INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 1, No 2, 2010

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1 Thickness Optimization of Vibrating Shells for Minimum Volume Mallika.A 1,Ramana Rao.N.V 2 1 Associate Professor, Dept. of Civil Engineering, VNR Vignana Jyothi Institute of Engineering and Technology,Hyderabad,Andhra Pradesh, India 2 Professor & Principal, Dept. of Civil Engineering, JNTU College of Engineering, Hyderabad,Andhra Pradesh, India ABSTRACT Shell structures are the most often used structural elements in nature and technology. This paper presents a new technique for shape optimization of curved shell structures that takes advantage of the geometric modeling and automatic meshing capabilities of an existing parametric/associative CAD system. Using the method of inverse technique, a hanging model is created and inverted over 180. The surfaces are discretized by shell elements involving shell structures subjected to free vibration. An iteratively adopted configuration with different mesh sizes, free vibration analysis is performed and optimum shape is obtained for minimum volume with a constraint on fundamental frequency. Key words: Shells, shape optimization, inverse technique, finite element method, finite element mesh, free vibration analysis 1. Introduction Optimization theory encompasses the quantitative study of optima and methods for finding them. Classical optimization methods 1 often encounter great difficulty when faced with the challenge of solving complex problems that abound in the real world. Vitally important applications in business, engineering, economics and science cannot be tackled with any reasonable hope of success, within practical time horizons by solution methods developed with the predominant focus of academic research in the past three decades. The various technical papers 1 13,19, Ph.D. theses 14 16, review papers 17,18,20 and books published on structural optimization reflect the increasing interest in this area. A few authors have developed computer aided optimum design tools. The shell vibration research was reported as early as 1888 by Lord Rayleigh and A. E. H. Love, but a significant amount of efforts by other researchers began in 1930s and thereafter. Leissa 16 published a monograph on the vibrations of shells compiling the previous works done on this topic. Subsequently, some other review articles were published in the literature by others. Qatu 17,18 published two review articles in 2002 and covered the period of N.Camprubí,K.U.Bletzinger 2 have adopted inverse method to obtain deflection free optimum shapes. The present work deals with the free vibration analysis of paraboloid shell structures obtained using inverse method, essentially the extension of the authors 2. To the author s knowledge no attempt was made to optimize the inverse models further as they are inherently satisfying minimum strain energy and no bending criteria (Ref 2 ). 211

2 Keeping this in view, in the present paper, author tried to optimize the deflection free inverse models under free vibrations to obtain minimum volume criteria. 2. Structural analysis and shape optimization The aim of the optimal structural design is to obtain a design, a set of values for the design variables, which minimizes an objective function and complies with the constraints that depend on the design variables. The design variables of a structure can be properties of the cross section of the elements such as surface areas, thicknesses, inertia moments, co ordinates etc. Depending upon the design variables, three optimization methodologies generally used in form finding process are 1. Using the coordinates of the nodes of the FE mesh 2. Defining a parametric model using some variables of geometry 3. Stating a set of predefined geometries with the optimum shape being a linear combination of them with the design variables as combination factors. In this paper the first method is used in obtaining the deflection free inverse model (fig.1) and the second approach is used to achieve the specific objective functions. Figure 1:Inverse technique of approaching optimum shell form Generally, the objective functions used in optimization under static loading are strain energy minimization or maximizing the stiffness, stress leveling, minimization of weight or volume etc. and in optimization under dynamic loading are maximization of fundamental frequency, maximizing the sum of few frequencies etc. The constraints considered are commonly of two types, equality and inequality constraints and are explicit functions of the design variables. An illustrative example from reference author 12, a paraboloid with clamped corners is considered for checking the validity of the inverse model generated in ANSYS. The inverse models are then optimized for two different objective functions (minimizing the volume and maximizing the fundamental frequency). In the present study free vibration analysis is performed on inverse models using subspace iteration method to extract the mode shapes, using ANSYS software. To verify the accuracy of the finite element solution, the discretization error can be evaluated by using error estimation techniques. Based on such error estimates a further refinement of the mesh can be carried out. If the estimated error in the finite element solution is within the acceptable limits then the adaptive analysis is complete, otherwise further refinement 212

3 is carried out.sensitivity analysis consists of the systematic calculation of the derivatives of the response of the finite element model with respect to the, the design variables such as thickness etc. 3. Algorithm for shape optimization General approach for shape optimization of shell structures should include The use of minimum number of design variables. The need for convenient geometric representation of boundaries. The accurate estimation of discretization errors. The incorporation of automatic mesh generation, re meshing, and refinement capabilities. The inclusion of efficient sensitivity calculations. The requirement of efficient optimization procedures. The benefits of graphical display of changes in shape, mesh, error distributions, stress distributions, etc. to allow users to obtain an insight into the structural behavior. 4. Mathematical statement of the structural optimization problem The structural optimization problem can be expressed in mathematical form as Minimize (or maximize): F(s) Subject to constraints g j (s) 0, j=1 m h k (s) = 0, k=1.n l u s i s i s i, i = l...n in which, s is the vector of design variables, F(s) is the objective function to be minimized or maximized, g j (s) are inequality constraints and h k (s) are equality l u constraints. s i and s i are lower and upper bounds. 4.1 Design Optimization Method in ANSYS In the present problem ANSYS which is robust and efficient optimization module is built in is used for structural analysis and optimization. The design optimization terms used in ANSYS are: Design variables are independent quantities, varied to achieve the optimum design. State variables are quantities that constrain the design. Objective function is the dependent variable that we are attempting to minimize. Feasible design is a design that satisfies all specified constraints (those on the state variables as well as on the design variables). If any one of the constraints is not satisfied, the design is considered infeasible. The best design is the one, which satisfies all constraints and produces the minimum objective function value. The optimization methods are traditional techniques that strive for either minimization or maximization of an objective function subject to constraints. The ANSYS program always tries to minimize the objective function as it efficiently handles the minimization problems. If we need to maximize a quantity, restate the problem and minimize the 213

4 quantity x1 = 1/x.In the present paper one of the objective function is to maximize the fundamental frequency, is converted into the problem of minimizing the time period and optimized. The ANSYS optimization procedure offers several methods and tools that in various ways attempt to address the mathematical problem stated above. The optimization methods transform the constrained problem into an unconstrained one that is eventually minimized. The design tools, on the other hand, do not directly perform minimization. Use of the tools offer alternate means for understanding design space and the behaviour of the dependent variables.the optimization method employed in the present study is Sub problem approximation method. 4.2 Sub problem approximation method This is an advanced zero order method, which requires only the values of the dependent variables (state variables and the objective function), and not their derivatives. It is a general method, which can be applied efficiently to a wide range of engineering problems. There are two concepts that play a key role in the sub problem approximation method: the use of approximations for the objective function and the conversion of the constrained optimization problem to an unconstrained problem. State variables and limits on design variables are used to constrain the design and make the optimization problem a constrained one. The ANSYS program converts this problem to an unconstrained optimization problem because minimization techniques for the latter are more efficient. The conversion is done by adding penalties to the objective function approximation to account for the imposed constraints. 4.3 Convergence Checking At the end of each loop, a check for convergence is made. The problem is said to be converged if the current, previous, or best design is feasible and any of the following conditions are satisfied: The change in objective function from the best feasible design to the current design is less than the objective function tolerance. The change in objective function between the last two designs is less than the objective function tolerance. The changes in all design variables from the current design to the best feasible design are less then their respective tolerances. The changes in all design variables between the last two designs are less than their respective tolerances. 5. Validation of the inverse model generated in ANSYS A flat plate of size 5m x 5m is modeled with clamped corners and meshed into 2 x 2 size using 4 noded shell elements. A concentrated load of 10kN 12 is applied at the centre node and the analysis is carried out in ANSYS software. Using inverse technique a deflection free model is generated for 5m x5m ground plan and the maximum deflection is found to be mm. Same model is considered for analysis as per reference author 12 using FEM.. The maximum deflection obtained from theoretical finite element 214

5 formulation is mm and from ANSYS it is found to be mm,which are very close. 6. Illustrative example 6.1 Problem Definition In the present paper a flat plate with a central concentrated load is considered initially and the deflection profile is inverted by to get the deflection free inverse model for a specified loading. Free vibration analysis is carried out on the inverse model using subspace iteration method in ANSYS software. In the present paper the objective function consider is Minimizing the volume with a constraint that the fundamental frequency of the structure remains same as that of initial value. The design variables considered are thickness of the shell defined as a function of x and z coordinates of each node. 6.2 Initial Geometry T1 T4 T2 T3 Figure 2: (a) Shell 63 element (b) Design (Thickness)variables Taking the symmetry of the structure as an advantage, a quarter of a shell has been modeled in ANSYS applying symmetry boundary conditions. Initially quarter of the flat plate (5m x5m) is modeled using 4 key points. It is discretized into number of finite elements using 4 noded shell 63 elements. Shell 63 element in ANSYS has the advantage of taking different thicknesses at 4 nodes (figure 2(a)). Four thickness variables T1,T2,T3,T4 at four corners of the quarter plate are considered as design variables using a thickness function(fig.2(b)).the thickness of the plate between the nodes is considered to vary smoothly. Initially thickness of the plate is assumed as 10mm uniform throughout the plate area. Material properties are considered as that of isotropic steel. A concentrated load of 10k N 12 is applied at the centre node of the plate. 6.3 Structural Analysis and Optimization 215

6 Static analysis is performed and the hanging model from the deflected profile of the plate is created. It is inverted 1 (fig.1) by to form a paraboloid. The same concentrated load is applied at the centre node of the paraboloid and the structure is analysed. The deflections are compared after analysis and found to be almost negligible in all the cases. The plate element is resized into more number of finite elements of different sizes to reduce the estimated error to lie between 3 8%, for all boundary conditions considered. The various boundary conditions considered in the present study are Case 1: fixed corners Case 2: simply supported corners Case 3: fixed Edges Case 4: simply supported edges Generation of inverse models is shown in figure 3, figure 4, figure 5, figure 6 for the above mentioned boundary conditions respectively. Modal analysis is performed on the deflection free models using subspace iteration method in ANSYS and first five modes are extracted for each boundary condition. (a)flat plate loaded (b)hanging model (c) inverse model Figure 3: Figure showing the generation of Inverse model for fixed corner boundary Condition (Only quarter plate is considered because of symmetry) (a)flat plate loaded (b)hanging model (c) inverse model Figure 4: Figure showing the generation of Inverse model for simply supported corner boundary condition (Only quarter plate is considered because of symmetry) 216

7 (a)flat plate loaded (b)hanging model (c) inverse model Figure 5: Figure showing the generation of Inverse model for Fixed Edge Boundary condition (Only quarter plate is considered because of symmetry) (a)flat plate loaded (b)hanging model (c) inverse model Figure 6: Figure showing the generation of Inverse model for simply supported edge boundary condition (Only quarter plate is considered because of symmetry) Taking advantage of the built in optimization module in ANSYS design optimization is carried out until an optimum value is obtained by checking the convergence of the results. 6.4 Discussion of the results Optimization performed with the objective function minimizing the volume of the shell structure keeping the initial fundamental frequency constant showed considerable improvement over the initial shape of the inverse model. The results are shown in table1. 217

8 Table 1: Optimized parameters for various boundary conditions of the inverse model Boundary condition Thickness Variables(mm) Max. Volume deflection x10 9 T1 T2 T3 T4 (mm) (mm 3 ) Corners fixed Corners simply supported Edges fixed Edges simply supported For case 1 with fixed corner boundary the initial volume m 3 was reduced to m 3 showing 11.15% reduction in volume. For case 2 with simply supported corners, the initial volume of m 3 was reduced to m 3 showing volume reduction by 3.19 % For case 3 with fixed edge boundary condition the volume reduction was very much negligible. For case 4 with simply supported edge boundary condition the volume was reduced from m 3 to m 3, was minimized by 3.7%. In all the cases the estimated structural percentage error lies between 3 8%.The displacement contours along with maximum deflection are shown in figure 10. Case1: max.deflection: mm Case 2: max.deflection: mm Case3: max.deflection: mm Case 4: max.deflection: mm Figure 10: Contours showing the deflection profile of the inverse models for boundary condition cases CONCLUSIONS The inverse models created from the flat plate loaded downwards are showing better structural behavior as the mechanical background of the shape bears direct relation between load and geometry Inverse models obtained from flat plate loaded centrally with a concentrated load, for various boundary conditions did not show significant increase in the 218

9 fundamental frequency when optimization is performed keeping the initial volume constant which reveals that deflection free shape leads to an increase in the stiffness of the structure. Different optimum shapes can be obtained for various magnitudes and types of the loads. For each range of iteration various local optima are obtained and they need not necessarily be considered as global optimum values. The study of inverse technique can be extended for various shapes of shell structures and dynamic characters of shell can be studied for different types of loads. 8. References 1. M. E. Botkin,1982, Shape optimization of plate and shell structures AIAA Journal vol.20 no.2 ( ). 2. N. Camprubí, M. Bischoff, K. U. Bletzinger,2002, Shape optimization of shell structures Technische Universität München,Statusseminar 3. S. Justin, M.G. Rajendran J,Raja Murugadoss,2004, Modified genetic algorithm using tabu research for optimal lay up of shells paper published in international conference(wise). 4. T. Lindby and J. L. T. Santos,1999, Shape optimization of three dimensional shell structures with the shape parametrization of a CAD system Springer link journal, Volume 18, Numbers Bhattacharya,B and Ramaswamy,G.S,1978, Analysis of Funicular shells by the Finite Element method Journal of structural eng.vol 6, N03, D.Rupesh kumar and N.V.Ramana Rao,2008, Shape Optimization with prescribed movement direction of design variables with stress levelling as objective, Journal of structural eng.vol 34,Number5 7. D.Rupesh kumar and N.V.Ramana Rao,2008,, Shape Optimization with prescribed movement direction of design variables with strain energy minimization as objective, Journal of structural eng.vol 34,Number6 8. E.Hinton, M.Özakça. and N.V.R.Rao,1995, Free vibration analysis and shape optimization of variable thickness prismatic folded plates and curved shells Part I:finite strip formulation, J. Sound and Vibration, Vol.181(4), pp E.Hinton and N.V.R Rao,1993, Analysis and shape optimization of variable thickness prismatic folded plates and curved shells. Part 1: finite strip formulation, Thin Walled Structures, Vol.17, pp

10 10. Hinton E., Rao N.V.R. and Sienz J.,1992, Finite element structural shape and thickness optimization of axisymmetric shells, Engineering Computations, V. 9, pp Timoshenko,S. and Woinowsky Krieger,1959, Theory of plates and shells. New York: McGraw Hill, G.S.Ramaswamy,1984, Design and construction of concrete shell roofs. Krieger Pub Co. 13. Rao S.S.,2003, Engineering optimization: Theory and practice, New Age New International, Delhi. 14. Ozakca M.,1997, Analysis and optimal design of structures with adaptivity, Ph.D. thesis, No. C/Ph/168/93, Department of Civil Engineering, University College of Swansea, Swansea. 15. Seinz J.,1994, Integrated structural modelling, adaptive analysis, and shape optimization, Ph.D. thesis, No. C/Ph/181/94, Department of Civil Engineering, University College of Swansea, Swansea. 16. A. W. Leissa,,1973, Vibration of Shells, NASA SP 288, Government Printing Office, Washington, DC. 17. M. S. Qatu, , Recent research advances in the dynamic behavior of shells,, Part 1: laminated composite shells, Applied Mechanics Reviews, 55, M. S. Qatu, , Recent research advances in the dynamic behavior of shells, Part 2: homogeneous shells, Applied Mechanics Reviews, 55, , Kandasamy, and A. V. Singh, 2006, Free vibration analysis of skewed open circular cylindrical Shells, Journal of Sound and Vibration, 290, Ding Y.,1986, Shape optimization of structures: A literature survey, Computers and Structures, V. 24, pp