REPRINT 132
Finite Element Displacement Analysis
of Plate Bending Based
on Rectangular. Elements
By Civil Engineer HARALD ~N5TEEN
Norwegian Building Research Institute
~
Norges byggforslmingsinstitutt
NORWEGIAN BUILDING RESEARCH INSTITUTE
NUl
00000
OSLO 1966
University of Newcastle upon Tyne Department of Civil Engineering
WORKING SESSION No.4.
PAPER No. 14.
INTERNATIONAL SYMPOSIUM
The use oj Electronic Digital Computers in Structural Engineering
FINITE ELEMENT DISPLACEMENT ANALYSIS OF PLATE BENDING
BASED ON RECTANGULAR ELEMENTS
Harald Hansteen
1
SYNOPSIS
This paper describes a finite element procedure which ensures complete continuity of displacements between adjacent elements. The reliability of the procedure is illustrated by numerical solutions given
at the end of the paper.
INTRODUCTION
In plate bending problems few of the cases met with in practice can
be solved by analytically exact methods. Very often one has to rely on
approximate procedures of variable accuracy. It is natural, therefore,
that investigators have shown great interest in examining the possibilities offered by the finite element method, and as a result a number of
papers on the subject have been published in the recent years. The major
part of these works is
concerned with plate elements of rectangular form.
Numerical results reported are yet too few to give reason for any conclusions concerning the accuracy and reliability of the approach in general.
However, encouraging results have been obtained in particular cases,
Zienkiewicz
(8). Some investigators have wanted to utilize the advantage
of triangular elements in representing plates of arbitrary shape. Numerical examples carried through by this procedure have not proved.to be
1
Civil engineer, Norwegian Building Research Institute, Oslo, Norway.
A",., .'~
•••. ~ _.1. •
as accurate as for rectangular elements. The results show a converging
character as the element size decreases; however, the values towards
which they converge greatly depend. on the displacement patterns selected for the elements and have generally little resemblance to the correct
values, Clough (4).
Melosh (7) hap shown that a necessary condition for the finite element
displacement method to converge towards the correct solution is that the
selected displacement patterns must be continuous in the internal of the
elements, and maintain continuity with displacements of adjacent elements.
This condition has not been fully accomplished by the procedures published
so far, all of which fail to establish continuity in normal slopes along
the lines connecting the nodal points. Clough (4) claims this circumstance
to be the basic reason for the lack of accuracy of his results. One
should anticipate a similar tendency to oceur for the procedures applying
rectangular elements. When such a tendency is not as yet found, the reason
is probably that in the limited number of test examples examined the discrepancy in normal slope between the elements has been of little significanoe.
ANALYSIS OF ELEMENT STIFFNESS IN BENDING
The fundamental idea of the finite element method is to represent the
actual structure by a finite number of individual elements, interconnected
at a finite number of nodal points. The stiffness of the idealized structure is obtained by adding the stiffnesses possessed by the individual
elements. To obtain a good computational model it is therefore of great
importance to be careful when deciding the· stiffness properties of the
elements.
To be able.to derive the element stiffnesses, one has to assume a finite number of displacement modes [degrees of freedom] for the eiement.
The final dispiacement ·pattern of the element Will be l:lJnited to a linear
I
combination of these displacement modes. These :lJnposed restpictloDs to the
deformation of the element is equivalent to the introduction of constraints,
and it is fairly obvious; therefore, that the computed stiffnes~~~ w1l~.be
greater than the true ones, provided that the selected displacement patterns satisfy the Melosh conditions referenced above. Further· it should be
emphasized that the Melosh conditions are necessary, but not sufficient,
to ensure convergency of the results. Another important condition must
be that the selected displace~nts should be able to describe all the
essential modes of the true solution in all points of the
plate. If this is not achieved the finite element s.olution will converge
towards values for the displacements which are smaller than the true values.
In Fig. 1a are shown the four rectangular elements interconnected at
nodal point number k. The outer edges of the elements are supposed to be
completely fixed, and the object is to study the stiffness properties of
the elements due to generalized unit displacements of point k. The general displacement pattern of the built-in plate is composed by four essentially different modes of displacements, as described in Table 1. To the
right in the Table are given the corresponding values of the nodal displacements:
Displ. at node k.
Displacement patterns
.
1. Positive displ. of all four elements
wk
wk,x
w
k,y
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
w
k,xy
2. Positive displ. of elements I and IV,
negative displ. of elements II and III
3. Positive displ. of elements I and II.
negative displ. of elements III and IV
4. Positive displ. of elements I and III,
negative displ. of elements II and IV
Table 1
Here w is the normal displacement of node k and the comma denotes park
tial differentation.
Fig. lb shows element III provided with, an internal coordinate system
(x,y,z), z positive downwards, and with internal numbering of the nodes
from 1 to 4 [internal node 1 corresponds to the global numbering k]. The
side lengths of the element are a and b, and a new set of dimensionless
coordinates (~,~) which may substitute coordinates (x,y) are defined as
follows:
5=
x/a, "l = y/b
(1 )
The simplest polynomial representation of the displacements satisfying
both the Melosh conditions and the displacement patterns of Table 1, will
for node 1 be:
(5,1) is adopted to describe the continuous normal displace1
ments corresponding to displacements at node 1. The equation may be written
The notation
V
on matrix form, yielding
(3)
where IA 1 is a row vector and"" 1 a column vector as follows:
(4)
.., 1 = { wI' aw 1 ,x' bW 1 ,y' abw 1 ,xy}
(5)
Similarily one can find the displacement patterns corresponding to unit
displacements at nodes 2 - 4. Hence the total displacement of an element
is expressed by:
v(s,')
=
[A 1,A 2 ,A y "'4 ] "'1
=
A(S'7)
(6 )
'W
""2
1>01
3
'''''4
The "'i matrices are easily obtained from Eq. (5) by substituting the actual nodal point number. The IA i matrices will be
A2
= [ 4<t+~)2(1+S)(~7)2(1+'l)
,
+2(~+~)2(1+S)(~7)2(~+'l)
,
+(~+~l(i+5)(i+'l)2(t+?)
,
2( ~+5)2 (i+s) a+?
= [
= [
4(~+5)2(1+5)(~+?)2(1+7)
2(i+s)2( 1+s)(i+'(')2( ~'1)
,
2( ~+5h~+5) (i+')2( 1+?)
]
l( 1+?)
(~+S)2(~5)(t+?)2(i+?) ]
4(~5)2(1+5)(~+7)2(1+'Z) , +2(~5)2( t+S) (~+?)2( 1+'()
2( ~§)2( 1+:5)(~+'7)2( i+?) , +(~S)2(~+5)(~+?)2(~?) ]
(7a- c)
curvatures,~,
Differentiating Eq. (6) the internal element
may be ex-
pressed qy the generalized displacements at the nodes, as follows:
(8)
Introducing now the elastic characteristics Df the finite element material defined by the stress-strain matrix ID , the internal moments,M ,
in the element are expressed by
where
[)
Et 3
~
12(1'l-\?)
0
"
0
E
~ [~odulus
"
~
t
(10)
0
"
0
2( 1+,,)
of elasticity
Pcisson's ratio
thickness of the plate
The elastic energy stDred in an element upon deformation, Wi' should
equalize the external work, W , performed by the load on the element.
e
This load consists of the reactive nodal forces from adjacent elements,
S, and distributed and concentrated loads at the surface of the plate.
For brevity Dnly distributed surface loads,
derivation below; however, there are no
q(S'~)'
are considered in the
di~ficulties
in taking into ac-
count also concentrated forces or moments arbitrarily situated on the elements. The expressions for work then are:
Wi ~
is )(.t M dV ~ -kwt [J
B t[)1B
Vol
Vol
We
~t
\,}S +
-k lre~ t
q d A
dV]w
~ t"hs
~ ~ "...h~......
+ {ret't q d A]
(11)
~ ~ ....t[S +5]
(12)
Equating the two expressions one obtains
(13 )
where
j
i3 t(S,?)[JIB (S.,,/) dV
(14 )
Vol
j At(5''7)q(~,?)dA
(15 )
Area
~
is the 16 by 16 stiffness matrix of an element and 5
a 16 by 1 mat-
rix representing the external load on the plate. is has generally nouzero elements and may be interpreted as generallized loadings [both vertical forces and moments] acting in the nodal points. [See also Zienkiewicz
(8)].
If the size of the elements is small compared to the dimensions of the
plate the substitution of distributed loads by concentrated vertical forces at the nodes is a reasonably good approximation. When using relatively large elements, however, one should anticipate improvement of the
accuracy by doing the more detailed analysis shown above. As the latter
procedure completely corresponds to the actual loads on the plate, andas
the elements ors are usually easy to calculate, there should be little
reason not to prefer this procedure in all cases.
The stiffness of the assembled structure is obtained from Eq. (13) by
adding the stiffnesses of the individual elements meeting in each nodal
point. The final equation relating generalized
nodal loads,
~
• to the
corresponding nodal displacements, r , may be written.
R ; D< ..
~
(16 )
being the stiffness matrix of the assembled structure. After inserting
the actual boundary conditions in Eq. (16),
th~
nodal displacements are'
found by solving the system of simultaneous equations. With the nodal displacements known the plate moments may be computed from Eq. (9) ..
EXAMPLES OF NUMERICAL SOLUTIONS
The numerical results shown below are intended to illustrate the accuracy of the procedure. The moment values
repo~ted
are the mean values of
the moments at each nodal point. In the cases where distributed loads
occur the corresponding nodal loads have been computed from Eq. (15).
Example l : The first example is the standard problem of a simply sup-'
ported quadratic plate og uniform thickness shown in Fig. 2. Two loading
cases are considered:
~ase
1: A uniform load allover the plate.
case 2: A concentrated load in the middle of the plate.
To study the convergence of the solution four different finite
elemen~
meshes of successively reduced sizes have been employed as shown in the
figure. The accuracy of the various solutions is exemplified by comparing
the values obtained for the central point of the plate to the exact values.
The results are given in the table at the bottom of Fig. 2.
Example 2: Fig. 3 shows the same plate as in example 1, this time partially loaded by a uniform load over a square 1/8 by 1/8 of the side lengths.
The applied finite element mesh, as well as the position of the loaded area
-~;
[shaded in the figure], are shown in the figure. In the accompaning table
,
are given the computed and the exact displacements and moments at the nodal ..•
points of the section A-A.
,
Example 3: The third example is a uniformly loaded plate completely
fixed at three edges' and free at the fourth edge. The results are cumpared to a nearly exact solution of the problem proposed by Hellan (6).
Hellan has given the displacement values with three significant digits.
These digits are verified by the finite element solution. The necessary
data of the example and the moments in significant sections of the plate
are shown in Fig. 4.
CONCLUSIONS
The numerical examples cover some of the loading cases and'ed!e conditions that frequently oocur in practical problems. It ia intere!ting to note
that the degree of accuracy obtained for the displacements and the moments in the various caaes do not depend on the type of the problem.
This indicates that equally accurate results may be expected for other
types of problems.
In example 1 are tested tbe convergence properties of the procedure.
That the results do co;"';"~e toward the correct solution is clearly demonstrated by the resultfr. Another interesting feature is that even the
1
results obtained for the very coarse mesh n = 1 are well within the limits of accuracy needed in practical computations. One should, however,
be careful not to draw any conclusions from one simple example. Further
investigations are needed to clarify the connection between the size of
the finite elements and the accuracy of the results.
The results of Examples 2 and 3 are close to the exact values. The
greatest errors occur at points where the gradient of the curvatures are .
steep, as in point 3 of Example 2. The maximum values seem to be very
accurate. An exception is the moment value in point 1 of Example 3. However, in this particular point Hellan (6) points out that the reference
value is less reliable.
Note. - During the preparation of this paper it came to the author's
knowledge that a report, that in parts may develop along similar lines
as the present work, have been made by Butlin (3). As this report has
not been available for the author he has not been able to check the points
of similarity.
REFERENCES
1. Argyris, J. H. "Matrix Displacement Analysis of Anisotropic Shells
by Triangular Elements" Journal of the Royal AerCftau.tlcal Society,
Vol. 69, Nov. 1965.
2. Argyris, J. H. "The Trondheim Lectures on the Matrix Theory of
Structures". John Wiley
&
Sons, Ltd., London 1966.
3. ButUn, G. A. "On the Finite Element Technique in Plate Bending
Analysis; a Derivation of a Basic Stiffness 'Matrix" Internal re-
P9rt at Engineering Dep., Churchill College, Cambridge.
4. Clough, R. W. "The Finite Element Method in Strutural Mechanics" in
"Stress Analysis." John Wiley & Sons, Ltd., London 1965.
5. Clough, R. W. and Tocher, J. L. "Analysis of Thin Arch Dama by the
Finite Element Method': Int. Symp. on the Theory of Arch Dams, Southampton Univ., Pergamon Press, 1964.
6. Hellan, K. "The Rectangular Plate with One Free Edge", Trondhelm 1960.
7. Melosh, R. J. "Basis for the Derivation of Matrices for the Direct
Stiffness Method", J." Am. Aero. Astro., 1, 1631, 1963.
Displacements and lDOments at the central point.
3
4
exact
common
factor
44376
44353
44344
44343
4
3
+6
qa /Et x10 '
4,68
4,66
4,73
4,75
4,79
qa 2xl0+2 .
1209
1252
1260
1263
1265
Pa 2 /Et3x10+4
1
2
w
45002
M
w
n =
I
case
case II
Fig. 2.
y
,
Simply
I
---+--- --X
I,
,
I..
M~sh SIZ~S
a
.1
support~d ~dg~s
Loads
Cas~ I: uniform load
9
Ca$~lI: conc~ntrat~d load
in c~ntral point
v
=0,3
Analysis of this
lJuart~r
P
Dis~lacements
Point
2
f.e.m.
6447
and moments along section A - A
3
4
5
6
12312
13735
11793
6499
.w,
!
exact
6453
12320
13737
11793
6501
f.e.m.
0,27
1,08
2,70
2,62
0,94
exact
0,26
1,19
2,70
2,63
0,94
foe .m.
0,75
1,85
2,70
2,48
1,18
exact
0,73
1,83
2,70
2,49
1,15
r.
/ ..
common
factor
4 ~7
qa /Eexl0·
MJli
..
qa 2xl0 +3
My
f.e.m. ='finite element method
Fig: 3.
Simply supported edges
:j--+--t-e~~::t-__~!!~i form load
A-.4-~--+-~
.It
~-+--rH-t---t-- -
'X
V
1
A---·
2
•
:: 0,3
34567
II • • •
• ---A
9
8.
Zienkiewicz, O. C. "Finite Element Procedures in the Solution of
Plate and Shell Problems"in Stress Analysis". John Wiley & Sons Ltd.,
London, 1965.
9. Zienkiewicz, O. C. and Cheung, Y. K. "Finite Element Method of Analysis
of Arch Dam Shells and Comparison with Finite Difference Procedure,s".
Intern. Symp. on the Theory of Arch Dams, Southampton Univ., Pergamon
Press, 1964.
-
41><
,
I
.....
-I~
~
.:.:
•
~
~.
!sf
~
Moments.
1
Point
2
3
4
f.e.m.
+ 6,89
+ 5,18
+ 3,25
+ 1,20
Hellan (6 )
+ 7,14
+ 5,18
+ 3,22
+ 1,15
5
°
°
Mx
Point
5
6
f.e .m.
0
• 1,20
+ 3,20
+ 4,64
+ 5,15
Hellan (6)
°
+ 1,13
+ 3,17
+ 4,64
+ 5,16
Point
9.
10
11
12
13
0
0,09
1,16
1,98
2,56
+ 1,29
0,40
0,71
°
0,11
1,17
1,98
2,54
+ 1.32
0,35
0,68
°
8
7
common
factor
9
My
Mx
My
f.e.ro.
+
Mx
Hellan (6)
My
5}15
°
+ 5,16
f.e.ro. = finite element method
Fig. 4.
1
Upp~r ~dg~ fre~.
13
12
11
10
2
3
4
~
8 9
~f---_~a
oth~r edg~s
fixed
Uniform load CJ
v.0.2
--x
I
.. 1
qa 2x 10+2