By Dr. Steve Naomis, Dr. Paul C. M. Lau (auth.)

This ebook offers a style that is able to comparing the deformation features of skinny shell constructions A unfastened vibration research is selected as a handy technique of learning the displacement behaviour of the shell, permitting it to deform obviously with out enforcing any specific loading stipulations. The strain-displacement equations for skinny shells of arbitrary geometry are constructed. those relationships are expressed generally curvilinear coordinates and are formulated solely within the framework of tensor calculus. The ensuing conception isn't limited to shell constructions characterised by means of any specific geometric shape, loading or boundary stipulations. the entire displacement and pressure equations constructed by way of Flugge are approximated through the curvilinear finite distinction technique and are utilized to computing the usual frequencies and mode shapes of basic skinny shells. This strategy allows either the displacement parts and geometric homes of the shell to be approximated numerically and effectively. the choice of a suitable displacement box to approximate the deformation of the shell inside of each one finite distinction mesh is mentioned intimately. furthermore, comparisons are made among using moment and third-order finite distinction interpolation meshes.

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**Example text**

2. 2. 2. 2. 4, the Christoffel symbol is introduced to represent the components of the derivative of a base vector.. 3 is a measure of the rate of change of the base vector 8 3 along a parametric curve. r ~3 is a plane symmetric tensor and is expressed by the symbol bal! which is referred to as the curvature tensor. 16) The above formula is simple and particularly useful in the numerical computation of the curvature tensor where it is often possible to express the base vectors 8 as a functions of the curvilinear coordinates xa.

7a) 2j g 3 j d j are the contravariant components of a symmetric second order tensor called the stress tensor. It is important to note that the quantities a lj dA l • ~j clA2 and a3i dA3 are not exactly forces since gi is generally not a dimensionless unit vector. 5. 5 THE CONSTITUTIVE EQUATIONS For an elastic material whose stress-strain behaviour is essentially linear, Hooke's law may be written in the form =: Eijk! 1) kI where Eijld is a fourth order tensor referred to as the elastic modulus.

Let 4> be a scalar expressed as the product of a second order tensor A.. 43, h. 51 to be used to define the covariant derivative of A.. 52b) + Ai I r jkl (2. 1. 52d) Njl i k A/'k A ij I k Aij A/ fik. 1 Curvilinear Coordinates On a Surface A surface can be defined as the locus of a point whose coordinates are functions of two independent parameters. 1) The variables xu. constitute a system of curvilinear coordinates while the curves on a surface along which one parameter remains constant are called parametric curves.