 # The Direct Stiffness Method for Truss Analysis with Python

## Build your own finite element truss analysis software using Python and tackle large scale structures

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## What you'll learn

You’ll learn how to use the Direct Stiffness Method to build complete structural models that can be solved using Python
You’ll build your own truss analysis programme that can be used to identify displacements, reactions and internal member forces for any truss structure
You’ll learn about common models of elastic behaviour such as plane stress and plane strain and how they apply to real-world structures
You’ll learn about stiffness matrices and how the Principle of Minimum Potential Energy can be used to determine the stiffness matrix of any structural element

## Description

Welcome to this DegreeTutors course on the Direct Stiffness Method for Truss Analysis with Python. The aim of this course is to equip you with the tools and knowledge to build your own truss analysis software using the Direct Stiffness Method. In this course you’ll develop a truss solver that can determine axial forces and deflected shapes for pin-jointed truss structures. ### By the time you complete this course you’ll have written a piece of software to deploy on your own analysis projects.

Understanding structural analysis theory and hand-analysis techniques is an essential requirement of any competent engineer. It informs our intuition of structural behaviour and provides a foundation from which we can analyse complex structures.

However, most large structural analyses leverage the speed of structural analysis software. And modern structural analysis software is dominated by matrix analysis methods like the direct stiffness method. In this course you’ll implement these techniques to build your own version of a structural analysis software.

This course is broken into 9 sections that build progressively towards our goal of developing a generalised truss solver.

### Section 1 – Introduction and course overview

The main goal of this brief introductory section is to get your coding environment set up and to give you an idea of how we approach using and learning Python in this course.

### Section 2 – Modelling Elastic Behaviour

We’ll start by focusing on fundamental models of elastic behaviour. If we don’t understand the material behaviour we can’t understand the overall structural behaviour. So the lectures in this section are very important in developing a complete understanding of structural behaviour.

### Section 3 – Finite Element Equations & Stiffness Matrices

We’ll expand our focus from material behaviour to modelling structural stiffness and in particular we’ll develop a stiffness matrix for an axially loaded bar element. The aim of this section is to build a stiffness matrix that we can combine with other element stiffness matrices to model a complete structure.

### Section 4 – Direct Stiffness Method: Step-by-Step

In this section we shift gears and put what we’ve learned so far to work. We’ll walk our way step-by-step through the direct stiffness method starting with very simple two bar truss. This will allow you to see the process in action on a very simple structure.

### Section 5 – Direct Stiffness Method in Python

Now that you understand conceptually how the direct stiffness method works, we’ll implement it in Python using a Jupyter notebook. This is our first step along the road to building a completely general truss analysis notebook.

### Section 6 – Direct Stiffness Method on Larger Structures

In section six we’re going to analyse another larger truss structure. As in section 4, we’ll walk our way through the solution step-by-step to make sure you fully understand the analysis procedure. This section is about getting the reps in and giving you more practice on a larger structure.

### Section 7 – Optimising for Larger Structures in Python

We’ll again implement our solution in code by porting our solution from section 6 into a Jupyter notebook. In this section we’ll continue taking steps towards our overall goal of a generalised truss solver by further generalising our analysis code.

### Section 8 – Building a Generalised Truss Solver in Python

In section 8 we bring together all of the code we’ve worked on so far and completely generalise it. This means we’ll be writing code in this section to analyse any truss structure. This is where we finally achieve the goal of developing a generalised truss solver.

### Section 9 – Taking your Solver for a Test Drive

In the final section of the course we take a victory lap and take your new truss solver for a test drive. This short section is simply about making sure you can input structural data correctly into your solver and admiring your handy work as your code takes over and completes your structural analysis.

## Who this course is for

• Engineering students who want an introduction to computer-based methods of structural analysis
• Engineering students who’ve studied the Direct Stiffness Method but got lost along the way
• Engineers and students who want to see how they can leverage Python in their work
• Engineers and students who want a tools to analyse realistic structures without resorting to commercial software

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!

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## Course content

### Welcome and Setup Introduction and course overview (Preview)
(06:40) Our approach to Python (Preview)
(02:02) Getting started with Jupyter Notebooks
(12:04)

### Modelling Elastic Behaviour Section overview (Preview)
(01:08) Stress and strain in 2D (Preview)
(14:48) Strain & displacement in 2D
(12:25) 3 dimensions and matrix notation
(19:08) Plane Stress condition
(16:41) Plane Strain condition
(06:52) Material matrix summary
(08:40)

### Finite Element Equations & Stiffness Matrices Section overview (Preview)
(01:39) Finite Element equations & the Direct Stiffness method (Preview)
(17:52) The Principle of Minimum Potential Energy
(15:18) The Finite Element equations
(13:48) Stiffness matrix for bar element
(13:13) Finite element equation review
(14:16) Transformation from local to global coordinates
(16:31)

### Direct Stiffness Method: Step-by-Step Section overview (Preview)
(01:38) Analysis procedure overview (Preview)
(06:07) Calculating element stiffness matrices
(04:37) Building the primary stiffness matrix
(06:24) Reducing to structure stiffness matrix
(04:48) Solve for unknown displacements
(01:39) Solve for unknown reactions
(02:43) Finding element forces
(05:31)

### Direct Stiffness Method in Python Section overview (Preview)
(01:24) Element stiffness matrices (Preview)
(14:48) Whole structure stiffness matrix
(11:52) Reactions, element forces and nodal displacements
(12:58) Visualising our output
(13:00) Refactor – Tidying up with functions
(17:33)

### Direct Stiffness Method on Larger Structures Section overview (Preview)
(00:57) Calculating element stiffness matrices
(04:28) Building the primary stiffness matrix
(03:47) Reducing to structure stiffness matrix
(01:44) Solve for unknown displacements
(01:05) Solve for unknown reactions
(01:19) Finding element forces
(03:35)

### Optimising for Larger Structures in Python Section overview (Preview)
(01:06) Building the primary stiffness matrix
(19:09) Extracting the structure stiffness matrix
(03:00) Displacements and reactions
(09:29) Calculating member forces
(04:58) Visualising our output

(10:16)

### Building a Generalised Truss Solver in Python Section overview (Preview)
(01:06) Establishing input data
(07:16) Calculating member orientation and length
(27:43) Primary and Structure Stiffness matrices
(10:07) Displacements, reactions and member forces
(16:30) Automating the output visualisation
(10:04) Automating the text summary
(10:35)

### Taking your Solver for a Test Drive Section overview (Preview)
(00:45) Test Question #1
(09:44) Test Question #2
(05:58) Test Question #3
(04:07) Course wrap up and debrief
(01:38)

## Student Feedback

### I Y Aydin

5/5

Dr Sean is an expert on this topic. I’am really so sorry that i have noticed his courses so late! I am civil engineer and master degree student in Computational Mechanics in Germany . And I can really say i would be very happy if i could become his Phd student. 🙂 I’am glad to take this course. Feel free to buy this course because its really worth it.
I hope Dr Sean accept Phd student i would be glad if i become his phd student.

September 2020