Abstract
This paper is devoted to the behavior of a non-homogeneous simply supported beam under three-point bending. The individual shear deformation function of a planar cross-section is adopted, and longitudinal displacements, strains, and stresses for two parts of the beam are explained. By applying the principle of stationary potential energy, a system of two differential equations of equilibrium is derived and solved analytically. The positions of the neutral axis, shear coefficients, and deflections are then calculated for three different beam families. Additionally, the bending problem of these beams is studied numerically using the finite element method (FEM). The results of both analytical and numerical calculations are presented in tables and figures. The main contribution of this paper lies in the development of an analytical model incorporating the individual shear deformation function and a numerical FEM model for this beam.
Keywords:
non-homogeneous beam, shear deformation theory, three-point bendingReferences
1. Manjunatha B.S., Kant T., New theories for symmetric/unsymmetric composite and sandwich beams with C0 finite elements, Composite Structures, 23(1): 61–73, 1993, https://doi.org/10.1016/0263-8223%2893%2990075-2
2. Reddy J.N., Analysis of functionally graded plates, International Journal of Numerical Methods in Engineering, 47(1–3): 663–684, 2000, https://doi.org/10.1002/%28sici%291097-0207%2820000110/30%2947%3A1/3<663.
3. Reddy J.N., Mechanics of Laminated Composite Plates and Shells: Theory and Analysis,. 2nd ed., CRC Press, Boca Raton, London, New York, Washington, 2004.
4. Zenkour A.M., Generalized shear deformation theory for bending analysis of functionally graded plates, Applied Mathematical Modelling, 30(1): 67–84, 2006, https://doi.org/10.1016/j.apm.2005.03.009
5. Altenbach H., Eremeyev V.A., Direct approach-based analysis of plates composed of functionally graded materials, Archive of Applied Mechanics, 78(10): 775–794, 2008, https://doi.org/10.1007/s00419-007-0192-3
6. Shen H.S., Functionally Graded Materials: Nonlinear Analysis of Plates and Shells, CRC Press, Boca Raton, 2009.
7. Carrera E., Brischetto S., A survey with numerical assessment of classical and refined theories for the analysis of sandwich plates, Applied Mechanics Reviews, 62(1): 010803, 2009, https://doi.org/10.1115/1.3013824
8. Carrera E., Giunta G., Petrolo M., Beam Structures. Classical and Advanced Theories, John Wiley & Sons, Ltd., 2011.
9. Lezgy-Nazargah M., Shariyat M., Beheshti-Aval S.B., A refined high-order global-local theory for finite element bending and vibration analyses of laminated composite beams. Acta Mechanica, 217(3): 219–242, 2011, https://doi.org/10.1007/s00707 010 0391 9.
10. Weps M., Naumenko K., Altenbach H., Unsymmetric three-layer laminate with soft core for photovoltaic modules, Composite Structures, 105: 332–339, 2013, https://doi.org/10.1016/j.compstruct.2013.05.029
11. Zenkour A.M., A simple four-unknown refined theory for bending analysis of functionally graded plates, Applied Mathematical Modelling, 37(20–21): 9041–9051, 2013, https://doi.org/10.1016/j.apm.2013.04.022
12. Magnucki K., Smyczyński M., Jasion P., Deflection and strength of a sandwich beam with thin binding layers between faces and a core, Archives of Mechanics, 65(4): 301–311, 2013.
13. Chen D., Yang J., Kitipornchai S., Elastic buckling and static bending of shear deformable functionally graded porous beam, Composite Structures, 133: 54–61, 2015, https://doi.org/10.1016/j.compstruct.2015.07.052
14. Sayyad A.S., Ghugal Y.M., Bending, buckling and free vibration of laminated composite and sandwich beams: a critical review of literature, Composite Structures, 171: 486–504, 2017, https://doi.org/10.1016/j.compstruct.2017.03.053
15. Vo T.P., Thai H.-T., Nguyen T.-K., Lanc D., Karamanli A., Flexural analysis of laminated composite and sandwich beams using a four-unknown shear and normal deformation theory, Composite Structures, 176: 388–397, 2017, https://doi.org/10.1016/j.compstruct.2017.05.041
16. Abrate S., Di Sciuva M., Equivalent single layer theories for composite and sandwich structures: a review, Composite Structures, 179: 482–494, 2017, https://doi.org/10.1016/j.compstruct.2017.07.090
17. Magnucka-Blandzi E., Kędzia P., Smyczyński M., Unsymmetrical sandwich beams under three-point bending – Analytical studies, Composite Structures, 202: 539–544, 2018, https://doi.org/10.1016/j.compstruct.2018.02.086
18. Magnucki K., Lewiński J., Magnucka-Blandzi E., Kędzia P., Bending, buckling and free vibration of a beam with unsymmetrically varying mechanical properties, Journal of Theoretical and Applied Mechanics, 56(4): 1163–1178, 2018, https://doi.org/10.15632/jtam-pl.56.4.1163
19. Magnucki K., Witkowski D., Magnucka-Blandzi E., Buckling and free vibrations of rectangular plates with symmetrically varying mechanical properties – Analytical and FEM studies, Composite Structures, 220: 355–361, 2019, https://doi.org/10.1016/j.compstruct.2019.03.082
20. Sayyad A.S., Ghugal Y.M., Modeling and analysis of functionally graded sandwich beams: A review, Mechanics of Advanced Materials and Structures, 26(21): 1776–1795, 2019, https://doi.org/10.1080/15376494.2018.1447178
21. Meksi R., Benyoucef S., Mahmoudi A., Tounsi A., Bedia E.A.A., Mahmoud S.R., An analytical solution for bending, buckling and vibration responses of FGM sandwich plates, Journal of Sandwich Structures and Materials, 21(2): 727–757, 2019, https://doi.org/10.1177/1099636217698443
22. Magnucki K., Magnucka-Blandzi E., Lewiński J., Milecki S., Analytical and numerical studies of an unsymmetrical sandwich beam - bending, buckling and free vibration, Engineering Transactions, 67(4): 491–512, 2019, https://doi.org/10.24423/EngTrans.1015.20190725
23. Genovese D., Elishakoff I., Shear deformable rod theories and fundamental principles of mechanics, Archive of Applied Mechanics, 89(10): 1995–2003, 2019, https://doi.org/10.1007/s00419-019-01556-7
24. Magnucki K., Lewinski J., Magnucka-Blandzi E., Bending of two-layer beams under uniformly distributed load – Analytical and numerical FEM studies, Composite Structures, 235: 111777, 2020, https://doi.org/10.1016/j.compstruct.2019.111777
25. Magnucki K., Lewinski J., Magnucka-Blandzi E., An improved shear deformation theory for bending beams with symmetrically varying mechanical properties in the depth direction, Acta Mechanica, 231(10): 4381–4395, 2020, https://doi.org/10.1007/s00707-020-02763-y
26. Malikan M., Eremeyev V.A., A new hyperbolic-polynomial higher-order elasticity theory for mechanics of thick FGM beams with imperfection in the material composition, Composite Structures, 249: 112486, 2020, https://doi.org/10.1016/j.compstruct.2020.112486
27. Malikan M., Eremeyev V.A., The effect of shear deformations’ rotary inertia on the vibrating response of multi-physics composite beam-like actuators, Composite Structures, 297: 115951, 2020, https://doi.org/10.1016/j.compstruct.2020.115951
28. Sedighi H.M, Malikan M., Valipour A., Żur K.K., Nonlocal vibration of carbon/boron –nitride nano-hetero-structure in thermal and magnetic fields by means of nonlinear finite element method, Journal of Computational Design and Engineering, 7(5): 591–602, 2020, https://doi.org/10.1093/jcde/qwaa041
29. Masjedi P.K., Weaver P.M., Variable stiffness composite beams subject to non-uniformly distributed loads: An analytical solution, Composite Structures, 256: 112975, 2021, https://doi.org/10.1016/j.compstruct.2020.112975
30. Garg A., Belarbi M.-O., Chalak H.D., Chakrabarti A., A review of the analysis of sandwich FGM structures, Composite Structures, 258: 113427, 2021, https://doi.org/10.1016/j.compstruct.2020.113427
31. Magnucki K., Magnucka-Blandzi E., Wittenbeck L., Three models of a sandwich beam: bending, buckling and free vibrations, Engineering Transactions, 70(2): 97–122, 2022, https://doi.org/10.24423/EngTrans.1416.20220331
2. Reddy J.N., Analysis of functionally graded plates, International Journal of Numerical Methods in Engineering, 47(1–3): 663–684, 2000, https://doi.org/10.1002/%28sici%291097-0207%2820000110/30%2947%3A1/3<663.
3. Reddy J.N., Mechanics of Laminated Composite Plates and Shells: Theory and Analysis,. 2nd ed., CRC Press, Boca Raton, London, New York, Washington, 2004.
4. Zenkour A.M., Generalized shear deformation theory for bending analysis of functionally graded plates, Applied Mathematical Modelling, 30(1): 67–84, 2006, https://doi.org/10.1016/j.apm.2005.03.009
5. Altenbach H., Eremeyev V.A., Direct approach-based analysis of plates composed of functionally graded materials, Archive of Applied Mechanics, 78(10): 775–794, 2008, https://doi.org/10.1007/s00419-007-0192-3
6. Shen H.S., Functionally Graded Materials: Nonlinear Analysis of Plates and Shells, CRC Press, Boca Raton, 2009.
7. Carrera E., Brischetto S., A survey with numerical assessment of classical and refined theories for the analysis of sandwich plates, Applied Mechanics Reviews, 62(1): 010803, 2009, https://doi.org/10.1115/1.3013824
8. Carrera E., Giunta G., Petrolo M., Beam Structures. Classical and Advanced Theories, John Wiley & Sons, Ltd., 2011.
9. Lezgy-Nazargah M., Shariyat M., Beheshti-Aval S.B., A refined high-order global-local theory for finite element bending and vibration analyses of laminated composite beams. Acta Mechanica, 217(3): 219–242, 2011, https://doi.org/10.1007/s00707 010 0391 9.
10. Weps M., Naumenko K., Altenbach H., Unsymmetric three-layer laminate with soft core for photovoltaic modules, Composite Structures, 105: 332–339, 2013, https://doi.org/10.1016/j.compstruct.2013.05.029
11. Zenkour A.M., A simple four-unknown refined theory for bending analysis of functionally graded plates, Applied Mathematical Modelling, 37(20–21): 9041–9051, 2013, https://doi.org/10.1016/j.apm.2013.04.022
12. Magnucki K., Smyczyński M., Jasion P., Deflection and strength of a sandwich beam with thin binding layers between faces and a core, Archives of Mechanics, 65(4): 301–311, 2013.
13. Chen D., Yang J., Kitipornchai S., Elastic buckling and static bending of shear deformable functionally graded porous beam, Composite Structures, 133: 54–61, 2015, https://doi.org/10.1016/j.compstruct.2015.07.052
14. Sayyad A.S., Ghugal Y.M., Bending, buckling and free vibration of laminated composite and sandwich beams: a critical review of literature, Composite Structures, 171: 486–504, 2017, https://doi.org/10.1016/j.compstruct.2017.03.053
15. Vo T.P., Thai H.-T., Nguyen T.-K., Lanc D., Karamanli A., Flexural analysis of laminated composite and sandwich beams using a four-unknown shear and normal deformation theory, Composite Structures, 176: 388–397, 2017, https://doi.org/10.1016/j.compstruct.2017.05.041
16. Abrate S., Di Sciuva M., Equivalent single layer theories for composite and sandwich structures: a review, Composite Structures, 179: 482–494, 2017, https://doi.org/10.1016/j.compstruct.2017.07.090
17. Magnucka-Blandzi E., Kędzia P., Smyczyński M., Unsymmetrical sandwich beams under three-point bending – Analytical studies, Composite Structures, 202: 539–544, 2018, https://doi.org/10.1016/j.compstruct.2018.02.086
18. Magnucki K., Lewiński J., Magnucka-Blandzi E., Kędzia P., Bending, buckling and free vibration of a beam with unsymmetrically varying mechanical properties, Journal of Theoretical and Applied Mechanics, 56(4): 1163–1178, 2018, https://doi.org/10.15632/jtam-pl.56.4.1163
19. Magnucki K., Witkowski D., Magnucka-Blandzi E., Buckling and free vibrations of rectangular plates with symmetrically varying mechanical properties – Analytical and FEM studies, Composite Structures, 220: 355–361, 2019, https://doi.org/10.1016/j.compstruct.2019.03.082
20. Sayyad A.S., Ghugal Y.M., Modeling and analysis of functionally graded sandwich beams: A review, Mechanics of Advanced Materials and Structures, 26(21): 1776–1795, 2019, https://doi.org/10.1080/15376494.2018.1447178
21. Meksi R., Benyoucef S., Mahmoudi A., Tounsi A., Bedia E.A.A., Mahmoud S.R., An analytical solution for bending, buckling and vibration responses of FGM sandwich plates, Journal of Sandwich Structures and Materials, 21(2): 727–757, 2019, https://doi.org/10.1177/1099636217698443
22. Magnucki K., Magnucka-Blandzi E., Lewiński J., Milecki S., Analytical and numerical studies of an unsymmetrical sandwich beam - bending, buckling and free vibration, Engineering Transactions, 67(4): 491–512, 2019, https://doi.org/10.24423/EngTrans.1015.20190725
23. Genovese D., Elishakoff I., Shear deformable rod theories and fundamental principles of mechanics, Archive of Applied Mechanics, 89(10): 1995–2003, 2019, https://doi.org/10.1007/s00419-019-01556-7
24. Magnucki K., Lewinski J., Magnucka-Blandzi E., Bending of two-layer beams under uniformly distributed load – Analytical and numerical FEM studies, Composite Structures, 235: 111777, 2020, https://doi.org/10.1016/j.compstruct.2019.111777
25. Magnucki K., Lewinski J., Magnucka-Blandzi E., An improved shear deformation theory for bending beams with symmetrically varying mechanical properties in the depth direction, Acta Mechanica, 231(10): 4381–4395, 2020, https://doi.org/10.1007/s00707-020-02763-y
26. Malikan M., Eremeyev V.A., A new hyperbolic-polynomial higher-order elasticity theory for mechanics of thick FGM beams with imperfection in the material composition, Composite Structures, 249: 112486, 2020, https://doi.org/10.1016/j.compstruct.2020.112486
27. Malikan M., Eremeyev V.A., The effect of shear deformations’ rotary inertia on the vibrating response of multi-physics composite beam-like actuators, Composite Structures, 297: 115951, 2020, https://doi.org/10.1016/j.compstruct.2020.115951
28. Sedighi H.M, Malikan M., Valipour A., Żur K.K., Nonlocal vibration of carbon/boron –nitride nano-hetero-structure in thermal and magnetic fields by means of nonlinear finite element method, Journal of Computational Design and Engineering, 7(5): 591–602, 2020, https://doi.org/10.1093/jcde/qwaa041
29. Masjedi P.K., Weaver P.M., Variable stiffness composite beams subject to non-uniformly distributed loads: An analytical solution, Composite Structures, 256: 112975, 2021, https://doi.org/10.1016/j.compstruct.2020.112975
30. Garg A., Belarbi M.-O., Chalak H.D., Chakrabarti A., A review of the analysis of sandwich FGM structures, Composite Structures, 258: 113427, 2021, https://doi.org/10.1016/j.compstruct.2020.113427
31. Magnucki K., Magnucka-Blandzi E., Wittenbeck L., Three models of a sandwich beam: bending, buckling and free vibrations, Engineering Transactions, 70(2): 97–122, 2022, https://doi.org/10.24423/EngTrans.1416.20220331
