Abstract
The Kantorovich variational method was used in this study to solve the flexural problem of Kirchhoff-Love plates with two opposite edges x=±a/2 clamped and the other two edges y=±b/2 simply supported, for the case of uniformly distributed transverse load over the entire plate domain. The plate considered was assumed homogeneous, and isotropic. The total potential energy functional for the Kirchhoff-Love plate was found as the sum of the potential energy of the applied distributed load and the strain energy functional of the plate. The symmetrical nature of the plate and the load was used to choose the deflection function as a single series of infinite terms involving the cosine function of the y coordinate variable with the unknown function being dependent on the x coordinate variable. Integration with respect to the y coordinate variable simplified the total potential energy functional to be dependent on the unknown function of x, and its derivatives. Euler-Lagrange differential equations were used to find the differential equations of equilibrium of the plate as a system of homogeneous fourth order ordinary differential equations (ODE) in the unknown function. The system of fourth order ODE was solved using the method of differential D operators, or the method of trial functions to obtain the general solution as the linear superposition of the homogeneous and particular solutions. The demands of symmetry of unknown function about the point x= 0, and the boundary conditions at the clamped edges were used to obtain all the four integration constants; thus, completely determining the deflection. Bending moment distributions were determined using the bending moment deflection relations. The deflection was determined at the centre of the plate. The bending moment values were also determined at the centre of the plate, and at the middle of the clamped edges. Expressions obtained for the deflection at the centre, and bending moment values were found to be single series of infinite terms with highly converging properties. The solutions obtained converged to the exact solutions with the use of only four terms of the series.
Highlights
Plates are three dimensional structures that have one dimension that is very small relative to the other two dimensions [1, 2]
The analysis of plates employs the three-dimensional theory of elasticity [1]. They are governed by the simultaneous requirements of the generalized Hooke’s law of linear elasticity, the strain – displacement relations, and differential equations of equilibrium, subject to the boundary conditions imposed by the restraints and the load [3,4,5,6]
The general aim is to use the Kantorovich variational method to solve the flexural problem of Kirchhoff-Love plates with CSCS edges for the case of uniformly distributed transverse load
Summary
Plates are three dimensional structures that have one (transverse) dimension that is very small relative to the other two (in-plane) dimensions [1, 2] They are commonly used in structural applications in buildings, machine parts, ships, aircrafts, aerospace structures and naval facilities. They are commonly subjected to transversely applied distributed and/or point forces and carry such forces by the development of bending moments in the two in-plane directions and a twisting moment. Three broad theories exist for modelling plates They are: (i) Thin plate theories (8...10 ≤ ⁄h ≤ 8...100), where is the least in-plane dimension, and h is the plate thickness: – small deformation thin plate theories (Kirchhoff-Love plate theory); – large deformation thin plate theories (von Karman Plate theory). They are: (i) Thin plate theories (8...10 ≤ ⁄h ≤ 8...100), where is the least in-plane dimension, and h is the plate thickness: – small deformation thin plate theories (Kirchhoff-Love plate theory); – large deformation thin plate theories (von Karman Plate theory). (ii) Moderately thick plate theories (Reissner [9] plate theory, Mindlin [10] plate theory) and, (iii) Thick plate theories ( ⁄h ≤ 8...10)
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