- Email: [email protected]
Buckling analysis of open top tanks subjected to harmonic settlement
Thin-Walled Structures 63 (2013) 37–43
Contents lists available at SciVerse ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Buckling analysis of open top tanks subjected to harmonic settlement Jianguo Gong, Jun Tao, Jian Zhao, Sheng Zeng, Tao Jin n Institute of Process Equipment, Zhejiang University, Hangzhou 310027, China
a r t i c l e i n f o
abstract
Article history: Received 15 March 2012 Received in revised form 26 September 2012 Accepted 26 September 2012 Available online 23 November 2012
This work addresses the buckling behavior of the open top tanks subjected to harmonic settlement. First of all, the buckling behavior and the critical harmonic settlement of an open top tank for various wave numbers are investigated. The results present that buckling occurs on the upper shell for a small wave number, while it changes to other places of the shell for large wave numbers. Also, with the wave number increasing, the buckling point is closer to the base of the shell. Besides, for the original tank, the critical harmonic settlement decreases greatly when the wave number is small, while the critical harmonic settlement decreases slightly when the wave number is large. Then, the parametric studies for the buckling behavior of the open top tank are conducted, composed of the height-to-radius (h/r) ratio and the radius-to-thickness (r/t) ratio. Regarding the h/r ratio, for a certain wave number, the critical harmonic settlement versus the h/r ratio is monotonically decreasing. Also, with the wave number increasing, the critical harmonic settlement decreases more and more slightly. Regarding the r/t ratio, for a certain wave number, the critical harmonic settlement versus the r/t ratio is monotonically decreasing. Moreover, with the increasing wave number, the critical harmonic settlement decreases more and more slightly. Finally, the buckling results of the open top tank are compared with that of the conical roof tank. The results illustrate that the open top tank can hold a larger harmonic settlement than the conical roof tank for no restraint from the roof for a small wave number. However, for large wave numbers, there is a small difference in settlement sustaining capacity for open top tanks and conical roof tanks. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Buckling Tank Open top Harmonic settlement Geometrical nonlinearity
1. Introduction Large scale oil tanks are important devices for oil storage and are usually constructed in the coastal areas or on the island. Due to the poor soil conditions, the tanks withstand large settlement, which can be resolved into three components: uniform settlement, planar settlement and differential settlement. For the uniform settlement or the planar settlement, its effect on the structure of tanks is limited if the value of the settlement is not very large. For the differential settlement, it is also named harmonic settlement, because it is usually simplified as the harmonic forms for various wave numbers. This settlement is usually smaller, but is more important. Even a small differential settlement beneath the tank wall can cause a large distortion at the upper shell or the roof, especially for tanks without a roof [1,2]. Thus, the differential settlement is the most dangerous settlement type. Moreover, studying the buckling performance of the tank subjected to harmonic settlement is important and necessary. For tanks subjected to harmonic settlement, a short literature review is provided as follows. In an early stage, the researchers
n
Corresponding author. E-mail address: [email protected] (T. Jin).
0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.09.009
are interested in the theoretical analysis of the radial displacement at the top and the harmonic settlement at the base. Many theories are adopted to identify the relationship of the two. At this time, the open top and floating roof tanks are the main research topic. For instance, Malik et al. [3] assumed that the settlement beneath the tank wall was in harmonic form. Based on an inextensional theory, an equation between the radial displacement and the harmonic settlement was proposed. It was concluded that the radial displacement predicted by this theory is adequate for a small wave number. However, great discrepancy exists for large wave numbers. Then, Kamyab and Palmer [4–7] put forward the membrane theory and the modified Donnel large deflection equation to make the wave number reasonable over a wider range. In their work, the assumption of harmonic settlement was adopted and the primary wind girder was included in the analyses. Moreover, the validity of the three analytical solutions mentioned above was discussed in detail and three critical harmonic number n1, n2, n3 were presented. It was concluded that these three solutions are adequate to a certain range of wave numbers. Exactly speaking, the inextensional theory, membrane theory and modified Donnel large deflection equation are valid for non1, n1 on on2 and n2 onon3, respectively. Additionally, D’Orazio et al. [8] presented a new method called fold line model to pursuit for the relation between the
38
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
radial displacement and the harmonic settlement. It was revealed that the radial displacement predicted agrees well with field measurement. However, determining the fold line of the settlement is greatly dependent on a person’s judgment. Along with the development at full speed of computer technology, many investigations are on the basis of the numeric methods, of which the finite element method is the most used [2,9–18]. In this stage, the static behavior of open top and floating roof tanks is performed by finite element method and is compared with the result predicted by the classical theories stated above. For example, Jonaidi and Ansourian [10] argued that the typical magnitude of maximum error for the range n considered is in the range of 10–18% depending on the shell geometry and Iratio. Moreover, other types of tanks are involved in the analyses, such as tanks with a roof. Meanwhile, various analysis types are carried out, including the linear bifurcation analysis and the buckling analysis. For example, Jonaidi and Ansourian [11] studied the linear buckling of closed-top cylindrical shells subjected to edge vertical deformation. In their work, the radial and circumferential constraints were imposed at the top of shell. It was concluded that for uniform shells, the shear buckling mode is observed for a small wave number and the mode is dominated by axial buckling with the increasing wave number; the shear buckling mode occurs in the upper region of the shell at high wave number for the tapped shells. However, the assumption of imposing the radial and circumferential deflections at the top edge is not reasonable. The results will be more accurate if the model of the roof is built. Godoy and Sosa [12] analyzed the buckling strength of conical roof tanks induced by the localized support settlements. A linear analysis, a geometrically nonlinear analysis and a bifurcation buckling analysis were performed using the finite element computer package ABAQUS. It was revealed that the equilibrium path is highly nonlinear and that the shell displays a plateau for a settlement of the order of half the shell thickness; the geometric nonlinearity should be included, because linear results provide a poor indication of the real shell displacements. However, the effects of some factors are not considered, including the heightto-radius ratio and slope of the conical roof. Later, Godoy and Sosa [13] carried out an experiment for a small flat roof tank subjected to harmonic settlement. It was displayed that the equilibrium path is nonlinear and the shell displays a stable symmetric bifurcation behavior; moreover, the results of the experiments are consistent with that of the computational results. Cao and Zhao [14] studied the buckling behavior of the fixed-roof tank under harmonic settlement. Three types of buckling were involved: elastic bifurcation analysis, geometrical nonlinear analysis of the perfect shell and geometrical nonlinear analysis of the imperfect shell. Moreover, the effects of the wave number, radiusto-thickness ratio, height-to-radius ratio, and the geometric imperfection on the buckling strength were investigated. It was indicated that both the critical harmonic settlement and the buckling mode are closely related to the geometric parameters mentioned above. However, the finite element model of the roof is not built directly, which is replaced by restraining the radial and the circumferential displacement of the joints between the shell and the roof. Thus, the results will be more accurate if the finite element model of the roof is considered. To make the results more reasonable, Gong et al. [15] reported the buckling behavior of conical roof tanks subjected to harmonic settlement. The geometrical nonlinear behavior was included in the analyses. The buckling modes, settlement–displacement curves and critical harmonic settlements for various wave numbers were provided. Besides, the parametric studies were conducted, including the height-to-radius ratio, radius-to-thickness ratio, slope and thickness of the conical roof. The work mentioned above focuses on the buckling behavior of conical roof tanks or fixed roof tanks.
However, the buckling behavior of open top or floating roof tanks subjected to harmonic settlement is rarely reported. Portela and Godoy [16] discussed the buckling behavior of an open top tank under wind pressures and imperfection sensitivity was conducted. It was revealed that buckling occurs at the windward and the imperfection of the shell has a great influence on the buckling load. Nevertheless, the harmonic settlement at the base is not included. Holst and Rotter [2,17] studied the effect of local settlement on the elastic buckling of a thin cylindrical shell under axial compression. It was concluded that the local settlement with a small amplitude can cause a severe loss of buckling strength, and the growth of the dimple appears to protect the shell from the greater strength loss. However, the main attention of their work is on the axial buckling load. Although the buckling behavior of open top or floating roof tanks is rarely studied, several tank failures due to support settlement in practical engineering are reported. For example, Jia [19] mentioned two floating roof tanks damaged by the differential settlement in one petrochemical plant located in Shanghai. For one tank (Tank 124#), it is partially located on a pre-consolidated soil. For the other tank (Tank 125#), part of the foundation is close to a river and the foundation treatment was not carried out before construction. Finally, both tanks lost effectiveness due to the continuously increasing differential settlement. Jiang [20] reported another two tank failures. For each tank, only part of the tank locates on ground consolidated by the earlier presence of a tank. As time goes on, the differential settlement beneath the tank wall caused the failure of the tank. Thus, investigating the buckling behavior of open top and floating roof tanks subjected to the harmonic settlement is urgent and required. In this work, the open top tank is chosen as a research subject. Considering that a floating roof nearly has no structural effect on the tank, an open top tank is a floating roof tank. Thus, the open top tank in following also means floating roof tank. The geometrical nonlinear buckling analyses (GNA) for open top tanks are conducted in following, which are also recommended by Rotter and Schmidt [21] for determining the load carrying capacity. Ref. [21] is a very important reference book for engineers and researchers, which provide the designer of a metal shell structure with an extended guide to the design of the shell against buckling failure. The structure of this work is as follows: Firstly, the finite element model of an open top tank is introduced in Section 2. Then, the buckling behavior of the open top tank subjected to harmonic settlement is described in Section 3. Then, the parametric studies for the buckling behavior, composed of the height-to-radius (h/r) ratio and the radius-to-thickness (r/t) ratio, are discussed in Section 4. Finally, comparisons of buckling behavior for open top tanks and conical roof tanks are reported in Section 5. 2. Finite element model of open top tank The tank discussed in following is based on the tank adopted by Godoy and Sosa [12]. However, the tank mentioned in Ref. [12] is a conical roof tank, not an open top tank. To study the buckling behavior of the open top tank, the conical roof of the tank discussed in Ref. [12] is omitted directly, while other structure parameters of the tank remain unchanged. In this way, we can compare the buckling results of the open top tank with that of the conical roof tank. In addition, for simplifying the calculations, the variations of the shell thickness are neglected. The structure parameters of an open top tank are listed in Table 1. Table 1 Structure parameters of open top tank. Inner radius r (m) Height of tank h (m) Shell thickness t (mm)
15.12 12.191 8.92
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
ABAQUS, a general purpose finite element computer package, is used to conduct all the analyses [2,10–13,15–17,22]. Firstly, the geometric model of the open top tank is built. Since the open top tank model is symmetrical about the plane Z ¼0 (see Fig. 1), the model of half of the structure is applied to reduce the computation effort. Secondly, the geometric model of the open top tank is discretized by the 4-node shell element S4. After the discretization of the tank model, 6, 560 shell elements are generated and the finite element mesh model is shown as Fig. 1. Then, the material properties for the open top tank are set. The model is assumed to have the properties of steel, with elastic modulus E¼2.06 1011 Pa and Poisson’s ratio v ¼0.3. Finally, the boundary conditions of the open top tank are given as follows: (1) The symmetry constraints are applied in the symmetry plane Z ¼0 (see Fig. 1). (2) The radial displacement and the circumferential displacement are constrained at the base of the shell, while the axial displacement (Y-axis) is u ¼un cos(nj), where, un is the amplitude of the nth harmonic settlement; n is wave number; j is the angle of the point at the base of the shell. In addition, to obtain the buckling point of the whole model, the arc-length method is employed [22–25], which can track not only the process of pre-buckling stage but also the process of the post-buckling stage. Moreover, considering that the tank model may produce a large deformation when buckling happens, a large displacement analysis for the model is necessary.
3. General buckling behavior The main goal of the buckling analysis is to obtain the critical harmonic settlement at the base of the shell for different wave
Fig. 1. Finite element mesh model of open top tank.
39
numbers. Fig. 2 shows the typical settlement–displacement curves for n¼10, 14, 18, 22 with the typical tank geometry r/t¼1700 and h/r¼0.8. In Fig. 2, the vertical axis stands for the load factor of the harmonic settlement un normalized with respect to the shell thickness, while the horizontal axis represents the out-of-plane radial displacement w normalized by the shell thickness. For a small wave number (e.g. n¼10) (see Fig. 2a), first of all, the load factor changes linearly. Then, the load factor increase nonlinearly when the linear range is got across. Finally, the buckling point of the open top tank is obtained and the tank falls into the post-buckling stage, in which the load factor displays a strong nonlinearity, presenting an odd behavior shown in zoom in Fig. 2a). For this odd behavior, we can understand it like this. When the buckling point A1 is reached, the shell falls into the post-buckling stage. At this time, the response shows a negative stiffness and the structure must release strain energy to remain in equilibrium. After achieving a limit point B1, the load factor increases with the radial displacement increasing. Later, a new critical point A2 is obtained. Then, the curve starts to decrease with the increasing radial displacement again. This is why the equilibrium path shows the odd behavior in zoom in Fig. 2a). Moreover, the stiffing path with large settlement during the post-buckling stage is not evident, which is quite different from that for wave number n¼14, 18, 22. This reveals that the tank does not have a good settlement sustaining capacity for a small wave number. For large wave numbers (e.g. n¼14, 18, 22) (see Fig. 2b), the buckling path is highly unstable with the displacement increasing, which is a clear sign of the instability of the tank. Moreover, the load factor shows a little different behavior, which is different from that of the wave number n¼10. Exactly speaking, the post-buckling balance path reveals a good settlement sustaining capacity of the tank for large wave numbers. Fig. 3 shows the buckling modes of the open top tank for the wave number n¼6, 10, 18, 22. For a small wave number, buckling happens on the upper shell (e.g. n¼6, 10). However, for large wave numbers, buckling happens on the middle part of the shell (e.g. n¼ 18) or the bottom part of the shell (e.g. n¼22). In other words, the buckling location is closer to the base with the wave number on the increase. The main cause for this lies in that the harmonic settlement at the base becomes more and more difficult to be transferred to the upper shell. The harmonic settlement can be transferred to the upper shell for a small wave number, but can only be transferred to other places on the shell for large wave numbers. That is why buckling happens on the upper shell for a small wave number but occurs at the lower shell for large wave numbers. As is described at the beginning of this section, the main goal for the buckling analysis is to acquire the critical harmonic settlement at the base of the shell. For the critical harmonic settlement, it corresponds to the limit point A or A1 in Fig. 2. Here, we will focus on the relation of the critical harmonic settlement and the wave number, as is shown in Fig. 4. The vertical axis
Fig. 2. Typical settlement–displacement curves for various wave numbers (h/r¼ 0.8 and r/t ¼1700).
40
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
Fig. 3. Buckling modes for various wave numbers (h/r ¼0.8 and r/t ¼1700).
Fig. 4. Critical harmonic settlement–wave number curve (h/r ¼ 0.8 and r/t ¼1700).
Fig. 5. Critical harmonic settlement versus wave number for various h/r ratios (r/t ¼1700).
represents the load factor of the critical harmonic settlement normalized with respect to the shell thickness, while the horizontal axis stands for the wave number. In Fig. 4, on the whole, the critical harmonic settlement decreases more and more slightly with the increasing wave number. Specifically, when the wave number is small, the critical harmonic settlement decreases significantly. However, when the wave number is large, the critical harmonic settlement decreases slightly. This means the tank can stand a larger harmonic settlement when the wave number is small, while it can only hold a smaller harmonic settlement when the wave number is large.
numbers for constant r/t ¼1700. It can be observed that for a certain h/r ratio, the critical harmonic settlement decreases versus the wave number is monotonically decreasing. Take the ratio h/r ¼0.5 for example. The critical harmonic settlement decreases greatly for a small wave number, particularly for a wave number smaller than or equal to 4. However, the critical harmonic settlement decreases slightly for large wave numbers, especially for a wave number larger than or equal to 10. Also, with the h/r ratio increasing, the critical harmonic settlement decreases more and more slightly. For example, for the ratio h/r ¼0.5, the critical harmonic settlement at n ¼2 is 183.6, which is more than 2, 200 times than that at n ¼22. However, for the ratio h/r ¼2.0, the critical harmonic settlement at n¼ 2 is about 126.2, which is more than 1, 500 times than that at n¼ 22. Fig. 6 displays the critical harmonic settlements of different wave numbers under various h/r ratios. For a certain wave number, the critical harmonic settlement versus the h/r ratio decreases monotonically. Also, with the wave number on the increase, the critical harmonic settlement decreases more and more slightly. For example, when the wave number n is 2, the critical harmonic settlement at h/r ¼0.5 is 183.6, which is about 1.45 times than that at h/r ¼2.0. Nevertheless, when wave number n is 22, the critical harmonic settlement at h/r¼ 0.5 is about 0.0805, which is about 1.0 times than that at h/r¼ 2.0.
4. Parametric study In this section, two factors which are related to the critical harmonic settlement at the base of the shell will be discussed, composed of the height-to-radius (h/r) ratio and the radius-tothickness (r/t) ratio. 4.1. Height-to-radius ratio (h/r) Height-to-radius ratios ranging from 0.5 to 2.0 are studied in this section. Fig. 5 demonstrates the results for various wave
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
Fig. 6. Critical harmonic settlement versus h/r ratio for various wave numbers (r/t¼ 1700).
41
Fig. 8. Critical harmonic settlement versus r/t ratio for various wave numbers (h/r¼ 0.8).
top tank has the same structure parameters as the conical roof tank (except for the conical roof), comparing the results of the open top tank with that of the conical roof tank is necessary. For the conical roof tank mentioned in Ref. [12], the buckling results are given by Gong et al. [15]. Here, four factors will be discussed below: the buckling mode, the wave number, the height-to-radius (h/r) ratio and the radius-to-thickness (r/t) ratio. 5.1. Buckling modes
Fig. 7. Critical harmonic settlement versus wave number for various r/t ratios (h/r ¼0.8).
4.2. Radius-to-thickness ratio (r/t) Radius-to-thickness ratios ranging from 500 to 2000 are studied in this section. Fig. 7 presents the results for various wave numbers for constant h/r ¼0.8. It can be observed that, for a certain r/t ratio, the critical harmonic settlement decreases greatly for a small wave number. However, the critical harmonic settlement decreases slightly for large wave numbers. Also, with the increasing r/t ratio, the critical harmonic settlement decreases more and more slightly. Fig. 8 explains the critical harmonic settlements of different wave numbers under various r/t ratios. For a certain wave number, the critical harmonic settlement versus the r/t ratio is monotonically decreasing. Moreover, with the increasing wave number, the critical harmonic settlement decreases more and more slightly. Specially, for a small wave number, the critical harmonic settlement decreases greatly with the r/t ratio, but decreases slightly with the r/t ratio for large wave numbers.
5. Discussion Through the analyses mentioned above, the buckling results of the open top tank have been obtained. Considering that the open
In this work, buckling modes for open top tanks happens at the top edge of the shell for a small wave number, while it changes to the lower shell with the wave number increasing. The behavior is different from that for conical roof tanks. For conical roof tanks, buckling occurs at the conical roof when the wave number is small, but it happens on the lower shell for large wave numbers. Although the buckling modes for these two types of tanks are different from each other, the explanations are nearly the same, which are shown below: for a small wave number, the base settlement can be transferred to the upper shell or the roof, while the settlement can only be transferred to lower part of the shell for large wave numbers. 5.2. Wave number In this work, for the original tank, the critical harmonic settlement decreases significantly for a small wave number, but decreases slightly for large wave numbers. The buckling behavior is much like that of the conical roof tank. Although the tendency of the two types of tanks is similar to each other, the values of the critical harmonic settlement for both tanks have a big difference. For instance, when the structure parameters of the tank is the same (h/r ¼0.8 and r/t ¼1 700), the critical harmonic settlement of the open top tank at n ¼2 is 161.4, which is more than 300 times than that of the conical roof tank at n ¼2. This is because for the open top tank, the top edge of the shell is unrestrained and the upper shell can deflect freely. However, for the conical roof tank, the conical roof restrains the deformation of the upper shell. Thus, the open top tank can hold a larger critical harmonic settlement than the conical roof tank for a small wave number. However, for large wave numbers, there is a small difference for both tanks. For example, the critical harmonic settlement of the open top tank for wave number n¼14 is 0.187, which is only 1.3 times than that of the conical roof tank for wave number n¼14. This is because buckling occurs at the lower shell for large wave numbers. At this
42
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
time, the stiffening effect of conical roof on the lower shell is limited. Thus, the critical harmonic settlements for both types of tanks have a small difference for large wave numbers. 5.3. Height-to-radius ratio (h/r) In this work, for a certain wave number, the critical harmonic settlement versus the h/r ratio decreases monotonically. Also, with the wave number on the increase, the critical harmonic settlement decreases more and more slightly. The behavior is quite different from that of the conical roof tank. For the conical roof tank, the critical harmonic settlement versus the h/r ratio is complex when the wave number is smaller than 6, while the critical harmonic settlement increases with the h/r ratio on the increase when the wave number is larger than or equal to 6. 5.4. Radius-to-thickness ratio (r/t) In this work, for a certain wave number, the critical harmonic settlement versus the r/t ratio is monotonically decreasing. Moreover, with the increasing wave number, the critical harmonic settlement decreases more and more slightly. The behavior is similar to that of the conical roof tank for a wave number larger than or equal to 6. However, for a wave number smaller than 6, the open top tank displays a different behavior. The critical harmonic settlement versus the r/t ratio is monotonically increasing when the wave number n is 2 and 3, while the critical harmonic settlement increases firstly, and then decreases when the wave number n is 4 and 5. 5.5. Summary For the open top tank and the conical roof tank, the main difference is the conical roof. For a small wave number, buckling occurs at the upper shell. For the existence of the conical roof, the deformation of the top edge of the shell is restrained and the upper shell cannot deflect freely. Thus, the conical roof tank can only deflect in a small range before attaining the buckling point. This means the open top tank can hold a larger harmonic settlement than the conical roof tank for a small wave number. However, for large wave numbers, there is a small difference in settlement sustaining capacity for open top tanks and conical roof tanks.
6. Conclusions This work deals with the buckling behavior of the open top tanks subjected to the harmonic settlement by means of numerical studies. The following conclusions can be drawn from the results of the investigation. 1) Buckling occurs on the upper part of the shell for a small wave number, while it changes to the middle or the lower part of the shell for large wave numbers. In other words, the buckling location is closer to the base of the shell with the wave number on the increase. 2) Regarding the h/r ratio, for a certain wave number, the critical harmonic settlement versus the h/r ratio decreases monotonically. Also, with the wave number on the increase, the critical harmonic settlement decreases more and more slightly. 3) Regarding the r/t ratio, for a certain wave number, the critical harmonic settlement versus the r/t ratio is monotonically decreasing. Moreover, with the increasing wave number, the critical harmonic settlement decreases more and more slightly.
4) When the open top tank has the same structure parameters as the conical roof tank (except for the conical roof), the open top tank can hold a larger critical harmonic settlement than the conical roof tank for no restraint from the roof for a small wave number. However, for large wave numbers, there is a small difference in the settlement sustaining capacity for open top tanks and conical roof tanks.
Acknowledgments During the writing stages of the work, Ms. Sun gave the authors valuable advice. Without her help, it would have been difficult to finish the writing of this work. The authors gratefully acknowledge Prof. W.M. Tao and Associate Prof. Y.M. Xu of Zhejiang University, for their kind assistance with the explanation of the equilibrium path of tanks. References [1] Teng JG, Rotter JM. Buckling of thin metal shells. London: Spon Press; 2004. [2] Holst JMFG, Rotter JM. Axially compressed cylindrical shells with local settlement. Thin-Walled Structures 2005;43(5):811–25. [3] Malik Z, Morton J, Ruiz C. Ovalization of cylindrical tanks as a result of foundation settlement. Journal of Strain Analysis for Engineering Design 1977;12(4):339–42. [4] Kamyab H, Palmer SC. Analysis of displacements and stresses in oil storage tanks caused by differential settlement. Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science 1989;203(1):61–70. [5] Kamyab H, Palmer SC. Displacements in oil storage tanks caused by localized differential settlement. Journal of Pressure Vessel Technology, Transactions of the ASME 1991;113(1):71–80. [6] Palmer SC. Structural effects of foundation tilt on storage tanks. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 1992;206(E2):83–92. [7] Palmer SC. Stress in storage tanks caused by differential settlement. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 1994;208(E1):5–16. [8] D’ Orazio TB, Duncan JM, Bell RA. Distortion of steel tanks due to settlement of their walls. Journal of Geotechnical Engineering-ASCE 1989;115(6): 871–90. [9] Zhao Y, Cao QS, Xie XY. Floating roof steel tanks under harmonic settlement: FE parametric study and design criterion. Journal of Zhejiang University, Science A 2006;7(3):398–406. [10] Jonaidi M, Ansourian P. Harmonic settlement effects on uniform and tapered tank shells. Thin-Walled Structures 1998;31(1–3):237–55. [11] Jonaidi M, Ansourian P. Buckling of cylindrical shells subjected to edge vertical deformation. Advances in Steel Structures. Oxford: Elsevier; 1999 p. 679–86. [12] Godoy LA, Sosa EM. Localized support settlements of thin-walled storage tanks. Thin-Walled Structures 2003;41(10):941–55. [13] Godoy LA, Sosa EM Deflections of thin-walled storage tanks with roof due to localized support settlement. In: Proceedings II International Conference on Advances in Structural Engineering and Mechanics. Seoul (Korea): Techno-Press; 2002. [14] Cao QS, Zhao Y. Buckling strength of cylindrical steel tanks under harmonic settlement. Thin-Walled Structures 2010;48(6):391–400. [15] Gong J, Cui W, Zeng S. Buckling analysis of large scale oil tanks with a conical roof subjected to harmonic settlement. Thin-walled Structures 2012;52(7): 143–8. [16] Portela G, Godoy LA. Wind pressures and buckling of cylindrical steel tanks with a conical roof. Journal of Constructional Steel Research 2005;61(6): 786–807. [17] Holst JMFG, Rotter JM Nonlinear response and buckling of cylindrical tanks due to foundation settlement. In: Krupka V, editor. Proceedings International Conference on Design, Inspection and Maintenance of Cylindrical Steel Tanks and Pipelines. Prague(Czech Republic); 2003. pp. 29–35. [18] Rotter JM, Cai M, Holst JMFG. Buckling of thin cylindrical shells under locally elevated compressive stresses. Journal of Pressure Vessel Technology 2011;133(1) p. 011204–11. [19] Jia QS. Analysis and treatment for foundation engineering accidents of tanks. Ground Improvement 2006;17(2):10–21 [in Chinese]. [20] Jiang JR. Case study of earth undercutting method applied to correct unevenly settled foundation of 20 000 m3 vertical steel storage tank. Petroleum Engineering Construction 2011;37(4):92–5 [in Chinese]. [21] Rotter JM, Schmidt H. Buckling of Steel Shells: European Design Recommendations. Fifth ed. Brussels: European Convention for Constructional Steelwork; 2008. [22] ABAQUS user’s manual. USA: Hibbit, Karlsson and Sorensen Inc; 2004.
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
[23] Riks E. An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures 1979;15(7):529–51. [24] Ramm E. Strategies for tracing the nonlinear response near limit points. In: Non-linear Finite Element Analysis in Structural Mechanics. New York: Springer; 1981. p. 68–89.
43
[25] Crisfield MA. An arc-length method including line searches and accelerations. International Journal for Numerical Methods in Engineering 1983;19(9):1269–89.
Contents lists available at SciVerse ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Buckling analysis of open top tanks subjected to harmonic settlement Jianguo Gong, Jun Tao, Jian Zhao, Sheng Zeng, Tao Jin n Institute of Process Equipment, Zhejiang University, Hangzhou 310027, China
a r t i c l e i n f o
abstract
Article history: Received 15 March 2012 Received in revised form 26 September 2012 Accepted 26 September 2012 Available online 23 November 2012
This work addresses the buckling behavior of the open top tanks subjected to harmonic settlement. First of all, the buckling behavior and the critical harmonic settlement of an open top tank for various wave numbers are investigated. The results present that buckling occurs on the upper shell for a small wave number, while it changes to other places of the shell for large wave numbers. Also, with the wave number increasing, the buckling point is closer to the base of the shell. Besides, for the original tank, the critical harmonic settlement decreases greatly when the wave number is small, while the critical harmonic settlement decreases slightly when the wave number is large. Then, the parametric studies for the buckling behavior of the open top tank are conducted, composed of the height-to-radius (h/r) ratio and the radius-to-thickness (r/t) ratio. Regarding the h/r ratio, for a certain wave number, the critical harmonic settlement versus the h/r ratio is monotonically decreasing. Also, with the wave number increasing, the critical harmonic settlement decreases more and more slightly. Regarding the r/t ratio, for a certain wave number, the critical harmonic settlement versus the r/t ratio is monotonically decreasing. Moreover, with the increasing wave number, the critical harmonic settlement decreases more and more slightly. Finally, the buckling results of the open top tank are compared with that of the conical roof tank. The results illustrate that the open top tank can hold a larger harmonic settlement than the conical roof tank for no restraint from the roof for a small wave number. However, for large wave numbers, there is a small difference in settlement sustaining capacity for open top tanks and conical roof tanks. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Buckling Tank Open top Harmonic settlement Geometrical nonlinearity
1. Introduction Large scale oil tanks are important devices for oil storage and are usually constructed in the coastal areas or on the island. Due to the poor soil conditions, the tanks withstand large settlement, which can be resolved into three components: uniform settlement, planar settlement and differential settlement. For the uniform settlement or the planar settlement, its effect on the structure of tanks is limited if the value of the settlement is not very large. For the differential settlement, it is also named harmonic settlement, because it is usually simplified as the harmonic forms for various wave numbers. This settlement is usually smaller, but is more important. Even a small differential settlement beneath the tank wall can cause a large distortion at the upper shell or the roof, especially for tanks without a roof [1,2]. Thus, the differential settlement is the most dangerous settlement type. Moreover, studying the buckling performance of the tank subjected to harmonic settlement is important and necessary. For tanks subjected to harmonic settlement, a short literature review is provided as follows. In an early stage, the researchers
n
Corresponding author. E-mail address: [email protected] (T. Jin).
0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.09.009
are interested in the theoretical analysis of the radial displacement at the top and the harmonic settlement at the base. Many theories are adopted to identify the relationship of the two. At this time, the open top and floating roof tanks are the main research topic. For instance, Malik et al. [3] assumed that the settlement beneath the tank wall was in harmonic form. Based on an inextensional theory, an equation between the radial displacement and the harmonic settlement was proposed. It was concluded that the radial displacement predicted by this theory is adequate for a small wave number. However, great discrepancy exists for large wave numbers. Then, Kamyab and Palmer [4–7] put forward the membrane theory and the modified Donnel large deflection equation to make the wave number reasonable over a wider range. In their work, the assumption of harmonic settlement was adopted and the primary wind girder was included in the analyses. Moreover, the validity of the three analytical solutions mentioned above was discussed in detail and three critical harmonic number n1, n2, n3 were presented. It was concluded that these three solutions are adequate to a certain range of wave numbers. Exactly speaking, the inextensional theory, membrane theory and modified Donnel large deflection equation are valid for non1, n1 on on2 and n2 onon3, respectively. Additionally, D’Orazio et al. [8] presented a new method called fold line model to pursuit for the relation between the
38
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
radial displacement and the harmonic settlement. It was revealed that the radial displacement predicted agrees well with field measurement. However, determining the fold line of the settlement is greatly dependent on a person’s judgment. Along with the development at full speed of computer technology, many investigations are on the basis of the numeric methods, of which the finite element method is the most used [2,9–18]. In this stage, the static behavior of open top and floating roof tanks is performed by finite element method and is compared with the result predicted by the classical theories stated above. For example, Jonaidi and Ansourian [10] argued that the typical magnitude of maximum error for the range n considered is in the range of 10–18% depending on the shell geometry and Iratio. Moreover, other types of tanks are involved in the analyses, such as tanks with a roof. Meanwhile, various analysis types are carried out, including the linear bifurcation analysis and the buckling analysis. For example, Jonaidi and Ansourian [11] studied the linear buckling of closed-top cylindrical shells subjected to edge vertical deformation. In their work, the radial and circumferential constraints were imposed at the top of shell. It was concluded that for uniform shells, the shear buckling mode is observed for a small wave number and the mode is dominated by axial buckling with the increasing wave number; the shear buckling mode occurs in the upper region of the shell at high wave number for the tapped shells. However, the assumption of imposing the radial and circumferential deflections at the top edge is not reasonable. The results will be more accurate if the model of the roof is built. Godoy and Sosa [12] analyzed the buckling strength of conical roof tanks induced by the localized support settlements. A linear analysis, a geometrically nonlinear analysis and a bifurcation buckling analysis were performed using the finite element computer package ABAQUS. It was revealed that the equilibrium path is highly nonlinear and that the shell displays a plateau for a settlement of the order of half the shell thickness; the geometric nonlinearity should be included, because linear results provide a poor indication of the real shell displacements. However, the effects of some factors are not considered, including the heightto-radius ratio and slope of the conical roof. Later, Godoy and Sosa [13] carried out an experiment for a small flat roof tank subjected to harmonic settlement. It was displayed that the equilibrium path is nonlinear and the shell displays a stable symmetric bifurcation behavior; moreover, the results of the experiments are consistent with that of the computational results. Cao and Zhao [14] studied the buckling behavior of the fixed-roof tank under harmonic settlement. Three types of buckling were involved: elastic bifurcation analysis, geometrical nonlinear analysis of the perfect shell and geometrical nonlinear analysis of the imperfect shell. Moreover, the effects of the wave number, radiusto-thickness ratio, height-to-radius ratio, and the geometric imperfection on the buckling strength were investigated. It was indicated that both the critical harmonic settlement and the buckling mode are closely related to the geometric parameters mentioned above. However, the finite element model of the roof is not built directly, which is replaced by restraining the radial and the circumferential displacement of the joints between the shell and the roof. Thus, the results will be more accurate if the finite element model of the roof is considered. To make the results more reasonable, Gong et al. [15] reported the buckling behavior of conical roof tanks subjected to harmonic settlement. The geometrical nonlinear behavior was included in the analyses. The buckling modes, settlement–displacement curves and critical harmonic settlements for various wave numbers were provided. Besides, the parametric studies were conducted, including the height-to-radius ratio, radius-to-thickness ratio, slope and thickness of the conical roof. The work mentioned above focuses on the buckling behavior of conical roof tanks or fixed roof tanks.
However, the buckling behavior of open top or floating roof tanks subjected to harmonic settlement is rarely reported. Portela and Godoy [16] discussed the buckling behavior of an open top tank under wind pressures and imperfection sensitivity was conducted. It was revealed that buckling occurs at the windward and the imperfection of the shell has a great influence on the buckling load. Nevertheless, the harmonic settlement at the base is not included. Holst and Rotter [2,17] studied the effect of local settlement on the elastic buckling of a thin cylindrical shell under axial compression. It was concluded that the local settlement with a small amplitude can cause a severe loss of buckling strength, and the growth of the dimple appears to protect the shell from the greater strength loss. However, the main attention of their work is on the axial buckling load. Although the buckling behavior of open top or floating roof tanks is rarely studied, several tank failures due to support settlement in practical engineering are reported. For example, Jia [19] mentioned two floating roof tanks damaged by the differential settlement in one petrochemical plant located in Shanghai. For one tank (Tank 124#), it is partially located on a pre-consolidated soil. For the other tank (Tank 125#), part of the foundation is close to a river and the foundation treatment was not carried out before construction. Finally, both tanks lost effectiveness due to the continuously increasing differential settlement. Jiang [20] reported another two tank failures. For each tank, only part of the tank locates on ground consolidated by the earlier presence of a tank. As time goes on, the differential settlement beneath the tank wall caused the failure of the tank. Thus, investigating the buckling behavior of open top and floating roof tanks subjected to the harmonic settlement is urgent and required. In this work, the open top tank is chosen as a research subject. Considering that a floating roof nearly has no structural effect on the tank, an open top tank is a floating roof tank. Thus, the open top tank in following also means floating roof tank. The geometrical nonlinear buckling analyses (GNA) for open top tanks are conducted in following, which are also recommended by Rotter and Schmidt [21] for determining the load carrying capacity. Ref. [21] is a very important reference book for engineers and researchers, which provide the designer of a metal shell structure with an extended guide to the design of the shell against buckling failure. The structure of this work is as follows: Firstly, the finite element model of an open top tank is introduced in Section 2. Then, the buckling behavior of the open top tank subjected to harmonic settlement is described in Section 3. Then, the parametric studies for the buckling behavior, composed of the height-to-radius (h/r) ratio and the radius-to-thickness (r/t) ratio, are discussed in Section 4. Finally, comparisons of buckling behavior for open top tanks and conical roof tanks are reported in Section 5. 2. Finite element model of open top tank The tank discussed in following is based on the tank adopted by Godoy and Sosa [12]. However, the tank mentioned in Ref. [12] is a conical roof tank, not an open top tank. To study the buckling behavior of the open top tank, the conical roof of the tank discussed in Ref. [12] is omitted directly, while other structure parameters of the tank remain unchanged. In this way, we can compare the buckling results of the open top tank with that of the conical roof tank. In addition, for simplifying the calculations, the variations of the shell thickness are neglected. The structure parameters of an open top tank are listed in Table 1. Table 1 Structure parameters of open top tank. Inner radius r (m) Height of tank h (m) Shell thickness t (mm)
15.12 12.191 8.92
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
ABAQUS, a general purpose finite element computer package, is used to conduct all the analyses [2,10–13,15–17,22]. Firstly, the geometric model of the open top tank is built. Since the open top tank model is symmetrical about the plane Z ¼0 (see Fig. 1), the model of half of the structure is applied to reduce the computation effort. Secondly, the geometric model of the open top tank is discretized by the 4-node shell element S4. After the discretization of the tank model, 6, 560 shell elements are generated and the finite element mesh model is shown as Fig. 1. Then, the material properties for the open top tank are set. The model is assumed to have the properties of steel, with elastic modulus E¼2.06 1011 Pa and Poisson’s ratio v ¼0.3. Finally, the boundary conditions of the open top tank are given as follows: (1) The symmetry constraints are applied in the symmetry plane Z ¼0 (see Fig. 1). (2) The radial displacement and the circumferential displacement are constrained at the base of the shell, while the axial displacement (Y-axis) is u ¼un cos(nj), where, un is the amplitude of the nth harmonic settlement; n is wave number; j is the angle of the point at the base of the shell. In addition, to obtain the buckling point of the whole model, the arc-length method is employed [22–25], which can track not only the process of pre-buckling stage but also the process of the post-buckling stage. Moreover, considering that the tank model may produce a large deformation when buckling happens, a large displacement analysis for the model is necessary.
3. General buckling behavior The main goal of the buckling analysis is to obtain the critical harmonic settlement at the base of the shell for different wave
Fig. 1. Finite element mesh model of open top tank.
39
numbers. Fig. 2 shows the typical settlement–displacement curves for n¼10, 14, 18, 22 with the typical tank geometry r/t¼1700 and h/r¼0.8. In Fig. 2, the vertical axis stands for the load factor of the harmonic settlement un normalized with respect to the shell thickness, while the horizontal axis represents the out-of-plane radial displacement w normalized by the shell thickness. For a small wave number (e.g. n¼10) (see Fig. 2a), first of all, the load factor changes linearly. Then, the load factor increase nonlinearly when the linear range is got across. Finally, the buckling point of the open top tank is obtained and the tank falls into the post-buckling stage, in which the load factor displays a strong nonlinearity, presenting an odd behavior shown in zoom in Fig. 2a). For this odd behavior, we can understand it like this. When the buckling point A1 is reached, the shell falls into the post-buckling stage. At this time, the response shows a negative stiffness and the structure must release strain energy to remain in equilibrium. After achieving a limit point B1, the load factor increases with the radial displacement increasing. Later, a new critical point A2 is obtained. Then, the curve starts to decrease with the increasing radial displacement again. This is why the equilibrium path shows the odd behavior in zoom in Fig. 2a). Moreover, the stiffing path with large settlement during the post-buckling stage is not evident, which is quite different from that for wave number n¼14, 18, 22. This reveals that the tank does not have a good settlement sustaining capacity for a small wave number. For large wave numbers (e.g. n¼14, 18, 22) (see Fig. 2b), the buckling path is highly unstable with the displacement increasing, which is a clear sign of the instability of the tank. Moreover, the load factor shows a little different behavior, which is different from that of the wave number n¼10. Exactly speaking, the post-buckling balance path reveals a good settlement sustaining capacity of the tank for large wave numbers. Fig. 3 shows the buckling modes of the open top tank for the wave number n¼6, 10, 18, 22. For a small wave number, buckling happens on the upper shell (e.g. n¼6, 10). However, for large wave numbers, buckling happens on the middle part of the shell (e.g. n¼ 18) or the bottom part of the shell (e.g. n¼22). In other words, the buckling location is closer to the base with the wave number on the increase. The main cause for this lies in that the harmonic settlement at the base becomes more and more difficult to be transferred to the upper shell. The harmonic settlement can be transferred to the upper shell for a small wave number, but can only be transferred to other places on the shell for large wave numbers. That is why buckling happens on the upper shell for a small wave number but occurs at the lower shell for large wave numbers. As is described at the beginning of this section, the main goal for the buckling analysis is to acquire the critical harmonic settlement at the base of the shell. For the critical harmonic settlement, it corresponds to the limit point A or A1 in Fig. 2. Here, we will focus on the relation of the critical harmonic settlement and the wave number, as is shown in Fig. 4. The vertical axis
Fig. 2. Typical settlement–displacement curves for various wave numbers (h/r¼ 0.8 and r/t ¼1700).
40
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
Fig. 3. Buckling modes for various wave numbers (h/r ¼0.8 and r/t ¼1700).
Fig. 4. Critical harmonic settlement–wave number curve (h/r ¼ 0.8 and r/t ¼1700).
Fig. 5. Critical harmonic settlement versus wave number for various h/r ratios (r/t ¼1700).
represents the load factor of the critical harmonic settlement normalized with respect to the shell thickness, while the horizontal axis stands for the wave number. In Fig. 4, on the whole, the critical harmonic settlement decreases more and more slightly with the increasing wave number. Specifically, when the wave number is small, the critical harmonic settlement decreases significantly. However, when the wave number is large, the critical harmonic settlement decreases slightly. This means the tank can stand a larger harmonic settlement when the wave number is small, while it can only hold a smaller harmonic settlement when the wave number is large.
numbers for constant r/t ¼1700. It can be observed that for a certain h/r ratio, the critical harmonic settlement decreases versus the wave number is monotonically decreasing. Take the ratio h/r ¼0.5 for example. The critical harmonic settlement decreases greatly for a small wave number, particularly for a wave number smaller than or equal to 4. However, the critical harmonic settlement decreases slightly for large wave numbers, especially for a wave number larger than or equal to 10. Also, with the h/r ratio increasing, the critical harmonic settlement decreases more and more slightly. For example, for the ratio h/r ¼0.5, the critical harmonic settlement at n ¼2 is 183.6, which is more than 2, 200 times than that at n ¼22. However, for the ratio h/r ¼2.0, the critical harmonic settlement at n¼ 2 is about 126.2, which is more than 1, 500 times than that at n¼ 22. Fig. 6 displays the critical harmonic settlements of different wave numbers under various h/r ratios. For a certain wave number, the critical harmonic settlement versus the h/r ratio decreases monotonically. Also, with the wave number on the increase, the critical harmonic settlement decreases more and more slightly. For example, when the wave number n is 2, the critical harmonic settlement at h/r ¼0.5 is 183.6, which is about 1.45 times than that at h/r ¼2.0. Nevertheless, when wave number n is 22, the critical harmonic settlement at h/r¼ 0.5 is about 0.0805, which is about 1.0 times than that at h/r¼ 2.0.
4. Parametric study In this section, two factors which are related to the critical harmonic settlement at the base of the shell will be discussed, composed of the height-to-radius (h/r) ratio and the radius-tothickness (r/t) ratio. 4.1. Height-to-radius ratio (h/r) Height-to-radius ratios ranging from 0.5 to 2.0 are studied in this section. Fig. 5 demonstrates the results for various wave
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
Fig. 6. Critical harmonic settlement versus h/r ratio for various wave numbers (r/t¼ 1700).
41
Fig. 8. Critical harmonic settlement versus r/t ratio for various wave numbers (h/r¼ 0.8).
top tank has the same structure parameters as the conical roof tank (except for the conical roof), comparing the results of the open top tank with that of the conical roof tank is necessary. For the conical roof tank mentioned in Ref. [12], the buckling results are given by Gong et al. [15]. Here, four factors will be discussed below: the buckling mode, the wave number, the height-to-radius (h/r) ratio and the radius-to-thickness (r/t) ratio. 5.1. Buckling modes
Fig. 7. Critical harmonic settlement versus wave number for various r/t ratios (h/r ¼0.8).
4.2. Radius-to-thickness ratio (r/t) Radius-to-thickness ratios ranging from 500 to 2000 are studied in this section. Fig. 7 presents the results for various wave numbers for constant h/r ¼0.8. It can be observed that, for a certain r/t ratio, the critical harmonic settlement decreases greatly for a small wave number. However, the critical harmonic settlement decreases slightly for large wave numbers. Also, with the increasing r/t ratio, the critical harmonic settlement decreases more and more slightly. Fig. 8 explains the critical harmonic settlements of different wave numbers under various r/t ratios. For a certain wave number, the critical harmonic settlement versus the r/t ratio is monotonically decreasing. Moreover, with the increasing wave number, the critical harmonic settlement decreases more and more slightly. Specially, for a small wave number, the critical harmonic settlement decreases greatly with the r/t ratio, but decreases slightly with the r/t ratio for large wave numbers.
5. Discussion Through the analyses mentioned above, the buckling results of the open top tank have been obtained. Considering that the open
In this work, buckling modes for open top tanks happens at the top edge of the shell for a small wave number, while it changes to the lower shell with the wave number increasing. The behavior is different from that for conical roof tanks. For conical roof tanks, buckling occurs at the conical roof when the wave number is small, but it happens on the lower shell for large wave numbers. Although the buckling modes for these two types of tanks are different from each other, the explanations are nearly the same, which are shown below: for a small wave number, the base settlement can be transferred to the upper shell or the roof, while the settlement can only be transferred to lower part of the shell for large wave numbers. 5.2. Wave number In this work, for the original tank, the critical harmonic settlement decreases significantly for a small wave number, but decreases slightly for large wave numbers. The buckling behavior is much like that of the conical roof tank. Although the tendency of the two types of tanks is similar to each other, the values of the critical harmonic settlement for both tanks have a big difference. For instance, when the structure parameters of the tank is the same (h/r ¼0.8 and r/t ¼1 700), the critical harmonic settlement of the open top tank at n ¼2 is 161.4, which is more than 300 times than that of the conical roof tank at n ¼2. This is because for the open top tank, the top edge of the shell is unrestrained and the upper shell can deflect freely. However, for the conical roof tank, the conical roof restrains the deformation of the upper shell. Thus, the open top tank can hold a larger critical harmonic settlement than the conical roof tank for a small wave number. However, for large wave numbers, there is a small difference for both tanks. For example, the critical harmonic settlement of the open top tank for wave number n¼14 is 0.187, which is only 1.3 times than that of the conical roof tank for wave number n¼14. This is because buckling occurs at the lower shell for large wave numbers. At this
42
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
time, the stiffening effect of conical roof on the lower shell is limited. Thus, the critical harmonic settlements for both types of tanks have a small difference for large wave numbers. 5.3. Height-to-radius ratio (h/r) In this work, for a certain wave number, the critical harmonic settlement versus the h/r ratio decreases monotonically. Also, with the wave number on the increase, the critical harmonic settlement decreases more and more slightly. The behavior is quite different from that of the conical roof tank. For the conical roof tank, the critical harmonic settlement versus the h/r ratio is complex when the wave number is smaller than 6, while the critical harmonic settlement increases with the h/r ratio on the increase when the wave number is larger than or equal to 6. 5.4. Radius-to-thickness ratio (r/t) In this work, for a certain wave number, the critical harmonic settlement versus the r/t ratio is monotonically decreasing. Moreover, with the increasing wave number, the critical harmonic settlement decreases more and more slightly. The behavior is similar to that of the conical roof tank for a wave number larger than or equal to 6. However, for a wave number smaller than 6, the open top tank displays a different behavior. The critical harmonic settlement versus the r/t ratio is monotonically increasing when the wave number n is 2 and 3, while the critical harmonic settlement increases firstly, and then decreases when the wave number n is 4 and 5. 5.5. Summary For the open top tank and the conical roof tank, the main difference is the conical roof. For a small wave number, buckling occurs at the upper shell. For the existence of the conical roof, the deformation of the top edge of the shell is restrained and the upper shell cannot deflect freely. Thus, the conical roof tank can only deflect in a small range before attaining the buckling point. This means the open top tank can hold a larger harmonic settlement than the conical roof tank for a small wave number. However, for large wave numbers, there is a small difference in settlement sustaining capacity for open top tanks and conical roof tanks.
6. Conclusions This work deals with the buckling behavior of the open top tanks subjected to the harmonic settlement by means of numerical studies. The following conclusions can be drawn from the results of the investigation. 1) Buckling occurs on the upper part of the shell for a small wave number, while it changes to the middle or the lower part of the shell for large wave numbers. In other words, the buckling location is closer to the base of the shell with the wave number on the increase. 2) Regarding the h/r ratio, for a certain wave number, the critical harmonic settlement versus the h/r ratio decreases monotonically. Also, with the wave number on the increase, the critical harmonic settlement decreases more and more slightly. 3) Regarding the r/t ratio, for a certain wave number, the critical harmonic settlement versus the r/t ratio is monotonically decreasing. Moreover, with the increasing wave number, the critical harmonic settlement decreases more and more slightly.
4) When the open top tank has the same structure parameters as the conical roof tank (except for the conical roof), the open top tank can hold a larger critical harmonic settlement than the conical roof tank for no restraint from the roof for a small wave number. However, for large wave numbers, there is a small difference in the settlement sustaining capacity for open top tanks and conical roof tanks.
Acknowledgments During the writing stages of the work, Ms. Sun gave the authors valuable advice. Without her help, it would have been difficult to finish the writing of this work. The authors gratefully acknowledge Prof. W.M. Tao and Associate Prof. Y.M. Xu of Zhejiang University, for their kind assistance with the explanation of the equilibrium path of tanks. References [1] Teng JG, Rotter JM. Buckling of thin metal shells. London: Spon Press; 2004. [2] Holst JMFG, Rotter JM. Axially compressed cylindrical shells with local settlement. Thin-Walled Structures 2005;43(5):811–25. [3] Malik Z, Morton J, Ruiz C. Ovalization of cylindrical tanks as a result of foundation settlement. Journal of Strain Analysis for Engineering Design 1977;12(4):339–42. [4] Kamyab H, Palmer SC. Analysis of displacements and stresses in oil storage tanks caused by differential settlement. Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science 1989;203(1):61–70. [5] Kamyab H, Palmer SC. Displacements in oil storage tanks caused by localized differential settlement. Journal of Pressure Vessel Technology, Transactions of the ASME 1991;113(1):71–80. [6] Palmer SC. Structural effects of foundation tilt on storage tanks. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 1992;206(E2):83–92. [7] Palmer SC. Stress in storage tanks caused by differential settlement. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 1994;208(E1):5–16. [8] D’ Orazio TB, Duncan JM, Bell RA. Distortion of steel tanks due to settlement of their walls. Journal of Geotechnical Engineering-ASCE 1989;115(6): 871–90. [9] Zhao Y, Cao QS, Xie XY. Floating roof steel tanks under harmonic settlement: FE parametric study and design criterion. Journal of Zhejiang University, Science A 2006;7(3):398–406. [10] Jonaidi M, Ansourian P. Harmonic settlement effects on uniform and tapered tank shells. Thin-Walled Structures 1998;31(1–3):237–55. [11] Jonaidi M, Ansourian P. Buckling of cylindrical shells subjected to edge vertical deformation. Advances in Steel Structures. Oxford: Elsevier; 1999 p. 679–86. [12] Godoy LA, Sosa EM. Localized support settlements of thin-walled storage tanks. Thin-Walled Structures 2003;41(10):941–55. [13] Godoy LA, Sosa EM Deflections of thin-walled storage tanks with roof due to localized support settlement. In: Proceedings II International Conference on Advances in Structural Engineering and Mechanics. Seoul (Korea): Techno-Press; 2002. [14] Cao QS, Zhao Y. Buckling strength of cylindrical steel tanks under harmonic settlement. Thin-Walled Structures 2010;48(6):391–400. [15] Gong J, Cui W, Zeng S. Buckling analysis of large scale oil tanks with a conical roof subjected to harmonic settlement. Thin-walled Structures 2012;52(7): 143–8. [16] Portela G, Godoy LA. Wind pressures and buckling of cylindrical steel tanks with a conical roof. Journal of Constructional Steel Research 2005;61(6): 786–807. [17] Holst JMFG, Rotter JM Nonlinear response and buckling of cylindrical tanks due to foundation settlement. In: Krupka V, editor. Proceedings International Conference on Design, Inspection and Maintenance of Cylindrical Steel Tanks and Pipelines. Prague(Czech Republic); 2003. pp. 29–35. [18] Rotter JM, Cai M, Holst JMFG. Buckling of thin cylindrical shells under locally elevated compressive stresses. Journal of Pressure Vessel Technology 2011;133(1) p. 011204–11. [19] Jia QS. Analysis and treatment for foundation engineering accidents of tanks. Ground Improvement 2006;17(2):10–21 [in Chinese]. [20] Jiang JR. Case study of earth undercutting method applied to correct unevenly settled foundation of 20 000 m3 vertical steel storage tank. Petroleum Engineering Construction 2011;37(4):92–5 [in Chinese]. [21] Rotter JM, Schmidt H. Buckling of Steel Shells: European Design Recommendations. Fifth ed. Brussels: European Convention for Constructional Steelwork; 2008. [22] ABAQUS user’s manual. USA: Hibbit, Karlsson and Sorensen Inc; 2004.
J. Gong et al. / Thin-Walled Structures 63 (2013) 37–43
[23] Riks E. An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures 1979;15(7):529–51. [24] Ramm E. Strategies for tracing the nonlinear response near limit points. In: Non-linear Finite Element Analysis in Structural Mechanics. New York: Springer; 1981. p. 68–89.
43
[25] Crisfield MA. An arc-length method including line searches and accelerations. International Journal for Numerical Methods in Engineering 1983;19(9):1269–89.