Framework Of Carrera Unified Formulation Research Articles

Mesh objective characteristic element length for higher-order finite beam elements

The use of fracture energy regularization techniques can effectively mitigate the mesh dependency of numerical solutions caused by the strain softening behavior of quasi-brittle materials. However, the successful regularization depends on the correct estimation of the crack bandwidth in Finite Element solutions. This paper aims to present an enhanced crack band formulation to overcome the strain localization instability especially for the higher-order elements developed in the framework of Carrera Unified Formulation (CUF). Besides, a modified Mazars damage method incorporating fracture energy regularization is employed to describe the nonlinear damage behavior of the concrete. To evaluate the efficiency of the proposed crack band formulation, three experimental concrete benchmarks are selected for the numerical damage analysis. By comparing numerical and experimental results, the proposed method can guarantee mesh objectivity despite varying finite element numbers and orders, indicating perseved fracture energy consumption within proposed higher-order beam models.

Free vibration analysis of composite beams and laminated reinforced panels by refined dynamic stiffness method and CUF-based component-wise theory

This paper presents a comprehensive analysis of the free vibration behavior of composite beams and laminated reinforced panels. Employing high-order theories with displacement variables only, the investigation combines dynamic stiffness method (DSM) and the Carrera unified formulation (CUF). The 3D displacement field can be expanded in the framework of CUF as any order of generic unknown variables over the cross-section, in the case of beam theories. Specifically, Lagrange expansions (LE) of cross-sectional displacement variables are considered, enabling the refinement modeling of complex cross-sections with different layers (layer-wise) and components (component-wise). The governing differential equations and natural boundary conditions are derived using the principle of virtual displacements. Subsequently, an exact dynamic stiffness matrix is developed by establishing a relationship between the amplitudes of harmonically varying loads and the corresponding responses. The Wittrick–Williams algorithm is employed to solve the transcendental eigenvalue problem resulting from this approach. The Lagrange-based CUF(LE)-DSM outperforms other polynomials (such as Taylor)-based CUF in analyzing composite structures, enabling detailed analysis with various geometries, lamination schemes and boundary conditions. It delivers accurate results in reinforced panel analysis, requiring only 13% of the DOFs compared to 3D FE models, thereby reducing computational costs significantly, as confirmed by comparative studies and validations.

High-order effects on the vibration of membrane structures

This study analyzed pretension membranes, focusing on high-order effects. Using a combination of the Carrera Unified Formulation (CUF) and the finite element method, the research employs the unique framework of CUF to integrate innovative structural theories in a seamlessly scalable way. The analysis methodically explores the complex effects of shear and bending stiffness on crucial factors such as thickness, geometry, and materials. A comprehensive numerical examination of the subject is undertaken on numerous cases, and the assessment includes established analytical solutions as well as findings provided by commercial software tools. Particular emphasis is placed on mode aberration, underscoring the critical need for more refined structural models. Furthermore, the results show the crucial role of adopting high-order theory in the dynamic analysis of membrane structure, especially for application in space.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

1D higher-order theories for quasi-static progressive failure analysis of composites based on a full 3D Hashin orthotropic damage model

This work investigates the crack propagation in composites by adopting a novel full three-dimensional (3D) Hashin-based orthotropic damage model combined with higher-order one-dimensional (1D) finite elements based on the Carrera Unified Formulation (CUF). Previous literature has proven that CUF provides structural formulations with great accuracy and improved computational efficiency. Moreover, a Layer-Wise (LW) formulation can be implemented within the CUF framework, allowing an accurate description of the 3D stress state in composite laminate, representing crucial information for progressive failure analysis. A Newton–Raphson predictor–corrector algorithm is used for the numerical solution of classical case tests, i.e., compact tension and three-point bending tests. The obtained results are compared with experimental outcomes and with solutions from well-established 2D damage models and a 3D Abaqus numerical model, demonstrating the capability of the proposed method to efficiently capture both the failure load and shape of the crack pattern.

Thermomechanical analysis of carbon fiber-reinforced polymer representative volume elements with voids

This work presents results of numerical simulations to investigate the effect of different void percentages on composite materials’ Coefficient of Thermal Expansion (CTE) and local stress fields. A random distribution of voids is considered within the Representative Volume Element (RVE) matrix, and different types of microstructures are considered, including square-packed and randomly distributed fibers. The use of a higher-order beam model within the framework of Carrera Unified Formulation (CUF) leads to a Component-Wise (CW) approach, resulting in an accurate, 3D description of the cross-section although using a 1D formulation. Numerical results for different fiber volume fractions and void concentration percentages demonstrate the agreement of the computed effective coefficients of thermal expansion of the present micromechanical thermoelastic model with references from the literature. The local stress fields are affected by voids, with higher effects over the matrix. Furthermore, higher void fractions lead to higher variability of stress peaks.

Flutter analysis of delaminated composite box-beam using higher-order kinematics

The flutter analysis of delaminated box-beam has been performed using the Lagrange polynomial in Carrera Unified Formulation (CUF) framework. In this formulation, the cross-sectional displacement field of each layer can be defined by the arbitrary displacement variables using the Layerwise (LW) approach. The cross-section of the present structural model is discretized using the quadratic elements. The unsteady aerodynamic contribution has been obtained using Theodorsen’s lift function. Subsequently, the structural and aeroelastic model has been defined using the Principle of Virtual Displacement (PVD) in the CUF framework. For verification of the present model, free variation analysis has been performed for delaminated and healthy composite structures. Furthermore, a composite box-beam model with single and multiple delaminations has been developed to perform free vibration and flutter analysis. This study supports the capability of the CUF model to perform flutter analysis of the complex model with multi-delaminated composite box-beam.

Optimum material distribution of porous functionally graded plates using Carrera unified formulation based on isogeometric analysis

The current literature deals with the multi-objective optimal material distribution of porous functionally graded plates subjected to free vibration in the framework of Carrera Unified Formulation (CUF) based on isogeometric analysis (IGA). The objective function is to maximize the natural frequencies and minimize the weight of the PFG plate, considering the material properties and porosity through the thickness of the plate. The governing equation of the PFG plate is obtained by using the principle of virtual displacement (PVD) based on the CUF. A non-uniform rational B-splines (NURBS)-based isogeometric finite element model associated with CUF is utilized for the plate free vibration analysis. The Legendre polynomials are used to approximate the thickness direction of plates. The rule of mixture is used to estimate the effective material properties. The Multi-Objective Particle Swarm Optimization (MOPSO) algorithm is employed as an optimizer. Design variables are the power and porosity indices. For verification, several structural design examples are compared with those available in the literature. Several numerical examples for optimizing the PFG plates, including maximization of the first natural frequency and minimization of mass, illustrate the effectiveness and reliability of the proposed method. The results show that the optimum values for the power and porosity indices depend on the objective function and boundary conditions. Furthermore, the clamped boundary condition leads to less mass than the simply-supported boundary condition for each desired first natural frequency.

Flutter analysis of laminated composite structures using Carrera Unified Formulation

In present work, the flutter analysis of laminated composite structures has been performed using the p-k method in Carrera Unified Formulation (CUF). In the framework of CUF, a hierarchical kinematic finite element model is used to compute the flutter condition of laminated composite plate and box-beam structures as it is very accurate and computationally efficient. The CUF refined theories are based on the Lagrange and Taylor-like cross-sectional displacement fields. In CUF, the order of the expansion can be chosen arbitrary, which is an independent parameter in the formulation. The governing equation is based on the principle of virtual displacement and defined in the form of “fundamental nuclei” using CUF. Theodorsen theory was used to define the aerodynamics loading conditions and the p-k method was used to compute the flutter conditions. Flutter conditions of different types of laminated composite structures with Lagrange and Taylor expansion were performed. A similar model was developed in MSC-Nastran and computed results were compared with literature and CUF model. The results indicate that the analyzed model has good agreement with reference and MSC-Nastran. The study suggests that the CUF models can produce accurate results with a low computational cost.

Shell finite element models with local kinematic refinements based on Reissner’s Mixed Variational Theorem with layer-wise descriptions

This work presents refined shell finite element (FE) models based on Reissner’s Mixed Variational Theorem (RMVT) with layer-wise (LW) Node-Dependent Kinematics (NDK) for the analysis of multi-layered composite structures. LW descriptions of displacements and transverse stresses employing hierarchical Legendre polynomials (HLP) are adopted. Developed in the framework of Carrera Unified Formulation (CUF), adaptable refinement through hierarchical shell elements with NDK is extended to the RMVT formulations. The numerical accuracy and efficiency are assessed by comparing the FE solutions for bending shells against references available in the literature or from modeling with commercial software.

Layerwise mixed elements with node-dependent kinematics for global–local stress analysis of multilayered plates using high-order Legendre expansions

Carrera Unified Formulation (CUF) is taken a step further to render node-dependent kinematics (NDK) capabilities to new layerwise finite elements based on Reissner’s Mixed Variational Theorem (RMVT), especially suited for global–local stress analysis of multilayered plates, ensuring high numerical accuracy and computational efficiency, all together. In the framework of CUF, as introduced originally for multilayered structures, any degree of kinematic refinement can be considered in agreement with Equivalent Single-Layer (ESL) or Layer-Wise (LW) theories to develop advanced finite element models, whether based on the Principle of Virtual Displacements (PVD) or RMVT. The degree of kinematic refinement, which usually holds equally for the entire element, can be taken a step further, by being assigned locally to each of its nodes, making full use of CUF to render NDK capabilities to the elements. Besides, even though the elements can adopt any type of nodal shape functions, high p-order hierarchical Legendre expansions (HLE) can also be combined with NDK, achieving excellent convergence rates. These capabilities combined, explored first under the PVD, are for once integrated in the proposed elements under RMVT to further benefit accurate stress analysis. These elements can be applied throughout the entire mesh, adapting to local, transitional and global regions straightforwardly, providing high numerical accuracy, locally, with minimal computational efforts, globally. The numerical results focus on stress analysis of multilayered composite plates, including local effects, to demonstrate the predictive capabilities of the proposed RMVT-based LW elements with NDK and HLE combined, considering well-known benchmark three-dimensional exact solutions for assessment.

Hygro-thermo-mechanical modelling of multilayered plates: Hybrid composite laminates, fibre metal laminates and sandwich plates

A new layerwise mixed model is developed for the coupled hygro-thermo-mechanical static analysis of multilayered plates, addressing hybrid composite laminates, fibre metal laminates and sandwich plates under hygro-thermo-mechanical loadings. This model is based on a mixed least-squares formulation, i.e. a strong form model, with a layerwise variable description for displacements, transverse stresses and in-plane strains, along with temperature, transverse heat flux and in-plane components of the thermal gradient, as well as moisture, transverse moisture flux and in-plane components of the moisture gradient, all taken as independent variables. This mixed formulation ensures that the interlaminar C0 continuity requirements, where the material properties change, are fully fulfilled a priori by all independent variables. The effects of hygrothermal environments in the behaviour of multilayered plates are here demonstrated considering hybrid composite laminates, fibre metal laminates and sandwich plates with different side-to-thickness ratios under a series of hygro-thermo-mechanical loadings. Three-dimensional (3D) exact solutions are used to assess the results by the present model and some further by an alternative weak form model in the framework of Carrera Unified Formulation (CUF). The present model is shown to predict a highly accurate quasi-3D hygro-thermo-mechanical description of the through-thickness distributions of displacements and stresses, temperature and heat flux, moisture and moisture flux, all together.

Evaluation of shear and membrane locking in refined hierarchical shell finite elements for laminated structures

Shear and membrane locking phenomena are fundamental issues of shell finite element models. A family of refined shell elements for laminated structures has been developed in the framework of Carrera Unified Formulation, including hierarchical elements based on higher-order Legendre polynomial expansions. These hierarchical elements were reported to be relatively less prone to locking phenomena, yet an exhaustive evaluation of them regarding the mitigation of shear and membrane locking on laminated shells is still essential. In the present article, numerically efficient integration schemes for hierarchical elements, including also reduced and selective integration procedures, are discussed and evaluated through single-element p-version finite element models. Both shear and membrane locking are assessed quantitatively through the estimation of strain energy components. The numerical results show that the fully integrated hierarchical shell elements can overcome the shear and membrane locking effectively when a sufficiently high polynomial degree is reached. Reduced and selective integration schemes can help with the mitigation of locking on lower-order hierarchical shell elements.

A global/local approach based on CUF for the accurate and efficient analysis of metallic and composite structures

The design and analysis of aerospace structures requires a detailed evaluation of stresses. Nevertheless, the complexity of large structures and the use of composite materials can significantly increase the computational costs of the models. The computational burden of such analyses can be reduced by a suitable global/local approach developed in a very general Finite Element framework. Generally, a global/local modelling approach aims at using a finer mesh in the “local” zones where a detailed evaluation of stress/strain field is required, whereas a coarse mesh is used in the rest of the structure. This work proposes a global/local methodology to set up a high-order beam model in the Carrera Unified Formulation framework only for a reduced region of the global model. The methodology makes use of two steps. In the first step, a static analysis of the global structure is done by means of a commercial software in order to identify the critical regions deserving more accurate investigations. In the second step, thus, a high-order beam model is employed for the local region based on the information from the previous global analysis. Linear elastic static analysis are considered in this work, and the attention is mainly focussed on the capability of the method to provide stable solutions and accurate 3D stress fields in the local region, even in the case of laminated composite structures. Hence, the effectiveness of the proposed approach is proven through some meaningful benchmarks.

Free vibration analysis of cross-ply laminated beam structures using refined beam theories and B-spline basis functions

In this article, 1 D refined beam theories based on Carrera Unified Formulation (CUF) and the Isogeometric approach (IGA) are synthesized for the free vibration analysis of laminated beam structures. The B-spline basis functions utilized in IGA are employed to approximate the displacement field due to their interesting features in analysis. Taylor expansions are utilized in the framework of CUF which presents finite element matrices in the form of fundamental nuclei that is independent of type and order of expansion. Higher-order B-spline basis functions attenuate the effect of shear locking properly and higher-order theories presented by CUF are free from Poisson locking phenomenon and utilizations of shear correction factor. Also, equivalent single layer (ESL) models have been utilized as a solution field description approach. Several numerical results including composite beams are carried out and it is shown that coupling IGA and CUF ends in a suitable methodology to analyze laminated beam structures.

B-spline finite element approach for the analysis of thin-walled beam structures based on 1D refined theories using Carrera unified formulation

In the current study, 1D refined beam theories on the basis of Carrera Unified Formulation (CUF) are combined with Isogeometric approach (IGA) for the static and free vibration analysis of thin-walled beam structures. The B-spline basis functions utilized in IGA are employed to approximate displacement field due to their interesting attributes in analysis. N-Order Taylor-like expansions are utilized in the framework of CUF which presents finite element matrices in the form of fundamental nuclei that is independent of the type and order of expansion. Higher-order B-spline basis functions attenuate the effect of shear locking properly and higher-order theories presented by CUF are free from Poisson locking phenomenon and utilizations of shear correction factor. Several numerical results including both solid and thin-walled structures are investigated and it is shown that coupling IGA and CUF ends in a suitable methodology to analyze beam structures.

Effect of Nonstructural Masses on Civil Structures by CUF-Based Finite Element Models

This paper proposes an application of already established higher-order models and, namely, investigates the free vibration analysis of civil engineering structures subjected to nonstructural masses. The refined one-dimensional theories adopted are based on the Carrera unified formulation (CUF). The stiffness and mass matrices are obtained by means of the principle of virtual displacements in conjunction with the finite element method (FEM). According to CUF, in fact, their formulation is performed in terms of fundamental nuclei, which depend neither on the adopted class of beam theory nor on the FEM approximation along the axis. In order to account for nonstructural localised masses, the fundamental nucleus of the mass matrix has been opportunely modified. In this work, Lagrange polynomials have been employed in the framework of CUF to develop pure translational displacement-based refined beam models. The models obtained using this approach are referred, in the literature, to as component wise (CW), which allows to model each structural component as a 1D element. The free vibration analysis has been carried out for different cases of nonstructural mass distribution in typical civil buildings. The results obtained are compared with classical Solid/Shell FEM solutions from the commercial code MSC Nastran. The capabilities of the CW models are demonstrated, since this formulation is able to replicate the 3D solid results with enhanced performances in terms of computational efforts. Moreover, the importance of taking into account the correct distribution of the nonstructural masses is shown to be of fundamental importance in vibration analysis of civil buildings.

An extension of Carrera unified formulation in polar coordinate for analysis of circular sandwich plate with FGM core using GDQ method

The Carrera Unified Formulation (CUF) is a technique that, in addition to eliminate the imperfections of other theories, allows us to achieve a large class of models, such as equivalent single layer, layer wise, and mixed formulation in a unified manner. For this reason, this formulation can be very effective in different analyses, and according to the desired analysis, an appropriate model could be chosen. In this research, for the first time, the Carrera Unified Formulation is extended in the polar coordinates for analyzing the sandwich circular plate with the functionally graded material core. In order to apply variations in the properties of the functionally graded material, the variable kinematic method is used in the frame work of CUF, which ultimately leads to a reduction in degrees of freedom and an increase in the accuracy of the results. The functionally graded material is modeled as a mixture of ceramics and metal, whose properties change according to a power distribution in the direction of thickness. In this research, the generalized differential quadrature (GDQ) method is used to solve the governing equations. The obtained results are compared with the existing three-dimensional results as well as the generalized Zig-Zag theory, which indicates the high accuracy of the CUF formulation for circular plates in the polar coordinates. In addition, new results are provided for different geometries, thickness ratios, boundary conditions as well as deflection, radial displacements, annular and radial stresses, and transverse stresses along the thickness.

Free vibration and buckling analysis of composite laminated plates using layerwise models based on isogeometric approach and Carrera unified formulation

ABSTRACTIn this paper, layerwise (LW) theory has been utilized along with that of equivalent single layer (ESL) for free vibration and linearized buckling analysis of composite laminated plates. To this end, the isogeometric approach and Carrera unified formulation (CUF) have been combined. In particular, Taylor-like and Legendre-like polynomial expansions have been utilized in the framework of CUF to approximate the solution field in ESL and LW models, respectively. Indeed, CUF provides an ideal tool that facilitates the implementation of higher orders of the solution field expansion. As ESL model cannot inherently provide interlaminar continuity, they are not suitable for analyzing thick laminated plates. However, the LW model not only presents a three-dimensional (3D)-like accurate mathematical model using two-dimensionalplate theories but also considers interlaminar continuity requirements and obviates the need for the use of shear correction factor. In addition, the nonuniform rational B-spline basis functions have been employed to approximate the solution field, due to their interesting attributes in the analysis. These functions are able to describe the exact geometry of the structure and make it technically feasible to provide refinement process during analysis. The presented numerical results confirm the validity of the proposed methodology.

Exact solutions for free vibration analysis of laminated, box and sandwich beams by refined layer-wise theory

The present work addresses a closed-form solution for the free vibration analysis of simply supported composite laminated beams via a refined one-dimensional (1D) model, which employs the Carrera Unified Formulation (CUF). In the framework of CUF, the 3D displacement field can be expanded as any order of generic unknown variables over the cross section, in the case of beam theories. Particularly, Lagrange expansions of cross-sectional displacement variables in conjunction with layer-wise (LW) theory are adopted in this analysis, which makes it possible to refine the kinematic fields of complex cross section by arbitrary order and accuracy. As a consequence, the governing equations can be derived using the principle of virtual work in a unified form and can be solved by a Navier-type, closed-form solution. Numerical investigations are carried out to test the performance of this novel method, including composite and sandwich beams ranging from simple to complex configurations of the cross section. The results are compared with those available in the literature as well as the 3D finite element method (FEM) solutions computed by commercial codes. The present CUF model is proved to be able of achieving high accurate results with less computational costs. Besides, they may serve as benchmarks for future assessments in this field.