FEA-DONE

FEA analysis – mech3004 – Loc Nguyen

DEPARTMENT OF MECHANICAL ENGINEERING

COURSEWORK SUBMISSION

CODE AND TITLE OF COURSEWORK

Course code:

H300

Title: MECH3004 – APPLIED MECHANICS

FINITE ELEMENT MODELLING ASYSMENT

STUDENT NAME: Loc Nguyen

DEGREE AND YEAR: BENG MECHANICAL ENGINEERING YEAR 3 (2015)

DATE COURSEWORK DUE FOR SUBMISSION: 20/11/2015

LECTURER: DR BELE

RECEIVED DATE AND INITIALS:

I confirm that this is all my own work (if submitted electronically, submission will be taken as confirmation that this is your own work, and will also act as student signature)

Signed: LOC NGUYEN

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Table of content

1. Introduction – page 32. Theory – page 3

3. Methodology – page 44. Results – page 5-7

5. Discussion – page 7-86. Conclusion – page 87. Reference – page 8

8. Appendix – page 9- 14

WORD COUNT: 1497 excluding figures and tables

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1. Introduction

This report aims to demonstrate the following:

Finite element method based solution for a cantilever beam with notches and shoulder fillets subjected to a moment at its end.

Such geometry and boundary conditions is created in ANSYS and the solution of stress concentration factors from ANSYS would be compared with those obtained from “Roarks formulae for stress and strain”

During the analysis using ANSYS, the convergence of principal and Von Mises stresses with finer mesh sizes is also studied

The important of understanding of elasticity theory including St Venant’s principle during interpreting results.

2. Theory

Finite element analysis (FEA)

FEA is widely used today in industry to solve complex problems in structure analysis. It operates by dividing the complex structure into a series of smaller parts, such part is called element and nodes in between.

From the theory, it can be said that with more elements - finer mesh size, it would be better to solve the problem. In this assignment, a studying of effects of mesh size to the consistency in computational solving time and accuracy of results comparing to the theoretical solutions is done to show the characteristic of mesh size.

Stress concentration

When a large stress gradient exists at a localise area of a structure, it is called stress concentration. The localised stress exceeds the average or nominal stress in a material. In our report, the nominal stress is replaced by the maximum value of applied pressure at the end of the beam and the stress concentration factor Kf is defined as:

K f=SEQV (stress)

Pmax

St Venant’s Principle

St Venant’s principle states that at sections distant from the surface of loading, the localised effect is negligible and statically equivalent systems of forces produce the same stresses on the same area. [1]

The ratios of length in this assignment were chosen to fit this principle, therefore applying forces instead of pressures on the loaded end would give the same result.

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3. Methodology

ANSYS setup:

A model of cantilever beam (presented in appendix A) is created in ANSYS Mechanical. The beam is then constrained its displacement on the left end to be 0 and a moment is applied on the right end by defining a varying pressure gradient.

The element type is chosen as Quad 8 Node Plane183 with element behaviour K3 set to “Plane Stress with thickness” and the thickness is set to be 1. The material is chosen as a linear, elastic, isotropic material with = 1 and = 0.3.

To analyse the beam, the U-notches are called point 1,2 and the shoulders are called point 3 and 4.

Convergence study with mesh sizes and comparison with Roark’s Formula

The beam was meshed with a random Global Size of 5 and no Smart Size chosen at first. It is clear shown that the mesh at notches and shoulder are coarse with large element size so the solution would not have the required accuracy.

A convergence study is then conducted by improving the mesh – decreasing the mesh Global Size then using the Smart Size mesh to determine the effects on the von-Mises stress. Once the correct mesh is found, maximum von-Mises stress at notches and shoulders would be obtained to calculate stress concentration factor. The convergence of the stress results as the mesh size gets fine would be validated and discussed later.

The calculated stress concentration factor will be compared with the theoretical result from Roark’s Formula. (formulae are shown in appendix B)

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4. Result

Stress concentration factor calculated by using Roark’s formula:

Location Stress concentration factors, Kt

Notches 2.16Shoulders

1.76

Table 1: result of theoretical stress concentration factor

Convergence study on global size – numerical result can be found in appendix C

The effect of gradually decreasing of the mesh Global Size on the stress concentration factor are shown below in following graphs:

00.20.40.60.811.21.4

1.45

1.5

1.55

1.6

1.65

1.7

Kt on mesh global size at top notch

Global Size

STre

ss co

ncen

trati

on fa

ctor

Kt

Fig 1: Stress concentration on global size at top notch

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00.20.40.60.811.200.20.40.60.811.21.41.61.82

Kt on mesh global size at bottom shoulder

Global size

Stre

ss co

ncen

trati

on fa

ctor

Kt

Fig 2: Stress concentration on global size at bottom shoulderFrom appendix C, it can be said that the stress concentration difference at the notches can not reach higher than 20%, however the stress concentration at the shoulders converge to the theoretical value as the mesh size decreases.

It is also observed that the maximum value of von-Mises stresses at point 1 and 2 are really close to each other. Same can be applied for stresses at point 3 and 4. This can be said due to the symmetry along the axis of the bar

The chosen mesh global size is 0.3 (about half of radius r value) to achieve a balance between percentage difference in both notches and shoulders. To achieve faster convergence, Smart Size is used to create better mesh near the notches and shoulders which would help to achieve a better result.

Convergence study on Smart Size – numerical result can be found in appendix D

0123451.571.5751.581.5851.591.5951.61.6051.611.615

Stress concentration factor Kt against Smart Size at top notch

Global Size 0.3 Global Size 0.25 Global Size 0.2

Smart Size

Smar

t Con

cent

ratio

n fa

ctor

Kt

Fig 3: stress concentration factor kt against smart size at top notch

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From the above graph, it can be seen that using global size of 0.2, the value of Kt converged to 1.59 immediately.

However, by using the mesh global size of 0.3 with smart size 4, it would give the closest stress concentration factor to the theoretical ones. So it would be chose as the final mesh size

Point

Mesh size Smart Size SEQV KT % DIFFERENCE

1 0.3 4 80.139 1.60278 25.802 0.3 4 80.461 1.60922 25.503 0.3 4 88.642 1.77284 -0.734 0.3 4 87.134 1.74268 0.98

Table 2: final mesh size and its % difference to the theoretical values:

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Following are 2 pictures showing the mesh at the top notch with initial mesh size and the final chosen mesh size:

Fig 4: Mesh size 1 compared with Mesh size 0.3 smart size 4

Extra study on varied value of pressure and length of the bar:

As the pressure and length of the bar changes, it would give no effect on the stress concentration factors. Study was done with pressure of 50 and 500, the ratio between maximum von-Mises stress and pressure would keep the stress concentration factor the same.

As the length L of the rectangle bar increases from 15 to 30, there were also no significant effects on the stress concentration factor.

5. Discussion on results:

Convergence study:

It can be assumed that the choice of element type Plane183 would provide a faster convergence, as this element has 8-node comparing to 4-node of Plane182. Plane183 is capable of representing deformations more accurately even at a coarser mesh while Plane182 is incapable of creating a degenerated triangular element. [2]

Smart Size option also gives faster convergence due to the fact that the greatest difference using different global mesh size is at the curvature of the notches. The curvatures are better drawn with Smart Size option, because this created smaller mesh elements than the global size near the arc of the shoulder and the notch. [3]

FEA analysis

The maximum stress appears at the predicted points. For the notches, as for different mesh size, the location of maximum stress alternates

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between top and bottom notches. For the shoulders, the stress in the bottom is greater than the top, which means that the compression at the bottom is higher than the tension at the top. However, its difference is very small which is reasonable as the material should behave the same with tension or compression. [4]

Comparing result with Roark table

Location ANSYS Kt Roark Kt % differenceNotch 1.6 2.16 25%Shoulder 1.77 1.76 <1%

Table 3: comparing result between Ansys and Roark’s

It can be seen that there is less than 1% difference in shoulder stress concentration factor however in the notch, the minimum percentage difference possible is 23%. This shows a limitation in ANSYS in recreating valid results for stress concentration.

There are two types of errors can occur in FEA analysis: computational errors, due to round-off errors in floating point calculations and discretisation errors due to limitations of how certain geometries can be represented with the given element type.

6. Conclusion

The results of this report proved that the results obtained from FEA analysis done in ANSYS are close to the experimental results from Roark’s formula with a few limitations in some cases. Finding the correct mesh is a crucial step in this analysis to get a balance in accuracy and constant of computational time. Using the minimum possible mesh size will not help to achieve the best result.

7. Reference:

[1] P.P. Benham, R.Crawford, and C.G Armstrong, Mechanics of Engineering Materials. Pearson Edcuation Limited, 2nd ed., 1996

[2] ANSYS, “Mechanical apdl element reference,” Southpointe 275 Technology Drive, Canonsburg PA 15317, Release 14 2011

[3] ANSYS, “Modeling and meshing guide,” Southpointe 275 Technology Drive, Canonsburg PA 15317, Release 14 2011

[4] J.M. Gere, Mechanics of Materials. Brooks/Cole – Thomson Learning, 6th ed., 2004

[5] W.C.Young and R.G.Budynas, Roark’s Formulas for Stress and Strain. United States: McGraw Hill, 7th ed., 2002

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Appendix A – Set up in ANSYS

Geometry:

Fig 5: geometry of the beam

Where:

hD

=0.15

hr=2

Dimension L is obtained from the relationship of L and D in Roark’s Formula for Stress and Strain, case 5B:

LD

> 0.8

[r / (D−2h ) ]1/4

L10

> 0.8

[0.75 /(10−3 ) ]1 /4

L = 15 > 13.98

D = 10h = 1.5r = 0.75L = 15

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Moments:

Linearly varying pressure is set along right end of the bar to stimulate the moment applied on the right end

Fig 6: linearly varying pressure

Material properties:

Young’s Modulus EX = 1Poisson’s Ratio PRXY = 0.3

Appendix B – Theoretical stress concentration factor [5]

For the notches with hr=2 and

hD

=0.15

K tn=3.738−7.334 (0.3 )+7.88 (0.3 )2−3.2845 (0.3 )3=2.16

For the shoulders with hr=2 and

hD

=0.15

K ts=2.359−2.171 (0.3 )+0.471 (0.3 )2−0.341 (0.3 )3=1.76

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These values can then be used to calculate the maximum von-Mises stress by:

K t=σmaxσnom

=σ maxPmax

Appendix C – Convergence study with global size

Mesh Size

Point and node

von-Mises SEQV stress

Stress concentration factor Kt

% Difference to Roark Kt

1 Point 1 (214) 74.488 1.48976 30.98Point 2 (54) 75.38 1.5076 30.16Point 3 (44) 71.485 1.4297 18.73Point 4 (98) 71.82 1.4364 18.35

0.75 Point 1 (272) 79.383 1.58766 26.45Point 2 (72) 77.654 1.55308 28.05Point 3 (184) 74.558 1.49116 15.24Point 4 (156) 75.013 1.50026 14.72



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0.3 Point 1 (672) 79.762 1.59524 26.10Point 2 (176) 80.821 1.61642 25.12Point 3 (554) 86.794 1.73588 1.331Point 4 (310) 88.461 1.76922 -0.563

0.2 Point 1 (1000) 79.411 1.58822 26.42Point 2 (260) 79.533 1.59066 26.31Point 3 (826) 89.801 1.79602 -2.086Point 4 (458) 88.91 1.7782 -1.073

Table 4: result from ANSYS

As the node global size gets lower than 0.19, the computation time takes longer and the limitation of node prevents the study to go further.

Appendix D - Convergence study with smart size

Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.3 4 20 80.139 1.60278

25.79

0.3 3 20 77.526 1.55052

28.21

0.3 2 20 78.221 1.56442

27.57

Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.25 4 20 80.459 1.60918

25.50

0.25 3 20 79.882 1.59764

26.03

0.25 2 20 79.222 1.58444

26.64

Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.2 4 20 79.7 1.594 26.20

0.2 3 20 79.7 1.594 26.20

0.2 2 20 79.7 1.594 26.20Table 5: ANSYS result at top notch

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Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.3 419 80.461

1.60922 25.50

0.3 319 77.838

1.55676 27.93

0.3 2 1977.838

1.55676 27.93

Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.25 419 78.707

1.57414 27.12

0.25 319 78.954

1.57908 26.89

0.25 2 1979.328

1.58656 26.55

Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.2 4 1979.429

1.58858 26.45

0.2 319 79.429

1.58858 26.45

0.2 219 79.429

1.58858 26.45

Table 6: ANSYS result at bottom notch

Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.3 4211 88.642

1.77284 -0.73

0.3 3215 84.126

1.68252 4.40

0.3 2215 84.126

1.68252 4.40

Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.25 4249 89.307

1.78614 -1.49

0.25 3249 87.612

1.75224 0.44

0.25 2 253 88.26 1.7652 -0.30Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.2 4307 89.698

1.79396 -1.93

0.2 3307 89.698

1.79396 -1.93

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0.2 2307 89.698

1.79396 -1.93

Table 7: ANSYS result at top shoulder

Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.3 4543 87.134

1.74268 0.98

0.3 3547 85.576

1.71152 2.75

0.3 2 750 85.65 1.713 2.67Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.25 4892 88.762

1.77524 -0.87

0.25 3 892 88.09 1.7618 -0.100.25 2

896 86.1621.7232

4 2.09Mesh size

Smart size

Node

von- Mises stress

Kt % difference

0.2 4 1112 89.13 1.7826 -1.28

0.2 3 1112 89.13 1.7826 -1.28

0.2 2 1112 89.13 1.7826 -1.28

Table 8: ANSYS result at bottom shoulder

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Appendix E – Contour Plot at mesh size 0.3 smart size 4

Fig 7: contour plot

Fig 8: close zoom at top notch and bottom shoulder

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FEA-DONE

Documents

localised stress

nominal stress

theory page

plane stress

results page

appendix page

methodology page

introduction page