Finite Element Modeling of Variable Membrane Thickness for Field Fabricated Spherical (LNG) Pressure Vessels

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Engineering, 013, 5, 469-474 http://dx.doi.org/10.436/eng.013.55056 ulished Online May 013 (http://www.scirp.

1 Engineering, 013, 5, ulished Online May 013 ( Finite Element Modeling of Variale Memrane hickness for Field Faricated Spherical (LNG) ressure Vessels Oludele deyefa, Oluleke Oluwole Department of Mechanical Engineering, University of Iadan, Iadan, Nigeria Received January 9, 013; revised March, 013; accepted March 9, 013 Copyright 013 Oludele deyefa, Oluleke Oluwole. his is an open access article distriuted under the Creative Commons ttriution License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited. BSRC his study investigated thickness requirements for field faricated (large) spherical liquefied natural gas (LNG) pressure vessels using the finite element method. In the FEM modeling, 3-dimenisonal analysis was used to determine thickness requirements at different sections of a 5-m radius spherical vessels ased on the allowale stress of the material as given in SME Section II art D. Shallow triangular element ased on shallow shell formation was employed using area coordinate system which had een proved etter than the gloal coordinate system in an earlier work of the authors applied to shop uilt vessels. his element has five degrees of freedom at each corner node-five of which are the essential external degrees of freedom excluding nodal degree of freedom associated with in plane shell rotation. Set of equations resulting from Finite Element nalysis were solved with computer programme code written in FORRN 90 while the thickness requirements of each section of spherical pressure vessels sujected to different loading conditions were determined. he results showed memrane thickness decreasing from the ase upwards for LNG vessels ut constant thickness for compressed gas vessels. he otained results were validated using values otained from SME Section VIII art UG. he results showed no significant difference ( > 0.05) with values otained through SME Section VIII art UG. Keywords: LNG; FEM; Field Faricated ressure Vessels; Shell hickness; Modeling 1. Introduction Over the past few decades, world consumption of LNG has increased more than five-fold and it is predicted that this growth will continue to e very strong. he growing demand from large markets such as China and India comined with the increasing popularity in a large numer of other smaller markets has resulted in the development of many new LNG facilities throughout the world. here are significant natural gas reserves gloally and exploration companies are rapidly developing facilities for exporting the natural gas with corresponding receiving facilities eing planned and uilt in emerging markets. With a timeframe of some 5-10 years required for planning and construction, there is currently much activity underway in the LNG supply chain in preparation for current and predicted demands. Because of its unexampled advantages such as less floor area covering, highpressure capaility and transport facilitates, Spherical pressure vessels used for storage of gas and liquefied gas more widely than other storage tanks in the oil, gas, chemical and other fields. Since the 1990s, the numer of nuclear fuel storage and LNG storage facilities have increased, interest in the safety of these facilities have also increased as well as researches related to these fields [1,]. Single-curvature polyhedron hydro-ulging technology is a new technology for manufacturing spherical vessels and it has a good application foreground. his technology has een used in practice, ut the designing and manufacturing of polyhedral are ased on experiences, and the final quality of spherical vessels cannot e forecast quantitatively. Dong and other workers [3] in their paper, used the FEM code, MRC to simulate the hydro-ulging process of a singlecurvature polyhedron, including loading and offloading. he distriutions of stress and strain were simulated as well as other important data in their work [3]. Comparing with experimental data, their work showed that singlecurvature polyhedron hydro-ulging process could e simulated well y the finite element method code. he authors [4-6] have worked on shop uilt spherical vessels Copyright 013 SciRes.

2 470 O. DEYEF, O. OLUWOLE design and stress modeling in spherical vessels using gloal and area coordinate systems in the finite element modeling. Different design specifications and selection procedures have een outlined [7,8]. Construction of shop uilt spherical [9] and field faricated cylindrical vessels have eing carried out [10], however, there is still intense interest in the designing of spherical pressure vessels [11-13]. Work has proved the area coordinate system for triangular shell elements reliale and easy to use than the gloal coordinate system [5,6]. hus the area coordinates system was applied in this modeling.. Methodology.1. Finite Element Modeling Finite element analysis was used in this work. he analysis of this system required transformation into a discrete mathematical system [14]. he simplified structural model consisted of discrete structural elements (4 elements) as opposed to the system used in an earlier method [4,5] which needed fine meshing to converge to a stress level. he approximate ehavior of each element was expressed in terms of selected generalized stress and strain variales using elasticity theory. he elements were then assemled y enforcing equilirium of forces and compatiility of displacements at the nodes on the model. hese conditions were expressed as a set of nonhomogenous linear equations in which the variales were element forces and structural displacements and the constant terms were the applied loads [14]... Displacement Functions he use of shallow triangular element (Figure 1) and area coordinates was made use of in this work to represent the transverse displacement, w, as a polynomial function of degree 3 as given y [15]. Linear polynomial equations were then used to represent the memrane displacements u and v using area coordinates, resulting in a constant strain triangle for the memrane action. he assumed displacement equations are: u a1l1al a3l3 (1) v a4l1a5l a6l3 () w a L a L a L a LL a L L a L L a 1 L L LL L 31 L 13 L 13 L a 1 L L LL L 31 L 13 L 13 L a 1 L L LL L 31 L 13 L 13 L (3) where w x (4) y w y (5) x l i l k j li and li is the length of the side opposite node i. he modified interpolation for displacement is taken as a (6) to determine constants a s, known displacements at nodes are sustituted and the equations ecome 1 a C (7) where is the nodal degrees of freedom, C 1 is inverse of transformation matrix and [a] is vector of independent constants..3. Strain-Displacement Equations Strain-displacement relationships for shallow thin shells as given y [16] are simplified for the shallow shell and expressed as follows in curvilinear coordinates. u w v w w x, y, k x x r y r x w w ky, k xy y xy he aove strain Equation (8) can e written in matrix form after necessary sustitutions of u, v and w in Equations (1)-(3) into the aove strain equations..4. Stresses in a Curved riangular Element Stress varies from point to point along the shell profile and also through the thickness of the shell making it an (8) Copyright 013 SciRes.

3 O. DEYEF, O. OLUWOLE 471 the work done y the external load on the system. In the finite element method, the potential energy of a shell is expressed as: n (14) where k k 1 k is the potential energy of the k th element..6. Stiffness Matrix Figure 1. Shallow triangular element. unknown function of two variales [14]. It is represented as shown elow [4]: 6M N, m (9) t t where: M is the moment per unit length, M and is the ending stress at the surface. N is force per unit length and m which is memrane stress..5. Strain Energy he strain energy equation for an isotropic linear shell as given y [17] was adopted in this work; U t t E 1v 1 x y vxy 1v xy dd xdy (10) where, t thickness of the shell, v oisson s ratio and E Modulus of elasticity and are the strain and shear strain notations. fter sustitution for strains in the aove expression and integration with respect to, the strain energy can e separated into the memrane energy U m and the ending energy. U U Um U (11) Et 1 U e e ve e 1ve dxdy m 1 v x y x y x y 3 x y x y 4 1 v xy (1) Et 1 U k k vk k 1 v k dxdy he potential energy, U W (13) where W represents 1 m m m 1 k t C B D Bd C 1 1 d k t C B D B C m (15) (16) k m and k are element stiffness matrices due to memrane and ending stresses respectively D m and D are elasticity matrices for memrane and ending stresses respectively B m and B are strain matrices for memrane and ending stresses respectively. herefore, element total stiffness matrix is k k k (17) he element stiffness matrices were then comined to give the system stiffness matrix. he stiffness matrices k and k m in terms of area coordinates were using three Gauss quadrature points. o integrate explicitly, the integral equation elow as it is in [6] was very useful. a c ac!!! LLL 1 3d (18) ac! where is the area of triangular element.7. Consistent Load Vector It is well known fact that the est and accurate approach for dealing with distriuted loads in FEM is the use of a consistent load vector which is derived y equating the work done y the distriuted load through the displacement of the element to the work done y the nodal generalized loads through the nodal displacements. he shallow triangular shell element acted upon y a distriuted load q per unit area in the direction of w, has work done y this load given y: where w is: 1 qwdd x y (19) 1 w a C and for the present element is given as (0) L L L LL L L L L (1) Copyright 013 SciRes.

4 47 O. DEYEF, O. OLUWOLE where 1 L L LL L 31 L 13 L 13 L 1 L L LL L 31 L 13 L 13 L 1 L L LL L 31 L 13 L 13 L and is as defined in Section.. he work done y the consistent nodal generalized force through the nodal displacements is given y: F () Hence, from Equations (19)-(), the nodal forces were otained 1 F C qdd x y (3) Equation (3) gives the nodal forces on a single element; and the nodal forces for the whole structure were otained y assemling the elements nodal forces..8. Boundary Conditions Before the system equations are ready for solution, they must e modified to account for the oundary conditions of the prolem. For this system, it was assumed that displacements in all directions with the exception of radial direction are zero. lso, due to the symmetry nature of the system, 1/6 of the spherical vessel (Figure ) was used therey reducing computing time. Shown (Figure ) is sample of meshing of 1/6 of the spherical vessel with 4 elements and 6 nodes as used in this work. Figure 3 shows the arrangement for the elements in the spherical shell. Course 1 takes thickness of Element 1. he lower and upper portions of Course take thicknesses of Elements and 3 respectively while Course 3 has thickness corresponding to the thickness of Element Cases Considered Case 1: Storage ank Storing LNG For a spherical vessel with the following simulation parameters, the thickness of each element corresponding to the memrane stress developed at the centroid was determined (ale 1). Memrane stresses at the centroid were delierately programmed to e within the range of 0.5% and 0.8% less than the spherical vessel construction material allowale stress given y SME standard to avoid a rise over the allowale memrane stress. Design Internal ressure = 600 KN/m Density of Stored roduct = 560 Kg/m 3 Material of Construction = 516M Grade 70 Material llowale Stress = 138 MN/m Specified Minimum Yield stress = 60 Ma Figure. ypical spherical mesh. Figure 3. 3-course version spherical vessel. Material Factor of safety = 1.88 Radius of Spherical Vessel = 5.0 metres Case : Storage ank Storing Compressed Gas hickness of each element corresponding to the memrane stress developed at the centroid was determined for spherical vessels storing compressed gas (ale ). Mem- rane stresses at the centroid were delierately programmed to e within the range of 0.5% and 0.8% less than the spherical vessel construction material allowale stress to avoid a rise over the allowale memrane stress. Internal Design ressure = 600 KN/m Material of Construction = 516M Grade 70 Material llowale Stress = 138 MN/m Specified Minimum Yield stress = 60 Ma Material Factor of safety = Radius of Spherical Vessel = 5.0 metres 3. Results and Discussions ales 1 and show the thicknesses and developed stress values otained for storage vessels storing LNG and compressed gas respectively. hese values were explicitly rought out in Figures 4-7. It could e oserved that for the storage tank storing LNG that the memrane thickness is thicker at the ase decreasing upwards (Figure 4). For the tank storing compressed gas there is uniform thickness throughout the tank (Figure 6). It Could e seen as well that the safety factor given for the stress developed at the centroid of each element has not exceeded the material allowale stress for oth tanks (Figures 5 and 7) and hence oth tanks are operating at stresses elow the Copyright 013 SciRes.

5 O. DEYEF, O. OLUWOLE 473 ale 1. Element thicknesses for spherical vessel storing liquefied product (LNG). tfe (m) t SME (m) % Deviation Developed Stress (Ma) Factor of Safety Element Element Element Element ale. Element thicknesses for spherical vessel storing compressed gas. tfe (m) t SME (m) % Deviation Developed Stress (Ma) Factor of Safety Element Element Element Element Figure 4. Element thicknesses for spherical vessel storing liquefied product (LNG). Figure 5. Developed stresses in spherical vessel storing liquefied product (LNG). Figure 6. Element thicknesses for spherical vessel storing compressed gas. specified minimum yield stress given y SME code. It could also e seen from ales 1 and that the thicknesses calculated for each element are in close agreement Figure 7. Developed stresses in spherical vessel storing compressed gas. with SME values. he maximum percentage deviation from the thicknesses using FE model and SME is.64%. his percentage value is reasonale as the results showed no significant difference ( > 0.05). he wisdom in delierately programming the allowale stress to e within the range of 0.5% and 0.8% less than the spherical vessel construction material allowale stress could e seen here ecause all the stress values fall a little elow the allowale memrane stress. he FE model in this research work also proved that it is possile to otain reasonale results with few elements using area coordinates as opposed to the large numer of elements needed for the model developed y deyefa et al. [5] using gloal coordinates. hus the use of area coordinates allow an easy modelling of variale vessel thickness which would otherwise would have ecome impossile using gloal coordinates. REFERENCES [1] H. M. Koh, J. K. Kim and J. H. ark, Fluid-Structure Interaction nalysis of 3-D Rectangular anks y a Variationally Coupled BEM-FEM and Comparison with est Results, Earthquake Engineering and Structural Dynamics, Vol. 7, No., 1998, pp doi:10.100/(sici) (19980)7:<109::id-e QE714>3.0.CO;-M Copyright 013 SciRes.

6 474 O. DEYEF, O. OLUWOLE [] Y.-S. Choun and C.-B. Yun, Sloshing nalysis of Rectangular anks with a Sumerged Structure y Using GUIDELINE storage_tank_rev_.pdf Small-mplitude Water Wave heory, Earthquake En- [9] Wilco, Spherical Compressed Natural Gas Vessels, 01. gineering and Structural Dynamics, Vol. 8, No. 7, 1999, pp doi:10.100/(sici) (199907)8:7<763::id-e [10] Y.-M. Yang, J.-H. Kim, H.-S. Seo, K. Lee and I.-S. Yoon, QE841>3.0.CO;-W Development of the World s Largest ove-ground Full Containment LNG Storage ank, 3rd World Gas Con- [3] J. Dong, et al., Numerical Calculation and nalysis of ference, msterdam, 5-9 June 006, pp Single Curvature olyhedron Hydro-Bulging rocess for Manufacturing Spherical Vessels, Institute of Nuclear [11] EGE, LG Spherical Storage ank Design, 01. Energy echnology, singhua University, Beijing, [4] O.. deyefa and O. O. Oluwole, Finite Element Mod- [1] Wikipedia, Storage ank, 01. eling of Stress Distriution in Spherical Liquefied Natural Gas (LNG) ressure Vessels, roceedings of the Nigerian Institute of Industrial Engineers, Nigerian Institute of Industrial Engineers, uja, 011, pp [13] Wikipedia, ressure Vessel, [14] S. I. Mahadeva, nalysis of a ressure Vessel Junction [5] O.. deyefa and O. O. Oluwole, Finite Element na- y the Finite Element Method, h.d. Dissertation, exas lysis of Von-Mises Stress Distriution in a Spherical ech University, Luock, 197. Shell of Liquified Natural Gas (LNG) ressure Vessels, [15] O. C. Zienkiewicz and R. L. aylor, he Finite Element Engineering, Vol. 3, No. 10, 011, pp Method Set, 5th Edition, Vol. : Solid Mechanics, But- [6] O.. deyefa and O. O. Oluwole, Finite Element Modeling of Shop Built Spherical (LNG) ressure Vessels, Engineering, Unpulished, 013. terworth-heinemann, Oxford, 000. [7] IJ, Design Recommendation for Storage anks, [8] Kolmetz, Storage anks Selection and Sizing, 011. [16] E. Reissner, On Some rolems in Shell heory, roceedings of 1st Symposium on Naval Structural Mechanics, California, ugust [17] H. L. Langhaar, Energy Methods in pplied Mechanics, Wiley & Sons, New York, 196. Copyright 013 SciRes.

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