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Inelastic stability of conical tanks
Thin-Walled Structures 31 (1998) 343–359
Inelastic stability of conical tanks A.A. El Damattya,*, M. El-Attarb, R.M. Korolb a
Department of Civil and Environmental Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 b Department of Civil Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7
Abstract The aim of this investigation is to study the effect of different imperfection shapes on the inelastic stability of liquid-filled conical tanks and to determine the critical imperfection shape that would lead to the minimum inelastic limit load. The study is carried out numerically using a self-developed shell element used to simulate a number of conical tanks having an imperfection shape in the form of Fourier series of equal coefficients. The Fourier analysis of the buckling modes indicates that the existence of axisymmetric imperfection will lead to the critical inelastic limit load for conical tanks. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Tanks; Conical; Stability; Imperfections; Buckling; Liquid-filled
Notation Ck Dk E ET k Lb r s t w0
Fourier coefficients for the cos k imperfection terms Fourier coefficients for the sin k imperfection terms modulus of elasticity of the wall material tangent modulus circumferential wave number buckling wavelength radius at base of cone distance measured along generator of cone thickness of cone wall amplitude of imperfection wave
* Corresponding author. 0263-8231/98/$—see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 8 ) 0 0 0 2 0 - 2
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␥ v y
specific weight of water Poisson’s ratio of the wall material angle measured in the circumferential plane angle of inclination of cone generator with the vertical meridional membrane stress along bottom edge of the tank yield stress of the wall material a dimensionless quantity used for design curves a dimensionless quantity used for design curves
1. Introduction Conical shaped steel vessels having cylindrical upper section caps are widely used as superstructures for elevated water tanks. The authors are aware that approximately 200 of such containment vessels exist in Canada alone. The cross-sectional elevation, shown in Fig. 1, represents a typical conical tank consisting of curved steel panels welded together along circumferential and longitudinal edges. The steel vessel is welded to a steel plate at its bottom, which is then anchored to a concrete slab supported by reinforced concrete towers. Indeed, a lack of adequate design guidelines for conical tanks in North America may have contributed to the collapse of the water tower in Fredericton, Canada in December of 1990. It is imperative that stability analyses and related design formulae consider the effects of both geometric imperfections and residual stresses due to welding. A recent study done by El Damatty et al. [1] has shown that geometric
Fig. 1.
Elevated conical tank.
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imperfections have a significant effect on the stability of liquid-filled conical tanks but that the effect of residual stress is somewhat less important. Unfortunately, field measurements of geometric imperfections existing in prototype shell structures are rare in civil engineering structures. Due to a lack of pertinent statistical information, a conservative approach is to consider the effect of various imperfection shapes in order to determine the one leading to the minimum buckling capacity of the structure. In the study done by El Damatty et al. [1], an elastic stability analysis of liquidfilled conical tanks having various imperfection shapes was considered and the axisymmetric imperfection shape was found to be the critical one. By extending the analysis to include nonlinear material behaviour, it was found that for conical tanks having practical dimensions, yielding precedes the elastic buckling and that inelastic buckling governs the failure of these liquid-filled structures. For the inelastic stability of conical tanks, while axisymmetric imperfections tend to be more significant in decreasing the stability of the structure, non-axisymmetric imperfections cause circumferential bending which can influence the onset of yielding of the structure and hence reduce its inelastic buckling capacity [2]. To the best of the authors’ knowledge, no study has been done to consider the effect of various imperfection shapes on the inelastic stability of liquid-filled conical tanks. This study starts by conducting stability analyses for a number of small-scale conical models that were physically tested by Vandepitte et al. [3]. In this context, the data bank of geometric imperfections for the small-scale models was used in the analyses. After verifying the numerical model, several large-scale liquid-filled conical tanks were simulated to be considered in an inelastic stability analysis. An efficient means for studying the effects of axisymmetric and non-axisymmetric imperfection shapes is by adopting the use of Fourier analysis. This was done and the critical case was determined for a given geometry and loading.
2. Finite-element formulation In order to study numerically the inelastic stability of conical tanks having both axisymmetric and non-axisymmetric imperfection shapes, a suitable shell element which includes both the geometric and material non-linearity effects has to be used. The isoparametric shell elements commonly used in commercial finite-element computer programs result in overly stiff solutions when used to model thin plates and shells. Meanwhile, the predicted transverse shear stresses are found to be excessively large due to the presence of spurious transverse shear modes. Several attempts have been made to overcome this problem but none have been successful in general application in the nonlinear range until very recently. However, Koziey and Mirza [4] have now formulated a consistent subparametric triangular shell element that has proven to be free of superior shear modes. In the consistent subparametric shell element, the incremental displacement ⌬ui is expressed in terms of the incremental global displacement degrees of freedom ⌬Un, ⌬Vn and ⌬Wn directed along axes x, y and z, respectively, and four incremental rotations; ⌬␣ and ⌬ about the local y⬘-axis, and ⌬ and ⌬ about the local x⬘-
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axis. Axes x⬘ and y⬘ are located in a plane tangent to the mid-surface of the shell (see Fig. 2). The subparametric shell element is free from the spurious shear modes because the displacements are approximated using cubic interpolation functions and the rotations are approximated using quadratic interpolation functions. The consistent shell element has 13 nodes, ten nodes of which are used to achieve complete cubic polynomial for the displacements, while six nodes are used to obtain a complete quadratic polynomial for the rotations as shown in Fig. 2. Rotations ⌬␣ and ⌬ are constant through the element thickness resulting in a linear variation of ⌬u, ⌬v and ⌬w through the thickness. On the other hand, rotations ⌬␣ and ⌬ vary quadratically which leads to a cubic variation of ⌬u, ⌬v and ⌬w through the thickness. El Damatty et al. [5] extended the element formulation to include a large displacement analysis along with a strain-hardening plasticity model using Von Mises yield criterion and its associated flow rule. The total Lagrangian approach is used to formulate the nonlinear stiffness matrix. In addition, the incremental load method is employed and the Newton–Raphson method is used to solve iteratively the following equilibrium equation: [KL ⫹ Ks]{⌬U} ⫽ {R} ⫺ {F}
(1)
where [KL] is the sum of the linear and the initial strain stiffness matrix; [KS] is the initial stress stiffness matrix; {R} is the external load vector; {F} is the unbalanced load vector; and {⌬U} is the vector of incremental degrees of freedom. In the case of perfect structures, the initial strains at the beginning of the first iteration of the
Fig. 2. Coordinates and degrees of freedom of the consistent shell element.
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first load increment are set equal to zero, while for imperfect structures, the strains due to the assumed imperfection shape are introduced in the model at the first iteration of the first load increment.
3. Causes of failure of conical tanks and analysis procedure The way in which hydrostatically loaded conical tanks suffer from instability can be briefly stated as a local buckling phenomenon resulting from the combined effects of meridional stresses due to the weight of the fluid and partially stabilized by tensile hoop stresses [3]. Complicating the problem are geometric imperfections of fabrication and the residual stresses due to the welding process. However, an earlier study by El Damatty et al. [1] has shown that residual stresses have a relatively small effect and, therefore, will not be considered here. In the numerical analysis, the following assumptions are made: 1. Tanks are assumed to be full cones (the cylindrical part is omitted). Some preliminary analyses were conducted to compare the buckling strength of a conical tank with an upper cylindrical cap and an equivalent full cone having the same volume capacity (both structures have the same thickness, bottom radius, and angle of inclination). Analyses indicate that the equivalent full cone has almost the same buckling strength of the conical tank with cylindrical cap. 2. The cone is assumed to be simply supported at its bottom edge. A preliminary analysis was done to compare the buckling capacity of simply supported cones to those fixed at their bottom edges. The analysis indicates that assuming fixed boundary conditions at the bottom increases the buckling capacity by about 25%. 3. Boundary conditions were assumed to be free at the top of the cone, in spite of the presence of a cylindrical roof in practical situations. This assumption will be justified later. To arrive at a state of instability, the hydrostatic pressure acting on the walls of the tanks is multiplied by a load factor p. As the parameter p increases progressively, the stiffness of the tank will correspondingly decrease until the structure reaches its limit load at the critical load factor pcr. A critical load factor equal to 1.0 means that the tank is on the verge of failure under the weight of the existing fluid. If it is smaller than 1.0, then the structure can not sustain the hydrostatic load. When the critical load factor is larger than 1.0, then the tank is safe with a factor of safety proportional to the load factor. The analyses considered in this study include perfect, axisymmetrically imperfect and non-axisymmetrically imperfect tanks. For both perfect conical shells and those having an axisymmetric imperfection shape, symmetry about two axes exists. Consequently, only one-quarter of the cone needs to be modeled. In the analysis undertaken, 128 elements were used, as shown in Fig. 3. For the case of tanks having a non-axisymmetric imperfection shape, the full cone has to be modeled, and 512 shell elements were used in order to maintain an equivalent degree of accuracy. As shown in Fig. 3, a fine mesh was used near the bottom of the tank where higher stress
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Fig. 3.
Finite-element mesh for the quarter cone.
concentrations are anticipated. The element length le at the bottom layer of the cone was chosen to be less than one-quarter of the first longitudinal buckling wavelength in order to assure the detection of the lowest buckling mode. 4. Comparison between finite element and experimental results As a response to the failure of an elevated conical tank in Belgium, Vandepitte et al. [3] conducted an extensive experimental program to study the stability of liquid-
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filled conical tanks. The experimental programs included testing a large number of small conical tank models made of mylar, brass, aluminium, or steel. All models were full cones open at the top and having a wide range of inclination angle v and slenderness r/(t cos v). Here, r is the bottom radius of the tank, t is the wall thickness and v is the angle between the generator and the vertical, as shown in Fig. 4. For each model the imperfection amplitude was measured and the tank classified accordingly to a “good” or “poor” geometry. Each cone was filled with water, and the height of water at which the tank failed was recorded. In the current study, an attempt was made to duplicate numerically a number of tests conducted by Vandepitte et al. [3]. Twelve steel tanks (Tanks S1 to S12) having dimensions within the range used in their experiments were selected for the numerical work and are presented in Table 1. The thickness of the 12 tanks was taken equal to 0.31 mm, similar to the thickness of the tested steel specimens. Material properties of all tanks were taken as follows: modulus of elasticity, E ⫽ 2 ⫻ 105 MPa; tangent modulus ET ⫽ 0.03E ⫽ 6 ⫻ 103 MPa; yield stress, y ⫽ 300 MPa; and Poisson’s ratio, ⫽ 0.3. For each tank, three sets of analyses were conducted: elastic stability analysis of perfect tanks, elastic stability analysis of imperfect tanks, and inelastic stability analysis of imperfect tanks, and the following procedure was used in analyzing each tank:
Fig. 4.
Parameters describing the dimensions of conical tanks.
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Table 1 Dimensions and results for perfect elastic stability analysis of small scale conical tanks Tank
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12
Radius(mm)
121 161 242 282 107 142 214 249 90 120 179 209
Anglev
30 30 30 30 40 40 40 40 50 50 50 50
Vandepitte et al. [3]
Current study
Lb (mm)
Lb (mm)
23.7 27.3 33.5 36.2 23.7 27.3 33.5 36.2 23.7 27.3 33.5 36.2
hcr (mm) 2261 2223 2144 2105 1636 1614 1566 1542 1159 1146 1119 1104
25 26 34 35 23 29 35 38 25 30 35 38
hcr (mm) 2261 2223 2190 2150 1650 1614 1580 1560 1200 1200 1165 1150
1. A certain height, h, of the tank was selected, and the tank was assumed to be filled with water. 2. The nonlinear analysis was then conducted by multiplying the hydrostatic pressure by a load factor, p. By increasing p gradually, the tank was determined to have failed at pcr. 3. Steps (1) and (2) were then repeated for various heights h. The relation between h and pcr was subsequently plotted and the height hcr corresponding to a pcr equal to unity evaluated. 4.1. Elastic stability analysis of perfect tanks Due to symmetry, the consistent shell element described in the previous sections was used to model one-quarter of the 12 tanks. Table 1 shows a comparison between the buckling wavelength lb evaluated by the current analysis and one based on a formula suggested by Vandepitte et al. [3]. Fig. 5 shows the buckling mode along the generator of a typical tank and illustrates how the buckling wavelength Lb is measured. A comparison between the buckling wavelengths shows excellent agreement. In the same table, the critical water heights hcr of the tanks obtained from the current analysis is compared to those from a nonlinear finite difference analysis conducted by Vandepitte et al. [3] Again, excellent agreement in the elastic stability analysis of the perfect tanks is evident. 4.2. Assumed imperfection shapes Vandepitte et al. [3] registered the imperfection shapes of the cones tested and classified them as “good” or “poor” cones according to the maximum amplitude of
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Fig. 5.
351
Buckling mode for a typical perfect conical tank along the generator.
the measured imperfection, w0. Accordingly, a tank was considered a poor cone if: 0.004Lb ⬍ w0 ⬍ 0.01Lb. In order to duplicate the behaviour of the poor cones tested by Vandepitte et al. [3], the maximum amplitude of imperfection assumed in the finite element analysis of the 12 cones was taken equal to 0.01Lb. The profile of the imperfection shape in the meridional direction was taken as a sine wave having a wavelength equal to the buckling wavelength of the perfect tank shown in Fig. 5. The variation of imperfections in the circumferential direction was based on the available data bank for those measured in small-scale models. In this data bank, the imperfection shape was expressed as a double Fourier series whose coefficients are reported by Arbocz and Abramovich [6]. The averages of Fourier coefficients of six measurements of stainless steel shells reported in the data bank were used in the analysis. Based on the above, the imperfection shape along the surface of a tank was assumed as follows:
冉 冊冘
2s w¯(s, ) ⫽ w0 sin ( C cos k ⫹ Lb k ⫽ 0 k N
冘 N
Dj sin j)
(2)
j⫽1
where is the angle in the circumferential plane and s is the distance along the generator. The cut-off limit for the Fourier terms used in the above expansion was chosen to be 10% of the value of the highest term, i.e. terms Ck or Dj having an absolute value less than 10% of the absolute maximum of Ck and Dj were neglected. The above condition was satisfied by taking N ⫽ 8 in Eq. (2). Coefficients Ck and Dj were normalized such the maximum amplitude of imperfection equals to 0.01Lb.
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4.3. Results of the stability analysis of imperfect tanks Elastic as well as inelastic analyses have been carried out for the 12 selected tanks in order to find the critical height of water hcr at which each tank buckles. The results of the analyses are shown in Fig. 6 in the form of a relation between the dimensionless parameters and , which were used by Vandepitte et al. [3]. These dimensionless parameters were defined as:
⫽ 1000
r t
冢
冣
1/2
2␥r√1 ⫺ E sin 2v
2
and ⫽
t sin v r2
(3)
where is the meridian membrane stress along the bottom edge of the shell. The relationship connecting and is shown on a log–log scale. The top line represents the results of the analysis of perfect cones, which are virtually identical, based on both the current study and that by Vandepitte et al. [3]. The lower line represents the limit that has a 99% probability of being surpassed by the experimental results conducted by Vandepitte et al. [3]. The results of the current finite element analysis are shown to lie between the two lines, i.e. between the theoretical perfect analysis and the bottom edge of the experimental results. In general, the elastic analy-
Fig. 6. Comparison between the finite-element analysis of small cones and the design charts suggested by Vandepitte et al. [3].
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sis resulted in higher values for hcr and correspondingly higher compared to those resulting from the inelastic analysis. This indicates that yielding precedes elastic buckling in the cases examined. It is noted that the results of the analysis of imperfect tanks lie in the upper third of the gap between the two limits. There are two possible reasons for this. The first is that the lower limit line is a result of hundreds of tested models while only 30 of them are steel tanks. It is expected that the steel specimens do not correspond to the lower limit, which in most cases resulted from the aluminum specimens [7]. The second reason is that experimental values of v as low as 10° were used, while in the present analysis the minimum value of v was taken as 30°. Indeed, Vandepitte et al. [3] suggested that the lower limit line corresponds to small values of v and that it would be situated above its current position if v were higher. 5. Critical imperfection shape for inelastic stability analysis The stability of liquid-filled steel conical tanks as thin-walled shell structures is greatly affected by the presence of initial geometric imperfections. Although data banks are available from the aerospace industry, they can not be used directly in the analysis and design of conical tanks due to the differences in quality control and the method of fabrication. Although many researchers suggest that elastic stability of hydrostatically loaded conical tanks is more sensitive to axisymmetric than to nonaxisymmetric imperfections [1], some researchers contradict this conclusion and suggest that non-axisymmetric imperfections are more critical [8]. Meanwhile, to the best of the authors’ knowledge, no study has been done to determine the critical imperfection shape for inelastic stability analysis of liquid-filled conical tanks. This is important since analyses have shown that liquid-filled conical tanks generally fail by inelastic buckling. Along the generator of a conical shell, an imperfection wave having the same wavelength Lb of the buckling wave (see Fig. 5) localized at the bottom of the structure is obviously the critical one. On the other hand, it is hard to judge which profile of imperfection along the circumference is critical. An axisymmetric imperfection has the tendency to match the axisymmetrically applied load (due to hydrostatic pressure) and also to match the geometry of the structure. Meanwhile, a nonaxisymmetric imperfection shape introduces extra circumferential bending stresses and might lead to early yielding of the structure. In this part of the analysis, a procedure was adopted to study the effect of both axisymmetric and non-axisymmetric imperfections with a variation in wave number. The procedure is based on Fourier analysis of the displacements resulting from the finite element analysis of conical tanks excited by various equal weighted imperfection modes. In this procedure, the steps described below were followed: 1. An imperfection shape similar to the one described by Eq. (2) is applied to the tank. All the coefficients Ck and Dj are set equal to unity, i.e. all circumferential imperfection modes are excited equally. Note that k ⫽ 0 corresponds to axisymmetric imperfection shape.
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2. The tank is modeled using the consistent shell element. An inelastic stability analysis is performed with the vessel filled with water. The displacement shape of the structure prior to the inelastic buckling is recorded. 3. Global displacements, prior to buckling are resolved to obtain the displacement w(h, ) normal to the surface at a certain height h from the bottom of the tank. A Fourier analysis is performed to express w(h, ) in the form:
冘 N
w(h, ) ⫽
冘 N
Ak cos k ⫹
k⫽0
Bj sin j
(4)
j⫽1
4. The coefficients Ak and Bj above are evaluated. A large value of Ak or Bj means that the associated imperfection mode has a significant contribution to the inelastic stability. Thirteen full-scale tanks (T1 to T13) were modeled for the analysis conducted in this part. Dimensions of the tanks are shown in Table 2, and were selected such that the critical load factor of all imperfect tanks would be between 1.0 and 2.0 to ensure that the considered dimensions represent practical values. Material properties of the tank wall were similar to those described in the small-scale model. Results of the analysis indicated that conical tanks buckled inelastically, i.e. yielding always precedes elastic buckling. The Fourier coefficients Ak and Bj were calculated for the 13 tanks and are plotted in Fig. 7 for tanks T2, T11 and T12 as a demonstration. This figure shows that the axisymmetric imperfection is typically the most critical imperfection mode for liquid-filled conical tanks. The second significant imperfection shape is the one involving the function sin 2. These coefficients were found to range from 0.6 to 0.9 for different tanks. Other terms that have a significant Table 2 Critical load factors for perfect and imperfect tanks Tank r(m) h(m) v t(mm)
T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13
4.0 4.0 4.0 4.5 4.5 4.5 5.0 5.0 5.0 4.5 4.5 4.5 4.5
7 8 9 7 8 9 7 8 9 8 9 7 8
45 45 45 45 45 45 45 45 45 40 40 50 50
10 12 12 10 12 12 10 12 14 10 12 12 14
Perfect analysis
Inelastic imperfect analysis
Poor cones
Elastic
Inelastic All kinds
Axisymmetric Axisymmetric Vandepitte et al. [3] + sin 2
3.80 3.95 3.05 3.55 3.60 2.80 3.20 3.40 3.50 3.85 4.25 3.20 3.20
2.60 2.60 2.05 2.45 2.45 1.95 2.30 2.35 2.30 2.55 2.65 2.30 2.20
1.50 1.45 1.25 1.40 1.40 1.15 1.35 1.35 1.30 1.50 1.50 1.30 1.30
2.05 2.00 1.55 1.60 1.70 1.30 1.40 1.40 1.35 1.80 2.15 1.40 1.40
1.75 1.45 1.25 1.60 1.50 1.15 1.50 1.45 1.35 1.60 1.55 1.30 1.30
1.33 1.31 0.95 1.29 1.28 0.93 1.25 1.25 1.23 1.45 1.50 1.14 1.07
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Fig. 7.
Fourier coefficients for the buckling modes of different tanks.
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contribution are cos k with k ⫽ 4, 6, 8 and sin k with k ⫽ 6, 8. The remaining terms in the Fourier expansion were found to be very small. It should be noted that the above observations were found in the Fourier analysis of the 13 tanks. To verify the accuracy of the Fourier analysis, the simulated displacement from the Fourier expansion using Eq. (4) and the actual displacement calculated from the finite element analysis along the circumference in the critical buckling plane are plotted in Fig. 8 for tank T2 as an example. It is clear from the figure that the Fourier expansion is in excellent agreement with the actual displacement. Fig. 9 shows a typical buckling mode for the analyzed tanks where horizontal displacements prior to inelastic buckling are plotted along the generator of the tank. It is evident that large bending deformations occur which introduce additional stresses that caused early yielding prior to the collapse of both tanks. It is also evident that buckling is localized near the bottom of the tank. These plots also show that the horizontal displacements near the top of the tank are very small. This suggests that the boundary condition near the tank top will not affect the results of the analysis.
6. Parametric study In order to determine which imperfection shape ought to be used for design purposes, results from three sets of analyses were considered: tanks having the imperfection described by Eq. (2) with equal amplitude coefficients; tanks having axisymmetric imperfection only; and vessels having axisymmetric ⫹ sin 2 imperfections. In all cases the imperfection amplitude was normalized to a maximum value of 0.01Lb to correspond to the “poor” cone of Vandepitte et al. [3]. Results of these analyses together with those for elastic and inelastic perfect tanks, together with
Fig. 8.
Comparison between actual and simulated displacements.
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Fig. 9. Horizontal displacements along tank generator near buckling.
the predicted load factors (based on design curves of Vandepitte) are presented in Table 2. Comparison of the load factors for the different tanks indicates that the axisymmetric imperfection case produces the lowest critical load factors. However, these load factors are higher than the load factors calculated from the experimental results of Vandepitte et al. [3]. Adding the sin 2 imperfection to the axisymmetric case led to higher load factors compared to those resulting from the case of axisymmetric imperfection alone. Therefore, a conservative inelastic stability analysis of liquidfilled conical tanks appears to be one in which only axisymmetric imperfections are considered. In Fig. 10 the maximum transverse displacements are plotted versus the load factor for tank T5. The figure shows the behaviour of perfect as well as imperfect tanks having axisymmetric imperfections. Although the load factor for the perfect tanks is much higher than for the imperfect tanks, the maximum transverse displacement at the point of buckling is slightly higher for imperfect tanks. It is also evident that there is a loss of stiffness of the tanks near failure, which results in a large increase in deflection with only a slight increase in loading. Having established that axisymmetric imperfections are critical for inelastic buck-
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Fig. 10.
Load deflection curves for perfect and imperfect tanks.
ling, inelastic stability analyses were conducted on the small-scale tanks S1 to S12 with axisymmetric imperfections imposed. The critical water height hcr at buckling was calculated for each tank. The results are plotted as a relation between and in Fig. 6. It is evident from this figure that the resulting critical height, considering only axisymmetric imperfections, is less than the case where the imperfection shape is based on the imperfection data bank. This indicates that the axisymmetric imperfection is always a conservative assumption. Fig. 6 also shows that for all tanks, the lower bound of the experimental results conducted by Vandepitte et al. [3] is conservative.
7. Summary and conclusions An inelastic stability analysis of liquid-filled conical tanks under hydrostatic loading was performed using the consistent shell element. The results of stability analyses of small-scale tanks are compared to those of the available experimental work. The results obtained from the above analyses are shown to be within the range of earlier experimental results. A procedure has been adopted to determine the imperfection shape that is most critical. This was achieved by assuming an imperfection shape in the form of a Fourier expansion having equal coefficients, and then performing a Fourier analysis for the buckling mode resulting from the inelastic stability analysis. Results obtained from the analysis lead to the conclusion that the inelastic insta-
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bility of liquid-filled conical tanks is most sensitive to axisymmetric imperfections. Compared to other non-axisymmetric functions, the sin 2 imperfection shape has also a significant contribution to the inelastic stability of the tanks. However, it is more conservative to consider the axisymmetric imperfections only when conducting inelastic stability analysis of conical tanks. Results of the various analyses conducted on thin walled conical tanks indicate that yielding usually precedes elastic buckling. As such, these structures will generally fail by inelastic buckling.
References [1] El Damatty AA, Korol RM, Mirza FA. Stability of imperfect steel conical tanks under hydrostatic loading. ASCE 1997;123(ST6):703–12. [2] Wunderlich W, Rensch HJ, Bochum H. Analysis of elastic–plastic buckling and imperfection-sensitivity of shells of revolution. In: Ramm E, editor. Buckling of shells. Berlin: Springer-Verlag, 1982: 137–74. [3] Vandepitte D, Rathe J, Verhegghe B, Paridaens R, Verschaeve C. Experimental investigation of hydrostatically loaded conical shells and practical evaluation of buckling load. In: Ramm E, editor. Buckling of shells. Berlin: Springer-Verlag, 1982: 375–99. [4] Koziey B, Mirza FA. Consistent thick shell element. Computers and Structures 1997;65(12):531–41. [5] El Damatty AA, Mirza FA, Korol RM. Large displacement extension of consistent shell element for static and dynamic analysis. Computers and Structures 1997;62(6):943–60. [6] Arbocz J, Abramovich H. The initial imperfection data bank at the Delft University of Technology— part I. Report LR-290, The Netherlands: Delft University of Technology, 1979. [7] Ellinas CP. Buckling of offshore structures—a state of the art review of the buckling of offshore structures. London: Grenada Publishing, 1984. [8] Bushnell D. Computerized buckling analysis of shells. Dordrecht, The Netherlands: Martinus Nijhoff Publishers, 1985.
Inelastic stability of conical tanks A.A. El Damattya,*, M. El-Attarb, R.M. Korolb a
Department of Civil and Environmental Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 b Department of Civil Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7
Abstract The aim of this investigation is to study the effect of different imperfection shapes on the inelastic stability of liquid-filled conical tanks and to determine the critical imperfection shape that would lead to the minimum inelastic limit load. The study is carried out numerically using a self-developed shell element used to simulate a number of conical tanks having an imperfection shape in the form of Fourier series of equal coefficients. The Fourier analysis of the buckling modes indicates that the existence of axisymmetric imperfection will lead to the critical inelastic limit load for conical tanks. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Tanks; Conical; Stability; Imperfections; Buckling; Liquid-filled
Notation Ck Dk E ET k Lb r s t w0
Fourier coefficients for the cos k imperfection terms Fourier coefficients for the sin k imperfection terms modulus of elasticity of the wall material tangent modulus circumferential wave number buckling wavelength radius at base of cone distance measured along generator of cone thickness of cone wall amplitude of imperfection wave
* Corresponding author. 0263-8231/98/$—see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 8 ) 0 0 0 2 0 - 2
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␥ v y
specific weight of water Poisson’s ratio of the wall material angle measured in the circumferential plane angle of inclination of cone generator with the vertical meridional membrane stress along bottom edge of the tank yield stress of the wall material a dimensionless quantity used for design curves a dimensionless quantity used for design curves
1. Introduction Conical shaped steel vessels having cylindrical upper section caps are widely used as superstructures for elevated water tanks. The authors are aware that approximately 200 of such containment vessels exist in Canada alone. The cross-sectional elevation, shown in Fig. 1, represents a typical conical tank consisting of curved steel panels welded together along circumferential and longitudinal edges. The steel vessel is welded to a steel plate at its bottom, which is then anchored to a concrete slab supported by reinforced concrete towers. Indeed, a lack of adequate design guidelines for conical tanks in North America may have contributed to the collapse of the water tower in Fredericton, Canada in December of 1990. It is imperative that stability analyses and related design formulae consider the effects of both geometric imperfections and residual stresses due to welding. A recent study done by El Damatty et al. [1] has shown that geometric
Fig. 1.
Elevated conical tank.
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imperfections have a significant effect on the stability of liquid-filled conical tanks but that the effect of residual stress is somewhat less important. Unfortunately, field measurements of geometric imperfections existing in prototype shell structures are rare in civil engineering structures. Due to a lack of pertinent statistical information, a conservative approach is to consider the effect of various imperfection shapes in order to determine the one leading to the minimum buckling capacity of the structure. In the study done by El Damatty et al. [1], an elastic stability analysis of liquidfilled conical tanks having various imperfection shapes was considered and the axisymmetric imperfection shape was found to be the critical one. By extending the analysis to include nonlinear material behaviour, it was found that for conical tanks having practical dimensions, yielding precedes the elastic buckling and that inelastic buckling governs the failure of these liquid-filled structures. For the inelastic stability of conical tanks, while axisymmetric imperfections tend to be more significant in decreasing the stability of the structure, non-axisymmetric imperfections cause circumferential bending which can influence the onset of yielding of the structure and hence reduce its inelastic buckling capacity [2]. To the best of the authors’ knowledge, no study has been done to consider the effect of various imperfection shapes on the inelastic stability of liquid-filled conical tanks. This study starts by conducting stability analyses for a number of small-scale conical models that were physically tested by Vandepitte et al. [3]. In this context, the data bank of geometric imperfections for the small-scale models was used in the analyses. After verifying the numerical model, several large-scale liquid-filled conical tanks were simulated to be considered in an inelastic stability analysis. An efficient means for studying the effects of axisymmetric and non-axisymmetric imperfection shapes is by adopting the use of Fourier analysis. This was done and the critical case was determined for a given geometry and loading.
2. Finite-element formulation In order to study numerically the inelastic stability of conical tanks having both axisymmetric and non-axisymmetric imperfection shapes, a suitable shell element which includes both the geometric and material non-linearity effects has to be used. The isoparametric shell elements commonly used in commercial finite-element computer programs result in overly stiff solutions when used to model thin plates and shells. Meanwhile, the predicted transverse shear stresses are found to be excessively large due to the presence of spurious transverse shear modes. Several attempts have been made to overcome this problem but none have been successful in general application in the nonlinear range until very recently. However, Koziey and Mirza [4] have now formulated a consistent subparametric triangular shell element that has proven to be free of superior shear modes. In the consistent subparametric shell element, the incremental displacement ⌬ui is expressed in terms of the incremental global displacement degrees of freedom ⌬Un, ⌬Vn and ⌬Wn directed along axes x, y and z, respectively, and four incremental rotations; ⌬␣ and ⌬ about the local y⬘-axis, and ⌬ and ⌬ about the local x⬘-
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axis. Axes x⬘ and y⬘ are located in a plane tangent to the mid-surface of the shell (see Fig. 2). The subparametric shell element is free from the spurious shear modes because the displacements are approximated using cubic interpolation functions and the rotations are approximated using quadratic interpolation functions. The consistent shell element has 13 nodes, ten nodes of which are used to achieve complete cubic polynomial for the displacements, while six nodes are used to obtain a complete quadratic polynomial for the rotations as shown in Fig. 2. Rotations ⌬␣ and ⌬ are constant through the element thickness resulting in a linear variation of ⌬u, ⌬v and ⌬w through the thickness. On the other hand, rotations ⌬␣ and ⌬ vary quadratically which leads to a cubic variation of ⌬u, ⌬v and ⌬w through the thickness. El Damatty et al. [5] extended the element formulation to include a large displacement analysis along with a strain-hardening plasticity model using Von Mises yield criterion and its associated flow rule. The total Lagrangian approach is used to formulate the nonlinear stiffness matrix. In addition, the incremental load method is employed and the Newton–Raphson method is used to solve iteratively the following equilibrium equation: [KL ⫹ Ks]{⌬U} ⫽ {R} ⫺ {F}
(1)
where [KL] is the sum of the linear and the initial strain stiffness matrix; [KS] is the initial stress stiffness matrix; {R} is the external load vector; {F} is the unbalanced load vector; and {⌬U} is the vector of incremental degrees of freedom. In the case of perfect structures, the initial strains at the beginning of the first iteration of the
Fig. 2. Coordinates and degrees of freedom of the consistent shell element.
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first load increment are set equal to zero, while for imperfect structures, the strains due to the assumed imperfection shape are introduced in the model at the first iteration of the first load increment.
3. Causes of failure of conical tanks and analysis procedure The way in which hydrostatically loaded conical tanks suffer from instability can be briefly stated as a local buckling phenomenon resulting from the combined effects of meridional stresses due to the weight of the fluid and partially stabilized by tensile hoop stresses [3]. Complicating the problem are geometric imperfections of fabrication and the residual stresses due to the welding process. However, an earlier study by El Damatty et al. [1] has shown that residual stresses have a relatively small effect and, therefore, will not be considered here. In the numerical analysis, the following assumptions are made: 1. Tanks are assumed to be full cones (the cylindrical part is omitted). Some preliminary analyses were conducted to compare the buckling strength of a conical tank with an upper cylindrical cap and an equivalent full cone having the same volume capacity (both structures have the same thickness, bottom radius, and angle of inclination). Analyses indicate that the equivalent full cone has almost the same buckling strength of the conical tank with cylindrical cap. 2. The cone is assumed to be simply supported at its bottom edge. A preliminary analysis was done to compare the buckling capacity of simply supported cones to those fixed at their bottom edges. The analysis indicates that assuming fixed boundary conditions at the bottom increases the buckling capacity by about 25%. 3. Boundary conditions were assumed to be free at the top of the cone, in spite of the presence of a cylindrical roof in practical situations. This assumption will be justified later. To arrive at a state of instability, the hydrostatic pressure acting on the walls of the tanks is multiplied by a load factor p. As the parameter p increases progressively, the stiffness of the tank will correspondingly decrease until the structure reaches its limit load at the critical load factor pcr. A critical load factor equal to 1.0 means that the tank is on the verge of failure under the weight of the existing fluid. If it is smaller than 1.0, then the structure can not sustain the hydrostatic load. When the critical load factor is larger than 1.0, then the tank is safe with a factor of safety proportional to the load factor. The analyses considered in this study include perfect, axisymmetrically imperfect and non-axisymmetrically imperfect tanks. For both perfect conical shells and those having an axisymmetric imperfection shape, symmetry about two axes exists. Consequently, only one-quarter of the cone needs to be modeled. In the analysis undertaken, 128 elements were used, as shown in Fig. 3. For the case of tanks having a non-axisymmetric imperfection shape, the full cone has to be modeled, and 512 shell elements were used in order to maintain an equivalent degree of accuracy. As shown in Fig. 3, a fine mesh was used near the bottom of the tank where higher stress
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Fig. 3.
Finite-element mesh for the quarter cone.
concentrations are anticipated. The element length le at the bottom layer of the cone was chosen to be less than one-quarter of the first longitudinal buckling wavelength in order to assure the detection of the lowest buckling mode. 4. Comparison between finite element and experimental results As a response to the failure of an elevated conical tank in Belgium, Vandepitte et al. [3] conducted an extensive experimental program to study the stability of liquid-
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filled conical tanks. The experimental programs included testing a large number of small conical tank models made of mylar, brass, aluminium, or steel. All models were full cones open at the top and having a wide range of inclination angle v and slenderness r/(t cos v). Here, r is the bottom radius of the tank, t is the wall thickness and v is the angle between the generator and the vertical, as shown in Fig. 4. For each model the imperfection amplitude was measured and the tank classified accordingly to a “good” or “poor” geometry. Each cone was filled with water, and the height of water at which the tank failed was recorded. In the current study, an attempt was made to duplicate numerically a number of tests conducted by Vandepitte et al. [3]. Twelve steel tanks (Tanks S1 to S12) having dimensions within the range used in their experiments were selected for the numerical work and are presented in Table 1. The thickness of the 12 tanks was taken equal to 0.31 mm, similar to the thickness of the tested steel specimens. Material properties of all tanks were taken as follows: modulus of elasticity, E ⫽ 2 ⫻ 105 MPa; tangent modulus ET ⫽ 0.03E ⫽ 6 ⫻ 103 MPa; yield stress, y ⫽ 300 MPa; and Poisson’s ratio, ⫽ 0.3. For each tank, three sets of analyses were conducted: elastic stability analysis of perfect tanks, elastic stability analysis of imperfect tanks, and inelastic stability analysis of imperfect tanks, and the following procedure was used in analyzing each tank:
Fig. 4.
Parameters describing the dimensions of conical tanks.
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Table 1 Dimensions and results for perfect elastic stability analysis of small scale conical tanks Tank
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12
Radius(mm)
121 161 242 282 107 142 214 249 90 120 179 209
Anglev
30 30 30 30 40 40 40 40 50 50 50 50
Vandepitte et al. [3]
Current study
Lb (mm)
Lb (mm)
23.7 27.3 33.5 36.2 23.7 27.3 33.5 36.2 23.7 27.3 33.5 36.2
hcr (mm) 2261 2223 2144 2105 1636 1614 1566 1542 1159 1146 1119 1104
25 26 34 35 23 29 35 38 25 30 35 38
hcr (mm) 2261 2223 2190 2150 1650 1614 1580 1560 1200 1200 1165 1150
1. A certain height, h, of the tank was selected, and the tank was assumed to be filled with water. 2. The nonlinear analysis was then conducted by multiplying the hydrostatic pressure by a load factor, p. By increasing p gradually, the tank was determined to have failed at pcr. 3. Steps (1) and (2) were then repeated for various heights h. The relation between h and pcr was subsequently plotted and the height hcr corresponding to a pcr equal to unity evaluated. 4.1. Elastic stability analysis of perfect tanks Due to symmetry, the consistent shell element described in the previous sections was used to model one-quarter of the 12 tanks. Table 1 shows a comparison between the buckling wavelength lb evaluated by the current analysis and one based on a formula suggested by Vandepitte et al. [3]. Fig. 5 shows the buckling mode along the generator of a typical tank and illustrates how the buckling wavelength Lb is measured. A comparison between the buckling wavelengths shows excellent agreement. In the same table, the critical water heights hcr of the tanks obtained from the current analysis is compared to those from a nonlinear finite difference analysis conducted by Vandepitte et al. [3] Again, excellent agreement in the elastic stability analysis of the perfect tanks is evident. 4.2. Assumed imperfection shapes Vandepitte et al. [3] registered the imperfection shapes of the cones tested and classified them as “good” or “poor” cones according to the maximum amplitude of
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Fig. 5.
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Buckling mode for a typical perfect conical tank along the generator.
the measured imperfection, w0. Accordingly, a tank was considered a poor cone if: 0.004Lb ⬍ w0 ⬍ 0.01Lb. In order to duplicate the behaviour of the poor cones tested by Vandepitte et al. [3], the maximum amplitude of imperfection assumed in the finite element analysis of the 12 cones was taken equal to 0.01Lb. The profile of the imperfection shape in the meridional direction was taken as a sine wave having a wavelength equal to the buckling wavelength of the perfect tank shown in Fig. 5. The variation of imperfections in the circumferential direction was based on the available data bank for those measured in small-scale models. In this data bank, the imperfection shape was expressed as a double Fourier series whose coefficients are reported by Arbocz and Abramovich [6]. The averages of Fourier coefficients of six measurements of stainless steel shells reported in the data bank were used in the analysis. Based on the above, the imperfection shape along the surface of a tank was assumed as follows:
冉 冊冘
2s w¯(s, ) ⫽ w0 sin ( C cos k ⫹ Lb k ⫽ 0 k N
冘 N
Dj sin j)
(2)
j⫽1
where is the angle in the circumferential plane and s is the distance along the generator. The cut-off limit for the Fourier terms used in the above expansion was chosen to be 10% of the value of the highest term, i.e. terms Ck or Dj having an absolute value less than 10% of the absolute maximum of Ck and Dj were neglected. The above condition was satisfied by taking N ⫽ 8 in Eq. (2). Coefficients Ck and Dj were normalized such the maximum amplitude of imperfection equals to 0.01Lb.
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4.3. Results of the stability analysis of imperfect tanks Elastic as well as inelastic analyses have been carried out for the 12 selected tanks in order to find the critical height of water hcr at which each tank buckles. The results of the analyses are shown in Fig. 6 in the form of a relation between the dimensionless parameters and , which were used by Vandepitte et al. [3]. These dimensionless parameters were defined as:
⫽ 1000
r t
冢
冣
1/2
2␥r√1 ⫺ E sin 2v
2
and ⫽
t sin v r2
(3)
where is the meridian membrane stress along the bottom edge of the shell. The relationship connecting and is shown on a log–log scale. The top line represents the results of the analysis of perfect cones, which are virtually identical, based on both the current study and that by Vandepitte et al. [3]. The lower line represents the limit that has a 99% probability of being surpassed by the experimental results conducted by Vandepitte et al. [3]. The results of the current finite element analysis are shown to lie between the two lines, i.e. between the theoretical perfect analysis and the bottom edge of the experimental results. In general, the elastic analy-
Fig. 6. Comparison between the finite-element analysis of small cones and the design charts suggested by Vandepitte et al. [3].
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sis resulted in higher values for hcr and correspondingly higher compared to those resulting from the inelastic analysis. This indicates that yielding precedes elastic buckling in the cases examined. It is noted that the results of the analysis of imperfect tanks lie in the upper third of the gap between the two limits. There are two possible reasons for this. The first is that the lower limit line is a result of hundreds of tested models while only 30 of them are steel tanks. It is expected that the steel specimens do not correspond to the lower limit, which in most cases resulted from the aluminum specimens [7]. The second reason is that experimental values of v as low as 10° were used, while in the present analysis the minimum value of v was taken as 30°. Indeed, Vandepitte et al. [3] suggested that the lower limit line corresponds to small values of v and that it would be situated above its current position if v were higher. 5. Critical imperfection shape for inelastic stability analysis The stability of liquid-filled steel conical tanks as thin-walled shell structures is greatly affected by the presence of initial geometric imperfections. Although data banks are available from the aerospace industry, they can not be used directly in the analysis and design of conical tanks due to the differences in quality control and the method of fabrication. Although many researchers suggest that elastic stability of hydrostatically loaded conical tanks is more sensitive to axisymmetric than to nonaxisymmetric imperfections [1], some researchers contradict this conclusion and suggest that non-axisymmetric imperfections are more critical [8]. Meanwhile, to the best of the authors’ knowledge, no study has been done to determine the critical imperfection shape for inelastic stability analysis of liquid-filled conical tanks. This is important since analyses have shown that liquid-filled conical tanks generally fail by inelastic buckling. Along the generator of a conical shell, an imperfection wave having the same wavelength Lb of the buckling wave (see Fig. 5) localized at the bottom of the structure is obviously the critical one. On the other hand, it is hard to judge which profile of imperfection along the circumference is critical. An axisymmetric imperfection has the tendency to match the axisymmetrically applied load (due to hydrostatic pressure) and also to match the geometry of the structure. Meanwhile, a nonaxisymmetric imperfection shape introduces extra circumferential bending stresses and might lead to early yielding of the structure. In this part of the analysis, a procedure was adopted to study the effect of both axisymmetric and non-axisymmetric imperfections with a variation in wave number. The procedure is based on Fourier analysis of the displacements resulting from the finite element analysis of conical tanks excited by various equal weighted imperfection modes. In this procedure, the steps described below were followed: 1. An imperfection shape similar to the one described by Eq. (2) is applied to the tank. All the coefficients Ck and Dj are set equal to unity, i.e. all circumferential imperfection modes are excited equally. Note that k ⫽ 0 corresponds to axisymmetric imperfection shape.
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2. The tank is modeled using the consistent shell element. An inelastic stability analysis is performed with the vessel filled with water. The displacement shape of the structure prior to the inelastic buckling is recorded. 3. Global displacements, prior to buckling are resolved to obtain the displacement w(h, ) normal to the surface at a certain height h from the bottom of the tank. A Fourier analysis is performed to express w(h, ) in the form:
冘 N
w(h, ) ⫽
冘 N
Ak cos k ⫹
k⫽0
Bj sin j
(4)
j⫽1
4. The coefficients Ak and Bj above are evaluated. A large value of Ak or Bj means that the associated imperfection mode has a significant contribution to the inelastic stability. Thirteen full-scale tanks (T1 to T13) were modeled for the analysis conducted in this part. Dimensions of the tanks are shown in Table 2, and were selected such that the critical load factor of all imperfect tanks would be between 1.0 and 2.0 to ensure that the considered dimensions represent practical values. Material properties of the tank wall were similar to those described in the small-scale model. Results of the analysis indicated that conical tanks buckled inelastically, i.e. yielding always precedes elastic buckling. The Fourier coefficients Ak and Bj were calculated for the 13 tanks and are plotted in Fig. 7 for tanks T2, T11 and T12 as a demonstration. This figure shows that the axisymmetric imperfection is typically the most critical imperfection mode for liquid-filled conical tanks. The second significant imperfection shape is the one involving the function sin 2. These coefficients were found to range from 0.6 to 0.9 for different tanks. Other terms that have a significant Table 2 Critical load factors for perfect and imperfect tanks Tank r(m) h(m) v t(mm)
T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13
4.0 4.0 4.0 4.5 4.5 4.5 5.0 5.0 5.0 4.5 4.5 4.5 4.5
7 8 9 7 8 9 7 8 9 8 9 7 8
45 45 45 45 45 45 45 45 45 40 40 50 50
10 12 12 10 12 12 10 12 14 10 12 12 14
Perfect analysis
Inelastic imperfect analysis
Poor cones
Elastic
Inelastic All kinds
Axisymmetric Axisymmetric Vandepitte et al. [3] + sin 2
3.80 3.95 3.05 3.55 3.60 2.80 3.20 3.40 3.50 3.85 4.25 3.20 3.20
2.60 2.60 2.05 2.45 2.45 1.95 2.30 2.35 2.30 2.55 2.65 2.30 2.20
1.50 1.45 1.25 1.40 1.40 1.15 1.35 1.35 1.30 1.50 1.50 1.30 1.30
2.05 2.00 1.55 1.60 1.70 1.30 1.40 1.40 1.35 1.80 2.15 1.40 1.40
1.75 1.45 1.25 1.60 1.50 1.15 1.50 1.45 1.35 1.60 1.55 1.30 1.30
1.33 1.31 0.95 1.29 1.28 0.93 1.25 1.25 1.23 1.45 1.50 1.14 1.07
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Fig. 7.
Fourier coefficients for the buckling modes of different tanks.
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contribution are cos k with k ⫽ 4, 6, 8 and sin k with k ⫽ 6, 8. The remaining terms in the Fourier expansion were found to be very small. It should be noted that the above observations were found in the Fourier analysis of the 13 tanks. To verify the accuracy of the Fourier analysis, the simulated displacement from the Fourier expansion using Eq. (4) and the actual displacement calculated from the finite element analysis along the circumference in the critical buckling plane are plotted in Fig. 8 for tank T2 as an example. It is clear from the figure that the Fourier expansion is in excellent agreement with the actual displacement. Fig. 9 shows a typical buckling mode for the analyzed tanks where horizontal displacements prior to inelastic buckling are plotted along the generator of the tank. It is evident that large bending deformations occur which introduce additional stresses that caused early yielding prior to the collapse of both tanks. It is also evident that buckling is localized near the bottom of the tank. These plots also show that the horizontal displacements near the top of the tank are very small. This suggests that the boundary condition near the tank top will not affect the results of the analysis.
6. Parametric study In order to determine which imperfection shape ought to be used for design purposes, results from three sets of analyses were considered: tanks having the imperfection described by Eq. (2) with equal amplitude coefficients; tanks having axisymmetric imperfection only; and vessels having axisymmetric ⫹ sin 2 imperfections. In all cases the imperfection amplitude was normalized to a maximum value of 0.01Lb to correspond to the “poor” cone of Vandepitte et al. [3]. Results of these analyses together with those for elastic and inelastic perfect tanks, together with
Fig. 8.
Comparison between actual and simulated displacements.
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Fig. 9. Horizontal displacements along tank generator near buckling.
the predicted load factors (based on design curves of Vandepitte) are presented in Table 2. Comparison of the load factors for the different tanks indicates that the axisymmetric imperfection case produces the lowest critical load factors. However, these load factors are higher than the load factors calculated from the experimental results of Vandepitte et al. [3]. Adding the sin 2 imperfection to the axisymmetric case led to higher load factors compared to those resulting from the case of axisymmetric imperfection alone. Therefore, a conservative inelastic stability analysis of liquidfilled conical tanks appears to be one in which only axisymmetric imperfections are considered. In Fig. 10 the maximum transverse displacements are plotted versus the load factor for tank T5. The figure shows the behaviour of perfect as well as imperfect tanks having axisymmetric imperfections. Although the load factor for the perfect tanks is much higher than for the imperfect tanks, the maximum transverse displacement at the point of buckling is slightly higher for imperfect tanks. It is also evident that there is a loss of stiffness of the tanks near failure, which results in a large increase in deflection with only a slight increase in loading. Having established that axisymmetric imperfections are critical for inelastic buck-
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Fig. 10.
Load deflection curves for perfect and imperfect tanks.
ling, inelastic stability analyses were conducted on the small-scale tanks S1 to S12 with axisymmetric imperfections imposed. The critical water height hcr at buckling was calculated for each tank. The results are plotted as a relation between and in Fig. 6. It is evident from this figure that the resulting critical height, considering only axisymmetric imperfections, is less than the case where the imperfection shape is based on the imperfection data bank. This indicates that the axisymmetric imperfection is always a conservative assumption. Fig. 6 also shows that for all tanks, the lower bound of the experimental results conducted by Vandepitte et al. [3] is conservative.
7. Summary and conclusions An inelastic stability analysis of liquid-filled conical tanks under hydrostatic loading was performed using the consistent shell element. The results of stability analyses of small-scale tanks are compared to those of the available experimental work. The results obtained from the above analyses are shown to be within the range of earlier experimental results. A procedure has been adopted to determine the imperfection shape that is most critical. This was achieved by assuming an imperfection shape in the form of a Fourier expansion having equal coefficients, and then performing a Fourier analysis for the buckling mode resulting from the inelastic stability analysis. Results obtained from the analysis lead to the conclusion that the inelastic insta-
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bility of liquid-filled conical tanks is most sensitive to axisymmetric imperfections. Compared to other non-axisymmetric functions, the sin 2 imperfection shape has also a significant contribution to the inelastic stability of the tanks. However, it is more conservative to consider the axisymmetric imperfections only when conducting inelastic stability analysis of conical tanks. Results of the various analyses conducted on thin walled conical tanks indicate that yielding usually precedes elastic buckling. As such, these structures will generally fail by inelastic buckling.
References [1] El Damatty AA, Korol RM, Mirza FA. Stability of imperfect steel conical tanks under hydrostatic loading. ASCE 1997;123(ST6):703–12. [2] Wunderlich W, Rensch HJ, Bochum H. Analysis of elastic–plastic buckling and imperfection-sensitivity of shells of revolution. In: Ramm E, editor. Buckling of shells. Berlin: Springer-Verlag, 1982: 137–74. [3] Vandepitte D, Rathe J, Verhegghe B, Paridaens R, Verschaeve C. Experimental investigation of hydrostatically loaded conical shells and practical evaluation of buckling load. In: Ramm E, editor. Buckling of shells. Berlin: Springer-Verlag, 1982: 375–99. [4] Koziey B, Mirza FA. Consistent thick shell element. Computers and Structures 1997;65(12):531–41. [5] El Damatty AA, Mirza FA, Korol RM. Large displacement extension of consistent shell element for static and dynamic analysis. Computers and Structures 1997;62(6):943–60. [6] Arbocz J, Abramovich H. The initial imperfection data bank at the Delft University of Technology— part I. Report LR-290, The Netherlands: Delft University of Technology, 1979. [7] Ellinas CP. Buckling of offshore structures—a state of the art review of the buckling of offshore structures. London: Grenada Publishing, 1984. [8] Bushnell D. Computerized buckling analysis of shells. Dordrecht, The Netherlands: Martinus Nijhoff Publishers, 1985.