Beam Element with a 3D Response for Shear Effects
Publication: Journal of Engineering Mechanics
Volume 144, Issue 1
Abstract
This paper presents a beam element for which the cross section remains neither plane nor orthogonal to the beam axis. The cross section with arbitrary geometry is modeled by two-dimensional (2D) finite elements. The element kinematics is defined by the displacement shapes of the cross sections and the axial functions of their corresponding averaged movements. The deformed displacement shapes are obtained by minimizing the potential energy of a beam slice submitted to the compatibility constraints of the kinematics framework. The recursive procedure that yields the higher-order deformed displacement shapes is initiated with the rigid-body components of the displacement field. The mechanical model is validated through examples of linear-elastic materials with isotropic and nonisotropic cross sections. Higher-order beam elements yield three-dimensional (3D) stresses and strains that correspond to elasticity solutions and 3D solid element models, for concentrated and distributed loads. The beam elements do not require shear coefficients. The first-order formulation is extended to nonlinear materials.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The writer thanks Prof. Benjamin Ernani Diaz, Prof. Maria Paola Santisi d’Avila, and Prof. Jose Kimio Ando for their valuable comments.
References
Alpdogan, C., Dong, S. B., and Taciroglu, E. (2010). “A method of analysis for end and transitional effects in anisotropic cylinders.” Int. J. Solids Struct., 47(7–8), 947–956.
Antman, S. S. (1974). “Kirchhoff’s problem for nonlinearly elastic rods.” Quart. Appl. Math., 32(3), 221–240.
Bathe, K. J. (1982). Finite element procedures in engineering analysis, Prentice-Hall, Englewood Cliffs, NJ.
Bickford, W. B. (1982). “A consistent higher order beam theory.” Dev. Theor. Appl. Mech., 11, 137–150.
Challamel, N. (2011). “Higher-order shear beam theories and enriched continuum.” Mech. Res. Commun., 38(5), 388–392.
Challamel, N., and Ameur, M. (2013). “Out-of-plane buckling of microstructured beams: Gradient elasticity approach.” J. Eng. Mech., 1036–1046.
Cowper, G. R. (1966). “The shear coefficient in Timoshenko’s beam theory.” J. Appl. Mech., 33(2), 335–340.
Dong, S. B., Alpdogan, C., and Taciroglu, E. (2010). “Much ado about shear correction factors in Timoshenko beam theory.” Int. J. Solids Struct., 47(13), 1651–1665.
Dong, S. B., Kosmatka, J. B., and Lin, H. C. (2001). “On Saint-Venant’s problem for an inhomogeneous, anisotropic cylinder—Part I: Methodology for Saint-Venant solutions.” J. Appl. Mech., 68(3), 376–381.
Han, S., and Bauchau, O. A. (2015). “On the solution of Almansi–Michell’s problem.” Int. J. Solids Struct., 75–76, 156–171.
Heyliger, P. R., and Reddy, J. N. (1988). “A higher order beam finite element for bending and vibration problems.” J. Sound Vib., 126(2), 309–326.
Ieşan, D. (1986). “On Saint-Venant’s problem.” Arch. Ration. Mech. Anal., 91(4), 363–373.
Ieşan, D. (1987). Saint-Venant’s problem, Springer, Berlin.
Jouravski, D. I. (1856). “Remarques sur la résistance d’un corps prismatique et d’une pièce composée en bois ou en tôle de fer à une force perpendiculaire à leur longueur’.” Ann. Ponts Chaussees, 12, 328–351 (in French).
Kaneko, T. (1975). “On Timoshenko’s correction for shear in vibrating beams.” J. Phys. D Appl. Phys., 8(16), 1927–1936.
Kazic, M., and Dong, S. B. (1990). “Analysis of restrained torsion.” J. Eng. Mech., 870–891.
Levinson, M. (1981). “A new rectangular beam theory.” J. Sound Vib., 74(1), 81–87.
Lin, H. C., and Dong, S. B. (2006). “On the Almansi-Michell problems for an inhomogeneous, anisotropic cylinder.” J. Mech., 22(1), 51–57.
Lin, H. C., Dong, S. B., and Kosmatka, J. B. (2001). “On Saint-Venant’s problem for an inhomogeneous, anisotropic cylinder—Part III: End effects.” J. Appl. Mech., 68(3), 392–398.
Mason, W. E., and Herrmann, L. R. (1968). “Elastic shear analysis of general prismatic beams.” J. Eng. Mech. Div., 94(4), 965–983.
Michell, J. H. (1901). “The theory of uniformly loaded beams.” Quart. J. Pure Appl. Math., 32, 28–42.
Rankine, W. J. M. R. (1858). A manual of applied mechanics, R. Griffin, Glasgow, U.K.
Reddy, J. N. (1984a). “A simple higher-order theory for laminated composite plates.” J. Appl. Mech., 51(4), 745–752.
Reddy, J. N. (1984b). Energy and variational methods in applied mechanics, Wiley, New York.
Reissner, E. (1972). “On one-dimensional finite-strain beam theory: The plane problem.” Zeitschrift für angewandte Mathematik und Physik ZAMP, 23(5), 795–804.
Reissner, E. (1973). “On one-dimensional large-displacement finite-strain beam theory.” Stud. Appl. Math., 52(2), 87–95.
Reissner, E. (1983). “Further considerations on the problem of torsion and flexure of prismatical beams.” Int. J. Solids Struct., 19(5), 385–392.
Saint-Venant, A. J. C. B. (1856). “Mémoire sur la flexion des prismes.” J. Math. Pure Appl., 1, 89–189 (in French).
Schramm, U., Kitis, L., Kang, W., and Pilkey, W. D. (1994). “On the shear deformation coefficient in beam theory.” Finite Elem. Anal. Des., 16(2), 141–162.
Schulz, M., and d’Avila, M. P. S. (2010). “Analysis of reinforced concrete shells with transverse shear forces.” J. Struct. Eng., 837–848.
Schulz, M., and Filippou, F. C. (1998). “Generalized warping torsion formulation.” J. Eng. Mech., 339–347.
Simo, J. C. (1985). “A finite strain beam formulation. The three-dimensional dynamic problem. I.” Comput. Meth. Appl. Mech. Eng., 49(1), 55–70.
Simo, J. C., and Vu-Quoc, L. (1986). “A three-dimensional finite-strain rod model. II: Computational aspects.” Comput. Meth. Appl. Mech. Eng., 58(1), 79–116.
Stephen, N. G. (1980). “Timoshenko’s shear coefficient from a beam subjected to gravity loading.” J. Appl. Mech., 47(1), 121–127.
Timoshenko, S. P. (1921). “On the correction for shear of the differential equation for transverse vibrations of prismatic bars.” Philos. Mag., 41(245), 744–746.
Timoshenko, S. P. (1922). “On the transverse vibrations of bars of uniform cross-section.” Philos. Mag., 43(253), 125–131.
Timoshenko, S. P. (1945). “Theory of bending, torsion and buckling of thin-walled members of open cross section.” J. Franklin Inst., 239(4), 249–268.
Vlasov, V. Z. (1961). Thin-walled elastic beams, Monson, Jerusalem.
Wang, C. M., Reddy, J. N., and Lee, K. H. (2000). Shear deformable beams and plates: Relationships with classical solutions, Elsevier, Oxford, U.K.
Willam, K., Pramono, E., and Sture, S. (1989). “Fundamental issues of smeared crack models.” Fracture of Concrete and Rock: SEM-RILEM Int. Conf., S. P. Shah and S. E. Swartz, eds., Springer, New York, 142–157.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 144 • Issue 1 • January 2018
Copyright
©2017 American Society of Civil Engineers.
History
Received: Sep 6, 2016
Accepted: Jun 7, 2017
Published online: Oct 31, 2017
Published in print: Jan 1, 2018
Discussion open until: Mar 31, 2018
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.