Discussion of ‘‘Exact Finite Element for Nonuniform Torsion of Open Sections’’ by Magdi Mohareb and Farhood Nowzartash February 2003, Vol. 129, No. 2, pp. 215–223. DOI: 10.1061/ASCE0733-94452003129:2215 Katy Saade ´ 1 ; Bernard Espion, M.ASCE 2 ; and Guy Warze ´ e 3 1 Graduate Research Assistant, Dept. of Continuum Mechanics and Dept. of Civil Engineering, Univ. Libre de Bruxelles ULB, C.P. 194/5, Av. F. D. Roosevelt 50, 1050 Brussels, Belgium. E-mail: ksaade@ smc.ulb.ac.be 2 Associate Professor, Dept. of Civil Engineering, Univ. Libre de Brux- elles ULB, Brussels, Belgium. E-mail: bespion@ulb.ac.be 3 Professor, Dept. of Continuum Mechanics, Univ. Libre de Bruxelles ULB, Brussels, Belgium. E-mail: gwarzee@smc.ulb.ac.be The authors present a finite-element formulation for the study of the elastic nonuniform torsion of open cross sections based on Vlassov assumptions. The paper includes a general introduction that reviews solutions in the literature and proposes to develop an exact finite-element formulation that satisfies the internal equilib- rium conditions. The assumptions and the variational formulation are clearly developed in order to compare approximate beam fi- nite elements based on polynomial shape functions and the exact homogeneous formulation that leads to exact results for the nodal angle of twist, its derivative and the corresponding internal forces. Although the developments are very clear and useful, the dis- cussers want to complete the state of the art and mention that Dvorkin et al. 1989, De Ville 1989, and Batoz and Dhatt 1990 did similar work 14 years ago. To the best knowledge of the discussers, the major studies of De Ville 1989 and Batoz and Dhatt 1990 were only published in French, which may explain why they eluded the authors’ attention. But the common point of all these studies with the discussed work is the use of the same nonpolynomial interpolation that leads to the exact static solution by using the minimum number of elements. Identical assumptions and kinematics are used in order to formulate the same variational formulation, equilibrium equation, and boundary conditions. The unified scope is reached by the same method: the equilibrium equation Eq. 12 in the paper under discussion, Equation 3.6.104a in Batoz and Dhatt 1990 and Equation 3.52 in De Ville 1989 is solved under the assumption of no member twisting moment or bimoments. The twisting angle is related to the de- grees of freedom by including hyperbolic or exponential func- tions that result from the analytical solution of the equilibrium equation. Similar calculations are done in order to get the shape function vector. The authors give the explicit expression of the stiffness matrix in Eqs. 33 – 42. Additional mathematical op- erations using the definition of hyperbolic cosines and sines are necessary to reduce this tedious form and to obtain a simpler formulation as presented by Batoz and Dhatt 1990 in Equation 3.6.112 or by De Ville 1989 in Equation 7.22. The given formu- lation is discussed in the three works. The uniform de Saint Venant torsion is presented as a limit case. The polynomial func- tions are found to also be a particular case of the general torsional problem that includes both uniform torsion de Saint Venant and nonuniform torsion Vlassov. It is also important to note that Gunnlaugsson et al. 1982 and Shakourzadeh et al. 1995 went further by formulating a Bensco- ter beam theory for both open and closed cross sections with hyperbolic shape functions. The same previously listed develop- ments are based on enriched kinematics including the influence of the transverse shear strain. The Vlassov hypothesis that the trans- verse shear strain must vanish at the centerline is relaxed. The hyperbolic functions are again used in order to obtain the exact solution of the nonuniform torsion of open and closed cross sec- tions. It is interesting to note that the stiffness matrix obtained by the previous formulations based on Vlassov kinematics can be de- rived from the one Eq. 22 in Shakourzadeh et al. 1995 based on Benscoter kinematics by neglecting the ratio of the Saint- Venant torsional constant to the sectorial moment of inertia. This is justified by the well-known difference of torsional behavior between closed and open cross sections. Thin open sections have very small torsional rigidity and exhibit large amounts of warping effects. Closed profiles resist to uniform torsion by a global rigid- ity related to the circulation of a flow along the contour and by its local rigidity specific to each thin wall as if the beam were con- stituted by the assembly of longitudinal bands. The de Saint Ve- nant part in the total torsional behavior is more important for closed cross sections than for open cross sections. References Batoz, J. L., and Dhatt, G. 1990. ‘‘Mode ´lisation des structures par e ´le ´- ments finis Modelling of structures with finite elements.’’ Herme `s edition, Paris in French. De Ville De Goyet, V. 1989. ‘‘L’analyse statique non line ´aire par la me ´thode des e ´le ´ments finis des poutres a ` section non syme ´trique Nonlinear static analysis of beams with unsymmetrical cross sections by the finite element method.’’ PhD thesis, Univ. de Lie `ge, Belgium in French. Dvorkin, E. N., Celentano, D., Cuitino, A., and Gioia, G. 1989. ‘‘A Vlassov beam element.’’ Comput. Struct., 331, 187–196. Gunnlaugsson, G. A., and Pedersen, P. T. 1982. ‘‘A finite element for- mulation for beams with thin walled cross-sections.’’ Comput. Struct., 156, 691–699. Shakourzadeh, H., Guo, Y. C., and Batoz, J. L. 1995. ‘‘A torsional bending element for thin-walled beams with open and closed cross sections.’’ Comput. Struct., 556, 1045–1054. Closure to ‘‘Exact Finite Element for Nonuniform Torsion of Open Sections’’ by Magdi Mohareb and Farhood Nowzartash February 2003, Vol. 129, No. 2, pp. 215–223. DOI: 10.1061/ASCE0733-94452003129:2215 Magdi Mohareb, M.ASCE, 1 and Farhood Nowzartash 2 1 Associate Professor, Dept. of Civil Engineering, Univ. of Ottawa, 161 Louis Pasteur St. A025, P.O. Box 450, Stn. A, Ottawa ON, Canada K1N 6N5. E-mail: mmohareb@uottawa.ca 2 Assistant Professor, Dept. of Civil Engineering, Bahai Institute for Higher Education, Iran, Tehran. DISCUSSIONS AND CLOSURES 1420 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / SEPTEMBER 2004