Linear Static Analysis of a Plate with Hole · PDF fileLinear Static Analysis of a Plate with...

Linear Static Analysis of a Plate with Hole

Fergyanto E. Gunawan ([email protected])

Department of Mechanical Engineering

Toyohashi University of Technology

Objectives:

� Modeling a symmetric structure

� Bottom-up and top-down approaches in modeling

� ANSYS-APDL

� Sensitivity analysis.

DISCLAIMER: In this module, majority of the content is taken from Appendix E MSC.Nastran 120 Exercise Workbook. Though the

reference tailors the need of Nastran and Patran; in this report, the content is modi�ed to suit the ANSYS Interface. This document

was composed in LATEX; vector graphics were generated by Adobe Illustrator 10, Matlab and ANSYS. A bitmap graphic, Fig. 9, is

reproduced from a picture in p. 136 of Design of Machine Element [1].


Model Description:

The following �gure shows a uni-axial loaded plate with hole. For given data in Fig. 1, in addition to those

in Table 1, calculate the maximum stress around the hole and plot the stress along the ligament.

10.0

4.0

R1.0

100 psi

Figure 1: The plate with a circular hole; unit is in.

Table 1: Model properties

Plate thickness, t : 0.125 in.

Young modulus, E : 10.0E+06 psi

Poisson ratio, � : 0.3

References

[1] M. F. Spotts. Design of Machine Elements. Prentice Hall, 1998.

[2] Nicholas M. Baran. Finite element analysis on microcomputer. McGraw-Hill, Inc., 1988.

[3] MSC/NASTRAN for Windows User's Guide, 1997.

[4] R. E. Peterson. Stress Concentration Design Factors. New York: John Wiley & Sons, Inc. 1974.

FEG 2


Pre-Processing Phase:

CREATE A NEW FOLDER

1. De�ne some parameters: ANSYS Pulldown Menu

Parameters B Scalar Parameters

Selection: LENGTH = 5.0

Accept

Selection: WIDTH = 2.0

Accept

Selection: RAD = 1.0

Accept

Selection: PI = ACOS(-1.0)

Accept

Close

2. Select an element type: ANSYS Main Menu

Preprocessor B Element Type B Add/Edit/Delete

Add

Solid 8node 82Plane

OK

Options

Element behavior K3 : Plane strs w/thk

OK

Close

FEG 3


3. De�ne the plate thickness: ANSYS Main Menu

Preprocessor B Real Constants B Add/Edit/Delete

Add

Type 1 PLANE82

OK

Thickness THK 0.125

OK

Close

4. De�ne material properties: ANSYS Main Menu

Preprocessor B Material Props B Material Models

Structural B Linear B Elastic B Isotropic

EX 10.0E+06

PRXY 0.3

OK

Material B Exit

5. Create ten keypoints: ANSYS Main Menu

Preprocessor B Modeling B Create B Keypoints B In Active CS

NPT Keypoint number: 1

X, Y, Z Location in active CS: RAD 0 0

Apply


FEG 4


X, Y, Z Location in active CS: RAD*COS(PI/8) RAD*SIN(PI/8) 0

Apply


X, Y, Z Location in active CS: RAD*COS(PI/4) RAD*SIN(PI/4) 0

Apply


X, Y, Z Location in active CS: RAD*COS(3/8*PI) RAD*SIN(3/8*PI) 0

Apply


X, Y, Z Location in active CS: 0 RAD 0

Apply


X, Y, Z Location in active CS: 0 WIDTH 0

Apply


X, Y, Z Location in active CS: WIDTH WIDTH 0

Apply


X, Y, Z Location in active CS: WIDTH 0 0

Apply


X, Y, Z Location in active CS: LENGTH 0 0

FEG 5


Apply


X, Y, Z Location in active CS: LENGTH WIDTH 0

OK

6. Create lines: ANSYS Main Menu

Preprocessor B Modeling B Create B Lines B Arcs B Through 3 KPs

< Pick Keypoints: 1, 3, and then 2 >

Apply

< Pick Keypoints: 3, 5, and then 4 >

OK

Preprocessor B Modeling B Create B Lines B Lines B Straight line

< Pick Keypoints: 1, and then 8 >








Cancel

7. Show the line number: ANSYS Pulldown Menu

PlotCtrls B Numbering

LINE Line numbers : � on

FEG 6


OK

Plot B Lines

You should see a model similar to that in Fig. 2. In the next steps, we will create three areas; those

areas are surrounded by lines, for Area 1: L2, L4, L6, L5; for Area 2: L1, L3, L7, L4; and for Area 3:

L7, L8, L9, L10.

Figure 2: Model of plate with hole having line numbers turned on.

8. Create three areas: ANSYS Main Menu

Preprocessor B Modeling B Create B Areas B Arbitrary B By Lines

< Pick lines: L2, L4, L6, and then L5 >

Apply


Apply


FEG 7


OK

9. Control the mesh density: ANSYS Main Menu

Preprocessor B Meshing B MeshTool

Lines : Set

< Pick lines: 1, 2, 3, 4, 5, 6, 7, and then 9 >

Apply

NDIV No. of element divisions 4

Apply

< Pick lines: 7 and 9 >

NDIV No. of element divisions 6

Ok

10. Show the area number: ANSYS Pulldown Menu


AREA Area numbers : � on

OK

Plot B Areas

11. Mesh the model: ANSYS Main Menu


Shape : � Quad

Shape : � Mapped

Mesh

< Pick areas: A1, A2, and then A3 >

FEG 8


NDIV = 4

PLANE42 PLANE82

PLANE42 PLANE82

Figure 3: MeshTool of ANSYS and some of its functionality.

OK

FEG 9


Figure 4: The plate with hole model having area numbers turned on.

Important note I The results, especially the stress, of a �nite element analysis strongly depends on the

mesh. Concerning the mesh, Nicholas M. Baran [2] suggests followings:

Mesh Transition: You can use a tringle or control the mesh spacing ratio. However,

in using the mesh spacing ratio, keep l2 � 2l1 and l4 � 2l3.

l1l2

l3

l4

Element Aspect Ratio: Aspect ratio (l=w) should be kept less than 3, if possible.

w

l

w

l

Excessiveaspect ratio

Element Skewness: Try to keep the skew angle, �, less than 30 degrees. Nastran

issues a warning if the angle is greater than 30 degrees [3].

θ

FEG 10


Figure 5: The �nite element mesh of the plate with hole model.

Important note I The general guidelines are:

� Fine mesh in the area of interest

� Fine mesh in the region that has high stress gradient.

� Use quadratic element instead of linear element if possible.

Solution Phase:

1. De�ne the analysis type: ANSYS Main Menu

Solution B Analysis Type B New Analysis

[ANTYPE] Type of analysis: � Static

2. Apply the boundary condition: ANSYS Main Menu

Solution B De�ne Loads B Apply B Structural B Displacement B Symmetry B.C

< Pick Lines: L5, L3, L8 >

OK

FEG 11


Important Notes I Following �gures show several types of symmetry structures:

(a) Axial Symmetry (b) Planar Symmetry

(c) Cyclic Symmetry (d) Repetitive Symmetry

However, the symmetricalness does not only about the geometry, but also the constraints

and the loading conditions. For an example, see following:

Symmetry point

Anti-symmetry point

3. Apply uniform stress: ANSYS Main Menu

Solution B De�ne Loads B Apply B Structural B Pressure B On Lines

< Pick Line: L9 >.

OK

VALUE Load PRES: -1.0

OK

4. Solve the problem: ANSYS Main Menu

Solution B Solve B Current LS

OK

Close

FEG 12


Post Processing Phase:

1. Plot deformation: ANSYS Main Menu

General Postproc B Plot Results B Deformed Shape

KUND Items to be plotted: � Def + undef edge

OK

The result is shown Fig. 6.

Figure 6: The deformation and undeformed plate with hole.

2. Plot the von Misses stress: ANSYS Main Menu

General PostProc B Plot Results B Contour Plot B Nodal Solu

Nodal Solution B Stress B von Misses stress

Undisplaced shape key: Deformed shape with undeformed edge

FEG 13


OK

The results are shown in Fig. 7.

Figure 7: The von Misses stress in the loaded plate with hole.

3. Select the nodes along ligament: ANSYS Pulldown Menu

Plot B Nodes

Select B Entities

Nodes

OK

Select Nodes: � Box

< Click and drag your mouse to form a box; for example, see Fig. 8. >

OK

4. List the results of stresses: ANSYS Main Menu

FEG 14


Figure 8: Nodes selection using a box.

General PostProc B List Results B Nodal Solution

Nodal Solution B Stress B X-Component of stress

OK

You should obtain:

NODE SX SY SZ SXY SYZ SXZ

2 4.2351 0.84957E-01 0.0000 -0.19128E-01 0.0000 0.0000

10 0.67527 -0.74895E-02 0.0000 0.15242E-01 0.0000 0.0000

12 2.4619 0.43886 0.0000 -0.24530E-02 0.0000 0.0000

14 1.7877 0.28282 0.0000 0.83778E-02 0.0000 0.0000

16 1.3116 0.91777E-01 0.0000 0.99075E-02 0.0000 0.0000

The exact solution can be seen in Refs. [1, 4] that the stress concentration factor, K is de�ned as

K =Highest value of actual stress on hole; �max

Nominal stress for minimum cross section: (1)

For d=W = 0:50, where d is the hole diameter, and W is the plate width, the K is 2.169 (see Fig. 9).

Meanwhile, the nominal stress for the minimum cross section is 4=2 � 1 psi = 2.0 psi; therefore, the

highest theoretical stress is

�max = 2:169� 2:0 = 4:338 psi: (2)

ANSYS provides you, at Node 2, �x = 4.2351 psi, or 2.37 % lower than the exact solution.

5. In addition, you also needs those nodes locations: ANSYS Pulldown Menu

List B Nodes

Sort �rst by: NODE Number

OK

FEG 15


2.169

Figure 9: Stress concentration factor for a at bar with a transverse hole in axial tension [1].

You should obtain, after removing the midside nodes

NODE X Y Z THXY THYZ THZX

2 0.0000 1.0000 0.0000 0.00 0.00 0.00

10 0.0000 2.0000 0.0000 0.00 0.00 0.00

12 0.0000 1.2500 0.0000 0.00 0.00 0.00

14 0.0000 1.5000 0.0000 0.00 0.00 0.00

16 0.0000 1.7500 0.0000 0.00 0.00 0.00

Therefore, we can now plot the stress along the ligament:

0 2 4 6 80

0.5

1

1.5

2

Y−

Axi

s (in

)

SX (psi)

FEMPeterson

Figure 10: The normal stress along the plate ligament.

Figure 10 concludes the �rst part of this module. In the second part, we will discusss the top-down

approach in creating a �nite element model. In addition, we also discuss about ANSYS-APDL code for the

present problem.

FEG 16


Top-down Modeling Approach

Two approachs in modeling:

Bottom-up approach: keypoints I lines I area I volume I meshing

Top-down approach: primitives I boolean operations I meshing

We study the top-down approach in this section. In the next section, the APDL code also will be based on

the present log �le.

CREATE A NEW FOLDER

Pre-Processing Phase:

1. Select element type, de�ne the thickness, and also de�ne material properties. Repeat the previous

pre-processing phase, the �rst four steps.

2. Create two rectangles: ANSYS Main Menu

Preprocessor B Modeling B Create B Areas B Rectangle B By 2 Corners

WP X 0.0

WP Y 0.0

Width WIDTH

Height WIDTH

Apply

WP X WIDTH

WP Y 0.0

Width LENGTH-WIDTH

Height WIDTH

OK

3. Turn on the keypoint number, the area numbers and the line numbers: ANSYS Pulldown Menu

FEG 17



KEYPOINT Keypoint numbers: � On

AREA Area Numbers � On

LINE Line numbers � On

OK

4. Create a circle: ANSYS Main Menu

Preprocessor B Modeling B Create B Area B Circl B Solid Circle

WP X 0.0

WP Y 0.0

Radius RAD

OK

5. Substract Area A1 to Area A3: ANSYS Main Menu

Preprocessing B Modeling B Operate B Booleans B Substract B Areas

< Pick Area A1 >

OK

< Pick Area A3 >

OK

You should see the new area which has a number of A4

6. Merges coincident Keypoints: ANSYS Main Menu

Preprocessor B Numbering B Ctrls B Merge Items

FEG 18


Label Type of item to be merge Keypoints

OK

7. Create a keypoint: ANSYS Main Menu

Preprocessing B Modeling B Create B Keypoints B In Active CS

NPT Keypoint number 100

X, Y, Z Location in active CS 0.0 0.0 0.0

8. Create a line: ANSYS Main Menu

Preprocessing B Modeling B Create B Lines B Lines B Strainght Line

< Pick Node 100 and 8 >

OK

OK

9. Cut the area into two areas: ANSYS Main Menu

Preprocessing B Modeling B Operate B Booleans B Divide B Area by Line

< Pick Area A4 >

OK

< Pick Line L1 >

OK

You should obtain A1, A2, and A3

10. Merges coincident Keypoints: ANSYS Main Menu

FEG 19


Preprocessor B Numbering B Ctrls B Merge Items

Label Type of item to be merge Keypoints

OK

Figure 11: Model with the line numbers turned on.

11. Create a new parameter: ANSYS Pulldown Menu

Parameter B Scalar Parameters

Selection: NOE = 4

OK

12. Control the mesh density: ANSYS Pulldown Menu


Lines Set

< Pick Lines L10, L11, L13, L14, L3, L4, L2, L6 >

Apply

NDIV No. of element divisions NOE

FEG 20


Apply

< Pick Lines 5, and 7 >

Apply

NDIV No. of element divisions NOE*3/2

OK

Shape Quad

Shape Mapped

Mesh

< Pick Area A1, A2, A3 >

OK

Close

See the previous solution phase.

Mesh Sensitivity Study

The stress depends on the mesh in a �nite element analysis; although, physically it should not. To study

how the stress depend on the mesh, we perform a mesh sensitivity study. In this study, we analyse the

structure for a number of the mesh-size. In short, we change NOE, and see how it a�ect the maximum �x.

The important lines from your log �le are reproduced following.

1 *SET,LENGTH,5

2 *SET,WIDTH,2

3 *SET,RAD,1

4 *SET,PI,ACOS(-1.0)

5 /PREP7

6 ET,1,PLANE82

7 KEYOPT,1,3,3

8 KEYOPT,1,5,0

9 KEYOPT,1,6,0

10 R,1,0.125,

11

12 MPTEMP,,,,,,,,

13 MPTEMP,1,0

14 MPDATA,EX,1,,10E6

15 MPDATA,PRXY,1,,0.3

FEG 21


16

17 BLC4,0,0,WIDTH,WIDTH

18 BLC4,WIDTH,0.0,LENGTH-WIDTH,WIDTH

19 CYL4,0.0,0.0,RAD

20 ASBA, 1, 3

21 K,100,0.0,0.0,0.0,

22 LSTR, 100, 3

23 ASBL, 4, 1

24 NUMMRG, KP

25 *SET,NOE,4

26 LESIZE,_Y1, , ,NOE, , , , ,1

27 LESIZE,_Y1, , ,NOE*3/2, , , , ,1

28 MSHAPE,0,2D

29 MSHKEY,1

30 AMESH,_Y1

31 FINISH

32 /SOL

33 ANTYPE,0

34 DL,P51X, ,SYMM

35 SFL,P51X,PRES,-1.0,

36 SOLVE

The commands that need to be retouched are shown in lines 26, 27, 34, and 35. Those funny stu�s, Y1 and

P51X, are the problem. For the command LESIZE in lines 26 and 27, the complete command is (see manual)

LESIZE, NL1, SIZE, ANGSIZ, NDIV, SPACE, KFORC, LAYER1, LAYER2, KYNDIV

where NL1 is the line number, and NDIV is the number of elements. We modi�ed those two commands with

following commands:

LESIZE, 10, , ,NOE, , , , ,1

LESIZE, 11, , ,NOE, , , , ,1

LESIZE, 13, , ,NOE, , , , ,1

LESIZE, 14, , ,NOE, , , , ,1

LESIZE, 3, , ,NOE, , , , ,1

LESIZE, 4, , ,NOE, , , , ,1

LESIZE, 2, , ,NOE, , , , ,1

LESIZE, 5, , ,NOE*3/2, , , , ,1

LESIZE, 6, , ,NOE*3/2, , , , ,1

For the symmetry constraints, the complete command is

DL, LINE, AREA, Lab, Value1, Value2

where LINE is the line number; therefore, we rewrite line 34 with

FEG 22


DL, 14, ,SYMM

DL, 13, ,SYMM

DL, 5, ,SYMM

The last is SFL command; as you may guess, it should be replaced with

SFL, 6, PRES,-1.0,

When we put everything together, we have a working APDL code:

/clear

*set, length, 5

*set, width, 2

*set, rad, 1

*set, pi, acos(-1.0)

*set, NOE, 128

/prep7

et, 1, plane82 ! Eight-node plane element

keyopt, 1, 3, 3 ! Plane stress with thickness

keyopt, 1, 5, 0

keyopt, 1, 6, 0

r, 1, 0.125, ! Thickness

mptemp,,,,,,,,

mptemp, 1, 0

mpdata, ex, 1,, 10e6 ! Young’s modulus

mpdata, prxy, 1,, 0.3 ! Poisson’s ratio

blc4, 0, 0, width, width ! Create a rectangle

blc4, width, 0.0, length-width, width ! Create a rectangle

cyl4, 0.0, 0.0, rad ! Create a circle

asba, 1, 3 ! Cut the rectangle

k, 100, 0.0, 0.0, 0.0, ! Create a point

lstr, 100, 3 ! Create a line

asbl, 4, 1 ! Divided an area

nummrg, kp ! Merge nodes

lesize, 10, , , NOE, , , , ,1 ! NOE along L10

lesize, 11, , , NOE, , , , ,1

lesize, 13, , , NOE, , , , ,1

lesize, 14, , , NOE, , , , ,1

lesize, 3, , , NOE, , , , ,1

lesize, 4, , , NOE, , , , ,1

lesize, 2, , , NOE, , , , ,1

lesize, 5, , , NOE*3/2, , , , ,1

lesize, 6, , , NOE*3/2, , , , ,1

mshape, 0, 2d

FEG 23


mshkey, 1

amesh, 1 ! Mesh Area 1



finish

/sol

antype, 0 ! Static analysis

dl, 14, ,symm ! Symmetric constraints

dl, 13, ,symm !

dl, 5, ,symm !

sfl, 6, pres,-1.0, ! Apply uniform tensile stress

solve ! Solve

finish

Within the miracle of APDL and a set of untold bedtime stories, following lines may save your time and

may keep your laziness above everything:

/post1

xStressExact = 2.169*2.0

myNode = node(0.0, RAD, 0.0)

*get, xStress, node, myNode, S, X

error = ABS(xStressExact - xStress)/xStressExact*100

*cfopen, sensitivityStudyResult, dat, , append

*vwrite, NOE, xStress, xStressExact, error

(F15.0, 2X, E15.4, 2X, E15.5, 2X, E15.4)

*cfclos

finish

which produce a nice formatted data in the �le: sensitivityStudyResult.dat; such as:

4. 0.4235E+01 0.43380E+01 0.2372E+01

8. 0.4307E+01 0.43380E+01 0.7178E+00

16. 0.4336E+01 0.43380E+01 0.4887E-01

32. 0.4345E+01 0.43380E+01 0.1673E+00

64. 0.4348E+01 0.43380E+01 0.2287E+00

128. 0.4349E+01 0.43380E+01 0.2452E+00

or graphically

Finally, you know what you see, grasp the reality that ANSYS may dissappoint you.

FEG 24


0 50 100 1504.22

4.24

4.26

4.28

4.3

4.32

4.34

4.36

Number of elements (NOE)

SX

(ps

i)

FE SolutionExact Solution

Figure 12: E�ect of mesh density to the maximum stress.

FEG 25

Linear Static Analysis of a Plate with Hole · PDF fileLinear Static Analysis of a Plate with...

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