Equivalent Single-layer Model Research Articles

Equivalent single-layer model for hierarchical diamond honeycomb sandwich panels using variational asymptotic method

Sandwich panels with a hierarchical core facilitate the optimal utilization of limited materials to enhance bearing capacity in strategic locations. Nevertheless, the complex sub-structural cells have hindered extensive investigation into their static and modal characteristics. The objective of this study is to obtain the effective plate properties by homogenizing the representative unit cell and subsequently applying them to a two-dimensional equivalent single layer model (2D-ESL) based on the variational asymptotic method (VAM). Furthermore, three-point bending test of 3D-printed specimen and 3D detailed finite element results are employed to confirm the predictions of 2D-ESL under various load and boundary conditions. The effects of material factors (such as fiber volume fraction and layup configuration) and structural parameters (including aspect ratio, hierarchy ratio, and slenderness ratios of ribs and ligaments) on the equivalent stiffness and modal characteristics are also examined. Moreover, sandwich panels with different hierarchical configurations (including 2, 3 and infinite segments within the inclined ribs) are investigated to provide valuable insights for selecting appropriate hierarchies based on specific conditions. Notably, the transition from polyline to arc-shaped segments has been proven to effectively mitigate local stress concentration, leading to a more uniform distribution of local displacement.

Research on Decision Optimization and the Risk Measurement of the Power Generation Side Based on Quantile Data-Driven IGDT

In an environment marked by dual carbon goals and substantial fluctuations in coal market prices, coal power generation enterprises face an urgent imperative to make scientifically informed decisions regarding production management amidst significant market uncertainties. To tackle this challenge, this paper proposes a methodology for optimizing electricity generation side market decisions and assessing risks using quantile data-driven information-gap decision theory (QDD-IGDT). Initially, a dual-layer decision optimization model for electricity production is formulated, taking into account coal procurement and blending processes. This model optimizes the selection of spot coal and long-term contract coal prices and simplifies the dual-layer structure into an equivalent single-layer model using the McCormick envelope and Karush–Kuhn–Tucker (KKT) conditions. Subsequently, a quantile dataset is generated utilizing a short-term coal price interval prediction model based on the quantile regression neural network (QRNN). Interval constraints on expected costs are introduced to develop an uncertainty decision risk measurement model grounded in QDD-IGDT, quantifying decision risks arising from coal market uncertainties to bolster decision robustness. Lastly, case simulations are executed by using real production data from a power generation enterprise, and the dual-layer decision optimization model is solved by employing the McCormick–KKT–Gurobi approach. Additionally, decision risks associated with coal market uncertainties are assessed through a one-dimensional search under interval constraints on expected cost volatility. The findings demonstrate the effectiveness of the proposed research methodology in cost optimization within the context of coal market uncertainties, underscoring its validity and economic efficiency.

Equivalent single layer approach for ultimate strength analysis of box girder under bending load

The objective of this study is to assess the ultimate strength of box girder using equivalent single layer (ESL) approach. Box girders, composed of stiffened panels, are subjected to a four-point bending load for the ultimate strength analysis. The results of ESL were validated using FEM and experiments. Modeling of ESL includes a single plate with the same stiffness as the actual stiffened panel topology, as opposed to traditional FEM modeling and test specimen, where the stiffener is explicitly modeled. For ESL approach, VUGENS subroutine in Abaqus software is used to define the non-linear stiffness properties. Material and geometric non-linearities were considered during simulations. The ESL stiffness matrix was derived from unit cell simulations under six loading conditions. In this case, a unit cell is a periodic structure that represents the entire stiffened panel model. Contact interaction is utilized to simulate the controlled movement of a hollow cylinder contacting a box girder, a precise modeling approach essential for comprehensive analysis. The explicit dynamic analysis is used to allow the large deformation and non-linear material behavior on stiffened panels. The effect of initial imperfection, residual stress, and stiffener slenderness on the ultimate strength was studied. The ultimate strength of a box girder is determined by its maximum bending moment. According to the results, the ESL model achieves results similar to the traditional FEM model and experiment. The accuracy of ESL to FEM in the ultimate strength assessment varies depending on the magnitude of residual stress and initial imperfection applied to the stiffened panel.

A homogenization method for natural frequencies and damping of sandwich panels based on representative volume elements

The existing representative volume element (RVE) homogenization method, grounded in finite element analysis, assumes a planar configuration for the sections of sandwich panels, implying rigidity. To accommodate non-planar displacements, this study introduces a novel homogenization method reliant on the displacements obtained from four-point bending and four-point torsion. The present method is employed to investigate the mode shapes, natural frequencies, and loss factors of sandwich panels utilizing an equivalent single-layer shell model. Validation of the present method is conducted through the analysis of lattice sandwich panels and corrugated sandwich panels incorporating viscoelastic layers. Comparative analyses encompassing the results of the experiment, the conventional solid model method, the present method, and the conventional plane method are undertaken. The universality of the present method is substantiated. The present method enhances the precision of natural frequency prediction.

A new finite element for the analysis of functionally graded shells

This paper presents the development of a finite element model for functionally graded shells. The formulation solves the equations of the Stress Approach Model for Functionally Graded shells (SAM-FG) developed and validated by Rubio Rascón et al. (2020). SAM-FG involves only 5 generalized displacements and its equations differ from those of classical first order shear deformation theories mainly in the expressions of the generalized stiffnesses and the approximation of stresses. A 10-node isoparametric triangular element using cubic shape functions and 5 degrees of freedom per node is proposed. The equations of the numerical method were implemented in a program called FESAM-FG. The finite element code enables to calculate and visualize generalized fields on the middle surface of the shell and stress components at a given position across the thickness of the structure. To test the accuracy and convergence of FESAM-FG results, the numerical tool is applied to the structural analysis of four families of functionally graded shells with various distributions of material properties and thickness-to-radius ratios. Results are compared to those obtained by solid finite elements, a higher order shear deformation theory developed by Viola et al. (2014) and two shell models used for laminated composite structures: an equivalent single layer model and a layerwise model, both using 30 fictitious layers.

Buckling and Wrinkling of Sandwich Structures Reinforced by Functionally Graded Carbon Nanotubes

Soft-core sandwich structures subjected to coplanar compression loading might exhibit wrinkling behaviors, which will reduce significantly the support capability of the structure. In order to prevent the wrinkling behaviors of sandwich structures, this work attempts to improve stiffness of face sheets reinforced by carbon nanotubes (CNTs). To achieve such a goal, the following work is to be required. First, an equivalent single-layer model will be proposed to accurately analyze the wrinkling behaviors of sandwich structures because only layerwise models possess the capability to accurately predict wrinkling behaviors of sandwich structures, but these models are often computationally expensive. Second, the influence of distributions of the CNTs through thickness on the buckling and wrinkling behaviors for the sandwich beams will be investigated in detail. Moreover, the translation mechanism between the buckling deformation and the wrinkling deformation has been also explored. The proposed model has been verified by using the quasi three-dimensional (3D) elasticity solutions and the 3D finite-element results (3D-FEM). Third, by selecting the proper distributions of the CNTs through the thickness of sandwich structures, the wrinkling behaviors of sandwich structures can be effectively prevented. Thus, a scheme resisting wrinkling deformation of sandwich structures has been proposed, which can provide a reference for other investigators.

A hybrid analytical method for predicting vibrations in vacuum and water of the sandwich plate with z-direction reinforcements and cavities

This paper presents a multi-functional sandwich plate with z-direction reinforcements and cavities that has the potential to meet the pressure resistance and the acoustic stealth requirement of the underwater equipment in deep sea. In order to predict its free and forced vibrations in both vacuum and water, a hybrid analytical method is proposed by combining a multi-level homogenization model and a layerwise model. The core with z-direction reinforcements and cavities is homogenized and effective elastic constants are calculated by the Mori-Tanaka formula. Vibration equations of the sandwich plate are derived utilizing the layerwise model based on the first-order shear deformation theory. The water effect is taken into account by the Euler's equation. Vibration results are solved under simply supported boundaries. Validation studies and parameter analyses are then conducted. It is found that different from the existing theoretical approach based on the equivalent single-layer model that is only applicable for the sandwich with a hard core material, the present hybrid analytical method is well suitable for vibration predictions of the sandwich plate with z-direction reinforcements and cavities, no matter the hard or soft core. Besides, due to the homogenization of the three-phase core and the neglecting of local vibrations in the layerwise model, the present analytical method only predicts accurate free vibrations when the half wave numbers are less than numbers of cavities in x and y directions, and also just shows the high precision in force vibration calculations as the maximum analysis frequency is less than the natural frequency of the mode corresponding to numbers of cavities.

Equivalent mechanical model of resin-coated aramid paper of Nomex honeycomb

Nomex honeycomb, as the core part of sandwich structure, has been widely used in the aviation and aerospace industries. Ultrasonic cutting, as a technology that can reduce machining defects, has been highlighted in recent years, but some problems in the cutting process still need to be investigated significantly. Finite element method (FEM) offers an alternative way of in-depth understanding of the problems. However, the multi-layer structure of the Nomex honeycomb cell wall, which is the resin-coated aramid paper, makes it difficult to establish a detailed multi-layered finite element (FE) model at a macro-scale level. Therefore, a novel segmented material mechanical model, which is applicable to the equivalent single-layer model, is proposed based on the analysis of the mechanical behaviour of the multi-layer structure of the resin-coated aramid paper. Moreover, the calculating method of equivalent mechanical parameters of the multi-layer structure are proposed theoretically. The proposed method is first verified by FE models at a micro-scale level in both single-layer and multi-layer approaches. Additionally, simulations and experiments are performed to verify that the proposed method performs well in the simulation of ultrasonic cutting for Nomex honeycomb, and the difficulty of modelling and CPU time of simulation are reduced significantly.

Ultimate strength assessment of stiffened panels using Equivalent Single Layer approach under combined in-plane compression and shear

Equivalent Single Layer (ESL) approach is extended to model ultimate strength of stiffened panels under a combination of in-plane compression and shear. Ultimate strength under this combined loading depends on the loading path and could be lower than for only uni-axial compression since shear load can produce axial forces. Thus, to account for this effect, the recent ESL model is extended to include A13 stiffness component as non-zero value. Procedure to accurately obtain A13 is presented. In ESL approach, a stiffened panel is transformed into a two-dimensional (2D) single layer with the same stiffness obtained from unit cell simulations. To obtain non-linear stiffness matrix of ESL, elastic–plastic material properties and initial imperfection were applied to the unit cell. ESL responses were validated by comparing numerical and experimental results from the literature. Several stiffened panel configurations were analyzed to obtain different collapse modes. Combined loads were applied for shear to compression ratio of 0, 1, and 2. Lateral pressure loading was also considered in the simulations. Analyses were carried out based on the load sequences consisting of: 1) compression and shear loaded simultaneously and 2) shear applied first, followed by compression. The results show that the modified ESL can well capture the effect of shear load on ultimate strength in comparison to a detailed 3D FEM model of stiffened panels. The accuracy of the ESL varies depending on the collapse mode of stiffened panels.

Thermomechanical loading and cut-out effect on static and dynamic responses of multilayered structure with TD properties

The influences of cut-out on the static and dynamic deflection of the multilayered composites under thermomechanical are predicted numerically. In this regard, a mathematical model is prepared using the third-order kinematics associated with the isoparametric finite element steps. Moreover, the solution accuracy of the derived model has been tested frequently by solving a series of examples (obtained from MATLAB code) similar to the published one. In addition, the model verification is extended further by comparing the numerical results with the in-house experimental data recorded under the static and dynamic loading conditions. The equivalent single-layer theoretical model (prepared for the cut-out abided laminated structure) exactness is maintained by considering the variation of in-plane displacement as cubic. In contrast, the state space variable along with the thickness is constant. Additionally, the experimental comparisons are made considering the influences of different cut-out geometry, temperature increment, and loading intensities. The effects of cut-out and the laminated structural geometrical parameters that affect the final design consideration due to the thermomechanical loading as well as the temperature-dependent properties are computed using the derived model, and the outcomes are discussed in detail.

Large deformations of piezoelectric laminated beams based on the absolute nodal coordinate formulation

Piezoelectric beams can be deformed into many patterns and have potential application prospect in the field of smart foldable structures. Knowing the large deformation actuation mechanism of such structures is the basis in the soft machine design. In this paper, using the Equivalent Single-layer model, a new flexible piezoelectric laminated beam element based on the Absolute Nodal Coordinate Formulation (ANCF) is established. This beam element contains three layers, in which the base layer is made of linear elastic material, and the top and bottom layers are both made of piezoelectric materials. The constitutive equation of the piezoelectric material is introduced to derive the elastic force and the piezoelectric force. The correctness of the new beam element are validated by comparing with the analytical solution and the results from the equivalent bending moment equation. A group of large deformation patterns of the laminated beam are given to prove the applicability of the proposed element. Results show that when various voltages are applied to the piezoelectric actuators that located in arbitrary positions of the base beam, the flexible beam can achieve the desired shape smoothly. Dynamic responses of a highly flexible pendulum with and without piezoelectric layers are also investigated. Research in this work can be helpful to the understanding of complicated coupled nonlinear mechanics of flexible or soft structures with covered smart piezoelectric materials.

Buckling and free vibration of laser-welded web-core sandwich panels: Extreme sensitivity to variation of weld rotational stiffness

T-joints in laser-welded web-core sandwich panels connect face-plates with web-plates. Rotational stiffness of the T-joints affects transverse shear stiffness of the panels. In this paper we describe the effect of geometric and material parameters on the rotational stiffness of the T-joints, and show that laser weld thickness bears the highest influence. The analysis is performed using detailed FEM models of the joints that include laser welds. The effect is translated to critical buckling and first natural frequency of laser-welded web-core sandwich panels. Structural analyses are performed using equivalent single layer model with the first order shear deformation theory. The models used in this research are validated using experiments and numerical results from literature. The most sensitive and the least sensitive panels to variation of the rotational stiffness are outlined by means of global optimization. We describe them in terms of stiffness coefficients and design variables. The analysis reveals that insensitive panels have small height and small distance between web-plates, and as such have low structural performance for their area density. Panels with the best structural performance are generally quite sensitive to variation of the rotational stiffness, but the sensitivity can be even higher in other panels. It was found that rotational stiffness of T-joints is important in design and optimization of the panels, since the best performing panels of the same area density have different cross-sections for different rotational stiffness.

Theoretical modelling and experimental verification of modal responses of skewed laminated sandwich structure with epoxy-filled softcore

The experimental eigenvalue responses of the epoxy-filled skew sandwich structure are computed first-time in this research to show the suitability of equivalent type single-layer higher-order theory (including through-thickness stretching term effect) for the analysis. The sandwich shell model is formulated mathematically for the variable geometrical configurations considering the effect of skew angles. Further, the motion equation of the vibrated structure is solved numerically with the advent of the linear isoparametric finite element technique. Firstly, the numerical solution accuracy is established by verifying the modal values with the already published data including their element sensitivity test. Also, a few experimental frequencies (first-five mode) are recorded (impact type vibration analyzer) using the in-house fabricated sandwich plate components considering the experimental material properties for the comparison purpose. Moreover, a simulation model is prepared using the commercial package to show the efficacy of the presently proposed single layer theory for the analysis of the sandwich structure with and without the skew angle effect. The present comparison indicates that the proposed equivalent single-layer model is capable of solving the modal responses considering different structural input parameters (number and stacking sequences of the face sheet layers and variable aspect/thickness ratios) with adequate accuracy. Lastly, the verified model is explored to show its applicability by solving different numerical examples due to the change in their basic input parameters affecting the geometry, material properties and the stiffness.

An adaptive shell element for explicit dynamic analysis of failure in laminated composites Part 2: Progressive failure and model validation

To enable modelling of the progressive failure of large, laminated composite components under crash or impact loading, it is key to have a numerical methodology that is both efficient and numerically robust. A potential way is to adopt an adaptive method where the structure is initially represented by an equivalent single-layer shell model, which during the analysis is adaptively transformed to a high-resolution layer-wise model in areas where higher accuracy is required. Such a method was recently developed and implemented in the commercial finite element solver LS-DYNA, aiming at explicit crash analysis (Främby, Fagerström and Karlsson: An adaptive shell element for explicit dynamic analysis of failure in laminated composites - Part 1: Adaptive kinematics and numerical implementation, 2020). In the current work, the method is extended to the case of interacting inter- and intralaminar damage evolution. As a key part, we demonstrate the importance of properly regularising the intralaminar failure described by a smeared-crack model, and show that neglecting to account for the crack-versus-mesh orientation may lead to significant errors in the predicted energy dissipation. We also validate the adaptive approach against a four-point beam bending test with matrix-induced delamination growth, and simultaneously show the capability of the proposed method to – at lower computational expense – replicate the results from a refined, non-adaptive model.

Free vibration analysis of composite laminated beams with curvilinear fibers via refined theories

In this article, free vibration finite element analysis of variable stiffness composite laminate (VSCL) beams is carried out. The variable stiffness of laminated beams is acquired by using laminates with curvilinear fibers instead of straight fibers. The analysis is based on the higher-order theories that have been presented by Carrera Unified Formulation (CUF). Equivalent Single Layer (ESL) model, which is independent of the number of constitutive layers, has been utilized to consider a number of unknown variables. Several numerical results for the analysis of VSCL beams are presented and the effect of the fiber orientation angle and various boundary conditions on the natural frequencies are demonstrated.

Free vibration analysis of axially moving laminated beams with axial tension based on 1D refined theories using Carrera unified formulation

In this paper, free vibration finite element analysis of axially moving laminated composite beams subjected to axial tension is studied. It is assumed that the beam has a constant axial velocity and is subject to uniform axial tension. The analysis is based on higher-order theories that have been presented by Carrera Unified Formulation (CUF). In the CUF technique, the three dimensional (3D) displacement fields are expressed as the approximation of the arbitrary order of the displacement unknowns over the cross-section. This higher-order expansion is considered in equivalent single layer (ESL) model. The governing equations of motion are obtained via Hamilton\' s principle. Finally, several numerical examples are presented and the effect of the ply-angle, travelling speed and axial tension on the natural frequencies and beam stability are demonstrated.

Efficient 3D Stress Capture of Variable-Stiffness and Sandwich Beam Structures

Accurate modeling of composite structures is important for their safe application under different loading conditions. To provide accurate predictions of three-dimensional (3D) stress fields in an efficient computational framework, in this study, a modeling approach that builds upon the recently developed hierarchical Serendipity Lagrange finite elements (FEs) is employed. The approach provides layerwise (LW) and equivalent single-layer (ESL) models for analyzing constant- and variable-stiffness laminated beam structures. To enhance the capability of the ESL model, two zig-zag (ZZ) functions, namely, Murakami’s ZZ function (MZZF) and the refined ZZ theory function (RZT), are implemented. For constant-stiffness laminated and sandwich beams, the RZT ZZ function predicts the structural response more accurately than the MZZF. However, for variable-stiffness laminated structures, the applicability of RZT remains unknown and its accuracy is therefore tested within the present formulation. Results obtained are validated against 3D closed-form and 3D FE solutions available from the literature. For similar levels of accuracy, significant gains in computational efficiency are achieved over 3D FE and LW models by using the ESL approach with RZT ZZ functions.

Nonlinear finite element analysis of lattice core sandwich beams

A geometrically nonlinear finite element model is developed for the bending analysis of micropolar Timoshenko beams using the principle of virtual displacements and linear Lagrange interpolation functions. The nonlinearity enters the model via a nonlinear von Kármán strain term that allows the micropolar beam to undergo moderate rotations. The nonlinear micropolar Timoshenko beam is used as an equivalent single layer model to study four different lattice core sandwich beams. A two-scale energy method is used to derive the micropolar constitutive equations for web, hexagonal, Y-frame and corrugated core topologies. Various bending cases are studied numerically using the developed 1-D finite element model. Reduced integration techniques are used to overcome the shear and membrane locking. The present 1-D results are in good agreement with the corresponding 2-D finite element beam frame results for global bending.

Low-Terahertz Modeling of Graphene/Dielectric Multilayers Using an Equivalent Single Layer in Reverberation Environment

Low-terahertz field transmission through a graphene/dielectric laminate (GDL) is investigated by modeling each graphene/dielectric bilayer (GDB) as a homogeneous uniaxial anisotropic medium. The effective medium approximation is applied to define the transverse effective complex conductivity of the GDB, which is expressed as a function of the thickness and permittivity of the dielectric layer, and of the graphene sheet conductivity. The graphene/dielectric multilayered structure is modeled as an equivalent single layer (ESL), having the total thickness of the multilayer and characterized by the effective complex conductivity of the GDB. The model is applied to simulate two different sets of chemically doped graphene samples, characterized by measured values of the dc sheet resistance. The transmissivity, reflectivity, and shielding effectiveness of the resulting GDLs are computed in a reverberation environment up to 10 THz by applying the transfer matrix method to both the GDL and to the ESL. The comparison of the obtained results proves the high accuracy of the homogenization-based ESL model.

Buckling of sandwich panels with transversely flexible core: Correction of the equivalent single-layer model using thick-faces effect

Modeling a periodic structure as a homogeneous continuum allows for an effective structural analysis. This approach represents a sandwich panel as a two-dimensional plate of equivalent stiffness. Known as the equivalent single-layer, the method is used here to analyze bifurcation buckling of three types of sandwich panels with unidirectional stiffeners in the core: truss-core, web-core and corrugated-core panels made of an isotropic material. The transverse shear stiffnesses of these panels can differ by several orders of magnitude, which cause incorrect buckling analysis when using the equivalent single-layer model with the first-order shear deformation theory. Analytical solution of the problem predicts critical buckling loads that feature infinite number of half-waves in the direction perpendicular to the stiffeners. Finite element model also predicts buckling modes that have non-physical, saw-tooth shape with infinite curvature at nodes. However, such unrealistic behavior is not observed when using detailed three-dimensional finite element models. The error in the prediction of the critical buckling load is up to 85% for the cases considered here. The correction of the equivalent single-layer model is proposed by modeling the thick-faces effect to ensure finite curvature. This is performed in the finite element setting by introducing an additional plate with tied deflections to the equivalent single-layer plate. The extra plate is represented with bending and transverse shear stiffness of the face plates. As a result, global buckling is predicted accurately. Guidelines are proposed to identify the sandwich panels where ordinary model is incorrect. Truss-core and web-core sandwich panels need the correction. Corrugated-core panels without a gap between plates in the core have smaller shear orthotropy and do not need the correction. Modeling the thick-faces effect ensures correct results for all cases considered in this study, and thus one should resort to this approach in case of uncertainty whether the ordinary equivalent single-layer model is valid.