# Cantilever beam

## 1 Benchmark

The table above Figure 6.6 in Bell (1987b) analyses the tip displacement of a cantilever beam subjected to a point load at its tip. The beam has an HE300B cross section, with a given shear area
$$A_S = 0.029\text{ m}^2$$
, a second moment of inertia
$$I = 0.2517\text{ m}^4$$
, an Elastic modulus
$$E = 210\cdot10^9\text{ Pa}$$
and a Poisson ratio of
$$\nu = 0.3$$
. The height of the cross section is
$$h = 0.3\text{ m}$$
.
For different ratios of length
$$l$$
over height
$$h$$
of the beam, the table gives the expected proportion of displacement with and without considering Shear deformation, which in Ashes can be modelled by Timoshenko and Euler-Bernoulli elements, respectively.

The data from the table is reproduced below.

 $$l/h$$ 2 5 10 20 $$\sigma$$ 2.88 1.3 1.08 1.02

where
$$\sigma$$
is the ratio of the displacement witth Timoshenko elements over the displacement with Euler-Bernoulli elements.

For this test, we remove gravity loads and we apply a load
$$P = 100\text{ kN}$$
at the tip of the beam, as shown in the figure below:

The analytical solution of the tip displacement for an Euler-Bernoulli beam can be found for example in Wikipedia as
$$w_E = \frac{Pl^3}{3EI}$$

By using the characteristics of the cross-section and the Euler-Bernoulli analytical solution, we can find the expected tip displacements for different beam lengths and beam theories, as summarised in the table below:

 $$l\text{ }(m)$$ 0.6 1.5 3 6 $$w_E\text{ }(m)$$ $$1.36\cdot10^{-4}$$ $$0.00213$$ $$0.0170$$ $$0.136$$ $$w_T\text{ }(m)$$ $$3.92\cdot10^{-4}$$ $$0.00277$$ $$0.0184$$ $$0.139$$

The load cases are run with the Linear and the Nonlinear solvers, so there is a total of 16 simulations.

## 2 Results

The test is considered passed if the results from Ashes lie within 0.5% of the Benchmark values

The report for this test can be found on the following link:

The test is also run assuming a static simulation. The results are expected to be the same and can be found in this report: