Linear Buckling

1. Flat Isotropic Plate

THis test calculates the linear buckling load of a flat panel, using different element types. One half of the plate is modelled.

Flat isotropic plate

Figure 128. Flat isotropic plate

The plate has a length a=1 [m], width b=0.25 [m], and thickness t=0.001 [m], and is meshed 20 by 10 elements or 10 by 5 elements, depending on the element type. The material is isotropic linear elastic, with elasticity modulus e=72*109 [N/m2] and ν=0.3.

All 4 edges are simply supported. Edge 3 has either symmetric or anti-symmetric boundary conditions. A uniformly distributed force of 1000 [N/m] in the x-direction is applied along edge 4.

The analytical solution has been obtained from Roark's Formulas[39]. The buckling loads λ obtained with this model are summarised in the tables below:

Table 24. Results with symmetric BC's

Type λ1 λ2 λ3
analytical 1.041 1.222 1.627
Q4.S.MITC.E4 20x10 elements 1.047 1.246 1.643
Q8.S.MITC 20x10 elements 1.042 1.224 1.631
Q9.S.MITC 10x5 elements 1.041 1.223 1.633

Table 25. Results with anti-symmetric BC's

Type λ1 λ2 λ3
analytical 4.170 4.376 4.519
Q4.S.MITC.E4 20x10 elements 4.261 4.580 4.582
Q8.S.MITC 20x10 elements 4.223 4.473 4.550
Q9.S.MITC 10x5 elements 4.169 4.395 4.520

The test is also executed with irregular meshes in the y-direction and symmetry conditions along edge 1 and 3

[39] R. Roark; Formulas for Stress & Strain, Sixth edition, Warren C. Young, McGraw-Hill Book Company