Analysis of critical imposed load of plate using variational calculus

Abstract

This work studied the critical load analysis of rectangular plates, carrying uniformly distributed loads utilizing direct variational energy calculus. The aim of this study is to establish the techniques for calculating the critical lateral imposed loads of the plate before deflection attains the specified maximum threshold, qiw as well as its corresponding critical lateral imposed load before the plate reaches an elastic yield point. The formulated potential energy by the static elastic theory of the plate was minimized to get the shear deformation and coefficient of deflection. The plates under consideration are clamped at the first and second edges, free of support at the third edge and simply supported at the fourth edge (CCFS). From the numerical analysis obtained, it is found that the critical lateral imposed loads (qiw and qip) increase as the thickness (t) of plate increases, and decrease as the length to width ratio increases. This suggests that as the thickness increases, the allowable deflection improves the safety of the plate, whereas an increase in the span (length) of the plate increases the failure tendency of the plate structure.

Exact Three-Dimensional Stability Analysis of Plate Using an Direct Variational Energy Method

“…Plate structures are used in structural engineering, mechanical engineering, and aerospace engineering etc., in making retaining walls, floor slabs, bridges, railways, ships etc. [2,3]. The plates are usually under the action of axially compressive or tensile loads acting in the mid-plane of the plate.…”

“…Saffron, and M. Eisenberger (2005) [14] used both first order shear deformation theory (FSDT) and higher order shear deformation theory (HSDT) in their analysis of isotropic plate while Reissner (1979) [15] applied the same to anisotropic plate material. Onyeka et al (2021) and Senjanović et al (2013) [17,18] utilized the analytical approach to get the exact polynomial displacement function from the governing equation. They did not apply trigonometric function which gives closer form solution than polynomial whose exact function tends to infinity.…”

“…They did not apply trigonometric function which gives closer form solution than polynomial whose exact function tends to infinity. Meanwhile, Reissner (1979), Ibearugbulem (2016), Onyeka et al (2021) and Senjanović et al (2013) [15][16][17][18] did not consider the effect of the in-plane load on the stability of the structure. The HSDT or moderately thick plate theory developed by previous scholars is an improvement of the FSDT, due to its avoidance of shear correction factor and has been applied successfully by different researchers in the stability analysis of thick plates [19][20][21][22][23][24][25].…”

Analytical Solution for the Static Bending Elastic Analysis of Thick Rectangular Plate Structures Using 3-D Plate Theory

“…Bending is the deformation of a plate towards its surface perpendicularly with the influence of moments and external forces [12]. Plate failure occurs when deformations grow beyond the critical applied load [13]. The stability and safety of plates depend on the value of this critical load, which becomes a major concern in the investigation of plate structures.…”

A Study on Stability Analysis of 3-D Shear Deformable Isotropic Plate Elastically Restrained against Rotation and Simply Supported in the two Adjacent Edges using Exact Displacement Potential

“…Considering the span-depth ratio (a/t), rectangular plates with 50 ≤ a/t ≤ 100 are classified as thin plate, 20 ≤ a/t ≤ 50 as moderately thick and a/t ≤ 20 as thick plate [5]. The unique properties of thick plates such as light weight, economy, ability to tailor the structural properties and its ability to withstand heavy loads [6], have attracted more research interest.…”