The use of polynomial shape function in the buckling analysis of ccfc rectangular plate

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 12 | Dec-2013, Available @ http://www.ijret.org 652
THE USE OF POLYNOMIAL SHAPE FUNCTION IN THE BUCKLING
ANALYSIS OF CCFC RECTANGULAR PLATE
Ezeh, J. C.1
, Ibearugbulem, O. M.2
, Nwadike, A.N.3
, Maduh, U. J.4
1
Associate Professor, Civil Engineering Department, Federal Unitech Owerri, Imo State, Nigeria,
jcezeh2003@yahoo.com
2
Civil Engineering Department, Federal Unitech Owerri, Imo State, Nigeria, ibeowums@yahoo.co.uk
3
Civil Engineering Department, Federal Unitech Owerri, Imo State, Nigeria, nwadikeamarachukwu@yahoo.com
4
Civil Engineering Department, Federal Unitech Owerri, Imo State, Nigeria,ujmaduh@gmail.com
Abstract
The use of polynomial series function in the buckling analysis of a CCFC is presented. The polynomial series shape function was
truncated at the fifth orthogonal terms, which satisfied all the boundary conditions of the plate to obtain a peculiar shape
function, which was applied in Ritz method. The peculiar shape function is substituted into the total potential energy functional,
which was minimized, and the critical buckling load of the plate was obtained. The critical buckling load is a function of a
coefficient, “K”. The values of K from earlier and the present studies were compared within the range of aspect ratios from 0.1 to
2.0. A graph of critical buckling load against aspect ratio was plotted. It was discovered that for aspect ratios of 0.4, 0.5 and 1.0,
the critical buckling loads coefficients were 26.94, 17.39 and 4.83. It was also observed from the behavior of the graph that as
aspect ratio increases from 0.1 to 2.0, the critical buckling load decreases.
Keywords: Total Potential Energy Functional, Shape Function, Polynomial Series Shape Function, Critical Buckling
load, Boundary Condition, Ritz Method.
--------------------------------------------------------------------***----------------------------------------------------------------------
NOTATION
W is the shape function (deflection function).
V is the shear force.
𝜇 is Poisson’s ratio
First partial derivative of deflection with respect to R is:
𝑤′ 𝑅
=
𝜕𝑤
𝜕𝑅
.
First partial derivative of deflection with respect to Q is:
𝑤′ 𝑄
=
𝜕𝑤
𝜕𝑄
Second partial derivative of deflection with respect to R and
Q is: 𝑤′ 𝑅′𝑄
=
𝜕2 𝑤
𝜕𝑅𝜕𝑄
Second partial derivative of deflection with respect to R
is: 𝑤′′ 𝑅
=
𝜕2 𝑤
𝜕𝑅2
Second partial derivative of deflection with respect to Q
is: 𝑤′′ 𝑄
=
𝜕2 𝑤
𝜕𝑄2
Third partial derivative of deflection with respect to R and Q
is: 𝑤" 𝑅′𝑄
=
𝜕3 𝑤
𝜕𝑅2 𝜕𝑄
𝐷 =
𝐸𝑡3
12(1−𝜇2)
is flexural rigidity, t is the plate thickness. E is
Young’s modulus.
1.INTRODUCTION
Since the origin of plate, Fourier and trigonometric series
has been used to treat plate problems as shape function.
There are many approaches many scholars employed in
solving the problem of plates like energy method, numerical
method, and equilibrium method. These approaches give
approximate results in most difficult situations.
C.Erdem and Ismail used numerical method to analyze an
isotropic rectangular plate with four clamped edges under
distributed loads of an exact solution of the governing
equation in terms of trigonometric and hyperbolic
function[1]. They also compared the obtained results with
those earlier reported, which shows reasonable agreement
with other available results, but with a simpler and practical
approach. Robert and Govindjee used the classical double
cosine series expansion and an exploit of the Sherman-
Woodburg Formular to solve the problem of clamped
rectangular plate [2]. Ibearugbulem used Taylor-Mclaurin's
series as shape function to analyze the instability of axially
compressed thin rectangular plate with four edges clamped
[5]. He truncated the Taylor-Mclaurins series at the fifth
term which satisfied all the boundary conditions of the plate.
Upon substituting the resulted shape function into the total
potential energy, he obtained the critical buckling load of
the plate. Some scholars used finite element method in
analyzing the problems of rectangular plate like Han and
Petty[3], Tajdari et al [11], An- chien Wu et al [10], Ye [9],
Rakesh [8], Sandeep Singh et al [7], and Ganapathi et al [4]
Although the buckling analysis of rectangular plates has
received the attention of many researchers for several
years, its treatment has left much to be done. Most of the
available solutions do not satisfy exactly the prescribed
boundary conditions or the governing differential agnation
or both. In the past, there were no reference books to which

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Engineers and students can turn to for a clear and orderly
exposition of the Isotropic rectangular plate with clamped –
clamped – free – clamped edge subjected to dynamic
boundary conditions, which are also known as non-essential
boundary condition. Ibearugbulemsolved CCFC rectangular
plate using Taylor – Mclaurin’s shape function with the aid
of eigen value computer programme[6].
In order to provide the solutions to this problem, this paper
is presented to address the issue. The plate is subjected to in-
plane load in one axis (x-axis) of the principal plane as
shown in figure 1.
2. TOTAL POTENTIAL ENERGY FUNCTIONAL
FOR BUCKLING OF PLATE.
Ibearugbulem [6] derived the total potential energy
functional for a rectangular isotropic plate subjected to in-
plane load in x – direction and the polynomial shape
function as follows;
πx
=
D
2b2
b3
a3
w′′ R 2
+
a
b
w′′ Q 2
+
2b
a
w′ R′Q 2
1
0
1
0
∂R ∂Q
−
bNx
2a
w′ R 2
1
0
1
0
∂R ∂Q (1)
𝑊 = am Rm
. bnQn
4
n=0
4
m=0
(2)
Where “a” and “b” are plate dimensions (lengths) along x
and y directions. Nx is the in-plane load in x-direction, D is
flexural rigidity. W is the shape function.πxis Total potential
energy functional with load along x – axis; R =
x
a
; Q =
y
b
; 0 ≤ R ≤ 1; 0 ≤ Q ≤ 1 (R and Q are dimensionless
quantities).
3. SHAPE FUNCTION RESULTING FROM
POLYNOMIAL SERIES.
The boundary conditions for CCFC plate are;
w R = 0 = 0 ; w (Q = 0) = 0; w(R = 1) = 0 (3)
w′R
R = 0 = 0 ; w′Q
(Q = 0) = 0; w′R
R = 1 = 0 (4)
VQ Q = 1
= w′′′Q
+ 2 − μ w′′R′Q
Q = 1 = 0 (5)
Applying these boundary conditions in equation 2 gave;
w = A R2
− 2R3
+ R4
4Q2
− 4Q3
+ Q4
(6)
4. APPLICATION OF RITZ METHOD
Partial derivative of equation (6) with respect to either R or
Q or both gave the following equations;
w′R
= A 2R − 6R2
+ 4R3
4Q2
− 4Q3
+ Q4
(7)
w′′R
= A 2 − 12R + 12R2
4Q2
− 4Q3
+ Q4
(8)
w′Q
= A R2
− 2R3
+ R4
8Q − 12Q2
+ 4Q3
(9)
w′′Q
= A R2
− 2R3
+ R4
8 − 24Q + 12Q2
(10)
w′R′Q
= A 2R − 6R2
+ 4R3
8Q − 12Q2
+ 4Q3
11
Integrating the square of these equations (7), (8), (10), and
(11) partially with respect to R and Q in a closed domain
respectively gave:
w′ R 2
∂R ∂Q
1
0
1
0
= A2
0.019047619 0.406349206 = 0.007739984876A2
(12)
w′′ R 2
1
0
1
0
∂R ∂Q
= A2
0.8 0.40634920 = 0.325079365A2
13
w′′Q 2
∂R ∂Q
1
0
1
0
= A2
1.587301587x10−3
12.8 = 0.02031746A2
(14)
w′ R′Q 2
1
0
1
0
∂R ∂Q
= A2
0.019047619 1.219047619 = 0.023219954A2
15
Substituting equations (12), (13), (14) and (15) into equation
(1), and minimizing it gave the following result;
π x
∂A
=
DA2
Pa2
0.325079365 + 0.02031746P4
+ 0.023219954P2
−
NxA2
2P
0.007739984876 = 0 (16)
Making Nx the subject of the equation (16) with aspect ratio
of P = a/b
Nx =
Dπ2
b2
4.255489716
P2
+ 0.265968103P2
+ 0.303963542 17
5. RESULT AND CONCLUSIONS
A graph of critical buckling load against aspect ratio was
plotted. From the graph, it was observed that as the aspect
ratio increases from 0.1 to 2.0, the critical buckling load
decreases. The graph was divided into two segments ranging
from 0.1 to 0.7 In the first segment, the graph is of 6th
degree polynomial with an equation of Y = 21892X6
–
60455X5
+ 68130X4
– 40190X3
+ 13164X2
– 23034X +
b NX NX
y
x
P = a/b
CCFC
a

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1752. In the second segment, the graph is of 5th degree
polynomial with an equation of Y = -5.704X5
+ 44.58X4
-
139.9X3
+ 223.4X2
– 184.7X + 67.20.
a: First Segment
b: Second Segment
Figure 2: Graph of critical buckling load against aspect
ratio.
The average percentage difference between the solutions
from Ibearugbulem and the present study shown on table 1
was 32.112%. This value is more than the required
percentage because Ibearugbulem used a rigorous computer
matrix program to analyze the plate which has so many
disadvantages. It was observed that as the aspect ratio
increases from 0.1 to 2.0, the present work and
Ibearugbulem start converging.
Table 1: K values for different aspect ratios for CCFC
rectangular plate
Aspect
Ratio
P =
a
b
Z From
Ibearugbul
em (2012)
Z From
Present
Study
Differe
nces
Percentage
Difference
0.1 159.16 425.86 266.69 167.56
0.2 54.24 106.70 52.42 96.72
0.3 35.41 47.61 12.20 34.46
0.4 27.29 26.94 0.35 1.27
0.5 17.73 17.39 0.34 1.90
0.6 12.54 12.22 0.32 2.55
0.7 9.42 9.12 0.30 3.20
0.8 7.41 7.12 0.29 3.88
0.9 6.05 5.77 0.28 4.58
1 5. 08 4.83 0.25 5.01
6. CONCLUSION
The polynomial series as a shape function was applied in
this paper for the buckling behavior of an isotropic
rectangular plate.It was also deduced that the polynomial
series satisfies all the non-essential (Dynamics) boundary
conditions of the graph.
The shape function derived in this approach is a good
alternative to the shape function assumed using
trigonometric series. Therefore, the present method can
accurately predict the critical buckling load of isotropic
rectangular plates.
REFERENCE
[1]. C. ErdemImrak and Ismail Gerdemeli (2006), "The
problem of isotropic rectangular plate with four
clamped edge". Journal of Sadhana, vol. 32, Part 3,
Pp 181-186.
[2]. Robert L.Taylor and Sanjay Govindjee
(2002),"Solution of Clamped Rectangular Plate
Problem”. Journal of Structural Engineering
Mechanic and Materials. Vol.9.
[3]. Han, W.,Petyt, M.(1997)."Geometrically Nonlinear
Vibration Analysis of Thin Rectangular Plates
Using the Hierarchical Finite Element Method".
y = 21892x6 -
60455x5 +
68130x4 -
40190x3 +
13164x2 -
23034x + 1752.
0
50
100
150
200
250
300
350
400
450
0 0.5 1
Criticalbucklingloadcoefficient
Aspect Ratio
CCFC PLATE
y = -5.704x5 + 44.58x4 -
139.9x3 + 223.4x2 - 184.7x +
67.20
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3
Criticalbucklingloadcoefficient
Aspect Ratio
CCFC PLATE
Poly. (CCFC
PLATE)

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Journals of Computers & Structures, vol. 63, issue
2, Pp 295 - 308.
[4]. Ganapathi, M., Varadan, T.K, and Sharma, B.S
(1991)." Nonlinear Flexural Vibration of
Laminated Orthotropic Plate". Journal of
Computers and Structures. Vol. 39, Issue 6, Pp 685
- 688.
[5]. Ibearugbulem, O.M, Osadebe, N.N., Ezeh, J.C.,
Onwuka, D. O (2013). “Instability of Axially
Compressed CCCC Thin Rectangular Plate Using
Taylor-Mclaurins Series Shape Function on Ritz
Method”. Journal of Academic Research
International, vol. 4, No. 1., Pp 346-351
[6]. Ibearugbulem, O.M. (2012). “Application of a
Direct Variational Principal in Elastic Stability of
Rectangular Flat Thin Plates”. Ph.D. Thesis
submitted to postgraduate school, Federal
University of Technology, Owerri, Nigeria.
[7]. Sandeep Singh et al (2012). “Buckling Analysis of
Thin Rectangular Plates with Cutouts Subjected to
Partial edge Compression Using FEM”. Journal of
Engineering, Design and Technology, vol.10,
issue1.
[8]. Rakesh TimappaNaik (2010).“Elastic Buckling
Studies of Thin Plates and Cold-Formed Steel
Members in Shear”. Report Submitted to the
Faculty of Virginia Polytechnic Institute and State
University in Partial Fulfillment of the
Requirements for the Degree of Master of Science
in Civil Engineering.
[9]. Ye Jianqiao (1994). “Large Deflection of Imperfect
Plates by Iterative BE-FE Method”. Journal of
Engineering Mechanics, vol. 120, No. 3,Pp 431-
445.
[10]. An-Chien Wu, Pao-Chun Lin and Keg-Chyuan Tsai
(2013), “High-Mode Buckling-Restrained Brace
Core Plate”,Journals of the International
Association for Earthquake Engineering.
[11]. Tajdari, M., Nezamabadi, A.R., Naeemi, M and
Pirali, P (2011), “The Effects of Plate-Support
Condition on Buckling Strength of Rectangular
Perforated Plates under Linearly Varying in-plane
Normal Load”, Journal of World Academy of
Science, Engineering and Technology, vol. 54, Pp
479-486.

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