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Lab report carried out from performing finite element analysis on a plate with a hole at its center....

Description

Finite element analysis report. Digital stress analysis of a uniform plate with a hole at its centre. Jonathan Barker. E3 Mechanical engineering. Student number: @00523823.

Contents Abstract. .................................................................................................................................................. 3 1.

2.

3.

4.

Introduction. ................................................................................................................................... 3 1.1.

Aims and objectives. ............................................................................................................... 4

1.2.

Scope of report. ...................................................................................................................... 4

Methodology................................................................................................................................... 5 2.1.

Data analysis tests................................................................................................................... 7

2.2.

Mesh density........................................................................................................................... 8

2.3.

Equations used. ..................................................................................................................... 10

Results. .......................................................................................................................................... 11 3.1.

Tables. ................................................................................................................................... 11

3.2.

Graphs. .................................................................................................................................. 15

3.3.

Corresponding diagrams and visual plots. ............................................................................ 19

Discussion of results. ..................................................................................................................... 23 4.1. R1. .............................................................................................................................................. 23 4.2. R2. .............................................................................................................................................. 23 4.3. R3. .............................................................................................................................................. 23 4.4. R4. .............................................................................................................................................. 23 4.5. R5 ............................................................................................................................................... 24 4.6. R6. .............................................................................................................................................. 24 4.7. R7. .............................................................................................................................................. 24 4.8. Overall evaluation of results. ..................................................................................................... 24

5.

Conclusions. .................................................................................................................................. 25

6.

References. ...................................................................................................................................26

Abstract. In this report, several aspects of digital analysis have been carried out upon a symmetrical and uniformly loaded plate of given vertical length, thickness, and containing a hole at its centre of given diameter. The purpose of this report is to show the relationship between analytic mesh density used in the simulation software ANSYS and structural deformation quantities, I.E., longitudinal stress, displacement in the x direction, and the stress concentration factor (K). Using collected and tabulated data from studying a simulated model on ANSYS, graphically presented data in corelation to the given analysis tasks, and visual plotting images taken from ANSYS; this report shows that there is a direct correlation in the accuracy of data values and the mesh density. As the mesh density becomes finer, the accuracy of the data increases and converges to a point. Another aspect of this report discusses the comparison of two differently set bodies of identical geometry however the software is programmed to have one body be set to a standard 4 noded quadrilateral, and another to be set with 8 nodes. Comparatively, it can be surmised that both plates demonstrate conversion to a point but the data for each does trend in different ways. Furthermore, the percentage error and inconsistency within the 4 noded body is far greater than when compared to the 8 noded body.

1. Introduction. The widespread use of finite element analysis throughout the world of manufacturing engineering is because its application can detect problems within crucial components of a design and the process can digitally simulate specific and required conditions that the part must undergo without physical testing processes that would waste physical resources and time, therefore making finite element analysis of components a crucial aspect of manufacture engineering testing. In this computerised laboratory experiment, the task set is to design and set up two relatively simple components in ANSYS mechanical APDL. The components being two plates of a certain length, thickness, and diameter with a hole in the centre of the plate of a given radius, with a given load attached to the plate. The one aspect in which the plates differed was in the number of nodes associated with the separating of the plate. One body would be set with 4 noded quadrilaterals and another will be set with 8 noded quadrilaterals. Then in undergoing the analysis, the comparative accuracy of the data produced by each body can be compared. Each body was subjected to 7 different tests to analyse different aspects of the part. Using ANSYS, quantities like the longitudinal stress and the displacement in the x direction could be plotted onto the part and show the areas of high/low pressure or displacement. ANSYS also allows the user to change the mesh analysis density to observe and see different trends in the accuracy of measuring said quantities. This feature is used to make the mesh density increasingly finer, experimentally it is hoped that the value for longitudinal stress will converge more accurately to a final result and then subsequently this convergence can be shown graphically (see 3. results and diagrams) for both bodies for 5 out of 7 of the analysis tests.

1.1.

Aims and objectives.

The aims of this project are to compare and contrast the two differently noded bodies when undergoing the same set of tests, ultimately showing which type of base setting for the body is the best suited for this kind of analysis. Aims more concentrated on the individual analysis of each plate is the comparison of different mesh densities as a means to demonstrate the conversion of analytical data. Theoretically, the finer the mesh density is the greater the accuracy is of the results due to an increased surface area of analysis and calculation. Further increasing the mesh will only cause slight changes to the final result as the data will now be trending in steady state fashion, therefore another aim for this project is to show this converging trend across the scope of the different varieties of analysis and seeing if both bodies display this converging trend within their own sets of data. As a more absolute and practical objective in this experiment, the analysis of this style of component is important to show the designs potential flaws and subsequently meeting the main objective of conducting finite element analysis in the first place. Another secondary aim of this project is to compare empirical values of calculated stress concentration K (see 2. Methodology) with generated and simulated values of K extrapolated from ANSYS and the absolute error between those two values, furthermore, exploring the usefulness and practicality of ANSYS being used in conducting this type of finite element analysis.

1.2.

Scope of report.

This report will show the data analysis results via tables and graphs; the results of the various different tests carried out on the two bodies. To coincide with these results there will be example images taken from ANSYS to show what the physical plotting looked like when carrying out the analysis. For example, the quantity of longitudinal stress can be shown distributed about the plate in relation to the simple set of data, further adding to the understanding of how the component will react to the force being applied to it. These results will then be discussed and reviewed to determine if the aims and objectives for the task have been met and therefore, determining what type of body is best utilised in this type of analysis and is there a trend of convergence in the data set. The designed component is a simple one however this type of analysis is vitally important on the smaller, simpler components as these are the parts that are more likely to cause some sort of structural failure within the internals of a larger component. Therefore meaning, this analysis and comparative study done on what is a simple component is still very important as smaller components failing lead to much larger issues within fully designed parts.

2. Methodology Initially the model of the plate had to be constructed and set to the correct parameters. Each member undertaking the project was given individual values for the vertical length of the plate; the plate thickness, the diameter of the hole at the plates centre, and the plates Youngs modulus and Poisson ratio. Both bodies shared the same dimensions but different base nodal settings (4 nodes or 8 nodes). For myself, these values were as follows: Vertical length 60 mm Plate thickness Hole diameter

3 mm

Youngs modulus Poisson ratio

81kN/mm^2

Uniform force applied

150N

37 mm

0.3

The horizontal length of the plate was to be determined still for this problem, using St. Venant’s principle [01] I determined that an appropriate horizontal length would be 4 times that of the vertical length of the plate, 240 mm precisely. Below is a Solidworks stetch of the plate that is fully dimensioned in 2D, excluding the 3 mm in thickness.

Fig 1. Solidworks sketch of plate with dimensions.

From this point a model had to be built up on ANSYS with all appropriate nodes and sectioned quadrilaterals. This process was assisted by a demonstration set up video provided by finite element analysis lecturer Dr. Mojtaba Moatamedi [2] and in following that video the model could be built effectively with three separate quadrilaterals occupying a ¼ section of the plate, then using rotation and symmetry the full plate was formed.

Fig 2. ¼ section of plate to be rotated forming the full plate, 3 differently coloured quadrilaterals separate this section and represent the coarsest mesh density to be analysed.

Fig 3. Fully rotated plate on ANSYS, illustrating the relationship and symmetry between the 3 quadrilaterals once rotated.

2.1.

Data analysis tests.

The key given for the two bodies was simply; the 4 noded body was A1 and the 8 noded body was A2. Both bodies underwent the same set of tests set out in the project brief [3] produced by Dr. Moatamedi. These tests were accompanied with a labelled diagram of a general plate, both are shown below.

•

R1: Plot the longitudinal stress at point A against the mesh density

•

R2. Do the same for the longitudinal stress at C.

•

R3. Plot the longitudinal deflection of Point C against mesh density.

•

R4. For each analysis take a point on the edge of the hole that is connected to two elements and calculate the percentage error in the stress values that the stress fields of the two elements.

•

R5. Do the same as above for a point as near to the hole as possible that is connected to four elements.

•

R6. For the finest FE mesh only, calculate the longitudinal stress concentration factor at A and compare it with the (empirical) hand calculated value.

•

R7. For the finest FE mesh only, plot the longitudinal stress between Points A and B, using nodal values.

Fig 4. Diagram taken from [3], description of tests has also been paraphrased from the same source.

2.2.

Mesh density.

For the first 5 tests the mesh density of each body was varied from 1-8 via doubling I.E., 1, 2, 4, 8. The last two tests only relied on the finest mesh of 8. To better illustrate the difference meshing makes to the number of elements present on the body, here are images taken from ANSYS displaying all 4 different structural textures of the mesh density used during the analysis undertook in this project.

Fig 5. Plate with a mesh density of 1.

Fig 6. Plate with a mesh density of 2.

Fig 7. Plate with a mesh density of 4.

Fig 8. Plate with a mesh density of 8.

Note- mesh density as a quantity relates to how many elements exist in each of the three separated quadrilaterals as shown in Fig 2.

2.3.

Equations used.

All hand calculation equations used in this experiment were associated with R6 to calculate empirically the value of the stress concentration K, and then using associated equations determining K using data shown by the ANSYS simulation. Stated below are all the equations used within this section of testing.

d

d

d

(1) K = 3 − 3.13 ( D) + 3.66( D)2 − 1.53( )3 D

wD

(2) σnom = t(D−d) σ

(3) K = σpeak nom

Key: • • • • • •

d= diameter of hole in the plate D= Vertical length of the plate t= thickness of plate K= stress concentration σpeak= peak tensile stress σnom= nominal tensile stress

Equation (1) was taken from the book “Stress concentration factors” by Peterson. R.E [4] and is the formula used to calculate the value for stress concentration empirically without using data from the ANSYS simulation. The value of (2) will remain constant as the dimensions are uniform either empirically or within the simulation, therefore the answer for σnom will be consistent. Using these equations, with rearrangement of (3), two values of K can be produced and compared side by side for both bodies.

3. Results. 3.1.

Tables.

R1: Mesh density 1 2 4 8

Longitudinal stress at A/MPa (4 nodes) 71.53 107.79 129.43 137.26

Longitudinal stress at A/MPa (8 nodes) 129.71 136.57 137.96 138.13

R2: Mesh density 1 2 4 8

Longitudinal stress at C/MPa (4 nodes)

Longitudinal stress at C/MPa (8 nodes)

8.25 -0.42 -5.44 -5.69

-12.82 -5.92 -5.35 -5.79

Displacement at C/mm (4 nodes)

Displacement at C/mm (8 nodes)

0.015 0.021 0.022 0.022

0.0228 0.0225 0.0225 0.0225

R3: Mesh density 1 2 4 8

R4/A1:

Mesh density

Connected elements. Element 1 stress/ MPa

Element 2 stress/ MPa

E1+E2

Ref

1 2 4 8

59.99 50.68 32.63 19.46

30.78 22.34 11.23 5.94

90.77 28.34 21.4 13.52

45.39 36.51 21.93 12.70

Error (%) 64% 78% 98% 106%

R4/A2:

Mesh density

Connected elements. Element 1 stress/ MPa

Element 2 stress/ MPa

E1+E2

Ref

1 2 4 8

72.97 13.2 7.41 15.98

43.30 9.92 7.15 15.89

116.27 23.12 14.56 31.872

58.135 11.56 7.28 15.936

Error (%) 51% 28% 4% 1%

R5/A1.1: Element 1 stress/ MPa Node 7 25 80

Mesh density 2 49.21 4 57.81 8 54.04

Element 2 stress/ MPa

Element 3 stress/MPa

Element 4 stress/ MPa

E1+E2+E3+E4

Ref

43.16 48.39 44.03

21.17 48.41 44.62

52.9 34.69 32.27

166.44 189.30 174.96

41.61 47.33 43.74

R5/A1.2:

Node 7 25 80

Mesh density 2 4 8

Error 1

Error 2

Error 3

Error 4

Error 5

Error 6

Average error (%)

14.54 19.9 22.89

68.12 19.86 21.54

8.87 48.84 47.77

52.84 0.042 1.35

23.4 28.95 26.89

74.98 28.99 28.24

40.46 24.43 24.78

R5/A2.1: Element 1 stress/ MPa Node 12 97 291

Mesh density 2 44.03 4 17.7 8 8.3

Element 2 stress/ MPa

Element 3 stress/MPa

Element 4 stress/ MPa

E1-E2E3-E4

Ref

49.35 18.38 8.39

55.2 18.77 8.34

45.08 18.24 8.23

193.66 73.09 33.26

48.42 18.27 8.32

R5/A2.2: Error 1

Error 2

Error 3

Error 4

Error 5

Error 6

Average error (%)

Node

Mesh density

12

2

10.99

23.07

2.17

12.08

8.82

20.90

13.01

97

4

3.72

5.86

2.96

2.13

0.77

2.90

3.06

291

8

1.08

0.48

0.84

0.60

1.92

1.32

1.04

R6/A1: Mesh density 8

Bending stress nominal/MPa 130.43

Bending stress peak (generated)/MPa 137.26

Bending stress nominal/MPa 130.43

Bending stress peak (generated)/MPa 138.13

Bending stress peak (empirical)/MPa 240

Empirical value of K

Generated value of K

Average

Error (%)

1.84

1.749

1.794

5%

R6/A2: Mesh density 8

Bending stress peak (empirical)/MPa 240

R7/A1: A

B

Nodes 2 9 8 7 6 5 4 3

Longitudinal stress/MPa 137.26 105.19 85.87 71.62 60.19 50.11 40.3 29.82

Empirical value K

Generated Average value of K

Error (%)

1.84

1.737

6%

1.789

R7/A2

A

B

Nodes 2 16

Longitudinal stress/MPa 138.13 107.97

14 12

87.59 72.55

10 8 6 4

60.48 49.8 39.35 29.1

3.2.

Graphs.

Longitudinal stress at A/MPa

R1, 4 nodes 160 140 120 100 80 60 40 20 0 0

1

2

3

4

5

6

7

8

9

6

7

8

9

6

7

8

9

Mesh density

Lonitudinal stress at A/...