 # TUTORIALS

## Tutorial articles to guide you through engineering analysis and design topics from the ground up Structural Analysis

## 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. Structural Analysis

## Macauley’s Method for Fast Beam Deflections

In this tutorial we’re going to revisit the topic of deflection calculation and introduce Macauley’s Method. We’ll again rely on integrating the differential equation of the deflection curve, except this time we’ll speed the whole process up considerably by using Macauley’s method. This technique is also sometimes known as the Singularity Function method. We’ll work our way through some examples that will leave you well placed to apply Macauley’s Method on your own. We’ll also write a couple of helper Python scripts along the way to make our lives a little easier. Structural Analysis

## Truss Analysis using the Method of Joints and Method of Sections

In this tutorial we’re going to focus on trusses, also known as pin-jointed structures. We’ll briefly discuss their key features and methods of analysis. We’re going to start at the very beginning by briefly considering what exactly a truss is – but we’ll very quickly move on to truss analysis and demonstrate the joint resolution method and method of sections with some worked examples. Structural Analysis

## Finite Element Analysis and Structural Behaviour Modelling Case Study

In this post we will use the Tintagel footbridge as a case study to explore structural behaviour and show how we can build up an understanding of the structure through analysis of increasingly refined finite element models models. We’ll apply this iterative approach by starting with a simple beam model and incrementally working towards a full 3D finite element model. Throughout this post we’ll make use of finite element analysis codes developed in DegreeTutors courses. Structural Analysis

## Calculating Beam Deflection from First Principles

In this tutorial we’re going to explore beam deflection and see how we can calculate the deflection of any beam from first principles using the differential equation of the deflection curve. We’ll work our way through a numerical example before discussing how we can use superposition along with tabulated formulae to speed up the process of calculating beam deflection. Structural Analysis

## Truss Analysis using the Direct Stiffness Method

In this tutorial we examine the Direct Stiffness Method and work our way through a detailed truss analysis. By the end of this complete introduction, you should understand the basic ideas behind why the method works, how to implement it for truss analysis and you should understand the power and scalability of the technique. Once understood, the direct stiffness method opens the door to structural analysis of large scale complex structures. Structural Dynamics

## Understanding Structural Dynamics and Inertia

By the end of this post you’ll understand when and why a dynamic analysis is performed instead of a (usually more straightforward) static analysis. Although we typically encounter static loading, dynamic loads occur with sufficient frequency that we need to understand how to assess their influence on a structure. Typical forms of dynamic loading can include loading due to earthquakes, wind, vehicle or pedestrian traffic or wave loading on coastal or offshore structures. Structural Analysis

## Shear and Moment Diagrams – An Ultimate Guide

Shear force and bending moment diagrams tell us about the underlying state of stress in the structure. Determining shear and moment diagrams is an essential skill for any engineer. Unfortunately it’s probably the one structural analysis skill most students struggle with most. So in this post we’ll give you a thorough introduction to shear forces, bending moments and how to draw shear and moment diagrams. By the end of this post you’ll know a lot more about shear forces and moment moments then when you started. Mechanics of Materials

## 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. Mechanics of Materials

## 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. Structural Analysis

## Structural Analysis and Stability – Asymmetrically Propped Structures (2/2)

In part two of this tutorial series we’ll consider how to analyse the lateral stability of a multi-storey structure with an asymmetrical arrangement of stabilising elements. Asymmetrically propped structures undergo twisting or rotation about the centre of stiffness in addition to direct lateral translation. We will consider how to determine the additional forces induced by this twisting. Finally we’ll compare our hand analysis results to a simple finite element analysis model. Structural Analysis

## Structural Analysis and Stability – Symmetrical Structures (1/2)

All structures typically experience some form of lateral loading during their design life. Typical sources of lateral loading include forces due to wind blowing against the structure, hydrostatic forces due to groundwater (acting against basement walls for example) or inertia forces due to ground motion (earthquakes). In this, the first of a two-part series on structural stability, we will introduce common lateral stability schemes before diving into some numerical examples in this and the next post. Structural Analysis

## 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 Structural Analysis

## 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. Structural Analysis

## 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. Structural Analysis

## Structural Analysis

Structural analysis is the process of using mathematical and mechanical principles to determine the magnitude of internal forces in a structure. One of the main roles of a civil or structural engineer is carrying out structural analysis as the first step to designing a safe structure. In this tutorial we’ll demonstrate exactly how this process works for a real world structure. 