Finite Element Analysis of 3D Structures using Python

Section 1: Welcome and Preliminaries

In section one we’ll get our bearings and take a look at what the journey ahead looks like. This will give you a good idea of what to expect and how our solver will take shape. As this course is the fourth in a series, I’ll also say a few words on prerequisites and what you should consider covering before working through the course. Both of the short lectures in this section can be watched below before enrolling – take a look.

Section 2: The 3D Beam Stiffness Matrix

In this section we get right into the thick of it and start to work out how we’re going to model our 3D beam elements. By the end of this section we will have fully derived the 12×12 beam element stiffness matrix. We’ll start by briefly reviewing the 2D beam element before tackling the additional degrees of freedom in our 3D element.

Section 3: Location and Rotation in 3D Space

If there is one thing that has the potential to confuse you when trying to perform 3D frame analysis it’s how to deal with rotations in 3D space. Understanding how to handle element orientation, rotations and transitions between reference frames is critical. We’re going to tackle this head-on in this section. By the end of the section you will understand exactly how we’re capturing the orientation of our beam elements. The output at the end of this section will be our element transformation matrix.

Section 4: Generating frame data in Blender

In this section we take a break from theory and dive into Blender for some light relief. In this section we’ll generate our structural model and use Blender’s subdivide tools to easily ‘mesh’ our structure. By the end of this section we’ll have all of our model data loaded up inside our Jupyter Notebook and be ready to start writing our solver.

Section 5: Building the Transformation Matrix

In this section we’re going to focus on capturing the orientation of each element within our structure. This is one example of something that we didn’t need to spend too much time on for purely axially loaded 3D members. But for 3D beam elements, whether they bend about a major or minor axis has a huge impact on their behaviour. We’ll start by defining a default orientation for all members. Once we have a predictable orientation for each member, we’ll work out a strategy for calculating and storing each member’s transformation matrix.

Section 6: Building and Solving the Stiffness Matrix

At this point in our development, we have a lot of the necessary foundation work done. In this section we can actually solve the structure. If you’ve covered any of the prerequisite courses, you will be on familiar ground in this section. By the end of this section, our structure will be solved and all of our data will be just waiting to be visualised which we’ll take care of in the next section.

Section 7: Visualising Results in 3D

All of the data in the world is of no value to us unless we can visualise it and ultimately interpret it. This section is going to focus on building the standard plots you would expect to see in any structural analysis software. Once this section is complete we will have a fully functioning solver and a complete workflow where we build our models in Blender, import them and analyse them in our Jupyter Notebook solver and optimally review the deflections back in Blender. The next sections in the course are focused on implementing features that enhance the usability of our solver and turn it into a versatile and valuable analysis tool.

Section 8: Implementing Bar Bracing elements:

In this section we’re going to focus on implementing purely axially loaded elements within our code. We’ll refer to these as bar elements, as opposed to beam elements that also resist shear, bending and torsion. We typically encounter these types of elements as bracing members within structures. The portal frame structure we’ve been working on up to now is a classic example of a structure that would use this as a lateral stability system. After this section is complete, our solver will be capable of handling some of the most common forms of framed structure.

Section 9: Implementing Pinned Members:

In this section we’ll implement the ability to model a new type of element – beam elements that have a pin at node i or node j. These pins will act as rotational releases for major and minor axis bending at the respective node. Practically these pins allow our solver to handle simply supported or simply restrained structural members within our models. A practical example of this might be a beam that has a simple fin-plate shear connection on one or both ends. We wouldn’t want to model moment transmission through these connections. After completing this section we will be able to model the required rotational releases.

Section 10: Implementing Distributed Loading:

The ability to handle distributed loading is probably the last big omission in our solver, so we’ll put that right in this section. As usual we’ll start in Blender and identify and export the members within our model that we want to apply uniformly distributed loading to. In contrast to the manual process of determining equivalent nodal actions when we did this for 2D frames, we’ll automate the process here making for a much smoother workflow. Once this section is complete, we’ll have a well rounded solver to put to work.

Section 11: Bridge Model Case Study:

The purpose of this is to take a non-standard structure, unlike what we’ve been modelling up to this point – and see how to approach its modelling and analysis using the tools we’ve developed. One of the benefits of doing this is that we’ll identify good opportunities to further improve our solver – which you should consider doing when you’ve finished this course! I’ll also demonstrate how to make use of some of Blender’s fantastic mesh editing tools that make our lives so much easier!