- Email: [email protected]
Bending capacity of corroded welded hollow spherical joints
Thin-Walled Structures 127 (2018) 523–539
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Full length article
Bending capacity of corroded welded hollow spherical joints ⁎
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 joint Corrosion Bending stiffness Corroded pattern Corroded thickness
Welded hollow spherical joints (WHSJ) have been widely used in space lattice structures. Corrosion is inevitable in the service life of WHSJ, which significantly reduces their bending capacity and seriously threatens their structural safety. This study aims to investigate the influence of corrosion that occurred at different places on the bending stiffness and bending capacity of WHSJ. The effects of corroded location and size, corroded thickness, and diameter and thickness of spherical body were investigated through a series of nonlinear numerical analyses. Two types of corroded shape were investigated. The equation used for predicting the residual bending stiffness of corroded WHSJ was established. Results indicated that corrosions in different places had distinct influence on the bending capacity of WHSJ. Loading capacity could be significantly reduced when corrosion occurred at other areas. The derived results would provide a foundation for estimating the residual bending stiffness and strength of structures connected by corroded WHSJ.
1. Introduction Welded hollow spherical joints (WHSJ) have been widely used in space lattice structures. In 1965, WSHJ were developed and first applied on the project of Science and Technology Hall in Tianjin [1,2]. Several researchers have studied their mechanical behavior and their influence on integral structures [3–5]. Han [6,7] studied the ultimate bearing capacity of WHSJ, and Wang [8] investigated the axial flexibility and flexural stiffness of WHSJ by finite element (FE) approach. Gu [9] studied the influence of WHSJ on single-layer latticed domes by using a refined FE analysis (FEA). Zhao [10] investigated the influence of welding residual stress on the ultimate loading capacity of WHSJ. Liu [11] examined the post-fire behavior of WHSJ through experimental work. Corrosion is a type of fatal injury for steel structures in their service phase. However, the abovementioned studies did not consider the influence of corrosion, which reduces the loading capacity of WHSJ. Corrosion is inevitable for the service phase of steel structures, especially for swimming pools and marine structures. Corrosion that occurred at the roof structure of a swimming pool in Tianjin University is shown in Fig. 1. The corrosion constantly initially occurred at the weld of steel pipe and spherical body and then expanded to other locations. Corrosion that occurred at WHSJ would significantly reduce their loading capacity and seriously threaten their structural safety. Numerous researchers have investigated the influence of corrosion on steel plates [12–14]. Ok et al. [15] conducted over 256 nonlinear FEAs on panels with various locations and sizes of pitting corrosion and ⁎
adopted the multi-variable regression method to derive new formulas in predicting the ultimate strength of unstiffened plates with localized corrosion. Nakai et al. [16] conducted a series of tests to investigate the effect of pitting corrosion on the strength of web plates subjected to patch loading. Huang et al. [17] developed an assessment formula for predicting the ultimate strength of hull plates with pitting corrosion damages under biaxial in-plane compression loading. Sultana et al. [18] utilized FEA to investigate the effect of random corrosion on the compressive strength capacity of marine structural units. Saad-Eldeen [19–21] performed a series of investigations on the influence of corrosion on box girders. The thickness of structural elements is uniformly reduced in general (uniform) corrosion. Then, the bending stiffness and ultimate strength of structural elements are influenced. Several existing studies have indicated that bending stiffness has a significant influence on the mechanical behavior of structures connected by semi-rigid connections [22,23]. Thus, the influence of corrosion on bending stiffness of WHSJ should be estimated in detail. Saad-Eldeen et al. [24] investigated the effect of corrosion degradation on the ultimate strength of corroded steel box girders and observed a significant reduction in stiffness. Ultimate strength calculations are typically conducted by excluding thickness loss, which will result in the decrease of the slenderness of plate and column. A considerable amount of research on loading capacity has been conducted by using empirical formulas, IACS rules, and FE methods [25]. The present study aims to investigated the influence of corrosion on the bending capacity of WHSJ. Effects of corroded location and size,
Corresponding author. E-mail address: [email protected] (Z. Zhao).
https://doi.org/10.1016/j.tws.2018.03.007 Received 17 December 2017; Received in revised form 28 January 2018; Accepted 6 March 2018 0263-8231/ © 2018 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 1. Corrosion of WHSJ.
These conclusions can be used as basis on estimating the residual loading capacity of overall structures. Relevant research on this aspect has yet to be conducted by researchers. The present work will thus provide an important reference for engineers in the maintenance of existing structures. Bending stiffness has a significant influence on the mechanical behavior of reticulated shell structures. Thus, the proposed calculating methods and formulas on estimating the residual stiffness of WHSJ will provide a foundation for estimating the mechanical behavior of reticulated shell structures connected by corroded WHSJ. 2. Establishment of FE model A refined numerical model was established based on ANSYS code to investigate the influence of corrosion on the loading capacity of WHSJ. Element SHELL181 was adopted, and the corrosion was simulated by reducing the element thickness [29]. Fig. 2 shows the FE model of the spherical body of WHSJ constituted by shell elements, which is shear deformable and has four nodes with five independent degrees of freedom per node (three for translation and two for flexural rotation). The material of steel was Q345 whose yield strength, elastic modulus, Poisson ratio, and density were 345 MPa, 210 GPa, 0.3, and 7800 kg/m3, respectively. The ideal elastoplastic model was adopted. Corroded zones of WHSJ may exhibit different patterns under practical conditions. The distribution patterns of corroded zones were categorized into three types, namely, latitudinal pattern (Pattern I), longitudinal pattern (Pattern II), and pit corrosion. Detailed information on the different patterns is shown in Fig. 3. Symbol Tc and Hc indicate the corroded thickness and the height of corroded zone,
Fig. 2. Numerical model of corroded WHSJ.
corroded thickness, and diameter and thickness of spherical body were investigated through a series of nonlinear numerical analyses. FE method was adopted, and thickness loss was considered by specifying the section of shell elements. The derived results would provide a foundation for estimating the residual strength of structures connected by corroded WHSJ. The shape of corrosion was idealized. However, the derived results revealed the influence of corrosion on the bending capacity and stiffness of WHSJ. WHSJ are the key component for reticulated shell structures. Thus, the detailed location of corrosion on WHSJ directly influences the safety of reticulated shell structures. The derived conclusions provide information on the influence of corrosion on WHSJ.
Fig. 3. Distribution patterns of corroded zone.
524
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 4. Schematic of pit corrosion.
Fig. 5. Comparisons of force-displacement curves derived by FEA and experiment.
Fig. 7. Influence of element size.
respectively. Symbol α and Dc indicate the central angle of Pattern II and the diameter of Pattern III, respectively. In addition, pit corrosion is represented by the combination of Patterns I and II, as shown in Fig. 4. The thickness element located in the corroded zone was adjusted to model the corrosion influence.
minimum and maximum multipliers for the arc-length radius were set. The axial force was applied, and the force-displacement curves were derived. The comparisons of force-displacement curves derived by FEA and experiment are shown in Fig. 5. The results derived by FEA agreed well with that derived by the experiment. Comparison of the failure mode is shown in Fig. 6. The figure shows that the FEA model can accurately predict the failure mode of WHSJ caused by axial force.
2.1. Validation of FE model The experimental results derived by Liu [26] were adopted to validate the reliability of the established FE model in the present work. The external diameter and thickness of 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 based on the stress–strain curves derived by Liu [26], and the multilinear isotropic hardening model was adopted in ANSYS. The element size was set as 10 mm. The arc-length method was activated by “ARCLEN” command, and the
2.2. Effect of corroded location for Pattern I Corrosion was assumed to occur at different locations of WHSJ in investigating the influence of corroded location on the bending capacity of WHSJ. The zones were named as Zones I, II, and III. Zone I was located at the middle part of the spherical body, and Zone III was located at the conjunction of steel pipe and the spherical body, as shown
Fig. 6. Comparison of failure mode under axial force.
525
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 8. Load–displacement curves (Hc = 40 mm).
Fig. 9. Load–displacement curves (Hc = 80 mm).
Fig. 10. Load–displacement curves (Hc = 120 mm).
in Fig. 8. Zone II was found between Zones I and III. The distribution of corroded shape was assumed to be consistent in Pattern I in this subsection. The thicknesses of steel pipe and spherical body were set as 16 mm. The spherical body diameter was 0.56 m. Symbol Hc represents the height of corroded zone, and was set to 40, 80, and 120 mm in investigating its influence on loading capacity. Different mesh sizes have evident influences on the reliability of FEA [26–28]. Thus, the sensitivity analysis on mesh size was initially performed. Corrosion was assumed to occur at Zone I. The values of Hc and Tc were 40 mm and 15 mm, respectively. The element size was set as 5, 10, and 20 mm. The moment–rotation curves are shown in Fig. 7. The figure shows that the element size had nearly no influence on the bending capacity and bending rigidity of WHSJ. The post-buckling behavior was slightly influenced; however, this phenomenon was not the focus of the present work. Thus, the element size was set to 10 mm for future analysis.
Fig. 11. Changing tendency of reduction factor along with Tc.
526
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 12. Failure mode with different values of Tc (Hc = 4 0 mm, Zone I).
Fig. 13. Failure mode with different corrosion locations (Pattern I).
Fig. 14. Influence of corrosion height on loading capacity (Zone I).
capacity was not influenced when corrosion thickness was less than 14 mm, that is, 87.5% of the spherical body thickness. The results shown in Fig. 12 indicated that the failure mode changed when Tc increased from 14 mm to 15 mm. The failure was caused by the yield that occurred at the conjunction of steel pipe and spherical body when Tc = 14 mm. By contrast, the failure was caused by the buckling that occurred at the corrosion zone when Tc = 15 mm. The results also indicated that the bending capacity was nearly not influenced by Hc when
The influence of corroded locations was also investigated, and the results are shown in Figs. 8–10. From the results, the bending capacity of WHSJ was the most sensitive to corrosion that occurred at Zone III. The bending capacity decreased immediately after corrosion occurred in this zone, and the results indicated that the bending capacity was significantly influenced by Tc. The bending capacity of WHSJ was the most insensitive to corrosion that occurred at Zone I. The bending capacity was influenced when Tc reached 15 mm and the bending
527
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 15. Influence of corrosion height on loading capacity (Zone II).
height Hc. The influence of corrosion on bending capacity was also not influenced by Hc. This finding was different for the corrosion that occurred at Zone II, and the influence of corrosion on bending capacity was closely related to Hc. (Fig. 12). Failure modes corresponding to different corrosion locations are shown in Fig. 13. The buckling occurred at Zones I and II when corrosion thickness reached a certain value, and the bending capacity suddenly reduced by a large degree. This result was caused by failure mode transformation. The buckling still occurred at Zone III although the corrosion thickness at Zones I and II did not reach a certain degree. Thus, the bending capacity appeared not to be influenced by Tc. Parametrical analysis was conducted to investigate the influence of Hc systematically at Zones I and II. The results are shown in Figs. 14 and 15. Tc was set as 10, 13, and 15 m. Corrosion height Hc was set as 40, 80, 120, 160, 200, and 240 mm. The results in Fig. 14(a) and (b) show that the bending capacity was not influenced by corrosion although Hc reached 40% of the spherical body diameter when Tc reached 54% of the spherical body thickness. Buckling occurred at Zone I when Tc reached 93% of the spherical body thickness, and the reduction factor slightly decreased with the increase of Hc, as shown in Fig. 16. For the corrosion that occurred at Zone II, the change of reduction factor with Hc became slow with the increase of Tc.
Fig. 16. Changing tendency of reduction factor along with Hc.
2.3. Effect of corroded location for Pattern II The shape of corroded zone was assumed as Pattern II in this subsection. The size of corroded zone was represented by α (Fig. 3), and the value of α was set as 30°, 60°, and 90°. The influence of bending moment direction on corroded WHSJ was also investigated. The included angle between the bending moment and center line of corroded zone (Fig. 17) was investigated through parametrical analysis. The included angle β was changed from 90° to 270° with a step of 30° considering symmetrical characteristic. Corroded thickness was set as 4, 10, and 15 mm. Force-displacement curves corresponding to different corroded thicknesses are shown in Fig. 18. α and Tc had large influence on the loading capacity of WHSJ. Buckling occurred when Tc reached 15 mm. This finding was observed from the load–displacement curves when Tc was equal to 15 mm. The loading capacity and initial stiffness of WHSJ did not influence the value of β when Tc was less than 10 mm, and they were significantly influenced by β when Tc was 15 mm. The loading capacity changed from 75 kN to 100 kN when α = 30° and Tc= 15 mm. The changing amplitude reached 30% of the loading capacity. Thus, the relative location between the bending moment direction and corroded zone also
Fig. 17. Schematic of bending moment direction and corrosion zone.
corrosion occurred at Zone I. The bending capacity started to be influenced when Tc at Zone II increased to 14 mm. The parameter “reduction factor,” that is, the ratio of residual bending capacity of corroded WHSJ and bending capacity of perfect WHSJ, was introduced to quantify the influence of corrosion thickness. The results are shown in Fig. 11. From the results in Zone III, the reduction factor nearly decreased linearly with the increase of corrosion thickness, and the reduction factor was not influenced by corrosion
528
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 18. Load–displacement curves (D = 0.56 m).
Fig. 19. Failure mode with different values of α (Pattern II, β = 90°).
529
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 20. Influence of the diameter of WHSJ on bending capacity.
finding was reflected by the snap-through of load–displacement curves when Tc = 15 mm.
had a significant influence on loading capacity. In addition, the changing amplitude decreased with the increase of α. The failure modes of WHSJ that correspond to different conditions are shown in Fig. 19. Corrosion was located in the compression zone when β = 90° and was located in the tension zone when β = 270°. Buckling occurred at the corrosion location when Tc reached a certain degree (about 93% of total thickness). Buckling occurred when Tc increased to 15 mm regardless of whether β = 90° or β = 270°. The failure was caused by the yield that occurred at the conjunction of steel pipe and spherical body when Tc was less than or equal to 14 mm. This
3. Effects of diameter and thickness 3.1. Effects of diameter and thickness for Pattern I The influence of the diameter of WHSJ on bending capacity that corresponds to different values of Tc was investigated for Pattern I in this subsection. The thicknesses of the spherical body and steel pipe
530
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 21. Influence of the thickness of WHSJ on loading capacity.
indicated the ratio of the initial bending stiffness of corroded WHSJ. The results indicated that the spherical body diameter nearly had no influence on the initial bending stiffness. The suggested ratio of diameter to thickness of WHSJ for truss structures was between 25 and 45 based on the Technical Specification for Space Frame Structures (JGJ7-2010) [30]. Then, the suggested thickness was between 10 mm and 19 mm when the diameter of WHSJ was 0.48 m. The influence of spherical body thickness on bending capacity was investigated. The results derived with different spherical body thicknesses are shown in Fig. 21. The ultimate strength of
were set as 8 mm, and the spherical body diameter was set as 0.48, 0.60, and 0.9 0 m. Corroded thickness was set as 1, 2, 3, 4, 5 and 6 mm. The results derived in different conditions are shown in Fig. 20. From the results, the bending capacity of WHSJ decreased nearly linearly with the increase of corroded thickness for different diameters. This finding is obviously observed in Fig. 20(d). The initial stiffness in the loading direction decreased with the increase of corroded thickness. The results shown in Fig. 20(d) indicated that reduction factor was not influenced by diameter. The influence of corrosion on initial bending stiffness is shown in Fig. 20(e). The initial bending stiffness factor (Iini)
531
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 22. Influence of WHSJ diameter on loading capacity.
3.3. Effect of corrosion on initial bending stiffness
corroded WHSJ decreased linearly with the increase of Tc. The results shown in Fig. 21(d) indicated that the spherical body thickness had influence on reduction factor. The influence of spherical body thickness on initial bending stiffness is shown in Fig. 21(e). The initial bending stiffness factor was only related to Tc/T and was nearly not influenced by the spherical body thickness.
3.3.1. Pattern I The results derived in the above section indicated that the initial stiffness of corroded WHSJ was not influenced by the thickness and diameter of the spherical body. The bending capacity was significantly influenced by the corrosion that occurred at Zone III. Thus, the analysis conducted in this section was based on the corrosion that occurred at Zone III. The initial bending stiffness was mainly influenced by Hc and Tc. Thus, the changing tendency of initial bending stiffness factor along with Hc and Tc was investigated in this section. The diameter and thickness of the spherical body was set with different values to investigate their influence on initial bending stiffness factor. The initial bending stiffness factor (Iini) indicated the ratio of the initial bending stiffness of corroded WHSJ. The contour of initial bending stiffness factor is shown in Fig. 24. The initial bending stiffness factor was mainly influenced by Tc, and Hc slightly influenced Iini. Iini did not change with the change of Hc when Hc was larger than 20 mm (approximately 3.3% of the spherical body diameter). The contour of Iini was nearly not influenced by the thickness and diameter of the spherical body. Iini was mainly influenced by Tc, and the results shown in Fig. 24(a) were used for curve fitting analysis. Then, Iini was represented by Tc through Eq. (1). The T accuracy of the proposed equation was then validated by the results on Joints I, II, and III. The results are shown in Fig. 25. The results derived by FEA was absolutely in agreement with those derived by the proposed equation.
3.2. Effect of diameter and thickness for Pattern II The influence of the diameter of WHSJ on loading capacity corresponding to different values of Tc was investigated for Pattern II in this subsection. The thicknesses of spherical body and steel pipe were set as 14 mm, and the spherical body diameter was set as 0.48, 0.60, and 0.80 m. The corroded thickness was set as 2, 4, 6, 8, 10, and 12 mm. The value of α was set as 90°. The results derived under different conditions are shown in Fig. 22. From the results, the bending capacity of WHSJ decreased rapidly with the increase of corroded thickness, which was different from Pattern I. The changing tendency of reduction factor with diameter is shown in Fig. 22(d). The reduction factor decreased rapidly with the increase of Tc, and the spherical body diameter nearly had no influence on reduction factor. The influence of spherical body thickness on ultimate strength in the corroded zone of Pattern II was also investigated. The diameter of WHSJ was set as 0.60 m and was kept unchanged. The spherical body thickness was set as 8, 12, and 16 mm. The results are shown in Fig. 23. The spherical body thickness had no influence on the reduction factor.
532
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 23. Influence of the diameter of WHSJ on loading capacity.
Iini = 1.02 − 0.26 ×
Tc T 2 − 0.768 × ⎛ c ⎞ T ⎝T ⎠
remained within 2%. The accuracy of results for different sizes of WHSJ indicated that the proposed method can be efficiently adopted for estimating the residual stiffness of WHSJ.
(1)
α 2 360
( ) + 0.0812 × ( ) − 0.4198 × ( ) − 1.415 × ( ) × ( ) + 0.219 × ( ) + 0.1992 ( ) + 0.476 × ( ) × ( ) − 0.24 ( ) × ( ) − 0.3137 ( )
3.3.2. Pattern II The changing tendency of the initial bending stiffness factor along with α and Tc was investigated. The diameter and thickness of the spherical body was set with different values to investigate their influence on the initial bending stiffness factor, as shown in Fig. 26. The initial bending stiffness factor (Iini) indicated the ratio of the initial bending stiffness of corroded WHSJ. From the results, the initial bending stiffness was significantly influenced by α and Tc. The residual initial bending stiffness of WHSJ was determined by α and Tc. The contour of Iini was nearly not influenced by the spherical body diameter. The contour of Iini appeared to be only related to α and Tc. The value of Iini was predicted by α and Tc. Thus, the curve fitting method was adopted to represent Iini by α and Tc, as shown in Eq. (1). Parameters α and Tc were independent from each other. T 360 Only the data shown in Fig. 26(a) were utilized in the curve fitting analysis. The contour of Iini derived by the proposed equation is shown in Fig. 27. The results derived by the proposed equation agreed well with that derived by FEA. Then, error analysis was conducted to estimate the accuracy of the proposed method. The results are shown in Fig. 28. Generally, the error
Iini = 0.9651 − 0.2536 × Tc T
α 2 360
α 360
Tc T
Tc 2 T
α 360
Tc T
α 360
α 3 360
Tc 2 T
Tc 3 T
(2)
4. Influence of corrosion on buckling capacity of integral structure The multi-scaled refined numerical model for K6-type reticulated shell structures was established, as shown in Fig. 29, and the influence of corrosion on the buckling capacity of joints located at different places was investigated. The thicknesses of spherical joint and steel pipe were set as 13 mm and 8 mm, respectively. Additional detailed information can be found in [10]. The corrosion type was assumed to be in Pattern I, and corrosion occurred at the conjunction of spherical body and steel pipe. The corroded thickness was set as 0, 3, 5, 8, 10, and 12 mm. The corrosion was assumed to only occur on the studied joint. Vertical load was also
533
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 24. Change of initial bending stiffness factor along with Hc and Tc (Pattern I).
534
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
applied at Joint I when its mechanical behavior was studied. Four joints located at typical locations were investigated. Load–displacement curves of each joint are shown in Fig. 30. The corrosion had no influence on the initial stiffness of integral structure. However, the buckling capacity was significantly reduced. Buckling capacity was reduced by 40% when Tc reached 12 mm, that is, 85.7% of the spherical body thickness. The reduction factor of buckling capacity is shown in Fig. 30. The reduction factor of different joints was nearly the same. 5. Conclusions Bending capacity and bending stiffness of corroded WHSJ were investigated in this study. The shape of corroded zone was divided into two types, namely, latitudinal (Pattern I) and longitudinal (Pattern II)
Fig. 25. Validation of the proposed method.
Fig. 26. Change of bending stiffness factor along with α and Tc (Pattern II).
535
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 27. Contour of Iini derived by the proposed equation (Pattern II).
Fig. 28. Tolerance analysis (Pattern II).
536
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 29. Numerical model of integral structure.
(4) The equations for estimating the residual bending stiffness (Patterns I and II) were proposed, and their accuracy were validated. The initial bending stiffness of corroded WHSJ was accurately predicted by the proposed method. (5) The influence of corrosion on the buckling capacity of integral structure was investigated based on a refined numerical model. The results indicated that the corrosion had no influence on the initial stiffness of integral structure. Buckling capacity was reduced by 40% when Tc reached 85.7% of the spherical body thickness.
patterns. Other corroded shapes could be represented by combining the two types. The effects of corroded location and the diameter and thickness of spherical body on bending capacity and bending stiffness were studied. The conclusions can be summarized as follows: (1) The corrosion in different places caused distinct influence on the bending capacity of WHSJ. Ultimate strength was nearly not influenced when Tc in Zone I reached 87.5% of the spherical body thickness. (2) The reduction factor of bending capacity caused by corrosion in Zone III nearly decreased linearly with the increase of corrosion thickness, and the reduction factor was not influenced by corrosion height Hc. (3) The influence of corrosion on bending stiffness was not influenced by the spherical body diameter, and the initial bending stiffness factor was only related to Tc/T and was nearly not influenced by the spherical body thickness.
Acknowledgments This work was financially supported by the State Key Research Development Program of China (Grant Nos. 2016YFC0801404 and 2016YFC0600704) and the Project Funded by China Postdoctoral Science Foundation (No. 2017M621156).
537
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 30. Load–displacement curves corresponding to different joints.
[9] Lei Gu, Maoqiang Ding, Xueyi Fu, Xianchuan Chen, A refined finite element analysis on single layer latticed domes with welded hollow spherical joints, J. Build. Struct. 32 (8) (2011) 42–50. [10] Zhong-wei Zhao, Han Zhu, Zhi-hua Chen, Mechanical behavior of single-layer reticulated shell connected by welded hollow spherical joints with considering welding residual stress, Weld. World 60 (1) (2016) 61–69. [11] Hongbo Liu, Jie Lu, Zhihua Chen, Residual behavior of welded hollow spherical joints after exposure to elevated temperatures, J. Constr. Steel. Res. 137 (2017) 102–118. [12] M. Witkowska, C. Guedes Soares, Ultimate strength of locally damaged panels, Thin-Wall Struct. 97 (2015) 225–240. [13] X.H. Shi, J. Zhang, C. Guedes Soares, Experimental study on collapse of cracked stiffened plate with initial imperfections under compression, Thin-Wall Struct. 114 (2017) 39–51. [14] Khedmati MR, Nouri ZHME, Analytical simulation of nonlinear elastic–plastic average stress–average strain relationships for un-corroded/both-sides randomly corroded steel plates under uniaxial compression, Thin-Wall Struct. 86 (2015) 132–141. [15] D. Ok, Y. Pu, A. Incecik, Computation of ultimate strength of locally corroded unstiffened plates under uniaxial compression, Mar. Struct. 20 (2007) 100–114.
References [1] X.L. Liu, Design and Construction of Plate Grid Structures, Press of Tianjin University, Tianjin, China, 2000. [2] Z.S. Makowski, Analysis, Design and Construction of Double-Layer Grids, Applied Science Publishers Ltd, London, 1981. [3] Y. Ding, L. Qi, Z.X. Li, Mechanical calculation model for welded hollow spherical joint in spatial latticed structures, Adv. Steel Constr. 7 (4) (2011) 330–343. [4] X. Li, Load-carrying capacity and practical design method of welded hollow spherical joints in space latticed structures, Adv. Steel Constr. 6 (4) (2010) 976–1000. [5] Qinghua HAN, Quanzhi ZHOU, Yue CHEN, Baolin YU, Ultimate bearing capacity of hollow spherical joints welded with circular pipes under eccentric loads, Trans. Tianjin Univ. 13 (1) (2007) 28–34. [6] Han Qinghua, Liu Xiliang, Ultimate bearing capacity of the welded hollow spherical joints in spatial reticulated structures, Eng. Struct. 26 (1) (2004) 73–82. [7] Han Qinghua, Liu Yiming, Xu Ying, Stiffness characteristics of joints and influence on the stability of single-layer latticed domes, Thin-Walled Struct. 107 (2016) 514–525. [8] Xing Wang, Shi-lin Dong, Hai-ying Wan, Finite element analysis of welded spherical joints’ stiffness, J. Zhejiang Univ. 34 (1) (2000) 77–82.
538
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
[16] T. Nakai, H. Matsushita, N. Yamamoto, Effect of pitting corrosion on strength of web plates subjected to patch loading, Thin-Wall Struct. 44 (2006) 10–19. [17] Y. Huang, Y. Zhang, G. Liu, Q. Zhang, Ultimate strength assessment of hull structural plate with pitting corrosion damnification under biaxial compression, Ocean Eng. 37 (2010) 1503–1512. [18] S. Sultana, Y. Wang, A.J. Sobey, J.A. Wharton, R.A. Shenoi, Influence of corrosion on the ultimate compressive strength of steel plates and stiffened panels, Thin-Wall Struct. 96 (2015) 95–104. [19] S. Saad-Eldeen, Y. Garbatov, C. Guedes Soares, Strength assessment of a severely corroded box girder subjected to bending moment, J. Constr. Steel Res. 92 (2014) 90–102. [20] S. Saad-Eldeen, Y. Garbatov, C. Guedes Soares, Effect of corrosion severity on the ultimate strength of a steel box girder, Eng. Struct. 49 (2013) 560–571. [21] S. Saad-Eldeen, Y. Garbatov, C. Guedes Soares, Analysis of plate deflections during ultimate strength experiments of corroded box girders, Thin-Wall Struct. 54 (2012) 164–176. [22] Zhongwei Zhao, Haiqing Liu, Bing Liang, Novel numerical method for the analysis of semi-rigid jointed lattice shell structures considering plasticity, Adv. Eng. Softw. 114 (2017) 208–214. [23] Zhongwei Zhao, Zhihua Chen, Xiangyu Yan, Xu Hao, Bingzhen Zhao, Simplified
[24]
[25]
[26]
[27]
[28]
[29] [30]
539
numerical method for latticed shells that considers member geometric imperfection and semi-rigid joints, Adv. Struct. Eng. 19 (4) (2016) 689–702. S. Saad-Eldeen, Y. Garbatov, C. Guedes Soares, Effect of corrosion degradation on the ultimate strength of steel box girders, Corros. Eng. Sci. Technol. 47 (2012) 272–283. D.K. Kim, S.J. Kim, H.B. Kim, X.M. Zhang, C.G. Li, J.K. Paik, Ultimate strength performance of bulk carriers with various corrosion additions, Ships Offshore Struct. 10 (2015) 59–78. Hongbo Liu, Jie Lu, Zhihua Chen, Residual behavior of welded hollow spherical joints after exposure to elevated temperatures, J. Constr. Steel Res. 137 (2017) 102–118. B. Faggiano, A. Marzo, A. Formisano, et al., Review: innovative steel connections for the retrofit of timber floors in ancient buildings: a numerical investigation, Comput. Struct. 87 (1–2) (2009) 1–13. A. Formisano, G. De Matteis, F.M. Mazzolani, Experimental and numerical researches on aluminium alloy systems for structural applications in civil engineering fields, Key Eng. Mater. 710 (2016) 256–261. ANSYS user’s manual. SAS IP inc, 1998. Technical specification for space frame structures (JGJ7-2010), China, 2010.
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Full length article
Bending capacity of corroded welded hollow spherical joints ⁎
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 joint Corrosion Bending stiffness Corroded pattern Corroded thickness
Welded hollow spherical joints (WHSJ) have been widely used in space lattice structures. Corrosion is inevitable in the service life of WHSJ, which significantly reduces their bending capacity and seriously threatens their structural safety. This study aims to investigate the influence of corrosion that occurred at different places on the bending stiffness and bending capacity of WHSJ. The effects of corroded location and size, corroded thickness, and diameter and thickness of spherical body were investigated through a series of nonlinear numerical analyses. Two types of corroded shape were investigated. The equation used for predicting the residual bending stiffness of corroded WHSJ was established. Results indicated that corrosions in different places had distinct influence on the bending capacity of WHSJ. Loading capacity could be significantly reduced when corrosion occurred at other areas. The derived results would provide a foundation for estimating the residual bending stiffness and strength of structures connected by corroded WHSJ.
1. Introduction Welded hollow spherical joints (WHSJ) have been widely used in space lattice structures. In 1965, WSHJ were developed and first applied on the project of Science and Technology Hall in Tianjin [1,2]. Several researchers have studied their mechanical behavior and their influence on integral structures [3–5]. Han [6,7] studied the ultimate bearing capacity of WHSJ, and Wang [8] investigated the axial flexibility and flexural stiffness of WHSJ by finite element (FE) approach. Gu [9] studied the influence of WHSJ on single-layer latticed domes by using a refined FE analysis (FEA). Zhao [10] investigated the influence of welding residual stress on the ultimate loading capacity of WHSJ. Liu [11] examined the post-fire behavior of WHSJ through experimental work. Corrosion is a type of fatal injury for steel structures in their service phase. However, the abovementioned studies did not consider the influence of corrosion, which reduces the loading capacity of WHSJ. Corrosion is inevitable for the service phase of steel structures, especially for swimming pools and marine structures. Corrosion that occurred at the roof structure of a swimming pool in Tianjin University is shown in Fig. 1. The corrosion constantly initially occurred at the weld of steel pipe and spherical body and then expanded to other locations. Corrosion that occurred at WHSJ would significantly reduce their loading capacity and seriously threaten their structural safety. Numerous researchers have investigated the influence of corrosion on steel plates [12–14]. Ok et al. [15] conducted over 256 nonlinear FEAs on panels with various locations and sizes of pitting corrosion and ⁎
adopted the multi-variable regression method to derive new formulas in predicting the ultimate strength of unstiffened plates with localized corrosion. Nakai et al. [16] conducted a series of tests to investigate the effect of pitting corrosion on the strength of web plates subjected to patch loading. Huang et al. [17] developed an assessment formula for predicting the ultimate strength of hull plates with pitting corrosion damages under biaxial in-plane compression loading. Sultana et al. [18] utilized FEA to investigate the effect of random corrosion on the compressive strength capacity of marine structural units. Saad-Eldeen [19–21] performed a series of investigations on the influence of corrosion on box girders. The thickness of structural elements is uniformly reduced in general (uniform) corrosion. Then, the bending stiffness and ultimate strength of structural elements are influenced. Several existing studies have indicated that bending stiffness has a significant influence on the mechanical behavior of structures connected by semi-rigid connections [22,23]. Thus, the influence of corrosion on bending stiffness of WHSJ should be estimated in detail. Saad-Eldeen et al. [24] investigated the effect of corrosion degradation on the ultimate strength of corroded steel box girders and observed a significant reduction in stiffness. Ultimate strength calculations are typically conducted by excluding thickness loss, which will result in the decrease of the slenderness of plate and column. A considerable amount of research on loading capacity has been conducted by using empirical formulas, IACS rules, and FE methods [25]. The present study aims to investigated the influence of corrosion on the bending capacity of WHSJ. Effects of corroded location and size,
Corresponding author. E-mail address: [email protected] (Z. Zhao).
https://doi.org/10.1016/j.tws.2018.03.007 Received 17 December 2017; Received in revised form 28 January 2018; Accepted 6 March 2018 0263-8231/ © 2018 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 1. Corrosion of WHSJ.
These conclusions can be used as basis on estimating the residual loading capacity of overall structures. Relevant research on this aspect has yet to be conducted by researchers. The present work will thus provide an important reference for engineers in the maintenance of existing structures. Bending stiffness has a significant influence on the mechanical behavior of reticulated shell structures. Thus, the proposed calculating methods and formulas on estimating the residual stiffness of WHSJ will provide a foundation for estimating the mechanical behavior of reticulated shell structures connected by corroded WHSJ. 2. Establishment of FE model A refined numerical model was established based on ANSYS code to investigate the influence of corrosion on the loading capacity of WHSJ. Element SHELL181 was adopted, and the corrosion was simulated by reducing the element thickness [29]. Fig. 2 shows the FE model of the spherical body of WHSJ constituted by shell elements, which is shear deformable and has four nodes with five independent degrees of freedom per node (three for translation and two for flexural rotation). The material of steel was Q345 whose yield strength, elastic modulus, Poisson ratio, and density were 345 MPa, 210 GPa, 0.3, and 7800 kg/m3, respectively. The ideal elastoplastic model was adopted. Corroded zones of WHSJ may exhibit different patterns under practical conditions. The distribution patterns of corroded zones were categorized into three types, namely, latitudinal pattern (Pattern I), longitudinal pattern (Pattern II), and pit corrosion. Detailed information on the different patterns is shown in Fig. 3. Symbol Tc and Hc indicate the corroded thickness and the height of corroded zone,
Fig. 2. Numerical model of corroded WHSJ.
corroded thickness, and diameter and thickness of spherical body were investigated through a series of nonlinear numerical analyses. FE method was adopted, and thickness loss was considered by specifying the section of shell elements. The derived results would provide a foundation for estimating the residual strength of structures connected by corroded WHSJ. The shape of corrosion was idealized. However, the derived results revealed the influence of corrosion on the bending capacity and stiffness of WHSJ. WHSJ are the key component for reticulated shell structures. Thus, the detailed location of corrosion on WHSJ directly influences the safety of reticulated shell structures. The derived conclusions provide information on the influence of corrosion on WHSJ.
Fig. 3. Distribution patterns of corroded zone.
524
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 4. Schematic of pit corrosion.
Fig. 5. Comparisons of force-displacement curves derived by FEA and experiment.
Fig. 7. Influence of element size.
respectively. Symbol α and Dc indicate the central angle of Pattern II and the diameter of Pattern III, respectively. In addition, pit corrosion is represented by the combination of Patterns I and II, as shown in Fig. 4. The thickness element located in the corroded zone was adjusted to model the corrosion influence.
minimum and maximum multipliers for the arc-length radius were set. The axial force was applied, and the force-displacement curves were derived. The comparisons of force-displacement curves derived by FEA and experiment are shown in Fig. 5. The results derived by FEA agreed well with that derived by the experiment. Comparison of the failure mode is shown in Fig. 6. The figure shows that the FEA model can accurately predict the failure mode of WHSJ caused by axial force.
2.1. Validation of FE model The experimental results derived by Liu [26] were adopted to validate the reliability of the established FE model in the present work. The external diameter and thickness of 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 based on the stress–strain curves derived by Liu [26], and the multilinear isotropic hardening model was adopted in ANSYS. The element size was set as 10 mm. The arc-length method was activated by “ARCLEN” command, and the
2.2. Effect of corroded location for Pattern I Corrosion was assumed to occur at different locations of WHSJ in investigating the influence of corroded location on the bending capacity of WHSJ. The zones were named as Zones I, II, and III. Zone I was located at the middle part of the spherical body, and Zone III was located at the conjunction of steel pipe and the spherical body, as shown
Fig. 6. Comparison of failure mode under axial force.
525
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 8. Load–displacement curves (Hc = 40 mm).
Fig. 9. Load–displacement curves (Hc = 80 mm).
Fig. 10. Load–displacement curves (Hc = 120 mm).
in Fig. 8. Zone II was found between Zones I and III. The distribution of corroded shape was assumed to be consistent in Pattern I in this subsection. The thicknesses of steel pipe and spherical body were set as 16 mm. The spherical body diameter was 0.56 m. Symbol Hc represents the height of corroded zone, and was set to 40, 80, and 120 mm in investigating its influence on loading capacity. Different mesh sizes have evident influences on the reliability of FEA [26–28]. Thus, the sensitivity analysis on mesh size was initially performed. Corrosion was assumed to occur at Zone I. The values of Hc and Tc were 40 mm and 15 mm, respectively. The element size was set as 5, 10, and 20 mm. The moment–rotation curves are shown in Fig. 7. The figure shows that the element size had nearly no influence on the bending capacity and bending rigidity of WHSJ. The post-buckling behavior was slightly influenced; however, this phenomenon was not the focus of the present work. Thus, the element size was set to 10 mm for future analysis.
Fig. 11. Changing tendency of reduction factor along with Tc.
526
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 12. Failure mode with different values of Tc (Hc = 4 0 mm, Zone I).
Fig. 13. Failure mode with different corrosion locations (Pattern I).
Fig. 14. Influence of corrosion height on loading capacity (Zone I).
capacity was not influenced when corrosion thickness was less than 14 mm, that is, 87.5% of the spherical body thickness. The results shown in Fig. 12 indicated that the failure mode changed when Tc increased from 14 mm to 15 mm. The failure was caused by the yield that occurred at the conjunction of steel pipe and spherical body when Tc = 14 mm. By contrast, the failure was caused by the buckling that occurred at the corrosion zone when Tc = 15 mm. The results also indicated that the bending capacity was nearly not influenced by Hc when
The influence of corroded locations was also investigated, and the results are shown in Figs. 8–10. From the results, the bending capacity of WHSJ was the most sensitive to corrosion that occurred at Zone III. The bending capacity decreased immediately after corrosion occurred in this zone, and the results indicated that the bending capacity was significantly influenced by Tc. The bending capacity of WHSJ was the most insensitive to corrosion that occurred at Zone I. The bending capacity was influenced when Tc reached 15 mm and the bending
527
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 15. Influence of corrosion height on loading capacity (Zone II).
height Hc. The influence of corrosion on bending capacity was also not influenced by Hc. This finding was different for the corrosion that occurred at Zone II, and the influence of corrosion on bending capacity was closely related to Hc. (Fig. 12). Failure modes corresponding to different corrosion locations are shown in Fig. 13. The buckling occurred at Zones I and II when corrosion thickness reached a certain value, and the bending capacity suddenly reduced by a large degree. This result was caused by failure mode transformation. The buckling still occurred at Zone III although the corrosion thickness at Zones I and II did not reach a certain degree. Thus, the bending capacity appeared not to be influenced by Tc. Parametrical analysis was conducted to investigate the influence of Hc systematically at Zones I and II. The results are shown in Figs. 14 and 15. Tc was set as 10, 13, and 15 m. Corrosion height Hc was set as 40, 80, 120, 160, 200, and 240 mm. The results in Fig. 14(a) and (b) show that the bending capacity was not influenced by corrosion although Hc reached 40% of the spherical body diameter when Tc reached 54% of the spherical body thickness. Buckling occurred at Zone I when Tc reached 93% of the spherical body thickness, and the reduction factor slightly decreased with the increase of Hc, as shown in Fig. 16. For the corrosion that occurred at Zone II, the change of reduction factor with Hc became slow with the increase of Tc.
Fig. 16. Changing tendency of reduction factor along with Hc.
2.3. Effect of corroded location for Pattern II The shape of corroded zone was assumed as Pattern II in this subsection. The size of corroded zone was represented by α (Fig. 3), and the value of α was set as 30°, 60°, and 90°. The influence of bending moment direction on corroded WHSJ was also investigated. The included angle between the bending moment and center line of corroded zone (Fig. 17) was investigated through parametrical analysis. The included angle β was changed from 90° to 270° with a step of 30° considering symmetrical characteristic. Corroded thickness was set as 4, 10, and 15 mm. Force-displacement curves corresponding to different corroded thicknesses are shown in Fig. 18. α and Tc had large influence on the loading capacity of WHSJ. Buckling occurred when Tc reached 15 mm. This finding was observed from the load–displacement curves when Tc was equal to 15 mm. The loading capacity and initial stiffness of WHSJ did not influence the value of β when Tc was less than 10 mm, and they were significantly influenced by β when Tc was 15 mm. The loading capacity changed from 75 kN to 100 kN when α = 30° and Tc= 15 mm. The changing amplitude reached 30% of the loading capacity. Thus, the relative location between the bending moment direction and corroded zone also
Fig. 17. Schematic of bending moment direction and corrosion zone.
corrosion occurred at Zone I. The bending capacity started to be influenced when Tc at Zone II increased to 14 mm. The parameter “reduction factor,” that is, the ratio of residual bending capacity of corroded WHSJ and bending capacity of perfect WHSJ, was introduced to quantify the influence of corrosion thickness. The results are shown in Fig. 11. From the results in Zone III, the reduction factor nearly decreased linearly with the increase of corrosion thickness, and the reduction factor was not influenced by corrosion
528
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 18. Load–displacement curves (D = 0.56 m).
Fig. 19. Failure mode with different values of α (Pattern II, β = 90°).
529
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 20. Influence of the diameter of WHSJ on bending capacity.
finding was reflected by the snap-through of load–displacement curves when Tc = 15 mm.
had a significant influence on loading capacity. In addition, the changing amplitude decreased with the increase of α. The failure modes of WHSJ that correspond to different conditions are shown in Fig. 19. Corrosion was located in the compression zone when β = 90° and was located in the tension zone when β = 270°. Buckling occurred at the corrosion location when Tc reached a certain degree (about 93% of total thickness). Buckling occurred when Tc increased to 15 mm regardless of whether β = 90° or β = 270°. The failure was caused by the yield that occurred at the conjunction of steel pipe and spherical body when Tc was less than or equal to 14 mm. This
3. Effects of diameter and thickness 3.1. Effects of diameter and thickness for Pattern I The influence of the diameter of WHSJ on bending capacity that corresponds to different values of Tc was investigated for Pattern I in this subsection. The thicknesses of the spherical body and steel pipe
530
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 21. Influence of the thickness of WHSJ on loading capacity.
indicated the ratio of the initial bending stiffness of corroded WHSJ. The results indicated that the spherical body diameter nearly had no influence on the initial bending stiffness. The suggested ratio of diameter to thickness of WHSJ for truss structures was between 25 and 45 based on the Technical Specification for Space Frame Structures (JGJ7-2010) [30]. Then, the suggested thickness was between 10 mm and 19 mm when the diameter of WHSJ was 0.48 m. The influence of spherical body thickness on bending capacity was investigated. The results derived with different spherical body thicknesses are shown in Fig. 21. The ultimate strength of
were set as 8 mm, and the spherical body diameter was set as 0.48, 0.60, and 0.9 0 m. Corroded thickness was set as 1, 2, 3, 4, 5 and 6 mm. The results derived in different conditions are shown in Fig. 20. From the results, the bending capacity of WHSJ decreased nearly linearly with the increase of corroded thickness for different diameters. This finding is obviously observed in Fig. 20(d). The initial stiffness in the loading direction decreased with the increase of corroded thickness. The results shown in Fig. 20(d) indicated that reduction factor was not influenced by diameter. The influence of corrosion on initial bending stiffness is shown in Fig. 20(e). The initial bending stiffness factor (Iini)
531
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 22. Influence of WHSJ diameter on loading capacity.
3.3. Effect of corrosion on initial bending stiffness
corroded WHSJ decreased linearly with the increase of Tc. The results shown in Fig. 21(d) indicated that the spherical body thickness had influence on reduction factor. The influence of spherical body thickness on initial bending stiffness is shown in Fig. 21(e). The initial bending stiffness factor was only related to Tc/T and was nearly not influenced by the spherical body thickness.
3.3.1. Pattern I The results derived in the above section indicated that the initial stiffness of corroded WHSJ was not influenced by the thickness and diameter of the spherical body. The bending capacity was significantly influenced by the corrosion that occurred at Zone III. Thus, the analysis conducted in this section was based on the corrosion that occurred at Zone III. The initial bending stiffness was mainly influenced by Hc and Tc. Thus, the changing tendency of initial bending stiffness factor along with Hc and Tc was investigated in this section. The diameter and thickness of the spherical body was set with different values to investigate their influence on initial bending stiffness factor. The initial bending stiffness factor (Iini) indicated the ratio of the initial bending stiffness of corroded WHSJ. The contour of initial bending stiffness factor is shown in Fig. 24. The initial bending stiffness factor was mainly influenced by Tc, and Hc slightly influenced Iini. Iini did not change with the change of Hc when Hc was larger than 20 mm (approximately 3.3% of the spherical body diameter). The contour of Iini was nearly not influenced by the thickness and diameter of the spherical body. Iini was mainly influenced by Tc, and the results shown in Fig. 24(a) were used for curve fitting analysis. Then, Iini was represented by Tc through Eq. (1). The T accuracy of the proposed equation was then validated by the results on Joints I, II, and III. The results are shown in Fig. 25. The results derived by FEA was absolutely in agreement with those derived by the proposed equation.
3.2. Effect of diameter and thickness for Pattern II The influence of the diameter of WHSJ on loading capacity corresponding to different values of Tc was investigated for Pattern II in this subsection. The thicknesses of spherical body and steel pipe were set as 14 mm, and the spherical body diameter was set as 0.48, 0.60, and 0.80 m. The corroded thickness was set as 2, 4, 6, 8, 10, and 12 mm. The value of α was set as 90°. The results derived under different conditions are shown in Fig. 22. From the results, the bending capacity of WHSJ decreased rapidly with the increase of corroded thickness, which was different from Pattern I. The changing tendency of reduction factor with diameter is shown in Fig. 22(d). The reduction factor decreased rapidly with the increase of Tc, and the spherical body diameter nearly had no influence on reduction factor. The influence of spherical body thickness on ultimate strength in the corroded zone of Pattern II was also investigated. The diameter of WHSJ was set as 0.60 m and was kept unchanged. The spherical body thickness was set as 8, 12, and 16 mm. The results are shown in Fig. 23. The spherical body thickness had no influence on the reduction factor.
532
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 23. Influence of the diameter of WHSJ on loading capacity.
Iini = 1.02 − 0.26 ×
Tc T 2 − 0.768 × ⎛ c ⎞ T ⎝T ⎠
remained within 2%. The accuracy of results for different sizes of WHSJ indicated that the proposed method can be efficiently adopted for estimating the residual stiffness of WHSJ.
(1)
α 2 360
( ) + 0.0812 × ( ) − 0.4198 × ( ) − 1.415 × ( ) × ( ) + 0.219 × ( ) + 0.1992 ( ) + 0.476 × ( ) × ( ) − 0.24 ( ) × ( ) − 0.3137 ( )
3.3.2. Pattern II The changing tendency of the initial bending stiffness factor along with α and Tc was investigated. The diameter and thickness of the spherical body was set with different values to investigate their influence on the initial bending stiffness factor, as shown in Fig. 26. The initial bending stiffness factor (Iini) indicated the ratio of the initial bending stiffness of corroded WHSJ. From the results, the initial bending stiffness was significantly influenced by α and Tc. The residual initial bending stiffness of WHSJ was determined by α and Tc. The contour of Iini was nearly not influenced by the spherical body diameter. The contour of Iini appeared to be only related to α and Tc. The value of Iini was predicted by α and Tc. Thus, the curve fitting method was adopted to represent Iini by α and Tc, as shown in Eq. (1). Parameters α and Tc were independent from each other. T 360 Only the data shown in Fig. 26(a) were utilized in the curve fitting analysis. The contour of Iini derived by the proposed equation is shown in Fig. 27. The results derived by the proposed equation agreed well with that derived by FEA. Then, error analysis was conducted to estimate the accuracy of the proposed method. The results are shown in Fig. 28. Generally, the error
Iini = 0.9651 − 0.2536 × Tc T
α 2 360
α 360
Tc T
Tc 2 T
α 360
Tc T
α 360
α 3 360
Tc 2 T
Tc 3 T
(2)
4. Influence of corrosion on buckling capacity of integral structure The multi-scaled refined numerical model for K6-type reticulated shell structures was established, as shown in Fig. 29, and the influence of corrosion on the buckling capacity of joints located at different places was investigated. The thicknesses of spherical joint and steel pipe were set as 13 mm and 8 mm, respectively. Additional detailed information can be found in [10]. The corrosion type was assumed to be in Pattern I, and corrosion occurred at the conjunction of spherical body and steel pipe. The corroded thickness was set as 0, 3, 5, 8, 10, and 12 mm. The corrosion was assumed to only occur on the studied joint. Vertical load was also
533
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 24. Change of initial bending stiffness factor along with Hc and Tc (Pattern I).
534
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
applied at Joint I when its mechanical behavior was studied. Four joints located at typical locations were investigated. Load–displacement curves of each joint are shown in Fig. 30. The corrosion had no influence on the initial stiffness of integral structure. However, the buckling capacity was significantly reduced. Buckling capacity was reduced by 40% when Tc reached 12 mm, that is, 85.7% of the spherical body thickness. The reduction factor of buckling capacity is shown in Fig. 30. The reduction factor of different joints was nearly the same. 5. Conclusions Bending capacity and bending stiffness of corroded WHSJ were investigated in this study. The shape of corroded zone was divided into two types, namely, latitudinal (Pattern I) and longitudinal (Pattern II)
Fig. 25. Validation of the proposed method.
Fig. 26. Change of bending stiffness factor along with α and Tc (Pattern II).
535
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 27. Contour of Iini derived by the proposed equation (Pattern II).
Fig. 28. Tolerance analysis (Pattern II).
536
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 29. Numerical model of integral structure.
(4) The equations for estimating the residual bending stiffness (Patterns I and II) were proposed, and their accuracy were validated. The initial bending stiffness of corroded WHSJ was accurately predicted by the proposed method. (5) The influence of corrosion on the buckling capacity of integral structure was investigated based on a refined numerical model. The results indicated that the corrosion had no influence on the initial stiffness of integral structure. Buckling capacity was reduced by 40% when Tc reached 85.7% of the spherical body thickness.
patterns. Other corroded shapes could be represented by combining the two types. The effects of corroded location and the diameter and thickness of spherical body on bending capacity and bending stiffness were studied. The conclusions can be summarized as follows: (1) The corrosion in different places caused distinct influence on the bending capacity of WHSJ. Ultimate strength was nearly not influenced when Tc in Zone I reached 87.5% of the spherical body thickness. (2) The reduction factor of bending capacity caused by corrosion in Zone III nearly decreased linearly with the increase of corrosion thickness, and the reduction factor was not influenced by corrosion height Hc. (3) The influence of corrosion on bending stiffness was not influenced by the spherical body diameter, and the initial bending stiffness factor was only related to Tc/T and was nearly not influenced by the spherical body thickness.
Acknowledgments This work was financially supported by the State Key Research Development Program of China (Grant Nos. 2016YFC0801404 and 2016YFC0600704) and the Project Funded by China Postdoctoral Science Foundation (No. 2017M621156).
537
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
Fig. 30. Load–displacement curves corresponding to different joints.
[9] Lei Gu, Maoqiang Ding, Xueyi Fu, Xianchuan Chen, A refined finite element analysis on single layer latticed domes with welded hollow spherical joints, J. Build. Struct. 32 (8) (2011) 42–50. [10] Zhong-wei Zhao, Han Zhu, Zhi-hua Chen, Mechanical behavior of single-layer reticulated shell connected by welded hollow spherical joints with considering welding residual stress, Weld. World 60 (1) (2016) 61–69. [11] Hongbo Liu, Jie Lu, Zhihua Chen, Residual behavior of welded hollow spherical joints after exposure to elevated temperatures, J. Constr. Steel. Res. 137 (2017) 102–118. [12] M. Witkowska, C. Guedes Soares, Ultimate strength of locally damaged panels, Thin-Wall Struct. 97 (2015) 225–240. [13] X.H. Shi, J. Zhang, C. Guedes Soares, Experimental study on collapse of cracked stiffened plate with initial imperfections under compression, Thin-Wall Struct. 114 (2017) 39–51. [14] Khedmati MR, Nouri ZHME, Analytical simulation of nonlinear elastic–plastic average stress–average strain relationships for un-corroded/both-sides randomly corroded steel plates under uniaxial compression, Thin-Wall Struct. 86 (2015) 132–141. [15] D. Ok, Y. Pu, A. Incecik, Computation of ultimate strength of locally corroded unstiffened plates under uniaxial compression, Mar. Struct. 20 (2007) 100–114.
References [1] X.L. Liu, Design and Construction of Plate Grid Structures, Press of Tianjin University, Tianjin, China, 2000. [2] Z.S. Makowski, Analysis, Design and Construction of Double-Layer Grids, Applied Science Publishers Ltd, London, 1981. [3] Y. Ding, L. Qi, Z.X. Li, Mechanical calculation model for welded hollow spherical joint in spatial latticed structures, Adv. Steel Constr. 7 (4) (2011) 330–343. [4] X. Li, Load-carrying capacity and practical design method of welded hollow spherical joints in space latticed structures, Adv. Steel Constr. 6 (4) (2010) 976–1000. [5] Qinghua HAN, Quanzhi ZHOU, Yue CHEN, Baolin YU, Ultimate bearing capacity of hollow spherical joints welded with circular pipes under eccentric loads, Trans. Tianjin Univ. 13 (1) (2007) 28–34. [6] Han Qinghua, Liu Xiliang, Ultimate bearing capacity of the welded hollow spherical joints in spatial reticulated structures, Eng. Struct. 26 (1) (2004) 73–82. [7] Han Qinghua, Liu Yiming, Xu Ying, Stiffness characteristics of joints and influence on the stability of single-layer latticed domes, Thin-Walled Struct. 107 (2016) 514–525. [8] Xing Wang, Shi-lin Dong, Hai-ying Wan, Finite element analysis of welded spherical joints’ stiffness, J. Zhejiang Univ. 34 (1) (2000) 77–82.
538
Thin-Walled Structures 127 (2018) 523–539
Z. Zhao et al.
[16] T. Nakai, H. Matsushita, N. Yamamoto, Effect of pitting corrosion on strength of web plates subjected to patch loading, Thin-Wall Struct. 44 (2006) 10–19. [17] Y. Huang, Y. Zhang, G. Liu, Q. Zhang, Ultimate strength assessment of hull structural plate with pitting corrosion damnification under biaxial compression, Ocean Eng. 37 (2010) 1503–1512. [18] S. Sultana, Y. Wang, A.J. Sobey, J.A. Wharton, R.A. Shenoi, Influence of corrosion on the ultimate compressive strength of steel plates and stiffened panels, Thin-Wall Struct. 96 (2015) 95–104. [19] S. Saad-Eldeen, Y. Garbatov, C. Guedes Soares, Strength assessment of a severely corroded box girder subjected to bending moment, J. Constr. Steel Res. 92 (2014) 90–102. [20] S. Saad-Eldeen, Y. Garbatov, C. Guedes Soares, Effect of corrosion severity on the ultimate strength of a steel box girder, Eng. Struct. 49 (2013) 560–571. [21] S. Saad-Eldeen, Y. Garbatov, C. Guedes Soares, Analysis of plate deflections during ultimate strength experiments of corroded box girders, Thin-Wall Struct. 54 (2012) 164–176. [22] Zhongwei Zhao, Haiqing Liu, Bing Liang, Novel numerical method for the analysis of semi-rigid jointed lattice shell structures considering plasticity, Adv. Eng. Softw. 114 (2017) 208–214. [23] Zhongwei Zhao, Zhihua Chen, Xiangyu Yan, Xu Hao, Bingzhen Zhao, Simplified
[24]
[25]
[26]
[27]
[28]
[29] [30]
539
numerical method for latticed shells that considers member geometric imperfection and semi-rigid joints, Adv. Struct. Eng. 19 (4) (2016) 689–702. S. Saad-Eldeen, Y. Garbatov, C. Guedes Soares, Effect of corrosion degradation on the ultimate strength of steel box girders, Corros. Eng. Sci. Technol. 47 (2012) 272–283. D.K. Kim, S.J. Kim, H.B. Kim, X.M. Zhang, C.G. Li, J.K. Paik, Ultimate strength performance of bulk carriers with various corrosion additions, Ships Offshore Struct. 10 (2015) 59–78. Hongbo Liu, Jie Lu, Zhihua Chen, Residual behavior of welded hollow spherical joints after exposure to elevated temperatures, J. Constr. Steel Res. 137 (2017) 102–118. B. Faggiano, A. Marzo, A. Formisano, et al., Review: innovative steel connections for the retrofit of timber floors in ancient buildings: a numerical investigation, Comput. Struct. 87 (1–2) (2009) 1–13. A. Formisano, G. De Matteis, F.M. Mazzolani, Experimental and numerical researches on aluminium alloy systems for structural applications in civil engineering fields, Key Eng. Mater. 710 (2016) 256–261. ANSYS user’s manual. SAS IP inc, 1998. Technical specification for space frame structures (JGJ7-2010), China, 2010.