This paper proposes one-dimensional formulations based on hierarchical expansions of the unknown displacement variables for the analysis of multilayered structures made of anisotropic composite layers. The hierarchical technique shows variable kinematic properties and it is based on the Carrera unified formulation. Two different classes of refined theories are proposed: the first expands the unknown variables in terms of power polynomials of the cross-sectional coordinates (it consists of a Taylor-like expansion); the second class of onedimensional theories uses Lagrange polynomials (Lagrange expansion) and subdomain discretizations of the cross section, and it leads to only pure displacements as the unknown variables. Taylor-like expansion is used to develop equivalent single-layer formulations, and Lagrange expansion is used to construct both equivalent single-layer and layerwise descriptions. The finite element method is employed to develop numerical applications. Using the Carrera unified formulation, finite element matrices are obtained in terms of a few fundamental nuclei that are formally independent of all the considered one-dimensional formulations. A number of numerical examples are given concerning on beams, plates, and more complex structures. Comparisons with results from plate and solid models are provided. The following has been concluded: 1) The proposed formulation represents a reliable, compact, and accurate method to develop refined one-dimensional models. 2) The present one-dimensional models are very effective at detecting both global and local responses of composite structures. 3) Shell-and solidlike results are obtained with a significant reduction in the computational costs.

Refined One-Dimensional Formulations for Laminated Structure Analysis

Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory

This paper is a review of the most influential approaches to developing beam models that have been proposed over the last few decades. Essentially, primary attention has been paid to isotropic structures while a few extensions to composites have been given for the sake of completeness. Classical models - Da Vinci, Euler-Bernoulli, and Timoshenko - are described first. All those approaches that are aimed at the improvement of classical theories are then presented by considering the following main techniques: shear correction factors, warping functions, Saint-Venant based solutions and decomposition methods, variational asymptotic methods, the Generalized Beam Theory and the Carrera Unified Formulation (CUF). Special attention has been paid to the latter by carrying out a detailed review of the applications of 1D CUF and by giving numerical examples of static, dynamic and aeroelastic problems. Deep and thin-walled structures have been considered for aerospace, mechanical and civil engineering applications. Furthermore, a brief overview of two recently introduced methods, namely the mixed axiomatic/asymptotic approach and the component-wise approach, has been provided together with numerical assessments. The review presented in this paper shows that the development of advanced beam models is still extremely appealing, due to the computational efficiency of beams compared to 2D and 3D structural models. Although most of the techniques that have recently been developed are focused on a given number of applications, 1D CUF offers the breakthrough advantage of being able to deal with a vast variety of structural problems with no need for ad hoc formulations, including problems that can notoriously be dealt with exclusively by means of 2D or 3D models, such as complete aircraft wings, civil engineering constructions, as well as multiscale and wave propagation analyses. Moreover, 1D CUF leads to a complete 3D geometrical and material modeling with no need of artificial reference axes/surfaces, reduced constitutive equations or homogenization techniques

Recent developments on refined theories for beams with applications

Adopting the scaling functions of B-spline wavelet on the interval (BSWI) as trial functions, a new finite element method (FEM) of BSWI is presented. Instead of traditional polynomial interpolation, scaling functions at the certain scale have been adopted to form the shape functions and construct wavelet-based elements. Unlike the process of wavelets added directly in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space via the corresponding transformation matrix. The transformation matrix is the key to construct wavelet-based elements freely as long as we can ensure its non-singularity. Then, classes of C0 and C1 type elements are constructed. And the lifting scheme of BSWI elements is also discussed. The numerical examples indicate that the BSWI elements have higher efficiency and precision than traditional finite element method in solving 1D structural problems especially for geometric nonlinear, variable cross-section and loading cases.

The construction of 1D wavelet finite elements for structural analysis

In this paper a boundary element method is developed for the nonuniform torsion of simply or multiply connected prismatic bars of arbitrary cross section The bar is subjected to an arbitrarily distributed twisting moment, while its edges are restrained by the most general linear torsional boundary conditions Since warping is prevented, beside the Saint–Venant torsional shear stresses, the warping normal and shear stresses are also computed Three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions are formulated and solved employing a BEM approach Both the warping and the torsion constants using only boundary discretization together with the torsional shear stresses and the warping normal and shear stresses are computed Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy The magnitude of the warping shear stresses due to restrained warping is investigated by numerical examples with great practical interest

https://msvlab.hre.ntou.edu.tw/CM-Sapo-2003.pdf

Warping shear stresses in nonuniform torsion by BEM

Warping shear stresses in nonuniform torsion of composite bars by BEM

In this paper a boundary element method is developed for the solution of the general transverse shear loading problem of beams of arbitrary simply or multiply connected constant cross section. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross section, while its axes are not necessarily the principal ones. The transverse shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. Two boundary value problems that take into account the effect of Poisson’s ratio are formulated with respect to stress functions and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the transverse shear stresses at any interior point is accomplished by direct differentiation of these stress functions, while both the coordinates of the shear center and the shear deformation coefficients are obtained from these functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the proposed shear deformation coefficients compared with those obtained from a 3-D FEM solution of the ‘exact’ elastic beam theory is remarkable.

V. G. Mokos

Papers

Warping shear stresses in nonuniform torsion by BEM

Warping shear stresses in nonuniform torsion of composite bars by BEM

A BEM solution to transverse shear loading of beams

3-D beam element of composite cross section including warping and shear deformation effects

Secondary torsional moment deformation effect by BEM