Equivalent Plate Model Research Articles

Equivalent plate model of shape memory polymer composite based variable height corrugated structure

In this work, an equivalent plate model for a shape memory polymer composite based corrugation structure with varying height, for morphing application, is developed. A thermodynamically consistent constitutive relation based on temperature dependent phase transformation concept for shape memory polymer matrix is used in the composite model. The homogenization model is based on the equivalence of strain energy of a representative unit volume. Following validation, as an application, commonly used airfoil profile NACA 63210 was considered to demonstrate the model utility. Parametric studies have been undertaken to investigate the influence of corrugation parameters like thickness and number of corrugations.

VAM-based equivalent-homogenization model for 3D re-entrant auxetic honeycomb structures

The integration of two vertically crossed re-entrant honeycombs gives rise to a 3D re-entrant honeycomb (3D-RH), which displays intricate three-dimensional auxeticity. To simplify the modeling complexity, this study employs asymptotic analysis of the energy functional stored in the unit cell of the 3D-RH to develop unified constitutive models for 3D structures and panels with multiple length scales. Based on this, an equivalent 3D Cauchy continuum model is established for multi-layer 3D-RH, while a 2D equivalent plate model is developed for sandwich panel with single-layer 3D-RH (SP-3D-RH). The accuracy and effectiveness of these equivalent models are subsequently confirmed through investigations of the X-shaped compressive deformation of multi-layer 3D-RH, as well as the in-plane auxetic deformation and out-of-plane dome-shaped deformation of single-layer 3D-RH. Compared to traditional hexagonal and re-entrant honeycomb sandwich panels, the SP-3D-RH exhibited enhanced equivalent tensile stiffness while effectively mitigating maximum deformation and strain energy under bending and tension conditions. However, due to its relatively low equivalent shear stiffness, the SP-3D-RH exhibited significant deformation and strain energy when subjected to torsional loads, resulting in local stress concentration in the connect region between adjacent facesheets. Parameter analysis shows that the facesheet-to-strut thickness ratio had a substantial impact on the equivalent stiffness and in-plane behavior of the SP-3D-RH, while adjusting the horizontal strut length, strut depth, and re-entrant angle of the 3D-RH can improve the out-of-plane behavior.

Natural Frequency of Lightweight Composite Slabs Based on Experimental Study and Numerical Modelling

Recently, lightweight composite slabs have become increasingly popular. Lightweight composite slabs are an innovation that provides a better and more convenient construction method for floor systems. Under dynamic loads, lightweight composite slabs may experience meagre inertia forces due to poor stiffness or low mass. Compared to conventional composite slabs, lightweight composite slabs are 40% lighter and more susceptible to structural resonance. Therefore, the vibration behaviour must be controlled to avoid discomfort issues. This study investigates the natural frequency of lightweight composite slabs through experimental study and numerical modelling. In the experimental study, lightweight composite slabs were prepared for the hammer-impact test. The slab thickness ranges from 100 mm to 200 mm. In numerical modelling, lightweight composite slabs were modelled in SAP2000 using a unique technique called the simplified equivalent plate model. The effective material properties were derived from the rule of mixtures and depend exclusively on elastic properties with strength characteristics. The results of the experimental study and numerical modelling agree positively. The natural frequency decreased with slab thickness, signifying that the natural frequency is dominated by mass rather than stiffness. Overall, the natural frequency of lightweight composite slabs is around 27.23Hz to 31.45Hz.

2d Models of Steel Beams under Patch Loading for Buckling Coefficient Assessment

This paper proposes an equivalent 2d model incorporating translational and rotational springs to simulate constraints in the stability analysis of steel plates under in-plane patch loading. The main objective is to develop an equivalent stability analysis of these plates by accurately calibrating the level of constraint provided by the stiffeners and flanges. The calibration involves determining the spring constants of a 2d equivalent plate model to achieve the same linear elastic critical load and buckled shape as a 3d model. The paper uses the buckling coefficient as the objective function to calibrate the translational and rotational springs. The ultimate goal is to effectively replicate the constraint conditions imposed by the flanges and vertical stiffeners within the structure. The proposed simplified method yields results that closely align with those obtained from the original 3d model, incorporating finite-rigidity restraints.

Direct-Homogenization-Based Plate Models of Integrated Thermal Protection System for Reusable Launch Vehicles

Integrated thermal protection systems of reusable launch vehicles (RLVs) can have a corrugated core sandwich structure and experience location-dependent thickness-wise temperature gradient. The sandwich structure can be optimized depending on the location on RLV using finite element method simulations of RLV components. However, such analysis can be computationally challenging due to the disparate length scales between the RLV components and features of the sandwich structure. Two different equivalent plate models based on first-order shear and normal deformation theory, and first-order shear and second-order normal deformation theory (FSSNDT) have been utilized in this work to address this drawback. A direct homogenization technique involving unit cell and cantilever beam analysis has been developed to calibrate the equivalent plate properties for different thickness-wise temperature variations. The accuracy of the plate models has been evaluated by comparing their responses with a full-scale model for different thickness-wise temperature gradients and a uniform pressure. The comparisons clearly indicate that the FSSNDT-based plate model calibrated via direct homogenization captures both the displacements and strains in the in-plane and transverse directions accurately, and can be used to perform RLV component analysis efficiently to obtain location-specific optimal design of corrugated core sandwich structures.

Dynamic Characteristics and Effective Stiffness Properties of Sandwich Panels with Hierarchical Hexagonal Honeycomb.

The dynamic characteristics of sandwich panels with a hierarchical hexagonal honeycomb (SP-HHHs) show significant improvements due to their distinct hierarchy configurations. However, this also increases the complexity of structural analysis. To address this issue, the variational asymptotic method was utilized to homogenize the unit cell of the SP-HHH and obtain the equivalent stiffness, establishing a two-dimensional equivalent plate model (2D-EPM). The accuracy and effectiveness of the 2D-EPM were then verified through comparisons with the results from a detailed 3D FE model in terms of the free vibration and frequency- and time-domain forced vibration, as well as through local field recovery analysis at peak and trough times. Furthermore, the tailorability of the typical unit cell was utilized to perform a parametric analysis of the effects of the length and thickness ratios of the first-order hierarchy on the dynamic characteristics of the SP-HHH under periodic loading. The results reveal that the vertices serve as weak points in the SP-HHH, while the vertex cell pattern significantly influences the specific stiffness and stiffness characteristics of the panel. The SP-HHH with hexagonal vertex cells has superior specific stiffness compared to panels with circular and rectangular vertex cells, resulting in a more lightweight design and enhanced stiffness.

Twisting tubes as soft robotic valves

Soft materials and structures have been blooming with utmost popularity, showing great potential as soft machines for delicate design and new application. Among them, the simple structures of soft tubes are widely applied in engineering. However, a comprehensive understanding of their twisting behaviors is still lacking. In this study, the complete response of hyperelastic tubes under large-scale torsion is explored. The helical-buckling critical conditions and post-buckling characteristics of twisting tubes are revealed via simulations and experiments. Notably, it is found that a sequence of equal-length kinks along the tubing axis appear as the subsequent buckling of twist after the first helical buckling. And the factors determining the geometry of the periodic kink structure are discussed. To efficiently interpret and predict the kink responses, we employ an equivalent plate model based on finite element analysis. Additionally, the self-locking mechanism of kink for gas-blocking purpose and the pre-stretch effect on the twisting behaviours are investigated. Finally, mechanics design of twisting tubes served as soft valves and its application in soft robotics are demonstrated. This study with insights in extreme twist of elastic tubes is hoped to provide a reference to the off-the-shelf tubes widely applied in industry and inspire new flashes of soft machines.

Free and random vibration analyses of hourglass lattice sandwich panel using an equivalent model based on variational asymptotic method

The hourglass lattice sandwich panel (hourglass LSP) represents a novel type of lightweight and high-strength structure, showcasing new random vibration characteristics related to its distinct geometric configuration. In this paper, a two-dimensional equivalent plate model (2D-EPM) is constructed based on the equivalent stiffness obtained by the variational asymptotic homogenization of the unit cell. The accuracy of 2D-EPM in analyzing the random vibration characteristics (power spectral density (PSD) and root mean square (RMS) responses, as well as effective mass fraction) is validated by the three-dimensional direct numerical simulation and experiments, and its computational efficiency is greatly improved due to the reduced degree of freedoms. On this basis, the effects of structural parameters on the free and random vibration characteristics are systematically analyzed by using the novelty of unit-cell tailorability. The investigation revealed direct correlations between the displacement PSD and RMS with the aspect ratio and lattice rod radius, whereas the inclination angle and face-to-core thickness ratio exhibited inverse correlations. Furthermore, similarities in the low-frequency behaviors of hourglass, pyramid and auxetic LSPs were evident. Specifically, the displacement–RMS curve of the auxetic LSP was comparatively lower, and the hourglass LSP exhibited a similar response, with the second-lowest RMS curve implying correspondingly strong deformation resistance.

Equivalent plate dynamic modeling of space periodic truss structures

Antenna structures used in spaceflight always contain nonlinear joints distributed along their cross section; hence, an equivalent beam model (EBM) that uses a point to describe the cross section is not ideally suited to depicting connection nonlinearity in the structure. To address this difficulty, a novel orthotropic equivalent plate model (EPM) of a space periodic truss structure is proposed. Firstly, utilizing the Mindlin plate hypothesis, strain in the periodic truss element is expanded into a binary high-order Taylor series, and corrections are introduced to normal strain. Secondly, employing the principle of energy equivalence, the equivalent elastic and inertial matrices of the EPM are calculated for in-plane and out-of-plane vibration. Finally, the static condensation method is utilized to reduce the dimensions of the equivalent model to establish an orthotropic equivalent Mindlin plate dynamic model, and its inherent characteristics are analyzed. The simulations show that when in-plane shear strain is expanded to the third order, the EPM can more accurately represent torsional vibration of the periodic truss structure. By introducing normal strain correction coefficients in the longitudinal direction, lateral bending vibrations of the periodic truss structure can be simulated more precisely. Compared with the EBM, the EPM can more accurately depict mechanical characteristics along the cross section. The EPM broadens the scope of continuum equivalent modeling of periodic trusses, and is more useful in engineering practice. In addition, the frequencies and mode shapes of the analytical EPM are consistent with a model using the finite element method, which supports the accuracy and effectiveness of the EPM. The analytical EPM provides a convenient way to study the nonlinear connections and vibration control of antenna structures with multiple joints along their cross section.

Hydroelastic analysis and structural design of a modular floating structure applying ultra-high performance fiber-reinforced concrete

This paper proposes a hydroelastic and structural analysis method for modular floating structures (MFS) constructed from fiber-reinforced polymer (FRP) reinforced ultra-high performance concrete (UHPC) for floating photovoltaic (FPV) systems. Initial structural design, hydroelastic analysis of an equivalent plate model as well as structural analysis are performed in the approach. We investigate the hydroelastic responses of a continuous very large floating structure (VLFS) and three hinge-connected modular floating structures. The floating module is designed to be carbon fiber-reinforced polymer (CFRP) reinforced UHPC box-pontoon. Hinge connections are applied to neighboring modules to form the floating structure. In the hydroelastic analysis, an equivalent plate model was conducted, and the effects of the number of hinge connections and incident wave conditions on the bending moment and vertical deflection were studied. We found that the hinge connection had a great effect on reducing the bending moment. The aspect ratio of the module should also be considered. Cross-sectional analysis was performed to estimate the bending capacity. Nine cross-sectional trials were compared to verify their safety probability in the specific static analysis. Prestressing was found very effective in enhancing the bending capacity. Non-prestressed UHPC is not recommended owing to its limited bending capacity.

Buckling and post-buckling analysis of butterfly-shaped auxetic core sandwich plates based on variational asymptotic method

To further improve the buckling load, a novel type of butterfly-shaped auxetic core sandwich plate (butterfly CSP) was proposed based on the re-entrant honeycomb. Based on the periodic arrangement of the micro-structures within the butterfly CSP, the strain energy, constitutive relationship, and equivalent plate properties of the butterfly CSP were derived in detail using the variational asymptotic method, and the two-dimensional (2D) equivalent plate model (2D-EPM) of the butterfly CSP was developed. On this basis, the connection between the macro-plate model and the micro-unit-cell was established. A comparison with the results of the three-dimensional finite element model revealed that the 2D-EPM had high accuracy in predicting buckling and post-buckling behaviors as well as the local field distributions after buckling. Furthermore, the calculation efficiency was significantly improved due to the reduced number of elements and nodes. Finally, the effects of the structural and material parameters on the buckling behavior of the butterfly CSP were systematically analyzed. The influence of the core layer-to-facesheet thickness ratio on the buckling response was more significant than that of the core form, and the buckling load of the sandwich plate with a negative Poisson’s ratio was greater than its counterpart under the same conditions.

Multiscale structural analysis for plain weave composite plates using asymptotic homogenization method

Asymptotic homogenization method (AHM) is used to perform multiscale structural analysis for plain weave composite plates (PWCPs). First, AHM is used to obtain the effective elastic properties of yarns. Then, the effective plate properties of PWCPs are predicted using AHM. Finally, an equivalent plate model is established to perform the macroscale structure analysis, and recover the mesoscale fields. The accuracy and computational efficiency of the multiscale structural analysis are verified by comparing the results with those of direct numerical simulation. Furthermore, the influences of the geometrical parameters and laying angle of yarns on the equivalent plate stiffness are investigated.

A simplified semi-analytical model for predicting the global and local responses of hybrid honeycomb-like sandwich panel

The hybrid honeycomb-like sandwich panel (hybrid HSP) is composed of FRP facesheets and an aluminum/steel honeycomb-like core, which has excellent mechanical properties and lower cost. This paper presents a fast and accurate semi-analytical model for evaluating the mechanical behavior of hybrid HSP based on the variational asymptotic theory and considering two small parameters: thickness-to-width ratio and heterogeneity. The original HSP is decomposed into 2-D equivalent plate model (2D-EPM) analysis at the macro level and 2-D unit cell constitutive modeling at the micro level to obtain the equivalent stiffness matrix. The static and dynamic analysis is carried out by using the 2-D panel with the equivalent stiffness matrix, and the local stress, strain, and displacement fields are recovered by using the recovery relationship. Finally, the effects of material and geometric parameters on the effective performance of hybrid HSPs are systematically analyzed. Numerical examples show that the static and dynamic results of 2D-EPM are in good agreement with the 3D finite element results, with a relative error of less than 5%. Most importantly, calculation efficiency has been considerably improved. Compared with the traditional finite element model, the novelties of cell tailoring and local field recovery within the unit cell are very important features, which significantly facilitate the effective analysis of hybrid HSPs.

Equivalent orthotropic model for corrugated plates based on simplified constitutive relation

The transverse bending stiffness of a corrugated plate determined using an equivalent orthotropic model based on the Dirichlet boundary conditions and representative volume element method is the upper limit of the real solution, albeit often inaccurate in practice. Based on the Kirchhoff hypothesis, the constitutive relations of corrugated plates were reasonably simplified to propose a new corrugated plate equivalent orthotropic model. The resulting equivalent transverse bending stiffness agreed with the results of the most reasonable simplified expression available. Static response results obtained using the proposed model were more accurate than those obtained using the existing model without a simplified constitutive relation. Furthermore, for dynamic response, the proposed model exhibited improved accuracy when predicting low- and high-order natural frequencies. For the arc-and-tangent corrugated plate, two integral parameter expressions were calculated; the proposed equivalent plate model was verified by comparing the results of a full-scale test of the corrugated metal pipe arch culvert.

Forced vibration analysis of isogrid-stiffened composite plates using VAM-based equivalent plate model

The dynamic behaviors of isogrid-stiffened composite plates (SCPs) exhibit new dynamic characteristics as a result of the introduction of composite materials. To accurately and quickly characterize the forced vibration characteristics of the isogrid SCP, its equivalent plate properties were determined by homogenizing over the unit cell and input to a two-dimensional equivalent model (2D-EPM) obtained from the framework of the variational asymptotic approach. Experiments of free and forced vibrations, as well as numerical examples of free vibration under various boundary conditions and frequency-/time-domain forced vibration under single-/multi-point excitation, were performed to validate the accuracy and efficiency of the proposed model. The effects of structural parameters and stiffening schemes on dynamic characteristics were also investigated. Compared to traditional finite element models, the novelty of cell tailoring is a very important feature that significantly facilitates the investigation of the effect of geometric parameters on the dynamic performance of isogrid SCPs. The results showed that the dynamic performance of the antisymmetric angle-ply SCP was significantly better than that of the unidirectional SCP. The stiffener height had a stronger impact on the forced vibration response, and the inclined stiffeners could significantly improve the dynamic performance of stiffened composite plates. The modeling method has high accuracy and efficiency in forced vibration analysis, which is helpful in predicting the fatigue life of an isogrid SCP under a periodic load.

A VAM-Based Equivalent Model for Triangular Honeycomb Sandwich Panels: Comparison with Numerical and Experimental Data.

Due to their complex microstructures, the research on the static and dynamic behaviors of triangular honeycomb sandwich panels (triangular HSPs) is limited. In this study, the effective plate properties of triangular HSP was obtained by the homogenizing of the unit cell, and then the input to a VAM-based two-dimensional equivalent plate model (2D-EPM) to perform static and dynamic analyses. The accuracy of the proposed model for predicting the equivalent stiffness of the triangular HSP was verified by three-point bending experiments of 3D-printed specimens. Then, the static displacement, global buckling, and free vibrations predicted by 2D-EPM were verified with the results from three-dimensional finite element model simulations under various boundary conditions. The influences of structural parameters (including angle, core wall thickness, and cell side length of the unit cell) on the static and dynamic characteristics of triangular HSPs were also investigated, which can provide a useful tool for the modeling and evaluation of triangular HSPs under different conditions.

Thermal–Structural Coupling Analysis for Multiple Honeycomb Plates Connected by Hinges

The dynamical model of multiple honeycomb plates connected by hinges, including the thermal–structural coupling effect, is investigated in this paper. Under the assumption of small deformations and strains, the classical plate theory is used to establish the dynamical model of the honeycomb plate. The Chebyshev polynomial is adopted to describe the deformation of the plate and the linear torsional spring to represent the elastic hinge. Using the Rayleigh–Ritz method, the characteristic equation of the honeycomb sandwich structure is derived, from which the natural frequencies are obtained. Considering the thermal effect of the plate, the thermal–structural coupling dynamical equation is obtained. The thermal conduction in the thickness direction is calculated by the finite difference method. The present model is validated by comparing with the result generated by an equivalent orthotropic plate model established by the finite element (FE) software MSC Patran. The response analysis reveals the important effect of the metal skin and the hinge stiffness on the dynamic characteristics of the honeycomb sandwich structure. Numerical simulations reveal thermal–structural coupling features for the multi-honeycomb-plate structure from the perspective of the change in dynamic characteristics and the distribution of the temperature gradient. The thermal–structural coupling calculation method adopted herein can be used to solve the thermal conduction of the plate. It serves as the theoretical method for analyzing the dynamical behavior of the solar panel subjected to the solar heat.

Static, buckling, and free-vibration analysis of plain-woven composite plate with finite thickness using VAM-based equivalent model

In this article, to remove the periodicity requirement in the thickness direction of plain-woven composite plates with finite thickness, a novel equivalent plate model (2D-EPM) based on the variational asymptotic method was established to study the static and dynamic behavior of plain-woven composite plates under different boundary and load conditions. The effective plate properties were obtained by the constitutive modeling of the unit cell, and inputted into the 2D-EPM to perform static and dynamic analysis. The accuracy and computational efficiency of the equivalent model were verified by comparing with the static and dynamic results of a three-dimensional finite element model (3D-FEM). The influence of the geometric parameters (yarn width, height, and spacing) and yarn laying angle on the equivalent stiffness and vibration characteristics of the plate were also examined. The results of the proposed model more closely with the experimental data than those of the 3D-FEM. This was possibly because the periodic boundary conditions in the thickness direction overestimated the stiffness of the plate. In addition, the local field distributions within the unit cell were well captured. The non-interlaced parts of the yarns were most likely to fail or become damaged. Overall, this research work has shown the ability of the 2D-EPM to predict the static displacement, global buckling and free vibration of plain-woven composite plates with finite thickness.

Structural analysis of FRP laminate with triangle convexities (FLTC) using the VAM-based equivalent model

To comprehensively study the static and dynamic behaviors of fiber-reinforced polymer (FRP) laminates with triangle convexities (FLTC) under different boundary conditions and avoid structural damage, an equivalent model of the FLTC was developed based on the variational asymptotic method. A constitutive model of the unit cell was developed to obtain the constitutive relations that could be used as effective plate properties of the two-dimensional equivalent plate model (2D-EPM). The accuracy of the equivalent model and the obtained equivalent stiffness were verified by comparing with the global displacement, buckling modes, critical loads, free-vibration modes, and frequencies of a three-dimensional finite element model (3D-FEM). The influence of different layup configurations, structural arrangements and geometric parameters on the effective performance of the FLTC are discussed. The results showed that the critical load and natural frequencies obtained using the equivalent model were in good agreement with those obtained from the 3D-FEM, and the local field distributions within the unit cell of the FLTC were well captured. The free vibration and global buckling of the FLTC were closely related to the geometric sizes and layup configurations. Most importantly, the calculation efficiency was greatly improved. The research results provide a reference for the practical application of the FLTC in civil engineering.

Static and dynamic analysis of hemispherical convex-concave composite plate (HCCP) using variational asymptotic equivalent model

In order to deeply study the static and dynamic behavior of hemispherical convex-concave composite plate (HCCP) under different conditions and avoid structural damage, the equivalent model of HCCP was developed based on variational asymptotic method. The constitutive modeling of unit cell was carried out to obtain the constitutive relations that can be used as effective plate properties for the 2-D equivalent plate model (2D-EPM). The accuracy of the equivalent model and the obtained equivalent stiffness are verified by comparing with the static displacement, global buckling and free-vibration of 3D finite element model (3D-FEM). The influence of different layup configurations of laminates and geometric parameters on the equivalent stiffness, vibration characteristics and global buckling of HCCP are discussed. The results show that the buckling loads and natural frequencies obtained by using equivalent model are in good agreement with 3D-FEM. The vibration and buckling of HCCP are closely related to the geometric sizes and layup configurations of the laminate, and the local field distributions within the unit cell of HCCP are well captured. Most importantly, the calculation efficiency is greatly improved. The research results provide a reference for the practical application of HCCP in the civil engineering.