# Maximum stress

## 1 Test description

This test checks that the results from the ULS sensor are correct. The maximum stress computed by Ashes is compared to the analytical solution.

## 2 Model

This test uses the Tubular tower only template.
The csv file to run this test can be downloaded from here.

The model is shown in the figure below:

The model has a homogeneous circular hollow cross section of diameter
$$D = 3.87\text{ m}$$
and thickness
$$t = 0.025\text{ m}$$
. The length of the beam is
$$l = 100\text{ m}$$
and it is fixed onto the floor.
The material is steel, with an Elastic modulus
$$E = 2.1\cdot10^{11}\text{ Pa}$$

A force
$$F = 400\text{ kN}$$
is applied at the tip of the y-direction.
A static analysis is run for this test.

## 3 Analytical solution

With these input, it is expected that the maximum stress will occur at the bottom of the structure. The bending moment at the bottom will be
$$M = F\cdot l=40\text{ MNm}$$
. For a cylindrical cross section, the neutral axis goes trough the center of the circular cross section. Therefore, the distance from the neutral axis from the external side of the cross section is
$$c = D/2=1.935\text{ m}$$
The second moment of area of the cross section is
$$I = \pi/4\left[(D/2)^4-(D/2-t)^4\right]=0.5581\text{ m}^4$$

The maximum normal stress will thus be
$$\sigma_{max}=\frac{M\cdot c}{I}=138\text{ MPa}$$

From this input, it is also possible to compute the deflection of the tip, which will be
$$d = \frac{F\cdot l^3}{3EI} = 1.13\text{ m}$$

## 4 Results

The test is considered passed if both the maximum normal stress and the tip deflection are within 0.1% of the analytical solution.
The report for this test can be dowloaded from the following link: