Carrera Unified Formulation Research Articles

Carrera Unified Formulation (CUF) for the composite shells of revolution. Equivalent single layer models

Here, higher order models of elastic composite multilayer shells of revolution are developed using the variational principle of virtual power for the 3-D linear anisotropic theory of elasticity and generalized series in the shell thickness coordinates. Following the Unified Carrera Formula (CUF), the stress and strain tensors, as well as the displacement vector, were expanded into series in terms of the coordinates of the shell thickness. The higher-order cylindrical shell supported on the edges under axisymmetric loading, is considered and solved analytically using a Navier close form solution method. Also, composite axisymmetric circular plated as well as parabolic, hyperbolic and pseudo-spheric shell fixed ate the ends are considered. Numerical calculations were performed using the computer algebra software Mathematica. The resulting equations can be used for theoretical analysis and calculation of the stress-strain state, as well as for modeling thin-walled structures used in science, engineering, and technology. The results of calculation can be used as benchmark examples for finite element analysis of the higher order elastic shells.

An improved coupling of 3D state‐based peridynamics with high‐order 1D finite elements to reduce spurious effects at interfaces

AbstractPeridynamics (PD) is a nonlocal continuum theory capable of handling fracture mechanisms with ease. However, its use involves high computational costs. On the other hand, Carrera Unified Formulation (CUF) allows one to use one‐dimensional high‐order finite elements, resulting in excellent accuracy while improving computational efficiency. To address the high computational cost of solving fracture problems, a coupling technique between these two theories is necessary. Various approaches have been proposed to couple peridynamic grids with finite element meshes in the literature. However, most of these approaches are affected by arbitrary choices of blending functions and tuning parameters or exhibit spurious effects at the interfaces. To overcome these issues, we propose a simple coupling technique based on overlapping PD/CUF regions and continuity of the displacement field at the interfaces. This approach is verified through static analysis of classical beams and thin‐walled structures with applications in the aerospace industry.

Refined plate elements for the analysis of composite plate using Carrera unified formulation

Refined plate elements for the analysis of composite plate using Carrera unified formulation

Numerical approach to disbonds in bonded composite Single Lap Joints: Comparison between Carrera Unified Formulation and classical Finite Element modeling

The scope of this study is the monitoring of the adhesion integrity of composite Single Lap Joints (SLJs) through the assessment of the MUL2 CODE, software developed by the MUL2 Research Group - Department of Mechanical and Aerospace Engineering of Politecnico di Torino. The MUL2 CODE is implemented through the Carrera Unified Formulation (CUF) for 2D structures based on Hierarchical Legendre Expansion (HLE) polynomials. An efficient method for the Structural Health Monitoring (SHM) of bonded composite joints is experimentally verified and simulated by commercial FE software and by the CUF approach, in order to reduce the computational cost of analyses: using transient excitations via toneburst signals, the structural health of damaged SLJ is analytically, experimentally and numerically evaluated. The interaction mechanism between the waves traveling through the investigated specimens is numerically simulated with a simple Finite Elements (FE) model and it is solved via MUL2 CODE and commercial software Ansys Workbench (with 2D and 3D models), respectively. Experimental campaigns data are compared with analytical, CUF and Ansys results demonstrating the consistence of the MUL2 formulation that is computationally simpler, but very efficient for the joint analysis. The presented and discussed CUF application is able to quantify with a high accuracy the debonding extension in the damaged SLJ, simply tuning the excitation frequency of the SHM technique.

Refined Finite Elements for the Analysis of Metallic Plates Using Carrera Unified Formulation

Refined Finite Elements for the Analysis of Metallic Plates Using Carrera Unified Formulation

An analysis of the propagation of surface acoustic waves in a substrate covered by a metal T-plate with the Carrera unified formulation

In this work, the wave propagation of Rayleigh type through a periodic elastic element covered with a T-plate is analyzed. Viscous-spring artificial boundaries are used to satisfy boundary conditions of a periodic structure. By using the Carrera Unified Formulation (CUF), the various kinematics of the three-dimensional structure are consistently expressed and the propagation of waves within the model is obtained. Lagrange expansion is employed to generate the mathematical model of the structure. The numerical evaluation and the comparison with the results obtained by the analytical method and COMSOL involving different structures reveal that the results of this study are reliable and show that the model developed provides accurate results for the analysis of the periodic elastic structure of T-plates.

Three-Dimensional Vibration Model of Cylindrical Shells via Carrera Unified Formulation.

In this paper, we present a novel and unified model for studying the vibration of cylindrical shells based on the three-dimensional (3D) elastic theory and the Carrera Unified Formulation. Our approach represents a significant advancement in the field, as it enables us to accurately predict the vibrational behavior of cylindrical shells under arbitrary boundary conditions. To accomplish this, we expand the axial, circumferential, and radial displacements of the shell using Chebyshev polynomials and Taylor series, thereby reducing the dimensionality of the expansion and ensuring the precision and rigor of our results. In addition, we introduce three groups of artificial boundary surface springs to simulate the general end boundary conditions of the cylindrical shell and coupling springs to strongly couple the two surfaces of the cylindrical shell φ = 0 and φ = 2π to ensure continuity of displacements on these faces. Using the energy function of the entire cylindrical shell model, we obtain the characteristic equation of the system by finding the partial derivatives of the unknown coefficients of displacement in the energy function. By solving this equation, we can directly obtain the vibration characteristics of the cylindrical shell. We demonstrate the convergence, accuracy, and reliability of our approach by comparing our computational results with existing results in the literature and finite element results. Finally, we present simulation results of the frequency features of cylindrical shells with various geometrical and boundary parameters in the form of tables and figures. Overall, we believe that our novel approach has the potential to greatly enhance our understanding of cylindrical shells and pave the way for further advancements in the field of structural engineering. Our comprehensive model and simulation results contribute to the ongoing efforts to develop efficient and reliable techniques for analyzing the vibrational behavior of cylindrical shells.

Hygro-elasticity and moisture diffusivity analysis of periodically heterogeneous materials by hierarchical microscale models

The present paper focuses on developing a method to solve micromechanical analyses for the computation of moisture diffusivity and hygro-elastic characteristics of periodically heterogeneous materials and the recovery of the moisture flux and the stress over the Repeating Unit Cell (RUC). The model is based on the Mechanics of Structure Genome (MSG) to build a procedure capable of de-coupling the multiscale analysis into different steps on global and local levels, resulting in less demanding problems. On the other hand, the methodology uses the Carrera Unified Formulation (CUF) to engage a refined beam theory with high-fidelity capabilities. Hence, the longitudinal direction of the reinforcement is described with one-dimensional (1D) finite elements. Besides, for the cross-section, a hierarchical discretisation coupled with a non-isoparametric mapping technique allows reaching a high level of accuracy in the geometric description of the curvature of the fibre or inclusion. Finally, both hygro-elastic and moisture diffusivity problems are validated through numerical assessments showing excellent agreement with benchmarks in the literature.

Time-domain viscoelastic analysis of laminated composite plates by using a unified formulation

This paper presents the time-domain viscoelastic formulation based on the Carrera unified formulation (CUF) to obtain the response of composite structures submitted to bending loads. The governing equations are obtained through a modified form of the principle of virtual displacement (PVD) for a viscoelastic analysis. The problem is solved using the Laplace transform instead of direct integration to reduce the computational cost. The results demonstrate the strong capability of the time-domain viscoelastic formulation. Overall, the proposed methodology allows the direct implementation of new optimized high order deformation theories including the thickness stretching effect that may be implemented in futures works.

Vibration analysis of thermally loaded isotropic and composite beam and plate structures

This work proposed the use of the Carrera Unified Formulation (CUF) for the vibration and buckling analysis of structures subjected to thermal loads. In detail, the variation of natural frequencies for progressively large thermal loads is investigated. Here, particular attention is focused on the study of buckling thermal loads as degenerate cases of the vibration analysis and on the mode aberration caused by thermal stresses. From this standpoint, the use of CUF for the development of high-order beam and plate models is fundamental. Indeed, Lagrange-like (LE) polynomials are considered for developing the kinematic expansion and Layerwise (LW) theories are employed to characterize the complex phenomena that may appear in composite structures. A linearized formulation to study the natural frequencies variation as a function of the progressive increasing thermal loadings is adopted. Different isotropic and laminated composite structures have been analyzed and compared with the Abaqus solution to validate the presented methodology and provide some benchmark solutions. In addition, a parametric study was conducted to evaluate the stacking sequence and thickness effect in the vibration modes and thermal buckling loads. The results document the excellent accuracy and reliability of the presented methodology and show the potentialities of this numerical tool able to analyze cases that are difficult to study experimentally.

Advanced Finite Elements for Geometrically Nonlinear Analysis of Rectangular Plates under Various In-Plane Loadings Accounting for the Boundary Conditions of the Stiffeners

Highly flexible plate structures are subjected to various loading and edge conditions in many engineering applications, such as in the aerospace and civil industries. In this paper, the post-buckling response of rectangular plates subjected to axial and shear loadings is investigated by employing the Carrera Unified Formulation (CUF). As in previous works, only isolated plates have been considered, regardless of the effects of the stiffeners and boundary conditions imposed by the surroundings. However, in this research, different plate models are implemented, and the effects of boundary conditions imposed by the stiffeners are evaluated precisely. Comprehensive assessments are provided for the geometrically nonlinear equilibrium curves of the plate structures using the Newton–Raphson linearization method with the path-following constraint. Furthermore, the comparisons of different geometrically nonlinear assumptions according to the full Green–Lagrange nonlinear model and von Kármán nonlinear plate theory are presented. The importance of strain–displacement relationships and stiffeners’ effects in the nonlinear response of plate structures is highlighted. As a result, the accuracy and effectiveness of the presented CUF-2D model are demonstrated.

High-fidelity modeling approaches for the analysis of reinforced structures using one-, two- and three-dimensional elements

The present paper proposes a method for analyzing reinforced thin-walled structures based on high-order one-, two- and three-dimensional finite elements (FE). Refined finite elements are developed in the domain of the Carrera unified formulation (CUF). The node-dependent kinematic approach (NDK), which allows to connect in an easy manner elements with incompatible kinematics, has been used to connect elements with different dimensions without the need of ad hoc connection techniques. The formulation ensures the continuity of the displacement at the interface preventing the onset of singularities that lead to inaccurate results when beam, plate and solid elements have to be coupled to solve complex structures. The effectiveness of the present method has been confirmed by comparing the results with those from literature and with those obtained using commercial finite element codes. Static and free-vibration analyses of reinforced panels have been carried out to demonstrate the capabilities of the present models. The results show that the limits of classical structural models can be easily overcome using the present approach, and at the same time, a quasi three-dimensional solution can be obtained with a large computational cost saving.

Global/local buckling analysis of thin-walled I-section beams via hierarchical one-dimensional finite elements

Buckling instability is one of the factors that limit the ultimate load-carrying capacity of I-section beams, and the coupling of global buckling and local buckling makes it difficult to predict the mechanical behavior of I-section beams. Hierarchical one-dimensional finite elements for the global/local buckling analysis of I-section beams are presented in this paper. Within the framework of Carrera’s Unified Formulation (CUF), a model is built by using Lagrange polynomials to describe the three-dimensional displacement field as an arbitrary order approximation by displacement variables over the beam cross-section. One-dimensional finite elements are obtained by discretizing the governing equation along the beam length. Due to its high efficiency and step length adaptability, the Asymptotic Numerical Method (ANM) is adopted to solve the nonlinear governing equations. Several typical buckling problems in I-section beams, including global buckling, local buckling and global–local coupled buckling, are investigated. The numerical results demonstrate the validity and efficiency of the proposed model. The critical buckling load and displacement field can be precisely predicted, and a good agreement is found in comparison with results obtained via three-dimensional finite element solutions.

PROGRESSIVE DAMAGE ANALYSIS OF STEEL-REINFORCED CONCRETE BEAMS USING HIGHER-ORDER 1D FINITE ELEMENTS

The present work investigates progressive damage in steel-reinforced concrete structures. An elastic-perfectly plastic material response is considered for the reinforcing steel constituent, while the smeared-crack approach is applied to model the nonlinear behavior of concrete. The analysis employs one-dimensional numerical models based on higher-order finite elements derived using the Carrera unified formulation (CUF). A set of numerical assessments is presented to study the mechanical response of a steel-reinforced notched concrete beam loaded in tension. The predictions are found to be in very good agreement with reference experimental observations, thereby validating the numerical approach. It is shown that CUF allows for the explicit representation of the constituents within the composite beam, resulting in accurate solutions in a computationally efficient manner.

Weak-form differential quadrature element analysis of plate on a tensionless and frictional foundation using a higher-order kinematics

Plate-foundation systems, as a structure common in engineering practices, have received extensive scientific research. A novel technical scheme is proposed to improve the computational efficiency in predicting the nonlinear responses of a plate on the foundation. The core work of this technical scheme lies in two aspects. The first aspect is to create a novel plate-foundation system, where a higher order kinematical assumption common in Carrera Unified Formulation is used to model the plate, and a plate-foundation interaction model taking into account the tensionless and frictional characteristics is created. The adjustable kinematical parameters of the higher-order kinematical assumption endow the plate with flexible accuracy. The second aspect is to derive a two-dimensional (2D) differential quadrature (DQ) formula for 2D interpolation based on the one-dimensional (1D) DQ formula, and thus a weak-form differential quadrature element method (WDQEM) for nonlinear quasi-static analysis of the plate-foundation system is proposed by using the principle of virtual work and the 2D interpolation derived. The proposed WDQEM supports flexible nodal configuration in the element, making it degree-of-freedom-flexible and suitable to carry out both p- and h-version strategies in the numerical simulation. Several numerical examples are given to (i) verify the proposed technical scheme and the corresponding computer codes; (ii) to investigate the numerical performance of the WDQEM through comparison between the classical plate model and the 3D solid model, and (iii) to study some physical behaviors related to the tensionless and frictional characteristics of the plate-foundation system. The example results show that (i) the WDQEM converges much faster than the conventional finite element method due to the higher order approximation; (ii) the higher order kinematics, due to its capability of adjusting the accuracy of kinematical assumptions, can effectively reduce the computational scale; (iii) the smoothness problem in the post process of stresses analysis is greatly alleviated, (iv) the tensionless and frictional features of the plate-foundation system should be considered when evaluating both the vertical and horizontal bearing capacities of the plate.

A computational aeroelastic framework based on high-order structural models and high-fidelity aerodynamics

A computational framework for high-fidelity static aeroelastic analysis is presented. Aeroelastic analysis traditionally employs a beam stick representation for the structure and potential, inviscid and irrotational flow assumptions for the aerodynamics. The unique contribution of this work is the introduction of a high-order structural formulation coupled with a high-fidelity method for the aerodynamics. In more details, the Carrera Unified Formulation coupled with the Finite Element Method is implemented to model geometrically complex composite, laminated structures as equivalent bi-dimensional plates. The open-source software SU2 is then used for the solution of the aerodynamic fields. The in-house fluid-structure coupling algorithm is based on the Moving Least Square technique. The paper contains a thorough validation of each disciplinary solver of the aeroelastic framework, and provides a few application test cases. For an unswept, untapered and isotropic wing, it was found that the method provides results in agreement with predictions from models based on potential flow theory for moderate freestream velocities. Departures were reported for very low speed and in the high-subsonic regime, alerting the need of adopting high-fidelity flow solutions at these flow conditions. The computational framework was then applied to the static aeroelastic tailoring of a composite wing. The paper concludes providing an overview of future implementation steps towards a tool for the seamless analysis of composite structures subject to different flow conditions, from low to high speed.

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.

Stochastic characterization of multiscale material uncertainties on the fibre-matrix interface stress state of composite variable stiffness plates

This work analyses the stochastic response of fibre and matrix scale stresses of Variable Angle Tow (VAT) laminates affected by multiscale uncertainty defects. The aim is to evaluate the influence of the innermost constituents on the overall structural response via an accurate mechanical characterization of both macro- and microscales. The Carrera Unified Formulation (CUF) is employed to obtain two-dimensional (2D) and one-dimensional (1D) models for both scales. Indeed, 2D layer-wise (LW) and 1D component-wise (CW) approaches are adopted for the macroscale and the microscale, respectively. The use of 2D and 1D models proves to be convenient as a superior computational efficiency is reached, this aspect being of great importance as many analyses are necessary for uncertainty quantification. The numerical results demonstrate the validity of the proposed methodology to obtain an accurate description of the 3D stress state at the different scales. A special focus is made on the fibre-scale stresses and how they may vary when affected by multiscale uncertainty.

Adaptive mesh using non-conventional 1D and 2D finite elements based on CUF

When dealing with complex structures with a several number of degrees of freedom (DOFs), it is useful trying to reduce the mesh for computational cost reasons, seeking not affecting the accuracy in results. The Finite Element Method can become very costly in calculations and time because of the use of very fine 3D meshes. A possible solution can be the application of the Adaptive Mesh (AM), which allows to concentrate the very fine mesh only in regions where critical conditions are present, in terms of geometrical or loads constrains. By exploiting the Node-Dependent Kinematic approach of the Carrera Unified Formulation and using Lagrange expanding functions, this work presents the application of non-conventional 1D and 2D elements for mesh refinement, offering a new and convenient technique to apply in the framework of AM. The static analysis of some typical study cases is performed and the results are provided in terms of displacements and stresses. The presented elements allow us to combine them in order to obtain a mesh refinement employing much less degrees of freedom with respect to the use of classical 3D finite elements.

Displacement and strain data-driven damage detection in multi-component and heterogeneous composite structures

This work introduces the use of convolutional neural network (CNN) in combination with advanced structural theories for the damage detection of multi-component and composite structures. Well-established component-wise (CW) models based on the Carrera Unified Formulation (CUF) are developed first to demonstrate the effect of localized damages on the mechanical performance of thin-walled beams and laminates. Finite element Monte Carlo simulations of randomly damaged structures are then used to generate a large database of full-field displacement and strain images. These images are lately feed into a dedicated CNN for training purpose and for the prediction of location and intensity of structural damages occurring in unseen scenarios. The results demonstrate the validity of the present approach and suggest the importance of adopting opportune structural models to carry out localized damage detection by image-driven AI. Overall, the research provides good confidence for future investigation and experimental testing.