Seismic behavior of a lowrise horizontal cylindrical tank
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Abstract
Cylindrical storage tanks are widely used for various types of liquids, including hazardous contents, thus requiring suitable and careful design for seismic actions. The study herein presented deals with the dynamic analysis of a groundbased horizontal cylindrical tank containing butane and with its safety verification. The analyses are based on a detailed finite element (FE) model; a simplified onedegreeoffreedom idealization is also set up and used for verification of the FE results. Particular attention is paid to sloshing and asynchronous seismic input effects. Sloshing effects are investigated according to the current literature state of the art. An efficient methodology based on an “impulsiveconvective” decomposition of the containerfluid motion is adopted for the calculation of the seismic force. The effects of asynchronous ground motion are studied by suitable pseudostatic analyses. Comparison between seismic action effects, obtained with and without consideration of sloshing and asynchronous seismic input, shows a rather important influence of these conditions on the final results.
Keywords
Cylindrical tank Sloshing Asynchronous seismic action FE modeling Seismic behaviorIntroduction
Seismic loading can induce large damages in industrial facilities and their complex components (Babič and Dolšek 2016; Demartino et al. 2017a, b; Nuti et al. 2009). The loss of the structural integrity of these structures can have severe consequences on the population, the environment and the economy (Krausmann et al. 2010; Fiorentino et al. 2015; Rodrigues et al. 2017). Looking at power/chemical/petrochemical plants, storage tanks containers are widely employed. These hold liquids, compressed gases or mediums used for the short or longterm storage of heat or cold. Liquid storage tanks and piping systems are considered as critical components of those industrial facilities (Vathi et al. 2017; Bakalis et al. 2017).
The seismic response of tanks has been widely studied in the past starting from the pioneering studies of Housner (1957, 1963). In particular, Housner (1957) first presented the simplified formulae to compute the dynamic pressures developed on accelerated liquid containers and successively (Housner 1963) studied the dynamic behavior of groundsupported elevated water tanks considering equivalent spring–mass systems. Current practice for the seismic design of storage tanks is mainly based on Appendix E of API 650 (2007) standard and on Eurocode 8 (1998). Generally speaking, there are many different types of equipment used for the storage of liquids and gases. The characteristics of the different tanks adopted mainly depend on: (a) the quantity of fluid being stored, (b) the nature of the fluid, (c) the physical state of the fluid and (d) the temperature and pressure. In industrial plants, gases are usually stored under highpressure, often in liquid form since the volume is largely reduced. Within this framework, groundbased horizontal cylindrical tanks resting upon two supports are used mainly for storage of various liquids. The capacity of such tanks considerably exceeds those of horizontal tanks designed for land transportation. Under the conditions of normal exploitation, such tanks are loaded mainly with internal pressure being the sum of hydrostatic pressure and the uniform pressure caused by vapor of the medium contained therein (Magnucki et al. 2004).
For a cylindrical pressure vessel, there are two possible failure modes. One is the maximum stresses reaching the yield condition and then yielding zone spread leading to final plastic collapse and the other is elastic or elastic–plastic buckling leading to collapse. For the first problem it is mainly a stress analysis while for the second problem it is a stability analysis. In particular this instability appears usually in two forms: the elephant foot buckling and the diamond buckling (Niwa and Clough 1982; Haroun and Bhatia 1994; Hamdan 2000). The first form, which is an outward bulge located just above the tank base, results from the combined action of vertical compressive stresses, exceeding the critical stress, and hoop tension close to the yield limit. The second form is an elastic instability phenomenon due to the presence of high axial compressive stresses.
For sake of brevity, this study exclusively deals with stress analysis, while stability analysis is not documented.
More in detail, the present paper analyses the seismic performances of a groundbased horizontal cylindrical tank containing “butane”, focusing on the two topics of sloshing (1) and asynchronous seismic input (2).
As to the first aspect (1), it is worth noting that the seismic analysis of cylindrical storage tanks requires accounting for the fluid–structure interaction. This phenomenon, referred to as “liquid sloshing,” is generated by the presence of a free surface allowing for fluid motions and is generally caused by external tank excitation, significantly affecting in many cases the dynamic response (Hamdan 2000; Patkas and Karamanos 2007).
As to the second aspect (2), asynchronous motion denotes the differences in amplitude, phase and frequency content among ground motions recorded over extended areas (Nuti and Vanzi 2005; Lavorato et al. 2017). This spatiotemporal variation of ground motion is mainly attributed to (Zerva 2009; Koufoudi et al. 2018): (a) difference in arrival times of seismic waves at different locations; (b) loss of coherence of seismic waves (i.e. gradual reduction of its statistical dependence on distance and frequency), due to multiple reflection and refraction as they propagate through the highly inhomogeneous soil medium; (c) ground motion attenuation; (d) impact of local site effects. According to the current Italian and European technical codes (M.I.T 2008, Eurocode 8), if foundations are not properly interconnected with sufficiently rigid elements, asynchronous motion has to be accounted for.
The analysis herein presented comprises a sophisticated numerical FE modeling as well as a simplified model for the estimation of the dynamic properties of the tank structure. The paper is organized as follows: First, the steel cylindrical pressure vessel containing butane adopted for the case study is presented (“Casestudy” section). “Sloshing” and “Asynchronous seismic input and resulting structural demand” sections describe the mathematical model adopted for accounting for the sloshing and the asynchronous seismic input, while “Verification of finite element modeling” section focuses on the fundamental period of the structure. Results of the analyses are given in “Stress analysis” section; finally, conclusions are given in “Conclusions” section.
Casestudy
The research focuses on an existing groundbased horizontal cylindrical steel vessel resting upon two r.c. supports through steel saddles and containing pressurized butane (density ρ_{L} = 603 kg/m^{3}). The cylinder is 14.66 m long, with external diameter and thickness equal to 4 m and 12 mm, respectively. The lateral sides of the cylinder are constituted by curvilinear surfaces; a reinforcing steel ring can be observed at the middle of the cylinder (Fig. 2).
Geometrical properties of the cylindrical vessel
External diameter of the cylinder, m  4 
Thickness of the cylinder walls, m  0.012 
Height of the r.c. supports, m  3.35 
Width of the steel saddles, m  3.45 
Thickness of the steel saddles, m  0.02 
Mechanical properties of the cylindrical vessel
Design strength of cylinder steel (f_{d}) (MPa)  360 
Design strength of supporting saddles steel (f_{d}) (MPa)  230 
Design strength of r.c. supports (f_{d}) (MPa)  15 
By observing Table 2, it can be noted that a rather low design strength (15 MPa) is attributed to r.c. supports. This choice was made for sake of safety since, in the absence of suitable test survey on the tank under examination, test results on adjacent vessels had highlighted low quality concrete.

nominal expected life of the structure: V_{n} = 50 years;

utilization coefficient of the structure: 4th Class (C_{u}= 2);

reference period for the seismic action: V_{R} = 100 years;

behavior factor: q = 1.
Seismic zone is identified by the following characteristics: ground type: C; soil type T_{1} (S = 1.5).

design ground acceleration for the nocollapse requirement (ultimate limit state): a_{g} = 0.053 g;

maximum amplification factor of the acceleration response spectrum: F_{0} = 2.571;

upper period of the constant acceleration branch of the response spectrum: T _{C} ^{*} = 0.512 s.
The above values are representative of low seismicity areas in Italy (Vanzi et al. 2015; Fiorentino et al. 2018).
Sloshing
Seismic design provisions of liquidstorage tanks such as API 650 (2007) and Eurocode 8 (1998) are based on a mechanical springmass analogy initially developed by Graham and Rodriguez (1952), Jacobsen (1949) and Housner (1963) for rigid tanks and by Haroun and Housner (1982) for flexible tanks.
According to this analogy, a tank subjected to a seismic motion may be reduced to a simpler model with lumped masses and springs. More precisely a portion of the mass of the liquid content (M_{I}) is considered as rigidly connected to the tank walls while the remaining portion (M_{C}) is flexibly attached to the tank walls. The liquid (with mass M_{I}) that synchronizes with the vibration of the tank is called impulsive while the sloshing component of the fluid (with mass M_{C}), generating free surface waves and characterized by its own frequency of vibration, is referred to as convective component.
In Fig. 1, y_{2} = X(t) represents the motion of the external source while y_{1} = u_{1}(t) expresses the motion of the liquid mass associated to sloshing.
The total mass M_{T} is split into two parts m_{1} and m_{2}, corresponding to y_{1} and y_{2} and expressing the “convective” or “sloshing” motion (M_{1C}) and “impulsive” motion (M_{I}), respectively.
Dynamic properties of the cylindrical tank and seismic forces
M_{L} [ton]  M_{tank} [ton]  M_{1C} [ton]  M_{I} [ton]  T_{1C} [s]  T_{I} [s]  S_{A}(T_{1C}) [m/s^{2}]  S_{A}(T_{I}) [m/s^{2}]  F_{d} [kN]  F_{C,max} [kN]  F_{I,max} [kN] 

108  29.14  25.134  112  1.73  0.3  0.61  1.65  185.8  15.332  185.24 
Stiffness parameters of the SDOF model
M_{eq} (ton)  k_{st} (N/m)  E [N/m^{2}]  J [m^{4}]  H [m] 

135.63  4.15 × 10^{7}  30 × 10^{9}  0.038  4.35 
Pseudostatic analyses: maximum VonMises stresses
σ_{eff} [MPa]  

Soil differential displ. in x dir.  Soil differential displ. in y dir.  Soil differential displ. in x and y dir.  
Cylinder walls  Saddles  Cylinder walls  Saddles  Cylinder walls  Saddles 
52.93  53.67  45.23  31.24  55.18  58.77 
Application to the case study
On the basis of the above considerations, the seismic analysis of the cylindrical tank object of study was carried out under the most unfavorable hypothesis of maximum seismic force, that is with the cylinder filled with butane up to the “block level” (i.e. the maximum allowable liquid level in the tank for safety reasons) equal to 80% in height. The 80% filling height corresponds to the 85% filling volume. Table 3 shows the deriving values of the involved parameters.
From Table 3, by comparing the values of F_{D}, F_{C,max} and F_{I,max}, it can be deduced that the convective component of the fluid motion is negligible. So dynamic spectral analyses were carried out by modeling the liquid mass through its impulsive component only. In this way, an accuracy higher than 99% was obtained.
Verification of finite element modeling
 1.
a detailed finite element (FE) model;
 2.
a simplified methodology based on a SDOF analogy.
The structure was assumed perfectly constrained at the basis.
Assuming α = 3, a fundamental period in the longitudinal direction equal to 0.254 s was computed, in perfect accordance with the value obtained through the FE model. The values of the various quantities involved in the analysis are summarized in Table 4.
Asynchronous seismic input and resulting structural demand
According to the current Italian code (M.I.T 2008, section 7.2.5.1) and to Eurocode 8, if foundations are not properly interconnected with sufficiently rigid elements, the effects of relative horizontal displacements at the basis of the superstructure should be analyzed in order to account for possible asynchronous actions.
The cylindrical vessel under examination belongs to this typology and thus the effects of nonsynchronism were considered by introducing suitable soil differential displacements in the x and y directions at the foundation base. The differential displacements in the two directions were calculated according to section 3.2.5.2 of Italian code, excerpted from Nuti and Vanzi (2005), so obtaining in both directions d_{ij}(x) = 7 mm.
Stress analysis

dead load (G_{1});

internal pressure (P_{i} = 6 bar);

hydrostatic pressure (P_{H}, due to butane);

seismic spectral loads in all three directions.
Stress verifications (VonMises stresses; maximum values compared with strength values)
Element  σ _{eff}  f _{d}  Ratio  Verification 

MPa  MPa  β  
Cylinder lateral walls  134.3  360  2.68  Satisfied 
Cylinder vertical walls  22.8  230  10.09  Satisfied 
Supporting saddles  89.4  230  2.57  Satisfied 
Stress verifications in selected FE nodes (in terms of VonMises stresses)
Nodes (from Fig. 8)  1  2  3  4  7  8  9  10  11  12  13 

σ_{eff} (MPa)  134.3  118.9  105.5  93.4  22.8  20.7  22.6  89.4  78.4  37.5  11.7 
f_{d} (MPa)  360  360  360  360  230  230  230  230  230  230  230 
Ratio β  2.68  3.03  3.41  3.85  10.08  11.11  10.17  2.57  2.93  6.13  19.66 
As it emerges from Tables 6 to 7, in all elements maximum stresses are lower than the corresponding design limit strengths, that is the analyzed cylindrical tank has a good level of safety against seismic action. Verifications on RC supports are not documented due to the absence of experimental tests on steel reinforcement details. However, it can be argued that they behave like perfectly constrained shelves, mainly stressed in the x direction, and at most the low seismic forces acting on them can lead to the early formation of plastic hinges at the basis, with low ductility involvement (Fiore et al. 2016; Imperatore et al. 2012; Lavorato and Nuti 2010; Lavorato et al. 2015; Zhou et al. 2015).
Stress verifications including the effects of asynchronous ground motion (in terms of VonMises stresses)
Element  σ _{eff}  f _{d}  Ratio  Verification 

MPa  MPa  β  
Cylinder walls  148.4  360  2.43  Satisfied 
Supporting saddles  103.4  230  2.22  Satisfied 
It is worth to note that, when liquid storage tanks are founded on piles, in some cases soil–structure interaction could be beneficial, leading to a higher value of natural period and to reduced seismic spectral forces (Fiore et al. 2018). This effect could so diminish the stress increments due to asynchronous seismic motion.
Conclusions
In this study, the seismic behavior of a cylindrical pressure vessel containing butane was analyzed, accounting for the influence of sloshing effects and of asynchronous actions. Both a detailed FE model of the horizontal cylindrical tank and a simplified SDOF model were implemented. It was shown which are the most unfavorable load conditions to be considered under sloshing and asynchronous input effects: (1) seismic action with liquid in the sphere up to the “block level”; (2) nonsynchronism of ground motion in both x and y directions.
The high stress level associated with asynchronous seismic action was actually unexpected, given the low seismic design action. This was due to structural peculiarities, in which a very stiff and resistant system (the r.c. column) is connected to the tank via a welded steel saddle, which must accommodate, on a short length, the largest part of differential displacements. However, the possible steel yielding for higher seismic actions would not be a problematic issue for nonpressurized vessels; pressurized vessels, though, require larger reliability margins and simultaneous application of capacity design concepts, in order to satisfy structural safety when verified under large (larger than this paper design earthquake) seismic actions.
For the case of pressurized vessels, this is a reminder to check, even for low seismic action, for structural peculiarities (e.g. constraint behavior; coupling of very and moderately stiff/resistant components), which may become the weakest link for the whole structure, and, once identified, may quite often be structurally upgraded with modest efforts.
Notes
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