An Analytical Solution to Rectangular Isotropic Kirchhoff Plates Having Two Adjacent Edges Free Using Two Single Series and the Boundary Collocation Method

1. Introduction

This paper describes the application of Fogang’s [1] approach based on the boundary collocation method, used for the Kirchhoff plates supported at all corner points whereby the edges are arbitrarily supported (simply supported, clamped or free), to Kirchhoff plates supported at three corner points and having two adjacent edges free and the other edges also arbitrarily supported. The Kirchhoff–Love plate theory (KLPT) was developed in 1888 by Love using assumptions proposed by Kirchhoff [2]. Analytical methods such as the single trigonometric series of Lévy or the double trigonometric series of Navier are not applicable to the title problem because of the adjacent free edges. In this study the analysis was conducted using the boundary collocation method. This method, also called the generalized Trefftz [3] approach, consists of the use of trial functions which satisfy the governing differential equations of the problem. The unknown coefficients of those functions are determined by the satisfaction of the boundary conditions at collocation points along the boundary. About this method, Herrera [4] proposed a precise definition of Trefftz method and, starting with it, explained briefly a general theory while Piltner [5] presented a collection of personal choices to help future developers of numerical methods based on Trefftz trial functions. Many authors worked on analytical methods to plate bending problems but very few on plates having two adjacent edges free. Li et al. [6] developed a symplectic superposition method to derive analytic buckling solutions of biaxially loaded rectangular thin plates with two free adjacent edges that are characterized by having both the free edges and a free corner. Xu et al. [7] introduced the finite integral transform method to explore the accurate bending analysis of orthotropic rectangular thin plates with two adjacent edges free and the others clamped or simply supported: this method eliminated the need to preselect the deflection function, which made it more theoretical for calculating the mechanical responses of the plates. Babu et al. [8] extended the Levy's analytical solution approach for the analysis of rectangular strain gradient elastic plates under static loading with different boundary conditions at the edges using the method of superposition.

In this paper two single series were considered due to the possibility offered to satisfy the boundary conditions along the four edges, inspired from the Lévy solution that involves one single series and the satisfaction of the boundary conditions along two opposite edges (the other edges are simply supported). In addition the approach can be seen as a mixed analytical-numerical method: analytical since the efforts and deformations are described analytically throughout the plate and numerical since the boundary conditions are only satisfied at collocation points on the boundary.

2. Materials and Methods

2.1. Governing Equations of the Plate

xThe Kirchhoff–Love plate theory (KLPT) [1] is used for thin plates whereby shear deformations are not considered. In this section the equations of the KLPT are recalled. The governing equation of the isotropic Kirchhoff plate, derived by Lagrange, is given by

where w(x,y,z) is the mid-plane displacement in z-direction, q(x,y) the transverse distributed load, and D the flexural rigidity of the plate. The bending moments and twisting moments per unit length M_x and M_y, and M_xy , respectively, are given by

The shear forces per unit length are given by

The effective shear forces per unit length used along the free edges are expressed as follows:

In these equations, E is the elastic modulus of the plate material, h is the plate thickness, and ν is the Poisson’s ratio.

2.2. Plate Having Two Adjacent Edges Free And Supported At Three Corner Points

The rectangular plate and the axis convention (X, Y) are represented in Figure 1 below.

The plate dimensions in x- and y-direction were denoted by a and b, respectively. The rectangular plate was assumed to have the two adjacent edges x = a and y = b free, was supported at the three corner points (0, 0), (a, 0), and (0, b), and arbitrarily supported (simply supported, clamped or free) along the edges x = 0 and y = 0.

The displacement function was approximated with the sum of a particular solution w_p(x,y) to the governing differential equation (1) and two single series. The choice of two single series was on the one hand due to the possibility to satisfy the boundary conditions along the four edges: this was inspired from the Lévy solution that involves one single series and the satisfaction of the boundary conditions along two opposite edges, the other edges being simply supported. On the other hand two single series are "geometrically isotropic," in that the independent variables x and y are treated in the same fair way in terms of accuracy. The displacement function can then be taken

where m and n are chosen odd numbers in order to have deflections at the free edges and especially at the unsupported angle (a, b). Furthermore the term w_p(x,y), a particular solution to Equation (1), can be taken as the deflection of a plate strip parallel to the x-axis (or the y-axis) and subjected to the load q(x,y), and consequently

Setting αm = mπ/2a and βn = nπ/2b and substituting Equation (5) into (1) yield

Observing Equation (6) and given that (7) holds for any value of x and y, it results the following differential equations

The solutions to Equations (8a, b) are given by

where the coefficients A_Fm, B_Fm, C_Fm, D_Fm, A_Gn, B_Gn, C_Gn, and D_Gn are determined by satisfying the boundary conditions at selected collocation points. The collocation points should be suitably distributed along the edges.

The collocation points at the edges x = 0 and x = a are associated with the series having the function G_n(x) while those at y = 0 and y = b are associated with the series having the function F_m(y). Let us consider an approximate solution where the first and second series have M and N terms, respectively. It results then 4M + 4N unknown coefficients. Therefore M collocation points should be considered at each of the edges y = 0 and y = b and N collocation points at each of the edges x = 0 and x = a, as represented in Figure 2 for M = 4 and N = 5. Since two boundary conditions are set at each collocation point it results in 4M + 4N equations. So there are as many unknowns as equations.

The bending moments per unit length M_x and M_y, and the twisting moments per unit length M_xy are expressed using Equations (2a-c), (5), and (9a-b) as follows

The boundary conditions involving the bending moments at the free edges x = a and y = b are set observing that for odd values of m and n sin (mπ/2) = (-1)^(m-1)/2 and sin (nπ/2) = (-1)^(n-1)/2. The twisting moments per unit length M_xy using Equations (2c), (5), and (9a-b) are given by

Given that no twisting should occur at the unsupported angle (a, b) and observing that the series have zero twisting at this position (for odd values of m and n cos (mπ/2) = cos (nπ/2) = 0), the particular solution should then be twisting free at this position.

The effective shear forces per unit length V_x and V_y are expressed using Equations (4a-b), (5), and (9a-b) as follows

The boundary conditions involving the effective shear forces at the free edges x = a and y = b are set noting that for odd values of m and n cos (mπ/2) = cos (nπ/2) = 0. The slopes ∂w/∂x and ∂w/∂y used as boundary conditions are given by

The shear forces Q_y used by continuity equations are expressed using Equations (3b), (5), and (9a, b), as follows

As a recall the coefficients A_Fm, B_Fm, C_Fm, D_Fm, A_Gn, B_Gn, C_Gn, and D_Gn are determined by satisfying the boundary conditions at selected collocation points. Given a displacement function where the first and second series have M and N terms, respectively. The edges x = 0 and x = a should be discretized with N points (positions y_j) and the edges y = 0 and y = b with M points (positions x_j): the boundary conditions are then applied at these points whereby the positions y_j = k_Nb/N and x_j = k_Ma/M with k_N = 1, 2, 3, …N and k_M = 1, 2, 3, …M can be taken. Moreover by applying the boundary conditions the external running moments and running loads should be suitably distributed at the collocation points.

With the determination of the coefficients above the deflections are calculated using Equations (5) and (9a, b) and the efforts (bending moments M_x, M_y, and twisting moments M_xy , and effective shear forces V_x and V_y) using Equations (10a-c) and (11a-b).

3 Results and discussion

3.1. Rectangular Plate Having Two Adjacent Edges Free, Simply Supported Along The Other Edges And Subjected To A Uniformly Distributed Loading

Taking the particular solution w_p(x,y) as the deflection of a plate strip parallel to x-axis and simply supported at both ends, the displacement function is given by

The bending moments per unit length at the plate center, depending on the ratio b/a, are M_x,m = qa²N_xm, M_y,m = qa²N_ym whereby M and N are the number of terms of the first and second single series, respectively. Details of the analysis and results are presented in the supplementary material “Rectangular plate with two adjacent edges free SSFF” whereby the collocation points were the following

• Case M = 1 N = 1: y_j = b along the edges x = 0 and x = a and x_j = a along the edges y = 0 and y = b
• Case M = 2 N = 2: y_j = b/2 and b along the edges x = 0 and x = a and x_j = a/2 and a along the edges y = 0 and y = b
• Case M = 3 N = 3: y_j = b/3, 2b/3, and b along the edges x = 0 and x = a and x_j = a/3, 2a/3, and a along the edges y = 0 and y = b. Table 1 lists the results in Lecture notes [9] and those obtained in the present study.

As Table 1 shows, the results are in reasonable agreement with the exact results, the accuracy increasing the more terms of the series were taken. A powerful computation tool is therefore needed to consider higher number of terms of the series and so to approach the exact results.

4. Conclusions

In this paper, arbitrarily loaded isotropic rectangular Kirchhoff plates supported at three corner points, having two adjacent edges free and the other edges being simply supported, clamped or free, were analyzed. The deflection surface was approximated with the sum of a particular solution to the governing differential equation and two single series: the terms of the series were the product of an unknown function of an independent variable and a trigonometric function of the other independent variable, whereby the trigonometric functions satisfied the deflection-related boundary conditions. On the one hand the particular solution satisfied the nonhomogeneous differential equation and the single series were chosen so as to satisfy the homogeneous differential equation. On the other hand the boundary conditions were only satisfied at selected collocation points along the boundary, the number of collocation points in each direction corresponding to the number of terms of the associated series. However, due to the absence of symmetry of the system many unknown coefficients have to be calculated and so a powerful computation tool is needed to consider higher number of terms of the series and so to increase the accuracy. One numerical example was calculated and the results showed a reasonable agreement with the exact results, the accuracy increasing the more terms of series were considered: more examples with many terms will be presented in the next version of this paper. In future research cantilevered plates will be analyzed using this approach.