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## Tutorials

Tutorials to guide you through engineering analysis and design from the ground up

Building a Parametric Frame Analysis Pipeline with OpenSeesPy and OpsVis

In this second tutorial in our series focusing on OpenSeesPy, we’ll build a pipeline for the elastic analysis of a 2D portal frame structure using OpenSeesPy. Along the way, we’ll parameterise our code so we can specify the bay width and number of bays at the start of our pipeline. We’ll also use another library, OpsVis, to visualise our model and results without having to write our own plotting c...more

Dr Seán Carroll

An Introduction to OpenSees and OpenSeesPy for 2D Truss Analysis

In this tutorial, we’ll take a first look at OpenSeesPy, a Python library for performing finite element analysis based on the OpenSees framework. By the end of this tutorial, you’ll be able to perform 2D truss analysis using OpenSeesPy. I suspect you’ll also be very keen to explore OpenSeesPy further after you see how powerful it is! OpenSees is short for Open System for Earthquake Engineering Sim...more

Dr Seán Carroll

DegreeTutors is now EngineeringSkills

This is the new version of DegreeTutors.com and combines that site, the DegreeTutors Podia site and Labs, all into one platform. I've fully custom-coded and developed this platform to be the best place to learn engineering online! In this article, I'll introduce you to the new site, explain why I've spent so much time building it, point out some of the new features and improvements I've made and g...more

Dr Seán Carroll

Arch Analysis – How to Analyse Determinate and Indeterminate Arches

Arches are one of the most efficient and elegant structural forms. Their efficiency has seen them selected throughout history as the go-to form for spanning large distances. In this tutorial, we’ll explore the different methods for analysing both determinate 3-hinge arches and indeterminate 2-hinge and fixed arches. Through worked examples, we’ll develop the various methods, with particular attent...more

Callum Wilson

Influence Lines - A Complete Introduction for Civil and Structural Engineers

In this tutorial on influence lines, we will explore influence lines and the techniques that can be used to construct influence line diagrams for determinate and indeterminate beam structures. By the end of this tutorial, you will be confident in drawing and formulating influence lines for determinate structures. You will also develop an appreciation of how we can implement more advanced structura...more

Callum Wilson

Plastic Analysis of Frames – A Complete Guide – Part 2

In the second part of this series on plastic collapse analysis, we explore how what we learned in part one can be applied the frames. Plastic analysis methods are particularly useful in the analysis of indeterminate steel portal frames as it allows us to efficiently achieve relatively large spans by allowing for the formation of plastic hinges and moment redistribution within the frame. By the end...more

Callum Wilson

Plastic Analysis and Plastic Collapse – A Complete Guide – Part 1

Often in structural analysis, linear-elastic analysis is used, predicting failure when structural members reach their yield stress. This method, while effective, ignores the plasticity that some structural materials experience beyond their yield limit. This can lead to an underestimate of the structures safe working capacity. In part one of this tutorial series on plastic analysis, we explore the ...more

Callum Wilson

Calculating and Interpreting the Second Moment of Area

This tutorial focussed on the second moment of area, also known as the moment of inertia. By the end of this tutorial, you should be comfortable explaining what the second moment of area is, why it's important to engineers, how to calculate it and how to interpret the values in the context of civil and structural engineering. We’ll cover how to identify the location of the centroid of a cross-sect...more

Callum Wilson

How moment redistribution leads to more efficient designs

In this tutorial, we’ll discuss moment redistribution in reinforced concrete and how we can use it to our advantage to achieve more efficient designs. When designing any structural element, our first pass usually involves an elastic analysis. However, this approach can leave some structural capacity untapped. We’ll see how we can use the plastic behaviour of reinforced concrete at the ultimate lim...more

Dr Seán Carroll

Reinforced Concrete Fundamentals - Analysis and Design of Steel Reinforcement

In this tutorial, we’ll introduce the role of steel reinforcement in reinforced concrete design. We’ll see that steel plays a critical role in developing an internal moment of resistance and compensates for concrete’s inherent brittleness and weakness in tension. We’ll explore the fundamental mechanical model used to describe the behaviour of the cross-section under load. We’ll also see that to av...more

Dr Seán Carroll

How I Coded an Analytical Beam Calculator

In this post, guest author Vittorio Lora talks us through how he developed the idea for and ultimately built Beamsolver.com. A structural engineer by training, Vittorio has pivoted in his career to focus more on software development. But he couldn’t shake the desire to build the analytical beam calculator that he would have found so helpful as a student. Parameterised structural analysis problems ...more

Vittorio Lora

An Introduction to Concrete and its Strengths and Weaknesses

Concrete is one of the most important and ubiquitous materials in the construction industry globally. Twice as much concrete is used (by weight) as steel, wood, plastics and aluminium combined. Its global usage is estimated at 10 billion tons per year. Fundamentally, concrete is simply a mixture of cement, water, aggregate and sand. When mixed, they form a slurry that undergoes a chemical reaction...more

Dr Seán Carroll

3D Truss Calculator – Quick Start Guide

Welcome to this quick start guide on how to use the 3D truss analysis toolbox. In this tutorial, we’ll work through the solution of a sample 3D space frame (pin-jointed) structure. We’ll determine reaction forces, axial forces and nodal displacements. By the end of this tutorial, you'll be comfortable using the toolbox to analyse your own structures. In the video accompanying this tutorial, we’ll ...more

Dr Seán Carroll

An Introduction to Uniform Torsion in Cylindrical Shafts

In this tutorial, we introduce torsion. This is simply a bending moment applied about the longitudinal axis. Torsion will cause twisting about the longitudinal axis and is a very common form of loading. Our starting point will be to explore the concept of strain as it applies to circular bars and to derive an equation that relates the strain to the angle of twist in the bar. Next, we’ll tie shear ...more

Dr Seán Carroll

How to Analyse Indeterminate Beams using the Moment Distribution Method

In this tutorial we'll explore the moment distribution method. This is an excellent technique for quickly determining the shear force and bending moment diagrams for indeterminate beam and frame structures. In this tutorial, we'll focus on applying the moment distribution method to beams. We'll start by getting a clear understanding of the steps in the procedure before applying what we've learned ...more

Dr Seán Carroll

How to Apply the Virtual Work Method to Trusses

In this tutorial, you'll learn how to use the virtual work method to analyse trusses and calculate truss deflections. The virtual work method is based upon the Principle of Virtual Work which underpins many elegant and versatile analysis procedures. We'll focus here on how it can be applied to trusses. This tutorial will initially develop the underlying theory, starting with the concept of strain ...more

Dr Seán Carroll

How to Calculate Reactions using Free Body Diagrams

In this tutorial, we'll cover free body diagrams and how to use them to evaluate the forces acting on a structure in equilibrium. A free body diagram is a diagram in which only the forces imposed on an object are shown. Free body diagrams are a simple tool to help us identify all of the forces that influence an object or structure. Typically, one of the first steps in analysing a structure is to s...more

Dr Seán Carroll

How to Calculate Beam Deflection

In this tutorial, you'll learn how to calculate beam deflection from first principles using the differential equation of the deflection curve. We'll cover several calculation techniques, including one called Macauley's Method which greatly speeds up the calculation process. We'll work our way through a couple of numerical examples before discussing how we can use the principle of superposition and...more

Dr Seán Carroll

Forces, Moment of a Force and Force Systems

Welcome to the Fundamental Engineering Mechanics tutorial series. This is part one in a multi-part series aimed at anyone just starting out in the study of engineering. In this tutorial, we’ll start from the very beginning and discuss forces, moments or torques generated by forces and how to evaluate systems of forces and moments. Once you complete this tutorial and all of the worked examples, you...more

Dr Seán Carroll

A Complete Guide to Learning Structural Analysis

Learning how to use mathematics and mechanics to analyse structural behaviour is among the most challenging topics student engineers struggle with. Yet, for an engineer, particularly one involved in the analysis and design of structures, a firm grounding in fundamental structural behaviour is essential. In this post, I want to provide a complete structural analysis guide - a roadmap for learning s...more

Dr Seán Carroll

Free Truss Calculator – Quick Start Guide

This is a quick start guide for our free online truss calculator. Follow this short text tutorial or watch the Getting Started video to quickly orientate yourself with this handy free tool. We'll walk through the process of analysing a simple truss structure. By the end, you'll be comfortable using the truss calculator to quickly analyse your own truss structures. Students, in particular, should f...more

Dr Seán Carroll

Simulating crowd vibrations using the Duhamel Integral

In this Python mini-project, you'll learn about the Duhamel Integral and how it can be used to simulate the dynamic response of a single degree of freedom system. We'll discuss how to solve the integral and then write some Python code to implement our solution for any arbitrary loading. In the second half of this project, we're going to use our Duhamel Integral solver to build a crowd loading simu...more

Dr Seán Carroll

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...more

Dr Seán Carroll

Building a Beam Deflection Calculator in Python

In this project, we’ll build a beam deflection calculator that can generate beam deflections by directly integrating the bending moment diagram. The technique we’ll use for calculating deflection in this project is not limited to statically determinate structures, although you will need a complete bending moment diagram to integrate. This project builds on our previous Shear Force and Bending Mome...more

Dr Seán Carroll

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 di...more

Dr Seán Carroll

Building a Shear Force and Bending Moment Diagram Calculator in Python

In this project we're going to build a Shear Force and Bending Moment Diagram calculator using Python in the Jupyter Notebook development environment. Generating the shear force and bending moment diagram for a simple beam with anything other than basic loading can be a tedious and time-consuming process. Once you finish this project, you'll have a calculator that can produce shear force and bendi...more

Dr Seán Carroll

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

In this Python project we're going to build a Mohr's Circle calculator. By the end of this project, you will have built your own stress analysis Python code. Along the way we'll cover all of the fundamental topics that lead up to Mohr's circle of stress. You will learn about how we use the 2D stress element to represent the state of stress at a point, the purpose of stress transformation equations...more

Dr Seán Carroll

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.

Dr Seán Carroll

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 mak...more

Dr Seán Carroll

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 ...more

Dr Seán Carroll

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, vehicl...more

Dr Seán Carroll

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 explo...more

Dr Seán Carroll

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 collap...more

Dr Seán Carroll

Structural Analysis and Stability – Asymmetrically Propped Structures

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 ...more

Dr Seán Carroll

Structural Analysis and Stability – Symmetrical Structures

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 introdu...more

Dr Seán Carroll

Column Buckling - Realistic Buckling Behaviour

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...more

Dr Seán Carroll

Column Buckling Equations

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.

Dr Seán Carroll

Column Buckling and Stability

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...more

Dr Seán Carroll

Structural Analysis Case Study

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.

Dr Seán Carroll

What is a Truss?

A truss is a structure that consists of a collection of elements connected at pin joints or nodes. In theory, the pin joints provide no rotational resistance and behave as hinges. In practice this is not always the case. In this tutorial, we’ll discuss common forms of truss, their features, approximate methods of analysis and the key assumptions that relate to our analysis.

Dr Seán Carroll