Isotropic Plates Research Articles

Theory of thin elastic plate: history and current state of the problem

The article is an analytical review and is devoted to the theory of thin, isotropic elastic plates. There are basic relations of the theory based on the kinematic hypothesis confirming that the tangential displacements are distributed linearly along the thickness of the plate and its deflection does not depend on the normal coordinate. As a result, a system of equations of the sixth order with respect to two potential functions – the penetrating potential, which determines the plate deflection, and the boundary potential which makes it possible to set three boundary conditions on the plate edge and eliminate the known contradiction of Kirchhoff's plate theory was obtained. Problems that have no correct solution in the framework of Kirchhoff's theory – cylindrical bending of a plate with a free edge, bending of a rectangular plate with non-classical hinge fixing, torsion of a square plate by moments distributed along the contour, and bending of a plate by a rigid die – are considered. In conclusion, a brief historical review of the papers devoted to the theory of plate bending was presented.

Experiments for the Assessment of the Probabilistic Multistage Algorithm for Damage Detection in Flat and Curved CFRP Panels

The Probabilistic Multistage (PM) algorithm was developed by authors for identifying and localizing damages in isotropic plates. More specifically, PM algorithm consists of a fully automated probabilistic damage imaging methodology based on ultrasonic guided wave propagation and a 5-sensor array working in a pitch-catch approach. The algorithm processes the gathered data in three steps to identify key features associated with damage. The aim of this work is to assess the effectiveness of the PM algorithm on more complex structures. Specifically, the study investigates three cases of study, made of Carbon Fiber Reinforced Plastic (CFRP) composite, characterized by different geometries and layups. The algorithm is initially tested on panels with artificial damage in the form of a Teflon disk located in a specific location between the middle laminae of the panels, which is used to replicate the effect of delamination. In order to expand the experimental dataset without incurring additional costs or waste, new damage conditions are simulated by adding masses on the upper surface of the panels. Each plate is investigated considering three different damage sizes and 16 different damage locations. The proposed algorithm successfully detects damages both within and outside the sensor network. The PM algorithm produces a clear damage positioning map and a positioning (probabilistic field) range for the identified damage. This information can be used to assist operators in conducting inspections more efficiently by focusing on the highlighted areas, which may potentially lead to reduced maintenance and repair expenses.

Analysis of the thermomechanical behavior of a three-layer rectangular plate of the ceramic-metal-ceramic structure due to non-stationary convective heating

A rectangular isotropic layered plate is considered, which is convectively heated by the external environment. The thermal stress state of the plate is determined in two stages. At the first stage, the nonstationary temperature field is determined from the ratios of the non-stationary thermal conductivity problem. At the second stage, using the five-modal mathematical model of the shear theory of thermoelasticity, the parameters of the stress-strain state are determined.

Use of a Refined Theory for Three-Dimensional Bending Analysis of Isotropic Rectangular Thick Plates

In this paper, a refined plate theory (Alternative II theory) is presented for the three-dimensional bending analysis of an Isotropic thick plate. The theory has similarity to the first order shear deformation theory but requires no shear correction factors. The kinematics equations were developed based on the Alternative II Refined plate theory. Thereafter, using a complete three-dimensional constitutive relation, the total potential energy was developed. A governing equation and two compatibility equations were obtained by the variation of the total potential energy with respect to displacement and rotations respectively. Solving the governing and compatibility equations, a polynomial displacement function was obtained. The stiffness coefficients were then obtained using the displacement function. Thereafter, the equations for the in-plane normal and shear stresses, transverse normal and shear stresses as well as the lateral displacement were developed using the stiffness coefficients and the displacement function. Numerical values of the lateral displacement parameters were determined for a rectangular plate of aspect ratio 2.0, 1.0 and 0.5 for span to thickness ratios of 20, 10 and 7.14286. Also, numerical values of the lateral displacement and stresses were determined for a square plate for span to thickness ratios of 4, 10, 100 and 1000. The results from this work were compared with the work of previous researchers using simple percentage difference. It was observed that refined plate theories overestimate the lateral displacement of a plate. Hence, three-dimensional analysis is recommended for thick plate analysis.

MAGPIE: a web-based, open-source framework for plate vibration analysis

The analysis and simulation of plate vibrations are central to many branches of science and engineering, specifically acoustics and musical acoustics. Well-developed plate theories exist, along with numerous simulation software packages, but they are seldom available in open-source form and are not easily accessible to a greater audience. MAGPIE is a framework for interacting with a novel finite difference scheme for plates under general elastic boundary conditions. Free, clamped, and simply supported boundary conditions are obtainable by setting stiffness coefficients to very small or large values. Boundary conditions are parameterised which allow for intermediate possibilities. A web browser based implementation of the framework includes interactive elements. Users begin by providing parameters for an isotropic plate. These parameters are then used to generate visualisations of the mode shapes from the finite difference scheme simulation. Elastic boundary conditions can then be changed on a continuous scale to see their effect on mode shapes. Results are then available for export in common image and data formats. MAGPIE aims to make it easier to explain concepts such as mode shapes to a wider audience and also provide a tool for research that is accessible to students, scientists and musical instrument builders.

النموذج الرياضي المقارب لدراسة سلوك انبعاج صفيحة فولاذية

A mathematical study was conducted to investigate the buckling of isotropic steel plates using instability theories. The First-Order Shear Deformation Theory (FSDT) and Higher-Order Shear Deformation Theory (HSDT) were employed to account for shear effects. These latter theories were adopted because they consider shear deformation with high precision, unlike classical theories that neglect this important action. The equilibrium equations, derived from Hamilton's principle, were used, and Navier-type solutions were preferred to calculate buckling loads under uniaxial or biaxial loading. Parametric variations and different buckling modes were investigated, and the results showed excellent agreement with existing references. The main objective of this study in civil engineering is to continuously find mathematical formulations to solve plate buckling problems with extreme accuracy and to predict buckling behavior in simulations, mimicking what happens to metallic structures. Keywords: Plate Buckling-Elastic Instability-Isotropic Metal Plate-First-Order Shear Deformation Theory (FSDT)-Higher-Order Shear Deformation Theory (HSDT).

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.

Quintic and biquintic B-spline finite element solutions of clamped structures

Recent studies on the control of smart material partial differential equation systems point toward a need for considering higher-ordered polynomial splines in the formulation of bases for approximate solutions obtained via the Galerkin finite element method. However, basis modifications are not well documented in the literature, nor are they well understood by researchers who are often utilizing software packages in their basis and control designs. In this study, solutions are generated using modified bases with quintic and biquintic B-splines for three clamped structures: an ordinary differential equation cantilevered beam, a partial differential equation cantilevered beam, and an isotropic clamped plate. Convergent solutions are obtained and verified for all three models.

A general solution for electromechanical analysis of electroelastic composite plates with arbitrary holes

Stroh formalism provides an elegant method of solution for two-dimensional anisotropic elasticity problems through the generalized eigen relation associated with the dual coordinate system that gives the material eigen values and eigen vectors, and also the fundamental elasticity matrix and its associated Barnett-Lothe tensors. The extended Stroh-like formalism of Hwu and Hsieh has expanded the scope of basic Stroh formalism by addressing the electro-mechanical coupling effects in electro elastic plates. This paper presents an inclusive solution that addresses the two-dimensional problems in infinite electro-elastic, anisotropic, and isotropic plates with an arbitrary hole under remote arbitrary coupled electro-mechanical loading. This is achieved by adopting the Stroh formalism and its subsequent versions to consider the anisotropic and electro-elastic plates, and further, by incorporating the arbitrary biaxial loading condition into the boundary conditions, and the generalized mapping function into the basic formulation to accommodate all types of in plane loading and a variety of hole shapes respectively. Various equations of the solution are derived explicitly for easy understanding and computer implementation. The stress function is derived explicitly to satisfy the condition of traction free hole with remote electro-mechanical loading under generalized plane stress-open circuit condition. The stresses and electric displacements around the hole boundary are obtained by taking the derivative of the stress function with respect to unit normal along the hole boundary. The solution presented can be degenerated to the anisotropic or isotropic case by choosing the corresponding material properties, or appropriate values of complex parameters respectively. This general solution has exactly reproduced the results of other solutions in the literature for piezoelectric plates given by different methods. New results of stresses and electric displacements are obtained for various arbitrary holes in [PZT5H/45/-45/PZT5H]s graphite/epoxy plate and PZT 4 plates. Results for holes in piezoelectric laminates and PZT 4 piezo layer are also presented by FEM.

Guided Wave Phased Array Ultrasonic Imaging for Thin Plate Inspection

This study focuses on the application of phased array ultrasonic testing with guided waves for Structural Health Monitoring (SHM) in thin plates. Addressing the limitations of conventional ultrasonic testing efficiently by inspecting thin, long structures. This study employs phased array technology to thin isotropic plates at high frequency(2MHz), thereby elevating imaging and defect detection in structural components. Through the utilization of the full matrix capture (FMC) scanning approach in 3-D finite element (FE) simulations, A0 and S0 modes were generated on steel plates with through-hole geometries. The process involved capturing reflected signals to form an FMC matrix that includes all A-scan signals. Subsequently, the total focusing method (TFM) algorithm was applied to FMC data for the reconstruction of focused images of A0 and S0 modes separately. A key improvement in our approach is integrating a time-domain filtering method into the FMC data. This enhancement addresses the dispersive nature of guided waves, thus improving the accuracy in identifying defect locations and sizes within the images. Our finite element simulations on steel specimens, with varied through-hole locations, revealed the method's potential in accurately imaging defects located deeper within the structure. Additionally, the study expanded to evaluate the efficacy of phased array ultrasonic testing (PAUT) in SHM, particularly in conjunction with guided waves to study the flaw detection sensitivity of the defect through the TFM algorithm on isotropic steel plates up to wavelength(λ)/16. This whole analysis considered crucial factors such as ultrasonic wave mode, frequency, and group velocity for the structure under inspection. This study represents a significant contribution to the field of SHM, introducing a sophisticated, real-time monitoring methodology that promises to enhance structural integrity assessment. Keywords: SHM, PAUT, GW, FMC, TFM FE Simulations, Time Domain Filtering, Ultrasonic Testing.

ЖҰҚА ИЗОТРОПТЫ ПЛАСТИНАЛАРДЫҢ ИІЛУІН ТАЛДАУ

In the field of modern construction, thin isotropic plates are widely used. The development of calculation methods and mathematical models for thin isotropic plates of rectangular shape is an important issue. The article examines the bending of a rectangular thin isotropic plate. The method of separation of variables is employed to derive the plate calculation theory. New formulas for the function of deflection and internal forces of the plate are determined. A simplified version of the classical theory is presented, defining the solution to the plate bending problem using simple polynomials. The values of the deflection function and internal forces of a rectangular thin isotropic plate are presented in the form of tables and diagrams. The obtained calculation results are compared with Navier's method and the finite element method. The results obtained completely coincide with the results obtained by other methods and thereby confirm the correctness of the proposed version of the theory.

О деформировании решетчатой пластинки глаза

Several problems pertaining to deformation of the circular elastic isotropic plate of variable rigidity in the presence of elastic support on the edge were study. In the first problem Timoshenko model was considered. In the second problem the deformation of a multilayer plate with piecewise constant Lame coefficient along the thickness coordinate which are assumed to be constant within one layer was researched, the Zig-zag method is used for modeling. The basis for constructing a solution for both problems is the use of Lagrange’s variation principle of plates, numerical solution constructed using the Ritz method. During the research work the deflection and inflection points of the lamina cribrosa were compared for a healthy eye and for an eye with different stages of primary open-angle glaucoma for constant Lame coefficient as well as for their exponentially decreasing above. The problem of deflection of the lamina cribrosa when tacking into account the prelaminar layer was solved based on the Zig-zag method, the solution to this problem was also compared for a healthy eye and for an eye with various stages of primary open-angle glaucoma.

Does VIMC-FMC work well on highly ductile bimetal joints? An investigation on mixed mode I/II fracture of AA1050/Cu bimetal weakened by U-notches

In this study, it is attempted to check if the virtual isotropic material concept (VIMC), originally suggested for fiber-reinforced composite laminates, can be used as an appropriate simplifying tool in fracture estimation of notched bimetals. First, the aluminum alloy AA1050 and the copper are cladded together using the roll-bonding technique. Then, several U-notched bimetal specimens are manufactured to perform fracture tests under mixed mode I/II loading conditions. Due to the highly ductile behavior of the AA1050/Cu bimetal, the fictitious material concept (FMC) is adopted to equate the bimetal with a fictitious material having perfectly linear elastic behavior with the aim to avoid the elastoplastic fracture analysis. Finally, the VIMC and FMC are coupled with three famous brittle fracture models, i.e., the U-notch maximum tangential stress (UMTS), U-notch mean stress (UMS), and U-notch strain energy density (USED) criteria, for predicting the experimentally measured strength of the notched bimetal specimens. The results indicate that the maximum discrepancy between the theoretical predictions and the experimental results in all the combined failure criteria is less than 8%. Therefore, the VIMC-FMC combination is found to be quite effective as a converter to solve the complex problem of highly ductile fracture of notched inhomogeneous anisotropic bimetals under mixed mode I/II loading using only the linear elastic analysis of the fictitious homogeneous isotropic plates. Also, found in this research is that not only VIMC works well on fiber-reinforced composite laminates, but also on bimetal joints.

A class of elastic isotropic plate lattice materials with near-isotropic yield stress

Thoughtfully engineered lattice materials offer a canvas for tailoring a diverse range of mechanical attributes. Yet, their pronounced anisotropy and inherently unstable nonlinear mechanical behavior curtail their suitability for energy absorption applications. Here, we propose the adoption of lightweight plate lattice structures for energy absorption and loading support. This involves strategically integrating plates into the rhombic dodecahedron framework, facilitating elastic isotropy and nearly yield isotropy. The proposed plate lattices showcase remarkable properties, including an exceptionally high bulk modulus that reaches the HS bound. Moreover, they boast an enhanced energy absorption capacity, surpassing that of the rhombic dodecahedron truss lattice by up to 2.58 times. The effects of relative density and loading direction on the mechanical properties were thoroughly explored through numerical simulations. Simulation predictions were rigorously validated through experimental verification. The failure modes of plate samples transition from being dependent on the loading direction, leading to collapse, to becoming stable regardless of the loading direction as the relative density increases from 0.1 to 0.2. It is noteworthy that plate lattices demonstrate SEA values comparable to those of shell lattices, even surpassing the stiffest isotropic plate lattice. These characteristics underscore their potential for applications in loading support and energy absorption.

Effect of quadratically varying electric potential on the multiple cracks in piezoelectric materials

A basic solution to the problem of three collinear equal cracks in a transversely isotropic piezoelectric material plate is obtained under the influence of electromechanical forces. Stroh formalism and complex variable methods are used to derive the mathematical expressions for SIF (stress intensity factor), COD (crack opening displacement), and COP (crack opening potential). The length of the saturation zones is investigated at each crack tip under the influence of quadratically varying electric displacement. Furthermore, a numerical study is carried out to discuss the effect of quadratically varying electrical potential on the saturation zone, COD, and COP. The results obtained numerically are reported graphically.

The Dispersion Calculator - a free software for calculating dispersion curves of guided waves

The nondestructive inspection of components by means of guided waves is an emerging technology in fields like aerospace or pipeline construction and maintenance. The guided waves' capability of propagating many meters in a structure is utilized for swift inspection tasks as well as structural health monitoring. The German Aerospace Center uses Lamb waves for the quality assurance of large-scale aerospace vehicle components made from carbon composites. However, the use of guided waves in NDT requires knowledge of the dispersion curves. DISPERSE is the most renown software for the calculation of dispersion curves. This paper presents the open source Dispersion Calculator (DC). The MATLAB® -based DC is an interactive and fully validated software for the computation of dispersion curves and mode shapes of guided waves in isotropic plates and multilayered anisotropic composites. Damping caused by viscoelasticity and fluid-loading can be considered. DC features the capability of calculating laminates consisting of several hundreds of layers so that even the largest specimens, like rocket booster pressure vessels, can be calculated. This is made possible through the implementation of the stiffness matrix method. DC is also able to distinguish the different mode families, like symmetric, antisymmetric, and nonsymmetric Lamb, shear horizontal, and Scholte waves, depending on the symmetry and coupling properties of a given layup. Lastly, DC features a highly efficient and robust dispersion curve tracing algorithm. DC is available at GitHub, and it has gained many users world-wide since its initial release in 2018.

Determination and analysis of stress concentration factor in finite plate with different polygonal discontinuities under uniaxial compression using finite element analysis (FEA)

Stress concentration is the accumulation of stress in a body due to a sudden change in its geometry. Stresses and stress concentrations are created at the nearer area of the structural discontinuities. Any structural discontinuity compromises the strength of the structure. Numerical approaches can be used to determine the maximum stress concentration for various complex geometries without regard to time or cost constraints. The current study gives a thorough finite element analysis of stress concentration in a structural steel plate with a polygonal cutout. In contrast to orientation and area, this study looks at an isotropic finite plate for the stress concentration factor (SCF) with central polygonal cutouts (triangular, square, pentagonal, and hexagonal). ANSYS, a finite element program, calculates the von Mises stresses, and based on the results, stress concentrations are calculated. The orientation of polygonal cutouts in a square plate and the constant area of polygonal cutouts in a square plate are two parameters considered in the calculation. These results are verified by literature for the orientation of the hole, which concludes that a square hole position parallels the loading direction showing a 50% reduction in stress concentration factor.

Vibration response of ultra-shallow floor beam (USFB) composite floors

This paper examines the vibration response of the steel-concrete composite Ultra Shallow Floor Beam (USFB®) flooring system which incorporates asymmetric steel perforated beams to accommodate the concrete floor slab within the depth of the flanges while allowing reinforcement and/or service ducts to pass through the web openings. This is a lightweight flooring system that can accommodate long spans, thus becoming susceptible to floor vibrations due to external resonant dynamic loads. To investigate the influence of slab thickness and boundary conditions on the natural frequencies of the USFB flooring system, parametric studies are conducted using a finite element model and five floor spans. The model was first validated against an experimental test conducted by the authors. Emphasis is placed on the fundamental frequency to predict the possibility of resonance of this complex flooring system with typical human-induced dynamic loads in building structures. To further facilitate the practical numerical modelling and vibration analysis of buildings with USFB floors in standard commercial structural software, an analytical method of deriving equivalent isotropic plate properties is developed and its accuracy is numerically verified vis-à-vis with detailed ABAQUS models.

A neuro-fuzzy based prediction modeling and mechanical characterization of multiwalled-carbon nanotubes filled Luffa cylindrica hybrid core and isotropic skin-based sandwich plate

The study focuses on mechanical characterization and adaptive neuro-fuzzy inference system (ANFIS)-based prediction modeling of MWCNT-filled Luffa cylindrica hybrid core-based sandwich plates. Experiments determine natural frequencies using modal impact hammer tests with a Fast Fourier Transform (FFT) analyzer. Elastic properties from tensile tests utilized for numerical simulations of natural frequencies in ABAQUS, showing high consistency with experimental results. SEM analysis characterizes the fiber-matrix bond in the hybrid core. ANFIS modeling establishes input–output relationships for unseen data sets. Comparisons of numerical simulations with ANFIS predictions demonstrate an accuracy within a 5% margin of error, providing insights into dynamic behavior and potential applications.

Hysteretic behavior of the segmented buckling‐resistant braces with LYP160

AbstractThe goal was to evaluate the hysteretic performance of buckling‐resistant braces with low yield point steel LYP160, the monotonic tensile and cyclic loading tests of LYP160 test specimens were conducted and the cyclic constitutive relationship was obtained. According to the load–displacement curves of the specimens, the low‐yield point steel was characterized by good ductility and energy absorption ability. With consideration of the Chaboche model for the materials, the cyclic hardening parameters of low‐yield point steel were obtained. On this basis, the hysteretic properties of buckling‐resistant braces under cyclic loads were simulated and analyzed. After the analysis and comparison of buckling‐resistant braces specimens with isotropic core plate and segmented variable section core plate, it can be found that: when the conventional buckling‐resistant braces with an isotropic core plate were loaded to L/100, the lateral deformation of the buckling‐resistant brace (BRB) would reach 17 mm. Additionally, serious squeezing could be observed on the lateral restraining members. The conventional BRB would become ineffective due to the accumulation of deformation at both ends of the BRB. When the segmented buckling‐resistant brace was applied, the core plate with variable section would buckle first in the middle area, other parts could continue to consume energy thanks to the action of the limit plate. It would avoid the situation that other areas would be unable to consume energy after the core plate yields at one area first. Under the action of cyclic loads, no stiffness degradation was noted in the segmented buckling‐resistant brace. Segmented buckling‐resistant braces demonstrated superior ductility and energy dissipation capacity.