Static Bending Analysis Research Articles

A new first-order mixed beam element for static bending analysis of functionally graded graphene oxide powder-reinforced composite beams

In this paper, a new first-order mixed beam element is established and applied to investigate the static bending of the functionally graded graphene oxide powder-reinforced composite beams. The proposed beam element is developed via a mixed finite element formulation based on first-order shear deformation theory. The new element consists of two nodes and three degrees of freedom per node. In addition, the present beam element uses linear shape functions. The proposed beam element is free of shear locking without using selective or reduced integration. The comparison study demonstrates that the present beam element can predict exact solutions with very few elements, so, the computation cost is reduced. Then the proposed beam element is used to analyze the static bending of functionally graded graphene oxide powder-reinforced composite beams. A comprehensive parameter study is carried out to demonstrate the effects of some parameters such as the boundary conditions, graphene oxide powder weight fraction, graphene distribution patterns, graphene oxide powder dimensions, and the slender ratio of the beams.

A New Beam Finite Element for Static Bending Analysis of Slender Transversely Cracked Beams on Two-Parametric Soils

This paper derives an original finite element for the static bending analysis of a transversely cracked uniform beam resting on a two-parametric elastic foundation. In the simplified computational model based on the Euler–Bernoulli theory of small displacements, the crack is represented by a linear rotational spring connecting two elastic members. The derivations of approximate transverse displacement functions, stiffness matrix coefficients, and the load vector for a linearly distributed load along the entire beam element are based on novel cubic polynomial interpolation functions, including the second soil parameter. Moreover, all derived expressions are obtained in closed forms, which allow easy implementation in existing finite element software. Two numerical examples are presented in order to substantiate the discussed approach. They cover both possible analytical solution forms that may occur (depending on the problem parameters) from the same governing differential equation of the considered problem. Therefore, several response parameters are studied for each example (with additional emphasis on their convergence) and compared with the corresponding analytical solution, thus proving the quality of the obtained finite element.

A unified size-dependent plate theory for static bending and free vibration analyses of micro- and nano-scale plates based on the consistent couple stress theory

Based on the consistent couple stress theory (CCST), the authors develop a unified formulation of various shear deformation plate theories for static bending and free vibration analyses of simply-supported, micro-/nano-scale plates embedded in an elastic medium. The CCST-based classical plate theory (CPT), first-order shear deformation plate theory (SDPT), and Reddy's refined SDPT, as well as the CCST-based sinusoidal, exponential, and hyperbolic SDPTs can be obtained by assigning specific through-thickness distributions of the shear deformations into the unified formulation, and they are thus included as special cases of the unified consistent couple stress plate theory. The unified CCST-based plate theory is applied to static bending and free vibration analyses of simply-supported, functionally graded microplates (FGMPs) and multilayered graphene sheets (MLGSs) embedded in an elastic medium. The material length scale parameter, aspect ratio, and shear deformation effects on the deformation, stress, and frequency parameters of FGMPs and MLGSs are investigated. It is shown that the solutions for the deformation, the in-plane stress, and the frequency parameters of the FGMPs/MLGSs obtained using the MCST and the CCST are almost identical to each other. The material length scale parameter effects on static bending and free vibration behaviors of the FGMP/MLGS are significant.

Static Bending of Isotropic Circular Cylindrical Shells Based on the Higher Order Shear Deformation Theory of Reddy and Liu

In this paper, a displacement based shear deformation theory formulated on the cubic in-plane displacement field equation of Reddy and Liu is presented for the static bending analysis of isotropic circular cylindrical shells. The adopted displacement field accounts for a quadratic (parabolic) distribution of the transverse shear through the shell thickness as well as satisfies the need for a stress free upper and lower boundary surfaces of the shell. The equations of static equilibrium are obtained on application of the principle of virtual work. Numerical results of the bending analysis for the displacements and stresses are presented for the simply supported shell. A comparison made to those of the Kirchhoff-Love theory for varying shell length to mean – radius of curvature ratios, shows good agreement for thin shells irrespective of the shell length to radius of curvature ratio (<i>l</i> / <i>a</i>). The transverse sharing effect is found to be noticeable in the deformation of thick shells, however, this effect diminishes with a continuous increase in <i>l</i> / <i>a</i> ratios.

Static bending analysis of functionally graded sandwich beams using a novel mixed beam element based on first-order shear deformation theory

In this paper, the static bending behavior of the functionally graded sandwich beams is investigated using a novel mixed beam element based on the first-order shear deformation theory. The proposed beam element consists of two nodes and three degrees of freedom per node as the traditional Timoshenko beam element. By introducing a special process, selective or reduced integration are not required to calculate the element stiffness matrix and nodal force vector. The efficiency and accuracy of the proposed element are verified via some comparison studies. Then the proposed element is applied to study the bending behavior of the functionally graded sandwich beams with various boundary conditions. The influences of some parameters such as the power-law index, the slender ratio and the boundary conditions are studied carefully. The comparison studies and numerical results present that the new beam element is compatible with the static bending analysis of the functionally graded sandwich beams with a very coarse mesh. Besides, the numerical results show that the stress concentration may occur on the separated surface of the core layer and two face sheets, which should be noticed in practical applications of the FG sandwich beams.

A refined quasi-3D logarithmic shear deformation theory-based effective meshfree method for analysis of functionally graded plates resting on the elastic foundation

In this paper, a new refined quasi-3-dimensional logarithmic shear deformation theory (RQ-3DLSDT) and an advanced moving Kriging interpolation (AMKI) based-meshless method is combined for the first time to study the static bending, free vibration, and compressive buckling analysis of isotropic and sandwich functionally graded plates laid on elastic foundations. The RQ-3DLSDT considering the effect of thickness stretching based only on four-unknown variables to determine the displacement field leading to reduce computational efforts. The weak forms of the equilibrium equation are discretized and numerically solved by the AMKI mesh-free method employed a proposed quadric correlation function for getting stable solutions.

Bending of Symmetric Sandwich FGM Beams with Shear Connectors

This paper carries out the static bending analysis of symmetric three-layer functionally graded sandwich beams, in which each layer is made from different functionally graded materials, and they are connected by shear connectors due to sliding movement. The finite element formulations are based on Timoshenko’s first-order shear deformation beam theory (FSDT) and the finite element method to establish the equilibrium equation of beams. The calculation program is coded in the MATLAB environment, and then verification examples are given out to compare the numerical data of present work with those of exact open sources. The impact of several geometrical and material parameters on the mechanical response of the structure, such as the height-to-length ratio, boundary conditions, volume fraction index, and especially the shear coefficient of connectors, is being explored. When designing and using these types of structures in engineering practice, the computed results can be utilized as a valid reference.

A quasi-three-dimensional isogeometric model for porous sandwich functionally graded plates reinforced with graphene nanoplatelets

The purpose of this study is to present a quasi-three-dimensional (quasi-3D) shear deformation theory for static bending and free vibration analyses of porous sandwich functionally graded (FG) plates with graphene nanoplatelets (GPLs) reinforcement. In addition, we propose a novel sandwich plate model with various outstanding features in terms of structural performance. The quasi-3D theory-based isogeometric analysis (IGA) in conjunction with refined plate theory (RPT) is first exploited to capture adequately the thickness stretching effect for porous sandwich FG plate structures reinforced with GPLs. The Non-Uniform Rational B-Splines (NURBS)-based IGA is employed in order to describe exactly the geometry models as well as approximate the unknown field with higher-order derivatives and continuity requirements while the RPT model includes only four essential variables. The sandwich FG plates consist of a core layer containing internal pores reinforced by GPLs and two functionally graded materials (FGMs) skin layers. Effective mechanical properties can be evaluated by employing the Halpin-Tsai model along with the rule of mixture. Various combinations of two porosity distributions and three GPL dispersions in the core layer are thoroughly investigated. Several numerical investigations are conducted to examine the effects of several key parameters on the static bending and free vibration behaviors of sandwich FG plate structures.

Correction to: Static Bending and Vibration Analysis of Piezoelectric Semiconductor Beams Considering Surface Effects

Correction to: Static Bending and Vibration Analysis of Piezoelectric Semiconductor Beams Considering Surface Effects

Static bending and vibration analysis of piezoelectric semiconductor beams considering surface effects

Static bending and vibration analysis of piezoelectric semiconductor beams considering surface effects

Isogeometric analysis of size-dependent Bernoulli–Euler beam based on a reformulated strain gradient elasticity theory

In this paper, we present an efficient and accurate numerical approach for static bending and free vibration analyses of microstructure-dependent Bernoulli-Euler beams. The current approach includes strain gradient, couple stress (rotation gradient) and velocity gradient effects simultaneously through a reformulated strain gradient elasticity, which applies only one material parameter for each gradient effect. Based on the Hamilton’s principle and an Isogeometric Analysis (IGA) approach, the governing equations are derived and solved, respectively, which effectively fulfills the higher continuity requirements in the present microstructure-dependent Bernoulli-Euler beam formulation. The static bending and free vibration analyses of simply supported microstructure-dependent beams are studied by directly applying the current approach and compared with the corresponding analytical solutions. The physical mesh convergence and numerical results prove the high performance and accuracy of the present numerical approach. In addition, the cantilever and clamped microbeams are also carried out to show the applicability of the present approach.

Bending, axial buckling and shear buckling analyses of FG-porous plates based on a refined plate theory

ABSTRACT In this paper, numerical solutions are presented for static bending and mechanical buckling analyses of functionally graded porous (FGP) plates. The plate is modelled based on a refined plate theory and three porosity distributions with the same total mass density are considered. The set of the governing equations is derived using minimum potential energy and is solved numerically using the differential quadrature method (DQM). Convergence and accuracy of the solution are confirmed and the effects of porosity parameter, porosity distribution pattern, thickness and aspect ratio of the plate and boundary conditions on the deflection, intensity and distribution of stress and critical buckling load are investigated. It is shown by the numerical examples that an increment in the porosity parameter leads to a reduction in the critical buckling load and an increase in static deflection of the plate. Numerical results reveal that the effect of the porosity parameter on the distribution and intensity of the stress components strongly is dependent on the porosity distribution pattern.

A New Sinusoidal Shear Deformation Theory for Static Bending Analysis of Functionally Graded Plates Resting on Winkler–Pasternak Foundations

In this article, a new sinusoidal shear deformation theory was developed for static bending analysis of functionally graded plates resting on elastic foundations. The proposed theory used an undefined integral term to reduce the number of the unknown to four without any shear correction factors. The high accuracy and efficiency of the proposed theory were proved thanks to the comparisons of the present results with other available solutions. And then, the proposed theory was successfully applied to investigate the bending behavior of the functionally graded plates resting on Winkler–Pasternak foundations. The governing equations of motion were established by using Hamilton’s principle, and the Navier’s solution technique was employed to solve these equations. The effects of some factors of the geometrics, the materials properties, and the elastic foundation parameters on the bending behaviors of the FGM plates were investigated intensely. Also, some novel results and special phenomenon were carried out.

Static bending and free vibration analysis of functionally graded porous plates laid on elastic foundation using the meshless method

This paper presents a numerical approach for static bending and free vibration analysis of the functionally graded porous plates (FGPP) resting on the elastic foundation using the refined quasi-3D sinusoidal shear deformation theory (RQSSDT) combined with the Moving Kriging–interpolation meshfree method. The plate theory considers both shear deformation and thickness-stretching effects by the sinusoidal distribution of the in-plane displacements, satisfies the stress-free boundary conditions on the top and bottom surfaces of the plate without shear correction coefficient. The advantage of the plate theory is that the displacement field of plate is approximated by only four variables leading to reduce computational efforts. Comparison studies are performed for the square FGPP with simply supported all edges to verify the accuracy of the present approach. The effect of the aspect ratio, volume fraction exponent, and elastic foundation parameters on the static deflections and natural frequency of FGPP are also investigated and discussed.
 Keywords:
 meshless method; Moving Kriging interpolation; refined quasi-3D theory; porous functionally\break graded plate; Pasternak foundation.

Formulation of a New Mixed Four-Node Quadrilateral Element for Static Bending Analysis of Variable Thickness Functionally Graded Material Plates

A new mixed four-node quadrilateral element (MiQ4) is established in this paper to investigate functionally graded material (FGM) plates with variable thickness. The proposed element is developed based on the first-order shear deformation and mixed finite element technique, so the new element does not need any selective or reduced numerical integration. Numerous basic tests have been carried out to demonstrate the accuracy and convergence of the proposed element. Besides, the numerical examples show that the present element is free of shear locking and is insensitive to the mesh distortion, especially for the case of very thin plates. The present element can be applied to analyze plates with arbitrary geometries; it leads to reducing the computation cost. Several parameter studies are performed to show the roles of some parameters such as the power-law index, side-to-thickness ratio, boundary conditions (BCs), and variation of the plate thickness on the static bending behavior of the FGM plates.

Effect of porosity on the bending of functionally graded plates integrated with PFRC layer

This paper will present the analysis of static bending of functionally graded porous plates attached to a piezoelectric fiber-reinforced composite layer. The plate is exposed to sinusoidal electrical and mechanical loads. The displacement field is presented by using a sinusoidal shear deformation theory. The virtual work principle will be used for the derivation of the equilibrium equations and Navier’s method for the solution. Displacements and stresses are confirmed by comparing them with corresponding literature. Lastly, numerical results confirm that the porosity factor, applied voltage, volume fraction exponent, Young’s modulus ratio, aspect ratio, and side-to-thickness ratio have a huge influence on the electromechanical bending of functionally graded porous plates.

The Third‐Order Shear Deformation Theory for Modeling the Static Bending and Dynamic Responses of Piezoelectric Bidirectional Functionally Graded Plates

This work is the first exploration of the static bending and dynamic response analyses of piezoelectric bidirectional functionally graded plates by combining the third‐order shear deformation theory of Reddy and the finite element approach, which can numerically model mechanical relations of the structure. The present approach and mechanical model are confirmed through the verification examples. The geometrical and material study is conducted to evaluate the effects of the feedback coefficients, volume fraction parameter, and constraint conditions on the static and dynamic behaviors of piezoelectric bidirectional functionally graded structures, and this work presents a wide variety of static and dynamic behaviors of the plate with many interesting results. There are many meanings that have not been mentioned by any work, especially the working performance of the structure is better than that when the feedback parameter of the piezoelectric component is added, that is, the piezoelectric layer increases the working efficiency. Numerical investigations are the important basis for calculating and designing related materials and structures in technical practice.

Finite Element Modeling for Static Bending Behaviors of Rotating FGM Porous Beams with Geometrical Imperfections Resting on Elastic Foundation and Subjected to Axial Compression

The static bending analysis of the FG porous beam resting on the two‐parameter elastic foundation is initially carried out using a combination of Reddy’s high‐order shear deformation theory and the finite element technique, where the initial geometrical imperfection and rotation movement in one fixed axis are calculated. Through the power‐law distribution function with porosities, material characteristics vary constantly from one surface to the next in the direction of thickness, and the beam is concurrently impacted by an acting force perpendicular to the beam axis and an axial compressive force. The stiffness matrix of the beam element changes as a result, and the static bending response of this beam is significantly different from that of ordinary beams. Comparison cases with published findings are used to verify the computational theory. The calculations clearly reveal many innovations for rotating beams that are influenced by many different kinds of loads, which may be used to the designing, manufacturing, and usage of these structures in reality.

Thermal vibration and nonlinear buckling of micro-plates under partial excitation

In this study, a finite element formulation is proposed to study bending, thermal vibration, and buckling behavior of a modified couple stress-based micro-plate under partial piezoelectric excitation. To this end, the micro-plate is modeled using the classical plate theory (CPT) in conjunction with von Kármán nonlinear strains. The modified couple stress theory is employed to take into account the size-dependent behavior of the system. The nonlinear equations of motion are derived using Hamilton's principle. In order to obtain numerical solutions to the problem, the displacement finite element model is developed. A parameter study is conducted to study the effects of different parameters, such as temperature rise, material length parameter scale (MLSP), boundary conditions, aspect ratio, and piezoelectric layers configurations on nonlinear behavior of micro-plates. The results are divided into three separate sections: static bending analysis, thermal vibration analysis, and thermal buckling analysis; in each section, the accuracy of the model is checked by comparing the results with the ones reported in the literature. Obtained results show that the bending behavior of the micro-plate is influenced by the size effects and piezoelectric configurations; furthermore, it is observed that applied voltage to the piezoelectric layers and temperature rise affect the fundamental frequency of the system. Finally, the effect of piezoelectric layers on the buckling behavior of the model is studied; the results show that the fully covered model is less prone to buckling under the thermal loading cases. The proposed model provides good control over the mechanical performance of plate-like components, which makes the model widely applicable in the design and optimization of Micro-Electro-Mechanical-Systems (MEMS).

Nonlinear Static Bending and Free Vibration Analysis of Bidirectional Functionally Graded Material Plates

Bidirectional functionally graded material (2D-FGM) plates have mechanical properties that vary continuously in both the thickness and one-edge directions; these plates are more and more widely used in design and engineering applications. When these structures are subjected to strong loads, they can be largely deformed; therefore, nonlinear calculations, in this case, are necessary. In this paper, nonlinear static bending and nonlinear free vibration behaviors of 2D-FGM plates are studied by using the finite element method based on the third-order shear deformation theory; the Newton-Raphson method is used to solve this problem. The accuracy of this approach is confirmed by comparing the results with respect to other papers. The effects of some numerical aspect ratios such as volume fraction index and thickness-to-length ratio on nonlinear static bending and free vibration of the plates are explored. This study shows that there is a big difference between the numerical results obtained from the nonlinear problem and those from the linear one.