Implementation

Attention

This page shows a preview of the assignment. Please fork and clone the assignment to work on it locally from GitHub

After the workshop, the solution will be added to this preview and to the GitHub-repository

In this notebook you will continue to implement the matrix method and check it with some sanity checks.

Exercise (0)

Check whether your implementation of last week was correct using the provided solution

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

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

%load_ext autoreload
%autoreload 2

1. The Node class

The Node class from last week is unchanged and complete

2. The Element class

The implementation is incomplete:

• The function add_distributed_load should compute the equivalent load vector for a constant load \(q\) and moves those loads to the nodes belonging to the element. Remember to use the add_load function of the Node class to store the equivalent loads (remember we have two nodes per element). Also keep local/global transformations in mind and store self.q = q for later use;

• The function bending_moments receives the nodal displacements of the element in the global coordinate system (u_global) and uses it to compute the value of the bending moment at num_points equally-spaced points along the element length. Keep local/global transformations in mind and use the ODE approach in SymPy / Maple / pen and paper to compute an expression for \(M\). Do the same for for \(w\) in the function full_displacement.

Exercise (2.1)

Add the missing pieces to the code, before you perform the checks below.

Having made your implementations, it is now time to verify the first addition of your code with a simple sanity check. We would like to solve the following simply-supported beam:

Choose appropriate values yourself.

Exercise (2.2)

Use the code blocks below to set up this problem. After you’ve added the load, print the element using print(YOUR ELEMENT). Do the shown values for the nodal loads correspond with what you’d expect?

#YOUR CODE HERE

print(#YOUR ELEMENT HERE

Exercise (2.3)

Now solve the nodal displacements. Once you are done, compare the rotation at the right end of the beam. Does it match a solution you already know?

#YOUR CODE HERE

Exercise (2.4)

Calculate the bending moment at midspan and plot the moment distribution using plot_moment_diagram. Do the values and shape match with what you’d expect?

u_elem = con.full_disp(#YOUR CODE HERE)
#YOUR CODE HERE

Exercise (2.5)

Calculate the deflection at midspan and plot the deflected structure using plot_displaced. Do the values and shape match with what you’d expect?

#YOUR CODE HERE

3. The Constrainer class

We’re going to expand our Constrainer class, but the implementation is incomplete:

• The constrainer class should be able to handle non-zero boundary conditions too. constrain should be adapted to do so + the docstring of the class itself. Furthermore, the assert statement of fix_dof should be removed.

• The function support_reactions is incomplete. Since the constrainer is always first going to get constrain called, here we already have access to self.free_dofs. Together with self.cons_dofs, you should have all you need to compute reactions. Note that f is also passed as argument. Make sure you take into account the contribution of equivalent element loads that go directly into the supports without deforming the structure.

Exercise (3.1)

Add the missing pieces to the code and docstring, before you perform the checks below.

We’re going to verify our implementation. Therefore, we’re going to solve an extension bar, supported at both ends, with a load \(q\).

Choose appropriate values yourself.

Exercise (3.2)

Can you say on beforehand what will be the displacements? And what will be the support reactions?

Use the code blocks below to set up and solve this problem and check the required quantities to make sure your implementation is correct.

#YOUR CODE HERE

Again, we’re going to verify our implementation. Therefore, we’re going solve a beam, with a load \(F\) and support displacement \(\bar w\) for the right support.

Choose appropriate values yourself.

Exercise (3.3)

Use the code blocks below to set up and solve this problem and check the required quantities to make sure your implementation is correct.

#YOUR CODE HERE