Static bending analysis of two-directional functionally graded beam using simple Timoshenko beam elements

This article presents the static bending of two-directional functionally graded (FG) beam by using simple Timoshenko beam elements. The Matlab code developed based on the finite element formulation is validated by solving two-directional FG beam problems under distributed load and two boundary conditions. Numerical results which are in terms of maximum normalized transverse deflections are compared with the analytical solutions and the results from previous studies. Besides, the shapes of transverse deflection and rotation along the length of beams are also depicted in this article to provide specific views about the static behavior of proposed structure

A Notion in Modeling Concrete Members

In this paper, the influence of aggregate size on width of fracture process zone wc is considered. Some researchers observed that the greater the grains of aggregate, the wider the fracture process zone (FPZ). The average value of the FPZ width taken from tests performed by Woliński was 26.6 mm and it did not depend on maximum aggregate size Dmax. There are no consistent conclusions as to whether the width of FPZ depends on aggregate size, and there are no standard methods of FPZ width measurement. The problem arises how to choose the width of FPZ in numerical modeling of concrete structures. For example, Bažant and Oh proposed to take wc = 3Dmax in numerical calculations. To discuss this problem, the authors’ own numerical simulations concerning bent concrete members with different widths of FPZ: 5, 10, 20, 26.5, 50 and 100 mm were performed. On the basis of the comparison of obtained results, significant differences dependent on wc have been observed. Taking into account the minimum potential energy in a member, it can be said that the most rational thing to do is to take the smallest elongation within the localized microcracking. This condition takes place in the analyzed beam when wc = 50 mm. The assumption wc = 3Dmax does not fit this criterion. Also, the width from the experiment performed by Woliński is not in good relation to obtained numerical results. The main conclusion from this paper is that the width of FPZ does have an influence on obtained numerical results performed by crack band model. The problem of estimating the width of FPZ in numerical simulations exists and requires further research

Static bending analysis of two-directional functionally graded beam using simple Timoshenko beam elements

This article presents the static bending of two-directional functionally graded (FG) beam by using simple Timoshenko beam elements. The Matlab code developed based on the finite element formulation is validated by solving two-directional FG beam problems under distributed load and two boundary conditions. Numerical results which are in terms of maximum normalized transverse deflections are compared with the analytical solutions and the results from previous studies. Besides, the shapes of transverse deflection and rotation along the length of beams are also depicted in this article to provide specific views about the static behavior of proposed structure

A study of functionally graded porous beam based on simple beam theory

In this article, the bending behaviors of functionally graded porous (FGP) beams are determined associated with uniform load. The simple beam theory is carried out with various boundary conditions. Two types of porosity are also applied to study the influences of material properties on bending behaviors. The results obtained in this article are presented and compared with other results in the references to verify the correctness in implementing the formula and writing the Matlab code. Last but not least, this article can help researchers to have an overview of the bending characteristics of the functionally graded porous beams

A study of functionally graded porous beam based on simple beam theory

In this article, the bending behaviors of functionally graded porous (FGP) beams are determined associated with uniform load. The simple beam theory is carried out with various boundary conditions. Two types of porosity are also applied to study the influences of material properties on bending behaviors. The results obtained in this article are presented and compared with other results in the references to verify the correctness in implementing the formula and writing the Matlab code. Last but not least, this article can help researchers to have an overview of the bending characteristics of the functionally graded porous beams

Approximated transverse deflection of sandwich beam with 2D-FG and ceramic face sheets and 1D-FG core

The approximated numerical deflection of a sandwich beam with two directional functionally graded (2D-FG) and ceramic face sheets and one directional functionally graded (1D-FG) core, namely SW2D1DC, is presented under uniform load and various boundary conditions. The finite element code written in Matlab is applied in this article to investigate the influences of material properties on transverse deflections. The results of this article are given and compared with other results in the references to verify the feasibility of the application. This study also provides some more information about the characteristics of SW2D1DC beams

Approximated transverse deflection of sandwich beam with 2D-FG and ceramic face sheets and 1D-FG core

The approximated numerical deflection of a sandwich beam with two directional functionally graded (2D-FG) and ceramic face sheets and one directional functionally graded (1D-FG) core, namely SW2D1DC, is presented under uniform load and various boundary conditions. The finite element code written in Matlab is applied in this article to investigate the influences of material properties on transverse deflections. The results of this article are given and compared with other results in the references to verify the feasibility of the application. This study also provides some more information about the characteristics of SW2D1DC beams

Exact Solutions for Fixed-Fixed Anisotropic Beams under Uniform Load by Using Maple

The approximate solutions of stresses and displacements were obtained for fixed-fixed anisotropic beams subjected to uniform load. A stress function involving un-known coefficients was constructed, and the general expressions of stress and displacement were obtained by means of airy stress function method. Two types of the description for the fixed end boundary condition were considered. The introduced unknown coefficients in stress function were determined by using the boundary conditions. The approximate solutions for stresses and displacements were finally obtained. Numerical tests show that the solutions agree with the FEM results. These solutions are achieved by using Maple software

A NEW C0 THIRD-ORDER SHEAR DEFORMATION THEORY FOR THE NONLINEAR FREE VIBRATION ANALYSIS OF STIFFENED FUNCTIONALLY GRADED PLATES

Nonlinear free vibration of stiffened functionally graded plates is presented by using the finite element method based on the new C0 third-order shear deformation theory. The material properties are assumed to be graded in the thickness direction by a power-law distribution. Based on the Von Karman theory and the third-order shear deformation theory, the nonlinear governing equations of motion are derived from the Hamilton’s principle. An iterative procedure based on the Newton-Raphson method is employed in computing the natural frequencies and mode shape. The comparison between these solutions and the other available ones suggests that this procedure is characterized by accuracy and efficiency

A COMBINED STRAIN ELEMENT TO FUNCTIONALLY GRADED STRUCTURES IN THERMAL ENVIRONMENT

Functionally graded materials are commonly used in a thermal environment to change the properties of constituent materials. They inherently withstand high temperature gradients due to a low thermal conductivity, core ductility, low thermal expansion coefficient, and many others. It is essential to thoroughly study mechanical responses of them and to develop new effective approaches for an accurate prediction of solutions. In this paper, a new four-node quadrilateral element based on a combined strain strategy and first-order shear deformation theory is presented to achieve the behaviour of functionally graded plate/shell structures in a thermal environment. The main notion of the combined strain strategy is based on the combination of the membrane strain and the shear strain related to tying points as well as bending strain with respect to a cell-based smoothed finite element method. Due to the finite element analysis, the first-order shear deformation theory (FSDT) is simple to implement and apply for structures, but the shear correction factors are used to achieve the accuracy of solutions. The author assumes that the temperature distribution is uniform throughout the structure. The rule of mixtures is also considered to describe the variation of material compositions across the thickness. Many desirable characteristics and the enforcement of this element are verified and proved through various numerical examples. Numerical solutions and a comparison with other available solutions suggest that the procedure based on this new combined strain element is accurate and efficient