Non-linear Finite Element Analysis of 2D Catenary & Cable Structures using Python

Non-linear Finite Element Analysis of 2D Catenary & Cable Structures using Python

Build an iterative solution toolbox to analyse structures that exhibit geometric non-linearity due to large deflections.

⏰ 10 h 49 min | 62 lessons
Published: April 2022
After completing this course...
You will understand the concept of geometric non-linearity and when it should be considered.
You will learn how to modify the element stiffness matrix to account for large deflections and changes in geometry.
You will implement a Newton-Raphson solution algorithm that seeks to converge on the final state of the structure.
You will build a workflow that uses open-source modelling tools to quickly generate structural geometry.
COURSE OVERVIEW

After completing this course, you’ll have built an iterative numerical solver for cable and truss structures that exhibit geometric nonlinearity due to large deformations.

This course focuses on building the understanding and tools to analyse structures that undergo large deflections when loaded. Large changes to the geometry of a structure can alter the internal stress distribution. This is known as geometric non-linearity and requires a more sophisticated solution strategy.

We’ll place particular emphasis on cable and catenary structures as these are classic examples of structures whose deformation under load can lead to geometric non-linearity. However, the code developed can be equally deployed to flexible truss structures.

The tools developed in this course are not meant as a replacement for commercial non-linear solvers (we’re not going to be rebuilding SAP2000! 🙂) – the objective here is to build your understanding of the behaviour and the best way to do this is by implementing what you learn by building your own solver.

Cable-supported truss
Progressive loading and non-linear response

Your final code will be capable of handling structures like the one above that consist of a mixture of axially loaded cable (tension only) and bar (tension and compression) elements. Our solver implements an iterative algorithm, so a solution that converges is not always guaranteed! We’ll be leaving the relative comfort and certainty of linear analysis behind – welcome to geometric non-linearity!

Course prerequisites

Much of what we cover in this course is about how to implement an iterative solver for the tricky problem of geometric non-linearity. This essentially involves implementing and updating the direct stiffness method in a loop. So, you should have a good understanding of the direct stiffness method for axially loaded elements.

Even if you have a theoretical understanding of the direct stiffness method, I still recommend that you complete my course on applying the direct stiffness method to truss structures.

That course is called, The Direct Stiffness Method for Truss Analysis with Python and it’s my entry point into matrix-based structural analysis.

A lot of the code we write in this course, for example how to build the overall structure stiffness matrix and basically all of the nuts and bolts of the basic stiffness method, is written and explained in more detail in that course. You’ll lower the cognitive load on yourself if you encounter this code in a less demanding context first.

The next question to address is how much Python do you need to know? Technically none, however, because there is a lot to contend with in this course, I would say trying to learn Python on top is going to be quite demanding. It’s not that the coding we’ll do is any more complex than what we’ve done in previous courses, it’s just about managing the cognitive demand.

This is another great reason to take the prerequisite course. Get familiar with Python and how I implement the direct stiffness method in that introductory course first. This will leave you in a great position to tackle this course and really focus on what’s new and challenging about analysing non-linear structures.

We’ll develop our Python code using the versatile Jupyter development environment. When you complete this course you’ll have a range of standalone code notebooks to deploy on your own projects. The code developed within each section of the course is also provided for download as a reference.

Course Breakdown

Section 1: Introduction and Course Breakdowns

In this short introductory section, we’ll take a tour of the course, section by section, to give you an idea of what to expect as you move through the course. We’ll also briefly discuss the course prerequisites.

Section 2: ‘Heavy’ cables – the linear solution

In section 2 we’ll start our analysis of cable behaviour. Cables very often undergo changes in their geometry under loading, either due to cable extension, directly applied loads or flex in the cable supports – they are the classic example of a non-linear structure. In this section, we’re going to establish a linear (closed-form) cable solution that ignores non-linear effects. This will provide a baseline case to test our non-linear code against later.

Section 3: Getting Comfortable with Non-linearity

In section 3, we’ll start to talk about non-linear structural behaviour, before focusing on geometric non-linearity in particular. The aim of this section is to take any mystery out of the term ‘geometric non-linearity’. We’ll also explore, at concept level, how we might set about solving for the behaviour of a non-linear structure. We’ll introduce the Newton Raphson method which informs the main architecture of the solver code we’ll write later on.

6-bar catenary

Section 4: The Non-linear Stiffness Matrix

One of the central elements of a matrix-based non-linear structural analysis is a stiffness matrix that can capture the influence of large deflections of the structure. In section 4 we’ll focus on deriving a form of non-linear element stiffness matrix that can do just that. Familiarity with the material in the prerequisite course will be helpful here.

Section 5: Building our 2D Solver Toolbox

Now that we have a stiffness matrix that can reflect progressive stiffening or softening due to large deformations and we understand, conceptually at least, how to iterate towards a solution, it’s time to build out the code that can bring these ideas to life. In section 5, we’ll do the bulk of our code development. In bitesize chunks, we’ll build our solver, bit-by-bit. By the end of this section, you’ll have a functioning non-linear solver.

Section 6: Visualising the Results

After building a solver, next, we need to turn our attention to visualising the output results. This is the focus of section 6. We’ll build out the data visualisation that brings our solution to life and allows us to explore how the structure evolves towards its final equilibrium state.

Section 7: ‘Heavy’ Cables – the Non-linear Solution

In section 7 we’ll return to a conversation we started at the very beginning of the course. We’ll use our new solver to simulate the cable we kicked the course off with. We’ll compare our code’s results with our earlier linear solution. We’ll be able to closely approximate the linear solution but also observe the emergence of non-linear behaviour as the axial stiffness of our cable is progressively reduced.

6-bar catenary

Section 8: Modelling Initial Geometry in Blender

In section 8 we’ll take a break from pure coding and detour over to Blender, the free 3D modelling tool we’ll be using to generate structural geometry. If you’ve taken some of my other structural analysis courses you’ll be familiar with how we use Blender. If you’re completely new to Blender, I’ve included an appendix section at the end of the course to help you get set up and familiar with Blender.

Section 9: Mixing Cables and Bars in the Same Model

In the final section of the course, we’ll expand our code to handle structures that consist of both bar and cable elements. One of the features of a cable is its inability to resist compression. Our code doesn’t yet capture this behaviour – we’ll address that in this final section of the course. We’ll also complete the modelling and analysis of a cable-stayed lattice tower – a classic example of a conventional and commonly found structure that utilises geometrically non-linear cables.

Progressive loading and non-linear response

Who this course is for

  • Students and professional engineers who are familiar with methods of linear structural analysis and want to learn about non-linear behaviour.
  • Anyone who has taken my linear 2D truss analysis course and wants to extend their analysis capabilities beyond linear structures.
  • Students and professional engineers who want to learn more about how to implement iterative methods of analysis in Python.

The codes developed in this course are for educational purposes only and are not tested or certified for use beyond the educational scope of this course. Always employ your own engineering judgement first and foremost, regardless of what the computer says!

Section 2
‘Heavy’ Cables - the Linear Solution
3. Section 2 overview
02:09 (Preview)
4. Deriving a linear heavy cable equation
10:06
5. Accounting for cable self-weight
09:37
6. Problem-specific boundary conditions
11:12
7. Solving for max cable tension 📂
23:34 (Preview)
Section 3
Getting Comfortable with Non-linearity
8. Section 3 overview
01:05 (Preview)
9. What is non-linear structural behaviour?
12:50
10. Large deflections and geometric non-linearity
23:24
11. An iterative solution strategy
14:24
Section 4
The Non-linear Stiffness Matrix
12. Section 4 overview
01:30 (Preview)
13. Building the transformation matrix
09:24
14. The linear stiffness matrix
09:40
15. Additional force due to large deflections
12:14
16. The local non-linear stiffness matrix
08:22
17. The global non-linear stiffness matrix
13:04
Section 5
Building our 2D Solver Toolbox
18. Section 5 overview
01:33 (Preview)
19. Initial setup and data import
29:36
20. Plotting the initial configuration
17:07
21. Blocking out the main convergence loop
28:32
22. Building the transformation matrices
09:39
23. Adding pre-tension to each member
08:08
24. Building the stiffness matrix
14:48
25. Solving for displacements
07:08
26. Updating the internal force system
07:10
27. Building a convergence test function
06:39
28. Calculating axial forces
04:34
29. Allowing for smaller external force increments
24:56
30. Generating a text summary output
12:11
31. Adding self-weight calculation 📂
14:46
Section 6
Visualising the Results
32. Section 6 overview
00:57 (Preview)
33. Plot setup and data selection
09:26
34. Plotting the undeformed structure
05:09
35. Building a colour scale
16:23
36. Plotting the deformed structure
05:46
37. Adding axial force labels
05:18
38. Plotting the applied forces
14:21
39. Plotting the reactions 📂
04:51
Section 7
‘Heavy’ Cables - the Non-linear Solution
40. Section 7 overview
01:15 (Preview)
41. Exploring the convergence behaviour 📂
20:45
42. Modelling the cable with large axial stiffness
09:46
43. Introducing non-linearity by reducing the axial stiffness 📂
05:37
44. Linear vs. Non-linear comparison for a simple truss 📂
11:40
Section 8
Modelling Initial Geometry in Blender
45. Section 8 overview
01:15 (Preview)
46. Simulating initial catenary geometry
14:20
47. Basic geometry data export
08:41
48. Exporting cable definitions
05:28
49. Exporting restraint data
10:18
50. Exporting force location data 📂
05:26
Section 9
Mixing Cables and Bars in the Same Model
51. Section 9 overview
01:00 (Preview)
52. Modifying our code for different element types 📂
17:25
53. Analysing a combined cable and bar structure
09:45
54. Removing slack cable elements 📂
20:53
55. Antenna tower - modelling and analysis 📂
15:46 (Preview)
56. Course wrap up & Certificate of Completion
02:54
Section 10
Appendix - Introduction to Blender
57. How can Blender help us?
06:18
58. Downloading and installing Blender
03:22
59. Blender overview and interface
15:36
60. Object versus edit mode
10:06
61. Rectilinear modelling
10:50
62. Organic modelling 📂
18:26 (Preview)
Completion certificate
Completion certificate
  • Download your personalised Certificate of Completion once you’ve finished all course lectures.

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Ready to get started?
Fantastic introduction to the whats, whens and whys of non-linear behavior. Most of the class is obviously about catenary behavior, but fundamental for understanding other non-linearities as well.
John Carey
getting-started
Dr Seán Carroll
BEng (Hons), MSc, PhD, CEng MIEI, FHEA
Hi, I’m Seán, the founder of EngineeringSkills.com (formerly DegreeTutors.com). I hope you found this tutorial helpful. After spending 10 years as a university lecturer in structural engineering, I started this site to help more people understand engineering and get as much enjoyment from studying it as I do. Feel free to get in touch or follow me on any of the social accounts.
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