Uncertainty, Risk and Reliability Part 1 - Probabilistic Modelling of Structural Failure

Analytical Modelling of Plate and Shell Structures: Part 2 - Shells

Analytical Modelling of Plate and Shell Structures: Part 1 - Plates

Fundamentals of Reinforced Concrete Design to Eurocode 2

Modelling and Analysis of Non-linear Cablenet Structures using Python and Blender

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

Multi-Degree of Freedom Dynamics, Modal Analysis and Seismic Response Simulation in Python

Finite Element Analysis of Continuum Structures in Python

Finite Element Analysis of 3D Structures using Python

3D Space Frame Analysis using Python and Blender

Beam and Frame Analysis using the Direct Stiffness Method in Python

The Direct Stiffness Method for Truss Analysis with Python

Fundamentals of Engineering Structural Dynamics with Python

Analysing Hybrid Indeterminate Civil Engineering Structures

Mastering Virtual Work for Engineers

Fundamentals of Virtual Work for Civil Engineers

Moment Distribution Method: Analysis Bootcamp

Indeterminate Structures & The Moment Distribution Method

Shear Forces and Bending Moments: Analysis Bootcamp

Mastering Shear Force & Bending Moment Diagrams

Fundamentals of Structural Analysis

Black Friday 2024 - Complete Course Bundle

When capacity exceeds demand by a suitable margin of safety, we don’t expect failure to occur. Yet failures still occur. The partial factor approach gives us a binary outcome: pass or fail! It can’t tell us how likely failure is or how this likelihood changes as the capacity and demand variables change. This course aims to give you a clear understanding of how variability in strength and demand parameters is captured and how this allows us to calculate the probability of failure. We’ll start by exploring fundamental concepts in reliability theory, introducing the Limit State Function and the Safety or Reliability index. With a solid foundation established, we’ll begin discussing the First Order Second Moment method for estimating the probability of failure. After identifying the limitations of this method, we’ll tackle the first order reliability method. This is an iterative process that gives excellent results and is one of the most commonly used methods for assessing structural reliability today.  After completing this course, you’ll understand the often unseen and underappreciated significance that uncertainty plays in structural modelling. You’ll also have developed practical tools to estimate the probability of structural failure.

Welcome and course overview

The evolution of engineered construction and our understanding of it

Section overview - safety and reliability fundamentals

Modes of failure

Sources of uncertainty

How safe is safe enough?

Safety factors and partial safety factors - a semi-probabilistic approach

A probabilistic approach - the basic idea

Evaluating and visualising probability of failure in Python

Section overview - first order second moment method

Understanding the Taylor Series expansion

Taylor Series - expanding to more variables

The first order second moment method

Deriving the safety index

Example 1 - FOSM

Example 2 - FOSM

The lack of invariance problem

Section overview - first order reliability method

A strategy for non-linear limit state functions

Converting to standard normal space

Iterative algorithm to estimate the reliability index

Example 3 - FORM

Targeting a specific safety index

Example 4 - FORM

Dealing with non-normally distributed variables

Implementing the Normal Tail Approximation

Example 5 - FORM

FORM code refactor

Example 6 - FORM for reinforced concrete beam

Challenges estimating strength and loading variables

Intermission - part 1 recap and what next?

Uncover the role uncertainty plays in structural behaviour with the First Order Reliability Method

This is the second course in a two-part series focused on the analytical modelling of plate and shell structures. This time, we turn our attention to modelling shell behaviour and, specifically, membrane behaviour. This is the fundamental load-resisting mechanism for shell structure. This mechanical behaviour allows shells to span incredible distances with remarkably thin section sizes—structural efficiency is one of the defining characteristics of shell structures. In this course we’ll explore a model of membrane behaviour that allows us to predict stress distributions and displacements across a range of different shell geometries. If you’ve ever tried to unpack shell behaviour before, you’ll know that it can be a tough road, with some very intimidating mathematics along the way! One of the main aims of this course is to remove the mystery and give you strategies and techniques to cope with what, at first glance, can seem like impenetrable math.

Shell structures in the built environment (Recap)

Undeformed shell curvature (Recap)

Section Overview - Membrane Theory of Shells

Membrane theory and the membrane state of stress

Linear displacements and meridional strain

Linear displacements and hoop strain

Angular displacement

Equilibrium of the infinitesimal element

Equilibrium in the normal direction

Equilibrium in the vertical direction

A recap of our system of equations

Solving the system to find meridional, normal and angular displacements

Section Overview - Applying the Theory to Spherical Shells

Spherical shell under self-weight - force equations

Spherical shell under self-weight - evaluating forces

Spherical shell under self-weight - displacement equations

Spherical shell under self-weight - evaluating displacements

Spherical shell under snow loading

Spherical shell case study - membrane stresses

Spherical shell case study - membrane displacements

Truncated spherical shell (Skylight)

Truncated spherical shell - membrane forces

Section Overview - Liquid Retaining Shells

Evaluating normal pressure

Spherical tank - filled to the top

Spherical tank - membrane forces

Partially filled spherical tank - membrane forces

Section Overview - Zero & Negative Gaussian Curvature Shells

Cylindrical shell membrane forces

Cylindrical shell membrane displacements

Conical shell

Truncated conical shell

Geometry of the hyperboloid of revolution - the cooling tower

Visualising tower geometry

Cooling tower principal curvatures

Bridging between linear and angular coordinates

Cooling tower membrane force distribution

Section Overview - Multi-Shell Structures

Structure review and load path analysis

Truncated spherical roof (Level A-B)

Cylindrical walls (Level B-C)

Truncated inverted cone (Level C-D)

Spherical dome base (Level D-E)

Complete membrane force distribution

Course wrap-up and certificate of completion

Unlocking the Fundamentals of Shell Behaviour with Analytical Modelling and Membrane Theory

This course is part 1 in a two-part series exploring the analytical modelling of plate and shell structures. In this first part, we’ll be exploring 2D plates. By the end of this course, you will have a good understanding of how to analyse both circular and rectangular plates using purely analytical means - so no finite elements or numerical methods - just fundamental closed-form solutions. In addition to learning about the specifics of plate behaviour, you should also walk away with a better understanding of how to develop analytical models in the form of governing differential equations. You’ll also have a few new tools and techniques that you can use to unlock tricky math problems using Python.

Course overview

Section 2 overview

Geometric considerations and assumptions

Shell structures in the built environment

Fundamental load resistance mechanisms

Deformation curvature for 2D plates

Curvature and axis rotation

Undeformed shell curvature

Section 3 overview

Coordinates, limitations, assumptions and the plan

Establishing the radial and hoop strain

Jumping to stresses via Hooke’s Law

Determining the stress resultants

From equilibrium to the governing differential equation

Evaluating the shear force term

The influence of plate boundary conditions

Case study #1: Simply supported with edge moment

Case study #2: Simply supported with UDL

Case study #2: Further investigation & visualisation

Case study #3: Built-in with central point load

Case study #4: Built-in annular plate with UDL

Case study #5: Simply supported with linearly varying load

Case study #5: Code solution and plotting

Building a 3D visualisation of plate deflection

Section 4 overview

Setting up our derivation

Stress and strain analysis

Equilibrium and the governing differential equation

Stress distributions

Fixed and simply supported boundary conditions

Free edges

The corner effect

Navier’s solution for rectangular plate deflection

Determining plate moments

Case study #6: Navier’s solution for UDL

Case study #7: Navier’s solution for linearly varying loading

Case study #8: Navier’s solution for patch loading

Case study #9: Navier’s solution for point loading

Building a ‘Plate Calculator’ script

Course wrap up and certificate of completion

A practical guide to the analysis of circular and rectangular plates under load, from first principles.

The aim of this course is to provide an introduction to the behaviour of reinforced concrete in bending and shear. By the end of the course you should understand the mechanical models we use to characterise reinforced concere behaviour. You’ll also develop an understanding of the rules and analysis techniques prescribed by Eurocode 2 for the analysis and design reinforced concrete structures. In addition, towards the end of the course, we’ll also explore how we can use Python scripting to speed up repetitive manual calculations.

Who is this course for?

The optional role of Python in this course

The relevant codes for this course

Actions on structures

Ultimate limit state design

Serviceability limit state design

Worked example #1: Determining ULS actions

Worked example #2: Determining ULS actions

Material properties

Cross-section analysis

Ultimate moment capacity

Worked example #3: Basic section design

Worked example #4: Calculate section moment capacity

Worked example #5a: Over-reinforced sections

Worked example #5b: Automating the search for x

Doubly reinforced sections

Worked example #6: Doubly reinforced section design

Worked example #7: Calculating doubly reinforced moment capacity

Flanged beam design

Worked example #8: Flanged section moment capacity

Worked example #9: Flanged section design

Shear behaviour in beams

Reinforced shear resistance (Variable Strut Method)

Worked example #10: Bending and shear design

Worked example #11: Shear design example

Longitudinal shear in flanged beams

Worked example #12: Transverse shear reinforcement design

Section 5 overview

Designing singly reinforced sections

Expanding to doubly reinforced sections

Expanding to singly reforced flanged sections

Determining the correct analysis case

Under-reinforced section analysis

Over-reinforced section analysis

Doubly reinforced section analysis

Flanged section analysis

An introduction to ultimate limit state design for bending and shear with optional calculation automation using Python.

After completing this course, you’ll have developed a complete workflow for the modelling and analysis of non-linear 3D cablenet structures. We’ll expand beyond what we covered in our study of 2D non-linear cables by adapting the tools we built previously, for the analysis of non-linear 3D cablenet structures. However, instead of just modifying our existing code, we’ll spend time fleshing out a complete workflow that takes you from initial form-finding of cablenet geometry, right through to iterative stiffness method simulation to identify the cablenet deflections and tension profile.

Introduction and course overview

Course prerequisites

Moving to JupyterLab

Cablenet structures

Building the 3D transformation matrix

Extending the non-linear stiffness matrix

Generate a simple cablenet

Extending our Blender export scripts

Visualising axial forces and deflections in Blender

Updating our notebook structure

Basic updates to our preprocessing code

Implementing a basic 3D plot

Fixing the 3D scale problem

Implement support type visualisation

Implementing 3D arrows

Implementing 3D annotations

Building the transformation matrix

Calculating the pre-tension force vector

Learn how to combine parametric modelling, exploratory form-finding and iterative analysis techniques to simulate 3D tensile structures.

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