Applied Studies 240: Introduction to Structures

Part I: Introduction to Graphic Statics

Project 1: Learning the Basics by Designing a Series of Suspension Footbridges

Learning Plan

1. Review the Learning Outcomes.
2. Read the Introduction and Key Concepts.
3. Establish your Personal Archive if you have not already done so.
4. Submit evidence of your Personal Archive to your academic expert.
5. Answer the Focus Questions and the Discussion Forum Question.
6. Add materials as appropriate to your Personal Archive.
7. Complete Project 1. (Revisit the marking matrix.)
8. Do NOT submit Project 1 until you have also completed Project 2. Then submit them together as Collection 1.

Learning Materials

This course uses an approach called graphic statics to design structures (although you will also be exposed to numerical methods). In this first project, you will design a series of suspension footbridges as a means of introducing you to the concept of graphic statics.

This project-based, graphical approach to learning structures is based on your eText Form and Forces: Designing Efficient, Expressive Structures that uses a series of design exercises to introduce key structural concepts and principles. Your study package, however, also includes a number of essential components for your learning path. These include the following:

• Downloadable worksheets for your projects
• Interactive drawings on the eQUILIBRIUM website
• Statics Pad software

The worksheets are essential to the projects you will be asked to complete. Use of the eQUILIBRIUM website and the Statics Pad software is optional.

Instructions for how to use each of these resources are provided here, but you may need to configure your computer (as outlined below) to use the eQUILIBRIUM website and Statics Pad.

Worksheets

There is a worksheet for each of the projects you will be working on. These will help you with your design solution. Follow these instructions to download the worksheets from the Student Companion Site. The worksheets can be found under the pulldown menu called Browse by Resource.

eQUILIBRIUM Website

eQUILIBRIUM is an interactive tool for graphic statics-based structural design developed by the Block Research Group/ETH Swiss Federal Institute of Technology. You may find this useful in understanding the way loads and forces work in common structures. To access these interactive examples, go to the drawings page on the eQUILIBRIUM website.

These drawings can be viewed in any browser (Edge, Firefox, Chrome, or Opera). For optimal viewing, ensure your browser is up-to-date.

Statics Pad

The Statics Pad software will help you make precise and accurate graphical solutions for the projects you will be working on. It is only available for the Windows operating system. It is not, however, essential to designing these projects and if you prefer to generate your solutions by computer rather than by hand, any CAD software will have much of the same capabilities.

Follow these instructions to download the Statics Pad software from the Student Companion Site.

Personal Archive

You must create a Personal Archive for APST 240. This archive will be stored on your computer and can include items such as images, texts, scanned drawings, models, and websites. (See the Archives and Collections link on the course home page to learn how to develop your Personal Archive.) As part of the introduction to this course, submit evidence to your academic expert that you have created a Personal Archive. This can be a screenshot of your folders and subfolders, a diagram of your proposed system, or a written description of how you will organize your materials. You may also submit verification that you developed a Personal Archive for another course. You will not receive a mark for this activity, but you will use your Personal Archive in the final project.

Learning Outcomes

After successfully completing this project, you will acquire proficiency in the following areas:

1. Understanding of the basic loads and forces acting on a building;
2. Ability to apply graphic statics to calculate and analyze the forces in simple structures; and
3. Ability to integrate these concepts with your architectural designs.

Introduction

This project explores the fundamental concepts and principles of structural analysis, including methods of analysis, weight and mass, measuring forces, and modeling physical forces. It also demonstrates how to find graphical solutions to basic structural problems using scalars and vectors.

Key Concepts

Statics and Mechanics

Statics is the study of the loads and forces acting on physical systems that do not experience acceleration but rather are at rest because those loads and forces are in equilibrium—these systems are “static” (not in motion) rather than dynamic. One major objective of design is to ensure structures remain stable while at rest. Successful structural design requires a thorough understanding of the behaviour of objects subjected to the action of forces.

Statics is a branch of mechanics, and mechanics is the science of how forces affect physical systems.

Statics is also called the study of rigid body mechanics. A rigid body is one that does not deform under external forces or its deformation is so small as to be negligible. It is based on Newton’s Three Laws of Motion, often referred to as the laws of equilibrium, and are listed below.

The statics model, presented in this course, is a key concept for understanding the material world of design.

Figure 1.1. Western Concourse, King’s Cross Station with 52-metre span. (Architect: John McAslan; Engineering: Arup)

Part I of this course explores the basic units and concepts of structural analysis.

Units of Measure

To understand structures, you need to be able to quantify or measure key characteristics of matter. The critical “measurables” for structural analysis are length, mass, and force.

Length (metres or feet)

Length is the distance from one end of an object to another. Distances and other properties of an object are quantified as multiples of the unit length. A unit length is the basic value of a measuring system. The unit length of the metric or SI system is 1 metre. In construction, the imperial unit length of 1 foot is also often used.

Mass (kilograms)

Mass is the measure of matter’s ability to resist changes in velocity. In other words, mass measures how a body resists a change in its state of motion when a force is applied. The unit of mass in the metric or SI system is 1 kilogram.

Force (newtons)

Force is a push or pull exerted by one object on another. All forces have a direction, a magnitude, and a point of application. The unit of force in the metric or SI system is 1 newton. A newton is the force required to accelerate a 1-kilogram mass at a rate of 1 metre per second per second.

Weight

Weight is the force of gravity distributed over the volume of an object. It acts both remotely and externally to the object. To simplify things, the centre of gravity of the object (see definition below) is used as the point of application. Mass and weight are not the same thing—even though the two terms are sometimes (incorrectly) used interchangeably. On the moon, for example, an object weighs less (because the force of gravity is less) than it would on Earth, but its mass is the same. The reason for this is that weight is a force determined by mass and gravity. On Earth, you can calculate the weight of an object by multiplying its mass times the acceleration due to gravity. On Earth, the acceleration due to gravity is 9.8 m/s/s. This means that every second, the speed of a free-falling object on Earth increases by 9.8 metres per second.

Mass, Weight, Force, or Load?

Understanding all these measurables is challenging. Here is a succinct explanation of the differences between some of them:

Mass is a measure of the amount of material in an object, weight is the gravitational force acting on an object, force is a measure of the interaction between objects, and a load is a force exerted on an object (National Physical Laboratory, 2010).^[1]

Mathematical Models

Just as a physical model of a building is a simplified version of the real thing, mathematical models are simplified abstractions of the real world that help us understand it better. In structural analysis, some of these abstractions include particles, rigid bodies, and concentrated forces.

Particle

In the particle model, a particle has a mass, but its size can be ignored.

Rigid Body

A rigid body is a combination of particles remaining at a fixed distance from one another, before and after a load is applied. Material deformation is negligible in a rigid body. If the distance between the particles changed in a significant way because of a load, you would say the body had deformed and, therefore, was no longer rigid.

Concentrated Force

A concentrated force acts at a point on a body. Again, this is an abstraction to help with calculations since no force is so concentrated that it acts on a single point in the real world. In statics, we are often working with forces that act over a large surface area, so a load can be represented by a concentrated force if the area of its contact is small relative to the total area of the body it is acting upon. For example, the area of contact of a piece of mechanical equipment is relatively small in relation to the area of the roof it is sitting on, so it can be represented as a concentrated force.

Centre of Gravity

The centre of gravity is a single point in a body where (to simplify calculations) we imagine all its weight is concentrated. You can also think of it as the average location of a weight of an object. If a force is applied to the centre of gravity, the body will move in the direction of the force without rotating.

Figure 1.2. A mechanical unit (e.g., an air conditioner) sitting on a roof can be represented by a single concentrated force acting along the centre of gravity of the unit itself.

Vectors and Scalars

It’s easy to see from the previous example that a physical action has both direction and magnitude. To represent such actions, use vectors since they can represent both these qualities.

Figure 1.3. What a vector tells you.

A vector is simply a line with an arrowhead at one end, but it can tell you three things:

1. Its magnitude, or the length of the line segment. The length (or magnitude) of the vector represents the quantity of the action on the point of reference. In most of the questions you will be working on, this value represents the size of the force (usually measured in newtons).
2. Its direction, indicated by its line of action and measured as the angle between a reference axis and the line of action. The angle of the vector defines its direction from the point of reference. In this case, you can measure the angle to see that it is about 53° measured counterclockwise from the positive x-axis. If forces such as these are not resisted, then they will cause the object or body to move (or translate) along the line of action.
3. Its sense, which is the direction of the tip of the arrow. It identifies action toward or away from the point of reference. In this case, the sense is upward and to the right. If the arrow were at the other end of the vector and pointing in the opposite direction, it would be downward and to the left.

Vectors are very good for representing certain properties, such as forces or velocity, in which direction is critical. You don’t need them for directionless qualities such as age, temperature, or volume. A quantity in which direction is not important is called scalar and is represented by a line that does not have an arrowhead. A scalar value has only a magnitude. In physics, speed is considered a scalar value (such as 80 km/h), but velocity is a vector (80 km/h due north).

Cartesian Coordinates

To represent a vector, we usually use Cartesian coordinates with two coordinate axes (usually called x and y) and an ordered pair of numbers (x,y) to signify the distance along each axis. The values x and y are called the x-component and the y-component of the vector.

For example, because the diagram in Figure 1.3 is drawn in a Cartesian coordinate system, you can easily find other information about the vector and its properties:

The vector travels 3 units along the x-axis and 4 units up the y-axis, so the tip of the arrow has an x-coordinate that is 3 and a y-coordinate that is 4. This can be written as an ordered pair (3,4). This makes it easy to calculate the magnitude of the vector using the Pythagorean Theorem, or x² + y² = r², where r represents the magnitude. Substituting the values known for x and y gives

$3^2+4^2=r^2\text{or}$

$9+16=25\text{or}{r}^2=25$

If you take the square root of 25, you can see that r = 5, so the magnitude of this vector is 5.

(If you need a refresher on the Pythagorean Theorem, please see the OER on this topic. This document and all the other OERs referenced in this course can be found and downloaded at http://architecture.athabascau.ca/publications/index.php)

A good question to ask is 5 what? Metres? Newtons? In fact, it can be whatever you decide it should be, BUT no matter what you are measuring, it is essential that all other vectors be drawn at the same scale.

Sometimes, because the arrow is in the quadrant where both x- and y-coordinates have positive values (above 0), then its sense is said to be positive. If both the x- and y-coordinates have negative values, then the sense is said to be negative. This isn’t, however, a very useful way of looking at things because you can have vectors that are neither positive nor negative. A vector with a positive y value and a negative x value, for example, has neither a positive nor negative sense. Sometimes, as well, the negative sense is just the opposite of the existing sense. It is more precise to say that the sense of the vector in the illustration is upward to the right.

Static Equilibrium

The purpose of the structure of a building is to help ensure it doesn’t move. A body or a building that isn’t moving is said to be at rest or in static equilibrium. Equilibrium means that all the forces acting on the building are balanced by each other, so there is no net force that could start the building moving. In this respect, buildings are governed by Newton’s Laws of Equilibrium and are listed below.

Sir Isaac Newton (1643–1727), a famous British scientist, first published these laws in 1687 in his masterwork, Mathematical Principles of Natural Philosophy.

Newton’s Laws: The Three Laws of Equilibrium (also called Newton’s Laws of Motion)

1. A body at rest will remain at rest unless acted on by an unbalanced force.
2. A body acted on by an unbalanced force experiences an acceleration that has the same direction as the force and a magnitude that is directly proportional to the force. This relationship is often expressed as F=ma or Force equals mass times acceleration.
3. For every force of action, there is a reaction that is equal in magnitude, is opposite in direction, and has the same line of action. The mutual forces of action and reaction between two particles are equal, opposite, and collinear.

Loads, Forces, and Reactions

Every building is subjected to various kinds of loads in the form of weight or pressure. These loads can include the weight of snow on a roof (snow load) or the weight of people moving around a building. These loads cause forces to act on the structural components of the building such as beams, columns, and foundations. A force can be thought of as a push or a pull. These loads and forces also cause internal stress on the structural components. These kinds of stresses include bending and shear.

In the metric system, these loads are often measured in N/m² or newtons per square metre. A force of 1 newton acting on an area of 1 m by 1 m, or 1 N/m², can also be called 1 Pa or pascal. (If you are having trouble with any of the units in these projects, please see the OER entitled “Units of Measurement in Architecture.”

To find out more about the various kinds of loads that act on buildings, please see the OER entitled “Loads on Buildings.”

Note that all these forces are really just pulls (tension) or pushes (compression)—sometimes combined together. For example, in bending, the top of the beam is being pushed together (through compression) and the bottom of the beam is being pulled apart (through tension). Here are links to two short videos that animate the way these forces work:

• Ryan, V. (2016, October 29). Structural forces in action. [Video file].
• Raultecnologia. (2012, September 6). Structures and forces. [Video file].

To find out more about the various kinds of forces that act on structural components, please see the OER entitled “Basic Forces.”

You will be using this chart to explore these different kinds of forces in Project 1.

Stress and Strain – What’s the Difference?

Even when a structure is balanced and in static equilibrium, a force will still cause stress and deformation in the components of that structure. Stress is a force that can cause an object to change and strain is the change in that object. Stress can exist without strain, but strain cannot exist without stress. Strain is an object’s response to stress.

Free-Body Diagrams

The first step in solving any structural problem is to identify all the forces acting on an object. To simplify this process, the part of the structure you want to analyze is isolated as if it were floating in space—in this sense, it is a free body. To create a free-body diagram, do the following:

1. Identify all the forces acting on the free body.
2. Identify the direction and sense of each force and draw vectors to represent them. In some cases, you will also know the magnitude of those forces, so draw them to scale.

Figure 1.4. The external forces acting on a jet in flight (left) can be represented by a free-body diagram (right).

Transmissibility

Transmissibility is the idea that you can move the point of application of an external force to anywhere along its line of action and it will not change the external reactions. In other words, forces with an identical magnitude and direction that are applied along the same line of action will result in the same acceleration and moment. This is important because sometimes it can greatly simplify the structural analysis by being able to move a force along its line of action. (See Figure 1.5 or visit the Adaptive Maps website to see more on this topic.)

Figure 1.5. The principle of transmissibility of forces.

BUT, transmissibility does NOT apply to internal forces such as stress. In those cases, the point of application DOES matter. This is because the difference between the stresses may cause different changes to the geometry of the object (such as deformations), and these changes may affect the reaction forces.

Figure 1.6. Transmissibility does NOT apply to internal forces.

Resultants

As many free-body diagrams demonstrate, an object can have multiple, different forces acting on it at the same time and each may have a different magnitude and direction. They can, however, be added together to create what is called a resultant force. A resultant force, or simply a resultant, is a single force that has the same effect on the object as all the individual forces acting together.

Designing with Loads and Forces

The purpose of all these definitions and principles is to help you understand how a building will act and react to the loads and forces it will experience. A failure to understand these sometimes intricate inter-relationships can be catastrophic. The history of architecture is littered with structural failures from the collapsed pyramid at Meidum, Egypt, which failed more than 4500 years ago, to that of the World Trade Towers due to terrorist attacks in 2001 in which almost 3000 people died.

Take some time and read the 2007 paper by Dr. Natarajan Krishnamurthy, “Forensic Engineering in Structural Design and Construction.” In particular, review Table 2 on page 12 of this article. It suggests that some 43%—almost half!—of structural failures are due to mistakes in planning and design. One day, part of your job will be to help ensure these mistakes don’t occur on a building you are designing. Preventing these mistakes begins with a good understanding of what a structure can, and cannot, do. Just because you can model a building in three-dimensional software doesn’t mean it will stand up. You will return to this issue in your final project.

Required Reading

• Chapter 1: Designing a Series of Suspension Footbridges in Form and Forces: Designing Efficient, Expressive Structures
• Fagnan, J. Calculating a Numerical Solution.

Focus Questions

You do not need to submit final written answers for these questions, but you should be able to answer them to your own satisfaction and add resources to your Personal Archive that support your answers.

1. Do you prefer to solve structural problems numerically or graphically? Why?
2. Could you use any of the designs you will create in Projects 1c and 1d in any of your design projects?

Discussion Forum Question

Given that architects on major projects always work with structural engineers, how much knowledge of structures do you think an architect needs? Submit your response here.

Evaluation

Your work will be evaluated using the marking matrix outlined in the Evaluation and Grading section of the Course Orientation.

Identifying a Suitable Project

Project-Based Learning

This course uses an approach called project-based learning that relates your learning activities to real-world situations. Specifically, we want to relate your learning about structures to buildings you are designing and to your design studios in particular. To this end, you are asked to identify a project of your own that you can use to apply your knowledge of structures. It is important to pick a suitable project, so please read the next section carefully.

ADST 300 is a prerequisite or a corequisite for APST 240. In that studio, you will be designing an emergency shelter; this would be an ideal project to use for this course as well. You are, however, welcome to use any building project you have completed or are working on from any of your design studios or a building project you are working on at work. The project you choose should have the following characteristics:

• It should be relatively simple since you will be asked to analyze parts and pieces of it.
• It should have a precise location since you will be asked to quantify weather-based loads and material transportation options.

Whichever building you choose will be used throughout this course. In Project 1e, you will identify and quantify its live and dead loads; in Project 4b, you will look at the embodied carbon in its structure; in Project 5b, you will diagram the flow of forces in one of its façades; in Project 6b, you will design a framing system; in Project 7d, you will draw bending and moment diagrams for a part of its structure; in Project 8c, you will design a floor deck; in Project 9b, you will design columns and beams; and in Project 10b, you will look at the use of wood and how to minimize the use of materials; and finally, in Project 11, you will be asked to assemble all these smaller projects into one Collection about your building.

Project Description

Graphic Statics, Suspended Structures, and Catenaries

In this series of projects, you will begin designing structures through graphic statics. Your work here will consist of five small projects:

1. Understanding the forces in a building
2. Drawing suspension footbridges
3. Completing the design of Bridge #8 using Worksheet 01A
4. Completing the design of an aerial walkway using Worksheet 01B
5. Identifying and quantifying the loads in one of your own designs

Each of these is described in more detail below, but to complete any of them, you will first need to read Chapter 1: Designing a Series of Suspension Footbridges carefully.

Project 1a: Understanding the Forces in a Building

To understand structures, you have to understand the forces acting on a building. This exercise provides a hands-on appreciation of those key forces. For this project, you will need something like modeling clay (Plasticine).

Compression

Roll a column of Plasticine about 3 cm in diameter and 15 cm in length. Stand it upright and hit it with your fist. Note what happens to the Plasticine. Take a picture of the result. This is compression in action.

Tension

Roll a column of Plasticine about 3 cm in diameter and 15 cm in length. Grab the ends with both hands and pull. Note what happens to the Plasticine. Take a picture of the result. This is tension in action.

Torsion

Roll a column of Plasticine about 3 cm in diameter and 15 cm in length. Grab the ends with both hands and twist your hands in opposite directions. Note what happens to the Plasticine. Take a picture of the result. This is torsion in action.

Bending

Roll a shape about 2 cm in diameter and 25 cm long. Pick it up so that one end lies in the palm of your left hand and one end is in the palm of your right hand. Note what happens to the middle of the Plasticine under the force of gravity. Take a picture of the result. This is bending in action.

Shear

Roll a shape that is identical to the one for the bending demonstration. Grip it with both hands near its centre. Your hands should be close together but not overlapping. Quickly pull up with your left hand and down with your right hand at the same time. Note what happens to the Plasticine. Take a picture of the result. This is shear in action.

Project 1b: Drawing Suspension Footbridges

Use the method of graphic statics to draw the force polygon for Bridge #1, Bridge #3, and Bridge #5 described in Chapter 1 of your eText. Compare and note how the forces change with the different arrangements.

Optional: Go to the drawings page on the eQUILIBRIUM website. Using interactive drawings Pedestrian Bridge 1 and Pedestrian Bridge 2, try to replicate the arrangements of Bridge #1, Bridge #3, and Bridge #5 described in your eText. The forces and lengths may differ with the numbers provided in the eText, but the resulting differences would be the same. This may be used as a substitute for the hand-drawn diagrams requested above.

Project 1c: Bridge #8

Use the method of graphic statics and Worksheet 01A to complete the design of this bridge. Now draw this by hand. Take a photo or scan your drawing to include in your Collection submission.

Project 1d: Aerial Walk

Use the method of graphic statics and Worksheet 01B to complete the design of this walkway.

Project 1e: Identifying and Quantifying Live and Dead Loads in Your Design

Using the design you identified earlier (see Identifying a Suitable Project) and using the OER entitled “Loads on Buildings,” identify and quantify the live and dead loads that will affect your design. You will have to do some research to find wind, rain, and snow loads for the site of your building. (For Canadian sites, the National Building Code is a good source of the loads for particular regions of the country.) Create a diagram that summarizes these loads. Identify locations where these loads cause tension, compression, bending, torsion, and/or shear. Over the course of the next few projects, you will explore the means to resist these loads.

Review Terms

You should be familiar with and able to define the following terms and concepts:

bending
Cartesian coordinates
centre of gravity
compression
concentrated force
force
free-body diagram (FBD)
length
line of action
load
mass
mathematical model
mechanics
particle
Pythagorean Theorem
reaction
resultants
rigid body
scalar
shear
static equilibrium
statics
strain
stress
tension
the Three Laws of Equilibrium (also known as Newton’s Laws of Motion)
torsion
transmissibility
vector
weight

Submission Requirements

For Collection 1, include the following items from Project 1:

• Project 1a – five images (one for each) of the forces investigated
• Project 1b – three images of suspension footbridges
• Project 1c – Worksheet 01A and one image of bridge design
• Project 1d – Worksheet 01B
• Project 1e – loading diagram

Footnote

^[1] National Physical Laboratory. (2010). What are the differences between mass, weight, force and load? (FAQ–Mass & Density). Retrieved from http://www.npl.co.uk/reference/faqs/what-are-the-differences-between-mass,-weight,-force-and-load-(faq-mass-and-density)