Clamped Plate Research Articles

Response of thin walled transversely stiffened clamped circular plates under uniform impulsive loads

Paper presents numerical models based on detailed characterization of various uncoupled damage models in a finite element analysis (FEA) setup to assess large deformations and observed ductile fracture failure modes of fully clamped circular plates as well as transversely stiffened circular plates in cross stiffened configurations under uniform impulsive loads. The numerical schemes have been based on a detailed definition of the constitutive properties based on inverse FEA simulations of the uniaxial tension test for the calibration of material and failure parameters. Experiments have been conducted using two types of freshly prepared mild steel uniaxial tension test specimens covering the envisaged range of stress tri-axialities to establish the steel plate material and fracture characteristics necessary for the calibration of the presented phenomenological damage models. Published results based on extensive experimental works by various authors on response of thin walled clamped circular plates under transversely applied uniform air blast loads, manifesting in large plastic deformations, thinning and tearing under various ductile fracture modes, have been used in the performance assessment of presented numerical models and procedure, which incorporate different damage models including temperature effects due to plastic dissipation. Based on evaluated performance of the discussed failure models within the FEA setup, a hybrid damage model with a tri-axiality based two part ductile fracture locus has been proposed, which shows very good correlation with the published experimental results, presented in terms of non-dimensional parameters over a wide range of impulsive loading. The validated numerical setup using the proposed hybrid damage model has also been used to predict the effect of transverse cross stiffened configurations of flat bar stiffeners on large deformations and ductile fracture of clamped circular plates under transverse uniform impulsive blast loads, along with some key insights in the evolution of plastic zones and damage progression.

Displacement and Stress Analysis of Thick Rectangular Clamped Plates Using Third-Order Energy Functional

This study analysed the stress and displacement responses of an all-clamped rectangular thick plate subjected to a uniformly distributed load using polynomial displacement functions. A polynomial shear deformation function was incorporated in the determination of the general governing equations; hence, there is no need for shear correction factors. Approximate Polynomial Displacement Functions w, u, and v for all the clamped thick rectangular plates subject to transverse loading were obtained. Numerical results were obtained using displacement equations obtained from the literature for the non-dimensional form of displacements and stresses of an all clamped (CCCC) plate at different aspect and span-depth ratios to determine the efficiency of this theory. The results obtained were validated as they showed good agreement with the results obtained by other researchers available in literature. Hence, this theory can be used as a reliable, concise, easy and dependable means for the stress and displacement analysis of thick plates. The results obtained also indicate that at a span-depth ratio of 100 and above, the classical plate theory (CPT) can be used to analyse plates, as the numerical values obtained are approximately equal to that of the CPT.

Delamination in GLARE laminates subjected to oblique low velocity impact considering friction

Study of oblique low velocity impact and the subsurface delamination caused in fibre metal laminates is not well reported in literatures. The present paper numerically investigates the oblique low velocity impact behaviour of clamped GLARE plates by a spherical impactor to study the influence of obliquity and coefficient of friction on the impact mechanism and on the extent of interfacial delamination. A three dimensional finite element formulation employing Newmark-β method has been developed for analyzing the oblique contact impact response of GLARE plate where Hertzian contact model is used for the normal component of contact and Mindlin and Deresiewicz theory is used for the tangential component. Results show that besides impactor velocity, obliquity of incidence and the coefficient of friction significantly influence the contact force history, contact duration and the delamination at the interfaces. In addition, the oblique angle also decides the instant and duration of reversal of tangential contact force where increasing obliquity leads to delayed instant and reduced duration of such reversal.

Dynamic response of UHMWPE plates under combined shock and fragment loading

Ultra-high molecular weight polyethylene (UHMWPE) fiber composite has been extensively used to construct lightweight protective structures against ballistic impacts, yet little is known about its performance when subjected to combined blast and fragment impacts. Built upon a recently developed laboratory-scale experimental technique to generate simulated combined loading through the impact of a fragment-foam composite projectile launched from a light gas gun, the dynamic responses of fully-clamped UHMWPE plates subjected to combined loading were characterized experimentally, with corresponding deformation and failure modes compared with those measured with simulated blast loading alone. Subsequently, to explore the underlying physical mechanisms, three-dimensional (3D) numerical simulations with the method of finite elements (FE) were systematically carried out. Numerical predictions compared favorably well with experimental measurements, thus validating the feasibility of the established FE model. Relative to the case of blast loading alone, combined blast and fragment loading led to larger maximum deflections of clamped UHMWPE plates. The position of the FSP in the foam sabot affected significantly the performance of a UHMWPE target, either enhancing or decreasing its ballistic resistance. When the blast loading and fragment impact arrived simultaneously at the target, its ballistic resistance was superior to that achieved when subjected to fragment impact alone, and benefited from the accelerated movement of the target due to simultaneous blast loading.

Large Deflection Analysis of Clamped Thin Rectangular Isotropic Plate under the Action of a Uniform Distributed Load

This present paper deals with large deflection of clamped thin rectangular isotropic plate under uniformly distributed load. The analysis of clamped plates is of great important and its structural behavior has not been completely understood. Previous studies have made a great in road in the analysis of clamped plates through varied approaches mostly assuming trigonometric series as displacement field. A variational method based on the principle of minimization of total potential energy has been employed through assumed displacement field. Therefore, the von Karman differential equation was decoupled through direct integration retaining its non-linear character. The aim of this present studies is to provide solution to analysis structural behavior of clamped (CCCC) thin rectangular isotropic plate under uniform distributed. The analysis was accomplished through a theoretical formulation of shape function based on polynomial series and subsequently, the formulated shape function was used on Ritz energy method. The resulting functional was minimized to obtain a general amplitude equation of the form K1∆3+K2∆+K3. Where K1, K2 and K3 are coefficients of amplitude equation and ∆ is the deflection coefficient (factor). Newton-Raphson method was used to evaluate the deflection coefficient. Values of ∆ from Timoshenko and that from present study were compared with an aspect ratios ranging from 1.0 to 1.5 with an increment of 0.1. From results obtained, the average percentage difference for the pervious and present studies is 3.7646%, The percentage difference for the plate was within acceptable limit of 0.05 or 5% level of significance in statistics. It is shown from the comparison that good agreement exists between the present and previous works and that this approach provides simply, direct, straightforward and highly accurate solutions for this family of problems.

The effect of tightening again on bolt loosening under transverse load: Experimental and finite element analysis

Bolt loosening is one of the most common failure modes in bolted joints. Tightening again is often used to compensate for the loss of clamping force in many critical applications. In this study, the effect of load control modes and lubrication conditions on bolt loosening were studied by experiments. Then, a series of experiments were carried out on the tightening again after bolt loosening under transverse load. Finally, in order to study the cause of fatigue fracture, FEA was carried out by using an accurate three-dimensional finite element model. The results show that the bolt loosening is mainly caused by relative motion between the two clamped plates. With oil lubrication, the bolt clamping force was able to reach the magnitude comparable to the initial preload when the nut is released to a specific angle (30° ∼ 60°) before tightening again. Nevertheless, the times of tightening again should be limited to avoid fatigue fracture of the bolt.

Time-domain model identification of structural dynamics from spatially dense 3D vision-based measurements

A novel approach is presented for the time-domain system identification of structural dynamic components exploiting the high spatial density of vision-based measurements. By using spatially dense measurements, the number of spatial measurement points is much larger than the dimensionality of the underlying dominant dynamics (i.e., the number of measurement points needed to meet observability of the targeted dynamics). This opens up the potential to develop new experimental identification methods that use this spatial overdetermination which were out of reach with conventional discrete sensors. The new approach presented in this paper directly extracts a relatively noise-free, low-order set of dynamic states by projecting the spatially dense time-domain measurements on a low-order dominant deformation motion basis. The basis is constructed by applying an over-complete singular value decomposition on the measurements. These dynamic states together with their numerically calculated first and second order time-derivatives are then used in a two-step identification of the structural dynamic parameters. Here, the structure of the model to identify is based on a-priori physical knowledge of the underlying set of partial differential equations, unlike a purely data-driven method. This approach has the benefits of providing a mathematical formulation that is straightforward to understand and implement and is time-efficient to solve. The presented approach is experimentally validated on a clamped plate as well as on a flexibly suspended plate which requires a correction of the rigid body motion. The identified models have shown to provide an accuracy up to 1 × 10−5 m with respect to the dominant measured motion components in both validation cases.

Lord Rayleigh’s Conjecture for Vibrating Clamped Plates in Positively Curved Spaces

We affirmatively solve the analogue of Lord Rayleigh’s conjecture on Riemannian manifolds with positive Ricci curvature for any clamped plates in 2 and 3 dimensions, and for sufficiently large clamped plates in dimensions beyond 3. These results complement those from the flat (Ashbaugh and Benguria in Duke Math J 78(1):1–17, 1995; Nadirashvili in Arch Ration Mech Anal 129(1):1–10, 1995) and negatively curved (Kristály in Adv Math 367:107113, 2020) cases that are valid only in 2 and 3 dimensions, and at the same time also provide the first positive answer to Lord Rayleigh’s conjecture in higher dimensions. The proofs rely on an Ashbaugh–Benguria–Nadirashvili–Talenti nodal-decomposition argument, on the Lévy–Gromov isoperimetric inequality, on fine properties of Gaussian hypergeometric functions and on sharp spectral gap estimates of fundamental tones for both small and large clamped spherical caps. Our results show that positive curvature enhances genuine differences between low- and high-dimensional settings, a tacitly accepted paradigm in the theory of vibrating clamped plates. In the limit case—when the Ricci curvature is non-negative we establish a Lord Rayleigh-type isoperimetric inequality that involves the asymptotic volume ratio of the non-compact complete Riemannian manifold; moreover, the inequality is strongly rigid in 2 and 3 dimensions, i.e., if equality holds for a given clamped plate then the manifold is isometric to the Euclidean space.

Non-linear analysis of orthotropic hyperelastic plate using VAM

An asymptotically accurate zeroth-order constitutive relation between 2D strains and stress tensor for an orthotropic hyperelastic material is derived using the Variational Asymptotic Method (VAM). Material nonlinearity is encompassed through the hyperelastic material model while geometric nonlinearity is by allowing finite displacements and rotations. Moderate strains of order 20% are considered. The small parameters such as the ratio of thickness to characteristic dimension, and maximum allowable strain (0.2) separate strain energy density into different orders. The analysis starts with the 3D strain energy density function and is divided into the thickness analysis (1D) and mid-surface analysis (2D). The 1D analysis leads to analytical expressions for warping functions, constitutive relations, and recovery relations. The 2D analysis is based on a non-linear finite element method that takes the constitutive relation as its input and obtains 2D displacements. The recovery relation converts them to 3D field variables i.e. displacements, strains, and stresses. The theory is applied to a square clamped plate subjected to uniformly distributed pressure. The results are accurate and obtained in a computationally efficient manner.

An approach on the vibro-acoustic properties of composite sandwich plates with foam core

In this paper, an approach is proposed and presented to tackle the vibro-acoustic properties of finite clamped composite sandwich plates with foam core. Composite sandwich plates are treated as being orthotropic and the apparent bending stiffnesses are calculated for the two principal directions. The apparent bending stiffnesses of composite sandwich plate are estimated by finite element calculation on beam elements cut from the considered composite sandwich plates. The sound transmission loss of clamped composite sandwich plates is predicted using orthotropic Kirchhoff’s plate theory, together with the obtained bending stiffnesses in two principal directions. Several sound transmission loss measurements were conducted in the laboratory on fiberglass/polyester composite sandwich plates with polyurethane foam core. The predicted sound transmission loss is compared with measured data and the agreement is reasonable.

Thermal Vibration of Rectangle Plate with Two-Dimensional Circular Thickness Effect

Frequencies comprising two-dimensional circular thickness impacts are computed for the first time, for clamped rectangle plate under nonhomogeneity and temperature field. The study also comprises the two-dimensional temperature and one-dimensional density effect, on frequencies computation, utilizing the Rayleigh–Ritz technique. The objective of the study is to reduce the frequencies of the plate as well as to provide the effect of various plate parameters on frequencies. Numerical illustration and comparative study of the frequencies with other available results well authenticate the objective of the study. A comparison of frequency modes done for the clamped rectangle and square plate corresponding to tapering and aspect ratio and nonhomogeneity concludes that frequency obtained for the present study is very less in comparison with other studies done. A convergence study of the obtained results is also presented with the help of figures. The convergence study done concludes that rate of convergence in case of circular variation is faster than the other variations.

Research on bubble pulsation and jet loading characteristics during a near-field underwater explosion

<p indent="0mm">To deeply study the bubble jet phenomenon occurring during the near-field underwater explosion, this study performed an underwater explosion test of small equivalent TNT exploding at the bottom of a clamped square plate. The load time–history curve of the bubble jet was successfully measured by the method employing an array wall pressure sensor. The experimental results show that the peak value of the bubble pulsation pressure at the bottom center of the clamped square plate is not significantly different from the peak value of the jet pressure. However, the duration of the jet pressure is 21.36 times that of the bubble pulsation pressure. The bubble pulsation pressure and its duration slightly attenuate, while the jet pressure and its duration attenuate faster with the increasing distance from the sensor to the center, indicating a quite concentrated range of jet load. Further, the numerical simulation of the whole process from the explosion to the bubble jet formation was performed via the Abaqus software. The bubble evolution process and the bubble jet load time-history curve obtained by the simulation are consistent with the experimental results. Finally, the distribution law of the specific impulse of the bubble jet, the evolution law of the bubble jet shape, and the change law of the head speed of the bubble jet during the near-field underwater explosion bubble pulsation process are obtained through numerical simulation. This work can provide a basis for the destructive strike of surface ship structures using bubble jets formed by a near-field underwater explosion.

Free vibration analysis of all round clamped thin isotropic rectangular plate by Ritz Direct Variational Method

This paper developed polynomial comparison functions for the free vibration analysis of clamped thin rectangular plates using the Ritz Direct Variational Method. The polynomials were derived systematically from a predefined formula, which could generate any number of trial functions for any set of plate’s classical boundary conditions. The method was implemented by means of a Mathematica computer programme developed by the authors. The frequency parameters so obtained agreed excellently with those available in the literature. The numerical values of the frequency parameters increased with the aspect ratio irrespective of the mode considered. In addition, the study showed that the more the number of polynomial coordinate functions in the shape function, the better the accuracy of the results. The convergence study corroborated the fact that a one-term approximation yields sufficient accuracy. The convergence was best for square plates, even though acceptable percentages of convergence were obtained for the other side ratios.

A study on an integrated aero-vibro-acoustic analysis procedure for a small-scale supersonic jet and surrounding thin plates

The goal of this paper is to examine the computational approaches for predicting both of the overall sound pressure level (OASPL) at a few locations and acceleration power spectral density (APSD) of surrounding thin plates due to the aero-acoustic pressure generated by a cold jet with M = 1.8. First, computational fluid dynamics (CFD), particularly delayed detached eddy simulation, are applied to predict the OASPL at the near-field and compute the acoustic properties. Second, the linearized boundary element method (BEM), that is, the Helmholtz-Kirchhoff method is utilized to propagate the pressure and obtain the OASPL at the far-field. Finally, the finite element method is implemented to predict the APSD for a clamped thin plate based on the optimal triangle membrane element, discrete Kirchhoff triangle plate bending element, and Newmark- β time integration scheme. Using the present CFD and BEM, the OASPLs are compared with the experimental results measured by microphones at both the near- and far-fields, respectively. Moreover, APSDs are compared with the experimental results obtained by an accelerometer at a few different locations. Although OASPLs are overestimated because of the coarse meshes in the higher-angle area and low order scheme of the present CFD analysis, the present integrated aero-vibro-acoustic analysis is capable of predicting the OASPL and APSD generated by a cold jet with M = 1.8.

Large deformations and failure of clamped circular steel plates under uniform impulsive loads using various phenomenological damage models

Paper presents numerical models based on a detailed characterization of various uncoupled damage models in a finite element analysis (FEA) setup to assess their results and comparative performance in simulation of structural response and failure of clamped circular plates subjected to a wide range of uniform impulse loading. The significance of numerical modeling of ductile fracture in simulating the various observed modes of failure in steel structures under intense blast loading has also been presented along with an overview of various phenomenological damage models in the published domain. Numerical schemes presented in the paper are founded on a detailed and well-defined material model using mechanical uniaxial tension tests (UTT) and its inverse FEA simulations. Elaborate UTT experiments have been conducted to establish the mechanical strength and ductile fracture characteristics of a particular grade of mild steel for its usage in the calibration of the discussed damage models. Relative performance of various simple as well as complex phenomenological damage models in simulating/ predicting the experimentally observed structural response and different failure modes of fully clamped circular steel plates over a broad range of uniform impulsive loads has also been presented. Published results based on extensive experimental works by various authors on response of thin walled clamped circular plates under transversely applied uniform air blast loads, manifesting in plastic deformations of large magnitude, inception of thinning and tearing under various ductile fracture modes, have been utilized in the validation of the numerical model and procedure. Based on the evaluated performance of the discussed numerical models, a hybrid damage model adopted within the FEA setup has been proposed, demonstrating a very good correlation of numerical predictions with experiments. The validated FEA setup incorporating the calibrated hybrid damage model has further been gainfully utilized towards critical insights in the evolution of plastic deformations, progression of damage and their correlation with experimentally observed failure modes over a wide range of loading, involving varying stress states and tri-axiality.

Water entry dynamic analysis of lightweight sandwich plates with lattice core: A semi-analytical approach and comparative study

This is the third part of water entry hydro-elastic analysis of clamped sandwich plates with lattice core (CSPLC). In the first and second parts analysis, the engineering prediction model is built to estimate the dynamic responses of CSPLC. To discuss the responses of CSPLC in more detail, the objective of the present work is to investigate the hydro-elastic responses of rectangular, clamped sandwich plates with lattice core (CSPLC) subjected to water entry via combined a novel semi-analytical approach and multi-physical numerical simulation. In the theoretical approach, the hydro-elastic characteristics are quantified by coupling a hydrodynamic water entry model with a first shear displacement model of elasticity for CSPLC. The fluid-structure interaction (FSI) effects of water entry are taken into account by dividing the total impact pressure into the rigid water entry pressure and the interacting pressure. To capture the precise responses of CSPLC, both the global displacement field and the local displacement field are considered in the analytical model. The dynamic governing equations are derived from the energy variation principle and solved by employing the modal approach. Furthermore, the 3D FSI finite element simulation is carried out to get the detailed hydro-elastic characteristics of CSPLC. This engineering semi-analytical model is found to yield accurate results for dynamic responses and offers significant savings in computational cost compared to the FSI simulations. As a comparative study, for a same geometrical stiffened panel, the FSI impact pressure and structural deflection are also given respectively.

Modal-based synthesis of passive electrical networks for multimodal piezoelectric damping

This work presents a new approach to design an electrical network which, when coupled to a structure through an array of piezoelectric transducers, provides multimodal vibration mitigation. The characteristics of the network are specified in terms of modal properties. On the one hand, the electrical resonance frequencies are chosen to be close to those of the targeted set of structural modes. On the other hand, the electrical mode shapes are selected to maximize the electromechanical coupling between the mechanical and electrical modes while guaranteeing the passivity of the network. The effectiveness of this modal-based synthesis is demonstrated using a free–free beam and a fully clamped plate.

Residual-based a posteriori error estimators for mixed finite element methods for fourth order elliptic singularly perturbed problems

We consider mixed finite element approximation of a singularly perturbed fourth-order elliptic problem with two different boundary conditions, and present a new measure of the error, whose components are balanced with respect to the perturbation parameter. With different boundary conditions, the simply supported plate model and the clamped plate model are considered. In particular, a balanced energy norm has been defined. Based on the new norm, residual-based a posteriori estimators are developed for both problems, which are uniform with respect to both the perturbation parameter and the mesh function. A novel analysis approach is introduced for the clamped plate model to address certain difficulty of the problem. Numerical examples are provided to confirm theoretical findings.

A geometrical approach to estimate the modal radiation efficiency of complex structures

The radiation efficiency of plates has been the study of many works. The rectangular simply supported plate is the most studied case, followed by the circular one. Analytical formulas and approximations for multiple boundary conditions, baffled and unbaffled cases have been the subject of study for decades. For a structure with a complex shape, numerical methods are appropriated such as FEM and BEM approaches. Semi-analytical based on Rayleigh’s integral also appear in the literature to calculate the radiation, mixed with the surface velocity calculated by FEM; they are easy to implement but they are resource consuming as large matrices are presented. In this work, we present a method for estimating the modal radiation efficiency of simply supported and clamped complex plates, based on the theory of Greenspan for circular radiators and with a geometrical approach.

Solid-shell approach based on first-order or higher-order plate and shell theories for the finite element analysis of thin to very thick structures

A new solid-shell approach dedicated to thin to very thick structures is presented. Plate or shell displacement fields are directly applied on a solid finite element model which contains several elements through the thickness. Any plate or shell theory based on kinematic assumptions can be used, in this study three theories have been considered. The classical first-order theory, the modified first-order theory and a higher-order theory lead to the FOSS, the MFOSS and the HOSS models respectively. Kinematic relations are imposed at nodes to meet through-the-thickness plate or shell displacement fields. From a numerical point of view, linear equations are applied on the assembled finite element model. The methodology relies on the slave and master nodes technique. Slave nodes are eliminated, leading to a reduction of the model size. Two examples are presented: a clamped square plate under distributed loading and a quarter of cylinder under pressure. For the thin and thick cases, displacements and stresses are observed. The FOSS model leads to unsatisfactory results. The HOSS model gives excellent results in the thin as well as the thick cases, by comparison with the solid approach. The MFOSS model is satisfactory for thin structures and leads to moderate errors in the thick case. This approach is also efficient from a computational cost point of view.