Reissner-Mindlin Plate Theory Research Articles

Error-controlled adaptive LR B-splines XIGA for assessment of fracture parameters in through-cracked Mindlin-Reissner plates

This paper presents an error-controlled adaptive extended isogeometric analysis (XIGA) for assessment of fracture behavior for through-cracked Mindlin-Reissner plates. The locally refined (LR) B-splines that facilitate the local refinement are used as the basis functions for the approximations, while the kinematics of plates are described in terms of the Mindlin-Reissner plate theory. Appropriate enrichment functions for cracked plates are incorporated into the formulation to represent geometrically displacement discontinuities across the crack faces and singularity of the stress in the vicinity at the crack tips, thus cracks are modeled topologically independent of the computational mesh. Here we extend the Zienkiewicz-Zhu method-based error estimation to the XIGA formulation for cracks in Mindlin-Reissner plates such that non-smoothnesses and stress singularity are reflected accurately. We conduct the error-controlled adaptivity for local mesh refinement to enhance the accuracy and performance. Fracture parameters (i.e., mixed-mode intensity factors) are calculated through the contour interaction integral method in the context of Mindlin-Reissner plate theory. Several numerical examples for through-cracked plates are studied to show the accuracy and effectiveness of the proposed formulation in modeling fracture of cracked moderately thick plates.

Free Vibration with Mindlin Plate Finite Element Based on the Strain Approach

This paper presents the development of a quadrilateral plate bending finite element for the computation of natural frequencies of plates with arbitrary geometry. This element is based on the Reissner–Mindlin plate theory using assumed strains rather than displacements and contains only the three physical degrees of freedom at each of the four corner nodes. Tests of convergence are first established for square plates with both simply supported and clamped on their edges where it is shown that a good rate of convergence is obtained using few elements. Other tests are then applied to rectangular, circular, skew and stepped plates, in addition to a rectangular plate with a central cutout as well. The numerical results obtained show that the present element has successfully passed patch tests and its results of the natural frequencies and the associated modes of vibration are in excellent agreement when compared with analytical and other available numerical solutions.

Analysis of wave propagation in functionally graded piezoelectric composite plates reinforced with graphene platelets

This paper presents a semi-analytical approach to investigate wave propagation characteristics in functionally graded graphene reinforced piezoelectric composite plates. Three patterns of graphene platelets (GPLs) describe the layer-wise variation of material properties in the thickness direction. Based on the Reissner-Mindlin plate theory and the isogeometric analysis, elastodynamic wave equation for the piezoelectric composite plate is derived by Hamilton’s principle and parameterized with the non-uniform rational B-splines (NURBS). The equation is transformed into a second-order polynomial eigenvalue problem with regard to wave dispersion. Then, the semi-analytical approach is validated by comparing with the existing results and the convergence on computing dispersion behaviors is also demonstrated. The effects of various distributions, volume fraction, size parameters and piezoelectricity of GPLs as well as different geometry parameters of the composite plate on dispersion characteristics are discussed in detail. The results show great potential of graphene reinforcements in design of smart composite structures and application for structural health monitoring.

Shape optimization of shell structure by using Genetic Algorithm (GA) method

Shell structure has been popular for various application in building, aerospace, oil and gas, ship and automobile industry. This type of element is light weight and can be provide great stiffness for large structure. Furthermore, it is very useful for building architecture and appearance. However, currently, construction is not only concerning structural strength but also focusing on building cost and its impact to environment. Finding the balance between structural strength, cost and aesthetic appearance is essentials this nowadays. Hence, construction framework to find the leeway between those parameters is greatly important. This paper will discuss the framework for shape optimization of shell element by using Genetic Algorithm (GA) method. The shape of structure will be developed from Non-Uniform Rationale Bi-Spline (NURBS). Finite element analysis based on Reissner-Mindlin plate theory is utilized to find the element stress within the structure. Genetic algorithm method will evaluate and perform search operation inside the result domain with constraints of structural weight and element stress. The framework then was implemented in numerical example to find its effectiveness on obtaining the most optimum shape of shell structure. Based on the numerical result, it is clear that the proposed frame work is able to reach the convergence to the optimum structure without violating the given constraints.

A novel dynamics model for railway ballastless track with medium-thick slabs

Motivated by the requirements for elaborated slab ballastless track dynamics analysis in practical engineering application, a novel dynamic model for the railway ballastless tracks with medium-thick slabs is proposed in this work based on the Reissner–Mindlin plate theory, and it is implemented into the coupled dynamics analysis of a vehicle and the ballastless track. First, an efficient and easily programmable computational algorithm is adopted to solve the transverse deflection of the Reissner–Mindlin plate, in which the displacements and shear strains are chosen as the independent variables and subsequently constructed by spline functions, resulting in no shear-locking effect. The involved partial differential equations are transformed into ordinary ones by using the energy variation principle. Further, a mathematical model for the ballastless track dynamics analysis is established, which can consider the effects of the shear deformation and moment of inertia involved in the medium-thick track slab. Experimental verification and comparative analysis with other models demonstrate the accuracy and efficiency of the proposed model. Finally, a spatially coupled dynamics model of a vehicle and the ballastless track is developed, and it is efficiently solved by using the hybrid explicit-implicit time integration method. Compared with the widely used modelling the track slab by elastic thin plate, the reliability and advantages of the proposed vehicle-slab track coupled dynamics model are demonstrated.

Nonlinear numerical analysis of composite slabs with steel decking

Abstract The composite slabs behavior is governed by longitudinal shear at the interface between the steel deck and concrete, which is developed in slabs under simple bending. The m-k method and the partial connection method, that are used in the evaluation of shear strength at the steel-concrete interface of composite slabs, are based on expensive and long-term experimental tests. The main objective of this work is to implement a finite element model for nonlinear numerical analysis of concrete slabs with steel decking. For this, flat shell elements are implemented, considering Reissner-Mindlin and Kirchoff plate theories, bar elements, considering the beam theory of Tymoshenko, and interface elements. In the numerical analyzes presented in the present work, the steel deck and the concrete slab, of thickness given by the total height of the slab less the height of the steel deck, are modeled with flat shell elements. The concrete rib is modeled with bar elements. The contact between steel deck and concrete is modeled through interface elements. The geometric and material nonlinearities are considered in the numerical analysis. The analyzed examples validate the numerical model suggested in this work, presenting the advantage of using a two-dimensional discretization of the problem while in comparative numerical models are uses a three-dimensional discretization of the concrete slab.

A new locking-free polygonal plate element for thin and thick plates based on Reissner-Mindlin plate theory and assumed shear strain fields

A new n-noded polygonal plate element is proposed for the analysis of plate structures comprising of thin and thick members. The formulation is based on the discrete Kirchhoff Mindlin theory. On each side of the polygonal element, discrete shear constraints are considered to relate the kinematical and the independent shear strains. The proposed element: (a) has proper rank; (b) passes patch test for both thin and thick plates; (c) is free from shear locking and (d) yields optimal convergence rates in L2-norm and H1-semi-norm. The accuracy and the convergence properties are demonstrated with a few benchmark examples.

Finite element analysis of aero-hydroelastic stability of arbitrary shape panels

A fluid-structure model based on Mindlin-Reissner plate theory and linearized incompressible potential flow theory is used to study the aero-hydroelastic stability of panels of arbitrary planforms. A finite-element procedure is developed to reduce the continuous system to a finite flow-structure system, which is solved in the frequency domain. The finite element method is crucial for the study of panels of arbitrary geometry and circumvents difficulties with the assumed mode shape approach. This paper is the first finite-element formulation of a fully coupled fluid-structure model of the aero-hydroelastic panel stability problem for incompressible flow. Circular, square, and triangular panels surrounded by a rigid baffle with different boundary conditions are studied. Eigenvalues of the system are surveyed as a function of the hydrodynamic loading at high and low density ratios. Critical flow velocity at divergence-onset and modal coalescence divergence or flutter are determined and compared for the three planforms. For simply supported or clamped panels of equal area and flexural rigidity, as the fluid velocity increases from zero the circular panel diverges first then the square and equilateral triangle, the later is the most stable. In order to compare the relative stability of the three geometries considered, a velocity ratio that consolidates divergence and flutter conditions for the three panels is introduced. Such a parameter offers a simple formula for the divergence velocity in terms of the panel geometry and flow and material properties, and is very useful to the designer in a wide range of engineering applications.

A unified polygonal locking-free thin/thick smoothed plate element

A novel cell-based smoothed finite element method is proposed for thin and thick plates based on Reissner-Mindlin plate theory and assumed shear strain fields. The domain is discretized with arbitrary polygons and on each side of the polygonal element, discrete shear constraints are considered to relate the kinematical and the independent shear strains. The plate is made of functionally graded material with effective properties computed using the rule of mixtures. The influence of various parameters, viz., the plate aspect ratio and the material gradient index on the static bending response and the first fundamental frequency is numerically studied. It is seen that the proposed element: (a) has proper rank; (b) does not require derivatives of shape functions and hence no isoparametric mapping required; (c) independent of shape and size of elements and (d) is free from shear locking.

A coupling FEM/BEM method with linear continuous elements for acoustic-structural interaction problems

The effects of dynamical load from acoustic waves to structural responses cannot be neglected for some cases such as a thin structure submerged in a heavy fluid. To deal with this kind of problem, a hybrid numerical method consisting of finite element method (FEM) and boundary element method (BEM) for the acoustic-structural coupling problems is proposed. The FEM is built on the first-order Reissner-Mindlin plate theory and the discrete shear gap (DSG) method is applied to construct a locking-free FEM. A collocation BEM with linear continuous element based on Burton-Miller formulation is adopted with an aim to produce a meshing conforming coupling method with the FEM not only on geometry but also on physical nodal quantities at the interface. In addition, analytical expressions of the singular integrals appeared in the Burton-Miller formulations are available to eliminate the numerical difficulties in the implementation of the BEM. Furthermore, the coupling matrix at the interface based on the conforming linear triangular mesh is obtained analytically to avoid the numerical quadrature. A solution scheme by absorbing the FEM equations to the BEM’s is proposed to take the advantage of less memory cost of the linear BEM. Optimized algorithms are designed to reduce the memory cost while keep the accuracy of the coupling method during the absorption of the structure effects to the acoustic domain. Numerical examples are setup to validate the accuracy and demonstrate the potential capability of the proposed method.

Fracture analysis of cracked thin-walled structures using a high-order XFEM and Irwin’s integral

We develop a novel fracture mechanics framework for cracked thin-walled structures based on the Mindlin-Reissner plate theory. The mixed interpolation of tensorial components (MITC) plate element within the extended finite element method (XFEM) is used for the discretization. High-order crack-tip enrichment functions are employed to resolve the linear elastic near field solutions and the stress intensity factors (SIFs) are extracted by using an analytical derivation of Irwin’s crack closure integral.The closed-form formulation of Irwin’s integral, given in terms of enriched degrees of freedom (DOFs), is shown to be valid for both structured and unstructured meshes. The SIFs of through-the-thickness cracks in thin-walled structures can thus be directly obtained upon the solution of the XFEM discrete system, without the need of constructing auxiliary fields for mode separation.The performance of the proposed approach is studied on several benchmark examples consisting of cracked plates and shells. We first consider plate examples with an inclined crack and two cracks approaching each other. Excellent accuracy in terms of SIFs, on structured and unstructured meshes is reported, even on rather coarse meshes. In particular, the case of two approaching cracks showcases the potential of Irwin’s integral in addressing crack coalescence problems. Finally, we demonstrate the accuracy of the proposed method on cylindrical-shell type problems with a crack subjected to either tension, torsion, or internal pressure.

Nonlinear aerothermoelastic analysis of composite laminated panels using a general higher-order shear deformation zig-zag theory

A general higher-order shear deformation zig-zag theory is proposed for predicting the nonlinear aerothermoelastic characteristics of composite laminated panels subjected to supersonic airflow. The von Kármán strains are employed to describe the structural nonlinearity of the panels, and the quasi-steady first-order piston theory is adopted to calculate the aerodynamic loads. The discretized equations governing the aerothermoelastic motion of the panels are established using the nonlinear finite element method. The proposed higher-order shear deformation zig-zag theory employs seven variables to represent the displacement field of the panel in a unified form, which is capable of accommodating various plate theories available in the literature, including the Kirchhoff plate theory, the Mindlin–Reissner plate theory and other higher-order shear deformation plate theories. The validity of the present model is confirmed by comparing the computed results with those solutions available in the literature. The aerothermoelastic behaviors (including the critical buckling, limit-cycle oscillation and flutter boundary) for composite laminated panels with different geometrical dimensions, temperature gradients and fiber orientations are examined. The discrepancies of the aerothermoelastic characteristics of composite laminated panels determined by different structural theories are discussed in detail. Physical insight into the mechanism of the differences among the aerothermoelastic behaviors of the panels determined by different plate theories is provided.

A quadratic piezoelectric multi-layer shell element for FE analysis of smart laminated composite plates induced by MFC actuators

Macro fiber composite (MFC) actuators developed by the NASA have been increasingly used in engineering structures due to their high actuation power, compatibility, and flexibility. In this study, an efficient two dimensional quadratic multi-layer shell element by using first order shear deformation theory (FOSDT) is developed to predict the linear strain–displacement static deformation of laminated composite plates induced by MFC actuators. FOSDT is adapted from the Reissner–Mindlin plate theory. An eight-node quadratic piezoelectric multi-layer shell element with five degrees of freedom is introduced to prevent locking effect and zero energy modes observed in nine-node degenerated shell element. Two types of MFC actuators are used: (1) MFC-d31 and (2) MFC-d33, which differ in their actuation forces. For result verification, the electro-mechanically coupled quadratic finite element (FE) model is compared with the ABAQUS results in various examples. Comparison of the results showed good agreement. The proposed quadratic FE formulation is simple and accurate, which eliminates the need for costly FE commercial software packages. It was observed that earlier studies have mostly emphasized on the effect of actuation power and MFC fiber orientations on mechanical shape deformation of smart composite plates. In this study, a more comprehensive, in-depth investigation is conducted into host structure performance such as boundary conditions, laminate stacking sequence configuration, and symmetry/asymmetry layups.

Effects of director orientation on the vibration of anisotropic nematic elastomer plates under various boundary conditions

Nematic elastomer (NE) is a kind of smart materials which is easily actuated by external fields and generates considerable force. In the present discussion, the vibration of a nematic plate is investigated with the aim to the prospective controlling for energy harvest and transferring. Based on the anisotropic Timoshenko, Reissner–Mindlin plate theory and linear hydrodynamics and viscoelasticity of nematic elastomer in low frequency limit, the governing equation is derived and solved by a hybrid analytical-numerical method under various boundary conditions. The effects of intrinsic parameters like anisotropic parameter, director rotation time and boundary condition on the vibration characteristics of NE plates are discussed in detail. The results give a well indication between the vibration performance and the director dissipation, which is definitely different from classical isotropic viscoelastic plate. This work provides effective information in the process of dynamic design and development of NE-based devices.

Adaptive analysis of plates and laminates using natural neighbor Galerkin meshless method

In this paper, natural neighbor Galerkin meshless method is employed for adaptive analysis of plates and laminates. The displacement field and strain field of plate are based on Reissner–Mindlin plate theory. The interpolation functions employed here were developed by Sibson and based on natural neighbor coordinates. An adaptive refinement strategy based on recovery energy norm which is in turn based on natural neighbors is employed for analysis of plates. The present adaptive procedure is applied to classical plate problems subjected to in-plane loads. In addition to that the laminated composite plates with cutouts subjected to transverse loads are investigated. Influence of the location of the cutout and the boundary conditions of the plate on the results have been studied. The results obtained with present adaptive analysis are accurate at lower computational effort when compare to that of no adaptivity. Further, the adaptive analysis provided accurate magnitude of maximum stresses and their locations in the laminate plates with and without cutout subjected to transverse loads. Additionally, failure prone areas in the geometry of the plates subjected to loads are revealed with the adaptive analysis.

Validation simulation for free vibration and buckling of cracked Mindlin plates using phase-field method

ABSTRACTThis article focus on validation phase-field method for simulation of free vibration and buckling of crack plates. The formula is derived from using Reissner-Mindlin plate theory. Validation simulation is carried out by numerically investigating free vibration and buckling of cracked plate with taking the configuration, material property, crack location, and other relevant assumptions as the same with the comparing references. The article shows that phase-field approach can be used to estimate the critical buckling load and frequencies of vibration mode. The article also demonstrates the significant advanced of phase-field method for plates with complex crack geometries.

An evaluation on the performance of two simple triangular bending plate elements

This paper will study and compare two different three-node triangular bending plate elements with three degree of freedom per node, i.e. MITC3 and DKMT. Both elements, which were developed based on Reissner-Mindlin plate theory and independent shear strain field, have simple formulation and have already been used widely. In this paper, numerical tests for circular plate case are conducted to verify the performance and show the convergence of these two triangular elements.

Three-dimensional flat shell-to-shell coupling: numerical challenges

Abstract The node-to-surface formulation is widely used in contact simulations with finite elements because it is relatively easy to implement using different types of element discretizations. This approach, however, has a number of well-known drawbacks, including locking due to over-constraint when this formulation is used as a twopass method. Most studies on the node-to-surface contact formulation, however, have been conducted using solid elements and little has been done to investigate the effectiveness of this approach for beam or shell elements. In this paper we show that locking can also be observed with the node-to-surface contact formulation when applied to plate and flat shell elements even with a singlepass implementation with distinct master/slave designations, which is the standard solution to locking with solid elements. In our study, we use the quadrilateral four node flat shell element for thin (Kirchhoff-Love) plate and thick (Reissner-Mindlin) plate theory, both in their standard forms and with improved formulations such as the linked interpolation [1] and the Discrete Kirchhoff [2] elements for thick and thin plates, respectively. The Lagrange multiplier method is used to enforce the node-to-surface constraints for all elements. The results show clear locking when compared to those obtained using a conforming mesh configuration.

Isogeometric Galerkin in rectangular plate bending problem based on UI approach

Abstract This paper presents the development of Isogeometric Analysis for plate bending problems based on unified and integrated (UI) approach, which is a modification of Reissner-Mindlin plate theory for solving thick to thin plate problems. In Reissner-Mindlin, the total displacement and two rotations are independent of each other, while in this UI approach the total displacement is split into bending displacement and shear displacement which causes the rotations, curvatures and shear deformations can be defined as first, second and third derivatives of bending displacement, respectively. The virtual work of Galerkin Method is used to define bending stiffness and shear stiffness of the element. Several convergence tests were conducted to observe the performance of unified and integrated approach in rectangular plate of different types of boundaries conditions. The result of thick and thin plate showed good results despite of low number of element with fourth degree of polynomial or increasing degree of polynomial with only one element.

A modified Kirchhoff plate theory for analyzing thermo-mechanical static and buckling responses of functionally graded material plates

In this paper, we present a modified Kirchhoff theory for analyzing thermo-mechanical static and buckling responses of functionally graded material (FGM) isotropic and sandwich plates. In comparison to the classical Kirchhoff theory, the proposed method is taken account into the shear deformation effects, then it can be applied for both moderately thick and thin plates. On the other hand, comparing to the first order shear deformation theory (FSDT) or Reissner-Mindlin plate theory, the number of independent unknowns reduces one variable and retains four degrees of freedom (Dofs) per node. The shear locking phenomenon is also negligible due to extending from the classical Kirchhoff theory. Furthermore, a new normalization of a quartic form is used to build the shape functions in the radial point interpolation method (RPIM). The novel imposing essential boundary conditions related to slopes of deflections has been proposed, then the enforcement of these gradient deflections becomes easily and directly as the same in finite element analysis. The obtained numerical results for the thermo-mechanical static and buckling problems from the proposed method are stable and well accurate prediction as comparing to the exact solutions and other published analyses.