5th International DAAAM Baltic Conference
“INDUSTRIAL ENGINEERING – ADDING INNOVATION CAPACITY OF LABOUR
FORCE AND ENTREPRENEURS”
20-22 April 2006, Tallinn, Estonia
STABILITY OF THE SHAPE OF ANTICLASTIC COMPOSITE
MATERIAL SHEET UNDER VARIABLE MOISTURE
CONDITIONS
Baikovs, A. & Rocēns, K.
Abstract: Due to the influence of the
environment, composite material sheets in
operating conditions change their original
shape, which often leads up to structural
inadequacy regarding the standard
requirements. Preservation of the initial
shape of composite material sheets by
using of reinforced plastic strengthening
has been investigated.
Calculation model is developed for
determination of the thickness of anticlastic
sheet rational strengthening, which
provides changes of the original bending
radius within the limits of preferable
intervals
under
variable
moisture
conditions. The dependence of bending
radii of sheets of layered timber composite
material on the strengthening thickness of
curved and concaved surfaces of the
composite material sheets has been
established.
Keywords: layered composite material,
anticlastic sheet, moisture deformations,
strengthening.
1. INTRODUCTION
Layered composite material sheets are
widely used in the construction and
transport machine-building industry, where
the part of structural elements during the
operating time and erection are under
variable moisture conditions, which often
leads up to the changes of the initial shape
of these elements. Searching after
structurally better and aesthetically more
attractive solutions of the erection of
various structures, non-standard anticlastic
elements are more frequently used. If the
anticlastic composite material sheets are
used in cable roof structures as load
bearing coverage elements or isolating
elements in interior decoration, then due to
variable moisture, undesirable changes of
the shape of elements are caused, which
can be partly prevented by using of
reinforced plastic strengthening.
The aim of the work is to develop and
approbate methods of calculation for
determination of the influence of initially
anticlastic layered strengthening on the
stability of sheets shape under variable
moisture conditions. Changes of the sheet
bending radii and longitudinal deformations in geometrical middle plane of the
sheet are assumed to be the main
characteristics of the shape stability.
2. METHOD OF CALCULATION
Method of calculation is made for the
element, which consists of orthotropic
layers in the case of plane stress using
statements
of
laminated
material
mechanics, discussed in the works [1-5].
The element layers are assumed to be tied
together by a thin glue layer, as a result of
which they deform together. Threshold of
the orthogonal coordinate system is placed
in the centre of geometrical middle plane.
The case with identical moisture content
for every element layer under any moisture
changes will be considered. In the
operating
conditions
the
moisture
distribution along the cross-section of the
element is variable, but the considered case
is more dangerous, because gives larger
effect on the changes of radii. It is assumed
that both radiuses of curvature of
anticlastic sheet practically together
become equal to  .
Reinforced plastic strengthening with equal
thicknesses is assumed to be on the top and
bottom planes of the composite sheet.
The calculation of shape stability
characteristics
of
the
strengthened
composite element under changing
moisture is carried out in several stages.
In the first stage it is defined how large the
moisture change should be, to straighten
out the anticlastic composite material sheet.
The calculations are carried out by
choosing the appropriate moisture change
(reducing the moisture until the anticlastic
composite material element straightens out)
or else being aware of a definite moisture
change, which has already provided the
required curvature of a plane composite
material sheet.
At the beginning the longitudinal

 n , caused by moisture
deformations

changes W , are determined at distances z0
and zn from geometrical middle plane of an
unstrengthened composite material sheet
(see fig.1):


   x 0  
kx 
 x  


   , where (1)
 0
  
   y       y    z n  k y 
  
  0

k 
 xy  n   xy 
 xy 
- middle plane deformations of the layered
element:


  x 0  
 Nx 
M x 

      
 0
(2)
   y    A N y   B  M y 


0
 ~ 
N 
M 
xy
 xy 
 xy 


- middle plane curvature of the layered
element:



 kx 
 Nx 
M x 
          .
 k y   B   N y   D  M y 
k 
 N 
M 
 xy 
 xy 
 xy 
Internal forces caused by moisture:
(3)

 Nx 
  m
Qij
Ny   

 N  n 1
 xy 
 x

 y

n
  xy
 



 W z n  z n 1  , (4)

 n
i,j = 1,2,6.
Internal moments caused by moisture:

M x 
   m 1
Qij
M y   

M  n 1 2
 xy 
 x

 y

n
  xy
 

 2

2
 W z n  z n 1

 n


(5)
Formulae used for construction of matrixes
  
A, B, D :


      1  
(6)
A  A1  A1BD  BA1B  BA1
 1  

B   A B ( D )

    1
D  D  BA1B , where


(7)
(8)
- the member „ij” of an axial strength
matrix for the whole element:
m

Aij   (Qij ) n ( z n  z n 1 ) ,
(9)

n 1
(here and hereafter i,j=1,2,6)
- the member „ij” of interacting strength
matrix for the whole element:

1 m
2
2
Bij   (Qij ) n ( z n  z n 1 )
2 n 1
(10)
- the member „ij” of bending strength
matrix for the whole element:

1 m
3
3
Dij   (Qij ) n ( z n  z n 1 ) .
3 n 1
(11)
Fig. 1. Calculable elements of the
anticlastic sheet: a.) before moisture
changes; b.) after moisture changes.
~
~
~
 kx 
 Nx 
M x 
 ~  ~  ~  ~  ~ .
 k y   B  N y   D  M y 
~ 
~ 
k~ 
N
M
 xy 
 xy 
 xy 
(19)
The reduced strength matrix of nth layer in
the direction of main axes:
(12)
Qij  n  [T 1 ]n [Qij ]n [T ]n , where
- the reduced strength matrix in the
directions of local main axes of nth layer:
Internal forces caused by moisture:
~
 x 
Nx 

 ~
~  m
 N y    Qij    y  W z n  z n 1 
~  n 1
n
N

 xy 
  xy 
(20)
Q11 Q12 0 
[Qij ]nˆ  Q12 Q22 0  ,
 0
0 Q66  n
or
using
characteristics:
(13)
technical
 E1
1   
12 21

 21E1

[Qij ]nˆ 
1   21 12

0


 12 E2
1   12 21
E2
1   21 12
0
deformative

0 

0 

G12 

 n
n
Internal moments caused by moisture:
~
M x 
 ~  m 1
 M y    Qij
~  n 1 2
M
 xy 
 x

 y
n
  xy
 

 ~ 2
2
 W z n  z n 1

n


(21)
(14)
- transformation matrixes:
 cos 
[T 1 ]n   sin 
sin  cos 
cos 
 sin  cos 
sin 
 cos 
[T ]n   sin 

 sin  cos 
cos 
sin  cos 
sin 
 2 sin  cos  
2 sin  cos  
(cos  2  sin  2 ) n
2 sin  cos  
 2 sin  cos  

(cos  2  sin  2 ) n
(15)
(16)
To calculate the longitudinal deformations
 n of strengthened sheet, which occur due

~
to the moisture changes W  W  W
established in the first stage, in the second
stage theoretically is inspected the
strengthened sheet without any beginning
curvature subjected to moisture changes
(fig.2).
Longitudinal deformations ~n , caused by
~
moisture changes W :
~
 kx 
  ~x     ~x 0  

 ~  , where (17)
 ~   ~0
   y       y    z n  k y 
 ~   ~ 0 
k~ 
xy
 xy  n 

 xy 
- middle plane deformations of the layered
element:
~
~
  ~x 0  
 Nx 
M x 

 ~~  ~ ~ 
~0
(18)
   y    A N y   B  M y 
~
~
0
 ~ 
N 
M 
xy
 xy 
 xy 


- middle plane curvature of the layered
element:
Fig.2. Calculable element strengthened
with reinforced plastic sheet: a.) before
moisture changes; b.) after moisture
changes.
Matrixes A’, B’, D’, Qij  n are established
analogically like the relevancies (6) – (16),
but the index „ n ” has to be replaced with
index „ n ”.
In the third stage the resultant longitudinal
deformations  n of anticlastic element,
which has been subjected to the moisture
action and changed moisture content by
W are calculated using relevance:

 n 1   n  ~n 1
(22)
To define the bending radii on the top and
bottom planes of the sheet, an equation
system (23) is constructed basing on the
relationship of right-angled triangle
between the angle and the sides, from
which the bending radius Rx Zn 1 (24) of the
bottom plane is figured out (see fig.3).
The difference between the bending radii

R x Zn 1 and Rx  of the sheet characterizes
Zn
the strengthening influence at definite
moisture changes. Analogical relevancies
are used to calculate the bending radius
.
Ry
Zn 1


h
tg x 
lx Zn 1  lx Z 1


Rx Zn 1
(23)
tg x 
b

 lx Zn 1

2
 b
h  (  l x Zn 1 )
2
, where (24)
R x Zn 1 
l x Zn 1  l x Z 1
l x
Zn
 x
Zn

b
2
(25)
Fig.3. Calculation scheme of bending
radius of the strengthened element in xz
plane.
3. EXAMPLE
In operating time under variable moisture
conditions the composite timber element is
subjected to undesirable changes of shape,
the prevention of which by applying a glass
fibre sheet is analytically approbated.
Cases of orthotropic composite element
with unsymmetrical structure consisting of
five glued together timber layers are
considered. The longitudinal fibers of the
layer are turned at 90 angles towards the
longitudinal fibers of adjacent layers
(90˚/0˚/90˚/0˚/90˚). The layer thicknesses

   
are t1  t2  t3  t4  1.6mm , t5  1.7mm .

Moisture change W  7% is the final and
initial moisture difference of the timber
layer, for example, for moisture changes in
the timber within the limits of 10% to 17%
if relative humidity of the air changes from
53 to 82% with the air temperature of 20C
(see [6] table 3-4). Moisture expansion
factors of layers are found in [7].
Glass fibre sheet with in two directions
oriented fibres, with epoxy binder is used
as the strengthening (henceforth GFRP).
Characteristic values of the GFRP
strengthening rigidity are found in [8], see
table 1. Cases of strengthening thickness
0.025, 0.050, 0.250, 0.640 and 1.77 mm
have been analytically considered.
Timber
GFRP
layer
E1 [MPa]
10000
42000
E2 [MPa]
400
42000
G12 [MPa]
500
15555
12
0.50
0.35
 21
0.02
0.35
1
0
0
2
0.0021
0
12
0.0010
0
Table 1. Deformative characteristics of
layer properties.
Characteristic
Index 1 in the table 1 indicates the
direction of the property parallely to the
direction of fibres, index 2 indicates the
property transversly to the fibers. For the
Poissons ratio the first index indicates the
direction of load, the other the direction of
transverse deformations.
Figure 4 shows the bending radii values
depending on the changes of the moisture
when the moisture in the composite
material decreases from 17% to 10%. The
40000
Rx, Ry
[cm]
20000
0
0
1
2
3
4
5
6
W [%]
7
-20000
-40000
tGFRP=0; Rx0
-60000
tGFRP=0.025mm; Rx
tGFRP=0.05mm; Rx
tGFRP=0.25mm; Rx
-80000
tGFRP=0.64mm; Rx
tGFRP=1.77mm; Rx
-100000
tGFRP=1.77mm; Ry
tGFRP=0.64mm; Ry
-120000
tGFRP=0.25mm; Ry
tGFRP=0.05mm; Ry
-140000
tGFRP=0.025mm; Ry
tGFRP=0; Ry0
-160000
Fig. 4. Changes of strengthened layered anticlastic composite material sheet
bending radii changes depending on the changes of material moisture.
bending radii „ Rx 0 ( W ) ” and „ Ry 0 ( W ) ”
of the unstrengthened sheet element
decreases most of all, if the moisture
difference increases. The remaining curves
show the changes of bending radii for the
sheets
strengthened
with
different
thickness of reinforced plastic if the
moisture difference decreases.
If the sheet is strengthened with 1.770 mm
thick strengthening sheets (ratio of
strengthening area to all strengthened
composite material area =0.302%), then
the change of bending radius of the
element in the direction of x axis is 5%, but
in that of y axis - 1.2% (see table 2).
Therefore, it may be presumed that in this
case the invariability of shape of the

element at moisture changes W  7% has
been provided with the precision of
admissible engineering calculations.
If thickness of the element is 0.640 mm
( = 0.135%), then changes of bending
radius of the element in the direction of x
axis is 5%, but in that of y axis 20.2% here only invariability of bending radius in
the direction of x axis is provided.
tGFRP
Rx , [%] Ry , [%]
[mm]
0.000
0.000


0.025
0.006
218.4
639.8
0.050
0.012
105.0
323.5
0.250
0.058
16.8
61.3
0.640
0.135
5.0
20.2
1.770
0.302
1.2
5.0
Table 2. Changes of bending radii
depending on the strengthening thickness if
material moisture changes by 7% for the
timber composite element strengthened
from the both sides.
- ratio of strengthening area to all
strengthened composite material area;
tGFRP – thickness of the strengthening layer
on the one side of the sheet.
In other cases the strengthening thickness
is chosen too thin, and invariability of the
shape is not provided because changes of
the bending radii exceed 5%.
4. CONCLUSIONS
Calculation methods for determination of
bending radii of initially anticlastic layered
sheets under variable moisture conditions
has been developed on condition that
reinforced plastic strengthening has been
used for preservation of its original shape.
Influence of the thickness of reinforced
strengthening layers on changes of bending
radii of initially anticlastic composite
material sheet due to moisture action has
been defined.
An opportunity to provide the original
shape of anticlastic timber composite
material sheets by using of reinforced glass
plastic strengthening under variable
moisture conditions and not exceeding the
difference of 5% has been demonstrated.
This work has been partly supported by the
European Social Fund within the National
Programme “Support for the carrying out
doctoral study programm’s and postdoctoral researches” project “Support for
the development of doctoral studies at Riga
Technical University”.
5. REFERENCES
1. Skudra A.M and Skudra A.A.
Introduction in the laminated material and
construction mechanics. Riga Technical
University 2002, Printed in Latvian
2. Ashton I.E., Halphin I.C., Petit P.H.
Primer on composite materials analysis.
Stamford, Technomik Publ. Co., 1969, 134
3. Tsai S.W. and Pegano N.I. Invariant
properties
of
composite materials.
Stamford, Technomic Publ. Co., 1968
4. Reissner E. and Stavsky I. Bending and
stretching of certain types of heterogenous
aelotropic elastic plates. Trans. ASME. Ser.
E, 1961, 28, N3, 402-408
5. Rocens K.A. and Steinerts K. Estimation
of the compliance and stiffness of
unbalanced monoclinic compositions.
Polymer mechanics, 1976, 6, 1030-1035.
Printed in Russian
6. Wood handbook. Wood as engineering
material. Forest Products Laboratory
USDA Forest Service, Madison, Wisconsin
1999
7. Structural Glued Laminated Timber.
Load and resistance factor design. Manual
for engineered wood construction. APA
The Engineered Wood Association 1996.
http://www.awc.org/pdf/
LRFD_Glulam.pdf, 62-63, Last visited in
2006.03.03,
8. Kelly A., Cahn R.W., Bever M.B..
Concise encyclopaedia of composite
materials. Revised edition. Elsevier
Science 1995
6. CORRESPONDING AUTHOR
M.sc. Andris Baikovs
Riga Technical University
Address: Azenes street 16, LV 1048 Riga,
Latvia
Phone: +371 9832031,
E-mail: abpasts@inbox.lv