On shear correction factors in the non-linear theory of elastic shells

Abstract

a b s t r a c tTheoretical values of two correction factors a s = 5/6 and a t = 7/10 are established for the respective transverse shear stress resultants and stress couples within the general, dynamically and kinematically exact, six-field theory of elastic shells. These values do not depend on the shell material symmetry, geometry of the base surface, the shell thickness, or any kind of kinematic and/or dynamic constraints. The analysis is based on the complementary energy density following from the transverse shear stresses acting only on the shell cross section. The appropriate quadratic and cubic distributions of the stresses across the thickness allow one to derive the consistent constitutive equations for the transverse shear stress resultants and stress couples with a s and a t as the respective correction factors. Four numerical examples of highly non-linear shell structures illustrate the influence of different values of a s and a t on the results. In particular, some influence of a t is noticed on the placement of bifurcation points. In dynamic problem of flight of three intersecting plates analysed with Newmark-type temporal algorithm, the value of a t influences the moment at which the relative error of total energy of the system begins to grow indefinitely leading to the solution failure.

On a consistent finite-strain plate theory based on three-dimensional energy principle

“…[30,31]). The overall two-dimensional field equation is obtained by an integration, but the concerned two-dimensional variable likeS (andx in a dynamic system) in (3.14) 1 is treated as a whole instead of being expanded in series.…”

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“…Hence, the deformations of the micropolar plate or shell are described by the translation vector v and the rotation vector θ which are defined at the base surface M. Using the direct approach the basics of the linear theory of micropolar shells are summarized in [Eremeyev and Zubov 2008;; see also Appendix D in [Eremeyev et al 2013]. The governing equations of the micropolar shells and plates coincide with the relations of the general 6-parameter nonlinear shell theory presented in [Chróścielewski et al 2004;Libai and Simmonds 1998] in the case of small deformations derived using the through-the-thickness integration procedure.…”

“…Obviously, in this theory of shells the action of the drilling moment M α3 is taken into account. For example, such possibility may be useful to describe the interaction of the shell and the rigid body or for the description of the deformations of multifolded plates, see [Chróścielewski et al 2004]. The static and kinematic boundary conditions take the form…”

“…This theory can be derived using a direct approach [Eremeyev et al 2013;Eremeyev and Zubov 2008;Rubin 2000] or based on the reduction of the three-dimensional motion equations [Chróścielewski et al 2004;Libai and Simmonds 1998]. The mathematical study of boundary-value problems was performed in [Bîrsan and Neff 2013;2014;Eremeyev and Lebedev 2011], while various solutions and finite-element calculations are presented in [Chróścielewski et al 2004;Eremeyev and Zubov 2008].…”

Novel Layerwise Shear Correction Factors for Zigzag Theories of Circular Sandwich Plates with Functionally Graded Layers

“…Extending an approach previously used for the composite beams, Nguyen et al (2008) proposed shear correction factors for the power-law FGM plates using the shear strain energy equivalence method. Chróścielewski, et al (2010) suggested using two shear correction factors for the geometrically nonlinear elastic regular and irregular shells. Menaa et al (2012) used a concept similar to Nguyen et al (2008), but presented an expression for plates made of sigmoid-law functionally graded materials.…”