Buckling Analysis of Plate

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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

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.

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.

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.

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.

• 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:

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 )

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

Buckling Analyses of Thin, isotropic Rectangular plate

• The governing differential equation of buckling of a thin

Rectangular plate is given by:

Boundary conditions for Rectangular PlatesFour 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

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

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

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:

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:

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

References:

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