Nonlinear buckling optimization technique to predict critical imperfection wavelength of combined liquid-filled steel conical tanks

Abstract

The shape of the imperfection induced by welding has an influence on the buckling resistance of thin shell structures, and many previous studies have come up with various models to estimate the critical imperfection shape. The aim of the current study is to assess the adequacy of three different approaches available in the literature, which consider that the imperfection wavelength matching the first buckling mode of a perfect tank to be the critical one. The first approach is based on buckling formulae calculated using a linear eigenvalue analysis performed on extensive experimental results of buckling of conical shells. The second approach assumes the critical wavelength, in view of the buckling mode profile detected from finite element analysis, as the distance between the inflection points of the elastic curve of the first buckling mode of a perfect tank. The third approach estimates the critical wavelength as double the distance between maximum and minimum points of the elastic curve. To determine the optimum wavelength that would lead to the minimum buckling capacity of the tank, the current study is conducted numerically by coupling a nonlinear finite element model, developed in house, and a direct search optimization technique. The results obtained from this numerical tool show good agreement with the first and the second approaches, which proves the adequacy of these two approaches in estimating the critical wavelength of the governing buckling mode, while the third approach yields a wavelength that overestimates the buckling capacity of the tank.

Keywords

imperfections optimization shell structures stability steel conical tanks

Introduction

Conical steel vessels are widely used as containments in elevated water tanks. The photos of two elevated conical tanks are shown in Figures 1 and 2. In the first figure, the vessel has a truncated conical shape and the tank is referred to as ‘pure conical tank’. The vessel of the tank shown in Figure 2 has a truncated conical shell with a superimposed cylindrical part. Such a type of tanks is referred to as ‘combined conical tank’. For those two configurations, the steel vessel consists of a number of curved panels, which are welded together along both the circumferential and the longitudinal directions. The steel vessel is typically welded at its bottom edge to a circular steel plate, which is in turn anchored to a heavy reinforced concrete circular slab supported by a reinforced concrete tower.

Figure 1.

Photo of a pure conical tank (http://www.san-con.com/specialty_structures.html).

Figure 2.

Photo of a typical elevated combined conical tank located in Canada (http://www.flikr.com/photos/greying_geezer/1093302118).

Several catastrophic failures of conical tanks happened during the past few decades in various locations around the globe. One of those failures occurred in Belgium in the 1970s while another failure happened in Fredericton, New Brunswick, Canada, in the early 1990s. A number of similarities exist between those two failure incidents. Both failures occurred when the structures were subjected to hydrostatic loading resulting from the contained fluid. Also, both failures were initiated by the buckling of the steel vessel as stated by Dawe et al. (1993), Korol (1991) and Vandepitte (1999). The buckling, which occurred at the bottom region of the steel vessels, was attributed to the use of an inadequate thickness for the steel shell in this region.

Following the collapse of the conical tank in Belgium, a research programme was initiated at Ghent University led by Professor Vandepitte. The research was mainly conducted experimentally. A large number of small-scale conical vessel models were constructed. The models had different dimensions and were made of different materials, with the majority of them made of Mylar. The experiments were conducted by gradually increasing the height of water inside the models. The water height at which each model buckled was detected. The experimental results were employed to develop a set of equations that can be used to assess the stability of conical tanks (Vandepitte et al., 1982). An expression for the wavelength of the buckling mode of conical tanks was also recommended based on the experimental results. It should be noted that the experimental programme was conducted strictly on pure conical vessel models. The method of construction of small-scale models is different than that of full-scale vessels. The latter involves both circumferential and longitudinal welding, which do not exist for small-scale models. As a result, the type of geometric imperfections that could exist in full-scale structures might be different than those existing in small-scale models. As thin shell structures, conical tanks are sensitive to geometric imperfections and their buckling capacities can be significantly affected by the shape and the magnitude of those imperfections.

The experimental programme conducted by Vandepitte et al. (1982) was reported in several studies (Paridaens et al., 1987; Vandepitte and Lagae, 1983; Vandepitte et al., 1988), which yielded design equations enabling practising structural engineers to estimate the buckling strength of liquid-filled steel conical shells while considering two different levels of expected initial imperfections, namely, ‘good cone’ and ‘poor cone’. The decision of using a specific level of initial imperfections, as assumed by Vandepitte, depends on the quality of welding. It is worth mentioning that these design equations were incorporated in the fourth edition of the European recommendations regarding shell buckling (European Convention for Constructional Steelwork, 1988).

Following the collapse of the Fredericton conical tank in the early 1990s, an extensive research programme focusing on the stability of such structures was initiated in Canada. The buckling behaviour of conical tanks, including the effect of geometric imperfections and residual stresses, was studied by El Damatty et al. (1997b, 1998). A simplified design procedure was then developed by El Damatty et al. (1999). It should be noted that all the above studies conducted either in Belgium or in Canada have focussed on pure conical tanks. Regarding combined conical tanks, which are the focus of the current study, the literature shows a number of studies that are concerned about their stability and optimum design. In terms of stability, a study was conducted by Hafeez et al. (2010) to characterize the buckling behaviour of combined conical tanks under the hydrostatic pressure. In their study, two different methods are used for estimating the critical wavelengths in view of the buckling mode profiles detected from finite element analysis. Numerical tools based on coupling finite element analysis and genetic algorithm optimization techniques were developed by El Ansary et al. (2010, 2011) to achieve optimum design of un-stiffened and stiffened, elevated, liquid-filled steel-combined conical tanks, respectively.

The main objective of the current study is to develop a numerical tool based on coupling a nonlinear finite element analysis model and a direct search optimization technique to predict the optimum buckling shape which is matching the critical imperfection wavelength for combined conical tanks having different practical geometric parameters. The results of this optimization tool are used to assess the adequacy of three different approaches available in the literature. First, the expression developed by Vandepitte et al. (1982) to determine the critical buckling wavelength of pure conical tanks is assessed when applied in case of combined tanks. Second, the adequacy of the two approaches developed by Hafeez et al. (2010) for combined conical tanks is also assessed. The first approach proposed by Hafeez et al. (2010) assumes the critical imperfection wavelength, in view of the buckling mode profile detected from finite element analysis, as the distance between the inflection points of the elastic curve of the first buckling mode of a perfect tank, while in the second approach, the wavelength is estimated as double the distance between maximum and minimum points of the elastic curve.

The outline of the current article is as follows. In the next section, a brief description of the causes of failure of conical tanks is provided. This is followed by a description of the numerical finite element model and the assumptions included in the analysis. In ‘Optimization technique’ section, the optimization technique is presented showing the objective function, design variables, guessing solutions and method of analysis. In ‘Analysis results’ section, detailed presentation and discussion of the numerical optimization results are given and compared to results reported in previous studies. Based on this comparison, recommendations are provided at the end of this section. Finally, in ‘Summary and conclusions’ section, the main conclusions drawn from the study are presented.

Causes of failure of conical tanks

Figure 3 shows the vertical projection of a combined conical vessel filled with water. In this figure, volume ‘A’ is bounded by an imaginary cylinder having a radius equivalent to the bottom radius $r_{b}$ of the vessel. Volume ‘B’ is bounded by the surface of the vessel and the surface of this imaginary cylinder. The weight of water inside volume ‘A’ is directly resisted by the vertical support underneath the vessel. The weight of water inside volume ‘B’ is supported by the inclined walls of the vessel. This leads to axial meridional compressive stresses acting along the longitudinal direction of the shell and hoop tensile stresses acting in the circumferential direction. The bottom region of the vessel is the most critical in terms of stress concentration since it supports the total weight of volume ‘B’ and has the smallest cross section. Compressive meridional stresses decrease rapidly when one moves upwards along the generatrix because the total compressive force acting on the corresponding parallel circle decreases and the perimeter of that circle increases.

Figure 3.

Cause of failure of conical tanks.

In addition, the boundary conditions at the bottom of the vessel restrain the displacements, leading to bending deformations and bending stresses in this region. As such, conical tanks can fail by instability due to a localized buckling effect near the base of the vessel. The buckling capacity of the vessel can be reduced due to the presence of geometric imperfections at this bottom region.

Finite element model

In this study, a nonlinear finite element model based on a degenerated consistent triangular shell element, developed by Koziey and Mirza (1997) is used. A sketch of the consistent shell element is shown in Figure 4. The element has 13 nodes, that is, three corner, three mid-side, six one-third-side, and one central nodes. As shown in the figure, each corner node has seven degrees of freedom, $u, v, w,$ α, β, Φ and Ψ. The displacements degrees of freedom $u, v, w$ are along the x, y and z global axes, respectively. The rotational degrees of freedom α, β, Φ and Ψ are associated with the corner and mid-side nodes where rotations α and Φ are about the local y′ axis and rotations β and Ψ are about the local x′ axis, where y′ and x′ are located in a plane tangent to the mid-surface. Rotations α and β are constant through the thickness of the element, thus providing linear variation of displacements which simulates bending deformation. While, rotations Φ and Ψ vary quadratically through the thickness that leads to a cubic variation of the through-thickness displacements, simulating shear deformations. The consistent shell element is an excellent tool to analyse plates and shell structures. It was successfully used in several previous studies and was validated against many experimental and numerical results (e.g. Azabi et al., 2016; Elansary et al., 2015, 2016; El Damatty et al., 1997b, 1998; Sweedan and El Damatty, 2009). More details about the features of this element and its linear formulation can be found in the study by Koziey and Mirza (1997). El Damatty et al. (1997a) have extended the formulation of this shell element to include the large deformation geometric nonlinear effect. Accordingly, this model can be used to predict the elastic buckling of shell structures by conducting a nonlinear elastic stability analysis. In the same study, El Damatty et al. (1997a) incorporated a nonlinear material model for steel in the finite element formulation. The material model can simulate either an elasto-plastic behaviour or a strain-hardening behaviour at the post-yield stage. Thus, the model can also be used to predict the inelastic buckling capacity of shell structures.

Figure 4.

Coordinate and degrees of freedom for a consistent shell element.

Modelling and assumptions

The conical and cylindrical parts of the steel vessels are simulated using an assembly of shell elements. The load acting on the structures results from the effect of hydrostatic pressure.

The following assumptions and considerations are accounted for in the modelling:

1. Due to double symmetry in both geometry and loading, only one-quarter of the vessel is considered in the analysis. A typical finite element mesh for one-quarter vessel is shown in Figure 5. A total of 320 elements are used to model the quarter tank, with eight and 20 rectangular divisions along the circumferential and longitudinal directions, respectively. As shown in the figure, a finer mesh is used at the bottom region where stress concentrations and buckling are anticipated. The length of the bottom four rows of elements $l_{e}$ along the tanks generator satisfies the following relation

l_{e} \leq \frac{l_{bv}}{4}

(1)

where $l_{bv}$ is the buckling wavelength of conical tanks as suggested by Vandepitte et al. (1982) and is given by

l_{bv} = 3.6 \sqrt{\frac{r_{b} t}{\cos θ_{v}}}

(2)

where $r_{b}$ and $t$ are the bottom radius and wall thickness of the tanks, respectively, and $θ_{v}$ is the angle of inclination of the tanks’ wall with the vertical direction.

Figure 5.

Finite element mesh for quarter tank.

It is worth mentioning that the mesh size described above is selected based on sensitivity analyses conducted in previous studies for similar tanks (El Ansary et al., 2010, 2011; Hafeez et al., 2010).

2. The boundary conditions at the base of the vessels are such that the displacements are restrained, and the rotations are free. In this region, a partial restraint to rotation is expected to be provided by welding. El Damatty et al. (1997b) have shown that the restraint of the rotation has a minor effect on the buckling capacity of conical vessels. However, the assumption of free rotation is conservative.

3. A ring beam and/or a roof cover often exist at the top of the structure. Under the axisymmetric loading resulting from hydrostatic pressure, those structure elements will constraint the radial deflection at the top edge of the vessel. However, in the numerical model, no restraint is imposed on the boundary condition at the top level of the vessel. This assumption was verified in a previous study by El Damatty et al. (1997b) where they showed that the radial displacement at the top edge under the hydrostatic pressure is almost zero. Furthermore, to have an accurate assessment of Hafeez et al. (2010) results, similar boundary conditions as that assumed in their study are imposed in the current analysis.

4. After reaching its yield strength value, the steel is assumed to have a strain-hardening behaviour. The unidirectional tangent modulus of steel in the post-yield stage is assumed to be equal to 2% of the Young’s Modulus of steel.

5. In the stability analysis of a shell structure, the first step involves the evaluation of the critical imperfection shape that leads to minimum buckling capacity of the structure. To determine such critical imperfection shape in case of combined conical tanks, a nonaxisymmetric imperfection shape can be defined by the following expression (El Damatty et al., 1997b)

W (s) = w_{o} \sin \frac{2 π s}{l_{I}} \cos (n θ)

(3)

where $w_{o}$ is the imperfection amplitude and $l_{I}$ is the imperfection wavelength and is equal to the wavelength of the first buckling mode of the perfect tank, as shown in Figure 6; s is a coordinate measured along the generator of the vessel; and n is an integer defining the circumferential wavelength of the imperfection shape and is equal to zero for case of axisymmetric imperfection shape.

Figure 6.

Axisymmetric imperfection shape along the generator of tank.

The ratio $w_{o} / l_{I}$ is a measure for the level of geometric imperfection, which is depended on the quality of construction. According to Vandepitte et al. (1982), conical tanks can be classified as

Good cone when $0 \leq \frac{w_{o}}{l_{I}} \leq 0.004$

Poor cone when $0.004 \leq \frac{w_{o}}{l_{I}} \leq 0.01$

In the current study, a low quality of construction is assumed for all tanks, so the classification of poor cone is applied in all analyses.

It should be mentioned that Hafeez et al. (2010) reported in their study that like the case of pure conical tanks analysed by El Damatty et al. (1997b), the presence of hydrostatic pressure tends to force combined conical tanks to buckle in an axisymmetric mode (n = 0), and, consequently equation (3) can be simplified as shown below to determine such critical imperfection shape in the current study

W (s) = w_{o} \sin \frac{2 π s}{l_{I}}

(4)

However, the built in-house programme proposed in this study is developed to be general enough to simulate any pattern of geometric imperfections, either axisymmetric or nonaxisymmetric. This has been done because this study is part of an on-going research project where the findings of this study will be used as the basis to predict the critical geometric imperfections in case of tanks stiffened with longitudinal stiffeners near the base of the steel vessel. In case of stiffened conical tanks, it is expected that the critical geometric imperfection is the nonaxisymmetric shape.

The critical imperfections, as defined by equation (4), are introduced into the finite element model as initial strains before the application of the hydrostatic loading where the wavelength $l_{I}$ is considered as the optimization design variable, as will be discussed in the next section. In the current study, this approach is recommended over other techniques (e.g. introducing an imperfect geometry of the mesh) because it is concluded that introducing the imperfections as initial strains in the built in-house programme is much easier than developing a preprocessor capable of generating an adaptive mesh that can accommodate changes in the imperfect geometry in each single iteration. The technique applied in the current study proves to be computationally efficient compared to the imperfect mesh approach.

Optimization technique

In the current study, the objective function is set to maximize the transverse displacement (perpendicular to the generator of the vessel) along the height of the tank as it was concluded by El Ansary et al. (2010) that the buckling capacities of the tanks are inversely proportional to the maximum deformation associated with a fixed value of loading. In order to achieve this objective function, a suitable and computationally efficient optimization technique is needed. Most of optimization techniques available in literature for solving general nonlinear optimization problems could be classified into two groups known as direct search and global search techniques. An unconstrained nonlinear optimization technique based on Nelder–Mead method or ‘Downhill Simplex method’ is applied in this study to find the maximum transverse displacement. This direct search approach begins the optimization procedure with a guess solution, which is often chosen randomly in the search space. The drawback of choosing a random guessing solution is that if it is not close enough to the global minimum solution, the optimization technique might be trapped in local minima. To overcome this drawback and weakness of the technique, guessing solutions based on scientific rationale are used in the current study for all studied tanks. Since the problem in hand considers only one design variable ‘critical imperfection wavelength’ $l_{I}$ , the analysis is conducted three times for each tank assuming three different guessing solutions obtained from the expression provided by Vandepitte et al. (1982) and the two approaches proposed by Hafeez et al. (2010). The following two sub-sections briefly describe the choice of the guessing solutions used in the direct search optimization technique according to the approaches available in the literature.

Guessing solution according to Vandepitte et al. (1982) wavelength I _bv

The extensive experimental investigation by Vandepitte et al. (1982) has led to equation (2), defined earlier for the buckling wavelength $l_{bv}$ . This expression is used to identify the guessing solution for the design variable $l_{I}$ of the studied tanks, and, consequently estimating the initial axisymmetric imperfection shape by substituting, in equation (4), $l_{bv}$ instead of $l_{I}$ . The level of geometric imperfection is considered in this study to follow the poor cone classification as defined by Vandepitte et al. (1982). As such, the imperfection amplitude $w_{o}$ is assumed to be equal to 0.01 $l_{bv} .$

Guessing solution according to Hafeez et al. (2010) approaches I _b1 and I _b2

In the study that was conducted by Hafeez et al. (2010), two methods are used for estimating the critical imperfection wavelengths in view of the buckling mode profiles detected from the finite element analysis. In the first approach, the critical imperfection wavelength $l_{b 1}$ is assumed to match the buckling wavelength, which is defined as the distance between the inflection points of the elastic curves of the buckling mode of the perfect tank as shown in Figure 7(a). In the second approach, the distance between maximum and minimum points of the elastic curve of the bottom region is evaluated (see Figure 7(b)). This distance is doubled to obtain the critical imperfection wavelength $l_{b 2}$ . It should be noted that the imperfection amplitude $w_{o}$ is assumed to be equal to 0.01 $l_{b 1}$ and 0.01 $l_{b 2}$ in the first and the second approaches, respectively. The wavelengths $l_{b 1}$ and $l_{b 2}$ reported by Hafeez et al. (2010), as shown in Tables 1 and 2 are implemented in the optimization algorithm as guessing solutions to predict the optimum buckling wavelength for all studied tanks.

Figure 7.

(a) Estimation of buckling wavelength by Hafeez et al.’s (2010) approach ‘(1)’. (b) Estimation of buckling wavelength by Hafeez et al.’s (2010) approach ‘(2)’.

Table 1.

Buckling load factors and buckling wavelengths for tanks with cap ratio $C = 0.3$ .

Tank	$r_{b}$ (m)	$H$ (m)	$θ_{V}^{°}$	$σ_{y}$ MPa	$t$ (mm)	$P_{cr}$ perfect	Buckling length (m)				$P_{cr}$ imperfect
							$l_{bv}$ ^a	$l_{b 1}$ ^b	$l_{b 2}$ ^c	$l_{bo}$ ^d	$l_{bv}$	$l_{b 1}$	$l_{b 2}$	$l_{bo}$
1	5.0	9.0	30	250	11	3.23	0.91	0.85	0.91	0.91	2.06	2.08	2.06	2.06
2	4.5	9.0	30	250	11	3.44	0.86	0.86	0.73	0.86	2.18	2.18	2.24	2.18
3	5.0	8.5	30	300	9	2.87	0.86	0.86	0.77	0.86	1.80	1.80	1.81	1.80
4	5.5	9.0	30	300	10	3.05	0.91	0.91	0.91	0.91	1.94	1.94	1.94	1.94
5	6.0	9.0	30	350	9	2.71	0.89	0.91	0.73	0.90	1.75	1.75	1.81	1.75
6	6.0	8.0	30	350	9	2.83	0.89	0.93	0.81	0.91	1.84	1.84	1.86	1.84
7	6.5	8.0	45	350	16	3.0	1.38	1.25	1.13	1.36	1.81	1.83	1.86	1.81
8	4.0	8.0	45	250	16	3.06	1.08	0.93	0.81	1.06	1.76	1.80	1.86	1.76
9	5.5	9.0	45	250	20	3.40	1.42	1.56	1.20	1.44	2.02	2.02	2.07	2.02
10	6.0	9.0	45	300	19	3.45	1.44	1.23	0.91	1.44	2.06	2.10	2.33	2.06
11	5.0	7.5	45	300	14	2.73	1.13	1.15	1.00	1.12	1.59	1.61	1.60	1.59
12	5.5	9.0	45	350	16	3.12	1.27	1.23	1.20	1.24	1.81	1.81	1.81	1.81
13	6.0	9.0	45	350	17	3.25	1.36	1.23	1.09	1.34	1.90	1.93	1.99	1.90
14	5.0	8.5	45	250	18	3.18	1.28	1.16	1.13	1.26	1.86	1.88	2.18	1.86
15	4.0	9.5	45	300	18	3.68	1.15	1.20	1.02	1.14	2.10	2.11	2.12	2.10
16	4.5	8.0	60	300	32	2.92	1.93	1.58	0.84	1.90	1.60	1.67	2.18	1.60
17	4.0	8.0	60	250	37	3.07	1.96	1.58	0.85	1.90	1.76	1.76	2.33	1.76

$l_{bv}$ : The buckling wavelength defined by Vandepitte et al. (1982).

$l_{b 1}$ : The buckling wavelength defined by Hafeez et al.’s (2010) ‘approach 1’.

$l_{b 2}$ : The buckling wavelength defined by Hafeez et al.’s (2010) ‘approach 2’.

$l_{bo}$ : Optimum buckling wavelength predicted in the current study.

Table 2.

Buckling load factors and buckling wavelengths for tanks with cap ratio $C = 0.5$ .

Tank	$r_{b}$ (m)	$H$ (m)	$θ_{V}^{°}$	$σ_{y}$ MPa	$t$ (mm)	$P_{cr}$ perfect	Buckling length (m)				$P_{cr}$ imperfect
							$l_{bv}$ ^a	$l_{b 1}$ ^b	$l_{b 2}$ ^c	$l_{bo}$ ^d	$l_{bv}$	$l_{b 1}$	$l_{b 2}$	$l_{bo}$
18	5.0	9.0	30	250	10	3.10	0.86	0.97	0.91	0.95	2.05	2.02	2.03	2.02
19	4.5	9.0	30	250	10	3.32	0.82	0.93	0.72	0.90	2.15	2.15	2.20	2.15
20	5.0	8.5	30	300	9	3.14	0.82	0.83	0.61	0.80	2.05	2.05	2.19	2.05
21	5.5	9.0	30	300	9	2.88	0.86	0.88	0.52	0.85	1.89	1.88	1.99	1.88
22	6.0	9.0	30	350	9	2.97	0.89	0.96	0.72	0.90	1.98	1.97	2.05	1.97
23	6.0	8.0	30	350	9	3.12	0.89	0.78	0.81	0.90	2.11	2.16	2.14	2.11
24	6.5	8.0	45	350	15	3.11	1.33	1.10	0.92	1.30	1.93	1.98	2.11	1.93
25	4.0	8.0	45	250	15	3.27	1.05	1.01	0.81	1.04	1.91	1.91	2.0	1.91
26	5.5	9.0	45	250	19	3.57	1.38	1.23	1.04	1.38	2.18	2.21	2.33	2.18
27	6.0	9.0	45	300	17	3.34	1.36	1.27	0.91	1.38	2.03	2.04	2.25	2.03
28	5.0	7.5	45	300	13	2.83	1.09	1.05	0.76	1.10	1.69	1.69	1.83	1.69
29	5.5	9.0	45	350	15	3.23	1.23	1.14	0.91	1.24	1.92	1.94	2.04	1.92
30	6.0	9.0	45	350	16	3.36	1.33	1.28	0.91	1.30	2.04	2.04	2.22	2.04
31	5.0	8.5	45	250	16	3.10	1.21	1.20	0.86	1.22	1.84	1.84	1.99	1.84
32	4.0	9.5	45	300	16	3.63	1.08	0.86	0.83	1.08	2.09	2.16	2.19	2.09
33	4.5	8.0	60	300	29	3.10	1.84	1.13	0.81	1.85	1.70	1.94	2.31	1.70
34	4.0	8.0	60	250	32	3.14	1.82	1.36	0.81	1.84	1.78	1.89	2.34	1.78

$l_{bv}$ : The buckling wavelength defined by Vandepitte et al. (1982).

$l_{b 1}$ : The buckling wavelength defined by Hafeez et al.’s (2010) ‘approach 1’.

$l_{b 2}$ : The buckling wavelength defined by Hafeez et al.’s (2010) ‘approach 2’.

$l_{bo}$ : Optimum buckling wavelength predicted in the current study.

Method of analysis

In the current study, 34 combined conical tanks covering a different practical geometric range are considered. The proposed geometric parameters of these tanks (bottom radius, $r_{b}$ ; total height, $H$ ; angle of inclination of the wall of the vessel with the vertical direction, $θ_{v}$ ; and cap ratio, $C = h_{cylinder} / H$ , where $h_{Cylinder}$ is the height of the top cylindrical part) and material properties are defined in Tables 1 and 2. For all studied tanks, modulus of elasticity $E = 2 \times 10^{5} MPa$ and Poisson’s ratio $ν = 0.3$ are assumed. Those tanks are divided equally into two different sets according to the cap ratio $C$ . The first set has a cap ratio $C = 0.3$ , as shown in Table 1, and the second set has a cap ratio $C = 0.5$ , as presented in Table 2. The required thickness of each tank is predicted by following the simplified design approach developed by Sweedan and El Damatty (2009) for hydrostatically loaded combined conical tanks. The proposed design approach is based on providing a magnification factor β that can be used to relate the maximum overall stresses developed in the tank wall to the theoretical membrane stresses resulting from static equilibrium of the shell under the internal hydrostatic pressure. Nonlinear multiple regression analyses were carried out using finite element results to determine the variation of this stress magnification factor β as function of the independent tank geometric parameters (bottom radius (r_b), tank height (H), height of cylindrical cap (h_cap), wall thickness (t), and angle of inclination of the wall of the vessel with the vertical direction $(θ_{v})$ ). At the end of their study, they provided illustrative numerical examples to demonstrate the application of this simplified design procedure following both mean values of regression coefficients and upper 95% confidence values. Similar steps are applied in the current study to predict the wall thicknesses provided in Tables 1 and 2 for the 34 studied tanks following the corresponding upper 95% regression coefficients, as presented in the study by Sweedan and El Damatty (2009). Following the step of predicting the wall thickness for each of the 34 tanks, a finite element model is developed considering the geometric and material properties associated with each tank as presented in Tables 1 and 2. In order to predict the optimum buckling wavelength $l_{bo}$ , which leads to the lowest buckling load for each of the studied tanks, a coupled finite element/optimization technique is applied. The methodology proposed in the current study can be summarized in the following steps:

A nonlinear finite element model is developed for each of the 34 studied tanks where this set of analysis considers only geometric nonlinearity. A direct search optimization technique is coupled with the finite element model to predict the optimum buckling wavelength of the tank of interest. This is done by maximizing the objective function (transverse deflection along tank generator) at a constant hydrostatic pressure ( $γ_{w} h$ ), as shown in Figure 8, where $γ_{w}$ is the specific weight of water and $h$ is the depth of the point at which the pressure is calculated, measured from the water surface. The strains due to the assumed imperfection shapes are introduced in the finite element model at the first iteration of the first load increment in the form of initial strains.

After estimating the optimum buckling wavelength of all studied tanks, another set of nonlinear finite element analysis, considering both geometric and material nonlinearities, is conducted by multiplying the hydrostatic pressure by a load factor $P$ . The analysis is conducted incrementally by gradually increasing the value of $P$ . As the load factor increases, the stiffness of the structure decreases till it buckles at a critical load factor $P_{cr}$ . If the value of $P_{cr}$ < 1.4, the structure cannot sustain the effect of the acting hydrostatic pressure. A value of $P_{cr}$ = 1.4 means that the structure is on verge of failure under the hydrostatic load. A value of $P_{cr}$ > 1.4 indicates that the structure has a certain margin of safety against instability failure. A factor of 1.4 is chosen as it is common in the limit state design code for steel structures that a load factor of 1.25 is used to magnify the dead loads while a reduction of 0.9 is applied to reduce the structure resistance. The hydrostatic load can be considered as a dead load case. Accordingly, a load factor $P_{cr} = 1.25 / 0.9 = 1.4$ is used as the critical load factor.

Figure 8.

Hydrostatic pressure distributions inside the conical vessel.

Analysis results

The optimum buckling wavelengths $l_{bo}$ predicted by the finite element/optimization technique for the 34 studied tanks are summarized in Tables 1 and 2 for cap ratios $C = 0.3$ and $C = 0.5$ , respectively. The tables show the geometric and material properties of each tank together with the assumed wavelength as reported in the literature by the three different approaches described earlier. Also, the table shows the critical load factors $P_{cr}$ at which the tanks buckle associated with each buckling wavelength. For each tank, the minimum value of buckling capacity is highlighted. It can be noticed from the results reported in Tables 1 and 2 that the values obtained by assuming imperfection wavelength as defined by Vandepitte et al.’s (1982) expression show a very good agreement with the optimization results, which proves the adequacy of applying Vandepitte’s expression in predicting the critical imperfection shape in combined conical tanks. On the contrary, the two approaches proposed by Hafeez et al. (2010) show some variations in the predicted buckling capacity of the tanks, especially for ‘approach (2)’. This appears clearly by comparing the critical buckling load factors predicted by assuming imperfection wavelengths according to Hafeez et al.’s (2010) ‘approach (2)’ to those obtained based on the optimum wavelength. Also, it can be noticed that the discrepancy in the results reported by Hafeez et al.’s (2010) ‘approach (2)’ increase with the increase of the angle of inclination of the tanks’ wall $θ_{v}$ , especially for tanks with cap ratio $C = 0.5$ . This conclusion shows the inadequacy of applying (Hafeez et al., 2010) ‘approach (2)’ especially in case of wide tanks ( $θ_{v} > 45 °$ ).

On the contrary, ‘approach (1)’ shows acceptable prediction of the critical imperfection wavelength in case of tall, narrow tanks only with angle $θ_{v}$ = 30°. It should be mentioned that the prediction of the critical imperfection wavelength using ‘approach (1)’ is dependent on the accuracy of specifying the inflection points in the estimated buckling wavelength. This may be the reason for the discrepancy in the wavelengths predicted by following this approach especially for wide tanks ( $θ_{v}$ ≥ 45°) compared to the optimum wavelength. As such, the results reported in this study prove the adequacy of utilizing, with confidence, the expression defined by Vandepitte et al. (1982) for conical shells in the case of combined conical tanks. This study is the first of its kind to test this expression in a comprehensive manner by comparing it to the optimum imperfection wavelength, which is predicted by screening all possible patterns through coupling a nonlinear finite element model with an efficient optimization technique. In the authors opinion, the outcomes of this article demonstrate significant impact to practitioners concerned about the design of such type of complex shell structures since it assures utilizing Vandepitte’s formulation for either pure or combined conical shells.

Summary and conclusions

The current study is conducted numerically through nonlinear finite element analysis and a direct search optimization technique to predict the critical geometric imperfection wavelength and the associated buckling capacity of combined liquid-filled conical tanks. The results obtained from the current study are used to assess the adequacy of three different approaches available in the literature to predict the critical geometric imperfection shape of such type of shell structures. The following conclusions can be drawn from the study:

The coupled finite element optimization technique developed in the current study is considered as an efficient numerical tool in estimating the critical imperfection wavelength that has a direct effect on reducing the buckling capacity of combined conical tanks. This numerical tool is used to assess the accuracy of three different approaches available in the literature to estimate the critical imperfection shape of conical tanks.

The optimization results show that the critical imperfection shape is axisymmetric and has an optimum wavelength equal to the buckling wavelength value predicted by Vandepitte et al. (1982) for conical shells.

The wavelength $l_{b 1}$ defined by Hafeez et al. (2010) as the distance between the inflection points of the elastic curves of the buckling mode of the perfect tank provides an accurate estimate for the imperfection wavelength of the governing axisymmetric buckling mode of combined conical tanks only for tall, narrow tanks with $θ_{v}$ = 30°.

The wavelength $l_{b 2}$ defined by Hafeez et al. (2010) as double the distance between maximum and minimum points of the elastic curve of the first buckling mode of the perfect tank leads to inaccurate prediction of the imperfection wavelength. As such, it is not recommended to apply this approach to estimate the critical imperfection wavelength of combined liquid-filled steel conical tanks.

Footnotes

Acknowledgements

The authors are grateful to SAN-CON Industries INC. for the valuable assistance with the photographs. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET: ).

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ayman M El Ansary

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Nonlinear buckling optimization technique to predict critical imperfection wavelength of combined liquid-filled steel conical tanks

Abstract

Keywords

Introduction

Causes of failure of conical tanks

Finite element model

Modelling and assumptions

Optimization technique

Guessing solution according to Vandepitte et al. (1982) wavelength I bv

Guessing solution according to Hafeez et al. (2010) approaches I b1 and I b2

Method of analysis

Analysis results

Summary and conclusions

Footnotes

Acknowledgements

Declaration of Conflicting Interests

Funding

ORCID iD

References

Guessing solution according to Vandepitte et al. (1982) wavelength I _bv

Guessing solution according to Hafeez et al. (2010) approaches I _b1 and I _b2