Wolf Altman

Vibration of thin shells of revolution based on a mixed finite element formulation

Abstract A modified Hellinger-Reissner functional for thin shells of revolution is presented. A mixed finite element formulation is developed from this functional which is free from line integrals and relaxed continuity terms. This formulation is applied to the problem of free vibration of spherical and conical shells. Bilinear trial functions are used for all field variables. The quadrilateral curved elements here presented satisfy the C0 continuity requirement of the functional. In all the results obtained the accuracy is quite good even for a reasonable

A thin cylindrical shell finite element based on a mixed formulation

Abstract A specialized functional for thin cylindrical shells derived from the Washizu-Hu variational principle using considerations of relaxed continuity requirements is presented. A mixed formulation for a cylindrical thin shell finite element is developed from this functional. The assumed fields for displacements and stress resultants are bilinear functions in the longitudinal and circunferential directions. The agreement between the present results and those obtained in previous formulations is good if the comparison is based on the precision related to the number of variables involved in the problem.

Vibration of thin cylindrical shells based on a mixed finite element formulation

Abstract A Hellinger-Reissner functional for thin circular cylindrical shells is presented. A mixed finite element formulation is developed from this functional, which is free from line integrals and relaxed continuity terms. This element is applied to the problem of vibration of rectangular cylindrical shells. Bilinear trial functions are used for all field variables. The element satisfies the compatibility and completeness requirements. The numerical results obtained in this work show that convergence is quite rapid and monotonic for a much smaller number of degrees of freedom than other finite element formulations.

Stability of annular sectors and plates subjected to follower loads based on a mixed finite element formulation

Abstract A compatible mixed finite element is developed utilizing a weak variational formulation and applied to vibration and stability analysis of annular sectors and plates subjected to follower loads. The relaxed continuity requirement of the normal rotation is taken into account in this formulation. The transverse displacement and moments are interpolated by bilinear polynomials. The nodal unknowns are the field variable values at each vertex of the trapezoidal curved element. Examples are presented and their results are discussed.

APPLICATION OF THE QUADRATIC FUNCTIONAL TO NONCONSERVATIVE PROBLEMS OF ELASTIC STABILITY

Abstract Two methods for solving nonconservative problems by applying the quadratic functional are presented. The corresponding approximate methods of solution are obtained by using suitable polinomial trial functions and are illustrated by solving Becks and Leipholzs rods.

Stability of cylindrical shell panels subjected to follower forces based on a mixed finite element formulation

Abstract A mixed finite element formulation is developed from a weak variational priniciple. This formulation is applied to stability analysis of cylindrical shell structures subjected to follower loading. Bilinear trial functions are used for all field variables. The rectangular curved elements presented here satisfy the continuity requirements for the field variables at the element interface. Two examples of a cantilevered cylindrical shell panel under different kinds of loading are solved.

Physical Components of Tensors

In the brand new book Physical Components of Tensors by the prominent Brazilian scientists Wolf Altman and Antonio Marmo de Oliveira appearing in the CRC Series in Applied and Computational Mechanics the reader can find a concise but comprehensive account of the fundamentals of tensor calculus in holonomic and anholonomic coordinate systems and a number of its applications in continuum and structural mechanics. Nowadays, it is hardly possible to imagine a modern description of the mechanical properties, statics and dynamics of solids and structures without using vector and tensor relations and tensor calculus. As pointed out by the authors “Anyone who lacks the knowledge of this mathematical tool is at a disadvantage in what concerns working effectively in this as well as several other fields of pure and applied mathematics”. However, this book is not just another, although excellent, textbook on classical tensor calculus. Definitely, it provides an exhaustive presentation of the theory of physical and nonholonomic components of tensors with application to the continuum mechanics and shell theory. In fact, this is what makes Altman and de Oliveira’s book unique. Doubtless, each graduate student, professor or researcher working in the field of mechanical or civil engineering, theoretical or experimental physics or applied mathematics would find it useful. The book is organized in five chapters. The first three chapters, which may be thought of as a self-contained introductory part of the book, namely, Chapter 1. Finite-Dimensional Vector Spaces, Chapter 2. Vector and Tensor Algebras and Chapter 3. Tensor Calculus introduce the mathematical background of the theory developed in Chapter 4. Physical and Anholonomic Components of Tensors and applied in Chapter 5. Deformation of Continuous Media. Here, it should be noted

Physical and Anholonomic Components of Tensors via Invariance of the Tensor Representation.

This article is concerned with the theory of physical and anholonomic components of tensors of single and double field based on the method of invariance of the tensor representation. It obtains formulas that describe a more general tensor calculus, which includes, as a particular case, the tensor calculus based on coordinate transformation. Explicit expressions of the usual connections are obtained from a study of the anholonomic connections. Also, expressions of the anholonomic components of the covariant derivative of a tensor are provided. As an application of this theory, the nonlinear kinematic equations of a body are written in terms of anholonomic components.