# TUTORIALS

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

## 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 analytical methods used to evaluate the ultimate capacity of a structure, the so-called plastic limit or plastic collapse limit.

## 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-section shape, how to calculate the different moments of inertia and how to use the parallel axis theorem to compute second moments of area for compound cross-section shapes.

## 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 limit state to develop more efficient designs by redistributing moments within the structure. We’ll do this by first explaining the moment redistribution behaviour in a statically indeterminate structure and then exploring what it means for the design of reinforced concrete sections.

## 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 avoid a brittle failure, we must limit the depth of the neutral axis at the ultimate limit state. After reading this tutorial, you’ll have a good understanding of how to perform concrete section analysis and basic design.

## 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 are notoriously difficult to solve algorithmically. Unlike numerical problems, solution techniques based on linear algebra just don’t scale well. Vittorio explains how it was actually the simple techniques we all learn first that ultimately unlocked the problem.

## 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 called hydration and gains strength slowly in a process called curing. In this article, we’ll discuss the properties of concrete, its constituents and what makes it such a versatile construction material.

## 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 also use the Blender modelling template file provided to model and analyse a structure from scratch. Like the 2D toolbox, students in particular, should find it helpful as a quick and easy tool for generating structural response data.

## 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 stress into the story and see how it relates to applied torque and torsional deformation. Finally, we’ll bring everything together with some numerical examples to demonstrate how to deploy the equations we’ve developed.

## 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 to a more challenging worked example at the end.

## 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 energy. This will make the jump to virtual work much easier to understand. We’ll bring it all together with a thorough worked example at the end.

## 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 sketch out its free body diagram, identifying all of the forces that must be considered in the analysis.

## 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 tabulated formulae to speed up the process even further.

## 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’ll be well prepared for the analysis of simple structures in later tutorials in the series.

## 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 structural analysis from the beginning; what to study, in what order and why.

## 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 find it helpful as a quick and easy tool to test manual solutions against.

## 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.

## 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.

## 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.

## 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.

## 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.