Carrera Unified Formulation Models Research Articles

A consistent crack bandwidth for higher-order beam theories: Application to concrete

Higher-order theories have a broad range of successful applications but also suffer from localization instability and mesh-size dependency when modeling quasi-brittle materials such as concrete with strain-softening behavior. To overcome the above difficulties, this paper proposes a fracture energy regularization method with a unified, consistent crack bandwidth specifically tailored for higher-order beam theories. The Carrera unified formulation (CUF) is applied to develop scalable structural theories and related finite elements. To evaluate the accuracy of the new crack bandwidth, three typical experimental quasi-static benchmarks of pure concrete structures are utilized. A modified Mazars damage model with tensile and compressive softening laws is implemented in these benchmarks. The comparison between numerical and experimental results demonstrates that the proposed method can accurately determine the correct crack bandwidth and preserve the dissipated energy per unit area of a fracture surface. Moreover, this robust estimation of crack bandwidth reduces the mesh dependency in general, ensuring the high efficiency of the CUF model.

Analysis of transversely isotropic compressible and nearly-incompressible soft material structures by high order unified finite elements

This work proposes high-order beam (1D) and plate (2D) finite element models for the large strain analysis of compressible and incompressible transversely isotropic hyperelastic media, defined within the Carrera Unified Formulation (CUF) framework. The strain energy density function adopted in fiber-reinforced hyperelastic materials modeling is presented and expressed in terms of invariants and pseudo-invariants of the right Cauchy-Green strain tensor. The explicit expression of the tangent elasticity tensor is derived through the assumption of coupled formulation of strain energy functions. Refined fully nonlinear beam and plate models are defined in a total Lagrangian formulation, deriving the governing equations of the nonlinear static analysis through the Principle of Virtual Displacements in terms of fundamental nuclei, in resulting expressions of internal and external forces vectors, and tangent stiffness matrix independent of kinematic models and approximation theories adopted. The iterative Newton-Raphson linearization scheme coupled with the arc-length constraint is adopted to obtain actual numerical solutions. Different benchmark analyses in hyperelasticity are performed to assess the capabilities of our proposed model, analyzing the three-dimensional stress field for moderate to large strain states and comparing actual numerical results with exact closed-form solutions or results available in the literature, demonstrating the capabilities and reliability of CUF models in the analysis of fiber-reinforced soft materials and structures.

A global–local approach to the high-fidelity impact analysis of composite structures based on node-dependent kinematics

The objective of the present work is to investigate progressive damage in fibre-reinforced composites under varying load conditions, and in particular transverse impact loads, using a global–local approach. The numerical models are built using higher-order structural theories based on the Carrera Unified Formulation (CUF). The Node-Dependent Kinematics (NDK) technique, an intrinsic feature of CUF models, is employed which enables the selective refinement of critical regions of interest within the structure and results in a global–local analysis. Progressive damage is governed by the CODAM2 material model, which is based on continuum damage mechanics. A series of numerical assessments are performed on composite laminates under varying load conditions, and predicted results of the global–local analysis are found to be in good agreement with experimental data, thereby validating the proposed approach. A comparison of its performance with reference high-fidelity CUF models of the full structure demonstrates the computational efficiency that can be achieved using the CUF-NDK global–local approach.

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.

Concrete damage analysis based on higher-order beam theories using fracture energy regularization

This paper presents the numerical damage analysis of concrete structures using higher-order beam theories based on Carrera Unified Formulation (CUF). The concrete constitutive relation is modeled using continuum damage mechanics based on a modified Mazars concrete damage model, in which both the tensile and compressive softening behaviors are regularized with classical fracture energy methodology. An expression is proposed to estimate the characteristic length in higher-order beam theories, to prevent mesh dependency. Both softening constitutive laws and fracture energy calculations are obtained according to Model Code 2010. To assess the efficiency of the proposed model, three classical benchmark quasi-static experiments are taken for validation. From the comparison between numerical and experimental results, the proposed CUF model using continuum damage mechanics can present 3D accuracy with low computational costs and reduce the mesh dependency.

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.

Flutter analysis of rotary laminated composite structures using higher-order kinematics

The flutter analysis of rotary laminated structures has been performed using Carrera Unified Formulation (CUF). CUF is a one-dimensional higher-order structural model, in which the displacement at the cross-section is the only unknown parameter. The cross-sectional displacement fields of the model are defined using the Lagrange- and Taylor expansion. The quasi-steady and unsteady theories (Theodorsen and Loewy) have been used to apply the aerodynamic force. Subsequently, the equation of motion is defined in ‘fundamental nucleus’ form using Hamilton’s principle. The engineering solution of the flutter condition has been obtained by using p-k method. The computational studies of flutter analysis have been done for various rotor hub, lamination sequence, and structural models. The regular, irregular plies and arbitrary fiber orientation of the orthotropic structures have also been considered. The present study suggests that the proposed CUF model can obtain an aeroelastic solution of rotary structure in a computationally and economically efficient manner.

Progressive damage analysis of composite laminates subjected to low-velocity impact using 2D layer-wise structural models

The present work deals with the progressive damage analysis of composite laminates subjected to low-velocity impact. We develop a numerical model using higher-order structural theories based on the Carrera Unified Formulation (CUF) with Lagrange polynomials and resulting in a 2D refined layer-wise model. To model damage, we use a combination of the continuum damage-based CODAM2 intralaminar damage model to account for fibre and matrix damage within the ply, and cohesive elements to account for delamination between successive composite plies. We carry out numerical assessments for the case of a linear elastic composite plate subjected to impact, to compare the current framework with standard approaches based on 3D finite element (FE) analysis. We, then, consider the elastoplastic analysis of a bimetallic laminated plate to compare the performance of the proposed layer-wise model and 3D-FE approaches, for the case of nonlinear impact analysis. The final assessment considers progressive damage due to low-velocity impact, and the results are compared with available literature data. The numerical predictions show a good correlation with reference experimental and simulation results, thus validating the current framework for impact analysis of composite structures. Comparisons of the proposed layer-wise structural models with those based on 3D finite elements demonstrate the improved computational efficiency of the CUF models in terms of model size and analysis time.

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.

The generation and validation of a CUF-based FEA model with laser-based experiments

Architectural structures today are increasingly complex and structural health monitoring plays an important role in guaranteeing their safety. How to improve the reliability of deformation analysis is, thus, one of the key problems. This article combines laser-based measurement technology and the Carrera unified formulation (CUF) method to investigate the deformation of engineering structures. Within this article, we simulate architectural structures using the CUF geometric model, which is consistent with the results of the laser tracker experiment. We aimed at constructing an intelligent and efficient CUF model which can be applied extensively in the monitoring of various constructs, such as tunnels and bridges. The innovation of this article is that high-accuracy laser tracker technology is integrated with an effective CUF model to investigate the load-displacement relationship considering lateral displacement.

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.

Contact analysis of laminated structures including transverse shear and stretching

This work presents contact problems of laminated structures via the Carrera Unified Formulation (CUF). The modeling approach makes use of higher-order 1D elements accounting for transverse shear and stretching. The current work considers normal, frictionless contact based on a node-to-node formulation, and the penalty approach to enforce the contact constraints. Numerical assessments compare classical beam theories, higher-order CUF, and 3D finite element models regarding solution accuracy, computational size, and time required for the analysis. The results show the validity of Layer-Wise CUF models to capture both global and local deformations accurately, which is a shortcoming of classical beam theories, and require at least an order of magnitude fewer degrees of freedom and computational time than a full 3D finite element analysis. Particularly relevant are the accurate distributions of transverse shear stress and stretching along the thickness in the perspective of failure analyses.

Computationally Efficient Concurrent Multiscale Framework for the Linear Analysis of Composite Structures

This paper presents a novel multiscale framework based on higher-order one-dimensional finite element models. The refined finite element models originate from the Carrera unified formulation (CUF): a novel and efficient methodology to develop higher-order structural theories hierarchically via a variable kinematic approach. The concurrent multiscale framework consists of a macroscale model to describe the structural level components interfaced with efficient CUF micromechanical models. Such micromechanical models can take into account the detailed architecture of the microstructure with high fidelity. The framework derives its efficiency from the capability of CUF models to detect accurate three-dimensional-like stress fields at reduced computational costs. This paper also shows the ability of the framework to interface with different classes of representative volume elements and the benefits of parallel implementations. The numerical cases focus on composite and sandwich structures and demonstrate the high fidelity and feasibility of the proposed framework. The efficiency of the framework stems from comparisons with the analysis time and memory requirement against traditional multiscale implementations.

Accurate stress fields of post-buckled laminated composite beams accounting for various kinematics

Highly flexible laminated composite structures, prone to suffering large-deflection and post-buckling, have been successfully employed in a number of scenarios. Therefore, accurate predictions of their stress distributions in the geometrically nonlinear analysis are of paramount importance for their design and failure evaluation. In this paper, for composite beams subjected to large-deflection and post-buckling, we investigate the effectiveness of different geometrically nonlinear strain approximations for the description of their nonlinear static response and for the determination of stress distributions. For this purpose, a unified formulation of geometrically nonlinear refined beam theory based on the Carrera Unified Formulation (CUF) and a total Lagrangian approach constitutes the basis of our analysis. Accordingly, various kinematics of one-dimensional structures are formulated via an appropriate index notation and an arbitrary cross-section expansion of the generalized variables, leading to lower- to higher-order beam models with only pure displacement variables for laminated composite beams. In view of the intrinsic scalable nature of CUF and by exploiting the principle of virtual work and a finite element approximation, nonlinear governing equations corresponding to various nonlinear strain assumptions can be straightforwardly and easily formulated in terms of fundamental nuclei, which are independent of the theory approximation order. Several numerical assessments are conducted, including large-deflection and post-buckling analyses of asymmetric and symmetric laminated beams under compression loadings. The numerical solutions are solved by using a Newton–Raphson linearization scheme along with a path-following method based on the arc-length constraint. Our numerical findings demonstrate the capabilities of the CUF model to calculate the large-deflection and post-buckling equilibrium curves as well as the stress distributions with high accuracy, which could be a basis to assess the validation ranges of various kinematics and different nonlinear strain approximations.

Wave propagation in compact, thin-walled, layered, and heterogeneous structures using variable kinematics finite elements

ABSTRACTThe article investigates wave propagation characteristics for a class of structures using higher-order one-dimensional (1D) models. 1D models are based on the Carrera Unified Formulation (CUF), a hierarchical formulation which provides a framework to obtain refined structural theories via a variable kinematics description. Theories are formulated by employing arbitrary expansions of the primary unknowns over the beam cross-section. Two classes of beam models are employed in the current work, namely Taylor Expansion (TE) and Lagrange Expansion (LE) models. Using the principle of virtual work and finite element method, the governing equations are formulated. The direct time integration of equation of motion is carried through an implicit scheme based on the Newmark method and a dissipative explicit method based on the Tchamwa–Wielgosz scheme. The framework is validated by comparing the response for the stress wave propagation in an isotropic beam to an analytical solution available in the literature. The capabilities of the proposed model are demonstrated by presenting results for wave propagation analysis of a sandwich beam and a layered annular cylinder structure. The ability of CUF models to detect 3D-like behavior with a reduced computational overhead is highlighted.

Static and free-vibration analyses of dental prosthesis and atherosclerotic human artery by refined finite element models.

Static and modal responses of representative biomechanical structures are investigated in this paper by employing higher-order theories of structures and finite element approximations. Refined models are implemented in the domain of the Carrera unified formulation (CUF), according to which low- to high-order kinematics can be postulated as arbitrary and, eventually, hierarchical expansions of the generalized displacement unknowns. By using CUF along with the principle of virtual work, the governing equations are expressed in terms of fundamental nuclei of finite element arrays. The fundamental nuclei are invariant of the theory approximation order and can be opportunely employed to implement variable kinematics theories of bio-structures. In this work, static and free-vibration analyses of an atherosclerotic plaque of a human artery and a dental prosthesis are discussed. The results from the proposed methodologies highlight a number of advantages of CUF models with respect to already established theories and commercial software tools. Namely, (i) CUF models can represent correctly the higher-order phenomena related to complex stress/strain field distributions and coupled mode shapes; (ii) bio-structures can be modeled in a component-wise sense by only employing the physical boundaries of the problem domain and without making any geometrical simplification. This latter aspect, in particular, can be currently accomplished only by using three-dimensional analysis, which may be computationally unbearable as complex bio-systems are considered.

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.

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.

Numerical modelling of the creep behaviour of GFRP sandwich panels using the Carrera Unified Formulation and Composite Creep Modelling

Sandwich panels often need to be designed to sustain significant permanent loads, raising the need to accurately account for long-term creep deformation. However, the accurate prediction of creep in sandwich panels is not trivial, particularly owing to their composite multi-layered nature, the multitude of possible face and core material combinations, and the possible existence of through-thickness shear reinforcement (ribs/webs). This paper presents numerical investigations on the creep behaviour of composite sandwich panels produced by vacuum infusion with glass-fibre reinforced polymer (GFRP) faces and ribs, and polyurethane (PUR) and polyethylene terephthalate (PET) foam cores. Carrera Unified Formulation (CUF) is implemented, for the first time using 1D elements with an equivalent single layer (ESL) methodology, to model the creep response of simple and ribbed panels by adopting a Composite Creep Modelling (CCM) approach. Previous experimental results from creep tests carried out on such sandwich panels and their constituent materials are used to obtain time-dependent constitutive relations for the materials in various layers and validate the numerical results. Additionally, results from analytical beam models using Timoshenko beam theory (TBT) with multi-layered sections are used to further validate the numerical outputs obtained with CUF. The developed numerical models were able to predict the experimental creep behaviour of the full-scale sandwich panels with reasonable accuracy. Differences observed between the CUF and TBT models mainly stem from the inherent approximations concerning the shear correction factors used with TBT, which contrast with the solutions provided by CUF, where such factors do not need to be considered when higher degrees of approximation are used.