Understanding non-linear structural behaviour and common geometric and material non-linearities

icon_non-linear | DegreeTutors.com


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.

Nonlinear-finite-element-analysis-of-2D-catenary-&-cable-structures-in-Python | DegreeTutors.com
After completing this course…
  • You will understand the concept of geometric non-linearity and when it should be considered.
  • You will understand how to modify the axially loaded element stiffness matrix to account for large deflections and changes in geometry.
  • You will have implemented a Newton-Raphson iterative solution algorithm that seeks to converge on the deformed state of the structure.
  • You will have a workflow that leverages open-source modelling tools in Blender to quickly generate the initial structural geometry.


P-delta effects | DegreeTutors.com

P-Delta Analysis and Geometric Non-linearity

In this tutorial, we’ll explore the P-Delta effect; a form of non-linear behaviour that can lead to large magnitude sway deflections in columns. Put simply, P-Delta describes the phenomenon whereby an additional or secondary moment is generated in a column due to the combination of axial load (P) and lateral sway, (Delta), of the column. This leads to non-linear structural behaviour and can result in lateral deflections far in excess of those arising from lateral loading alone. We’ll explore the phenomenon and write some code to help visualise the behaviour.

Propped cantilever plastic hinge 4 | DegreeTutors.com

Yielding, Plastic Deformation and Moment Redistribution in Beams (2/2)

In this tutorial, we’re going to work out exactly how to determine the plastic moment capacity of a cross-section. We’ll also explore the concept of moment redistribution with an illustrative example. By the end of this post you’ll be able to calculate the plastic moment capacity of any cross-section and understand in detail how moment redistribution occurs in a structure and ultimately how collapse can occur as a result of hinge formation.

Stress-strain-curve | DegreeTutors.com

The Stress-Strain Curve & Plastic Hinges in Beams (1/2)

So why is plastic behaviour so important to understand? It’s probably fair to say that most of our engineering analysis assumes linearly elastic behaviour. But in reality, if we limit our designs to purely elastic behaviour, we’re leaving a lot of structural capacity untapped. Structures very often have more load carrying capacity than a linearly elastic analysis suggests. In this post we’ll explore this reserve capacity.

Column Buckling with eccentric compression force

Column Buckling: Realistic Buckling Behaviour (3/3)

In this final post in this series on Column Buckling, we’ll look at more realistic buckling behaviour you’re likely to observe in reality. In particular we’ll explore the behaviour of columns subject to eccentric axial load and columns with an initial deformation, i.e. columns that don’t start out straight. It’s important to recognise that for a column with these characteristics, we do not observe the strict mathematical behaviour predicted for perfectly loaded perfectly straight columns

Column Buckling Equations (2/3)

In this post we’ll start to consider more realistic column structures. In particular we’ll determine an expression for the critical load for an axially loaded column with pinned ends. Then we’ll explore other support conditions. We’ll also introduce some other key concepts such as buckling modes and effective length.

2 bar idealised column structure

Column Buckling and Stability (1/3)

Long slender structural elements under the action of an axial load may fail due to buckling rather than direct compression. Buckling failure occurs when axial load induces a lateral deflection leading to a bending type failure. Buckling can also occur in plate and shell structures and is a relatively common cause of structural collapse. Depending on the geometry of the structural element, buckling can occur long before the material yields.

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

Nonlinear-finite-element-analysis-of-2D-catenary-&-cable-structures-in-Python | DegreeTutors.com
After completing this course, you’ll have built an iterative numerical solver for cable and truss structures that exhibit geometric non-linearity due to large deformations.

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

This course will build on the understanding developed in previous DegreeTutors courses and in particular our linear 2D analysis course, The Direct Stiffness Method for Truss Analysis with Python. It is strongly recommended that you complete this course first before tackling non-linear analysis.

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.

This course is divided into 9 sections:

  • Introduction and course overview
  • ‘Heavy’ cables – the linear solution
  • Getting comfortable with non-linearity
  • The non-linear stiffness matrix
  • Building our 2D solver toolbox
  • Visualising the results
  • ‘Heavy’ cables – the non-linear solution
  • Modelling initial geometry in Blender
  • Mixing cables and bars in the same model

The final code will be capable of handling structures like the one pictured 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!

You DO NOT need to be a Python programming guru to take this course. If you’ve taken the prerequisite course – or even if you’re just familiar with basic programming ideas like functions, loops and variables that will be plenty to get you started. 👍