Buckling Analysis of Square Plate

Buckling Analysis of a Square Plate

Let us consider a square plate with a side a and thickness h (see figure).

Buckling of a Square Plate,

The thickness of plate h is much smaller than the length of its side a.
The plate is uniformly compressed in a transversal direction.
Consider the case when the loaded edges of plate are simply-supported; non-loaded edges are clamped.

Buckling of a Square Plate, scheme of loading

Let us use the following data: plate side length a = 500 mm, thickness of plate h = 3 mm , applied distributed force P = 1 Pa.
Material characteristics assume default values: Young's modulus E = 2.1E+011 Pа, Poisson's ratio ν = 0.28.

Buckling of a Square Plate, the finite element model with applied loads and restraints

The finite element model with applied loads and restraints

Analytical solution for this problem is given by:

σcritical = K π2 D / a2 h ,

where E – Young’s modulus, D = E h3 / 12 (1-ν2) – cylindrical stiffness of plate, K – coefficient whose value depends on the type of the supports of the plate edges (in this case K = 7.69).
Thus, σcritical = K π2 D / a2 h = 0.5188E+008 Pa.
After carrying out calculation with the help of AutoFEM, the following results are obtained:

Table 1. Parameters of finite element mesh

Finite Element Type	Number of nodes	Number of finite elements
quadratic tetrahedron	5202	15000

Table 2. Result "Critical load"*

Numerical solution Critical load σ*critical, Pa	Analytical solution Critical load σcritical, Pa	Error δ = 100%\|σcritical-σcritical\| / \|σcritical\|
0.5277E+008	0.5188E+008	1.72

Buckling of a Square Plate, first buckling mode of the plate

Conclusions:

The relative error of the numerical solution compared with the analytical solution not exceed 1.72% for quadratic finite elements.

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.

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