Beams - Mechanics of Structures: Variational and Computational Methods - page 751

Normally the transverse shear calculations depend on material properties. Both the shear
center and the shear deformation coefficients usually depend on Poisson's ratio. The com-
puter results show that the Trefftz shear center, which does not depend on Poisson's ratio,
is virtually the same as the shear center obtained from the transverse shear boundary value
problem.
References
Bornscheuer, F.W., 1952, “Systematische Darstellung des Biege-und Verdrehvorganges unter beson-
derer Ber ucksichtigung der W olbkrafttorsion” Der Stahlbau , Vol. 21, p. 1.
Chang, P.Y., Thasanatorn, C. and Pilkey, W.D., 1975, Restrained warping stresses in thin-walled open
sections, J. Struct. Div. , ASCE, Vol. 101, pp. 2467-2472.
Copper, C., 1993, “Thermoelastic Solutions for Beams,” Ph.D. Thesis, University of Virginia.
Cowper, G.R., 1966, The shear coefficient in Timoshenko's beam theory, J. Appl. Mech. , Vol. 33, p. 2.
Goodier, J.N., 1938, On the problems of the beam and the plate in the theory of elasticity, Trans. Royal
Society of Canada , Vol. 32, pp. 65-88.
Herrmann, L.R., 1965, Elastic torsional analysis of irregular shapes, J. Eng. Mech. Div. , ASCE, Vol. 91,
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340.
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Mason, W.F. and Herrmann, L.R., 1968, Elastic shear analysis of general prismatic beams, J. Eng. Mech.
Div. , ASCE, Vol. 94, pp. 965-983.
Pilkey, W.D., 1994, Formulas for Stress, Strain, and Structural Matrices , Wiley, NY.
Pilkey, W.D., 2002, Analysis and Design of Elastic Beams, Computational Methods , Wiley, NY.
Pilkey, W.D. and Liu, Y., 1993, Field theory: A two-dimensional case for not using finite or boundary
elements, Fin. Elem. in Anal. & Design , Vol. 13, pp. 127-136.
Schramm, U., Kitis, L., Kang. W. and Pilkey, W.D., 1994, On the shear deformation coefficient in beam
theory, Fin. Elem. in Anal. & Design , Vol. 16, pp. 141-162.
Surana, K.S., 1979, Isoparametric elements for cross-sectional properties and stress analysis of beams,
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Trefftz, E., 1936, Uber den Schubmittelpunkt in einem durch eine Einzellast gebogenen Balken,
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and Computional Algorithms in Finite Element Analysis , Bathe, K.J., Oden, J.T. and Wunderlich,
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Problems
Displacements and Forces
12.1 Prove that a “plane cross-section remains plane” when a straight beam is subjected
only to bending moments at the end points. Assume that the shape of the cross-section
remains unchanged during deformation.