Laminated Plate Theory Research Articles

Theoretical and experimental research on the critical buckling pressure of the thin-walled metal liner installed in the composite overwrapped pressure vessel

Composite overwrapped pressure vessels (COPVs) are widely used in the field of high-pressure gas storage and transportation because of their light weight and high strength. In engineering, autofrettage is usually used to improve the fatigue life of composite pressure overwrapped vessels with metal liners. However, excessive pressure of autofrettage could cause buckling damage to the metal liner. In order to determine the upper limit of autofrettage (critical buckling pressure) and analyze its influence on the safety factor of COPVs, a theoretical calculation model of the critical buckling pressure about the metal liner was established based on classical laminated plate theory and the confined buckling theory. Then, a COPV with thin-walled welded metal liner was prepared by fiber winding process. In order to monitor and judge the buckling damage mode of the liner, the circumferential strain of the COPV's cylinder was measured by loading and unloading step-by-step. Finally, the influence of materials and dimensions of the metal liner on the pressure ratio (the ratio of the first layer failure pressure to the critical buckling pressure of the COPV's metal liner) was discussed. When the diameter to thickness ratio (R/t) of the metal liner was greater than 35, the pressure ratios of the 6063-T5 aluminum alloy liner and the S30408 stainless steel liner both exceeded 2.25. However, the pressure ratio of 6061-T6 aluminum alloy liner with R/t ≤ 60 was lower than 1.85, which shows that the metal liner with higher yield strength and lower elastic modulus has higher utilization rate of fiber winding layers. Due to the fact that the current design standards only ensure the reliability of COPVs through safety factors and various type testing with their evaluation methods, the proposed theoretical calculation method can reduce the uncertainty of COPVs during the design and testing rounds.

A novel approach for buckling optimization of stiffened piezolaminated composite plates

This study presents a novel approach for optimizing the critical buckling load of the stiffened piezolaminated composite plates using a self-adaptive special relativity search (SASRS) algorithm. The fiber orientations in the layers are considered as design variables. The stability equations are first derived based on the classical laminated plate theory (CLPT) with von Karman nonlinearity. The optimization problem is implemented in MATLAB. The optimization results are obtained to indicate the effecs of various boundary conditions, applied voltages, grid shapes, angles of diagonal ribs, and plate aspect ratios on the optimal design. Numerical investigations are carried out for 10-ply symmetrical stiffened piezolaminated plates. The robustness and efficiency of the proposed SASRS algorithm in maximizing the critical buckling load of the stiffened piezolaminated plates are demonstrated by comparing the optimized results from the standard special relativity search (SRS) algorithm. It is found that the proposed optimization method has better performance than the SRS algorithm.

Dynamic analysis of laminated plate and ballastless track slab on Pasternak foundation under moving load based on different shear deformation theory

In this work, different shear deformation theories are used for the first time to investigate the effect of shear deformation on the dynamic response of laminated plates and track slab on a Pasternak foundation under moving load. The Pasternak formulation was used to simulate the interaction between the slab and the elastic foundation. Using Hamilton’s principle, the governing equations are derived based on the third-order shear deformation theory (TSDT) in the higher-order shear deformation theory (HSDT). Additionally, governing equations based on the first-order shear deformation theory (FSDT) and the classical laminate plate theory (CLPT) are also derived. The analytical solution of the classical plate theory (CPT) is derived as an example for isotropic thin plates and used as a benchmark solution to verify the accuracy of the governing equations based on TSDT, FSDT, and CLPT. Then, the effect of the thickness-to-width ratio on the dynamic response of the laminated slab is investigated, along with the effects of the Pasternak foundation coefficient, the vertical load movement speed, and load eccentricity on the dynamic response of the laminated plate and the track slab. The effect of uncertainty in the modulus of elasticity on the analytical results is also examined. The results indicate that considering shear deformation is essential in the dynamic analysis of CRTSII plate ballast slabs, but higher-order shear deformation theory should be avoided due to its computational cost and complexity. This study provides a benchmark solution for further work.

Experimental study on the free and constrained vibration performance of composite wave springs

This article proposes a glass fiber reinforced plastic (GFRP) wave spring and conducts a study on its free and constrained vibration characteristics and compression performance. Initially, a stiffness prediction model for composite wave spring is established using laminated plate theory and Moore’s theorem. Additionally, the failure mode of the composite wave spring was predicted and experimentally confirmed using ABAQUS and the 3D-Hashin failure criterion. Finally, the vibration characteristics of composite wave spring and metal helical spring are investigated using the B&K testing system in both free and constrained states. The experimental stiffness and ultimate load of the composite wave spring are 20.67 N/mm and 1386.67 N. The composite wave spring exhibits good vibration reduction performance in both free and constrained states, with maximum vibration attenuation amplitudes of 122.71 and 144.93 dB. In comparison, the metal helical spring has maximum vibration attenuation amplitudes of 41.65 and 111.89 dB.

Characterizing nonlinear constitutive behaviors of fiber metal laminates

Abstract This study characterized the nonlinear tensile behavior of fiber metal laminates (FMLs). FMLs comprise layers of thin metallic sheets and fiber-reinforced composite layers, and a constitutive FML model includes the constitutive relationships of the FML’s constituent materials; however, nonlinear behavior is typically only considered for the metal components of an FML. In this study, a nonlinear constitutive relationship for the unidirectional fiber composites was modeled using a one-parameter plastic model. The nonlinear constitutive law for the metal was formulated using the J2 flow rule. These relationships were summed for each layer in accordance with laminated plate theory to obtain a constitutive FML model, which was then used for numerical predictions of nonlinear stress–strain curves. The model was validated by comparing its predictions with experimental results from the literature. Moreover, the effect of the inclusion of nonlinear fiber composite behavior on the model predictions was investigated. Results revealed that the difference between the model predictions and the experimental results was less than 4%. These predictions with nonlinear fiber composite behavior were substantially more accurate than those of the model without this behavior for FMLs with angle-ply fiber composites.

Nonlinear Vibrations of the Piezoelectric-Laminated Composite Cantilever Rectangular Plate Subjected to the Transverse and Parametric Excitation

The intelligent structure will be the key to the industrial manufacturing field in the future, piezoelectric materials are the most commonly used active materials in many mechatronic and vibration control systems. In this paper, we illustrated the nonlinear vibration behaviors of the piezoelectric-laminated composite cantilever rectangular plate subjected to transverse and parametric excitation. Two piezoelectric layers are attached to the upper and lower surfaces of graphite/epoxy composite laminated plate. The nonlinear dynamic equations are derived via the classical laminated plate theory, von Karman large deformation theory, the constitutive equation of the piezoelectric materials, and the Hamilton principle. Considering the Galerkin method, the nonlinear ordinary differential equations for the two-order vibration mode of the piezoelectric laminated composite cantilever rectangular plate are achieved. The multiple scale method is applied to obtain the average equations to study the nonlinear vibration characteristic of the stable motion. The primary resonance and the 1:1 internal resonance of the piezoelectric laminated composite cantilever rectangular plate are investigated. The amplitude-frequency response curves, force-amplitude response curves, time histories, phase curves, and bifurcation diagrams are given to investigate the nonlinear dynamic characteristic of the piezoelectric laminated composite cantilever rectangular plate. The detailed parametric analyses show that bubble response curves are separated from the amplitude-frequency response curves with the continuous increase of the external excitation due to the internal resonance. In addition, the results reveal the energy transform between the various order vibration modes of the system. This work is expected to provide theoretical guidance for the nonlinear vibration of the smart piezoelectric laminated composite structure.

Wavenumber dynamic stiffness formulation for exact dispersion analysis of moderately thick symmetric cross-ply laminated plate built-up waveguides

A wavenumber dynamic stiffness (WDS) formulation for exact dispersion analysis of moderately thick symmetric cross-ply laminated plate built-up waveguides is presented. The effects of shear deformation and rotatory inertia are considered in the formulation by adopting the first order shear deformation theory. The elemental WDS matrix is derived from the exact general solutions of the governing differential equations and can be assembled to model complex plate built-up waveguides using the assembly procedure similar to that of the finite element method. The nonlinear eigen-solution technique, the Wittrick–Williams (WW) algorithm, is employed to compute dispersion curves from the obtained WDS matrices. To enhance computational efficiency, the J0 count problem of the WW algorithm is addressed by developing explicit expressions for the J0 term. This eliminates the need for mesh refinement over the entire frequency/wavenumber range. The accuracy and efficiency of the proposed method are demonstrated through comparisons with analytical solutions based on the classical laminated plate theory and the first order shear deformation theory, as well as numerical results obtained from the wave and finite element method.

Analytical solutions for the free vibration of cross-ply composite laminated plates with arbitrary biaxial symmetric geometry

This paper presents an analytical model for the free vibration of composite laminated plates with arbitrary biaxial symmetric geometry under elastic boundary conditions, applicable to cross-ply fiber-reinforced composites. In this model, the domain segmentation integral method proposed by the authors is extended to the first-order shear deformation theory (FSDT) and the laminated plate theory. The energy integrals pertaining to the FSDT are analytically simplified by segmenting the integration domains and synthesizing orthogonal polynomial displacement functions over the plate domain coordinate intervals. This synthesis is achieved through the amalgamation of the penalty function method and the Gram-Schmidt orthogonalization process. Then, using the Rayleigh-Ritz procedure, a series solution for the free vibration problem of cross-ply fiber-reinforced composite laminated plates is obtained. The model provides an analytical mathematical relationship between the profile curve of composite laminated plates with arbitrary biaxial symmetric geometry and the strain energy, kinetic energy, and boundary potential energy during vibrational processes. The correctness and applicability of the method are validated by comparing the obtained results with those from the literatures. New results for laminated plates, such as hexagonal and astroid-shaped plates, were presented as references for future research. Taking a track shaped plate as an example, the effect of the number of layers on the vibration characteristics of cross-ply fiber-reinforced composite laminated plates with a certain thickness is studied.

Scale modeling of thermo-structural fire tests of multi-orientation wood laminates

The stacking sequence of laminated wood significantly impacts the composite mechanical behavior of the material, especially when scaling down thermo-mechanical tests on plywood. In previous research, we developed a scaling methodology for thermo-structural tests on samples with similar cross sections, however this paper focused on testing plywood samples with different stacking sequences between the scales. Plywood samples at ½-scale and ¼-scale were subjected to combined bending and thermal loading, with the loading scaled to have the same initial static bending stresses. While the ¼-scale 4-layer [0°/90°]s laminate and the ½-scale 8-layer [0°/90°/90°/0°]s laminate had an equal number of 0° and 90° layers, as the char front progresses, the sections behave differently. Thus, modeling becomes essential to extrapolating the data from the smaller ¼-scale test to predict the behavior of the larger ½-scale test. Reduced cross-sectional area models (RCAM) incorporating classical laminated plate theory were used to predict the mechanical response of the composite samples as the char front increased. Three methods were proposed for calibrating the RCAM models: Fourier number scaling, from detailed kinetics-based pyrolysis GPyro models, and fitting to data from fire exposure thermal response tests. The models calibrated with the experimental char measurements produced the most accurate predictions. The experimental char models validated to predict the behavior of the ¼-scale tests within 2.5%, were then able to predict the ½-scale test behavior within 4.5%.

Geometrically nonlinear bending analysis of laminated thin plates based on classical laminated plate theory and deep energy method

This paper establishes a geometrically nonlinear bending analysis framework using the deep energy method and the classical laminated plate theory (CLPT) for laminated plates. Inspired by the transfer learning technique, a load applied to a laminated plate can be divided into multiple load steps. The network parameters for the current load step, with the exception of the initial step, are initialized by inheriting values from their preceding steps. Including both von Kármán and Green-Lagrange strains, the plate strains are computed using the automatic differentiation and integrated along the thickness direction per laminate plate based on the constitutive theory. By combining the outputs of neural network, the external potential energy can be obtained, and the optimized network parameters are given by minimizing the total system potential energy of the laminated plate. In order to validate the proposed approach, several numerical examples are calculated, and the present solutions are compared with those given by the literature and the Finite Element Analysis (FEA). The results show that the proposed approach is indeed feasible, can reach high levels of precision under varying loads while offering a simplified calculation strategy.

Comparative analysis for optimal placement of piezoelectric actuators on smart thin plate

In contemporary smart structures, piezoelectric ceramic (PZT) actuators serve a crucial structural function. To construct a smart thin plate that integrates structural health monitoring (SHM) and active vibration control objectives, many placement methodologies were discussed. In current research, two positioning strategies of the attached PZT actuators are compared. The first strategy is a well-known controllability concept where PZT actuators’ optimum positions are obtained by maximizing the eigenvalues of the controllability Grammian matrix. The second strategy employs the energy concept to strategically position PZT actuators so that the utmost amount of exerted work is accomplished by the actuators. Thus, the optimal performance of actuators to provide energy for mitigating unwanted vibrations is maintained, as is the optimal excitation required for SHM in the designated modes. Kirchoff’s classical laminate plate theory (CLPT) is used to define displacements as well as the normal strains of this coupled electro-mechanical system, and then, using the Ritz solution, the modal eigenvalue problem is solved, and mode shapes are obtained. Consequently, normal strains are transformed into spatially dependent functions used to define both the virtual work and the controllability of the Grammian matrix for the PZT actuators as a function of their locations. Finally, iterative optimization based on a genetic algorithm (GA) is used to find the optimum actuator locations in the desired modes. The best location for the PZT actuator or actuators is explored for two specimen plates with different boundary conditions. The results for the two studied positioning strategies are then compared, showing the superiority of the proposed energy-based strategy. Further, a quantitative variance-based uncertainty analysis reveals a low variance of the output results for the proposed strategy.

Quasi-3D flexural analysis of laminated composite plates resting on elastic foundations using higher-order displacement model

This work examines the impact of transverse strains on the static, vibration, and buckling analysis of laminated composite plates that are symmetric and anti-symmetric and rest on elastic foundations. This laminated plate theory, which is called a quasi-3D plate theory since it considers the effects of both transverse shear and normal strains, is provided here. The significance of nonclassical influences on plate deformation, such as shear deformation and thickness stretching, is sufficiently highlighted by the current theory. The theory demonstrates realistic transverse shear stress distributions across the laminated plate's thickness and complies with the traction-free boundary conditions at the upper and bottom surfaces of the plate. Hamilton's principle provides the current theory's equations of motion. For simply supported edges, the laminated plates supported by elastic foundations are investigated. For the aim of verification, the numerical results of the displacements, stresses, frequencies, and buckling load derived using the present theory are, if possible, compared with established literature. The current results and those derived from the other hypotheses found in the literature show a good degree of consistency. Additionally, the distributions of the interlaminar stresses in thickness direction of laminated composite plates sitting on elastic foundations are demonstrated.

Numerical simulation of mechanical properties of composite high-pressure vessel for space application

Abstract A composite high-pressure vessel reinforced with T1000 carbon fiber and featuring an ultra-thin titanium lining for space applications is examined. Using the laminated plate theory for composite materials and the elastic-plastic theory for isotropic materials, a three-dimensional finite element model is developed. The pressure test results confirm the correctness of the finite element model. Under various internal pressure circumstances, the displacement and strain distribution of the metal liner and the laminated composite are compared and analyzed. It is shown that the composite vessel primarily deforms along the axial direction when the internal pressure is greater than the working pressure of the vessel. Only elastic deformation is present in the composite layer, but both plastic and elastic deformation are present in the ultra-thin titanium liner.

Flexural failure properties of fiber-reinforced hybrid laminated beam subject to three-point bending.

The present study investigates the flexural failure properties of a hybrid laminate beam subjected to three-point bending. A symmetrically laminated hybrid beam is constructed using high-strain and inexpensive glass fibre on the top layers and low-strain and expensive carbon fiber on the middle layers. Classical lamination plate theory is used to find the stress and strain distribution that occurs due to the bending moment on the compressive side. The theoretical failure limits of the laminated hybrid beam are analyzed considering the targeted span-to-depth ratios, volume fractions of the fibers and hybrid ratios using the Tsai-Wu failure criterion and Matlab codes. Using the graph of failure index versus hybrid ratios, the minimum thickness of carbon fiber needed for the delay of failure and cost efficiency of the laminated hybrid beam is identified by applying the linear interpolation method. The numerical results indicate that the failure index increases with the increasing loading span and decreases when the volume fraction of fiber increases. In particular, the placement of glass fiber on the top layer of the laminated hybrid beam might have contributed to obtaining higher strains and curvatures before the catastrophic failure properties of carbon fiber. The flexural stiffness of the laminates is found to increase when the hybrid ratio increases. Overall, it is noted that the theoretical analysis is one method that is less time-consuming and cost-effective than other alternative approaches, such as finite element methods and experimental tests to estimate the minimum thickness of high-stiffness and the expensive material needed to maintain the strength and stiffness of the hybrid composite structures over long periods.

Ultimate bearing behavior in the post-buckling stage of composite pressure hulls based on new detection methods

This study employs novel testing methods and analysis techniques to investigate the ultimate load-bearing characteristics and failure mechanisms of composite thin-walled pressure hulls. An analytical model based on the classical laminated plate theory, incorporating the Hashin damage failure criterion, is developed to determine their ultimate bearing capacity. Five carbon fiber cylindrical shells are manufactured to validate the failure mechanism and evaluate the model's predictions. Deep-sea pressure testing and ultrasonic testing are conducted on these shells. The resulting data is analyzed using both macroscopic and microscopic approaches to gain insights into their load-bearing behavior and failure mechanisms. The study demonstrates the significant influence of the thickness (t) to radius (R) ratio on the ultimate load-bearing capacity of these composite thin-walled pressure hulls. Failure primarily arises from compression failure in both the matrix and fibers. The load-bearing process encompasses uniform shrinkage, critical buckling, and post-buckling states. Material failure primarily occurs during the post-buckling stage, while no damage occurs during the initial buckling. On a microscale, failure at the interface between the fibers and matrix is identified as the primary contributor to overall structural failure. Enhancing the fracture energy of the interface is recommended to enhance the ultimate load-bearing capacity of the hulls.

Theoretical and numerical analysis of buckling stability of bi-directional corrugated sandwich panels

ABSTRACT Considering the biaxial stress state of underwater pressure-resistant structures, this paper proposed a bidirectional corrugated structure based on the cuttlefish bone's unidirectional corrugated wall structure. Building upon the calculation method for the equivalent stiffness of corrugated plates, formulas for the equivalent bending stiffnesses and equivalent torsional stiffness of bidirectional corrugated plates were derived. Kirchhoff's laminated plate theory was employed to derive the buckling critical loads for biaxial compression of bidirectional corrugated core panels with four simply supported edges. By comparing with finite element method results, the accuracy of various calculation methods in evaluating the equivalent stiffness of unidirectional and bidirectional corrugated plates is analyzed, and the proposed calculation method for the critical buckling load is validated. It is shown that the buckling resistance capacity of the bidirectional corrugated laminate structure can reach approximately 2.34 times that of an equivalent mass flat plate.

A novel finite element for the vibration analysis of tapered laminated plates

AbstractIn the present study, a new tapered finite element is developed by incorporating the in‐plane effect in our recently introduced element for the vibration analysis of internally tapered laminated plates. Stress and strain fields for a laminated plate with varying thickness along two orthogonal directions are expressed through extending classical laminated plate theory. Shape functions regarding bending and membrane behavior are extracted from the bending and axial basic displacement functions of the Euler‐Bernoulli beam, respectively. Several laminated plates with different taper configurations have been studied, and the results obtained from vibration analysis are compared with those of the literature. Since the element can capture the exact form of variation in thickness, the solution has shown to be competitively accurate with considerably fewer total degrees of freedom and elements in comparison with conventional finite elements.Highlight A new finite element for thin, tapered, laminated plates is introduced. Laminates' stiffness matrix is obtained for arbitrary thickness variation. A novel method is adopted to derive in‐plane shape functions. C1 continuity is guaranteed along the edges of elements. Calculation cost is considerably reduced.

Composite Plate Optimization: Integrating Analysis Techniques and Design Iterations for Enhanced Mechanical Performance

This essay delves into the comprehensive analysis of composite plate behavior under various loading conditions, focusing on uniaxial loading and the effects of layup design on safety factors. Through a comparative study between Classical Laminate Plate Theory (CLPT) calculations and Finite Element (FE) analysis, discrepancies in stress and strain predictions are identified, attributing them to assumptions made in each method. Furthermore, the essay explores failure prediction using Tsai-Wu criteria and Maximum Stress theory, determining the safety of laminates under different loading scenarios. Additionally, Finite Element analysis is employed to assess the impact of layup design variations on safety factors, leading to recommendations for optimized designs. The limitations of FE models compared to physical tests are discussed, along with alternative approaches such as advanced manufacturing techniques and novel fiber architectures to further enhance safety factors and alter failure modes.

A rapid and precise approach to predicting the cure‐induced warping of T‐shaped composite parts with inconsistent layup

AbstractThis work involved the precise prediction and in‐depth analysis of cure‐induced warping of T‐shaped parts with inconsistent layup. First, the warping mechanism of this kind of part was studied. Then, a finite element analysis (FEA) that considered the parts' structural characteristics and inconsistent layup using the curing mechanical constitutive model was developed to predict the warping. Following verification of the model's accuracy, the effects of material and structural characteristics on warping were investigated. When designing T‐shaped parts, selecting materials with lower thermal expansion coefficients can help minimize warping. Additionally, increasing the radius of the chamfer at the rib and flange connections is also effective. By taking into account the effects of structural parameters and inconsistent layup by using both classical laminated plate theory and artificial neural networks (ANN), a rapid and accurate surrogate model for warping prediction was finally developed. It was discovered that the two‐layer ANN model performed better than the other two models and could predict warping with accuracy and speed. For the parts analyzed in this work, this model is able to identify the fluctuations of warping captured by FEA. Moreover, by defining deviation as the difference between the predicted values of the surrogate model and the FEA, it can be observed that the majority of deviations for the validation set are within ±1.5 mm. This demonstrates that the two‐layer ANN possesses strong generalization ability and high accuracy in predicting warping.Highlights The inconsistent layup could significantly affect the T‐shaped part's warping. Cutting off a part's rib may significantly reduce its warping. A rapid and accurate model for warping prediction was developed. The layup and structure's influence could be identified by the two‐layer ANN.

Nonlinear vibrations and control of PFG-GR laminated composite cantilever rectangular variable cross-section plate with NPPF controller

This paper studies the nonlinear vibrations and control of the piezoelectric functionally graded graphene-reinforced laminated composite cantilever rectangular (PFG-GRLCCR) variable cross-section plate with the NPPF controller under the simultaneous resonant cases. The nonlinear dynamic governing equations of motion for the PFG-GRLCCR variable cross-section plate are obtained by using the extended Halpin-Tsai mode, linear constitutive equations of the piezoelectric materials, classical laminated plate theory, von Karman large deformation theory, and Hamilton principle. Considering Galerkin method, the nonlinear ordinary differential governing equations are obtained. The nonlinear positive position feedback (NPPF) controllers are designed according to the similarity principle. The multiple scale method is used to investigate the nonlinear vibration characteristics of the stable motion for the PFG-GRLCCR variable cross-section plate with the NPPF controllers. The amplitude-frequency response curves, force-amplitude response curves, time histories, phase portraits, and bifurcation diagrams are given to investigate the nonlinear dynamic characteristics of the PFG-GRLCCR variable cross-section plate with the NPPF controller. The results demonstrate that the NPPF controller can suppress the nonlinear vibrations of the structure well, and the effectiveness of the controllers is best when the NPPF controller has a similar nonlinear characteristic with the structure. The complex nonlinear vibration characteristics appear for the PFG-GRLCCR variable cross-section plate with the NPPF controller.