# Element weight

## 1 Test description

This test uses a one element model to compute the reaction force at the support at the bottom of the model. No other loads than gravity are applied, so the reaction force of the support node should be the same as the weight of the element.
Note that when computing the reaction force of an element, Ashes omits the bottom half of the element connected to the support.
We use the output Reaction force magnitude from the Support sensor.

The following load cases are tested:
1. One element model with a Circular hollow cross section
2. One element model with a Circular shape cross section
3. One element model with a Circular solid cross section
4. One element model with a Rectangular hollow cross section
5. One element model with a Rectangular shape cross section
6. One element model with a Circular hollow cross section on Mars

## 2 Analytical solution

In this section we compute the weigth of each model. When computing the reaction force, Ashes omits the lower half of the element connected to the support, so only half of the weight of each model should be compared to the output from Ashes.

### 2.1 Circular hollow cross section

The model is shown in the figure below: Note: circular hollow cross sections appear filled, but this only affects the visualization.

$$r = 2.5$$
m, the thickness is
$$t=0.2$$
m, so the structural area is
$$A=\pi r^2 - \pi(r-t)^2 = 3.0158 m^2$$
The length of the element is
$$l = 10$$
m and the material density is
$$d=8500 kg\cdot m^3$$
, so the mass of the element is
$$m = A\cdot l\cdot d = 256 346$$
kg.
The acceleration due to gravity is
$$g = 9.80665 \text{ m}\cdot \text{s}^{-2}$$
, so the weight is
$$W = 2513$$
kN.
The analytical value to compare to the output is
$$W/2 = 1257$$
kN.

### 2.2 Circular chape cross section

The model is shown in the figure below: The input linear mass is
$$m_l = 25000 kg/m^3$$
and the length of the element is
$$l=10 m$$
, so the mass of the element is
$$m = 250000$$
kg.
This gives a weigth of
$$W = g\cdot m = 2452\text{ kN}$$
.
The analytical value to compare to the output is
$$W/2 = 1226 \text{ kN}$$

### 2.3 Circular solid cross section

The model is shown in the figure below: The dimensions are the same as in the Circular hollow case, except that the cross section is now solid (i.e. filled).
This gives a half-weight of
$$W/2 = 8183\text{ kN}$$

### 2.4 Rectangular hollow cross section

The model is shown in the figure below: The rectangular hollow cross section has a height
$$a = 5\text{ m}$$
and a width
$$b = 1\text{ m}$$
. All other input are the same as for the Circular hollow cross section. This gives a structural area
$$A = a\cdot b-(a-2t)\cdot(b-2t) = 2.24\text{ m}^2$$
.
This gives a half-weight of
$$W/2 = 934\text{ kN}$$

### 2.5 Rectangular shape cross section

The model is shown in the figure below: The input are the same as in case 2, therefore the output to compare to is
$$W/2 = 1226\text{ kN}$$

### 2.6 Circular hollow cross section on Mars

The model is shown in the figure below: The characteristics of the model are the same as in the Circular hollow case, so the mass of the model is
$$m = 256346\text{ kg}$$

The acceleration due to gravity on Mars is
$$g_M = 3.72076\text{ m}\cdot\text{s}^{-2}$$
and the mass of Matt Damon is neglected, so the half-weight of the element is
$$W/2 = 477\text{ kN}$$
.

## 3 Results from the regression test

A test is considered passed if the results produced by Ashes are within 0.01% of the analytical solution.

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

In Ashes, the loads are applied with a ramp-up, i.e. they start at 0 and tere value is progressively increased until their value (see Analysis). Therefore, the output produced by Ashes will show a ramp-up period as well.
A passed test will therefore show a red horizontal line for the analytical results and a blue smoothly increasing curve for the results produced by Ashes, such as the one shown in the figure below: 