1

PLATE BENDING FREQUENCIES VIA THE FINITE ELEMENT METHOD

WITH RECTANGULAR ELEMENTS

Revision A

By Tom Irvine

Email: tomirvine@aol.com

May 6, 2011

Rectangular Plate in Bending

Figure 1.

Stiffness Matrix

Let u represent the out-of-plane displacement. The total strain energy of the plate is

dxdyyx

u12

y

u

x

u2

y

u

x

u

2

DU

b

0

a

0

22

2

2

2

22

2

22

2

2

(1)

a

b

Y

X 0

k l

m n

2

Note that the plate stiffness factor D is given by

)1(12

EhD

2

3

(2)

where

E = elastic modulus

h = plate thickness

= Poisson's ratio

The plate has one translational and two rotational degrees-of-freedom at each corner, for a total of 12

degrees-of-freedom. The two rotational components are about the x and y-axes respectively.

Let the four corners be labeled k, l, m, and n, as shown in Figure 1.

The rotation components θ at the corner 1 are

x

u

l

lx (3)

y

u

l

ly (4)

The displacement equation has 12 constants to enforce 12 continuity conditions.

312

311

310

29

28

37

265

24321

xyayxayaxyayxa

xayaxyaxayaxaa)t,y,x(u

(5)

The displacement equation can be written as

Z}A{)t,y,x(u T (6)

where

121110987654321T aaaaaaaaaaaa}A{ (7)

3

33322322T xyyxyxyyxxyxyxyx1}Z{ (8)

The strain energy is thus

Adxdy)y,x(SA2

DU

b

0

a

0

T (9)

where

22

2

2

2

22

2

22

2

2

yx

u12

y

u

x

u2

y

u

x

u)y,x(S

(10)

Again,

312

311

310

29

28

37

265

24321

xyayxayaxyayxa

xayaxyaxayaxaa)t,y,x(u

(11)

312

211

298

27542 yayxa3yaxya2xa3yaxa2a)t,y,x(u

x

(12)

212

311

2109

28653 xya3xaya3xya2xaya2xaa)t,y,x(u

y

(13)

4

The displacement and derivatives at each corner are

1a}t,0,0{u (14)

2a}t,0,0{x

u

(15)

3a}t,0,0{y

u

(16)

)a(a)a(a)a(a)a(}t,0,a{u 73

42

21 (17)

)a(a3)a(a2)a(}t,0,a{x

u7

242

(18)

)a(a)a(a)a(a)a(}t,0,a{y

u11

38

253

(19)

312

311

310

29

28

37

265

24321

ababaabaababaa

aabaabaaabaaaa}t,b,a{u

(20)

)a(b)a(ba3)a(b)a(ab2)a(a3)a(b)a(a2)a(}t,b,a{x

u12

311

29

287

2542

(21)

)a(ab3)a(a)a(b3)a(ab2)a(a)a(b2)a(a)a(}t,b,a{y

u12

211

310

298

2653

(22)

310

2631 b)a(b)a(b)a()a(}t,b,0{u (23)

312

2952 b)a(b)a(b)a()a(}t,b,0{

x

u

(24)

)a(b3)a(b2)a(}t,b,0{y

u10

263

(25)

5

The displacement vector for element i is

ynxnnymxmmylxllykxkki uuuu}u{ T (26)

The displacement vector can be written as

AB}u{ i (27)

where

00b3000b200100

b00b000b0010

00b000b00b01

ab3ab3ab22a0b2a0100

bba30bab2a30ba2010

abbababbaabababa1

0a00a00a0100

00000a300a2010

00000a00a0a1

000000000100

000000000010

000000000001

]B[

2

32

32

232

3222

33322322

32

2

32

(28)

Solve for A .

i}u{BA 1 (29)

Let

1B]c[ (30)

i}u{cA (31)

6

The strain energy is thus

i}u{cdxdy)y,x(Scu2

DU

b

0

a

0

TT (32)

The kinetic energy is

dxdyu2

hK

b

0

a

0

2

(33)

Recall the displacement equation

Z}A{)t,y,x(u T (34)

The velocity is

Z}A{u T (35)

}A{ZZ}A{]u[ TT2 (36)

Recall

i}u{cA (37)

ii }u{cZZc}u{]u[ TTT2 (38)

Let

TZZ)y,x(P (39)

ii }u{c)y,x(Pc}u{]u[ TT2 (40)

7

The kinetic energy is thus

ii }u{cdxdy)y,x(Pc}u{2

hK

b

0

a

0

TT

(41)

The virtual work due to the nodal forces and moment is

iiii }F{}u{}u{}F{W T (42)

where

ynxnnymxmmylxllykxkki MMFMMFMMFMMF}F{ T

(43)

Hamilton’s principle yields the equation of motion.

i}F{}u]{k[}u]{m[ ii (44)

The respective local mass and stiffness matrices are

]c[dxdy)y,x(P]c[h]m[b

0

a

0

T (45)

]c[dxdy)y,x(S]c[D]k[b

0

a

0

T (46)

The local matrices can then be assembled into global matrices. Then boundary conditions can be

applied.

8

The generalized eigenvalue problem is

0U}]M[]K[{ 2 (47)

where

K is the global stiffness matrix

M is the global mass matrix

is the natural frequency

U is the corresponding mode shape

Note that there is a natural frequency and mode shape for each degree-of-freedom.

References

1. W. Soedel, Vibrations of Shells and Plates, Third Edition (Dekker Mechanical Engineering), Third Edition, Marcel Dekker, New York, 2004.

2. R. Blevins, Formulas for Natural Frequency and Mode Shape, Krieger, Malabar, Florida,

1979. See Table 11-6.

9

APPENDIX A

Mass Matrix Core

33322322

3

3

3

2

2

3

2

2

T xyyxyxyyxxyxyxyx1

xy

yx

y

xy

yx

x

y

xy

x

y

x

1

}Z}{Z{

(A-1)

10

62446524334542334323

44264334256332452343

6436542335432433

52345423324432233222

43254233245322342232

3463324562345343

5335432234322322

4224432234322322

33532234522342232

423432233222

324322342232

33322322

T

yxyxxyyxyxyxxyyxyxxyyxxy

yxyxyxyxyxyxyxyxyxyxyxyx

xyyxyxyyxyxyxyyxyxyy

yxyxxyyxyxyxxyyxyxxyyxxy

yxyxyxyxyxyxyxyxyxyxyxyx

yxyxyxyxyxxyxyxxyxxx

xyyxyxyyxyxyxyyxyxyy

yxyxxyyxyxyxxyyxyxxyyxxy

yxyxyxyxyxxyxyxxyxxx

xyyxyxyyxyxyxyyxyxyy

yxyxxyyxyxxxyyxxxyxx

xyyxyxyyxxyxyxyx1

}Z}{Z{

(A-2)

11

APPENDIX B

Stiffness Matrix Core

Again,

33322322T xyyxyxyyxxyxyxyx1}Z{ (B-1)

The derivatives are

3222T

yyx30yxy2x30yx2010x

Z

(B-2)

0xy600y2x6002000x

ZT

2

2

(B-3)

2322T

xy3xy3xy2x0y2x0100y

Z

(B-4)

xy60y6x200200000y

ZT

2

2

(B-5)

22

T2

y3x30y2x20010000yx

Z

(B-6)

12

0xy600y2x6002000

0

xy6

0

0

y2

x6

0

0

2

0

0

0

x

Z

x

ZT

2

2

2

2

(B-7)

000000000000

0yx3600xy12yx3600xy12000

000000000000

000000000000

0xy1200y4xy1200y4000

0yx3600xy12x3600x12000

000000000000

000000000000

0xy1200y4x12004000

000000000000

000000000000

000000000000

x

Z

x

Z

2222

22

22

T

2

2

2

2

(B-8)

13

xy60y6x200200000

xy6

0

y6

x2

0

0

2

0

0

0

0

0

y

Z

y

ZT

2

2

2

2

(B-9)

2222

22

22

T

2

2

2

2

yx360xy36yx1200xy1200000

000000000000

xy360y36xy1200y1200000

yx120xy12x400x400000

000000000000

000000000000

xy120y12x400400000

000000000000

000000000000

000000000000

000000000000

000000000000

y

Z

y

Z

(B-10)

14

2222

2222

22

22

22

22

T

2

2

2

2T

2

2

2

2

yx360xy36yx1200xy1200000

0yx3600xy12yx3600xy12000

xy360y36xy1200y1200000

yx120xy12x400x400000

0xy1200y4xy1200y4000

0yx3600xy12x3600x12000

xy120y12x400400000

000000000000

0xy1200y4x12004000

000000000000

000000000000

000000000000

y

Z

y

Z

x

Z

x

Z

(B-11)

15

xy60y6x200200000

0

xy6

0

0

y2

x6

0

0

2

0

0

0

y

Z

x

ZT

2

2

2

2

(B-12)

000000000000

yx360xy36yx1200xy1200000

000000000000

000000000000

xy120y12xy400y400000

yx360xy36x1200x1200000

000000000000

000000000000

xy120y12x400400000

000000000000

000000000000

000000000000

y

Z

x

Z

2222

22

22

T

2

2

2

2

(B-13)

16

0xy600y2x6002000

xy6

0

y6

x2

0

0

2

0

0

0

0

0

x

Z

y

ZT

2

2

2

2

(B-14)

0yx3600xy12yx3600xy12000

000000000000

0xy3600y12xy3600y12000

0yx1200xy4x1200x4000

000000000000

000000000000

0xy1200y4x12004000

000000000000

000000000000

000000000000

000000000000

000000000000

x

Z

y

Z

2222

22

22

T

2

2

2

2

(B-15)

17

0yx3600xy12yx3600xy12000

yx360xy36yx1200xy1200000

0xy3600y12xy3600y12000

0yx1200xy4x1200x4000

xy120y12xy400y400000

yx360xy36x1200x1200000

0xy1200y4x12004000

000000000000

xy120y12x400400000

000000000000

000000000000

000000000000

x

Z

y

Z

y

Z

x

Z

2222

2222

22

22

22

22

T

2

2

2

2T

2

2

2

2

(B-16)

18

22

2

2

T22

y3x30y2x20010000

y3

x3

0

y2

x2

0

0

1

0

0

0

0

yx

Z

yx

Z

(B-17)

422322

224232

322

232

22

T22

y9yx90y6xy600y30000

yx9x90yx6x600x30000

000000000000

y6yx60y4xy400y20000

xy6x60xy4x400x20000

000000000000

000000000000

y3x30y2x20010000

000000000000

000000000000

000000000000

000000000000

yx

Z

yx

Z

(B-18)

19

Let 12

42222222322222

22224222223222

2222

3222222

2232222

2222

22

T22

T

2

2

2

2T

2

2

2

2T

2

2

2

2T

2

2

2

2

y9yx36yx9yx36xy36y6yx12xy6xy12yx36xy12y3xy12000

yx9yx36x9yx36xy36yx6yx12x6xy12yx36xy12x3xy12000

xy36xy36y36xy12y12xy36y120y12000

y6yx12yx6yx12xy12y4x4xy4xy4x12x4y2x4000

xy6xy12x6xy12y12xy4xy4x4y4xy12y4x2y4000

yx36yx36xy36x12xy12x36x120x12000

xy12xy12y12x4y4x12404000

y3x30y2x20010000

xy12xy12y12x4y4x12404000

000000000000

000000000000

000000000000

yx

Z

yx

Z

x

Z

y

Z

y

Z

x

Z

y

Z

y

Z

x

Z

x

Z

(B-19)

20

APPENDIX C

Example 1

Consider an aluminum plate, with dimensions 4 x 6 x 0.125 inches. The plate is simply-supported on

all sides.

The natural frequencies are

,3,2,1n,,3,2,1m,b

n

a

m

h

D22

(C-1)

inlbf1789

)2^3.01(12

3)^in125.0)(2^in/lbf07e0.1(

)1(12

EhD

2

3

(C-2)

sec^2/in^3 lbf05-3.238e

)in125.0(sec^2/in^4 lbf)386/1.0(h

(C-3)

21

,3,2,1n,,3,2,1m,in4

n

in6

m

sec^2/in^3 lbf05-3.238e

inlbf178922

mn

(C-4)

Table C-1. Natural Frequencies,

Hand Calculation

fn(Hz) m n

1054 1 1

2027 2 1

3243 1 2

3649 3 1

4216 2 2

Now perform a finite element analysis with 13 nodes along the X-axis and 9 nodes along the Y-axis.

The analysis is performed using the mass and stiffness matrices derived in the main text, as

implemented in Matlab script: plate_fea.m

The fundamental mode shape is shown in Figure C-1. The natural frequency comparison is shown in

Table C-2.

22

Figure C-1.

Table C-2. Natural Frequencies

Mode

Hand

Calculation

fn(Hz)

FEA

fn(Hz)

1 1054 1049

2 2027 2006

3 3243 3223

4 3649 3603

5 4216 4136

01

23

45

6

0

1

2

3

4

0

50

100

Mode 1 fn= 1049 Hz

23

APPENDIX D

Example 2

Repeat example 1 but change the boundary conditions such that the corners are pinned. The edges

are otherwise free.

The fundamental frequency is 291 Hz using the hand calculation formula from Reference 2.

Again, the analysis is performed using the mass and stiffness matrices derived in the main text, as

implemented in Matlab script: plate_fea.m

The natural frequency comparison is shown in Table D-1. The mode shapes are shown in the

following figures.

Table D-1. Natural Frequencies

Mode

Hand

Calculation

fn(Hz)

FEA

fn(Hz)

1 291 294

2 - 708

3 - 852

4 - 1110

5 - 1739

24

Figure D-1.

Figure D-2.

01

23

45

6

0

1

2

3

4

0

20

40

60

80

100

Mode 1 fn= 294.2 Hz

01

23

45

6

0

1

2

3

4

-150

-100

-50

0

50

100

150

Mode 2 fn= 708 Hz

25

Figure D-3.

Figure D-4.

0 1 2 3 4 5 6

0

1

2

3

4

-80

-60

-40

-20

0

20

40

60

80

Mode 3 fn= 852.3 Hz

01

23

45

6

0

0.5

1

1.5

2

2.5

3

3.5

4

-60

-40

-20

0

20

40

60

80

100

Mode 4 fn= 1110 Hz

26

Figure D-5.

0

1

2

3

4

5

6

0

0.5

1

1.5

2

2.5

3

3.5

4

-100

-50

0

50

100

Mode 5 fn= 1739 Hz