Bending Problems Research Articles

К ТОЧНОМУ РЕШЕНИЮ ЗАДАЧИ О РАСЧЕТЕ БАЛОЧНОЙ ПЛИТЫ И ПЛАСТИНКИ НА УПРУГОМ ОСНОВАНИИ ВИНКЛЕРА

The paper examines the problems of calculating a beam slab axisymmetrically loaded with round and rectangular plates on an elastic Winkler foundation in the traditional formulation without taking into account shear stresses and one-sided connections in the contact zone between the structure and the elastic foundation. The deflections of the structure are represented as a series of eigenfunctions of the differential equation of bending vibrations of a structure with free edges. The external load is also arranged in a series according to its own functions. Substituting these expansions into the differential equation for the deflection of a structure on the Winkler base and equating the coefficients for the same eigenfunctions, we find an expression for the deflections of the structure. Next, the forces in the structure are determined from the deflections. Solutions are given to problems of bending of a beam slab axisymmetrically loaded with round and square plates partially loaded with a uniformly distributed load.

Specified criterion for delamination upon bending of a composite beam

The interlayer strength in polymer fiber composites is characterized mostly by the strength of the matrix, which is much lower than fiber strength. For this reason, the analysis of fracture occurred through delamination is extremely important for assessing the operability of composite structural elements. When designing critical structures, it is necessary to know the interlayer shear strength, for which the method of bending a short beam has been standardized. The shear stresses and the interlayer shear strength in bending theory are traditionally assumed to be independent of the length and width of the beam. However, a large number of experimental studies prove the opposite fact that the geometry of the specimen affects the value of critical stresses. The linear fracture criterion proposed by the authors allows explanation and quantitatively description of the interlayer shear strength dependence on the geometry of the specimen. The influence of the heterogeneity of interlayer shear stresses across the beam on the critical stresses is analyzed. A strict solution of the bending problem showed that taking into account the specified shear stress distribution gives an insignificant correction to the determined value of the interlayer strength, which makes it possible to use a simplest parabolic distribution in height. The results of the analysis are confirmed in three-point bending tests of short composite beams of different widths. The results of fatigue tests of short beams made of carbon fiber reinforced plastic are analyzed. The relationship between tensile fatigue curves of polymer fiber composites and the fatigue curves obtained in cyclic three-point bending test of short beams has been revealed using the proposed linear fracture criterion. The estimation of the strength scale effect on the basis of the energy criterion of delamination with and without taking into account the refined distribution of interlayer shear stresses is presented.

The Role of Thin Plies in Traditional Composite Structures Under Different Loading Conditions

In the current paper, composites with both thin and thick plies are designed to meet specific criteria and tested under both in plane and out of plan critical loading conditions. The tests adopted are bolted joint, drop-weight impact, compression after impact, Charpy impact, bending, and bending after thermal aging. The results of the proposed design are compared with that of the traditional composites and show higher improvement in most of the load cases by using thin plies inside the lamination process with the traditional ones. The results showed that the two proposed alternatives with thin plies are of higher advantage for the bolted joints problem. On the other hand, the alternative with thin plies distributes to be surrounding each traditional ply is of high advantages for both the impact and the bending problems. The alternative with a core of thin plies at the middle of the laminate is of high advantage for the compression after impact.

Nonlinear deformation analysis of magneto-electro-elastic nanobeams resting on elastic foundation by using nonlocal modified couple stress theory

This article investigates a comprehensive analysis of the bending problem pertaining to magneto-electro-elastic (MEE) nanobeams on the Winkler-Pasternak foundation. Taken into account the influence of size effects of nano structures, a non-classical mechanical model is established to describe the bending behavior for MEE nanobeams. In this research, the nonlocal modified couple stress theory is utilized to accurately capture both the softening and hardening effects resulting from the size effect. The theory of third-order shear deformation and von Karman geometric nonlinear theory are employed in conjunction with the introduction of Maxwell's equation to develop the model of MEE nanobeam, the control equation of the nonlinear bending problem of MEE nanobeams is obtained by Minimum potential energy principle, and it is solved by Galerkin method. Finally, this study provides a detailed discussion on the influences of the ratio of nonlocal parameters and length-scale parameters, Winkler-Pasternak coefficient, external magnetic potential, external voltage and span-thickness ratio on nonlinear deflection of nanobeam.

Simulating body deformations with initial stresses using Hooke‐like isotropic hypoelasticity models based on corotational stress rates

AbstractWe use isotropic hypoelastic models based on corotational stress rates to simulate deformations of elastic bodies with initial stresses. Four material models based on different corotational stress rates are used: the Zaremba–Jaumann, Green–Naghdi, logarithmic, and Hill models. The main result of the study are new objective algorithms for integrating stresses that provide sufficiently accurate values of stresses for large time steps. In addition, a new approach to symmetrizing tangent stiffness matrices that has a clear mechanical interpretation was used in computations. All four material models were implemented in a homemade FE system for nonlinear analysis of deforming bodies. The developed algorithms were verified and validated by solving both uniform deformation problems that have exact solutions and applied problem of plate bending with non‐equilibrated initial stresses. The performance of the developed software was assessed by comparing numerical solutions obtained using this software with similar solutions obtained using the commercial MSC.Marc nonlinear FE system. Comparative analysis of the obtained solutions shows that our software is comparable in performance with one of the leading commercial software packages for solving problems of isotropic hypoelasticity with initial stresses.

An accurate and unified study on non-symmetric tensile and compressive responses of elastoplastic bars and beams until failure

A new and unified approach is proposed toward accurately analyzing overall elastoplastic responses of axially loaded bars and finite bending beams until failure. Results are presented for various constituent materials such as metals, reinforced concretes and reinforced composites, etc. Novelties in three respects are incorporated: (a) new uniaxial stress–strain functions are first presented in explicit forms for the purpose of accurately characterizing non-symmetric tensile and compressive behaviors of axially loaded bars from hardening to softening; (b) explicit solutions to both the varying neutral axis and the flexural moment of finite bending beams are then obtained directly in terms of these two uniaxial functions; and, hence, (c) the complex bending problem of various beams until failure is in a unified manner reduced to a simple issue of fitting two uniaxial functions to tension and compression data from uniaxial testing, thus bypassing the analytical and numerical complexity of various existing approaches. Numerical examples of model predictions are provided for typical constituent materials, including aluminium, meso-scopic heterogeneous concrete and ultra-high performance fiber-reinforced concrete with reinforcing rebars. Good agreement is achieved simultaneously with all experimental data for bars and beams made of these materials.

Analytical solution for the bending problem of micropolar plates based on the symplectic approach

Abstract An analytical solution for the bending problem of micropolar plates is derived based on the symplectic approach. By applying Legendre's transformation, we obtain the Hamiltonian canonical equation for the bending problem of a micropolar plate. According to the method of separation of variables, the homogenous Hamiltonian canonical equation can be transformed into an eigenvalue problem of the Hamiltonian operator matrix. We derive the eigensolutions of the eigenvalue problem for the simply-supported, free and clamped boundary conditions at the two opposite sides. Based on the adjoint symplectic orthogonal relation of the eigensolutions, the solution of the bending problem of micropolar plate is expressed as a series expansion of eigensolutions. Numerical results confirm the validity of the present approach for the bending problem of micropolar plates under various boundary conditions and demonstrate the capability of the proposed approach to capture the size-dependent behavior of micropolar plates.

Numerical modeling of nonlinear deformation processes for shells of medium thickness

When modeling a nonlinear isotropic eight-node finite element, the main kinematic and physical relationships are determined. In particular, isoparametric approximations of the geometry and an unknown displacement increment vector, covariant and contravariant components of basis vectors, metric tensors, strain tensors (Cauchy - Green and Almansi) and true Cauchy stresses in the initial and current configuration are introduced. Next, a variational equation is introduced in the stress rates in the actual configuration without taking into account body forces and the Seth material is considered, where the Almansi strain tensor is used as the finite strain tensor. Linearization of this variational equation, discretization of the obtained relations (stiffness matrix, matrix of geometric stiffness) is carried out. The resulting expressions are written as a system of linear algebraic equations. Several test cases are considered. The problem of bending a strip into a ring is presented. This problem is solved analytically, based on kinematic and physical relationships. Examples of nonlinear deformation of cylindrical and spherical shells are also shown. The method proposed in this paper for constructing a three-dimensional finite element of the nonlinear theory of elasticity, using the Seth material, makes it possible to obtain a special finite element, with which it is quite realistic to calculate the stress state of shells of medium thickness using a single-layer approximation in thickness. The obtained results of test cases demonstrate the operability of the proposed technique.

Numerical-and-Analytical Method for Solving Geometrically Nonlinear Bending Problems of Complex-Shaped Plates from Functionally Graded Materials

Numerical-and-Analytical Method for Solving Geometrically Nonlinear Bending Problems of Complex-Shaped Plates from Functionally Graded Materials

A refined shear deformation theory of an asymmetric sandwich beam with porous core: Linear bending problem

This paper is devoted to a sandwich beam with an asymmetric structure under uniformly distributed load. The first end of this beam is simply supported, while second end is clamped. The individual nonlinear deformation function of a plane beam cross section – the shear effect with consideration of the classical shear stress formula is analytically developed. Displacements, strains and stresses taking into account the individual nonlinear deformation function for successive layers of this beam are elaborated in detail. The system of two differential equation of equilibrium of this beam, based on the principle of stationary potential energy, is obtained. This system of equations is precisely solved analytically, according to the theory of differential equations. The individual nonlinear deformation functions, shear stresses and deflections of two cases of sample beams are determined in detail: the first with variable core porosity in the direction of the beam depth, the second with constant core porosity – standard sandwich structure. The calculations results specified in Tables and graphically presented in Figures.

Fractal Continuum Mapping Applied to Timoshenko Beams

In this work, a generalization of the Timoshenko beam theory is introduced, which is based on fractal continuum calculus. The mapping of the bending problem onto a non-differentiable self-similar beam into a corresponding problem for a fractal continuum is derived using local fractional differential operators. Consequently, the functions defined in the fractal continua beam are differentiable in the ordinary calculus sense. Therefore, the non-conventional local derivatives defined in the fractal continua beam can be expressed in terms of the ordinary derivatives, which are solved theoretically and numerically. Lastly, examples of classical beams with different boundary conditions are shown in order to check some details of the physical phenomenon under study.

Deep reduced-order least-square method—A parallel neural network structure for solving beam problems

In this paper, a novel physics-informed neural networks (PINNs) called deep reduced-order least-square (ROLS) is proposed. The deep ROLS is a combination of reduced-order and least-square methods in the context of PINN methodology. The key idea of the deep ROLS is to convert a higher-order PDE to a system of lower-order PDEs by using primary and secondary variables, then the loss function constituted by integrals of corresponding squared residuals over the problem domain is minimized. Specifically, the squared residuals are established based on L2 norm of respective reduced-order PDE terms, and the Gauss–Legendre quadrature rule is applied to approximate their integrals. A multi-network structure is also designed, in which each sub-network serves as one field variable corresponding to each reduced-order PDE. Moreover, this work proposes a scheme to handle both essential and natural boundary conditions (BCs) directly. The deep ROLS demonstrates its better advantages in comparison to original PINN method in terms of solution accuracy, computational cost and data efficiency for several beam bending problems having continuous and discontinuous solutions.

An analytical model for nanoscale flexoelectric doubly curved shells

The phenomenon of flexoelectricity, wherein the generation of electrical polarization results from strain gradients, has garnered significant attention in electromechanics. This phenomenon holds immense potential for diverse applications in nanoelectromechanical systems. Doubly curved shells present a particularly compelling structure for implementing flexoelectric devices, representing the most general scenario. To comprehensively understand the mechanical behavior of nanoscale flexoelectric doubly curved shells, we present a novel analytical model in this study. Our model incorporates a reformulated strain gradient elasticity theory, a modified expression of electric enthalpy density considering Maxwell’s self-field gradient, and a moderately thick shell configuration. To further elucidate this model, we establish complete formulations and analytical solutions of static bending and free vibration problems. This study reveals the crucial roles that various parameters such as principal curvature radii, length scale parameters, transverse structural sizes, and the Winkler–Pasternak foundation play in controlling the mechanical behavior of nanoscale flexoelectric doubly curved shells. These theoretical findings offer valuable insights into the design of nanodevices based on the direct flexoelectric effect.

Variant of strain gradient elasticity with simplified formulation of traction boundary value problems

AbstractIn this paper, we show that within the class of isotropic Mindlin‐Toupin gradient theories there exist a particular variant of the theory, which allows a completely simplified form of traction boundary value problems. This theory can be obtained assuming that the high‐grade part of the strain energy density depends only on the vector‐type quantities, that are the gradient of dilatation and the curl of small rotation. Such incomplete gradient theory becomes positive semi‐definite and at the same time it obeys the strong ellipticity conditions of the general Mindlin‐Toupin first strain gradient elasticity. Based on the variational approach it is shown that the equilibrium equations of the developed theory and its definition for the surface traction can be given only in terms of the total stresses (like in classical elasticity). Such formulation can be useful for derivation of the closed form solutions for the problems with traction‐type boundary conditions. Examples of the solutions for the problem of cylindrical bending and for the inplane crack tip fields are presented. It is shown that considered theory allows to obtain a regularized solution for the crack problems and at the same time it does not predict a non‐physical infinite increase of the material's rigidity under bending.

Curvature-based flexoelectric nanobeams: Analytical and numerical isogeometric analyses

In this paper, new curvature-based flexoelectric Euler-Bernoulli and Timoshenko beam models are derived based on the couple stress and its flexoelectric theory. These models consider both the microstructure and flexoelectric effects. The governing equations and boundary conditions are obtained using the variational formula of Hamilton's principle. Analytical solutions for the static bending and free vibration problems of Euler-Bernoulli and Timoshenko beam models with a simply supported nanobeam are presented. To extend the applicability of the curvature-based flexoelectric beam models, an isogeometric analysis (IGA) numerical approach which satisfy the needs of C1 continuity is developed. Numerical results are provided to verify the accuracy of the analytical and IGA numerical solutions, and the influence of flexoelectric effects, microstructure effects and boundary conditions on the mechanical behavior of nanobeams is analyzed.

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

APPLICATION OF DISCONTINUOUS SOLUTIONS IN BOUNDARY ELEMENT METHOD FOR THE BENDING PROBLEMS OF KIRCHHOFF PLATES WITH AN ARBITRARY CONTOUR

The discontinuous solutions represent a new direction for the indirect Boundary Element Method (BEM) in the indirect formulation. Discontinuous solutions represent those functions, which when crossing certain lines, the transverse deflection, the slope angle, the bending moment and the generalized shear force may have jumps. These can be used to solve important problems for which the existing methods do not have solutions or a satisfactory accuracy, such as: plates of arbitrary contour, presence of defects, mixed boundary conditions, contact problems, infinite domains etc. In this paper, we describe the methodology of application and the numerical implementation of discontinuous solutions for the bending problems of plates with an arbitrary contour in the classical theory (Kirchhoff). For this purpose, programming codes were developed in the Matlab language, which allowed to calculate the displacements and efforts in the plate. The obtained results were compared with the Finite Element Method (FEM) for different mesh densities.

Bending of Plates with Complex Shape Made from Materials that Differently Resist to Tension and Compression

A new numerical-analytical method for solving physically nonlinear bending problems of thin plates with complex shape made from materials that differently resist to tension and compression is developed. The uninterrupted parameter continuation method is used to formulate and linearize the problem of physically nonlinear bending. For the linearized problem, a functional in the Lagrange form, given on the kinematically possible displacement rates, is constructed. The main unknown problems (displacements, strains, stresses) were found from the solution of the initial problem, which was solved by the Runge-Kutta-Merson method with automatic step selection, by the parameter related to the load. The initial conditions are found from the solution of the problem of linear elastic deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta-Merson scheme are found from the solution of the variational problem for the functional in the Lagrange form. Variational problems are solved using the Ritz method in combination with the R-function method, which allows to submit an approximate solution in the form of a formula – a solution structure that exactly satisfies the boundary conditions and is invariant with respect to the shape of the domain where the approximate solution is sought. The test problem for the nonlinear elastic bending of a square hinged plate is solved. Satisfactory agreement with the three-dimensional solution is obtained. The bending problem of the plate of complex shape with combined fixation conditions is solved. The influence of the geometric shape and fixation conditions on the stress-strain state is studied. It is shown that failure to take into account the different behavior of the material under tensile and compression can lead to significant errors in the calculations of the stress-strain state parameters.

PROBLEM OF PLATE BENDING IN THE MOMENT ASYMMETRIC THEORY OF ELASTICITY

For a number of materials used in modern practice, calculations according to the classical theory of elasticity give incorrect results. To ensure the reliable operation of structures, there is a need for new theories. At present, of particular interest for practical applications is the asymmetric moment theory of elasticity. In the work, by the method of hypotheses, the three-dimensional equations of the moment asymmetric theory of elasticity are reduced to the equations of the theory of plates. The hypotheses of the theory of plates in the moment theory of elasticity are formulated on the basis of previously obtained our results of the reduction of three-dimensional equations to two-dimensional theories by a mathematical method. Just as in the classical theory of elasticity, the complete problem of the moment theory of plates is divided into two problems - a plane problem and a problem of plate bending. The equations of the plane problem have been obtained in many papers. The situation is different with the construction of the theory of plate bending in the moment theory of elasticity. In this work, for the first time, substantiated hypotheses are formulated and a consistent theory of plate bending is presented. A numerical calculation of the bending of a rectangular hinged plate is carried out according to the obtained applied theory. The calculation results are presented in the form of graphs.

Numerical and experimental modeling of the behavior of flexible shell elements of structures

This article is devoted to computer and experimental modeling of the behavior of flexible elements of shell structures, and the development of effective algorithms for solving emerging nonlinear boundary value problems of their calculation and optimization of parameters. At the same time, the probability of the results obtained using different approaches to the construction of a nonlinear theory is established. Their comparative analysis, error estimation, which in this case is given by calculation according to linear and corresponding nonlinear theories, is carried out. The results of calculated data and experimental studies of the behavior of real structural elements are compared.
 The results of a comparative analysis of the application of two mathematical models of deformation of flexible shell elements, obtained by direct integration of boundary value problems of shell mechanics, by the finite element method and experimental research, are presented.
 The problems of axisymmetric nonlinear bending of thin ring plates, corrugated membranes of a sinusoidal profile and bellows as a shell of rotation with a complex meridian shape are considered.
 Using the necessary optimality conditions of the principle of maximum L. S. Pontryagin obtained the results of the optimal design of a flexible corrugated membrane with a sinusoidal profile of the highest sensitivity. The results are presented in the form of tables, photos and graphs.