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.
- 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.
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.
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.
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.
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
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.
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.