3. Euler Buckling of Beam

The test computes the buckling load of a clamped beam subjected to compression. The beam is modelled with 2 and 3 node beams and with 4, 8, and 9 node MITC shell elements. The theoretical solution (Euler buckling) is readily obtained from the literature:[25]

P cr = 1 4 π 2 E J L 2

where E is the modulus of elasticity, J the moment of inertia, and L the length of the clamped beam.

For beam models the section properties are defined by constants, by section shape, and by section submodels.

One of the eccentricity tests computes the decrease of the buckling load when an eccentric compressive force is applied. The test is run for quadratic sections defined by constants, by section shape, and by section submodels (the different colours of the curves below).

Reduction of buckling load as function of beam section eccentricity ratio eccentricity/thickness for B2.S.RS (2 node beam) elements.

Figure 55. Reduction of buckling load as function of beam section eccentricity ratio eccentricity/thickness for B2.S.RS (2 node beam) elements.


Reduction of buckling load as function of beam section eccentricity ratio eccentricity/thickness for B3.S.RS (2 node beam) elements.

Figure 56. Reduction of buckling load as function of beam section eccentricity ratio eccentricity/thickness for B3.S.RS (2 node beam) elements.


One of the eccentricity tests rotates the beam by 45 degrees around the y-axis, testing the local beam reference frames. Local beam orientations are plotted below:

Beam orientation of eccentricity test (beam rotated 45 degrees around y-axis).

Figure 57. Beam orientation of eccentricity test (beam rotated 45 degrees around y-axis).




[25] R. J. Roark, W. C. Young, Formulas for Stress & Strain, Sixth edition, McGraw-Hill Book Company (1975).