Authors
-
Krzysztof MAGNUCKI
Łukasiewicz Research Network – Poznan Institute of Technology,
Poland
0000-0003-2251-4697
-
Ewa MAGNUCKA-BLANDZI
Institute of Mathematics, Poznan University of Technology,
Poland
0000-0002-6349-5579
-
Dawid WITKOWSKI
Łukasiewicz Research Network - Poznań Institute of Technology,
Poland
0000-0002-4847-5887
-
Natalia STEFAŃSKA
Łukasiewicz Research Network – Poznan Institute of Technology,
Poland
0000-0002-5377-9762
Abstract
This paper is devoted to the study of a homogeneous clamped beam with a monosymmetric cross section under uniformly distributed load or three-point bending. A nonlinear shear deformation theory of a plane beam cross section based on the classical shear stress formula known as the Zhuravsky shear stress is developed. The values of shear coefficients and maximum deflections of exemplary beams are analytically determined. Moreover, numerical FEM computations for these beams are carried out. The results of the research from both methods are shown in figures, specified in tables, and compared. The percentage relative differences between the analytical and numerical results prove that the proposed original shear deformation theory accurately describes the shear deformation problem of a beam’s planar cross‑section.
Keywords:
homogeneous beam, nonlinear shear deformation theory, bending, shear effectReferences
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21. Magnucki K., Stawecki W., Magnucka-Blandzi E., Bending of beams with consideration of a seventh-order shear deformation theory, Engineering Transactions, 68(2): 119–135, 2020, https://doi.org/10.24423/EngTrans.1129.20200214
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23. 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
24. Magnucki K., An individual shear deformation theory of beams with consideration of the Zhuravsky shear stress formula, [in:] Current Perspectives and New Directions in Mechanics, Modelling and Design of Structural Systems, Zingoni A. (Ed.), CRC Press, Taylor & Francis Group, Boca Raton, London, New York, 2022, https://doi.org/10.1201/9781003348443-112
2. Kurrer K.-E., The History of the Theory of Structures: From Arch Analysis to Computational Mechanics, Ernst & Sohn, Berlin, 2008.
3. Reddy J.N., Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, International Journal of Engineering Science, 48(11): 1507–1518, 2010, https://doi.org/10.1016/j.ijengsci.2010.09.020
4. Gao Y., Shang L., The exact theory of deep beams without ad hoc assumptions, Mechanics Research Communication, 37(6): 559–564, 2010, https://doi.org/10.1016/j.mechrescom.2010.07.018
5. Carrera E., Giunta G., Petrolo M., Beam Structures: Classical and Advanced Theories, John Wiley & Sons, 2011.
6. Challamel N., Higher-order shear beam theories and enriched continuum, Mechanics Research Communication, 38(5): 388–392, 2011, https://doi.org/10.1016/j.mechrescom.2011.05.004
7. Thai H-T., Vo T.P., A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams, International Journal of Engineering Science, 54: 58–66, 2012, https://doi.org/10.1016/j.ijengsci.2012.01.009
8. Thai H-T., Vo T.P., Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories, International Journal of Mechanical Sciences, 62(1): 57–66, 2012, https://doi.org/10.1016/j.ijmecsci.2012.05.014
9. Akgöz B., Civalek Ö., A size-dependent shear deformation beam model based on the strain gradient elasticity theory, International Journal of Engineering Science, 70: 1–14, 2013, https://doi.org/10.1016/j.ijengsci.2013.04.004
10. Reddy J.N., El-Borgi S., Eringen’s nonlocal theories of beams accounting for moderate rotations, International Journal of Engineering Science, 82: 159–177, 2014, https://doi.org/10.1016/j.ijengsci.2014.05.006
11. Pradhan K.K., Chakraverty S., Effects of different shear deformation theories on free vibration of functionally graded beams, International Journal of Mechanical Sciences, 82: 149–160, 2014, https://doi.org/10.1016/j.ijmecsci.2014.03.014
12. Senjanović I., Vladimir N., Tomić M., On new first-order shear deformation plate theories, Mechanics Research Communication, 73: 31–38, 2016, https://doi.org/10.1016/j.mechrescom.2016.02.005
13. Özütok A., Madenci E., Static analysis of laminated composite beams based on higher-order shear deformation theory by using mixed-type finite element method, International Journal of Mechanical Sciences, 130: 234–243, 2017, https://doi.org/10.1016/j.ijmecsci.2017.06.013
14. Faghidian S.A., On non-linear flexure of beams based on non-local elasticity theory, International Journal of Engineering Science, 124: 49–63, 2018, https://doi.org/10.1016/j.ijengsci.2017.12.002
15. Challamel N., Elishakoff I., A brief history of first-order shear-deformable beam and plate models, Mechanics Research Communication, 102: 103389, 2019, https://doi.org/10.1016/j.mechrescom.2019.06.005
16. Lin F., Tong L.H., Shen H-S., Lim C.W., Xiang Y., Assessment of first and third order shear deformation beam theories for the buckling and vibration analysis of nanobeams incorporating surface stress effects, International Journal of Mechanical Sciences, 186: 105873, 2020, https://doi.org/10.1016/j.ijmecsci.2020.105873
17. Magnucki K., Lewinski J., Far M., Michalak P., Three-point bending of an expanded-tapered sandwich beam – analytical and numerical FEM study, Mechanics Research Communication, 103: 103471, 2020, https://doi.org/10.1016/j.mechrescom.2019.103471
18. Magnucki K., Lewinski J., Magnucka-Blandzi E., Stawecka H., Three-point bending of an expanded-tapered beam with consideration of the shear effect, Journal of Theoretical and Applied Mechanics, 58(3): 661–672, 2020, https://doi.org/10.15632/jtam-pl/122203
19. Magnucki K., Paczos P., Wichniarek R., Three-point bending of beams with consideration of the shear effect, Steel and Composite Structures, An International Journal, 37(6): 733–740, 2020, https://doi.org/10.12989/scs.2020.37.6.733
20. 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
21. Magnucki K., Stawecki W., Magnucka-Blandzi E., Bending of beams with consideration of a seventh-order shear deformation theory, Engineering Transactions, 68(2): 119–135, 2020, https://doi.org/10.24423/EngTrans.1129.20200214
22. Magnucki K., Lewinski J., Magnucka-Blandzi E., A shear deformation theory of beams of bisymmetrical cross-sections based on the Zhuravsky shear stress formula, Engineering Transactions, 68(4): 353–370, 2020, https://doi.org/10.24423/EngTrans.1174.20201120
23. 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
24. Magnucki K., An individual shear deformation theory of beams with consideration of the Zhuravsky shear stress formula, [in:] Current Perspectives and New Directions in Mechanics, Modelling and Design of Structural Systems, Zingoni A. (Ed.), CRC Press, Taylor & Francis Group, Boca Raton, London, New York, 2022, https://doi.org/10.1201/9781003348443-112
