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Finite Element Analysis (FEA) Ali tayebisadrabadi
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FEA Report

Feb 08, 2017

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Page 1: FEA Report

Finite Element Analysis (FEA)

Ali tayebisadrabadi

Page 2: FEA Report

INTRODUCTION  In order to consider the effects of new technologies in society we would like

to mention the advances in industry. Numerical modeling has a variety of

applications and offers an efficient method for solving highly complex

engineering problems. Techniques such as finite element analysis enable us to

assess complex problems for which analytical solutions are not feasible. We cannot

have the analytical solutions for complex problems; however, we can provide the

numerical solutions for them by modelling in finite element software [1, 2],

simulation with graphical software and many more options. The aim of this project

is to investigate the effects of mesh density in the accuracy of the finite element

solution.

The application of finite element methods requires an understanding of its

background, applications and methodologies. In this problem we want to study

loading of a fixed-free cantilever beam with respect to the effect of variations in

elements’ size [3]. Maybe it is not true but this parameter might be called the mesh

density. In order to conduct finite element analysis, an awareness of the

fundamentals of the software is required. In general, finite element analysis

includes the following steps [4]:

- Model creation

- Model idealisation

- Symmetries

- Meshing and discretisation (simplification of the problem)

- Specifying initial conditions and limitations

- Applying loading and boundary conditions

- Methods of solution

Page 3: FEA Report

The above stages are common in finite element methods. However in the use of

specific finite element modelling software such as ABAQUS, further steps are

necessary as follows:

- Defining the type of analysis

- Creating or importing the geometry

- Defining material properties such as modulus of elasticity, poisson’s ratio

and any other required parameters

- Specifying mapped, sweep or simple meshing methods

- Follow the previous measured examples

- Post-processing of the model

- Extracting the desired results

Materials  &  Methods    In this report we will compare the results of analytical solutions with the results of

numerical modelling. The desired parameters for this model are the maximum

stress and the maximum deflection of the beam. Therefore, in the following

section, we will first describe the analytical solution of the model.

Analytical  Solution  Here, we have a fixed-free beam which is called a cantilever beam, and we would

like to study its behaviour upon applying vertical loading, assuming an elastic

model. We can imagine that on loading, due to deformation, we will have a

complete shape of a curve for beam. This would be because of this point that with

respect to the deformation in each point we will have a curved deflection for the

Page 4: FEA Report

point of beam. We can derive the radius of this curvature from the two following

equations [5]:

!!=

!!!!"!

[!!(!"!")!]!

!    

1𝑅=M(x)EI

The first equation is derived from the equation of a curved function with

respect to 2D variation of its motion (planar assumption). And the second equation

is with respect to the elastic behaviour of the beam. It is a general formulation for

curved beams and we use it to define an analytical solution for this problem. With

respect to low variation in square of dy/dx relatively to 1, and the combination of

equations we will have for torque:

𝑀(𝑥) = EId2ydx2

If we consider this equation we will find that moment of each point relates to its

modulus of elasticity, second moment of inertia, and the second order derivative of

the shape function. These magnitudes vary for different locations. Now we can

easily find the maximum value for deflection at the end of the beam [5].

𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙  𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 =𝐹𝐿!

3𝐸𝐼  &  𝐹 = 𝑘𝑥         =>    𝐾 = 3𝐸𝐼/𝐿!

Now if we consider the beam as a general linear spring with the equation of motion

F=kx, we can have the 3EI/L3 as the stiffness of the spring for the horizontal beam

in cantilever set up.

Page 5: FEA Report

The second goal of this project is to use the results of the analytical solution for

maximum stress to investigate the accuracy of the numerical solution under

varying mesh densities. Therefore, we will setup our equations to find the

maximum available stress at the model.

For a general cantilever beam under different types of loading and

geometrical conditions, we will have the following conditions:

- Define the type of loading (axial, torsion, momentous, shear force, … )

- Detect the critical points

- Define the stresses due to different loadings

- Combine the stresses with a unique failure criteria method

- Define the most critical point (the maximum stress)

If we want to consider the general path of finding the maximum stress in loading

and conduct an indepth analysis of beams we need to follow a definite path.

However, in this project the focus is on maximum normal stress.

There are also another option for our analysis which is to consider the model as a

complete 2 dimensional model which is probably not incorrect for this problem.

However, there might be inevitable problem in magnitudes of stress from 2D to 3D

transformation.

When we apply a point load at the end of the beam it will have definite magnitudes

of stress at the fixed end due to moment and point vertical load transfers to the

area.

Therefore normal stress will be finally:

𝑛𝑜𝑟𝑚𝑎𝑙  𝑠𝑡𝑟𝑒𝑠𝑠 = ±𝑀𝑐𝐼    (+𝑓𝑜𝑟  𝑡𝑒𝑛𝑠𝑖𝑜𝑛  &     − 𝑓𝑜𝑟  𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛  )

Page 6: FEA Report

There is also a transferred vertical point load at the fixed end which results

to shear stress which is:

𝑠ℎ𝑒𝑎𝑟  𝑠𝑡𝑟𝑒𝑠𝑠 = ±𝑉𝑄𝐼𝑡    𝑤ℎ𝑖𝑐ℎ  𝑚𝑖𝑔ℎ𝑡  𝑏𝑒  𝑛𝑒𝑔𝑙𝑖𝑔𝑎𝑏𝑙𝑒  

However this shear stress is zero for free surface which have the highest

magnitude of normal stress. In the following, the results of the analytical solution

are presented, which will be useful in analysing of the numerical solution.

Analytical  calculation    In the previous section on analytical solution, the following equation was

derived. This is used for the calculation of vertical deflection and the x component

of stress.

𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙  𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛  𝑤𝑖𝑙𝑙  𝑏𝑒  𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑  𝑡ℎ𝑟𝑜𝑢𝑔ℎ  𝑡ℎ𝑒   =𝐹𝐿!

3𝐸𝐼

=5 9.8 0.75 !

3 ∗ 113,800,000,000 ∗ 112 ∗ 0.001 ∗ 0.022

!= 0.068232

𝑎𝑛𝑑  𝑡ℎ𝑖𝑠  𝑤𝑖𝑙𝑙  𝑏𝑒  𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑒𝑑  𝑤𝑖𝑡ℎ  𝑥  𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡  𝑜𝑓  𝑠𝑡𝑟𝑒𝑠𝑠 =  = ±𝐹𝐿𝑐𝐼

= ±5(9.8) 0.75 (0.022)/2112 ∗ 0.001 ∗ 0.022

!= 4.5558  𝑒8  

Define  the  geometry  and  conditions  of  the  model:  There are many options for creating a geometrical model of the required

system. We can assume a complete 3D model or use a simpler 2D model and specify its thickness. After creating the model, other options are specified in modelling process, such as applying the elastic-plastic model conditions for the beam and also defining a failure criterion for the model.

Page 7: FEA Report

We would like to use a 2D or 3D model for simulating a fixed-free cantilever beam. This will help us to produce a complete model for our simulations. The behaviour of 2D & 3D models might have different results. This will be possible for when we use a 2D model with real constant. Under this condition, the element’s thickness will be imported into the analysis by the name of real constant in 2D analysis. Therefore, the thickness term is not entered in geometrical parameters. In this project we will describe the geometrical condition of the model of our fixed-free cantilever beam in ABAQUS.

 

Figure  1:  the  given  2D  geometry  for  the  model  f  cantilever  beam As it is obvious from the model we have a rectangular shape for the beam in

2D with 22*750 mm2 dimensions (Figure 1). The rectangular bar has the thickness

of 1 mm and therefore for the cross section of the beam we have another rectangle

of 1*22 mm2. The left side is completely fixed and the right side is under a load of

5 kg, which will be considered as a point load in our modelling processes. Further

information on the beam is given in the following table:

Table1: material properties and beam characteristics

Fixed free cantilever Beam Magnitude of Parameters

Module of Elasticity 113.8 GPa

Poison Ratio 0.342 (dimensionless)

Ultimate tensile stress 950 MPa (14% strain rate considered)

Yield stress/ Engineering stress 880 MPa

Page 8: FEA Report

The boundary conditions for this model are simple; with no displacement for

the fixed end and considering the middle axis of the beam to have no normal

tension (this is ideal assumption for 2D modelling).

   Simulation,  Modelling  &  Results    

In this research we will obtain the results for maximum vertical deflection,

maximum x component of stress, and also the time used for the process to be

solved with FEA. In this regard, we used ABAQUS commercial software to model

our geometry and will demonstrate the efficiency of the using ABAQUS software

for this project.

The following results will show the pictures of the model which are derived

under different mesh densities/ element numbers. By using smaller amounts for

element sizes the model can be idealized to higher number of elements. The

following table and the following figures can explain the conditions of the FEM

modeling and its results. As an example let us consider to the following table.

Table 1: information for the element types, numbers & etc...

Element Numbers Element Length Node numbers Element type

50 8.7234E+00 253 8-node linear brick, reduced integration, hourglass control

280 5.2707E+00 1140 8-node linear brick, reduced integration, hourglass control

1128 2.7187E+00 3969 8-node linear brick, reduced integration, hourglass control

4125 1.5000E+00 13536 8-node linear brick, reduced integration, hourglass control

11250 9.7407E-01 35682 8-node linear brick, reduced integration, hourglass control

Page 9: FEA Report

The following figure illustrates the variation in vertical deflection and x component

of stress in different locations along the modelled beam.

 

Figure  2:  results  of  a  model  with  280  elements  in  ABAQUS.  

In the following figure we can observe the maximum and minimum magnitudes for vertical deflection. This image is generated using the ABAQUS report generator toolbox.

 

Figure  3:  The  figure  shows  that  maximum  and  minimum  magnitude  for  vertical  deflection  were  highlighted  for  the  model

Page 10: FEA Report

There are also other parameters such as running time for the process, which might

be considered important. The following table demonstrates the time duration used

for each process.

# of elements Start time Finish time Used time

50 23:58:55 23:59:14 19

280 00:39:52 00:40:11 19

1128 00:59:26 00:59:46 20

4125 01:07:59 01:08:28 29

11250 01:16:01 01:16:59 58

The used time for different numbers of elements shows that we have a slight

increment for the used time of the process with increasing number of elements.

The derived data for the vertical deflection and x component of maximum stress is

given below. This is highly applicable for analysing the effects of mesh density

and changing element sizes.

Element numbers 50 280 1128 4125 11250

Vertical deflection (mm) 8.52e1 7.83e1 7.16e1 7.02e1 6.99e1

Von-mises stress 3.28e2 3.58e2 3.92e2 4.22e2 4.50e2

As it is obvious from the results of analytical solution that we need to increase our

element numbers. With increase in the number of elements, the results are in

better agreement with the analytical solution however, we will have more analysis

cost. Due to the greater time required for running the modelling process, we will

need to have a stronger processor. Therefore, it is very common to use a middle

point between these options, which might be called optimisation.

Page 11: FEA Report

 

Figure  4:  figure  shows  the  variation  of  different  parameters  with  changing  in  element  numbers  

In this graph we can find the trend of different parameters with variation in number

of elements used in the FEM model. We can observe the variation in maximum

deflection in the vertical direction which increases and gets closer to its value in

the analytical solution. Therefore we find it that approximately 4125 could be the

optimum number of element to be used for this simple model in 1000 kg loading.

In the following section we will illustrate the results of applying 1000 kg loading

on the same simple beam after meshing with more than 4000 elements.

1000   kg   loading   on   simple   beam:  The results of applying 1000 kg load on the

same simple beam were generated as below:

- Characteristic element length 1.5000E+00

- Number of element 8250/2 = 4125 (we are using 8 node elements)

- Total number of nodes: 13536

- Maximum stress: 8.44e10 pa

- Vertical deflection: 1.4e4 mm

0   2   4   6   8   10  

50  

280  

1128  

4125  

11250  

process  /me/10  

Maximum  stress  (X  component)*  e8  

ver/cal  deflec/on  (mm)  

Page 12: FEA Report

 

Figure  5:  results  of  modelling  with  1000kg  applied  load The results for vertical deflection and maximum stress might be a little uncertain.

For this amount of loading we need to define plasticity in our modelling. The

elastic assumption may not work for very well for this amount of loading.

Reference:          1. ABAQUS software 2. ABAQUS software report generator 3. LIANG Zu-feng, TANG Xiao-yan, (2007) “Analytical solution of fractionally damped

beam by Adomian decomposition method”, Applied Mathematics and Mechanics (English Edition), 28(2):219–228.

4. M. Fooladi, et al. (2009) “On the Analytical Solution of Kirchhoff Simplified Model for Beam by using of Homotopy Analysis Method” , World Applied Sciences Journal 6 (3): 297-302.

5. Jim Butterworth(1999) “Finite Element Analysis of Structural Steelwork Beam to Column Bolted Connections” Constructional research Unit.