Vladislav G. Sutyrin

Asymptotic theory for static behavior of elastic anisotropic I-beams

Abstract End effects for prismatic anisotropic beams with thin-walled, open cross-sections are analyzed by the variational-asymptotic method. The decay rates for disturbances at the ends of prismatic beams are evaluated, and the most influential end disturbances are incorporated into a refined beam theory. Thus, the foundations of Vlasovs theory, as well as restrictions on its applicability, are obtained from the variational-asymptotic point of view. Vlasovs theory is proved to be asymptotically correct for isotropic I-beams. The asymptotically correct generalization of Vlasovs theory for static behavior of anisotropic beams is presented. In light of this development, various published generalizations of Vlasovs theory for thin-walled anisotropic beams are discussed. Comparisons with a numerical 3-D analysis are provided, showing that the present approach gives the closest agreement of all published theories. The procedure can be applied to any thin-walled beam with open cross-sections.

Cross-sectional analysis of composite beams including large initial twist and curvature effects

An asymptotically exact methodology, based on geometrically nonlinear, three-dimensional elasticity, is presented for cross-sectional analysis of initially curved and twisted, nonhomogeneous, anisotropic beams. The analysis is subject only to the restrictions that the strain is small relative to unity and that the maximum dimension of the cross section is small relative to the wavelength of the deformation and to the minimum radius of curvature and/or twist. The final one-dimensional strain energy per unit length exhibits asymptotically correct second-order dependence of the initial curvature and twist parameters. Cross-sectional constants of the one-dimensional theory are obtained via finite element discretization over the cross-sectional plane. Numerical results obtained for both isotropic and composite beams are compared with published results from special purpose analyses for initially twisted, straight beams, as well as initially curved, untwisted beams. The agreement with previously published results is excellent.

On asymptotically correct linear laminated plate theory

Abstract The focus of this paper is the development of asymptotically correct theories for laminated plates, the material properties of which vary through the thickness and for which each lamina is orthotropic. This work is based on the variational-asymptotical method, a mathematical technique by which the three-dimensional analysis of plate deformation can be split into two separate analyses: a one-dimensional through-the-thickness analysis and a two-dimensional “plate” analysis. The through-the-thickness analysis includes elastic constants for use in the plate theory and approximate closed-form recovering relations for all three-dimensional field variables expressed in terms of plate variables. In general, the specific type of plate theory that results from this procedure is determined by the procedure itself. However, in this paper only “Reissner-like” plate theories are considered, often called first-order shear deformation theories. This paper makes three main contributions: first it is shown that construction of an asymptotically correct Reissner-like theory for laminated plates of the type considered is not possible in general. Second, a new point of view on the variational-asymptotical method is presented, leading to an optimization procedure that permits a derived theory to be as close to asymptotical correctness as possible. Third, numerical results from such an optimum Reissner-like theory are presented. These results include comparisons of plate displacement as well as of three-dimensional field variables and are the best of all extant Reissner-like theories. Indeed, they even surpass results from theories that carry many more generalized displacement variables.

Refined theory of composite beams: The role of short-wavelength extrapolation

The present paper presents an asymptotically-correct beam theory with nonclassical sectional degrees of freedom. The basis for the theory is the variational-asymptotical method, a mathematical technique by which the three-dimensional analysis of composite beam deformation can be split into a linear, two-dimensional, cross-sectional analysis and a nonlinear, one-dimensional, beam analysis. The elastic constants used in the beam analysis are obtained from the cross-sectional analysis, which also yields approximate, closed-form expressions for three-dimensional distributions of displacement, strain, and stress. Such theories are known to be valid when a characteristic dimension of the cross section is small relative to the wavelength of the deformation. However, asymptotically-correct refined theories may differ according to how they are extrapolated into the short-wavelength regime. Thus, there is no unique asymptotically-correct refined theory of higher order than classical (Euler-Bernoulli-like) theory. Different short-wavelength extrapolations can be obtained by changing the meaning of the theorys one-dimensional variables. Numerical results for the stiffness constants of a refined beam theory and for deformations from the corresponding one-dimensional theory are presented. It is shown that a theory can be asymptotically correct and still have non-positive-definite strain energy density, which is completely inappropriate mathematically and physically. A refined beam theory, which appropriately possesses a positive-definite strain energy density and agrees quite well with experimental results, is constructed by using a certain short-wavelength extrapolation.

Refined theory of twisted and curved composite beams - The role of short-wavelength extrapolation

The present paper continues earlier work of the authors, in which an asymptotically-correct beam theory with nonclassical sectional degrees of freedom was developed. The basis for the theory is the variational-asymptotical method, a mathematical technique by which the three-dimensional analysis of composite beam deformation can be split into a linear, two-dimensional, cross-sectional analysis and a nonlinear, one-dimensional, beam analysis. The elastic constants used in the beam analysis are obtained from the crosssectional analysis, which also yields approximate, closedform expressions for three-dimensional distributions of displacement, strain, and stress. Such theories are known to be valid when a characteristic dimension of the cross section is small relative to the wavelength of the deformation. However, asymptotically-correct refined theories may differ according to how they are extrapolated into the shortwavelength regime. Thus, there is n o unique asymptoticallycorrect refined theory of higher order than classical (EulerBernoulli-like) theory. Different short-wavelength extrapolations can be obtained by changing the meaning of the theorys one-dimensional variables. Numerical results for the stiffness constants of refined beam theories and for deformations from corresponding one-dimensional theory are presented. It is shown that a theory can be asymptotically correct and still have non-positive-definite strain energy density, which is completely inappropriate mathematically and physically. A refined beam theory, which appropriately possesses a positive-definite strain energy density and agrees quite well with experimental results, is constructed by using a certain short-wavelength extrapolation.

Construction of Dynamical Theories for Elastic Anisotropic Beams

A general framework for constructing a refined beam theory is presented. Such refined theory is capable of describing end-effects and high-frequency vibrations. This is achieved by the introduction of new 1-D variables in addition to four classical ones associated with extension, torsion and bending in two orthogonal directions. These additional 1-D variables represent high-frequency modes of vibration with corresponding cross-sectional displacement field calculated by solving appropriate 2-D problems over the cross section. Next, the asymptotic analysis is conducted to ensure correct description of long wavelength vibrations. The need of the correct short-wave extrapolation is illustrated for various modes of vibrations. Timoshenko and Vlasov theories are cast in terms of this general framework. The numerical results of the applications of this theory will be reported later.

End effects in thin-walled beams

The nature of end effects for prismatic beams with thin-walled, open cross sections is analyzed by the variational-asymptotic method. This way, the decay rates for disturbances at the ends of prismatic beams are correctly evaluated. Thus, the foundations of Vlasovs theory, as well as restrictions on its applicability, are obtained from the variational-asymptotic point of view. Vlasovs theory is proved to be asymptotically correct for beams with thin-walled open cross-sections, such as I-beams. Finally, a generalization of Vlasovs theory for thin-walled anisotropic beams is provided.