International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Impact Factor (2012): 3.358 Volume 3 Issue 5, May 2014 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Variational Methods in a Thin Shell Problem Mangwiro Magodora 1 , TW Mapuwei 2 , J Masanganise 3 , K Basira 4 , CJ Chagwiza 5 1, 2,3,4,5 Bindura University of Science Education, Department of Mathematics & Physics, P Bag 1020, Bindura, Zimbabwe Abstract: Variational methods and their applications are considered in the solution of problems involving vibrations of thin shell segments consisting of partly positive and partly negative Gaussian curvature parts. The lower part of the spectrum of these shells when the boundary conditions exclude pure bending is investigated. The results obtained by using the Raleigh-Ritz approximation are compared with those obtained by applying the Shooting Method and it is observed that there is good agreement. Results obtained also indicate that the variational method used in the investigation gave large and less accurate results for the lowest frequency parameter while increasing the number of coordinate elements to two gave more accurate values. However, it is shown that increasing the number of elements has the disadvantage of increasing the computations needed to solve the problem. Keywords: Gaussian Curvature, Variational Method, Sturm-Liouville Problem, Raleigh-Ritz Approximation, Shooting Method Nomenclature � � , � � Lame’s coefficients � � , � � Orthogonal curvilinear coordinates � � , � � Linear operators � Young’s modulus � � , � = 12,3 Momentum terms ℎ Dimensionless thickness � Poisson’s ratio � Density � Acceleration due to gravity � Small geometric parameter � Frequency parameter � Number of waves in circular direction � � , � � Radii of convergence � Arc length � Angle in the circular direction Λ Eigenvalue Λ � Smallest eigenvalue �, �, � Coordinate axes � � , � � , � Stress projections � � (� � , � � ) Displacement projections � Natural frequency 1. Introduction A shell is a body bounded by two curved surfaces, where the distance between the surfaces is small in comparison with other body dimensions [24]. Shell structures have been constructed since ancient times. The Hagia Sophia in Istanbul and the Pantheon in Rome are well-known examples. Shells are very efficient in carrying loads acting perpendicular to their surface by in-plane membrane stresses [14]. Shell structures enjoy the unique position of having extremely high aesthetic value in various architectural designs. The understanding of the behaviour of shell structures enables designers or stress analysts to verify the accuracy of numerical structural analysis results for such structures [24]. Shells have a wide range of applications and uses in engineering. Cylindrical shells find extensive use in tanks, boiler gas, water conduits and aeroplane structures. Examples of shell structures in civil and architectural engineering are large-span roofs, water tanks, containment shells of nuclear power plants and concrete arch domes [24]. In mechanical engineering shell structures find use in piping systems, turbine disks and pressure vessels technology while in aeronautical and marine engineering shell forms are used in the construction of missiles, aircrafts, rockets, ships and submarines [24]. In the field of biomechanics shells are found in various biological forms such as the skull and eye, plant and animal shapes. Variational methods, also known as calculus of variations or energy methods are a branch of mathematics that involves finding stationary values of functionals. In other words calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. A functional is defined as an integral that has a specific value for each function from domain substituted into the functional, or a functional is an integral that implicitly contains differential equations that defines a problem. Variational solutions of shell problems are very useful when the desired result depends on overall rather than local conditions, for example in buckling and vibration problems or general magnitudes of deflections under transverse loads. In particular, an approximate solution of a differential equation can be obtained by using the Raleigh-Ritz method, which involves substituting an approximating function into the variational function, making sure the approximating function satisfies the boundary conditions. Several studies have recently been carried out involving thin shell theory. Among them are Timoshenko [21], Ventsel [24] and a host of many others. Due to their usefulness in the real world, shells deserve to be studied diligently and carrying out researches about shells is the best way to it. Other researchers of note who have studied shells include Masashi [13] and Aginam [2] et al, who have applied variational methods to analyse thin elastic shells with finite rotations and isotropic thin rectangular plates respectively. Also, Avramov & Brelavski [4] studied on vibrations of shells rectangular in the horizontal projection with two freely supported edges. Paper ID: 020131238 1919