mec computacional 2

8/3/2019 MEC COMPUTACIONAL 2

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Stress Concentration Factor Convergence Study of a Flat Plate with an

Elliptical Hole Under Elastic Loading Conditions

by

Dwight Snowberger

A Project Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

MASTER of ENGINEERING in MECHANICAL ENGINEERING

Approved:

_________________________________________

Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic Institute

Hartford, ConnecticutDecember 2008


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CONTENTS

LIST OF TABLES............................................................................................................. 3

LIST OF FIGURES ........................................................................................................... 4

ABSTRACT ...................................................................................................................... 6

1. Introduction.................................................................................................................. 7

2. Objectives .................................................................................................................... 8

3. Methodology................................................................................................................ 9

3.1 Schematics.....9

3.2 Stress Concentration Factor Equations for an Elliptical Hole .10

3.3 Boundary Conditions...11

3.4 Elements..113.5 FEA Model..12

4. Results........................................................................................................................ 14

5. Discussion.................................................................................................................. 18

6. Conclusion ................................................................................................................. 21

7. References.................................................................................................................. 22

8. Appendix A................................................................................................................ 23

9. Appendix B................................................................................................................ 26

10. Appendix C................................................................................................................ 32


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LIST OF TABLES

Table 1 Stress Concentration Factors for Various Ellipse radii

Table 2a Plane42 Element Type FEA Model Results/Data

Table 2b Plane82 Element Type FEA Model Results/Data


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LIST OF FIGURES

Figure 1. Flat Plate with an Elliptical Hole

Figure 2. FEA model Boundary Conditions

Figure 3. 4-noded Quad Element, (Reference 2)


Figure 5. FEA Model Size Control Labels

Figure 6 Magnified view of the stress distribution near the tip of the ellipse. (a/b = 1.25,

b = 0.8, Plane 42 element type shown)

Figure 7 Ellipse Short Radius, b vs Length of Element at the Right of Ellipse needed to

obtain an accuracy of +/- 1% from the calculated stress concentration factor

Figure 8 Ellipse Short Radius, b vs. # of Elements to the Right of the Ellipse needed to

obtain an accuracy of +/- 1% from the calculated stress concentration factor

Figure 9 Plane 42 FEA model results with a short radius of b = 0.1





















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ACKNOWLEDGMENT

I would like to thank my wife Amanda, and my son Steven for all of their support and

sacrifice throughout my graduate education experience.


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ABSTRACT

A flat plate with an elliptical hole made of steel (E=30e6psi) with dimensions of 10x20

inches with an elliptical hole of 1 for its long radius and a short radius that varied in

length from 1 to 0.1 in 0.1 increments, was loaded with a 1000psi pressure load in a

quartered FEA model created in ANSYS. A comparison was made between the equa-

tion for a flat plate with a circular hole and an elliptical hole equation with the ellipse

being a circle from Reference 1. The results were found to be the same, with a stress

concentration of 2.54 using the elliptical hole equation and 2.50 for the circular hole

equation. This showed that just the elliptical hole equation could be used to calculate the

stress concentration for all cases including when the ellipse was a circle.

Two different finite elements (Plane42 and Plane82 from the ANSYS element li-brary) were used on the model. These elements were used to show that increasing the

order of the element is one way to improve an FEA model. This was shown by measur-

ing the length of the element at the tip of the ellipse. The Plane42 (4-noded quad)

element had the shorter element length with 0.0095 while the Plane82 (8-noded quad)

had a length of 0.0147. Since the Plane82 element had a longer length than the Plane42

on the same model it meant that the Plane82 element was better at determining the stress

concentration factor that is within +/- 1% of the stress concentration as calculated from

the closed form solution.

Finally, it was also possible to produce a more accurate FEA model by increasing

the number of elements in a mesh. To show this, the number of elements used to mesh

the model were recorded and compared for each ellipse size. The Plane42 model ranged

from using 7 elements at the ellipse tip for the case of the ellipse being a circle to

needing 46 elements and an element scaling factor of 32 when the ellipse had a short

radius of 0.1. The Plane 82 model needed 4 elements with no scaling factor for thecircular hole case to using 38 elements with a scaling factor of 23 when the short radius

was 0.1. All of the values for the number of elements and element scaling factors were

recorded for each model only when the model produced a stress concentration that was

within +/- 1% of the stress concentration as calculated from the closed form solution.


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1. Introduction

Changes in geometry such as a circular hole or an elliptical hole cause increases in the

amount of stress created at these discontinuities. This stress increase is more commonly

known as the stress concentration factor. This factor is a ratio between the maximum

stress produced at the discontinuity divided by the nominal stress far away from the hole.

These factors have been well-studied and documented, with closed form solutions for

more common geometries available in such texts as Reference 1. For an elliptical hole

in a flat plate, the stress concentration will be different depending on the narrowness of

the ellipse.

FEA programs such as ANSYS can be used to approximate the stress concentration

factor as calculated using a closed form equation for a given geometry. The accuracy of

the model can be increased by two ways. One is by increasing the mesh density around

the discontinuity in order to better capture the increase in stress. The other method is to

increase the order of the element, such as using an 8-noded quad element vs. a 4-noded

quad element.


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2. Objectives

The objective of this project will be to study the effect an elliptical hole has on the stress

distribution of a flat plate as the sharpness of the ellipse increases from a circle to a

narrow crack. It will be shown that as the ellipse sharpness increases, more elements

will be needed to accurately capture the stress concentration factor to be within 1% of

the calculated value from Reference (1) for the specific geometry of a flat plate that is

10x20 inches and a 2-inch long diameter elliptical hole. Two element types will be

compared, which are the 4-noded quad (ANSYS Plane42) and the 8-noded quad

(ANSYS Plane82). Finally, the equations for a flat plate with a circular hole and an

elliptical hole will be compared for the case when the elliptical hole is a circle in order to

show that the results are similar enough that the elliptical equations can be used for the

case when the ellipse becomes a circle.


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3. Methodology

3.1 Schematics

This report will focus on the specific geometry of an elliptical hole in a flat plate in

Figure 1.

b = short radiusa = long radius

D = width of the flat plate

Figure 1 Flat Plate with an Elliptical Hole.

ba

1000psi

2

D = 10

20

1000psi


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The plates material will be ordinary steel with a Youngs Modulus of 30e6 psi and the

model will use a pressure load of 1000psi (nominal stress of the model). One important

point to note, this report is a study of the stress concentration factor under elastic loading

conditions. The materials modulus does not have any effect on the outcome of the

results, as long as the stress does not exceed the materials yield point (36,000 psi for

steel), the maximum stress will always be 1000psi multiplied by the stress concentration

factor.

3.2 Stress Concentration Factor Equations for an Elliptical Hole

Equation (1) (Reference 1) is the stress concentration factor for an elliptical hole in a flat

plate. Equation (1) is only valid if the a/b ratio is between 0.5 and 10. For this project

the ratio of a/b varies from 1 to 10.

(1)

where:

a = the long radius of ellipse (1)

b = the short radius of ellipse (will be varied from 1 to 0.1 in 0.1 increments)D = width of the flat plate (10 inches)

K = stress concentration factor for an elliptical hole in a flat plate.

The special case of where b = a for a circular hole, the elliptical hole equation (1) yields

a stress concentration of 2.54. For the same plate, using the equation for a circular hole

from Reference (1), the stress concentration is 2.50. This is only a 1.6% difference.

Therefore, the elliptical hole stress concentration equation (1) will be used for both the

circular and elliptical hole cases.


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3.3 Boundary Conditions for FEA model

Figure 2 shows the boundary conditions for the FEA model. The flat plate from Figure

1, was modeled as a quarter plate with the left vertical edge constrained in the x direction

and the bottom edge being constrained in the y direction. The right vertical edge of the

model will be a free edge and the top edge will be where the pressure load of 1000psi is

applied.

Figure 2 FEA model Boundary Conditions

3.4 Elements

There will be two element types used in this report. The first element is a 4-noded quad

element. In ANSYS the name of the element type is Plane42 shown in Figure 3.


1000psi

Free EdgeEdge constrained, dx = 0

Edge constrained, dy = 0x

Load Applied on Edge

Ellipse


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The second element to be used is an 8-noded quad element, called the Plane82 in

ANSYS, shown in Figure 4.


3.5 FEA Model

The 2-D FEA model in ANSYS (see Appendix A for log file code) was created to

accurately determine the stress concentration within 1% of the closed form solution of

equation (1) for an elliptical hole in a flat plate. In order to do this it was necessary to

set up manual controls for ANSYS to mesh the model and easily capture how many

elements were used at the point of highest stress, which is at the ellipse tip, to calculate

the stress concentration factor to be within 1% of the closed form solution calculated by

equation (1). Figure 5 is a schematic of the FEA models size control limits, which can

be changed by the user to either increase or decrease the mesh density at the hole and

around the whole model. These labels are referred to in the FEA model code of Appen-

dix A.

Figure 5. FEA Model Size Control Labels

Each highlighted edge

of the model will have

its own size controls.

Ellipse

Right Side of Flat PlateLeft Side of Ellipse

To of Plate

Right Side of

Ellipse Tip


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Additional manual mesh control of the FEA model is achieved by use of the LESIZE

command in ANSYS. Below is a sample line from the FEA model text file.

LESIZE,_Y1, , ,10, 3, , , ,1

This command will generate 10 elements on the line that it is assigned to, and the 3 will

size those elements based on a scaling factor where the first element will be 3 times

smaller than the last node in the line. This means that the elements will become gradu-

ally larger the farther away they are from the hole. The scaling factor command allows

the user to optimize the number of elements used in a model, since the farther away an

element is from the elliptical hole, the less its stress is going to change. Because the

greatest stresses will be produced on the right side of the ellipse tip, smaller elements

will be needed than at the top of the plate where larger elements can be used.


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4. Results

Table 1 shows the stress concentration factor for each of the 10 ellipses used. The first

column is the a/b ratio, which is the ratio between the long and short radius of the ellipse

(as seen in Figure 1), the 4 constants C1-C4 needed in equation (1), and the stress

concentration factor using equation (1). The last two columns of Table 1 are the stress

concentration factor tolerance, which will be used to determine if the FEA model from

ANSYS (Appendix A) has an adequate mesh density. This tolerance was chosen as +/-

1% from the calculated stress concentration (K) in equation (1).

a/b a b C1 C2 C3 C4 K K+1% K-1%

10.00 1.00 0.10 21.00 -25.25 32.17 -25.92 17.03 17.20 16.86

5.00 1.00 0.20 11.00 -12.81 14.50 -10.69 8.93 9.02 8.84

3.33 1.00 0.30 7.67 -8.67 9.14 -6.14 6.25 6.31 6.19

2.50 1.00 0.40 6.00 -6.59 6.66 -4.07 4.92 4.96 4.87

2.00 1.00 0.50 5.00 -5.35 5.28 -2.93 4.12 4.16 4.08

1.67 1.00 0.60 4.33 -4.52 4.42 -2.24 3.59 3.62 3.55

1.43 1.00 0.70 3.86 -3.92 3.85 -1.78 3.21 3.24 3.18

1.25 1.00 0.80 3.50 -3.48 3.44 -1.47 2.93 2.96 2.90

1.11 1.00 0.90 3.22 -3.13 3.15 -1.24 2.71 2.74 2.68

1.00 1.00 1.00 3.00 -2.86 2.93 -1.08 2.54 2.56 2.51

Table 1 Stress Concentration Factors for Various Ellipse radii

Table 2a and 2b show the results of the FEA models for the Plane 42 and the Plane 82

elements. This data was recorded for each model only when the maximum stress

produced at the ellipse tip divided by the nominal stress (1000psi) resulted in a stress

concentration factor that was within the +/- 1% tolerance from the stress concentration

calculated by equation (1).


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This is an explanation of Table 2a and 2bs columns, and Figure 5 gives a graphical

representation for the naming/location of the edges of the model.

b = short radius of the ellipse (inch),

Kroarks = Stress concentration factor calculated using equation (1).

K+1% and K-1% is the stress concentration factor, Kroarks +/- 1% in order to establish

a tolerance which will determine if the model has been meshed properlyGnom = Nominal stress in the plate far away from the hole. This will be the 1000psi

pressure load. (psi),

Gmax = Maximum stress created at the ellipse tip which is where the highest stresses

are produced. (psi),

Kmodel = Stress concentration factor obtained from the FEA model in ANSYS, whichis calculated by dividing Gmax/Gnom.

Right = Number of elements that were on the bottom edge of the model, which was to

the right of the ellipse tip.

Scale = Scaling factor used in the LESIZE command, which scaled the elements,

Left = Number of elements that were used to generate the mesh on the left vertical edge

of the model. This is also the side that was constrained to not move in the x-direction.Right Side = Number of elements that were applied to the right vertical edge of the

model. This was the unconstrained, free edge of the model.

Top = Number of elements that were applied to the top of model, which was the edge

that the 1000psi pressure load was applied.

Element Length on Right of Ellipse = Length of the element that is on the Right Side

edge of the model at the tip of the ellipse. Figure 6 shows the length dimension of the

element that was recorded for this column.

Figure 5. FEA Model Size Controls Labels, re-shown from page 12

Each edge of themodel will have its

own size controls.

Ellipse

Right SideLeft

To

Right


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Figure 6 Magnified view of the stress distribution at the tip of the ellipse. (a/b = 1.25, b =

0.8, Plane 42 element type shown)

Length of element

y

x

Ellipse


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5. Discussion

The first objective of this report is to show that the classical solution for a circular hole

in a flat plate is the same as using the closed form solution for an elliptical hole in a flat

plate with a short radius of b = a = 1 which is a circular hole. The results of the closed

form equation (Reference 1) for a flat plate with a circular hole produced a stress con-

centration of 2.50 and the elliptical hole equation (Reference 1) had a stress

concentration of 2.54. These results are within 1.6% of each other, which meant just the

elliptical equations were used to calculate the stress concentration factor for all cases

including when the ellipse become a circle.

The second objective was to show that it is possible to improve the results of an

FEA model by increasing the order of the elements used in the model. Based on the

results gathered from the FEA models from Tables 2a and 2b, the following figure was

created.

Length of Element at the Right of Ellipse vs Ellipse Short Radius

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b, Ellipse Short Radius (inch)

LengthofEllementtoRightofEllipse(inch)

Plane82

Plane42

Figure 7 Ellipse Short Radius, b vs Length of Element at the Right of Ellipse needed toobtain an accuracy of +/- 1% from the calculated stress concentration factor


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Figure 7 shows that as the elliptical hole became narrower a more refined mesh den-

sity was required. More importantly it shows that the Plane42 element type required the

most amount of refinement, because the element to the right of the ellipse tip (as shown

in Figure 6) needed to be smaller in order to capture the stress concentration that was

within +/- 1% of the actual stress concentration as calculated from closed form solutions.

In Figure 7, it can be seen that the Plane42 element type always used a smaller length of

element compared to the Plane82 element. The Plane42 element had a length of 0.0095

vs. the Plane82s length of 0.0147 when the ellipse had a short radius of 0.1.

The third objective of this report was to show that an FEA model could increase in

accuracy by increasing the number of elements used in a model. Figure 8 shows that as

the ellipse became narrower, more elements were needed at the ellipse tip in order to

obtain a stress concentration factor that was within +/- 1% of the actual stress concentra-

tion as calculated from closed form solutions. The figure also shows the Plane82

element was more efficient at meshing the model since it required less elements in order

to capture the stress concentration factor within the specified tolerance of +/-

1%.

Ellipse Short Radius, b vs. # of Elements to the Right of the Ellipse

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b, Ellipse Short Radius (inch)

#ofElementstotheRightoftheEllipse Plane82

Plane42

Figure 8 Ellipse Short Radius, b vs. # of Elements to the Right of the Ellipse needed toobtain an accuracy of +/- 1% from the calculated stress concentration factor


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Finally, this report shows the mesh density needed to calculate the stress concentration

factor within 1% of the closed form solution. The values needed to generate a mesh

density that falls within 1% of the closed form solution for the stress concentration factor

are shown in Tables 2a and 2b. These values are for the specific geometry of an elliptical

hole in a flat plate with an a/b ratio varying from 1 to 10, a width of 10 and subjected to

a pressure load of 1000psi, as can be seen in Figure 1, shown below for convenience.

Figure 1 Flat Plate with an Elliptical Hole.

ba

1000psi

2

D = 10

20

1000psi


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6. Conclusion

Discontinuities in a geometry such as an elliptical hole create an increase in the stress

distribution known as the stress concentration factor. It was shown that the results for

the stress concentration calculated using the elliptical hole equations when the ellipse

became a circle were the same. Because of this, the elliptical hole equation was used for

all cases including the circular hole case.

It was also shown that the accuracy of an FEA model can be increased by two

methods. The first method was to increase the order of the element used in the model. It

was shown that the 4-noded quad element (Plane42) needed smaller elements and more

of them in order to capture the stress concentration factor to be within +/- 1% of the

closed form solution for the stress concentration factor for this specific geometry, as

compared to the 8-noded quad element (Plane82). The second method was to increase

the number of elements in the model. This was done by showing that as the ellipse

became narrower, more elements were needed in order to obtain the stress concentration

factor within 1% of the closed form solution to equation (1) for this specific geometry.


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7. References

1 Young, Warren; Budynas, Richard,Roarks Formulas for Stress and Strain

2 ANSYS Help Menu, ANSYS INC., Release 10.0A1 UP20060105


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8. Appendix A

This appendix contains the FEA code used to create all of the ANSYS models for this

project.

/CLEAR,START

/PREP7!

ET,1,PLANE82

!

/REPLOT,RESIZE

!

MPTEMP,,,,,,,,

MPTEMP,1,0

MPDATA,EX,1,,30e6

MPDATA,PRXY,1,,.3

/REPLOT,RESIZE

CYL4,0,0,1

FLST,2,1,5,ORDE,1

FITEM,2,1

!

! ------------------ Ellipse Size ---------------

!

ARSCALE,P51X, , ,1,.1,1, ,0,1

!

RECTNG,0,5,0,10,

FLST,2,1,5,ORDE,1

FITEM,2,2

FLST,3,1,5,ORDE,1

FITEM,3,1

ASBA,P51X,P51X,SEPO,DELETE,DELETE

!

FLST,5,1,4,ORDE,1

FITEM,5,10CM,_Y,LINE

LSEL, , , ,P51X

CM,_Y1,LINE

CMSEL,,_Y

!

!--------- Right Side Ellipse (Horizontal) -----------

!

LESIZE,_Y1, , ,38, 23, , , ,1

!

FLST,5,1,4,ORDE,1

FITEM,5,9

CM,_Y,LINE

LSEL, , , ,P51XCM,_Y1,LINE

CMSEL,,_Y

!

!------------ Ellipse ------------

!

LESIZE,_Y1, , ,32,12 , , , ,1

!

FLST,5,1,4,ORDE,1

FITEM,5,13

CM,_Y,LINE


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LSEL, , , ,P51X

CM,_Y1,LINE

CMSEL,,_Y

!

!---------------- Left Side of Ellipse (Vertical) -----------

!

LESIZE,_Y1, , ,38, 2, , , ,1

!FLST,5,1,4,ORDE,1

FITEM,5,11

CM,_Y,LINE

LSEL, , , ,P51X

CM,_Y1,LINE

CMSEL,,_Y

!

!------------- Right Side of Flat Plate -------------

!

LESIZE,_Y1, , ,22, 1, , , ,1

!

FLST,5,1,4,ORDE,1

FITEM,5,12

CM,_Y,LINE

LSEL, , , ,P51X

CM,_Y1,LINE

CMSEL,,_Y

!

!--------------- Top Side of Flat Plate -------------------

!

LESIZE,_Y1, , ,11,1 , , , ,1

!

MSHKEY,0

CM,_Y,AREA

ASEL, , , , 3

CM,_Y1,AREA

CHKMSH,'AREA'CMSEL,S,_Y

!

AMESH,_Y1

!

CMDELE,_Y

CMDELE,_Y1

CMDELE,_Y2

!

FINISH

/SOL

FLST,2,1,4,ORDE,1

FITEM,2,13

!/GO

DL,P51X, ,UX,0

FLST,2,1,4,ORDE,1

FITEM,2,10

!

/GO

DL,P51X, ,UY,0

FLST,2,1,4,ORDE,1

FITEM,2,12

/GO


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!

! ---------- Pressure Load (-1000psi) -----------

!

SFL,P51X,PRES,-1000,

/STATUS,SOLU

SOLVE

FINISH

/POST1!

/DSCALE,ALL,OFF

/EFACET,1

PLNSOL, S,EQV, 1,1.0

!


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9. Appendix B

This appendix shows plots of the final FEA models for the Plane42 element type.



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10. Appendix C

This appendix shows plots of the final FEA models for the Plane82 element type.



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