Reissner-Mindlin Theory Research Articles

Free vibration and buckling analysis of plates using B-spline wavelet on the interval Mindlin element

A finite element method (FEM) of B-spline wavelet on the interval (BSWI) is used in this paper to solve the free vibration and buckling problems of plates based on Reissner–Mindlin theory. By aid of the high accuracy of B-spline functions approximation for structural analysis, the proposed method could obtain a fast convergence and a satisfying numerical accuracy with fewer degrees of freedoms (DOF). The numerical examples demonstrate that the present BSWI method achieves the high accuracy compared to the exact solution and others existing approaches in the literatures. The BSWI finite element has potential to be used as a numerical method in analysis and design.

General anisotropic doubly-curved shell theory: A differential quadrature solution for free vibrations of shells and panels of revolution with a free-form meridian

The Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behavior of anisotropic doubly-curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature in the evaluation of the stress resultants three different approaches, specifically Qatu approach, Toorani–Lakis approach and Reissner–Mindlin approach, are considered and compared. An improvement of the Classical Reissner–Mindlin Theory (CRMT) using a different kinematical model is considered. By so doing a generalization of the theory of anisotropic doubly-curved shells and panels of revolution is proposed. Four different anisotropic shell theories, namely General First-order Shear Deformation Theory by Qatu (GFSDTQ), General First-order Shear Deformation Theory by Toorani–Lakis (GFSDTTL), General First-order Shear Deformation Theory by Reissner–Mindlin (GFSDTRM) and Classical Reissner–Mindlin Theory (CRMT), are compared in order to show the differences and the accuracy of these theories. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Results are obtained taking the meridian and circumferential coordinates into account, without using the Fourier modal expansion methodology. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed.

Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method

The Generalized Differential Quadrature (GDQ) Method is applied to study laminated composite shells and panels of revolution. The mechanical model is based on the so called First-order Shear Deformation Theory (FSDT) deduced from the three-dimensional theory, in order to analyze the above moderately thick structural elements. In order to include the effect of the initial curvature from the beginning of the theory formulation a generalization of the kinematical model is adopted for the Reissner–Mindlin theory. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The results are obtained taking the two co-ordinates into account, without using the Fourier expansion methodology, as done in semi-analytical methods. After the solution of the fundamental system of equations in terms of displacements and rotations, the generalized strains and stress resultants can be evaluated by applying the Differential Quadrature rule to the generalized displacements themselves. The transverse shear and normal stress profiles through the laminate thickness are reconstructed a posteriori by simply using local three-dimensional equilibrium equations. No preliminary recovery or regularization procedure on the extensional and flexural strain fields is needed when the Differential Quadrature technique is used. By using GDQ procedure through the thickness, the reconstruction procedure needs only to be corrected to properly account for the boundary equilibrium conditions. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained with 3D finite element methods. Stresses of several composite shell panels are evaluated. Very good agreement is observed without using mixed formulations and higher order kinematical models. Various examples of stress profiles for different shell elements are presented to illustrate the validity and the accuracy of GDQ method.

Dynamic simulation of damage‐fracture transition in smoothed particles hydrodynamics shells

SUMMARYThis paper presents a meshless method for the modeling of shell‐type structures in fast dynamics. The model is based on the Mindlin–Reissner theory and takes into account material and geometric nonlinearities. The phenomena that occur prior to rupture are dealt with using damage laws, while the rupture itself is represented through the introduction of sharp discontinuities. The method does not represent cracks explicitly, which makes the treatment of multicracking easier. The time discretization is carried out in the framework of explicit dynamics, and the spatial discretization is handled through the smoothed particles hydrodynamics method and the use of moving least square functions. The capabilities of the method are demonstrated using cracking, puncturing and fragmentation examples. Copyright © 2012 John Wiley & Sons, Ltd.

Homogenization of cellular sandwich panels

The Bending-Gradient plate theory originally presented in Lebée and Sab (International Journal of Solids and Structures 48 (2011) 2878–2888) is applied to cellular sandwich panels. This theory is the extension of the Reissner–Mindlin theory to heterogeneous plates. Its application clarifies common assumptions made in sandwich theory. It also enables to define a direct homogenization scheme for deriving the shear forces stiffness of sandwich panels. Finally, the conventional bounds used for estimating sandwich panels stiffnesses are justified.

A triangular finite element with drilling degrees of freedom for static and dynamic analysis of smart laminated structures

In this work a new three-node finite element with piezoelectric coupling is presented. It can be used for the static and dynamic analysis of light-weight smart structures, i.e. laminated composite structures with piezoelectric patches attached to its surface or embedded within the laminated layers. The element is a continuum-based degenareted plate element based on the Reissner–Mindlin theory. It uses the Allman formulation for the in-plane strains together with the generalization of the discrete Kirchhoff technique to include the transverse shear strains. It has six mechanical degrees of freedom per node with no difficulties in the drilling rotations and one electrical degree of freedom per finite element. The electric field is assumed constant across the thickness of each piezoelectric layer. The bending and membrane consistent mass matrices are derived for applications on structural dynamics. Since this finite element was firstly developed to allow the active vibration control of the flexible multibody components, which formulation may be based on the diagonalization of the mass matrix, a comparison of a set of numerical results is obtained using a mass matrix diagonalization. The numerical results obtained by this finite element correlated well with other published results.

An improved treatment of mixed interpolation functions in eight-node assumed natural strain shell element for vibration analysis

In this article, we investigate the vibration analysis of plates and shells, using an eight-node shell element that allows for the effects of transverse shear deformation and rotary inertia. The natural frequencies of plates and shells are presented, and the forced vibration analysis of plates and shells subjected to arbitrary loading is carried out. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To improve the eight-node shell element for free and forced vibration analysis, a new combination of sampling points for assumed natural strain method was applied. The refined first-order shear deformation theory based on Reissner–Mindlin theory, which directly addresses the transverse shear deformation without a shear correction factor, is adopted for the development of a new eight-node assumed strain shell element with rotary inertia effect. In order to validate the finite element numerical solutions, the reference solutions of plates based on the first-ord...

New nine-node Lagrangian quadrilateral plate element based on Mindlin-Reissner theory using IFM

This paper presents a new nine-node Lagrangian quadrilateral plate bending element (MQP9) using the Integrated Force Method (IFM) for the analysis of thin and moderately thick plate bending problems. Three degrees of freedom: transverse displacement w and two rotations theta(x) and theta(y) are considered at each node of the element. The Mindlin-Reissner theory has been employed in the formulation which accounts the effect of shear deformation. Many standard plate bending benchmark problems have been analyzed using the new element MQP9 for various grid sizes via Integrated Force Method to estimate defections and bending moments. These results of the new element MQP9 are compared with those of similar displacement-based plate bending elements available in the literature. The results are also compared with exact solutions. It is observed that the presented new element MQP9 is free from shear locking and produced, in general, excellent results in all plate bending benchmark problems considered.

Homogenization of thick periodic plates: Application of the Bending-Gradient plate theory to a folded core sandwich panel

In a previous paper from the authors, the bounds from Kelsey et al. (1958) were applied to a sandwich panel including a folded core in order to estimate its shear forces stiffness (Lebée and Sab, 2010b). The main outcome was the large discrepancy of the bounds. Recently, Lebée and Sab (2011a) suggested a new plate theory for thick plates – the Bending-Gradient plate theory – which is the extension to heterogeneous plates of the well-known Reissner–Mindlin theory. In the present work, we provide the Bending-Gradient homogenization scheme and apply it to a sandwich panel including the chevron pattern. It turns out that the shear forces stiffness of the sandwich panel is strongly influenced by a skin distortion phenomenon which cannot be neglected in conventional design. Detailed analysis of this effect is provided.

Closed form solutions for element equilibrium and flexibility matrices of eight node rectangular plate bending element using integrated force method

Closed form solutions for equilibrium and flexibility matrices of the Mindlin-Reissner theory based eight-node rectangular plate bending element (MRP8) using Integrated Force Method (IFM) are presented in this paper. Though these closed form solutions of equilibrium and flexibility matrices are applicable to plate bending problems with square/rectangular boundaries, they reduce the computational time significantly and give more exact solutions. Presented closed form solutions are validated by solving large number of standard square/rectangular plate bending benchmark problems for deflections and moments and the results are compared with those of similar displacement-based eight-node quadrilateral plate bending elements available in the literature. The results are also compared with the exact solutions.

Parametric study on supersonic flutter of angle-ply laminated plates using shear deformable finite element method

The influence of fiber orientation, flow yaw angle and length-to-thickness ratio on flutter characteristics of angle-ply laminated plates in supersonic flow is studied by finite element approach. The structural model is established using the Reissner-Mindlin theory in which the transverse shear deformation is considered. The aerodynamic pressure is evaluated by the quasi-steady first-order piston theory. The equations of motion are formulated based on the principle of virtual work. With the harmonic motion assumption, the flutter boundary is determined by solving a series of complex eigenvalue problems. Numerical study shows that (1) The flutter dynamic pressure and the coalescence of flutter modes depend on fiber orientation, flow yaw angle and length-to-thickness ratio; (2) The laminated plate with all fibers aligned with the flow direction gives the highest flutter dynamic pressure, but a slight yawing of the flow from the fiber orientation results in a sharp decrease of the flutter dynamic pressure; (3) The angle-ply laminated plate with fiber orientation angle equal to flow yaw angle gives high flutter dynamic pressure, but not the maximum flutter dynamic pressure; (4) With the decrease of length-to-thickness ratio, an adverse effect due to mode transition on the flutter dynamic pressure is found.

FRACTURE OF THIN PIPES WITH SPH SHELL FORMULATION

We study the dynamic fracture of thin-walled structure mainly due to impact and explosive loading. Therefore, we make use of a meshless smoothed particle hydrodynamics (SPH) shell formulation based on Mindlin–Reissner's theory. The formulation is an extension of the continuum-corrected and stabilized SPH method, so that thin structure can be modeled using only one particle characterizing mean position of shell surface. Fracture is based on separation of particles. We study tearing of pre-notched plates, fracture due to impact loading and dynamic fracture of cylindrical shells.

A Bending-Gradient model for thick plates, Part II: Closed-form solutions for cylindrical bending of laminates

In the first part ( Lebée and Sab, 2010a) of this two-part paper we have presented a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called Bending-Gradient plate theory is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case when the plate is homogeneous. Moreover, we demonstrated that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In this paper, the Bending-Gradient theory is applied to laminated plates and its predictions are compared to those of Reissner–Mindlin theory and to full 3D ( Pagano, 1969) exact solutions. The main conclusion is that the Bending-Gradient gives good predictions of deflection, shear stress distributions and in-plane displacement distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/ h goes to infinity.

A Bending-Gradient model for thick plates. Part I: Theory

This is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In part two ( Lebée and Sab, submitted for publication), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner–Mindlin theory and to full 3D Pagano’s exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/ h goes to infinity.

FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations

In this paper, the generalized differential quadrature (GDQ) method is applied to study the dynamic behavior of functionally graded materials (FGMs) and laminated doubly curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner–Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Firstly, the differential quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner–Mindlin and the Toorani–Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A two-constituent functionally graded lamina consists of ceramic and metal those are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of shell and panel structures considered.

Efficient analysis of laminated composite and sandwich plates with interfacial imperfections

An enhanced first-order shear deformation theory is developed for laminated composite plates that have interfacial imperfections. The proposed methodology is as follows: First, higher-order zigzag displacements are introduced to account for the interfacial imperfections, which are modeled as spring-layers. This makes it possible to describe such imperfections as a five-degree-of-freedom problem, which makes the proposed theory efficient yet accurate. Second, the relationship between the derived displacements and the Reissner–Mindlin displacements are systematically established by using the least square approximation, which renders a so-called ‘effective shear modulus’ via strain energy transformation. Finally, the governing equations based on the transformed strain energy, which are the same as the Reissner–Mindlin equations except in the case of the transverse shear modulus, are solved either analytically or numerically. Once the governing equations are solved, the displacements of the Reissner–Mindlin-like theory can be improved by using the derived relationships. In other words, the higher-order in-plane zigzag displacements are recovered in terms of the obtained solutions. One advantage of the proposed theory is that it has the same number of degrees of freedom as the Reissner–Mindlin theory, which implies that any finite elements developed for such a theory can be used without any modifications. The accuracy of the present theory is assessed by comparing its analytical solutions with available data from the literature. The effects of interfacial imperfections on stresses and shear correction factors are also explored via numerical examples.

2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution

In this paper, the Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behaviour of laminated composite doubly-curved shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to analyse the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner–Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships . The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The discretization of the system by means of the Differential Quadrature (DQ) technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner–Mindlin and Toorani–Lakis theory are presented. Furthermore, GDQ results are compared with those presented in literature and the ones obtained by using commercial programs such as Abaqus , Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed.

A finite element formulation for piezoelectric shell structures considering geometrical and material non‐linearities

AbstractIn this paper, we present a non‐linear finite element formulation for piezoelectric shell structures. Based on a mixed multi‐field variational formulation, an electro‐mechanical coupled shell element is developed considering geometrically and materially non‐linear behavior of ferroelectric ceramics. The mixed formulation includes the independent fields of displacements, electric potential, strains, electric field, stresses, and dielectric displacements. Besides the mechanical degrees of freedom, the shell counts only one electrical degree of freedom. This is the difference in the electric potential in the thickness direction of the shell. Incorporating non‐linear kinematic assumptions, structures with large deformations and stability problems can be analyzed. According to a Reissner–Mindlin theory, the shell element accounts for constant transversal shear strains. The formulation incorporates a three‐dimensional transversal isotropic material law, thus the kinematic in the thickness direction of the shell is considered. The normal zero stress condition and the normal zero dielectric displacement condition of shells are enforced by the independent resultant stress and the resultant dielectric displacement fields. Accounting for material non‐linearities, the ferroelectric hysteresis phenomena are considered using the Preisach model. As a special aspect, the formulation includes temperature‐dependent effects and thus the change of the piezoelectric material parameters due to the temperature. This enables the element to describe temperature‐dependent hysteresis curves. Copyright © 2011 John Wiley & Sons, Ltd.

A geometrically exact formulation for thin multi-layered laminated composite plates: Theory and experiment

Abstract A geometrically exact approach is employed to formulate the equations of motion of thin multi-layered isotropic and laminated composite plates subject to excitations that cause large strains, displacements, and rotations. The linearization of the obtained semi-intrinsic theory leads to the Mindlin–Reissner theory while an ad hoc truncated kinematic approximation delivers, as a by-product, the Foppl–von Karman theory of plates. An experimental validation is sought for fully clamped plates which are either of the isotropic single-layered type or of the multi-layered laminated composite type. To this end, nonlinear equilibrium paths are constructed both theoretically and experimentally when the plates are subject to a quasi-statically increasing central point load. The comparisons between the experimentally obtained results and those furnished by the geometrically exact theory as well as by the Foppl–von Karman (FVK) theory show the high accuracy of the proposed nonlinear theory while the FVK theory becomes increasingly inaccurate at deflection amplitudes of the order of the plates thickness.

Guidelines and Recommendations to Construct Theories for Metallic and Composite Plates

This work has evaluated the refinement of some classical theories, such as the Kirchhoff and Reissner-Mindlin theories, adding generalized displacement variables (up to fourth-order) to the Taylor-type expansion in the thickness plate direction. Isotropic, orthotropic, and laminated plates have been analyzed, varying the thickness ratio, orthotropic ratio, and stacking sequence of the layout. Higher-order theories have been implemented according to the compact scheme known as the Carrera unified formulation. The results have been restricted to simply-supported orthotropic plates subjected to harmonic distributions of transverse pressure for which closed-form solutions are available. For a given plate problem (isotropic, orthotropic, or laminated), the effectiveness of each employed generalized displacement variable has been established comparing the error obtained accounting for and removing the variable in the plate governing equations. A number of theories have therefore been constructed imposing a given error with respect to the available best results. Guidelines and recommendations that are focused on the proper selection of the displacement variables that have to be retained in refined plate theories are then furnished. It has been found that the terms that have to be used according to a given error vary from problem to problem, but they also vary when the variable that has to be evaluated (displacement, stress components) is changed. Diagrams (errors in terms of geometrical and orthotopic ratios) and graphical schemes have been built to establish the appropriate theories with respect to the data of the problem under consideration.