Finite Element Analysis of the Beams

Under Thermal Loading

Mohammad Tawfik, PhD

Aerospace Engineering Department

Cairo University

2

Table of Contents

Finite Element Analysis of the Beams Under Thermal Loading............................. 1

1. Derivation of the Finite Element Model .............................................................. 3

1.1. The Displacement Functions .......................................................................... 3

1.2. Displacement Function in terms of Nodal Displacement ............................... 4

1.3. Nonlinear Strain Displacement Relation ........................................................ 6

1.4. Inplane Forces and Bending Moments in terms of Nodal Displacements ..... 8

1.5. Deriving the Element Matrices Using Principal of Virtual Work .................. 9

1.5.1. Virtual work done by external forces ................................................. 12

2. Solution Procedures and Results of Panel Subjected to Thermal Loading ....... 14

References ............................................................................................................. 17

3

1. Derivation of the Finite Element Model

In this section, the equation of motion with the consideration of large deflection are

derived for a plate subject to external forces and thermal loading. The thermal loading

is accounted for as a constant temperature distribution. The element used in this study is

the rectangular 4-node Bogner-Fox-Schmidt (BFS) C1 conforming element (for the

bending DOF’s). The C1 type of elements conserves the continuity of all first

derivatives between elements.

1.1. The Displacement Functions

The displacement vector at each node for FE model is

T

ux

ww

( 1.1)

The above displacement vector includes the membrane inplane displacement u and

transverse displacement vector

T

x

ww

.

The 4-term polynomial for the transverse displacement function is assumed in the

form

3

4

2

321),( xaxaxaayxw ( 1.2)

or in matrix form

}{

xx1 32

aH

axw

w

( 1.3)

4

where Taaaaa 4321}{ is the transverse displacement coefficient vector. In

addition, the two-term polynomial for the inplane displacement functions can be written

as

xbbxu 21)( ( 1.4)

or in matrix form

}{

1

bH

bxu

u

( 1.5)

where Tbbb 21 is the inplane displacement coefficient vector. The coordinates

and connection order of a unit 4-node rectangular plate element are shown in Figure 1.1.

Figure 1.1. Node Numbering Scheme

1.2. Displacement Function in terms of Nodal Displacement

The transverse displacement vector at a node of the panel can be expressed by

16

15

1

222222

2322322322

3232223222

3332232233322322

29664330220010000

3322332020100

3232302302010

1

a

a

a

yxxyyxxyyxyx

yxyxyxyxxyxyxyxyx

yxxyyxxyyyxyxyxyx

yxyxyxyxxyyxyxyyxxyxyxyx

yx

w

y

wx

ww

( 1.6)

5

Substituting the nodal coordinates into equation (2.7), we obtain the nodal bending

displacement vector {wb} in terms of {a} as follows,

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

2

2

32

32

222222

2322322322

3232223222

3332232233322322

2

32

2

32

4

2

4

4

4

3

2

3

3

3

2

2

2

2

2

1

2

1

1

1

0000300200010000

0000003000200100

0000000000010

0000000000001

9664330220010000

3322332020100

3232302302010

1

0000030020010000

0000000000100

0000000003002010

0000000000001

0000000000010000

0000000000000100

0000000000000010

0000000000000001

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

bb

bb

bbb

bbb

baabbaabbaba

babababaababababa

baabbaabbbabababa

babababaabbababbaabababa

aa

aaa

aa

aaa

yx

w

y

wx

w

w

yx

w

y

wx

w

w

yx

w

y

wx

w

w

yx

w

y

wx

w

w

( 1.7)

or

aTw bb ( 1.8)

From equation (2.9), we can obtain

bb wTa1

( 1.9)

Substituting equation (2.10) into equation (2.3) then

bwbbw wNwTHw 1

( 1.10)

where the shape function for bending is

1 bww THN ( 1.11)

Similarly, the inplane displacement {u, v} can be expressed by

bxyyx

xyyx

v

u

10000

00001 ( 1.12)

6

Substituting the nodal coordinates into the equation (2.13), we can obtain the

inplane nodal displacement {wm} of the panel

8

7

6

5

4

3

2

1

4

4

3

3

2

2

1

1

0010000

0000001

10000

00001

0010000

0000001

00010000

00000001

b

b

b

b

b

b

b

b

b

b

abba

abba

a

a

v

u

v

u

v

u

v

u

( 1.13)

or

bTw mm ( 1.14)

From (2.15),

mm wTb1

( 1.15)

Substituting equation (2.16) into equation (2.6) gives

mummu wNwTHu 1

( 1.16)

where the inplane shape functions are

1 muu THN ( 1.17)

1.3. Nonlinear Strain Displacement Relation

The von Karman large deflection strain-displacement relation for the deflections u,

and w can be written as follows

2

22

2

1

x

wz

x

w

x

ux

( 1.18)

or

7

zm ( 1.19)

where

m = membrane inplane linear strain vector,

= membrane inplane nonlinear strain vector,

z = bending strain vector.

The inplane linear strain can be written in terms of the nodal displacements as

follows

mmmmuu

m wBwTx

Hb

x

H

x

u

1 ( 1.20)

The inplane nonlinear strain can be written as follows

bbbww

w

wBwTx

Ha

x

H

ax

HG

x

w

x

w

2

1][

2

1}{

2

1

}{2

1

2

1

2

1

1

( 1.21)

where the slope matrix and slope vector are

x

w

( 1.22)

x

wG

( 1.23)

and

Combining equations (2.23) and (2.26), the inplane strain can be written as follows

8

bmmm ww BB2

1 ( 1.24)

The strain due to bending can be written in terms of curvatures as follows

}{}{}{1

2

2

2

2

2

2

bbbbww wBwT

x

Ha

x

H

x

w

( 1.25)

Thus, the nonlinear strain-nodal displacement relation can be written as

bbbmm

m

wzww

z

BBB

2

1

}{

( 1.26)

1.4. Inplane Forces and Bending Moments in terms of Nodal Displacements

In this section, the derivation of the relation presenting the inplane forces {N} and

bending moments {M} in terms of nodal displacements for global equilibrium will be

derived. Constitutive equation can be written in the form

T

T

M

N

D

A

M

N

0

0 ( 1.27)

where 14

,

EAQhA extensional matrix ( 1.28) (a)

EIQh

D 12

3

flexural matrix (c)

2/

2/),,(

h

hT dzzyxTQN inplane thermal loads (d)

2/

2/),,(

h

hT zdzzyxTQM thermal bending moment (e)

and

9

h thickness of the panel,

{} thermal expansion coefficient vector,

T(x,y,z) temperature increase distribution above the ambient temperature

For constant temperature distribution in the Z-direction, the inplane and bending loading

due to temperature can be written in the following form

TEANT

02/

2/

h

hT zdzQTM for isotropic Beam.

with

EQ ][ ( 1.29)

Expanding equation (2.36) gives

Tm

Tbmm

Tm

NNN

NwAwA

NAN

BB2

1

][

( 1.30)

bb wDDM B][}]{[ ( 1.31)

1.5. Deriving the Element Matrices Using Principal of Virtual Work

Principal of virtual work states that

0int extWWW ( 1.32)

Virtual work done by internal stresses can be written as

V A

TT

ijij dAMNdVW }{}{int ( 1.33)

where

10

TTT

b

T

m

T

m

TT

m

T

ww

BB

( 1.34)

and

T

b

T

b

Tw B ( 1.35)

Note that

GG

2

1

Substituting equations (2.39), (2.40), (2.43), and (2.44) into equation (2.42), the

virtual work done by internal stresses can be expressed as follows

dA

wDw

NwAwA

ww

WA

bb

T

b

T

b

Tbmm

TTT

b

T

m

T

m

BB

BB

BB

][

2

1

*

int

( 1.36)

The terms of the expansion of equation (2.45) are listed as follows

mm

T

m

T

m wAw BB ( 1.37) (a)

b

T

m

T

m wAw BB2

1 (b)

T

T

m

T

m Nw B (c)

mm

TTT

b wAw BB (d)

b

TTT

b wAw BB2

1 (e)

T

TTT

b Nw B (f)

bb

T

b

T

b wDw BB ][ (g)

11

Terms (a) and (g) of equation (2.46) can be written in the matrix form as

m

b

m

b

mbw

w

k

kww

0

0 ( 1.38)

Where the linear stiffness matrices are

dADkA

b

T

bb BB ][][ ( 1.39)

dAAkA

m

T

mm BB][ ( 1.40)

While terms (b) + (d) of equation (2.46) can be written as

Abm

TT

b

mm

TTT

bb

T

m

T

m

Amm

TTT

b

mm

TTT

bb

T

m

T

m

A

mm

TTT

bb

T

m

T

m

mbmbbmbmbnmb

m

b

mb

bmnm

mb

dA

wBNw

wAwwAw

dA

wAw

wAwwAw

dAwAwwAw

wnwwnwwnw

w

w

n

nnww

B

BBBB

BB

BBBB

BBBB

2

1

2

1

2

1

2

1

2

1

2

1

2

1

12

11

2

11

2

1

01

11

2

1

Note that

bmmxmm

T

mm

TwBNGNN

x

wNwA

B

Thus,

dAAnnA

T

m

T

bmmb BB]1[]1[ ( 1.41)

dABNnA

m

T

nm B1 ( 1.42)

The first order nonlinear stiffness matrices, nmmb nn 1&1 , are linearly dependent on

the node DOF {wm}([Nm]) and {wb}([]).

12

The second order nonlinear stiffness can be derived from term (e):

b

TTT

bb

T

b wAwwnw BB2

12

3

1

Thus,

dAAnA

TT

BB2

3]2[ ( 1.43)

Also

bT

TT

bT

TT

bT

TT

b

Tx

TT

bTx

TT

bT

TTT

b

wBNwbCNwGNw

x

wNwN

x

wwNw

BBB

BBB

Thus,

dABNkA

T

T

TN B][ ( 1.44)

where

TxT NN ( 1.45)

Term (f) of equation (2.46) can be written in matrix form as follows

dANpA

T

T

mTm B ( 1.46)

1.5.1. Virtual work done by external forces

For the static problem, we may write:

A

Surfaceiiext

dAyxpw

dSuTW

),(

( 1.47)

13

where T is the surface traction per unit area and p(x,y,t) is the external load vector. The

right hand side of equation (2.57) can be rewritten as b

T

b pw where

dAyxpNpA

T

wb ),( ( 1.48)

Finally, we may write

m

b

mb

bmnm

TN

m

b

W

W

N

N

NN

K

K

K

00

02

3

1

01

11

2

1

00

0

0

0

Tm

b

P

P 0

0 ( 1.49)

Equation (2.59) presents the static nonlinear deflection of a panel with thermal

loading, which can be written in the form

TTN PPWNNKK

2

3

11

2

1 ( 1.50)

Where

K is the linear stiffness matrix,

TNK is the thermal geometric stiffness matrix,

1N is the first order nonlinear stiffness matrix,

2N is the second order nonlinear stiffness matrix,

P is the external load vector,

TP is the thermal load vector,

14

2. Solution Procedures and Results of Panel Subjected to

Thermal Loading

The solution of the thermal loading problem of the panel involves the solution of the

thermal-buckling problem and the post-buckling deflection. In this chapter, the solution

procedure for predicting the behavior of panel will be presented.

For the case of constant temperature distribution, the linear part of equation 2.60 can

be written as follows

0 WKTK TN ( 2.1)

Which is an Eigenvalue problem in the critical temperature crT .

Equation (2.60) that describes the nonlinear relation between the deflections and the

applied loads can be also utilized for the solution of the post-buckling deflection. Recall

TTN PWNNKK

2

3

11

2

1 (2.60)

Introducing the error function W as follows

023

11

2

1

TTN PWNNKKW ( 2.2)

which can be written using truncated Taylor expansion as follows

WdW

WdWWW

( 2.3)

15

where

tan21 KNNKK

dW

WdTN

( 2.4)

Thus, the iterative procedures for the determination of the post-buckling displacement

can be expressed as follows

TiiiTNi PWNNKKW

2

3

11

2

1 ( 2.5)

iiiWWK 1tan ( 2.6)

ii WKWi

1

tan1 ( 2.7)

11 iii WWW ( 2.8)

Convergence occur in the above procedure, when the maximum value of the 1iW

becomes less than a given tolerance tol ; i.e. toliW 1max .

Figure 2.1 presents the variation of the maximum transverse displacement of the

panel when heated beyond the buckling temperature. Notice that the rate of increase of

the buckling deformation is very high just after buckling, then it decreases as the

temperature increases indicating the increase in stiffness due to the increasing influence

of the nonlinear terms.

16

0

0.5

1

1.5

2

2.5

6 11 16 21 26

Temperature Increase (C)

Wm

ax

/Th

ick

ne

ss

Figure 2.1. Variation of maximum deflection of the plate with temperature increase.

17

References

1 Mei, C., “A Finite Element Approach for Non-linear panel-flutter,” AIAA Journal,

Vol. 15, No. 8, 1977, pp. 1107-1110.

2 Zhou, R. C., Xue, D. Y., and Mei, C., “On Analysis of Nonlinear Panel-flutter at

Supersonic Speeds,” Proceedings of the First Industry/Academy Symposium On

Research For Future Supersonic And Hypersonic Vehicles, Vol. 1, Greensboro,

North Carolina, 1994, pp. 343-348.

3 Xue, D. Y., and Mei, C., “Finite Element Non-linear Panel-flutter with Arbitrary

Temperature in Supersonic Flow,” AIAA Journal, Vol. 31, No. 1, 1993, pp. 154-162.

4 Frampton, K. D., Clark, R. L., and Dowell, E. H., “State-Space Modeling For

Aeroelastic Panels With Linearized Potential Flow Aerodynamic Loading,” Journal

Of Aircraft, Vol. 33, No. 4, 1996, pp. 816-822.

5 Dowell, E. H., “Panel Flutter: A Review of The Aeroelastic Stability of Plates and

Shells,” AIAA Journal, Vol. 8, No. 3, 1970, pp. 385-399.

6 Bismarck-Nasr, M. N., “Finite Element analysis of Aeroelasticity of Plates and

Shells,” Applied Mechanics Review, Vol. 45, No. 12, 1992, pp. 461-482.

7 Bismarck-Nasr, M. N., “Finite Elements in Aeroelasticity of Plates and Shells,”

Applied Mechanics Review, Vol. 49, No. 10, 1996, pp. S17-S24.

8 Mei, C., Abdel-Motagaly, K., and Chen, R., “Review of Nonlinear Panel Flutter at

Supersonic and Hypersonic Speeds,” Applied Mechanics Review, Vol. 52, No. 10,

1999, pp. 321-332.

18

9 Sarma, B. S., and Varadan, T. K. “Non-linear Panel-flutter by Finite Element

Method,” AIAA Journal, Vol. 26, No. 5, 1988, pp. 566-574.

10 Yang, T. Y., and Sung, S. H. “Finite Element Panel-flutter in Three-Dimensional

Supersonic Unsteady Potential Flow,” AIAA Journal, Vol. 15, No. 12, 1977, pp.

1677-1683.

11 Ashley, H., and Zartarian, G., “Piston Theory – A New Aerodynamic Tool for the

Aeroelastician,” Journal of Aeronautical Sciences, Vol. 23, No. 12, 1956, pp. 1109-

1118.

12 Dixon, I. R., and Mei, C., “Finite Element Analysis of Large-Amplitude Panel-

flutter of Thin Laminates,” AIAA Journal, Vol. 31, No. 4, 1993, pp. 701-707.

13 Abdel-Motagaly, K., Chen, R., and Mei, C. “Nonlinear Flutter of Composite Panels

Under Yawed Supersonic Flow Using Finite Elements,” AIAA Journal, Vol. 37, No

9, 1999, pp. 1025-1032.

14 Zhong, Z. “Reduction of Thermal Deflection And Random Response Of Composite

Structures With Embedded Shape memory Alloy At Elevated Temperature”, PhD

Dissertation, 1998, Old Dominion University, Aerospace Department, Norfolk,

Virginia.

15 Frampton, Kenneth D., Clark, Robert L., and Dowell, Earl H. “Active Control Of

Panel-flutter With Linearized Potential Flow Aerodynamics”, AIAA Paper 95-1079-

CP, February 1995.

19

16 Gray, C. E., Mei, C., and Shore, C. P., “Finite Element Method for Large-Amplitude

Two-Dimensional Panel-flutter at Hypersonic Speeds,” AIAA Journal, Vol. 29, No.

2, 1991, pp. 290-298.

17 Benamar, R, Bennouna, M. M. K., and White R. G. “The effect of large vibration

amplitudes on the mode shapes and natural frequencies of thin elastic structures

PART II: Fully Clamped Rectangular Isotropic Plates”, Journal of Sound and

Vibration, Vol. 164, No. 2, 1993, pp. 295-316.

18 Benamar, R, Bennouna, M. M. K., and White R. G. “The effect of large vibration

amplitudes on the mode shapes and natural frequencies of thin elastic structures

PARTIII: Fully Clamped Rectangular Isotropic Plates – Measurements of The Mode

Shape Amplitude Dependence And The Spatial Distribution Of Harmonic

Distortion”, Journal of Sound and Vibration, Vol. 175, No. 3, 1994, pp. 377-395.

19 Liu, D. D., Yao, Z. X., Sarhaddi, D., and Chavez, F., “From Piston Theory to

Uniform Hypersonic-Supersonic Lifting Surface Method,” Journal of Aircraft, Vol.

34, No. 3, 1997, pp. 304-312.

20 Lee, I., Lee, D.-M., and Oh, I.-K, “Supersonic Flutter Analysis of Stiffened

Laminated Plates Subject to Thermal Load,” Journal of Sound and Vibration, Vol.

224, No. 1, 1999, pp. 49-67.

21 Surace, G. and Udrescu, R., “Finite-Element Analysis of The Fluttering Panels

Excited by External Forces,” Journal of Sound and Vibration, Vol. 224, No. 5, 1999,

pp. 917-935.

20

22 Bismarch-Nasr, M. N. and Bones, A., “Damping Effects in Nonlinear Panel Flutter,”

AIAA Journal, Vol. 38, No. 4, 2000, pp. 711-713.

23 Young, T. H. and Lee, C. W., “Dynamic Stability of Skew Plates Subjected to

Aerodynamic and Random In-Plane Forces,” Journal of Sound and Vibration, Vol.

250, No. 3, 2002, pp. 401-414.

24 Zhou, R.C., Lai, Z., Xue, D. Y., Huang, J. K., and Mei, C., “Suppression Of

Nonlinear Panel-flutter with Piezoelectric Actuators Using Finite Element Method”,

AIAA Journal, Vol. 33, No. 6, 1995, pp. 1098-1105.

25 Frampton, K. D., Clark, R. L., and Dowell, E. H., “Active Control Of Panel-flutter

With Piezoelectric Transducers,” Journal Of Aircraft, Vol. 33, No. 4, 1996, pp. 768-

774.

26 Dongi, F., Dinkler, D., and Kroplin, B. “Active Panel-flutter Suppression Using

Self-Sensing Piezoactuators,” AIAA Journal, Vol. 34, No. 6, 1996, pp. 1224-1230.

27 Scott, R. C., and Weisshaar, T. A., “Controlling Panel-flutter Using Adaptive

Materials,” AIAA Paper 91-1067-CP, 1991.

28 Suzuki, S. and Degali, T., “Supersonic Panel-flutter Suppression Using Shape

Memory Alloys,” International Journal of Intelligent Mechanics: Design and

Production, Vol. 3, No. 1, 1998, pp. 1-10.

29 Tawfik, M., Ro, J. J., and Mei, C., “Thermal post-buckling and aeroelastic

behaviour of shape memory alloy reinforced plates,” Smart Materials and Structures,

Vol 11, No. 2, 2002, pp. 297-307.

21

30 Xue, D.Y., “Finite Element Frequency Domain Solution of Nonlinear Panel-flutter

With Temperature Effects And Fatigue Life Analysis”, PhD Dissertation, 1991, Old

Dominion University, Mechanical Engineering Department, Norfolk, Virginia.