Sensitivity analysis of the stress state in shell courses of welded tanks for oil storage

AbstractThe paper deals with the analysis of reliability and safety of a welded tank for the storage of oil, which is located in the Czech Republic. The oil tank has a capacity of 125 thousand cubic meters. It is one of the largest tanks of its kind in the world. Safety is ensured by a steel outer intercepting shell and a double bottom. The tank was modelled in the programme ANSYS. The computational model was developed using the finite element method – elements SHELL181. A nonlinear contact problem was analysed for the simulation of the interaction between the bottom plate and foundation. The normative approach in design and check of tanks according to standards API 650, CSN EN 14015, EEMUA 159 and API 653 is mentioned. The dominant loading of the filled tank is from oil. The normative solution is based on the shell theory, which considers constant wall thickness. For real tanks sheet thicknesses of individual courses increase with increasing depth. Stochastic sensitivity analysis was used to study the ef...


Introduction
Czech Republic has crude oil reserves for 94 days. The EU law requiring reserves for 90 days is thus met. Crude oil is brought into the Czech Republic via the Druzhba pipeline from Russia and the IKL pipeline from Bavarian Vohburg near Ingolstadt (where it is transported via the TAL pipeline from the region of the Caspian Sea, the Middle East and North Africa). Given the current situation in Ukraine and Crimea oil supply to the Czech Republic is monitored daily and the current state of reserves is evaluated.
Storage of the oil is in large cylindrical tanks, which were constructed not far from the village Nelahozeves. The tank illustrated in Figure 1 has a height of 27 m and a capacity of 125 thousand cubic meters. It is one of the largest storage tanks of its kind in the world.
There is a floating roof on the surface of the stored liquid (crude oil), which prevents evaporation of the oil. The floating pontoon roof moves along with the oil level in the tank. In a drained tank the floating roof rests on supports on the bottom of the tank, see Figure 2. A full tank contains crude oil worth 57 million Euros. The actual tank type is acc. to international standards and the structural design of some of its parts is based on calculations carried out in the Czech Republic. Safety is ensured by its double steel coversteel outer intercepting shell and double bottom, see Figure 3. The steel outer intercepting shell is a safety mechanism intended to prevent the leakage of oil in case of damage to the inner shell (tank shell). The structural design consisting of the shell of steel outer intercepting shell and steel tank allows the erection of tanks closer to each other leading to savings on otherwise needed space for containment dikes.
Fire is one of the biggest operation safety risks of oil storage tanks. In the event of fire the tanks are extinguished with foam from the fire protection system, which is part of the tank. Fire is most probable to occur between the roof and the tank, which is fitted with special rubber double seal.
Oil is a strategically important raw material and thus increased attention should be paid to its transportation and storage. The dominant load of a cylindrical tank is the loading of its inner shell by hydrostatic pressure of the oil. Plate thicknesses of individual courses increase from the top downwards. The plate of the bottom shell course has the greatest thickness because the hydrostatic pressure is highest there. Plate thicknesses of individual courses are designed acc. to standards API 650 or ČSN EN 14015, standards EE-MUA 159 and API 653 are for inspection (service). The design criteria of the mentioned standards are based on the allowable stress design concept.

Allowable stress method
The principal factor, which determines the thickness of the tank shell, is the internal loading caused by the head of liquid. The minimum acceptable thickness for welded tank shells may be calculated from the following formula listed in standard EEMUA 159 for the basic control of tank shells: where: t min is the minimum acceptable thickness in mm from the above equation (1), which should not be less that 2.5 mm or 50% of the original shell plate thickness, whichever is greater. This applies to inspections. The thickness is limited to 40 mm in design (reason for the introduction of maximum thickness is fear of brittle fracture for larger thicknesses and acc. to the tank diameter from 5 to 12 mm (minimum thickness for stability); D is the nominal diameter [m]; H is the height from the point under consideration to the maximum filling height [m], see Figure 2; W specify gravity of the content; P design vapour pressure in mbar; S maximum allowable stress in MPa; E original joint efficiency for the tank; E = 0.85 if original E is unknown; E = 1.0 when evaluating the retirement thickness in a corroded plate, when away from welds or joints by at least the greater of 25 mm or twice the plate thickness. A similar formula is given in standards API 650, ČSN EN 14015 and API 653. The design and control formulae in the mentioned standards are based on the shell theory, which considers constant wall thickness. Therefore the stress in (1) is evaluated 0.3 m from the bottom edge of the respective course at the point where the shell theory can be applied. This method of calculation is known as the "one foot" method (Long, Garner 2004).

Analysis of the stress state using the ANSYS program
The stress and deformation of a fully filled tank, which is shown in Figure 1, was analysed. The tank has nine courses. 17 mm, 12 mm, 11 mm, 10 mm. The tank is made of steel grade S355 apart from the top two courses, which are made of steel S235. The density of oil is considered as 880 kgm -3 .
The computational model was developed in the ANSYS programme. Four models A, B, C and D were developed. The model is located in a cylindrical coordinate system, where axis Z is the axial direction (cylinder axis -Z coincides with the axis of the tank shell), axis X is the radial direction and axis Y is the circumferential direction. A sector of part of the tank of angle 1° was modelled. Meshing was performed using elements SHELL181. Models A and B (Figs 4 to 6) were modelled from part of the tank bottom. Placement on the foundations was simulated as a contact problem. For models C and D (Figs 7 to 9) connection of the shell to the bottom of the tank was substituted by boundary conditions (fixed end for C and rotation about the circumferential axis was allowed for D). Models A, C and D have their shell plates aligned to the centreline. Model B has shell plates aligned to the inner side of the plate. Stress distribution along the height of the tank was studied: SEQV is the reduced stress acc. to von Mises, SINT is the reduced stress acc. to Tresca (τ max ), SY is the circumferential stress.

Stochastic sensitivity analysis
In further studies the effects of the variability of the plate thickness and Young's modulus of the course on stress along the height of the tank will be analysed. The effect of loading due to hydrostatic pressure on the walls of the full tank on Mises stress is studied. In order to study the effect of the stiffness of one course on the stress in adjacent courses, the thicknesses of the courses and Young's modulus were considered as random quantities, see Table 1. Other input material and geometric characteristics are considered with values from the previous chapter.
Input random quantities were considered acc. to results of experimental research (Melcher et al. 2004;Kala et al. 2009;Soares 1988). It may be noted that the data of experimental research has been the source for a number of reliability studies, see e.g. (Kala 2012;Gottvald, Kala 2012). Analysis was performed using 100 simulation runs of Latin Hypercube Sampling method (LHS), which is a method of type Monte Carlo (McKey et al. 1979;Iman, Conover 1980). It was observed that the random variability of Young's modulus has a minimum effect on the stress in the tank. The effect of the random variability of the thickness of the plate of the i th course on the stress along the height of the tank is depicted in Figure 12. A negative correlation means that with increasing plate thickness the stress decreases at the point at which the correlation coefficient is plotted, see Figure 12. Results for the top course are irrelevant because they are located where the stress is zero; therefore they are not shown in Figure 12. It may be added that correlation indicates dependence, but the contrary is not true. Sensitivity analysis based on correlation requires that the output is monotonically dependent on each input variable. This is a problem of models of the so-called "black box" type where using simple statistical analysis of the LHS method cannot detect cases of non-correlation or "false insensitivity", and which need additional studies confirming the monotonic dependence of the output on the input. Correlation as an indicator of sensitivity can be used in transparent models, which is the case of the presented study or studies for e.g. ).

Conclusions
From the analysis of the stress state along the height of the tank it was found that the reduced stress acc. to von Mises (SEQV), reduced stress acc. to Tresca (SINT) and circumferential stress SY have approximately the same course, see Figure 10. Stress SY has the relatively highest values and displays the relatively most erratic behaviour. These results were evaluated for model A, for which part of the bottom was also modelled. Comparison of stress SEQV of all four models showed that the effect of the boundary conditions of the stiffness of the bottom is not significant with the exception of the first course, where the difference was up to 100 MPa. It is debatable if modelling the joint of the wall-bottom using a hinge is sufficient for the analysis of the stress of the first course.
Sensitivity analysis based on the evaluation of the correlation between the random plate thicknesses and stress SEQV showed that the variability of the thickness of one course affects the stress in adjacent courses, see Figure 12. This effect is more significant in the bottom courses. The load is partially transferred to the Fig. 12. Correlation between thickness t i and von Mises stress state SEQV for model A adjacent courses due to differences in the stiffness of adjacent courses. The stiffness of adjacent courses has the greatest influence on stress in the bottom courses. The influence zone is higher for the bottom courses than the upper courses. Equation (1), which is the basis of reliability assessment of standards API 650 and ČSN EN 14015 is based on the assumption that the stiffness of one course has no influence on the stress in adjacent courses. This assumption is only partially true for real tanks.
The obtained results of the stochastic sensitivity analysis open the debate on the detailed modelling of tanks using the finite element method. The AN-SYS programme offers a wide range of shell finite elements, which are suitable for the analysis of stress of thin steel plates. Generally, the more detailed the applied computational model, the more reliable are the conclusions, which are obtained through the analysis of stress. It is always necessary to be aware of conventional and practically verified methods used for design and assessment, which are often the basis of reliable standardized procedures.