An analogy between analytical, approximate and numerical methods in nonlinear buckling of functionally graded columns

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Kaveh Salmalian ◽  
Ali Alijani ◽  
Habib Ramezannejad Azarboni
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Finite Difference ◽  
Stiffness Matrix ◽  
Functionally Graded ◽  
The Finite Element Method ◽  
Content Type ◽  
Tangent Stiffness Matrix ◽  
Initial Assumption ◽  
Post Buckling

Purpose The purpose of this study is to investigate the post-buckling analysis of functionally graded columns by using three analytical, approximate and numerical methods. A pre-defined function as an initial assumption for the post-buckling path is introduced to solve the differential equation. The finite difference method is used to approximate the lateral deflection of the column based on the differential equation. Moreover, the finite element method is used to derive the tangent stiffness matrix of the column. Design/methodology/approach The non-linear buckling analysis of functionally graded materials is carried out by using three analytical, finite difference and finite element methods. The elastic deformation and Euler-Bernoulli beam theory are considered to establish the constitutive and kinematics relations, respectively. The governing differential equation of the post-buckling problem is derived through the energy method and the calculus variation. Findings An incremental iterative solution and the perturbation of the displacement vector at the critical buckling point are performed to determine the post-buckling path. The convergence of the finite element results and the effects of geometric and material characteristics on the post-buckling path are investigated. Originality/value The key point of the research is to compare three methods and to detect error sources by considering the derivation process of relations. This comparison shows that a non-incremental solution in the analytical and finite difference methods and an initial assumption in the analytical method lead to an error in results. However, the post-buckling path in the finite element method is traced by the updated tangent stiffness matrix in each load step without any initial limitation.

scholarly journals Analysis of the Functionally Step-Variable Graded Plate Under In-Plane Compression

Materials ◽  
2019 ◽  
Vol 12 (24) ◽  
pp. 4090 ◽  
Author(s):  
Leszek Czechowski ◽  
Zbigniew Kołakowski
Keyword(s):  
Mechanical Properties ◽  
Finite Element Method ◽  
Finite Element ◽  
Functionally Graded ◽  
The Finite Element Method ◽  
Graded Materials ◽  
Number Of Layers ◽  
Critical Equilibrium ◽  
Post Buckling ◽  
Plane Compression

A study of the pre- and post-buckling state of square plates built from functionally graded materials (FGMs) and pure ceramics is presented. In contrast to the theoretical approach, the structure under consideration contains a finite number of layers with a step-variable change in mechanical properties across the thickness. An influence of ceramics content on a wall and a number of finite layers of the step-variable FGM on the buckling and post-critical state was scrutinized. The problem was solved using the finite element method and the asymptotic nonlinear Koiter’s theory. The investigations were conducted for several boundary conditions and material distributions to assess the behavior of the plate and to compare critical forces and post-critical equilibrium paths.

Development of Bifurcation Analysis Method for Rate Dependent Materials

2019 ◽  
Vol 794 ◽  
pp. 220-225
Author(s):  
Daiki Towata ◽  
Yuichi Tadano
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Bifurcation Analysis ◽  
Stiffness Matrix ◽  
Computational Method ◽  
Analysis Method ◽  
The Finite Element Method ◽  
Tangent Stiffness Matrix ◽  
Rate Dependent ◽  
Element Method

In this study, a novel numerical method to analyze the bifurcation problemof a rate dependent material using the finite element method is proposed. The consistent stiffness matrix, which is required for a bifurcation analysis using the finite element method, for a rate dependent material is generally hard to compute, therefore, a computational method to calculate the tangent stiffness matrix based on a numerical differential is introduced so that exact bifurcation analyses for the rate dependent material can be conducted. A numerical example of the proposed method is demonstrated, and the adequacy of the proposed method is discussed.

Development of a Model for Analysing Radial Flow of Slightly Compressible Fluids

2009 ◽  
Vol 62-64 ◽  
pp. 629-636
Author(s):  
John A. Akpobi ◽  
E.D. Akpobi
Keyword(s):  
Differential Equation ◽  
Finite Element Method ◽  
Finite Element ◽  
Finite Difference ◽  
Radial Flow ◽  
Compressible Fluids ◽  
The Finite Element Method ◽  
Time Approximation ◽  
Slightly Compressible ◽  
Fluid Properties

In this work, we develop a finite element-finite difference method to solve the differential equation governing the radial flow of slightly compressible fluids. The finite element method is used to carry out spatial approximations so as to study the variation of fluid properties at the various nodes to which effect we divided the entire radial domain of the fluid into a mesh of four radial 1-D quadratic elements which exposes nine nodes to intense study. Time approximation is done with the aid of the Crank-Nicolson finite difference scheme.

scholarly journals Numerical equilibrium analysis of a stack of steel post pallets

2018 ◽  
Vol 4 (21) ◽  
pp. 271-280
Author(s):  
Józef Pelc
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Quadratic Form ◽  
Stiffness Matrix ◽  
Equilibrium Analysis ◽  
Critical Number ◽  
The Finite Element Method ◽  
Tangent Stiffness Matrix ◽  
Forklift Truck ◽  
The Impact

A method for analyzing the equilibrium of a stack of loaded post pallets is presented. The finite element method was used to investigate the behavior of the bottom pallet in the stack during the addition of successive pallets. The stack was regarded as a self-stable multi-storey structure without bracings which is subjected to the weight of loaded pallets, horizontal forces resulting from sway and bow imperfections, and the impact of a forklift truck. The definite quadratic form of the tangent stiffness matrix after every increment in load was determined by nonlinear analysis to indicate the loss of post stability. An analysis of the stacking process of the evaluated pallets did not reveal a buckling trend in the posts of the bottommost pallet and demonstrated that the loss of equilibrium can lead to the collapse of the entire stack when a critical number of pallets is reached.

An efficient finite element formulation for nonlinear analysis of clustered tensegrity

2016 ◽  
Vol 33 (1) ◽  
pp. 252-273 ◽  
Author(s):  
Liang Zhang ◽  
Qiang Gao ◽  
Yin Liu ◽  
Hongwu Zhang
Keyword(s):  
Finite Element ◽  
Nonlinear Analysis ◽  
Stiffness Matrix ◽  
Body Motion ◽  
Finite Element Formulation ◽  
Force Density ◽  
Element Formulation ◽  
Content Type ◽  
Tangent Stiffness ◽  
Tangent Stiffness Matrix

Purpose – The purpose of this paper is to propose an efficient finite element formulation for nonlinear analysis of clustered tensegrity that consists of classical cables, clustered cables and bars. Design/methodology/approach – The derivation of the finite element formulation is based on the co-rotational approach, which decomposes a geometrically nonlinear deformation into a large rigid body motion and a small-strain deformation. A tangent stiffness matrix of a clustered cable is proposed and the Newton-Raphson scheme is employed to solve the nonlinear equation. Findings – The derived tangent stiffness matrix, including an additional stiffness terms that describes the slide effect of pulleys, can regress to the stiffness matrix of a classical cable, which is convenient for the implementation of finite element procedure. Two typical numerical examples show that the proposed formulation is accurate and requires less iteration than the force density method. Originality/value – The co-rotational formulation of a clustered cable is originally proposed, although some mature methods, such as the TL, Force Density and Dynamic Relaxation method, have been applied to nonlinear analysis of clustered tensegrity. The proposed co-rotational formulation proved efficient.

OpenMp solvers for parallel finite element and meshless analyses

2014 ◽  
Vol 31 (1) ◽  
pp. 2-17 ◽  
Author(s):  
S.H. Ju
Keyword(s):  
Finite Element ◽  
Stiffness Matrix ◽  
Preconditioned Conjugate Gradient ◽  
Content Type ◽  
Minimum Requirement ◽  
Source Codes ◽  
Parallel Solution ◽  
Parallel Finite Element ◽  
Fortran 90 ◽  
Global Stiffness Matrix

Purpose – This paper develops C++ and Fortran-90 solvers to establish parallel solution procedures in a finite element or meshless analysis program using shared memory computers. The paper aims to discuss these issues. Design/methodology/approach – The stiffness matrix can be symmetrical or unsymmetrical, and the solution schemes include sky-line Cholesky and parallel preconditioned conjugate gradient-like methods. Findings – By using the features of C++ or Fortran-90, the stiffness matrix and its auxiliary arrays can be encapsulated into a class or module as private arrays. This class or module will handle how to allocate, renumber, assemble, parallelize and solve these complicated arrays automatically. Practical implications – The source codes can be obtained online at http//myweb.ncku.edu.tw/∼juju. The major advantage of the scheme is that it is simple and systematic, so an efficient parallel finite element or meshless program can be established easily. Originality/value – With the minimum requirement of computer memory, an object-oriented C++ class and a Fortran-90 module were established to allocate, renumber, assemble, parallel, and solve the global stiffness matrix, so that the programmer does not need to handle them directly.

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