Abstract
Static inconsistencies that arise when modelling the flexural behaviour of beams, plates and shells with clamped boundary conditions using a certain class of axiomatic, higher-order shear deformation theory are discussed. The inconsistencies pertain to displacement-based theories that enforce conditions of vanishing shear strain at the top and bottom surfaces a priori. First it is shown that the essential boundary condition of vanishing Kirchhoff rotation perpendicular to an edge (w,x=0 or w,y=0) is physically inaccurate, as the rotation at a clamped edge may in fact be non-zero due to the presence of transverse shear rotation. As a result, the shear force derived from constitutive equations erroneously vanishes at a clamped edge. In effect, this boundary condition overconstrains the structure leading to underpredictions in transverse bending deflection and overpredictions of axial stresses compared to high-fidelity 3D finite element solutions for thick and highly orthotropic plates. Generalised higher-order theories written in the form of a power series, as in Carrera’s Unified Formulation, do not produce this inconsistency. It is shown that the condition of vanishing shear tractions at the top and bottom surfaces need not be applied a priori, as the transverse shear strains inherently vanish if the order of the theory is sufficient to capture all higher-order effects. Finally, the transverse deflection of the generalised higher-order theories is expanded in a power series of a non-dimensional parameter and used to derive a material and geometry dependent shear correction factor that provides more accurate solutions of bending deflection than the classical value of 5/6.
Highlights
The presence of these terms leads to essential boundary conditions dw;x 1⁄4 0 and dw;y 1⁄4 0 and associated natural boundary conditions on the higher-order moments, when the governing field equations are derived in a variationally consistent manner using the principle of virtual displacements
In the following two examples, we investigate the effect this has on the governing field equations and boundary conditions as derived in a variationally consistent manner from the principle of virtual displacements (PVD)
The results show that Reddy’s higher-order plate theory (RHOT) always leads to an underprediction of w compared to 3D FEM which arises due to the stiffening effect of w;x 1⁄4 0 at the boundary
Summary
In practical applications composite laminates are typically modelled as thin plates and shells because the thickness dimension t is an order of magnitude smaller than representative in-plane dimensions lx and ly. Using this framework Reissner [45,46] developed an enhanced theory from the piecewise-linear in-plane stresses of CPT and parabolic transverse shear equations derived from Cauchy’s 3D equilibrium equations This approach has been applied to symmetrically laminated straight-fibre composites [47], variable stiffness composites [48] and sandwich panels where ZZ effects are important [49]. The advantage of these mixed variational statements is that both axial and transverse shear stresses can generally be captured accurately directly without the need for further post-processing steps. Extended reviews of ZZ theories and the application of RMVT to layered structures are provided by Carrera [10,5,53]
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