# Effect of Uncertainty in Localized Imperfection on the Ultimate Compressive Strength of Cold-Formed Stainless Steel Hollow Sections

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Statistical Analysis of the Maximum Localized Imperfection (ω)

_{max}) are available for the particular case of RHS and SHS stainless steel specimens. A statistical analysis of experimental results of the ω

_{max}from the literature [19,20,21,22,25,26,27,28,29,30,31] is carried out in this section. A total of 161 cold-formed stainless steel RHS and SHS samples are collected. A summary of the samples is shown in Table 1. The studied samples refer to the stainless steel grades commonly used in construction. In these references, some studies [21,22] provided the pattern of ω in the transverse direction (cross-sectional), in which all the reported patterns are very close to a half-sine wave. Few of them reported the variation of localized imperfection in the longitudinal direction. The distribution of ω in the longitudinal direction for two tubes reported in the work of [19] is shown in Figure 5. It is observed that ω in the longitudinal direction has a considerable variability and its characterization in a definite closed-form is not feasible.

_{max}among the samples collected in the literature was identified by statistical distribution tests (Anderson–Darling method), as well as from probability plots. Both distribution tests and probability plots were performed using the statistical software Minitab 18 [32].

_{max}.

_{max}is shown in Figure 6c. The log-normal distribution is fitted to the histogram. Comparison of the cumulative probability (CDF) curve against the log-normal distribution is shown in Figure 6d, in which CDF determines the probability that an observation will be less than or equal to a certain value.

_{max}/b, 0.5 mm}, where ω

_{max}/b ≤ 0.008; b is the side (straight side of the cross-section) length; and ω

_{max}represents the maximum deviation from the straight side.

## 3. Fourier Series-Based 3D Models with Random ω

_{i}, y

_{j}, z

_{ij}) (I = 1, 2, …, m; j = 1, 2, …, n) on the surface, z

_{ij}governs localized imperfection (ω). All z

_{ij}elements comprise an n × m matrix [

**Z**], which can be determined by

_{1}(x

_{i}) and f

_{2}(x

_{i}) are functions that are decomposed into Fourier series.

**S**] is an m × m diagonal matrix,

_{1}(x

_{i}) and f

_{2}(x

_{i}) are two functions that are decomposed into Fourier series with random coefficients. [

**F**] and [

_{1}**F**] govern two curved surfaces, as shown in Figure 7a, where L and B are the length and width of the surface, respectively. Localized imperfection (ω) is determined by matrix ([

_{2}**F**] − [

_{2}**F**]) [

_{1}**S**]. It consists of two components: the transverse variation and longitudinal variation, as shown in Figure 7b. The shape and magnitude of ω in the longitudinal direction depends on the curve along the longitudinal centerline. It is determined by the function [f

_{2}(x

_{i}) − f

_{1}(x

_{i})] sin (π/2). The shape of ω in the transverse direction is modelled by a half-sine-wave, as its shape in the transverse direction reported in most literatures is convexity/concavity. The half-sine-wave is determined by the function [f

_{2}(x

_{i}) − f

_{1}(x

_{i})] sin (πy

_{j}/B), as shown in Figure 7b, where the two half-sine waves correspond to (x

_{a}, y

_{j}) and (x

_{b}, y

_{j}) (j = 0, 1, …, m). The generated surface with random ω is determined by [

**F**] + ([

_{1}**F**] − [

_{2}**F**]) [

_{1}**S**], as shown in Figure 7c.

## 4. Case Study of Stainless Steel Columns with Cold-Formed RHS and SHS

_{l}) higher than 0.776. This is to ensure that the columns undergo cross-sectional local buckling reduction before they reach the ultimate compressive strength. According to the work of [24], the nominal compressive strength of a column with RHS or SHS is determined by min{P

_{ne}, P

_{nl}}, where P

_{ne}and P

_{nl}are the nominal global buckling strength and local buckling strength, respectively. The interaction between global and local buckling depends on λ

_{l}and is determined by

_{nl}/P

_{ne}versus λ

_{l}is shown in Figure 9. It should be pointed out that, although Equations (10) and (11) are developed based on carbon steel members, the two equations are applicable to stainless steel members with cold-formed RHS and SHS and give an accurate prediction [40,41].

_{1}, b

_{2}, and t are the depth, width, and thickness of the hollow cross-section, respectively; R is external radius of the round corner; L is the length of the column; λ

_{c}and λ

_{l}are member slenderness and cross-sectional slenderness, respectively; ω

_{g}is the amplitude of global member imperfection (out-of-straightness). ω

_{g}is not reported for some cases of stub columns (λ

_{c}≤ 0.2), while the shape of ω

_{g}is adopted as a half-sine wave for other columns.

## 5. Generation of 3D Models of the Studied Columns and Finite Element (FE) Analysis

#### 5.1. Generation of 3D Model with Random ω Using MATLAB

_{2}(x) and f

_{2}(x) was performed in Matlab. For the stub columns (λ

_{c}≤ 0.2), Fourier series expansion of function f

_{1}(x) generated a straight line. For other columns, f

_{1}(x) generated half-sine-waves, where the magnitude of the half-sine wave was taken as the corresponding ω

_{g}, shown in the above Table 2. For all columns, coefficients of Fourier series terms of function f

_{2}(x) were defined as random. The maximum amplitude of the modelled ω for each column was limited to min{0.008b, 0.5}. For each column, 50 models with random values of localized imperfection

**ω**were produced. The developed Matlab program automatically created a Python script associated with an Input file operated in Abaqus. It is worth pointing out that the distribution of the generated random ω

_{max}followed a log-normal distribution as the experimental data of ω

_{max}. This was explicitly set in the developed Matlab program.

#### 5.2. FE Analysis Using ABAQUS

_{max}, geometrically and materially nonlinear analysis with imperfections (GMNIA) was carried out to determine the ultimate compressive strength of the column.

## 6. Effect of Uncertainty in ω on the Ultimate Compressive Strength of the Studied Columns

_{u-EXP}is the ultimate compressive strength obtained from experiment; P

_{u-rand}is the predicted ultimate compressive strength for each model (each column have 50 models); µ and COV are the mean value and coefficients of variation, respectively; and |ε

_{max}| is the maximum value of relative error for each set of 50 models.

_{u-rand})/P

_{u-EXP}and COV (P

_{u-rand}) versus λ

_{l}are plotted in Figure 10a,b, respectively. A plot of

**|**ε

_{max}

**|**against λ

_{l}is shown in Figure 10c. It is observed that the value of µ(P

_{u-rand})/P

_{u-EXP}ranges between 0.931 and 1.103 for all the columns, except the column with λ

_{l}= 1.20. Compared with P

_{u-EXP}, the value of µ(P

_{u-rand}) for most columns with relatively lower cross-sectional slenderness (λ

_{l}< 1.0) is overestimated, while the value of µ(P

_{u-rand}) for columns with higher cross-sectional slenderness seems to be underestimated. For the column with λ

_{l}= 1.20, the value of µ(P

_{u-rand})/P

_{u-EXP}is 1.142. It indicates that most predicted results (from the 50 models with random ω) significantly overestimate the experimental result. This may be because the value of the actual maximum localized imperfection for this column, which is not reported in the work of [28], is relatively larger compared with the modelled ω, whose maximum value is min{0.008b, 0.5}.

_{u-rand}) and

**|**ε

_{max}

**|**increase as λ

_{l}increases. One explanation is that, the larger the cross-sectional slenderness, the more sensitive the column is to initial localized imperfection. Consequently, the change in the value of modelled localized imperfection can result in a larger discrepancy in the ultimate compressive strength.

_{l}< 1.0), the values of COV (P

_{u-rand}) and

**|**ε

_{max}

**|**are less than 0.13 and 0.12, respectively. The result indicates that uncertainty in ω has no considerable influence on the ultimate compressive strength of these columns. This may be because the columns with relatively lower cross-sectional slenderness are still not sensitive to initial localized imperfection. Besides, the result indicates that ω can statistically be modelled as deterministic for these columns, such as using measured ω in the experimental study.

_{l}≥ 1.2, COV (P

_{u-rand}) are around 0.139–0.238 and the maximum value of

**|**ε

_{max}

**|**is 17.5%. It demonstrates that random ω results in largely scattered ultimate compressive strength for the columns with larger cross-sectional slenderness, and it is important to consider the effect of uncertainty in ω on these columns. The distribution of P

_{u-rand}for a typical column (R1L1200) is shown in Figure 11. In the figure, P

_{u-rand}is normalized by P

_{u-EXP}. It is found that the distribution of P

_{u-rand}/P

_{u-EXP}can be fitted by a normal distribution.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Classification of stainless steels groups based on the content of chromium (Cr) and nickel (Ni).

**Figure 2.**Cold-formed stainless steel rectangular hollow section (RHS) and square hollow section (SHS) and their applications in construction industry. (

**a**) Cold-formed stainless steel RHS and SHS; (

**b**) facade of the building of department of Chemistry, ETH Zürich (Switzerland); (

**c**) support frame of Marqués de Riscal Vineyard (Spain).

**Figure 4.**Localized imperfection (ω). (

**a**) Random ω along longitudinal centerline of the surface; (

**b**) ω in transverse direction (convexity/concavity); (

**c**) idealized ω obtained from buckle analysis.

**Figure 5.**Distribution of ω in the longitudinal direction for two tubes reported in the work of [19]. (

**a**) RHS 100 × 40 × 2; (

**b**) SHS 60 × 60 × 3.

**Figure 6.**Identifying probability distribution for ω

_{max}. (

**a**) Goodness of fit test results for 16 different distribution tests; (

**b**) probability plot of ω

_{max}; (

**c**) histogram of ω

_{max}; (

**d**) comparison of cumulative probability (CDF) curve against the log-normal distribution. AD, Anderson–Darling.

**Figure 7.**Development of 3D surface with random ω. (

**a**) Two surfaces determined by [F

_{1}] and [F

_{2}]; (

**b**) surface determined by ([F

_{2}] − [F

_{1}]) [S]; (

**c**) surface determined by [F

_{1}] + ([F

_{2}] − [F

_{1}]) [S].

**Figure 8.**Generated 3D model. (

**a**) Surface with random ω and half-sine edges; (

**b**) member with random ω and half-sine edges.

**Figure 10.**Predicted results against cross-sectional slenderness. (

**a**) µ(P

_{u-rand})/P

_{u-EXP}versus λ

_{l}; (

**b**) coefficients of variation (COV) (P

_{u-rand}) versus λ

_{l}; (

**c**)

**|**ε

_{max}

**|**versus λ

_{l}.

Reference | Stainless Steel Groups | Grade | No. of Samples with Measured ω |
---|---|---|---|

B.F. Zheng et al., 2016 [25] | Austenitic | EN1.4301 | 4 |

I. Arrayago. et al., 2016 [26] | Ferritic | EN1.4003 | 12 |

B. Young and W.M. Lui, 2005 [21] | Duplex | EN1.4162 | 5 |

O. Zhao et al., 2015 [20] | Austenitic | EN1.4301 | 10 |

Austenitic | EN1.4571 | 6 | |

Austenitic | EN1.4307 | 6 | |

Austenitic | EN1.4404 | 6 | |

Duplex | EN1.4162 | 6 | |

M. Theofanous and L. Gardner, 2009 [27] | Duplex | EN1.4162 | 8 |

W.M. Lui et al., 2014 [22] | Duplex | EN1.4462 | 10 |

Y. Huang and B. Young, 2013 [28] | Duplex | EN1.4162 | 22 |

S. Afshan and L. Gardner, 2013 [29] | Ferritic | EN1.4003 | 6 |

Ferritic | EN1.4509 | 2 | |

M. Bock et al., 2015 [30] | Ferritic | EN1.4003 | 8 |

I. Arrayago and E. Real, 2015 [31] | Ferritic | EN1.4003 | 26 |

O. Zhao et al., 2016 [19] | Ferritic | EN1.4003 | 24 |

Total: 161 |

Reference | Specimen | b_{1} (mm) | b_{2} (mm) | t (mm) | R (mm) | L (mm) | λ_{c} | λ_{l} | ω_{g} |
---|---|---|---|---|---|---|---|---|---|

[37] | SHS2L300 | 50.1 | 50.3 | 1.58 | 2.8 | 300 | 0.14 | 0.8 | - |

RHS1L3000 | 140.1 | 79.9 | 3.01 | 10.0 | 3000 | 0.71 | 0.9 | 0.927 | |

[28] | C5L200 | 100.1 | 50.1 | 2.5 | 3.7 | 200 | 0.18 | 1.0 | - |

C6L200 | 150.0 | 50.1 | 2.5 | 4.3 | 200 | 0.18 | 1.5 | - | |

C6L550 | 150.1 | 50.2 | 2.49 | 4.5 | 550 | 0.48 | 1.4 | 0.5 | |

C5L900R | 100.1 | 50.4 | 2.49 | 3.5 | 900 | 0.79 | 0.8 | 0.857 | |

C6L900 | 150.4 | 50.3 | 2.47 | 4.7 | 900 | 0.79 | 1.3 | 0.857 | |

C6L1200 | 149.9 | 50.5 | 2.46 | 4.5 | 1200 | 1.04 | 1.2 | 1.143 | |

C6L1550 | 150.5 | 50.3 | 2.49 | 4.5 | 1550 | 1.35 | 1.0 | 1.476 | |

[29] | RHS 120 × 80 × 3-SC2 | 120.0 | 80.0 | 2.83 | 6.7 | 362 | 0.16 | 0.82 | - |

RHS 120 × 80 × 3-1077 | 120.0 | 79.9 | 2.87 | 6.8 | 1077 | 0.35 | 0.8 | 0.95 | |

RHS 120 × 80 × 3-1577 | 120.0 | 79.9 | 2.81 | 6.4 | 1577 | 0.51 | 0.8 | 0.96 | |

[38] | R1L1200 | 120.1 | 40.1 | 1.94 | 5.0 | 1199 | 0.48 | 1.1 | 0.254 |

R1L2000 | 120.2 | 40.0 | 1.95 | 5.1 | 2000 | 0.80 | 1.0 | 0.444 | |

R3L2000 | 120.0 | 80.0 | 2.80 | 6.7 | 2000 | 0.44 | 0.8 | 0.381 | |

[39] | SHS100 × 100 × 2-LC-2 m | 99.8 | 99.9 | 1.86 | 3.2 | 2000 | 0.73 | 1.0 | 0.1 |

RHS100 × 50 × 2-LCJ-2 m | 99.8 | 49.8 | 1.83 | 3.7 | 2000 | 0.80 | 0.9 | 0.6 | |

RHS100 × 50 × 2-LC-1 m | 99.8 | 50.0 | 1.82 | 3.6 | 1000 | 0.69 | 1.0 | 0.1 | |

RHS120 × 80 × 3-LC-1 m | 120.0 | 80.2 | 2.86 | 5.7 | 1001 | 0.45 | 0.8 | 1 | |

[21] | 160 × 80 × 3 | 160.1 | 80.8 | 2.87 | 9.0 | 600 | 0.09 | 1.2 | - |

200 × 110 × 4 | 196.2 | 108.5 | 4.01 | 13.0 | 600 | 0.07 | 1.1 | - |

**Table 3.**Experimental results and predicted results for the studied columns. COV, coefficients of variation.

Specimen | λ_{l} | P_{u-EXP} (kN) | µ(P_{u-rand}) (kN) | µ(P_{u-rand}) /P_{u-EXP} | COV(P_{u-rand}) | |ε_{max}| |
---|---|---|---|---|---|---|

SHS2L300 | 0.84 | 175.7 | 177.8 | 0.985 | 0.086 | 0.043 |

RHS1L3000 | 0.88 | 513.5 | 454.7 | 0.980 | 0.073 | 0.077 |

C5L200 | 0.95 | 370.1 | 387.5 | 1.103 | 0.036 | 0.065 |

C6L200 | 1.47 | 404.1 | 413.2 | 0.931 | 0.175 | 0.171 |

C6L550 | 1.41 | 353.2 | 388.1 | 1.026 | 0.212 | 0.175 |

C5L900R | 0.84 | 336.0 | 326.0 | 1.007 | 0.055 | 0.029 |

C6L900 | 1.32 | 333.5 | 345.2 | 0.988 | 0.139 | 0.098 |

C6L1200 | 1.20 | 284.5 | 300.7 | 1.142 | 0.108 | 0.185 |

C6L1550 | 1.02 | 230.0 | 249.2 | 1.008 | 0.095 | 0.102 |

RHS 120 × 80 × 3-SC2 | 0.82 | 441.0 | 434.2 | 1.072 | 0.093 | 0.086 |

RHS 120 × 80 × 3-1077 | 0.79 | 463.0 | 432.1 | 1.018 | 0.109 | 0.075 |

RHS 120 × 80 × 3-1577 | 0.79 | 382.0 | 401.5 | 0.973 | 0.045 | 0.058 |

R1L1200 | 1.07 | 167.0 | 153.5 | 1.02 | 0.129 | 0.115 |

R1L2000 | 0.97 | 141.3 | 137.9 | 0.999 | 0.088 | 0.055 |

R3L2000 | 0.79 | 394.0 | 355.7 | 1.071 | 0.071 | 0.047 |

SHS100 × 100 × 2-LC-2 m | 1.04 | 176.0 | 183.0 | 1.066 | 0.162 | 0.096 |

RHS100 × 50 × 2-LCJ-2 m | 0.92 | 157.0 | 145.3 | 1.041 | 0.085 | 0.117 |

RHS100 × 50 × 2-LC-1 m | 0.96 | 163.0 | 151.4 | 1.090 | 0.133 | 0.102 |

RHS120 × 80 × 3-LC-1 m | 0.79 | 448.0 | 415.5 | 1.053 | 0.079 | 0.086 |

160 × 80 × 3 | 1.24 | 537.3 | 505.0 | 0.939 | 0.158 | 0.139 |

200 × 110 × 4 | 1.07 | 957.0 | 928.0 | 0.958 | 0.081 | 0.070 |

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**MDPI and ACS Style**

Shen, Y.; Chacón, R.
Effect of Uncertainty in Localized Imperfection on the Ultimate Compressive Strength of Cold-Formed Stainless Steel Hollow Sections. *Appl. Sci.* **2019**, *9*, 3827.
https://doi.org/10.3390/app9183827

**AMA Style**

Shen Y, Chacón R.
Effect of Uncertainty in Localized Imperfection on the Ultimate Compressive Strength of Cold-Formed Stainless Steel Hollow Sections. *Applied Sciences*. 2019; 9(18):3827.
https://doi.org/10.3390/app9183827

**Chicago/Turabian Style**

Shen, Yanfei, and Rolando Chacón.
2019. "Effect of Uncertainty in Localized Imperfection on the Ultimate Compressive Strength of Cold-Formed Stainless Steel Hollow Sections" *Applied Sciences* 9, no. 18: 3827.
https://doi.org/10.3390/app9183827