On the 2D models of plates and shells including the transversal shear

Abstract

The 2D Kirchhoff–Love (KL) theory and the Timoshenko–Reissner (TR) theory for thin shells made of the transversally isotropic, homogeneous material are discussed. For the cyclic‐symmetric deformations of shells of revolution the asymptotic analysis of stress–strain states is performed. Two simple linear problems for double‐periodic deformations of plates are studied applying the exact 3D theory and the 2D approximate theories. From these problems it follows that the KL theory is asymptotically correct because it gives the first term of asymptotic expansion of the 3D solution. The TR theory is asymptotically incorrect. It gives correctly the first term only and incorrectly the second term. But if the transversal shear modulus is comparatively small then this theory gives the main part of the second term. The case of the extremely small shear modulus is discussed. As an example the multi‐layered plate with alternating hard and soft isotropic layers is studied.

The vibrations and stability of a prestressed plate on an elastic foundation

“…(7.1) and (7.2), while the approximate values of the parameter p can be found from the two-dimensional models considered in Sections 4-6. The parameter u (see (6.2)) corresponds to 10 , the parameter v (see (6.3)) corresponds to 20 , while the parameter k (see (4.4)) or the parameter t 1 from (5.11) corresponds to 30 . Moreover, the parameter t 2 from (5.11) is close to the value of 21 , with certain assumptions, 12 which is confirmed by the numerical results presented below.…”

Bending of a two-layer beam with non-rigid contact between the layers

“…Then, the problem has a periodic solution, to construct which we use an algorithm 5,6 proposed earlier. We put (2.1) and henceforth omit the hat over the letters.…”

Nonclassical theories for bending analysis of orthotropic circular plate

“…Some of these theories are reviewed in (Altenbach 1998, Reddy andWang 2000). Asymptotic correctness of the Timoshenko-Reissner theory for thin shells with small shear modulus is discussed in (Tovstik and Tovstik 2007).…”