Static Bending Analysis Research Articles

An improved first-order strain-based finite element model with in-plane drilling rotation for static bending and free vibration analysis of functionally graded plates

This work aims to develop a novel assumed strain finite element model based on an improved first-order shear deformation theory (IFSDT) for static and free vibration analysis of functionally graded (FG) plates. This IFSDT contains only five unknowns, and its distribution of the shear strain functions is quadratic across the plate thickness, satisfying the stress-free boundary conditions at the plate’s upper and lower surfaces without any shear correction coefficients commonly used in the standard first-order shear deformation theory. In the framework of the strain approach and using the IFSDT, a four-node quadrilateral plate element containing six degrees of freedom per node is formulated. This new element does not require any shear-locking treatment and its displacement fields are assumed using polynomial functions with 24 coefficients. A simple power law distribution is used to represent the FG plate properties which are considered to vary continuously through thickness. The membrane-bending coupling due to the plate heterogeneity across the thickness is avoided by introducing the concept of the neutral plane. Numerical results on the deflections, stresses, and frequencies of circular and rectangular FG plates with different types of boundary conditions and for a wide range of materials are examined to assess the accuracy of the present model. An excellent efficiency in the analysis of FG plates is demonstrated from the results obtained of the suggested element which is free from any shear locking.

Manufacturing Technology, Experimental and Numerical Analysis of Static Bending of Three-Layer Composite Plate with Honeycomb Structure

The use of thin-walled constructions with honeycomb structure manufactured using additive technologies has several advantages compared to the constructions manufactured using traditional technologies. First of all, this is explained by the fact that additive technologies greatly simplify the production of honeycomb structures. A technology for manufacturing a three-layer composite plate with a honeycomb structure obtained using additive technologies is proposed in the paper. PLA plastic was chosen as the material for the honeycomb structure production. Printing was carried out using a "Delta" printer with parallel kinematic chains using FDM technologies. The printing temperature was 215 °C, table temperature – 60 °C. A fundamentally new scheme of an experimental plant for studying the bending of a three-layer plate is proposed. The aim of the research is to conduct tests of samples of a three-layer honeycomb panel for static bending when one edge of the sample is rigidly pinched. For the tests, a certified TiraTest 2300 universal testing machine (UTM), which allows to perform tensile and compression tests at a given traverse speed and measure the load with a relative error of 1%, was used. The UTM allows to load the sample and measure the load and displacement of the traverse. A sixteen-channel strain gauge station is used to measure the surface deformations of the housings. Transverse displacements of a three-layer plate were obtained as experimental data. The bending of the three-layer plate is modeled in the commercial package ANSYS. The results of experimental and numerical analysis coincide well.

Deriving Ritz formulation for the Static Flexural Solutions of Sinusoidal Shear Deformable Beams

This paper presents the Ritz variational method for the static bending analysis of sinusoidal shear deformable beams. The theory accounts for transverse shear deformation and satisfies transverse shear stress-free conditions at the top and bottom surfaces of the beam. The total potential energy functional for the thick beam bending problem is formulated and minimized using a Ritz procedure. The problem considered simply supported boundary conditions and two cases of loading – uniformly distributed load over the span and point load at the center. The function is a function of two unknown displacement functions constructed in terms of unknown generalized displacement parameters and shape functions that satisfy the boundary conditions. Ritz minimization of the functional is used to find the generalized displacement parameters and then the displacements w(x) and The displacements and stresses are found for the loading distributions considered. It is found that the results obtained agree remarkably well with the exact results obtained using the theory of elasticity. The differences between the present results and the exact solutions are less than 0.3% for maximum transverse displacement for both cases of loading considered. For uniform load over the beam, the result for l/t = 10 is 0.112% greater than the exact solution. For the point load at the center, the result for l/t = 10 is 4.468% greater than the exact solution. The increased difference in the point load case is due to the singular nature of the point load problem.

Static bending analysis of BDFG nanobeams by nonlocal couple stress theory and nonlocal strain gradient theory

This paper presents analytical solutions for the bending behavior of bi-directional functionally graded (BDFG) micro and nanobeams, wherein the material properties vary along both the thickness and axial directions, following power-law and exponential-law profiles, respectively. This study employs two size-dependent theories, nonlocal modified couple stress theory (NCST) and nonlocal strain gradient theory (NSGT), to account for size effects inherent in nanoscale structures. The governing differential equations are derived using Hamilton's principle, and the Laplace transform technique is utilized for their solution. The study critically compares the size effects captured by NCST and NSGT and assesses the influence of material gradation parameters in both directions. Additionally, the impacts of nonlocal and length scale parameters are thoroughly investigated. The findings indicate that NSGT tends to overestimate size effects compared to NCST. This research enhances the understanding of the mechanical behavior of BDFG nanobeams, offering valuable insights for the design and analysis of nanoscale structures across diverse applications.

Static bending analysis of pressurized cylindrical shell made of graphene origami auxetic metamaterials based on higher-order shear deformation theory

Static bending responses of a pressurized composite cylindrical shell made of a copper matrix reinforced with functionally graded graphene origami are studied in this paper. The kinematic relations are extended based on a new higher-order shear and normal deformation theory in the axisymmetric framework. The constitutive relations are extended for the composite cylindrical shell where the effective modulus of elasticity, Poisson's ratio, thermal expansion coefficient and density are estimated using the Halpin-Tsai micromechanical model and the rule of mixture. Some modified coefficients are employed for correction of the mentioned material properties in terms of the volume fraction and the folding degree of graphene origami, characteristics of copper and graphene nanoplatelets and thermal loads. The principle of virtual work is used to derive governing equations through computation of strain energy and external work. The static bending results including radial and axial displacements, circumferential strain and stress are presented along the longitudinal and radial directions in terms of volume fraction, folding degree and distribution of graphene origami. The results show an increase in radial displacement and circumferential strain with an increase in folding degree and a decrease in volume fraction of graphene origami. The main novelty of this work is investigating the effect of foldability parameter and various distribution of graphene origami on static results of short cylindrical shell.

Linear static, geometric nonlinear static and buckling analyses of sandwich composite beams based on higher-order refined zigzag theory

This study aims at developing finite element formulations based on higher-order refined zigzag theory (HRZT) to analyze sandwich composite beams under linear static bending, geometric nonlinearity and buckling. In HRZT, the higher-order terms associated with average shear strain and rotation due to zigzag are added based on refined zigzag theory (RZT). In linear static bending analysis, both C0 and C1 elements are developed. For the C0 element, an extra node is used in order to eliminate shear locking. For geometric nonlinear static analysis, the two-node C0 element is developed to solve the bending response with lateral loading and buckling response. The numerical results are compared with those calculated by 2D exact solutions, analytical solutions of HRZT and 2D models using commercial software. Various beam theories such as first-order shear deformation theory (FSDT), higher-order shear deformation theory (HSDT), RZT, and other types of higher-order zigzag theory are also introduced for comparisons. Through comprehensive comparisons with existing works on FEM or zigzag theories, the HRZT beam elements developed in this study exhibit superior accuracy and are free from shear locking in linear static analysis. Furthermore, the study also demonstrates that these elements achieve high accuracy in geometric nonlinear analysis and buckling analysis.

Finite-element modeling for static bending and free vibration analyses of double-layer non-uniform thickness FG plates taking into account sliding interactions

Finite-element modeling for static bending and free vibration analyses of double-layer non-uniform thickness FG plates taking into account sliding interactions

Static bending analysis of a transversely cracked strip tapered footing on a two-parameter soil using a new beam finite element

In this paper, a new beam Euler–Bernoulli finite element for the transverse static bending analysis of cracked slender strip tapered footings on an elastic two-parameter soil is presented. Standard Hermitian cubic interpolation functions are selected to derive the closed-form expressions of complete stiffness matrix and the load vector. The efficiency of the proposed finite element is verified on an example with several width tapering variations of a simple cracked footing with the results of governing differential equation. Another novelty of this study is improved bending moment functions with included discontinuity conditions at the crack location. These functions now accurately describe the bending moments in the vicinity of the crack of the finite element.

Static bending and stability analysis of sandwich conical shell structures with variable thickness core

In this paper, static bending and buckling behavior of the simply supported truncated sandwich conical shell with variable thickness core are examined. The stiffness of the system changes as a result of changes in the porous core’s thickness, which varies along the generator direction. Additionally, there are five various styles of porosity distribution schemes for the porous foam core that are taken into consideration. Hamilton’s principle, FSDT and Galerkin method are utilized to obtain a set of matrix equations for calculating the deflections and critical buckling loads of the system under thermal environment. The static properties and mode are investigated.

New Enriched Beam Element for Static Bending Analysis of Functionally Graded Porous Beams Resting on Elastic Foundations

New Enriched Beam Element for Static Bending Analysis of Functionally Graded Porous Beams Resting on Elastic Foundations

Shear locking free polygonal elements for the analysis of functionally graded plates using (n + 1) integration scheme and Reissner-Mindlin theory

A new shear-locking free polygonal elements is proposed for static bending and free vibration analysis of thin/thick functionally graded material plates. The domain is discretized with arbitrary polygons and Wachspress interpolants are employed for approximating the unknowns. The plate kinematics is based on Reissner-Mindlin plate theory and shear locking syndrome is suppressed by employing the novel n + 1 integration scheme based on Richardson extrapolation technique. Within this, a two stage sampling technique is introduced with the Weighted Richardson scheme to compute the system matrices. From the systematic study, it is inferred that the proposed scheme passes patch tests and yields accurate results.

Geometrically nonlinear bending mesh-free analysis of functionally graded porous sandwich beam

In this work, geometrically nonlinear static bending analysis of sandwich beams with a porous core and two skins made of functionally graded materials using a mesh-free method is presented. The material types of the core and face sheets of the beam are chosen so that the material continuity between the layers is guaranteed. The nonlinear governing equation including geometric nonlinearity is established via the principle of virtual work. This equation is discretized into a system of algebraic equations by the mesh-free approach, which is based on the C1 point interpolation method and polynomial basis functions, and then solved by the direct iterative method. The convergence of the mesh-free method is tested to determine the sufficient mesh level for the analysis. Comparative and comprehensive studies are performed to investigate both the correctness and the influences of several important parameters and boundary conditions on the linear and nonlinear deflections of the beam.

Nonlinear static analysis of functionally graded porous sandwich plates resting on Kerr foundation

The main purpose of this work is to explore the nonlinear static bending analysis of functionally graded porous sandwich plates with a ceramic core (FSCC) subjected to a transverse uniform load resting on Kerr foundation (KF) using the mixed interpolation of tensorial components for the four-node rectangular element (called MITC4). Porosity appearing in two skin layers is assumed according to the uneven porosity distribution with a porosity factor Ω . The advantage of this element is fast convergence, high accuracy and cancels the shear-locking phenomenon. The present study considers the geometrical nonlinearity resulting from mid-plane stretching based on the von Kármán geometrical nonlinearity. The first-order shear deformation theory (FSDT) is employed to approximate the displacement field of the plate due to its simplicity and efficiency. The current formulation’s authenticity is evaluated by comparing it with previously published works. For the first time, the nonlinear bending of FSCC subjected to a transverse uniform load resting on the Kerr foundation is investigated. Furthermore, the influence of input parameters on the nonlinear bending behavior of FSCC is fully provided.

High-Precision Isogeometric Static Bending Analysis of Functionally Graded Plates Using a New Quasi-3D Spectral Displacement Formulation

A new quasi-three-dimensional (3D) shear deformation theory, called the spectral displacement formulation (SDF), is proposed for high-precision static bending analyses of functionally graded plates. The main idea is to expand unknown displacement fields into Chebyshev series of a unique form in the thickness direction; the truncation numbers are set to be adjustable to meet various application requirements. Specifically, 3D elasticity solutions and traction-free boundary conditions can be approached by increasing the number of Chebyshev bases. The SDF is also an extension of the classical plate theory and naturally avoids the shear locking problem, making it versatile for functionally graded material (FGM) plates of arbitrary thicknesses. The C1 continuity requirement for the discretization of the generalized displacements is conveniently fulfilled by the nonuniform rational B-splines (NURBS)-based isogeometric method. Numerical examples demonstrate the excellent performance of the proposed method for the displacement and stress analyses of functionally graded plates. The high precision and versatility of the present method have manifested its great potential applications in strain-based or stress-based reliability analysis, optimization design, fatigue analysis, and fracture analysis of FGM plates, and other related fields.

Analysis of Berger nonlinear elastic static plate bending of rectangular plates

This paper deals with a nonlinear static plate model based on Berger theory, which is a specific case of a generalization of the Woinowsky-Krieger mathematical model of beam bending. It is considered a plate bending with forces acting in the middle plane of the plate and a contact problem with an elastic foundation, where the normal compliance condition is employed. A variational equation of the problem and a functional of the total potential energy corresponding to the variational equation are derived. Under additional assumptions on the data (e.g., clamped plate), the existence and uniqueness of the solution are proved. A numerical solution is based on the Galerkin method and Courant approximation. The theory is illustrated by a numerical example.

Large deformation analysis of a hyperplastic beam using experimental / FEM/ meshless collocation method

In the present study, nonlinear numerical and experimental static bending analyses of hyperelastic beams are surveyed. Timoshenko's beam theory is used to obtain the Cauchy-Green deformation tensor. The governing equations are extracted using the neo-Hookean strain energy and Euler–Lagrange relation. Afterward, the nonlinear governing equations are solved numerically using the meshless collocation method (MCM). To validate the present investigation, a comparison study is accomplished between the acquired results from MCM and the experimental test of digital image correlation (DIC) and finite element method (FEM). The hyperelastic beam is made of natural rubber. Also, loading and boundary conditions are considered as concentrated force and clamped–clamped, respectively. In addition, the displacement contour is extracted by the DIC. The uniaxial tensile test based on the ASTM D412 standard is performed to extract the mechanical properties of natural rubber. The hyperelastic theory and neo-Hookean strain energy function are utilized to model the nonlinear behavior of rubber in the FEM. The error for the maximum displacement of system obtained from both numerical methods is less than 12% compared to the DIC. Additionally, it can be observed that in large deformations, unlike the FEM, MCM is able to predict the system's response with high accuracy.

Analytic solutions for static bending and free vibration analysis of FG nanobeams in thermal environment

Beam, plate, and shell structures manufactured from functionally graded materials (FGMs) are becoming increasingly prevalent in practice. The manufacture of these structures is not free from flaws, one of which is that the thickness distribution of the material does not change continuously. This is the first study to apply two distinct analytical solutions to examine the static and free vibration responses of imperfect FG nanobeams in terms of material distribution and temperature influence. The calculation formulas are based on the novel shear strain theory, which, despite being simple, accounts for both bending and shear strain, rendering the calculation procedure highly convenient. Either of the two precise solutions is computed using direct integrals and can be computed for various boundary conditions, which is not possible with Navier’s solutions. The results of the verification indicate that the approaches given in the article guarantee the needed precision. This study also analyzes the effect of several geometrical, material, nonlocal, and imperfection characteristics on the static bending and free oscillation response of imperfect FG nanobeams.

Finite-Element Modeling for Static Bending Analysis of Rotating Two-Layer FGM Beams with Shear Connectors Resting on Imperfect Elastic Foundations

The static bending of rotating two-layer functionally graded material (FGM) beams with shear connectors resting on imperfect elastic foundations was performed for the first time in this study. Timoshenko beam theory and the finite-element method are used to derive finite-element formulations. The precision of the current approach and mechanical model is shown by comparing the calculated findings of this study to those of previous precise papers. A wide variety of parameter studies are carried out to capture the impact of geometric and material characteristics on the structure’s static bending behaviors, such as the distance d, rotational speed, volume fraction index, elastic foundation parameters, and boundary conditions. This paper’s theory and mechanical models are fascinating since mechanical systems involving rotational motion are pretty standard in engineering practice. Therefore, the computed results of this work aim to contribute to our understanding of structures of this kind.

3D static bending analysis of functionally graded piezoelectric microplates resting on an elastic medium subjected to electro-mechanical loads using a size-dependent Hermitian C 2 finite layer method based on the consistent couple stress theory

Based on the consistent couple stress theory (CCST), we develop a size-dependent Hermitian C 2 finite layer method (FLM) for carrying out the three-dimensional (3D) static bending analysis of a simply-supported, functionally graded (FG) piezoelectric microplate which is placed under closed-circuit surface conditions. The microplate of interest is assumed to be resting on a Winkler-Pasternak foundation and subjected to either sinusoidal or uniformly distributed electro-mechanical loads. By setting the material length scale parameter at zero and ignoring the piezoelectric and flexoelectric effects, we reduce the formulation of the Hermitian C 2 FLM for analyzing FG piezoelectric microplates to that for analyzing FG piezoelectric macroplates and FG elastic microplates, respectively. The accuracy and the convergence rate of the Hermitian C 2 FLM are assessed by comparing the solutions it produces with the exact and approximate 3D solutions of FG piezoelectric macroplates and FG elastic microplates reported in the literature. Because the Hermitian C 2 FLM requires that the first-order and second-order derivatives of the primary variables must be continuous at each nodal plane, which in turn leads to their solutions converging rapidly and being able to obtain the accurate results of the electric and elastic variables induced in the microplates, especially for the transverse shear and normal stresses and the electric displacements. The effects of piezoelectricity, flexoelectricity, and the material length scale parameter on the deformations and stresses induced in the FG piezoelectric microplates are significant. Highlights A CCST-based Hermitian C 2 finite layer method (FLM) is developed for analyzing functionally graded (FG) piezoelectric microplates. The current FLM can be reduced to that for analyzing FG elastic microplates by ignoring the piezoelectric and flexoelectric effects. The current FLM can be reduced to that for analyzing FG piezoelectric macroplates by setting the material length scale parameter at zero. Implementing the current FLM reveals that its solutions converge rapidly and are in excellent agreement with those obtained using the exact 3D models. The effects of piezoelectricity, flexoelectricity, and the material length scale parameter on the static bending behavior of FG piezoelectric microplates are significant.

A novel higher-order refined zigzag theory for static bending analysis in sandwich composite beam

Sandwich composite beams are widely used in variety fields. A typical sandwich composite beam includes a soft core covered by two stiff face layers. For such beam structures, large stiffness difference between two adjacent layers can result in shear deformations and zigzag displacement phenomenon. In this research, a novel higher-order refined zigzag theory (HRZT) is presented for solving the static bending problems of a sandwich composite beam with a soft core. The HRZT is derived based on the refined zigzag theory (RZT) by adding the higher-order zigzag terms. Unlike RZT, the kinematics assumption of HRZT can obtain a continuous shear stress distribution across the thickness. The governing equations of HRZT are derived by variational principle and the general solutions of which are derived in exact forms. The HRZT, RZT and FEM with commercial software are used to solve the static bending responses of sandwich composite beams with cantilevered and simply-supported boundary conditions. By comparing the FEM results, both the displacements and shear stresses calculated by HRZT are verified with high accuracy. In the present study, it is shown that HRZT is able to preserve the advantages of the RZT on the modelling of the zigzag displacement, but also improves the shear stress distributions that are continuous across the interface between two layers.