december 19 2008 - pbworkssumit-tripathi-projects.pbworks.com/f/fem_plate.pdfmae 529: modal analysis...

MAE 529:

Modal analysis of

Rectangular plate

with all four edges

simply supported

December 19

2008 Project Report

By Sumit Tripathi

Contents Objective: ................................................................................................................................................ 3

Model particulars: ................................................................................................................................... 3

(a) Plate specifications................................................................................................................... 3

(b) Elements used in analysis ......................................................................................................... 3

(c) Assumptions/Methods ............................................................................................................. 3

Shell 93 Element (K I R C H O F F – L O V E)............................................................................................... 4

(a) Thickness ratio = ¼, thickness=16m .......................................................................................... 4

Mode shapes with Shell93 .............................................................................................................. 8

Mode frequency calculations ......................................................................................................... 12

(b) Shell 93 Element (Thickness ratio = 1/16) ............................................................................... 13

Mesh geometry ............................................................................................................................. 13

All 6 mode shapes with finest mesh ............................................................................................... 15

(c) Shell 93 Element (Thickness ratio = 1/64) ............................................................................... 17

All 6 mode shapes with finest mesh for shell93, thickness ratio 1/64 ............................................. 18

Mode frequency calculations ......................................................................................................... 18

Comparison of FEM and analytical solution for the three thickness ratios. ......................................... 19

Shell63 (K I R C H O F F – L O V E shell theory) ........................................................................................ 20

(a) Analysis with Shell63( elements (Thickness ratio=1/4) ............................................................ 20

Mesh geometry ............................................................................................................................. 20

Shell281 (M I N D L I N- R E I S S N E R ) .................................................................................................. 22

(a) Analysis with shell281 (thickness ratio ¼) ............................................................................... 22

Comparision Between Kirchoff- Love and Mindlin-Reissner shell theory ................................................ 23

Solid 186 ............................................................................................................................................... 24

(a) Analysis with Solid 20 nodes brick element (thickness ratio=1/4) ........................................... 24

Mesh geometry ............................................................................................................................. 24

Mode shapes with Solid186 ........................................................................................................... 26

Mode Frequency calculations ........................................................................................................ 28

(b) Solid186, 20 nodes brick element (thickness ratio=1/16) ........................................................ 30

Mesh Geometry ............................................................................................................................. 30

Frequency Calculations .................................................................................................................. 32

(c) Solid186, 20 nodes brick element (thickness ratio=1/64) ........................................................ 33

Mesh Geometry ............................................................................................................................. 33

All 6 mode shapes with coarsest mesh for solid186, 64/1 ...................................................... 34

Frequency Calculations .................................................................................................................. 36

Comparison of Analytical and FEM solutions for Shell93 and Solid186 for three thicknesses .................. 37

Analysis with Solid185 (thickness ratio 1/64) ......................................................................................... 39

Conclusions: .......................................................................................................................................... 40

Objective:

To perform modal analysis of a rectangular steel plate with all four edges simply supported. We

will perform the analysis using different types of elements and evaluate performance of each

element as compared to other in the context of mode shapes.

Model particulars:

(a) Plate specifications

(As per Person number 35904604)

E= 11 GPa

Density=220 Kg/m^3

Poisson’s Ratio= 0.15

(Close to Oak Wood)

Length / breadth = 1

2a= 128m

2b=128m

Thickness=16m ,4m & 1m

(b) Elements used in analysis shell93, shell63, shell281 & solid186, solid185

(c) Assumptions/Methods

All the graphs plotted in this analysis are log-log values.

Mode extraction method used: Block Lanczos

Note: (I have also used Subspace method , however its performance slows down as compared to Block

Lanczos with high mesh density)

Exact frequency calculation from FEM solutions

21

2

2

1

)(2

)(

nnn

nnnex

Where 21 ,, nnn are three frequencies associated with three finest meshes.

Shell 93 Element (K I R C H O F F – L O V E)

The element has six degrees of freedom at each node: translations in the nodal x, y, and z

directions and rotations about the nodal x, y, and z-axes. The deformation shapes are quadratic

in both in-plane directions. The element has plasticity, stress stiffening, large deflection, and

large strain capabilities.

Because of mid nodes, this element gives quadratic convergence.

I have modeled my plate so that all edges of middle plane have UX, UY and UZ equal to zero.

(a) Thickness ratio = ¼, thickness=16m

Shell93, Frequency Convergence thickness ratio=1/4

h 1 0.5 0.25 0.125 Exact

Frequencies for all 6 modes with different mesh (Hz)

Mode 1 5.7621 5.7613 5.7612 5.7612 5.7612

Mode 2 14.1000 14.0830 14.0820 14.0811 14.0730

Mode 3 14.1000 14.0830 14.0820 14.0811 14.0730

Mode 4 21.2730 21.2420 21.2400 21.2390 21.2380

Mode 5 26.8290 26.6660 26.6550 26.6530 26.6526

Mode 6 26.9070 26.7410 26.7300 26.7280 26.7276

Convergence ex

feex

Mode 1 0.0001562174523 0.0000173574925 0.0000000000024 0.0000000000024

Mode 2 0.0019185674619 0.0007105805366 0.0006395224822 0.0005755702332

Mode 3 0.0019185674619 0.0007105805366 0.0006395224822 0.0005755702332

Mode 4 0.0016479894504 0.0001883416493 0.0000941708234 0.0000470854105

Mode 5 0.0066201698400 0.0005044335963 0.0000917151995 0.0000166754909

Mode 6 0.0067138367395 0.0005030181087 0.0000914578380 0.0000166286979

X Y Z Dispacement vector sum Convergence

h 1 0.5 0.25 0.125 Exact

Maximum displacements at nodes (m)

Mode 1 0.000254 0.000260 0.000260 0.000260 0.000260

Mode 2 0.000260 0.000265 0.000284 0.000271 0.000277

Mode 3 0.000260 0.000265 0.000284 0.000271 0.000277

Mode 4 0.000250 0.000255 0.000261 0.000261 0.000261

Mode 5 0.000266 0.000284 0.000287 0.000287 0.000287

Mode 6 0.000322 0.000367 0.000367 0.000367 0.000367

Convergence

Uex

UfeUex

Mode 1 0.0259777662565 0.0000192001217 0.0000192001241 0.0000192001217

Mode 2 0.0584063587257 0.0434340216736 0.0281366426405 0.0182269711294

Mode 3 0.0584530555793 0.0433723920178 0.0281251424212 0.0182379527486

Mode 4 0.0429234488318 0.0210027398037 0.0001933953941 0.0000017808047

Mode 5 0.0736331919200 0.0083287797669 0.0016831680365 0.0003401524255

Mode 6 0.1230199435372 0.0002904839281 0.0001452419640 0.0000726209820

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2Frequency Convergence comparison b/w 6 modes with shell93--ratio xi=1/4

log(h)

log(||(w

ex-w

fe||/

||wex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

However, if we try to make mesh dense in the areas of maximum displacements, we will get better

convergence.

New Mesh

h1

h2

h3

h4

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-6

-5

-4

-3

-2

-1

0Displacement vector sum Convergence comparison b/w 6 modes with shell93--ratio xi=1/4

log(h)

log(||(U

ex-U

fe||/

||Uex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

The above mesh have stiff elements. Since, shell93 have quadratic convergence it will show proper

results even with stiff elements.

Comparison between two meshes:

Clearly second mesh gives better convergence.

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-12

-11

-10

-9

-8

-7

-6

-5

-4

-3


log(h)

log(||(w

ex-w

fe||/||w

ex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-14

-12

-10

-8

-6

-4

-2

0Frequency Convergence comparison b/w 6 modes with shell93--ratio xi=1/4

log(h)

log(||(w

ex-w

fe||/||w

ex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Shell93, Frequency Convergence(new mesh) thickness ratio=1/4

h 1 0.5 0.25 0.125 Exact


Mode 1 5.7623 5.7611 5.7612 5.7612 5.7612

Mode 2 14.1610 14.0860 14.0830 14.0820 14.0815

Mode 3 14.1610 14.0860 14.0830 14.0820 14.0815

Mode 4 21.3860 21.2500 21.2420 21.2400 21.2393

Mode 5 27.1480 26.6880 26.6610 26.6550 26.6533

Mode 6 27.2370 26.7640 26.7360 26.7300 26.7284

Mode 1 0.0001909324456 0.0000173574939 0.0000000000010 0.0000000000010

Mode 2 0.0056457053582 0.0003195682279 0.0001065227427 0.0000355075810

Mode 3 0.0056457053582 0.0003195682279 0.0001065227427 0.0000355075810

Mode 4 0.0069054270379 0.0005022128755 0.0001255532189 0.0000313883047

Mode 5 0.0185610994087 0.0013024392596 0.0002894309465 0.0000643179881

Mode 6 0.0190298355170 0.0013332789139 0.0002857026245 0.0000612219910

Mode shapes with Shell93

Mode 1

h1

h2

h3

h4

ex

feex

Mode 2

h1

h2

h3

h4

Mode 3

h1

h2

h3

h4

Mode 4

h1

h2

h3

h4

Mode 5

h1

h2

h3

h4

Mode 6

h1

h2

h3

h4

All 6 mode shapes with finest mesh

1

2

3

4

5

6

Mode frequency calculations

Let us try to derive mode frequencies from fundamental frequency.

2^

2^

2^

2^0

b

n

a

mmn

Where,

m= number of crests /troughs in x direction (length of 1st axis),

n= number of crests /troughs in y direction(length of 2nd

axis)

a=half length, b=half breadth

for my particular case a=b=64; Hz7612.50

2nd

natural frequency

= 5.7612*(1^2 + 2^2)/(1^2 + 1^2)

=14.40 Hz, which is quite close to 14.0815Hz, the exact frequency from F.E. model for 2nd

mode.

3rd natural frequency

= 5.7612*(2^2 + 1^2)/(1^2 + 1^2)

=14.40 Hz, which is quite close to 14.0815Hz, the exact frequency from F.E. model for 3rd mode.

4th natural frequency

= 5.7612*(2^2 + 2^2)/(1^2 + 1^2)

=23.044 Hz, which is quite close to 21.2393Hz, the exact frequency from F.E. model for 4th mode.


The plane of symmetry in this case switches to half diagonals of the square plate.

=( 5.7612*(3^2 + 2^2)/(1^2 + 1^2))/sqrt(2)

=26.479 Hz, which is quite close to 26.6533Hz, the exact frequency from F.E. model for 5th mode.

Analytical calculation for 6th mode remains same as 5

th mode.

(b) Shell 93 Element (Thickness ratio = 1/16)

Mesh geometry

Shell93,Frequency Convergence for thickness ratio=1/16

h 1 0.5 0.25 0.125 Exact


Mode 1 1.5551 1.5538 1.5537 1.5538 1.5538

Mode 2 3.9759 3.8972 3.8949 3.8949 3.8949

Mode 3 3.9759 3.8972 3.8949 3.8949 3.8949

Mode 4 6.2807 6.1816 6.1736 6.1735 6.1735

Mode 5 8.0391 7.7973 7.7827 7.7821 7.7821

Mode 6 8.0492 7.7994 7.7845 7.7838 7.7838

Mode 1 0.0008688656471 0.0000321802086 0.0000321802097 0.0000321802086

Mode 2 0.0207964260956 0.0005905158026 0.0000000000001 0.0000000000001

Mode 3 0.0207964260956 0.0005905158026 0.0000000000001 0.0000000000001

Mode 4 0.0173647505959 0.0013122649201 0.0000164033115 0.0000002050414

Mode 5 0.0330279183736 0.0019565110441 0.0000804045635 0.0000033042972

Mode 6 0.0341010411069 0.0020086045830 0.0000943639737 0.0000044332068

ex

feex

In above graph we can see that mode 2(green) and mode 3(red) totally coincided. Because mode 2 has

m=1 , n=2 whereas, mode 3 has m=2, n=1. Therefore, numerical values of frequencies for these two

modes are same.

All 6 mode shapes with finest mesh

1

2

3

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-14

-12

-10

-8

-6

-4

-2


log(h)

log(||(w

ex-w

fe||/

||wex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

4

5

6

Fundamental frequency = 1.5538 Hz

2nd

natural frequency

= 1.5538 *(1^2 + 2^2)/(1^2 + 1^2)

=3.88 Hz, which is quite close to 3.8949 Hz, the exact frequency from F.E. model for 2nd

mode.


= 1.5538 *(2^2 + 1^2)/(1^2 + 1^2)

=3.88 Hz, which is quite close to 3.8949 Hz, the exact frequency from F.E. model for 3rd mode.


= 1.5538 *(2^2 + 2^2)/(1^2 + 1^2)

=6.2152 Hz, which is quite close to 6.1735 Hz, the exact frequency from F.E. model for 4th mode.



=( 1.5538 *(3^2 + 2^2)/(1^2 + 1^2))/sqrt(2)


(c) Shell 93 Element (Thickness ratio = 1/64)

Mesh geometry remains same.


h 1 0.5 0.25 0.125 Exact


Mode 1 0.3964 0.3944 0.3942 0.3942 0.3942

Mode 2 1.0338 0.9893 0.9868 0.9867 0.9867

Mode 3 1.0338 0.9893 0.9868 0.9867 0.9867

Mode 4 1.6233 1.5825 1.5763 1.5760 1.5760

Mode 5 2.0809 1.9845 1.9752 1.9750 1.9750

Mode 6 2.0908 1.9857 1.9755 1.9750 1.9750

Mode 1 0.0057083417901 0.0005581489750 0.0000000000001 0.0000000000001

Mode 2 0.0477508239586 0.0026705795692 0.0001165729177 0.0000050885004

Mode 3 0.0477508239586 0.0026705795692 0.0001165729177 0.0000050885004

Mode 4 0.0300226600318 0.0041340845809 0.0002000363507 0.0000096791782

Mode 5 0.0536225981307 0.0048123629153 0.0001034916756 0.0000022256274

Mode 6 0.0586467264352 0.0054308421094 0.0002662177505 0.0000130498897

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-14

-12

-10

-8

-6

-4

-2


log(h)

log(||(w

ex-w

fe||/

||wex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

ex

feex

All 6 mode shapes with finest mesh for shell93, thickness ratio 1/64

1

2

3

4

5

6

Mode frequency calculations

Fundamental frequency = 0.3942 Hz

2nd

natural frequency

= 0.3942 *(1^2 + 2^2)/(1^2 + 1^2)


mode.


= 0.3942 *(2^2 + 1^2)/(1^2 + 1^2)

=0.9855 Hz, which is quite close to 0.9867 Hz, the exact frequency from F.E. model for 3rd

mode.


= 0.3942 *(2^2 + 2^2)/(1^2 + 1^2)




=( 0.3942 *(3^2 + 2^2)/(1^2 + 1^2))/sqrt(2)


Comparison of FEM and analytical solution for the three thickness ratios.

Thickness ratio = 1/4 Thickness ratio = 1/16

Thickness ratio = 1/64

It can be concluded from these

three graphs that as thickness

reduces and comes to moderate

level, shell93 element performs

better, since FEM solution

follows analytical solution more

closely.

2 2.5 3 3.5 4 4.5 5 5.5 612

14

16

18

20

22

24

26

28Comparision of analytical and FEM solution for Thickness ratio 1/4

Modes

Natu

ral fr

equeccie

s (

Hz)

FEM

Analytical

2 2.5 3 3.5 4 4.5 5 5.5 62

4

6

8

10

12

14

16


Modes

Natu

ral fr

equeccie

s (

Hz)

FEM

Analytical

2 2.5 3 3.5 4 4.5 5 5.5 60

2

4

6

8

10

12

14


Modes

Natu

ral fr

equeccie

s (

Hz)

FEM

Analytical

Shell63 (K I R C H O F F – L O V E shell theory)

(a) Analysis with Shell63( elements (Thickness ratio=1/4) This element does not have mid nodes.

The element has six degrees of freedom at each node: translations in the nodal x, y, and z

directions and rotations about the nodal x, y, and z-axes. Stress stiffening and large deflection capabilities

are included.

Mesh geometry


h 1 0.5 0.25 0.125 Exact


Mode 1 6.3196 6.3296 6.3329 6.3338 6.3341

Mode 2 15.7370 15.7990 15.8260 15.8330 15.8354

Mode 3 15.7530 15.8020 15.8270 15.8330 15.8349

Mode 4 25.2860 25.3060 25.3290 25.3350 25.3371

Mode 5 31.2480 31.5200 31.6320 31.6610 31.6711

Mode 6 31.3370 31.5630 31.6420 31.6630 31.6706

Mode 1 0.0022951033191 0.0007163564100 0.0001953699300 0.0000532827082

Mode 2 0.0062170636135 0.0023017975491 0.0005967623275 0.0001547161589

Mode 3 0.0051717891533 0.0020773574684 0.0004985657925 0.0001196557902

Mode 4 0.0020175004819 0.0012281447122 0.0003203855772 0.0000835788463

Mode 5 0.0133601957466 0.0047719332416 0.0012355898572 0.0003199295166

Mode 6 0.0105335362119 0.0033975812444 0.0009031545080 0.0002400790464

ex

feex

Clearly order of convergence for shell93 is better than shell63.

Mode 5- Convergence Shell63/shell93 = 0.0003199295166/0.0000643179881 = 4.97

This implies that shell93 converges faster than shell63, which was expected as shell93 have mid nodes.

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-14

-12

-10

-8

-6

-4

-2


log(h)

log(||(w

ex-w

fe||/

||wex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5


log(h)

log(||(w

ex-w

fe||/

||wex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Shell281 (M I N D L I N- R E I S S N E R )

(a) Analysis with shell281 (thickness ratio ¼) This elements also have mid nodes hence exhibits quadratic convergence. It is an 8-node element with

six degrees of freedom at each node: translations in the x, y, and z axes, and rotations about the x, y, and

z-axes.

The modeling of shell281 is governed by the first order shear deformation theory (

Mindlin-Reissner shell theory).

Mesh geometry remains same as in case of Shell93.


h 1 0.5 0.25 0.125 Exact


Mode 1 5.6975 5.6966 5.6965 5.6965 5.6965

Mode 2 13.8330 13.7600 13.7570 13.7570 13.7570

Mode 3 13.8330 13.7600 13.7570 13.7570 13.7570

Mode 4 20.7010 20.5690 20.5610 20.5610 20.5610

Mode 5 26.1890 25.7610 25.7310 25.7290 25.7289

Mode 6 26.2800 25.8380 25.8080 25.8060 25.8059

Mode 1 0.0001755463934 0.0000175546441 0.0000000000053 0.0000000000053

Mode 2 0.0055244602749 0.0002180708005 0.0000000000002 0.0000000000002

Mode 3 0.0055244602749 0.0002180708005 0.0000000000002 0.0000000000002

Mode 4 0.0068090073442 0.0003890861341 0.0000000000002 0.0000000000002

Mode 5 0.0178843100021 0.0012492920678 0.0000832861378 0.0000055524091

Mode 6 0.0183734589601 0.0012455644068 0.0000830376271 0.0000055358418

ex

feex

Comparision Between Kirchoff- Love and Mindlin-Reissner shell theory

Kirchoff- Love shell theory –shell93 Mindlin-Reissner shell theory –shell281

Mode 6- Convergence Shell93/shell281 = 0.0000130498897/ 0.0000055358418= 2.357

It is evident from above graph that shell281 have faster convergence than shell93.

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-14

-12

-10

-8

-6

-4

-2


log(h)

log(||(w

ex-w

fe||/||w

ex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-14

-12

-10

-8

-6

-4

-2


log(h)

log(||(w

ex-w

fe||/||w

ex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Solid 186 SOLID186 is a higher order 3-D 20-node solid element that exhibits quadratic displacement behavior.

The element is defined by 20 nodes having three degrees of freedom per node: translations in the nodal x,

y, and z directions. It also has mixed formulation capability.

(a) Analysis with Solid 20 nodes brick element (thickness ratio=1/4)

Mesh geometry

While creating volumes for solid brick elements area is extruded in two equal halves so that one middle

plane is created. All the edges of this middle plane have UX, UY, UZ=0

Solid186,Frequency Convergence thickness ratio=1/4

h 1 0.25 0.125 Exact


Mode 1 5.6668 5.6339 5.6302 5.6297

Mode 2 13.6600 13.5380 13.5290 13.5283

Mode 3 13.6600 13.5380 13.5290 13.5283

Mode 4 20.3220 20.0560 20.0370 20.0355

Mode 5 25.5400 25.1590 25.1370 25.1357

Mode 6 25.6200 25.2540 25.2300 25.2283

Mode 1 0.0065844770441 0.0007405034974 0.0000832785088

Mode 2 0.0097364027904 0.0007182592222 0.0000529863361

Mode 3 0.0097364027904 0.0007182592222 0.0000529863361

Mode 4 0.0142976710614 0.0010212622187 0.0000729473013

Mode 5 0.0160866403013 0.0009288873665 0.0000536365409

Mode 6 0.0155255790277 0.0010180707559 0.0000667587381

Here, I used only 3 mesh refinements, because the forth mesh refinement was exceeding the ansys limit

of number of elements for ‘ANSYS’. However, it is not required to refine mesh any further since

frequencies and mode shapes are already converged.

ex

feex

For solid elements above mesh gives much better performance than the mesh having more number of

volumes in initial geometry.

I tried many combinations, some of them is as follows

Both of these geometries gave erroneous results. The best one is the previous mesh with no partitions on

edges.

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1Frequency Convergence comparison b/w 6 modes with solid186--ratio xi=1/4

log(h)

log(||(w

ex-w

fe||/

||wex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode shapes with Solid186

Mode 1

h=1

h=0.5

h=0.25

Mode 2

h=1

h=0.5

h=0.25

Mode 3

h=1

h=0.5

h=0.25

Mode 4

h=1

h=0.5

h=0.25

Mode 5

h=1

h=0.5

h=0.25

Mode 6

h=1

h=0.5

h=0.25

All 6 mode shapes with finest mesh for solid186, 4/1

1

2

3

4

5

6

Mode Frequency calculations

Using,

2^

2^

2^

2^0

b

n

a

mmn

and Hz6297.50

2nd

natural frequency

= 5.6297*(1^2 + 2^2)/(1^2 + 1^2)


mode.


= 5.6297*(2^2 + 1^2)/(1^2 + 1^2)



= 5.6297*(2^2 + 2^2)/(1^2 + 1^2)

=22.5188 Hz, which is close to 20.0355 Hz, the exact frequency from F.E. model for 4th mode.



=( 5.6297*(3^2 + 2^2)/(1^2 + 1^2))/sqrt(2)



th mode.

(b) Solid186, 20 nodes brick element (thickness ratio=1/16)

Mesh Geometry

h1

h2

h3

h4

Note: Thickness is refined only once

Solid186,Frequency Convergence for thickness ratio=1/16

h 1 0.5 0.25 0.125 Exact


Mode 1 1.5538 1.5531 1.5523 1.5523 1.5523

Mode 2 3.8916 3.8883 3.8864 3.8862 3.8862

Mode 3 3.8916 3.8883 3.8864 3.8862 3.8862

Mode 4 6.1641 6.1565 6.1515 6.1511 6.1511

Mode 5 7.7800 7.7557 7.7505 7.7501 7.7501

Mode 6 7.7815 7.7569 7.7516 7.7510 7.7509

Mode 1 0.0009663080592 0.0005153642983 0.0000000000002 0.0000000000002

Mode 2 0.0013955952471 0.0005464315447 0.0000575191100 0.0000060546431

Mode 3 0.0013955952471 0.0005464315447 0.0000575191100 0.0000060546431

Mode 4 0.0021191098043 0.0008835514528 0.0000706841163 0.0000056547294

Mode 5 0.0038623323670 0.0007268754677 0.0000559134975 0.0000043010382

Mode 6 0.0039448971625 0.0007710817709 0.0000872922759 0.0000098821444

ex

feex


1

2

3

4

5

6

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-14

-12

-10

-8

-6

-4


log(h)

log(||(w

ex-w

fe||/

||wex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Frequency Calculations

Hz5523.10

2nd

natural frequency

=1.5523*(1^2 + 2^2)/(1^2 + 1^2)


mode.


=1.5523*(2^2 + 1^2)/(1^2 + 1^2)



=1.5523*(2^2 + 2^2)/(1^2 + 1^2)

=6.2092 Hz, which is close to 6.1511 Hz, the exact frequency from F.E. model for 4th mode



=(1.5523*(3^2 + 2^2)/(1^2 + 1^2))/sqrt(2)



th mode.

(c) Solid186, 20 nodes brick element (thickness ratio=1/64)

Mesh Geometry

h1

h2

h3

h4


h 1 0.5 0.25 0.125 Exact


Mode 1 0.3937 0.3943 0.3942 0.3942 0.3942

Mode 2 0.9875 0.9868 0.9867 0.9867 0.9867

Mode 3 0.9875 0.9868 0.9867 0.9867 0.9867

Mode 4 1.5842 1.5762 1.5761 1.5761 1.5761

Mode 5 1.9816 1.9752 1.9748 1.9747 1.9747

Mode 6 1.9816 1.9752 1.9748 1.9747 1.9747

Mode 1 0.0014205266089 0.0001014661870 0.0000507330938 0.0000253665472

Mode 2 0.0008209348515 0.0001216199773 0.0000608099883 0.0000304049938

Mode 3 0.0008209348515 0.0001216199773 0.0000608099883 0.0000304049938

Mode 4 0.0051392678127 0.0000634477505 0.0000000000003 0.0000000000003

Mode 5 0.0035111411209 0.0002700877785 0.0000675219446 0.0000168804862

Mode 6 0.0035111411209 0.0002700877785 0.0000675219446 0.0000168804862

ex

feex

The FEM calculations had problem with 64/1 . The coarse mesh gave erroneous results because the

elements were cuboids with very high aspect ratios on faces.

All 6 mode shapes with coarsest mesh for solid186, 64/1

1

2

3

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-14

-12

-10

-8

-6

-4


log(h)

log(||(w

ex-w

fe||/

||wex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

4

5

6

We can see that mode shapes 5 and 6 are flipped as compared to previous solutions.

However, the problem was resolved with first mesh refinement on length and breadth section

leaving thickness unrefined throughout.


1

2

3

4

5

6

Frequency Calculations

Hz3942.00

2nd

natural frequency

=0.3942*(1^2 + 2^2)/(1^2 + 1^2)

=0.9855 Hz, which is close to 0.9867 Hz, the exact frequency from F.E. model for 2nd

mode.


=0.3942*(2^2 + 1^2)/(1^2 + 1^2)

=0.9855 Hz, which is close to 0.9867 Hz, the exact frequency from F.E. model for 3rd

mode.


=0.3942*(2^2 + 2^2)/(1^2 + 1^2)

=1.5768 Hz, which is close to 1.5761 Hz, the exact frequency from F.E. model for 4th mode



=(0.3942*(3^2 + 2^2)/(1^2 + 1^2))/sqrt(2)

=1.8118 Hz, which is close to 1.9747 Hz, the exact frequency from F.E. model for 5th mode.


th mode.

Comparison of Analytical and FEM solutions for Shell93 and Solid186 for

three thicknesses SOLID186

4/1 16/1 64/1

SHELL93

We can clearly see that trend for Solid186 and Shell93 are almost similar.

However, if we plot the difference of analytical and FEM solutions for Solid186 and

Shell93, we can compare them much better.

2 2.5 3 3.5 4 4.5 5 5.5 612

14

16

18

20

22

24

26

28

SOLID186 Thickness ratio 1/4

Modes

Natu

ral fr

equeccie

s (

Hz)

FEM

Analytical

2 2.5 3 3.5 4 4.5 5 5.5 62

4

6

8

10

12

14

16

18

SOLID186, Thickness ratio 1/16

Modes

Natu

ral fr

equeccie

s (

Hz)

FEM

Analytical

2 2.5 3 3.5 4 4.5 5 5.5 60

2

4

6

8

10

12

14

16

SOLID186 Thickness ratio 1/64

Modes

Natu

ral fr

equeccie

s (

Hz)

FEM

Analytical

2 2.5 3 3.5 4 4.5 5 5.5 612

14

16

18

20

22

24

26

28

SHELL93, Thickness ratio 1/4

Modes

Natu

ral fr

equeccie

s (

Hz)

FEM

Analytical

2 2.5 3 3.5 4 4.5 5 5.5 62

4

6

8

10

12

14

16

18


Modes

Natu

ral fr

equeccie

s (

Hz)

FEM

Analytical

2 2.5 3 3.5 4 4.5 5 5.5 60

2

4

6

8

10

12

14

16


Modes

Natu

ral fr

equeccie

s (

Hz)

FEM

Analytical

Clearly Shell 93 difference always remains below Solid186, which concludes that performance of shell

element is better.

2 2.5 3 3.5 4 4.5 5 5.5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Difference with analytical solution, Thickness ratio 1/4

Modes

ab

s((

Wa

na

lytic-W

fem

) / W

an

aly

tic)

SHELL93

SOLID186

Analysis with Solid185 (thickness ratio 1/64) SOLID185 is used for 3-D modeling of solid structures. It is defined by eight nodes having three degrees

of freedom at each node: translations in the nodal x, y, and z directions.

Mesh geometry remains same as in case of Solid186.


h 1 0.5 0.25 0.125 Exact


Mode 1 1.0438 0.7490 0.6134 0.5393 0.4498

Mode 2 3.0023 2.1138 1.7000 1.4702 1.1832

Mode 3 3.0023 2.1138 1.7000 1.4702 1.1832

Mode 4 4.1908 2.9999 2.4550 2.1575 1.7998

Mode 5 6.5835 4.5773 3.6415 3.1187 2.4569

Mode 6 6.5835 4.5773 3.6415 3.1187 2.4569

Mode 1 1.3204845469648 0.6650443678293 0.3637015048852 0.1989021951836

Mode 2 1.5374413046126 0.7865114844253 0.4367818731777 0.2425627705564

Mode 3 1.5374413046126 0.7865114844253 0.4367818731777 0.2425627705564

Mode 4 1.3285398787336 0.6668384991440 0.3640749742987 0.1987746464560

Mode 5 1.6795868380279 0.8630322524045 0.4821471057460 0.2693593790169

Mode 6 1.6795868380279 0.8630322524045 0.4821471057460 0.2693593790169

Clearly, solid185 has

very slow

convergence as

compared to solid186

for 64/1 (low

thickness ratio).

Although, it gave

reasonable

convergence result

with 16/1 &

4/1 ,

(not presented in this

analysis for solid185)

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4Frequency Convergence comparison b/w 6 modes with solid185--ratio xi=1/64

log(h)

log(|

|(w

ex-w

fe|| /

||w

ex||)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

ex

feex

The poor convergence of SOLID185 in the case of 64/1 can be explained by presence of stiff

elements in modeling of mesh for 64/1 . Since, this element does not have any mid node the shape

function will be of first order only (it will not have quadratic elements), which will cause non-

smoothening of curves at the stiff corners. This will lead to numerical imprecision.

Conclusions:

With Block Lanczos method, we can achieve faster convergence to FEM problems as compared

to subspace.

Shell elements give better performance than solid elements.

Shell93 gives better performance with moderate thick plates.

Mindlin-Reissner shell theory exhibits better results as compared to Kirchoff – love for square

plate with all edges simply supported.

Elements, which do not have mid nodes give poor convergence with stiff mesh elements.

Elements, with mid nodes exhibit much better results with stiff mesh elements.

december 19 2008 - pbworkssumit-tripathi-projects.pbworks.com/f/fem_plate.pdfmae 529: modal analysis...

Documents