This document presents the derivation of a finite element model for analyzing beams under thermal loading. It describes: 1) The displacement functions used in the model, including a 4-term polynomial for transverse displacement and 2-term polynomial for in-plane displacement. 2) Deriving the element matrices using the principle of virtual work, accounting for external and thermal loading. 3) The procedures and results of applying the model to analyze a panel subjected to thermal loading.

1. Finite Element Analysis of the Beams
Under Thermal Loading
Mohammad Tawfik, PhD
Aerospace Engineering Department
Cairo University

2. 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. 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
u
x
w
w








(1.1)
The above displacement vector includes the membrane inplane displacement u and
transverse displacement vector
T
x
w
w








.
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. 4
where  T
aaaaa 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    T
bbb 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
2
9664330220010000
3322332020100
3232302302010
1
a
a
a
yxxyyxxyyxyx
yxyxyxyxxyxyxyxyx
yxxyyxxyyyxyxyxyx
yxyxyxyxxyyxyxyyxxyxyxyx
yx
w
y
w
x
w
w

(1.6)

5. 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
w
x
w
w
yx
w
y
w
x
w
w
yx
w
y
w
x
w
w
yx
w
y
w
x
w
w
(1.7)
or
    aTw bb  (1.8)
From equation (2.9), we can obtain
     bb wTa
1
 (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
 b
xyyx
xyyx
v
u













10000
00001
(1.12)

6. 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 wTb
1
 (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
w
z
x
w
x
u
x













 (1.18)
or

7. 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
           mmmm
uu
m wBwT
x
H
b
x
H
x
u










1
 (1.20)
The inplane nonlinear strain can be written as follows
      
             bbb
ww
w
wBwT
x
H
a
x
H
a
x
H
G
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
w
G


 (1.23)
and
Combining equations (2.23) and (2.26), the inplane strain can be written as follows

8. 8
          bmmm ww   BB
2
1
 (1.24)
The strain due to bending can be written in terms of curvatures as follows
          }{}{}{
1
2
2
2
2
2
2
bbbb
ww
wBwT
x
H
a
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)
EIQ
h
D 
12
3
flexural matrix (c)
  

2/
2/
),,(
h
h
T dzzyxTQN  inplane thermal loads (d)
  

2/
2/
),,(
h
h
T zdzzyxTQM  thermal bending moment (e)
and

9. 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 
   0
2/
2/
 
h
h
T zdzQTM  for isotropic Beam.
with
EQ ][ (1.29)
Expanding equation (2.36) gives
     
          
     Tm
Tbmm
Tm
NNN
NwAwA
NAN











BB
2
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. 10
     
         TTT
b
T
m
T
m
TT
m
T
ww 



BB 

(1.34)
and
     T
b
T
b
T
w 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
W
A
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  BB
2
1
(b)
     T
T
m
T
m Nw  B (c)
         mm
TTT
b wAw BB   (d)
            b
TTT
b wAw   BB
2
1
(e)
       T
TTT
b Nw   B (f)
      bb
T
b
T
b wDw BB ][ (g)

11. 11
Terms (a) and (g) of equation (2.46) can be written in the matrix form as
 












m
b
m
b
mb
w
w
k
k
ww
0
0
 (1.38)
Where the linear stiffness matrices are
   dADk
A
b
T
bb  BB ][][ (1.39)
    dAAk
A
m
T
mm  BB][ (1.40)
While terms (b) + (d) of equation (2.46) can be written as
 
           
                  
                   
          
                     
        



















































A
bm
TT
b
mm
TTT
bb
T
m
T
m
A
mm
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
nn
ww












B
BBBB
BB
BBBB
BBBB
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
2
1
1
2
1
1
2
1
01
11
2
1
Note that
                bmmxmm
T
mm
T
wBNGNN
x
w
NwA  


B
Thus,
      dAAnn
A
T
m
T
bmmb   BB]1[]1[ (1.41)
       dABNn
A
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. 12
The second order nonlinear stiffness can be derived from term (e):
               b
TTT
bb
T
b wAwwnw   BB
2
1
2
3
1

Thus,
        dAAn
A
TT
   BB
2
3
]2[ (1.43)
Also
               
                      bT
TT
bT
TT
bT
TT
b
Tx
TT
bTx
TT
bT
TTT
b
wBNwbCNwGNw
x
w
NwN
x
w
wNw













BBB
BBB
Thus,
     dABNk
A
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
      dANp
A
T
T
mTm    B (1.46)
1.5.1. Virtual work done by external forces
For the static problem, we may write:
 



A
Surface
iiext
dAyxpw
dSuTW
),(

(1.47)

13. 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
    dAyxpNp
A
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
1
1
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. 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
1
1
2
1
(2.60)
Introducing the error function   W as follows
               02
3
1
1
2
1






  TTN PWNNKKW (2.2)
which can be written using truncated Taylor expansion as follows
         W
dW
Wd
WWW 

 (2.3)

15. 15
where
             tan21 KNNKK
dW
Wd
TN 

 (2.4)
Thus, the iterative procedures for the determination of the post-buckling displacement
can be expressed as follows
              TiiiTNi PWNNKKW  





 2
3
1
1
2
1
(2.5)
     iii
WWK 1tan  (2.6)
      ii WKW i



1
tan1 (2.7)
      11   iii WWW  (2.8)
Convergence occur in the above procedure, when the maximum value of the   1iW
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. 16
0
0.5
1
1.5
2
2.5
6 11 16 21 26
Temperature Increase (C)
Wmax/Thickness
Figure 2.1. Variation of maximum deflection of the plate with temperature increase.

17. 17
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18. 18
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19. 19
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20. 20
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21. 21
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