This article explains a mathematically consistent approach for solving the equations of Timoshenko’s beam theory for statically loaded beams. Theoretic sections 3.4 – 3.5 give a good description of the shear deformation and the primary approach for calculating deflections of beams under bending, taking into account both causes for deflection: bending moment and shear force. Values for the shear correction factor are discussed in section 4. This work was started to check the validity of an equation for deflection of a symmetrically loaded short rectangular beam with span/height ratio = 3 under four-point bending with upper-span/span ratio = 1/3. The exact solution is not presented here, but we can confirm that the presented theory, when applied for the mentioned loading scheme, leads to thi s equation using a shear correction factor k = 5/6.
- S.P. Timoshenko, History of Strength of Materials. State publ. house for technical-theor. Literat. Moscow. (1957) (in Russian)
- S.P. Timoshenko, J.N. Goodier, Theory of Elasticity. McGraw-Hill Book company Inc, (1951).
- S.P. Timoshenko Strength of Materials. Part I. Elementary theory and problems. Publ. D. van Nostrand company. II ed. (1948)
- S.P. Timoshenko, Phil. Mag. 41, 744-746 (1921).
- N.G. Stephen, J. Appl.Mech. Mar., 47 (1): 121-127 (1980)
- I. Kisiov, Strength of materials, III ed. Technika. (1963) (in Bulgarian “Съпротивление на материалите”)
- EN ISO 178:2019. Plastics – determination of flexural properties
- G.M. Gere, B.G Goodno, Mechanics of materials 8-th ed., SI. Cengage Learning (2013)
- S.P. Mandgakov, Mandichev GI, Veselinov KV et. al. Collection of Problems and Methodological Guidelines on Strength of Materials. Ed. by S.P Nedelchev. Technika. (1989) (in Bulgarian “СБОРНИК от задачи и методически указания по съпротивление на материалите”)
- K.T. Chan, K.F. Lai, N.G. Stephen, K. Young. J. of Sound and Vibr. 330 (14), pp. 3488-3497 (2011)
- H. Hadavinia, K. Gordnian, J. Karwatzki and A. Aboutorabi, SDHM, 2 No 4, pp.197-206 (2006)
- N.G. Stephen, J. of Sound and Vibr. 292, pp. 372–389 (2006)
- X.-F. Li, J. of S. and Vibr. 318 No 4-5, pp. 1210–1229 (2008)
- Jae-Hoon Kang, J. of Sound and Vibr. 333 No 14, pp. 3332–3337 (2014)
- Marie Brons, J.J. Thomsen, J. of Sound and Vibr. 459, 114856 (2019)
- G.R. Cowper, J. Appl. Mech., 33 No 2, pp. 335–340 (1966)
- T. Kaneko, J. Phys. D (Appl. Phys.) 8, No 16, pp. 1927-1936 (1975).
- A. Dı´az-de-Anda, J. Flores, et al., J. of Sound and Vibr. 331, No 26, pp. 5732–5744 (2012)
- T.H.G. Megson Aircraft Structures for engineering students. IV ed. ELSEVIER (2007)
- Ke Gong, Appl. Math. and Mech. (Engl. Ed.), 21, No 9 pp. 1091-1098 (2000)
- Yuwaraj M.Ghugal, Rajneesh Sharma, Lat. Amer. J. of Solids and Structures, 8, No 2, pp. 183 – 195 (2011)