Timoshenko blade

This test uses the same benchmark as the Cantilever beam based on the book from Bell (1987b), but a blade is used instead of a support section.

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.

$$l/h$$
251020
$$\sigma$$
2.881.301.081.02

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

For this test, we create a blade with the same structural characteristics as those listed above. 
We remove gravity and aerodynamic loads and we apply a load 
$$P = 100\text{ kN}$$
 at the tip of the blade, 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 blade lengths and beam theories, as summarised in the table below:

$$l\text{ }(m)$$
0.61.536
$$w_E\text{ }(m)$$
$$1.36\cdot10^{-4}$$

$$0.00213$$

$$0.0170$$

$$0.136$$
$$w_T\text{ }(m)$$
$$3.92\cdot10^{-4}$$
$$0.00277$$
$$0.0184$$
$$0.139$$


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:

https://www.simis.io/downloads/open/benchmarks/current/Timoshenko Blade.pdf