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A new mixed model based on the enhanced-Refined Zigzag Theory for the analysis of thick multilayered composite plates
Full text linkThe Refined Zigzag Theory (RZT) has been widely used in the numerical analysis of multilayered
and sandwich plates in the last decay. It has been demonstrated its high accuracy in predicting global quantities, such as maximum displacement, frequencies and buckling loads, and local quantities such
as through-the-thickness distribution of displacements and in-plane stresses [1,2]. Moreover, the C0
continuity conditions make this theory appealing to finite element formulations [3]. The standard RZT,
due to the derivation of the zigzag functions, cannot be used to investigate the structural behaviour
of angle-ply laminated plates. This drawback has been recently solved by introducing a new set of
generalized zigzag functions that allow the coupling effect between the local contribution of the zigzag
displacements [4]. The newly developed theory has been named enhanced Refined Zigzag Theory (en-
RZT) and has been demonstrated to be very accurate in the prediction of displacements, frequencies,
buckling loads and stresses. The predictive capabilities of standard RZT for transverse shear stress
distributions can be improved using the Reissner’s Mixed Variational Theorem (RMVT). In the mixed
RZT, named RZT(m) [5], the assumed transverse shear stresses are derived from the integration of local
three-dimensional equilibrium equations. Following the variational statement described by Auricchio
and Sacco [6], the purpose of this work is to implement a mixed variational formulation for the en-RZT,
in order to improve the accuracy of the predicted transverse stress distributions. The assumed kinematic
field is cubic for the in-plane displacements and parabolic for the transverse one. Using an appropriate
procedure enforcing the transverse shear stresses null on both the top and bottom surface, a new set
of enhanced piecewise cubic zigzag functions are obtained. The transverse normal stress is assumed as
a smeared cubic function along the laminate thickness. The assumed transverse shear stresses profile
is derived from the integration of local three-dimensional equilibrium equations. The variational functional
is the sum of three contributions: (1) one related to the membrane-bending deformation with a
full displacement formulation, (2) the Hellinger-Reissner functional for the transverse normal and shear
terms and (3) a penalty functional adopted to enforce the compatibility between the strains coming
from the displacement field and new “strain” independent variables. The entire formulation is developed
and the governing equations are derived for cases with existing analytical solutions. Finally, to assess
the proposed model’s predictive capabilities, results are compared with an exact three-dimensional solution,
when available, or high-fidelity finite elements 3D models. References: [1] Tessler A, Di Sciuva
M, Gherlone M. Refined Zigzag Theory for Laminated Composite and Sandwich Plates. NASA/TP-
2009-215561 2009:1–53. [2] Iurlaro L, Gherlone M, Di Sciuva M, Tessler A. Assessment of the Refined
Zigzag Theory for bending, vibration, and buckling of sandwich plates: a comparative study of different
theories. Composite Structures 2013;106:777–92. https://doi.org/10.1016/j.compstruct.2013.07.019.
[3] Di Sciuva M, Gherlone M, Iurlaro L, Tessler A. A class of higher-order C0 composite and sandwich
beam elements based on the Refined Zigzag Theory. Composite Structures 2015;132:784–803.
https://doi.org/10.1016/j.compstruct.2015.06.071. [4] Sorrenti M, Di Sciuva M. An enhancement
of the warping shear functions of Refined Zigzag Theory. Journal of Applied Mechanics 2021;88:7.
https://doi.org/10.1115/1.4050908. [5] Iurlaro L, Gherlone M, Di Sciuva M, Tessler A. A Multi-scale
Refined Zigzag Theory for Multilayered Composite and Sandwich Plates with Improved Transverse Shear
Stresses, Ibiza, Spain: 2013. [6] Auricchio F, Sacco E. Refined First-Order Shear Deformation Theory
Models for Composite Laminates. J Appl Mech 2003;70:381–90. https://doi.org/10.1115/1.1572901
Tria and quad plate finite elements based on RZT (m) for the analysis of multilayered sandwich structures
Full text linkAim of the paper is to develop and to assess a class of plate finite elements for the analysis of multilayered composite and sandwich structures. The adopted model is the mixed Refined Zigzag Theory (RZT(m)), based on the kinematics of the Refined Zigzag Theory (RZT) and on the assumption of transverse shear stresses coming from integration of indefinite equilibrium equations. A triangular and quadrilateral flat finite element are developed by means of the Reissner’s Mixed Variational Theorem and an interpolation strategy to eliminate shear locking. Several numerical examples are discussed to demonstrate the accuracy of RZT(m) and related finite elements for static response, free-vibrations and critical load problems of sandwich structures
Qualche considerazione sui concetti di equilibrio e stabilità e sulla loro trattazione linearizzata
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Material and strain sensing uncertainties quantification for the shape sensing of a composite wing box
Full text linkThe shape sensing, i.e. the reconstruction of the displacement field of a structure from discrete strain measures, is a key tool for the support and development of the modern Structural Health Monitoring frameworks and has received a huge attention in the last few decades. The parallel increase in the use of composite materials in the aerospace industry has consequently generated the need to investigate the applicability of the shape sensing methods to this peculiar kind of materials. In fact, the manufacturing complexity of the composite materials can result in a significant variability in the characteristics of the material. Therefore, a study on the propagation of this kind of uncertainty on the performance of the shape sensing methods is paramount. The uncertainties in the strain measurements can influence the shape sensing results and must be also considered. This paper, for the first time, investigates the propagation of these two sources of inputs’ uncertainty on the performance of three shape sensing methods, the inverse Finite Element Method (iFEM), the Modal Method (MM) and the Ko's Displacement theory. Using the Monte Carlo Simulation (MCS) with Latin Hypercube Sampling (LHS), the robustness of the three methods with respect to the inputs’ variability is evaluated on the reconstruction of the displacement field of a composite wing box. The MM shows a significant robustness and the iFEM, although more affected by the uncertainties, is the method that achieves the best accuracy. The Ko's displacement theory, on the other hand, is the less accurate and the less robust
Numerical and experimental predictions of the static behaviour of thick sandwich beams using a mixed {3,2}-RZT formulation
Full text linkThis paper presents a numerical and experimental assessment of the static behaviour of thick sandwich beams using the mixed {3,2}-Refined Zigzag Theory (RZT(m) {3,2}). The displacement field of the RZT(m) {3,2} assumes a piecewise continuous cubic zigzag distribution for the axial contribution and a smoothed parabolic variation for the transverse one. At the same time, the out-of-plane stresses are assumed continuous a-priori: the transverse normal stress is given as a third-order power series expansion of the thickness coordinate, whereas the transverse shear one is derived through the integration of Cauchy's equation. The equilibrium equations and consistent boundary conditions are derived through a mixed variational statement based on the Hellinger-Reissner (HR) theorem and a penalty functional to enforce the strain compatibilities between the assumed independent stress fields and those obtained with the constitutive equations. Based on the proposed model, a simple C0-continuous two-node beam finite element is formulated (2B-RZT(m){3,2}). Firstly, the analytical and FE model accuracies of the presented formulation are addressed, and comparisons with the available three-dimensional elasticity solutions are performed. Subsequently, an experimental campaign is conducted to evaluate the static response of various thick sandwich beam specimens in three- and four-point bending configurations. The thick beam specimens are equipped with Distributed Fibre Optic Sensors (DFOS) embedded in the sandwich layup to measure axial deformation at the sandwich interfaces directly. Finally, the experimental data are compared with the available numerical models, highlighting the formulated numerical model's performances and limitations
Shape sensing with inverse Finite Element Method for slender structures
No full textThe methodology known as "shape sensing" allows the reconstruction of the displacement field of a structure starting from strain measurements, with considerable implications for structural monitoring, as well as for the control and implementation of smart structures. An approach to shape sensing is based on the inverse Finite Element Method (iFEM) that uses a variational principle enforcing a least-squares compatibility between measured and analytical strain measures. The structural response is reconstructed without the knowledge of the mechanical properties and load conditions but based only on the relationship between displacements and strains. In order to efficiently apply iFEM to the most common structural typologies of civil engineering, its formulation according to the kinematical assumptions of the Bernoulli-Euler theory is presented. Two beam inverse finite elements are formulated for different loading conditions. Depending on the type of element, the relationship between the minimum number of required measurement stations and the interpolation order is defined. Several examples representing common applications of civil engineering and involving beams and frames are presented. To simulate the experimental strain data at the station points and to verify the accuracy of the displacements obtained with the iFEM shape sensing procedure, a direct FEM analysis of the considered structures is performed using the LUSAS software
Experimental and numerical investigation of the Refined Zigzag Theory for accurate buckling analysis of highly heterogeneous sandwich beams
Full text linkThe Refined Zigzag Theory (RZT) is a structural theory developed for the analysis of composite multilayer and sandwich beams. However, the accuracy of RZT for buckling analysis of sandwich beams has not been experimentally investigated, and for RZT and Timoshenko Beam Theory (TBT) the effect of the degree of heterogeneity on their accuracy requires further study. The aim of this work was to validate the use of the RZT for predicting the critical buckling loads of sandwich beams, even with highly heterogeneous material properties, and to assess the use of the TBT for the same application. Buckling experiments were conducted on five foam-core sandwich beams, which varied in geometry and included highly heterogeneous configurations. For each beam, two finite element (FE) models were analyzed using RZT- and TBT-beam FEs. The comparison between the numerical and the experimental results highlighted a major capability of RZT to correctly predict the critical buckling load for all the beams considered. The dependence of the TBT results on the beam characteristics was further investigated through a parametric analysis, which showed the dominant effect to be a close to linear relationship between the TBT error and the beam face-to-core thickness ratio. The work demonstrated the outstanding accuracy of the RZT predictions, including the superior capabilities with respect to TBT, and has application for rapid and accurate analysis of industrial structures
Shape sensing of beams with complex cross-sections using the inverse Finite Element Method
No full textThis paper presents the iFEM formulation for beams with complex cross-sectional profiles having at most one plane of symmetry. As the iFEM has already been successfully used for the shape sensing of beams with circular or rectangular cross sections, this work paves the way for formulating a generalized one-dimensional beam element, applicable for any kind of cross-section. Challenges arising due to the asymmetry of the profile, in regard to sensor placement on the surface of the beam and accurate prediction of transverse shear strain measures from experimental strain measurements are addressed. The new formulation is illustrated with the specific case of a beam with a symmetric airfoil profile (NACA 0016), subjected to symmetrically and asymmetricallyapplied static load conditions
Multilayered triangular and quadrilateral flat shell elements based on the Refined Zigzag Theory
Full text linkThe paper presents a class of C0-continuous, flat shell elements based on the Refined Zigzag Theory (RZT) for the analysis of multilayered and curved composite and sandwich structures. The use of the interdependent interpolation strategy allows eliminating the shear-locking phenomenon and introducing the drilling rotation necessary to complete the set of classical nodal degrees of freedom (three displacements and three rotations). Additional kinematic variables are present in RZT, the zigzag rotations around the in-plane axes that measure the normal distortion typical of multilayered structures. An additional “drilling” zigzag rotation is therefore included among the nodal degrees of freedom in order to properly model curved and built-up structures. A stabilization procedure is adopted to suppress spurious zero-energy modes. A three-node triangular and a four-node quadrilateral flat shell element are formulated with 9 degrees of freedom per node. Example problems involving flat and curved multilayered structures are presented and discussed in order to assess the accuracy and convergence properties of the presented elements. Both static response predictions and free vibrations analyses are considered and the comparison is made with analytic RZT solutions, high-fidelity 3D finite element models and FSDT-based flat shell elements
Buckling analysis of angle-ply multilayered and sandwich plates using the enhanced Refined Zigzag Theory
Full text linkThe recent enhancement of the standard Refined Zigzag Theory (RZT), herein named the enhanced Refined Zigzag Theory (en-RZT), has extended the range of applicability of the RZT to angle-ply multilayered and sandwich plates. The aim of the present investigation is to assess the numerical performances of the en-RZT for the buckling analysis of angle-ply multilayered and sandwich rectangular plates under in-plane normal loads. The linearized stability equations are obtained using the Ritz method in conjunction with the principle of virtual work, by means of Gram–Schmidt orthogonal polynomials. In order to assess the accuracy of the en-RZT, buckling loads of angle-ply laminated and sandwich plates are evaluated and compared with the numerical results available in open literature. The numerical investigation highlights the high accuracy of the en-RZT in predicting buckling loads. The study contains a parametric analysis aimed to investigate the influence of various design parameters, such as plate aspect ratio, thickness, lamina orientations, in-plane load combinations and boundary conditions on the buckling loads
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