Investigation of compression of cylindrical packages/Cilindro formos kartoniniu pakuociu tyrimai.
Kibirkstis, E. ; Bivainis, V. ; Ragulskis, L. 等
1. Introduction
In this paper some of the more important results investigating the
possibilities of improving the ecological level of packages, which were
obtained while performing the joint Lithuanian-Ukrainian research
project [1], are analyzed. One of the main requirements is to produce
packages in such a way that their volume and weight would be small while
at the same time the strength parameters of the package would remain
sufficient. In this paper the influence of the package shape on
mechanical characteristics in the process of its compression is
investigated.
In the process of analyzing the research papers of similar
character one can note that the research papers [2-4] where the results
of experimental and numerical investigation of cardboard packages of
rectangular shape are presented. In the paper [5] the relationship
between the deformations of plastic packages of cut conical shape under
axial load is investigated. Experimental and theoretical investigations
of static axial compression of circular plastic PVC pipes and two-layer
hollow bars are presented in the research papers [6, 7]. The behaviour
of deformation of their shape is being very similar to the deformation
of the surface of cardboard packages of cylindrical shape. In the paper
[8] the behaviour of cardboard as of an anisotropic material under the
action of loading is analyzed, and the investigation is performed by
using the finite element method. Similar investigations of elements
buckling are presented in the paper [9].
However, the research papers investigating resistance to
compression of the paper/paperboard packages of cylindrical shape were
not found. Thus the purpose of this paper is to perform numerical and
experimental investigations of cylindrical packages by determining the
peculiarities of deformation of the cylinder surface.
The analysis of stability of cylindrical packages using three
dimensional elements is based on the relationships described in [10,
11]. The analysis of stability of a shell type structure using a three
dimensional element consists of two stages. Static problem by assuming
the displacements at the lower and upper edges of the analyzed structure
to be given is solved. Stability of the structure because of the
additional stiffness due to the static compression determined in the
previous stage of analysis is investigated. The model for the analysis
of axial loading of a package is based on the analysis of an
axi-symmetric elastic structure. Geometric nonlinearity is taken into
account by the method of initial strains [10-12]. The force is increased
by small steps and thus graphical relationship of displacement-force is
calculated.
2. Model for the analysis of the shell type structure stability
The nodal coordinates of a three dimensional element are obtained
from the nodal coordinates of a two dimensional element. So a node of a
two dimensional element corresponds to three nodes of a three
dimensional element: on the lower surface, on the middle surface
(coinciding with the node of the two dimensional element) and on the
upper surface.
First, the static problem is analyzed. It is assumed that the
displacements on the lower and upper boundary of the analyzed structure
are given and they produce the loading vector. Thus the vector of
displacements is determined by solving the system of linear algebraic
equations. In the second stage of the analysis, the eigenproblem of
stability of the structure with additional stiffness due to the static
compression is solved.
Further x, y and z denote the axes of the system of coordinates. In
order to obtain the coordinates of the nodes on the lower surface of the
shell and on the upper surface of the shell for each node of the two
dimensional element, the derivatives of x, y and z with respect to the
local coordinates [zeta] and [eta] are calculated as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
where N are the shape functions of the analyzed two dimensional
finite element, [x.sub.i], [y.sub.i], [z.sub.i] are the nodal
coordinates of the finite element.
The vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is
the unit vector in the direction of vector product of [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII.].
So the coordinates of the node on the lower surface of the shell
are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], where h
denotes thickness of the shell.
The coordinates of the node on the upper surface of the shell are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
3. Model for the analysis of axial loading of the packages
Further x denotes radial coordinate and y denotes axial coordinate
of the cylindrical system of coordinates. The element has two nodal
degrees of freedom: the displacements u and v in the directions of the
axes x and y.
The vector of displacements {[delta]} is determined by solving the
system of linear algebraic equations. The force is increased by small
steps.
The following derivatives are calculated
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)
where [N.sub.i] are the shape functions of the finite element.
Then the following matrix is introduced
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)
[FIGURE 1 OMITTED]
So the geometric nonlinearity can be taken into account by the
method of initial strains
{[[EPSILON].SUB.0]} = -1/2[A][G]{[DELTA]} (5)
This produces the load vector
{F} = [integral][[B].sup.T][D]{[[epsilon].sub.0]}2[pi]xdxdy (6)
where the matrix [B] is defined from
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)
and [D] is the matrix of elastic constants.
4. Results of analysis of the shell type structure stability
A cylindrical structure is analyzed. For the static problem the
following boundary conditions are assumed: on the lower boundary all the
displacements are equal to 0. On the upper boundary all the
displacements are equal to 0 except the displacement in the direction of
the z axis which is equal to 1. For the eigenproblem on the lower and
the upper boundaries all the displacements are assumed equal to zero.
Dimensions of the package and mechanical characteristics of the
cardboard "Galerie Card 280" are presented in Table.
Calculations were performed using isotropic model by assuming the
physical parameters equal to the averaged values of the corresponding
parameters of the orthotropic model.
The first eigenmode of stability is presented in Fig. 1, a, the
second--in Fig. 1, b, the sixth--in Fig. 1, f.
The first and the second, the third and the fourth, the fifth and
the sixth eigenmodes correspond to the same eigenvalue respectively.
A thin rectangular structure parallel to the axial coordinate is
analyzed. On the lower boundary all the displacements are assumed equal
to zero. The node at the center of the upper boundary is loaded by a
force in the direction of the axial coordinate, the value of the force
is negative (it is acting in opposite direction than the y axis).
Graphical relationship from numerical investigations of package
walls elements deformation under axial compression is obtained and
presented in Fig. 2. The force is represented in the vertical direction,
while the displacement of the same degree of freedom is represented in
the horizontal direction.
[FIGURE 2 OMITTED]
5. Method of experimental investigations
In order to perform the experimental tests, cardboard packages of
cylindrical and square geometrical shape were produced. For production
of the cylindrical packages, two types of cardboard were used; they were
chosen on the basis of the recommendations provided by the producer and
they are most suitable for the production of packages of the
investigated size.
Continuing the research of work [2], cardboard package with the
walls of the same height and same area of lateral walls were produced
for the experiment. The main characteristics of the samples and of the
types of used cardboards are presented in the Table.
The setup used for experimental investigations was of the same type
as described in the research paper [4].
Empty (not filled) packages were investigated under the action of
static axial load. The direction of cardboard of the side walls of the
packages coincided with the direction of compression. During the
experimental investigation the highest value of the axial compression,
at which the structure of the package looses stability and starts to
buckle, was registered. The diagram of the experimental setup is
presented in Fig. 3.
From the estimates of the values of the axial compression force
obtained in the course of experimental investigations, the maximum
value, at which the cylindrical cardboard package loses stability and
starts to buckle, is determined. The relationships of package
deformation due to axial load are presented graphically in Fig. 4, when
the package deforms by the interval of 1 mm. The packages were deformed
up to 20 mm. In order to compare the behaviour of the geometrical shape
of the experimental samples during the process of compression, the
sample was photographed by the digital camera with a fixed interval of
photographing.
The tests were carried out at the ambient temperature of 20[+ OR
-]2[degrees] C and air humidity 65[+ or -]2%.
[FIGURE 3 OMITTED]
6. Results of experimental investigations and their analysis
The obtained relationship of the deformation of samples of packages
of cylindrical and rectangular shape from axial load F is presented in
Fig. 4.
[FIGURE 4 OMITTED]
[TABLE OMITTED]
Packages of cylindrical shape made of the cardboard "Galerie
Card 280" start to buckle under the vertical axial load at about
554 N, while packages made of the cardboard "Alaska 275" start
to buckle under the axial compression of about 433 N. The latter package
starts to buckle under the axial compression force which is about 22%
lower. Packages of rectangular shape produced from the cardboard
"Galerie Card 280" lose stability under the action of the
axial load at about 205 N. As it is seen from the results of
experimental investigations, the shape of the package has substantial
influence on the allowable vertical axi-symmetric compression load.
Under axial load per perimeter, area, moment of inertia and radius of
inertia units, the package of cylindrical shape resists the axial load,
which is about 2.4 times higher when compared with the package of
rectangular type (see Fig. 4, curves 1 and 3, Table 1).
The character of variation of the compression force in the region
of elastic zone is almost linear.
In Fig. 5 the photos of the compression process of the cylindrical
package are presented, which show the behaviour of the package during
its deformation, and in Fig. 5, e the photo of the deformed rectangular
package is presented.
When analyzing the obtained results of numerical and experimental
investigation (Figs. 2 and 4), it is possible to make a conclusion that
the deformation of the material of the package in the region of elastic
zone is described by the numerical model with acceptable precision.
However, the model does not take into account plastic deformations and
thus is not applicable in the region of plastic deformations. In the
investigations of packages the zone of plastic deformations is of
secondary importance because the package with plastic irreversible
deformations is unsuitable for use.
By comparing the image of the outer walls of the cylindrical and
rectangular packages deformed during the process of axial compression,
one can note that the character of deformation in the vertical direction
of the packages presented in Fig. 5, d and in Fig. 5, e is different.
By comparing the obtained results of numerical (Fig. 1) and
experimental (Fig. 5, a-d) investigations of the cylindrical package,
one can see that the numerical model describing the axial compression of
the cylindrical package should be improved because the photos of
deformation of the walls obtained during the experimental investigations
(Fig. 5, a-d) indicate the development of deformations in the upper and
lower parts of the package.
[FIGURE 5 OMITTED]
7. Conclusions
Results of the experimental investigations indicated that the shape
of the package has substantial influence on the allowable vertical axial
load. The package of cylindrical shape under axial load per perimeter,
area, moment of inertia and radius of inertia units resists the axial
load, which is about 2.4 times higher when compared with the package of
rectangular type of the similar mass and the same base perimeter and
height.
It is determined that the character of surface deformation of the
packages of cylindrical shape under the action of vertical axial load is
essentially different from the image of the deformation of the walls of
the packages of rectangular type.
As indicated by the results of experimental investigations, the
presented numerical model of the investigation of cylindrical package
does not fully describe the process of deformation under axial
compression in the vertical direction. Thus more complicated models
might be applied for the solution of this problem.
The problem for the analysis of package stability as of a
cylindrical shell type structure is solved using three dimensional
elements. The nodal coordinates of the three dimensional elements are
obtained from the two dimensional ones which represent the middle
surface of the shell.
The static problem by assuming the displacements at the edges of
the analyzed structure to be given is solved. Then the stability of the
investigated structure because of the additional stiffness due to the
static compression determined previously is analyzed.
The axial loading of an axi-symmetric package is analyzed.
Geometric nonlinearity is taken into account by the method of initial
strains.
The graphical relationship of displacement--force is obtained.
The obtained results are used in the process of design of package
elements.
References
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Printing and Packaging Production Technologies, Considering their
Ecological and Operational Qualities (No. V-23/2007, No. V-36/2008).
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Experimental study of paperboard package resistance to compression.
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(10.) Zienkiewicz, O.,C. The Finite Element Method in Engineering
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E. Kibirkstis *, V. Bivainis **, L. Ragulskis ***, A. Dabkevicius
****
* Kaunas University of Technology, Studentu_ 56, 51424 Kaunas,
Lithuania, E-mail:
[email protected]
** Kaunas University of Technology, Studentu_ 56, 51424 Kaunas,
Lithuania, E-mail:
[email protected]
*** VytautasMagnus University, Vileikos 8, 44404 Kaunas, Lithuania,
E-mail:
[email protected]
**** Kaunas University of Technology, Studentu_ 56, 51424 Kaunas,
Lithuania, E-mail:
[email protected]