THE appearance of a treatise in English upon the mathematical theory of elasticity is an event the potential importance of which may be judged by the that the author, in his frequent suggestions for collateral reading, refers to only three such, those of Southwell, Timoshenko, and Love. In spirit and content Sokolnikoff}s book differs greatly from each and all of these. It may be described by a possible sub-title: “A pure mathematician surveys topics related to certain problems in the mathematical theory of elasticity”. It is symptomatic of the change in outlook of American mathematics over the past few decades. Mathematical Theory Of Elasticity Prof. I. S. Sokolnikoff with the collaboration of Asst. Prof. R. D. Speche. Pp. xi + 373. (New York and London: McGraw-Hill Book Co., Inc., 1946.) 22s. 6d.

Mathematical Theory Of Elasticity

Beam theories are exploited worldwide to analyze civil, mechanical, automotive, and aerospace structures. Many beam approaches have been proposed during the last centuries by eminent scientists such as Euler, Bernoulli, Navier, Timoshenko, Vlasov, etc. Most of these models are problem dependent: they provide reliable results for a given problem, for instance a given section and cannot be applied to a different one. Beam Structures: Classical and Advanced Theories proposes a new original unified approach to beam theory that includes practically all classical and advanced models for beams and which has become established and recognised globally as the most important contribution to the field in the last quarter of a century. The Carrera Unified Formulation (CUF) has hierarchical properties, that is, the error can be reduced by increasing the number of the unknown variables. This formulation is extremely suitable for computer implementations and can deal with most typical engineering challenges. It overcomes the problem of classical formulae that require different formulas for tension, bending, shear and torsion; it can be applied to any beam geometries and loading conditions, reaching a high level of accuracy with low computational cost, and can tackle problems that in most cases are solved by employing plate/shell and 3D formulations.

Beam Structures: Classical and Advanced Theories

On Timoshenko-like modeling of initially curved and twisted composite beams

A formulation is presented for the nonlinear dynamics of initially curved and twisted anisotropic beams. When the applied loads at the ends of, and distributed along, the beam are independent of the deformation, neither displacement nor rotation variables appear: an intrinsic formulation. Like well-known special cases of these equations governing nonlinear dynamics of rigid bodies and nonlinear statics of beams, the complete set of intrinsic equations has a maximum degree of nonlinearity equal to two. Advantages of such a formulation are demonstrated with a simple example. When the initial curvature and twist are constant along the beam, two space-time conservation laws are shown to exist, one being a work-energy relation and the other a generalized impulse-momentum relation. These laws can be used, for example, as benchmarks to check the accuracy of any proposed solution, including time-marching and finite element schemes. The structure of the intrinsic equations suggests parallel approaches to spatial and temporal discretization. A particularly simple spatial discretization scheme is presented for the special case of the nonlinear static behavior of end-loaded beams that, by virtue of the Kirchhoff analogy, leads to a time-marching scheme for the dynamics of a pivoted rigid body in a gravity field. This time-marching scheme conserves both the angular momentum about a vertical line passing through the pivot and total mechanical energy, whereas the analogous spatial discretization scheme for the nonlinear static behavior of end-loaded beams satisfies analogous integrals of deformation along the beam span. Remarkably, a straightforward generalization of these discretization schemes is shown to satisfy both space-time conservation laws for the nonlinear dynamics of beams when the applied loads are constant within a space-time element.

Geometrically Exact, Intrinsic Theory for Dynamics of Curved and Twisted Anisotropic Beams

On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results

The computer program VABS (Variational Asymptotic Beam Section Analysis) uses the variational asymptotic method to split a three-dimensional nonlinear elasticity problem into a twodimensional linear cross-sectional analysis and a one-dimensional, nonlinear beam problem. This is accomplished by taking advantage of certain small parameters inherent to beam-like structures. VABS is able to calculate the one-dimensional cross-sectional stiffness constants, with transverse shear and Vlasov refinements, for initially twisted and curved beams with arbitrary geometry and material properties. Several validation cases are presented. First, an elliptic bar is modeled with transverse shear refinement using the variational asymptotic method, and the solution is shown to be identical to that obtained from the theory of elasticity. The shear center locations calculated by VABS for various cross sections agree well with those obtained from common engineering analyses. Comparisons with other composite beam theories prove that it is unnecessary to introduce ad hoc kinematic assumptions to build an accurate beam model. For numerical validation, values of the one-dimensional variables are extracted from an ABAQUS model and compared with results from a one-dimensional beam analysis using cross-sectional constants from VABS. Furthermore, point-wise three-dimensional stress and strain fields are recovered using VABS, and the correlation with the three-dimensional results from ABAQUS is excellent. Finally, classical theory is shown to be insufficient for general-purpose beam modeling. Appropriate refined theories are recommended for some classes of problems.

Validation of the Variational Asymptotic Beam Sectional Analysis

Variational asymptotic beam sectional analysis – An updated version

Variational asymptotic method for unit cell homogenization of periodically heterogeneous materials

The generalized Timoshenko theory for composite beams embedded in the computer program VABS has the same structure as Timoshenko’s original theory for isotropic beams without the restrictive assumptions of the original theory. An overview of this theory is presented to show its general and rigorous framework. Certain theoretical details missing from previous developments are supplied, such as the proof of a kinematical identity and the expression of the recovery theory in terms of sectional stress resultants. It has been demonstrated that the VABS generalized Timoshenko theory reproduces the elasticity solution for the flexure problem of isotropic prisms. Numerical results are presented in support of the long-term validation effort, focusing especially on calculation of sectional stiffnesses (including shear correction factors) and shear center location, making use of the VABS model for composite beam analysis (including buckling and vibration), and recovering three-dimensional field variables over the section. The accuracy of the VABS generalized Timoshenko theory is demonstrated, and some of its practical advantages over three-dimensional finite element analysis are exhibited. Presented at the 44th Structures, Structural Dynamics and Materials Conference, Norfolk, Virginia, April 7 – 10, 2003 Assistant Professor, Department of Mechanical and Aerospace Engineering, Utah State University, Logan, Utah 84322-4130. Formerly, Post Doctoral Fellow, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia. Email: wenbin.yu@usu.edu. Professor, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia, 303320150. Email: dewey.hodges@ae.gatech.edu. Fellow, AHS.

Wenbin Yu

Papers

On Timoshenko-like modeling of initially curved and twisted composite beams

Validation of the Variational Asymptotic Beam Sectional Analysis

Variational asymptotic beam sectional analysis – An updated version

Variational asymptotic method for unit cell homogenization of periodically heterogeneous materials

Generalized Timoshenko Theory of the Variational Asymptotic Beam Sectional Analysis