IJETAE_0313_04

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 3, March 2013)

20

Pure Bending Analysis of Thin Rectangular Flat Plates Using

Ordinary Finite Difference Method Ezeh, J. C.

1, Ibearugbulem, O. M.

2, Onyechere, C. I.

3

1,2,3 Department of Civil Engineering, Federal University of Technology, Owerri, Nigeria.

Abstract— Analytical methods search for the universal

mathematical expressions representing the general and exact

solutions of a problem. Unfortunately, exact solutions are only

possible for a few particular cases which frequently represent

simplification of reality. Numerical methods on the other

hand aim to provide approximate solutions in the form of a

set of numbers, to the mathematical equations governing a

problem. In this study, a numerical analysis using Ordinary

Finite Difference Method (OFDM) on pure bending of thin

rectangular flat plates was implemented. The Analysis was

accomplished through a theoretical transformation of the

partial differential equations for the plate into finite

difference form to fit the chosen grid pattern. The finite

difference forms were evaluated at the grid points in order to

obtain a set of simultaneous linear equations. After using the

boundary conditions, SSSS and CCCC respectively, the

unknown functional values in form of deflections were

determined. Visual Basic (VB) software was developed to

simplify the determination of the deflections. The resulting

maximum deflections Wmax, were compared with exact

solutions from previous studies within the range of aspect

ratios of 1.0 to 2.0 as shown on tables 1 and 2. The tables show

that solutions from the present study approximates closely

and rapidly to the exact solutions. Hence, Ordinary finite

difference method (OFDM) provides a very simple and

approximate solution for thin rectangular flat plates.

Keywords- Approximate Solutions; Boundary Conditions;

Maximum Deflection; Pure Bending; Thin plates; Visual

Basic.

I. INTRODUCTION

The pure bending analysis of thin rectangular flat plates

is of interest in the field of mechanics, civil and aerospace

engineering. Except for plates of simple boundary

conditions, the governing plate equation ∇ 4w = Po/D

yields plate deflections only with considerable difficulty

(Ugural, 1999).In the past years, plate problems have been

treated by the use of Fourier series or trigonometric series

as the shape function of the deformed plate. However, no

matter the approach used, the use of trigonometric series

(double Fourier series and single Fourier series) has been

predominant. Most times, when it is becoming intractable

to use the trigonometric series, trial and error means of

getting a shape function that would approximate the

deformed shape of the plate would be used(Ibearugbulem

et al., 2011).

As a result of the slow convergence of the double

trigonometric series, researchers proposed an efficient

method of making the convergence of the double Fourier

series faster (Jiu et al, 2007; Ibearugbulem et al, 2011).

Timoshenko and Woinowsky-Krieger (1987), using single

trigonometric functions, got exact solutions for pure

bending analysis of SSSS, CSCS and CCCC thin rectangular flat plates. Ventsel and Krauthammer (2001),

used ordinary finite difference method to obtain exact

solutions for a thin rectangular flat plate simply supported

at all the four edges.

In this paper, ordinary finite difference method (OFDM)

was used to obtain solutions for the pure bending analysis

of thin rectangular flat plates carrying uniformly distributed

load with the following boundary conditions (Figures 1 -

2): (i) SSSS and (ii) CCCC. An interactive finite difference

method based software program was written in Visual

Basic and provided to make the solution easy.

II. THIN PLATE THEORY

The governing differential equation for deflection of thin

plates was given by Ugural (1999) as;

∇ ( ) ( )

(1)

Where; W is the deflection.

x, y are the coordinates of the plate.

Po is the applied uniformly distributed static load.

∇ is the biharmonic differential operator (i.e., ∇ = ∇ ∇ )

∇ ( )

( )

b

Y

S

S

X

S S

a

Figure 1: SSSS plate

b

Y

C

C

X

C C

a

Figure 2: CCCC plate



21

h

( ) is the flexural rigidity of the plate.

E, and h are the Young’s modulus, Poisson’s ratio, and

the thickness of the plate (Ventsel and Krauthammer, 2001,

Jiu et al, 2007, Awele et al., 2003).

III. BOUNDARY CONDITIONS

For a thin rectangular flat plate with edge lengths ‘a’ and

‘b’, there are eight boundary conditions for every case.

Two cases are discussed below: (i) fully simply supported,

SSSS. (ii) fully clamped, CCCC. The boundary conditions

for these plates are:

SSSS plate: W(0,y) =0; W(a,y) = 0; W(x,0) = 0; W(x,

b) = 0; ( ) ; ( ) ; ( ) ;

( ) (3)

CCCC Plate: W(0,y) =0; W(a,y) = 0; W(x,0) = 0; W(x,b) =

0; ( ) ; ( ) ; ( ) ;

( ) (4)

IV. ORDINARY FINITE DIFFERENCE COEFFICIENTS AND

PATTERNS

Where P = M / N is the aspect ratio. M and N are the

spans of each rectangular panel.

Point i(0,0) is taken as the origin, using the Central

Difference, the Ordinary Finite-difference coefficients for

the differentials are given below;

[ ( ) ( )]

[ ] (5)

[ ( ) ( )]

[ ] (6)

[ ( ) ( ) ( )]

[

] (7)

=

[ ( ) ( ) ( )]

[

] (8)

⁄ [ ( ) ( ) ( ) ( )

( )]

⁄ [ ] ( )

⁄ [ ( ) ( ) ( ) ( )

( )

⁄ [ ] ( )

[

]

∇

[ ]

[ ]

[

] ( )

The Patterns are given as:

Figure 4: Pattern for

.

-1 0 1


.

1 -2 1


.

-1

0

1


.

1

-2

1

Figure 3: Diagram showing a discretized Rectangular Plate

M M M M

N

N

N

N

15 16 17 18 19

5 6 7 8 9

1 2 i 3 4

10 11 12 13 14

20 21 22 23 24

x

y



22

The derived coefficients and pattern for The derived

coefficients and pattern for the governing equation are

applied at each of the internal nodes with the appropriate

boundary conditions to generate the required matrix

equation from which the deflections and moments were

determined. An Ordinary Finite-difference based Visual

Basic program was written and used to ease the solution.

The program generates the required matrix equation as well

as the deflections for both SSSS and CCCC plates when

one enters the grid size, n and the Poisson’s ratio . The

grid size here means the number of the internal nodes along

each row of the discretized plate.

V. RESULTS AND DISCUSSIONS

The uniformly distributed load is represented by P while

the maximum deflection is represented by Wmax. Wmax=

KPa4/D. Where K is a numerical value.

Table 1

Values of K at different aspect ratios for SSSS plate;

Aspect

ratio,a/b

K from PRESENT

STUDY K from

Timoshenk

o(1987)

% DIFFERENCE

n =7 n =9 n = 7 n = 9

1 0.00406 0.00406 0.00406 0.00000 0.00000

1.1 0.00486 0.00486 0.00485 0.20619 0.20619

1.2 0.00564 0.00564 0.00564 0.00000 0.00000

1.3 0.00638 0.00639 0.00638 0.00000 0.15674

1.4 0.00707 0.00708 0.00705 0.28369 0.42553

1.5 0.0077 0.00772 0.00772 -0.25907 0.00000

1.6 0.00829 0.0083 0.0083 -0.12048 0.00000

1.7 0.00882 0.00883 0.00883 -0.11325 0.00000

1.8 0.0093 0.00931 0.00931 -0.10741 0.00000

1.9 0.00973 0.00973 0.00974 -0.10267 -0.10267

2 0.01011 0.01012 0.01013 -0.19743 -0.09872

Average % Difference -0.03731 0.05337

The average percentage difference between the solution

from Timoshenko and Krieger (1987), (in this case, exact

method) and the present study according to Table1 is -

0.03731 for grid size n = 7 and 0.05337 for grid size n = 9.

This means that the solution from the present study is a

very close approximation of the exact solution. The

closeness of this method to the exact solution increases as

the grid size ‘n’ increases.

Figure 7: Pattern for .

1

-4-4/P2

6+6/P4

+8/P2

-4-4/P2

1

-4/P4

-4/P2

-4/P4

-4/P2

1/P

4

1/P4

2/P2

2/P2

2/P2

2/P2



23

Table 2

Values of K at different aspect ratios for CCCC Plate.ѵ =0.3

a/b

Present Study. Timoshe

nko n = 9 n = 11 n = 13

1 0.00137 0.00134 0.00132 0.00126

1.1 0.00163 0.0016 0.00157 0.0015

1.2 0.00187 0.00182 0.0018 0.00172

1.3 0.00207 0.00202 0.00199 0.00191

1.4 0.00223 0.00218 0.00215 0.00207

1.5 0.00237 0.00232 0.00229 0.0022

1.6 0.00248 0.00243 0.00239 0.0023

1.7 0.00256 0.00251 0.00248 0.00238

1.8 0.00263 0.00258 0.00254 0.00245

1.9 0.00268 0.00263 0.00259 0.00249

2 0.00272 0.00267 0.00263 0.00254

The average percentage difference between the solution

from Timoshenko and Krieger and the present study

according to table 2 is 7.94588 for grid size n = 9; 5.68351

for n= 11 and 4.14286 for n =13. This means that the

solution from this present study was a very close

approximation of the exact solution.

Also, as the grid size n increases, the solution from the

present study gets closer to the exact solution. However,

the results show that the solutions from this method are

upper bound.

REFERENCES

[1] Awele, M., Ayodele, J.C., and Osaisai, F.E. (2003): Numerical Methods for Scientists and Engineers.Ibadan, Nigeria: Beaver

publications

[2] Ibearugbulem, O.M., Osadebe, N.N., Ezeh, J.C., and Onwuka, D.O. (2011): Buckling Analysis of Axially Compressed SSSS Thin

Rectangular Plates using Taylor-Mclaurin Shape Function.

International Journal of Civil and Structural Engineering Vol. 2, No. 2 Pp.667 – 671; ISSN 0976 – 4399.

[3] Jiu, H. W.; Liu, A. Q. and Chen, H. L. (2007): Exact Solutions for

Free- Vibration Analysis of Rectangular Plates using Bessel Functions; Journal of Applied Mechanics ASME, Vol. 74.

[4] Timoshenko, S. and Woinowsky-Krieger, S. (1987): Theory of

Plates and Shells, 2nd Edition, McGraw-Hill Books Company, USA.

[5] Ugural, A. C. (1999): Stresses in Plates and Shells, 2nd Ed.,

McGraw-Hills Books Co., Singapore.

[6] Ventsel E. and Krauthammer K. (2001): Thin Plates and Shells; Marcel Decker Inc., New

[7] York.

[8] Willims, D. J. and Aalami, B. (1979): Thin Plate Design for In-Plane Loading; Granada Publishing Ltd., London.

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