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--- | ||
jupytext: | ||
cell_markers: '"""' | ||
text_representation: | ||
extension: .md | ||
format_name: myst | ||
format_version: 0.13 | ||
jupytext_version: 1.16.1 | ||
kernelspec: | ||
display_name: base | ||
language: python | ||
name: python3 | ||
--- | ||
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```{index} Transformations; Class exercise using analytical formulas | ||
``` | ||
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```{custom_download_link} intro.ipynb | ||
:text: ".ipynb" | ||
``` | ||
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```{custom_download_link} intro.md | ||
:text: ".md" | ||
``` | ||
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(lesson13.1)= | ||
# Lesson November 25th | ||
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During today's lesson you'll work on a complex exercise on the topic of the Transforming tensors. Please ask your questions regarding the [homework](homework13.1) as well! | ||
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## Exercise Transforming tensors | ||
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Given is the following structure and cross-section: | ||
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```{figure} intro_data/structure.svg | ||
:align: center | ||
``` | ||
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1. Find the relevant cross-sectional properties. | ||
2. Find the normal and shear stresses just below $\text{G}$, in $\text{H}$, in $\text{I}$ and just right of $\text{C}$ in cross-section $\text{A}$. | ||
3. Find the principal values of the stresses in the points just below $\text{G}$, in $\text{H}$, in $\text{I}$ and just right of $\text{C}$ in cross-section $\text{A}$. | ||
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````{admonition} Solution assignment 1 | ||
:class: tip, dropdown | ||
Normal force centre is given by: | ||
```{figure} intro_data/NC.svg | ||
:align: center | ||
``` | ||
- $A \approx 17500 \text{ mm}^2$ | ||
- $I_{zz} \approx 655 \cdot 10^6 \text{ mm}^4$ | ||
```` | ||
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````{admonition} Solution assignment 2 | ||
:class: tip, dropdown | ||
- $\sigma_\text{just below G} = +6.73 \text{ MPa}$ | ||
- $\tau_\text{just below G} = +0.164 \text{ MPa}$ | ||
- $\sigma_\text{H} = +6.73 \text{ MPa}$ | ||
- $\tau_\text{H} = 0 \text{ MPa}$ | ||
- $\sigma_\text{I} = -2 \text{ MPa}$ | ||
- $\tau_\text{I} = +0.35 \text{ MPa}$ | ||
- $\sigma_\text{just right of C} = -8.53 \text{ MPa}$ | ||
- $\tau_\text{just right of C} = -0.12 \text{ MPa}$ | ||
```` | ||
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````{admonition} Solution assignment 3 | ||
:class: tip, dropdown | ||
- $\sigma_\text{1, just below G} = +6.73 \text{ MPa}$ | ||
- $\sigma_\text{2, just below G} = -0.0040 \text{ MPa}$ | ||
- $\sigma_\text{1, H} = +6.73 \text{ MPa}$ | ||
- $\sigma_\text{2, H} = 0 \text{ MPa}$ | ||
- $\sigma_\text{1, I} = 0.59 \text{ MPa}$ | ||
- $\sigma_\text{2, I} = -2.06 \text{ MPa}$ | ||
- $\sigma_\text{1, just right of C} = 0.0018 \text{ MPa}$ | ||
- $\sigma_\text{2, just right of C} = -8.5 \text{ MPa}$ | ||
```` | ||
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Test of adding more text | ||
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```{code-cell} | ||
:tags: [remove-cell] | ||
Izz = 655e6 | ||
tau_G = -3e3 * -250*10*286 / 20 / Izz | ||
print(tau_G) | ||
tau_C = -3e3 * -250*20*(286-500) / 40 / Izz | ||
print(tau_C) | ||
sigma_G = +6.73 | ||
tau_G = -tau_G | ||
sigma_G_1 = 1/2 * (sigma_G + 0) + ((1/2 * (sigma_G + 0))**2 + tau_G**2)**0.5 | ||
sigma_G_2 = 1/2 * (sigma_G + 0) - ((1/2 * (sigma_G + 0))**2 + tau_G**2)**0.5 | ||
print(sigma_G_1, sigma_G_2) | ||
sigma_H = +6.73 | ||
tau_G = 0 | ||
sigma_H_1 = 1/2 * (sigma_H + 0) + ((1/2 * (sigma_H + 0))**2 + tau_G**2)**0.5 | ||
sigma_H_2 = 1/2 * (sigma_H + 0) - ((1/2 * (sigma_H + 0))**2 + tau_G**2)**0.5 | ||
print(sigma_H_1, sigma_H_2) | ||
sigma_I = -2 | ||
tau_I = +0.35 | ||
sigma_I_1 = 1/2 * (sigma_I + 0) + ((1/2 * (sigma_I + 0))**2 + tau_I**2)**0.5 | ||
sigma_I_2 = 1/2 * (sigma_I + 0) - ((1/2 * (sigma_I + 0))**2 + tau_I**2)**0.5 | ||
print(sigma_I_1, sigma_I_2) | ||
sigma_C = -8.53 | ||
tau_C = -tau_C | ||
sigma_C_1 = 1/2 * (sigma_C + 0) + ((1/2 * (sigma_C))**2 + tau_C**2)**0.5 | ||
sigma_C_2 = 1/2 * (sigma_C + 0) - ((1/2 * (sigma_C))**2 + tau_C**2)**0.5 | ||
print(sigma_C_1, sigma_C_2) | ||
``` |
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