A brief history of first-order shear-deformable beam and plate models

This paper presents a brief history of beam and plate models in elasticity, that take into account the rotary inertia and shear contribution. We show that Bresse (1859) already in the mid XIXth century rigorously derived the set of equations for curved shear and axially extensible beams in dynamics, although without shear correction factor. When restricted to straight beams, Bresse (1859) obtained a two-field beam kinematics composed of independent deflection and rotation variables. These equations have been generalized by Timoshenko (1913; 1916; 1920; 1921; 1922), without direct reference to the works of Bresse, by a slight modification of the shear stiffness, that incorporates a shear correction factor generally different from unity. The calibration of the shear correction factor with respect to three-dimensional elasticity solutions has been elaborated jointly by Ehrenfest and Timoshenko in the second decade of the XXth century (the calibration of the shear correction factor for rectangular cross section has been finalized by Timoshenko in 1922, even if the basic results were available in his letter dated from 1913, in collaboration with Ehrenfest). Whereas Bresse-Timoshenko-Ehrenfest equations have been elaborated mainly during more than half a century between 1859 and 1922 (even if researches on shear beam theories were still active), the Uflyand-Mindlin plate theory has been built in a more compact period, mainly between 1944 and 1951. Whereas Reissner (1944, 1945) developed a static (or stress-based) formulation, the kinematic theory of Uflyand-Mindlin plate model has been first elaborated by Bolle (1947) and Hencky (1947) for static setting, before the full generalization to dynamics by Uflyand a year later, in 1948, and its complete variational derivation by Mindlin (1951). A lot of efforts were spent since almost half of the century to calibrate the shear correction factor and to enrich the kinematics of Uflyand-Mindlin plate theory. It is quite surprising that the shear correction factors of both the Bresse-Timoshenko-Ehrenfest beam model and its Uflyand-Mindlin plate analogy were already implicitly available in the paper of Timoshenko (1922), performed, in his own testimony with Ehrenfest. (C) 2019 Elsevier Ltd. All rights reserved.