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11111 111111V#Ittijl M Jtimmin Linii 1U11 I 1 I L I M Ryszard WALENTYNSKI ■ iii m l JTT 1111111K ■ 1 13 ' I 1] l i i i i] 11111 111111V#iTTijl M jTiMMiN linii 1U11 i 1 i l i m 2 E h (- 3 ( - 1 + V2) 181® 4E h3H (apr) (a«1) (bęj^ Jari) (br3) ( y Pq) 3 (1 + v )^ !§M| 4 E h 3 H v (ai:i) \(Y p g ) 4 E h 3 (a qi 4 E h 3 H f 8 E h 3 ( 4 E h 3 H ' 8 E h 3 H (e 4 E h 3 (a p r ) fbr ) (Ppg) 3 (3 8 E h 3 v ( a i ; i ) (Ł (Ppg) - 3 + 3 v 2 (l + v) 8 E h 3 (ąPJ) (b^1) (pP9) ~ 8 E h 3 v (ągi) (b1*) (ppq) 3 (1 + v) + -3 + 3 v 2 2 E h 3 (a«) (a«1) (c?pq5 ( 2Bh 3 v (a ^ ) (ai3) {i9pq; 3 (1 + v) + 3 - 3 v 2 ^m&RWHKTwó-, V TOUTKHWKI \ >1 , Gliwice 2003 0 5 Rysz ENTYŃSKIENTYNSKI APPLICATION OF COMPUTER ALGEBRA IN SYMBOLIC COMPUTATIONS AND BOUNDARY-VALUE PROBLEMS OF THE THEORY OF SHELLS 2 Eh (-3 + 5 h 2 K) (aPJ) (a1*1) (ypq) 3 (1 + v ) 2 Eh (-3 + 5 h 2 K) v ( a p q ) (ai j ) (TPq) 3 (-1 + v2) 4 E h 3 H ( a p r ) ( a q i ) (b ) ( b r j ) ( T p a ) 3 (1 + v ' 4 E h 3 H v ( a l j 4 E h 3 ( a qj 8 E h 3 (ap:i) (bq i) ( Ppq) 8 E h 3 v (apq) (blj) (ppq) 3 (1 + v) -3 + 3 v 2 2 E h 3 (aPJ) (aqi) (Opq) 2 E h 3 v (apq) (a^) ((9pq) 3 (1 + v ) 3 - 3 v 2 WYDAWNICTWO POLITECHNIKI ŚLĄSKIEJ GLIWICE 2003 OPINIODAWCY Prof. dr hab. inż. Piotr KONDERLA — Politechnika Wrocławska Dr hab. inż. Bogumił WRANA — Profesor Politechniki Krakowskiej CONTENTS KOLEGIUM REDAKCYJNE REDAKTOR NACZELNY — Prof. dr hab. inż. Andrzej BUCHACZ NOMENCLATURE 13 REDAKTOR DZIAŁU — Dr inż. Marianna GLENSZCZYK SEKRETARZ REDAKCJI — Mgr Elżbieta LEŚKO 1. INTRODUCTION 23 1.1. SYMBOLIC COMPUTATION - A NEW SCIENTIFIC TREND 23 1.2. TOOLS OF TENSOR ANALYSIS ASSISTANCE 26 1.3. COMPUTER ALGEBRA IN MECHANICS, DIFFERENTIAL GEOMETRY AND REDAKCJA THE THEORY OF SHELLS 27 Mgr Feliks LIPSKI 1.4. PROGRESS TENDENCIES IN THE THEORY OF SHELLS 27 1.5. AIM AND CONTENT OF THE CONTRIBUTION 29 1.5.1. Part I: Symbolic computations 30 REDAKCJA TECHNICZNA 1.5.2. Part II: Approximation of boundary value problems by means of the Least Squares Alicja NOWACKA Method 31 1. SYMBOLIC COMPUTATIONS 33 2. BASIC RELATIONS OF THE THEORY OF SHELLS 35 2.1. GEOMETRICAL DESCRIPTION 35 ZESZYTY NAUKOWE N r kol. 1587 2.1.1. Reference surface 35 BUDOWNICTWO z. 100 2.1.2. Parallel surface 37 2.2. GEOMETRICAL RELATIONS 39 PL ISSN 0434-0779 2.2.1. Kinematic relations 39 2.2.2. Deplanation 40 2.2.3. Strains 4 0 © Copyright 2003 by 2.2.4. Influence of temperature 41 Ryszard WALENTYNSKI 2.3. CONSTITUTIVE RELATIONS 42 [email protected] 2.3.1. Stress tensor 42 Utwór w całości ani we fragmentach nie może być powielany ani rozpowszechniany za 2.3.2. Internal forces 42 pomocą urządzeń elektronicznych, mechanicznych, kopiujących, nagrywających i innych, 2.4. EQUATIONS OF EQUILIBRIUM 43 w tym również nie może być umieszczany ani rozpowszechniany w postaci cyfrowej zarówno w Internecie, jak i sieciach lokalnych bez pisemnej zgody posiadacza praw autorskich. 2.5. PHYSICAL COMPONENTS 44 3 4 C o n ten ts C ontents 5 3. EQUATIONS OF EQUILIBRIUM 45 n. BOUNDARY-VALUE PROBLEMS 71 3.1. CHANGING COVARIANT DERIVATIVES IN EQUATIONS TO (ORDINARY) PARTIAL DERIVATIVES 45 7. DESCRIPTION OF THE REFINED LEAST SQUARES METHOD 7 3 3.2. SUMMATIONS 46 7.1. CLASSICAL APPROACH 73 3.3. TRANSFORMATION TO MATHEMATICS DIFFERENTIAL EQUATIONS 47 7.2. ONE-DIMENSIONAL PROBLEMS 7 4 3.4. NONLINEAR EQUATIONS OF THE SECOND-ORDER THEORY 48 7.2.1. Functional 74 4. CONSTITUTIVE RELATIONS 49 7.2.2. Application of the Ritz method 74 4.1. DEFINITIONS 49 7.2.3. System of linear algebraic equations 76 4.2. EVALUATION 50 7.3. NONLINEAR TASKS 7 7 4.3. ELEMENTS OF SIMPLIFICATION 54 7.4. MULTIDIMENSIONAL PROBLEMS 7 8 4.3.1. Grouping of terras 5 4 7.5. FEATURES OF THE REFINED LEAST SQUARES METHOD 7 8 4.3.2. Moderated simplification 55 7.5.1. Advantages of the method 78 4.3.3. Replacement 55 7.5.2. Disadvantages of the method 79 4.3.4. Result of simplifications 56 8 . ELEMENTS OF THE IMPLEMENTATION OF THE METHOD 8 0 4.4. SATISFACTION OF THE LAST EQUATION OF EQUILIBRIUM 57 8.1. TRANSLATION INTO THE MATHEMATICS L A N G U A G E 8 0 5. DESCRIPTION OF AN ARBITRARY SHELL 58 8.1.1. Polynomial approximation 80 8.1.2. Weighted differential equations 82 5.1. GEOMETRICAL DESCRIPTION OF THE REFERENCE SURFACE 58 8.1.3. Extraction of terms from equations 82 5.2. GEOMETRICAL PROPERTIES 59 8.1.4. Weighted boundary conditions 83 5.3. KINEMATIC RELATIONS 60 8.1.5. Guessing weights 83 5.4. STRAINS 61 8.1.6. Extraction of terms from boundary conditions 84 6. FINAL RESULTS OF SYMBOLIC COMPUTATIONS 62 8.1.7. Coefficients of the system matrix 84 8.1.8. Function of integration 84 6.1. CYLINDRICAL SHELL 62 8.1.9. The matrix of the system of linear algebraic equations 85 6.2. INTERNAL FORCES 63 8.1.10. Vector of free elements 85 6.2.1. Stretching forces 63 8.2. SOME COMPUTATIONAL ASPECTS OF SOLVING THE SYSTEM OF LINEAR 6.2.2. Moments 64 ALGEBRAIC EQUATIONS 86 6.2.3. Transverse forces 64 6.3. PARTIAL DIFFERENTIAL EQUATIONS 65 9. CHIMNEY EXPOSED TO AN ANTISYMMETRICAL LOAD 8 8 6.4. VARIABLE SEPARATION 66 9.1. DESCRIPTION OF THE PROBLEM 8 8 6.4.1. Stretching forces 67 9.1.1. Numerical data of the problem 88 6.4.2. Moments 68 9.1.2. Differential equations 89 6.4.3. Transverse forces 68 9.1.3. Engineering interpretation 89 6.4.4. Ordinary differential equations 68 9.2. ONE-STEP APPROACH 9 0 6 C o n ten ts C ontents 9.3. TWO-STEP APPROACH 90 11.3. ILLUSTRATING EXAMPLES 151 9.3.1. Grouping of the boundary conditions 90 11.3.1. Illustration o f the first case 151 9.3.2. Computational experiment - basis of the two-step approach 91 11.3.2. Illustration o f the third case 152 9.4. STEP ONE-BASE SOLUTION 92 11.4. ATTEMPT OF A MATHEMATICAL EXPLANATION 153 9.4.1. Rotations 94 11.4.1. Explicit tasks 153 9.4.2. Displacements 96 11.4.2. Im plicit tasks (shells) 154 9.4.3. Stretching and shear forces 99 11.5. EXTENSION OF MATHEMATICS POSSIBILITIES 156 9.4.4. Moments 102 11.6. ESTIMATION OF THE QUALITY OF THE GLOBAL CONVERGENCE 157 9.4.5. Transverse forces 104 9.4.6. Global equilibrium 104 12. REISSNER-MINDLIN PLATE 159 9.5. STEP TWO - REFINEMENT WITHIN BOUNDARY LAYERS 105 12.1. NOTATIONS AND VALUES OF DATA 159 9.5.1. Free edge 105 12.2. PROBLEM DESCRIPTION 160 9.5.2. Fixed edge 112 12.3. APPROXIMATION 162 12.4. PHYSICAL INTERPRETATION 166 10. CYLINDRICAL SHELL SUBJECTED TO A SINUSOIDAL LOAD 11 9 12.5. SOME REMARKS ON THE ANALYSIS OF ERROR AND CONVERGENCE 166 10.1. DESCRIPTION OF THE PROBLEM 119 10.1.1. Numerical data of the problem 119 13. CONCLUSIONS 168 10.1.2. Differential equations 121 13.1. SYMBOLIC COMPUTATIONS 168 10.1.3. Boundary conditions 121 13.2. BOUNDARY VALUE PROBLEMS 169 10.2. SHORT SHELL 122 13.3. ROLE OF COMPUTER ALGEBRA AND CONTRIBUTION TO ITS DEVELOPMENT 170 10.3. ERROR AND CONVERGENCE ANALYSIS 125 10.3.1. Estimation of the global error 126 14. FURTHER DEVELOPMENTS 171 10.3.2. Estimation of the local error 130 10.3.3. A few remarks on convergence and weights 134 ACKNOWLEDGEMENTS 172 10.4. LONG SHELL-ONE-STEP APPROACH 135 APPENDIX. BRIEF REVIEW OF THE AUTHOR’S CONTRIBUTIONS 173 10.5. LONG SHELL-TWO-STEP APPROACH 136 10.5.1. Base solution 136 BIBLIOGRAPHY 176 10.5.2. Boundary-layer refinement 140 SUMMARY 195 11. PHYSICAL INTERPRETATION AND MATHEMATICAL EXPLANATION 1 4 6 11.1. PHYSICAL INTERPRETATION OF THE BOUNDARY-CONDITION PHENOMENON 147 11.2. EXPLICIT PROBLEM — AS A BASE OF MATHEMATICAL EXPLANATION 148 11.2.1. Differential equation of a straight bar 149 11.2.2. Solutions of the equation 150 Spis treści 9 3. RÓWNANIA RÓWNOWAGI 45 3.1. ZAMIANA POCHODNYCH KOWARIANTNY CH W RÓWNANIACH NA POCHODNE CZĄSTKOWE (ZWYKŁE) 45 SPIS TREŚCI 3.2. SUMOWANIA 46 3.3. TRANSFORMACJA NA RÓWNANIA RÓŻNICZKOWE MATHEMATICS 47 3.4. NIELINIOWE RÓWNANIA TEORII DRUGIEGO RZĘDU 48 OZNACZENIA 13 4. ZWIĄZKI KONSTYTUTYWNE „ 49 4.1. DEFINICJE 49 1. WPROWADZENIE 23 4.2. OBLICZENIA 50 1.1. ALGEBRA KOMPUTEROWA - NOWY TREND NAUKOWY 23 4.3. ELEMENTY UPROSZCZENIA 54 1.2. NARZĘDZIA WSPOMAGANIA ANALIZY TENSOROWEJ 26 4.3.1. Grupowanie składników 54 1.3. ALGEBRA KOMPUTEROWA W MECHANICE, GEOMETRII RÓŻNICZKOWEJ 4.3.2. Upraszczanie kontrolowane 55 I TEORII POWłOK 27 4.3.3. Podstawianie 55 1.4. TENDENCJE ROZWOJOWE W TEORII POWŁOK 27 4.3.4. Wynik uproszczeń 56 1.5. CEL I ZAKRES PRACY 29 4.4. SPEŁNIENIE OSTATNIEGO RÓWNANIA RÓWNOWAGI 57 1.5.1. Część I: 2^gadnienia symboliczne 30 1.5.2. Część II: aproksymacja zadań brzegowych z użyciem Metody Najmniejszych Kwadratów 31 5.
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