Buckling Analysis of a Compressed Straight Beam

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 crosssection – 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.

Material characteristics assume default values: Young's modulus E = 2.1E+011 Pа, Poisson's ratio ν = 0.28.
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.

The analytical solution to determine the critical load appears as:

Pcritical= π2 E J / ( μ L)2

where Е – the Young's modulus, J – the moment of inertia, L – the beam length, μ – the length factor that depends on the support arrangements and the beam loading method. In this case, μ = 2.

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

Table 1. Parameters of finite element mesh

Table 2. Result "Critical load"*

 

Conclusions:

The relative error of the numerical solution compared to the analytical solution is equal to 0.43% 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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