Classical 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.

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.

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.

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.

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.

A comprehensive formulation for determining static characteristics of mosaic multi-stable composite laminates under large deformation and large rotation

Multi-stable composite laminates are composite materials that exhibit multi-stable states, making them highly suitable for use in morphing structures. These materials are capable of maintaining each stable state without expending any energy. As a result, they are used extensively in numerous applications and garnered the interest of scholars and aerospace organizations. In the context of practical applications, such as morphing structures, it is insufficient for designers to rely solely on common bi-stable composite laminates that exhibit large deformations and medium rotations to achieve their desired objectives. Consequently, based on the objectives of the design, there are two potential resolutions to address this limitation. A designer may utilize mosaic multi-stable composite laminates to achieve a morphing structure that exhibits high flexibility, significant deformation, and substantial rotation. The utilization of a series connection between a bi-stable composite laminate and a symmetric composite laminate results in the formation of a mosaic bi-stable composite laminate with variable stiffness. Furthermore, the amalgamation of two asymmetric composite laminates with inverted orientations engenders a mosaic tri-stable composite laminate. The present research examines the static characteristics of mosaic bi-stable and tri-stable composite laminates. It also seeks to analyze the factors affecting the behavior of these types of laminates. A geometrically exact model was formulated for this objective. Apart from the geometrically exact model, a widely used and uncomplicated model relying on the conventional Classical Laminated-Plate Theory (CLPT) and Von-Karman nonlinear strains was employed. The proposed models were validated through finite element simulations. The system's static equations were derived using the virtual work principle and the Rayleigh-Ritz method. The present study examines and explores quasi-static snap-through behavior between stable states through the application of concentrated forces. The findings indicate a high level of concurrence between the outcomes derived from the geometrically exact model and the finite element analyses, particularly in composite laminates exhibiting significant deformations and rotations.

Stiffness and progressive failure prediction of 2-D tri-axially braided composites

This paper studies the mechanical properties of the 2-D tri-axial braided composite. A volume element is established, so that the classical laminate plate theory and the series-parallel model can be adopted to study a bi-directional stress-strain response. Failure criteria are given and the failure strength is determined. The finite element simulation is used to verify the reliability of the present theoretical model.

The buckling performance of a piezoelectric laminated composite plate via static method

Piezoelectric materials are widely used as actuators, due to the advantages of quick response, high sensitivity and linear strain-electric field relationship. The previous work on the piezoelectric material plate structures are not enough, however such structures play a very important role in the practical design. In this paper, the buckling performance of piezoelectric laminated composite plate (PLCP) is analyzed based on static method to parametric study the buckling control. The stress components of the matrix layer are formulated based on electro-mechanical coupling theory and Kirchhoff’s classical laminated plate theory. Buckling differential governing equation of PLCP is obtained by using the equilibrium conditions. The solution of the governing equation is assumed as a sum of a series of trigonometric shape functions, and then its expression is obtained by using static method. The effectiveness of the developed method is validated by the comparison with finite element method. Especially, the developed method can be used for engineering applications more easily, and it does not require to rebuild the calculation model as finite element method during the calculation and analysis of PLCP. The buckling performance of PLCP and its influencing factors are numerically analyzed through the developed method. The buckling performance of PLCP is reasonably increased by parametric studying different loads, laying angle, laying sequence, height of the matrix plate, and layer size. This paper is a valuable reference for the design and analysis of PLCP.

Nonlinear vibrations of FG-GRLCC rectangular variable cross-section plate subjected to transverse and parametric excitations in thermal environment

This paper investigates the linear and nonlinear vibration responses for the functionally graded graphene-reinforced laminated composite cantilever (FG-GRLCC) rectangular variable cross-section plate subjected to the transverse and parametric excitations under the thermal environment. Halpin-Tsai model is used to calculate the material properties of the graphene-reinforced structure. Using the classical laminated plate theory, von Karman large deformation theory, Hamilton principle and Galerkin method, the dynamic model is given for the FG-GRLCC rectangular variable cross-section plate. The natural frequencies and mode shapes are analyzed for the FG-GRLCC rectangular variable cross-section plate under the thermal environment through using Rayleigh-Ritz method. The averaged equations of the system are obtained based on the multiple scale perturbation (MSP) under the primary, 1/2 sub-harmonic and 1:1 internal resonances. The comparisons between the theoretical algorithm and finite element method are proposed to illustrate the accuracy of the present model. The results about the frequency veering phenomenon and mode shape interaction have been illustrated. The amplitude-frequency response curves and force-amplitude response curves are depicted for the FG-GRLCC rectangular variable cross-section plate. The nonlinear and chaotic vibrations for the FG-GRLCC rectangular variable cross-section plate is studied by using the bifurcation diagrams and max Lyapunov exponents.

Innovative approaches to deflection analysis in strengthened reinforced concrete Slabs: Combining experiments and theoretical research

Structures often experience high-stress conditions, leading to premature deterioration and costly replacements before their intended design life ends. To address this issue, strengthening techniques are employed to mitigate the degradation of reinforced concrete (RC) slabs. This study focuses on investigating the flexural strengthening of RC slabs. Specifically, it examines the impact of different externally bonded plates and slab thickness on the deflection of RC slabs during the pre-crack elastic stage. Additionally, the study seeks to predict measured displacement values using approximate methods like the Classical Laminated Plate Theory (CLPT) and the First-order Shear Deformation Theory (FSDT). The experiment utilized six square slabs measuring 1100 mm by 1100 mm, with thicknesses of 60 mm and 90 mm. For the slabs with a thickness of 60 mm, two-way flexural reinforcing bars, each with a 10 mm diameter, were placed at 150 mm intervals, while the slabs with a thickness of 90 mm had reinforcing bars spaced at 100 mm intervals. Two of the RC slabs were strengthened with carbon plates, two with steel plates, and the remaining two served as control samples. The results indicated that steel plate strengthening improved RC slabs by 47 %, while carbon plates achieved a 43 % enhancement. Furthermore, increasing the slab thickness significantly reduced the deflection of RC slabs. A good agreement was noted when the deflection capacities predicted by the analytical model were compared to the experimental findings.

An analytical solution for thermomechanical buckling of variable angle tow composite laminates with elastically restrained edges

The inherent anisotropy and variable stiffness characteristics exhibited by Variable Angle Tow (VAT) composite materials offer significant possibilities for enhancing the performance of corresponding structures. Through appropriate selection of the curved fiber paths, it becomes feasible to design structures that are both lightweight and efficient. However, conducting mechanical analysis on VAT composite structures is of considerable challenges, particularly in seeking analytical solutions. In view of this, this paper aims to investigate the thermomechanical buckling behavior of VAT composite laminates with elastically restrained edges and presents an analytical solution for the studied problem. The analysis is based on the assumption that the fiber orientation angle of each lamina varies linearly or nonlinearly along the x direction, and the fundamental equations for thermomechanical buckling of VAT laminates are derived using the classical laminated plate theory. The analytical solution is obtained through the combined utilization of Taylor series and Frobenius series. Furthermore, a comprehensive parameter analysis is conducted to assess the impact of different factors, such as fiber paths and elastic support stiffness, on the critical thermomechanical buckling load and mode of VAT laminates. The resulting findings serve as a valuable reference for VAT laminate design, while the proposed analytical solution can be employed as a benchmark for validating other numerical results.