CHAPTER - 5 CONCLUSIONS AND FUTURE SCOPE
5.2 Future Scope of Work
• The IRSA method can be further extended to the individual elements of the structural model by extending the programming capabilities of the Software.
• The IRSA method can be extended further to the uncertainty problems for dynamic analysis of shell elements to reduce the error in analysis.
• As the IRSA method is proved successful in determining linear dynamic response of the Three-Dimensional structures due to seismic loading there is more scope for future expansion of the method to non-linear analysis.
REFERENCES
1. Acéves A., Ercolani N., Jones C., Lega J. and Moloney J., “Introduction to singular perturbation methods Nonlinear oscillations”,
http://math.arizona.edu/~ntna2007/Perturbation_Methods.pdf
2. Adhiakri S., Friswell M. I., “Random Eigen-value Problems in Structural Dynamics”, http://michael.friswell.com/PDF_Files/C177.pdf
3. Akpan N. Ikot , Louis E. Akpabio, Ita O. Akpan, Michael I. Umo and Eno E.
Ituen, “Quantum Damped Mechanical Oscillator”, International Journal of Optics, Volume 2010 (2010), Article ID 275910, 6 pages,
doi:10.1155/2010/275910
4. Alexander Kiselev, Christian Remling, Barry Simon, “Effective Perturbation Methods for One-Dimensional Schrödinger Operators”,
www.math.caltech.edu/papers/bsimon/p268.pdf
5. Andrias keese, “A Review of Recent Developments in the Numerical Solution of Stochastic Partial Differential Equations (Stochastic Finite Elements)”, Institute of Scientific Computing, Informatikbericht Nr.: 2003-06, October 2003.
6. Andrzej Pownuk, “General interval FEM program based on sensitivity analysis”, The University of Texas at El Paso, Texas, Research Report No.
2007-06.
7. Babuška I., and Oden T. J., “Verification and Validation in Computational Engineering and Science: Basic Concepts”, Computer Methods in Applied Mechanics and Engineering, 193:4057–4066, 2004.
8. Belytschko T., Moës N., Usui S., and Parimi C., “Arbitrary Discontinuities in Finite Elements”, International Journal of Numerical Methods in Engineering, 50(4):993–1013, 2001.
9. Ben-Haim Y., and Elishakoff I., “Convex Models of Uncertainty in Applied Mechanics”, Elsevier Science, Amsterdam, 1990.
10. Berleant D., “Automatically verified reasoning with both intervals and probability density functions”, Interval Computations (2):48–70, 1993.
11. Bulleit W. M., “Unecrtainty in Structural Engineering”, Practice Periodical on Structural Design and Construction,Vol. 13, No. 1, February 1, 2008.
12. Caesar B., Eckert L., Hoppe A., “Design Parameter Update of Dynamic Mathematical Models in the Presence of Test Noise and Mode Pairing Problems”, Proc. Internat, Conf. ‘Spacecraft Structures and Mechanical Testing’, Noordwijk, The Netherlands, ESA SP-321, Vol.1, pp.491-499, 1995.
13. Casey L. Richardson, Jan Hegemann, Eftychios Sifakis, Jeffrey Hellrung, Joseph M. Teran, “An XFEM method for modelling geometrically elaborate crack propagation in brittle materials” ,
http://www.math.ucla.edu/~jteran/papers/RHSHT10.pdf
14. Chandrasekaran S. and Anubhab Roy, “Seismic Evaluation of Multi-Storey RC Frame Using Modal Pushover Analysis”, Nonlinear Dynamics (2006) 43:
329–342, Springer.
15. Chen H., and Belytschko T., “An Enriched Finite Element Method for Elastodynamic Crack Propagation”, International Journal of Numerical Methods in Engineering, 58(12):1873–1905, 2003.
16. Chen S. H., Lian H. D., and Yang X.W., “Interval static displacement analysis for structures with interval parameters”, Int. J. Numer. Methods Engrg. 53:393–407, 2002.
17. Chen S., Qiu Z., Song D. , “A New Method for Computing the Upper and Lower Bounds on Frequencies of Structures with Interval Parameters”, Mechanics Research Communications, Vol.22, No.5, pp.431-439, 1995.
18. Chessa J., and Belytschko T., “An Extended Finite Element Method for Two-Phase Fluids”, J. Appl. Mech., 70(1):10– 17, 2003.
19. Cheushov I. D., “Introduction to the Theory of Infinite-Dimensional Dissipative Systems”, ACTA Scientific Publishing House, 2002, Page 11.
20. Collins J.D., Hart G.C., Hasselman T.K., and Kennedy B., “Statistical Identification of Structures”, AIAA Journal, Vol. 12, No. 2 (1974), pp. 185-190, 1974.
21. Corliss G., Foley C., and Kearfott R. B., “Formulation for reliable analysis of structural frames”, In: Muhanna, R. L., and Mullen R. L. (eds.), Proc. NSF Workshop on Reliable Engineering Computing, Savannah, Georgia, 2004.
22. Dempster A. P., “Upper and lower probabilities induced by a multi-valued mapping”. Ann. Mat. Stat. 38:325–339, 1967.
23. Desmet W., Sas P., Vandepitte D., “An indirect Treftz method for the steady-state dynamic analysis of coupled vibro-acoustic systems”, K.U. Leuven,
Department of Mechanical Engineering, Division PMA, Celestijnenlaan 300 B, B-3001 Leuven, Belgium
24. Desmet W., Sas P., Vandepitte D., “On the numerical properties of a wave based prediction technique for coupled vibro-acoustic analysis”, Department of Mechanical Engineering, Division PMA, K.U.Leuven, Belgium, ISMA, 2003.
25. Dessombz O., Thouverez F., Laıne J.P., and Jezequel L., “Analysis of mechanical systems using interval computations applied to finite elements methods”, J. Sound. Vib., 238(5):949–968, 2001.
26. Dimarogonas A.D., “Interval Analysis of Vibrating Systems”, Journal of Sound and Vibration, Vol.183, No.4, pp.739-749 (1995), 1995.
27. Elishakoff I., and Ren Y., “The Bird’s Eye View on Finite Element Method for Stochastic Structures”, Computer Methods in Applied Mechanics and Engineering, 168:51–61, 1999.
28. Elishakoff I., Duan D., “Application of Mathematical Theory of Interval Analysis to Uncertain Vibrations”, Proc. of NOISE-CON 94, Ft. Lauderdale, Florida, pp.519-524, 1994.
29. Elishakoff I., Ren Y.J., “Large variation finite element method for stochastic problems”, Oxford University Press, Oxford, UK, 2003.
30. Eric Vanden-Eijnden, “Introduction to regular perturbation theory”,
Introduction to Math Modeling, http://www.cims.nyu.edu/~eve2/reg_pert.pdf 31. Etman, L.F.P., “Design and analysis of computer experiments: The method of
Sacks et al.”, Engineering Mechanics report WFW 94.098, Eindhoven University of Technology, 1994.
32. Friswell M. I., Prells U., Penny J. E. T., “Determining Uncertainty bounds for Eigen-values”, http://michael.friswell.com/PDF_Files/C186.pdf
33. Friswell M.I., Mottershead J.E., “Finite Element Model Updating in Structural Dynamics”, Kluwer Academic Press, 1995, Dordrecht.
34. Guan J. W., Bell D. A., “A Generalization of the Dempster-Shafer Theory”, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.798443.pdf 35. Guyader J. L., Parizet E., “Uncertainty of vibroacoustic behaviour of
industrially identical structures. A new challenge for structural acoustic people”, Fifth International Congress on Sound and Vibration, Adelaide, Australia, Specialist keynote paper, 1997.
36. Haddad Khodaparast H., Mottershead J. E., Badcock K. J., “Interval model updating in structural dynamics”, Mechanical Systems and Signal Processing, 2010.
37. Haddad Khodaparast H., Mottershead J.E., “Efficient Methods in Stochastic Model Updating”, in Proceedings of International Conference on Noise and Vibration, ISMA2008, 2008 September 27-30, Leuven, Belgium (2008), pp.
1855-1869.
38. Haddad Khodaparast H., Mottershead J.E., Badcock K.J., “Interval Model Updating: Method and Application”, University of Liverpool, School of Engineering, Liverpool, United Kingdom,
www.cfd4aircraft.com/research_themes/modelupdating/paper140.pdf 39. Haddad Khodaparast H., Mottershead J.E., Friswell M.I., “Perturbation
methods for the estimation of parameter variability in stochastic model
updating”, Mechanical Systems and Signal Processing, Vol. 22, No. 8 (2008), pp. 1751-1773.
40. Hansen E., “Global Optimization Using Interval Analysis”, Marcel Dekker, Inc., New York, 1992.
41. Höhn W., and Mittelmann H. D., “Some Remarks on the Discrete Maximum Principle for Finite Elements of Higher- Order”, Computing, 27:145–154, 1981.
42. Indrajit Chowdhury, Shambhu P. Das Gupta, “Dynamics of Structure and Foundation – A unified Approach”, Vol.1&2, CRC Press, Taylor & Francis Group, London, UK, 2009.
43. IS 456: 2000, “Plain and Reinforced Concrete-Code of Practice”, Fourth Revision, Bureau of Indian Standards.
44. IS: 875 (Part 1) – 1987, “Code of Practice for Design Loads (Other than Earthquake) for Buildings and Structures, Part 1 Dead Loads — Unit Weights of Building Materials and Stored Materials”, Second Revision, Bureau of Indian Standards.
45. IS: 875 (Part 2) – 1997, “Code of Practice for Design Loads (Other than Earthquake) for Buildings and Structures, Part 2 Imposed Loads”, Second Revision, Bureau of Indian Standards.
46. IS1893 (Part 1): 2002,”Criteria for Earthquake Resistant Design of
Structures, Part 1 General Provisions and Buildings”, Fifth Revision, Bureau of Indian Standards.
47. Iwona Skalna, M.V. Rama Rao, Andrzej Pownuk, “Systems of fuzzy equations in structural mechanics”, The University of Texas at El Paso, Texas, Research Report No. 2007-01, http://www.math.utep.edu/preprints/2007/2007-01.pdf 48. José Fonseca, Cris Mares, Michael Friswell, John Mottershead, “Review of
Parameter Uncertainty Propagation Methods in Structural Dynamic Analysis”, http://michael.friswell.com/PDF_Files/C149.pdf
49. José R. Fonseca, Cris Mares, Michael I. Friswell, John E. Mottershead, “The Propagation of Parameter Uncertainty through Structural Dynamics Models”, http://michael.friswell.com/PDF_Files/C159.pdf
50. Karátson J., and Korotov S., “Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions”, Numer. Math. 99:669–698, 2005.
51. Klimke Andreas W., “Uncertainty Modeling using Fuzzy Arithmetic and Sparse Grids”, Doctoral Thesis, Submitted to University of Stuttgart, Germany, 2006.
52. Lophaven S. N, Nielsen H. B, Sndergaard J., DACE, “A MATLAB Kriging Toolbox”, Technical Report I, Technical University of Denmark, 2002, MM-TR-2002-12, DK-2800 Kgs. Lyngby Denmark.
53. Maglaras G., Nikolaidis E.,Haftka R.T. and Cudney H.H., “Analytical-experimental comparison of probabilistic methods and fuzzy set based
methods for designing under uncertainty”, Structural Optimization 13, 69-80, Springer-Verlag, 1997.
54. Manohar CS, Gupta S, “Modeling and Evaluation of Structural reliability:
current status and future directions”, Department of Civil engineering, IISc, Bangalore, India.
55. Marco Prandina, “Modelling of Structural Damping”, CFD Method
Development and Application for Computational Aero-elasticity and Flight Dynamics, http://www.cfd4aircraft.com/
56. McWilliam S., “Anti-optimisation of uncertain structures using interval analysis”, Comput. Struct., 79:421–430, 2000.
57. Mehdi Modares, Mullen Robert L.,“Static Analysis of Uncertain Structures Using Interval Eigenvalue Decomposition”, REC, 2008.
58. Mehdi Modarreszadeh, “Dynamic analysis of structures with interval
uncertainty”, Doctoral Thesis, Case Western Reserve University, Ohio, 2005.
59. Michael I. Friswell, José R. Fonseca, John E. Mottershead, Arthur W. Lees,
“Quantification of Uncertainties Using Inverse Methods”, http://michael.friswell.com/PDF_Files/C178.pdf
60. Michael Spiegel, Ross Gore, Paul F. Reynolds, “Quantifying and analyzing uncertainty in simulations to enable user understanding”, In Proceedings of the Modeling, Simulation, & Gaming Student Capstone Conference, Suffolk, VA, 2008. The Virginia Modeling, Analysis, & Simulation Center (VMASC).
61. Moens D., Vandipitte D., “Non-probabilistic approaches for non-deterministic dynamic FE analysis of imprecisely defined structures”, PROCEEDINGS OF ISMA2004.
62. Möller B., Graf W., and Beer M., “Fuzzy structural analysis using α level-optimization”, Comput. Mech., 26(6):547–565, 2000.
63. Moore R. E., “Methods and Applications of Interval Analysis”, SIAM, Philadelphia, 1979.
64. Mottershead J.E., Friswell M.I., “Model updating in structural dynamics: a survey”, Journal of Sound and Vibration, Vol. 167, No. 2 (1993), pp. 609-625.
65. Muhanna Rafi, Kreinovich Vladik, Solin Pavel, Chessa Jack, Araiza Roberto and Xiang Gang, "Interval Finite Element Methods: New Directions",
Departmental Technical Reports (CS). Paper 130.http://digitalcommons.utep.edu/cs_techrep/130
66. Muhanna R. L. and Mullen R. L., “Development of interval based methods for fuzziness in continuum mechanics”, In: Proc. ISUMA-NAFIPS’95, 1995, pp.
23–45.
67. Muhanna R. L. and Mullen R. L., “Formulation of fuzzy finite element methods for mechanics problems”, Compu. Aided Civ. Infrastruct. Engrg., 14:107–117, 1999.
68. Muhanna R. L. and Mullen R. L., “Uncertainty in mechanics problems-interval-based approach”, J. Engrg. Mech. 127(6):557–566, 2001.
69. Muhanna R. L., Mullen R. L., and Zhang H., “Penalty-Based Solution for the Interval Finite-Element Methods”, ASCE, Engineering Mechanics,
131(10):1102–1111, 2005.
70. Mullen R. L. and Muhanna R. L., “Bounds of structural response for all possible loadings”, J. Struct. Engrg., ASCE, 125(1):98–106, 1999.
71. Mullen R. L. and Muhanna R. L., “Structural analysis with fuzzy-based load uncertainty”, In: Proc. 7th ASCE EMD/STD Joint Spec. Conf. on
Probabilistic Mech. and Struct. Reliability. Mass., 1996, pp. 310–313.
72. Neumaier A. and Pownuk A., “Linear systems with large uncertainties, with applications to truss structures”, Reliable Computing.
73. Nirjhar Dhang, “Structural Dynamics- An Overview”, Chapter 12, SE101 (Lecture Notes), Department of Civil Engineering, IIT, Kharagpur.
74. Online References, http://hpfem.org/~pavel/public/proceed.html
75. Pantelides C. P. and Ganzerli S., “Comparison of fuzzy set and convex model theories in structural design”, Mech. Systems Signal Process, 15(3):499–511, 2001.
76. Paszynski M., Kurtz J., Demkowicz L., “Parallel, Fully Automatic hp-Adaptive 2D Finite Element Package”, the University of Texas at Austin, TICAM Report 04-07, 2004.
77. Popova E. D., Datcheva M., Iankov R., and Schanz T., “Mechanical models with interval parameters”, In K. Gürlebeck L. Hempel C. Könke (Eds.) IKM2003: Digital Proc. of 16th Intern. Conf. on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering, ISSN 1611-4086.
78. Prells U., Friswell M. I., “A Different View on Parametric Modelling”, http://michael.friswell.com/PDF_Files/C141.pdf
79. Qui Z., Wang X., Friswell M.I., “Eigen-value bounds of structures with uncertain-but-bounded parameters”, Journal of Sound and Vibration, Vol.
282, No. 1-2 (2005), pp. 297-312.
80. Rama Rao M. V. and Ramesh Reddy R., “Analysis of Cable-stayed bridge with multiple uncertainties – A fuzzy finite element approach”, Journal of Structural Engineering, Vol.33, NO.6, February-March 2007 p.523-525
81. Rama Rao M. V., “Analysis of Cable-stayed Bridges by FuzzyFinite Element Modeling”, Doctoral Thesis, College of Engineering, Osmania University, 2004
82. Rama Rao M. V., Andrzej Pownuk, “Stress distribution in a reinforced concrete flexural member with uncertain structural parameters, part-I”, Research Report No. 2007-5
83. Rama Rao M. V., Vandewalle S., Munck M. De and Moens D., “Dynamic Analysis of a Cable-Stayed Bridge with Uncertain Structural Parameters”, Safety, Reliability and Risk of Structures, Infrastructures and Engineering Systems – Furuta, Frangopol & Shinozuka (eds), 2010, Taylor & Francis Group, London, ISBN 978-0-415-47557-0
84. Ronaldo Fernandes Nunes, “Dynamic Analysis of Non-Deterministic Problems using Fuzzy Set Based Methods”, Doctoral Thesis, University of Compinas, SP – Brazil, 2005.
85. Scott Ferson, Vladik Kreinovich, Janos Hajagos, William Oberkampf and Lev Ginzburg, “Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty”, SAND2007-0939, Newyork, 2007, http://www.ramas.com/intstats.pdf
86. Shahariz Abdul Aziz, Jeyakody Parthiban, “Fuzzy Logic”,
http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/sbaa/report.html 87. Shorter P J, Mace B R, “Stochastic Energy Flow Models”, Department of
Mechanical Engineering, University of Auckland, New Zealand,
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.34.3186&rep=rep1
&type=pdf
88. Solin P., “Partial Differential Equations and the Finite Element Methods”, J.
Wiley & Sons, Hoboken, New Jersey, 2005.
89. Solin P., Vejchodsky T., and R. Araiza, “Discrete Conservation of
Nonnegativity or Elliptic Problems Solved by the hp-FEM”, University of Texas at El Paso, Department of Computer Science, Technical Report UTEP-CS-05-29, August 2005, http://www.cs.utep.edu/vladik/2005/tr05-29.pdf 90. Sondipon Adhikari, “A non-parametric approach for uncertainty
quantification in elastodynamics”, 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Newport, Rhode Island, USA, May 2006.
91. Sondipon Adhikari, Michael I. Friswell1, and Kuldeep P. Lonkar,
“Uncertainty in Structural Dynamics: Experimental Case Studies on Beams and Plates”, ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, 2007.
92. Štemberk P., Kruis J., “Fuzzy Dynamic Analysis of a 2D Frame”, Acta Polytechnica, Vol. 44 No. 5–6/2004,
http://ctn.cvut.cz/ap/download.php?id=424
93. Stephen Schrader, William M. Riggs and Robert P. Smith, “Choice over Uncertainty and Ambiguity in Technical Problem Solving”, Working Paper, Alfred P. Sloan School of management, February 1993.
94. Sunaga Teruo, “Theory of an Interval Algebra and its Application to Numerical Analysis”, RAAG Memoirs, Vol.2, 1958.
95. The centre for REC, http://savannah.gatech.edu/rec/publications.html 96. Worden K., Manson G., Lord T. M., Friswell M. I., “Some Thoughts on
Uncertainty Quantification and Propegation”, http://michael.friswell.com/PDF_Files/C162.pdf
97. Young Rosalind Cecily, “The algebra of many-valued quantities”, Mahematische Annalen, Verlag Von Julius Springer, Berlin, 1931.
98. Zadeh L. A., “Fuzzy Sets as a Basis for a Theory of Possibility”, Fuzzy Sets and Systems, 1:3–28, 1978,
http://gen.lib.rus.ec/get?md5=345abc786a9cff62a1d04212493c7d07
99. Zadeh L.A., “Fuzzy sets”, Information and Control, Vol.8, pp.338-353, 1965.
APPENDIX - A
SOFTWARE MODEL ANALYSIS MODE SHAPE OUTPUT Mode Shapes for Model 1(E=24.75GPa, wl = 3.6kN/m2):
Figure A.1.1: 3D Mode Shape 1for Structure Model 1 Figure A.1.2: 3D Mode Shape 2for Structure Model 1
Figure A.1.3: 3D Mode Shape 3for Structure Model 1 Figure A.1.4: 3D Mode Shape 4 for Structure Model 1
Figure A.1.5: 3D Mode Shape 5 for Structure Model 1 Figure A.1.6: 3D Mode Shape 6 for Structure Model 1
Figure A.1.7: 3D Mode Shape 7 for Structure Model 1 Figure A.1.8: 3D Mode Shape 8 for Structure Model 1
Figure A.1.9: 3D Mode Shape 9 for Structure Model 1 Figure A.1.10: 3D Mode Shape 10 for Structure Model 1
Figure A.1.11: 3D Mode Shape 11 for Structure Model 1 Figure A.1.12: 3D Mode Shape 12 for Structure Model 1
Mode Shapes for Model 2(E=24.75GPa, wl = 4.0kN/m2):
Figure A.2.1: 3D Mode Shape 1 for Structure Model 2 Figure A.2.2: 3D Mode Shape 2 for Structure Model 2
Figure A.2.3: 3D Mode Shape 3 for Structure Model 2 Figure A.2.4: 3D Mode Shape 4 for Structure Model 2
Figure A.2.5: 3D Mode Shape 5 for Structure Model 2 Figure A.2.6: 3D Mode Shape 6 for Structure Model 2
Figure A.2.7: 3D Mode Shape 7 for Structure Model 2 Figure A.2.8: 3D Mode Shape 8 for Structure Model 2
Figure A.2.9: 3D Mode Shape 9 for Structure Model 2 Figure A.2.10: 3D Mode Shape 10 for Structure Model 2
Figure A.2.11: 3D Mode Shape 11 for Structure Model 2 Figure A.2.12: 3D Mode Shape 12 for Structure Model 2
Mode Shapes for Model 3(E=24.75GPa, wl = 4.4kN/m2):
Figure A.3.1: 3D Mode Shape 1 for Structure Model 3 Figure A.3.2: 3D Mode Shape 2 for Structure Model 3
Figure A.3.3: 3D Mode Shape 3 for Structure Model 3 Figure A.3.4: 3D Mode Shape 4 for Structure Model 3
Figure A.3.5: 3D Mode Shape 5 for Structure Model 3 Figure A.3.6: 3D Mode Shape 6 for Structure Model 3
Figure A.3.7: 3D Mode Shape 7 for Structure Model 3 Figure A.3.8: 3D Mode Shape 8 for Structure Model 3
Figure A.3.9: 3D Mode Shape 9 for Structure Model 3 Figure A.3.10: 3D Mode Shape 10 for Structure Model 3
Figure A.3.11: 3D Mode Shape 11 for Structure Model 3 Figure A.3.12: 3D Mode Shape 12 for Structure Model 3
Mode Shapes for Model 4(E=25.00GPa, wl = 3.6kN/m2):
Figure A.4.1: 3D Mode Shape 1 for Structure Model 4 Figure A.4.2: 3D Mode Shape 2 for Structure Model 4
Figure A.4.3: 3D Mode Shape 3 for Structure Model 4 Figure A.4.4: 3D Mode Shape 4 for Structure Model 4
Figure A.4.5: 3D Mode Shape 5 for Structure Model 4 Figure A.4.6: 3D Mode Shape 6 for Structure Model 4
Figure A.4.7: 3D Mode Shape 7 for Structure Model 4 Figure A.4.8: 3D Mode Shape 8 for Structure Model 4
Figure A.4.9: 3D Mode Shape 9 for Structure Model 4 Figure A.4.10: 3D Mode Shape 10 for Structure Model 4
Figure A.4.11: 3D Mode Shape 11 for Structure Model 4 Figure A.4.12: 3D Mode Shape 12 for Structure Model 4
Mode Shapes for Model 5(E=25.00GPa, wl = 4.0kN/m2):
Figure A.5.1: 3D Mode Shape 1 for Structure Model 5 Figure A.5.2: 3D Mode Shape 2 for Structure Model 5
Figure A.5.3: 3D Mode Shape 3 for Structure Model 5 Figure A.5.4: 3D Mode Shape 4 for Structure Model 5
Figure A.5.5: 3D Mode Shape 5 for Structure Model 5 Figure A.5.6: 3D Mode Shape 6 for Structure Model 5
Figure A.5.7: 3D Mode Shape 7 for Structure Model 5 Figure A.5.8: 3D Mode Shape 8 for Structure Model 5
Figure A.5.9: 3D Mode Shape 9 for Structure Model 5 Figure A.5.10: 3D Mode Shape 10 for Structure Model 5
Figure A.5.11: 3D Mode Shape 11 for Structure Model 5 Figure A.5.12: 3D Mode Shape 12 for Structure Model 5
Mode Shapes for Model 6(E=25.00GPa, wl = 4.4kN/m2):
Figure A.6.1: 3D Mode Shape 1 for Structure Model 6 Figure A.6.2: 3D Mode Shape 2 for Structure Model 6
Figure A.6.3: 3D Mode Shape 3 for Structure Model 6 Figure A.6.4: 3D Mode Shape 4 for Structure Model 6
Figure A.6.5: 3D Mode Shape 5 for Structure Model 6 Figure A.6.6: 3D Mode Shape 6 for Structure Model 6
Figure A.6.7: 3D Mode Shape 7 for Structure Model 6 Figure A.6.8: 3D Mode Shape 8 for Structure Model 6
Figure A.6.9: 3D Mode Shape 9 for Structure Model 6 Figure A.6.10: 3D Mode Shape 10 for Structure Model 6
Figure A.6.11: 3D Mode Shape 11 for Structure Model 6 Figure A.6.12: 3D Mode Shape 12 for Structure Model 6
Mode Shapes for Model 7(E=25.25GPa, wl = 3.6kN/m2):
Figure A.7.1: 3D Mode Shape 1 for Structure Model 7 Figure A.7.2: 3D Mode Shape 2 for Structure Model 7
Figure A.7.3: 3D Mode Shape 3 for Structure Model 7 Figure A.7.4: 3D Mode Shape 4 for Structure Model 7
Figure A.7.5: 3D Mode Shape 5 for Structure Model 7 Figure A.7.6: 3D Mode Shape 6 for Structure Model 7
Figure A.7.7: 3D Mode Shape 7 for Structure Model 7 Figure A.7.8: 3D Mode Shape 8 for Structure Model 7
Figure A.7.9: 3D Mode Shape 9 for Structure Model 7 Figure A.7.10: 3D Mode Shape 10 for Structure Model 7
Figure A.7.11: 3D Mode Shape 11 for Structure Model 7 Figure A.7.12: 3D Mode Shape 12 for Structure Model 7
Mode Shapes for Model 8(E=25.25GPa, wl = 4.0kN/m2):
Figure A.8.1: 3D Mode Shape 1 for Structure Model 8 Figure A.8.2: 3D Mode Shape 2 for Structure Model 8
Figure A.8.3: 3D Mode Shape 3 for Structure Model 8 Figure A.8.4: 3D Mode Shape 4 for Structure Model 8
Figure A.8.5: 3D Mode Shape 5 for Structure Model 8 Figure A.8.6: 3D Mode Shape 6 for Structure Model 8
Figure A.8.7: 3D Mode Shape 7 for Structure Model 8 Figure A.8.8: 3D Mode Shape 8 for Structure Model 8
Figure A.8.9: 3D Mode Shape 9 for Structure Model 8 Figure A.8.10: 3D Mode Shape 10 for Structure Model 8
Figure A.8.11: 3D Mode Shape 11 for Structure Model 8 Figure A.8.12: 3D Mode Shape 12 for Structure Model 8
Mode Shapes for Model 9(E=25.25GPa, wl = 4.4kN/m2):
Figure A.9.1: 3D Mode Shape 1 for Structure Model 9 Figure A.9.2: 3D Mode Shape 2 for Structure Model 9
Figure A.9.3: 3D Mode Shape 3 for Structure Model 9 Figure A.9.4: 3D Mode Shape 4 for Structure Model 9
Figure A.9.5: 3D Mode Shape 5 for Structure Model 9 Figure A.9.6: 3D Mode Shape 6 for Structure Model 9
Figure A.9.7: 3D Mode Shape 7 for Structure Model 9 Figure A.9.8: 3D Mode Shape 8 for Structure Model 9
Figure A.9.9: 3D Mode Shape 9 for Structure Model 9 Figure A.9.10: 3D Mode Shape 10 for Structure Model 9
Figure A.9.11: 3D Mode Shape 11 for Structure Model 9 Figure A.9.12: 3D Mode Shape 12 for Structure Model 9