1D Carrera Unified Formulation Research Articles

Vibration suppression of thick plates with attached piezoelectric patches using 1D Carrera Unified Formulation (CUF) based finite element models

The first six vibration modes of a thick plate with 10 piezoelectric sensor-actuator pairs are suppressed using advanced beam elements based on the one-dimensional Carrera Unified Formulation (CUF). A three-dimensional model is developed by combining one-dimensional beam elements with two-dimensional Lagrange expansions in this paper. For the reduction of vibration of a thick cantilever plate, a PIDF type control is designed and tuned. Based on the results of the simulation, it can be concluded that the simulation of thick plate models with impulse inputs is efficient as well after model order reduction.

GA Optimization of Variable Angle Tow Composites in Buckling and Free Vibration Analysis through Layerwise Theory

In the current research, variable angle tow composites are used to improve the buckling and free vibration behavior of a structure. A one-dimensional (1D) Carrera Unified Formulation (CUF) is employed to determine the buckling loads and natural frequencies in Variable Angle Tow (VAT) square plates by taking advantage of the layerwise theory (LW). Subsequently, the Genetic Algorithm (GA) optimization method is applied to maximize the first critical buckling load and first natural frequency using the definition of linear fiber orientation angles. To show the power of the genetic algorithm for the VAT structure, a surrogate model using Response Surface (RS) method was used to demonstrate the convergence of the GA approach. The results showed the cost reduction for optimized VAT performance through GA optimization in combination with the 1D CUF procedure. Additionally, a Latin hypercube sampling (LHS) method with RS was used for buckling analysis. The capability of LHS sampling confirmed that it could be employed for the next stages of research along with GA.

Evaluation of bending and post-buckling behavior of thin-walled FG beams in geometrical nonlinear regime with CUF

In this paper, the nonlinear behavior of beams with one-dimensional (1D) Functionally Graded Materials (FGM) is investigated by Carrera Unified Formulation (CUF). In the study, kinematic variants of CUF are extended with Lagrange Extension (LE), thus adopting Layer-Wise (LW) approach. In the CUF formulation combined with geometric nonlinear equations, the Lagrangian approximation and Newton–Raphson linearization scheme are used with the method based on arc length constraint. It is assumed that the variation of material properties in the thickness direction follows an exponential grading. The displacement and stress distributions of the Functionally Graded (FG) cantilever beam under transverse/axial loading are investigated for different compositional gradient exponents. The three-dimensional (3D) stress distributions of FG beams are investigated with the 1D CUF model and the results are confirmed by the literature. The effect of the compositional gradient exponent on the mechanical behavior is expressed and it is emphasized that the combined formulation is time-efficient and highly sensitive in determining the nonlinear behavior.

Nonlinear analysis of compact and thin-walled metallic structures including localized plasticity under contact conditions

This work presents the numerical analysis of elastoplastic contact problems of compact and thin-walled metallic structures. The emphasis is on the use of higher-order 1D elements with pure displacement variables and based on the Carrera Unified Formulation (CUF) to capture localized effects and cross-sectional distortions. Contact interactions are normal and frictionless via a node-to-node contact algorithm with the penalty approach for contact enforcement. The analysis considers the material nonlinearity via the von Mises constitutive law. Numerical assessments compare the CUF solutions with 3D finite element analysis concerning the solution quality, computational size, and analysis time. The results show the ability of 1D CUF models of accurately evaluating localized deformations and plasticity. The CUF results are in good agreement with reference 3D finite element solutions, and require an order of magnitude fewer degrees of freedom and analysis time, making them computationally efficient.

Computationally efficient, high-fidelity micromechanics framework using refined 1D models

A novel micromechanical framework based on higher-order refined beam models is presented. The micromechanical framework is developed within the scheme of the Carrera Unified Formulation (CUF), a hierarchical formulation which provides a framework to obtain refined structural theories via a variable kinematic description. The Component-Wise approach (CW), a recent extension of one-dimensional (1D) CUF models, is utilized to model components within the representative volume element (RVE). CW models employ Lagrange-type polynomials to interpolate the kinematic field over the element cross-sections of the beams and efficiently handle the analysis of multi-component structures such as RVE. The governing equations are derived in the weak form using finite element method. The framework derives its efficiency from the ability of CUF models to produce accurate 3D displacement and stress fields at a reduced computational cost. Three different cases of micromechanical homogenization are presented to demonstrate the efficiency and high-fidelity of the proposed framework. The results are validated through published literature results and via the commercial software ABAQUS. The capability of CUF-CW models to accurately predict the overall elastic moduli along with the recovery of local 3D fields is highlighted.

Application of aerospace structural models to marine engineering

The large container ships and fast patrol boats are complex marine structures. Therefore, their global mechanical behaviour has long been modeled mostly by refined beam theories. Important issues of cross section warping and bending-torsion coupling have been addressed by introducing special functions in these theories with inherent assumptions and thus compromising their robustness. The 3D solid Finite Element (FE) models, on the other hand, are accurate enough but pose high computational cost.In this work, different marine vessel structures have been analysed using the well-known Carrera Unified Formulation (CUF). According to CUF, the governing equations (and consequently the finite element arrays) are written in terms of fundamental nuclei that do not depend on the problem characteristics and the approximation order. Thus, refined models can be developed in an automatic manner. In the present work, a particular class of 1D CUF models that was initially devised for the analysis of aircraft structures has been employed for the analysis of marine structures. This class, which was called Component-Wise (CW), allows one to model complex 3D features, such as inclined hull walls, floors and girders in the form of components. Realistic ship geometries were used to demonstrate the efficacy of the CUF approach. With the same level of accuracy achieved, 1D CUF beam elements require far less number of Degrees of Freedom (DoFs) compared to a 3D solid FE solution.

3D thermoelastic analysis of rotating disks having arbitrary profile based on a variable kinematic 1D finite element method

ABSTRACTA variable kinematic 1D finite element (FE) method is presented for 3D thermoelastic analysis of rotating disks with variable thickness. The principle of minimum potential energy is used to derive general governing equations of the disks subjected to body forces, surface forces, concentrated forces, and thermal loads. To solve the equations, the 1D Carrera unified formulation (CUF), which enables to go beyond the kinematic assumptions of classical beam theories, is employed. Based on the 1D CUF, the disk is considered as a beam, which can be discretized into a finite number of 1D elements along its axis. The displacement field over the beam’s cross section is approximated by Lagrange expansions. This methodology leads to an FE formulation that is invariant with respect to the order of expansions used over the cross sections, and thus the 3D problem reduces to a 1D problem. The effect of the cross section discretization on displacement and stress fields is investigated. Results obtained from this method are in good agreement with the reference analytical and finite difference solutions. The proposed innovative method can be very effective in the thermoelastic analysis of rotating disks.

A refined finite element method for stress analysis of rotors and rotating disks with variable thickness

In this paper, a refined finite element method (FEM) based on the Carrera unified formulation (CUF) is extended for stress analysis of rotors and rotating disks with variable thickness. The variational form of the 3D equilibrium equations is obtained using the principle of minimum potential energy and solved by this method. Employing the 1D CUF, a rotor is assumed to be a beam along its axis. In this case, the geometry of the rotor can be discretized into a finite number of 1D beam elements along its axis, while the Lagrange polynomial expansions may be employed to approximate the displacement field over the cross section of the beam. Therefore, the FEM matrices and vectors can be written in terms of fundamental nuclei, whose forms are independent of the order of the beam theories. The validity and capabilities of the presented procedure are investigated in a number of numerical examples, and some conclusions are reported and compared well with the analytical and 3D finite element solutions. The results obtained by the 1D CUF models are in close agreement with the reference solutions. Moreover, it is verified that the innovative procedure presented in this paper can be used as an accurate tool of structural analysis for complex rotors to reduce the computational costs.

Evaluation of energy and failure parameters in composite structures via a Component-Wise approach

This paper deals with the static analysis of fiber reinforced composites via the Component-Wise approach (CW). The main aim of this work is the investigation of the CW capabilities for the evaluation of integral quantities such as the strain energy, or integral failure indexes. Such quantities are evaluated in the global structures and local volumes. The integral failure indexes, in particular, are proposed as alternatives to point-wise failure indexes. The CW approach has been recently developed as an extension of the 1D Carrera Unified Formulation (CUF). The CUF provides hierarchical higher-order structural models with arbitrary expansion orders. In this work, Lagrange-type polynomials are used to interpolate the displacement field over the element cross-sections. The CW makes use of the 1D CUF finite elements to model simultaneously different scale components (fiber, matrix, laminae and laminates) with a reduced computational cost. CW models do not require the homogenization of the material characteristics nor the definition of mathematical lines or surfaces. In other words, the material characteristics of each component, e.g. fibers and matrix, are employed, and the problem unknowns are placed above the physical surface of the body. In the perspective of failure analyses, the integral evaluation of failure parameters is introduced to determine critical portions of the structure where failure could take place. Integral quantities are evaluated using 3D integration sub-domains that may cover macro- and micro-volumes of the structure. The integral quantities can be evaluated directly on fiber and matrix portions. Numerical results are provided for different configurations and compared with solid finite element models. The results prove the accuracy of the CW approach and its computational efficiency. In particular, 3D local effects can be detected. The use of the integral failure index provides qualitatively reliable results; however, experimental campaigns should be carried out to relate such indexes to the failure occurrence.

High-Fidelity and Computationally Efficient Component-Wise Structural Models: An Overview of Applications and Perspectives

The Component-Wise approach (CW) is a novel structural modeling strategy that stemmed from the Carrera Unified Formulation (CUF). This work presents an overview of the enhanced capabilities of the CW for the static and dynamic analysis of structures, such as aircraft wings, civil buildings, and composite plates. The CW makes use of the advanced 1D CUF models. Such models exploit Lagrange polynomial expansions (LE) to model the displacement field above the cross-section of the structure. The use of LE allows the improvement of the 1D model capabilities. LE models provide 3D-like accuracies with far fewer computational costs. The use of LE leads to the CW. Although LE are 1D elements, every component of an engineering structure can be modeled via LE elements independently of their geometry, e.g. 2D transverse stiffeners and panels, and of their scale, e.g. fiber/matrix cells. The use of the same type of finite elements facilitates the finite element modeling to a great extent. For instance, no interface techniques are necessary. Moreover, in a CW model, the displacement unknowns are placed along the physical surfaces of the structure with no need for artificial lines and surfaces. Such a feature is promising in a CAD/FEM coupling scenario. The CW approach can be considered as an accurate and computationally cheap analysis tool for many structural problems. Such as progressive failure analyses, multiscale, impact problems and health-monitoring.

Recent developments on refined theories for beams with applications

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

Static analyses of FGM beams by various theories and finite elements

The 1D Carrera Unified Formulation (CUF) is here used to perform static analyses of functionally graded (FG) structures. The hierarchical feature of CUF allows one to automatically generate an infinite number of displacement theories that may include any kind of functions of the cross-section coordinates (x, z), among which those used to describe the variation of the mechanical properties of FG materials. The governing equations are derived by means of the Principle of Virtual Displacements in a weak form and solved by means of the Finite Element method (FEM). The equations are written in terms of “fundamental nuclei”, whose forms do not depend on the used expansions. Trigonometric, polynomial, exponential and miscellaneous expansions are here used and evaluated for various structural problems. Resulting theories are assessed by considering several aspect-ratios, gradation laws, loading and boundary conditions. The results are compared with 1-, 2- and 3-D solutions both in terms of displacements and stress distributions. The comparisons confirm that the 1D CUF elements are valuable tools for the study of FG structures.

Free Vibration Response of Thin and Thick Nonhomogeneous Shells by Refined One-Dimensional Analysis

The free vibration analysis of thin- and thick-walled layered structures via a refined one-dimensional (1D) approach is addressed in this paper. Carrera unified formulation (CUF) is employed to introduce higher-order 1D models with a variable order of expansion for the displacement unknowns over the cross section. Classical Euler–Bernoulli (EBBM) and Timoshenko (TBM) beam theories are obtained as particular cases. Different kinds of vibrational modes with increasing half-wave numbers are investigated for short and relatively short cylindrical shells with different cross section geometries and laminations. Numerical results of natural frequencies and modal shapes are provided by using the finite element method (FEM), which permits various boundary conditions to be handled with ease. The analyses highlight that the refinement of the displacement field by means of higher-order terms is fundamental especially to capture vibrational modes that require warping and in-plane deformation to be detected. Classical beam models are not able to predict the realistic dynamic behavior of shells. Comparisons with three-dimensional elasticity solutions and solid finite element solutions prove that CUF provides accuracy in the free vibration analysis of even short, nonhomogeneous thin- and thick-walled shell structures, despite its 1D approach. The results clearly show that bending, radial, axial, and also shell lobe-type modes can be accurately evaluated by variable kinematic 1D CUF models with a remarkably lower computational effort compared to solid FE models.

Nonhomogeneous atherosclerotic plaque analysis via enhanced 1D structural models

The static analysis of structures with arbitrary cross-section geometry and material lamination via a refined one-dimensional (1D) approach is presented in this paper. Higher-order 1D models with a variable order of expansion for the displacement field are developed on the basis of Carrera Unified Formulation (CUF). Classical Euler-Bernoulli and Timoshenko beam theories are obtained as particular cases of the first-order model. Numerical results of displacement, strain and stress are provided by using the finite element method (FEM) along the longitudinal direction for different configurations in excellent agreement with three-dimensional (3D) finite element solutions. In particular, a layered thin-walled cylinder is considered as first assessment with a laminated conventional cross-section. An atherosclerotic plaque is introduced as a typical structure with arbitrary cross-section geometry and studied for both the homogeneous and nonhomogeneous material cases through the 1D variable kinematic models. The analyses highlight limitations of classical beam theories and the importance of higher-order terms in accurately detecting in-plane cross-section deformation without introducing additional numerical problems. Comparisons with 3D finite element solutions prove that 1D CUF provides remarkable three-dimensional accuracy in the analysis of even short and nonhomogeneous structures with arbitrary geometry through a significant reduction in computational cost.

Static Aeroelastic Response of Wing-Structures Accounting for In-Plane Cross-Section Deformation

In this paper, the aeroelastic static response of flexible wings with arbitrary cross-section geometry via a coupled CUF-XFLR5 approach is presented. Refined structural one-dimensional (1D) models, with a variable order of expansion for the displacement field, are developed on the basis of the Carrera Unified Formulation (CUF), taking into account cross-sectional deformability. A three-dimensional (3D) Panel Method is employed for the aerodynamic analysis, providing more accuracy with respect to the Vortex Lattice Method (VLM). A straight wing with an airfoil cross-section is modeled as a clamped beam, by means of the finite element method (FEM). Numerical results present the variation of wing aerodynamic parameters, and the equilibrium aeroelastic response is evaluated in terms of displacements and in-plane cross-section deformation. Aeroelastic coupled analyses are based on an iterative procedure, as well as a linear coupling approach for different free stream velocities. A convergent trend of displacements and aerodynamic coefficients is achieved as the structural model accuracy increases. Comparisons with 3D finite element solutions prove that an accurate description of the in-plane cross-section deformation is provided by the proposed 1D CUF model, through a significant reduction in computational cost.

Free vibration of FGM layered beams by various theories and finite elements

The Carrera Unified Formulation (CUF) is used to perform free-vibrational analyses of functionally graded (FG) structures. CUF is a hierarchical formulation for obtaining refined structural theories that account for variable kinematic description. These theories can be obtained by expanding the unknown displacement variables over the beam section axes by adopting any kind of function. The number of the terms in the expansions is a free parameter of the analysis. For Taylor-like expansions, the linear case can result in classical beam theories. For the first time in the 1D CUF framework, the Finite Element method is used to solve the governing equations of functionally graded beams which are derived in a weak form by means of the Principle of Virtual Displacements. These equations are written in terms of f¨undamental nuclei.¨ Their forms do not depend on the expansions used. Several structures are considered, including a sandwich beam with FG core, laminated beams, thin- and thick-walled boxes as well as sandwich cylinders. The results are shown in terms of natural frequencies and compared with those available in existing literature.

Dynamic response of thin-walled structures by variable kinematic one-dimensional models

This paper investigates the accuracy capabilities of using variable kinematic modeling in compact and thin-walled beam-like structures with dynamic loadings. Carrera Unified Formulation (CUF) is employed to introduce refined one-dimensional (1D) models with a variable order of expansion for the displacement unknowns over the beam cross-section. Classical Euler–Bernoulli and Timoshenko beam theories are obtained as particular cases of these variable kinematic models while a higher order expansion permits the detection of in-plane cross-section deformation, since it leads to shell-like solutions. Finite element (FE) method is used to provide numerical results and the Newmark method is implemented as a time integration scheme. Some assessments with closed form solutions are discussed and comparisons with shell-type results obtained with commercial FE software are made. Further analyses address both compact and thin-walled cross-sections. In particular, the case of a deformable thin-walled cylinder loaded by time-dependent internal forces is discussed. The results clearly show that finite elements which are formulated in the CUF framework do not introduce additional numerical problems with respect to classical beam theories. Comparisons with elasticity solutions prove that the present 1D CUF model offers an accuracy in analyzing thin-walled structures which is typical of shell or three-dimensional models with a remarkable reduction in the computational cost required.

Flutter analysis of composite lifting surfaces by the 1D Carrera Unified Formulation and the doublet lattice method

Flutter analyses of composite lifting surfaces are presented in this paper. Results were obtained through an advanced aeroelastic formulation based on higher-order 1D structural models coupled with the doublet lattice method. The g-method was used to compute flutter conditions. The Carrera Unified Formulation (CUF) was exploited to build 1D structural models. Refined theories were obtained by using Taylor-like expansions of the cross-section displacement field. In the CUF framework, the order (N) of the expansion is considered as a free-parameter, this means that N can be considered as an input of the analysis. Convergence studies on N can be straightforwardly conducted in order to establish the proper 1D theory for a given problem. Flutter analyses were conducted on several structural configurations. The effect of the stacking sequence and the effect of the sweep angle were analyzed. Results show the enhanced capabilities of CUF 1D in dealing with the flutter analysis of typical composite lifting surfaces with plate-like accuracy and low computational costs.