Nonlinear Bending Analysis Research Articles

Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material

Size-dependent nonlinear Euler-Bernoulli and Timoshenko beam models, which account for the through-thickness power-law variation of two-constituent functionally graded (FG) materials, are derived to investigate the nonlinear bending and free vibration behaviors in the framework of the nonlocal strain gradient theory. The nonlinearity due to the stretching effect of the mid-plane of the FG beam is the source of nonlinearity of the considered bending and free vibration problems. The size-dependent equations of motion and boundary conditions are derived by employing the Hamilton’s principle. The beam models contain material length scale and nonlocal parameters to consider the effects of both inter-atomic long-range force and microstructure deformation mechanism. In the case of hinged-hinged boundary conditions, the analytical solutions for the nonlinear bending deflection and free vibration frequencies of nonlocal strain gradient Euler-Bernoulli and Timoshenko beams are deduced. The influences of the through-thickness power-law variation of a two-constituent FG material and size-dependent parameters on nonlinear bending deflection and free vibration frequencies are investigated. Due to the intrinsic stiffening effect brought by the stretching effect of the mid-plane of the beam, the nonlinear bending deflections are smaller than their linear counterparts under the action of the same force, while the nonlinear vibration frequencies are higher than their linear counterparts for the same amplitude of the nonlinear oscillator. The nonlinear bending deflections and free vibration frequencies can be affected significantly by the through-thickness grading of FG materials in the beam. When the nonlocal parameter is smaller than the material characteristic parameter, the nonlinear FG beam reveals a stiffness-hardening effect. When the material characteristic parameter is smaller than the nonlocal parameter, the FG beam reveals a stiffness-softening effect.

Nonlinear static analysis of single layer annular/circular graphene sheets embedded in Winkler–Pasternak elastic matrix based on non-local theory of Eringen

Abstract Nonlinear bending analysis of orthotropic annular/circular graphene sheets has been studied based on the non-local elasticity theory. The first order shear deformation theory (FSDT) is applied in combination with the nonlinear Von-Karman strain field. The obtained differential equations are solved by using two methods, first the differential quadrature method (DQM) and a new semi-analytical polynomial method (SAPM) which is innovated by the authors. Applying the DQM or SAPM, the differential equations are transformed to nonlinear algebraic equations system. Then the Newton–Raphson iterative scheme is used. First, the obtained results from DQM and SAPM are compared and it is concluded that although the SAPM’s formulation is considerably simpler than DQM, however, the SAPM’s results are so close to DQM. The results are validated with available papers. Finally, the effects of small scale parameter on the results, the comparison between local and non-local theories, and linear to nonlinear analyses are investigated.

Nonlinear bending analysis of variable cross-section laminated plates using the dynamic relaxation method

This paper tries to analyze the laminated plates with variable cross section, using the Dynamic relaxation method for solving the governing equations of the thin composite plate, obtained from the CPT theory. Comprehensive comparison and parametric studies prove the accuracy and efficiency of the utilized approach with interesting specifications such as fully vector calculations, independency to the lamina scheme (angle and number of plies) and boundary conditions so that the laminated plates with uniform, variable one direction and variable two direction of cross section could be analyzed. Results show that the behaviour of the composite plates depends on both lamina scheme (the stacking sequences and the number of the plies), and the cross section variation of the laminated plate. In this manner, utilizing the laminated plates with variable two direction of cross section could absorb more potential energy in comparison with uniform and variable one direction of cross section, so that they are useful for using in passive control mechanisms in which the kinetic energy should be removed from the structure by transforming to the potential energy.

Nonlinear bending analysis of bilayer orthotropic graphene sheets resting on Winkler–Pasternak elastic foundation based on non-local continuum mechanics

In this paper, the nonlinear bending behavior of bilayer orthotropic rectangular graphene sheets resting on a two parameter elastic foundation is studied subjected to uniform transverse loads using the non-local elasticity theory. The non-local theory consider the small scale effects. Considering the non-local differential constitutive relations of Eringen theory based on first order shear deformation theory (FSDT) and using the von-Karman strain field, the nonlinear formulations are derived. Equilibrium partial differential equations are expressed in terms of generalized displacements and rotations. Because of nonlinear partial differential equations, if it is not impossible but it is too complicated to find an analytical solution, so, the differential quadrature method (DQM) that is a high accurate numerical method is investigated to solve the governing equations. The Newton–Raphson iterative scheme is applied to solve the obtained nonlinear algebraic equations system. Different boundary conditions including clamped, simply supports and free edges are considered. Since there is not any researches available for nonlinear bending of bilayer rectangular graphene sheets with FSD theory, so considering the monolayer, the results are compared with available papers. Finally, the small scale effect parameter due to various types of conditions such as thickness ratio, boundary conditions, stiffness of elastic foundation, the van der Waals interactions between the layers, nonlinear to linear FSDT analysis and the differences between non-local and local elasticity theories are investigated.

Nonlinear bending and postbuckling analysis of matrix cracked hybrid laminated plates containing carbon nanotube reinforced composite layers in thermal environments

The nonlinear bending and postbuckling behaviors of a hybrid laminated plate resting on a Pasternak elastic foundation in thermal environments are investigated in this paper. The plate is composed of conventional fiber reinforced composite (FRC) layers and carbon nanotube reinforced composite (CNTRC) layers. The CNTRC layer consists of reinforcing carbon nanotubes either uniformly distributed (UD) or functionally graded (FG) along the thickness direction. Transverse matrix cracking is introduced in the FRC and UD CNTRC layers and modeled by a refined self-consistent method. The governing equations of the plate are based on a higher order shear deformation plate theory with a von Kármán-type of kinematic nonlinearity and solved by a two-step perturbation technique. The plate-foundation interaction and thermal effects are also included. The material properties of both CNTRC and FRC layers are assumed to be temperature-dependent and are estimated by a micro-mechanical model. A parametric study is conducted to investigate the effects of matrix crack density, foundation stiffness, temperature rise and in-plane boundary conditions on the nonlinear bending and postbuckling behaviors of antisymmetric hybrid laminated plates containing CNTRC layers.

Geometrically non-linear bending analysis of thick two-directional functionally graded annular sector and rectangular plates with variable thickness resting on non-linear elastic foundation

Non-linear analysis of two-directional functionally graded annular sector plates has not been performed yet. The present paper focuses on the non-linear bending analysis of variable thickness two-directional functionally graded circular/annular sector plates resting on the non-linear elastic foundation using the generalized differential quadrature (GDQ) and the Newton–Raphson iterative methods. The material properties vary simultaneously along transverse and radial directions according to a power-law distribution of the volume fraction of the constituents. Based on higher-order shear deformation theory (HSDT) with nine degree of freedom in the displacement field and von Kármán's non-linearity, the equilibrium equations are derived using the principle of minimum total potential energy. The concept of physical neutral surface is applied to the HSDT. The elastic foundation is modeled as shear deformable with hardening/softening cubic non-linearity. Rectangular plates are also analyzed based on the HSDT by a proper change in the geometry of annular sector plates. The results of present study are compared with those available in the literature and close agreement is observed. The effects of power-law indices, thickness variation, coefficients of foundation, various boundary conditions and geometrical parameters on linear and non-linear static behaviors of circular/annular sector plates under uniform and non-uniform loading are comprehensively investigated. Predictions of the first-order shear deformation theory are also obtained and compared with those of the HSDT.

Nonlinear Bending Analysis for Steel Purlin with Perforation

This paper presents a numerical investigation on the effect of perforation to the steel purlin on bending behaviour. In order to reduce the weight of steel sections, web profile with openings has been introduced in the construction industry, so does the purlin. However, there is no research done on the C-purlin with perforation. Hence, 33 nonlinear analyses were carried out to investigate effect of perforations to the steel purlin on bending behaviour. Yield moment of the steel purlin with and without perforation will be determined. And, effect of various perforation shapes and size on bending behaviour is investigated. Finite element model was developed using software LUSAS. C-channel steel section without opening is used as a control specimen. The results are expressed in term of yield moment. The results show that the edge distance from the opening to the support plays an important role in affecting the moment capacity of C-purlin. Increasing the opening spacing would increase the moment capacity. Number of openings also affects the moment capacity of the section. Octagonal shape with 0.40H and e= 350 mm shows the highest yield moment value with 16.318% higher than that of section without opening. In conclusion, perforation increases the moment capacity of the C-steel purlin.

Nonlinear Bending Analysis of Sector Graphene Sheet Embedded in Elastic Matrix Based on Nonlocal Continuum Mechanics

The nonlinear bending behavior of sector graphene sheets is studied subjected to uniform transverse loads resting on a Winkler-Pasternak elastic foundation using the nonlocal elasticity theory. Considering the nonlocal differential constitutive relations of Eringen theory based on first order shear deformation theory and using the von-Karman strain field, the equilibrium partial differential equations are derived. the nonlinear partial differential equations system is solved using the differential quadrature method (DQM) and a new semi analytical polynomial method (SAPM). By using the DQM or SAPM, the partial differential equations are converted to nonlinear algebraic equations, then the Newton–Raphson iterative scheme is applied to solve the resulting nonlinear algebraic equations system. The obtained results from DQM and SAPM are compared and observed the SAPM results are so close to DQM. Whereas, the SAPM’s formulations are considerably simpler than the DQM. Different boundary conditions including clamped, simply supported and free edges are considered. The obtained results are validated with available researches, then the small scale effects is investigated on the results due to various conditions such as outer radius to thickness ratio, boundary conditions, linear to nonlinear analysis, nonlocal to local analysis ratio, angle of the sector and stiffness value of elastic foundation.

Large deflection analysis of plates and cylindrical shells by an efficient four-node flat element with mesh distortions

This study reports an improved finite element computational model using a flat four-node element for nonlinear bending analysis of plates and cylindrical shells with element distortions. The von Karman’s large deflection theory and the total Lagrangian approach are employed in the formulation to describe small strain geometric nonlinearity with large deformations using the first-order shear deformation theory. The most important feature of the developed element is the evaluation of linear membrane bending and nonlinear geometric stiffness matrices based on integration along the boundary of smoothing elements. This technique can give more accurate numerical integrations even with badly shaped elements or coarse meshes when compared to other flat elements using domain integration techniques. The accuracy and predictive capability of the present model is demonstrated by several numerical investigations and comparative studies with analytical/experimental and other numerical solutions available in the literature.

Nonlinear bending analysis of radial-stiffened annular laminated sector plates with dynamic relaxation method

In this paper the nonlinear bending of laminated stiffened annular sector plates under mechanical loading with various boundary conditions is investigated. The plates are made of layers with orthotropic properties and different fiber orientations, which the aforementioned fibers are placed in a Cartesian coordinate system. Based on first-order shear deformation plate theory (FSDT) and von Karman relations for large deflection, nonlinear equilibrium equations are developed. Dynamic relaxation (DR) numerical method combined with the finite difference discretization technique is used to solve the plate nonlinear partial differential equations and FORTRAN program is developed to generate the numerical results. Effects of the plate thickness-to-radius ratio, boundary condition, stiffener depth, plate lay-ups and the sector angle are discussed.

Nonlinear bending analysis of FG-CNT reinforced composite thick plates resting on Pasternak foundations using the element-free IMLS-Ritz method

With very limited literature being available on the nonlinear bending behaviors of functionally graded carbon nanotube (FG-CNT) reinforced composite thick plates, this paper fills the apparent void by providing solutions to this problem based on the first-order shear deformation theory (FSDT). The plate considered rests on elastic foundations under transversely distributed loads. The analysis is carried out using the element-free IMLS-Ritz method. The arc-length iterative algorithm and the modified Newton–Raphson method are employed to obtain the nonlinear responses of FG-CNT reinforced composite plates. Convergence and comparison studies on a few example problems are performed to validate the numerical stability and accuracy of the IMLS-Ritz method. In this study, the characteristics of nonlinear bending influenced by foundation stiffness, transverse shear deformation, CNT distribution, CNT volume fraction and boundary conditions are examined.

Nonlinear bending analysis of FGM circular plates based on physical neutral surface and higher-order shear deformation theory

In this paper, a model of the FGM circular plates is successfully established by physical neutral surface and higher-order shear deformation theory. The primary material properties are assumed to be temperature-dependent and vary along the thickness, while Poisson's ratio depends weakly on temperature change and position and is assumed to be a constant. It is worth noting that the displacements have special forms, there are no stretching–bending couplings in constitutive equations, and governing equations have the simple forms, so the solution procedure is similar to that of the homogeneous isotropic plate. The validity of this new model is confirmed by comparison with related researchers' results. Multi-term Ritz method is novel for nonlinear bending analysis of FGM circular plates, and influences played by different supported boundaries, thermal environmental conditions and volume fraction index are discussed in detail.

Nonlinear bending analysis of laminated composite stiffened plates

This paper deals with the geometric nonlinear bending analysis of laminated composite stiffened plates subjected to uniform transverse loading. The eight-noded degenerated shell element and three-noded degenerated curved beam element with five degrees of freedom per node are adopted in the present analysis to model the plate and stiffeners respectively. The Green-Lagrange strain displacement relationship is adopted and the total Lagrangian approach is taken in the formulation. The convergence study of the present formulation is carried out first and the results are compared with the results published in the literature. The stiffener element is reformulated taking the torsional rigidity in an efficient manner. The effects of lamination angle, depth of stiffener and number of layers, on the bending response of the composite stiffened plates are considered and the results are discussed.

Nonlinear bending analysis of functionally graded CNT-reinforced composite Mindlin polymeric temperature-dependent plate resting on orthotropic elastomeric medium using GDQM

This research deals with the nonlinear bending analysis of nanocomposite polymeric temperature-dependent plates reinforced by single-walled carbon nanotubes (SWCNTs) subjected to a transverse uniform load. For the carbon nanotube-reinforced composite (CNTRC) plate, uniform distribution and three types of functionally graded (FG) distribution patterns of SWCNT reinforcements are assumed. The material properties of FG-CNTRC plate are graded in the thickness direction and estimated based on the rule of mixture. The CNTRC is located in an orthotropic temperature-dependent elastomeric medium. Based on orthotropic Mindlin plate theory along with von Karman geometric nonlinearity, the governing equations are derived using Hamilton’s principle and discretized by generalized differential quadrature method. The influences of the volume fractions of carbon nanotubes, elastomeric medium, aspect ratio, temperature, boundary conditions and applied force are considered on the nonlinear bending of the plate. Results indicate that CNT distribution close to top and bottom is more efficient than that distributed nearby the mid-plane for increasing the stiffness of plates. Furthermore, considering elastomeric medium decreases deflection of the FG-CNTRC plate. Results show good agreement with the solutions obtained by the finite element method and ANSYS software.

Nonlinear bending of nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments

Nonlinear bending analysis is presented for nanocomposite cylindrical panels subjected to a transverse uniform or sinusoidal load resting on elastic foundations in thermal environments. Carbon nanotubes are used to reinforce the cylindrical panels in two distinguished patterns, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements. The material properties of CNTRCs are assumed to be temperature-dependent and are estimated by a micromechanical model. The governing equations of the panel are derived based on a higher-order shear deformation theory with a von Kármán-type of kinematic nonlinearity and are solved by a two-step perturbation technique. The nonlinear bending behaviors of the CNTRC panels with different CNT volume fraction distributions, foundation stiffnesses, temperature rise, and the character of in-plane boundary conditions are studied in details.

Post-buckling analysis of FGM annular sector plates based on three dimensional elasticity graded finite elements

In this article, post-buckling and non-linear bending analysis of functionally graded annular sector plates based on three dimensional theory of elasticity in conjunction with non-linear Green strain tensor is considered. In-plane normal compressive loads have been applied to either radial, circumferential, or all edges of annular sector plates. Material properties are graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents while Poisson׳s ratio is assumed to be constant. The governing equations are developed based on the principle of minimum total potential energy and solved based on graded finite element method. Non-linear equilibrium equations are solved based on iterative Newton–Raphson method. The effects of material gradient exponent, different sector angles, thickness ratio, loading condition and two different boundary conditions on the post-buckling behavior of FGM annular sector plates have been investigated. Results denote that due to the stretching–bending coupling effects of the FGMs, the post-buckling behavior of movable simply supported FGM plates is not of the bifurcation-type buckling. Moreover, FGM annular sector plates subjected to uniaxial compression at radial edges show a non-linear bending behavior with unique and stable equilibrium paths following a flattening feature.

Nonlinear bending behavior of orthotropic Mindlin plate resting on orthotropic Pasternak foundation using GDQM

Rectangular plates resting on elastic foundations are operational activities of large transportation aircraft on runways, footings, foundation of spillway dam, civil building in cold regions, and bridge structures. Hence, in the present work, nonlinear bending analysis of embedded rectangular plates is investigated based on orthotropic Mindlin plate theory. The elastic medium is simulated by orthotropic Pasternak foundation. Adopting the nonlinear strain–displacement relation, the governing equations are derived based on energy method and Hamilton’s principle. The generalized differential quadrature method is performed for the case when all four ends are clamped supported. The influences of the plate thickness, shear-locking, elastic medium constants, and applied force on the nonlinear bending of the rectangular plate are studied. Results indicate that increasing the plate thickness decreases the deflection of the plate. It is also observed that increasing the applied force increases the deflection of the plate. Furthermore, considering elastic medium decreases deflection of the plate, and the effect of the Pasternak-type is higher than the Winkler-type on the maximum deflection of the plate. Also, it is found that the present results have good agreement with previous researches.

Nonlinear bending of FGM cylindrical panels resting on elastic foundations in thermal environments

Abstract A nonlinear bending analysis is presented for a simply supported, functionally graded cylindrical panel resting on an elastic foundation in thermal environments. The panel is exposed to elevated temperature and is subjected to a transverse uniform or sinusoidal load. Material properties of functionally graded materials (FGMs) are assumed to be temperature-dependent, and graded in the thickness direction based on Mori-Tanaka micromechanics model. The formulations are based on a higher order shear deformation shell theory with a von Karman-type of kinematic nonlinearity and include shell panel–foundation interaction and the thermal effects. A two-step perturbation technique is employed to determine the load-deflection and load-bending moment curves. The numerical illustrations concern nonlinear bending response of FGM cylindrical panels with two constituent materials resting on Pasternak elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The effects of the volume fraction index, temperature variation, foundation stiffness as well as the character of in-plane boundary conditions are also examined.

Nonlinear bending of functionally graded tapered beams subjected to thermal and mechanical loading

Non-linear bending analysis of tapered functionally graded (FG) beam subjected to thermal and mechanical load with general boundary condition is studied. The governing equations are derived and a discussion is made about the possibility of obtaining analytical solution. In the case of no axial force along the beam, a closed form solution is presented for the problem. For the general case with axial force, the Galerkin technique is employed to overcome the shortcoming of the analytical solution. Moreover, the Generalized Differential Quadrature (GDQ) method is also implemented to discretize and solve the governing equations in the general form and validate the results obtained from two other methods. The effects of various thermal and mechanical loading on the nonlinear bending of tapered FG beam are investigated by implementing different analytical and numerical approaches.

Nonlinear bending analysis of ring-stiffened circular and annular general angle-ply laminated plates with various boundary conditions

In this study, elastic large deflection analysis of axisymmetric ring-stiffened circular and annular general angle-ply laminated plates subjected to transverse uniform load is studied. Based on first order shear deformation theory (FSDT) and large deflection von-Karman relations, the governing equations are derived. The dynamic relaxation (DR) method in conjunction with the central finite difference discretization technique is used to solve the nonlinear equilibrium equations. A detailed parametric study is carried out to investigate the influences of plate thicknesses, stiffener width, stiffener depth, fiber orientation, stacking sequence and different types of boundary conditions. Also, some linear and nonlinear analysis is provided to consider the effect of nonlinearity on the results.