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.
|
$$l/h$$
|
2 | 5 | 10 | 20 |
|
$$\sigma$$
|
2.88 | 1.30 | 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 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:
|
$$l\text{ }(m)$$
|
0.6 | 1.5 | 3 | 6 |
|
$$w_E\text{ }(m)$$
|
$$1.36\cdot10^{-4}$$
|
$$0.00213$$
|
$$0.0170$$
|
$$0.136$$
|
|
$$w_T\text{ }(m)$$
|
$$3.92\cdot10^{-4}$$
|
$$0.00277$$
|
$$0.0184$$
|
$$0.139$$
|
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: