc. 4.2. Finite element simulation The CFS channel section is modelled and analysed in the generalpurpose finite element package Abaqus [20]. The four-noded linear shell element, S4R, with reduced integration, is used to model the member. Shell elements are the usual choice for such “thin” members with curved geometries which are expected to undergo large overall deformations. Fixed-fixed boundary conditions are imposed at the ends of the member. This is only an ideal boundary condition and does not necessarily reflect realistic situations. However, the focus is on studying the effect of Type 1 GI on a lipped channel of varying slenderness, irrespective of the actual boundary conditions. The effect of the boundary condition is incorporated through the effective slenderness ratio, which is normally practised for such structural systems. The boundary condition as well as the axial force are applied at the centroid of the end cross-sections. The constraint and force effects are transferred to the CFS member by connecting (all degrees of freedom of) the centroid with the nodes on the cross-section through “couplings” (Fig. 2). ‘Equation constraints’ are used to avoid the torsional behaviour of a member. The worst aspect ratio of S4R elements is kept within 3.0 to achieve a better numerical accuracy [5]. For the inelastic modelling of steel, an elastic-perfectly-plastic (EPP) material model is adopted, with yield stress (Fy) = 388 MPa, elastic modulus (E) = 203 GPa, and Poisson's ratio (ν) = 0.3 [5]. The effect of residual stresses and the rounded corners are not included here, as practised by many researchers, including Young and Yan [21]. The FE simulation is a two step process: eigenvalue analysis (to obtain the fundamental mode shape) followed by a second-order “buckling” analysis with an initial perturbation. The subspace iteration eigensolver is opted in Abaqus to find the mode shapes and eigenvalues. Fig. 3 shows sample buckling modes (both local and global). The Type 1 geometric imperfection is adopted on the lowest eigenmode (which corresponds to a local buckling mode) for that section, which gives the maximum imperfection magnitude at the centre of the web, similar to Fig. 1. The selected mode shape (scaled to the amplitude of GI) is then perturbed as imperfection into the perfect member (Fig. 4) to perform a nonlinear buckling analysis. In order to understand the post-(local) buckling behaviour of the
Fig. 2. Coupling of the centroid to the cross-section.
CFS section, an incremental static analysis incorporating geometric and material nonlinearities is performed until the member reaches its critical load. The Static,Riks iterative scheme in Abaqus is adopted here to conduct this “buckling” analysis. This technique uses the arc-length method to measure the progress of a solution in the load-displacement space, leading to converged solutions irrespective of the type of response [22]. As mentioned earlier, the initial geometry of the section is perturbed with the local buckling mode scaled to the amplitude of the GI. An axial compressive load, applied at the centroid of one end section, is incrementally increased and the lateral displacement of the member is monitored at the mid-depth of the web at the mid-span of member. Fig. 5 shows sample load-deformation curves obtained from such analyses. These are results for a member with λ c = 1.00 . Two load-deformation plots are presented for: (a) a CFS section with a specific GI amplitude of d1/ t = 7.31, and (b) the same CFS section without any geometric imperfection. For the member without imperfection, the lateral deformation is small and follows the compression load linearly for a large amount of compression. Afterwards, it displays a gradually reducing stiffness and reaches a maximum load carrying capacity of 112.7 kN. Ahead of this point, the stiffness becomes negative indicating an unstable/buckled state of the member. On the other hand, the imperfect member shows a noticeable deformation from the very beginning with a nonlinear load-deformation curve. This indicates how local GI influences the post-(local) buckling behaviour of the CFS section. The maximum compressive load carried by the imperfect member is only 60.08 kN. The maximum load carried by any member – with or without imperfection – is termed as its critical load (Pn), which gives the buckling capacity of the member. Fig. 6 shows the deformed shape of
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Fig. 3. Sample buckling modes (deformation scale = 3), (a) Local (central part of the member), (b) Global (full length of the member).
(a) Local (central part of the member)
(b) Global (full length of the member) 5. Statistical analysis considering the variation in imperfection
the imperfect member beyond the critical load point. This deformed shape also shows the co-existence of local and overall (flexural) buckling modes, confirming the interaction of different modes observed by previous researchers [15]. However, the most significant aspect of Fig. 5 is the reduction in the critical load, Pn, by about 47% on account of local (Type 1) GI. The finite element analysis (FEA) approach adopted here is validated with the experimental results reported by Young and Rasmussen [12], where they performed compression tests on fixed ended CFS lipped channels. Table 1 shows measured dimensions of the specimens tested. In our model, material characteristics (including nonlinearity) are incorporated based on the measured properties reported by Young and Rasmussen [12]. However, the magnitude of the local geometric imperfection is considered as 25% of the plate thickness of the member, as reported by Young and Rasmussen [12]. The ultimate load obtained using the FEA is compared with the experimental result and is reported in Table 2. It is clear from this comparison that our FE modelling approach is able to estimate the buckling strength with suitable accuracy. Moreover, the FE analysis approach adopted here is also compared to the numerical analysis performed by Young and Yan [21] for the same lipped channel section. Zeinoddini [23] considered a fixed-fixed member of length 8 ft. (2.44 m), with a normalised Type 1 imperfection magnitude, d1/ t = 0.31. The same section with the same imperfection is modelled and analysed in this paper using the method described earlier in this section. The critical load for buckling (Pn) is found to be within 2.77% error from the value obtained by Zeinoddini [23]. On the basis of these comparisons, the analysis methodology adopted for imperfect CFS sections in this work is deemed satisfactory.
5.1. Statistical modelling of the imperfection Imperfection statistics have been collected for CFS sections over last few decades. In this work, two different sets of statistical data – generalised (‘Stat-1’) and particular (‘Stat-2’) – are used for modelling the imperfection amplitude. Schafer and Peköz [6] provided the statistics of Type 1 local imperfection amplitude of CFS sections, based on data collected through different research studies. This statistics is for CFS sections in general and termed as Stat-1 in this work. Later, Zeinoddini and Schafer [5] presented the Type 1 imperfection statistics specifically for the lipped channel Section 362S162-68. These statistics, with quantile values and parameters, are reproduced in Table 3. It should be noted, following the general practice, these statistics are provided for the normalised imperfection d1/ t . In the present work, these statistics are modelled with common probability distributions (keeping the mean and standard deviation the same) with the Kolmogorov-Smirnov goodness-of-fit test (K-S test). The suitability of modelling the imperfection statistics with normal and lognormal distributions are checked. Both distributions are found suitable to model these statistics with an acceptable significance level. However, the lognormal distribution is found to be a better fit for Stat-1, while the normal distribution is found better for Stat-2 (with a significance level less than 0.05). The comparison between the two probability models with respect to the original data is provided through their cumulative distribution functions (CDF) in Fig. 7. Considering that both models provide a good fit, the following probability models are chosen for the uncertainty propagation:
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(a) Full member
(a) Full member
(b) Inset Fig. 6. Deformations after buckling (deformation scale = 2), (a) Full member, (b) Inset.
(b) Inset Table 1 Measured dimensions of specimen.
Fig. 4. FE model of the CFS member perturbed with initial imperfection, (a) Full member, (b), Inset.
Specimen ID
Web (mm)
Flanges (mm)
Lips (mm)
Thickness (mm)
Length (mm)
L36F0280 L48F0300
97.3 97.1
37.0 49.0
12.5 12.2
1.48 1.47
280 300
Table 2 Comparison of FE analysis and experimental results. Specimen ID
PFEM (kN)
PExp (kN)
PFEM /PExp
L36F0280 L48F0300
104.0 104.1
100.2 102.3
1.038 1.018
Table 3 Statistics of Type 1 (normalised) imperfection (d1/t).
Fig. 5. “Buckling” curves at the mid-span (for λ c = 1.00).
d1/ t ∼ LN (0.50, 0.66) For Stat–1
(2)
d1/ t ∼ LN (0.48, 0.24) For Stat–2
(3)
CDF
Stat-1
Stat-2
0.25 0.50 0.75 0.95 0.99
0.14 0.34 0.66 1.35 3.87
0.36 0.47 0.55 – –
Mean Std. dev.
0.50 0.66
0.48 0.24
random samples of d1/ t are generated based on the lognormal distribution model of Eq. (2)-(3). These imperfection samples are used to create N imperfect CFS members. Each of these specimens is then analysed for its critical load (Pn), using its finite element model, as described in Section 4.2. The N values of Pn obtained provide the
5.2. Monte Carlo simulation A “crude” Monte Carlo simulation (MCS)-based approach is adopted to obtain the effect of local geometric imperfections on the buckling strength. For each statistical data (Stat-1 and Stat-2), adequate (say N) 451
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Fig. 7. CDF of the observed data and trial distributions for normalised imperfection, (a) Stat-1, (b) Stat-2.
Fig. 8. Convergence of the mean and standard deviation of Pn for Stat-1 (λ c = 1.27) .
Fig. 9. Convergence of the mean and standard deviation of Pn for Stat-2 (λ c = 1.27) .
conditions, and coupling constraints) 2. Perform an eigenvalue buckling analysis and find the buckling modes 3. Select the local buckling mode corresponding to Type 1 imperfection 4. Generate a random sample of d1/ t based on the selected probabilistic model (Eq. (2)-(3)) 5. Perturb the FE model with the Type 1 mode scaled to this d1/ t 6. Perform an incremental iterative (Static,Riks) analysis to obtain Pn
statistics of buckling strength of an imperfect channel section with Type 1 local imperfection. The required number of MCS samples, N, are decided based on the convergence (to an acceptable tolerance) of the first two moments of Pn: mean (μPn) and standard deviation (σPn). A step-wise description of the overall analysis method for a specific member, with a certain length or λc, is as follows: 1. Create a FE model of the CFS member (with proper boundary 452
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Fig. 10. Frequency histograms and empirical CDFs of Pn for Stat-1.
reported for each λc in Table 4. The K-S test is used again to check if any common probability distribution can be used to model the variation of Pn. For this purpose, normal, lognormal, gamma and Gumbel distributions – with the sampled mean and standard deviation values for each data-set – are tried. For the seven λc values, none of these distributions provides a fit with an acceptable significance level (αc). Table 4 also shows the 5 percentile Pn values obtained for each data-set. The 5 percentile value is considered purely from a design specification perspective. Most design guidelines use a 5 percentile value to define characteristic/nominal material strength/member capacity. The same concept can also be used here to define a design value (Pd) for the buckling strength including the effect of local geometric imperfections. In addition, Table 5 shows the reduction in Pn of an imperfect member from that of the same member without any imperfection. This reduction is evaluated at the mean Pn value, and also at its 5 percentile or design value (Pd). The percentage reduction is tabulated for the seven selected λc values, which shows that the reduction due to imperfection can be as high as 35.91% for the design strength of the member. Table 5 also shows that this reduction is higher in the (low-to-) intermediate slenderness range, and it decreases both towards high and low slenderness zones. This trend remains true for both the generalised and the particular statistics of imperfection used in this study. For high
from the load-deformation plot 7. Store Pn vs. d1 data 8. Repeat steps 4–7 for all samples, until μ Pn and σPn converge 9. Model the statistics of Pn for further analyses This process is repeated for each of the λc values mentioned in Section 4 to study the effect of this imperfection on CFS sections of different lengths. Also, this study is conducted for both the generalised (Stat-1) and the section-specific (Stat-2) imperfection statistics. It is found that the mean (μPn) and the standard deviation (σPn) of Pn converge – to an acceptable level of accuracy – by 1000 simulations for all cases of λc tested here. For illustration, convergence of these two parameters over the number of analysis conducted are shown in Figs. 8 and 9 for λ c = 1.27 .
6. Statistical analysis of the buckling strength Statistical analysis of 1000 MCS samples of Pn is carried out individually for each λc (and repeated for each set of statistics). Figs. 10 and 11 summarise the two sets of Pn statistics in the form of frequency histograms and empirical CDFs. Based on the sampled data, mean (μ Pn ) and standard deviation (σPn ) of Pn are obtained. These values are 453
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Fig. 11. Frequency histograms and empirical CDFs of Pn for Stat-2.
Table 4 Statistics of the buckling strength (Pn) . Stat-1
λc
0.50 0.75 1.00 1.27 1.53 2.05 2.50
Mean (kN)
106.2 100.5 87.98 65.62 49.38 27.99 18.91
Table 5 Percentage reduction in Pn from that of the member without imperfection. Stat-2
Std. dev. (kN)
5 percentile (kN)
Mean (kN)
4.820 6.189 8.071 3.383 2.542 1.306 0.1601
96.73 88.38 72.16 58.99 44.39 25.43 18.60
104.9 98.50 84.29 64.25 48.49 27.86 18.86
Stat-1 Std. dev. (kN)
5 percentile (kN)
3.169 3.828 4.791 1.706 1.478 0.8174 0.0921
98.68 90.99 74.90 60.91 45.59 26.25 18.69
Stat-2
λc
At μ Pn
At Pd
At μ Pn
At Pd
0.50 0.75 1.00 1.27 1.53 2.05 2.50
11.63 16.03 21.86 8.416 6.752 4.575 0.7141
19.49 26.16 35.91 17.67 16.16 13.27 2.362
12.70 17.71 25.13 10.32 8.436 5.014 0.9764
17.87 23.98 33.47 14.99 13.90 10.47 1.864
Pd values obtained (through FE simulations) are compared to the corresponding “Euler load”, Pe, which is the elastic flexural buckling strength of the member without considering the effects of local and distortional buckling (and any geometric imperfection):
slendernesses, elastic flexural buckling governs the failure of a member (with the overall flexural buckling mode being the lowest eigenmode), and the failure is less affected by local imperfections. For intermediate or low slenderness, significant local buckling takes place before the member undergoes overall buckling. In this range, we see the maximum effect of the local geometric imperfection. If the slenderness is reduced further, yielding of the material or inelastic buckling prevails. This also reduces the effects of local imperfection on the member failure.
Pe =
π 2EI Le2
(4)
where, E = Young's modulus, I = minimum moment of inertia of the member cross-section, Le = effective length of the member based on the boundary/support conditions. Performing simple regression analyses, 454
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Fig. 12. Suitability of Eqs. (4)–(5), (a) Stat-1, (b) Stat-2.
definition of λc used here (Eq. (1)), torsional effects are excluded. Fig. 13 gives a comparison of the critical strength values obtained in this work (for the imperfect CFS lipped channel) for both statistics. Based on the analyses conducted at seven discrete values of the nondimensional slenderness ratio, the comparison is presented in the form of “strength curves”, where Pn values (normalised by Ae Fy ) are plotted against λc. The 5 percentile values correspond to the design strength recommended in this work (for Pd), which are compared against the design strength recommended in AISI. It is observed that geometric imperfections have significant effect on the buckling strength, and the buckling strength values predicted by AISI through Eq. (7) and (8) are found to be approximately 6.0–9.0% higher (that is, uncoservative) compared to the values found in this work. Although the generalised and the particular statistics yield different results, their difference is only marginal. At the very high λc value of 2.5, the effect of local GI becomes negligible, as these curves almost coincide. Fig. 13. Normalised buckling strength of the lipped CFS channel.
7. Summary and conclusions
the design strength of an imperfect member, Pd, can be easily related to the Euler load using a simple bilinear function of the nondimensional slenderness ratio. For Stat-1, we get
Pd = (0.1669λc3 − 0.9851λc2 + 2.041λ c − 0.5956) Pe
This study focuses on the effects of local geometric imperfections on the buckling capacity (Pn) of a lipped channel CFS column. The emphasis is on a statistical understanding of the overall member behaviour (in terms of its critical load) in the presence of Type 1 local geometric imperfection. Using the statistics of its amplitude, reported by previous researchers, the statistics of Pn is obtained here using FE analyses within a Monte Carlo simulation based framework. A numerical investigation of a fixed-ended channel member is carried out, using a cross-validated finite element model of the member, for 12,000 structural analyses to obtain the statistics of Pn for a random imperfection magnitude. It is observed that the local imperfection can have a significant effect on the buckling strength of the member, and this cannot be ignored in its design process. On average, the reduction can be up to 25.13% from the strength of a member without imperfection. At the suggested design level, this reduction can be as severe as 35.91%. A 5 percentile value of the buckling strength is recommended as its design value (Pd). Also, the column strength values recommended by AISI are found to be approximately 6.0–9.0% higher (that is, unconservative) than the design value obtained from the statistical analysis presented here. Simple equations, based on the nondimensional slenderness (λc), are suggested (for the selected member) to obtain this value on the basis of the Euler load (elastic flexural buckling strength not considering the local, distortional and torsional effects). Strength curves, akin to the ones available in design standards, are also provided for the selected column member with imperfection. These simple expressions (or, the strength curves) can be used by any designer to obtain the design strength of the imperfect CFS member without going through the computation-intensive numerical and statistical route of obtaining this. It is also observed that both imperfection statistics (generalised and particular for this section) result in a similar statistics of Pn. The variation in results for these two statistics, for the selected range of λc, is only about 2.0–3.5%. This enables users to opt for the generalised
(5)
and for Stat-2,
Pd = (0.1612λc3 − 0.9778λc2 + 2.063λ c − 0.6002) Pe
(6)
2
These approximations correspond to R values higher than 0.98 for both statistics. Fig. 12 shows that based on the statistics obtained at seven different λc values, these relations can be considered adequate. Using these equations, one can easily obtain the design capacity of the imperfect lipped channel CFS, given the length and boundary conditions, without going through a rigorous finite element and statistical analysis procedure. It should be noted here that as the nondimensional slenderness increases, the Pd/ Pe ratio approximates 1.0, signifying that at very high λc the member fails at Euler's load. The design strength (Pd) obtained this way is also compared with codal provisions for CFS lipped channel columns. As per AISI, the buckling strength is obtained by multiplying the effective area (Ae) with the critical stress Fn, which is defined for two different ranges of λc: 2
Fn = 0. 658 λc Fy
for λ c ≤ 1.5
(7)
and
Fn =
0.877 Fy λc2
for λ c > 1.5
(8)
where Fy is the yield stress of the material. Eq. (8) represents the range of slenderness where generally the overall member behaviour is governed only by flexural buckling, and can be understood as a modification of Euler's equation. Eq. (7), however, represents the range where the effects of material yielding (inelastic buckling) predominates, along with those of local and overall torsional buckling. However, as per the 455
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compression, Trans. Am. Soc. Mech. Eng. 54 (1932) 53–57 (APM-54-5). [11] C. Schenk, G. Schuëller, Buckling analysis of cylindrical shells with random geometric imperfections, Int. J. Non-Linear Mech. 38 (7) (2003) 1119–1132. [12] B. Young, K. Rasmussen, Design of lipped channel coloumns, J. Struct. Eng. 124 (2) (1998) 140–148. [13] D. Dubina, V. Ungureanu, Effect of imperfections on numerical simulation of instability behaviour of cold-formed steel members, Thin-Walled Struct. 40 (3) (2002) 239–262. [14] J. Bonada, M. Casafont, F. Roure, M. Pastor, Selection of the initial geometrical imperfection in nonlinear FE analysis of cold-formed steel rack columns, ThinWalled Struct. 51 (0) (2012) 99–111. [15] J. Loughlan, N. Yidris, K. Jones, The failure of thin-walled lipped channel compression members due to coupled local-distortional interactions and material yielding, Thin-Walled Struct. 61 (2012) 14–21. [16] P.B. Dinis, E.M. Batista, D. Camotim, E. dos Santos, Local-distortional-global interaction in lipped channel columns: experimental results, numerical simulations and design considerations, Thin-Walled Struct. 61 (2012) 2–13. [17] T.V. Galambos, Guide to Stability Design Criteria for Metal Structures, 5th ed., John Wiley and Sons, Inc, New York, USA, 1998. [18] P. Nandini, V. Kalyanaraman, Strength of cold-formed lipped channel beams under interaction of local, distortional and lateral torsional buckling, Thin-Walled Struct. 48 (10–11) (2010) 872–877. [19] K. Sachidananda, K.D. Singh, Numerical study of fixed ended lean duplex stainless steel (LDSS) flat oval hollow stub column under pure axial compression, ThinWalled Struct. 96 (2015) 105–119. [20] Simulia. Abaqus FEA User’s Manual, Version 6.9. Dassault Systèmes, VélizyVillacoublay, France, 2009. [21] B. Young, J. Yan, Channel columns undergoing local, distortional, and overall buckling, J. Struct. Eng. 128 (6) (2002) 728–736. [22] D.C.Y. Yap, G.J Hancock, Interaction buckling and post-buckling in the distortional mode of thin-walled sections. Research Report No. 870; School of Civil Engineering, University of Sydney; Sydney, Australia, 2006. [23] V.M. Zeinoddini, Geometric imperfections in cold formed steel members. Ph.D. thesis; Baltimore, United States, 2011.
statistics in the case of unavailability of imperfection data for a particular CFS channel. The probabilistic techniques used here requires a large number of simulations, making the process computationally very expensive for each and every compression member. Future studies should focus on reducing the computational demand of such a work, so that strength curves can be obtained for any new section easily. References [1] C. Yu, B.W. Schafer, Distortional Buckling Tests on Cold-formed Steel Members in Bending, Johns Hopkins University, Baltimore, USA, 2005 Tech. Rep.. [2] G.J. Hancock, T.M. Murray, D.S. Ellifritt, Cold-Formed Steel Structures to the AISI Specification, Marcel Dekker, Inc, New York, USA, 2001. [3] AISI. AISI-S100, North American specification for the design of cold-formed steel structural members. American Iron and Steel Institute, Washington, USA, 2007. [4] SA/SNZ. AS/NZS 4600, Cold-Formed Steel Structures. Standards Australia/ Standards New Zealand, Sydney, Australia, 2005. [5] V.M. Zeinoddini, B. Schafer, Simulation of geometric imperfections in cold-formed steel members using spectral representation approach, Thin-Walled Struct. 60 (2012) 105–117. [6] B.W. Schafer, T. Peköz, Computational modeling of cold-formed steel: characterizing geometric imperfections and residual stresses, J. Constr. Steel Res. 47 (3) (1998) 193–210. [7] B.W. Schafer, Local, distortional, and euler buckling of thin-walled columns, J. Struct. Eng. 128 (2002) 3. [8] M.V.A. Kumar, V. Kalyanaraman, Design strength of locally buckling stub-lipped channel columns, J. Struct. Eng. 138 (2012) 11. [9] G. Winter, Strength of thin steel compression flanges. Tech. Rep.; Cornell Univ. Engg. Exp. Station, New York, USA, 1947. [10] T. von Karman, E.E. Sechler, L.H. Donnell, The strength of thin plates in
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Probabilistic estimation of the buckling strength of a CFS lipped-channel section with Type 1 imperfection
MARK
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Hashmi S.S. Ahmeda, Siddhartha Ghosha, , Mohit Mangalb a b
Structural Safety, Risk & Reliability Lab, Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India Department of Civil and Environmental Engineering, Hong Kong University of Science & Technology, Kowloon, Hong Kong
A R T I C L E I N F O
A B S T R A C T
Keywords: Cold-formed steel Uncertainty Imperfection Local buckling Post-buckling Lipped channel
Local geometric imperfections in a cold-formed steel (CFS) member can significantly alter the force-carrying capacity of the member. These are the dents and undulations which occur during cold-rolling, handling, transportation and erection of CFS members. The buckling strength of a lipped channel section with Type 1 local imperfection is obtained and characterised statistically using finite element analyses and Monte Carlo simulations. The reduction in strength due to these imperfections are found to be significant. The quantification of reduction in strength due to these imperfections is found for different values of non-dimensional slenderness ratio. Based on the statistical analysis, design equations and strength curves are recommended for the buckling strength of geometrically imperfect members. Legitimacy of using a generalised statistics of imperfection, in the case of unavailability of specific data for a particular section, is also verified.
1. Geometric imperfections in cold-formed steel sections Cold-formed steel (CFS) sections are fabricated from thin steel sheets using either press-braking or roll-forming process, by passing the sheet through a number of dies. The characteristic that differentiates CFS sections from hot-rolled ‘structural’ steel sections is that the shape of the cross-section, instead of the thickness of the section, is used for carrying loads [1]. The force carrying capacity of CFS sections depends largely on the shape achieved through a cold-forming process. Deviations from the target shape may affect a CFS section's capacity significantly. However, due to a multiplicity of reasons it is almost impossible to maintain the perfection in the cross-sectional dimensions of a CFS section that is finally used in construction. One major reason is that cold-rolling mills do not adhere to a very strict quality in the coldrolling process, which results in geometrically imperfect sections coming out of these mills. The other significant reason is the loads during handling, transportation, and erection (those loads, for which the section is not typically designed), which cause visible deformations in these thin members. Geometric imperfections (GIs) in CFS sections include global and local deviations. Whereas global behaviours, such as bow, camber and twist, are categorised under global GI, local deviations are characterised by dents and undulations in the member elements (flange, web etc.). As mentioned earlier, these deviations can significantly alter the force-carrying capacity of a section. CFS sections are characterised by the presence of different instability
⁎
Corresponding author. E-mail address: [email protected] (S. Ghosh).
http://dx.doi.org/10.1016/j.tws.2017.07.001 Received 19 October 2015; Received in revised form 7 June 2017; Accepted 1 July 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
modes, such as local, distortional and global buckling, prior to the ultimate ‘failure’ of the member. The design of these sections is generally governed by the ‘post-(local/distortional) buckling’ behaviour. The strength limit states are defined by ‘overall’ buckling, which includes flexural, torsional and torsional-flexural modes. Owing to typical proportions of CFS member elements, these buckling modes (in real structures) are usually elastic [2]. However, some stocky members fail by inelastic (overall) buckling as well. Design standards, such as AISI S100 [3] or AS/NZS 4600 [4], include the effect of inelasticity in the post-(local) buckling behaviour of CFS members. In this article, we focus on the effects of Type 1 local geometric imperfections on the buckling strength of a lipped channel (‘C’) section, in a statistical sense. A lipped C section is considered for this study because these are the most commonly used sections in structures made of coldformed steel, other than ‘Z’ sections. 2. Previous studies on geometric imperfections in CFS sections A local (or, cross-sectional) GI is understood as local unevenness or undulation in the elements of cross-section distributed over the length of the member, which is generally referred as local geometric imperfection or cross-sectional geometric imperfection [5]. Schafer and Peköz [6] categorised the local GIs into two groups: 1. Type 1: Maximum local imperfection in a stiffened element, such as a web
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imperfection magnitude in nonlinear FE analyses of CFS was presented by Bonada et al. [14]. Loughlan et al. [15] examined the failure mechanism of fixed-ended CFS lipped channels with due consideration to the influence of geometric imperfections. Results from the finite element simulations compared favourably with the tests performed, when the local geometric imperfection was properly modelled. They also reported the substantial influence of geometric imperfection on the load-deformation and buckling behaviour from the onset of loading. Dinis et al. [16] analysed the coupled instabilities in CFS lipped channel columns through finite element analyses. They showed the influence of GI on the elastic-plastic post-buckling behaviour of fixed-ended lipped channel columns. They also found the strength estimations based on the direct strength method (DSM) to be inaccurate for these cases. So far, research on the effect of GI on the behaviour of CFS sections primarily looked at aspects such as numerical modelling of GI, selection of GI, impact of model interaction, etc. However, very little work has been reported on the probabilistic performance assessment of the CFS member with geometric imperfections. Considering this fact the main focus of this work is set on the probabilistic estimation of buckling strength of a CFS member with local geometric imperfection.
Fig. 1. Type 1 geometric imperfection of the lipped channel cross-section.
2. Type 2: Maximum deviation from straightness for a lip-stiffened or an unstiffened flange
3. Objective and scope of the work
Fig. 1 illustrates the Type 1 deformation for the case of a lipped channel section. The local deviations are characterised by ‘dents’ and ‘irregular undulations’ in the plate. However, they found that little or no information was available on the specific location of these dents, which made it difficult to find which eigenmode(s) a specific imperfection would trigger. In their detailed statistical treatment of geometric imperfections based on the collected data, Schafer and Peköz [6] considered the amplitude of the imperfection as a random variable, and recommended a direct probabilistic simulation of both the imperfection magnitude and its distribution along the length of the member as the most robust way to assess the effect of geometric imperfections. However, they also pointed to the difficulty with the practical implementation of such an approach, and suggested more affordable but less accurate approaches. Traditionally, the effective width method [3] has been used to evaluate the strength of the CFS member. This method includes the effect of local buckling [2], but not the effect of interaction of different elements of a cross-section [7,8]. In addition, this method is very tedious. The semi-empirical approach adopted from Winter [9], which is a modification of the theoretical equation originally proposed by Karman et al. [10], does not explicitly account for geometric imperfections, either. However, since the modification of Karman et al. [10]'s equation was based on experimental results of real sections, one can expect that the equation(s) in AISI S100 can somewhat represent the imperfection, although not rigorously but in a very limited manner. However, as highlighted before, local geometric imperfections are very random in nature [6,11]. Therefore, a rigorous treatment of this randomness is necessary for a proper assessment of a CFS column's load carrying capacity. In this line, many researchers measured and characterised geometric imperfections in CFS sections [6] and tried to quantify their effect(s) on the capacity of a CFS member. Young and Rasmussen [12] performed compressive tests on two cross-sections of lipped channels, with two different boundary conditions. Prior to the test, detailed measurement of geometric imperfections was performed and reported. Dubina and Ungureanu [13] studied the effect of geometric imperfections on the buckling strength of CFS channels through numerical simulations. They pointed out the necessity of the codification of both size and shape of imperfections for numerical analysis. Also on the basis of their study, they showed that the different shapes of local imperfection have different effects on the buckling strength of the member. The importance of the selection of an initial geometrical
The primary objectives of this work are (i) to find the effect of Type 1 local geometric imperfections on the buckling strength of a CFS section, (ii) to statistically quantify this strength, and (iii) to compare the findings with the recommendations given in design codes (which do not explicitly account for such imperfection effects). Due to the presence of uncertainty in local imperfection magnitudes, a probabilistic framework is preferred to analyse the effect of these imperfections. Additionally, a comparative study between the effects of two different types of imperfection statistics – section-independent ‘generalised’ statistics and section-specific ‘particular’ statistics – is performed. This is to assess the legitimacy of using the generalised statistics of imperfection, in the case that a particular statistic is unavailable. As an initiatory work in this area from a statistical perspective, our study is limited to a single lipped channel section with a Type 1 deformation (Fig. 1). In this work, the yielding of the material is included in the member behaviour, however, torsional and flexural-torsional effects are avoided using suitable constraints.
4. Critical load analysis of the channel section A finite element (FE) approach is adopted for the buckling analysis of the member with some geometric imperfection, in order to properly capture the failure behaviour, including the interaction with local and distortional buckling modes. Second-order inelastic analyses (that is, including both material and geometric nonlinearity) are used to arrive at the ‘critical load’ for the member [17]. Since this load is obtained from the load-deformation curve of the member, the analysis lets us observe the sequence of local/distortional buckling and the overall member buckling. The lipped channel section selected for this study is 362S162-68 with the following dimensions: depth of the web = 92 mm (3.625 in), width of the flange = 41.3 mm (1.625 in), and thickness = 1.7 mm (0.068 in). Six different lengths of the section are considered in order to study the behaviour for different nondimensional slenderness ratios (λc) in the range of 0.5–2.5, where λc is defined as [3]
λc =
Fy Fe
(1)
where Fy is the material yield stress, and Fe is the minimum critical elastic column buckling stress (considering only the flexural mode). 448
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4.1. Incorporation of the imperfection into the FE model Generally, two alternative approaches are adopted in order to incorporate the initial imperfection into the finite element (FE) model for computational simulation of a CFS member's behaviour. One approach would be to measure the imperfections physically and model the very intricate imperfect geometry. This would be accurate but highly impractical as measurements for each and individual section would be prohibitively expensive. Alternatively, buckling shapes can be used to describe the imperfections into the FE model, which is the more common practice followed by researchers in general. In this modal approach of imperfection modelling, a single buckling mode or superposition of various buckling modes, scaled suitably, emulates the imperfection into a perfect member. This technique is quite common and has been adopted by various researchers [5,14,18,19]. The buckling shape is used to describe the distribution of the imperfection over the length of the member with measured statistics of maximum imperfection. In the current paper, the Type 1 local GI is adopted as a perturbation on the lowest eigenmode for the chosen channel cross-section. For this section, this lowest eigenmode corresponds to a local buckling similar to the deviation depicted in Fig. 1, and the deviation at the centre of the web (at the mid-span) is scaled to the maximum imperfection magnitude (d1). Obviously, this modelling also allows overall flexural buckling effects, with which the local buckling may interact before failure, depending on a member's λc. 4.2. Finite element simulation The CFS channel section is modelled and analysed in the generalpurpose finite element package Abaqus [20]. The four-noded linear shell element, S4R, with reduced integration, is used to model the member. Shell elements are the usual choice for such “thin” members with curved geometries which are expected to undergo large overall deformations. Fixed-fixed boundary conditions are imposed at the ends of the member. This is only an ideal boundary condition and does not necessarily reflect realistic situations. However, the focus is on studying the effect of Type 1 GI on a lipped channel of varying slenderness, irrespective of the actual boundary conditions. The effect of the boundary condition is incorporated through the effective slenderness ratio, which is normally practised for such structural systems. The boundary condition as well as the axial force are applied at the centroid of the end cross-sections. The constraint and force effects are transferred to the CFS member by connecting (all degrees of freedom of) the centroid with the nodes on the cross-section through “couplings” (Fig. 2). ‘Equation constraints’ are used to avoid the torsional behaviour of a member. The worst aspect ratio of S4R elements is kept within 3.0 to achieve a better numerical accuracy [5]. For the inelastic modelling of steel, an elastic-perfectly-plastic (EPP) material model is adopted, with yield stress (Fy) = 388 MPa, elastic modulus (E) = 203 GPa, and Poisson's ratio (ν) = 0.3 [5]. The effect of residual stresses and the rounded corners are not included here, as practised by many researchers, including Young and Yan [21]. The FE simulation is a two step process: eigenvalue analysis (to obtain the fundamental mode shape) followed by a second-order “buckling” analysis with an initial perturbation. The subspace iteration eigensolver is opted in Abaqus to find the mode shapes and eigenvalues. Fig. 3 shows sample buckling modes (both local and global). The Type 1 geometric imperfection is adopted on the lowest eigenmode (which corresponds to a local buckling mode) for that section, which gives the maximum imperfection magnitude at the centre of the web, similar to Fig. 1. The selected mode shape (scaled to the amplitude of GI) is then perturbed as imperfection into the perfect member (Fig. 4) to perform a nonlinear buckling analysis. In order to understand the post-(local) buckling behaviour of the
Fig. 2. Coupling of the centroid to the cross-section.
CFS section, an incremental static analysis incorporating geometric and material nonlinearities is performed until the member reaches its critical load. The Static,Riks iterative scheme in Abaqus is adopted here to conduct this “buckling” analysis. This technique uses the arc-length method to measure the progress of a solution in the load-displacement space, leading to converged solutions irrespective of the type of response [22]. As mentioned earlier, the initial geometry of the section is perturbed with the local buckling mode scaled to the amplitude of the GI. An axial compressive load, applied at the centroid of one end section, is incrementally increased and the lateral displacement of the member is monitored at the mid-depth of the web at the mid-span of member. Fig. 5 shows sample load-deformation curves obtained from such analyses. These are results for a member with λ c = 1.00 . Two load-deformation plots are presented for: (a) a CFS section with a specific GI amplitude of d1/ t = 7.31, and (b) the same CFS section without any geometric imperfection. For the member without imperfection, the lateral deformation is small and follows the compression load linearly for a large amount of compression. Afterwards, it displays a gradually reducing stiffness and reaches a maximum load carrying capacity of 112.7 kN. Ahead of this point, the stiffness becomes negative indicating an unstable/buckled state of the member. On the other hand, the imperfect member shows a noticeable deformation from the very beginning with a nonlinear load-deformation curve. This indicates how local GI influences the post-(local) buckling behaviour of the CFS section. The maximum compressive load carried by the imperfect member is only 60.08 kN. The maximum load carried by any member – with or without imperfection – is termed as its critical load (Pn), which gives the buckling capacity of the member. Fig. 6 shows the deformed shape of
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Fig. 3. Sample buckling modes (deformation scale = 3), (a) Local (central part of the member), (b) Global (full length of the member).
(a) Local (central part of the member)
(b) Global (full length of the member) 5. Statistical analysis considering the variation in imperfection
the imperfect member beyond the critical load point. This deformed shape also shows the co-existence of local and overall (flexural) buckling modes, confirming the interaction of different modes observed by previous researchers [15]. However, the most significant aspect of Fig. 5 is the reduction in the critical load, Pn, by about 47% on account of local (Type 1) GI. The finite element analysis (FEA) approach adopted here is validated with the experimental results reported by Young and Rasmussen [12], where they performed compression tests on fixed ended CFS lipped channels. Table 1 shows measured dimensions of the specimens tested. In our model, material characteristics (including nonlinearity) are incorporated based on the measured properties reported by Young and Rasmussen [12]. However, the magnitude of the local geometric imperfection is considered as 25% of the plate thickness of the member, as reported by Young and Rasmussen [12]. The ultimate load obtained using the FEA is compared with the experimental result and is reported in Table 2. It is clear from this comparison that our FE modelling approach is able to estimate the buckling strength with suitable accuracy. Moreover, the FE analysis approach adopted here is also compared to the numerical analysis performed by Young and Yan [21] for the same lipped channel section. Zeinoddini [23] considered a fixed-fixed member of length 8 ft. (2.44 m), with a normalised Type 1 imperfection magnitude, d1/ t = 0.31. The same section with the same imperfection is modelled and analysed in this paper using the method described earlier in this section. The critical load for buckling (Pn) is found to be within 2.77% error from the value obtained by Zeinoddini [23]. On the basis of these comparisons, the analysis methodology adopted for imperfect CFS sections in this work is deemed satisfactory.
5.1. Statistical modelling of the imperfection Imperfection statistics have been collected for CFS sections over last few decades. In this work, two different sets of statistical data – generalised (‘Stat-1’) and particular (‘Stat-2’) – are used for modelling the imperfection amplitude. Schafer and Peköz [6] provided the statistics of Type 1 local imperfection amplitude of CFS sections, based on data collected through different research studies. This statistics is for CFS sections in general and termed as Stat-1 in this work. Later, Zeinoddini and Schafer [5] presented the Type 1 imperfection statistics specifically for the lipped channel Section 362S162-68. These statistics, with quantile values and parameters, are reproduced in Table 3. It should be noted, following the general practice, these statistics are provided for the normalised imperfection d1/ t . In the present work, these statistics are modelled with common probability distributions (keeping the mean and standard deviation the same) with the Kolmogorov-Smirnov goodness-of-fit test (K-S test). The suitability of modelling the imperfection statistics with normal and lognormal distributions are checked. Both distributions are found suitable to model these statistics with an acceptable significance level. However, the lognormal distribution is found to be a better fit for Stat-1, while the normal distribution is found better for Stat-2 (with a significance level less than 0.05). The comparison between the two probability models with respect to the original data is provided through their cumulative distribution functions (CDF) in Fig. 7. Considering that both models provide a good fit, the following probability models are chosen for the uncertainty propagation:
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(a) Full member
(a) Full member
(b) Inset Fig. 6. Deformations after buckling (deformation scale = 2), (a) Full member, (b) Inset.
(b) Inset Table 1 Measured dimensions of specimen.
Fig. 4. FE model of the CFS member perturbed with initial imperfection, (a) Full member, (b), Inset.
Specimen ID
Web (mm)
Flanges (mm)
Lips (mm)
Thickness (mm)
Length (mm)
L36F0280 L48F0300
97.3 97.1
37.0 49.0
12.5 12.2
1.48 1.47
280 300
Table 2 Comparison of FE analysis and experimental results. Specimen ID
PFEM (kN)
PExp (kN)
PFEM /PExp
L36F0280 L48F0300
104.0 104.1
100.2 102.3
1.038 1.018
Table 3 Statistics of Type 1 (normalised) imperfection (d1/t).
Fig. 5. “Buckling” curves at the mid-span (for λ c = 1.00).
d1/ t ∼ LN (0.50, 0.66) For Stat–1
(2)
d1/ t ∼ LN (0.48, 0.24) For Stat–2
(3)
CDF
Stat-1
Stat-2
0.25 0.50 0.75 0.95 0.99
0.14 0.34 0.66 1.35 3.87
0.36 0.47 0.55 – –
Mean Std. dev.
0.50 0.66
0.48 0.24
random samples of d1/ t are generated based on the lognormal distribution model of Eq. (2)-(3). These imperfection samples are used to create N imperfect CFS members. Each of these specimens is then analysed for its critical load (Pn), using its finite element model, as described in Section 4.2. The N values of Pn obtained provide the
5.2. Monte Carlo simulation A “crude” Monte Carlo simulation (MCS)-based approach is adopted to obtain the effect of local geometric imperfections on the buckling strength. For each statistical data (Stat-1 and Stat-2), adequate (say N) 451
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Fig. 7. CDF of the observed data and trial distributions for normalised imperfection, (a) Stat-1, (b) Stat-2.
Fig. 8. Convergence of the mean and standard deviation of Pn for Stat-1 (λ c = 1.27) .
Fig. 9. Convergence of the mean and standard deviation of Pn for Stat-2 (λ c = 1.27) .
conditions, and coupling constraints) 2. Perform an eigenvalue buckling analysis and find the buckling modes 3. Select the local buckling mode corresponding to Type 1 imperfection 4. Generate a random sample of d1/ t based on the selected probabilistic model (Eq. (2)-(3)) 5. Perturb the FE model with the Type 1 mode scaled to this d1/ t 6. Perform an incremental iterative (Static,Riks) analysis to obtain Pn
statistics of buckling strength of an imperfect channel section with Type 1 local imperfection. The required number of MCS samples, N, are decided based on the convergence (to an acceptable tolerance) of the first two moments of Pn: mean (μPn) and standard deviation (σPn). A step-wise description of the overall analysis method for a specific member, with a certain length or λc, is as follows: 1. Create a FE model of the CFS member (with proper boundary 452
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Fig. 10. Frequency histograms and empirical CDFs of Pn for Stat-1.
reported for each λc in Table 4. The K-S test is used again to check if any common probability distribution can be used to model the variation of Pn. For this purpose, normal, lognormal, gamma and Gumbel distributions – with the sampled mean and standard deviation values for each data-set – are tried. For the seven λc values, none of these distributions provides a fit with an acceptable significance level (αc). Table 4 also shows the 5 percentile Pn values obtained for each data-set. The 5 percentile value is considered purely from a design specification perspective. Most design guidelines use a 5 percentile value to define characteristic/nominal material strength/member capacity. The same concept can also be used here to define a design value (Pd) for the buckling strength including the effect of local geometric imperfections. In addition, Table 5 shows the reduction in Pn of an imperfect member from that of the same member without any imperfection. This reduction is evaluated at the mean Pn value, and also at its 5 percentile or design value (Pd). The percentage reduction is tabulated for the seven selected λc values, which shows that the reduction due to imperfection can be as high as 35.91% for the design strength of the member. Table 5 also shows that this reduction is higher in the (low-to-) intermediate slenderness range, and it decreases both towards high and low slenderness zones. This trend remains true for both the generalised and the particular statistics of imperfection used in this study. For high
from the load-deformation plot 7. Store Pn vs. d1 data 8. Repeat steps 4–7 for all samples, until μ Pn and σPn converge 9. Model the statistics of Pn for further analyses This process is repeated for each of the λc values mentioned in Section 4 to study the effect of this imperfection on CFS sections of different lengths. Also, this study is conducted for both the generalised (Stat-1) and the section-specific (Stat-2) imperfection statistics. It is found that the mean (μPn) and the standard deviation (σPn) of Pn converge – to an acceptable level of accuracy – by 1000 simulations for all cases of λc tested here. For illustration, convergence of these two parameters over the number of analysis conducted are shown in Figs. 8 and 9 for λ c = 1.27 .
6. Statistical analysis of the buckling strength Statistical analysis of 1000 MCS samples of Pn is carried out individually for each λc (and repeated for each set of statistics). Figs. 10 and 11 summarise the two sets of Pn statistics in the form of frequency histograms and empirical CDFs. Based on the sampled data, mean (μ Pn ) and standard deviation (σPn ) of Pn are obtained. These values are 453
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Fig. 11. Frequency histograms and empirical CDFs of Pn for Stat-2.
Table 4 Statistics of the buckling strength (Pn) . Stat-1
λc
0.50 0.75 1.00 1.27 1.53 2.05 2.50
Mean (kN)
106.2 100.5 87.98 65.62 49.38 27.99 18.91
Table 5 Percentage reduction in Pn from that of the member without imperfection. Stat-2
Std. dev. (kN)
5 percentile (kN)
Mean (kN)
4.820 6.189 8.071 3.383 2.542 1.306 0.1601
96.73 88.38 72.16 58.99 44.39 25.43 18.60
104.9 98.50 84.29 64.25 48.49 27.86 18.86
Stat-1 Std. dev. (kN)
5 percentile (kN)
3.169 3.828 4.791 1.706 1.478 0.8174 0.0921
98.68 90.99 74.90 60.91 45.59 26.25 18.69
Stat-2
λc
At μ Pn
At Pd
At μ Pn
At Pd
0.50 0.75 1.00 1.27 1.53 2.05 2.50
11.63 16.03 21.86 8.416 6.752 4.575 0.7141
19.49 26.16 35.91 17.67 16.16 13.27 2.362
12.70 17.71 25.13 10.32 8.436 5.014 0.9764
17.87 23.98 33.47 14.99 13.90 10.47 1.864
Pd values obtained (through FE simulations) are compared to the corresponding “Euler load”, Pe, which is the elastic flexural buckling strength of the member without considering the effects of local and distortional buckling (and any geometric imperfection):
slendernesses, elastic flexural buckling governs the failure of a member (with the overall flexural buckling mode being the lowest eigenmode), and the failure is less affected by local imperfections. For intermediate or low slenderness, significant local buckling takes place before the member undergoes overall buckling. In this range, we see the maximum effect of the local geometric imperfection. If the slenderness is reduced further, yielding of the material or inelastic buckling prevails. This also reduces the effects of local imperfection on the member failure.
Pe =
π 2EI Le2
(4)
where, E = Young's modulus, I = minimum moment of inertia of the member cross-section, Le = effective length of the member based on the boundary/support conditions. Performing simple regression analyses, 454
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Fig. 12. Suitability of Eqs. (4)–(5), (a) Stat-1, (b) Stat-2.
definition of λc used here (Eq. (1)), torsional effects are excluded. Fig. 13 gives a comparison of the critical strength values obtained in this work (for the imperfect CFS lipped channel) for both statistics. Based on the analyses conducted at seven discrete values of the nondimensional slenderness ratio, the comparison is presented in the form of “strength curves”, where Pn values (normalised by Ae Fy ) are plotted against λc. The 5 percentile values correspond to the design strength recommended in this work (for Pd), which are compared against the design strength recommended in AISI. It is observed that geometric imperfections have significant effect on the buckling strength, and the buckling strength values predicted by AISI through Eq. (7) and (8) are found to be approximately 6.0–9.0% higher (that is, uncoservative) compared to the values found in this work. Although the generalised and the particular statistics yield different results, their difference is only marginal. At the very high λc value of 2.5, the effect of local GI becomes negligible, as these curves almost coincide. Fig. 13. Normalised buckling strength of the lipped CFS channel.
7. Summary and conclusions
the design strength of an imperfect member, Pd, can be easily related to the Euler load using a simple bilinear function of the nondimensional slenderness ratio. For Stat-1, we get
Pd = (0.1669λc3 − 0.9851λc2 + 2.041λ c − 0.5956) Pe
This study focuses on the effects of local geometric imperfections on the buckling capacity (Pn) of a lipped channel CFS column. The emphasis is on a statistical understanding of the overall member behaviour (in terms of its critical load) in the presence of Type 1 local geometric imperfection. Using the statistics of its amplitude, reported by previous researchers, the statistics of Pn is obtained here using FE analyses within a Monte Carlo simulation based framework. A numerical investigation of a fixed-ended channel member is carried out, using a cross-validated finite element model of the member, for 12,000 structural analyses to obtain the statistics of Pn for a random imperfection magnitude. It is observed that the local imperfection can have a significant effect on the buckling strength of the member, and this cannot be ignored in its design process. On average, the reduction can be up to 25.13% from the strength of a member without imperfection. At the suggested design level, this reduction can be as severe as 35.91%. A 5 percentile value of the buckling strength is recommended as its design value (Pd). Also, the column strength values recommended by AISI are found to be approximately 6.0–9.0% higher (that is, unconservative) than the design value obtained from the statistical analysis presented here. Simple equations, based on the nondimensional slenderness (λc), are suggested (for the selected member) to obtain this value on the basis of the Euler load (elastic flexural buckling strength not considering the local, distortional and torsional effects). Strength curves, akin to the ones available in design standards, are also provided for the selected column member with imperfection. These simple expressions (or, the strength curves) can be used by any designer to obtain the design strength of the imperfect CFS member without going through the computation-intensive numerical and statistical route of obtaining this. It is also observed that both imperfection statistics (generalised and particular for this section) result in a similar statistics of Pn. The variation in results for these two statistics, for the selected range of λc, is only about 2.0–3.5%. This enables users to opt for the generalised
(5)
and for Stat-2,
Pd = (0.1612λc3 − 0.9778λc2 + 2.063λ c − 0.6002) Pe
(6)
2
These approximations correspond to R values higher than 0.98 for both statistics. Fig. 12 shows that based on the statistics obtained at seven different λc values, these relations can be considered adequate. Using these equations, one can easily obtain the design capacity of the imperfect lipped channel CFS, given the length and boundary conditions, without going through a rigorous finite element and statistical analysis procedure. It should be noted here that as the nondimensional slenderness increases, the Pd/ Pe ratio approximates 1.0, signifying that at very high λc the member fails at Euler's load. The design strength (Pd) obtained this way is also compared with codal provisions for CFS lipped channel columns. As per AISI, the buckling strength is obtained by multiplying the effective area (Ae) with the critical stress Fn, which is defined for two different ranges of λc: 2
Fn = 0. 658 λc Fy
for λ c ≤ 1.5
(7)
and
Fn =
0.877 Fy λc2
for λ c > 1.5
(8)
where Fy is the yield stress of the material. Eq. (8) represents the range of slenderness where generally the overall member behaviour is governed only by flexural buckling, and can be understood as a modification of Euler's equation. Eq. (7), however, represents the range where the effects of material yielding (inelastic buckling) predominates, along with those of local and overall torsional buckling. However, as per the 455
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