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.     

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.     

Buckling of fixed-roof aboveground oil storage tanks under heat induced by an external fire

This paper presents computational modeling and results of steel storage tanks under heat induced by an adjacent fire. In this research, modeling is restricted to the structural behavior of the tank, with emphasis on thermal buckling of the shell. Two tanks that buckled under a huge fire in Bayamón, Puerto Rico in 2009, are investigated in detail: a small tank with a self-supported conical roof, and a large tank in which the conical roof is supported by a set of rafters and columns. For a tank that is empty, the results show that a relatively low temperature is enough to produce static buckling of the shell. In pre-buckling states, the cylindrical shell has thermal expansion; at the critical state the displacements reverse and inwards displacements are observed at advanced post-buckling states. Parametric studies are performed to understand the influence of the shell thickness, the level of fluid stored in the tank, the area affected by fire in the circumferential direction, and the temperature gradient through the thickness. The buckling modes are compared with real deflection of tanks that were affected by fire.     

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.     

The effects of long-term corrosion on the dynamic characteristics of ground based cylindrical liquid storage tanks

Results of numerical investigations on the effects of material degradation due to corrosion on the dynamic characteristics of ground-based, anchored, steel liquid storage tanks are presented. Internal corrosion is considered as a time-dependent constant thinning of the wall, at locations in contact with residual water, water condensate, atmospheric oxygen and acid gases. Dynamic analyses are performed on numerical tank-liquid models, having different aspect ratios and wall thicknesses at different stages of wall thinning at the specified locations. The aim of the analyses is to determine the corrosion effects on the natural periods and mode shapes of vibration. Steady-state, harmonic base excitation analyses are also carried out to determine the corrosion effects on the hydrodynamic pressures produced in the liquid. It is found that progressive corrosion has significant effects on the tank fundamental period and its associated mode shape of vibration as well as the magnitude and location of the maximum hydrodynamic pressure and that as design provisions should cover the service life of the tank, the errors associated with the current code provisions for design of such tanks cannot be ignored.     

Collapse design of large steel digester tanks

This paper deals with practical problems encountered in the design of seismically loaded liquid-filled vertical tanks. Global stability, buckling under combined action of axial compression and internal pressure and the effect of flexible boundaries on axial buckling are studied. The question of the mutual interaction of global and local failure modes is considered. Numerical studies on the basis of finite element-discretized-shell models are carried out which show the typical features of the different failure modes. Both the effect of internal pressure on axial buckling (elephant foot failure) as well as the flexibility of the base ring results in noteworthy reductions of the axial buckling resistance. Global stability effects turned out to play no major role in the present tank design.     

Elephant's foot buckling in pressurised cylindrical shells

Metal cylindrical bins, silos and tanks are thin shell structures subject to internal pressure from stored materials together with axial compression from the frictional drag of stored materials on the walls and horizontal loads. The governing failure mode is frequently buckling under axial compression. The internal pressure exerted by the stored fluids or solids can significantly enhance the buckling strength, but high internal pressures lead to severe local bending near the base. Local yielding then precipitates an early elastic‐plastic buckling failure. This failure mode, commonly known as “elephant's foot buckling”, has received relatively little attention to date and until recently was often ignored in tank and silo design. This problem is an unusual buckling condition, because it involves very high tensile stresses in one direction, coupled with rather small compressive stresses in the orthogonal direction. Thus, although it is a buckling failure involving considerable plasticity, it occurs at low buckling stresses and under conditions that appear to be classically “slender”. The normal concatenation of “slender” with “elastic” in buckling formulations does not apply at all here. This paper describes alternative approaches to the formulation of design rules for the elastic‐plastic instability and collapse of axially‐loaded internally‐pressurised thin cylindrical shells adjacent to the base support. The differences between the different approaches arise from different conceptual models for the manner in which an elastic‐plastic slender structure instability should be treated.     

谐波沉降下开顶罐的屈曲分析

研究有关谐波沉降下开顶罐的屈曲性能。首先,研究各种谐波数下开顶罐的极限谐波沉降和屈曲性能。结果显示受小波数影响的上壳体发生屈曲,而受大波数影响时则在壳体的其他地方发生屈曲。随着谐波数的增加,在壳体底部也出现越来越多的屈曲点。此外,对开顶罐来说,受小波数影响时其极限谐波沉降大大降低,而受大波数影响时其极限谐波沉降降低极少。对开顶罐的屈曲性能进行数值研究,考察高径比(h/r)和径厚比(r/t)的影响。可知:在某种特定的谐波数下,极限谐波沉降与高径比(h/r)为单调递减的关系。同样的,随着谐波数的增加,极限谐波沉降的降低变化越来越小。而在某种特定的谐波数下,极限谐波沉降与径厚比(r/t)也为单调递减的关系。此外,随着谐波数的增加,极限谐波沉降的降低变化也越来越小。最后,将开顶罐和锥形顶罐的屈曲结果进行比较。结果显示,受小波数影响的开顶罐比不受锥形顶约束的锥形顶罐更能承受大的谐波沉降。然而,受大波数影响时,开顶罐与锥形顶罐的沉降持续能力有所不同。     

Buckling strength of the cylindrical shell and tank subjected to axially compressive loads

This paper aims to develop practical design equations and charts estimating the buckling strength of the cylindrical shell and tank subjected to axially compressive loads. Both geometrically perfect and imperfect shells and tanks are studied. Numerical analysis is used to evaluate buckling strength. The modeling method, appropriate element type and necessary number of elements to use in numerical analysis are recommended. According to the results of the parametric study of the perfect shell, the buckling strength decreases significantly as the diameter-to-thickness ratio increases, while it decreases slightly as the height-to-diameter ratio increases. These results are different from those in the case of columns. The buckling strength of the perfect tank placed on an extremely soft foundation and a stiff foundation increases by up to 1.6% and 5.6%, respectively, compared with that of the perfect shell. The buckling strength of the shell and tank decreases significantly as the amplitude of initial geometric imperfection increases. Convenient and sufficiently accurate design equations and charts used for estimating buckling strength are provided.     

Numerical and experimental investigations of the response of aluminum cylinders with a cutout subject to axial compression

Understanding how a cutout influences the load bearing capacity and buckling behavior of a cylindrical shell is critical in the design of structural components used in automobiles, aircrafts, and marine applications. Numerical simulation and analysis of moderately thick and thin unstiffened aluminum cylindrical shells (D/t=45, 450 and L/D=2, 5, 10), having a square cutout, subjected to axial compression were systematically carried out in this paper. The investigation examined the influence of the cutout size, cutout location, and the shell aspect ratio (L/D) on the prebuckling, buckling, and postbuckling responses of the cylindrical shells.An experimental investigation on the moderately thick-walled shells was also carried out. A good correlation was observed between the results obtained from the finite element simulation and the experiments. Furthermore, empirical equations, in the form of a ‘buckling load reduction factor’ were developed using the least square regression method. These simple equations could be used to predict the buckling capacities of several specific types of cylindrical shells with a cutout.     

Elastic buckling of horizontal barrelled shells filled with liquid—Numerical analysis

In the paper an alternative for horizontal cylindrical tank is presented. The proposed solution has the form of a barrelled tank in which the classical cylindrical shell is replaced by the barrelled one. The geometry of this shape is described. The buckling behaviour of the horizontal barrelled shell filled with liquid is analysed. The liquid level and the negative internal pressure which appear during emptying of the tank are also taken into consideration. On the basis of numerical analyses number of plots as well as analytical formulae are elaborated as a tool for design of barrelled tank with respect of stability criterion. Analyses are made on three families of barrelled shells of different capacities. In each family the capacity and length are constant. Advantages of barrelled tanks in comparison with cylindrical one are discussed.     

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.     

Experiments on dented cylindrical shells under peripheral pressure

In spite of numerous papers in the literature on the buckling behavior of cylindrical shell structures, the effect of local large imperfections caused by physical contacts has not been exhaustively examined yet. To this end, this paper reports on an experimental program on the buckling and post-buckling response of thin cylindrical shells with local dent imperfections under uniform external pressure. The results of this study can be used in practical structures with similar geometric features, i.e. D/t ratio.     

Critical sizes of ground and underground horizontal cylindrical tanks

The paper deals with ground and underground horizontal cylindrical tanks supported at both ends. The ground tanks are loaded with internal hydrostatic pressure and small negative pressure. On the other hand, the underground tanks are located in water containing soil and loaded with external hydrostatic pressure. Critical states of both structures are determined based on solving the equation of stability of cylindrical shell. The problem so defined has been converted to calculation of critical thicknesses of walls for the family of circular cylindrical tanks of different capacities. Critical sizes of the structures have been determined as functions of dimensionless critical thickness and dimensionless tank length.     

Tragverhalten und Bemessung windbelasteter Kreiszylinderschalen mit diskreter Verankerung

Structural behaviour and design of wind‐loaded cylindrical shells with discrete anchorage. Stubby cylindrical shells are often used as storage tanks. Because of their large diameters, in general they are not anchored continuously, but discretely. Nevertheless, the previous investigations of the stress and strain state of the shell were done almost exclusively by assuming a continuous anchorage. In the contribution, it is analyzed for the critical load case wind on the emptied tank, how the internal forces and the buckling strength verification are changed, if the discrete anchorage is realistically taken into consideration at the tensile zone of the base cross section. The differences are quantified. From it, it can be concluded, that a continuously anchorage may be used in the mechanical model, if the interaction buckling strength verification of the stress design is only exploited up to 90%.     

Parametric study on buckling behaviour of dented short carbon steel cylindrical shell subjected to uniform axial compression

Generally, thin cylindrical shells are susceptible for geometrical imperfections like non-circularity, non-cylindricity, dents, swellings, etc. All these geometrical imperfections decrease the static buckling strength of thin cylindrical shells, but in this paper only effect of a dent on strength of a short (Lc/Rc∼1, Rc/t=117, 175, 280) cylindrical shell is considered for analysis. The dent is modeled on the FE surface of perfect cylindrical shell for different angles of inclination and sizes at half the height of cylindrical shell. The cylindrical shells with a dent are analyzed using non-linear static buckling analysis. From the results it is found that in case of shorter dents, size and angle of inclination of dents do not have much effect on static buckling strength of thin cylindrical shells, whereas in the case of long dents, size and angle of inclination of dents have significant effect. But both short and long dents reduce the static buckling strength drastically. It is also found that the reduction in buckling strength of thin cylindrical shell with a dent of same size and orientation increases with increase in shell thickness.     

Challenges in the computation of lower-bound buckling loads for tanks under wind pressures

This paper reports on the implementation of a lower-bound approach for the buckling of imperfection-sensitive shells using general purpose finite element codes. The stability of cylindrical steel tanks under wind pressure is evaluated for two tank configurations: conical roof tanks and open top tanks. For both tank configurations, several geometric relations are considered in order to find the variation of the knock-down factor as the geometry changes. The reduced energy method is implemented to compute a lower-bound for critical wind pressures and the results are compared with the static non-linear analysis carried out on the same models. An alternative way to implement the reduced energy method is presented to improve the results obtained with the proposed methodology.     

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.     

Elastic-plastic collapse of non-uniform cylindrical shells subjected to uniform external pressure

The objective of this paper is to derive analytical solutions for the elastic buckling and plastic collapse pressures of a cylindrical shell with reduced thickness over part of its circumference. The section of reduced thickness is used to represent a corroded region in a pipe. The proposed solutions are extensions of Timoshenko's solutions for the elastic-plastic collapse of a linear elastic, perfectly plastic cylindrical shell subjected to uniform external pressure. A modified interaction formula for the fully plastic membrane forces and bending moments in the non-uniform cylinder has been proposed for plastic collapse. A parametric study shows that the elastic buckling pressure decreases smoothly with corrosion angle when the corrosion depth is less than 0.5t. When the corrosion depth is greater than 0.5t, the elastic buckling strength first decreases very rapidly with corrosion angle. Furthermore, the elastic buckling pressure decreases uniformly with corrosion depth when the corrosion angles are greater than 30°, while the elastic buckling strength decreases more rapidly at higher corrosion depths when corrosion angles are less than 30°. Another parametric study on a steel pipe shows that the initial and fully plastic yield pressures both decrease monotonically with corrosion depth for a given corrosion angle and imperfection.