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jupytext:
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    extension: .md
    format_name: myst
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kernelspec:
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```{margin}

::::::{attention}
This page shows a preview of the assignment. Please fork and clone the assignment to work on it locally from [GitHub](https://github.com/CIEM5000-2026/practice-assignments)
::::::

::::::{versionadded} v2026.2.0 After workshop 2
Solutions workshop 2 in text and downloads 
::::::

```

# Apply

In this notebook you will solve a 2-element frame at the end of the notebook.

+++

Our matrix method implementation is now completely stored in a local package, consisting of three classes.

+++

```{custom_download_link} ./Workshop_2_Apply_stripped.ipynb
:text: ".ipynb"
:replace_default: "True"
```

```{custom_download_link} ./Workshop_2_Apply_stripped_sol.ipynb
:text: ".ipynb solution"
:replace_default: "False"
```

```{custom_download_link} ./Workshop_2_Apply.md
:text: ".md:myst"
:replace_default: "False"
```

```{custom_download_link} https://github.com/CIEM5000-2026/practice-assignments
:text: "All files practice assignments"
:replace_default: "False"
```

```{custom_download_link} https://github.com/CIEM5000-2026/practice-assignments/tree/solution_workshop_2
:text: "All files practice assignments with solutions workshop 2"
:replace_default: "False"
```


## Two-element frame

+++

```{figure} https://raw.githubusercontent.com/ibcmrocha/public/main/twoelemframe.png
:class: sticky-margin
:align: center
:width: 400
```


With:
- $EI = 1500$
- $EA = 1000$
- $q = 9$
- $L = 5$
- $\bar\varphi = 0.15$

```{exercise-start} Workshop 2 - Apply
:label: exercise_ws_2
:nonumber: true

The final example for the workshops is the two-element frame above. Here you should make use of all the new code you implemented:
    
- Set up the problem and compute a solution for `u_free`. Remember to consider the prescribed horizontal displacement $\bar{u}$ at the right end of the structure.
- Compute and plot bending moment lines for both elements (in the local and global coordinate systems)
- Compute reactions at both supports
```

```{code-cell} ipython3
:tags: [thebe-remove-input-init]

import matplotlib as plt
import numpy as np
sys.path.insert(1, '/matrixmethod_solution')
import matrixmethod_solution as mm
%config InlineBackend.figure_formats = ['svg']
```

```{code-cell} ipython3
:tags: [remove-cell]

import matplotlib as plt
import numpy as np
import matrixmethod_solution as mm
%config InlineBackend.figure_formats = ['svg']
```

```{code-cell} ipython3
:tags: [disable-execution-cell]

import numpy as np
import matplotlib as plt
import matrixmethod as mm
%config InlineBackend.figure_formats = ['svg']
```

```{code-cell} ipython3
#YOUR CODE HERE
```

```{code-cell} ipython3
for elem in elems:
    u_elem = con.full_disp(#YOUR CODE HERE)[#YOUR CODE HERE.global_dofs()]
    elem.plot_displaced #YOUR CODE HERE
```

```{exercise-end}
```

::::::{hint}
:class: dropdown

For the given parameter values, if your implementation is fully correct, you should get the following nodal displacements and support reactions:

$$
\mathbf{u}_\mathrm{free} = \left[-0.09274451, -0.13310939,  0.51159348, -0.01644455\right]
$$

$$
\mathbf{f}_\mathrm{cons} = \left[27.35024439, -63.82451092,  17.64975561, -71.17548908, -36.75489076\right]
$$

You should also get the following moment lines for the two elements:

- in local coordinate system:
![](https://raw.githubusercontent.com/ibcmrocha/public/main/moments_local.svg)

- in global coordinate system:
![](https://raw.githubusercontent.com/ibcmrocha/public/main/moments_global.svg)

And the following displacements:
- in local coordinate system:
![](https://raw.githubusercontent.com/ibcmrocha/public/main/displacements_local.svg)

- in global coordinate system:
![](https://raw.githubusercontent.com/ibcmrocha/public/main/displacements_global.svg)
::::::

+++

```{solution-start} exercise_ws_2
:class: dropdown
```

```{code-cell} ipython3
:tags: [thebe-init]

EI = 1500
EA = 1000
q = 9
L = 5
phibar = 0.15

mm.Node.clear()
mm.Element.clear()

nodes = []

nodes.append(mm.Node(0,0))
nodes.append(mm.Node(L,-L))
nodes.append(mm.Node(2*L,0))

elems = []

elems.append(mm.Element(nodes[0], nodes[1]))
elems.append(mm.Element(nodes[1], nodes[2]))

section = {}
section['EI'] = EI
section['EA'] = EA

for elem in elems:
    elem.set_section(section)
    print(elem)

con = mm.Constrainer()

con.fix_dof (nodes[0], 0)
con.fix_dof (nodes[0], 1)
con.fix_dof (nodes[2], 0)
con.fix_dof (nodes[2], 1)
con.fix_dof (nodes[2], 2, phibar)

elems[0].add_distributed_load([0,q])
elems[1].add_distributed_load([0,2*q])

print(con)

global_k = np.zeros ((3*len(nodes), 3*len(nodes)))
global_f = np.zeros (3*len(nodes))

for e in elems:
    elmat = e.stiffness()
    idofs = e.global_dofs()
    
    global_k[np.ix_(idofs,idofs)] += elmat

for n in nodes:
    global_f[n.dofs] += n.p

Kc, Fc = con.constrain ( global_k, global_f )
u_free = np.matmul ( np.linalg.inv(Kc), Fc )
print(u_free)

print(con.support_reactions(global_k,u_free,global_f))
```

```{code-cell} ipython3
:tags: [thebe-init]

for elem in elems:
    u_elem = con.full_disp(u_free)[elem.global_dofs()]
    elem.plot_displaced(u_elem,num_points=51,global_c=False,scale=1)
```

```{code-cell} ipython3
:tags: [thebe-init]

for elem in elems:
    u_elem = con.full_disp(u_free)[elem.global_dofs()]
    elem.plot_displaced(u_elem,num_points=51,global_c=True,scale=1)
```

```{code-cell} ipython3
:tags: [thebe-init]

for elem in elems:
    u_elem = con.full_disp(u_free)[elem.global_dofs()]
    elem.plot_moment_diagram(u_elem,num_points=20,global_c=False)
```

```{code-cell} ipython3
:tags: [thebe-init]

for elem in elems:
    u_elem = con.full_disp(u_free)[elem.global_dofs()]
    elem.plot_moment_diagram(u_elem,num_points=20,global_c=True,scale=0.05)
```

```{solution-end}
```