Accounting for geometric nonlinearity in finite element strength calculations of thin-walled shell-type structures

Relevance. Currently, in connection with the wider spread of large-span thinwalled structures such as shells, an urgent issue is the development of computational algorithms for the strength calculation of such objects in a geometrically nonlinear formulation. Despite a significant number of publications on this issue, a rather important aspect remains the need to improve finite element models of such shells that would combine the relative simplicity of the resolving equations, allowance for shear deformations, compactness of the stiffness matrix being formed, the facilitated possibility of modeling and changing boundary conditions and etc. The aim of the work is to develop a finite element algorithm for calculating a thin shell with allowance for shear deformations in a geometrically nonlinear formulation using a finite element with a limited number of variable nodal parameters. Methods. As research tools, the numerical finite element method was chosen. The basic geometric relations between the increment of deformations and the increment of the components of the displacement vector and the increment of the components of the normal vector angle are obtained in two versions of the normal angle of the reference. The stiffness matrix and the column of nodal forces of the quadrangular finite element at the loading step were obtained by minimizing the Lagrange functional. Results. On the example of calculating a cylindrical panel rigidly clamped at the edges under the action of a concentrated force, the efficiency of the developed algorithm was shown in a geometrically nonlinear setting, taking into account the transverse shear strain.

Finite Element Analysis of Functionally Graded Beams using Different Beam Theories

The present study deals with buckling, free vibration, and bending analysis of Functionally Graded (FG) and porous FG beams based on various beam theories. Equation of motion and boundary conditions are derived from Hamilton’s principle, and the finite element method is adopted to solve problems numerically. The FG beams are graded through the thickness direction, and the material distribution is controlled by power-law volume fraction. The effects of the different values of the power-law index, porosity exponent, and different boundary conditions on bending, natural frequencies and buckling characteristics are also studied. A new function is introduced to approximate the transverse shear strain in higher-order shear deformation theory. Furthermore, shifting the position of the neutral axis is taken into account. The results obtained numerically are validated with results obtained from ANSYS and those available in the previous work. The results of this study specify the crucial role of slenderness ratio, material distribution, and porosity condition on the characteristic of FG beams. The deflection results obtained by the proposed function have a maximum of six percent difference when the results are compared with ANSYS. It also has better results in comparison with the Reddy formulae, especially when the beam becomes slender. Doi: 10.28991/cej-2020-03091604 Full Text: PDF

Creasing severity and reverse-side cracking

Crease cracking can be detrimental to the functionality and appearance of paperboard-based pack-aging. The effect of creasing severity on the degree of reverse-side crease cracking (bead-side of the crease) of paperboard was investigated. Samples were creased with a range of rule and channel geometries, and the cracking degree was quantified as the percent of cracked length relative to the total length of the crease. The cracking degree was typically below 5% at low crease penetration depths, but was exponentially higher beyond a critical penetration depth. A rule and channel combination with a wider clearance shifted the critical depth to larger values. The creasing severity parameter, termed the creasing draw, converged the cracking degree data from different rule and channel combinations to a single curve. The creasing draw was derived from the same analytical expres-sions as the transverse shear strain and quantifies the length of paper that is drawn into the channel during creasing. The critical draw is defined as the draw at which cracking becomes greater than 5%, which corresponds with the point at which cracking becomes exponential. The critical draw is a material/system parameter that defines the level below which cracking is minimal.

Dynamic analysis of functionally graded carbon nanotube–reinforced shell structures with piezoelectric layers under dynamic loads

This research makes a first attempt to investigate the dynamic characteristics of functionally graded carbon nanotube–reinforced composite plates and shell structures with surface-bonded piezoelectric layers. A variational formulation is derived based on the linear double director shell theory to ensure realistic parabolic variation of transverse shear strain along the thickness direction. The assumed natural strains method is adopted to enhance the accuracy of the four-node piezoelectric shell element developed in this study. Numerical studies are conducted to validate the efficiency and numerical stability of the proposed model to predict the behavior of piezolaminated composite shell structures. Furthermore, dynamic responses are extended to functionally graded carbon nanotube–reinforced composite shells covered by two active layers. The host structure is reinforced by single-walled carbon nanotubes, which are assumed to be graded through the thickness direction with different types of distributions and embedded in a polymer matrix. The effect of the volume fractions, distribution type, and geometrical parameters of the carbon nanotubes is examined.

Modeling, Analyses, and Optimization of Planar Active Frame Structures Composed of Piezoelectric Beams

Frame structures are widely used in engineering applications, especially in space structures. For special use such as shape and vibration control of such structures, piezoelectric patches are usually placed on the beam surfaces to form active frame structures. To perform shape control or vibration control tasks, modeling methods for the formed active frame structures need to be studied. This paper develops a new distributed model of an active frame structure composed of multilayer piezoelectric beam components. First, the governing equations of a beam, bonded with piezoelectric patches, are developed via the generalized Hamilton principle, by considering the transverse shear strain. Then, the analytical solutions of the governing equations and the generalized element stiffness matrix are derived through the distributed transfer function formulation. Finally, the analytical solution of the entire system is obtained by the technique for assembling element stiffness matrix. In numerical simulations, buckling and vibration of an active frame structure are both studied. In addition, a novel Improved Ant Lion algorithm is proposed for optimal design of the frame structures. The optimization examples confirm that the proposed algorithm is more efficient than other existing popular algorithms such as Genetic Algorithm (GA) and Ant Lion Optimization (ALO) algorithm.

Free Vibration Analysis of Smart Laminated Functionally Graded CNT Reinforced Composite Plates via New Four-Variable Refined Plate Theory

This paper presents a new four-variable refined plate theory for free vibration analysis of laminated piezoelectric functionally graded carbon nanotube-reinforced composite plates (PFG-CNTRC). The present theory includes a parabolic distribution of transverse shear strain through the thickness and satisfies zero traction boundary conditions at both free surfaces of the plates. Thus, no shear correction factor is required. The distribution of carbon nanotubes across the thickness of each FG-CNT layer can be functionally graded or uniformly distributed. Additionally, the electric potential in piezoelectric layers is assumed to be quadratically distributed across the thickness. Equations of motion for PFG-CNTRC rectangular plates are derived using both Maxwell’s equation and Hamilton’s principle. Using the Navier technique, natural frequencies of the simply supported hybrid plate with closed circuit and open circuit of electrical boundary conditions are calculated. New parametric studies regarding the effect of the volume fraction, the CNTs distribution, the number of layers, CNT fiber orientation and thickness of the piezoelectric layer on the free vibration response of hybrid plates are performed.

Static analysis of carbon nanotube-reinforced FG shells using an efficient solid-shell element with parabolic transverse shear strain

Purpose This paper aims to study the static behavior of carbon nanotubes (CNTs) reinforced functionally graded shells using an efficient solid-shell element with parabolic transverse shear strain. Four different types of reinforcement along the thickness are considered. Design/methodology/approach Furthermore, the developed solid-shell element allows an efficient and accurate analysis of CNT-reinforced functionally graded shells under linear static conditions. Findings The validity and accuracy of the developed solid-shell element are illustrated through the solution of deflection and stress distribution problems of shell structures taken from the literature. The influences of some geometrical and material parameters on the static behavior of shell structures are discussed. Originality/value The finite element formulation is based on a modified first-order enhanced solid-shell element formulation with an imposed parabolic shear strain distribution through the shell thickness in the compatible strain part. This formulation guarantees a zero transverse shear stress on the top and bottom surfaces of the shell and the shear correction factors is no longer needed.

A Refined Simple First-Order Shear Deformation Theory for Static Bending and Free Vibration Analysis of Advanced Composite Plates

A refined simple first-order shear deformation theory is developed to investigate the static bending and free vibration of advanced composite plates such as functionally graded plates. By introducing the new distribution shape function, the transverse shear strain and shear stress have a parabolic distribution across the thickness of the plates, and they equal zero at the surfaces of the plates. Hence, the new refined theory needs no shear correction factor. The Navier solution is applied to investigate the static bending and free vibration of simply supported advanced composite plates. The proposed theory shows an improvement in calculating the deflections and frequencies of advanced composite plates. The formulation and transformation of the present theory are as simple as the simple first-order shear deformation. The comparisons of deflection, axial stresses, transverse shear stresses, and frequencies of the plates obtained by the proposed theory with published results of different theories are carried out to show the efficiency and accuracy of the new theory. In addition, some discussions on the influence of various parameters such as the power-law index, the slenderness ratio, and the aspect ratio are carried out, which are useful for the design and testing of advanced composite structures.