Abstract
It was found that additional shear stress arising from the bending moment should be involved in the formulation due to the effect of variable cross section, which is quite different from the conventional method of analysis for the prismatic members. This paper focuses on the analytical formulation, properties and distribution regularity of the shear stress for the elastic tapered beams with rectangular cross-section. Currently, an analytical expression for the shear stress in elastic tapered beams is derived theoretically based on the theory of elasticity. Most notably, it is strictly proved in mathematics that the additional shear stresses arising from bending moment are self-balanced which only change the distribution of shear stress in the section but will not affect the shear stress resultant. This new finding will be beneficial to promoting the development and consummation of non-prismatic beam theory. The correctness of the theoretical formula have been verified through comparisons with finite element (FE) results of two groups with a total of 12 three-dimensional FE models. The validity of self-balanced properties of additional shear stress in tapered beams under pure bending is additionally verified by using Free Body Cuts in ABAQUS program.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auricchio F, Balduzzi G, Lovadina C (2015) The dimensional reduction approach for 2D non-prismatic beam modelling: A solution based on Hellinger-Reissner principle. International Journal of Solids and Structures 15:264–276, DOI: https://doi.org/10.1016/j.ijsolstr.2015.03.004
Balduzzi G, Aminbaghai M, Auricchio F, Füssl J (2018) Planar Timoshenkolike model lor multilayer non-prismatic beams. International Journal of Mechanics and Materials in Design 14(1):51–70, DOI: https://doi.org/10.1007/s10999-016-9360-3
Balduzzi G, Aminbaghai M, Sacco E, Füssl J, Eberhardsteiner J, Auricchio F (2016) Non-prismatic beams: A simple and effective Timoshenkolike model. International Journal of Solids and Structures 90:236–250, DOI: https://doi.org/10.1016/j.ijsolstr.2016.02.017
Balduzzi G, Hochreiner G, Füssl J (2017) Stress recovery from one dimensional models for tapered bi-symmetric thin-walled I beams: Deficiencies in modern engineering tools and procedures. Thin Walled Structures 119:934–944, DOI: https://doi.org/10.1016/j.tws.2017.06.031
Beltempo A, Balduzzi G, Alfano G, Auricchio F (2015) Analytical derivation of a general 2D non-prismatic beam model based on the Hellinger-Reissner principle. Engineering Structures 101:88–98, DOI: https://doi.org/10.1016/j.engstruct.2015.06.020
Bertolini P, Eder M, Taglialegne L, Valvo P (2019) Stresses in constant tapered beams with thin-walled rectangular and circular cross sections. Thin Walled Structures 137:527–540, DOI: https://doi.org/10.1016/j.tws.2019.01.008
Bleich F (1932) Stahlhochbauten, chapter 16:80–85. Verlag von Julius Springer, Berlin, Germany
El-Mezaini N, Balkaya C, Çitipitioğlu E (1991) Analysis of frames with nonprismatic members. Journal of Structural Engineering 117(6): 1573–1592, DOI: https://doi.org/10.1061/(ASCE)0733-9445(1991)117:6(1573)
Gere JM, Goodno BJ (2010) Mechanics of materials, 7th edition. Cengage Learning, Toronto, Canada
Gere JM, Timoshenko SP (1984) Mechanics of materials, 2nd SI edition. Van Nostrand Reinhold Co., New York, NY, USA
Hassanein MF, Kharoob OF (2014) Shear buckling behavior of tapered bridge girders with steel corrugated webs. Engineering Structures 74:157–169, DOI: https://doi.org/10.1016/j.engstruct.2014.05.021
Hassanein MF, Kharoob OF (2015) Linearly tapered bridge girder panels with steel corrugated webs near intermediate supports of continuous bridges. Thin Walled Structures 88:119–128, DOI: https://doi.org/10.1016/j.tws.2014.11.021
Hibbeler RC, Fan SC (2011) Mechanics of materials, 8th edition. Prentice Hall, Englewood, NJ, USA
Hodges DH, Ho JC, Yu W (2008) The effect of taper on section constants for in-plane deformation of an isotropic strip. Journal of Mechanics of Materials and Structures 3:425–440, DOI: https://doi.org/10.2140/jomms.2008.3.425
Hodges DH, Rajagopal A, Ho JC, Yu W (2010) Stress and strain recovery for the in-plane deformation of an isotropic tapered strip-beam. Journal of Mechanics of Materials and Structures 5:963–975, DOI: https://doi.org/10.2140/jomms.2010.5.963
Oden JT, Ripperger EA (1981) Mechanics of elastic structures. Hemisphere Publishing Corporation, New York, NY, USA
Orr JJ, Ibell TJ, Darby AP, Evernden M (2014) Shear behaviour of non-prismatic steel reinforced concrete beams. Engineering Structures 71:48–59, DOI: https://doi.org/10.1016/j.engstruct.2014.04.016
Paglietti A, Carta G (2009) Remarks on the current theory of shear strength of variable depth beams. The Open Civil Engineering Journal 3:28–33, DOI: https://doi.org/10.2174/1874149500903010028
Rajagopal A, Hodges DH (2015) Variational asymptotic analysis for plates of variable thickness. International Journal of Solids and Structures 75:81–87, DOI: https://doi.org/10.1016/j.ijsolstr.2015.08.002
Rajagopal A, Hodges DH, Yu W (2012) Asymptotic beam theory for planar deformation of initially curved isotropic strips. Thin-Walled Structures 50:106–115, DOI: https://doi.org/10.1016/j.tws.2011.08.012
Timoshenko SP (1956) Strength of materials. Part II. Advanced theory and problems. David Van Nostrand, New York, NY, USA
Timoshenko SP, Goodier JN (1951) Theory of elasticity, 2nd edition. McGraw-Hill, New York, NY, USA
Trahair N, Ansourian P (2016a) In-plane behaviour of web-tapered beams. Engineering Structures 108:47–52, DOI: https://doi.org/10.1016/j.engstruct.2015.11.010
Trahair N, Ansourian P (2016b) In-plane behaviour of mono-symmetric tapered beams. Engineering Structures 108:53–58, DOI: https://doi.org/10.1016/j.engstruct.2015.11.011
Zhou M, Liu Z, Zhang J, Shirato H (2017) Stress analysis of linear elastic nonprismatic concrete-encased beams with corrugated steel webs. Journal of Bridge Engineeirng 22(6), DOI: https://doi.org/10.1061/(ASCE)BE.1943-5592.0001042
Zhou M, Zhang J, Zhong J, Zhao Y (2016) Shear stress calculation and distribution in variable cross sections of box girders with corrugated steel webs. Journal of Structural Engineering 142(6), DOI: https://doi.org/10.1061/(ASCE)ST.1943-541X.0001477
Acknowledgements
This study was supported by the National Natural Science Foundation of China under Grant No. 51808559. The authors also express their gratitude for the funding provided by the Natural Science Foundation of Hunan Province (Grant 2019JJ50770).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhou, M., Fu, H. & An, L. Distribution and Properties of Shear Stress in Elastic Beams with Variable Cross Section: Theoretical Analysis and Finite Element Modelling. KSCE J Civ Eng 24, 1240–1254 (2020). https://doi.org/10.1007/s12205-020-0772-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12205-020-0772-0