Marco Petrolo

Beam Structures: Classical and Advanced Theories

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.

Unified Formulation Applied to Free Vibrations Finite Element Analysis of Beams with Arbitrary Section

This paper presents hierarchical finite elements on the basis of the Carrera Unified Formulation for free vibrations analysis of beam with arbitrary section geometries. The displacement components are expanded in terms of the section coordinates, (x, y), using a set of 1-D generalized displacement variables. N-order Taylor type expansions are employed. N is a free parameter of the formulation, it is supposed to be as high as 4. Linear (2 nodes), quadratic (3 nodes) and cubic (4 nodes) approximations along the beam axis, (z), are introduced to develop finite element matrices. These are obtained in terms of a few fundamental nuclei whose form is independent of both N and the number of element nodes. Natural frequencies and vibration modes are computed. Convergence and assessment with available results is first made considering different type of beam elements and expansion orders. Additional analyses consider different beam sections (square, annular and airfoil shaped) as well as boundary conditions (simply supported and cantilever beams). It has mainly been concluded that the proposed model is capable of detecting 3-D effects on the vibration modes as well as predicting shell-type vibration modes in case of thin walled beam sections.

Refined one-dimensional formulations for laminated structure analysis

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.

Guidelines and Recommendations to Construct Theories for Metallic and Composite Plates

This work has evaluated the refinement of some classical theories, such as the Kirchhoff and Reissner-Mindlin theories, adding generalized displacement variables (up to fourth-order) to the Taylor-type expansion in the thickness plate direction. Isotropic, orthotropic, and laminated plates have been analyzed, varying the thickness ratio, orthotropic ratio, and stacking sequence of the layout. Higher-order theories have been implemented according to the compact scheme known as the Carrera unified formulation. The results have been restricted to simply-supported orthotropic plates subjected to harmonic distributions of transverse pressure for which closed-form solutions are available. For a given plate problem (isotropic, orthotropic, or laminated), the effectiveness of each employed generalized displacement variable has been established comparing the error obtained accounting for and removing the variable in the plate governing equations. A number of theories have therefore been constructed imposing a given error with respect to the available best results. Guidelines and recommendations that are focused on the proper selection of the displacement variables that have to be retained in refined plate theories are then furnished. It has been found that the terms that have to be used according to a given error vary from problem to problem, but they also vary when the variable that has to be evaluated (displacement, stress components) is changed. Diagrams (errors in terms of geometrical and orthotopic ratios) and graphical schemes have been built to establish the appropriate theories with respect to the data of the problem under consideration.

Performance of CUF Approach to Analyze the Structural Behavior of Slender Bodies

This paper deals with the accurate evaluation of complete three-dimensional (3D) stress fields in beam structures with compact and bridge-like sections. A refined beam finite-element (FE) formulation is employed, which permits any-order expansions for the three displacement components over the section domain by means of the Carrera Unified Formulation (CUF). Classical (Euler-Bernoulli and Timoshenko) beam theories are considered as particular cases. Comparisons with 3D solid FE analyses are provided. End effects caused by the boundary conditions are investigated. Bending and torsional loadings are considered. The proposed formulation has shown its capability of leading to quasi-3D stress fields over the beam domain. Higher-order beam theories are necessary for the case of bridge-like sections. Various theories are also compared in terms of shear correction factors on the basis of definitions found in the open literature. It has been confirmed that different theories could lead to very different values of shear correction factors, the accuracy of which is subordinate to a great extent to the section geometries and loading conditions. However, an accurate evaluation of shear correction factors is obtained by means of the present higher-order theories.

Advanced Beam Formulations for Free-Vibration Analysis of Conventional and Joined Wings

This work extends advanced beam models to carry out a more accurate free-vibration analysis of conventional (straight, or with sweep/dihedral angles) and joined wings. The beam models are obtained by assuming higher-order (up to fourth) expansions for the unknown displacement variables over the cross-section. Higher-order terms permit bending/torsion modes to be coupled and capture any other vibration modes that require in-plane and warping deformation of the beam sections to be detected. Classical beam analyses, based on the Euler- Bernoulli and on Timoshenko beam theories, are obtained as particular cases. Numerical solutions are obtained by using the finite element (FE) method, which permits various boundary conditions and different wing/section geometries to be handled with ease. A comparison with other shell/solid FE solutions is given to examine the beam model. The capability of the beam model to detect bending, torsion, mixed and other vibration modes is shown by considering conventional and joined wings with different beam axis geometries as well as with various sections (compact, plate-type, thin-walled airfoil-type). The accuracy and the limitations of classical beam theories have been highlighted for a number of problems. It has been concluded that the proposed beam model could lead to quasi-three-dimensional dynamic responses of classical and nonclassical beam geometries. It provides better results than classical beam approaches, and it is much more computationally efficient than shell/solid modeling approaches. DOI: 10.1061/(ASCE)AS.1943-5525.0000130.

Refined 1D Finite Elements for the Analysis of Secondary, Primary, and Complete Civil Engineering Structures

This paper proposes the use of an advanced one-dimensional (1D) variable kinematic model to analyze typical civil engineering structures. The model is referred to as component-wise (CW), and it was originally introduced for the analysis of multilayered plate and shell structures. The CW approach is based on the Carrera unified formulation (CUF). CUF can be used for the straightforward development of a large variety of classical and refined beam theories that are able to capture three-dimensional stress/strain states and nonclassical phenomena such as the in-plane warping of the cross section. CUF can be seen as a hierarchical formulation because it has variable kinematic features; 1D models with arbitrarily chosen accuracy can be obtained, including the classical beam theories. CUF models are formulated in terms of fundamental nuclei, whose expressions are formally independent of the adopted hierarchical scheme. Lagrange polynomials are used in the proposed CW models to expand the displacement field of the beam above the cross section. The finite-element method was used in this work to obtain numerical solutions. The conducted numerical investigation shows that CW models can be successfully applied to any geometries with no restrictions on the ratio between the cross-sectional dimensions and the length of the beam. A very short C-shaped beam was therefore used to analyze a classical portal frame. Similarly, analyses of truss structures and a full industrial construction were carried out. Classical beam/plate/solid finite elements as well as combinations of them were used to obtain solutions from a commercial code for comparison purposes. The results show the effectiveness of the CW approach both in terms of accuracy and computational efficiency.

Component-wise Method Applied to Vibration of Wing Structures

This paper proposes advanced approaches to the free vibration analysis of reinforced-shell wing structures. These approaches exploit a hierarchical, one-dimensional (1D) formulation, which leads to accurate and computationally efficient finite element (FE) models. This formulation is based on the unified formulation (UF), which has been recently proposed by the first author and his coworkers. In the study presented in this paper, UF was used to model the displacement field above the cross-section of reinforced-shell wing structures. Taylor-like (TE) and Lagrange-like (LE) polynomial expansions were adopted above the cross-section. A classical 1D FE formulation along the wings span was used to develop numerical applications. Particular attention was given to the component-wise (CW) models obtained by means of the LE formulation. According to the CW approach, each wings component (i.e. spar caps, panels, webs, etc.) can be modeled by means of the same 1D formulation. It was shown that MSC/PATRANVR can be used as pre- and postprocessor for the CW models, whereas MSC/NASTRANVR DMAP alters can be used to solve the eigenvalue problems. A number of typical aeronautical structures were analyzed and CW results were compared to classical beam theories (Euler-Bernoulli and Timoshenko), refined models (TE) and classical solid/shell FE solutions from the commercial code MSC/NASTRANVR. The results highlight the enhanced capabilities of the proposed formulation. In fact, the CW approach is clearly the natural tool to analyze wing structures, since it leads to results that can only be obtained through 3D elasticity (solid) elements whose computational costs are at least one-order of magnitude higher than CW models

Classical, refined and component-wise analysis of reinforced-shell structures

This paper compares early and very recent approaches to the static analysis of reinforced-shell wing structures. Early approaches were those based on the pure semimonocoque theory along with the beam assumptions of the Euler–Bernoulli and Timoshenko type. The recent approaches are based on a hierarchical, one-dimensional formulation. These are obtained by adopting various polynomial expansions of the displacement field above the cross-section of the structure according to the unified formulation which was recently proposed by the first author. Two classes were developed in the unified formulation framework. In the first class, Taylor expansion models were developed by exploiting N-order Taylor-like polynomials; classical beam theories (Euler–Bernoulli and Timoshenko) were obtained as special cases of Taylor expansion. In the second class, Lagrange expansion models were built by means of four- and nine-point Lagrange-type polynomials over the cross-section of the wing. The component-wise approach was obtai...

Guidelines and Recommendations on the Use of Higher Order Finite Elements for Bending Analysis of Plates

This paper compares and evaluates various plate finite elements to analyse the static response of thick and thin plates subjected to different loading and boundary conditions. Plate elements are based on different assumptions for the displacement distribution along the thickness direction. Classical (Kirchhoff and Reissner-Mindlin), refined (Reddy and Kant), and other higher-order displacement fields are implemented up to fourth-order expansion. The Unified Formulation UF by the first author is used to derive finite element matrices in terms of fundamental nuclei which consist of 3×3 arrays. The MITC4 shear-locking free type formulation is used for the FE approximation. Accuracy of a given plate element is established in terms of the error vs. thickness-to-length parameter. A significant number of finite elements for plates are implemented and compared using displacement and stress variables for various plate problems. Reduced models that are able to detect the 3D solution are built and a Best Plate Diagram (BPD) is introduced to give guidelines for the construction of plate theories based on a given accuracy and number of terms. It is concluded that the UF is a valuable tool to establish, for a given plate problem, the most accurate FE able to furnish results within a certain accuracy range. This allows us to obtain guidelines and recommendations in building refined elements in the bending analysis of plates for various geometries, loadings, and boundary conditions.