Engineering Transactions, 68, 2, pp. 119–135, 2020
10.24423/EngTrans.1129.20200214

Bending of Beams with Consideration of a Seventh-Order Shear Deformation Theory

Krzysztof MAGNUCKI
Łukasiewicz Research Network - Institute of Rail Vehicles “TABOR”
Poland

Włodzimierz STAWECKI
Łukasiewicz Research Network - Institute of Rail Vehicles “TABOR”
Poland

Ewa MAGNUCKA-BLANDZI
Poznań University of Technology
Poland

The subject of the paper is a simply- supported prismatic beam with bisymmetrical crosssections under non-uniformly distributed load. The shapes of the cross-sections and the nonuniformly distributed load are described analytically. The individual seventh-order shear deformation theory-hypothesis of the planar beam cross-sections is assumed. Based on the principle of stationary potential energy two differential equations of equilibrium are obtained. The system of the equations is analytically solved, and the shear and deflection coefficients of the beam are derived. Moreover, the shear stress patterns for selected cross-sections are determined and compared with stresses determined by Zhuravsky’s formula. The results of example calculations are presented in tables and figures.
Keywords: nonlinear hypothesis; shear deformation theory; shear stresses
Full Text: PDF
Copyright © The Author(s). This is an open-access article distributed under the terms of the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0).

References

Gere J.M., Timoshenko S.P., Mechanics of materials, 2nd Ed., PWS-KENT Publishing Company, Boston 1984.

Rychter Z., A family of shear-deformation beam theories and a refined Bernoulli-Euler theory, International Journal of Engineering Sciences, 31(4): 559–567, 1993, doi: 10.1016/0020-7225(93)90049-Z.

Wang C.M., Reddy J.N., Lee K.H., Shear deformable beams and plates: Relationships with classical solutions, Elsevier, Amsterdam, Lausanne, New York, Shannon, Singapore, Tokyo, 2000.

Hutchinson J.R., Shear coefficients for Timoshenko beam theory, ASME, Journal of Applied Mechanics, 68 (1): 87–92, 2001, doi: 10.1115/1.1349417.

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, doi: 10.1016/j.ijengsci.2010.09.020.

Shi G., Voyiadjis G.Z., A sixth-order theory of shear deformable beams with variational consistent boundary conditions, ASME, Journal of Applied Mechanics, 78(2): 021019, 2011, doi: 10.1115/1.4002594.

Beck A.T., da Silva Jr. C.R.A., Timoshenko versus Euler beam theory: Pitfalls of a deterministic approach, Structural Safety, 33(1): 19–25, 2011, doi: 10.1016/j.strusafe.2010.04.006.

Kim N-I., Shear deformable doubly- and mono-symmetric composite I-beams, International Journal of Mechanical Sciences, 53(1): 31–41, 2011, doi: 10.1016/j.ijmecsci.2010.10.004.

Magnucka-Blandzi E., Dynamic stability and static stress state of a sandwich beam with a metal foam core using three modified Timoshenko hypothesis, Mechanics of Advanced Materials and Structures, 18(2): 147–158, 2011, doi: 10.1080/15376494.2010.496065.

Magnucka-Blandzi E., Magnucki K., Wittenbeck L., Mathematical modelling of shearing effect for sandwich beams with sinusoidal corrugated cores, Applied Mathematical Modelling, 39(9): 2796–2808, 2015, doi: 10.1016/j.apm.2014.10.069.

Schneider P., Kienzler R., On exact rod/beam/shaft-theories and the coupling among them due to arbitrary material anisotropies, International Journal of Solids and Structures, 56–57: 265–279, 2015, doi: 10.1016/j.ijsolstr.2014.10.022.

Senjanović I., Vladimir N., Neven H., Tomić M., New first order shear deformation beam theory with in-plane shear influence, Engineering Structures, 110: 169–183, 2016, 10.1016/j.engstruct.2015.11.032.

Endo M., ‘One-half order shear deformation theory’ as a new naming for the transverse, but not in-plane rotational, shear deformable structural models, International Journal of Mechanical Sciences, 122: 384–391, 2017, doi: 10.1016/j.ijmecsci.2016.10.016.

Kienzler R., Schneider P., Second-order linear plate theories: Partial differential equations, stress resultants and displacements, International Journal of Solids and Structures, 115–116: 14–26, 2017, doi: 10.1016/j.ijsolstr.2017.01.004.

Adámek V., The limits of Timoshenko beam theory applied to impact problems of layered beams, International Journal of Mechanical Sciences, 145: 128–137, 2018, doi: 10.1016/j.ijmecsci.2018.07.001.

Magnucki K., Witkowski D., LewiŃski J., Bending and free vibrations of porous beams with symmetrically varying mechanical properties – Shear effect, Mechanics of Advanced Materials and Structures,27(4): 325–332, 2020, doi: 10.1080/15376494.2018.1472350.

Magnucki K, Lewiński J., Analytical modeling of I-beam as a sandwich structure, Engineering Transactions, 66(4): 357–373, 2018, doi: 10.24423/EngTrans.898.20180809.

Magnucki K., Bending of symmetrically sandwich beams and I-beams – Analytical study, International Journal of Mechanical Sciences, 150: 411–419, 2019, doi: 10.1016/j.ijmecsci.2018.10.020.

Magnucki K., Lewinski J., Cichy R., Bending of beams with bisymmetrical cross sections under non-uniformly distributed load – Analytical and numerical-FEM studies, Archive of Applied Mechanics, 89(10): 2103–2114, 2019, doi: 10.1007/s00419-019-01566-5.




DOI: 10.24423/EngTrans.1129.20200214