Static Analysis of a Round Plate Clamped along the Contour

Static Analysis of a Round Plate Clamped Along the Contour

We need to find the maximum deflection of a round plate of radius R and thickness h, which is clamped (fixed) along the contour and is loaded with a uniform pressure q distributed on the top face of the plate.

Static Analysis of a Round Plate Clamped Along the Contour

Because of the symmetry of this study, we will work with one quarter of the plate.
Assume the plate radius R = 0.2 m, thickness h = 0.003 m, and the pressure q =10 kN / m2. Material characteristics: the Young's modulus E=2.1E+011 Pa, the Poisson's ratio ν = 0.28.
Next, we need to apply boundary conditions. The side surface of the plate will be fully restrained, whereas the free faces introduced after discarding 3/4 of the plate are subjected to partial restraints in the normal to the faces direction, because the points in the sections cannot have extra displacements in the normal direction due to the symmetry. Pressure in the amount of 10 kN / m2 is applied to the top face of the plate.

Static Analysis of a Round Plate Clamped Along the Contour, the finite element model with applied loads and restraints

The finite element model with applied loads and restraints

There is an analytical solution for this study. The deflection at the plate center is calculated by the formula:

w = q . R4 / 64 . D = 4.8762E-004 m ,

where q – is the pressure amount, R – the plate radius, D = ( E . h3 ) / (12 . (1-ν2) ) – flexural rigidity.
The stress on the plate contour is calculated by the formula:

σ = 0.75 . q . (R / h)2 = 3.3333E+007 Pa.

After carrying out calculation with the help of AutoFEM, the following results are obtained:

Table 1.Parameters of the finite element mesh

Finite Element Type	Number of Nodes	Number of Finite Elements
quadratic tetrahedron	3556	10942

Table 2.Result "Displacement, magnitude"*

Numerical Solution Displacement w*, m	Analytical Solution Displacement w, m	Error δ = 100%* \|w* - w\| / \|w\|
4.8339E-004	4.8762E-004	0.87

Static Analysis of a Round Plate Clamped Along the Contour, Result "Displacement, magnitude"

Table 3.Result "Normal Stress OX"*

Numerical Solution Stress σX, Pa	Analytical Solution Stress σ, Pa	Error δ = 100%* \|σX - σ\| / \|σ\|
3.6070E+007	3.3333E+007	8.21

Static Analysis of a Round Plate Clamped Along the Contour, Result "Normal Stress OX"

Dependence of the relative error on the number of finite elements

Conclusions:

The relative error of the numerical solution compared to the analytical solution is 0.87% for displacements and 8.21% for stresses when using quadratic finite elements.

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