Figure 1. General plate-bending problem

Analysis of simply supported isotropic thin-plate bending problem by finite difference method

Emma Consuelo O. Como1, Jaan Ruy Conrad P. Nogot2 and Gilbert M. Oca3 1,2Materials Physics Research Group, Physics Division, IMSP, CAS, UP Los Baños

3Computational Physics Research Group, Physics Division, IMSP,CAS, UP Los Baños October 7, 2010

Abstract

Introduction

For problems involving complicated geometries, loadings, and material properties, it is generally not possible to obtain analytical mathematical solutions. Hence, one needs to rely on numerical methods, such as finite difference method (FDM)4, for acceptable approximate solutions. A typical problem area of interest that is solvable by use of the FDM is the structural analysis of the plate-bending problem (see figure 1).

A plate can be considered the two-dimensional extension of a beam in simple bending. Both plates and beams support loads transverse or perpendicular to their plane and through bending action [2].

The small transverse (out-of-plane) displacement z of a thin plate (see figure 2) is governed by the Classical Plate Equation,

where D is the flexural rigidity of the plate, w its deflection in the transverse (z) direction and q, the applied transverse load.

1 E-mail Address: yamadaemma@gmail,com 2 E-mail Address: jrcpnogot@gmail.com 3 E-mail Address: boyfriendnibluefairy@gmail.com 4 FDM is one of the three basic discretization methods, where one replaces the domain by a finite point set and replaces derivatives by differences [1]

(1)

An isotropic thin-plate supported along its edges is subjected to an areal load, bending the plate in the process. The resulting deflection along the transverse direction (z-direction) of the plate is computed and analyzed in this study. A second-order differential equation is first defined to simplify the Classical Plate Equation to a Poisson equation counterpart. This simplified equation is then solved using the finite differential method, ensuing the deflection of the plate along the z-direction. This deflection increases when the applied transverse load is increased, the plate is made thinner and the plate is made of a material of a relatively smaller flexural rigidity.

2

The flexural rigidity is given by

where h is the plate’s thickness, E is the Young’s Modulus, and v is the Poisson’s ratio5 of the plate material [3].

The differential operator in eq. 1 is called the Laplacian differential operator ∆, where

When D is constant throughout the plate, the plate equation can be simplified to,

where is called the biharmonic differential operator [4].

This small deflection theory assumes that w is small in comparison to the thickness of the plate h, and the strains and the midplane slopes are much smaller than 1.

5 Poisson's ratio is the ratio of the relative contraction strain, or transverse strain normal to the applied load, to the relative extension strain, or axial strain in the direction of the applied load. Ranges from engineering materials from .2 to .5 [5] A plate is called thin when its thickness h is at least one order of magnitude smaller than the span or diameter of the plate [6]

(2)

(3)

(4)

Figure 2. Simply supported square plate with load q

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Figure 3. Discrete Finite Differential Grid

Figure 2 shows a thin-plate with an area load q acting on it. The plate is assumed to be isotropic6 and is supported along the edges. No deflections at the plate’s edges and slope normal to the boundary are zero. Given the foregoing boundary conditions, the deflection along the z-direction is determined.

Methodology

The deflections of a square plate, with supported edges subjected to an areal load q, in the z direction is given by

where D is called the flexural rigidity and q is the areal load.

The above equation can be simplified by defining another equation u which is equal to

Equation 5 then becomes

Equation 7 is solved using finite difference method. A region, shown in figure 3, is partitioned into a grid with points satisfying the boundary conditions specified in figure 2. When applying the finite difference method in the nine discretized domain points, nine equations are produced. The solutions u1, u2,…, u9 of the nine equations are solved numerically using Gaussian elimination method [7][8][9]. The computed ui’s are correspondingly equated in expression 6 to finally solve for z. The same method, utilizing finite difference method and Gaussian elimination, was used to work out

6

Isotropic materials are materials having no directional variation in their mechanical properties (e.g. wood, and many crystals); homogeneous.

(5)

(6)

(7)

4

the z values for equation 6.

The algorithm7, implementing the method, is presented here.

1. Compute for the Areal load divided by the Flexural rigidity (q/D). 2. Initialize the coefficient matrix and constant matrix produced by FDM in equation 7. 3. Solve ui’s using Gaussian elimination method. 4. Initialize new coefficient matrix produced by FDM in equation 6 with elements of the constant

matrix replaced by ui’s. 5. Solve z values using Gaussian elimination method.

The behavior of equation 5 was examined by varying the areal load and changing the flexural rigidity (changing the material) and the thickness of the plate.

Results and Discussion

Figure 4 shows the deflection along the z-direction of a 1 cm thick copper thin-plate under different loads. As the load is varied from 50 N to 500 N to 1000 N, the deflection of the plate is observed to have also increased. One sees a direct proportionality of the deflection and the applied transverse force (load), and these results satisfy the governing equation (4). Being simply-supported, it is also observed from the following figure that there is no deflection along the plate’s edges, satisfying the specified boundary condition.

7 For the Fortran 90 program, email or contact any of the authors.

Figure 4. Deflection of a copper plate along the z axis when a load, q, of (a) 50N, (b) 500N, (c) 1000N is applied

4a

4c

4b

5

Figure 5. Deflection of a (a) aluminum, (b) copper thin-plate under 500 N

Figure 5, on the other hand, shows the deflection along the z direction of a 1 cm thin (a) aluminum and (b) copper plate both subjected to 500 N. Aluminum has a E value of 7.0 x 1010 Pa and a Poisson’s ratio (v) of 0.334. Copper has 11 x 1010 and 0.355, respectively [10]. Based from the obtained z deflection, it appeared that for a constant load, aluminum plate bends relatively greater than copper plate. It can be observed that the flexural rigidity of aluminum is 6565.8, which is smaller than copper’s 10,488.5. From the governing equation (4), the flexural rigidity is inversely proportional to the deflection, and the foregoing findings satisfy this claim.

Shown in figure 6 are the corresponding deflections of a copper thin-plate when its thickness is varied. It is observed that as the thickness of the plate is decreased, the deflection along the z-direction increases. This follows that the thinner the plate, the more it can be bend by an introduction of a transverse load. The foregoing observations imply an inverse relationship between the thickness of the material and its deflection along the transverse direction, complying with the governing equation for thin-plates.

Moreover, it is observed for all three cases that the deflections of the nine discretized domain points (see figure 3) vary from each other. However, and more importantly, a trend of the deflection can be deduced from these three cases. Domain point 33 is observed to have the greatest deflection along the transverse direction compared to the other eight points’, which appear to decrease substantially, relative to domain point 33. There are no deflections along the plate’s edges for the three cases because the plate is simply supported. This Gaussian-like surface was observed because the plate is initially conceived as isotropic and homogenous and the load is subjected in the plate’s center.

5a 5b

6

Figure 6. Deflection of a copper plate of (a) 1mm, (b) 1cm, (c) 0.1m copper thin-plate under 500 N loads

Summary and Conclusion

The study attempted to compute for the deflection along the transverse direction of a simply supported isotropic thin-plate resulting from an applied transverse load. A second order differential equation was first defined to simplify the given plate equation to an equivalent Poisson equation, thru the property of the biharmonic differential operator, and by using the finite difference method to solve the simplified plate equation, this objective was successfully meet.

To validate the results of the study, the areal load, the flexural rigidity and the thickness of the plate were varied and the corresponding trends were analyzed. An increase in the deflection along the transverse direction of the thin-plate can be accounted by an increase in the load applied to the plate, a decrease in the thickness of the plate and the usage of a plate made from a material with a relatively smaller flexural rigidity D. These results, all and all, comply with the Classical Plate equation.

6c

6a 6b

7

References

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