Pure Bending Analysis of Isotropic Thin Rectangular Plates Using Third-Order Energy Functional

Abstract

Studies previously carried out on pure bending analysis of isotropic thin rectangular plate using total potential energy functional have mainly used second order (Ritz energy function) and fourth order (Galerkin and work error energy functional). The objective of this study is pure bending analysis of isotropic thin rectangular CCCC, CSCS, CSSS and SCCC plates. In this paper, third order energy functional is used for analysis. The Rayleigh Ritz energy method of direct variation approach for plate analysis is adopted. The third order energy functional method is derived from first principle by using equations and principles of theory of elasticity. Polynomial series is used to formulate the approximate shape functions for the plate with various boundary conditions. Direct variational calculus was applied on the third-order energy function to obtain the coefficient of deflection which was used to determine the stiffness component for the plates. The coefficients of deflection for CCCC, CSCS, CSSS and SCCC are obtained for various aspect ratios (ranging from 1.0 to 2.0 at the increment of 0.1). The various coefficient of deflection is used to obtain the deflection at the center of the plates. The central deflection of the various plate with aspect ratio (ranging from 1.0 to 2.0 at the increment of 0.1) from this studies were compared with the values of Szilard (2004) and Timoshenko and Krieger (1959). From the comparison, it was observed that the maximum percentage difference recorded for CCCC, CSCS and CSSS plates was 3.58, 4.91 and 1.79 respectively. While for SCCC there was no significant difference. These values of percentage differences are relatively small, which indicates that the method of this present study is considered adequate, satisfactory and reliable for analyzing pure bending isotropic thin rectangular plate.

Spring-Value in Kirchhoff-Love Plate: Displacement, Buckling, Pure-Shear, Vibration

“…At s * = 1, "N cr ≡ 8.73 here, and 8.986" in Ref. [2]; only the present method is able to track the strength up to s * = ∞ as demonstrated in Table 5 for CSCC; Oba, et al, [20] supplied the comparable displacements; omitted in Timoshenko's works.…”

“… " may be interpreted as principal loading curvature that must align with "(∂ 2 w/∂x 2 )" approximately or else find (χ) by equation (20). So, Figure 4, explaining Mohr's loading curvatures will supply the critical solution curvature.…”

Tuning the hydraulic resistance by swelling-induced buckling of membranes in high-aspect-ratio microfluidic devices

“…k depends on the boundary conditions at the edges of the membrane, the shape of the membrane, and the stress distribution through the membrane, which makes it difficult to estimate a sufficient and precise theoretical value for our experimental case. 19,21,35…”

“…k depends on the boundary conditions at the edges of the membrane, the shape of the membrane, and the stress distribution through the membrane, which makes it difficult to estimate a sufficient and precise theoretical value for our experimental case. 19,21,35 By comparing σ and σ b , we see that γ that causes the membrane to buckle is depending on the inverse of the aspect ratio by γ − 1 ∝ (b/h m ) −2 . This dependence may provide potential access for controlled buckling of spatially selective parts or membranes in microfluidic devices at defined γ b by adjusting the aspect ratio.…”