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Experimental study on damaged cylindrical shells under compression
Thin-Walled Structures 80 (2014) 13–21
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Experimental study on damaged cylindrical shells under compression Tohid Ghanbari Ghazijahani n, Hui Jiao, Damien Holloway School of Engineering, University of Tasmania, Hobart, Australia
art ic l e i nf o
a b s t r a c t
Article history: Received 23 January 2014 Received in revised form 25 February 2014 Accepted 26 February 2014
Sensitivity to initial imperfections under compressive loading has been extensively studied in shell structures. However, due to the existence of a wide range of imperfections with various shapes and amplitudes, the real behavior of such structures needs to be further investigated when they face with a damaged area. This study presents an experimental program in which buckling and failure response of damaged shell specimens are analyzed. The results of this study can be generalized for many kinds of cylindrical shells to full scale of applications with similar D/t ratios. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Thin-walled cylindrical shells Tests Compression Initial damage Dent
1. Introduction Axial compressive stresses in cylindrical shell structures arise from various causes such as in towers and chimneys caused by the weight of the structures [1]. Such structures are often vulnerable to physical contacts of the other elements during their service life. A wide range of references can be found in the literature in regard to the buckling and failure response of cylindrical shell structures with normal fabrication-related imperfections under compressive stresses, e.g. [2–9]. However, quite a few studies reflect such structures with large imperfections caused by a collision. Local bulges, dents and unilateral corrugations were studied and it was found that local dents significantly reduced the critical load [10]. The effect of localized imperfections on the buckling of cylindrical shells under axial compression was studied and a considerable reduction of the critical load due to the imperfections was observed [11]. Two new approaches were proposed for the numerical and analytical stability analyses of imperfect shells [12]. Theoretical buckling stress of such shells was defined by Eq. (1), [1]. In this equation E is Young's modulus and “ν“ is Poisson's ratio. “t“ and “r“ are respectively the thickness and the radius of the shells. This equation mostly overestimates considerably the buckling load of thin shell due to the occurrence of local deformations resulted from the geometric imperfections in experimental models. It should be noted that, the number of full buckling waves around
n
Corresponding author. Tel.: þ 61 469 311 896. E-mail addresses: [email protected], [email protected] (T. Ghanbari Ghazijahani). http://dx.doi.org/10.1016/j.tws.2014.02.029 0263-8231 & 2014 Elsevier Ltd. All rights reserved.
the circumstance and half-wavelength of the buckling waves were also defined theoretically by Eqs. (2) and (3) respectively [1]. Et
t r
ð1Þ
0:25 0:5 r 0:5 r C 0:909 t t
ð2Þ
pffiffiffiffi ðrtÞ0:5 C1:728 rt
ð3Þ
s ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 0:605 E 2 r
n¼ λ¼
3ð1 ν Þ
3 ð1 ν2 Þ 4 π
ð12ð1 ν2 ÞÞ0:25
Analytical methods were developed for the estimation of the upper critical loads for non-reinforced cylindrical shells with axisymmetric dents (bulges) located on the shell structures. An investigation has been conducted on the buckling of steel cylindrical shells with a single local dent [13,14]. It was found that even a single dent strongly influences the magnitude of the critical load. Refs. [15,16] are highly relevant papers to this work. Axial compression was applied to the damaged specimens and the buckling behavior was studied [15]. It was found that damaged shells reached almost half of the capacity of the intact specimens in the case of eccentric loading. For undamaged specimens failure was quite abrupt and without warning whereas the damaged structures had a phase of stable growth of a dent before the catastrophic failure ensued. Parametric study on the buckling behavior of dented short carbon steel cylindrical shell subjected to uniform axial compression was conducted [16]. Cylindrical shells with longitudinal dents were believed to have higher buckling strengths than cylindrical shells with circumferential dents.
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Table 1 Specifications of the specimens. Specimen d (mm)
Orientation of the dent
HD (mm)
HD/H WD (mm)
D/t¼ 607.08, t ¼ 0.25 (mm), L/D ¼ 1.44
LCC.1 LCC.2 LCC.3 LCC.4 LCC.5 LCC.6 LCC.7 LCC.8 LCC.9 LCC.10 LCC.11 LCC.12 LCC.13
– – 0.45 2.3 3 6.2 3 6.5 2 3.3 3.2 8 1.8
– – Horizontal Horizontal Horizontal Horizontal Diagonal Diagonal Vertical Vertical Horizontal Horizontal Horizontal
– – 110 110 110 110 110 110 110 110 55 55 20
– – 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.25 0.25 0.09
– – 19 39 63 75 60 75 75 90 56 72 35
D/t¼ 339.52, t ¼ 0.25 (mm), L/D ¼ 1.36
SCC.1 SCC.2 SCC.3 SCC.4 SCC.5 SCC.6 SCC.7 SCC.8 SCC.9 SCC.10 SCC.11 SCC.12 SCC.13 SCC.14
– – 0.4 0.8 1.2 1.8 2.5 0.5 0.9 2.1 1 2.7 1.2 1.2
– – Horizontal Horizontal Horizontal Horizontal Horizontal Horizontal Horizontal Horizontal Horizontal Horizontal Vertical Diagonal
– – 15 15 15 15 15 30 30 30 57.8 57.8 57.8 57.8
– – 0.130 0.130 0.130 0.130 0.130 0.260 0.260 0.260 0.5 0.5 0.5 0.5
– – 14 21 26 30 38 16 26 32 22 37 44 42
The present study was performed on 27 locally dented shell specimens as very limited experimental data are found on the dented shells under compression. Dent imperfections of different depths, locations and orientations were modeled. This paper focuses on the following points:
Experimental modeling of dented shells; Buckling of intact and damaged specimens; Failure modes of such structures; Capacity assessment of dented shells and; Comparison of the results with other works and existing standards.
2. Experimental program 2.1. System set-up 2.1.1. Apparatus The test apparatus for the present experimental program was a MTS-810 machine. Two end plates were fabricated in which the specimen ends were placed in grooves with a slight tolerance in order to restrain the specimens in the radial direction while allowing rotation of the walls about the circumferential line. End plates were gripped by two jaws of the MTS machine. Before the tests the two top and bottom plates were calibrated accurately to ensure that the axial loading was uniformly applied. The vertical distance between the two plates before the tests was adjusted such that no additional weight of the plates was applied to the top portion of the specimens. 2.1.2. Specimens In this study, 27 precisely-fabricated specimens with two different D/t ratios were tested (see Table 1 and Fig. 1). LCC.1 to
Fig. 1. Test set-up for cylindrical shell specimens.
LCC.13 (D¼ 151.77 mm, L ¼218.55 mm and t¼ 0.25 mm), were the specimens with higher D/t ratio and SCC.1 to SCC.15 were the specimens with lower D/t ratio (D ¼84.88 mm, L ¼115.44 mm and t¼0.25 mm). The Young's modulus as the main material property in buckling tests of thin shells and Poisson's ratio were obtained from a tensile coupon test as 210 GPa and 0.3 respectively. The specimens had top and bottom circular caps to simulate the practical instances of the tanks with two end caps. Fig. 2 presents two practical instances of the steel tanks with circular flat roofs and local geometric defects on the surface of such structures. 2.1.3. Indentation and measurements Indentation was conducted by means of a steel indenter with a circular sharp end (see Figs. 3–5). Indentation was made through closely moving the indenter over the body of the shell specimens to achieve a dent with the desired depth and corresponding width. A finger LVDT was utilized in order to have accurate recordings of the dented area (Fig. 5). A small ball-shaped tip of this LVDT made it capable to move along and/or across the dented region and accurately detect the geometric depressions of the dent. CEA-06240UZ-120 Micro-Measurements USA made strain gauges were employed in order to record the strain values in the critical areas of the test samples. End shortening of the specimens was recorded by the MTS machine.
3. Test results 3.1. Buckling and failure 3.1.1. Intact specimens Intact specimens in this study are considered as control specimens against which the other locally imperfect specimens were
T. Ghanbari Ghazijahani et al. / Thin-Walled Structures 80 (2014) 13–21
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Fig. 4. Illustration of a typical dented specimen.
Fig. 2. (a) Steel reservoirs with circular flat roofs [17], and (b) local geometric defects on the body of a steel silo [17].
Fig. 3. Horizontal, vertical and diagonal dents.
evaluated. Thus, for each set of LCC and SCC specimens the intact models were repeated to ensure the accuracy of the results and the buckling/failure modes. The buckling mode of LCC.1, LCC.2, SCC.1 and SCC.2 specimens were all in a diamond mode of buckling near the edge area of the specimens. In this mode of buckling, initially several small diamond-shaped deformations occurred in one tier, after which the deformations deepened, until final failure of the specimens accompanied by covering the whole circumferential direction with the above mentioned deformations. The position of the buckling can be largely attributed to the end effects of the tested models or initial imperfections of the edge area of the specimens.
3.1.2. Dented specimens As can be seen in Table 1, LCC.3–LCC.6 were the specimens with the horizontal dents at the mid-height of the specimens. For these specimens buckling initiated at the edge area similar to the intact specimens. However, for the specimens with larger dents (i.e. LCC.5 and LCC.6) initial buckling was accompanied by development of the deformations in the dented area. Note that the effect of the dent on initial buckling was not considerable for small
dents, but deformations in the dented area were significant in the post-buckling phase, for all tests. Fig. 6 shows the progress of the buckling in the specimens with large dents, i.e. LCC.5 and LCC.6. Specimens LCC.7 and LCC.8 were indented diagonally with different depths of the dent. As can be seen in Fig. 7, buckling occurred concurrently at the edge zone of the specimens and both ends of the diagonal dent in such a way that buckling was initiated at the edge area of the diagonal dent followed by development of the deformations in an inclined manner which eventually met the edge deformations. As regards the specimens with vertical dents, the buckling phenomenon was similar to the diagonal dents. However, in these specimens the initial buckling wave was initiated at and/or adjacent to the vertically dented zone (see Figs. 8 and 9). As discussed before, the top and bottom edge zones were quite vulnerable to buckling initiation. For this reason, specimens LCC11–LCC3 and SCC3–SCC10 were prepared with dent imperfection located near the top edge to evaluate the effect of the dent at the most sensitive area to the imperfections. The dented area near the edge zone helped trigger the buckling concurrently at the top zone together with the area right adjacent to the pre-existing lobe of the dent. It is interesting to note that the diamond mode buckling tier was quite dependent on the value of HD parameter, which is the distance of the center of the dent from the top end. As can be seen in Fig. 10, for lower values of HD, the diamond mode occurred in one tier whereas two tiers of diamond mode occurred for dents with higher values of HD. Moreover, for SCC specimens with the horizontal dents, the buckling mode was similar to the LCC specimens as can be seen in Fig. 11. Since the material of the dented area changed due to the effect of local loads, i.e. indentation, buckling mostly occurred at the zone immediately adjacent to the dented segment. This may be due to the fact that the material of the dented zone entered the inelastic region so that it became brittle in comparison to the intact elements of the shells. On this basis, the area adjacent to the dented zone was affected by the initial buckling owing to the geometric irregularities in the nearby dented zone. Note that unlike the dented zone, the material was not severely changed into inelastic in the adjacent area of a dent; however the geometric imperfections fairly existed in that area due to the effects of the dent which by and large makes the adjacent area geometrically irregular. The adjacent area and the buckling lobes were marked in Fig. 11. 3.2. Buckling capacity of the dented specimens: The buckling capacity of the present specimens was affected by the pre-existing dent imperfections. Fig. 12(a) shows the axial load
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Fig. 5. (a) Measurement of the dented area by a finger transducer and (b) indenter.
Fig. 6. Buckling initiation and failure in: (a) LCC.5 and (b) LCC.6.
Fig. 7. Buckling initiation and failure in LCC.7.
capacity of the present specimens versus the parameter ζ which was defined as ζ¼ d(D/t). As can be seen, SCC specimens were more sensitive to the initial dent than LCC specimens. Moreover, data scatter is seen more in LCC specimens in comparison with SCC specimens indicating that the specimens with higher D/t ratio were more vulnerable to the local effects before the ultimate load capacity was reached. Thus, SCC specimens showed a stable decrease against the axial loading when they were imposed to the local dents. In general, the buckling behavior of shells under axial stresses greatly depends on the amplitude and the shape of the geometrical imperfections as many imperfection forms may not have a very deleterious influence so that the decrease in strength may not be easily deduced considering the depth of dents [1]. Parameter χ was defined as “χ¼ dWD” in which WD was included as another geometrical parameter of a dent to show the effect of the dent considering both depth and width of the dented area (see Fig. 13). A decreasing trend of the buckling capacity is seen in Fig. 13 in
which SCC specimens present a higher decrease with less scatter in data in comparison with LCC specimens. The capacity of different LCC specimens considering various orientations is also seen in Fig. 12(b). Based on the results, no big difference was seen in this study between the specimens with different forms of the imperfection, i.e. horizontal, vertical and diagonal. Figs. 14 and 15 show end shortening of the present LCC and SCC specimens. After some initial stiffening very early in the loading, a nearly linear pre-buckling behavior is seen after which the typical bifurcation is detected. As can be seen in Figs. 14 and 15 a clear second bifurcation after the first bifurcation was seen in the form of a steeply falling post-buckling path as the mode switched from one circumferentially deformed buckling to another in dynamic jumps of the diamond mode [1]. In these specimens, the second bifurcation point can be attributed to the development of large deformations at the dented area after the initial bifurcation occurred at the edge of the shells.
T. Ghanbari Ghazijahani et al. / Thin-Walled Structures 80 (2014) 13–21
17
Fig. 8. Typical deformation progress in vertical and horizontal dents.
Fig. 9. Vertical dent at mid-height of the specimen and failure mode.
Fig. 10. (a) Horizontally dented SCC specimens with dents 15 mm from the top end and intact specimen, (b) Three horizontally dented specimens with the dent at 30 mm from the top end, d ¼ 0.5, 0.9 and 2.1 from left to right. Two tiers of buckling are seen.
Strain values of the specimens at the dented area (S.2) and an equivalent intact zone (S.1) are presented in Figs. 16 and 17. As can be seen, the strain values follow a linear and/or a nonlinear trend depending on where they were placed, which may or may not be
exactly at the zone of the initiation of the buckling. As it is hard to determine the area at which the initiation of the buckling in such structures occurs, strain values may show a great scatter depending on a point on which they were placed on the surface of the shells.
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4. Comparison of the results with other studies and standards Previously published studies largely reflected the effect of small amplitude imperfections on the buckling behavior of shell structures. As mentioned before, quite a few studies are found regarding a local large amplitude imperfection as a dent. Among those studies, the most relevant investigation is adopted herein for comparison with the current results [15]. In this study, sufficient tests of dented shells with nearly similar D/t ratio were conducted.
Fig. 11. Post-buckling of horizontal dents with different depths of 1 mm and 2.7 mm at the mid-height in SCC specimens. The area adjacent to the dented zone is marked.
Above all, similar WD, as an important geometric feature of the dented zone, was reported in this study. Fig. 18 shows a capacity ratio which we define it as the maximum load of the specimens in the present study divided by the load in the mentioned reference in which “S” is taken to be equivalent to WD of the current study. A range of approximately 0.5–1 was achieved which is regarded as a good agreement between the results of the two studies. Consistency of the results was greater for the specimens with larger dents, i.e. S¼50 mm, indicating that larger imperfections mostly governed the buckling behavior in the two studies. It is noteworthy that very few design codes and standards have defined the allowable imperfection tolerated during the welding and/or construction of the shell structures. Fig. 19 shows a schematic illustration for the imperfection tolerance limits in different directions presented in three different codes [21–23]. Table 2 shows the main geometrical parameters of the dents for the present specimens considering equivalent geometric specifications mentioned in the above standards. It should be mentioned that the geometric specifications tolerated by different standards are far less than the case of dent imperfections as a large local defect. Notwithstanding this, all the current specimens showed a stable pre-buckling and post-buckling behavior on the one hand, and a quite smooth decrease in the buckling capacity on the other hand. This phenomenon was reported in [18–20] for shells under
Fig. 12. (a) Capacity versus ζ for LCC and SCC specimens and (b) LCC specimens with different dent orientations.
Fig. 13. Capacity versus χ for LCC and SCC specimens.
T. Ghanbari Ghazijahani et al. / Thin-Walled Structures 80 (2014) 13–21
19
Fig. 14. End shortening for the specimens LCC and SCC specimens.
Fig. 15. End shortening for the specimens SCC.1–SCC.10.
Fig. 16. Strain values at the dented area and the edge area of the specimens SCC.7 and SCC.10.
Fig. 17. Strain values at the dented area and the edge area of the specimens LCC.3 and LCC.4.
external pressure as well. In these references computations of tv and lmQ revealed that weld-induced imperfections similar to the dents of current specimens far exceeded the amounts allowed by the codes. In such cases, the standards simply suggest that the structure cannot be used. However, there is no need to stop using
these structures and their function should continue with special care [19]. What's more, despite extensive data in the literature for shells under compression, comparisons must be drawn with a great deal of care for design purposes, for different imperfection types in order to generalize the buckling behavior [1]. Therefore,
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further research is still certainly required to improve the understanding of various forms of the imperfections, particularly local large dents under different load and boundary conditions.
5. Conclusion In this study, 27 cylindrical shell specimens with two different D/t ratios were tested under axial loading. A local dent-shaped imperfection with different geometries and positions was introduced on the surface of the specimens. The results of the current study can be summarized as follows:
Typically buckling occurred at the edge zone of the specimens
5.1. Buckling mode
Top and bottom edge zones were quite vulnerable to the
buckling phenomenon. The dented area near the edge zone helped trigger the buckling which occurred concurrently at the dented zone together with the area right adjacent to the preexisting lobe of the dent. Diamond-shaped buckling tier(s) were quite dependent on the proximity of the dent to the cylinder ends (HD values). For lower values of HD, the diamond mode occurred in one tier, whereas two tiers in the diamond mode were seen for dents with higher HD. In severely dented specimens, initial buckling was accompanied by the development of the deformations at the dented area; whereas, for small dents the effect of the dent was not considerable. After the initiation of buckling, the dented area was largely affected by the deformations in the post-buckling phase.
and concurrently initiated at both ends of the dent in an inclined manner leading to meet the edge of the specimens. For vertically dented specimens, initial buckling occurred at and/or adjacent to the diamond waves of the edge area of the specimens. The most significant zone of a dent occurrence was the edge area of the present specimens. The material of the dented zone entered the inelastic region so that it became brittle in comparison to the intact elements of the shells. The area adjacent to the dented zone was affected by the initial buckling owing to the geometric irregularities at nearby zone of a dent. Note that the material did not severely change into inelastic in the adjacent area of a dent, however the geometric imperfections fairly existed due to the irregularities of the dented area.
5.2. Buckling strength
SCC specimens were more sensitive to the initial dent than LCC specimens.
Specimens with higher D/t ratio were more vulnerable to the
local effects before the ultimate load capacity was reached so that data scatter is seen more in LCC specimens in comparison with SCC specimens. A second bifurcation point on the post-buckling path can be attributed to the development of large deformations at the dented area after the initial bifurcation occurred at the edge area. A good agreement is obtained with the previous study on the dented shells with similar D/t ratio.
Table 2 Main parameters of the dents considering the mentioned standards.
Fig. 18. Comparison of the data with the results of Ref. [15] in which “S” is corresponding to WD.
Specimen
d/t
d/WD (%)
WD/t
Specimen
d/t
d/WD (%)
WD/t
LCC.3 LCC.4 LCC.5 LCC.6 LCC.7 LCC.8 LCC.9 LCC.10 LCC.11 LCC.12 LCC.13
1.8 9.2 12 24.8 12 26 8 13.2 12.8 32 7.2
2.37 5.9 47.62 8.27 5 8.67 2.67 3.67 5.71 11.11 5.14
76 156 25.2 300 240 300 300 360 224 288 140
SCC.3 SCC.4 SCC.5 SCC.6 SCC.7 SCC.8 SCC.9 SCC.10 SCC.11 SCC.12 SCC.13 SCC.14
1.6 3.2 4.8 7.2 10 2 3.6 8.4 4 10.8 4.8 4.8
2.86 3.81 4.62 6 65.79 3.13 3.46 6.56 4.55 7.3 2.73 2.86
56 84 104 120 15.2 64 104 128 88 148 176 168
DIN 18800-4
tr<1% of Lmx
ESSCRec, 2008
L=Ir:tr=W tr<1% of Lmx
ENV 1993-1-6, 2007
Lmx=Lg:tr=∆w0 tr≤0.006Lmx OR (0.01 Lmx, 0.016 Lmx for different classes)
Fig. 19. Geometric tolerances in different codes: (a) Initial dimple in vertical direction and (b) Initial dimple in circumferential direction [18–20].
T. Ghanbari Ghazijahani et al. / Thin-Walled Structures 80 (2014) 13–21
Although the geometric specifications tolerated by different standards are far less than the case of the dent imperfections, all the specimens showed a stable pre-buckling and postbuckling behavior on the one hand, and a smooth decrease in the buckling capacity on the other hand. To sum up, one may note that despite extensive data in the literature for shells under compression, comparisons with different imperfection types must be drawn with a great deal of care in order to generalize the buckling behavior for design purposes. Therefore, further research is still required to improve the understanding of various forms of the imperfections particularly local large dents under different load and boundary conditions. Acknowledgments Efforts of Mr. Andrew Bylett, Mr. Peter Seward and Mr. David Morley for preparation of the specimens and the test rig and Mr. Calverly Gerard for LabVIEW program are greatly appreciated. References [1] Teng J-G, Rotter JM. Buckling of thin metal shells. London: Taylor & Francis, Spon Press, Taylor & Francis, Group; 2004. [2] Brush DO, Almroth BO. Buckling of bars, plates, and shells, vol. 6. New York: McGraw-Hill; 1975. [3] Calladine CR. Theory of shell structures. Cambridge University Press; 1989. [4] Deml M, Wunderlich W. Direct evaluation of the ‘worst’imperfection shape in shell buckling. Comput MethodsAppl. Mech Eng 1997;149:201–22. [5] Hutchinson J, Muggeridge D, Tennyson R. Effect of a local axisymmetric imperfection on the buckling behaviorof a circular cylindrical shell under axial compression. AIAA J 1971;9:48–52. [6] Holst JMF, Rotter JM, Calladine CR. Imperfections in cylindrical shells resulting from fabrication misfits. J Eng Mech 1999;125:410–8.
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[7] Holst J. Local dimpling of the shell surface of a tank due to shrinkage effects. Struct Granul Solids 2011:153. [8] Ghanbari Ghazijahani T, Showkati H. Bending experiments on thin cylindrical shells. Materials with complex behaviour II. Germany: Springer; 2012; 119–39. [9] Ghanbari Ghazijahani T, Showkati H. Experiments on cylindrical shells under pure bending and external pressure. J Constru Steel Res 2013;88:109–22. [10] Gavrilenko G. Stability of cylindrical shells with local imperfections. Int Appl Mech 2002;38:1496–500. [11] Jamal M, Lahlou L, Midani M, Zahrouni H, Limam A, Damil N, et al. A semianalytical buckling analysis of imperfect cylindrical shells under axial compression. Int J Solids Struct 2003;40:1311–27. [12] Gavrilenko G. Numerical and analytical approaches to the stability analysis of imperfect shells. Int Appl Mech 2003;39:1029–45. [13] Gavrilenko G, Krasovskii V. Stability of circular cylindrical shells with a single local dent. Strength Mater 2004;36:260–8. [14] Ghanbari Ghazijahani T, Showkati H. Locally imperfect conical shells under uniform external pressure. Strength Mater 2013;45:369–77. [15] Hambly E, Calladine CR. Buckling experiments on damaged cylindrical shells. Int J Solids Struct 1996;33:3539–48. [16] Prabu B, Raviprakash A, Venkatraman A. Parametric study on buckling behaviour of dented short carbon steel cylindrical shell subjected to uniform axial compression. Thin-Walled Struct 2010;48:639–49. [17] Available:〈http://www.cstindustries.com/products/trough-deck-roofs-covers/〉 〈http://www.structuremag.org/article.aspx?articleID=446〉. [18] Niloufari A, Showkati H, Maali M, Fatemi S Mahdi. Experimental investigation on the effect of geometric imperfections on the buckling and post-buckling behavior of steel tanks under hydrostatic pressure. Thin-Walled Struct 2014;74:59–69. [19] Fatemi SM, Showkati H, Maali M. Experiments on imperfect cylindrical shells under uniform external pressure. Thin-Walled Struct 2013;65:14–25. [20] Maali M, Showkati H, Fatemi S Mahdi. Investigation of the buckling behavior of conical shells under weld-induced imperfections. Thin-Walled Struct 2012;57:13–24. [21] DIN 18800. Stahlbauten.Teil 4: Stabilitätsfalle, Schalenbeulen 1990. [22] ECCS EDR5. European Recommendations for steel construction. Buckling of shells, 5th ed. In: Rotter JM, Schmidt H, editors. European convention for constructional steelwork. Brussels; 2008384 pp. [23] EN1993-1-6: Eurocode 3Design of steel structures, Part 1.6: General rules— Strength and stability of shell structures. Eurocode 3 Part 1.6, CEN, Brussels. 2007.
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Experimental study on damaged cylindrical shells under compression Tohid Ghanbari Ghazijahani n, Hui Jiao, Damien Holloway School of Engineering, University of Tasmania, Hobart, Australia
art ic l e i nf o
a b s t r a c t
Article history: Received 23 January 2014 Received in revised form 25 February 2014 Accepted 26 February 2014
Sensitivity to initial imperfections under compressive loading has been extensively studied in shell structures. However, due to the existence of a wide range of imperfections with various shapes and amplitudes, the real behavior of such structures needs to be further investigated when they face with a damaged area. This study presents an experimental program in which buckling and failure response of damaged shell specimens are analyzed. The results of this study can be generalized for many kinds of cylindrical shells to full scale of applications with similar D/t ratios. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Thin-walled cylindrical shells Tests Compression Initial damage Dent
1. Introduction Axial compressive stresses in cylindrical shell structures arise from various causes such as in towers and chimneys caused by the weight of the structures [1]. Such structures are often vulnerable to physical contacts of the other elements during their service life. A wide range of references can be found in the literature in regard to the buckling and failure response of cylindrical shell structures with normal fabrication-related imperfections under compressive stresses, e.g. [2–9]. However, quite a few studies reflect such structures with large imperfections caused by a collision. Local bulges, dents and unilateral corrugations were studied and it was found that local dents significantly reduced the critical load [10]. The effect of localized imperfections on the buckling of cylindrical shells under axial compression was studied and a considerable reduction of the critical load due to the imperfections was observed [11]. Two new approaches were proposed for the numerical and analytical stability analyses of imperfect shells [12]. Theoretical buckling stress of such shells was defined by Eq. (1), [1]. In this equation E is Young's modulus and “ν“ is Poisson's ratio. “t“ and “r“ are respectively the thickness and the radius of the shells. This equation mostly overestimates considerably the buckling load of thin shell due to the occurrence of local deformations resulted from the geometric imperfections in experimental models. It should be noted that, the number of full buckling waves around
n
Corresponding author. Tel.: þ 61 469 311 896. E-mail addresses: [email protected], [email protected] (T. Ghanbari Ghazijahani). http://dx.doi.org/10.1016/j.tws.2014.02.029 0263-8231 & 2014 Elsevier Ltd. All rights reserved.
the circumstance and half-wavelength of the buckling waves were also defined theoretically by Eqs. (2) and (3) respectively [1]. Et
t r
ð1Þ
0:25 0:5 r 0:5 r C 0:909 t t
ð2Þ
pffiffiffiffi ðrtÞ0:5 C1:728 rt
ð3Þ
s ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 0:605 E 2 r
n¼ λ¼
3ð1 ν Þ
3 ð1 ν2 Þ 4 π
ð12ð1 ν2 ÞÞ0:25
Analytical methods were developed for the estimation of the upper critical loads for non-reinforced cylindrical shells with axisymmetric dents (bulges) located on the shell structures. An investigation has been conducted on the buckling of steel cylindrical shells with a single local dent [13,14]. It was found that even a single dent strongly influences the magnitude of the critical load. Refs. [15,16] are highly relevant papers to this work. Axial compression was applied to the damaged specimens and the buckling behavior was studied [15]. It was found that damaged shells reached almost half of the capacity of the intact specimens in the case of eccentric loading. For undamaged specimens failure was quite abrupt and without warning whereas the damaged structures had a phase of stable growth of a dent before the catastrophic failure ensued. Parametric study on the buckling behavior of dented short carbon steel cylindrical shell subjected to uniform axial compression was conducted [16]. Cylindrical shells with longitudinal dents were believed to have higher buckling strengths than cylindrical shells with circumferential dents.
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T. Ghanbari Ghazijahani et al. / Thin-Walled Structures 80 (2014) 13–21
Table 1 Specifications of the specimens. Specimen d (mm)
Orientation of the dent
HD (mm)
HD/H WD (mm)
D/t¼ 607.08, t ¼ 0.25 (mm), L/D ¼ 1.44
LCC.1 LCC.2 LCC.3 LCC.4 LCC.5 LCC.6 LCC.7 LCC.8 LCC.9 LCC.10 LCC.11 LCC.12 LCC.13
– – 0.45 2.3 3 6.2 3 6.5 2 3.3 3.2 8 1.8
– – Horizontal Horizontal Horizontal Horizontal Diagonal Diagonal Vertical Vertical Horizontal Horizontal Horizontal
– – 110 110 110 110 110 110 110 110 55 55 20
– – 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.25 0.25 0.09
– – 19 39 63 75 60 75 75 90 56 72 35
D/t¼ 339.52, t ¼ 0.25 (mm), L/D ¼ 1.36
SCC.1 SCC.2 SCC.3 SCC.4 SCC.5 SCC.6 SCC.7 SCC.8 SCC.9 SCC.10 SCC.11 SCC.12 SCC.13 SCC.14
– – 0.4 0.8 1.2 1.8 2.5 0.5 0.9 2.1 1 2.7 1.2 1.2
– – Horizontal Horizontal Horizontal Horizontal Horizontal Horizontal Horizontal Horizontal Horizontal Horizontal Vertical Diagonal
– – 15 15 15 15 15 30 30 30 57.8 57.8 57.8 57.8
– – 0.130 0.130 0.130 0.130 0.130 0.260 0.260 0.260 0.5 0.5 0.5 0.5
– – 14 21 26 30 38 16 26 32 22 37 44 42
The present study was performed on 27 locally dented shell specimens as very limited experimental data are found on the dented shells under compression. Dent imperfections of different depths, locations and orientations were modeled. This paper focuses on the following points:
Experimental modeling of dented shells; Buckling of intact and damaged specimens; Failure modes of such structures; Capacity assessment of dented shells and; Comparison of the results with other works and existing standards.
2. Experimental program 2.1. System set-up 2.1.1. Apparatus The test apparatus for the present experimental program was a MTS-810 machine. Two end plates were fabricated in which the specimen ends were placed in grooves with a slight tolerance in order to restrain the specimens in the radial direction while allowing rotation of the walls about the circumferential line. End plates were gripped by two jaws of the MTS machine. Before the tests the two top and bottom plates were calibrated accurately to ensure that the axial loading was uniformly applied. The vertical distance between the two plates before the tests was adjusted such that no additional weight of the plates was applied to the top portion of the specimens. 2.1.2. Specimens In this study, 27 precisely-fabricated specimens with two different D/t ratios were tested (see Table 1 and Fig. 1). LCC.1 to
Fig. 1. Test set-up for cylindrical shell specimens.
LCC.13 (D¼ 151.77 mm, L ¼218.55 mm and t¼ 0.25 mm), were the specimens with higher D/t ratio and SCC.1 to SCC.15 were the specimens with lower D/t ratio (D ¼84.88 mm, L ¼115.44 mm and t¼0.25 mm). The Young's modulus as the main material property in buckling tests of thin shells and Poisson's ratio were obtained from a tensile coupon test as 210 GPa and 0.3 respectively. The specimens had top and bottom circular caps to simulate the practical instances of the tanks with two end caps. Fig. 2 presents two practical instances of the steel tanks with circular flat roofs and local geometric defects on the surface of such structures. 2.1.3. Indentation and measurements Indentation was conducted by means of a steel indenter with a circular sharp end (see Figs. 3–5). Indentation was made through closely moving the indenter over the body of the shell specimens to achieve a dent with the desired depth and corresponding width. A finger LVDT was utilized in order to have accurate recordings of the dented area (Fig. 5). A small ball-shaped tip of this LVDT made it capable to move along and/or across the dented region and accurately detect the geometric depressions of the dent. CEA-06240UZ-120 Micro-Measurements USA made strain gauges were employed in order to record the strain values in the critical areas of the test samples. End shortening of the specimens was recorded by the MTS machine.
3. Test results 3.1. Buckling and failure 3.1.1. Intact specimens Intact specimens in this study are considered as control specimens against which the other locally imperfect specimens were
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Fig. 4. Illustration of a typical dented specimen.
Fig. 2. (a) Steel reservoirs with circular flat roofs [17], and (b) local geometric defects on the body of a steel silo [17].
Fig. 3. Horizontal, vertical and diagonal dents.
evaluated. Thus, for each set of LCC and SCC specimens the intact models were repeated to ensure the accuracy of the results and the buckling/failure modes. The buckling mode of LCC.1, LCC.2, SCC.1 and SCC.2 specimens were all in a diamond mode of buckling near the edge area of the specimens. In this mode of buckling, initially several small diamond-shaped deformations occurred in one tier, after which the deformations deepened, until final failure of the specimens accompanied by covering the whole circumferential direction with the above mentioned deformations. The position of the buckling can be largely attributed to the end effects of the tested models or initial imperfections of the edge area of the specimens.
3.1.2. Dented specimens As can be seen in Table 1, LCC.3–LCC.6 were the specimens with the horizontal dents at the mid-height of the specimens. For these specimens buckling initiated at the edge area similar to the intact specimens. However, for the specimens with larger dents (i.e. LCC.5 and LCC.6) initial buckling was accompanied by development of the deformations in the dented area. Note that the effect of the dent on initial buckling was not considerable for small
dents, but deformations in the dented area were significant in the post-buckling phase, for all tests. Fig. 6 shows the progress of the buckling in the specimens with large dents, i.e. LCC.5 and LCC.6. Specimens LCC.7 and LCC.8 were indented diagonally with different depths of the dent. As can be seen in Fig. 7, buckling occurred concurrently at the edge zone of the specimens and both ends of the diagonal dent in such a way that buckling was initiated at the edge area of the diagonal dent followed by development of the deformations in an inclined manner which eventually met the edge deformations. As regards the specimens with vertical dents, the buckling phenomenon was similar to the diagonal dents. However, in these specimens the initial buckling wave was initiated at and/or adjacent to the vertically dented zone (see Figs. 8 and 9). As discussed before, the top and bottom edge zones were quite vulnerable to buckling initiation. For this reason, specimens LCC11–LCC3 and SCC3–SCC10 were prepared with dent imperfection located near the top edge to evaluate the effect of the dent at the most sensitive area to the imperfections. The dented area near the edge zone helped trigger the buckling concurrently at the top zone together with the area right adjacent to the pre-existing lobe of the dent. It is interesting to note that the diamond mode buckling tier was quite dependent on the value of HD parameter, which is the distance of the center of the dent from the top end. As can be seen in Fig. 10, for lower values of HD, the diamond mode occurred in one tier whereas two tiers of diamond mode occurred for dents with higher values of HD. Moreover, for SCC specimens with the horizontal dents, the buckling mode was similar to the LCC specimens as can be seen in Fig. 11. Since the material of the dented area changed due to the effect of local loads, i.e. indentation, buckling mostly occurred at the zone immediately adjacent to the dented segment. This may be due to the fact that the material of the dented zone entered the inelastic region so that it became brittle in comparison to the intact elements of the shells. On this basis, the area adjacent to the dented zone was affected by the initial buckling owing to the geometric irregularities in the nearby dented zone. Note that unlike the dented zone, the material was not severely changed into inelastic in the adjacent area of a dent; however the geometric imperfections fairly existed in that area due to the effects of the dent which by and large makes the adjacent area geometrically irregular. The adjacent area and the buckling lobes were marked in Fig. 11. 3.2. Buckling capacity of the dented specimens: The buckling capacity of the present specimens was affected by the pre-existing dent imperfections. Fig. 12(a) shows the axial load
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Fig. 5. (a) Measurement of the dented area by a finger transducer and (b) indenter.
Fig. 6. Buckling initiation and failure in: (a) LCC.5 and (b) LCC.6.
Fig. 7. Buckling initiation and failure in LCC.7.
capacity of the present specimens versus the parameter ζ which was defined as ζ¼ d(D/t). As can be seen, SCC specimens were more sensitive to the initial dent than LCC specimens. Moreover, data scatter is seen more in LCC specimens in comparison with SCC specimens indicating that the specimens with higher D/t ratio were more vulnerable to the local effects before the ultimate load capacity was reached. Thus, SCC specimens showed a stable decrease against the axial loading when they were imposed to the local dents. In general, the buckling behavior of shells under axial stresses greatly depends on the amplitude and the shape of the geometrical imperfections as many imperfection forms may not have a very deleterious influence so that the decrease in strength may not be easily deduced considering the depth of dents [1]. Parameter χ was defined as “χ¼ dWD” in which WD was included as another geometrical parameter of a dent to show the effect of the dent considering both depth and width of the dented area (see Fig. 13). A decreasing trend of the buckling capacity is seen in Fig. 13 in
which SCC specimens present a higher decrease with less scatter in data in comparison with LCC specimens. The capacity of different LCC specimens considering various orientations is also seen in Fig. 12(b). Based on the results, no big difference was seen in this study between the specimens with different forms of the imperfection, i.e. horizontal, vertical and diagonal. Figs. 14 and 15 show end shortening of the present LCC and SCC specimens. After some initial stiffening very early in the loading, a nearly linear pre-buckling behavior is seen after which the typical bifurcation is detected. As can be seen in Figs. 14 and 15 a clear second bifurcation after the first bifurcation was seen in the form of a steeply falling post-buckling path as the mode switched from one circumferentially deformed buckling to another in dynamic jumps of the diamond mode [1]. In these specimens, the second bifurcation point can be attributed to the development of large deformations at the dented area after the initial bifurcation occurred at the edge of the shells.
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Fig. 8. Typical deformation progress in vertical and horizontal dents.
Fig. 9. Vertical dent at mid-height of the specimen and failure mode.
Fig. 10. (a) Horizontally dented SCC specimens with dents 15 mm from the top end and intact specimen, (b) Three horizontally dented specimens with the dent at 30 mm from the top end, d ¼ 0.5, 0.9 and 2.1 from left to right. Two tiers of buckling are seen.
Strain values of the specimens at the dented area (S.2) and an equivalent intact zone (S.1) are presented in Figs. 16 and 17. As can be seen, the strain values follow a linear and/or a nonlinear trend depending on where they were placed, which may or may not be
exactly at the zone of the initiation of the buckling. As it is hard to determine the area at which the initiation of the buckling in such structures occurs, strain values may show a great scatter depending on a point on which they were placed on the surface of the shells.
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4. Comparison of the results with other studies and standards Previously published studies largely reflected the effect of small amplitude imperfections on the buckling behavior of shell structures. As mentioned before, quite a few studies are found regarding a local large amplitude imperfection as a dent. Among those studies, the most relevant investigation is adopted herein for comparison with the current results [15]. In this study, sufficient tests of dented shells with nearly similar D/t ratio were conducted.
Fig. 11. Post-buckling of horizontal dents with different depths of 1 mm and 2.7 mm at the mid-height in SCC specimens. The area adjacent to the dented zone is marked.
Above all, similar WD, as an important geometric feature of the dented zone, was reported in this study. Fig. 18 shows a capacity ratio which we define it as the maximum load of the specimens in the present study divided by the load in the mentioned reference in which “S” is taken to be equivalent to WD of the current study. A range of approximately 0.5–1 was achieved which is regarded as a good agreement between the results of the two studies. Consistency of the results was greater for the specimens with larger dents, i.e. S¼50 mm, indicating that larger imperfections mostly governed the buckling behavior in the two studies. It is noteworthy that very few design codes and standards have defined the allowable imperfection tolerated during the welding and/or construction of the shell structures. Fig. 19 shows a schematic illustration for the imperfection tolerance limits in different directions presented in three different codes [21–23]. Table 2 shows the main geometrical parameters of the dents for the present specimens considering equivalent geometric specifications mentioned in the above standards. It should be mentioned that the geometric specifications tolerated by different standards are far less than the case of dent imperfections as a large local defect. Notwithstanding this, all the current specimens showed a stable pre-buckling and post-buckling behavior on the one hand, and a quite smooth decrease in the buckling capacity on the other hand. This phenomenon was reported in [18–20] for shells under
Fig. 12. (a) Capacity versus ζ for LCC and SCC specimens and (b) LCC specimens with different dent orientations.
Fig. 13. Capacity versus χ for LCC and SCC specimens.
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Fig. 14. End shortening for the specimens LCC and SCC specimens.
Fig. 15. End shortening for the specimens SCC.1–SCC.10.
Fig. 16. Strain values at the dented area and the edge area of the specimens SCC.7 and SCC.10.
Fig. 17. Strain values at the dented area and the edge area of the specimens LCC.3 and LCC.4.
external pressure as well. In these references computations of tv and lmQ revealed that weld-induced imperfections similar to the dents of current specimens far exceeded the amounts allowed by the codes. In such cases, the standards simply suggest that the structure cannot be used. However, there is no need to stop using
these structures and their function should continue with special care [19]. What's more, despite extensive data in the literature for shells under compression, comparisons must be drawn with a great deal of care for design purposes, for different imperfection types in order to generalize the buckling behavior [1]. Therefore,
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further research is still certainly required to improve the understanding of various forms of the imperfections, particularly local large dents under different load and boundary conditions.
5. Conclusion In this study, 27 cylindrical shell specimens with two different D/t ratios were tested under axial loading. A local dent-shaped imperfection with different geometries and positions was introduced on the surface of the specimens. The results of the current study can be summarized as follows:
Typically buckling occurred at the edge zone of the specimens
5.1. Buckling mode
Top and bottom edge zones were quite vulnerable to the
buckling phenomenon. The dented area near the edge zone helped trigger the buckling which occurred concurrently at the dented zone together with the area right adjacent to the preexisting lobe of the dent. Diamond-shaped buckling tier(s) were quite dependent on the proximity of the dent to the cylinder ends (HD values). For lower values of HD, the diamond mode occurred in one tier, whereas two tiers in the diamond mode were seen for dents with higher HD. In severely dented specimens, initial buckling was accompanied by the development of the deformations at the dented area; whereas, for small dents the effect of the dent was not considerable. After the initiation of buckling, the dented area was largely affected by the deformations in the post-buckling phase.
and concurrently initiated at both ends of the dent in an inclined manner leading to meet the edge of the specimens. For vertically dented specimens, initial buckling occurred at and/or adjacent to the diamond waves of the edge area of the specimens. The most significant zone of a dent occurrence was the edge area of the present specimens. The material of the dented zone entered the inelastic region so that it became brittle in comparison to the intact elements of the shells. The area adjacent to the dented zone was affected by the initial buckling owing to the geometric irregularities at nearby zone of a dent. Note that the material did not severely change into inelastic in the adjacent area of a dent, however the geometric imperfections fairly existed due to the irregularities of the dented area.
5.2. Buckling strength
SCC specimens were more sensitive to the initial dent than LCC specimens.
Specimens with higher D/t ratio were more vulnerable to the
local effects before the ultimate load capacity was reached so that data scatter is seen more in LCC specimens in comparison with SCC specimens. A second bifurcation point on the post-buckling path can be attributed to the development of large deformations at the dented area after the initial bifurcation occurred at the edge area. A good agreement is obtained with the previous study on the dented shells with similar D/t ratio.
Table 2 Main parameters of the dents considering the mentioned standards.
Fig. 18. Comparison of the data with the results of Ref. [15] in which “S” is corresponding to WD.
Specimen
d/t
d/WD (%)
WD/t
Specimen
d/t
d/WD (%)
WD/t
LCC.3 LCC.4 LCC.5 LCC.6 LCC.7 LCC.8 LCC.9 LCC.10 LCC.11 LCC.12 LCC.13
1.8 9.2 12 24.8 12 26 8 13.2 12.8 32 7.2
2.37 5.9 47.62 8.27 5 8.67 2.67 3.67 5.71 11.11 5.14
76 156 25.2 300 240 300 300 360 224 288 140
SCC.3 SCC.4 SCC.5 SCC.6 SCC.7 SCC.8 SCC.9 SCC.10 SCC.11 SCC.12 SCC.13 SCC.14
1.6 3.2 4.8 7.2 10 2 3.6 8.4 4 10.8 4.8 4.8
2.86 3.81 4.62 6 65.79 3.13 3.46 6.56 4.55 7.3 2.73 2.86
56 84 104 120 15.2 64 104 128 88 148 176 168
DIN 18800-4
tr<1% of Lmx
ESSCRec, 2008
L=Ir:tr=W tr<1% of Lmx
ENV 1993-1-6, 2007
Lmx=Lg:tr=∆w0 tr≤0.006Lmx OR (0.01 Lmx, 0.016 Lmx for different classes)
Fig. 19. Geometric tolerances in different codes: (a) Initial dimple in vertical direction and (b) Initial dimple in circumferential direction [18–20].
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Although the geometric specifications tolerated by different standards are far less than the case of the dent imperfections, all the specimens showed a stable pre-buckling and postbuckling behavior on the one hand, and a smooth decrease in the buckling capacity on the other hand. To sum up, one may note that despite extensive data in the literature for shells under compression, comparisons with different imperfection types must be drawn with a great deal of care in order to generalize the buckling behavior for design purposes. Therefore, further research is still required to improve the understanding of various forms of the imperfections particularly local large dents under different load and boundary conditions. Acknowledgments Efforts of Mr. Andrew Bylett, Mr. Peter Seward and Mr. David Morley for preparation of the specimens and the test rig and Mr. Calverly Gerard for LabVIEW program are greatly appreciated. References [1] Teng J-G, Rotter JM. Buckling of thin metal shells. London: Taylor & Francis, Spon Press, Taylor & Francis, Group; 2004. [2] Brush DO, Almroth BO. Buckling of bars, plates, and shells, vol. 6. New York: McGraw-Hill; 1975. [3] Calladine CR. Theory of shell structures. Cambridge University Press; 1989. [4] Deml M, Wunderlich W. Direct evaluation of the ‘worst’imperfection shape in shell buckling. Comput MethodsAppl. Mech Eng 1997;149:201–22. [5] Hutchinson J, Muggeridge D, Tennyson R. Effect of a local axisymmetric imperfection on the buckling behaviorof a circular cylindrical shell under axial compression. AIAA J 1971;9:48–52. [6] Holst JMF, Rotter JM, Calladine CR. Imperfections in cylindrical shells resulting from fabrication misfits. J Eng Mech 1999;125:410–8.
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[7] Holst J. Local dimpling of the shell surface of a tank due to shrinkage effects. Struct Granul Solids 2011:153. [8] Ghanbari Ghazijahani T, Showkati H. Bending experiments on thin cylindrical shells. Materials with complex behaviour II. Germany: Springer; 2012; 119–39. [9] Ghanbari Ghazijahani T, Showkati H. Experiments on cylindrical shells under pure bending and external pressure. J Constru Steel Res 2013;88:109–22. [10] Gavrilenko G. Stability of cylindrical shells with local imperfections. Int Appl Mech 2002;38:1496–500. [11] Jamal M, Lahlou L, Midani M, Zahrouni H, Limam A, Damil N, et al. A semianalytical buckling analysis of imperfect cylindrical shells under axial compression. Int J Solids Struct 2003;40:1311–27. [12] Gavrilenko G. Numerical and analytical approaches to the stability analysis of imperfect shells. Int Appl Mech 2003;39:1029–45. [13] Gavrilenko G, Krasovskii V. Stability of circular cylindrical shells with a single local dent. Strength Mater 2004;36:260–8. [14] Ghanbari Ghazijahani T, Showkati H. Locally imperfect conical shells under uniform external pressure. Strength Mater 2013;45:369–77. [15] Hambly E, Calladine CR. Buckling experiments on damaged cylindrical shells. Int J Solids Struct 1996;33:3539–48. [16] Prabu B, Raviprakash A, Venkatraman A. Parametric study on buckling behaviour of dented short carbon steel cylindrical shell subjected to uniform axial compression. Thin-Walled Struct 2010;48:639–49. [17] Available:〈http://www.cstindustries.com/products/trough-deck-roofs-covers/〉 〈http://www.structuremag.org/article.aspx?articleID=446〉. [18] Niloufari A, Showkati H, Maali M, Fatemi S Mahdi. Experimental investigation on the effect of geometric imperfections on the buckling and post-buckling behavior of steel tanks under hydrostatic pressure. Thin-Walled Struct 2014;74:59–69. [19] Fatemi SM, Showkati H, Maali M. Experiments on imperfect cylindrical shells under uniform external pressure. Thin-Walled Struct 2013;65:14–25. [20] Maali M, Showkati H, Fatemi S Mahdi. Investigation of the buckling behavior of conical shells under weld-induced imperfections. Thin-Walled Struct 2012;57:13–24. [21] DIN 18800. Stahlbauten.Teil 4: Stabilitätsfalle, Schalenbeulen 1990. [22] ECCS EDR5. European Recommendations for steel construction. Buckling of shells, 5th ed. In: Rotter JM, Schmidt H, editors. European convention for constructional steelwork. Brussels; 2008384 pp. [23] EN1993-1-6: Eurocode 3Design of steel structures, Part 1.6: General rules— Strength and stability of shell structures. Eurocode 3 Part 1.6, CEN, Brussels. 2007.