Analysis of a Cantilever Beam using Element Free Galerkin Method and Finite Element Method

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Analysis of a Cantilever Beam using Element Free Galerkin Method and Finite Element Method

Corresponding Author: Remya. C. R INTRODUCTION

A mesh is an open space between the strands of a net that is formed by connecting nodes in a predefined manner [1]. In mesh free methods, there is no connectivity existing between the nodes so we can add or delete nodal points whenever it is necessary without worrying about their relations. This provides some advantages such as it can be used for analyzing large deformation problems with moderately high accuracy. The main drawback of FEM is that the process of mesh generation and remeshing was computationally heavy and require more time. This problem does not occur in case of meshless methods. The main features of mesh free methods are,

1. Can be used to capture maximum stress concentrated at nodes.

2. Can be effectively used for crack growth problems.

ABSRACT

Nowadays, numerical methods are used to solve many of the engineering problems with the help of computers. Among them two commonly used conventional mesh based methods are Finite Difference Method (FDM) and Finite Element Method (FEM), but these methods have some limitations; first of all, it was difficult to handle complex problems and geometries such as problems with deformed boundaries, free surface, large deformation problems etc., Secondly the process of grid generation was tedious and time consuming.

Innorder to overcome these difficulties a new method called meshless method was introduced.

Element Free Galerkin Method is one of the meshless methods and is found to be superior to FEM.

In the present study, analysis of a cantilever beam was done under static loading condition using EFG and FE methods and the results were compared and finally validated using analytical solutions. From this study I concluded that the EFGM was an effective alternative to FEM and is more flexible than FEM.

Key words: EFGM, FEM, Meshless Methods

Remya C R^#1, Suji P ^#2

#1 M Tech Student, Sri Vellappally Natesan College of Engineering, Mavelikara, 9497112620

#2 Asst. Professor, Dept. of Civil Engineering, Sri Vellappally Natesan College of Engineering, Mavelikara, 9562446045

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etc.,

4. No need of mesh.

5. Can handle large deformation problems [1].

Recently a blended Finite-Element and Meshfree Galerkin approximation scheme was adopted to solve the inelastic response of the plane frames [2] and was found to be a better choice to solve engineering problems. In this area numerous studies were done to evaluate the performance of EFGM and from those studies it was found that it is a best alternative to FEM. The main problem in EFGM was, the MLS method used to construct shape function does not satisfy Kronecker δ property[3], even though MLS method was widely used, because it reduces much computational effort. So an improved interpolating moving least square method was found to be a better one to solve many problems.

MATERIALS AND METHODS

The study was conducted in the following steps,

1. A detailed study was done to understand the theoretical background of EFG and FE methods.

2. An algorithm was developed and analyzed in MATLAB to obtain the parameters such as displacements and stresses using EFGM.

3. Then the same beam was analyzed using FEM in ANSYS V.14.

4. Comparison of these two results was carried out and then validation was done with its analytical solutions.

The methodology flow chart for the analysis is shown in Fig 1.

Fig 1 Overview of methodology Geometry creation

Mesh generation

Shape function based on element

Shape function based on nodes

Discretized system equations

Solution for field variables Node generation

FEM EFGM Analytical Solution

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©2017 RS Publication, [email protected] Page 112 The modeling was done with The Timoshenko beam (allows shear deformation) and has a dimension of 48m x 12m x 1m. Material properties; Youngs modulus (E) =30 GPa and Poissons ratio (μ) = 0.3. One end of the beam was fixed and at the other end a point load of 1000 N was applied. The beam is shown in Fig 2.

Fig 2 Timoshenko beam FINITE ELEMENT ANALYSIS

FEM is a computer aided numerical method for obtaining approximate numerical solution to the abstract equations of calculus that predict the response of physical systems subjected to external influences. ANSYS is finite element analysis software which enables to perform various tasks [4]. Here ANSYS V.14 was used. Beam element 189 was used which is a linear beam element suitable for slender sections and was based on Timoshenko beam theory.

MATLAB ANALYSIS

MATLAB is a high performance platform for doing engineering calculations especially matrix calculations. The name MATLAB is Derived from MATrix LABoratory. Here in MATLAB an algorithm for EFGM was developed and analysed to obtain the value of displacements and stresses at various nodal points. Moving Least Square Method (MLS) was used to construct the shape functions from scattered data (nodal points).

ANALYTICAL SOLUTIONS

The exact analytical solutions for the Timoshenko beam was obtained by using the following equations; [5]

For displacement,

For stress,

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Uy = Displacement in y direction D = Depth of beam

L = Total length of the beam P = Load applied at the free end I = moment of inertia

σx and σy = Stresses in x and y direction respectively

RESULTS AND DISCUSSIONS Finite Element Method

The displacement values at various points obtained from ANSYS V.14 are listed below in table 2 and the stress contour for the beam was shown in Fig 3. From this contour we can see that the maximum stress is 2000 and is at the support where maximum bending occurs.

Table 2 Displacement from FEM

Distance from fixed end (m) Displacement in y direction (mm)

0 0

4 0.00020

8 0.00050

12 0.00089

16 0.00150

20 0.00220

24 0.00287

28 0.00384

32 0.00484

36 0.00579

40 0.00687

44 0.00786

48 0.00910

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Element Free Galerkin Method

In this paper at first, the analysis of the Timoshenko beam was done in MATLAB platform using EFGM. The displacement values obtained at various points were listed in table1.

Table 1 Displacement from EFGM

Distance from fixed end (m) Displacement in y direction (mm)

0 0.00000

4 0.00010

8 0.00040

12 0.00090

16 0.00140

20 0.00210

24 0.00290

28 0.00370

32 0.00470

36 0.00570

40 0.00670

44 0.00780

48 0.00890

The stress is found to be 2000 N/m² at support and zero at free end.

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Here analytical equations are used to validate above two methods. The displacements obtained are listed below (Table 3).

Table 3 Analytical displacements

Distance from fixed end (m) Displacements in y direction (u y)

0 0.00000

4 0.00011

8 0.00039

12 0.00082

16 0.00128

20 0.00206

24 0.00285

28 0.00372

32 0.00466

36 0.00567

40 0.00672

44 0.00780

48 0.00889

From the analytical equation the stress obtained is 2000 N/m² at support and zero at the free end.

Comparison and Validation

First of all the EFGM was compared with the FE results and found that, two results were matches each other but there was a slight variation. Then the results were validated using analytical solutions, which is the exact solution for the problem. After validation we can see that the EF solution has high convergence rate than FEM. The comparison graph is shown in Fig 4.

Fig 4 Comparison Graph

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0 4 8 12 16 20 24 28 32 36 40 44 48

displacements (mm)

Distance from free end (m)

ANALYTICAL EFGM

FEM

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Important conclusions drawn from the study and are listed below,

1. EFGM was capable of analyzing structures and has a high convergence rate (more accurate).

2. Validation using analytical solution shows that the results incorporate with EFGM, but there was a slight variation in case of displacements.

3. The percentage of error between analytical and EFG solution was 0.112% and between analytical and FEM was 2.362%.

4. FEM also gives a better result but compared to other two methods but is less accurate and difficult to analyze complex geometrical problems.

5. From all the three analysis, the stress values were found to be same and are maximum at support.

REFERENCE

[1] G. R. Liu, Meshfree methods: moving beyond the finite element method, CRC Press, 2003.

[2] L. Louie, Meshfree method for Inelastic Frame analysis, Journal of Structural Engineering, pp 676-684, 2009.

[3] M. M. Hamed and B. Idir, appling element free Galerkin method on beam and plate, International Journal of Chemical, Molecular, Nuclear, Materials and Metallurgical Engineering, Vol 10, pp 627-635,2016.

[4] S. Moaveni, Finite element analysis theory and application with ANSYS, 1999.

[5] S. S. Pandey, P. K. Kasundra and S. D. Daxini, Introduction of Meshfree Methods and Implementation of Element Free Galerkin (EFG) Method to Beam Problem, International Journal on Theoretical and Applied Research in Mechanical Engineering, Vol 2(3), pp.85-89, 2013.