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Cantilever Beam (slender)

To validate the implementation of the beam elements, the following cantilever beam is analysed:

  • Young's modulus: \(E=210\) GPa
  • Poisson's ratio: \(\nu=0.3\)
  • Thickness: 10 mm
  • Width: 10 mm
  • Length: 200 mm

The cantilever beam is fixed on the left end and a concentrated load of 1 kN is applied in vertical direction on the right side of the beam. The maximum deflection \(w\) is used for results comparison. This problem is well known, and results can be easily compared with an analytical solution.

From above geometrical properties, the cross sectional area is \(A=100\) mm², the bending moment of inertia \(I_{yy}=I_{zz}=833.33\) mm\(^4\) and the torsion moment of inertia is \(J=1666.66\) mm\(^4\).

Analytical solution

For the linear problem, the analytical solution gives:

\[ w = \dfrac{FL^3}{3EI} \]

For the present beam, the theoretical deflection is determined to w = 15.238 mm.

Numerical solution

For the numerical solution 2-node linear and 3-node quadratic beam elements are used. The Young's modulus and the cross-sectional area as well as the moment of interias \(I_{yy}, I_{zz}, J\) are prescribed using the *Beam properties keyword of the *Material definition:

*Material, name = beam, phases=1
*Beam properties
1d-4, 210d6, 80.769d6, 8.33d-10, 8.3333d-10, 16.6667d-10, 0.8333, 0.8333

Note that the properties are given in m and kN/m²:

  • Young's modulus: \(E=210 \cdot 10^6\) kPa
  • Shear modulus: \(G=80.77 \cdot 10^6\) kPa
  • Bending moments of inertia \(I_{yy}=I_{zz}=8.33 \cdot 10^{-10}\) m\(^4\)
  • Torsion moments of inertia \(J=16.67 \cdot 10^{-10}\) m\(^4\)
  • Shear correction factors \(\kappa_y=\kappa_z=5/6\)

Different discretisations (number of elements) have been used.

Input files

The complete input files can be downloaded here

Results

The results of the numerical simulations are summarised in the table below:

Element Number of
elements/nodes		 Max. deflection Difference to
analytical solution
u2-beam 1/2 15.274 mm +0.2 %
u2-beam 5/6 15.274 mm +0.2 %
u3-beam 1/3 11.606 mm -23.8 %
u3-beam 2/5 14.497 mm -4.8 %
u3-beam 4/9 15.127 mm -0.7 %

From above results it becomes evident that the 3-node (quadratic) beam element suffers from shear locking. This is especially pronounced for slender structures. Refining the spatial resolution reduces the amount of shear locking.