The document discusses using the differential quadrature method to analyze buckling in thin plates. It provides an overview of buckling and introduces the differential quadrature method as an efficient numerical technique. The method transforms differential equations into algebraic equations using sampling points. The document applies the method to analyze buckling in isotropic rectangular plates with different boundary conditions and aspect ratios. Results show the differential quadrature method provides accurate results using fewer grid points compared to other methods like finite element analysis.

1. Buckling Analysis of Plate
:

2. Table of Content
 Buckling
 Scope of the Work
 Need for Differential Quadrature Method
 Differential Quadrature Method
Choice for sampling grid points
 Numerical application & discussion
 Buckling analyses of Thin, isotropic rectangular plates
Boundary conditions for Rectangular plate
Four edges Clamped ( C-C-C-C )
Four Edges Simply Supported ( S-S-S-S )
 Results
 Future Scope of the work
 Conclusion

3. BUCKLING
 In science buckling is a mathematical instability,
leading to a failure mode.
 Buckling is characterized by a sudden failure of a
structural member subjected to high compressive
stress, where the actual compressive stress at the
point of failure is less than the ultimate compressive
stresses that the material is capable of withstanding
 The plates can buckle in any direction with
varied boundary conditions. To analyze the
buckling of plates both isotropic and composite
we adopt differential Quadrature method to
break down the problem in form of differential
equations.

4. Scope of the work
• The method of differential Quadrature is a numerical solution
technique for differential systems by means of a polynomial-
collocation approach at a finite number of points.
• An inherent advantage of the approach is its basic simplicity
and small computational effort with easy programmability.
• Numerical examples have shown the accuracy, efficiency and
great potential of this method for structural analysis.
• Finite element method is too lengthy to solve the engineering
problems as compared to Differential Quadrature method.
• DQ method is less time consuming and we don’t have to solve
equation for each and every grid point.

5. Need for Differential Quadrature method
• Engineering system includes two main stages:
Construction of a mathematical model for a given physical phenomenon
and the solution to this mathematical equation.
• Approximate numerical methods have been widely used to
solve partial differential equations. The most commonly used
numerical methods for such applications are the finite element,
finite difference , and boundary element method to adequate
accuracy.
• In seeking a more efficient numerical method that requires
fewer grid points yet achieves acceptable accuracy, the method
of Differential Quadrature was introduced by Bellman.

6. Differential Quadrature Method
• It is an effective numerical technique for the solution of non-
linear partial equations.
• The basic idea of the DQM is that the derivative of a function
with respect to a space variable at a given sampling points in
the domain of that variable.
• DQM transforms the given differential equation into a set of
analogous algebraic equations in terms of the unknown values
at the reselected sampling points in the field domain.

8. • the first order derivative approximation formula given above equation can
be expressed in closed form by the following linear transformations for the
partial derivatives with respect to x and y:
• The approximation formulae for higher order partial derivatives are
obtained by iterating the linear transformations given by above equations:

9. Choice of Sampling Grid Points
• A decisive factor for the accuracy of the differential quadrature
solution is the choice of the sampling or grid points.
• In DQM the sampling points in various coordinate directions
may be different in number as well as in their type:
Type-I: Xi=(i-1)/(N-1) i=1,2,3……..N ( uniform type )
Yi=Xi
Type-II Xi=1/2[1-cos((i-1)/N-1)*pi)] i=1,2,3….N
Yi=Xi
( Chebyshev-Gauss-Lobatto type )

10. Numerical Applications & Results
• To verify the analytical formulation presented by other method
isotropic rectangular plates are considered Plates of different
types of boundary conditions are selected as test samples to
demonstrate the applicability and accuracy of DQ method.
• The results are obtained for each case using various numbers
of grid points.
• Several test samples for different support conditions are
selected to demonstrate the convergence properties, accuracy
and the simplicity in numerical implementation of DQ
procedures.
• Grid point are chooses as:
Xi=1/2[1-cos((i-1)/N-1)*pi)] i=1,2,3….N
Yi=Xi

11. Buckling Analyses of Thin, isotropic Rectangular plate
• The governing differential equation of buckling of a thin
Rectangular plate is given by:

12. Boundary conditions for Rectangular Plates
Four edges clamped (C-C-C-C )
• The boundary conditions for a plate clamped on all four edges
are that the displacement and rotation must be zero on the
edges.
 w(X,0)=w(X,1)=0 & w(0,Y)=w(1,Y)=0
 dw/dY (X,0)= dw/dy (X,1)=0
• Applying the differential Quadrature to these boundary
conditions:
 w1i=wNi=0 & wi1=wiN=0
 w1j=wnj=0 & wj1=wjN=0

13. Four edges Simply Supported (S-S-S-S) :
 w(X,0)=w(X,1)=0 & w(0,Y)=w(1,Y)=0
 second derivative of deflection in both direction is zero as
moment is zero at edges.
 Applying Differential Quadrature to these equations:
 w1j=wNj=0 & wi1=wiN=0
 w1j=wNj=0 & wi1=wiN=0

14. Vibration of Plates
• Plates belong to basic structural elements in civil and mechanical
engineering.
• The conventional differential quadrature method has also been
applied to the vibration analysis of plates.
• Very accurate results can be obtained applying a grid with points
densely concentrated near boundaries.
• The dimensionless governing equation for free vibration of the plate
is as follows:
W denotes dimensionless mode shape function
X = x/a and Y = y/b are dimensionless coordinates
a and b are lengths of the plate edges
a/b is the aspect ratio and is the dimensionless frequency

15. Mode sequence 0.4 0.667 1.0 1.5 2.5
1. 11.4487 14.2561 19.7392 32.0762 71.5546
2. 16.1862 27.4156 49.3480 61.6850 101.1634
3. 24.0818 43.8649 49.3480 98.6960 150.5115
4. 35.1355 49.3480 78.9568 111.0331 219.5967
5. 41.0576 57.0244 98.6961 128.3049 256.6097
6. 45.7950 78.9569 98.6961 177.6529 286.2185
7. 49.3217 80.0526 128.3049 180.1183 308.2603
8. 53.6907 93.2130 128.3049 209.7292 335.566
9. 64.7443 106.3724 167.7813 239.3380 404.6518
Vibration Results comparison on the basis of Aspect ratio:
S-S-S-S end conditions:

16. Mode sequence 0.4 0.667 1.0 1.5 2.5
1. 16.8475 19.9512 27.0541 44.8903 105.2970
2. 21.3573 34.0199 60.5385 76.5448 133.4833
3. 29.2255 54.3636 60.7861 122.3181 183.6596
4. 40.4933 57.5077 92.8361 129.3924 253.030
5. 51.4504 67.7898 114.5563 152.5270 321.5650
6. 55.0961 90.0506 114.7038 202.6139 344.3503
7. 55.9631 90.4848 145.7807 203.5907 349.7692
8. 63.6114 108.6286 146.0805 244.4143 397.5713
9. 72.9689 121.8564 188.4604 274.1769 456.0433
C-C-C-C end condition:

17. Results
Type of
support
N=11 N=18 N=22 Standard
C-C-C-C 14.39 14.33 14.88 14.8
S-S-S-S 4.03 4.00 4.00 4,0
S-C-S-C 7.82 7.82

18. References:
References:
• Bert Cw, Wang,X And Striz,A Z, Convergence Of The Dq Method In The Analyses Of Anisotropic Plate, Journal Of Sound
And Vibration
•
• Krowiak Artur , Methods Based On The Differential Quadrature In Vibration Analysis Of Plates, Journal Of Theoretical And
Applied Mechanics
•
• Xinwei Wang, Lifei Gan, Yihui Zhang, Buckling Analysis Of A Laminate Plate, Engineering Structures Application Of
Differential Quadrature (DQ) And Harmonic Differential Quadrature (HDQ) For Buckling Analysis Of Thin Isotropic Plates
And Elastic Columns
•
• Artur Krowiak, Journal Of Theorotical And Applied Mechanics , Methods Based On The Differential Quadrature In Vibration
Analysis Of Plates
•
• E. Kormaníková, I. Mamuzic, Buckling Analyses Of Laminated Plate
•
• O¨Mer Civalek , Dokuz Eylu¨ L Xinwei Wang , Lifei Gan, Yihui Zhang Advances In Engineering Software Differential
Quadrature Analysis Of The Buckling Of Thin Rectangular Plates With Cosine-Distributed Compressive Loads On Two
Opposite Sides
•
• Faruk Civan And C. M. Sliepcevich , Journal Of Mathematical Analysis And Applications, Differential Quadrature For
Multidimentional Problem