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Buckling of cylindrical shells with measured settlement under axial compression
Thin-Walled Structures 123 (2018) 351–359
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Buckling of cylindrical shells with measured settlement under axial compression
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Zhiping Chena, , Haigui Fanb, Jian Chenga, Peng Jiaoa, Feng Xua, Chenchao Zhenga a b
Institute of Process Equipment, Department of Chemical and Biological Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang 310027, PR China School of Chemical Machinery and Safety Engineering, Dalian University of Technology, 2 Linggong Road, Dalian, Liaoning 116023, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cylindrical shells Measured settlement Axial buckling Geometrical nonlinearity
Buckling behavior of cylindrical shells with measured settlement under axial compression has been researched and analyzed in this paper. Using the method of Fourier series expansion, the measured settlement data is transformed into differential settlement. Applying the finite element simulation method, the differential settlement is applied to bottom of the cylindrical shell in order to simulate behavior of the cylindrical shell under differential settlement. Based on the deformed cylindrical shell caused by settlement, the finite element simulation method is applied to research buckling behavior of the cylindrical shell under axial compression considering the geometric nonlinearity and large deformation. The meridional membrane stress distribution in the circumferential direction of the cylindrical shell under differential settlement is obtained and compared with the differential settlement distribution. Effects of liquid storage on buckling behavior of the cylindrical shell are researched by applying hydrostatic pressure to the inner surface. Parametric analysis is carried out to research the effects of diameter-thickness ratio and height-diameter ratio of the cylindrical shell on the axial buckling capability. Results show that the cylindrical shell will be subjected to deformation and meridional membrane stress because of the differential settlement and the critical axial buckling load will be decreased then. Liquid storage is helpful for the cylindrical shell to remain stable. The cylindrical shell with larger height-diameter ratio and smaller diameter-thickness ratio is more capable to resist buckling under axial compression following differential settlement.
1. Introduction As a typical thin walled structure, cylindrical shells are widely used in various industries, such as steel silos, oil storage tanks and nuclear reactors [1–3]. These structures may be subjected to many kinds of axial compression, like gravity, vacuum effects and uplift caused by earthquake. The axial buckling capacity should be regarded as a controlling factor in the design stage of cylindrical shells in order to avoid potential failure. A collapsed steel silo caused by axial compression has been shown in Fig. 1. Buckling of cylindrical shells under axial compression has attracted extensive attention in recent years. Results of previous researches show that initial imperfections of the cylindrical shell have great influence on the axial buckling strength [4–6]. The researches of initial imperfections are mainly about the dimensional discrepancy caused in the manufacturing process, longitudinal and girth welds, local imperfection and damage [7–10]. In engineering, uneven soil deposition, especially for the coastal soft soil foundation, will lead to differential settlement of the cylindrical shell structures [11]. Deformation and stress ⁎
redistribution of the cylindrical shell caused by settlement may weaken its mechanical properties. The cylindrical shell will be unable to continue working as usual when the differential settlement reaches a critical value. Zhao et al. [12–14] and Gong et al. [15–17] had done some researches about buckling of cylindrical shells under settlement applying finite element simulation. However, for most cylindrical shells in service, settlement will not cause instability or failure directly. But the deformation and stress redistribution caused by settlement will have great influence on the axial buckling strength of the cylindrical shell. Compared to initial imperfections, deformation and stress of the cylindrical shells caused by settlement are much larger. Holst and Rotter [18] researched buckling behavior of axially compressed cylindrical shells with local settlement. The local settlement was assumed as local uplift in the form of quasi-continuous with a sinusoidal variation beneath the cylindrical shell. The initiation and development of deformation caused by local settlement and its effect on buckling of the cylindrical shell under axial compression had been studied then. However, in engineering, settlement beneath the cylindrical shell is distributed randomly in the circumferential direction. Several
Corresponding author. E-mail address: [email protected] (Z. Chen).
https://doi.org/10.1016/j.tws.2017.11.006 Received 29 March 2016; Received in revised form 16 October 2017; Accepted 7 November 2017 Available online 01 December 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
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Fig. 2. Measured settlement data of the tank.
theoretical research of settlement [19–21], the discrete settlement data obtained from the measuring points can be transformed into differential settlement expression applying the method of Fourier series expansion, as shown in Eq. (1).
Fig. 1. A collapsed steel silo caused by axial compression.
u = 0.088 × − 0.872 − 0.706 + 1.011
measuring points are located on basement of the cylindrical shell in order to obtain settlement data. Review of previous literature shows that research about buckling of cylindrical shells with measured settlement under axial compression is quite limited. It is quite significant to research the measured settlement beneath the cylindrical shell and buckling behavior of the cylindrical shell under axial compression. This paper focuses on the measured settlement in engineering to research the effect of differential settlement on the cylindrical shell's deformation. Based on that, the axial compression is applied to the deformed cylindrical shell and the critical axial buckling load coefficient will be obtained. The influence mechanism of settlement on axial buckling behavior is researched. Effects of differential settlement amplitudes on the critical axial buckling load coefficient of the cylindrical shell are also studied. Meanwhile, structure parameters and liquid storage of the cylindrical shell are also determining factors to the deformation and meridional membrane stress under settlement. The axial buckling capacity of the cylindrical shell with different liquid storages is analyzed. Parametric analysis of structure parameters has been carried out to research effects of structure parameters on the axial buckling strength of the cylindrical shell with measured settlement under axial compression.
cos(2θ ) − 0.045 × sin(2θ ) + 0.944 × cos(3θ ) × sin(3θ ) + cos(4θ ) − 2.375 × sin(4θ ) + 1.746 × cos(5θ ) × sin(5θ ) − 0.088 × cos(6θ ) − 1.545 × sin(6θ ) × cos(7θ ) − 0.601 × sin(7θ ) − 0.188 × cos(8θ )
(1)
where u represents the differential settlement beneath the cylindrical shell with unit of mm and θ is the circumferential coordinate with the unit of rad. Diagram of the differential settlement distributed in the circumferential direction beneath the cylindrical shell is shown in Fig. 3. The horizontal coordinate is central angle, which satisfies φ = θ × 180/π. The differential settlement u derived from measured settlement data has been shown in Eq. (1). In order to research behavior of the cylindrical shell under different settlements, the settlement amplitude is defined as λ = u0/u. When λ is selected as different values, different differential settlement expressions will be obtained by u0 = λu. The commercial finite element package ABAQUS is applied in the simulation analysis. The finite element model of the cylindrical shell is
2. Behavior of cylindrical shell under measured settlement As the foundation property changes differently in the circumferential direction, the cylindrical shell will be subjected to differential settlement. In order to reveal the deformation behavior of the cylindrical shell under settlement in engineering, the measured settlement data of an oil storage tank is selected in this paper [14]. The oil storage tank is a typical thin walled structure and 16 measuring points have been located evenly at the bottom along circumferential direction to record settlement of the tank. The measuring point number and the corresponding settlement data are shown in Fig. 2, with the unit of mm. The measured data reflects settlement of some discrete points in the circumferential direction beneath the cylindrical shell. In order to analyze deformation behavior of the cylindrical shell under settlement, the measured settlement data needs to be transformed into the continuous distributed form in circumferential direction. According to the
Fig. 3. Differential settlement distributed in the circumferential direction.
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Fig. 6. Radial displacement vs settlement curves of the cylindrical shell.
represents settlement amplitude. Fig. 6 shows that the radial displacement presents approximately linear variation as the settlement increases. It indicates that the cylindrical shell doesn’t buckle or lose stability under the settlement as shown in Eq. (1). The maximum outward and inward displacements are 2.3t and 1.1t respectively. Previous research shows that axial buckling strength of the cylindrical shell is quite sensitive to initial imperfections. The critical axial buckling load will be weakened seriously compared to the perfect cylindrical shell because of initial imperfections. The deformation and stress caused by settlement can be regarded as initial imperfections when the cylindrical shell is under axial compression. Effects of differential settlement on the axial buckling behavior will be researched based on the deformed cylindrical shell then.
Fig. 4. Finite element model of the cylindrical shell.
shown in Fig. 4, with the diameter of 24.75 m, height of 19.8 m and thickness of 15 mm. As a typical thin walled structure, the 4-noded doubly curved S4R shell element has been chosen in the cylindrical shell model. The shell material is isotropic with Young's modulus E = 206 GPa and Poisson ratio v = 0.3. Radial displacement of the cylindrical shell's bottom is restrained and the top boundaries are fully restrained except the axial displacement. The differential settlements with different amplitudes are applied to the cylindrical shell's bottom as axial displacement boundaries. Behavior of the cylindrical shell under differential settlement is performed as geometric nonlinear and static. The radial displacement nephogram of the cylindrical shell under different settlement amplitudes is shown in Fig. 5. The scale factor is selected as 20. As it can be seen, outward bulges and inward depressions have occurred to the cylindrical shell because of the differential settlement. When the settlement amplitude reaches 1.0, outward bulges are mainly concentrated in the bottom areas, while inward depressions are mainly distributed in the middle and bottom areas. The maximum outward displacement point and maximum inward displacement point of the cylindrical shell with settlement amplitude λ = 1.0 in Fig. 5 are selected to obtain the relation curves of radial displacement and settlement. As shown in Fig. 6, the horizontal coordinate represents the ratio of radial displacement ω to thickness t and the vertical coordinate
3. Buckling behavior of axially compressed cylindrical shell following settlement A reference point is created at center of the top boundary and set as coupling constraint with the top boundary of the cylindrical shell. The bottom of the cylindrical shell is set as fixed support and the top is fully restrained except the axial displacement. The uniform axial compression on the deformed cylindrical shell is simulated by applying downward displacement to the reference point. The relation curves of maximum radial displacement and axial load coefficient of the cylindrical shell can be obtained by using the Riks method to determine the
Fig. 5. Radial displacement nephogram under different settlement amplitudes.
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Fig. 7. Radial displacement nephogram of the axially compressed shell following the settlement amplitude λ = 1.0.
critical value, the maximum radial displacement begins to increase rapidly along with slight decrease of axial pressure. It means that the cylindrical shell has been subjected to classic primary buckling. Eight points on the load-displacement curve, i.e. P0-P7, are selected to describe the buckling process. P0 represents that the settlement is just finished and the axial compression is about to begin. P1 is in the initial stage of axial compression. P2 is critical buckling point. When the axial pressure increases to P2, the cylindrical shell begins to buckle and the vertical coordinate value corresponding to P2 is defined as critical axial buckling load coefficient, which is presented by μcr. The points of P3–P7 are in the post-buckling stage of the cylindrical shell. An obvious depression has occurred to the cylindrical shell under axial compression and the maximum radial displacement point of the depression is selected. As shown in Fig. 9, the path going through this point along the axial direction of the cylindrical shell is established as Path 1, and the path going through this point along the circumferential direction of the cylindrical shell is established as Path 2. The radial displacements of Path 1 at different points from P0 to P7 are plotted in Fig. 10. In Fig. 10, the horizontal coordinate is ratio of radial displacement to thickness of the cylindrical shell. Positive means outward deformation and negative means inward deformation. The vertical coordinate is height of the cylindrical shell. As it can be seen that in the initial stage of the axial compression process, the radial displacement is mainly concentrated in the middle part of the cylindrical shell. As the axial compression continues, displacement in the middle part decreases gradually and displacement in the bottom part begins to increase, following the displacement mutation at x = 0.1h. A local deformation at x = 0.1h of the cylindrical shell begins to form at P2. With the axial compression continues, the local deformation experiences a rapid
buckling equilibrium paths considering geometric nonlinear and large deformation. Fig. 7 shows the radial displacement nephogram of the cylindrical shell as axial compression increases when the settlement amplitude is 1.0. As it can be seen that in the initial stage, the radial displacements are mainly caused by settlement. They are mainly located at the concentration areas of the differential settlement in the circumferential direction. The radial displacements at the top areas of the cylindrical shell are generally small. As the axial compression increases gradually, the radial displacements become more and more concentrated until an obvious depression occurs to the cylindrical shell at the bottom region. The radial displacements caused by differential settlement are mainly located at the bottom areas, leading to the initial imperfections at the bottom areas larger than other areas. Thus, the bottom areas of the cylindrical shell are more likely to be subjected to local buckling under axial compression. Fig. 8 shows the load-displacement curve of the cylindrical shell under axial compression following the settlement amplitude λ = 1.0. The horizontal coordinate is the ratio of maximum radial displacement ωm to thickness t of the cylindrical shell. The vertical coordinate is axial load coefficient, which is the ratio of axial pressure in this cylindrical shell to the classical critical buckling pressure of perfect cylindrical shell, i.e. μ = σ/σcl, σ represents axial pressure, σcl = 1.21Et/D represents the critical buckling pressure of perfect cylindrical shell under axial compression, D is diameter. It can be seen that the maximum radial displacement varies linearly approximately as the axial pressure increases in the initial stage. When the axial pressure reaches the
Fig. 8. Load-displacement curve under axial compression when the settlement amplitude is 1.0.
Fig. 9. Diagram of Path 1 and Path 2.
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increase. It indicates that buckling of the cylindrical shell occurs around x = 0.1h at Path 1. When a cylindrical shell is subjected to axial pressure, the meridional membrane stress of the shell will increase. When the meridional membrane stress reaches critical value, even a small increase of stress will cause a very large deformation, leading to buckling of the cylindrical shell. As we know, meridional membrane stresses throughout the cylindrical shell are quite sensitive to deformations and imperfections. Because of the deformation caused by differential settlement, meridional membrane stress distribution of the cylindrical shell will be influenced and the axial buckling strength will be affected then. In order to analyze buckling mechanism of the cylindrical shell with differential settlement under axial compression, the meridional membrane stress distribution along Path 2 of the cylindrical shell under differential settlement is obtained. Relation of differential settlement and meridional stress distributions is shown in Fig. 11. The right vertical coordinate in Fig. 11 is ratio of meridional stress of this cylindrical shell σx to the classical critical buckling pressure of perfect cylindrical shell σcl. Positive means that the meridional stress is tensile stress, negative means compressive stress. For the differential settlement distribution, positive means downward displacement, negative means uplift. As we can see, the distribution characteristics of differential settlement and meridional stress are similar. The positive parts and negative parts of them are almost the same, and larger differential settlement produces larger meridional stress. It indicates that
downward displacement parts of the differential settlement will cause tensile stress to the corresponding parts of Path 2 and uplift parts will cause compressive stress. Fig. 11 shows that the cylindrical shell has been subjected to local tensile stresses and compressive stresses under differential settlement before axial compression. Based on that, when the axial compression is applied to the cylindrical shell, the local tensile stresses caused by differential settlement and the global compressive stress caused by axial compression will cancel out gradually and the total stresses of these parts will decrease gradually then. However, the local compressive stresses caused by differential settlement will be superimposed by the global compressive stress caused by axial compression, leading to the total stresses of these parts larger than the meridional membrane stress of perfect cylindrical shell under the same axial compression. Because the global compressive stress caused by axial compression is uniformly distributed in the circumferential direction, the location of maximum local compressive stress caused by differential settlement will still be the location of maximum total compressive stress after superimposed by uniform axial pressure. It will be most likely to be subjected to buckling for the location with maximum meridional compressive stress of the cylindrical shell under axial compression. In Fig. 11, the maximum local meridional compressive stress locates at φ = 18.8, which is consistent with the location of depression in Fig. 7 when the cylindrical shell buckles. Due to the superposition of local meridional compressive stresses caused by differential settlement, the uniform axial compression load applying to the cylindrical shell when it buckles will be smaller than the critical axial buckling load of perfect cylindrical shell. It verifies that differential settlement will produce local meridional compressive stresses and the axial buckling capacity of the cylindrical shell will be reduced then. The critical axial buckling load and displacement mutation position of the cylindrical shell are both dependent on the maximum meridional compressive stress caused by differential settlement. In order to analyze the effects of different settlements on the critical axial buckling load, the load-displacement curves of the cylindrical shell under axial compression following different settlement amplitudes are presented in Fig. 12. When the settlement amplitude is zero, i.e. λ = 0, it means that there is no differential settlement and the cylindrical shell remains perfect. The axial load coefficient is 1.01 when buckling occurs, it indicates that the error between critical buckling loads for the perfect cylindrical shell calculated by finite element method and theoretical formula is only 1%, which verifies the rationality and accuracy of the finite element method. Fig. 12 shows that the cylindrical shells with different settlement amplitudes are all subjected to classic primary buckling when the axial pressures increase to critical values. The
Fig. 11. Relation of differential settlement and meridional stress distributions.
Fig. 12. Load-displacement curves under axial compression with different settlement amplitudes.
Fig. 10. Radial displacement variation of Path 1 during axial compression.
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Fig. 14. Meridional stress distributions of the cylindrical shell under different settlement amplitudes.
Fig. 13. Critical axial buckling load coefficient vs settlement amplitude curve.
critical axial buckling load coefficient of each curve, i.e. μcr, reflects the axial buckling capacity of the cylindrical shell following a certain differential settlement. Fig. 12 shows that there are different critical axial buckling load coefficients corresponding to the cylindrical shell with different settlements. The relation curve of critical axial buckling load coefficient and settlement amplitude has been plotted in Fig. 13. The horizontal coordinate is settlement amplitude and the vertical coordinate is critical axial buckling load coefficient of the cylindrical shell. As it can be seen that the critical axial buckling load coefficient decreases almost linearly as the settlement amplitude increases. It indicates that the axial buckling capacity reduction of the cylindrical shell caused by differential settlement will become more serious as the settlement amplitude increases. For example, when the settlement amplitude is 1.0, the critical axial buckling load coefficient is 0.27. It means that the critical axial buckling load of the cylindrical shell following the differential settlement in Eq. (1) is just 0.27 times of that corresponding to the perfect cylindrical shell. Three different settlement amplitudes, including λ = 0.2, λ = 0.6 and λ = 1.0, are taken for examples. For each settlement amplitude, when the cylindrical shell following that settlement buckles under axial compression, the path going through the maximum radial displacement point along the circumferential direction will be established, like Path 2 in Fig. 9. In order to research effects of different settlements on stress redistribution of the cylindrical shell before axial compression, the meridional stress distribution curves along the paths of the cylindrical shell under different settlement amplitudes are shown in Fig. 14. As we can see, the meridional stress distribution characteristics in the circumferential direction of the cylindrical shell under different settlement amplitudes are almost the same. But the meridional stress magnitude is different corresponding to different settlement amplitudes. Larger settlement amplitudes will produce larger meridional stresses of the cylindrical shell, leading to reduction of axial buckling capacity and critical axial buckling load coefficient. This is consistent with the result obtained from Fig. 13. It also explains the influencing mechanism of different settlements on the axial buckling capacity of the cylindrical shell. Buckling research of the cylindrical shell with measured settlement under axial compression shows that the settlement will cause deformation and stress redistribution, leading to meridional tensile and compressive stresses of the cylindrical shell. The meridional compressive stresses caused by settlement will reduce the axial buckling capacity of the cylindrical shell. The critical axial buckling load and radial displacement mutation position of the cylindrical shell depend on the maximum meridional compressive stress caused by settlement. As the
settlement becomes larger, meridional compressive stresses will increase gradually and the ability to resist buckling of the cylindrical shell will decrease then. Thus the settlement is an important factor that needs to be taken into account in the buckling analysis of axially compressed cylindrical shell. The critical axial buckling load will be predicted effectively using the measured settlement data beneath the cylindrical shell. It is helpful to the analysis and research of axial buckling capability and service life of the cylindrical shell in engineering. 4. Effects of liquid storage in the cylindrical shell There are always different materials stored in cylindrical shells in engineering. Variation of liquid storage will produce different pressures on the inner surface of the cylindrical shell and the axial buckling behavior may be affected. The liquid is assumed to be water and six different liquid heights, i.e. h = 0, h = 3.96 m, h = 7.92 m, h = 11.88 m, h = 15.84 m and h = 19.8 m, are selected to research the effects of liquid storage in the cylindrical shell. The hydrostatic pressures will be applied to inner surface of the cylindrical shell for the six different cases. For example, when the liquid height is 11.88 m, the hydrostatic pressure ps = 9800 × (11.88-x) (Pa) will be applied to inner surface from h = 0 to h = 11.88 m, where x represents the axial coordinate. The relation curves of critical axial buckling load coefficient and liquid height of the cylindrical shell following different settlement amplitudes are shown in Fig. 15, where horizontal coordinate is the ratio of liquid height to the cylindrical shell height. As it can be seen in Fig. 15, for different settlement amplitudes, the critical axial buckling load coefficient will increase gradually as the liquid height rises. It indicates that liquid in the cylindrical shell is helpful to improve the buckling capability of axially compressed cylindrical shell with settlement. Taking the settlement amplitude λ = 0.6 as an example, when the liquid height is zero, the critical axial buckling load coefficient is 0.523; when the liquid height is 1.0, the critical axial buckling load coefficient becomes 0.913, which has increased 39%. Meanwhile, when the settlement amplitude is larger, the improvement effects of liquid on the axial buckling capacity of the cylindrical shell will become more obvious. For example, when the settlement amplitude is 0.2, as the liquid height rises from zero to 1.0, the critical axial buckling load coefficient will increase from 0.802 to 0.982, whose difference is 0.18. While when the settlement amplitude is 1.0, as the liquid height rises from zero to 1.0, the critical axial buckling load coefficient will increase from 0.271 to 0.848. The difference is 0.577, which is larger than that of the settlement amplitude λ = 0.2. Analysis shows that liquid in the cylindrical shell is helpful to improve the axial buckling capacity. As we have researched in the empty 356
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Fig. 16. Critical axial buckling load coefficient vs diameter-thickness ratio curves.
Fig. 15. Critical axial buckling load coefficient vs liquid height curves.
2062.5, the critical axial buckling load coefficient will decrease from 0.877 to 0.758, whose difference is 0.119. While when the settlement amplitude is 1.0, as the diameter-thickness ratio increases from 1375 to 2062.5 in the same way, the critical axial buckling load coefficient will decrease from 0.397 to 0.073. The difference is 0.324, which is larger than that of the settlement amplitude λ = 0.2. Therefore, when the settlement beneath the cylindrical shell is larger, the critical axial buckling load will be more sensitive to the diameter-thickness ratio variation. The meridional stress distribution in the circumferential direction of the cylindrical shells with three different diameter-thickness ratios under settlement amplitude λ = 1.0 are shown in Fig. 17. As we can see that under the same differential settlement, the cylindrical shell with larger diameter-thickness ratio will produce greater maximum meridional compressive stress and the critical axial buckling load coefficient will become smaller then. It indicates that differential settlement has greater effect on the axial buckling capacity of the cylindrical shell with larger diameter-thickness ratio.
cylindrical shell, inward deformation begins to grow rapidly and an obvious depression is produced when the axial pressure reaches critical value, leading to buckling of the cylindrical shell. However, when there is liquid stored in the cylindrical shell, hydrostatic pressure will be generated on inner surface. The inward deformation growth caused by settlement and axial compression will be hindered by the hydrostatic pressure and the axial buckling capacity of the cylindrical shell will be improved then. As liquid height rises, hydrostatic pressure will become larger and the improvement effects on the axial buckling capacity will be more obvious. Thus, in engineering, liquid stored in the cylindrical shell needs to be analyzed according to actual situation. While in this paper, the cylindrical shell is assumed to be empty for simplification. 5. Parametric analysis The above analysis shows that settlement beneath the cylindrical shell will decrease the critical axial buckling load and reduce the axial buckling capability. But in engineering, due to the structure size diversity of cylindrical shells, effects of settlement on the critical axial buckling load will be different. Parametric analysis is carried out to research effect of structure size on the critical buckling load of the cylindrical shell with differential settlement under axial compression.
5.2. Height-diameter ratio Seven cylindrical shells with different height-diameter ratios are selected to research the relation of critical axial buckling load coefficient and height-diameter ratio. Relation curves corresponding to different settlement amplitudes have been presented in Fig. 18. The
5.1. Diameter-thickness ratio Seven cylindrical shells with different diameter-thickness ratios are selected to research the relation of critical axial buckling load coefficient and diameter-thickness ratio. Relation curves corresponding to different settlement amplitudes have been presented in Fig. 16. The horizontal coordinate is diameter-thickness ratio. Fig. 16 shows that for different settlement amplitudes, the critical axial buckling load coefficient will decrease as the diameter-thickness ratio increases. It indicates that effect of settlement on the axial buckling strength of the cylindrical shell with larger diameter-thickness ratio are greater than that with smaller diameter-thickness ratio. Taking the settlement amplitude λ = 0.6 as an example, the critical axial buckling load coefficient is 0.603 when the diameter-thickness ratio is 1375. While when the diameter-thickness ratio is 2062.5, the critical axial buckling load coefficient has decreased to 0.425. It verifies that settlement has smaller effect on the critical axial buckling load of the cylindrical shell with smaller diameter-thickness ratio. Meanwhile, for the curves of different settlement amplitudes, reduction magnitudes of the critical axial buckling load coefficient as diameter-thickness ratio increases are also different. For example, when the settlement amplitude is 0.2, as the diameter-thickness ratio increases from 1375 to
Fig. 17. Meridional stress distribution of different diameter-thickness ratios.
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influence on the buckling strength of the cylindrical shell with differential settlement under axial compression. The same settlement will cause different meridional stresses to the cylindrical shells with different structure sizes. Results show that the cylindrical shell with larger diameter-thickness ratio and smaller height-diameter ratio will be subjected to greater meridional compressive stress under settlement and the axial buckling capacity will be affected more seriously. Therefore, a reasonable selection about structure size of the cylindrical shell, such as smaller diameter-thickness ratio and larger height-diameter ratio, will help to increase its axial buckling capability following differential settlement. 6. Conclusions
Fig. 18. Critical axial buckling load coefficient vs height-diameter ratio curves.
This paper presents deformation behavior of the cylindrical shell under differential settlement based on the measured settlement data. Then, buckling behavior of the deformed cylindrical shell under axial compression is researched. Stress distribution of the cylindrical shell under differential settlement and influencing mechanism of settlement on the axial buckling capacity are analyzed. The relations of critical axial buckling load and settlement amplitude, liquid in the cylindrical shell and structure parameters are obtained. The following conclusions may be drawn from this study
Fig. 19. Meridional stress distribution of different height-diameter ratios.
(1) The cylindrical shell will be subjected to deformation because of differential settlement. The deformation caused by differential settlement can be regarded as initial imperfection of the cylindrical shell under axial compression. (2) The cylindrical shell following settlement will be subjected to classic primary buckling under axial compression. The critical axial buckling load can reflect the buckling capacity of the cylindrical shell. The meridional compressive stress caused by differential settlement is crucial to the buckling capacity reduction of the cylindrical shell. As differential settlement beneath the cylindrical shell becomes larger, the meridional stress will grow bigger and the critical axial buckling load will decrease then. It verifies the influencing mechanism of settlement on the buckling capacity of the cylindrical shell. (3) Liquid will generate hydrostatic pressure on inner surface of the cylindrical shell. The inward deformation growth caused by settlement and axial compression will be hindered by the hydrostatic pressure and the axial buckling capacity of the cylindrical shell will be improved. It is helpful to analyze stability of the cylindrical shell with liquid storage. (4) Structure parameters of the cylindrical shell, such as diameterthickness ratio and height-diameter ratio, will have great influence on the critical axial buckling load. The cylindrical shell with smaller diameter-thickness ratio and larger height-diameter ratio is more capable of resisting buckling under axial compression. Thus, a reasonable selection about structure size of the cylindrical shell according to the actual situation in engineering will help to improve the axial buckling capability following differential settlement, the stability and safety of the cylindrical shell will also be enhanced.
horizontal coordinate is height-diameter ratio. Fig. 18 shows that for different settlement amplitudes, the critical axial buckling load coefficient will increase as the height-diameter ratio increases. It indicates that effect of settlement on the axial buckling capacity of the cylindrical shell with smaller height-diameter ratio are greater than that with larger height-diameter ratio. Taking the settlement amplitude λ = 0.4 for an example, the critical axial buckling load coefficient is 0.721 when the height-diameter ratio is 1.2. While when the height-diameter ratio is 0.6, the critical axial buckling load coefficient is 0.631. It verifies that differential settlement has smaller effect on the critical axial buckling load of the cylindrical shell with larger height-diameter ratio. Similarly, the meridional stress distribution in the circumferential direction of the cylindrical shells with three different height-diameter ratios under settlement amplitude λ = 1.0 are plotted in Fig. 19. It shows that under the same differential settlement, the cylindrical shell with smaller height-diameter ratio will be subjected to greater maximum meridional compressive stress and the critical axial buckling load coefficient will become smaller then. It indicates that differential settlement has greater effect on the axial buckling capacity of the cylindrical shell with smaller height-diameter ratio. Figs. 16–19 have also verified that the meridional compressive stress caused by differential settlement plays a decisive role in the axial buckling capacity of the cylindrical shell. The parametric analysis shows that structure size has great
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Full length article
Buckling of cylindrical shells with measured settlement under axial compression
T
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Zhiping Chena, , Haigui Fanb, Jian Chenga, Peng Jiaoa, Feng Xua, Chenchao Zhenga a b
Institute of Process Equipment, Department of Chemical and Biological Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang 310027, PR China School of Chemical Machinery and Safety Engineering, Dalian University of Technology, 2 Linggong Road, Dalian, Liaoning 116023, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cylindrical shells Measured settlement Axial buckling Geometrical nonlinearity
Buckling behavior of cylindrical shells with measured settlement under axial compression has been researched and analyzed in this paper. Using the method of Fourier series expansion, the measured settlement data is transformed into differential settlement. Applying the finite element simulation method, the differential settlement is applied to bottom of the cylindrical shell in order to simulate behavior of the cylindrical shell under differential settlement. Based on the deformed cylindrical shell caused by settlement, the finite element simulation method is applied to research buckling behavior of the cylindrical shell under axial compression considering the geometric nonlinearity and large deformation. The meridional membrane stress distribution in the circumferential direction of the cylindrical shell under differential settlement is obtained and compared with the differential settlement distribution. Effects of liquid storage on buckling behavior of the cylindrical shell are researched by applying hydrostatic pressure to the inner surface. Parametric analysis is carried out to research the effects of diameter-thickness ratio and height-diameter ratio of the cylindrical shell on the axial buckling capability. Results show that the cylindrical shell will be subjected to deformation and meridional membrane stress because of the differential settlement and the critical axial buckling load will be decreased then. Liquid storage is helpful for the cylindrical shell to remain stable. The cylindrical shell with larger height-diameter ratio and smaller diameter-thickness ratio is more capable to resist buckling under axial compression following differential settlement.
1. Introduction As a typical thin walled structure, cylindrical shells are widely used in various industries, such as steel silos, oil storage tanks and nuclear reactors [1–3]. These structures may be subjected to many kinds of axial compression, like gravity, vacuum effects and uplift caused by earthquake. The axial buckling capacity should be regarded as a controlling factor in the design stage of cylindrical shells in order to avoid potential failure. A collapsed steel silo caused by axial compression has been shown in Fig. 1. Buckling of cylindrical shells under axial compression has attracted extensive attention in recent years. Results of previous researches show that initial imperfections of the cylindrical shell have great influence on the axial buckling strength [4–6]. The researches of initial imperfections are mainly about the dimensional discrepancy caused in the manufacturing process, longitudinal and girth welds, local imperfection and damage [7–10]. In engineering, uneven soil deposition, especially for the coastal soft soil foundation, will lead to differential settlement of the cylindrical shell structures [11]. Deformation and stress ⁎
redistribution of the cylindrical shell caused by settlement may weaken its mechanical properties. The cylindrical shell will be unable to continue working as usual when the differential settlement reaches a critical value. Zhao et al. [12–14] and Gong et al. [15–17] had done some researches about buckling of cylindrical shells under settlement applying finite element simulation. However, for most cylindrical shells in service, settlement will not cause instability or failure directly. But the deformation and stress redistribution caused by settlement will have great influence on the axial buckling strength of the cylindrical shell. Compared to initial imperfections, deformation and stress of the cylindrical shells caused by settlement are much larger. Holst and Rotter [18] researched buckling behavior of axially compressed cylindrical shells with local settlement. The local settlement was assumed as local uplift in the form of quasi-continuous with a sinusoidal variation beneath the cylindrical shell. The initiation and development of deformation caused by local settlement and its effect on buckling of the cylindrical shell under axial compression had been studied then. However, in engineering, settlement beneath the cylindrical shell is distributed randomly in the circumferential direction. Several
Corresponding author. E-mail address: [email protected] (Z. Chen).
https://doi.org/10.1016/j.tws.2017.11.006 Received 29 March 2016; Received in revised form 16 October 2017; Accepted 7 November 2017 Available online 01 December 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
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Fig. 2. Measured settlement data of the tank.
theoretical research of settlement [19–21], the discrete settlement data obtained from the measuring points can be transformed into differential settlement expression applying the method of Fourier series expansion, as shown in Eq. (1).
Fig. 1. A collapsed steel silo caused by axial compression.
u = 0.088 × − 0.872 − 0.706 + 1.011
measuring points are located on basement of the cylindrical shell in order to obtain settlement data. Review of previous literature shows that research about buckling of cylindrical shells with measured settlement under axial compression is quite limited. It is quite significant to research the measured settlement beneath the cylindrical shell and buckling behavior of the cylindrical shell under axial compression. This paper focuses on the measured settlement in engineering to research the effect of differential settlement on the cylindrical shell's deformation. Based on that, the axial compression is applied to the deformed cylindrical shell and the critical axial buckling load coefficient will be obtained. The influence mechanism of settlement on axial buckling behavior is researched. Effects of differential settlement amplitudes on the critical axial buckling load coefficient of the cylindrical shell are also studied. Meanwhile, structure parameters and liquid storage of the cylindrical shell are also determining factors to the deformation and meridional membrane stress under settlement. The axial buckling capacity of the cylindrical shell with different liquid storages is analyzed. Parametric analysis of structure parameters has been carried out to research effects of structure parameters on the axial buckling strength of the cylindrical shell with measured settlement under axial compression.
cos(2θ ) − 0.045 × sin(2θ ) + 0.944 × cos(3θ ) × sin(3θ ) + cos(4θ ) − 2.375 × sin(4θ ) + 1.746 × cos(5θ ) × sin(5θ ) − 0.088 × cos(6θ ) − 1.545 × sin(6θ ) × cos(7θ ) − 0.601 × sin(7θ ) − 0.188 × cos(8θ )
(1)
where u represents the differential settlement beneath the cylindrical shell with unit of mm and θ is the circumferential coordinate with the unit of rad. Diagram of the differential settlement distributed in the circumferential direction beneath the cylindrical shell is shown in Fig. 3. The horizontal coordinate is central angle, which satisfies φ = θ × 180/π. The differential settlement u derived from measured settlement data has been shown in Eq. (1). In order to research behavior of the cylindrical shell under different settlements, the settlement amplitude is defined as λ = u0/u. When λ is selected as different values, different differential settlement expressions will be obtained by u0 = λu. The commercial finite element package ABAQUS is applied in the simulation analysis. The finite element model of the cylindrical shell is
2. Behavior of cylindrical shell under measured settlement As the foundation property changes differently in the circumferential direction, the cylindrical shell will be subjected to differential settlement. In order to reveal the deformation behavior of the cylindrical shell under settlement in engineering, the measured settlement data of an oil storage tank is selected in this paper [14]. The oil storage tank is a typical thin walled structure and 16 measuring points have been located evenly at the bottom along circumferential direction to record settlement of the tank. The measuring point number and the corresponding settlement data are shown in Fig. 2, with the unit of mm. The measured data reflects settlement of some discrete points in the circumferential direction beneath the cylindrical shell. In order to analyze deformation behavior of the cylindrical shell under settlement, the measured settlement data needs to be transformed into the continuous distributed form in circumferential direction. According to the
Fig. 3. Differential settlement distributed in the circumferential direction.
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Fig. 6. Radial displacement vs settlement curves of the cylindrical shell.
represents settlement amplitude. Fig. 6 shows that the radial displacement presents approximately linear variation as the settlement increases. It indicates that the cylindrical shell doesn’t buckle or lose stability under the settlement as shown in Eq. (1). The maximum outward and inward displacements are 2.3t and 1.1t respectively. Previous research shows that axial buckling strength of the cylindrical shell is quite sensitive to initial imperfections. The critical axial buckling load will be weakened seriously compared to the perfect cylindrical shell because of initial imperfections. The deformation and stress caused by settlement can be regarded as initial imperfections when the cylindrical shell is under axial compression. Effects of differential settlement on the axial buckling behavior will be researched based on the deformed cylindrical shell then.
Fig. 4. Finite element model of the cylindrical shell.
shown in Fig. 4, with the diameter of 24.75 m, height of 19.8 m and thickness of 15 mm. As a typical thin walled structure, the 4-noded doubly curved S4R shell element has been chosen in the cylindrical shell model. The shell material is isotropic with Young's modulus E = 206 GPa and Poisson ratio v = 0.3. Radial displacement of the cylindrical shell's bottom is restrained and the top boundaries are fully restrained except the axial displacement. The differential settlements with different amplitudes are applied to the cylindrical shell's bottom as axial displacement boundaries. Behavior of the cylindrical shell under differential settlement is performed as geometric nonlinear and static. The radial displacement nephogram of the cylindrical shell under different settlement amplitudes is shown in Fig. 5. The scale factor is selected as 20. As it can be seen, outward bulges and inward depressions have occurred to the cylindrical shell because of the differential settlement. When the settlement amplitude reaches 1.0, outward bulges are mainly concentrated in the bottom areas, while inward depressions are mainly distributed in the middle and bottom areas. The maximum outward displacement point and maximum inward displacement point of the cylindrical shell with settlement amplitude λ = 1.0 in Fig. 5 are selected to obtain the relation curves of radial displacement and settlement. As shown in Fig. 6, the horizontal coordinate represents the ratio of radial displacement ω to thickness t and the vertical coordinate
3. Buckling behavior of axially compressed cylindrical shell following settlement A reference point is created at center of the top boundary and set as coupling constraint with the top boundary of the cylindrical shell. The bottom of the cylindrical shell is set as fixed support and the top is fully restrained except the axial displacement. The uniform axial compression on the deformed cylindrical shell is simulated by applying downward displacement to the reference point. The relation curves of maximum radial displacement and axial load coefficient of the cylindrical shell can be obtained by using the Riks method to determine the
Fig. 5. Radial displacement nephogram under different settlement amplitudes.
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Fig. 7. Radial displacement nephogram of the axially compressed shell following the settlement amplitude λ = 1.0.
critical value, the maximum radial displacement begins to increase rapidly along with slight decrease of axial pressure. It means that the cylindrical shell has been subjected to classic primary buckling. Eight points on the load-displacement curve, i.e. P0-P7, are selected to describe the buckling process. P0 represents that the settlement is just finished and the axial compression is about to begin. P1 is in the initial stage of axial compression. P2 is critical buckling point. When the axial pressure increases to P2, the cylindrical shell begins to buckle and the vertical coordinate value corresponding to P2 is defined as critical axial buckling load coefficient, which is presented by μcr. The points of P3–P7 are in the post-buckling stage of the cylindrical shell. An obvious depression has occurred to the cylindrical shell under axial compression and the maximum radial displacement point of the depression is selected. As shown in Fig. 9, the path going through this point along the axial direction of the cylindrical shell is established as Path 1, and the path going through this point along the circumferential direction of the cylindrical shell is established as Path 2. The radial displacements of Path 1 at different points from P0 to P7 are plotted in Fig. 10. In Fig. 10, the horizontal coordinate is ratio of radial displacement to thickness of the cylindrical shell. Positive means outward deformation and negative means inward deformation. The vertical coordinate is height of the cylindrical shell. As it can be seen that in the initial stage of the axial compression process, the radial displacement is mainly concentrated in the middle part of the cylindrical shell. As the axial compression continues, displacement in the middle part decreases gradually and displacement in the bottom part begins to increase, following the displacement mutation at x = 0.1h. A local deformation at x = 0.1h of the cylindrical shell begins to form at P2. With the axial compression continues, the local deformation experiences a rapid
buckling equilibrium paths considering geometric nonlinear and large deformation. Fig. 7 shows the radial displacement nephogram of the cylindrical shell as axial compression increases when the settlement amplitude is 1.0. As it can be seen that in the initial stage, the radial displacements are mainly caused by settlement. They are mainly located at the concentration areas of the differential settlement in the circumferential direction. The radial displacements at the top areas of the cylindrical shell are generally small. As the axial compression increases gradually, the radial displacements become more and more concentrated until an obvious depression occurs to the cylindrical shell at the bottom region. The radial displacements caused by differential settlement are mainly located at the bottom areas, leading to the initial imperfections at the bottom areas larger than other areas. Thus, the bottom areas of the cylindrical shell are more likely to be subjected to local buckling under axial compression. Fig. 8 shows the load-displacement curve of the cylindrical shell under axial compression following the settlement amplitude λ = 1.0. The horizontal coordinate is the ratio of maximum radial displacement ωm to thickness t of the cylindrical shell. The vertical coordinate is axial load coefficient, which is the ratio of axial pressure in this cylindrical shell to the classical critical buckling pressure of perfect cylindrical shell, i.e. μ = σ/σcl, σ represents axial pressure, σcl = 1.21Et/D represents the critical buckling pressure of perfect cylindrical shell under axial compression, D is diameter. It can be seen that the maximum radial displacement varies linearly approximately as the axial pressure increases in the initial stage. When the axial pressure reaches the
Fig. 8. Load-displacement curve under axial compression when the settlement amplitude is 1.0.
Fig. 9. Diagram of Path 1 and Path 2.
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increase. It indicates that buckling of the cylindrical shell occurs around x = 0.1h at Path 1. When a cylindrical shell is subjected to axial pressure, the meridional membrane stress of the shell will increase. When the meridional membrane stress reaches critical value, even a small increase of stress will cause a very large deformation, leading to buckling of the cylindrical shell. As we know, meridional membrane stresses throughout the cylindrical shell are quite sensitive to deformations and imperfections. Because of the deformation caused by differential settlement, meridional membrane stress distribution of the cylindrical shell will be influenced and the axial buckling strength will be affected then. In order to analyze buckling mechanism of the cylindrical shell with differential settlement under axial compression, the meridional membrane stress distribution along Path 2 of the cylindrical shell under differential settlement is obtained. Relation of differential settlement and meridional stress distributions is shown in Fig. 11. The right vertical coordinate in Fig. 11 is ratio of meridional stress of this cylindrical shell σx to the classical critical buckling pressure of perfect cylindrical shell σcl. Positive means that the meridional stress is tensile stress, negative means compressive stress. For the differential settlement distribution, positive means downward displacement, negative means uplift. As we can see, the distribution characteristics of differential settlement and meridional stress are similar. The positive parts and negative parts of them are almost the same, and larger differential settlement produces larger meridional stress. It indicates that
downward displacement parts of the differential settlement will cause tensile stress to the corresponding parts of Path 2 and uplift parts will cause compressive stress. Fig. 11 shows that the cylindrical shell has been subjected to local tensile stresses and compressive stresses under differential settlement before axial compression. Based on that, when the axial compression is applied to the cylindrical shell, the local tensile stresses caused by differential settlement and the global compressive stress caused by axial compression will cancel out gradually and the total stresses of these parts will decrease gradually then. However, the local compressive stresses caused by differential settlement will be superimposed by the global compressive stress caused by axial compression, leading to the total stresses of these parts larger than the meridional membrane stress of perfect cylindrical shell under the same axial compression. Because the global compressive stress caused by axial compression is uniformly distributed in the circumferential direction, the location of maximum local compressive stress caused by differential settlement will still be the location of maximum total compressive stress after superimposed by uniform axial pressure. It will be most likely to be subjected to buckling for the location with maximum meridional compressive stress of the cylindrical shell under axial compression. In Fig. 11, the maximum local meridional compressive stress locates at φ = 18.8, which is consistent with the location of depression in Fig. 7 when the cylindrical shell buckles. Due to the superposition of local meridional compressive stresses caused by differential settlement, the uniform axial compression load applying to the cylindrical shell when it buckles will be smaller than the critical axial buckling load of perfect cylindrical shell. It verifies that differential settlement will produce local meridional compressive stresses and the axial buckling capacity of the cylindrical shell will be reduced then. The critical axial buckling load and displacement mutation position of the cylindrical shell are both dependent on the maximum meridional compressive stress caused by differential settlement. In order to analyze the effects of different settlements on the critical axial buckling load, the load-displacement curves of the cylindrical shell under axial compression following different settlement amplitudes are presented in Fig. 12. When the settlement amplitude is zero, i.e. λ = 0, it means that there is no differential settlement and the cylindrical shell remains perfect. The axial load coefficient is 1.01 when buckling occurs, it indicates that the error between critical buckling loads for the perfect cylindrical shell calculated by finite element method and theoretical formula is only 1%, which verifies the rationality and accuracy of the finite element method. Fig. 12 shows that the cylindrical shells with different settlement amplitudes are all subjected to classic primary buckling when the axial pressures increase to critical values. The
Fig. 11. Relation of differential settlement and meridional stress distributions.
Fig. 12. Load-displacement curves under axial compression with different settlement amplitudes.
Fig. 10. Radial displacement variation of Path 1 during axial compression.
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Fig. 14. Meridional stress distributions of the cylindrical shell under different settlement amplitudes.
Fig. 13. Critical axial buckling load coefficient vs settlement amplitude curve.
critical axial buckling load coefficient of each curve, i.e. μcr, reflects the axial buckling capacity of the cylindrical shell following a certain differential settlement. Fig. 12 shows that there are different critical axial buckling load coefficients corresponding to the cylindrical shell with different settlements. The relation curve of critical axial buckling load coefficient and settlement amplitude has been plotted in Fig. 13. The horizontal coordinate is settlement amplitude and the vertical coordinate is critical axial buckling load coefficient of the cylindrical shell. As it can be seen that the critical axial buckling load coefficient decreases almost linearly as the settlement amplitude increases. It indicates that the axial buckling capacity reduction of the cylindrical shell caused by differential settlement will become more serious as the settlement amplitude increases. For example, when the settlement amplitude is 1.0, the critical axial buckling load coefficient is 0.27. It means that the critical axial buckling load of the cylindrical shell following the differential settlement in Eq. (1) is just 0.27 times of that corresponding to the perfect cylindrical shell. Three different settlement amplitudes, including λ = 0.2, λ = 0.6 and λ = 1.0, are taken for examples. For each settlement amplitude, when the cylindrical shell following that settlement buckles under axial compression, the path going through the maximum radial displacement point along the circumferential direction will be established, like Path 2 in Fig. 9. In order to research effects of different settlements on stress redistribution of the cylindrical shell before axial compression, the meridional stress distribution curves along the paths of the cylindrical shell under different settlement amplitudes are shown in Fig. 14. As we can see, the meridional stress distribution characteristics in the circumferential direction of the cylindrical shell under different settlement amplitudes are almost the same. But the meridional stress magnitude is different corresponding to different settlement amplitudes. Larger settlement amplitudes will produce larger meridional stresses of the cylindrical shell, leading to reduction of axial buckling capacity and critical axial buckling load coefficient. This is consistent with the result obtained from Fig. 13. It also explains the influencing mechanism of different settlements on the axial buckling capacity of the cylindrical shell. Buckling research of the cylindrical shell with measured settlement under axial compression shows that the settlement will cause deformation and stress redistribution, leading to meridional tensile and compressive stresses of the cylindrical shell. The meridional compressive stresses caused by settlement will reduce the axial buckling capacity of the cylindrical shell. The critical axial buckling load and radial displacement mutation position of the cylindrical shell depend on the maximum meridional compressive stress caused by settlement. As the
settlement becomes larger, meridional compressive stresses will increase gradually and the ability to resist buckling of the cylindrical shell will decrease then. Thus the settlement is an important factor that needs to be taken into account in the buckling analysis of axially compressed cylindrical shell. The critical axial buckling load will be predicted effectively using the measured settlement data beneath the cylindrical shell. It is helpful to the analysis and research of axial buckling capability and service life of the cylindrical shell in engineering. 4. Effects of liquid storage in the cylindrical shell There are always different materials stored in cylindrical shells in engineering. Variation of liquid storage will produce different pressures on the inner surface of the cylindrical shell and the axial buckling behavior may be affected. The liquid is assumed to be water and six different liquid heights, i.e. h = 0, h = 3.96 m, h = 7.92 m, h = 11.88 m, h = 15.84 m and h = 19.8 m, are selected to research the effects of liquid storage in the cylindrical shell. The hydrostatic pressures will be applied to inner surface of the cylindrical shell for the six different cases. For example, when the liquid height is 11.88 m, the hydrostatic pressure ps = 9800 × (11.88-x) (Pa) will be applied to inner surface from h = 0 to h = 11.88 m, where x represents the axial coordinate. The relation curves of critical axial buckling load coefficient and liquid height of the cylindrical shell following different settlement amplitudes are shown in Fig. 15, where horizontal coordinate is the ratio of liquid height to the cylindrical shell height. As it can be seen in Fig. 15, for different settlement amplitudes, the critical axial buckling load coefficient will increase gradually as the liquid height rises. It indicates that liquid in the cylindrical shell is helpful to improve the buckling capability of axially compressed cylindrical shell with settlement. Taking the settlement amplitude λ = 0.6 as an example, when the liquid height is zero, the critical axial buckling load coefficient is 0.523; when the liquid height is 1.0, the critical axial buckling load coefficient becomes 0.913, which has increased 39%. Meanwhile, when the settlement amplitude is larger, the improvement effects of liquid on the axial buckling capacity of the cylindrical shell will become more obvious. For example, when the settlement amplitude is 0.2, as the liquid height rises from zero to 1.0, the critical axial buckling load coefficient will increase from 0.802 to 0.982, whose difference is 0.18. While when the settlement amplitude is 1.0, as the liquid height rises from zero to 1.0, the critical axial buckling load coefficient will increase from 0.271 to 0.848. The difference is 0.577, which is larger than that of the settlement amplitude λ = 0.2. Analysis shows that liquid in the cylindrical shell is helpful to improve the axial buckling capacity. As we have researched in the empty 356
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Fig. 16. Critical axial buckling load coefficient vs diameter-thickness ratio curves.
Fig. 15. Critical axial buckling load coefficient vs liquid height curves.
2062.5, the critical axial buckling load coefficient will decrease from 0.877 to 0.758, whose difference is 0.119. While when the settlement amplitude is 1.0, as the diameter-thickness ratio increases from 1375 to 2062.5 in the same way, the critical axial buckling load coefficient will decrease from 0.397 to 0.073. The difference is 0.324, which is larger than that of the settlement amplitude λ = 0.2. Therefore, when the settlement beneath the cylindrical shell is larger, the critical axial buckling load will be more sensitive to the diameter-thickness ratio variation. The meridional stress distribution in the circumferential direction of the cylindrical shells with three different diameter-thickness ratios under settlement amplitude λ = 1.0 are shown in Fig. 17. As we can see that under the same differential settlement, the cylindrical shell with larger diameter-thickness ratio will produce greater maximum meridional compressive stress and the critical axial buckling load coefficient will become smaller then. It indicates that differential settlement has greater effect on the axial buckling capacity of the cylindrical shell with larger diameter-thickness ratio.
cylindrical shell, inward deformation begins to grow rapidly and an obvious depression is produced when the axial pressure reaches critical value, leading to buckling of the cylindrical shell. However, when there is liquid stored in the cylindrical shell, hydrostatic pressure will be generated on inner surface. The inward deformation growth caused by settlement and axial compression will be hindered by the hydrostatic pressure and the axial buckling capacity of the cylindrical shell will be improved then. As liquid height rises, hydrostatic pressure will become larger and the improvement effects on the axial buckling capacity will be more obvious. Thus, in engineering, liquid stored in the cylindrical shell needs to be analyzed according to actual situation. While in this paper, the cylindrical shell is assumed to be empty for simplification. 5. Parametric analysis The above analysis shows that settlement beneath the cylindrical shell will decrease the critical axial buckling load and reduce the axial buckling capability. But in engineering, due to the structure size diversity of cylindrical shells, effects of settlement on the critical axial buckling load will be different. Parametric analysis is carried out to research effect of structure size on the critical buckling load of the cylindrical shell with differential settlement under axial compression.
5.2. Height-diameter ratio Seven cylindrical shells with different height-diameter ratios are selected to research the relation of critical axial buckling load coefficient and height-diameter ratio. Relation curves corresponding to different settlement amplitudes have been presented in Fig. 18. The
5.1. Diameter-thickness ratio Seven cylindrical shells with different diameter-thickness ratios are selected to research the relation of critical axial buckling load coefficient and diameter-thickness ratio. Relation curves corresponding to different settlement amplitudes have been presented in Fig. 16. The horizontal coordinate is diameter-thickness ratio. Fig. 16 shows that for different settlement amplitudes, the critical axial buckling load coefficient will decrease as the diameter-thickness ratio increases. It indicates that effect of settlement on the axial buckling strength of the cylindrical shell with larger diameter-thickness ratio are greater than that with smaller diameter-thickness ratio. Taking the settlement amplitude λ = 0.6 as an example, the critical axial buckling load coefficient is 0.603 when the diameter-thickness ratio is 1375. While when the diameter-thickness ratio is 2062.5, the critical axial buckling load coefficient has decreased to 0.425. It verifies that settlement has smaller effect on the critical axial buckling load of the cylindrical shell with smaller diameter-thickness ratio. Meanwhile, for the curves of different settlement amplitudes, reduction magnitudes of the critical axial buckling load coefficient as diameter-thickness ratio increases are also different. For example, when the settlement amplitude is 0.2, as the diameter-thickness ratio increases from 1375 to
Fig. 17. Meridional stress distribution of different diameter-thickness ratios.
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influence on the buckling strength of the cylindrical shell with differential settlement under axial compression. The same settlement will cause different meridional stresses to the cylindrical shells with different structure sizes. Results show that the cylindrical shell with larger diameter-thickness ratio and smaller height-diameter ratio will be subjected to greater meridional compressive stress under settlement and the axial buckling capacity will be affected more seriously. Therefore, a reasonable selection about structure size of the cylindrical shell, such as smaller diameter-thickness ratio and larger height-diameter ratio, will help to increase its axial buckling capability following differential settlement. 6. Conclusions
Fig. 18. Critical axial buckling load coefficient vs height-diameter ratio curves.
This paper presents deformation behavior of the cylindrical shell under differential settlement based on the measured settlement data. Then, buckling behavior of the deformed cylindrical shell under axial compression is researched. Stress distribution of the cylindrical shell under differential settlement and influencing mechanism of settlement on the axial buckling capacity are analyzed. The relations of critical axial buckling load and settlement amplitude, liquid in the cylindrical shell and structure parameters are obtained. The following conclusions may be drawn from this study
Fig. 19. Meridional stress distribution of different height-diameter ratios.
(1) The cylindrical shell will be subjected to deformation because of differential settlement. The deformation caused by differential settlement can be regarded as initial imperfection of the cylindrical shell under axial compression. (2) The cylindrical shell following settlement will be subjected to classic primary buckling under axial compression. The critical axial buckling load can reflect the buckling capacity of the cylindrical shell. The meridional compressive stress caused by differential settlement is crucial to the buckling capacity reduction of the cylindrical shell. As differential settlement beneath the cylindrical shell becomes larger, the meridional stress will grow bigger and the critical axial buckling load will decrease then. It verifies the influencing mechanism of settlement on the buckling capacity of the cylindrical shell. (3) Liquid will generate hydrostatic pressure on inner surface of the cylindrical shell. The inward deformation growth caused by settlement and axial compression will be hindered by the hydrostatic pressure and the axial buckling capacity of the cylindrical shell will be improved. It is helpful to analyze stability of the cylindrical shell with liquid storage. (4) Structure parameters of the cylindrical shell, such as diameterthickness ratio and height-diameter ratio, will have great influence on the critical axial buckling load. The cylindrical shell with smaller diameter-thickness ratio and larger height-diameter ratio is more capable of resisting buckling under axial compression. Thus, a reasonable selection about structure size of the cylindrical shell according to the actual situation in engineering will help to improve the axial buckling capability following differential settlement, the stability and safety of the cylindrical shell will also be enhanced.
horizontal coordinate is height-diameter ratio. Fig. 18 shows that for different settlement amplitudes, the critical axial buckling load coefficient will increase as the height-diameter ratio increases. It indicates that effect of settlement on the axial buckling capacity of the cylindrical shell with smaller height-diameter ratio are greater than that with larger height-diameter ratio. Taking the settlement amplitude λ = 0.4 for an example, the critical axial buckling load coefficient is 0.721 when the height-diameter ratio is 1.2. While when the height-diameter ratio is 0.6, the critical axial buckling load coefficient is 0.631. It verifies that differential settlement has smaller effect on the critical axial buckling load of the cylindrical shell with larger height-diameter ratio. Similarly, the meridional stress distribution in the circumferential direction of the cylindrical shells with three different height-diameter ratios under settlement amplitude λ = 1.0 are plotted in Fig. 19. It shows that under the same differential settlement, the cylindrical shell with smaller height-diameter ratio will be subjected to greater maximum meridional compressive stress and the critical axial buckling load coefficient will become smaller then. It indicates that differential settlement has greater effect on the axial buckling capacity of the cylindrical shell with smaller height-diameter ratio. Figs. 16–19 have also verified that the meridional compressive stress caused by differential settlement plays a decisive role in the axial buckling capacity of the cylindrical shell. The parametric analysis shows that structure size has great
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