Behavior of stiffened liquid-filled conical tanks

Unreinforced steel conical-shaped containment vessels are frequently used in water tower applications. The failure of one of these structures in Fredericton, New Brunswick, Canada, several years ago, raises the question of whether there are adequate safety provisions for existing conical tanks. The aim of this investigation is to study the effect of welding rectangular-shaped longitudinal stiffeners to enhance the buckling capacity of existing conical tanks and to improve the design of new structures. The investigation is carried out numerically using an in-house developed shell element model that includes the effects of geometric and material non-linearities and accounts for geometric imperfections. The study focuses on two cases of tanks reinforced by longitudinal stiffeners in the lower region: the case of stiffeners free at their bottom edge, which would correspond to the retrofit of existing tanks; and the second having stiffeners anchored to the bottom slab of the tank, which can duplicate the situation of a new design. An extensive parametric study is conducted to assess the typical behavior of the two cases and to determine the critical imperfection shape that leads to the minimum buckling capacity of such type of stiffened shell structures. Finally, a comparison between the buckling capacity of unstiffened and longitudinally stiffened conical tanks that have the same volume of steel is conducted, revealing a major benefit of including stiffeners.     

Stability of combined imperfect conical tanks under hydrostatic loading

Steel conical vessels with upper cylindrical caps are widely used as liquid containments in elevated water tanks. This type of structure for containing water is referred to as “combined conical tank”. A number of catastrophic failures of combined conical tanks occurred during the past decades in various locations around the globe. Previous studies available in the literature focused on pure conical tanks, where the vessels have no upper cylindrical caps. The current study focuses on characterizing the buckling behaviour of combined conical tanks under the effect of hydrostatic pressure. The study is conducted numerically using a three-dimensional finite element model developed in-house. The effects of geometric imperfection and residual stresses as well as the variation of the geometric and material parameters on the buckling capacity of combined conical tanks are investigated. Finally, a comparison between the buckling capacities of combined and equivalent pure conical tanks is conducted.     

A coupled finite element genetic algorithm for optimum design of stiffened liquid-filled steel conical tanks

In the current study an optimum design technique of stiffened liquid-filled steel conical tanks subjected to global and local buckling constraints is developed using a numerical tool that couples a non-linear finite element model developed in-house and a genetic algorithm optimization technique. This numerical tool is an extended version of an earlier one, adapted for the optimum design of unstiffened conical tanks. The design variables considered in the current study are the shell thicknesses, the geometry of the steel vessel as well as the dimensions and number of stiffeners. The developed numerical tool is capable of selecting the set of design variables that leads to optimum safe design. The analysis is conducted twice; first, case of stiffeners free at their bottom edge, which represents the case of retrofitting an existing tank. In the second case the stiffeners are assumed to be anchored to the bottom slab of the tank, which represents the situation of a newly designed tank. Finally, the optimum design of the stiffened tanks is compared to the optimum design of unstiffened tanks computed in a previous study.     

Wind pressures and buckling of cylindrical steel tanks with a conical roof

Tanks with a conical roof are studied in this paper under wind load, for a roof which is supported by rafters and columns. Buckling occurs in the form of deflections in the cylindrical shell and the buckling mode is localized in the windward region. Both bifurcation analysis and geometrically nonlinear analysis have been performed using finite element discretizations of the structure. The wind pressures have been obtained from wind tunnel experiments performed as part of the research, and have been obtained for tank geometries for which information was not previously available. The results show high imperfection sensitivity of tanks with a conical roof, and buckling loads for wind velocities in the same order as those expected to occur in the Caribbean region.     

A coupled finite element genetic algorithm technique for optimum design of steel conical tanks

Steel conical tanks having an upper cylindrical section and supported by a reinforced concrete shaft are widely used for water containment in elevated tanks. During the past few decades, a number of conical tanks have failed as a result of buckling of the steel vessel due to inadequate selection of the shell thickness. In the current study a powerful numerical tool that couples a non-linear finite element model and a genetic algorithm optimization technique is developed specifically for the analysis and design of steel conical tanks. The developed numerical tool is capable of selecting the set of design variables which satisfies the structure safety requirements while achieving a minimum structure weight and consequently minimum cost.     

Dynamic buckling of anchored steel tanks subjected to horizontal earthquake excitation

We investigate dynamic buckling of aboveground steel tanks with conical roofs and anchored to the foundation, subjected to horizontal components of real earthquake records. The study attempts to estimate the critical horizontal peak ground acceleration (Critical PGA), which induces elastic buckling at the top of the cylindrical shell, for the impulsive hydrodynamic response of the tank-liquid system. Finite elements models of three cone roof tanks with height to diameter ratios (H/D) of 0.40, 0.63 and 0.95 and with a liquid level of 90% of the height of the cylinder were used in this study. The tank models were subjected to accelerograms recorded during the 1986 El Salvador and 1966 Parkfield earthquakes, and dynamic buckling computations (including material and geometric non-linearity) were carried out using the finite element package ABAQUS. For the El Salvador accelerogram, the critical PGA for buckling at the top of the cylindrical shell decreased with the H/D ratio of the tank, while similar critical PGAs regardless of the H/D ratio were obtained for the tanks subjected to the Parkfield accelerogram. The elastic buckling at the top occurred as a critical state for the medium height and tallest models regardless of the accelerogram considered, because plasticity was reached for a PGA larger than the critical PGA. For the shortest model (H/D=0.40), depending on the accelerogram considered, plasticity was reached at the shell before buckling at the top of the shell.     

组合锥形液体贮罐的简化设计方法

组合圆锥-圆柱形壳形式的钢贮罐是一种常见的用于储存液体的压力罐。大部分此类压力罐的破坏都由于钢壳的失稳而引起。引起这些破坏的主要原因是没有适当的设计方法来对这类结构进行设计。在本次研究中,提出了简化的设计方法,以保证静水加载的组合钢罐在发生屈曲时的安全。应用非线性有限元模型进行数值模拟,对大变形和几何缺陷对钢罐稳定性的破坏作用进行分析。采用有限元计算结果与非线性回归分析建立函数,将壳的总应力与膜应力关联起来。采用算例说明了所提议设计方法的应用。     

Inelastic stability of conical tanks

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.     

Buckling strength of cylindrical steel tanks under harmonic settlement

Large vertical cylindrical steel tanks for bulk and fluid storage are usually constructed in soft foundations, so it is not surprising that tank foundations are susceptible to various types of settlement beneath the tank wall, which is usually decomposed as a Fourier series in harmonics. In this paper, buckling strength of cylindrical fixed-roof steel storage tanks under harmonic settlement is investigated through great deal of numerical analyses by the FE computer package ANSYS. Three types of buckling analyses are carried out which are the LBA, GNA, GNIA proposed also by Eurocode 3. The results show that the equilibrium path from both GNA and GNIA is highly nonlinear, and it seems ungrounded to establish design criterion on the principle of superposition based on the linear elastic theory. The influences of the harmonic wave number n, the radius-to-thickness ratio r/t, the height-to-radius ratio h/r, and the geometric imperfection δ₀/t on the buckling strength of the storage tanks are mainly investigated. The ultimate harmonic settlements for various tank geometries are addressed and plotted in each analysis together with the buckling modes. The buckling modes from GNA and GNIA agree well with the lowest linear bifurcation buckling modes from LBA, and take mainly two types of deformations: shearing buckling extending throughout the entire height for the lower wave number n=2–4 and the elephant's foot failure occurring at the upward settlement zone caused by the meridional compression for the higher wave number n>4. It is also indicated from the results that both the ultimate harmonic settlement and the buckling mode of the tank are closely correlative with the geometric parameters: the wave number n, the radius-to-thickness ratio r/t, the height-to-radius ratio h/r, and the initial geometric imperfection δ₀/t.     

Design and optimization of laminated conical shells for buckling

Optimum laminate configuration for the maximum buckling load of filament-wound laminated conical shells is investigated. In the case of a laminated conical shell, the thickness and the ply orientation (the design variables) are functions of the shell coordinates, influencing both the buckling load and the weight of the structure. Thus, optimization can be performed by maximization of the buckling load for a specific weight, or by minimization of the weight of the structure under the constraint of applied buckling load. Due to the complex nature of the problem a preliminary investigation is made into the characteristic behavior of the buckling load with respect to the volume as a function of the ply orientation.The exact buckling load is calculated by means of the computer code STAGS-A (Structural Analysis of General Shells [Almroth BO, Brogan FA, Meller E, Zele F, Petersen HT. Collapse analysis for shells of general shape, user's manual for STAGS-A computer code. Technical report AFFDL TR-71-8; 1973]) by adding a user written subroutine WALL, see Ref. [Goldfeld Y, Arbocz J. Buckling of laminated conical shells taking into account the variations of the stiffness coefficients. AIAA J 2004; 42(3):642–649]. The optimization problem is solved using response surface methodology.     

Buckling of cylindrical open-topped steel tanks under wind load

Vertical cylindrical welded steel tanks are typical thin-walled structures which are very susceptible to buckling under wind load. This paper investigates the buckling behavior of open-topped steel tanks under wind load by finite element simulation. The analyses cover six common practical tanks with volumes of 2×10³ m³ to 100×10³ m³ and height-to-diameter ratios H/D<1. The linear elastic bifurcation analyses are first carried out to examine the general buckling behavior of tanks under wind load, together with comparison to that of tanks under uniform pressure and windward positive pressure (only loaded by positive wind pressure in the windward region). The results show that for larger tanks in practical engineering, the stability carrying capacity of wind load is relatively lower. It is also indicated that the buckling behavior of tanks under wind load is governed by the windward positive pressure while wind pressure in other region of tank essentially has no influence on the buckling performance. The geometrically nonlinear analyses are then conducted to investigate the more realistic buckling behavior of tanks under wind load. It is found that the buckling behaviors of perfect tanks and imperfect tanks are much different. The weld induced imperfection only has little influence on the wind buckling behavior while the classical buckling mode imperfection has significant influence, leading to a considerable reduction of wind buckling resistance. The influences of thickness reduction of cylindrical wall, liquid stored in the tank and wind girder on the buckling behavior are also examined. It shows that the thickness reduction of cylindrical wall considerably reduces the wind buckling resistance while sufficient liquid stored in the tank and wind girder significantly increase the wind buckling resistance.     

Experimental behaviour of concrete-filled stiffened thin-walled steel tubular columns

An experimental study on the structural behaviour of concrete-filled stiffened thin-walled steel tubular columns is presented in this paper. The stiffening was achieved by welding longitudinal stiffeners on the inner surfaces of the steel tubes. Companion tests were also undertaken on 12 unstiffened concrete-filled steel tubular (CFST) columns, with or without steel fibres in the infill concrete. The test results showed that the local buckling of the tubes was effectively delayed by the stiffeners. The plate buckling initially occurred when the maximum load had almost reached for stiffened specimens, thus they had higher serviceability benefits compared to those of unstiffened ones. Some of the existing design codes were used to predict the load-carrying capacities of the tested composite columns.     

Buckling of steel shells subjected to non-uniform axial and pressure loading

The paper presents theoretical and numerical studies of the buckling of steel shells subjected simultaneously to axial and pressure loading. The type of combined non-uniform loading used for this analysis (horizontal pressure and wall frictional pressure which occurs in the silos) is given in the Eurocode 1 part 4. An Abaqus program was used to study the influence of the parameters affecting the strength of the shell. The results of this parametric study were used to develop a new semi-analytical method for the prediction of the critical stress of the stiffened and isotropic shells. The corresponding formulae are compared, in different examples of geometrical shells, with the results obtained by finite element simulation and by the formulae for buckling design given in recommendation EC3.     

Equivalent mechanical analog for dynamic analysis of pure conical tanks

This study is motivated by the lack of information regarding seismic analysis and design of liquid-filled conical tanks. The main challenge in this type of fluid–structure interaction problem is the estimation of the forces associated with the hydrodynamic pressure resulting from the vibration of the structure. A simplified mechanical analog that can be used to estimate the forces associated with horizontal ground excitation is developed in this study. The proposed mechanical analog takes into account the flexibility of the tank walls and simulates both the impulsive and sloshing components of the hydrodynamic pressure. Parameters of this analog are displayed in chart form as functions of the geometrical parameters of the tanks including the angle of inclination of the tanks’ walls. Finally, an example is provided to illustrate the application of the developed analog for evaluation of the response of liquid-filled conical tanks to horizontal ground excitation.     

Wind pressures and buckling of cylindrical steel tanks with a dome roof

An experimental/computational strategy is used in this paper to evaluate the buckling behavior of steel tanks with a dome roof under exposure to wind. First, wind tunnel experiments using small scale rigid models were carried out, from which pressure distributions due to wind on the cylindrical part and on the roof were obtained. Second, a computational model of the structure (using the pressures obtained in the experiments) was used to evaluate buckling loads and modes and to study the imperfection sensitivity of the tanks. The computational tools used were bifurcation buckling analysis (eigenvalue analysis) and geometrical nonlinear analysis (step-by-step incremental analysis). Geometric imperfections and changes in the buckling results due to reductions in the thickness were also included in the study to investigate reductions in the buckling strength of the shell. For the geometries considered, the results show low imperfection sensitivity of the tanks and buckling loads associated with wind speeds 45% higher than those specified by the ASCE 7-02 standard.     

Techniques for buckling experiments on steel silo transition junctions

Large elevated steel silos generally consist of a cylindrical vessel, a conical discharge hopper and a skirt which may either be supported on the ground or by a number of columns. The cone–cylinder–skirt junction is subject to a large circumferential compressive force due to the radial component of the meridional tension in the hopper, so either a ring is provided or the shell walls are locally thickened to strengthen the junction. Many theoretical studies have examined the buckling and collapse strengths of these junctions, but no previous experimental study has been reported. This has been due to the great difficulties associated with testing these thin-shell junctions at model scale. This paper first describes the development of an experimental facility for testing model steel silo transition junctions. Issues covered include the fabrication of quality model junctions using thin steel sheets, the loading method and the precise three-dimensional measurement of geometric imperfections and deformed shapes using a laser-displacement meter. Typical experimental results of a cone–cylinder–skirt–ring junction are next presented to demonstrate the capability of the developed facility. Procedures for processing the test results to determine both the buckling load and the number of buckling waves are also presented.     

A preliminary design formula for the strength of stiffened curved panels by design of experiment method

In bridge construction, the use of stiffened plates for box-girder or steel beams is common day to day practice. The advantages of the stiffening from the economical and mechanical points of view are unanimously recognized. For curved steel panels, however, applications are more recent and the literature on their mechanical behaviour including the influence of stiffeners is therefore limited. Their design with actual finite element software is possible but significantly time-consuming and this reduces the number of parameters which can be investigated to optimise each panel. The present paper is thus dedicated to the development of a preliminary design formula for the determination of the ultimate strength of stiffened cylindrical steel panels. This approximate formula is developed with the help of a design of experiment method which has been adapted from the current statistical knowledge. This method is first presented, and its feasibility as well as its efficiency are illustrated through an application to the reference case of unstiffened curved panels. Then, the case of stiffened curved panels is investigated and a preliminary design formula is developed. The ease of use of this formula for preliminary design is finally illustrated in a cost optimisation problem.     

Strength and ductility of stiffened thin-walled hollow steel structural stub columns filled with concrete

It is generally expected that inner-welded longitudinal stiffeners can be used to improve the structural performance of thin-walled hollow steel structural stub columns filled with concrete. Thirty-six specimens, including 30 stiffened stub columns and six unstiffened ones, were tested to investigate the improvement of ductile behaviour of such stiffened composite stub columns with various methods. The involved methods include increasing stiffener height, increasing stiffener number on each tube face, using saw-shaped stiffeners, welding binding or anchor bars on stiffeners, and adding steel fibres to concrete. It has been found that adding steel fibres to concrete is the most effective method in enhancing the ductility capacity, while the construction cost and difficulty will not be increased significantly.     

Numerical evaluation on shell buckling of empty thin-walled steel tanks under wind load according to current American and European design codes

Liquid storage steel tanks are vertical above-ground cylindrical shells and as typical thin-walled structures, they are very sensitive to buckling under wind loads, especially when they are empty or at low liquid level. Previous studies revealed discrepancies in buckling resistance of empty tanks between the design method proposed by the American Standard API 650 and the analytical formulas recommended by the European Standard EN1993-1-6 and EN1993-4-2. This study presents a comparison between the provisions of current design codes by performing all types of numerical buckling analyses recommended by Eurocodes (i.e. LBA-linear elastic bifurcation analysis, GNA-geometrically nonlinear elastic analysis of the perfect tank and GNIA-geometrically nonlinear elastic analysis of the imperfect tank). Such analyses are performed in order to evaluate the buckling resistance of two existing thin-walled steel tanks, with large diameters and variable wall thickness. In addition, a discussion is unfolded about the differences between computational and analytical methods and the conservatism that the latter method imposes. An influence study on the geometric imperfections and the boundary conditions is also conducted. Investigation on the boundary conditions at the foot of the tank highlights the sensitivity to the fixation of the vertical translational degree of freedom. Further, it is indicated that the imperfection magnitude recommended by the EN1993-1-6 is extremely unfavorable when applied to large diameter tanks. Comments and conclusions achieved could be helpful in order to evaluate the safety of the current design codes and shed more light towards the most accurate one.     

The effect of semi-rigid joints and an elastic bracing system on the buckling load of simple rectangular steel frames

A linear stability analysis for establishing the combined effect of joint flexibility and an elastic bracing system on the buckling load of steel plane frames is presented herein. Based on the beam-to-column model of Eurocode 3, the subsequent study shows that joint flexibility is a very important parameter that needs to be incorporated into the stability analysis of frames with semi-rigid connections. It was found that assuming flexible connections in such frames always leads to a reduction of their buckling load, which is proven to be significant in many characteristic cases. Numerical results for simple elastically braced or unbraced frames with semi-rigid connections, in tabular and graphical form, reveal the pronounced effect of joint flexibility and elastic bracing on their buckling load. The results presented herein can be readily used for the design of simple frames against buckling, while the flexible connection concept can be applied to all types of steel frames.