Reissner Mindlin Research Articles

Novel quadrilateral strain-based finite element for static, free vibration, and buckling analysis of plates

This paper focuses on the static analysis, free vibration, and buckling validation of Reissner–Mindlin plates using a newly developed quadrilateral finite element, termed FNSBP, which is based on the strain approach. The proposed element is equipped with three essential external degrees of freedom at each of its four corner nodes. The displacement field is formulated using assumed strain functions that inherently satisfy the compatibility equations. This new element was developed to improve upon the previously introduced SBRP strain-based rectangular plate element. The effectiveness and robustness of the FNSBP element are evaluated through various numerical examples involving both thick and thin plates of different geometries. Its performance is benchmarked against displacement-based elements from the literature and analytical solutions, demonstrating superior accuracy and convergence in many cases. The results indicate that the FNSBP element provides significant improvements in computational efficiency and accuracy, especially for complex plate configurations under different loading and boundary conditions, validating its applicability in structural analysis.

Enhancing performance of sandwich panel with three-dimensional orthogonal accordion cores

This study introduces a novel three-dimensional orthogonal accordion structure (3D-OAS) as the cellular core of sandwich panels, achieving multi-directional zero Poisson’s ratio through the orthogonal combination of two-dimensional accordion structures. To analyze its static characteristics effectively, a two-dimensional equivalent Reissner–Mindlin model (2D-ERM) was established utilizing the variational asymptotic method (VAM). The accuracy of 2D-ERM was confirmed by conducting three-point bending tests on 3D-printed specimens and analyzing the in-plane and out-of-plane deformation results of the 3D finite element model (3D-FEM). The comparison of global displacement contours and path-displacement curves between 3D-FEM and 2D-ERM showed a high level of agreement in predicting static deformation. The equivalent stiffness of SP-3D-OAS steadily increased as the inclined angle deviates by 90-degree, irrespective of whether it pertains to the convex or concave angle. Evaluation of deformability in sandwich panels with different cellular core forms revealed superior comprehensive performance in 3D-OAS, followed by 3D-YRS and 3D-XYAS, with a reduction of 16.41% and 17.35% in specific stiffness, respectively. Compared to the 3D-FEM, 2D-ERM significantly reduces computation time without compromising engineering accuracy. The research results provide a useful reference for optimal design of sandwich panels with multi-directional ZPR cellular core.

The Immersed Boundary Conformal Method for Kirchhoff–Love and Reissner–Mindlin shells

This work utilizes the Immersed Boundary Conformal Method (IBCM) to analyze linear elastic Kirchhoff–Love and Reissner–Mindlin shell structures within an immersed domain framework. Immersed boundary methods involve embedding complex geometries within a background grid, which allows for great flexibility in modeling intricate shapes and features despite the simplicity of the approach. The IBCM method introduces additional layers conformal to the boundaries, allowing for the strong imposition of Dirichlet boundary conditions and facilitating local refinement. In this study, the construction of boundary layers is combined with high-degree spline-based approximation spaces to further increase efficiency. The Nitsche method, employing non-symmetric average operators, is used to couple the boundary layers with the inner patch, while stabilizing the formulation with minimal penalty parameters. High-order quadrature rules are applied for integration over cut elements and patch interfaces. Numerical experiments demonstrate the efficiency and accuracy of the proposed formulation, highlighting its potential for complex shell structures modeled through Kirchhoff–Love and Reissner–Mindlin theories. These tests include the generation of conformal interfaces, the coupling of Kirchhoff–Love and Reissner–Mindlin theories, and the simulation of a cylindrical shell with a through-the-thickness crack.

Forced vibration characteristics of viscoelastic variable stiffness laminated composite plates using time and frequency domain approaches

The linear forced vibration characteristics of viscoelastic variable stiffness laminated composite plates (VVSLC) are studied using a generalized Maxwell model and finite element method. The integral form of the viscoelastic constitutive relation is converted to the incremental form for finite element formulation based on the Reissner–Mindlin plate theory. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response directly in the time domain is obtained using shooting technique coupled with Newmark’s time integration method. The implementation of shooting technique for Boltzmann integral-based viscoelasticity for curvilinear fibre composite plates is done for the first time in this study. For the comparison purpose, the response/resonance frequency/modal loss factor is also obtained using equivalent complex modulus based viscoelastic correspondence principle. It is observed that the variation in fibre orientation and boundary conditions leads to significant variations in response, stress/moment resultant amplitude and the damping factor of the VVSLC plates. Further, the present time domain based approach is capable of predicting damping factor at all forcing frequencies whereas complex eigenvalue analysis can predict damping factor only at discrete resonance frequency. Based on the detailed studies, it is found that the curvilinear fibre composite plates depict a significant reduction in response/moment resultants compared to straight fibre composite plates.

Numerical Analysis of the Cylindrical Shell Pipe with Preformed Holes Subjected to a Compressive Load Using Non-Uniform Rational B-Splines and T-Splines for an Isogeometric Analysis Approach

In this paper, we implement the finite detail technique primarily based on T-Splines for approximating solutions to the linear elasticity equations in the connected and bounded Lipschitz domain. Both theoretical and numerical analyses of the Dirichlet and Neumann boundary problems are presented. The Reissner–Mindlin (RM) hypothesis is considered for the investigation of the mechanical performance of a 3D cylindrical shell pipe without and with preformed hole problems under concentrated and compression loading in the linear elastic behavior for trimmed and untrimmed surfaces in structural engineering problems. Bézier extraction from T-Splines is integrated for an isogeometric analysis (IGA) approach. The numerical results obtained, particularly for the displacement and von Mises stress, are compared with and validated against the literature results, particularly with those for Non-Uniform Rational B-Spline (NURBS) IGA and the finite element method (FEM) Abaqus methods. The obtained results show that the computation time of the IGA based on the T-Spline method is shorter than that of the IGA NURBS and FEM Abaqus/CAE (computer-aided engineering) methods. Furthermore, the highlighted results confirm that the IGA approach based on the T-Spline method shows more success than numerical reference methods. We observed that the NURBS IGA method is very limited for studying trimmed surfaces. The T-Spline method shows its power and capability in computing trimmed and untrimmed surfaces.

Density-Based Isogeometric Topology Optimization of Shell Structures

Shell structures with high stiffness-to-weight ratios are desirable in various engineering applications. Topology optimization serves as a popular and effective tool for generating optimal shell structures. The solid isotropic material with penalization (SIMP) method is often chosen because of its simplicity and convenience. However, SIMP method is typically integrated with conventional Finite Element Analysis (FEA) which has limitations in computational accuracy. Achieving high accuracy with FEA necessitates a substantial number of elements, leading to computational burdens. In addition, the discrete representation of the material distribution function may result in rough boundaries. Owing to these limitations, this paper proposes an Isogeometric Analysis (IGA) based SIMP method for optimizing the topology of shell structures based on Reissner–Mindlin theory. This method uses Non-Uniform Rational B-Splines (NURBS) to represent both the shell structure and the material distribution function with the same basis functions, allowing for higher accuracy and smoother boundaries. The optimization model takes compliance as the objective function with a volume fraction constraint and the coefficients of the density function as design variables, resulting in an optimized shell structure defined by the material distribution function. To obtain fairing boundaries of the holes in the optimized shell structure, further process is conducted by fitting the boundaries with fair B-spline curves automatically. Furthermore, the proposed IGA-SIMP framework is applied to generate porous shell structures by imposing different local volume fraction constraints. Numerical examples are provided to demonstrate the feasibility and efficiency of the IGA-SIMP method, showing that it outperforms the FEA-SIMP method and produces smoother boundaries.

Topology optimization of curved thick shells using level set method and non-conforming multi-patch isogeometric analysis

We present a novel framework for topological shape optimization of curved non-conforming multi-patch and trimmed thick-shells subjected to external loads. Our method integrates the level set method (LSM) with a diffuse interface, a Hadamard shape derivative, and multi-patch isogeometric analysis (IGA) into a gradient descent algorithm to systematically capture the evolution of the shape. This integration enables us to directly manipulate CAD-compatible geometries and analysis techniques and to obtain the results as a CAD surface. The novelty lies in the utilization of multi-patch IGA models based on NURBS functions, which allows us to simultaneously maximize the stiffness and minimize the volume of the shell by searching for an optimal material distribution within its middle surface. The material is modeled under a small strain assumption in linear elasticity using a Reissner–Mindlin kinematic shell model in plane stress. The effectiveness of our approach is demonstrated on several curved conforming and non-conforming multi-patch geometries in 3D.

A three-dimensional meshless fluid–shell interaction framework based on smoothed particle hydrodynamics coupled with semi-meshless thin shell

Meshless methods are suitable for fluid–structure interaction simulations due to its Lagrangian feature and capability of handling large deformations. A three-dimensional meshless framework for the fluid–structure interaction simulation with shell structures is proposed in this study. The weakly compressible smoothed particle hydrodynamics is deployed in the fluid domain, where the boundary integral terms are incorporated to handle the one-layer shell boundary. On the other hand, a semi-meshless finite volume modeling method based on the Reissner–Mindlin shell is deployed in the one-layer shell domain. A one-way coupling method is employed for the interactive forcing between the fluid and shell particles near the interface, where the boundary integral method is employed to approximate the gradient and Laplacian operators in the fluid governing equations. Regarding the no-slip boundary, the original boundary integral method always leads to the artificial slipping phenomenon between the fluid and shell interface. To this end, a velocity penalty term with a relaxation factor is proposed to enforce the no-slip boundary condition, which substantially improves the accuracy of the original boundary integral method. In order to reduce computational costs, a multi-resolution strategy for the meshless framework is employed. A set of challenging cases with low and high Reynolds numbers are simulated by the present framework, and good accuracy is observed with the velocity penalty term in these cases. Overall, the new framework shows a satisfactory performance.

Bending, free vibration and buckling finite element analysis of porous functionally graded plates with various porosity distributions using an improved FSDT

Functionally graded materials (FGMs) are advanced composite materials with spatially varying properties, and their porosity distribution further enhances their complexity. The distribution pattern of porosity within a porous material plays a crucial role in determining the mechanical response of these structures. Therefore, the main objective of this study is to analyze the bending, free vibration, and buckling characteristics of porous FG plates by considering different porosity distributions and their effects on the overall behavior. To achieve this goal, a new finite element model is developed in the framework of an improved first-order shear deformation theory (IFSDT). In contrast to the conventional Mindlin–Reissner theory, the present IFSDT incorporates an improved mathematical formulation and provides a more realistic parabolic depiction of shear strain throughout the plate’s thickness without using any shear correction factors. In the present study, five types of porosity distribution functions are considered for the analysis. The material characteristics of the FGM porous plate change gradually in the thickness direction based on a power-law function. The governing equations are derived here using Hamilton’s principle, and a finite element method is employed for numerical analysis. Comparative analyses with previously published literature underscore the precision and simplicity of our developed finite element model. Moreover, the effects of various types of loads, porosity parameters, power-law index, side-to-thickness ratio, aspect ratio, porosity distributions and boundary conditions on the deflections, natural frequencies, and critical buckling loads are thoroughly analyzed in detail. Finally, the findings of this research contribute to the understanding of the mechanical behavior of FGMs and pave the way for designing and optimizing novel porous functionally graded structures.

Isogeometric method for buckling prediction and post-buckling analysis of variable stiffness composite underwater pressure shell

Composite cylindrical pressure hulls are thin-walled structures widely used for autonomous underwater vehicles. Buckling failure is one of the most important failure modes for these shells under external pressure. In existing buckling studies of cylindrical pressure hulls, FEM is the most popular analysis method but not efficient enough when dealing with structures with complex material distributions such as the variable stiffness composite shells. Motivated by this, an isogeometric method for buckling prediction and post-buckling analysis of variable stiffness composite underwater pressure shell is proposed in this article. In this method, based on Reissner–Mindlin shell formulas undergoing large deformation, a buckling analysis framework in IGA forms is established. Then a modified arc-length method is proposed based on kinematics used in the shell formulas, and is used to overcome the inaccuracy caused by the 2 degrees of freedom node rotation in the prediction step of arc-length method. In addition, the influence of bifurcation is considered, which may seriously affect the precision of buckling behavior simulation when the limit point is close to a bifurcation point. The computational accuracy of this framework has been validated in a series of constant stiffness composite shell cases involving buckling load prediction and post-buckling behavior simulation. For variable stiffness composite hull cases, this framework achieves the same precision as FEM with fewer elements. Therefore, the proposed framework provides an efficient way for the buckling design of underwater variable stiffness composite pressure hulls.

Isogeometric Topology Optimization of Multi-patch Shell Structures

Shell structures refer to structural elements that derive strength and load-bearing capacity from their thin and curved geometry. In practical applications, shell structures are commonly composed of multiple patches to represent intricate and diverse architectural configurations faithfully. Nevertheless, the design of multi-patch shell structures holds considerable promise. However, most of the previous work is devoted to the numerical analysis of multi-patch shell structures without further optimization design. The work proposes an inverse design framework, specifically focusing on multi-patch configurations based on Reissner–Mindlin theory. First, reparameterization and global refinement operations are employed on the provided multi-patch shell structures. Renumbering the indices of control points with shared degrees of freedom at the interface naturally ensures C0-continuity between patches. Subsequently, this study investigates the amalgamation of Isogeometric Analysis (IGA) and the Solid Isotropic Material with Penalization (SIMP) method for topology optimization of shell structures. The proposed approach is validated through numerical examples, emphasizing its capacity to enhance multi-patch shell structure design, showcasing robustness and efficiency.

A meshless computational framework for a modified dynamic system of vehicle coupled with plate structure

In previous simulations of train–bridge interaction systems (TBIS), the supporting system for the train are commonly treated as beam structures, leading to less accurate results, particularly for small-span cases. To address this limitation, a modified vertical TBIS is proposed. In the presented TBIS, the supporting system is considered as a Reissner–Mindlin plate, and the displacement field is described by first-order shear deformation theory (FSDT). To establish the model, radial point interpolation method (RPIM), a meshless method, is employed. Finally, a coupled dynamic equation is established to calculate various responses of the system. Several numerical examples are presented to illustrate the disparities between the system based on plate model and traditional beam model. The results indicate that the beam model yields higher estimates of the mid-span vertical displacement of the bridge, while the peak of the mid-span vertical acceleration is smaller compared to the plate model; additionally, it is observed that the carbody is primarily influenced by rail irregularities. Consequently, the proposed plate model offers distinct advantages over the beam model in providing comprehensive structural response information, thereby offering novel insights into bridge design and analysis. Additionally, this marks the inaugural application of the meshless method in the field of TBIS, which further extends the application scope of meshless methods.

Isogeometric Shape Optimization of Reissner–Mindlin Shell with Analytical Sensitivity and Application to Cellular Sandwich Structures

Structural shape optimization plays a significant role in structural design, as it can find an appropriate layout and shape to improve structural performance. Isogeometric analysis provides a promising framework for structural shape optimization, unifying the design model and analysis model in the optimization process. This paper presents an adjoint-based analytical sensitivity for isogeometric shape optimization of Reissner–Mindlin shell structures. The shell structures are modeled by multiple NURBS surfaces and design variables are associated with the position of control points. A multilevel approach is performed with a coarse mesh for the design model and a dense mesh for the analysis model. The sensitivity propagation is achieved through a transformation matrix between the design and analysis models. Structural compliance minimization problems with and without constraints are studied and the optimization history shows that the optimization can converge quickly within fewer iterations. The developed formulations are validated through several numerical examples and applied to the optimization of cellular sandwich structures, which are widely used in engineering applications. Numerical results show that optimized sandwich panels can achieve better performance in bending resistance.

Simultaneous analysis of continuously embedded Reissner–Mindlin shells in 3D bulk domains

AbstractA mechanical model and numerical method for the simultaneous analysis of Reissner–Mindlin shells with geometries implied by a continuous set of level sets (isosurfaces) over some three‐dimensional bulk domain is presented. A three‐dimensional mesh in the bulk domain is used in a tailored FEM formulation where the elements are by no means conforming to the level sets representing the shape of the individual shells. However, the shell geometries are bounded by the intersection curves of the level sets with the boundary of the bulk domain so that the boundaries are meshed conformingly. This results in a method which was coined Bulk Trace FEM before. The simultaneously considered, continuously embedded shells may be useful in the structural design process or for the continuous reinforcement of bulk domains. Numerical results confirm higher‐order convergence rates.

Governing equations of plate theories by direct manipulation of three-dimensional elasticity equilibrium

– This article shows that the governing equations of any plate theory can be directly obtained by manipulating the three-dimensional (3D) equilibrium indefinite equations. Stress components are transformed into plate stress resultants by implementing a few steps proposed in this article. Such a manipulation is shown for simple and complex deformation states. Various known plate theories are considered, such as membrane and bending ones. The most general equations are obtained by referring to higher-order theories based on the Carrera Unified Formulation. To show the method’s effectiveness, the differential equations of known (Reissner–Mindlin, Hildelbrand–Reissner–Thomas) and unprecedented plate theories are derived by simply working on the 3D-equilibrium equations. The introduced manipulation consists of the following steps: (1) The 3D equilibrium elasticity is formally written for each of the degrees of freedom of the considered plate theories; (2) the stress resultants are used to replace stress components in the latter equations; (3) the derivatives of the stress components along the plate reference surface coordinate x, y do not affect the nature of the indefinite equilibrium equations, and these derivatives are directly moved to the related stress resultants; (4) the derivatives over the thickness coordinate z should: (i) change the sign of the related stress resultants and (ii) be applied to the base functions of z used to define the stress resultants (as well as the assumption for the displacement along the thickness). In conclusion, this article presents a straightforward method that can effectively replace the use of Newton and Lagrange methods to derive the equilibrium equations of any plate theory. The four steps introduced in this article demonstrate the simplicity and efficiency of this approach.

Pseudo-equivalent model for sandwich panels with egg-shaped honeycomb-grid core

The novel sandwich panel with egg-shaped honeycomb-grid core (SP-EHC), formed by chamfering the grid walls, can achieve a balance between lightweight and high-strength, as well as promote effective ventilation and drainage due to the semi-closed honeycomb design. To determine the panel’s equivalent stiffness properties, an asymptotic analysis of the energy functional stored within the periodic unit cell is conducted. On this base, a 2D pseudo-equivalent model (2D-PEM) in the form of the Reissner–Mindlin model was formulated. The pseudo-excitation method was employed to analyze the self-power spectral density function and root mean square response of the 2D-PEM under random vibration. The accuracy and effectiveness of the model were validated through comparisons of the three-point bending experimental results of the 3D-printed specimen, exhibiting relative errors of 2.95%. In addition, free and random vibration analyses were conducted on the hybrid SP-EHC to further validate the model. Furthermore, the influence of material and structural parameters on specific stiffness, natural frequencies and first velocity RMS was systematically investigated. The findings reveal that enhancing the grid wall’s chamfering ratio is beneficial in achieving the desired objectives of lightweight and high-strength. In addition, compared to sandwich panels with orthogonal and pyramid grid cores, this improvement positively impacts the vibration characteristics of the sandwich panel. These findings offer valuable insights for the future design and optimization of such sandwich panels.

Correction: A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition

Correction: A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition

Compensating thermal bending of the runner disk in hydrodynamic thrust bearings: Simple approach for passively improving the performance of gas bearings

This paper presents a method for passively compensating thermal bending distortions of the thrust disk in gas thrust bearings by passive approaches. Thermal bending displays a prominent problem in hydrodynamic thrust bearings, especially in connection with air bearings. Axial temperature gradients cause the rotor disk to dish which in turn diminishes the lubricating gap topology and reduces the load capacity of the bearing. Consequently, the bearing performance is decreased and the bearing may even fail. The here proposed straightforward solutions are small changes in the rotor and runner disk design and are therefore rather simple and inexpensive to adopt. They comprise asymmetric mass distributions, recesses, disk cuts, and dual-material rotor disks. In order to show the benefit for the thrust bearing performance, detailed multiphysical numerical simulations of a representative rotor assembly with a foil thrust bearing are presented. The fully coupled finite element model calculates the deformations of the foil package (2D Reissner–Mindlin type shell equations), lubricating air film pressure (2D compressible Reynolds equation), air film temperatures (3D energy equation), and thermo-elastic deformations of the rotor and rotor disk (2D Navier-Lamé equations including centrifugal forces and thermal stresses). When the proposed design changes are applied, the model predicts reduced thermal bending, marked improvements in load capacity as well as decreased temperatures. The load capacity increase ranges up to 43% for an optimized design when compared against a standard symmetric design. Also, the sensitivity of different design parameters is discussed in detail. The suggested disk design changes may be applied to a broad variety of rotor-bearing systems. However, the design has to be tailored to the specific and individual machine operating conditions.

Phase‐field model for ductile fracture in the stress resultant geometrically exact shell

SummaryWe present a phase‐field fracture model for a stress resultant geometrically exact shell in finite deformation regime where the configuration manifold evolves according to deformation and fracture. The Reissner–Mindlin shell problem is first solved via the finite element method, where the independent unknown fields are the displacement and director. The phase‐field ductile fracture model is then coupled with the verified geometrically exact shell by enriching the three‐field elasto‐plastic free energy with a regularized crack surface energy. To capture the geometrical nonlinearity due to the large plastic deformation exhibited in the processing zone, the corresponding finite‐strain stress resultant elasto‐plasticity model is coupled with the phase field model. This coupling between the stress resultant elasto‐plasticity model and the phase‐field fracture model enables us to predict the different amounts of energy dissipation attributed to plastic deformation and fracture and hence simulate the crack propagation in the ductile regime properly. We introduce a mixed finite element model with displacement, director, and phase‐field order parameters as the nodal degree of freedoms formulated on the mid‐surface. An energy‐based arc‐length method is generalized to track the equilibrium path and mitigate instabilities arising from material nonlinearity. Four numerical examples are presented to validate the implementation of the model and demonstrate its capability to simulate ductile fracture in shell structures. These examples include a plane‐stress tension/shear fracture model, the Muscat‐Fenech and Atkins plate, axial tension of a notched cylindrical shell, and ductile fracture of a simply supported plate under uniform pressure.

A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition

A curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems.