تحلیل کمانشی ستون‌های نامنشوری با بار محوری غیریکنواخت با روش بدون شبکه پتروف گالرکین
نویسندگان
گروه مهندسی عمران، دانشکده مهندسی، دانشگاه صنعتی قوچان، قوچان
چکیده
استفاده از اعضای سازهای نامنشوری برای بهینه‌سازی هزینه و مقاومت در حال افزایش است. در شرایط یکسان، مقاطع اعضا در سازه‌های فولادی نازک‌تر از سازه‌های بتنی است، بنابراین تحلیل کمانشی در سازه‌های فولادی از اهمیت بیشتری برخوردار است. تعیین بار بحرانی ستون‌هایی که ممان اینرسی در طول آنها متغیر است با استفاده از روش تحلیلی، زمان زیادی از کاربر می‌گیرد. در این مقاله تحلیل کمانشی ستون‌های نامنشوری فولادی با ممان اینرسی متغیر در طول ستون با بهره‌جویی از روش بدون شبکه پتروف-گالرکین انجام شده است. در این روش نیازی به شبکه‌بندی مسئله نیست و با توزیع تعدادی گره به صورت منظم یا نامنظم، دامنه و مرزهای مسئله مشخص می‌شود. تغییرات ممان اینرسی در طول ستون با استفاده از یک تابع توانی با ضریب نامنشوری و توان متغیر مدل‌سازی شده است. برای گسسته‌سازی معادلات دیفرانسیل حاکم بر رفتار کمانشی ستون‌ها، از تابع درونیاب شعاعی و برای برقراری شرایط مرزی ستون از روش جریمه (پنالتی) استفاده شده است. مقایسه نتایج عددی روش پیشنهادی با نتایج روش‌های تحلیلی نشان می‌دهد که روش بدون شبکه در محاسبه بار کمانشی ستون‌های نامنشوری، از دقت بسیار بالایی برخوردار است. در ادامه، به کمانش ستون‌های ماهیچه‌ای با بار محوری متغیر در طول ستون به‌عنوان یک نمونه ستون پرکاربرد در مهندسی عمران برای نشان دادن کارایی روش پیشنهادی پرداخته شده است. در پایان، اثر پارامترهای مختلف از جمله غیریکنواخت بودن بار و نحوه تغییرات تدریجی ممان اینرسی در طول ستون، روی بار کمانشی ستون بررسی شده است.
کلیدواژه‌ها
عنوان مقاله [English]
Buckling Analysis of Non-Prismatic Columns Subjected to Non-Uniform Loading Using the Meshless Local Petrov-Galerkin Method
نویسندگان [English]
- F. F. Heidargheitaghi
- M. H. M. H. Ghadiri Rad
- M. Kazemi
Continuously varying cross-section members have found wide applications in engineering for cost and resistance optimization. Since steel structures generally have more slender members compared to concrete structures, buckling analysis of steel members is of more importance. Determining the critical load of functionally varying cross-section columns using the analytical solution is a time-consuming process. In this paper, buckling analysis of non-prismatic steel columns is conducted using the meshless local Petrov-Galerkin (MLPG) method. In meshless methods, the scattered nodes are used rather than the elements to model the problem domain and its boundaries. The change of the inertia moment within the length of a column is characterized by introducing a power function with variable taper ratio and exponent. The radial basis function is used to discretize the differential equation governing the buckling. The penalty method is used for the imposition of the boundary conditions. Numerical examples of the critical buckling load for prismatic and non-prismatic columns using the proposed method are compared with the analytical solution, and the effectiveness of the MLPG method for buckling analysis of non-prismatic columns is validated. Also, buckling analysis of muscle column members subjected to non-uniform axial load is carried out to show the efficiency of the proposed method. The effect of several parameters such as non-uniformity of the load and variation of the cross-section on the buckling load of the column is discussed in details.
کلیدواژه‌ها [English]
- Non-Prismatic Columns
- Meshless Local Petrov-Galerkin (MLPG) Method
- Buckling
- Radial Basis Function
- Eigen-Value
1. Szymczak, C., and Kujawa, M., “Flexural Buckling and Post-Buckling of Columns Made of Aluminium Alloy”, European Journal of Mechanics, Vol. 73, pp. 420-429, 2019.
2. Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, Courier Corporation, United States, 2009.
3. Arani, A. J., and Kolahchi, R., “Buckling Analysis of Embedded Concrete Columns Armed with Carbon Nanotubes”, Computers and Concrete, Vol. 17, No. 5, pp. 567-578, 2016.
4. Iremonger, M., “Finite Difference Buckling Analysis of Non-Uniform Columns”, Computers & Structures, Vol. 12, No. 5, pp. 741-748, 1980.
5. Ermopoulos, J. C., and Kounadis, A. N., “Stability of Frames with Tapered Built-Up Members”, Journal of Structural Engineering, Vol. 111, pp. 1979-1992, 1985.
6. Smith, W.G., “Analytic Solutions for Tapered Column Buckling”, Computers & structures, Vol. 28, pp. 677-681, 1988.
7. Williams, F.W., and Aston, G., “Exact or Lower Bound Tapered Column Buckling Loads”, Journal of Structural Engineering, Vol. 115, pp. 1088-1100, 1989.
8. Arbabi, F., and Li, F., “Buckling of Variable Cross-Section Columns: Integral-Equation Approach”, Journal of Structural Engineering, Vol. 117, pp. 2426-2441, 1991.
9. Siginer, A., “Buckling of Columns of Variable Flexural Rigidity”, Journal of Engineering Mechanics, Vol. 118, pp. 640-643, 1992.
10. Al-Gahtani, H. J., “Exact Stiffnesses for Tapered Members”, Journal of structural Engineering, Vol. 122, No. 10, pp. 1234-1239, 1996.
11. Bazeos, N., and Karabalis, D. L, “Efficient Computation of Buckling Loads for Plane Steel Frames with Tapered Members”, Engineering Structures, Vol. 28, pp. 771-775. 2006.
12. Raftoyiannis, I., Stamatopoulos, G. N., and Ermopoulos, J., “Buckling Behaviour of Doubly-Tapered Steel Columns Under Axial Compression and Biaxial Bending”, Proceedings of the International Colloquium on Stability and Ductility of Steel Structures, SDSS 2006, pp. 331-338, 2006.
13. Singh, K. V., and Li, G., “Buckling of Functionally Graded and Elastically Restrained Non-Uniform Columns”, Composites Part B: Engineering, Vol. 40, pp. 393-403, 2009.
14. Coşkun, S. B., and Atay, M. T., “Determination of Critical Buckling Load for Elastic Columns of Constant and Variable Cross-Sections Using Variational Iteration Method”, Computers & Mathematics with Applications, Vol. 58, pp. 2260-2266, 2009.
15. 15. Darbandi, S., Firouz-Abadi, R., and Haddadpour, H., “Buckling of Variable Section Columns Under Axial Loading”, Journal of Engineering Mechanics, Vol. 136, pp. 472-476, 2010.
16. 16. Wei, D. J., Yan, S. X., Zhang, Z. P., and Li, X. F., “Critical Load for Buckling of Non-Prismatic Columns Under Self-Weight and Tip Force”, Mechanics Research Communications, Vol. 37, pp. 554-558, 2010.
17. Serna, M., Ibáñez, J., and López, A., “Elastic Flexural Buckling of Non-Uniform Members: Closed-Form Expression and Equivalent Load Approach”, Journal of Constructional Steel Research, Vol. 67, pp. 1078-1085, 2011.
18. Huang, Y., and Luo, Q., “A simple Method to Determine the Critical Buckling Loads for Axially Inhomogeneous Beams with Elastic Restraint”, Computers & Mathematics with Applications, Vol. 61, pp. 2510-2517, 2011.
19. Jamali, S., “Determination of Critical Buckling Force for Elastic Columns with Constant Cross-Sectional Area and Variable Using Repetitive Change Method (VIM) ”, 3rd National Conference on Structures and Steel and First National Conference on Light Steel Structures (LSF), 2012.
20. Pinarbasi, S., “Buckling Analysis of Nonuniform Columns with Elastic End Restraints”, Journal of Mechanics of Materials and Structures, Vol. 7. pp. 485-507, 2012.
21. Taha, M., and Essam, M., “Stability Behavior and Free Vibration of Tapered Columns with Elastic End Restraints Using The DQM Method”, Ain Shams Engineering Journal, Vol. 4, pp. 515-521, 2013.
22. Zhang, B., Guo, Y., and Dou, C., “Ultimate Bearing Capacity of Asymmetrically Double Tapered Steel Columns with Tubular Cross-Section”, Journal of Constructional Steel Research, Vol. 89, pp. 52-62, 2013.
23. Rezaiee-Pajand, M., Shahabian, F., and Bambaeechee, M., “Stability of Semi-Rigid Portal Frames with Tapered Columns and Lateral Support”, Asian Journal of Civil Engineering (BHRC), Vol. 16, pp. 135-159, 2015.
24. Rezaiee-Pajand, M., Masoodi, A. R., and Bambaeechee, M., “Tapered Beam–Column Analysis by Analytical Solution”, Proceedings of the Institution of Civil Engineers-Structures and Buildings, Vol. 172, No. 11, pp. 789-804, 2019.
25. Rezaiee-Pajand, M., and Masoodi, A. R., “Exact Natural Frequencies and Buckling Load of Functionally Graded Material Tapered Beam-Columns Considering Semi-Rigid Connections”, Journal of Vibration and Control, Vol. 24, No. 9, pp. 1787-1808, 2018.
26. Lee, J. K., and Lee, B. K., “Free Vibration and Buckling of Tapered Columns Made of Axially Functionally Graded Materials”, Journal of Applied Mathematical Modelling, Vol. 75, pp. 73-87, 2019.
27. Lanc, D., Ivančić, I., and Katalenić, M., “Buckling Analysis of Columns Made of Functionally Graded Materials Via Rayleigh-Ritz Method”, Mathematical Modeling, Vol. 4, No. 1, pp. 18-21, 2020.
28. Ma, W. L., Jiang, Z. C., and Li, X. F., “Effect of Warping Shape on Buckling of Circular and Rectangular Columns Under Axial Compression”, Applied Mathematical Modelling, Vol. 88, pp. 1475-1490, 2021.
29. Dibajian, H., and Farzin, F., “Improving the Integration Method in the Galerkin Elementless Method with The Help Of Kriging Intermediation”, Numerical Methods in Engineering, Vol. 33. pp. 1-18, 2015.
30. Zhu, T., Zhang, J., and Atluri, S., “A Local Boundary Integral Equation (LBIE) Method in Computational Mechanics, and A Meshless Discretization Approach”, Computational mechanics, Vol. 21, pp. 223-235, 1998.
31. Organ, D., Fleming, M., Terry, T., and Belytschko, T. “Continuous Meshless Approximations for Nonconvex Bodies by Diffraction and Transparency”, Computational mechanics, Vol. 18, pp. 225-235, 1996.
32. Liu, W. K., and Jun, S., “Multiple‐Scale Reproducing Kernel Particle Methods for Large Deformation Problems”, International Journal for Numerical Methods in Engineering, Vol. 41, pp. 1339-1362, 1998.
33. Duarte, C. A., and Oden, J. T., “An hp Adaptive Method Using Clouds”, Computer Methods in Applied Mechanics and Engineering, Vol. 139. pp. 237-262, 1996.
34. Atluri, S., and Zhu, T., “A new Meshless Local Petrov-Galerkin (MLPG) Approach to Nonlinear Problems in Computer Modeling and Simulation”, Computer Modeling and Simulation in Engineering, Vol. 3, pp. 187-196, 1998.
35. Atluri, S., Cho, J., and Kim, H. G., “Analysis of Thin Beams, Using the Meshless Local Petrov–Galerkin Method, with Generalized Moving Least Squares Interpolations”, Computational Mechanics, Vol. 24. pp. 334-347, 1999.
36. Ching, H., and Batra, R., “Determination of Crack Tip Fields in Linear Elastostatics by the Meshless Local Petrov-Galerkin(MLPG) Method”, CMES- Computer Modeling in Engineering and Sciences, Vol. 2, No. 2, pp. 273-289, 2001.
37. Gu, Y., and Liu, G., “A Meshless Local Petrov-Galerkin (MLPG) Formulation for Static and Free Vibration Analyses of Thin Plates”, Computer Modeling in Engineering and Sciences, Vol. 2, pp. 463-476, 2001.
38. Long, S., and Atluri, S., “A Meshless Local Petrov-Galerkin Method for Solving the Bending Problem of a Thin Plate”, Computer Modeling in Engineering and Sciences, Vol. 3. pp. 53-64, 2002.
39. Raju, I., and Phillips, D., “Further Developments in the MLPG Method for Beam Problems”, Computer Modeling in Engineering and Sciences, Vol. 4, pp. 141-160, 2003.
40. Raju, I., “Analysis of a Column Subjected to Follower Forces by a Meshless Local Petrov-Galerkin Method”, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, 2004.
41. Sladek, J., Sladek, V., Zhang, C., Krivacek, J., and Wen, P. H., “Analysis of Orthotropic Thick Plates by Meshless Local Petrov–Galerkin (MLPG) Method”, International Journal for Numerical Methods In Engineering, Vol. 67, pp. 1830-1850, 2006.
42. Arjangpay, A., Darvizeh, M., Ansari, R., and Zarepour, G., “Axial buckling analysis of an isotropic cylindrical shell using the meshless local Petrov-Galerkin method”, Computational Methods in Civil Engineering, Vol. 2, pp. 219-230, 2011.
43. Ansari, R., and Arjangpay, A., “Nanoscale Vibration and Buckling of Single-Walled Carbon Nanotubes Using the Meshless Local Petrov–Galerkin Method”, Physica E: Low-Dimensional Systems and Nanostructures, Vol. 63, pp. 283-292, 2014.
44. Kargarnovin, M., and Ekhteraee Toussi, H., “Application of Meshless Galerking Method in Formulating Fracture Mechanic Problems”, Mechanical Engineering, Vol. 18, pp. 48-53, 2002.
45. Edalati, H., and Soltani, B., “Analysis of Thin Isotropic and Orthotropic Plates with Element-Free Galerkin Method and Various Geometric Shapes”, Journal of Computational Methods in Engineering, Vol. 34, pp. 143-157, 2016.
46. Liu, G.R., Dai, K.Y., Lim, K.M., and Gu, Y.T., “A Radial Point Interpolation Method for Simulation of Two-Dimensional Piezoelectric Structures”, Smart Materials and Structures, Vol. 12, pp. 171-180, 2003.
47. Liu, G., Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, 2002.
48. Wang, C. M., and Wang C. Y., Exact Solutions for Buckling of Structural Members, CRC press, 2004.
2. Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, Courier Corporation, United States, 2009.
3. Arani, A. J., and Kolahchi, R., “Buckling Analysis of Embedded Concrete Columns Armed with Carbon Nanotubes”, Computers and Concrete, Vol. 17, No. 5, pp. 567-578, 2016.
4. Iremonger, M., “Finite Difference Buckling Analysis of Non-Uniform Columns”, Computers & Structures, Vol. 12, No. 5, pp. 741-748, 1980.
5. Ermopoulos, J. C., and Kounadis, A. N., “Stability of Frames with Tapered Built-Up Members”, Journal of Structural Engineering, Vol. 111, pp. 1979-1992, 1985.
6. Smith, W.G., “Analytic Solutions for Tapered Column Buckling”, Computers & structures, Vol. 28, pp. 677-681, 1988.
7. Williams, F.W., and Aston, G., “Exact or Lower Bound Tapered Column Buckling Loads”, Journal of Structural Engineering, Vol. 115, pp. 1088-1100, 1989.
8. Arbabi, F., and Li, F., “Buckling of Variable Cross-Section Columns: Integral-Equation Approach”, Journal of Structural Engineering, Vol. 117, pp. 2426-2441, 1991.
9. Siginer, A., “Buckling of Columns of Variable Flexural Rigidity”, Journal of Engineering Mechanics, Vol. 118, pp. 640-643, 1992.
10. Al-Gahtani, H. J., “Exact Stiffnesses for Tapered Members”, Journal of structural Engineering, Vol. 122, No. 10, pp. 1234-1239, 1996.
11. Bazeos, N., and Karabalis, D. L, “Efficient Computation of Buckling Loads for Plane Steel Frames with Tapered Members”, Engineering Structures, Vol. 28, pp. 771-775. 2006.
12. Raftoyiannis, I., Stamatopoulos, G. N., and Ermopoulos, J., “Buckling Behaviour of Doubly-Tapered Steel Columns Under Axial Compression and Biaxial Bending”, Proceedings of the International Colloquium on Stability and Ductility of Steel Structures, SDSS 2006, pp. 331-338, 2006.
13. Singh, K. V., and Li, G., “Buckling of Functionally Graded and Elastically Restrained Non-Uniform Columns”, Composites Part B: Engineering, Vol. 40, pp. 393-403, 2009.
14. Coşkun, S. B., and Atay, M. T., “Determination of Critical Buckling Load for Elastic Columns of Constant and Variable Cross-Sections Using Variational Iteration Method”, Computers & Mathematics with Applications, Vol. 58, pp. 2260-2266, 2009.
15. 15. Darbandi, S., Firouz-Abadi, R., and Haddadpour, H., “Buckling of Variable Section Columns Under Axial Loading”, Journal of Engineering Mechanics, Vol. 136, pp. 472-476, 2010.
16. 16. Wei, D. J., Yan, S. X., Zhang, Z. P., and Li, X. F., “Critical Load for Buckling of Non-Prismatic Columns Under Self-Weight and Tip Force”, Mechanics Research Communications, Vol. 37, pp. 554-558, 2010.
17. Serna, M., Ibáñez, J., and López, A., “Elastic Flexural Buckling of Non-Uniform Members: Closed-Form Expression and Equivalent Load Approach”, Journal of Constructional Steel Research, Vol. 67, pp. 1078-1085, 2011.
18. Huang, Y., and Luo, Q., “A simple Method to Determine the Critical Buckling Loads for Axially Inhomogeneous Beams with Elastic Restraint”, Computers & Mathematics with Applications, Vol. 61, pp. 2510-2517, 2011.
19. Jamali, S., “Determination of Critical Buckling Force for Elastic Columns with Constant Cross-Sectional Area and Variable Using Repetitive Change Method (VIM) ”, 3rd National Conference on Structures and Steel and First National Conference on Light Steel Structures (LSF), 2012.
20. Pinarbasi, S., “Buckling Analysis of Nonuniform Columns with Elastic End Restraints”, Journal of Mechanics of Materials and Structures, Vol. 7. pp. 485-507, 2012.
21. Taha, M., and Essam, M., “Stability Behavior and Free Vibration of Tapered Columns with Elastic End Restraints Using The DQM Method”, Ain Shams Engineering Journal, Vol. 4, pp. 515-521, 2013.
22. Zhang, B., Guo, Y., and Dou, C., “Ultimate Bearing Capacity of Asymmetrically Double Tapered Steel Columns with Tubular Cross-Section”, Journal of Constructional Steel Research, Vol. 89, pp. 52-62, 2013.
23. Rezaiee-Pajand, M., Shahabian, F., and Bambaeechee, M., “Stability of Semi-Rigid Portal Frames with Tapered Columns and Lateral Support”, Asian Journal of Civil Engineering (BHRC), Vol. 16, pp. 135-159, 2015.
24. Rezaiee-Pajand, M., Masoodi, A. R., and Bambaeechee, M., “Tapered Beam–Column Analysis by Analytical Solution”, Proceedings of the Institution of Civil Engineers-Structures and Buildings, Vol. 172, No. 11, pp. 789-804, 2019.
25. Rezaiee-Pajand, M., and Masoodi, A. R., “Exact Natural Frequencies and Buckling Load of Functionally Graded Material Tapered Beam-Columns Considering Semi-Rigid Connections”, Journal of Vibration and Control, Vol. 24, No. 9, pp. 1787-1808, 2018.
26. Lee, J. K., and Lee, B. K., “Free Vibration and Buckling of Tapered Columns Made of Axially Functionally Graded Materials”, Journal of Applied Mathematical Modelling, Vol. 75, pp. 73-87, 2019.
27. Lanc, D., Ivančić, I., and Katalenić, M., “Buckling Analysis of Columns Made of Functionally Graded Materials Via Rayleigh-Ritz Method”, Mathematical Modeling, Vol. 4, No. 1, pp. 18-21, 2020.
28. Ma, W. L., Jiang, Z. C., and Li, X. F., “Effect of Warping Shape on Buckling of Circular and Rectangular Columns Under Axial Compression”, Applied Mathematical Modelling, Vol. 88, pp. 1475-1490, 2021.
29. Dibajian, H., and Farzin, F., “Improving the Integration Method in the Galerkin Elementless Method with The Help Of Kriging Intermediation”, Numerical Methods in Engineering, Vol. 33. pp. 1-18, 2015.
30. Zhu, T., Zhang, J., and Atluri, S., “A Local Boundary Integral Equation (LBIE) Method in Computational Mechanics, and A Meshless Discretization Approach”, Computational mechanics, Vol. 21, pp. 223-235, 1998.
31. Organ, D., Fleming, M., Terry, T., and Belytschko, T. “Continuous Meshless Approximations for Nonconvex Bodies by Diffraction and Transparency”, Computational mechanics, Vol. 18, pp. 225-235, 1996.
32. Liu, W. K., and Jun, S., “Multiple‐Scale Reproducing Kernel Particle Methods for Large Deformation Problems”, International Journal for Numerical Methods in Engineering, Vol. 41, pp. 1339-1362, 1998.
33. Duarte, C. A., and Oden, J. T., “An hp Adaptive Method Using Clouds”, Computer Methods in Applied Mechanics and Engineering, Vol. 139. pp. 237-262, 1996.
34. Atluri, S., and Zhu, T., “A new Meshless Local Petrov-Galerkin (MLPG) Approach to Nonlinear Problems in Computer Modeling and Simulation”, Computer Modeling and Simulation in Engineering, Vol. 3, pp. 187-196, 1998.
35. Atluri, S., Cho, J., and Kim, H. G., “Analysis of Thin Beams, Using the Meshless Local Petrov–Galerkin Method, with Generalized Moving Least Squares Interpolations”, Computational Mechanics, Vol. 24. pp. 334-347, 1999.
36. Ching, H., and Batra, R., “Determination of Crack Tip Fields in Linear Elastostatics by the Meshless Local Petrov-Galerkin(MLPG) Method”, CMES- Computer Modeling in Engineering and Sciences, Vol. 2, No. 2, pp. 273-289, 2001.
37. Gu, Y., and Liu, G., “A Meshless Local Petrov-Galerkin (MLPG) Formulation for Static and Free Vibration Analyses of Thin Plates”, Computer Modeling in Engineering and Sciences, Vol. 2, pp. 463-476, 2001.
38. Long, S., and Atluri, S., “A Meshless Local Petrov-Galerkin Method for Solving the Bending Problem of a Thin Plate”, Computer Modeling in Engineering and Sciences, Vol. 3. pp. 53-64, 2002.
39. Raju, I., and Phillips, D., “Further Developments in the MLPG Method for Beam Problems”, Computer Modeling in Engineering and Sciences, Vol. 4, pp. 141-160, 2003.
40. Raju, I., “Analysis of a Column Subjected to Follower Forces by a Meshless Local Petrov-Galerkin Method”, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, 2004.
41. Sladek, J., Sladek, V., Zhang, C., Krivacek, J., and Wen, P. H., “Analysis of Orthotropic Thick Plates by Meshless Local Petrov–Galerkin (MLPG) Method”, International Journal for Numerical Methods In Engineering, Vol. 67, pp. 1830-1850, 2006.
42. Arjangpay, A., Darvizeh, M., Ansari, R., and Zarepour, G., “Axial buckling analysis of an isotropic cylindrical shell using the meshless local Petrov-Galerkin method”, Computational Methods in Civil Engineering, Vol. 2, pp. 219-230, 2011.
43. Ansari, R., and Arjangpay, A., “Nanoscale Vibration and Buckling of Single-Walled Carbon Nanotubes Using the Meshless Local Petrov–Galerkin Method”, Physica E: Low-Dimensional Systems and Nanostructures, Vol. 63, pp. 283-292, 2014.
44. Kargarnovin, M., and Ekhteraee Toussi, H., “Application of Meshless Galerking Method in Formulating Fracture Mechanic Problems”, Mechanical Engineering, Vol. 18, pp. 48-53, 2002.
45. Edalati, H., and Soltani, B., “Analysis of Thin Isotropic and Orthotropic Plates with Element-Free Galerkin Method and Various Geometric Shapes”, Journal of Computational Methods in Engineering, Vol. 34, pp. 143-157, 2016.
46. Liu, G.R., Dai, K.Y., Lim, K.M., and Gu, Y.T., “A Radial Point Interpolation Method for Simulation of Two-Dimensional Piezoelectric Structures”, Smart Materials and Structures, Vol. 12, pp. 171-180, 2003.
47. Liu, G., Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, 2002.
48. Wang, C. M., and Wang C. Y., Exact Solutions for Buckling of Structural Members, CRC press, 2004.
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