Bending Solutions Research Articles

The bending-stretching effect of a nonlinear elastic beam under finite deformation

Second-order solutions for the pure bending of a beam under finite deformation are obtained in this paper. The equilibrium equations are solved analytically for the second-order displacements, from which the first Piola–Kirchhoff stresses, the curvatures and a bending-stretching parameter are calculated for a cross section with a transverse axis of symmetry. The displacements are dependent on the square of the inverse of elastic constant, whereas the stresses on the inverse of elastic constant. The curvatures and anticlastic curvatures are gradated in the transverse direction. The longitudinal stress is nonlinearly distributed across the section, and exhibits anomalous characteristics in materials with a negative Poisson's ratio. The shear stresses have vanishing force and moment resultants, and bending-twisting coupling does not occur. The normal stresses give rise to force-couple resultants. The longitudinal force resultant representing the bending-stretching effect is dependent on a reduced elastic parameter ξ. This effect is present as long as there is finite deformation, even if the material is linearly elastic. A small shear modulus μ and/or a large third-order elastic constant l, m or n will result in a large ξ. This study may be applicable to the elasticity design of soft medical and wearable robots, where elastic compatibility with tissues is a primary consideration.

An exact GITT solution for static bending of clamped parallelogram plate resting on an elastic foundation

PurposeThe purpose of this study is to propose generalised integral transform technique (GITT) to obtain the exact solutions for bending of clamped parallelogram plate resting on elastic foundation.Design/methodology/approachThe GITT is used to solve the bending problem of the full clamped parallelogram plate under an elastic foundation. The auxiliary problem was developed and the corresponding eigenfunction and eigenvalue were calculated simultaneously. The original partial differential governed equation has been represented by the transformed ordinary differential equation system and solved by the subroutine DBVPFD from International Mathematics and Statistics Library.FindingsThe GITT has been proven to be an efficient approach to solve the bending problem of the plate with different loads, boundary conditions and elastic foundations. The parametric study indicates that the elastic foundation modulus has significant contribution in reducing the vertical deflections and moments for both rectangular and parallelogram plates. With the increasing of aspect ratio (a/b) and the elastic foundation modulus, the trends of the deflection and moment reduction decreased significantly.Originality/valueThe present hybrid analytical-numerical methodology was first used to solve the mechanics problem of the clamped parallelogram plate resting on elastic foundation. Excellent convergence and high accuracy was observed by comparing with the published results. It exhibits potential application to investigate the mechanics problem of the composite plate with different boundary conditions in the shipbuilding and civil engineering.

Accurate bending analysis of rectangular thin plates with corner supports by a unified finite integral transform method

In this paper, for the first time, the double finite integral transform method is introduced to explore the analytical bending solutions of rectangular thin plates with corner supports. Introducing the concept of generalized simply supported edge, the general form analytical solutions for bending of the plates under consideration are obtained by applying the finite integral transform to the governing equation and some of the boundary conditions. The analytical bending solutions of plates under specific boundary conditions are then obtained elegantly by imposing the remaining boundary conditions. Compared with the conventional inverse/semi-inverse methods, the present method is straightforward without any predetermination of solution forms, which makes it very attractive for exploring new analytical bending solutions of plates with complex boundary conditions. Another advantage of the method is that the analytical solutions obtained converge rapidly due to utilization of the sum functions. Comprehensive analytical results obtained in this paper illuminate the validity and accuracy of the present method by comparison with those from the literature and the finite element method.

A Unified Representation of Folded Surfaces via Fourier Series

Folded surfaces find application in many technical fields, as they offer a variety of solutions to engineering design problems at different scales. An important realm of application is represented by structural design, where folded surfaces permit to achieve stiff yet lightweight shell structures. In despite of the great variety of fold patterns and their suitability as mechanical structures, the lack of a generalized geometrical representation is a limiting shortcoming for their efficient modelling and simulation. This paper bridges this gap introducing a continuous representation of folded surfaces, which is able to reduce the multiplicity of possible fold patterns to a unified mathematical description. This is achieved by using 2-dimensional Fourier series constructed on a Bravais Lattice. Within this framework, a typical design task like geometry manipulation turns out to be straightforward. At the same time, the continuous mathematical formulation applied to folded surfaces enables the direct derivation of differential geometry entities, which facilitates in turn the mechanical modelling of shell structures. In this study, we demonstrate the validity of the proposed approach by the use of two case studies. The first one achieves a consistent representation of an existing doubly-curved shell structure. In the second, we deliver an analytical solution for bending of a trapezoidal sheet.

Bending Solutions of FGM Reddy–Bickford Beams in Terms of Those of the Homogenous Euler–Bernoulli Beams

In this paper, correspondence relations between the solutions of the static bending of functionally graded material (FGM) Reddy–Bickford beams (RBBs) and those of the corresponding homogenous Euler–Bernoulli beams are presented. The effective material properties of the FGM beams are assumed to vary continuously in the thickness direction. Governing equations for the titled problem are formulated via the principle of virtual displacements based on the third-order shear deformation beam theory, in which the higher-order shear force and bending moment are included. General solutions of the displacements and the stress resultants of the FGM RBBs are derived analytically in terms of the deflection of the reference homogenous Euler–Bernoulli beam with the same geometry, loadings and end conditions, which realize a classical and homogenized expression of the bending response of the shear deformable non-homogeneous FGM beams. Particular solutions for the FGM RBBs under specified end constraints and load conditions are given to validate the theory and methodology. The key merit of this work is to be capable of obtaining the high-accuracy solutions of thick FGM beams in terms of the classical beam theory solutions without dealing with the solution of the complicated coupling differential equations with boundary conditions of the problem.

Solution for bending of a polygonal plate with cracks by the linear conjugate method

Stress–strain state of a polygonal multiply connected plate under bending is considered. The plate under the action of different force factors (bending moment M and uniformly distributed load of intensity q) is also carried out. Similar problems for a simple configuration plates were solved by different authors. But the bending problem for a plate with different length through cracks is considered first. Using the theory of a complex variable function and analytic functions (Kolosov-Muskheleshvili potential) the problem is solved by the linear conjugate method. Such plates (slabs) are widely used in shipbuilding (hull-deck of ships), in construction (interfloor slabs, balcony flooring slabs) in mechanical engineering, aircraft building, etc. Therewith, at first boundary condition for an entire plate (without cracks) under the action of the applied loads, (M and q) and then a problem on bending of a plate with cracks under the action of loads applied to crack faces and that are determined from the problem on entire plate (using the uniqueness of displacements $$v+iv$$ and uniqueness of the flexure w), are used. The obtained elliptic integrals F(x) and E(x) are calculated by the methods known from references. As a result for determining integral constants $$C_1 ,C_2 ,\ldots ,C_j $$ , we get a system of equations for the considered cases of applied loads. Some numerical examples are solved.

A semi-analytical solution for finite bending of a functionally graded hydrogel strip

In this paper, a semi-analytical formulation is developed to examine the swelling-induced finite bending of a functionally graded hydrogel strip, when the strip is embedded in a solvent bath of an assigned chemical potential. The cross-link density of the hydrogel polymeric network varies through the strip thickness either linearly or exponentially. As a result of inhomogeneous swelling ratio through the hydrogel strip thickness, the strip bends in a circle. In contrast to earlier solution methods, the initial configuration is mapped to the deformed state without assuming any intermediary virtual state, using a total deformation gradient tensor. The swelling response of the hydrogel is studied utilizing the Flory–Huggins model for the free energy changes due to the mixing and deformation of the hydrogel network. In order to validate the presented method, FEM is employed to solve the finite bending of the functionally graded hydrogel strip. Using the presented method, the effects of the hydrogel network cross-link density distribution on the radial and tangential stress fields, strip bending curvature, and semi-angle are studied. In contrast to hydrogel-based multilayers, continuous stress and deformation field are found for the functionally graded hydrogel strip. Also, multiple tangential stress-free axes are observed for functionally graded hydrogel strips under bending configuration.

Saint-Venant Problems for Two-dimensional Elastic Solids

Using the symplectic system method, the classical Saint-venant problems in elasticity are studied. Based on the Laplace transformation and integral transformation and the symplectic character, all the Saint-Venant solutions, including the tension, bending and local solutions, are obtained. The final solution of the problem can be directly analysed in the time domain solution space. As applications of the proposed method, some stress concentration boundary condition problems are discussed.

Static and Dynamic Behavior of Nanotubes-Reinforced Sandwich Plates Using (FSDT)

Based on the first order shear deformation plate theory (FSDT) in the present studie, static and dynamic behavior of carbon nanotube-reinforced composite sandwich plates has been analysed. Two types of sandwich plates, namely, the sandwich with face sheet reinforced and homogeneous core and the sandwich with homogeneous face sheet and reinforced core are considered. The face sheet or core plates are reinforced by single-walled carbon nanotubes with two types of distributions of uniaxially aligned reinforcement material which uniformly (UD-CNT) and functionally graded (FG-CNT). The analytical equations are derived and the exact solutions for bending and vibration analyses of such type’s plates are obtained. The mathematical models provided and the present solutions are numerically validated by comparison with some available results in the literature. Influence of Various parameters of reinforced sandwich plates such as aspect ratios, volume fraction, types of reinforcement and plate thickness on the bending and vibration analyses of carbon nanotube-reinforced composite sandwich plates are studied and discussed. The findings suggest that the (FG-CNT) face sheet reinforced sandwich plate has a high resistance against deflections compared to other types of reinforcement. It is also revealed that the reduction in the dimensionless natural frequency is most pronounced in core reinforced sandwich plate.

On Static Bending, Elastic Buckling and Free Vibration Analysis of Symmetric Functionally Graded Sandwich Beams

This article presents Navier type closed-form solutions for static bending, elastic buckling and free vibration analysis of symmetric functionally graded (FG) sandwich beams using a hyperbolic shear deformation theory. The beam has FG skins and isotropic core. Material properties of FG skins are varied through the thickness according to the power law distribution. The present theory accounts for a hyperbolic distribution of axial displacement whereas transverse displacement is constant through the thickness i.e effects of thickness stretching are neglected. The present theory gives hyperbolic cosine distribution of transverse shear stress through the thickness of the beam and satisfies zero traction boundary conditions on the top and bottom surfaces of the beam. The equations of the motion are obtained by using the Hamilton’s principle. Closed-form solutions for static, buckling and vibration analysis of simply supported FG sandwich beams are obtained using Navier’s solution technique. The non-dimensional numerical results are obtained for various power law index and skin-core-skin thickness ratios. The present results are compared with previously published results and found in excellent agreement.

On the nonlinear bending and post-buckling behavior of laminated sandwich cylindrical shells with FG or isogrid lattice cores

The nonlinear governing equations of three shell theories (Donnell, Love, and Sanders) with first-order approximation and von Karman’s geometric nonlinearity for laminated sandwich cylindrical shells with isotropic, functionally graded (FG) or isogrid lattice layers are decoupled. This uncoupling makes it possible to present a semi-analytical solution for the nonlinear bending and post-buckling behavior of short and long doubly simply supported, doubly clamped, and cantilever laminated sandwich cylindrical shells subjected to various types of thermo-mechanical loadings. The results for deflection, stress, critical axial traction, and mode shapes in FG shells are verified with those obtained from ABAQUS code. Finally, the case studies are presented for FG shells and laminated sandwich shells with different layups such as $$[\hbox {Al; ZrO}_2]$$ , $$[\hbox {Al; FG core; ZrO}_2]$$ , $$[\hbox {Al; Gr; ZrO}_2]$$ , $$[\hbox {Al; Gr; FG core; ZrO}_2]$$ , $$[\hbox {Al; isogrid lattice core; Al}]$$ . The closed-form solutions presented here for the kinetic parameters and critical axial loading in a nonlinear analysis can be used in the conceptual design of laminated sandwich cylindrical shells with arbitrary layups and boundary conditions. Furthermore, introducing an equivalent Young’s modulus through the shell thickness, a simple formula is presented for the calculation of critical load in long shells with simple and clamped ends under axial loading with a maximum error of 10%. Moreover, findings show that the boundary-layer type behavior is seen only in long cylindrical shells in the pre-buckling region. Under thermal loading, snap-through buckling is observed in clamped shells. However, in simply supported shells by increasing the temperature, the transverse deflection increases, and while $$\Delta T-w/h$$ curves do not show any buckling phenomenon, the $$N^{0}/N_{\mathrm{cr}}^{*} -\Delta T$$ curves show such a behavior.

An analytical solution for bending of axisymmetric circular/annular plates resting on a variable elastic foundation

In this paper, an analytical method is presented in order to determine the static bending response of an axisymmetric thin circular/annular plate with different boundary conditions resting on a spatially inhomogeneous Winkler foundation. To this end, infinite power series expansion of the deflection function is exploited to transform the governing differential equation into a new solvable system of recurrence relations. Singular points of the governing equation are effectively treated by applying the Frobenius theorem in the solution, which in turn permits the use of more-general analytical functions to describe the variation of the foundation modulus along the radius of the plate. Moreover, no special limitation is imposed on the transverse loading function as applied to the system. On employing the proposed method, the deflection response is obtained through an illustrative example for various boundary conditions along the plate edges, considering free, clamped, hinged, and elastically restrained boundaries. In addition, analytical results are validated and compared with those obtained using a finite element analysis, where an excellent agreement is found. Finally, the extension of the method to solve the more-general case of a variable two-parameter (Pasternak) foundation is indicated.

Symplectic superposition method-based new analytic bending solutions of cylindrical shell panels

Pursuing analytic bending solutions of cylindrical shell panels without two opposite simply supported edges is a classic but very difficult type of problems. The main challenge is on the mathematical complexity of the boundary value problems of the governing high-order partial differential equations. In this paper, the first endeavor is made on extending an up-to-date symplectic superposition method to bending of cylindrical panels, with focus on clamped panels and their variants. By introducing the problems into the Hamiltonian system (in physics) and the symplectic space (in mathematics), they come down to the symplectic eigen problems, which are analytically solved for fundamental analytic solutions of three types of subproblems, followed by superposition for final solutions. The new analytic solutions for the panels with four different combinations of boundary conditions are obtained, with comprehensive results tabulated to serve as benchmarks for future studies, all of which are well validated by the finite element method. The rigorous derivation by the present method without any assumptions/prior knowledge of solution forms may provide an exceptional route to more analytic solutions of some intractable shell problems.

Nonlinear analysis of viscoelastic micro-composite beam with geometrical imperfection using FEM: MSGT electro-magneto-elastic bending, buckling and vibration solutions

In this research, the nonlinear static, buckling and vibration analysis of viscoelastic micro-composite beam reinforced by various distributions of boron nitrid nanotube (BNNT) with initial geometrical imperfection by modified strain gradient theory (MSGT) using finite element method (FEM) are presented. The various distributions of BNNT are considered as UD, FG-V and FG-X and also, the extended rule of mixture is used to estimate the properties of micro-composite beam. The components of stress are dependent to mechanical, electrical and thermal terms and calculated using piezoelasticity theory. Then, the kinematic equations of micro-composite beam using the displacement fields are obtained. The governing equations of motion are derived using energy method and Hamilton\'s principle based on MSGT. Then, using FEM, these equations are solved. Finally the effects of different parameters such as initial geometrical imperfection, various distributions of nanotube, damping coefficient, piezoelectric constant, slenderness ratio, Winkler spring constant, Pasternak shear constant, various boundary conditions and three material length scale parameters on the behavior of nonlinear static, buckling and vibration of micro-composite beam are investigated. The results indicate that with an increase in the geometrical imperfection parameter, the stiffness of micro-composite beam increases and thus the non-dimensional nonlinear frequency of the micro structure reduces gradually.

An analytic solution for bending under tension of a two-layer strip of strain hardening materials

Abstract A semi-analytic solution for the process of plane strain bending under tension of a two-layer sheet of elastic-plastic, isotropic, incompressible, strain hardening material is provided. No restriction is imposed on the strain hardening law. A numerical technique is only necessary for solving transcendental equations and evaluating an ordinary integral to find the bending moment. The solution is constructed as a combination of three elementary analytic solutions. The general solution is simple but very cumbersome. In particular, the process usually consists of several stages. Several conditions control the range of validity of the solution in each stage. In order to find the correct solution, it is necessary to check all these conditions. However, for any given initial geometry of the sheet and any given constitutive equations it is straightforward to construct the solution.

An analytical solution for bending and free vibration responses of functionally graded beams with porosities Effect of the micromechanical models

In this paper, a new higher order shear deformation model is developed for static and free vibration analysis of functionally graded beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. Different patterns of porosity distributions (including even and uneven distribution patterns, and the logarithmic-uneven pattern) are considered. In addition, the effect of different micromechanical models on the bending and free vibration response of these beams is studied. Various micromechanical models are used to evaluate the mechanical characteristics of the FG beams whose properties vary continuously across the thickness according to a simple power law. Based on the present higher-order shear deformation model, the equations of motion are derived from Hamilton\'s principle. Navier type solution method was used to obtain displacement, stresses and frequencies, and the numerical results are compared with those available in the literature. A comprehensive parametric study is carried out to assess the effects of volume fraction index, porosity fraction index, micromechanical models, mode numbers, and geometry on the bending and natural frequencies of imperfect FG beams.

Two-dimensional generalized finite integral transform method for new analytic bending solutions of orthotropic rectangular thin foundation plates

In this paper, a two-dimensional generalized finite integral transform method is developed for new analytic bending solutions of orthotropic rectangular thin foundation plates. The vibrating beam functions are adopted as the integral kernels to construct the integral transform pairs. By imposing the transform to the governing equation, utilizing some inherent properties of the beam functions, the title problem is converted to that of solving a system of linear algebraic equations, by which the new analytic solutions are elegantly obtained in a straightforward way. Numerical examples validate the present method as well as the solutions yielded by satisfactory agreement with the literature and finite element analysis.

A new computationally efficient finite element formulation for nanoplates using second-order strain gradient Kirchhoff's plate theory

The strain gradient nonlocal theory is important to include the size effects of nanostructures in classical continuum theory with the corresponding development of computationally efficient numerical tool such as finite elements for the analysis of such structures with different boundary conditions. However, there is no literature on the finite element formulation of second-order strain gradient elastic plates. The weak form of the governing equation of motion of the Kirchhoff nanoplate using second-order positive/negative strain gradient nonlocal theories requires C2 continuity of transverse displacement. In this paper, a new computationally efficient nonconforming finite element formulation for the modelling of nanoplates using second-order positive/negative strain gradient nonlocal theories is presented. The performance of the developed finite element is compared with conforming finite element for rectangular isotropic Kirchhoff nanoplates with different boundary conditions. Analytical solution for static bending, free vibration, and buckling under biaxial in-plane compressive loading are also obtained for rectangular all edges simply supported isotropic Kirchhoff nanoplate for the comparison purpose. The nonconforming element is found to be computationally more efficient than the conforming element with better accuracy and convergence rate. The negative strain gradient model predicts results matching with the experimental results available in the literature.

Comparison of nano-plate bending behaviour by Eringen nonlocal plate, Hencky bar-net and continualised nonlocal plate models

This paper is concerned with the bending behaviour of small-scale simply supported plates as predicted by using the Eringen nonlocal plate model (ENM), the Hencky bar-net model (HBM) and the continualised nonlocal plate model (CNM). HBM comprises rigid beam segments connected by rotational and torsional springs. CNM is a nonlocal model derived by using a continualisation approach that does away with the unknown scale coefficient $$e_{0}$$ in ENM. The exact bending solutions for simply supported rectangular nano-plates are derived by using ENM, HBM and CNM. By making the segment length $$\ell $$ of HBM equal to the scale length of continualised and Eringen’s nonlocal plate model and noting the phenomenological similarities between ENM, HBM and CNM, the Eringen’s length scale value $$e_0 $$ is found to be dependent on the aspect ratio of the simply supported plate and independent of the applied transverse loading. For a very small scale length $$\ell $$ , $$e_0$$ of ENM converges to values ranging from $$1/\sqrt{8}$$ to $$1/\sqrt{6}$$ for square plate to longish rectangular plate when calibrated by either HBM or CNM.

Note on the generality of one plane strain bending universal solution

This note highlights an interesting result for the recently comprehensively examined universal solution describing plane strain bending and unbending of a sector of a circular tube made from a neo-Hookean material. We show that it is the most general and descriptive among all other exact plane bending solutions for incompressible, isotropic, non-linear elasticity. In particular, it recovers the existing results for bending of a rectangular block into a sector of a circular cylinder, straightening of a cylindrical sector, and closing of a cylindrical sector into an intact tube. By demonstrating this in the framework of incremental nonlinear elasticity, some new incidental buckling curves’ envelopes are uncovered.