FEA Y3 Report plate with hole PDF

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UNIVERSITY OF SALFORD School of Computing, Science and Engineering

Finite Element Analysis E3 Assignment: Plate with Hole

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Abstract The scope of this assignment is to find and investigate the stress concentration factor which is caused by a circular hole in the middle of a steel plate. A uniformly distributed load of 150N/mm2 is applied to the end of the plate. The element types used in this analysis are 4 and 8 Noded Quadrilateral elements. It was found out that the stress at certain points around the hole has a bigger value than the nominal stress which occurs on the rest of the plate. The stress concentration factor, k, from handbook calculation has a value of 2.13, while ANSYS calculation gives a value of 2.183. St. Venant’s Principle has an important role in the approach of the analysis in terms of model geometry and symmetry which is further discussed in the report. The stress values and stress profiles are analysed and plotted against mesh density in a series of tables and graphs.

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Contents Abstract................................................................................................................................................. 2 1.

Introduction................................................................................................................................... 4 1.1

Finite Element Analysis..........................................................................................................4

1.2

ANSYS.....................................................................................................................................5

1.3

Project aims and objectives...................................................................................................6

2.

Description of assignment.............................................................................................................7

3.

Theory........................................................................................................................................... 9

4.

3.1

Hand calculation....................................................................................................................9

3.2

Stress error estimation.........................................................................................................10

3.3

Convergent studies..............................................................................................................11

Project methodology...................................................................................................................13 4.1

Method................................................................................................................................13

4.2

Analyses...............................................................................................................................13

4.2.1

Task A1 – Four Noded Quadrilaterals...........................................................................13

4.2.2

Task A2 – Eight Noded Quadrilaterals..........................................................................14

4.3 5.

6.

7.

Results analysis.................................................................................................................... 14

Results......................................................................................................................................... 16 5.1

Tables...................................................................................................................................16

5.2

Graphs................................................................................................................................. 17

5.3

Calculations..........................................................................................................................21

5.3.1

Handbook calculations.................................................................................................21

5.3.2

ANSYS...........................................................................................................................21

Discussion....................................................................................................................................22 6.1

Symmetry, model and stress concentration.........................................................................22

6.2

Results analysis discussion...................................................................................................24

Conclusion................................................................................................................................... 26

References........................................................................................................................................... 27 Appendix............................................................................................................................................. 28

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1. Introduction 1.1 Finite Element Analysis Finite Element Analysis method is generally defined as a method used as an engineering tool. Akin, (2005) stated that it is rare to find a project that does not require some type of finite element analysis. The greatest advantage of FEA is its ability to handle truly arbitrary geometry. Another major advantage is the capability of working with nonhomogeneous and anisotropic materials. Almost all the structural analysis, whether static, dynamic, linear or nonlinear, is done by finite element techniques on large problems (Akin, 2005). FEA can analyze deeply the test results and can provide stresses and strains at different locations along the test specimen. Furthermore, the reasons behind the failure of the structure can be explained. The required steps for developing stress analysis are showed in the diagram below. These steps were created by Akin (2005) and can be applied to any related engineering project. Exact solutions for differential equations, however, are generally difficult to obtain. Numerical methods are adopted to obtain approximate solutions for differential equations. Differential equations are reduced to simultaneous linear algebraic equations and thus can be solved numerically (Yoshimoto et. al, 2006).

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Figure 1.1.1 – Stages of finite element analysis project (Source: Akin, 2005)

1.2 ANSYS Yoshimoto et. al., 2006 defines ANSYS as a general-purpose finite-element modeling package for numerically solving a wide range of mechanical problems. These problems include static/ dynamic, structural analysis, heat transfer and fluid problems, as well as acoustic and electromagnetic problems. A finite element analysis solution can be broken in three stages: the preprocessing (defining the problem) i.e. defining the key points/lines, etc., solution (assigning loads, constraints and solving) and the post processing which includes the further processing and viewing of the results. For this assignment, an academic version of ANSYS will be used and the analysis will be only 2D.

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1.3 Project aims and objectives The aims of this exercise is to find the stress concentration factor at Point A on the plate, to obtain the stress profile between Points A and B and find the stress value at Point C. Two types of analysis are used, 4 and 8 Noded Quadrilateral element types and a comparison between methods is discussed. Along with the main aim of the assignment which represents the analysis of the stress concentration factor at specific points, several objectives are included such as the design of the model in an engineering finite element analysis software (ANSYS), the comparison, analysis and discussion of the results. Furthermore, ways of reducing stress concentration will be investigated and discussed.

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2. Description of assignment 

The geometry is given by a long plate of thickness t and depth D. The length of the plate is considered to be infinite. The plate has a circular hole in it of diameter d, located with its centre at the exact centre of the plate.



Loading is uniformly distributed having a value of 150N/mm.



Material proprieties of the material are given by the Young’s modulus 83KN/mm2 and Poisson’s ratio of 0.43.



Geometric values: o Plate thickness, t = 3mm; o Depth of plate, D = 68mm; o Diameter of hole, d = 37.5mm.



Element type: 2D plane stress 4 and 8 noded quadrilateral elements.

Figure 2.1 – Plate with hole illustration

(source: Assignment brief)

The stress concentration is caused by the circular hole in the centre of the plate. The level of stress is higher around the hole in comparison with the nominal values across the plate. The size of the model is reduced; therefore, the symmetry of the plate is used in this

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analysis. The plate is divided in 4 parts, and each part is split into three geometrical areas as shown in the figure below.

Figure 2.2 – 3- Four sided areas (source: Assignment brief) The boundary conditions are dictated by the use of symmetry. There is no UX and UY movement as shown in the figure below.

Figure 2.3 – Boundary conditions (source: Assignment brief)

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3. Theory 3.1 Hand calculation Stress concentration can be defined as a large stress gradient which occurs in a small, localized area of a structure (Young & Budynas, 2002). Near the changes in geometry, the flow of stress is interfered with, causing high stress gradients where the maximum stress and strain may greatly exceed the average or nominal values based on simple calculations. The tangential stress throughout the plate is given by the equation below:

σθ=

[ ( ) ](

)

2 4 N θ a a (eq .3 .1.1) 1+ 2 − 1+3 4 cos 2 θ 2 2 r mm r

The figure below shows where the geometry of the plate and hole.

Figure 3.1.1 – Plate with a hole (source: Young and Budynas, 2002) The maximum stress is achieved when σ=3σ at r=a and θ is + or – 90. The tangential stress is –σ at θ=0° and θ=180°. When θ increases, the tangential stress also increases to 3σ at θ=90° and θ=270°. The stress concentration factor is defined as the ratio of the maximum stress to the

nominal stress. Therefore,

K=

σ max ¿ peak σ nom

. When the plate has a hole into it, it can be

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considered that

σ nom=σ ,

σ max =3 σ , therefore k has a value of 3. This analysis only

applies to a plate with a high width. (Young and Budynas, 2002) Furthermore, empirical value of stress concentration factor can be calculated using the equation proved in the figure below.

Figure 3.1.2 – Handbook calculation (source: Young and Budynas, 2002)

3.2 Stress error estimation The stress field is given by the variation in stress level that adjacent elements give for a common node. In the example below, node 5 is common to 4 elements 1,2,3 and 4. ANSYS software is designed to give the stress output at the nodes. The size of the discrepancies between the real values and the ones from the FE mesh should in reality be the same being a good indication of the accuracy with which the FE mesh is representing the real stress field.

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Figure 3.2.1 – 4 Noded Quadrilateral example (source: Practical Finite Element Analysis, 2016) If σ ij represents the stress at Node i from Element j, then the error E for the stress solution around Node 5 is found from the following equation:

E=

Max {|σ 51 −σ 52|, |σ 51 − σ 53|,|σ 51 −σ 54|,|σ 52 −σ 53|,|σ 52 −σ 54|,|σ 53−σ 54|}

Where,

σ ref σ ref

×100 % (eq .3 .2.1)

is the absolute maximum stress value that occurs anywhere in the

model.

3.3 Convergent studies To find the exact solution to a problem it is often necessary to repeat the F.E. analysis with a redefined mesh. To do this efficiently it is necessary to know what sort of error is present in the current analysis. To be able to reduce the residual force error, the mesh has to be refined successively. To achieve the finest mesh, a systematic convergence study needs to take place. The figure below shows a systematic convergence study of tip displacement against number of 1D linear rod elements used to model a tapering bar that is subject to an axial

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force. It can be noticed that the maximum number of linear elements is 6, so there is no point in going further with the mesh.

Figure 3.3.1 – Example of convergence graph (source: Finite Element Analysis, 2016) It is important for comparison purposes to keep doubling the mesh rather than, for example, doubling and then trebling the original mesh. The figure below shows an original 2 by 2 mesh refined to a 4 by 4, with a 3 by 3 in dashed lines. It can be seen that in the 3 by 3 mesh the centre node is lost. The pattern that retains all previous nodes is known as a reducible net. (Hampson, Finite Element Analysis, 2016)

Figure 3.3.2 – Mesh doubling example (source: Finite Element Analysis, 2016)

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4. Project methodology 4.1 Method The first step is to decide what length of the plate either side of the hole is necessary to model. Sufficient length has to be modelled so that at the extremes of the model, the level of stress goes down. According to Mottran and Shaw, 1996 St. Venant’s principle states that where a system of forces acts on a body it produces substantially different local effects. However, the stresses at different locations from the surface of the loading are basically the same.

4.2 Analyses Analyses of this project are divided into four and eight Noded Quadrilaterals. 4.2.1 Task A1 – Four Noded Quadrilaterals An initial coarse is created having one element in each geometrical area. The same step is followed when the mesh density is increased, i.e. 12, 48 and 192 elements respectively.

Figure 4.2.1.1 – Quad 4 Node 182

(source: Practical Finite Element Analysis, 2016) 13

4.2.2 Task A2 – Eight Noded Quadrilaterals The first part described in section 4.2.1 is repeated using eight noded quads instead of four noded quads. The figure below illustrates the difference between the four and eight noded quadrilaterals.

Figure 4.2.2.1 – Quad 8 Node 183 (source: Practical Finite Element Analysis, 2016)

4.3 Results analysis The analysis of the results comprises of convergence studies and error estimation, followed by a comparison of the stress concentration (obtained from finite element analysis) with those obtained from handbook calculation. The longitudinal profile Point A and Point B is also calculated. The following values and steps are to be followed for both series of analysis (four and eight Noded quadrilaterals): 

The longitudinal stress at Point A is plotted against mesh density where the mesh density is measured by the number of elements along the side of each geometrical area and is plotted logarithmically in base 2.



The same process is repeated at Point C. 14



The longitudinal deflection at Point C is extracted and a graph of deflection against mesh density is produced.



A point on the edge of a hole which is connected to two elements is chosen. The percentage error in the stress values that the stress fields of the two elements is calculated in each case. A graph of the percentage error against mesh density is plotted.



The same process is repeated for a chosen point which is connected to four elements. (in this case for eight noded quadrilaterals case)



The longitudinal stress concentration factor at A is calculated for the finest mesh only and the value is compared with the hand calculated value.



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

A series of figures showing the mesh density and the location of the key points of this assignment are attached to Appendix.

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5. Results The results section is mainly divided into tables, graphs and calculations. All sections follow the project methodology in recording and analysing the data. The overall results are illustrated graphically in a series of 6 graphs. The calculations are done to compare the concentration factor from hand calculation to ANSYS calculation.

5.1 Tables The table below shows the stress and deflection values for Four and Eight Noded Quadrilaterals meshes, both for tasks A1 and A2. The longitudinal stress (N/mm 2) is recorded at Point A and Point C, while at Point C the deflection (mm) is also recorded. Using the stress error estimation formulae, the percentage error at the edge for 2 and 4 elements is calculated.

Table 5.1.1 – Stresses and deflection for Four and Eight Noded Quadrilaterals meshes

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Table 5.1.2 – Longitudinal stress between Points A and B

5.2 Graphs The graphs below are plotted using the tables in section 5.1. Each graph has the mesh density calculated logarithmically in base 2 (A conversion table is attached to appendix). The Four Noded Quadrilaterals (A1 task) is marked with blue line on the graphs, while the Eight Noded Quadrilaterals (A2 task) corresponds to the orange line. Figure 5.2.6 shows the longitudinal stress profile between Points A and B which is also plotted for A1 and A2 task.

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Longitudinal stress at A against mesh density 260

Stress at point A (N/mm2)

240 220 200 180 160 140 120 100 0

1

2

3

Mesh Density (Logarithmically, in base 2) Four Noded Quadrilaterals

Eight Noded Quadrilaterals

Figure 5.2.1 – Longitudinal stress at A against mesh density

Longitudinal Stress at C against mesh density 15

Stress at point C (N/mm2)

10 5 0

0

0.5

1

1.5

2

-5 -10 -15 -20 Mesh density (Logarithmically, in base 2) Four Noded Quadrilaterals

Eight Noded Quadrilaterals

Figure 5.2.2 – Longitudinal Stress at C against mesh density

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2.5

3

Deflection at point C against mesh density 0.05

Deflection at point C (mm)

0.05 0.05 0.04 0.04 0.04 0.04 0.04 0.03 0.03 0.03

0

0.5

1

1.5

2

2.5

3

2.5

3

Mesh density (Logarithmically, in base 2) Four Noded Quadrilaterals

Eight Noded Quadrilaterals

Figure 5.2.3 – Deflection at C against mesh density

Percentage error against mesh density (2 elements) 45 40

Percentage error (%)

35 30 25 20 15 10 5 0

0

0.5

1

1.5

2

Mesh density (Logartihmically, in base 2) Four Noded Quadrilaterals

Eight Noded Quadrilaterals

Figure 5.2.4 – Percentage error against mesh density for 2 elements

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Percentage error against mesh density (4 elements) 300

Percenta...