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.

Welcome and course overview

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

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

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.

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

In this course, we’re going to build a full Isoparametric Finite Element solver. This unlocks our ability to model and analyse any 2D structural form. When we combine our finite element analysis solver with knowledge of the theory of plane stress and plane strain, this course will equip you with the ability to model 3D structures using 2D finite element models. We’ll build in the ability to simulate the influence of point load forces, distributed forces and body or self-weight forces. When you complete this course, you will have built your own 2D Finite Element solver, but more importantly, you’ll understand precisely how it works and what every single line of code does! This will give you the knowledge, experience and confidence to extend your solver and add the new features that matter to you.

Introduction and course overview

What you should cover before this course

Bar stiffness using generalised coordinates

Bar stiffness using shape functions

Quad elements and shape functions

The strain-displacement matrix, B

The Jacobian Matrix, J

Calculating the strain-displacement matrix

Accounting for plane-stress/strain

The equivalent surface force vector

The equivalent body force vector

Gauss numerical integration scheme

Numerical integration in 2D

Calculating the element stiffness matrix

Stiffness matrix convergence

Generating a simple quad mesh in Blender

Exporting mesh data from Blender

Exporting support and point load data

Importing support, loading and mesh data

Section 5 overview

Setting analysis parameters and plotting the structure

Building the global force vector

Defining the element stiffness matrix

Assigning the element stiffness coefficients

Building the structure stiffness matrix

Solving the structure

Visualising the deformed structure

Generating detailed text output

Euler-Bernoulli deflection validation

Section 6 overview

Generating the stress calculation grid

Visualising the stress calculation grid

Calculating the strain and stress fields

Visualising the normal stress distribution

Building normal strain/stress contour plots

Building shear stress/strain contour plots

Building a stress/strain interpolation query function

Visualising stress distributions at a vertical section

Euler-Bernoulli stress validation

Section 7 overview

Implementing self-weight in our analysis

Correcting self-weight reactions

Distributed loading - Interpolation limits

Distributed loading - Data export from Blender

Importing distributed loading data

Surface force vector - Setup

Surface force vector - Integration

Updating the global force vector and validating

Orthogonal surface forces

Section 8 overview

A recap of principal stresses

Calculating principal angles and stresses

Sorting principal stresses

Identifying max ten/comp principal stresses

Visualising the principal stress magnitude field

Visualising the principal stress vector field

Combining the magnitude and vector fields

The von Mises failure theory

Visualising the von Mises stress field

Perimeter stresses and updating plot edges - Planning

Perimeter stresses and updating plot edges - Coding

Section 9 overview

Generating 2D geometry in Blender

Meshing complex geometries

Exporting an irregular mesh

Generalising our solver

Stress distribution masks

Course wrap up

[BONUS] Cantilever model and analysis walkthrough

Understanding the 2D stress element

The stress transformation equations

Visualising transformation equations

Principal stresses and principal planes

Finding maximum shear stresses

2D stress analysis worked example

Stress analysis Python worked example

Understanding Mohr's circle of stress

Mohr's circle worked example in Python

Use the Isoparametric Finite Element Method to build an analysis tool for 2D structures in Python.

Finite Element Analysis of Continuum Structures in Python

Tools and techniques for modelling and understanding the underlying state of stress in structures

Stress Analysis

Go from the fundamentals of elastic failure theory to implementing section analysis in Python using sectionproperties | EngineeringSkills.com

Understanding Tresca and von Mises Elastic Failure Theories

Download the complete Jupyter Notebook file for this tutorial.

Go from the fundamentals of elastic failure theory to implementing section analysis in Python using sectionproperties

We're going to build a Mohr's Circle calculator using Python. Once complete, you will have your own stress analysis Python code.

Building a Mohr’s Circle Calculator for Stress Analysis in Python | EngineeringSkills.com

Once complete, you will have your own stress analysis Python code

Building a Mohr’s Circle Calculator for Stress Analysis in Python

Perform 2D stress analysis and use Mohr’s Circle to visualise the complete stress field

HUB / STRESS ANALYSIS

Tools and techniques for modelling and understanding the underlying state of stress in structures

Stress Analysis Courses

Stress Analysis Projects

Stress Analysis Tutorials