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Studies and Materials

in

Applied Computer

Science

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Chairman of Editorial Board

Janusz Kacprzyk Systems Research Institute, PAS

Editors-in-Chief

Jacek Czerniak Kazimierz Wielki University in Bydgoszcz Marek Macko Kazimierz Wielki University in Bydgoszcz

Secretary

Iwona Filipowicz Kazimierz Wielki University in Bydgoszcz

Editorial Office

Mariusz Dramski Maritime University in Szczecin

Piotr Dziurzański West Pomeranian University of Technology Łukasz Apiecionek Kazimierz Wielki University in Bydgoszcz Piotr Prokopowicz Kazimierz Wielki University in Bydgoszcz Hubert Zarzycki Kujawy and Pomorze University in Bydgoszcz Wiesław Urbaniak University of Economy in Bydgoszcz

Editorial Board

Stanisław Ambroszkiewicz Institute of Computer Science, PAS Rafał Angryk Montana State University, USA Zenon Biniek University of Information Technology Ryszard Budziński University of Szczecin

Joanna Chimiak-Opoka University of Innsbruck, Austria

Ryszard Choraś University of Technology and Life Sciences in Bydgoszcz Grzegorz Domek Kazimierz Wielki University in Bydgoszcz

Petro Filevych Lviv National University of Veterinary and Biotechnologies, Ukraina Piotr Gajewski Military University of Technology

Marek Hołyński President of Polish Information Processing Society Janusz Kacprzyk Systems Research Institute, PAS

Andrzej Kobyliński Warsaw School of Economics

Peter Kopacek Vienna University of Technology, Austria Józef Korbicz University of Zielona Góra

Jacek Koronacki Institute of Computer Science, PAS

Witold Kosiński Kazimierz Wielki University in Bydgoszcz, PJIIT Marek Kurzyński Wrocław University of Technology

Halina Kwaśnicka Wrocław University of Technology

Mirosław Majewski New York Institute of Technology, United Arab Emirates Andrzej Marciniak Poznań University of Technology

Marcin Paprzycki Systems Research Institute, PAS Witold Pedrycz Systems Research Institute, PAS

Andrzej Piegat West Pomeranian University of Technology Andrzej Polański Silesian University of Technology

Orest Popov West Pomeranian University of Technology Izabela Rojek Kazimierz Wielki University in Bydgoszcz Danuta Rutkowska Czestochowa University of Technology Leszek Rutkowski Czestochowa University of Technology Milan Sága University of Žilina, Slovakia

Roman Słowiński Systems Research Institute, PAS, Poznań University of Technology Włodzimierz Sosnowski Kazimierz Wielki University in Bydgoszcz, IFTR PAS

Andrzej Stateczny Maritime University of Szczecin Tomasz Szapiro Warsaw School of Economics

Janusz Szczepański Kazimierz Wielki University in Bydgoszcz, IFTR PAS Ryszard Tadeusiewicz AGH University of Science and Technology

Jan Węglarz Institute of Bioorganic Chemistry, PAS, Poznań University of Technology Sławomir Wierzchoń The Institute of Computer Science, PAS

Antoni Wiliński West Pomeranian University of Technology Andrzej Wiśniewski University of Podlasie

Studies and Materials in Applied Computer Science

Journal of young researchers, PhD students and students Endorsed by Polish Information Processing Society

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Studies and Materials

in

Applied Computer

Science

Journal of young researchers, PhD

students and students

(4)

Studies and Materials in Applied Computer Science

j o u r n a l o f y o un g r e s ea r c h e r s, P h D st u d e n t s a n d s tu d e n t s

© Copyright 2011 by the Foundation for Development of Mechatronics

© Copyright 2011 by Kazimierz Wielki University

Scientific patronage:

Faculty of Mathematics, Physics and Technical Sciences

Kazimierz Wielki University

ul. Chodkiewicza 30

85-064 Bydgoszcz, Poland

tel. (052) 34-19-331

fax. (052) 34-01-978

e-mail: simis@ukw.edu.pl

ISSN 1689-6300

ISBN 978-83-932977-2-6

Cover designed by: Łukasz Zawadzki

DTP by: Sebastian Szczepański

Publisher:

Foundation for Development of

Mechatronics

ul. Jeżynowa 19

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tel. +48 533-44-77-53

fax. +48 525-81-22-51

email: biuro@mechatronika.org.pl

Contact:

Jacek Czerniak, PhD. Eng.

Marek Macko, PhD. Eng.

Kazimierz Wielki University

ul. Chodkiewicza 30

85-064 Bydgoszcz, Poland

e-mail: jczerniak@ukw.edu.pl

mackomar@ukw.edu.pl

Printing (funded from resources of Ministry of Science and Higher Education within

IndexPlus programme):

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Edition of 510 copies

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C

ONTENTS

Foreword...7

Submitted Articles...11

SIGNIFICANCE OF CONDITION ATTRIBUTES IN CHILD WELL-BEING ANALYSIS

E

WA

A

DAMUS

,

P

RZEMYSŁAW

K

LĘSK

,

J

OANNA

K

OŁODZIEJCZYK

,

M

ARCIN

K

ORZEŃ

,

A

NDRZEJ

P

IEGAT

,

M

ARCIN

P

LUCIŃSKI

... 11

NUMERICAL STUDY OF LARGE SCALE TRUSS CONSTRUCTION WITH STRUCTURAL

UNCERTAINTIES

R

ÓBERT

B

EDNÁR

,

M

ILAN

S

ÁGA

,

M

ILAN

V

AŠKO

... 17

PROGRAMMING OF COMPOSITE PLATES DAMAGE CALCULATION

M

ARTIN

D

UDINSKY

,

D

ANIEL

R

IECKY

,

M

ILAN

Z

MINDAK

... 23

VIRTUAL SYSTEMS IN BUSINESS AND EDUCATIONAL PROCESSES

P

IOTR

Ł

OSOWSKI

... 31

DEVELOPING THE COMPUTERIZATION CONCEPT OF THE COMMERCIAL AGENCY

BY COMBINING ANALYTICAL PERSPECTIVES

S

ZYMON

J

EŻ

,

W

ALERY

S

USŁOW

... 41

THE SIMULATION AND THE EXPERIMENTAL VERIFICATION OF THE INFLUENCE OF

THE TIG – WELDING OF THE ALUMINIUM ALLOY AlMgSi07 ON CHANGES OF THE

MECHANICAL PROPERTIES IN COMPARISON WITH THE BASIC MATERIAL

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FOREWORD

Dear Readers,

We would like to express our thanks to all who have promoted SMACS in their societies, and we address

our special thanks to Members of the Scientific Council. We have been receiving information from

different universities that SMACS is promoted in wall display cases and during meetings with students. We

would also like to express special thanks to Deans and Managers of PhD studies for promotion of SMACS

among their young associates. We are pleased to inform that numbers 4 and 5 of SMACS journal issued in

2011 shall also be published in English. It was possible mainly thanks to the subsidy obtained from

MNiSW (Ministry of Science and Higher Education) within IndexPlus programme, which enabled us to

cover increased costs of the English edition. We are very glad that, thanks to the English edition, SMACS

shall become available to wide range of readers. Let us remind that, thanks to efforts of the editorial team,

who have successfully won further generous sponsors, authors can publish their papers in SMACS free of

charge. We would also like to invite p.t. sponsors who would like to support publishing of our journal in

following years.

SMACS Editors-in-chief

Jacek Czerniak, PhD. Eng.

Marek Macko, PhD. Eng.

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pp. 11-16

SUBMITTED ARTICLES

SIGNIFICANCE OF CONDITION ATTRIBUTES

IN CHILD WELL-BEING ANALYSIS

Ewa Adamus, Przemysław Klęsk, Joanna Kołodziejczyk, Marcin Korzeń, Andrzej Piegat, Marcin

Pluciński

Faculty of Computer Science and Information Systems West Pomeranian University of Technology

Żołnierska 49, 71-062 Szczecin, Poland

{eadamus, pklesk, jkolodziejczyk, mkorzen, apiegat, mplucinski}@wi.zut.edu.pl

Abstarct: In this study the significance of attributes in child well-being is presented. The main goal was to find features most specific for child-well being evaluation in Poland. The dataset was obtained from a survey based on a special questionnaire. To select important attributes three filter for individual attribute rank were used χ2_{, information gain and relief attribute evaluator and one filter-subset selector}

based on rough set theory. In the article the dataset is described in details. All the attributes are named, divided by category and for each a domain is given. Then methods of attribute selection applied in experiments are presented. Finally results on selecting attributes relevant for child well-being are discussed.

Keywords: Child well-being quality measurement, attribute selection.

1. INTRODUCTION

The most common reasons for using feature selection from a dataset is preparing data for learning methods. Those methods model data dependencies and huge amount of attributes can detract model quality and accuracy. The attribute selection used before learning results in lower model dimensionality, can remove noise from data and search for correlation in attributes. Another reason is important data characteristics obtainment by ranking attributes according to their significance in predicting a decision [3,4].

Two taxonomies are used for feature selection methods. The first distinguishes techniques due to the nature of the metric used to evaluate importance of attributes and calls them “filter” and “wrapper”. Wrappers use learning algorithms to evaluate attributes. Filters are independent of the learning

method and use general information from a dataset. The second taxonomy divides algorithms into those which evaluate (ranks) individual attributes and those which evaluate subset of attributes [4].

In this study the significance of attributes in child well-being is presented. The main goal was to find features most specific for child-well being evaluation in Poland. The dataset was obtained from a survey based on a special questionnaire [1,6]. A list of 81 attributes capable of influencing childhood was created and 557 records were collected. To select important attributes three filter for individual attribute rank were used χ2_{, information gain and}

relief attribute evaluator and one filter-subset selector based on rough set theory.

In the article the dataset is described in details. All the attributes are named, divided by category and for each a domain is given. Then methods of attribute selection applied in experiments are presented. Finally results on

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Ewa Adamus, Przemysław Klęsk, Joanna Kołodziejczyk i inni, Significance of condition attributes in child well-being analysis

selecting attributes relevant for child well-being are discussed.

2. DATASET ON CHILD WELL-BEING

The presented research is a part of a project on finding quality indicators of child well-being. The main idea was to gather data from which this information can be extracted by means of different data mining methods. Therefore a survey about childhood was performed. Each respondent answered 26 questions. The questionnaire was conducted mainly among students in Western Pomarania universities. The survey gathered 557 records.

There were different types of questions resulting in numerical, nominal and binary answers. As a result a dataset was obtained with 81 condition and 4 decision attributes.

Usually raw data need some preliminary processing which results in unified and better structure. Data cleaning, integration, discretisation are the most common initial process.

Mistakes or omissions during filling the questionnaire caused incomplete records. Dealing with missing values in a dataset is a part of data cleaning process. The dataset about childhood contained missing values for some attributes not exceeding 1% of the dataset size. As the number of records was not big even incomplete records were saved and missing values were replaced. Numerical attributes were substituted with the mean value and the most frequent value replaced an empty cell for nominal attributes. As preliminary experiments have shown, filling attributes in this way had practically no real influence on realised significance analysis.

All numerical attributes were discretised for two reasons: attribute standarisation and better learning algorithms performance. Discretisation was performed to ensure equal sample frequency for each nominal attribute value. The final list of 81 condition and 4 decision attributes is presented in Tablele 1.

Tablele 1 List of 81 condition and 4 decision attributes

No Attribute name Attribute values

1. sex F, M

2. age 18-20, 21-22, 23-24, 25 and over 3. home location village, small city, medium city,

big city 4. siblings 0, 1, 2, 3 and more 5. family mother and father, only mother,

only father, foster family, orphanage

6. father’s age at birth 18-24, 25-27, 28-31, 32 and over

7. mother’s age at birth 18-23, 24-26, 27-30, 31 and over 8. house very small, small, medium, large,

very large

father’s profession:

9. workman yes, no 10. office worker yes, no 11. freelancer yes, no 12. disability pensioner yes, no 13. pensioner yes, no 14. housekeeping yes, no 15. unemployed yes, no

mother’s profession:

16. workman yes, no 17. office worker yes, no 18. freelancer yes, no 19. disability pensioner yes, no 20. pensioner yes, no 21. housekeeping yes, no 22. unemployed yes, no

23. job problems very often, sometimes, rarely, never

childcare:

24. kindergarten yes, no 25. private kindergarten yes, no 26. nanny yes, no 27. family yes, no 28. friends, acquaintances yes, no 29. none yes, no 30. medical care rare, medium, often 31. medical care quality very weak, weak, neutral, good,

very good 32. private medical care never, rare, often 33. proportion of time

spent with parents mother, equally, more often with only mother, more often with father, only father

activities with father:

34. reading yes, no 35. painting yes, no 36. playing yes, no 37. watching TV yes, no 38. watching films yes, no 39. going to cinema yes, no 40. watching sport games yes, no 41. sport activities yes, no 42. listening to music yes, no 43. walks, picnic yes, no 44. playing music yes, no 45. playing board games yes, no 46. playing computer

games yes, no 47. learning yes, no 48. sharing a hobby yes, no

activities with mother:

49. reading yes, no 50. painting yes, no 51. playing yes, no

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pp. 11-16 52. watching TV yes, no

53. watching films yes, no 54. going to cinema yes, no 55. watching sport games yes, no 56. sport activities yes, no 57. listening to music yes, no 58. walks, picnic yes, no 59. playing music yes, no 60. playing board games yes, no 61. playing computer

games

yes, no 62. learning yes, no 63. sharing a hobby yes, no 64. number of books at

home few, many, a lot

afterschool activities:

65. sports yes, no 66. playing music yes, no 67. arts yes, no 68. dancing yes, no 69. studying foreign language yes, no 70. additional computer classes yes, no 71. mathematics and sciences yes, no 72. swimming pool yes, no 73. other yes, no

74. who chose activities? parents, parents and me, only me 75. pets never, rarely, often 76. school quality very poor, poor, average, good,

excellent 77. school safety very dangerous, dangerous,

neutral, safe, very safe 78. district safety very dangerous, dangerous,

neutral, safe, very safe 79. camps never, rarely, medium, often,

very often 80. contact with friends never, rarely, medium, often,

very often

81. contact with family never, rarely, medium, often, very often

82. intensity of education 1 … 5 83. intensity of

entertainment

1 … 5 84. health and physical

fitness

1 … 5 85. safety and living

conditions

1 … 5

All 81 condition attributes are subject to attributes significance analysis. In the study, the ranking of most influential features according to each decision attribute is done separately. Using different feature selection methods

various rankings can be obtained. A subset of attributes common for all rankings is the final experimental result.

3. ANALYSIS OF CONDITION ATTRIBUTES SIGNIFICANCE

Evaluation of attributes significance was performed with the application of following methods [2,3]:

 χ2_{Attribute Evaluator (χ}2_{AE): Evaluates the worth}

of an attribute by computing the value of the chi-squared statistic with respect to the class.

 Information Gain Attribute Evaluator (IG AE): Evaluates the worth of an attribute by measuring the information gain with respect to the class.

 Relief Attribute Evaluator (R AE): Evaluates the worth of an attribute by repeatedly sampling an instance and considering the value of the given attribute for the nearest instance of the same and different class. To perform the child well-being feature selection experiment, Weka (Waikato Environment for Knowledge Analysis) a powerful open-source Java-based machine learning workbench was used [2].

Moreover, attributes significance was analysed with the rough set theory application [5]. The theory defines the notion of condition attributes reducts and more exactly, in experiments the shortest reducts relative to decision attributes (i.e. such minimum subsets of condition attributes which enable a decision with the greatest quality) were searched. The quality of a relative reduct can be calculated exemplary as a value informing what part of the sample set (after reduction) includes samples that have consistent decision attributes.

The task of reduct finding is NP-hard, so finding shortest reducts in the space of 81 condition attributes was computationally too complex. A simplified (heuristic) method was applied to find suboptimal reducts known as a quick reduct algorithm [7,8].

4. RESULTS AND DISCUSSION

First each decision attribute i.e. ’intensity of education’, ’intensity of entertainment’, ’health and physical fitness’ and ’safety and living conditions’ were analysed separately. Applied filters resulted in different significant attribute lists. It issued from the fact that each method evaluated significance from another point of view. However some results were quite similar and this allowed for more general conclusions.

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Ewa Adamus, Przemysław Klęsk, Joanna Kołodziejczyk i inni, Significance of condition attributes in child well-being analysis

First ten attributes and best-found reducts are presented in each experiment. The ’attrib.’ columns from Tableles in the next sections contain the attribute number given in Tablele 1.

4.1. Attribute ranking for ’intensity of education’

χ2_{AE and IG AE placed attribute (76) i.e. ’school quality’}

as the first and most important feature connected with the decision attribute ’intensity of education’ (Tablele 2). R AE put (76) at the second position, but the difference between the first and second attribute is less than the displayed precision. Second the most important attribute from χ2_AE

and IG AE was ’medical care quality’ (31) however it was not indicated by R AE. Attributes (76) and (31) were also present in all found reducts. Another attribute that was common and ranked as the most important by R AE is ’number of books at home’ (64). There are three other common attributes in all three rankings: ’after school activity connected with studying mathematics and science’ (71), ’contacts with friends’ (80) and ’sports activities with father’ (41).

It should be noticed that differences in consecutive significance measures in all rankings are very small.

Tablele 2 Ranking of first ten most significant features that

influence ’intensity of education’

No attrib. χ2_{AE attrib. IG}_{AE attrib. R}_AE

1 76 66.8888 76 0.0641 64 0.0418 2 31 46.8861 31 0.0599 76 0.0418 3 33 44.4812 79 0.0451 71 0.0347 4 77 43.5276 77 0.0451 43 0.0312 5 80 38.3898 80 0.0431 2 0.0302 6 79 37.9393 64 0.0398 4 0.0265 7 64 30.8015 71 0.0348 39 0.0257 8 71 28.5901 41 0.0334 41 0.0253 9 78 26.7202 33 0.0330 47 0.0244 10 41 25.3914 78 0.0310 80 0.0237 reduct length attributes reduct quality

4 2, 31, 76, 78, 0.3806 5 2, 7, 31, 76, 78 0.6750 6 2, 7, 31, 76, 77, 81 0.8689 7 2, 3, 6, 31, 76, 77, 81 0.9587 8 2, 3, 6, 31, 64, 76, 77, 81 0.9928

4.2. Attribute ranking for ’intensity of entertainment’

Results are much more consistent (Tablele 3) for the ’intensity of entertainment’ attribute. All methods evaluated that the most important attribute was: (80) ’contacts with

friends’. Values of significance for the (80) attribute were several times greater than others and the attribute was present in all found reducts. That means that relationship with contemporaries is the basis in childhood entertainment evaluation.

influence ’intensity of entertainment

No attrib. χ2_{AE attrib. IG}_{AE attrib. R}_AE

4 3, 77, 79, 80 0.3824 5 3, 6, 77, 79, 80 0.7217 6 2, 3, 6, 77, 79, 80 0.9084 7 2, 7, 31, 47, 78, 79, 80 0.9641 8 2, 7, 27, 31, 47, 78, 79, 80 1

Some other attributes are common for all ranking lists. While ’contacts with family’ (81) are quite intuitive other are not directly related with the studied decision attribute. For example (76) ’school quality’ and (77) ’school safety’ according to common sense are associated with (80) because child’s friends come mostly from school. Quite unusual is the connection between (31) ’medical care quality’ and ’intensity of entertainment’.

4.3. Attribute ranking for ’health and physical fitness’

Based on results from all filters (Tablele 4) the most important attributes were: ’after school activities – sports’ (65) and ’contacts with friends’ (80) for decision attribute ’health and physical fitness’. (65) and (80) condition attributes take the two highest places. Third attributes in all lists were scored with significant values less than 50% of the second attribute value of importance. Therefore the rest of the list did not indicate links with the decision attribute so clearly.

Other important attribute, which was in the top four in all rankings, was ’camps’ (79). ’Medical care’ (30) is the attribute that is present in all lists, however it is not always scored as very significant.

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pp. 11-16

In all presented reducts the attributes ‘medical care quality’ (31) and ‘camps’ (79) are present. It means that they can also have significant influence on this decision attribute.

influence ’health and physical fitness’

4 2, 31, 65, 79 0.3303 5 2, 7, 31, 65, 79 0.6589 6 2, 7, 31, 65, 79, 80 0.8671 7 2, 6, 31, 65, 79, 80, 81 0.9659 8 2, 3, 6, 31, 65, 79, 80, 81 0.9928

4.4. Attribute ranking for ’safety and living conditions’

Most various results were obtained from analysing ’safety and living conditions’ decision attribute. Definitely the most important attribute was ’district safety’ (78) which was the second most important attribute in all lists and which was present in all found reducts (Tablele 5). Other features that can be significant are ’medical care quality’ (31), ’school safety’ (77), ’the size of house/residence’ (8) and ’parents job problems’ (23). All these attributes occurred in all lists and get high scores.

The first in the list and the most significant condition attribute indicated by χ2_{AE and IG AE is ’school quality’}

(76). However in the top ten attributes given by R AE this attribute is absent.

Unintuitive are connections between ’contact with friends’ (80) which is strongly supported by R AE and ’safety and living conditions’ decision attribute.

Tablele 5. Ranking of first ten most significant features that

influence ’safety and living conditions’

4 2, 31, 78, 79 0.3788 5 2, 6, 31, 78, 79 0.6984 6 2, 3, 6, 31, 78, 79 0.8833 7 2, 3, 6, 31, 75, 78, 79 0.9641 8 2, 3, 6, 31, 52, 75, 78, 79 1

4.5. Child well-being general evaluation

All presented results analyzed four different aspects of life separately. However the question about general childhood quality evaluation is a natural consequence of the presented research. To find the answer and set of most significant attributes for child well-being integrally, four decision attributes were joint into one resultant attribute.

In the survey respondents were asked to specify weights (in %) for all decision attributes in the general evaluation of child well-being quality. Each decision attribute can be scored by a numeric value from 1 to 5. Therefore the general evaluation can be calculated as:

Egeneral = round( wedu·Eedu + went·Eent + whealth·Ehealth +

wsafety·Esafety ) , (1)

where: E – decision attribute, w – weight in %.

The general evaluation proposed in (1) takes values from a set {1,2,3,4,5}. This additional attribute is also nominal and can also be examined with feature selection methods. Results will show the ranking of the most important attributes for child well-being quality evaluation.

Tablele 6. Ranking of first ten most significant features that

influence child well-being

No attrib. χ2_{AE attrib. IG}_{AE attrib. R}_AE

1 80 155.673 80 0.1642 80 0.1105 2 76 154.880 76 0.1011 76 0.0631 3 33 76.1039 77 0.0678 31 0.0475 4 77 72.5151 31 0.061 73 0.0457 5 31 69.9018 81 0.0577 43 0.0455 6 81 44.3828 79 0.0483 7 0.0449 7 78 40.3767 78 0.0480 10 0.0447 8 79 40.1450 33 0.0412 33 0.0437 9 23 34.0800 65 0.0405 65 0.0436 10 65 29.5536 8 0.0318 29 0.0434

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Ewa Adamus, Przemysław Klęsk, Joanna Kołodziejczyk i inni, Significance of condition attributes in child well-being analysis reduct length attributes reduct quality

4 2, 76, 78, 79 0.4764 5 2, 76, 78, 79, 80 0.7713 6 2, 31, 76, 78, 79, 80 0.9168 7 2, 31, 47, 76, 78, 79, 80 0.9773 8 2, 27, 31, 32, 76, 78, 79, 80 1

All filters indicated the same attributes: ’contacts with friends’ (80) and ’school quality’ (76) as the two most significant features for child well-being (Tablele 6). Both attributes were highly scored. They were present in almost all top ten attribute lists described previously.

Common for three lists are also attributes: ’medical care quality’ (31), ’sports as after school activity’ (65) and ’proportion of time spent with parents’ (33). All attributes significant for general childhood evaluation were indicated previously as important for each decision attribute independently. The only exception is the attribute (33).

5. SUMMARY

Presented research was aimed at indicating the set of significant attributes for child well-being estimation. The data-mining feature selection methods were used to obtain rankings and sets of the most influential aspect of persons childhood.

Research effects occurred conformable to earlier expectations, but some lower ranked attributes presented in Tableles 2-6 can be treated as a surprise.

The most interesting result concerns qualification which condition attributes has the real and greatest influence on all aspects of child well-being. Such qualification was possible thanks to the weighted aggregation of all known decision attributes. The most important occurred 'contact with friends' and 'school quality' and the significance of these attributes was distinctly greater than others.

On the base of found significant attributes such models as decision trees or rule models can be created and thanks to the attribute reduction they can be simpler and usually have greater quality and real accuracy.

References

1. Adamus E., Klęsk P., Kołodziejczyk J., Korzeń M., Piegat A. and Pluciński M.: Rules induction on

child well being. Metody Informatyki Stosowanej,

pp. 5-10, no. 4, 2010.

2. Bouckaert R.R., Frank E., Hall M., Kirkby R., Reutemann P., Seewald A. and Scuse D.: WEKA

Manual for version 3.6.0. University of Waikato,

Hamilton, New Zealand, 2008.

3. Cichosz P.: Learning systems. Wydawnictwa Naukowo-Techniczne, Warszawa, 2000 [in Polish].

4. Hall M.A., Holmes G.: Benchmarking Attribute

Selection Techniques for Discrete Class Data Mining. IEEE Trans. on Knowl. and Data Eng., pp.

1437-1447, vol. 15, no. 6, 2003.

5. Pawlak Z.: Rough Sets. International Journal of Computer and Information Sciences, pp. 341-356, vol. 11, no. 5, 1982.

6. Sen A.K.: Capability and Well-Being, pp. 30-54. The Quality of Life. Clarendon Press, Oxford, 1993. 7. Shen Q., Jensen R.: Rough Sets, their Extensions and

Applications. International Journal of Automation

and Computing, pp. 100-106, 04(1), 2007.

8. Wang X., Yang J., Teng X., Xia W., Jensen R.:

Feature selection based on rough sets and particle swarm optimization. Pattern Recognition Letters,

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pp. 17-22

NUMERICAL STUDY OF LARGE SCALE TRUSS CONSTRUCTION

WITH STRUCTURAL UNCERTAINTIES

Róbert Bednár, Milan Sága, Milan Vaško

Department of Applied Mechanics, Faculty of Mechanical Engineering, University of Žilina, Slovakia, robert.bednar@fstroj.uniza.sk, milan.vasko@fstroj.uniza.sk, milan.saga@fstroj.uniza.sk

Abstarct: The paper deals with efficiency and usability problem for the chosen solution methods for mechanical systems with structural uncertainties. They are significantly influencing the analysis results and the analysis itself. An application of the chosen approaches will be presented – the first one, a simple combination of only inf-values or only sup-values; the second one presents full combination of all inf-sup values; the third one uses the optimizing process as a tool for finding out an inf-sup solution and last one is Monte Carlo method as a comparison tool.

Keywords: Uncertain parameters, uncertainties and errors, Monte Carlo method, interval arithmetic, optimization, MATLAB

1. INTRODUCTION

In the last years there has been an increased interest in the modeling and analysis of engineering systems under uncertainties. To obtain reliable results for the solution of engineering problems, exact values for the parameters of the model equations should be available. Really, however, those values can often not be provided, and the models usually exhibit a rather high degree of uncertainty. Computational mechanics, for example, entails uncertainties in geometry, material and load parameters as well as in the model itself and in the analysis procedure too. For that reason, the responses, such as displacements, stresses, resonant frequency, or other dynamic characteristics, will usually exhibit any degree of uncertainty. It means that the obtained result using one specific value as a most significant value for uncertain parameter cannot be considered to be representative for the whole spectrum of possible results.

Uncertain parameters appear mostly as random variables and thus are described in the terms of stochastic approach. But without the knowledge of the probability density and the nature of distribution we are forced to use another approach, which could describe the parameters with the mentioned restrains and at the same time contain sufficient information about the character of the uncertainty.

Alternately to the use of probability methods we can use imprecise probabilities and the possibility theory [1], which involves the theory of interval numbers, fuzzy numbers and fuzzy sets [2, 3, 4, 9]. Without the information of the relevance of the data on the interval, we cannot use the fuzzy approach, but we are still able to use the interval approach to describe the uncertain parameters which are considered as unknown but bounded with lower and upper bounds.

Our short study proposes algorithms for modal and spectral interval computations of FE models and their efficiency analysis in view of the input uncertainty degree (20%, 50%) [5, 9].

2. UNCERTAINTIES AND ERRORS IN FINITE ELEMENT ANALYSIS

The accuracy of FEA is affected by errors and uncertainties, which may be related to the numerical tool itself or to the physics of the problem. The possible sources of uncertainties and errors in FEA include model uncertainty, discretization error, parameter uncertainty and rounding error [10]. The definitions of these uncertainty and errors are summarized and presented below.

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Róbert Bednár, Milan Sága, Milan Vaško, Numerical study of large scale truss construction with structural uncertainties

The mathematical model in FEA represents the physical system being analyzed. The actual problem is simplified and idealized, and is described by an accepted mathematical formulation such as the theory of elasticity, or thin-plate theory and so on [6, 7, 8]. The uncertainty about how well the mathematical model represents the true behavior of the real physical system is termed model uncertainty.

Typical model uncertainties in FEA are:

 the idealization of the boundary conditions,  the use of plane model rather than

three-dimensional model,

 the use of linear model rather than nonlinear model,

 the use of time-independent model rather than dynamic model.

Discretization terror

The esTablelished mathematical model is represented by an FE discretization. This involves selecting a mesh and elements. The computed solution of the FE model is in general only an approximation of the exact solution of the mathematical model, and the discrepancy is called discretization error. FEA solution is influenced by the factors, such as the number of elements used, the nature of element shape functions, integration rules used, and other formulation details of particular elements.

Parameter uncertainty

Parameter uncertainty arises because the precise data needed for the analysis are not available. This type of uncertainty is sometimes called parametric or data uncertainty. In FEA, the parameter uncertainty may exist in the geometrical, material or loading data. Parameter uncertainty may result from a lack of knowledge, an inherent variability in the parameters, or both.

Rounding error

FEA solution is limited in accuracy by the finite precision of computer arithmetic. When arithmetic operations are performed on floating point numbers, the exact result will not, in general, be represenTablele as a floating point number. The exact result will be rounded to the nearest floating point number, and this loss of information is referred to as rounding error. A more fundamental approach, however, is to use interval arithmetic. Interval arithmetic can rigorously bound the rounding error.

3. COMPUTATIONAL METHODS FOR INTERVAL ANALYSIS

If we want to use interval arithmetic approach, an uncertain number is represented by an interval of real numbers [2, 3]. The interval numbers derived from the experimental data or expert knowledge can then take into account the uncertainties in the model parameters, model inputs etc. Complete information about the uncertainties in the model may be included by this technique and one can demonstrate how these uncertainties are processed by the calculation procedure in MATLAB.

During the solving of the particular tasks using the interval arithmetic application on the solution of numerical mathematics and mechanical problems, the problem known as the overestimate effect is encountered. Its elimination is possible only in the case of meeting the specific assumptions, mainly related to the time efficiency of the computing procedures.

Considering uncertain parameters in interval form, some solution approaches already used or proposed by the authors are analyzed [5, 9]. The goal is to present algorithm description and comparison study of the following numerical methods:

Monte Carlo method (MC) also as a comparison tool, a simple combination of only inf-values or only sup-values (COM1),

a full combination of all inf-sup values (COM2),

a method which uses an optimization process as a tool for finding out a inf-sup solutions (OPT).

Monte Carlo method (MC) is a time consuming but reliable solution. Various combinations of the uncertain parameter deterministic values are generated and after the subsequent solution in the deterministic sense we obtain a complete set of results processed in an appropriate manner. Infimum and supremum calculation is following

     F F p , i ,...,m m . , m m , ... , i , p F F i i 100000 5000 and 1 where results all of max sup 100000 5000 and 1 where results all of min inf         ( 1)

Solution evaluation in marginal values of interval parameters (COM1) has its physical meaning for many engineering problems. We consider an approach where the extreme output values are obtained by the application of the extreme parameter values on input. That means that the infimum or supremum is obtained using the deterministic analysis for infimum or supremum of input uncertain parameters. Inf-sup calculation is

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pp. 23-30

 

F

min

of



F

 

p

,

F

 

p



inf



, (2)

 

F

max

of



F

 

p

,

F

 

p



sup



Solution evaluation for all marginal values of interval parameters (COM2) which is also based on the set of the deterministic analyses appears as the more suiTablele one. The marginal interval parameter values are considered again but the inf and sup are also combined. The method provides satisfying results and can be marked as reliable, even if there is still a doubt about the existence of the extreme solution for the uncertain parameter inner values. A solution for two interval numbers

p

₁



a

₁

b

₁ and

2 2 2

a

b

p



may be found in the following computational way

 





 





 

F F





a a

 

, F a b

 

, F b a

 

, F b b





. b b F , a b F , b a F , a a F F 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 of max sup , of min inf   (3)

The method of the inf and sup solution using the optimization techniques (OPT) is proposed by the authors as an alternative to the first and to the third method. It should eliminate a big amount of analyses in the first method and also eliminates the problem with the possibility of the inf and sup existence inside of the interval parameters for the deterministic values. Computational process for two interval numbers p1 and p2

may be found as follows

   

 

   

,

i.e.

find

so

that

 

max

.

sup

,

min

that

so

find

i.e.

,

inf









OPT OPT OPT OPT OPT OPT

F

p

(4)

4 SOLVING OF TRUSS STRUCTURE WITH INTERVAL PARAMETERS

Considering different uncertain parameters the numerical interval stress-strain study of a three-dimensional truss structure was performed. The geometry of the structure is presented on Figure. 1. The truss structure was loaded by forces F in all upper nodes of the structure. The truss structure consists of 70 nodes and 257 bars.

The certain and uncertain model parameters are defined as follows:

element mass density





7800

kg



m

3,

Young‘s modulus

_E

_{ .}

2

1 _

10

11

Pa

_,

cross-section areas

A



0. 0015

m

2, force

F



1000

N

.

The force, cross-section area and Young’s modulus were used as the uncertain parameters. The uncertainty degree was implemented for values of 20% and 50%.

Figure. 1 Analyzed truss structure, dimensions in [m]

The purpose of this study will be to compare the efficiency and exactness of the proposed methods MC, COM1, COM2 and OPT. The results of the MC analysis will be considered as reference values and will be used for the construction of the solution map. In the case of MC method, 10000 random inputs were generated; they were evaluated and properly processed to inf-sup solutions. The maximal and minimal inf-sup stress values are summarized in Table. 1 and maximal displacement shows Table. 2.

Table. 1

Stress inf/sup results for the chosen

bars [Pa]

Stress [Pa] Bar No. COM1 COM2 inf 2446582,18 2304062,83 200 sup 3117261,47 3310081,77 inf -4833204,26 -5132165,35 206 sup -3793339,61 -3572368,37 Stress [Pa] Bar No. OPT MMC inf 2304062,83 2546957,37 200 sup 3310081,77 3044004,29 inf -5132165,35 -4719621,57 206 sup -3572368,37 -3948967,81

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Table. 2 Displacement inf/sup result for the chosen node [m] Displacement [m] Node No. COM1 COM2 inf 0,000291311 0,000224461 39 sup 0,000303683 0,000394127 Displacement [m] Node No. OPT MMC inf 0,000224461 0,000130511 39 sup 0,000322467 0,000170684

Figure. 2 The maximal (bar No. 200), minimal

(bar No. 206) stress values

and maximal displacement (node No. 39)

Mapping of the generated input data by MC method for bars with max and min stresses and for node with max displacement is shown on Figs. 3–5. Stress solution on the bar No. 206 for various uncertainties is shown on Figs. 6– 7. Displacement solution on the node No. 39 for various uncertainties is shown on Figs. 8–9.

Figure. 3 Mapping of the generated input data for

bar No. 200

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pp. 23-30

node No. 39

Figure. 6 Stress solution on bar No. 206 (max

uncertainty 20%)

Figure. 7 Stress solution on bar No. 206 (max

uncertainty 50%)

Figure. 8 Displacement solution on node No. 39

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Figure. 9 Displacement solution on node No. 39

(max uncertainty 50%)

5. CONCLUSION

The paper presents methods and their applications in an interval structural analysis. The use of the interval arithmetic provides a new possibility of the quality and reliability appraisal of analyzed objects. It shows the stress-strain solution efficiency for solving problems including uncertain parameters with a various width of the interval.

The analyses results can be summarized as follows:  COM2 method provides decent results, but it is

limited due to the exponential growth of the analyses number for complicated problems,

 OPT method provides good results and is suiTablele for complicated problems because it does not need so many analyses as in the cases of the MC or COM2 methods,

 the cross-section area as uncertain parameter has the biggest influence on stress solution,

 the cross-section area and Young’s modulus as uncertain parameters have the biggest influence on displacement solution.

ACKNOWLEDGEMENT

The work has been supported by the grant projects VEGA1/0125/09 and 1/0727/10

.

REFERENCES

1. Dekýš,V., Sapietová, A., Kocúr, R.: On the reliability

estimation of the conveyer mechanism using the Monte Carlo method. Machine Dynamics Problems, 2006,

Vol. 30, No 3, pp. 58-64, ISSN 0239-7730.

2. Moore R., E.: Interval Analysis. Prentice Hall, Englewood Cliffs : New Jersey, 1966.

3. Neumaier, A.: Interval Methods for Systems of

Equations. Cambridge University Press : Cambridge,

1990.

4. Chen, S. H., Yang, X. W.: Interval Finite Element Method for Beam Structures. In: Finite Elem. Anal.

Des., 34, pp. 75-88, 2000.

5. Sága, M., Vaško, M.: Algorithmization of Interval

Structural Analysis. In: J. of Engineering, Annals of Faculty of Engineering Hunedoara, Tome VI, 2008,

ISSN 1584-2665, pp. 149–160.

6. Kopas, P., Melicher, R., Handrik, M.: FE analysis of

thermo-plastic material model using for bearing steels. In: Applied mechanics 2008, 10th_{International} Scientific Conference : April 7-10, 2008, Wisła,

Poland, 2008. ISBN 978-83-60102-49-7.

7. Jakubovičová L., Kopas P.: FE Analysis of

Deformation Aluminium Alloy EN AW 2007.T3 in Bending and Torsion Loading Process. Technolog,

No. 2, Vol. 3, 2011, pp. 7–10, ISSN 1337-8996. 8. Vaško, A.: Image analysis in materials engineering.

In: Konferencje, Poland, 2007, Nr 61, pp. 667-670. ISSN 1234-9895.

9. Vaško, M.: Application of Genetic Algorithms for

Solving of Mechanical Systems with Uncertain Parameters. PhD. thesis, Žilina, 2007.

10. Zhang, H.: Nondeterministic Linear Static Finite

Element Analysis: An Interval Approach. School of

Civil and Environmental Engineering : Georgia Institute of Technology, 2005.

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pp. 23-30

PROGRAMMING OF COMPOSITE PLATES DAMAGE CALCULATION

Martin Dudinsky, Daniel Riecky, Milan Zmindak

Faculty of Mechanical Engineering, University of Žilina Univerzitná 1, 010 26 Žilina, Slovakia

e-mail: martin.dudinsky@fstroj.uniza.sk, daniel.riecky@fstroj.uniza.sk, milan.zmindak@fstroj.uniza.sk

Abstract: The goal of this paper is to present the numerical results of elastic damage of thin unidirectional fiber-reinforced composite plates. The numerical implementation uses a layered shell finite element based on the Kirchhoff plate theory. Newton-Raphson method is used to solve the system of nonlinear equations and evolution of damage has been solved using return-mapping algorithm. The analysis is performed by finite element method and user own software is created in MATLAB programming language. One problem for two different materials was simulated in order to study the damage of laminated fiber reinforced composite plates.

Keywords: continuum damage mechanics, composite plate, finite element metod

1. INTRODUCTION

Composite materials are now common engineering materials used in a wide range of applications. They play an important role in the aviation, aerospace and automotive industry, and are also used in the construction of ships, submarines, nuclear and chemical facilities, etc.

The needs of explicit safety requirements for development of the Processes of design, manufacturing, maintenance, operation and requirement s are tightly linked to change and innovation. The challenges derived from new materials, new processes and new structural concepts are inherent to the use of composites for light structures. It is true that it is the responsibility of each individual discipline to implement processes that ensure the meeting of safety objectives. The damage to do so, for the most part results in structural integrity loss, so a coordinated set of requirements must be put in place, and monitored and enforced.

The meaning of the word damage is quite broad in everyday life. In continuum mechanics the term damage is referred to as the reduction of internal integrity of the material due to the generation, spreading and merging of small cracks, cavities and similar defects. In the initial stages of the deformation process the defects (microcracks, microcavities) are very small and relatively uniformly distributed in the microstructure of the material. If the

damage reaches the critical level (depends on the load type and material used), subsequent growth of defects will concentrate in some of the defects already present in material [7]. Damage is called elastic, if the material deforms only elastically (in macroscopic level) before the occurrence of damage, as well as during its evolution. This damage model can be used if the ability of the material to deform plastically is low. Fiber-reinforced polymer matrix composites can be considered as such materials. Composites represents a family of structural materials for which the accumulation of structural damage is complicated process. It involves fatigue damage initiation, damage growth contributions, continued accidental damage occurrences and, in addition, the contributions from property changes due to materials and manufacturing process failures. Unidirectional composites reinforced by long fibers are one kind of multicomponent composites. They can be considered as orthotropic and it is not necessary to perform homogenization in structural analysis. If multidirectional fiber-reinforced composites are analyzed, it is necessary to be perform homogenization in order to calculating the components of constitutive matrix, or tensor [15,8], which are required for further analysis of composite structures, for example damage and failure of composite structures.

Commercial Finite element method (FEM) software can perform analysis with many types of material nonlinearities, such as plasticity, hyperelasticity, viscoplasticity, etc. However, almost no software contains a module for damage analysis of composite materials.

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The goal of this paper is to present the numerical results of elastic damage of thin composite plates. The analysis was performed by user own software, created in MATLAB programming language. This software can perform numerical analysis of elastic damage using finite element method using layered plate finite elements based on the Kirchhoff plate theory. Locking effect was not removed, since this is a rather complicated issue [1]. This paper is organized in three sections: first, a general description of damage is provided, then the damage model used is examined. Finally, some numerical results obtained for damage of plates are presented.

2. THEORY AND NUMERICAL MODELING BACKROUND

A number of material modelling strategies exist to predict damage in laminated composites, subject to severe static or impulsive loads. Broadly, they can be classified as [10,13,16] :

 failure criteria approach (which can be based on the equivalent stress or strain),

 fracture mechanics approach (based on energy release rates),

 plasticity or yield surface approach,  damage mechanics approach

Strength-based failure criteria are commonly used with the FEM to predict failure events in composite structures. Numerous classical criteria have been derived to relate internal stresses and experimental measures of material strength to the onset of failure (maximum stress or strain, Hill, Hoffman, Tsai-Wu). These classical criteria implemented in most commercial FE codes are not able to physically capture the failure mode. Some of them cannot deal with materials having a different strength in tension and compression. The Hashin criteria are briefly reviewed in [9] and improvements are proposed by Puck and Schurmann [12] over Hashin’s theories are examined. However, few criteria can represent several relevant aspects of the failure process of laminated composites, e.g. the increase on apparent shear strength when applying moderate values of transverse compression, or detrimental effect of the in-plane shear stresses in failure by fiber kinking.

2.1 Damage mechanics

We consider a volume of material free of damage if no cracks or cavities can be observed at the microscopic scale. The opposite state is the fracture of the volume element.

Theory of damage describes the phenomena between the virgin state of material and the macroscopic onset of crack [6,14]. The volume element must be of sufficiently large size compared to the inhomogenities of the composite material. In Figure. 1 this volume is depicted. One section of this element is related to its normal and to its area S. Due to the presence of defects, an effective area for resistance can be found. Total area of defects, therefore, is:

S

_D





~

(1)

The local damage related to the direction n is defined as:

S

D

_

D

₍₂₎

For isotropic damage, the dependence on the normal n can be neglected, i.e.

n

D



_n



(3)

We note that damage D is a scalar assuming values between 0 and 1. For D = 0 the material is undamaged, for 0 < D < 1 the material is damaged, for

D = 1 complete failure occurs. The quantitative evaluation

of damage is not a trivial issue, it must be linked to a variable that is able to characterize the phenomenon. Several papers can be found in literature where the constitutive equations of the materials are a function of a scalar variable of damage [2,3]. For the formulation of a general multidimensional damage model it is necessary to generalize the scalar damage variables. It is therefore necessary to define corresponding tensorial damage variables that can be used in general states of deformation and damage [14].

Figure. 1 Representative volume element for

damage mechanics

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pp. 23-30

2.2 Numerical modeling

In composite laminates, defects tend to accumulate at the interface between plies or in intralaminar pockets that are rich in resin. In real structures, both mechanisms of failure are usually present. With respect to FEM simulation, a smeared approach is usually adopted for the simulation of matrix cracking, while delamination is modelled discretely [15]. One of the most powerful computational methods for structural analysis of composites is the FEM. The starting point would be a “validated” FE model, with a reasonably fine mesh, correct boundary conditions, material properties, etc. [1]. As a minimum requirement, the model is expected to produce stress and strains that have reasonable accuracy to those of the real structure prior to failure initiation. In spite of the great success of the finite and boundary element methods as effective numerical tools for the solution of boundary-value problems on complex domains, there is still a growing interest in the development of new advanced methods. Many meshless formulations are becoming popular due to their high adaptivity and a low cost to prepare input data for numerical analysis [5,11,4].

3. AMAGE MODEL USED

The model for fiber-reinforced lamina mentioned next was presented by Barbero and de Vivo [2] and is suiTablele for fiber - reinforced composite material with polymer matrix. On the lamina level these composites are considered as ideal homogenous and transversely isotropic. All parameters of this model can be easily identified from available experimental data. It is assumed that damage in principal directions is identical with the principal material directions throughout the damage process. Therefore the evolution of damage is solved in the lamina coordinate system. The model predicts the evolution of damage and its effect on stiffness and subsequent redistribution of stress.

3.1. Damage surface and damage potential

Damage surface is defined by tensors J and H [3]















33 22 11

0

0 J

J

H





H

₁

H

₂

H

₃



(4)

Damage surface is similar to the Tsai-Wu damage surface [6], which is commonly used for predicting failure of fiber-reinforced lamina with respect to experimental material strength values. Damage surface and damage potential have the form of [3]  



0



2 3 3 2 2 2 2 1 1 2 3 33 2 2 22 2 1 11 ,  J Y J Y J Y  H Y HY HY  g Y

(5)



,







J

11

Y

12



J

22

Y

22



J

33

Y

32











0



f Y

(6)

where the thermodynamic forces Y1, Y2 and Y3 can be

calculated by relations

0

1

3 2 6 2 2 2 1 66 2 1 2 2 2 1 12 2 2 4 2 22 2 2 2 2 6 2 2 2 1 66 2 1 2 2 2 1 12 2 1 4 1 11 2 1 1



















































Y

S

Y

S

Y



(7)

where stresses and components of matrix S are defined in the lamina coordinate system. Matrix S gives the strain-stress relations in the effective configuration [2].

Equation (5) and (6) can be written for different simple stress states: tension and compression in fiber direction, tension in a transverse direction, in-plane shear. Tensors J and H can be derived in terms of material strength values.

3.2. Hardening parameters

In the present model for damage isotropic hardening is considered and hardening function was used in the form of





























exp

1

2 1

_c

c





(8)

The hardening parameters γ0, c1 and c2 are determined by approximating the experimental stress-strain curves for in-plane shear loading. If this curve is not available, we can reconstruct it using function















₆ 6 12 6 6

tanh





F

G

F

(9)

(24)

where F6 is in-plane shear strength, G12 is the in-plane initial elasticity modulus and γ6 is the in-plane shear strain (in the lamina coordinate system). This function represents experimental data very well.

3.3. Critical damage level

The reaching of critical damage level is dependent on stresses in lamina. If in a point in lamina only normal stress occurs in the fiber direction and across the fibers (i.e., normal stress in lamina coordinate system), then simply comparing the values of damaged variables with critical values of damage variables for given material at this point is sufficient. The damage has reached critical level if at least one of the values D1, D2 in the point of lamina is greater or equal to the critical value. If in given point of lamina shear stress occurs (in lamina coordinate system), it is additionally necessary to compare the value of the product of (1 - D1) (1 - D2) with ks value from Table. 3 for given material. If the value of this product is less or equal to ks value, the damage has reached a critical level.

3.4. Implementation of numerical method

Newton-Raphson method was used for solving the system of nonlinear equations. Evolution of damage has been solved using return-mapping algorithm described in [2]. The input values are strains, strain increments in lamina coordinate system, state variables D1, D2, and δ in integration point from the start of last performed iteration,

Błąd! Nie można tworzyć obiektów przez edycję kodów pól.matrix (gives the stress-strain relations in effective

configuration [3]) and damage parameters related to damage model. The output variables are D1, D2, and δ, stresses and strains in lamina coordinate system in this integration point at the end of the last performed iteration. Another output is constitutive damage matrix Błąd! Nie można tworzyć

obiektów przez edycję kodów pól. in lamina coordinate

system, which reflects the effect of damage on the behavior of structure. Flowchart of numerical damage analysis is described in Figure. 2.

Figure. 2 Flowchart of numerical damage analysis of thin plates

4. NUMERICAL EXAMPLE

One problem for two different materials were simulated in order to study the damage of laminated fiber reinforced composite plates. The composites are reinforced by carbon fibers embedded in epoxy matrix. The simply supported composite plate with laminate stacking sequence [0, 45, -45, 90]S with dimensions 125×125×2.5 mm was

loaded by transverse force F = - 4000 N in the middle of the plate. Own program created in MATLAB language was used for this analysis.

Material properties, damage parameters and hardening parameters and critical damage values are given in Table. 1 to Table. 3. The parameters J33 and H3 are equal to

zero. The plate model was divided into 8×8 elements and was analyzed in fifty load substeps. The linear static analysis shows that the largest stress in absolute values are in parallel direction with fibers and transverse to fibers and they occur in the outer layers in the middle of the plate.

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pp. 23-30

Table. 1 Material properties

E1[GPa] E2[GPa] G12[GPa] υ12

M30/949 167 8.13 4.41 0.27

M40/948 228 7.99 4.97 0.292

Table. 2 Damage and hardening parameters

1

J

₂

H

₁

H

₂

γ

₀

c

₁

c

₂ M 30 / 94 9 0.95 2.10 -3 0.43 8 25.58 5.10-3 21.66 5.10-3 0. 6 0. 30 0. 39 5 M 40 / 94 8 2.20 8.10 -3 0.21 4 10.50 3.10-3 8.130.₁₀-3 ₁₂0. ₁₀0. ₃₉0. 5

Table. 3 Critical values of damage variables

cr 1t

D

cr 1c

D

cr c

D

₂ ks M30/949 0.105 0.111 0.5 0.944 M40/948 0.105 0.111 0.5 0.908

The largest shear stresses in absolute value are in the outer layers in corner nodes. The largest absolute stress values are in layers 2, 3, 6 and 7 in the middle of the plate. According to the results of linear static analysis can be expected that damage reaches the critical level in some of the above points. Figure. 3 shows the results of analysis of elastic damage for material M30/949. Figure. 3a shows the evolution of individual stress components in dependence on strains in lamina coordinate system in the midsurface of layer 1 (first layer from bottom) in integration point (IP) 1 (in element 1, nearest to the corner). Figure. 3b shows the evolution of individual stress components in the midsurface of layer 2 in IP 872 (in element 28, nearest to middle of plate). Figure. 4 plots described damage variables evolution in IPs.

The analysis results show that reaching the critical level is caused not by normal stresses in the lamina coordinate system, but by shear stresses (in the lamina coordinate system). The results of analysis of plate made from material M30/949 show that for a given load the critical level of damage is reached in layers 2 and 7 in the middle of

plate and its vicinity. In IPs that are closest to the center of plate in these layers, the critical level of damage was reached between 13th and 14th load substep. However, the model used for the damage does not correctly predicted failures [2]. In some cases, failures occur before the damage reaches a critical level. For given material load is F = -1096 already critical .

(a) (b)

Figure. 3 Stress and strain evolution for material

M30/949 in (a) IP 1, (b) IP 872

(a) (b)

Figure. 4 Damage variables

evolution for material M30/949 in

(a) IP 1, (b) IP 872

Figure. 5 shows the results of analysis of elastic damage for material M40/948. The results show that reaching the critical level of damage will be also caused by shear stresses in lamina coordinate system. However, the critical damage level was reached in the middle of plate and its vicinity in layers 1 and 8. The critical level of damage was reached between 12th and 13th load substep in nearest IPs.