Elastostatic Deformation Analysis of Thick Isotropic Beams by Using Different Beam Theories and a Meshless Method

Abstract

The elastostatic deformations of thick isotropic beams subjected to various sets of boundary conditions are presented by using different beam theories and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The analysis is based on the Euler-Bernoulli, Timoshenko and Reddy-Bickford beam theories. The performance of the SSPH method is investigated for the comparison of the different beam theories for the first time. For the numerical results, various numbers of nodes are used in the problem domain. Regarding to the computed results for RBT, various number of terms in the Taylor Series Expansions (TSEs) is employed. To validate the performance of the SSPH method, comparison studies in terms of transverse deflections are carried out with the analytical solutions. It is found that the SSPH method has provided satisfactory convergence rate and smaller L2 error.

Keywords

• Meshless Method, Element-Free, Beam, Euler-Bernoulli, Timoshenko, Reddy-Bickford

References

1. Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, fourth ed. Dover Publications, New York, 1927.
2. Timoshenko, S.P., Goodier, J.C., Theory of Elasticity, McGraw-Hill Co. Inc., New York, 1970.
3. Wang, C.M., Reddy, J.N., Lee, K.H., Shear Deformable Beams and Plates Relations with Classical Solutions, Elsevier Science Ltd., Oxford 2000.
4. Polizzotto, C., From the Euler-Bernoulli beam to the Timoshenko one through a sequence of Reddy-type shear deformable beam models of increasing order, European Journal of Mechanics A/Solids, 53, 62-74, 2015.
5. Levinson, M., A new rectangular beam theory, Journal of Sound and Vibration, 74, 81-87, 1981.
6. Bickford, W.B., A consistent higher order beam theory, Theor. Appl. Mech. 11, 137-150, 1982.
7. Heyliger, P.R., Reddy, J.N., A higher order beam finite element for bending and vibration problems, Journal of Sound and Vibration, 126 (2), 309-326, 1988.
8. Subramanian, P., Dynamic analysis of laminated composite beams using higher order theories and finite elements, Composite Structures, 73, 342-353, 2006.

9. Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci. 45, 288-307, 2007.
10. Carrera, E., Giunta, G., Refined beam theories based on a unified formulation, Int. J. Appl. Mech. 2 (1), 117-143, 2010.
11. Giunta, G., Biscani, F., Bellouettar, S., Ferreira, A.J.M., Carrera, E., Free vibration analysis of composite beams via refined theories, Composites Part B, 44, 540-552, 2013.
12. [12] Arya, H., A new zig-zag model for laminated composite beams: free vibration analysis, Journal of Sound and Vibration, 264, 485-490, 2003.
13. Jun, L., Hongxing, H., Dynamic stiffness analysis of laminated composite beams using trigonometric shear deformation theory, Composite Structures, 89, 433-442, 2009.
14. Kurama, M., Afaq, K.S., Mistou, S., Mechanical behavior of laminated composite beams by the new multi-layered laminated composite structures model with trigonometric shear stress continuity, Int. J. Solids Struct, 40, 1525-1546, 2003.
15. Donning, B.M., Liu, W.K., Meshless methods for shear-deformable beams and plates, Computer Methods in Applied Mechanics and Engineering, 152, 47-71, 1998.
16. Gu, Y.T., Liu, G.R., A local point interpolation method for static and dynamic analysis of thin beams, Computer Methods in Applied Mechanics and Engineering, 190, 42, 5515-5528, 2001.
17. Ferreira, A.J.M., Roque, C.M.C., Martins, P.A.L.S., Radial basis functions and higher-order shear deformation theories in the analysis of laminated composite beams and plates, Composite Structures, 66, 287-293, 2004.
18. Ferreira, A.J.M., Fasshauer, G.E., Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method, Computer Methods in Applied Mechanics and Engineering, 196, 134-146, 2006.
19. Moosavi, M.R., Delfanian, F., Khelil, A., The orthogonal meshless finite volume method for solving Euler–Bernoulli beam and thin plate problems, Thin-Walled Structures, 49, 923-932, 2011.
20. Roque, C.M.C., Figaldo, D.S., Ferreira, A.J.M., Reddy, J.N., A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method, Composite Structures, 96, 532-537, 2013.
21. Lucy LB, A numerical approach to the testing of the fission hypothesis, Astronomical Journal, 82, 1013–1024, 1977.
22. Chen JK, Beraun JE, Jin CJ, An improvement for tensile instability in smoothed particle hydrodynamics, Computational Mechanics, 23, 279–287, 1999.
23. Chen JK, Beraun JE, Jin CJ, Completeness of corrective smoothed particlemethod for linear elastodynamics, Computational Mechanics, 24, 273–285, 1999.
24. Liu WK, Jun S, Zhang YF, Reproducing kernel particle methods, International Journal for Numerical Methods in Fluids, 20, 1081–1106, 1995.
25. Liu WK, Jun S, Li S, Adee J, Belytschko T, Reproducing kernel particle methods for structural dynamics, International Journal for Numerical Methods in Engineering, 38, 1655–1679, 1995.
26. Chen JS, Pan C,Wu CT, Liu WK, Reproducing kernel particle methods for large deformation analysis of non-linear structures, Computer Methods in Applied Mechanics and Engineering, 139, 195–227, 1996.
27. Zhang GM, Batra RC, Modified smoothed particle hydrodynamics method and its application to transient problems, Computational Mechanics, 34, 137–146, 2004.
28. Batra RC, Zhang GM, Analysis of adiabatic shear bands in elasto-thermo- viscoplastic materials by modified smoothed particle hydrodynamics (MSPH) method, Journal of Computational Physics, 201, 172–190, 2004.
29. Zhang GM, Batra RC, Wave propagation in functionally graded materials by modified smoothed particle hydrodynamics (MSPH) method, Journal of Computational Physics, 222, 374–390, 2007.
30. Batra RC, Zhang GM, Search algorithm, and simulation of elastodynamic crack propagation by modified smoothed particle hydrodynamics (MSPH) method, Computational Mechanics, 40, 531–546, 2007.
31. Zhang GM, Batra RC, Symmetric smoothed particle hydrodynamics (SSPH) method and its application to elastic problems, Computational Mechanics, 43, 321-340, 2009.
32. Batra RC, Zhang GM, SSPH basis functions for meshless methods, and comparison of solutions with strong and weak formulations, Computational Mechanics, 41, 527-545, 2008.
33. Tsai CL, Guan YL, Batra RC, Ohanehi DC, Dillard JG, Nicoli E, Dillard DA, Comparison of the performance of SSPH and MLS basis functions for two-dimensional linear elastostatics problems including quasistatic crack propagation, Computational Mechanics, 51, 19-34, 2013.
34. Tsai CL, Guan YL, Ohanehi DC, Dillard JG, Dillard DA, Batra RC, Analysis of cohesive failure in adhesively bonded joints with the SSPH meshless method, International Journal of Adhesion & Adhesives, 51, 67-80, 2014.
35. Karamanli A, Mugan A, Solutions of two–dimensional heat transfer problems by using symmetric smoothed particle hydrodynamics method, Journal of Applied and Computational Mathematics 1, 1-6, 2012.
36. Karamanli A, Bending Deflection Analysis of a Semi-Trailer Chassis by Using Symmetric Smoothed Particle Hydrodynamics, International Journal of Engineering Technologies, 1, 4, 134-140, 2015.
37. Karamanli A, Mugan A, Strong from meshless implementation of Taylor series method, Applied Mathematics and Computation, 219, 9069-9080, 2013 .
38. Karamanli A, Different Implementation Approaches of the Strong Form Meshless Implementation of Taylor Series Method, International Journal of Engineering Technologies, Vol. 1, No:3, 2015.
39. Kaewumpai, S., Luadsong, A., Two-field-variable meshless method based on moving kriging interpolation for solving simply supported thin plates under various loads. Journal of King Saud University - Science, 1018–3647, 2014.
40. Yimnak, K., Luadsong, A., A local integral equation formulation based on moving kriging interpolation for solving coupled nonlinear reaction–diffusion equations. Advances in Mathematical Physics, 2014.
41. Zhuang, X., Zhu, H., Augarde, C., The meshless Shepard and least squares (MSLS) method, Computational Mechanics, 53, 343-357, 2014.
42. Fatahi, H., Nadjafi, JS, Shivanian, E, New spectral meshless radial point interpolation (SMRPI) method for the two-dimensional Fredholm integral equations on general domains with error analysis, Journal of Computational and Applied Mathematics, 264, 196-209, 2016.