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Cantilever Beam (IPE 200 profile)

To validate the implementation of the beam elements, the following cantilever beam with a IPE 200 profile is analysed:

  • Young's modulus: \(E=210\) GPa
  • Poisson's ratio: \(\nu=0.3\)
  • Bending moment of inertia: \(I_{zz}=1943\) cm\(^4\)
  • Bending moment of inertia: \(I_{yy}=142\) cm\(^4\)
  • Torsion moment of inertia: \(J=6.98\) cm\(^4\)
  • Cross sectional area: \(A=28.50\) cm\(^2\)
  • Length: 2 m

The cantilever beam is fixed on the left end and a concentrated load of 10 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.

Analytical solution

The total deflection according to Timoshenko is:

\[ w_{T} = w_{EB} + w_{shear} = \frac{FL^3}{3EI} + \frac{FL}{κGA} \]

For an IPE 200 profile with strong-axis bending, using the specialized analytical approach by Cowper, the shear correction factor is approximately \(\kappa = 0.4\) (accounting for the thin web of the I-section).

The Euler-Bernoulli deflection \(w_{EB}\) is 6.54 mm. Now calculating the shear contribution:

\[ w_{shear} = \frac{FL}{κGA} = \frac{10,000 \cdot 2}{0.4 \cdot 80.77 \cdot 10^9 \cdot 28.50 \cdot 10^{-4}} = 0.217 \cdot 10^{-3} ~\text{m} \]

Therefore, the total deflection incorporating shear effects is:

\[ w_{T} = 6.54 + 0.22 = 6.76 ~\text{mm} \]

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
2.85d-3, 210d6, 80.769d6, 19.43d-6, 1.42d-6, 6.98d-8, 0.4, 0.4

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
  • Cross sectional area: \(A=2.85 \cdot 10^{-3}\) cm\(^2\)
  • Bending moments of inertia \(I_{zz}=19.43 \cdot 10^{-6}\) m\(^4\)
  • Bending moments of inertia \(I_{yy}=1.42 \cdot 10^{-6}\) m\(^4\)
  • Torsion moments of inertia \(J=6.98 \cdot 10^{-8}\) m\(^4\)
  • Shear correction factors \(\kappa_y=\kappa_z=0.4\)

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 6.753 mm -0.1 %
u2-beam 5/6 6.753 mm -0.1 %
u3-beam 1/3 5.771 mm -14.6 %
u3-beam 2/5 6.641 mm -1.8 %
u3-beam 4/9 6.744 mm -0.2 %

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