تحلیل خمش ورق‌های کامپوزیت لایه‌ای غیرهمگن در صفحه با استفاده از توابع پایه متعادل شده بر مبنای تئوری تغییر شکل برشی مرتبه اول
نویسندگان
دانشکده مهندسی عمران، دانشگاه صنعتی اصفهان، اصفهان
چکیده
در این مقاله تحلیل خمش ورق کامپوزیت لایه‌ای غیرهمگن در صفحه به‌صورت عددی مورد بررسی قرار گرفته است. با توجه به ضخامت نسبتاً زیاد، از تئوری میندلین که تغییر شکل برشی در ضخامت را به‌صورت خطی در نظر می‌گیرد استفاده می‌شود. معادله دیفرانسیل حاکم بر تعادل مسئله به‌صورت انتگرال وزنی ارضاء می‌شود. توابع پایه برای تخمین پاسخ، چند جمله‌ای‌های چبی‌شف نوع اول بوده و وزن‌های مورد استفاده نیز از جنس توابع نمایی هستند. با توسعه فرمول‌بندی در یک ناحیه مجازی مستطیلی در برگیرنده سطح ورق، امکان محاسبه انتگرال وزنی به‌صورت ترکیب خطی تعدادی انتگرال یک‌بعدی و نرمال شده وجود دارد که سرعت عملیات را بسیار بالا می‌برد. به‌منظور صحت‌سنجی روش ارائه شده، مثال‌هایی از ورق کامپوزیت لایه‌ای همگن و ناهمگن با انواع جهت‌گیری الیاف و شرایط تکیه‌گاهی مورد بررسی قرار گرفته است. نتایج به‌دست آمده با جواب‌های حاصل از روش‌های تحلیلی و نیز حل عددی از نرم‌افزارهای تجاری تطابق خوبی دارد که کارایی روش پیشنهادی را نشان می‌دهد.
کلیدواژه‌ها
عنوان مقاله [English]
Static Analysis of in-Plane Heterogeneous Laminated Composite Plates Using Equilibrated Basis Functions Based on FSDT
نویسندگان [English]
- M. Azizpooryan
- N. Noormohammadi
In this paper, static analysis of in-plane heterogeneous laminated composite plates is numerically studied. The Mindlin’s theory which considers linear transverse shear deformation has been implemented. The governing partial differential equation is satisfied by a weighted residual integration. Chebyshev polynomials of the first kind are used as basis functions and exponential functions make up the weight functions of the integration. The emerging integrals may be composed of some pre-evaluated 1D normalized ones, which effectively paces up the solution progress. To verify the method, several examples of homogeneous as well as heterogeneous plates with various lamination schemes and boundary conditions have been solved. Results are compared with those from the literature or by commercial codes, which reveal excellent accuracy of the proposed method.
کلیدواژه‌ها [English]
- Equilibrated basis functions
- Moderately thick plate
- Heterogeneous
- composite
- Chebyshev
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2. Atluri, S. N., and Zhu, T., “A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics”, Computational Mechanics, Vol. 22, No. 2, pp. 117-127, 1998.
3. Trefftz, T., “Ein Gegenstuck Zum Ritzschen Verfahren”, Proceedings of 2nd International Congress on Applied Mechanics, Zurich, 1926.
4. Kupradze, V. D. and Aleksidze, M. A., “The Method of Functional Equations for the Approximate Solution of Certain Boundary Value Problems”, USSR Computational Mathematics and Mathematical Physics, Vol. 4, No .4, pp. 82-126, 1964.
5. Soghrati, S., “Implementation of Smooth Fundamental Solutions in Solving Some Governing Differential Equations in Solid Mechanics”, M.Sc. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2007. (in Persian).
6. Boroomand, B., Soghrati, S. and Movahedian, B., “Exponential Basis Functions in Solution of Static and Time Harmonic Elastic Problems in a Meshless Style”, International Journal for Numerical Methods in Engineering, Vol. 81, No. 8, pp. 971-1018, 2010.
7. Noormohammadi, N., “Solution of Solid Mechanics Problems Using Equilibrated Basis Functions and Mesh-Free Methods”, Ph.D. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2015. (in Persian).
8. Boroomand, B. and Noormohammadi, N., “Weakly Equilibrated Basis Functions for Elasticity Problems”, Engineering Analysis with Boundary Elements, Vol. 37, No. 12, pp. 1712-1727, 2013.
9. Bert, C. W., and Chen T. L. C., “Effect of Shear Deformation on Vibration of Antisymmetric Angle-Ply Laminated Rectangular Plates”, International Journal of Solids and Structures, Vol. 14, No. 6, pp. 465-473, 1978.
10. Swaminathan, K., and Ragounadin, D., “Analytical Solutions Using a Higher-Order Refined Theory for the Static Analysis of Antisymmetric Angle-Ply Composite and Sandwich Plates”, Composite Structures, Vol. 64, No. 3, pp. 405-417, 2004.
11. Reddy, J. N., Khdeir, A. A., and Librescu, L., “Le´vy Type Solutions for Symmetrically Laminated Rectangular Plates Using First-Order Shear Deformation Theory”, Journal of Applied Mechanics, Vol. 54, No. 3, pp. 740-742, 1987.
12. Khdeir, A. A., and Reddy, J. N., “Analytical Solutions of Refined Plate Theories of Cross-Ply Composite Laminates”, Journal of Pressure Vessels Technology, Vol. 113, No. 4, pp. 570-578, 1991.
13. Yuemei, L. and Rui, L., “Accurate Bending Analysis of Rectangular Plates with Two Adjacent Edges Free and the Others Clamped or Simply Supported Based on New Symplectic Approach”, Applied Mathematical Modeling, Vol.34, No.4, pp.856-865, 2010.
14. Urthaler, Y. and Reddy, J. N., “A Mixed Finite Element for the Nonlinear Bending Analysis of Laminated Composite Plates Based on FSDT”, Mechanics of Advanced Materials and Structures, Vol. 15, No. 5, pp. 355-354, 2008.
15. Bhar, A., Phoenix, S. S., and Satsangi, S. K., “Finite Element Analysis of Laminated Composite Stiffened Plates Using FSDT and HSDT: A Comparative Perspective”, Composite Structures, Vol. 92, No. 2, pp. 312-321, 2010.
16. Sladek, J., Sladek, V., Zhang, Ch., Krivacek, J., Wen, P. H., “Analysis of Orthotropic Thick Plates by Meshless Local Petrov–Galerkin (MLPG) Method.” International Journal for Numerical Methods in Engineering, Vol. 67, No. 13, pp. 1830-1850, 2006.
17. Jaberzadeh, E., Azhari, M., and Boroomand, B., “Free Vibration of Moving Laminated Composite Plates with and Without Skew Roller Using the Element-Free Galerkin Method”, Iranian Journal of Science and Technology: Transactions of Civil Engineering, Vol. 38, pp. 377-393, 2014.
18. Shahbazi, M., Boroomand, B., and Soghrati, S., “A Mesh-Free Method Using Exponential Basis Functions for Laminates Modeled by CLPT, FSDT and TSDT–Part I: Formulation.” Composite Structures, Vol. 93, No. 12, pp. 3112-3119, 2011.
19. Azhari, F., Boroomand, B., and Shahbazi, M., “Exponential Basis Functions in the Solution of Laminated Plates Using a Higher-Order Zig–Zag Theory.” Composite Structures, Vol. 105, pp. 398-407, 2013.
20. Motamedi Ghahfarokhi, A., “On Bending Problem of Laminated Composite Plates Using Exponential Basis Functions in Mesh-Less Local Form”, M.Sc. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2013 (in Persian).
21. Noormohammadi, N., and Boroomand, B., “A Fictitious Domain Method Using Equilibrated Basis Functions for Harmonic and Bi-Harmonic Problems in Physics”, Journal of Computational Physics, Vol. 272, pp. 189-217, 2014.
22. Dawe, D. J., and Roufaeil, O. L., “Rayleigh-Ritz Vibration Analysis of Mindlin Plates”, Journal of Sound and Vibration, Vol. 69, No. 3, pp. 345-359, 1980.
23. Yuhua, T., and Wang, X., “Buckling of Symmetrically Laminated Rectangular Plates Under Parabolic Edge Compressions”, International Journal of Mechanical Sciences, Vol. 53, No. 2, pp. 91-97, 2011.
24. Reddy, J. N., Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC press, 2003.
2. Atluri, S. N., and Zhu, T., “A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics”, Computational Mechanics, Vol. 22, No. 2, pp. 117-127, 1998.
3. Trefftz, T., “Ein Gegenstuck Zum Ritzschen Verfahren”, Proceedings of 2nd International Congress on Applied Mechanics, Zurich, 1926.
4. Kupradze, V. D. and Aleksidze, M. A., “The Method of Functional Equations for the Approximate Solution of Certain Boundary Value Problems”, USSR Computational Mathematics and Mathematical Physics, Vol. 4, No .4, pp. 82-126, 1964.
5. Soghrati, S., “Implementation of Smooth Fundamental Solutions in Solving Some Governing Differential Equations in Solid Mechanics”, M.Sc. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2007. (in Persian).
6. Boroomand, B., Soghrati, S. and Movahedian, B., “Exponential Basis Functions in Solution of Static and Time Harmonic Elastic Problems in a Meshless Style”, International Journal for Numerical Methods in Engineering, Vol. 81, No. 8, pp. 971-1018, 2010.
7. Noormohammadi, N., “Solution of Solid Mechanics Problems Using Equilibrated Basis Functions and Mesh-Free Methods”, Ph.D. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2015. (in Persian).
8. Boroomand, B. and Noormohammadi, N., “Weakly Equilibrated Basis Functions for Elasticity Problems”, Engineering Analysis with Boundary Elements, Vol. 37, No. 12, pp. 1712-1727, 2013.
9. Bert, C. W., and Chen T. L. C., “Effect of Shear Deformation on Vibration of Antisymmetric Angle-Ply Laminated Rectangular Plates”, International Journal of Solids and Structures, Vol. 14, No. 6, pp. 465-473, 1978.
10. Swaminathan, K., and Ragounadin, D., “Analytical Solutions Using a Higher-Order Refined Theory for the Static Analysis of Antisymmetric Angle-Ply Composite and Sandwich Plates”, Composite Structures, Vol. 64, No. 3, pp. 405-417, 2004.
11. Reddy, J. N., Khdeir, A. A., and Librescu, L., “Le´vy Type Solutions for Symmetrically Laminated Rectangular Plates Using First-Order Shear Deformation Theory”, Journal of Applied Mechanics, Vol. 54, No. 3, pp. 740-742, 1987.
12. Khdeir, A. A., and Reddy, J. N., “Analytical Solutions of Refined Plate Theories of Cross-Ply Composite Laminates”, Journal of Pressure Vessels Technology, Vol. 113, No. 4, pp. 570-578, 1991.
13. Yuemei, L. and Rui, L., “Accurate Bending Analysis of Rectangular Plates with Two Adjacent Edges Free and the Others Clamped or Simply Supported Based on New Symplectic Approach”, Applied Mathematical Modeling, Vol.34, No.4, pp.856-865, 2010.
14. Urthaler, Y. and Reddy, J. N., “A Mixed Finite Element for the Nonlinear Bending Analysis of Laminated Composite Plates Based on FSDT”, Mechanics of Advanced Materials and Structures, Vol. 15, No. 5, pp. 355-354, 2008.
15. Bhar, A., Phoenix, S. S., and Satsangi, S. K., “Finite Element Analysis of Laminated Composite Stiffened Plates Using FSDT and HSDT: A Comparative Perspective”, Composite Structures, Vol. 92, No. 2, pp. 312-321, 2010.
16. Sladek, J., Sladek, V., Zhang, Ch., Krivacek, J., Wen, P. H., “Analysis of Orthotropic Thick Plates by Meshless Local Petrov–Galerkin (MLPG) Method.” International Journal for Numerical Methods in Engineering, Vol. 67, No. 13, pp. 1830-1850, 2006.
17. Jaberzadeh, E., Azhari, M., and Boroomand, B., “Free Vibration of Moving Laminated Composite Plates with and Without Skew Roller Using the Element-Free Galerkin Method”, Iranian Journal of Science and Technology: Transactions of Civil Engineering, Vol. 38, pp. 377-393, 2014.
18. Shahbazi, M., Boroomand, B., and Soghrati, S., “A Mesh-Free Method Using Exponential Basis Functions for Laminates Modeled by CLPT, FSDT and TSDT–Part I: Formulation.” Composite Structures, Vol. 93, No. 12, pp. 3112-3119, 2011.
19. Azhari, F., Boroomand, B., and Shahbazi, M., “Exponential Basis Functions in the Solution of Laminated Plates Using a Higher-Order Zig–Zag Theory.” Composite Structures, Vol. 105, pp. 398-407, 2013.
20. Motamedi Ghahfarokhi, A., “On Bending Problem of Laminated Composite Plates Using Exponential Basis Functions in Mesh-Less Local Form”, M.Sc. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2013 (in Persian).
21. Noormohammadi, N., and Boroomand, B., “A Fictitious Domain Method Using Equilibrated Basis Functions for Harmonic and Bi-Harmonic Problems in Physics”, Journal of Computational Physics, Vol. 272, pp. 189-217, 2014.
22. Dawe, D. J., and Roufaeil, O. L., “Rayleigh-Ritz Vibration Analysis of Mindlin Plates”, Journal of Sound and Vibration, Vol. 69, No. 3, pp. 345-359, 1980.
23. Yuhua, T., and Wang, X., “Buckling of Symmetrically Laminated Rectangular Plates Under Parabolic Edge Compressions”, International Journal of Mechanical Sciences, Vol. 53, No. 2, pp. 91-97, 2011.
24. Reddy, J. N., Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC press, 2003.
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