FEA_LabReport

ME3602 FEA Lab Report

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Applications of Finite Element Modelling in Stress Analysis

Student Number: 1216434

1. Abstract

The stress analysis carried out in this report was done using ANSYS Workbench on a thin plate

consisting of double row of holes in it. The report demonstrates the study of localized stress or

stress concentration around the holes using nominal stress and stress concentration factor. The

analysis system used for this report was Static Structural Analysis which determined the total

deformation, maximum equivalent stress and maximum principal stress of the plate when it was

under an axial load in both positive and negative y-axis. The assumption made here is that load

applied was constant and uniformly distributed on the horizontal edges of the plate, taking into

account the effect of Poisson’s ratio. Using the Static Structural Analysis system the geometry

creation, mesh generation and refinement, applying boundary conditions and solving were carried

out. The results for maximum principal stress and number of elements were noted down in MS

Excel and convergence plot of the results were obtained. The results showed convergence after

the meshes were refined each time until no further refinement causes any variation. Both

theoretical and actual stress concentration factors, Kt and K, were determined and similar pattern

were obtained in the plot of K vs staggered angle between the rows of the holes.

2. Introduction to FEA Analysis Finite Element Modelling is an important numerical analysis technique in mechanical design where

a physical model is converted into a discrete mathematical model which to carry out mesh

generation and simulation for obtaining a certain set of results. Mechanical engineers use FEA

software such as ANSYS Workbench to solve static, structural, dynamic, linear, nonlinear, and

thermal problems. FEA analysis generally involves three stages: Pre-processor, Solver and Post-

processor. For a given static or dynamic problem under investigation, Pre-processor stage is

where the model creation and mesh generation can be carried out. The Solver stage takes over to

assemble a stiffness matrix from the mesh and proceed to solve the structural field where

displacement is first calculated and later other results such as stress, deformation and strains are

obtained. Linear problems can be solved easily using Hooke’s Law whereas non-linear problems

can be complicated to solve. The Post-processing stage is where the data for different set of

results are generated and displayed in a contour plot. Advantages of FEA analysis are: irregular

boundaries, general loads, different materials, boundary conditions, variable element size, easy

modification etc. Disadvantages of FEA include: approximate solutions, element density affects the

solution for a plate with a circular hole, poor shape quality of elements reduce accuracy of the

solution, errors in input data. [1]


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3. Pre-Processing 3.1 3D-modelling of the plate

Diameter of

hole (a) /

mm

Spacing

between

holes (b) /

mm

Length of

the plate /

mm

Width of

the plate /

mm

Elastic

Modulus /

GPa

Poisson’s

ratio

Yield

Stress /

MPa

2 32 250 100 210 0.32 250

Table_1: Parameters given for the geometry.

Figure_1 shows the schematic of the semi-finite plate containing the double-row of staggered

holes subject to axial tension.

Figure_1: Plate geometry.

The distance between the rows of holes, c, was variable from 0 mm to 45 mm and also the

thickness of the plate, t, had values of 3, 5 and 7 mm. The staggered angle, 𝜃, was dependent on

the values of c and b. Stress was applied on the plate by means of an axial tensile load of 1200 N

in the y-axis. The main tasks were: to determine the peak stress around the hole, and hence using

nominal stress, to calculate the stress concentration factor, SCF for different staggered angle and

different thickness of the plate. Figure_2 shows the 3D model of the plate created, with symmetry

of axis.

Figure_2: Extruded plate with double-row of holes.

c

b

a θ

σ

σ

w

l

𝜽

c


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The symmetry of axis was created and used to reduce the plate total surface area, hence

increasing the mesh elements and nodes in the half-plate to get more accurate results for better

mesh refinement. Also it is less time consuming to obtain the results for half-plate than a full plate.

3.2 Addition of material properties of the plate

The properties given were the Elastic modulus, Poisons ratio and Tensile or Yield strength. The

materials inside the ‘Engineering Data’ of the Static Structural Analysis system did not have the

exact properties which were given therefore a new material called ‘FEA_Material’ was added to the

material library and given parameters were inserted manually and used for the plate model for

meshing and simulation to make sure that peak stress and SCF calculation are solely based on the

input parameters only.

3.3 Mesh Discretization and Mesh-refinement study

Once the 3D model of the geometry is created, mesh can be generated in terms of elements and

nodes, with choice of different types of meshes such as prism mesh, tetrahedron mesh,

hexahedron mesh, quadrilateral mesh etc. The results obtained were affected by the choice of

mesh type. Four types mesh were generated, amongst which one was an ‘Auto Mesh’. The other

three were Prism mesh, Hexahedron mesh and Quadrilateral mesh. The figures below show the

different types of mesh generated.

Figure_3: Auto Mesh Figure_4: Quadrilateral Mesh

Figure_5: Tetrahedron/Prism Mesh Figure_6: Hexahedron Mesh


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The peak stress, 𝜎max, will be highest around the edges and inside of the holes hence a critical

mesh refinement study was carried out using Edge sizing and Inflation of the whole plate including

the holes. The horizontal and vertical edges of the plate were chosen for Edge sizing and certain

numbers of divisions were created on these edges. This will create an uniform appearance of the

mesh over the plate. To obtain as accurate results as possible, the edges of the holes were divided

into number of elements and Inflation was done on the inside faces of the holes.

Figure_7: Inflation and edge sizing of the holes.

Figure_8: Edge sizing of the horizontal edges of the plate.

Figure_9: Edge sizing of the holes.

The method chosen was called

‘Multizone’ which allowed the

user to choose between

hexahedron or prism mesh. The

meshes were refined by

increasing the Edge sizing

values of the edges of plate

same along with increasing the

number of divisions for edges of

the holes and changing the

Inflation settings. The numbers

of layer of circles around the

holes were increased while the

maximum thickness of the

inflation was decreased. This in

return increased elements and

nodes of the mesh hence

increasing the accuracy of the

results which were obtained

later. Although a certain inflation

settings did not generate the

mesh sometimes but a different

setting with a combination of

different edge sizing created a

denser mesh around the holes.

Figures on the left show how

edge sizing and inflation was

done. To obtain a convergence

plot of the results the meshes

were refined and three refined

meshes were obtained.


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3.4 Boundary Conditions

Figure_10: Forces and support on the plate.

4. Solving, Post-Processing and Discussion of Results

After applying the loads and support to the half-plate model, simulation was carried out and a

warning was obtained which indicated that the software used an iterative solver to solve the model.

Hence to get rid of that warning, a direct solver was used which did not change the results

obtained but made the solving process less time consuming. The simulation was run to obtain

results for total deformation, equivalent (von-Mises) stress, maximum principal stress and safety

factor. The simulation generated contour plots for these three results which are shown below.

4.1 Contour plots

Figure_11: Total Deformation of the half-plate.

The load of 0.6 kN was applied on

both horizontal faces of the plate in

the y-axis. To stop the plate from

experiencing rigid body motion, a

fixed support was used in one

vertex of the plate, which had

constrained the plate. There were

other types of support but they

were unable to stop the plate from

experiencing rigid body motion.

The contour plots on the left are displayed once

the solving process was carried out. The

deformation of the half-plate obtained was

expected to be as it is due to the fixed support

used at one vertex on the left vertical edge,

where it is deformed the minimum (blue area).

The deformation had a maximum value of

0.00049066 mm on the top horizontal face of

the plate due to the applied force of 0.6 kN. This

was elastic deformation which follows Hooke’s

Law meaning that if load applied was removed

from the half-plate, then the plate will return

back to original un-deformed shape. The holes

are seen to experience moderate deformation

(green area) by using the fixed support hence

they are not going to fail at this load.


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Figure_12: Equivalent stress around the hole.

Figure_13: Principal stress around the hole.

4.2 Convergence Plot of the Hexahedron Mesh

Figure_14: Convergence of results for Max. Principal Stress.

0

2

4

6

0 5000 10000 15000 20000 25000

Max Principal Stress/ MPa

Elements

Convergence_Plot_Hexa_Mesh

thickness 3 mm

thickness 5 mm

thickness 7 mm

The Equivalent (von-Mises) stress was highest

on the inner surface of the holes as seen in the

contour plot, with a value of 2.7934 MPa. The

von-Mises stress is a combination of principal

stresses which can be used to check whether

the plate will withstand a given load condition or

not. The plate will fail if the maximum equivalent

stress exceeds the yield stress of the material.

The safety factor tool can be used to determine

the yield stress of the plate. The safety factor of

the plate had a maximum value of 15 MPa,

hence yield stress was calculated to be (15 x

2.7934) = 41.9 MPa. So it can be concluded

that the plate will not fail.

The Maximum Principal Stress had a value of

3.0104 MPa on the inner face of the hole which

is the peak stress due to the presence of that

hole in that area of the plate. Minimum Principal

Stress was present in the y-direction which can

be seen in the blue region of the plot. Other

regions of the plate were within a green color

band as the maximum principal stress was

moderate there due to absence of holes, with a

value of 1.3111 MPa.


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For convergence of the results, Hexahedron mesh was used. Three refined hexahedron meshes

were used. The plot consists of three lines for three different thickness of the plate. Maximum

Principal stress converged at 5.10 MPa (for 3 mm), 2.99 MPa (for 5 mm) and 2.11 MPa (for 7 mm).

Therefore any further refinement of the meshes will not change the stress results. Although the

patterns for convergence plot were similar, the value of stress decreased for different thickness of

the plate.

Figure_15: Convergence of results for different mesh densities.

4.3 Analysis of stress concentration factor, K

Stress concentration factor is a dimensionless parameter that is determined using the maximum

principal stress, 𝜎max, and nominal stress for a given plate with certain thickness and number of

holes. The nominal stress is the stress calculated using the uniaxial load applied over the net area

of the plate. It is same for a certain thickness of the plate but changes when the plate gets

thicker/thinner. The value of K, either theoretical or experimental, depends on the principal stress,

nominal stress, diameter of the holes, horizontal pitch (spacing between the holes) and the

staggered angle between the holes. This can be demonstrated from Equation_1 and Equation_2

as shown below. The equations to calculate K will be different for staggered angles more than 60°.

𝑲𝒕𝒏 = 𝝈𝒎𝒂𝒙

𝝈 (𝟏 −

𝟐𝒂

𝒃𝐜𝐨𝐬 𝜽) --------------------- Equation_1 (for staggered angle < 60°)

𝑲𝒕𝒏 = 𝝈𝒎𝒂𝒙

𝝈 (𝟏 −

𝒂

𝒃) ---------------------------------Equation_2 (for staggered angle > 60°)

To calculate the theoretical K for a given staggered angle, linear interpolation method was used on

the chart of Ktn vs staggered angle for a plate with double-row of holes. Then using the values of K,

nominal stress, diameter of hole (a), horizontal pitch (b) and θ, theoretical 𝜎max can be determined

for a given staggered angle. To calculate experimental K for the plate under uniaxial load, the

value of 𝜎max will be the result obtained for the maximum principal stress.

0

1

2

3

4

5

6

0 5000 10000 15000 20000 25000 30000

Max Principal Stress/ MPa

Elements

Convergence_Plot_Different_Mesh

Hexa_Mesh

Prism_Mesh


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Figure_16: Variation of stress concentration factor with staggered angle between holes.

The plot for the K (7 mm) was closest to the theoretical plot of K vs staggered angle. This means it

has the lowest value of K in comparison with plate thickness of 3 and 5 mm for a certain value of

staggered angle. The plot tells us that increasing the staggered angle between the double-row of

holes will increase the stress concentration factor steadily until the values do not change anymore.

The peak value of theoretical K was 2.7 whereas it was 2.88 for 7 mm thickness, 2.92 for 5 mm

thickness and 2.98 for 3 mm thickness of the plate. The difference in K for different plate thickness

was due to the varied maximum principal stress which decreased with increasing thickness. But as

nominal stress also decreased with increasing thickness, the ratio of maximum principal stress

over nominal stress was not significantly different hence causing less variation in values of K which

is indicated by the plots being closer to one another. The percentage difference between the

values theoretical K and K (7 mm) was the least with the difference being 1-2% and was the

highest between K and K (3 mm) with the difference being around 4-5%.

Figure_17: Variation of experimental Maximum Principal Stress with staggered angle for different

plate thickness.

2

2.2

2.4

2.6

2.8

3

3.2

0 10 20 30 40 50 60 70 80

K

Staggered Angle / degree

K variation with staggered angle

Theoretical K

K (3 mm)

K (5 mm)

K (7 mm)

0

2

4

6

0 20 40 60 80

Max Principal Stress / MPa

Staggered Angle / degree

Max Principal Stress variation with Staggered Angle

Thickness 3 mm

Thickness 5 mm

Thickness 7 mm


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Figure_16 demonstrates the effect of changing staggered angle between the holes on the

maximum principal stress. The variation of results for a particular thickness of plate is very little.

The maximum principal stress was 5.10 MPa (for 3 mm), 3.01 MPa (for 5 mm) and 2.10 MPa (for 7

mm). This shows a gradual decrease in the stress value with increasing plate thickness. The

values of principal stresses obtained for this plot were experimental and were similar to the

theoretical values of maximum principal stresses. Therefore the mesh convergence was occurred

at the desired maximum principal stress for each plate thickness.

4.4 Recommendation for the best set of variable of ‘a’, ‘b’, ‘c’ and ‘t’

The value of stress concentration factor, K depends on a number of parameters such as diameter

of the holes (a), horizontal pitch between holes (b), the staggered angle, which changes by

changing the spacing between rows of holes (c) and the plate thickness (t).

To come up with an understanding of what values of these parameters will give the least value of

K, the chart in page-351 of reference [2] was used. If for example the plate under investigation for

this report was used in a mechanical product in real life, the peak stress around the holes will

determine when the structure will fail. This requires the hole to have minimum peak stress with the

least stress concentration factor value. Using the chart in reference [2], it can be seen that for

increasing ‘a/b’ the Ktn decreases. The two ways to increase ‘a/b’ will be: (i) increase ‘a’ (ii)

decrease ‘b’. Therefore chosen value of ‘a’ will be the given maximum value of 9 mm and of ‘b’ will

be the given minimum value of 12 mm.

Now to demonstrate how the value of ‘c’ affects the value of K, Figure_16 can be observed. In that

plot of maximum principal stress against staggered angle (different for different ‘c’) it can be seen

that maximum principal stress does not change significantly with increasing staggered angle. But

also, nominal stress will be same for each staggered angle therefore Equation_1 and Equation_2

are not enough to justify for choosing a certain value for staggered angle. Therefore using the

chart in reference [2] and Figure_15 it can be seen that smaller value of staggered angle will give

smaller value of K. Hence the spacing between the rows should be minimum so that staggered

angle will decrease, decreasing K, with constant ‘a’ and ‘b’.

To choose a suitable value for thickness (t), Figure_15 and Figure_16 can be analyzed, which

shows that the least value of K was obtained for the plate thickness of 7 mm. Therefore it can be

justified that increasing thickness will distribute the peak stress around the holes, hence reducing

the stress concentration factor, K.


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5. Conclusions

ANSYS 15.0 and Workbench was used to numerically model and simulate the problem under

investigation in this report which was a thin plate with double-row of staggered holes. The objective

was to carry out a stress analysis on the plate when it undergoes uniaxial tensile load of 1.2 kN.

The load applied was uniformly distributed on the horizontal faces of the plate in the y-direction. To

reduce the simulation run-time and apply more mesh densities around the holes, an axis of

symmetry was applied on the full plate of 250 mm length in the y-axis so that it generates a 3D

half-plate model. The thickness of the plate was 3, 5 and 7 mm. It was chosen as a design

parameter so that later the value can be changed without going back to the geometry frequently.

The vertical spacing between the holes was also chosen to be a design parameter hence can be

changed quickly in a way similar to changing thickness. The effect of changing thickness and

staggered angle were investigated in terms of determining peak stress around the holes and

subsequently the stress concentration factor. In order to achieve this, mesh generation was carried

out. The Auto mesh was generated without defining the mesh density of the plate. The other three

mesh types generated were manually defined. To get most accurate results of peak stress around

the hole, mesh density on the inner face of holes was increased using the tools Inflation and Edge

sizing. A direct solver was used to simulate the mesh which made the simulation less time

consuming. The use of correct boundary conditions and material properties ensured that results

obtained were not biased by any manual errors. Although in the assignment brief, the support for

the plate was not mentioned, therefore different types of support were tried and finally a fixed

support was used, which took into account the effect of Poisson’s ratio and stopped the plate from

experiencing rigid body motion and being under constrained. The maximum principal stress values

were obtained for different number of elements and convergence study was carried out. The

results were converged hence further refinement of meshes did not cause any variation. The stress

concentration factor variation with staggered angle was investigated and similar patterns were

obtained for different thickness of plate that looked similar to the pattern of theoretical SCF.

Results of the numerical analysis showed that SCF was affected by staggered angle and thickness

of the plate in the way predicted by theoretical analysis.

6. References

[1] FINITE ELEMENT METHOD. FINITE ELEMENT METHOD, [Online]. Lecture 8, 1-24. Available

at:

http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&cad=rja&uact=8&ved=0C

DEQFjAD&url=http%3A%2F%2Fmochsafarudin.files.wordpress.com%2F2008%2F10%2Flecture8.

pdf&ei=n0S1VLDECuuf7gbDsYDIBA&usg=AFQjCNHcWQJHjOvGYiJaPEoi8gnicMMeZg&bvm=bv.

83339334,d.ZGU [Accessed 14 January 2015].

[2] Pilkey, WDP, DFP, 2008. PETERSON’S STRESS. 3rd ed. Canada: JOHN WILEY & SONS, INC.

http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&cad=rja&uact=8&ved=0CDEQFjAD&url=http%3A%2F%2Fmochsafarudin.files.wordpress.com%2F2008%2F10%2Flecture8.pdf&ei=n0S1VLDECuuf7gbDsYDIBA&usg=AFQjCNHcWQJHjOvGYiJaPEoi8gnicMMeZg&bvm=bv.83339334,d.ZGU





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7. ANSYS Summary Report

Geometry

TABLE 2 Model (A4) > Geometry

Object Name Geometry

State Fully Defined

Definition

Source D:\FEA_Final\FEA_2014\Hexa_Mesh\FEA_2014_files\dp0\SYS-1\DM\SYS-1.agdb

Type DesignModeler

Length Unit Meters

Element Control Program Controlled

Display Style Body Color

Bounding Box

Length X 125. mm

Length Y 100. mm

Length Z 5. mm

Object Name Solid

State Meshed

Material

Assignment FEA_Steel

Nonlinear Effects Yes

Thermal Strain Effects Yes

Statistics

Nodes 29529

Elements 4141

Mesh Metric None

Symmetry

TABLE 5 Model (A4) > Symmetry

Object Name Symmetry

State Fully Defined

TABLE 6 Model (A4) > Symmetry > Symmetry Region

Object Name Symmetry Region Symmetry Region 2

State Fully Defined

Scope

Scoping Method Named Selection

Named Selection Symmetry:XYPlane Symmetry:YZPlane

Definition

Scope Mode Automatic

Type Symmetric

Coordinate System XYPlane YZPlane

Symmetry Normal Z Axis

Suppressed No


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Mesh

TABLE 7 Model (A4) > Mesh

TABLE 8 Model (A4) > Mesh > Mesh Controls

Object Name

MultiZone Edge Sizing

Edge Sizing 2

Edge Sizing 3

Edge Sizing 4

Edge Sizing 5

Edge Sizing 6

Inflation

State Fully Defined

Static Structural (A5)

TABLE 10 Model (A4) > Analysis

Object Name Static Structural (A5)

State Solved

Definition

Physics Type Structural

Analysis Type Static Structural

TABLE 11 Model (A4) > Static Structural (A5) > Analysis Settings

Object Name Analysis Settings

State Fully Defined

Solver Controls

Solver Type Direct

Weak Springs Program Controlled

TABLE 12 Model (A4) > Static Structural (A5) > Loads

Scope

Scoping Method Geometry Selection

Geometry 1 Face 1 Vertex 1 Face

Definition

Type Force Fixed Support Force

Define By Components Components

Coordinate System Global Coordinate System Global Coordinate System

X Component 0. N (ramped) 0. N (ramped)

Y Component 600. N (ramped) -600. N (ramped)

Z Component 0. N (ramped) 0. N (ramped)

Material Data

FEA_Steel

TABLE 16 FEA_Steel > Isotropic Elasticity

Temperature C Young's Modulus MPa Poisson's Ratio Bulk Modulus MPa Shear Modulus MPa

2.1e+005 0.32 1.9444e+005 79545

TABLE 17 FEA_Steel > Tensile Yield Strength

Tensile Yield Strength MPa


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