Optimal finite element modelling for modal analysis of liquid storage circular tanks

Transcription

1 Int. J. Structural Engineering, Vol. 5, No. 3, Optimal finite element modelling for modal analysis of liquid storage circular tanks Said A. Elkholy* Civil and Environmental Engineering Department, United Arab Emirates University, P.O. Box 15551, UAE and Fayoum University, P.O. Box 63514, Cairo, Egypt *Corresponding author Alaa A. Elsayed Fayoum Univresity, P.O. Box 63514, Cairo, Egypt Bilal El-Ariss Civil and Environmental Engineering Department, United Arab Emirates University, P.O. Box 15551, UAE Safaa A. Sadek Fayoum University, P.O. Box 63514, Cairo, Egypt Abstract: Model analysis of liquid storage vertical tanks is a complex task due to fluid-structure-soil interaction. Analysis of tanks subjected to earthquake excitations using finite element modelling (FEM) has become a preferred technique. The FEM is validated by comparing its results with experimental and theoretical results in the literature. However, most finite element studies in literature do not provide enough details on the selection of the used. The objective of this study is to provide optimal FEM options of parameters such as element types and number of which best predict the tank dynamic characteristics, natural frequencies and principal mode shapes. Coupled natural frequencies in sloshing modes were obtained for various tank height-to-diameter ratios, various tank wall thicknesses and various liquid depths. The FEM predictions compared well with literature available experimental and numerical results. A set of FEM options of parameters is recommended for elastic and inelastic analysis of such tanks. Copyright 2014 Inderscience Enterprises Ltd.

2 208 S.A. Elkholy et al. Keywords: liquid storage tank; finite ; modal analysis; FE modelling;; sloshing modes. Reference to this paper should be made as follows: Elkholy, S.A., Elsayed, A.A., El-Ariss, B. and Sadek, S.A. (2014) Optimal finite element modelling for modal analysis of liquid storage circular tanks, Int. J. Structural Engineering, Vol. 5, No. 3, pp Biographical notes: Said A. Elkholy is an Associate Professor of Structural Engineering in the Department of Engineering at Fayoum University. He obtained his PhD degree from Tokyo University, Japan and his MSc degree from Cairo University, Egypt. Alaa A. Elsayed is an Associate Professor of Structural Engineering in the Department of Civil Engineering at the Fayoum University. He obtained his PhD degree from the University of Windsor-Canada and his MSc degree from Cairo University, Egypt. Bilal El-Ariss is an Associate Professor of Structural Engineering in the Department of Civil and Environmental Engineering at the United Arab Emirates University. He obtained his PhD degree from Quebec, Canada and his MSc degree from Missouri, USA. Safaa A. Sadek is an Assistant Lecturer in the Civil Engineering Department, Fayoum University, Egypt. She obtained her MSc degree from Fayoum University, Egypt. 1 Introduction Damages to liquid storage tanks in past earthquakes have shown how critical they are to any society and; therefore, have attracted special attention to evaluating the seismic behaviour of such tanks. The most commonly used type of tanks is the vertical thin walled cylindrical tanks. Thin walled cylindrical liquid storage tanks have a simple geometry, but exhibit complex vibration behaviour. This vibration behaviour is a coupled system between the liquid and the tank structure because the tank walls are elastic and deformable. In the coupled system, various vibration modes simultaneously occur during earthquakes because the natural frequencies of most of the modes are in the exciting frequency range of the earthquakes. Seismic behaviour of elastic vertical tanks, whether they are fully or partially filled with liquids, is affected by the tank structural flexibility, fluid properties and soil characteristics. To fulfil the requirement of keeping these tanks functional after a major seismic excitation event, numerous theoretical and experimental studies have been performed to better analyse the seismic behaviour of these tanks. Housner (1963) developed the most commonly used analytical model in which hydrodynamic pressure induced by seismic excitations is separated into impulsive and convective components using lumped mass approximation. This model has been adopted with some modifications in most of the current codes and standards. Yang (1976) used a

3 Optimal FEM for modal analysis of liquid storage circular tanks 209 single degree of freedom model with different modes of vibrations to study the effects of wall flexibility on the pressure distribution in liquid and the corresponding forces in the tank structure. Veletsos and Yang (1977) developed flexible anchored tank linear models and found that the magnitude of the pressure was highly dependent on the wall flexibility. Minowa (1980, 1984) carried out experimental studies investigating the effect of flexibility of tank walls and hydrodynamic pressure acting on the wall of rectangular tanks. Haroun (1984) introduced a detailed analytical method and performed a series of experiments including ambient and forced vibration tests for rectangular tanks. He used the classical potential flow approach to estimate the hydrodynamic pressure assuming rigid tank walls. On the experimental part, three full scale water storage tanks were tested to determine the natural frequencies and mode shapes of vibrations. Also, Haroun and Tayel (1985) used the finite element method for analysing the dynamic response of liquid tanks subjected to vertical seismic ground motions. Veletsos and Tang (1986) analysed liquid storage tanks subjected to vertical ground motion on both rigid and flexible supporting foundations. Haroun and Abou-Izzeddine (1992) conducted a parametric study of numerous factors affecting the seismic soil tank interaction under vertical excitations. Dogangun et al. (1997) implemented a displacement-based fluid finite element model in the general purpose structural analysis computer code SAPIV to analytically study the effects of wall flexibility on the dynamic behaviour of rectangular storage tanks filled with liquid. Excluding the fluid sloshing of the liquid, Chen and Kianoush (2005) estimated the hydrodynamic pressure in two-dimensional flexible rectangular tanks using the sequential method. Kianoush and Chen (2006) analysed the dynamic behaviour of two-dimensional rectangular tanks subjected to vertical seismic excitations. Their model included impulsive and convective masses in their seismic analysis of two-dimensional rectangular tanks. Livaoglu (2008) studied the dynamic behaviour of a fluid rectangular tank-foundation system with a simple seismic analysis procedure using the interaction effects presented by Housner s two mass approximations for the liquid and the soil foundation system. Dutta et al. (2004) studied the dynamic behaviour of reinforced concrete elevated tanks with soil structure interaction (SSI). The investigation, in the initial phase, evaluates primary dynamic characteristics, impulsive lateral period and impulsive torsional-to-lateral period ratio of such systems incorporating the effect of SSI. Curadelli et al. (2010) studied the dynamic behaviour of elevated liquid storage containers because of the interest in their response to seismic loads (e.g., in the petro chemical industry) and in connection with the structural integrity and reliability analysis of diverse shell components (e.g., in nuclear reactors). Ruifu et al. (2011) used the multiple friction pendulum system (MFPS) to analyse the seismic response of an isolated vertical, cylindrical, extra-large liquefied natural gas (LNG) tanks for large displacements induced by earthquakes with long predominant periods. They compared their analysis results to those in the literature and they did not propose finite element modelling (FEM) in their methodology. Moslemi et al. (2011) considered fluid-structure interaction in liquid containing tanks by developing a finite element technique that implements such interaction. Utilising their developed technique, they were able conclude that investigating liquid sloshing effects in tanks of complex geometries such as conical tanks was possible. The they used for the liquid and tank wall were displacement-based fluid and shell, respectively. Amiri and Sabbagh-

4 210 S.A. Elkholy et al. Yazdi (2011, 2012) addressed experimental and numerical investigations of dynamic parameters, natural frequencies and mode shapes of fixed roof ground supported storage tanks. Three tall liquid storage tanks having similar height of m and different radii of 6.095, 8.00 and m were considered using the finite element package ANSYS. The finite element meshes of tank wall and roof were constructed using Shell181 quadrilateral shell and the fluid region was meshed by Fluid80 quadrilateral fluid. They concluded that the constructed models can serve as a baseline in structural dynamics for future analyses. They did not carry out further investigating on other element types that may yield more representative models of the real structures. Sezen et al. (2008) investigated the parameters influencing the dynamic behaviour of above-ground liquid tanks. They carried out simplified and finite element dynamic analyses of the tanks including the effect of liquefied gas-structure interaction using a ground motion recorded at a nearby site. The simplified proposed analysis was based on a three-mass model. In the ANSYS FEM the reinforced concrete columns were modelled using frame. Quadrilateral four-node-shell were used for stainless tank walls and eight-node-solid were used to model the thick reinforced concrete tank. Fluid inside the tank was modelled using eight-node-brick fluid. Jalali and Parvizi (2012) carried out an experimental work to investigate the effects of different parameters on modal properties of liquid-containing structures. Their experimental results were then used to construct analytical models for these structures using the finite element package, ANSYS. Shell element SHELL63 and fluid element FLUID80 from the ANSYS element library were used in their models which were capable of regenerating the experimental dynamic characteristics of the structures with an acceptable accuracy. Maheri and Abdollahi (2013) conducted a numerical study using ABAQUS to investigate the effects of material degradation due to corrosion on the dynamic buckling of three cone roof ground-based, cylindrical tanks with height-to-diameter ratios (H/D) of 0.40, 0.63 and 0.95, subjected to horizontal seismic-base excitations. They used four-sided doubly-curved shell to model the wall and three-sided shell (S3R) are utilised to model the roof. No liquid element was specified or identified in their study to model the liquid-tank interaction. Therefore, different simplified design provisions for tanks have been developed based on the simplified beam model analytical results reported by Veletsos, Haroun and Housner (Veletsos, 1984; Haroun and Housner, 1981) and on experimental research outputs reported in the literature. These simplified design provisions can be found in codes and standards such as American Petroleum Industry (API) Standard 650 Appendix E Seismic design of storage tanks (1980) or in Eurocode 8-Part 4 Tanks, Silos and Pipelines in Appendix B Seismic analysis procedure for anchored cylindrical tanks (1998). The finite element studies listed above did not provide enough details on the selection criteria of the element types used and did not investigate the effects of other element types available in the programme library on the performance of the tanks. The study also recommends the optimal ANSYS shell element type and number of that best predict the tank dynamic characteristics of natural frequencies and principal mode shapes.

5 Optimal FEM for modal analysis of liquid storage circular tanks Description of the finite element model There are two approaches for modelling liquid storage tanks by using mechanical and FE modelling techniques. The mechanical modelling technique has been defined as a simplified model that had been developed by different researchers and recommended by current major earthquake codes. The approaches for modelling tanks included approaches for modelling the fluid structure interaction (FSI) system and approaches for modelling the SSI system. The FE structural analysis programme, ANSYS (2006) was used in this study to produce the FE modelling needed for the tank dynamic analysis. For the FSI analysis, there are different FE approaches to represent fluid motion such as added mass approach, Lagrangian approach, Eulerian approach, Lagrangian-Eulerian in the FE method, smoothed particle hydrodynamic (SPH) methods and analytical methods such as Housner s two mass representations. The main objective of any FE modelling is to create a mathematical representation of the engineering system that reflects its actual geometry and behaviour. Building FE models in ANSYS requires familiarity with the ANSYS operating manual and element library. Each element in ANSYS has specific properties and behaviours to be defined according to the structure in the problem. 2.1 Modelling of the wall tank The 3D FE model of empty tank was established as surfaces using the pre-processor section in ANSYS. The selection of a suitable element for a given application is not a trivial matter and will directly influence the computational time and accuracy of the results. The ANSYS element library has many different types of shell. This study examined the library available shell properties and concluded that six have properties more appropriate for modelling liquid tank walls than other. These shell types are commonly used by several researchers for modelling tanks with different mesh sizes. Therefore, six different FE models were considered and analysed individually in this study to investigate the choice of each of these six and its total number in the mesh on its corresponding FE model performance and output. The best shell element would be the one whose FE model performance and results are the closest to, or have the least difference with, those results in the literature. The six shell considered in this paper are: 1 SHELL43 with stress stiffening, creep and large plastic strain and deflection capabilities, shown in Figure 1. It is a four-node element and is recommended for modelling moderately-to-thick shell structures. Each node has six degrees of freedom in the x, y and z nodal axes. Linear deformation shapes are used in both in-plane motions. For out-of-plane direction, the element uses a mixed interpolation of tensorial components. 2 SHELL63 elastic shell, shown in Figure 2. It is a four-node shell element and is known of its membrane and bending capabilities. Loads applied normal and in-plane are allowed. Each node of this element has six degrees of freedom in the x, y and z nodal axes. SHELL63 is also known of its large deflection and stress stiffening capabilities.

6 212 S.A. Elkholy et al. 3 SHELL93 is an eight-node shell element, shown in Figure 3. It is well suitable for modelling curved shells. Each node has six degrees of freedom in the nodal x, y and z axes. Deformation shapes are quadratic in both in-plane directions. SHELL93 has large plastic strain, stress stiffening and large deflection capabilities. 4 SHELL143 plastic shell, shown in Figure 4. SHELL143 is well suited to model nonlinear, flat or warped, thin to moderately-thick shell structures. It is a four-node element and each node has six degrees of freedom in the x, y and z nodal axes. SHELL143 has small plastic strain, stress stiffening, creep and large deflection capabilities. 5 SHELL181 finite strain shell, shown in Figure 5. It is well suited for modelling thin to moderately-thick shell structures. This element has four nodes and each node has six degrees of freedom at each node and in the x, y and z nodal axes. It is well-suited for linear, large rotation and/or large strain nonlinear applications. Change in shell thickness is accounted for in nonlinear analyses. It can be used instead of SHELL43 for many problems that have convergence difficulty with SHELL43. 6 SHELL281, shown in Figure 6. According to the authors knowledge, this is one of the most recently introduced by ANSYS. Element SHELL281 is an eight-node element with six degrees of freedom at each nodes: three translations and three rotations about the x, y and z nodal axes. The element is suitable for the analysis of thin to moderately-thick shell structures where it is recommended to use quadrilateral shaped. SHELL281 is recommended for modelling linear and nonlinear strain and large rotation applications. SHELL281 may be used for layered applications for modelling laminated composite shells or sandwich construction. Figure 1 SHELL43 elastic shell

7 Optimal FEM for modal analysis of liquid storage circular tanks 213 Figure 2 SHELL63 elastic shell Figure 3 SHELL93 eight-node structural shell Figure 4 SHELL143 plastic shell

8 214 S.A. Elkholy et al. Figure 5 SHELL181 finite strain shell Figure 6 SHELL281 geometry 2.2 Modelling of the fluid As for the fluid, the ANSYS library has different types of fluid that can be used to model the fluid in 2-D and 3-D problems. In this study, the fluid is divided into a number of 3-D fluid (FLUID80 3-D contained fluid) with eight nodes each node having three degrees of freedom. The fluid element FLUID80 is a modification of the 3-D structural solid element (SOLID45). FLUID80 element, shown in Figure 7, is recommended for modelling fluids within vessels with no net flow rate. The fluid element is suitable for computing hydrostatic pressures and fluid/solid interactions. Acceleration effects, such as in sloshing problems, as well as temperature effects, may be included.

9 Optimal FEM for modal analysis of liquid storage circular tanks 215 Figure 7 FLUID80 3-D contained-fluid element 2.3 Modelling of the FSI The fluid-structure interaction can be modelled by one of two ways. The first is called structure surrounded by water (SSW) and the second one is called water surrounded by structure (WSS). The latter is adopted in this research where the tank wall and fluid are used. In the WSS, the fluid domain is modelled inside the structure domain. 2.4 Coupling equations Fluid element at the boundary should not be attached directly to structural (shell ) but should have separate, coincident nodes that are coupled only in the direction normal to the interface. The FSI is accounted for by properly coupling the coincident nodes that lie in the common faces of the two domains (liquid and wall) in the radial direction. This means that the fluid is attached to the shell wall but can slip in the wall tangential directions and therefore only can exert normal pressures to the tank wall. 2.5 Boundary conditions The tanks studied in this research are assumed to have completely fixed-base. Therefore, all the nodes at the base are assumed to be fixed. 2.6 Master degrees of freedom Master degrees of freedom are required in modal analysis. These are degrees of freedom that describe the dynamic behaviour of the structure. The accuracy of the finite element analysis (FEA) depends greatly on the mesh used. A fairly course mesh is first employed and then refined in areas of interest or high stress. The solving time of the model is dependent on the number of which is determined by the mesh density. A low density is appropriate where the stress is uniform over a relatively large area. However, where there is high concentration of stresses over smaller areas a high mesh density is deployed. The model of a circular tank in this research was drawn in ANSYS using volumes, areas and lines and then the mesh is mapped (map meshing allows precise control of the meshing procedure).

10 216 S.A. Elkholy et al. 2.7 Analysis method The reduced method is widely referred to for modal analyses using the ANSYS fluid. This method is used for the eigenvalue and eigenvector extractions to calculate frequencies and modes shapes including the fluid modes. It utilises householderbisection-inverse iteration to evaluate the eigenvalues and eigenvectors. This method uses a small subset of degrees of freedom, referred to as master DOF which makes is suitable for fast calculations that yield an exact stiffness matrix and an approximate mass matrix [M]. The accuracy of the results, therefore, depends on how well [M] is approximated which in turn depends on the number and location of the master degrees of freedom. 3 Modelling and analysis results A verification example of liquid storage tank tested experimentally by Mazúch et al. (1996) and numerically analysed by Zienkiewicz (2005) using the FEA is presented. The tank different mode shapes and frequencies were experimentally obtained and documented by Mazúch et al. (1996). The different analytical frequencies of the same tank were obtained by Zienkiewicz (2005) without using computer software and without providing details on the types and number of used in the analysis. The example was modelled in this study using the computer programme ANSYS and the model frequency values for different mode shapes were compared to the tank experimental measurements and theoretical solutions obtained by Mazúch et al. (1996) and Zienkiewicz (2005) respectively. 3.1 A three-dimensional experimental tank A 3D flexible circular tank tested by Mazúch et al. (1996) was chosen to verify this study model results. The tank, shown in Figure 8, was tested in two different cases: empty tank case and filled with liquid tank case. The same tank was numerically analysed for both cases by Zienkiewicz (2005) using the FEA method. The available experimental and analytical results were used to verify the accuracy of the results produced by the ANSYS models proposed in this study. In addition, a parametric study was performed to evaluate the effect of some variables on the numerical simulations, such as element types and mesh sizes. Figure 8 Tank tested experimentally by Mazúch et al. (1996)

11 Optimal FEM for modal analysis of liquid storage circular tanks 217 The elastic cylindrical tank had a mean radius R = mm, a wall thickness h = 1.5 mm and a length L = 231 mm; the tank material characteristics were Young s modulus E = N/m 2, Poisson ratio υ = 0.3, density ρ s = 7,800 kg/m 3. Water was used as the contained fluid having a density of 1,000 kg/m 3. The tank was welded at the base to a 20 mm thick circular plate which was bolted to a heavy-base block.the tank was filled with water up to the level H. The boundary conditions of the shell were theoretically assumed to be of clamped-free type. Finite element analyses using ANSYS were performed to obtain the modal characteristics of the tank. The fluid region is divided into a number of three-dimensional contained fluid. The tank is modelled with different types of shell as mentioned above. 4 Empty tank case The experimental natural frequencies and mode shapes of the considered liquid storage tank were obtained from the vibration measurements by Mazúch et al. (1996). The analytical natural frequencies and mode shapes were computed as well by Zienkiewicz (2005). In this study, the empty tank was treated as a 3D empty tank and was modelled as surfaces using the pre-processor section in ANSYS. The bottom nodes of the tank shell are fixed in all six degrees of freedom. The model was drawn in ANSYS using volumes, areas and lines and then was map-meshed (map meshing allows precise control of the meshing procedure). In order to find the best numerical model for the experimental test, a total of 22 FE models of the tank were built in ANSYS, 19 models with four-node shell and three models with eight-node shell. To start modelling the tank, it was first decided to examine the effect of the model s element number increase along the tank circumferential perimeter (radial direction) and along its height on the convergence and accuracy of the tank frequency values obtained. First, the number of divisions along the tank height, N H, was chosen to be ten divisions and kept fixed while increasing the number of divisions along the radial direction, N R, from 6 to 30 divisions. Next, N H was increased to twenty divisions and kept fixed while increasing N R from 6 to 30 divisions. SHELL 43 were used in this model convergence test and the model frequency results were compared to the tank frequency measurements by Mazúch et al. (1996). Figure 9 shows the FE model of the empty tank using ANSYS. Figure 9 3D FE model of empty tank (see online version for colours)

12 218 S.A. Elkholy et al. Table 1 FE model natural frequencies for empty clamped-free cylindrical tanks (Hz) with varying N H and N R NH = 10 NH = 20 SHELL 43 ELEMENT NR = 6 NR = 12 NR = 18 NR = 24 NR = 30 NR = 6 NR = 12 NR = 18 NR = 24 NR = 30 Total no. of Total no. of ,692 2,688 3, ,392 2,412 3,648 5,100 Mode m n Experiment , , ,479 1,624 1, , , , , , ,495 1,486 1, ,628 1,677 1, , , , , , , , , ,851 1,924 1, ,821 1, ,815 1, , ,828 1, , ,969 2,060 2, , , , , , ,046 2, , ,151 2,486 2, , , , , , , , ,167.6

13 Optimal FEM for modal analysis of liquid storage circular tanks 219 Figure 10 Effects of element number increase, N H and N R, on the model result accuracy

14 220 S.A. Elkholy et al. The results of this model test are shown in Table 1 and in Figure 10. Figure 10 shows that increasing the numbers of in both directions, the height of the tank and the circumferential perimeter of the tank, minimises the difference in the model results and the experimental measurements which means increasing the results accuracy. However, it can be concluded that increasing the number of in the radial direction of the tank yields a faster increase in the accuracy of the model results than the accuracy increase when increasing the number of in the tank height direction. Number of in the tank-base would be equal to the total number of generated by ANSYS minus N H *4N R. The tank analysis began with ANSYS models of 252 using the different shell listed above separately. To investigate the effects of the element number in the model on the tank calculated dynamic properties, i.e., natural frequencies and principal mode shapes, the number of over the cylindrical shell surface was increased gradually from 252 to 7,168. The different types of shell above were used separately in the different models of the tank with the different element numbers to check the effects of choosing a particular individual shell element option on the validity and accuracy of the results, therefore to propose the optimal model element types and numbers for analysis of such tanks. In this analysis, a large number of different models were investigated (as shown in Table 2) to obtain the dynamic characteristics of the empty tank. The model mode shapes of the empty tank using ANSYS and those of experimental measurement are shown in Figure 11. A comparison shows good agreement of the results obtained from the FE analysis and the experiments. Therefore, the results demonstrate the validity of the model. The mode shapes of the cylindrical shell are identified by the number of axial modes, m and number of circumferential waves, n. Table 2 shows the experimental natural frequency measurements by Mazùch et al. (1996), the frequency analytical values from FEM by Zienkiewicz (2005) and the natural frequencies predicted by the ANSYS FE models in this study using different element types and numbers. The FE model-obtained frequency results were compared with their corresponding experimental measurements by Mazùch et al. while the analytical values by Zienkiewicz were used as a reference. This comparison is shown in Table 1 as error, or difference, percentages. The less the error or difference, the better the model result accuracy is. As it can be seen in Table 2, the obtained FE results display good pattern with the experimental results verifying that the current method of tank FE modelling can further be employed to study the FSI problems of liquid-containing structures. The table also shows that when using finer mesh, or increasing the number of, the frequencies obtained from the FE models tend to come close to the experimental measurements. The results of the final idealisation (7168 ) compared very well with the experimental measurements, except for the mode shape (m = 1 and n = 2) and they were adopted for the in-vacuum dynamic characteristics of the vertical clamped-free cylindrical shell. The mode shapes of the cylindrical shell are identified by the number of axial modes, m and the number of circumferential waves, n.

15 Optimal FEM for modal analysis of liquid storage circular tanks 221 Table 2a FE model natural frequencies for empty clamped-free cylindrical tanks (Hz) at different type of shells, 252 No. of Mode m n Experiment Shell 43 Shell 63 Shell 93 Shell 143 Shell 181 Shell 281 FEM (Zienkiewicz [27]) , ,479 1, , , , , , , ,628 1, , , , , , , ,851 1, , , , , , , ,969 2, , , , , , , ,151 2, , , , , , ,

16 222 S.A. Elkholy et al. Table 2b FE model natural frequencies for empty clamped-free cylindrical tanks (Hz) at different type of shells, 448 No. of Mode m n Experiment Shell 43 Shell 63 Shell 93 Shell 143 Shell 181 Shell 281 FEM (Zienkiewicz [27]) , ,479 1, , , , , , , ,628 1, , , , , , , ,851 1, , , , , , , ,969 2, , , , , , , ,151 2, , , , , , ,

17 Optimal FEM for modal analysis of liquid storage circular tanks 223 Table 2c FE model natural frequencies for empty clamped-free cylindrical tanks (Hz) at different type of shells, 1,792 No. of Mode m n Experiment Shell 43 Shell 63 Shell 93 Shell 143 Shell 181 Shell 281 FEM (Zienkiewicz [27]) ,479 1, , , , , , , ,628 1, , , , , , , ,851 1, , , , , , , ,969 2, , , , , , , ,151 2, , , , , , , ,792

18 224 S.A. Elkholy et al. Table 2d FE model natural frequencies for empty clamped-free cylindrical tanks (Hz) at different type of shells, 2,800 No. of Mode m n Experiment SHELL 43 % Difference SHELL163 % Difference SHELL 143 % Difference SHELL181 % Difference FEM % Difference ,479 1, , , , , ,624 1, , , , , ,851 1, , , , , ,969 2, , , , , ,151 2, , , , , ,800

19 Optimal FEM for modal analysis of liquid storage circular tanks 225 Table 2e FE model natural frequencies for empty clamped-free cylindrical tanks (Hz) at different type of shells, 7,168 No. of Mode m n Experiment SHELL 43 % Difference SHELL163 % Difference SHELL181 % Difference FEM % Difference ,479 1, , , , ,624 1, , , , ,851 1, , , , ,969 2, , , , ,151 2, , , , ,168

20 226 S.A. Elkholy et al. Figure 11 Predicted mode shapes for empty shell with different m and n: (a) m = 2, n = 4 (b) m = 2, n = 5 (c) m = 1, n = 5 (d) m = 1, n = 4 (see online version for colours) (a) (b) (c) (d) This table shows that SHELL93 tend to give the least errors which mean that the frequency values predicted by the FE models using these shell are more accurate than the values obtained by using other shell. SHELL 181 yields the highest error, or difference, in all modes between the analytical and experimental results. The other shell give result differences that are more than the result differences produced by SHELL93 and less than the result differences given by SHELL181. It should be noted that SHELL93 (eight-node large strain curved shell) is particularly well suited to analyse empty circular tanks since it is designed model curved shells. Figure 12 shows the effects of the element types on the accuracy of the empty tank natural frequency results by the FE model in comparison with those of experimental measurements. Table 2 and Figures 12 and 13 indicate that no particular SHELL element gives the highest result accuracy for all modes investigated in this paper. However, SHELL with the least result accuracy can be excluded and not recommended for ANSYS FE models used for the analysis of such tanks. It can be seen from Table 2 and Figures 12 and 13 that the option of using SHELL 181 element seems to be the least reliable option in modelling such tanks as it gives the largest difference between the ANSYS FE model results and the experimental values. Therefore, the option of deploying SHELL 181 element in predicting the frequency of the tank will have the least accuracy and should not be recommended. As for increasing the number of in the models, Table 2 shows that for any element used in this study the result differences, in general, decrease as the number of increases leading to a higher accuracy in the FE model output for any mode shape. However, for any mode shape Figures 12 and 13 indicate that as the number of any SHELL element becomes larger the increase in the result accuracy (or the decrease in

21 Optimal FEM for modal analysis of liquid storage circular tanks 227 the % difference) occurs rapidly up to a certain number of after which the accuracy becomes almost steady and almost unaffected by any further increase in the number of. This concludes that increasing the number of is a critical model tuning task and should be carefully sought to get the highest result accuracy with the least model time consumption. Therefore, it should be noted that increasing the element number randomly may not improve the results much but will likely increase the model time consumption. Figure 12 Effect of element types and numbers on the tank natural frequency

22 228 S.A. Elkholy et al. Figure 13 Effect of element numbers (mesh sizes) on the natural frequency (see online version for colours)

23 Optimal FEM for modal analysis of liquid storage circular tanks Full tank case The fluid region is divided into a number of 3-D contained fluid (FLUID80). The fluid movement at the bottom of the tank is considered to be constrained in the vertical direction. The vertical velocities of the fluid element nodes adjacent to each surface of the wetted shell coincide with those of the shell. The boundaries are assumed to be constrained in order to simulate the rigid walls. Figure 14 shows the 3D model of the tank filled with liquid using ANSYS and the shape of the shell meshing. Figure 14 A 3D model of the tank filled with liquid (see online version for colours) The FSI is considered by properly coupling the nodes at the interface of the fluid and the shell in the radial direction. Therefore, the fluid can exert only normal pressure on the tank wall and it can slip on the tank wall in the tangential directions. For the completely filled cylindrical tank, four different idealisations were considered as shown in Table 3. In the first idealisation, a total number of 792 were adopted. In the second idealisation, the number of over the cylindrical shell surface was increased first to 1,792 and then to 3,584. In the final test of idealisation, the number of was increased to 4,480 which were distributed over the shell structure. Figure 15 shows the effects of the element types on the empty tank natural frequency results by the FE model in comparison with those of experimental data. Table 3 and Figures 15 and 16 indicate that most of the shell in any mode shape either have high difference between their outputs and the measured values or have inconsistent behaviour whose result difference increases as the element number increases up to certain element number after which any further increase in the element number will yield a decrease in the difference. In general all of these shell except SHELL43 do not have a unified pattern in all mode shapes. This is attributed to the case studied which is the filled tank case. In this case, FLUID80 (3-D eight-node; with four nodes on each face) are utilised to model the liquid in the curved tank. When a shell element such as SHELL93 (eight-node shell) element or SHELL281 (eight-node shell) element is used to model the curved tank in conjunction with FLUID80 element, this will yield a model whose element nodes are not matching, particularly the mid-span nodes. This will yield inaccurate results. SHELL43 element is the only element among all the investigated in this study which gives the highest result accuracy as it yields consistent results and consistent behaviour patterns in all modes studied. The accuracy of its results increases rapidly as the number of increases up to certain number, after which the accuracy becomes almost steady and almost unaffected by any further increase in the number of. Therefore, SHELL43 seems to be the best and could be recommended for the analysis of filled tank case.

24 230 S.A. Elkholy et al. Table 3a FE model natural frequencies for full clamped-free cylindrical tanks (Hz) with different element numbers, (a) SHELL 43 Mode m n Experiment 792 1, ,027 1, , , , , ,094 1, , , , , ,245 1, , , , , ,299 1, , , , , ,546 1, , , , , ,748 1, , , , , ,584 4,480 Error FEM, [27] Error

25 Optimal FEM for modal analysis of liquid storage circular tanks 231 Table 3b FE model natural frequencies for full clamped-free cylindrical tanks (Hz) with different element numbers, SHELL 63 Mode m n Experiment ,792 3,584 4,480 Error FEM, [27] Error , ,094 1, ,245 1, ,299 1, ,546 1, ,748 1, , , , , , , , , , , , , , , , , , , , , , , ,

26 232 S.A. Elkholy et al. Table 3c FE model natural frequencies for full clamped-free cylindrical tanks (Hz) with different element numbers, SHELL 93 Mode m n Experiment 792 1,792 3,584 4,480 Error FEM, [27] Error , ,094 1, , , , , ,245 1, , , , , ,299 1, Not applicable #VALUE! Not applicable #VALUE! Not applicable #VALUE! 1, ,546 1, , , , , ,748 1, , , , ,

27 Optimal FEM for modal analysis of liquid storage circular tanks 233 Table 3d FE model natural frequencies for full clamped-free cylindrical tanks (Hz) with different element numbers, SHELL 143 Mode m n Experiment ,027 1, ,094 1, ,245 1, ,299 1, ,546 1, ,748 1, ,792 3,584 4,480 Error FEM, [27] Error , , , , , , , , , , , , , , , , , , , , , , ,

28 234 S.A. Elkholy et al. Table 3e FE model natural frequencies for full clamped-free cylindrical tanks (Hz) with different element numbers, SHELL 181 Mode m n Experiment 792 1, ,027 1, , , , , ,094 1, , , , , ,245 1, , , , , ,299 1, , , , , ,546 2, , , , , ,748 2, , , , , ,584 4,480 Error FEM, [27] Error

29 Optimal FEM for modal analysis of liquid storage circular tanks 235 Table 3f FE model natural frequencies for full clamped-free cylindrical tanks (Hz) with different element numbers, SHELL 281 Mode m n Experiment 792 1,792 3,584 4,480 Error FEM, [27] Error , , ,094 1, , , , , ,245 1, , , , , ,299 1, Not applicable #VALUE! Not applicable #VALUE! Not applicable #VALUE! 1, ,546 1, , , , , ,748 1, , , , ,

30 236 S.A. Elkholy et al. Figure 15 Effect of element types and element sizes on the natural frequency (full tank)

31 Optimal FEM for modal analysis of liquid storage circular tanks 237 Figure 16 Effect of element numbers (mesh sizes) on the natural frequency (see online version for colours) 6 Deviation in ANSYS model results Figure 17 shows the average, median, minimum and maximum differences between the experimentally measured frequency values and the model frequency results obtained in this study for both cases of empty tanks and filled tanks. Despite the large deviations between the maximum and minimum differences in both cases, Figure 17 reveals that the median and average difference values are leaning towards the minimum difference values. This indicates that the empty and filled tank frequency values produced in this study by the different ANSYS FE models using different shell and liquid are reasonable and can be described as reliable.

32 238 S.A. Elkholy et al. Figure 17 Deviation in ANSYS model results (see online version for colours) 7 Conclusions and recommendations Empty and liquid filled storage vertical thin tanks are analysed in this paper using ANSYS 3D finite element models. To predict the tank dynamic characteristics such as natural frequencies and principal mode shapes, a number of shell and liquid available in ANSYS element library were utilised extensively. A parametric study of different element numbers for each element type in the 3D FE models was carried out. Tank natural frequencies and mode shapes were obtained and compared with experimentally measured values and analytical results available in the literature. The paper investigated different ANSYS FE 3D models to recommend the optimal element types and element numbers which best predict the empty and filled tank dynamic characteristics of natural frequencies and principal mode shapes. ANSYS, a general-purpose FE programme, element library has many different types of shell six of which were most suitable for modelling the cylindrical tank walls therefore were investigated in this study. These are SHELL43 (four-node plastic large strain shell), SHELL63 (four-node elastic shell), SHELL93 (eight-node large strain curved shell), SHELL143 (eight-node plastic small strain shell) and SHELL181 (four-node finite strain shell) and SHELL281 (eight-node large strain nonlinear shell). As for modelling the fluid in the tank, fluid element FLUID80 (3-D eight-node contained fluid element) was used in this study. To investigate the effects of the element number in the model on the accuracy of the tank calculated dynamic natural frequencies and principal mode shapes, the number of over the cylindrical shell surface was increased gradually from 252 to 7,168. Accordingly, the number of liquid was increased. As it is indicated above, the results of the FE models are greatly influenced by the number of chosen and the meshing scheme utilised. Therefore, to find out the optimal model, it is highly recommended to investigate the optimal number of and corresponding meshing scheme. To fulfil this objective, the analysis should be performed by studying different models with different increasing number of and the optimal model will be the one with the least number of used and that would not cause a significant difference in the results. The study shows that increasing the number of in the radial direction of the tank wall yields a faster increase in the accuracy of the model results than increasing the

33 Optimal FEM for modal analysis of liquid storage circular tanks 239 number of along the height of the tank wall. However, any further increase in the element number beyond a specific element number results in an accuracy that increases at a slower rate and eventually becomes almost steady. For the case of an empty tank, it can be seen from the results obtained in this study that no particular shell element gives the highest result accuracy for all modes. However, SHELL 181 element seems to be the least reliable option in modelling such tanks in all mode shapes as it gives the largest difference between the ANSYS FE model results and the experimental measurements. Therefore, the option of using SHELL 181 element should not be recommended. On the other hand, SHELL93 tend to give the least errors which mean that the frequency values predicted by the FE models using these shell are more accurate than the values obtained by using other shell. SHELL93 element should be recommended for the analysis of empty circular thin tanks as well as structures with similar shapes such as chimneys, shafts and empty circular pipes. As for the number of in the models, the study concludes that increasing the number of is a critical model tuning task and should be carefully selected to get the highest result accuracy with the least model time consumption. The model results indicate that result accuracy increases rapidly as the element number increases but up to a certain number, beyond which the result accuracy becomes almost constant or steady and is not affected by any further increase in the element number in the model. For the case of a filled tank, some shell investigated in this study for any mode shapes have high difference between their results and those of the experimentally measured values. Other have inconsistent results whose difference with the experimental measurements increases as the element number increases in the model up to a certain element number after which any further increase in the element number will yield a decrease in the difference. In general all of these shell, except SHELL43 do not have a unified pattern in all mode shapes. This is attributed to mesh mapping. When a shell element such as SHELL93 (eight-node shell) element or SHELL281 (eight-node shell) element is used to model the curved tank in conjunction with FLUID80 (3-D eight-node with four nodes on each face) element, this will yield a model whose element nodes are not matching, particularly the mid-span nodes. This will yield inaccurate results. SHELL43 element is the only element among all investigated in this study that gives the highest result accuracy as it yields consistent results for all modes. The accuracy of its results increases rapidly as the number of increases up to certain number, after which the accuracy becomes almost steady and unaffected by any further increase in the number of. Therefore, SHELL43 seems to be the best and could be recommended for the analysis of filled tank case. Despite the large deviations between the maximum and minimum differences between the model results and the experimental measurements, the study reveals that the median and average difference values are leaning towards the minimum difference values. This indicates that the empty and filled tank frequency values produced in this study by the different ANSYS FE models using different shell and liquid can be described as reliable. To examine numerically the influence of the element types as well as meshing on the seismic behaviour of real steel tanks, a full-scale tank case and many parameters such as tank configuration, filling ratio and different ground motion characteristics must be considered and highly recommended in future research.