AP04_3web.vp


1 Introduction
Problems of structural optimization and analysis of stress

and strain often lead to the function optimization problem of
minimizing a given functional subject to a system of differen-
tial equations and inequalities with some boundary condi-
tions, and a system of integral equations and inequalities.
This class of structural optimization problems is called func-
tion structural optimization. The branch of mathematics that
deals with this problem is called Variational Calculus. The dif-
ficulty of solving this problem has led to the idea of transform-
ing the given function optimization problem with the
Galerkin method (based on the finite element method) to a
problem of parameter optimization. This paper deals with
both function optimization and parameter optimization of
laminate plate stiffness.

The paper aims to express the relation of a measure of
laminate plate stiffness with respect to the fiber orientation
of its plies. The inverse of the scalar product of the lateral
displacement of the central plane and the lateral loading of
the plate is the measure of laminate plate stiffness. In the case
of a simply supported rectangular laminate plate this mea-
sure of stiffness is maximized, and the optimum orientation of
its plies is searched.

We contemplate the problem of rectangular laminate
plate with common but fixed given dimensions, number of
plies, thickness of plies, mechanical properties of orthotropic
plies and lateral loading that is simply suported on its bound-
ary with the aim to specify an orientation of its plies that
maximizes the measure of stiffness, which is the inverse of the
measure of compliance. The measure of compliance is given
by the scalar product of the deflection function describing the
deformed middle plane of the plate and the lateral loading
over the projection of the plate into the middle plane.

We must search an actual deformed state with respect
to the common orientation of the plies. This solution is
used in the above expressed measure of compliance, and the
minimum of this expression is searched.

2 Preliminaries
We contemplate the problem of a rectangular laminate

plate with common but given
� dimensions
� number of plies

� thickness of plies
� mechanical properties of orthotropic plies
� lateral loading
that is simply suported on its boundary with the aim of speci-
fying an orientation of its plies that maximizes the measure of
stiffness

s
l

( )
( )

w
w

�
1

where

l( ) ( , ) ( , )w w x y q x y x y� � d d�
is the measure of compliance and where w(x,y) is the deflection
function describing the deformed middle plane of the plate,
q(x,y) is the lateral loading, and � is the projection of the plate
into middle plane x-y.

We must search an actual deformed state w(�) with respect
to the common orientationof the plies �. This solution is
used in the above expressed measure of compliance, and the
minimum of this expression is searched; i.e. we must solve the
problem

� �� �
�

� arg min ( )l w

3 Common rules
It is well-known that an actual deformed state minimizes

the potential energy
� �� u u u u� �a , l( ) ( ) ,

where a ,( )u u is elastic potential energy

� � � �a , Eijkl ij kl( ) ( ) ( ) ( )u u x u x u x� � ��
1
2

� �
�

�d

and l(u) is potential energy of external loads

l p u t ui i i i
t

( )u � �� �d d� �� �� .
It is also well-known that in the actual state �u the potential

energy � has the value

�( �) ( �)u u� � 	
1
2

0l .

Hence the problem of maximizing the stiffness measure is
transformed into the problem of searching a min-max point
of the potential energy


 � � �� � arg max min ( ) ( )E u u u u
E u

, a , l� �
� �E U

.

©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 21

Acta Polytechnica Vol. 44  No. 3/2004

Controlling Laminate Plate Elastic
Behavior

T. Mareš

This paper aims to express the relation of a measure of laminate plate stiffness with respect to the fiber orientation of its plies. The inverse of
the scalar product of the lateral displacement of the central plane and lateral loading of the plate is the measure of laminate plate stiffness. In
the case of a simply supported rectangular laminate plate this measure of stiffness is maximized, and the optimum orientation of its plies is
searched.

Keywords: mechanical properties, laminate, plate, composites.



4 Formulation of the problem
In the case introduced above there is the problem


 �� , � arg max min ( , )w w
w

� �

�

�
� �A W

� ,

� � ��
�

� �	
	 	 	 	 	

�
	

l l l lim jn mnop ko lp kl ij

N
E w w q x y wd

1

( , ) ( x y, )d�
��

where Emnop
	 is the elasticity tensor of 	-th orthotropic ply and


 �lik i k
	 	 	

	 	


 



 

�

�
�

�
�

�

�
�

cos sin
sin cos

is the tensor of transformation from the local coordinate sys-
tem of the 	-th ply into the global coordinate system.

It is not worthwhile to present the way of constructing this
relation.

5 The way of resolving
In this work the method of alternating fulfilment of neces-

sary conditions was used:

�� 	 	���
 	� 	���
 	 	 �	���
 	 	
� 

P c P c s P c s1
3 2 2

1

� � � � � �
�

�


 2
,

�
	

	
���

	 	 	
���

	
�

�� �� �

�


� � � � �

1

4
3

5
4 1 2

N

P c s P s w q ( , , ) ,�

�w w R R R
K

�� �
	

���
	

���
	 	

���
	

� 
� �


 
1 2 3
1

� � �
��

 tg tg2
,,

�
1

4 5 0 1 2

K

R R N


� � �

	
���

	 	
���

	
 
 	 �tg tg
3 4 ( , , , )�

In this P
	
���
, R
	

���
, � are known numbers, c	 	
� cos( ),
s 	 	
� sin( ) and 
	 are the searched ply orientations.

6 Results
First example:
� a square plate,
� six plies that are laid symmetrically with respect to the mid-

dle plane of the plate,
� continuous loading q � x y [N/mm2].

Graphic representation of the stiffness maximizing ply
orientation (first example – Fig. 1).

The previous three solution variants have the same value
as our measure of stiffness. This value of the measure of stiff-
ness is the maximum of all values. It is interesting that the first
variant is balanced and therefore not twisted. The same holds
for the folowing three solution variants.
Second example:
� a rectangular plate with side ratio 1:2,
� six plies that are laid symmetrically with respect to the mid-

dle plane of the of the plate,
� continuous loading of the plate q = x y [N/mm2].

Graphic representation of the stiffness maximizing ply
orientation (second example – Fig. 2).

22 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 44  No. 3/2004


3 =45°


2 =- 45°


1 =45°

	 � 3

	 � 1

	 � 2


3 x

y


3 =45°


2 =- 45°


1 =45°

	 � 3

	 � 1

	 � 2


3 x

y


3 = 45°-


2 =45°


1 =45°

	 � 3

	 � 1

	 � 2


3 x

y

First variant Second variant Third variant

Fig. 1: First example

	 � 3

	 � 1

	 � 2


3 =75°


2 =- 75°


1 =75°


3 x

y

	 � 3

	 � 1

	 � 2


3 =75°


2 =75°


1 =- 75°


3 x

y

	 � 3

	 � 1

	 � 2


3 = 75°-


2 =75°


1 =75°


3 x

y

First variant Second variant Third variant
Fig. 2: Second example



7 Conclusions
We contemplate the problem of a rectangular laminate

plate with common but fixed given dimensions, number of
plies, thickness of plies, mechanical properties of orthotropic
plies and lateral loading that is simply supported on its
boundary with the aim of specifying an orientation of its
plies that with maximize the measure of stiffness, which is
the inverse of the measure of compliance. The measure of
compliance is given by the scalar product of the deflection
function describing the deformed middle plane of the plate
and the lateral loading over the projection of the plate into
middle plane.

We must search an actual deformed state with respect to
the common orientation of the plies. This solution is used in
the above expressed measure of compliance, and the mini-
mum of this expression is searched.

It is not worthwhile to present the way of constructing of
this relation. It is also not worthwhile to present the way of re-
solving it. Therefore, we introduce only the result.

First example: a square plate with six plies that are laid
symmetrically with respect to the middle plane, of the plate.
This plate is continuously loaded with the value increasing as
the coordinates increase.

There are three solution variants that have the same value
as our measure of stiffness. This measure of stiffness value is
the maximum of all values. The optimal ply orientation is 45°
for all plies, the only difference is the orientation (plus, mi-
nus). It is interesting that the first variant is balanced and
therefore not twisted.

Second example: a rectangular plate with side ratio 1:2
with six plies that are laid symmetrically with whit respect to
the middle plane of the plate. This plate is also continuously
loaded with value increasing as the coordinates increase. The
same remark as for the previous example also holds for this

example. The only contrast is the optimal orientation of the
plies. In this example it is 75°.

8 Acknowledgment
This work has been supported by the Grant Agency of

Czech Republic under contract No. 106/01/0958.

References
[1] Alexejev V. M., Tichomirov V. M., Fomin S. V.: Mate-

matická teorie optimálních procesů. Praha: Academia 1991.
[2] Allaire G.: Shape optimization by the Homogenization

Method. New York: Springer-Verlag, 2002.
[3] Gürdal Z., Haftka R. T., Hajela P.: Design and Optimiza-

tion of Laminated Composite Materials. New York: John
Wiley & Sons, 1999.

[4] Washizu K.: Variational Methods in Elasticity and Plasticity.
2nd ed. Oxford: Pergamon Press, 1975. (In Russian:
Variacionnyje metody v teorii uprugosti i plastičnosti. Moskva:
Mir 1987.)

[5] Wilde D. J.: Globally Optimal Design. New York: John
Wiley & Sons, 1978.

Ing. Tomáš Mareš
phone:+420 224 352 525
e-mail:marest@sgi.fsid.cvut.cz

Department of Mechanics
Division of Strength of Materials

Czech Technical University in Prague
Faculty of Mechanical Engineering
Technická 4
166 07 Prague 6, Czech Republic

©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 23

Acta Polytechnica Vol. 44  No. 3/2004