Structural behaviour of single-deck floating roof tanks subject to seismic loading

During a seismic event, fluid storage tanks are subject to sloshing phenomena; their effect on the structural components of the tanks is one of the main aspects to be considered during the design. The movement in successive modal forms of the roof-fluid system determined by the earthquake can cause severe damage to both the containment shell and the floating roof. The fluid-structure interaction imposes extremely complex out-of-plane deformation fields, mainly in the presence of floating roofs with lower flexible resistance. Such deformations can cause severe structural damage and subsequent catastrophic events such as those experienced in the past in Japan and Turkey. In fact, recently recorded earthquakes in Japan and Turkey, characterized by oscillation periods between 3 and 8 sec (long-period ground motion), compatible with the natural first and second slosh frequencies of the fluid-structure system, led to conspicuous problems of instability failure with the consequent sinking of the floating roof. These events dictated a new starting point for the research into the seismic behaviour of roof-fluid-tank systems.


Introduction
The sloshing is one of the main issue when design liquid storage tanks [1], [2].In the case of fixed roof tanks, the sloshing of free surface during an earthquake induces an extensive hydrodynamic force on the walls and roof of the tank [3].On the other hand, the liquid storage tanks are generally covered with a FR (floating roof) to suppress oil vapor leakage from the free surface, the seismic loads could induce a large displacement of the FR due to the liquid sloshing.This FR motion possibly leads to severe structural damage and catastrophic fire accidents as experienced in Japan and Turkey during past earthquakes.
During the Tokachi-oki earthquake in 2003, several single-deck floating roofs (SDFR) located at Tomakomai were seriously damaged [4], the exposure of fuel oils to the atmosphere caused a large number of fires in the area.Other catastrophic events that brought greater attention to this phenomenon were found during the 1999 Kocaeli earthquake and the 1964 Niigata earthquake [5] in which the sinking of the roof led to the ignition of the hydrocarbons contained inside the tank.
After these catastrophes, the problem of sloshing has been studied in much more detail.The main theories have been formulated in Japan in the last two decades.To obtain an analytical solution for the calculation of stress states in floating roofs during a seismic event, a linear solution based on the velocity potential theory, assuming the fluid is incompressible and inviscid, was established in 1970s [6].Starting from this, Matsui presented the most complete solution for determining the displacement of a FR in a cylindrical storage tank subjected to seismic excitations.He proposed the explicit solutions for both single and double-deck type FRs, the double-deck FR was approximated by a uniform isotropic plate, and an SDFR was replaced by an inner deck with small bending stiffness and an outer pontoon with relatively large stiffness [7].
The developed analytical solutions do not consider the nonlinear effects of the free surface motion.To consider this effect, Yamauchi and Kamei used numerical methods to evaluate the nonlinearity of deformation during the sloshing phenomenon [8].
Starting from Matsui's analytical solution and considering nonlinearity effects, Yoshida et al. used axisymmetric finite element analysis to consider the hydrodynamic coupling of the fluid and a FR [9].
In this work an analytical procedure for stress evaluation in a specific floating roof single deck (SDFR) is presented.This method studies the relationship between vertical deck plate deformation and radial shrinkage.A specific application case will be described.Moreover, a mixed analytical/numerical procedure for the evaluation of the stress state in floating roofs of the Honeycomb type (HC) is also presented.First, the homogeneous mechanical characteristics of the HC panels are defined; in parallel, the out-of-plane displacement field is determined.The definition of the method is finalized with the analysis of a specific application case.
It was decided to consider only these two types of floating roofs (FR) for their very similar structural behavior.Both, in fact, due to their construction geometry impose extremely low stiffnesses.This leads to the simplifying assumption that the roof will not decisively influence the dynamic response of the system [8].
One of the most critical earthquakes for the sloshing phenomenon is the Tokachi-oki earthquake recorded in 2003 [8] Figure 1.Therefore, this specific case study is analyzed for the determination of the stress state in SD-type roofs.For the HC roof, a smaller 10 Kl tank was chosen as a case study.The earthquake considered for the analysis is the catastrophic earthquake Izmit of Mw 7.6 [10].Its accelerogram is shown in Figure 2.

Single-Deck roofs
Single-deck roofs (SD) have been evaluated via a fully analytical procedure in MATLAB.The main hypothesis, which is grounded in the proposed procedure, is to consider the structural stiffness of the roof small enough to have no influence on the natural frequencies and characteristic vibrational modes of the tank-fluid-roof system (Matsui [7]).The data available are spatial geometry of the roof, height of the free surface, density of the contained liquid and characteristic geometric parameters of the external pontoon.
Counting for the first two natural frequencies as the main cause of the global stress state of the system, a first computation of the angular frequencies of these two modes has been conducted: Under the assumption of non-viscous, incompressible, and irrotational fluid, the displacement of the free surface is entirely defined by a velocity potential function.The potential function ϕ which satisfies the Laplace equation, the wall condition, and the condition at the tank bottom, is: where x i (t) is the associated displacement of the i-th mode, obtained by solving the differential equation of the SDOF forced system.Once the wave profile for a given accelerogram was defined, the effects of nonlinearity were considered following the theory described by Goudarzi in [5].These effects, particularly present in conditions of high wave widths, decisively change the sloshing profile for the first mode (Figure 3).
To evaluate the stress state in the outer pontoon it has been hypothesized that this can be assimilated to a curved beam on an elastic foundation [8].The maximum circumferential inflectional moment of the pontoon in θ = 0 is: Where β n is the ratio between the stiffness of the pontoon and the elastic constant of the hypothetical foundation system.As a result of the second sloshing mode, the internal deck plate of the floating roof deforms considerably out of plan.The membrane force in the radial direction causes a radial contraction and a geometric ovalization in the pontoon [11].The pontoon considered as a beam on elastic foundation shows a deformation that can be seen as a superposition of both the ovalization and the radial contraction.In analogy with the approach proposed in [8], superimposing these two effects we can compute the bending moment and the normal stress associated with the second mode: Figure (4.a)shows these forces acting on the cross section of the pontoon [12].Once the forces are known, the stresses are acquired (the maximum tension is obtained through the SRSS method).

Case Study (SD)
The chosen application case is a tank with a diameter size of 78m and a capacity of 100kl, the fluid height is 15m with a density of 840 kg m 3 .The floating roof pontoon has the geometric characteristics shown in Figure (4.b)[12].
To determine the inertial properties of the closed section, a mirror model was created in a software CAD, and the results obtained are shown in Table 1.
Table 1: Inertia properties of the pontoon.

Geometry properties Value
Having defined the geometric properties and characterised the inertial properties of the pontoon, the differetial equation was solved for the two modes individually.In Figures 5 and  6 it is possible to evaluate the time course of the wave height for the two modes at the point r = R and θ = 0.

W 1st mode [m] W 2nd Mode[m]
Analytical procedure 1.09 0.66 Given the maximum displacements, for the two main modes of fluid oscillation, following the previously reported relationships, the seismic forces acting on the cross-section of the pontoon were defined.Table 3 shows the results for the case study analysed.Table 3: Seismic forces in the cross-section of the pontoon.

Seismic Force
After calculating these forces, the seismic stresses were calculated.The stresses from the first and second slosh modes were ultimately combined through the SRSS (square root of the sum of the roots).For a comparison with FEM a refined 3D model was defined (Figure 7) on which a vertical displacement field consistent with the solution of the differential problem was imposed.The vertical displacement field (Figure 8) led to the stress state shown in Figure 9.As can be seen from Table 5, the stress values obtained for this SD floating roof, in this specific seismic excitation state, are perfectly consistent with the FEM simulation.To answer this question and understand how size influences the effects of the first and second slosh modes, a second test case with the same geometrical and physical properties but with a diameter of 1/2 has been analyzed.Table 6 compares the results obtained: a decrease in diameter causes an increase in the effects related to the first mode and a decrease in those related to the second.

HoneyComb roofs
For HoneyComb (HC) roofs simulation with the above technique is not possible due to their complex geometry.In fact, on the entire surface of the roof, there are C-shaped reinforcing partitions and Honeycomb panels.This configuration did not allow for a fully analytical analysis to be performed.For this reason, it was necessary to develop a specific geometrical model, with FEM.First following IOP Publishing doi:10.1088/1757-899X/1306/1/0120029 the theory presented in [13], known as the geometric and physical properties of the HC panel, the mechanical properties in the three directions of an equivalent orthotropic material were computed.Therefore, it was possible to move from a complex geometry in the micro-scale to a macro-scale material with equivalent properties [14].
Having determined the characteristic properties of the homogenized material and the operative limits, the displacement field to be imposed on the roof has been defined.The hypothesis behind the procedure (as in the case of SD) is to consider the structural stiffness of the roof small enough not to clearly influence the natural frequencies and vibration modes characteristic of the tank-fluid-roof system.The presented method follow Matsui's approach [7].
Once the characteristic properties of the homogenized material and the displacement field generated by the seismic event have been defined, A FE Model has been defined.

Case Study (HC)
The application case is a tank with a diameter of 24m and a capacity of 10kl, the fluid height is 13.00m with a density of 740kg/m 3 .The roof is characterized by 40mm high aluminum HC panels, the cell has a regular hexagonal shape with a characteristic size of 15mm and 0.5mm thick walls.It can be seen in Figure (  With the relations presented in [13] the elastic characteristics and the plastic instability limits of the homogenized material were determined.The obtained values are reported in Table 7 e 8.    9 and 10.For the SD, the relationship between the vertical out-of-plane deformation and elliptical deformation of the pontoon is analyzed.A modus operandi is provided for the design of SDFR, evaluating the effects of the first and second sloshing mode on stresses.The following conclusions can be made. (i) The stress caused by the net radial compression of the pontoon is much higher than the force caused by its oval deformation.Furthermore, the compression stress caused by the contraction of deck plate due to the second sloshing mode dominates in pontoon plates.(ii) The second sloshing mode effects are more significant for tanks with storage capacity like that of the case study.Decreasing the storage capacity, the effects related to the first mode will become more important.
For the HC, the procedure studied presents an analytical-numerical development, the punctual characterization of the tension and deformation state required the use of a complete model.The following conclusions can be deduced from the results obtained.
(i) The connecting C-shapes and HC panels deform continuously, showing significant levels of plastic deformation.The zones of maximum plastic deformation coincide with the horizontal tangent zones of the vertical displacement field.(ii) The central core of the Honeycomb panel plasticizes (Figure 15) due to the formation of plastic hinges at the junction points of the base cell septa.Not reaching the PEEQ (Figure 16) value at which the densification process is activated leads to the conclusion that despite the activation of the plastic process, the roof will not suffer macroscopic damage.If the densification phenomenon were to be activated, the macroscopic spatial configuration of the roof would be lost.
The proposed methods can be used for the future design of SDFRs and HCFRs.They are also applicable for the selection of existing SDFRs and HCFRs for which repair is required as well as to determine the maximum filling level of the tanks.

Figure 3 :
Figure 3: Schematic shape of linear and nonlinear free surface profiles.

Figure 5 :
Figure 5: Time course of the wave for first slosh mode in r=R e θ = 0.

Figure 6 :
Figure 6: Time course of the wave for second slosh mode in r = R e θ = 0.
10.a) how the panels are connected to C-shaped aluminum connection elements.

Figure 16 :
Figure 16: Max equivalent plastic deformation for HC core plane -IZMIT earthquake.

Table 4 :
Seismic stress for individual slosh mode and stress Total.

Table 5 :
Comparison of seismic stress obtained from model and FEM.

Table 6 :
Results for tanks with the same input conditions but different diameters.

Table 7 :
Elastic characteristics of the Honeycomb core.

Table 9 :
Max equivalent plastic deformation for beam and ply.

Table 10 :
Max Von Mises stress for beam and ply.