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Bridge and Pircher

Buckling and Post-Buckling of Cylindrical Shells with Circumferential Weld Imperfections Subjected to Axial Load

Russell Q. BRIDGE, Martin PIRCHER Centre for Construction Technology Research University of Western Sydney, Australia

ABSTRACT: Steel silos and tanks are constructed from plates which are rolled to the correct curvature and welded together to form strakes. Several strakes of curved plates, placed on top of each other then form the completed structure. At each circumferential weld, a slight hourglass depression occurs forming axisymmetric imperfections which are known to be most deleterious. The detrimental influence of this particular type of imperfection on the buckling of silos and tanks is well known. This paper gives an overview over results gained from recent research into the influence of such imperfections on the buckling and post-buckling behaviour of silos and tanks.



Early research into the buckling of thin-walled shell structures was centered around applications commonly found in the aviation industry. Axisymmetric imperfections in cylindrical shells do not generally occur in aircraft structures and consequently they were dismissed as of little relevance to practical problems (Arbocz, 1974). Their influence on the buckling of shells was not investigated in depth until the 1970s despite Koiter's results (1945, 1963) which clearly showed that axisymmetric imperfections are extremely severe. Buckling of thin-walled shell structures suddenly became relevant for structural engineers in the 1950s when failures in thin walled steel tanks and silos became an issue. Manufacturing techniques had finally become sophisticated enough to allow the construction of steel silos thin enough to be imperfection sensitive. Small deviations from the nominal, or perfect geometry in such imperfection sensitive structures result in significant loss of strength. Calladine (1995) lists three criteria to define "imperfection-sensitive behaviour": (i) Buckling loads which fall short of the predictions of classical buckling theory (cl). (ii) Unpredictability of buckling strength, as evidenced by the wide scatter of experimentally observed buckling loads. (iii) Unstable, dynamic behaviour of the shell after the maximum load has been reached leading to catastrophic failure in some circumstances. Thin-walled circular cylindrical structures show all of the above mentioned behaviour patterns. Buckling loads in laboratory tests are usually scattered. Very small differences in the amplitude


Buckling and Post-Buckling of Cylindrical Shells with Circumferential Weld Imperfections

of imperfections can lead to enormous differences in strength. Buckling typically occurs suddenly and dynamic effects distinguish the behaviour immediately after the critical load has been reached. In Koiter's original theory (Koiter 1945) and later again in his special theory (Koiter 1963), axisymmetric imperfections in the shape of indefinitely extensive sinusoidal waves were used among other imperfection patterns, to determine the imperfection sensitivity of thin-walled cylindrical shells. Esslinger & Geier (1977) investigated cylinders with lapped joints and concluded that the axisymmetric portion of the imperfections governed the buckling of their specimens. Nowadays, steel silos and tanks are commonly constructed from plates which are rolled to the correct curvature and welded together to form strakes. Several strakes of curved plates, placed on top of each other then form the completed structure. At each circumferential weld, a slight hourglass depression occurs forming axisymmetric imperfections. Measurements on silos and tanks (Steinhardt & Schulz 1970, Ding et al. 1996) revealed that predominantly axisymmetric imperfections do occur in these structures. Research by Tennyson & Muggeridge (1969), Hutchinson et al. (1971) and Amazigo & Budianski (1972) investigated localised axisymmetric imperfections and demonstrated that a single axisymmetric imperfection can have a significant effect on the buckling strength of thin-walled cylinders. Two papers by Rotter & Teng (1989) and Teng & Rotter (1992) contributed more research on the topic of circumferential weld-induced imperfections using numerical methods. Until then only geometric imperfections in indefinitely long cylindrical shells were considered. Rotter (1996) did a limited study which included interaction between neighbouring welds and residual stresses for a small number of examples contradicting the only other study that takes residual stresses into account (Häfner 1982). The post-buckling behaviour of thin-walled cylinders under axial load has been the subject of large amounts of research ­ theoretical and experimental ­ since the early work summed up comprehensively in Hutchinson & Koiter (1970). This severly unstable post-buckling behaviour was observed experimentally and documented by Yamaki (1984) and Esslinger & Geier (1972). Esslinger (1967) offered a simple yet highly descriptive theoretical explanation for the initial post-buckling process. Increased computer capacity during the past two decades has also opened possibilities to study the post-buckling of cylinders numerically leading to interesting results for a wide range of problems which had been inaccessible before (eg. Guggenberger 1995). A review of recent advances is given in Teng (1996). This paper gives a review of research work that has been undertaken in recent years at the Centre for Construction Technology and Research at the University of Western Sydney, Australia. Several aspects of axisymmetric imperfections in shell-buckling and post-buckling problems have been investigated and results have been presented in several papers. The key factors affecting the buckling resistance include the weld imperfection shape, the magnitude of the weld imperfection, the interaction between the neighbouring weld imperfections and the residual stresses induced by the welding. A model for post-buckling behaviour has been developed that not only traces the response but also explains the changes in behaviour with increasing axial deformation.


Bridge and Pircher



A shape function based on elastic shell theory was derived (Eq. 1). In this equation w is the radial deformation, x is the axial coordinate starting at the weld, w0 is the amplitude of the imperfection, is the wavelength of the weld imperfection and is a measure of the roundness of the imperfection at the weld centre. w( x ) = w0 e


x x cos + sin


Ding et al. (1996) gave detailed imperfection maps gained from measurements taken on silos at Port Kembla, Australia. The data describing the localised weld imperfection was filtered out of these imperfection maps and was then used to calibrate the parameters w0, and in the context of Eq. 1. Details of the derivation of the shape function and the fitting process are given in Pircher et al. (2000) and Pircher et al. (2001) and an example for a fitted shape function is given in Figure 1. The mean amplitude of the depth of the imperfection w0 ranged from 0.176 t to 0.810 t (where t is the wall thickness of the silo). In most cases the imperfection was found to be inward facing but in some rare cases the opposite was the case. The half-wavelength was found to be close to the linear meridional bending half-wavelength 0 for the two lowest welds and closer to 20 for the other welds. A reason for this longer than expected half wavelength can be found in the fact that the strake height is not quite long enough to satisfy the requirements of a long cylinder which is the basis for Eq.1. The mean values for the roundness parameter ranged from 0.527 to 0.832 resulting in weld shapes well between the two extremes of = 0 and = 1 which are used in Rotter & Teng (1989) and correspond to pointed and perfectly rounded respectively.


Local axial coordinate, x - xw (mm)


Profile relative to overall meridian


Overall meridian 2 ( + x + x )


-500 w0 / t = 0.29 -1000 / 0 = 1.23 = 0.13 -1500 -5 -4 -3 -2 -1 0 1

Radial coordinate, w (mm)

Figure 1 Example for the fitting of the shape function to measured results 55

Buckling and Post-Buckling of Cylindrical Shells with Circumferential Weld Imperfections



Finite element models were used to study the influence of the shape function on the buckling behaviour of silos and tanks. Details on the modelling techniques are given in Pircher & Bridge (2000) and results are being published in Pircher & Bridge (2001a). The roundness of the imperfection at the weld centre represented by the parameter and the half-wavelength of the imperfection were shown to be significant influences on the buckling strength as shown in Figure 2. The amplitude w0 is a most important factor as that affects the buckling strength as indicated in Figure 3 where an amplitude of only one wall thickness t can reduce the buckling strength to 30% of the classical buckling strength cl for a perfect cylinder. The scatter in buckling strengths reported by other researchers can be explained by the differences in the shape functions used in their analyses as shown in The scatter in buckling strengths reported by other researchers can be explained by the differences in the shape functions used in their analyses as shown in Figure 4.




=0.0 =0.5 =1.0

/ cl


/0=1.85 /cl=0.333


0.3 0.5 1 1.5

/ 0




Figure 2 Buckling strength for variations of and for a long cylinder for w0/t = 1

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 0.5 1 1.5 2 2.5 3

long / cl

Imperfection amplitude w 0 /t

Figure 3 Variation in buckling strength for different imperfection amplitudes


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w0/t = 1.0


/ cl


Häfner A Häfner B Rotter & Teng B Rotter & Teng A

=1 Rotter final


=0 Rotter best Tennyson & Muggeridge Rotter closed

0.5 1 1.5 2 2.5 3



/ 0

Figure 4 Existing shape functions in the context of the new shape function



Strake heights in silos and tanks are dictated by the size of steel plates commercially available. This typically leads to strake heights which are too small to isolate the effects of neighbouring circumferential weld imperfections from each other. Rotter (1996) proposed a system of taking interaction between weld imperfection into account which is based on the assumptions that neighbouring imperfections are equally detrimental to the buckling strength. This system was also used for the work presented in Pircher & Bridge (2000) and Pircher & Bridge (2001a). Figure 5 illustrates how the buckling resistance is influenced not only by the amplitude of the imperfection w0 but also by the ratio between the strake height L and the linear meridional bending half wavelength 0.

1.30 1.20 1.10

/ long

1.00 0.90 0.80 0.70 0.00

w 0 /t = 0.5 w 0 /t = 1.0 w 0 /t = 1.5

2.00 4.00 6.00 8.00 10.00

Normalised strake height L / 0

Figure 5 Buckling strength for various imperfection amplitudes w0/t and half strake


Buckling and Post-Buckling of Cylindrical Shells with Circumferential Weld Imperfections



When two shell segments are welded together, the material along the weld is heated to the melting temperature, and stress relaxation at high temperature is so rapid that the heated structure is essentially stress free. However, when the structure is subsequently cooled to room temperature, large tensile stresses parallel to the weld develop as a result of thermal contraction close to the weld centre line, while compressive stresses develop farther away from the weld to keep equilibrium. Typically, circumferential tension stresses along the weld reach yield. For perfectly flat plates welded together this stress field can satisfy equilibrium accurately but in the case of circumferential welds in cylindrical structures these circumferential stresses are accompanied by stresses in the axial direction perpendicular to the weld and shear stresses to maintain equilibrium. Even at the point of buckling under axial stress, the area near the weld is still under considerable tension (Figure 1) which leads to an increase in buckling resistance. Compressive residual stresses further away from the weld typically range from 0.2 to 0.4 of the yield stress and are further increased by axial compression. With the increase in axial load the tensile stresses decrease and the compressive stresses further away from the weld are further increased until a point is reached where the structure buckles. The second contributing factor is the presence of axial stresses and the resulting bending moments. These bending moments can be expected to become relatively more influential in more thick-walled shells and erode the stabilising effect of the circumferential tensile membrane stresses. A discussion of these competing factors is given in Pircher & Bridge (2000) and Pircher & Bridge (2001a).


1.2 1 0.8 0.6

wt = width of the tension zone weld induced stresses (prior to loading)

/ yield

0.4 0.2 0

Stresses at buckling (with weld stresses)


-0.2 -0.4 -0.6

Stresses at buckling (initially stress free) Weld

0 25 50 75 100


Figure 6 Circumferential membrane stresses For the range of radius to thickness ratios (R/t-ratios) typical for steel silos and tanks, the weld induced residual stresses have been shown to have a stabilising effect (Figure 7). The strength gain is greater for more thin-walled shells and for smaller imperfection amplitudes.


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Increas in buckling strength %cl

10 8 6 4 2 0 0.0 0.5 1.0 1.5 2.0

R/t = 2000 R/t = 1500 R/t = 1000 R/t = 500



Imperfection amplitude w 0 /t

Figure 7 Influence of residual stresses for various imperfection amplitudes w0/t



A model for the post-buckling behaviour of silos and tanks was first developed for Pircher & Bridge (2001b) and then discussed in detail for a number of parameters in Pircher & Bridge (2001c). The quasi-static nature of this model neglects any dynamic effects and is therefore only an approximation of the actual process. The behaviour of the structure was found to be very unstable after reaching the critical load. The initial buckling mode undergoes a number of transitions until it reaches the post-buckling minimum where the load path finally becomes stable again. Upon reaching the post-buckling minimum there were a number of buckling modes (with fewer buckles around the circumference than the critical mode) that came very close to each other. This suggests that these modes are all possible and interaction between these modes is probable (Figure 8). At this stage, outwardfacing folds between the dominating inward buckles form (Figure 9). These folds carry the majority of the compressive load and can be interpreted as secondary columns, essentially changing the system from a shell to a folded plate structure. These folds become more pronounced with increasing radial deformation, thus increasing the second moment of area of these columns. The load carrying capacity of the structure rises again. If elastic material properties are assumed, the load can be further increased until these columns buckle and final collapse of the structure takes place in conjunction with the formation of secondary buckles. Teng & Song (2001) raised a number of modelling issues regarding the post-buckling analysis of thin-walled cylinders and some of these issues are also discussed for the present problem in Pircher & Bridge (2001c). In particular the difference between half-structure models and sectorial models are highlighted and the influence of imperfection amplitude, plasticity and residual stress fields on the post-buckling behaivour are studied.


Buckling and Post-Buckling of Cylindrical Shells with Circumferential Weld Imperfections

0.5 A 0.4 w 0/t =0.8 0.3 m =12 m =15 C w 0/t =1.0 B 0.2 m =16 0.1 D w 0/t =0.5

/ cl

0 0








u /t

Figure 8 Increase in load-bearing capacity after the post-buckling minimum

Figure 9 Change of load bearing system



An overview of recent research into the buckling and post-buckling behaviour of circular silos and tanks made from steel has been given in this paper. Circumferential weld-induced imperfections are known to be especially detremental to the buckling strength of these structures. Parameter studies regarding the shape of such localised circumferential imperfections, the strake height, weld-induced residual stress fields and the post-buckling behaviour are presented and references for detailed discussions of these topics are given.


Bridge and Pircher



Amazingo J.C., Budianski B. (1972) "Asymptotic Formulas for the Buckling Stresses of Axially Compressed Cylinders with Localized or Random Axisymmetric Imperfections", Journal of Applied Mechanics, v93, pp 179-184 Arbocz J. (1974) "The Effect of General Imperfections on the Buckling of Cylindrical Shells", In: Fung YC, Sechler EE editors. Thin-Shell Structures: Theory, Experiment and Design, Chapter 9, Prentice-Hall, Inc., Englewood Cliffs, N.J. Calladine C. R. (1995) "Understanding Imperfection-Sensitivity in the Buckling of Thin-Walled Shells", Thin-Walled Structures, v23, pp 215-235 Ding X., Coleman R., Rotter J.M. (1996) "Technique for Precise Measurement of Large-Scale Silos and Tanks", Journal of Surveying Engineering, v122, n1, pp 15-25 Esslinger M. (1967) "Eine Erklärung des Beulmechanismus Kreiszylinderschalen", Der Stahlbau, v12, pp 366-371 von dünnwandigen

Esslinger M. and Geier B. (1972) "Gerechnete Nachbeullasten als untere Grenze der experimentellen axialen Beullasten von Kreiszylindern", Der Stahlbau, v41, n12, pp 353-360 Esslinger M, Geier B. (1977) ,,Buckling Loads of Thin-walled Circular Cylinders with Axisymmetric Irregularities", Proceedings Steel Plated Structures: An International Symposium, London, pp 865-888 Guggenberger W. (1995) "Buckling and Postbuckling of Imperfect Cylindrical Shells Under External Pressure", Thin-Walled Structures, v23, pp 351-366 Häfner L. (1982) Einfluss einer Rundschweissnaht auf die Stabilität und Traglast des axialbelasteten Kreiszylinders, PhD-Thesis, Universität Stuttgart, Germany Hutchinson J.W., Tennyson R.C., Muggeridge D.B. (1971) "Effect of a Local Axisymmetric Imperfection on the Buckling of a Cylindrical Shell under Axial Compression", AIAA Journal, v9, n1, pp 48-52 Hutchinson J.W. and Koiter W. T. (1970) "Postbuckling Theory", Applied Mechanics Review, v23, pp 1353-1366 Koiter W.T. (1945) The Stability of Elastic Equilibrium. Dissertation at the Technische Hooge School, Delft. English translation by E Riks: Technical Report AFFDL-TR-70-25, Air Force Flight Dynamics Laboratory, Air Force Systems Command, Wright-Patterson Air Force Base, Ohio, 1970 Koiter W.T. (1963) "The Effect of Axisymmetric Imperfections on the Buckling of Cylindrical Shells under Axial Compression", Proceedings Koninklijke Nederlandse Akademie van Wetenschappen, pp 265-279


Buckling and Post-Buckling of Cylindrical Shells with Circumferential Weld Imperfections

Pircher, M., Berry, P.A., Bridge, R.Q. (2000) "The Properties of Circumferential Weld-Induced Imperfections in Silos and Tanks", Engineering Report CE17, School of Engineering & Industrial Design, University of Western Sydney Pircher, M. and Bridge, R.Q. (2000) "Buckling and Post-Buckling Behaviour of Silos and Tanks under Axial Load ­ Some New Aspects", Engineering Report CE18, School of Engineering & Industrial Design, University of Western Sydney Pircher M., Berry P.A., Ding X., Bridge R.Q. (2001) "The Shape of Circumferential WeldInduced Imperfections in Thin-Walled Steel Silos and Tanks", Thin-Walled Structures, awaiting publication Pircher M., Bridge R.Q. (2001a) "Buckling of Thin-Walled Silos and Tanks Under Axial Load ­ Some New Aspects", Journal of Structural Engineering, ASCE, v127, n10 Pircher, M. and Bridge, R.Q. (2001b) "The Influence of Circumferential Weld-induced Imperfections on the Buckling of Silos and Tanks", Journal of Constructional Steel Research, v57, pp 569-580 Pircher M., Bridge R.Q. (2001c) "On the Post-Buckling of Thin-Walled Silos and Tanks Under Axial Load", Advances in Structural Engineering, (submitted) Rotter, J.M. (1996) "Buckling and Collapse in Internally Pressurised Axially Compressed Silo Cylinders with Measured Axisymmetric Imperfections: Imperfections, Residual Streasses and Local Collapse", Proceedings Imperfections in Metal Silos Workshop, Lyon, France, pp 119-139 Rotter J. M. and Teng J. G. (1989) "Elastic Stability of Cylindrical Shells with Weld Depressions", Journal of Structural Engineering, ASCE, v115, n5, pp 1244-1263 Steinhardt O, Schulz U. (1970) Zur Beulstabilität von Kreiszylinderschalen Bericht der Versuchsanstalt fuer Stahl, Holz, Steine, Universitaet Karlsruhe Teng, J.G. (1996) "Buckling of Thin Shells: Recent Advances and Trends", Applied Mechanics Reviews ASME, v49, n4, pp 263-274 Teng J.G. and Rotter J.M. (1992) "Buckling of Pressurized Axisymmetrically Imperfect Cylinders under Axial Loads", Journal of Engineering Mechanics, v 118, n 2 pp 229-247 Teng, J.G. and Song, C.Y. (2001) "Numerical Models for Nonlinear Analysis of Elastic Shells with Eigenmode-Affine Imperfections", International Journal of Solids and Structures, v38, n18, pp 3263-3280 Tennyson R.C., Muggeridge D.B. (1969) "Buckling of Axisymmetric Imperfect Circular Cylindrical Shells under Axial Compression", AIAA Journal, v7, n11, pp 2127 ­ 2131 Yamaki N. (1984) Elastic Stability of Circular Cylindrical Shells, North-Holland, Amsterdam, Netherlands



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