Buckling Analysis of a Compressed Straight Beam (surface)

Let's review the buckling analysis of a straight beam compressed with an axial symmetrical load (the Euler's problem). A straight beam of the length l, width and height of the cross section – b and h respectively, is cantilevered at one end, and a compressing load P acting on the other end. Sought is the load factor corresponding to the start of the beam buckling. Assume the beam length equal to 0.5 m, and the crosssection dimensions b = 0.05 m, h = 0.02 m.

Buckling Analysis of a Compressed Straight Beam

Let's define the boundary conditions as follows. The bottom face is fully restrained, and the upper one is subjected to the distributed load in the amount of 1 N. Input data can be viewed here.

Buckling Analysis of a Compressed Straight Beam, the finite element model with applied loads and restraints

The finite element model with applied loads and restraints

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
linear triangle	306	500

Table 2. Result "Critical load"*

Numerical solution Critical load P*critical, Pa	Analytical solution Critical load Pcritical, Pa	Error δ = 100% \|Pcritical-Pcritical\| / \|Pcritical\|
6.9570E+004	6.9087E+004	0.7

Buckling Analysis of a Compressed Straight Beam, first buckling mode of the beam

First buckling mode of the beam

*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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