Probability distribution of the compression capacity of welded hollow spherical joints with randomly located corrosion

Probability distribution of the compression capacity of welded hollow spherical joints with randomly located corrosion

Thin-Walled Structures 137 (2019) 167–176 Contents lists available at ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate...

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Thin-Walled Structures 137 (2019) 167–176

Contents lists available at ScienceDirect

Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws

Full length article

Probability distribution of the compression capacity of welded hollow spherical joints with randomly located corrosion

T



Zhongwei Zhao , Haiqing Liu, Bing Liang School of Civil Engineering, Liaoning Technical University, Fuxin 123000, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Welded hollow spherical joints Randomly located corrosion Compression capacity Random analysis Normal distribution

Welded hollow spherical joints (WHSJs) are commonly used joints in reticulated shell structures. Corrosion that occurs on the surface of WHSJs can considerably reduce the compression capacity of WHSJs with randomly distributed corrosion. Randomly located corrosion is a typical corrosion type on steel structures. However, the influence of randomly located corrosion on the loading capacity of WHSJs has not yet been quantified. This work aims to construct a probabilistic distribution model of the compression capacity of WHSJs with randomly located corrosion and to establish the relationship between such model and the specified corrosion (including corroded thickness Tc, corrosion diameter Dc, and corrosion number n) and geometric parameters (including the thickness T and diameter D of the spherical body). The influences of the corrosion and geometric parameters on the random distribution of compression capacity were first analyzed. Then, the probabilistic distribution model of the compression capacity of WHSJs with randomly located corrosion was established. The analytical method presented in this study achieved high accuracy in predicting the probabilistic distribution model of compression capacity with specified Tc/T, n, and Dc.

1. Introduction Welded hollow spherical joints (WHSJs) are commonly used in reticulated shell structures. They were first adopted in a practical project a few decades ago [1,2]. Investigations on the mechanical behavior of WHSJs have elicited the attention of numerous researchers due to the high loading capacity of WHSJs. Research has focused on the mechanical behavior of WHSJs or the influences of WHSJs on the mechanical behavior of integral structures [3–5]. Studies on the mechanical behavior of WHSJs have been mostly conducted through finite element analysis (FEA) or experiment [6–9]. Research achievement on mechanical behavior has been valuable and adequate for designing space lattice shell structures. However, research on WHSJs has been conducted on the basis of perfect WHSJs. WHSJs were first applied approximately five decades ago. Corrosion is inevitable on steel structures, and WHSJs are also subject to corrosion, such as the WHSJs in the swimming pool shown in Fig. 1. Corrosion can reduce the effective thickness of WHSJs and then considerably decrease their ultimate loading capacity. Corrosion is highly relevant to the surrounding environment of WHSJs, and corroded zones occur randomly. Numerous studies have been performed on the influence of corrosion on the mechanical behavior of steel members. However, such studies have focused on steel plates and pipes [10–12]. Zhao [13–15] ⁎

investigated the influence of specific corrosion on the loading capacity of WHSJs. Stochastic probability theory is an effective method for predicting the ultimate loading capacity of WHSJs with randomly located corrosion. Sultana et al. [16] utilized FEA to investigate the effect of random corrosion on the compression capacity of marine structural units. SaadEldeen [17–19] conducted a series of investigations on the influence of corrosion on box girders. Yu [20] assessed the effect of local random pitting corrosion on the collapse pressure of a 2D ring under external pressure. Zhao and Zhai [21] performed a comprehensive set of experiments on aluminum alloy columns under axial and eccentric compressions from 1999 to 2016. A probabilistic model of the ultimate loading capacity of WHSJs should be established first before stochastic probability theory can be adopted. The objectives of the present work are as follows: (1) to construct a probabilistic distribution model of the compression capacity of WHSJs with randomly located corrosion and (2) to establish the relationship between such model and the specified corrosion and geometric parameters. The influences of the corrosion and geometric parameters on the random distribution of compression capacity were first analyzed. Then, the probabilistic distribution model of the compression capacity of WHSJs with randomly located corrosion was established.

Corresponding author. E-mail address: [email protected] (Z. Zhao).

https://doi.org/10.1016/j.tws.2019.01.031 Received 24 October 2018; Received in revised form 10 January 2019; Accepted 15 January 2019 Available online 18 January 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Corrosion of WHSJs.

2. Establishment of a finite element (FE) model 2.1. WHSJs with corrosion A refined numerical model was established on the basis of the ANSYS code to investigate the influence of corrosion on the loading capacity of WHSJs. The element Shell 181 was selected, and corrosion was simulated by reducing the thickness of this element [22], as shown in Fig. 2. Fig. 3 presents the FE model of the spherical body of a WHSJ with shell elements, which is shear deformable and comprises four nodes with five independent degrees of freedom per node (three for translation and two for flexural rotation). The corroded zone of a WHSJ may exhibit different patterns under practical conditions. The distribution pattern of the corroded zone was categorized into three types: latitudinal pattern (Pattern I), longitudinal pattern (Pattern II), and randomly located corrosion (Pattern III). The detailed information of the different patterns is provided in Fig. 2. Tc indicates the corroded thickness, Hc indicates the height of the corroded zone, α indicates the central angle of Pattern II, and Dc indicates the diameter of Pattern III. The thickness element located in the corroded zone was adjusted to model the influence of corrosion. The effect of randomly located corrosion on the compression capacity of WHSJs was investigated in this work. The loading method is illustrated in Fig. 3. Q235 and Q345 steel types are recommended for WHSJs according to the 《Technical specification for space frame structures》 [23]. Q235 and Q345 are abbreviations of steel grade designations. Q refers to yield strength (in Chinese Pinyin) and 235 and 345 denote the corresponding minimum nominal yield strengths of 235 N/mm2 and 345 N/mm2. Q345 is used in the present work. The density and Poisson's ratio of Q345 are 7.85 g/cm3 and 0.3, respectively. The chemical composition of Q345 is provided in Table 1.

Fig. 3. Numerical model of a WHSJ with randomly located corrosion.

between 13 mm and 24 mm in accordance with the 《Technical specification for space frame structures》 [23]. The thickness of the spherical body T was set as 13, 18, and 24 mm. Corrosion thickness Tc was set as different values to investigate its influence on bending capacity. The number of corrosions is represented by the symbol n, which indicates the severity of corrosion. The shape of the corrosion was circular with the same corroded thickness Tc. The corrosion center exhibited a uniform distribution over the surface. That is, the location of the corrosion complied with uniform distribution. The possible space of the location was the surface of the spherical body outside the steel pipe. Presumably, corrosion did not occur on the steel pipe and on the surface of the spherical body inside the steel pipe. The numerical model of a WHSJ with randomly located corrosion is shown in Fig. 4.

2.2. WHSJ with randomly located corrosion Randomly located corrosion was assumed to be distributed randomly on the surface of a spherical body. The diameter of the corrosion Dc was set as 80 mm. The recommended range of thickness for WHSJs is

Fig. 2. Distribution patterns of the corroded zone. 168

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Table 1 Chemical composition of Q345. Chemical element

C

Mn

Si

P

S

Al

V

Nb

Ti

Content (%)

≤ 0.2

1.0–1.6

≤ 0.55

≤ 0.035

≤ 0.035

≥ 0.015

0.02–0.15

0.015–0.06

0.02–0.2

3. Parametric analysis of compression capacity 3.1. Validation of the FE model The reliability of the FE model was validated by comparing the results of the model with the experimental results of Liu et al. [15,24]. The external diameter and thickness of the WHSJ were set as 0.2 m and 8 mm, respectively. The external diameter of the steel tube was 76 mm. The constitutive model was set in accordance with the stress–strain curves derived by Liu et al. [24], and the multilinear isotropic hardening command of ANSYS was used. The same element and mesh scale as those in [13] were adopted in this study. The detailed information about the validation process can be found in [13,24]. The comparison of the force–displacement curves derived via FEA and the experiment is shown in Fig. 5. The results derived via FEA agree well with those derived in the experiment. The comparison of the failure mode is depicted in Fig. 6. The FE model can also accurately predict the failure mode of the WHSJ caused by axial force.

Fig. 5. Comparison of the force–displacement curves derived via FEA and experiment.

body were systematically investigated in this subsection. The values of Tc/T, D, Dc, and fy were set as 0.875, 0.6 m, 80 mm, and 345 MPa, respectively. The value of T was set as 13, 18, and 24 mm. Analysis was conducted 50 times for each value of n according to the aforementioned results. The maximum number of corrosions was set as 150. The upper bound (FU) indicates the compression capacity of a perfect WHSJ with the same thickness T. The lower bound (FL) represents the compression capacity of a WHSJ with a thickness equal to T–Tc. Fu and Fl indicate the lower and upper bounds of the compression capacity of the WHSJ with the specified corrosion number n. The change tendencies of Fu and Fl with n are shown in Fig. 9. The results presented in Fig. 9 imply that the value of FL/FU was nearly not influenced by the value of T. That is, the ratio of the compression capacity of the WHSJ with randomly located corrosion to the compression capacity of the perfect WHSJ was only related to the value of Tc/T and not influenced by the specified value of T. The changing curves of Fu and Fl with n shown in Fig. 9 were extracted and transformed into a reduction factor by dividing FU as shown in Fig. 10. The reduction factor generally decreased with an increase in n. Fluctuation occurred for the reduction factor curves because computational time was insufficient. The reduction factor was nearly only relevant to the value of Tc/T when the value of D was specified. The influence of D on the reduction factor was also investigated. The value of D was set as 0.36, 0.6, and 0.9 m, and the value of T was set as 13 mm. Corrosion occurred uniformly on the entire surface of the WHSJ. The corroded WHSJ was equivalent to a perfect WHSJ with a thickness of T–Tc. The compression capacity and reduction factor that correspond to different values of D are shown in Fig. 11 a. The reduction factor was not influenced by the value of D. The fitting equation between R (reduction factor) and Tc/T was derived through nonlinear

3.2. Determination of the repeated number and probabilistic model Determining computational time in random analysis is an important parameter that can directly influence the probabilistic model of compression capacity. The accuracy of the results will increase with computational time. However, an increase in computational time indicates a considerable rise in computational power or time requirement, particularly when elastoplastic and parametric analyses are necessary. Therefore, the rational determination of repeated computational number (Nr) in random analysis is constantly the first issue that must be resolved. The influence of Nr on the probabilistic distribution model was examined in this subsection through parametric analysis. The value of Nr was set as 20, 30, 40, 50, 100, and 200. The values of Tc/T, D, Dc, T, and fy were set as 0.5, 0.6 m, 80 mm, 13 mm, and 345 MPa, respectively. The probabilistic distribution model was compared with the normal distribution model, as shown in Fig. 7. The compression capacity of the WHSJ with randomly located corrosion completely followed normal distribution. The error in the probabilistic distribution model of compression capacity decreased with an increase in Nr. The normal distribution parameter changed slightly when the value of Nr was higher than 50. Thus, computational time was set as 50 in the latter random analysis. The failure mode of the WHSJ with randomly distributed corrosion is shown in Fig. 8. 3.3. Influences of T and D on compression capacity The influences of the thickness (T) and diameter (D) of the spherical

Fig. 4. Numerical model of a WHSJ with randomly located corrosion (Dc = 0.08 m). 169

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Fig. 6. Comparison of failure modes under axial force.

curve fitting analysis, as illustrated in Fig. 11 b.

3.4. Influence of fy on compression capacity The effect of the yield strength of material fy on the compression capacity of the WHSJ with randomly located corrosion was estimated. The diameter of WHSJ was set as 0.6 m, and its thickness was set as 13 mm. The value of fy was set as 235 MPa and 345 MPa. The compression capacity of the WHSJ with randomly located corrosion that corresponds to different fy values is shown in Fig. 12. The ratio of FL to FU was not influenced by fy. The curves of Fu and Fl that correspond to different fy values were extracted and compared in Fig. 12 c. The curves of Fu and Fl were not influenced by fy. The results in Section 4 indicate that the compression capacity of the WHSJ with randomly located corrosion followed normal distribution. Therefore, the probabilistic distribution model was not influenced by fy.

Fig. 8. Failure mode of a WHSJ with randomly distributed corrosion.

Fig. 7. Influence of number of samples on probabilistic distribution. 170

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Fig. 9. Changing range of compression capacity with different Tc/T values (Dc = 80 mm).

the spherical body (T) exerts nearly no influence on the reduction factor of the WHSJ with randomly located corrosion. The upper and lower bounds of the reduction factor with different values of Tc/T were extracted as shown in Fig. 14. A nonlinear surface fitting technology was adopted to clarify the relationship among the reduction factor, n, and Tc/T. The values of the lower and upper bounds can be represented by n and Tc/T, respectively. The derived equations are shown as Eqs. (1) and (2). The upper and lower bounds of the reduction factor can be accurately computed using the two equations via n and Tc/T. The results derived through FEA and Eqs. (1) and (2) were compared to validate their accuracy, as shown in Fig. 15. Fig. 15 b illustrates that the error between the fitted and FE curves was relatively larger when the value of Tc/T was less than 0.125. The error remained at a low level when the value of Tc/T was higher than 0.125. The results predicted by the fitted curve tended toward a safe condition when the value of Tc/T was 0.875.

Fig. 10. Influence of T on the reduction factor.

3.5. Influence of Tc/T on compression capacity

Tc T + 0.063 × n2 − 0.083 × n × c T T Tc 2 Tc 3 2 0.017 × ( ) − 0.0027 × n + 0.019 × n × T T Tc 2 Tc 3 0.0025 × n × ( ) − 0.0096 × ( ) − 0.033 × n4 T T Tc T T 3 2 + 0.0051 × n × ( c )2 − 0.0024 × n × ( c )3 0.0042 × n × T T T Tc Tc 2 5 4 3 − 0.0027 × n × ( ) 0.013 × n − 0.0014 × n × T T Tc 3 2 0.0038 × n × ( ) (1) T

Ru = 0.83 − 0.15 × n − 0.12 × The effect of Tc/T on compression capacity was investigated in this subsection. The changing range of compression capacity with varying n is shown in Fig. 13. Analysis was conducted 50 times for each value of n. The value of Tc/T was set as 0.125, 0.25, 0.5, and 0.875 to clarify its influence on the changing range of compression capacity. The ratio of Fl to Fu for different ratios of Tc to T is shown in Fig. 13. The results presented in Figs. 13 and 9c imply that the changing range of compression capacity increased with an increase in Tc/T. The value of Fl is equal to the compression capacity of a perfect WHSJ with a thickness of T–Tc. The value of Fl can be reached when the value of n is beyond 70. The results shown in Fig. 13 indicate that the reduction speed of compression capacity with increasing n decreased after the value of n increased to a certain value. The results shown in Figs. 13 and 9c indicate that the thickness of

− − − + +

Fig. 11. Influence of D on compression capacity. 171

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Fig. 12. Influence of fy on compression capacity.

Tc T − 0.013 × n2 − 0.016 × n × c T T Tc 2 Tc 3 2 0.013 × ( ) + 0.027 × n − 0.019 × n × T T Tc 2 Tc 3 0.0052 × n × ( ) − 0.024 × ( ) + 0.037 × n4 T T Tc T T 3 2 − 0.0045 × n × ( c )2 + 0.013 × n × ( c )3 0.033 × n × T T T Tc Tc 2 5 4 3 + 0.0009 × n × ( ) 0.025 × n + 0.025 × n × T T Tc 3 2 0.003 × n × ( ) T

4. Probability distribution model with a specified Dc

Rl = 0.54 − 0.52 × n − 0.025 × + − − − +

Note: The values of n and Tc / T in Eq. (1) were normalized by and

Tc / T − 0.47 , 0.28

normalized by

The probability distribution of compression capacity was investigated in this section. The probabilistic distribution model can be established on the basis of the corrosion condition. The compression capacity of the WHSJ with randomly located corrosion was validated to follow normal distribution. Therefore, the major work required is to propose a method for determining the parameter of normal distribution. The computational time was 50 for each condition in Subsection 5.1. The probability of moment capacity that did not occur was less than 2%. The probability of moment capacity between μ–2.33σ and μ + 2.33σ was higher than 98% in accordance with the normal distribution. Eqs. (3) and (4) can be derived. The standard deviation can be obtained using Eq. (5).

(2)

n − 57.44 49.84

respectively. The values of n and Tc / T in Eq. (2) were n − 75 46.46

and

Tc / T − 0.42 , 0.27

respectively.

Ru = μ + 2.33σ

(3)

Rl = μ − 2.33σ

(4)

Fig. 13. Changing range of compression capacity with different Tc/T values (Dc = 80 mm). 172

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Fig. 14. Changing range of the reduction factor with n.

σ=

Ru − Rl 4.66

R + Rl μ= u 2

5. Influence of Dc on the probability distribution model

(5)

The aforementioned results were based on the specified condition where Dc = 80 mm. The method for determining the probabilistic model that corresponded to different Dc values was investigated. The value of Dc was set as 40, 60, 80, 100, and 120 mm. The value of Tc/T was set as 0.5. The changing range of compression capacity with n is shown in Fig. 17. The value of Dc considerably influenced the reduction factor. The results presented in Subsection 3.1 indicated that the compression capacity of the WHSJ with randomly located corrosion followed normal distribution. Therefore, the variation in normal distribution that corresponded to each value of n was computed to accurately describe the influence of Dc. The mean (μ) and standard deviation (σ) of the reduction factor that corresponded to each value of n were extracted as shown in Fig. 18. The influence of Dc on the mean of the reduction factor is shown in Fig. 18 a. The value of the reduction factor considerably decreased with an increase in Dc. The changing rule of μ with a variation in Dc was apparent. By contrast, the value of σ did not exhibit a clear relationship with a variation in Dc. Consequently, this relationship is difficult to describe using the proposed formulae. The normal distribution model of the reduction factor that corresponds to different combinations of Dc = 80 mm, Tc/T, and n can be determined on the basis of the results shown in Section 4. The normal distribution model of the reduction factor that corresponds to different values of Dc can also be determined if the relationship between the variations derived by different values of Dc and the value derived by Dc is equal to 80 mm. The symbols μR and σR were introduced to clarify the relationship between the μ and σ derived using different values of Dc and the value

(6)

The probabilistic model of compression capacity established through the aforementioned equation was validated by comparing the results of stochastic FEA. FEA was conducted 200 times under each condition, and the expectation (μ) and standard deviation (σ) were extracted and compared with those derived using the formulae presented in this work. The comparison of the results obtained using different methods is shown in Fig. 16. The probabilistic distribution model of the WHSJ with randomly located corrosion can be reflected by the regression formulae proposed in the preceding section. The parameters were changed to validate the applicability of the proposed method. The comparison of the results shown in Fig. 16 a and 16 e indicate that the thickness of the spherical body exerts no influence on the probabilistic model of compression capacity. A relatively larger error occurred between the FE curve (derived by stochastic FEA) and the fitted curve (derived through the formulae) under this condition. This error was caused by the error due to the surface fitting described in Subsection 3.4. The errors of expectation (μ) and standard deviation (σ) were less than 10% and 2%, respectively. The results derived by the fitted curves were safer. The comparison of the results shown in Fig. 16b, 16c, and 16 d indicates that the results derived by the fitted curves completely agree with those obtained via stochastic FEA. The method presented in this work can be fully adopted in the reliability analysis of reticulated shell structures connected by WHSJs with randomly located corrosion.

Fig. 15. Comparison of the results derived by FEA and Eqs. (1) and (2). 173

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Fig. 16. Distribution of the reduction factor.

derived with Dc = 80 mm. μR (σR) indicates the ratio of μ (σR) obtained by a specified Dc and the value obtained with Dc = 80 mm, as shown in Eqs. (7) and (8). The probability parameter that corresponds to different Dc values can be derived using the values of μR and σR. The contours of μR and σR are shown in Fig. 19. The values of μ and σ that correspond to a specified Dc can be derived by multiplying the μR and μ obtained using Eq. (6).

μR =

μ μ(Dc = 80mm)

(7)

σR =

σ σ(Dc = 80mm)

(8)

analysis was repeated 50 times for each value of n. The value of Dc was set as 50 mm and 90 mm, which differed from the Dc adopted for determining the contours of μR and σR. The results imply that the normal distribution models established using different methods are in good agreement. The values of μ obtained using FEA and the fitted curve were 0.84 and 0.81, respectively, for Dc = 50 mm and n = 50. The computational error for μ was 3.7%. The computational errors for μ and σ at Dc = 50 mm and n = 100 were minimal, and thus, could be ignored. The feasibility of the proposed method was validated.

6. Conclusions The influences of geometric parameters, including T, D, Tc/T, and fy, on the random distribution of compression capacity were systematically investigated. A probabilistic distribution model of the compression

The normal distribution models of the reduction factor derived using FEA and the presented fitted method were compared to validate the accuracy of the latter, as shown in Fig. 20. Stochastic numerical

Fig. 17. Influence of Dc on compression capacity. 174

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Fig. 18. Influence of Dc on probabilistic distribution variation (Tc/T = 0.5).

Fig. 19. Contours of μR and σR (Tc/T = 0.5).

when Dc = 80 mm. (4) A method for establishing a probabilistic model of the reduction factor that corresponded to different Dc values was proposed. The developed method exhibited high accuracy in predicting the probabilistic variation that corresponded to different values of Dc.

capacity of a WHSJ with randomly located corrosion was proposed. The equations for the reduction factor and the corrosion parameter were provided. The conclusions drawn can be summarized as follows. (1) The influences of T, D, fy, and Tc/T on the reduction factor were investigated. The reduction factor appeared relevant to the values of Tc/T and n only when the value of Dc was specified. (2) The compression capacity of the WHSJ with randomly located corrosion followed normal distribution. Stochastic FEA should be repeated at least 50 times to accurately predict the variation in normal distribution. (3) A method for determining the probabilistic model of the reduction factor was proposed. The formula for predicting the variation in normal distribution was provided using surface fitting technology. The values of μ and σ can be determined by specifying Tc/T and n

7. Future work The work reported in this article is conducted based on the premise that the diameter and depth of corrosion are the same, which disagrees with the actual condition. Tc and Dc should also be considered random variables.

Fig. 20. Comparison of the probabilistic model established using different methods. 175

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Acknowledgments [12]

The work described in this paper was financially supported by the project funded by the China Postdoctoral Science Foundation (No. 2017M621156) and the State Key Research Development Program of China (Grant Nos. 2016YFC0801404 and 2016YFC0600704).

[13] [14]

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