### Summary

Working individually or in groups, students explore the concept of stress (compression) through physical experience and math. They discover why it hurts more to poke themselves with mechanical pencil lead than with an eraser. Then they prove why this is so by using the basic equation for stress and applying the concepts to real engineering problems.### Engineering Connection

Civil engineers design many different types of structures to resist forces, such as buildings that withstand both gravity and the earthquake forces. Making sure structures can withstand these forces safely is the ultimate goal. Engineers perform careful analysis of the design of these structures to identify the stresses in individual structural components when the structures themselves are subjected to forces. Then, they determine whether or not each component can handle the stress, based on the material type. It is important for engineers to understand how geometry and configuration affect stress in a structural component, both mathematically and intuitively, to be able to design efficient and safe structures.

### Pre-Req Knowledge

Students should be familiar with:

- basic algebraic principles, such as order of operations, and substitution
- decimals, area calculations, and units of measurement
- the concept of force

Address these requirements by giving students more or less information at the beginning of the activity, as needed.

### Learning Objectives

After this activity, students should be able to:

- Explain the concept of stress, both intuitively and mathematically.
- Explain why stress is important when designing a structure.
- Relate a real experience to mathematical equations.

### More Curriculum Like This

**Fairly Fundamental Facts about Forces and Structures**

Students are introduced to the five fundamental loads: compression, tension, shear, bending and torsion. They learn about the different kinds of stress each force exerts on objects.

**Stressed and Strained**

Students are introduced to the concepts of stress and strain with examples that illustrate the characteristics and importance of these forces in our everyday lives. They explore the factors that affect stress, why engineers need to know about it, and the ways engineers describe the strength of mater...

**Designing Bridges**

Students learn about the types of possible loads, how to calculate ultimate load combinations, and investigate the different sizes for the beams (girders) and columns (piers) of simple bridge design. Additionally, they learn the steps that engineers use to design bridges.

**Strong as the Weakest Link**

To introduce the two types of stress that materials undergo — compression and tension — students examine compressive and tensile forces and learn about bridges and skyscrapers. They construct their own building structure using marshmallows and spaghetti to see which structure can hold the most weigh...

###
Educational Standards
Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*,
a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics;
within type by subtype, then by grade, *etc*.

Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*,
a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics;
within type by subtype, then by grade, *etc*.

###### Common Core State Standards - Math

- Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (Grade 7) More Details
- Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. (Grade 7) More Details
- Solve real-life and mathematical problems using numerical and algebraic expressions and equations. (Grade 7) More Details
- Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. (Grade 7) More Details
- Solve linear equations in one variable. (Grade 8) More Details
- Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. (Grade 8) More Details
- Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. (Grades 9 - 12) More Details
- Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (Grades 9 - 12) More Details
- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. (Grades 9 - 12) More Details

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###### International Technology and Engineering Educators Association - Technology

- The selection of designs for structures is based on factors such as building laws and codes, style, convenience, cost, climate, and function. (Grades 6 - 8) More Details
- Buildings generally contain a variety of subsystems. (Grades 6 - 8) More Details

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###### State Standards

###### Missouri - Math

- Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. (Grade 7) More Details
- Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. (Grade 7) More Details
- Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (Grade 7) More Details
- Solve real-life and mathematical problems using numerical and algebraic expressions and equations. (Grade 7) More Details
- Solve linear equations in one variable. (Grade 8) More Details
- Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. (Grade 8) More Details
- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. (Grades 9 - 12) More Details
- Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (Grades 9 - 12) More Details
- Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. (Grades 9 - 12) More Details

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### Materials List

Each group needs:

- mechanical pencil with an eraser (must have the lead diameter written on the pencil; the basic BIC
^{®}mechanical pencil works well) - ruler with millimeters marked
- (optional) calipers
- Feel the Stress Worksheet, one per person

### Introduction/Motivation

Civil engineers are responsible for making sure structures like bridges and buildings are safe for people to use and will not collapse while being used. To do this, structural engineers think of all the possible loads (forces that are applied to structures) that could possibly affect the structure. What kinds of loads affect structures? (Possible answers: Wind, snow, people, vehicles, earthquakes, furniture, etc.) Engineers then carefully analyze the structure to figure out how all those loads might affect the individual components of the structure, such as beams and columns.

Engineers use the term "internal force" to describe how the individual components are affected. Internal forces are caused by external forces that can be thought of as acting within the body of a material. (Example: Pulling a rubber band. The pull on the rubber band by your fingers is an external force. The elongation and thinning of the rubber band is the internal force called "tension.")

The basic types of internal forces include compression, tension, bending, torsion, and shear.

- Compression is a squeezing force that tends to cause shortening.
- Tension is a pulling force that tends to cause extension.
- Torsion is a twisting force.
- Bending is just what it sounds like — a force that tends to cause an object to "fold" or bend.
- Shear is a force that causes sliding or slipping of the planes of a material.

How do engineers quantify these internal forces? (Answer: Stress) What is stress? (Answer: Stress is the average amount of force per unit area in an object.) Intuitively, we can think of stress much like we think of pressure (an effect that is produced when a force is applied to a surface). Mathematically, stress is equal to force divided by cross-sectional area (write this on the board: stress = force/cross-sectional area). Engineers use this equation to figure out the amount of stress imposed on structural components that we subject to internal forces.

How, then, do structural engineers make sure these structural components are safe once they find out the amount of stress imposed on them? (Take student suggestions.) In other words, why do engineers care about the amount of stress imposed on a structural component? The answer is that different structural materials, such as steel, have been tested and have known stress limits, past which they fail. Engineers compare the total of all the stresses they think structural components will be subjected to, to the limit stresses of the structural materials used. This approach is the basis for a lot of what structural engineers do (to keep us all safe!).

Today, let's explore the concept of stress mathematically and reinforce it with a physical experience. Once the activity and calculations are completed, you will be able to answer some real structural engineering design questions and problems.

(As a class, go through an Example Stress Calculation using circular-shaped objects to make it similar to the activity. Possible examples might be calculating the stress in a circular column, calculating the stress imposed on a finger from a needle, or calculating the stress on the skin from a hypodermic needle).

### Vocabulary/Definitions

beam: Horizontal member of buildings that supports floors.

column: Long, vertical structural member that supports horizontal members of a structure.

compression: A type of internal force that can be thought of as squeezing or causing shortening.

cross-sectional area: A quantity expressing the two-dimensional size of a predefined surface, specifically the 2-D view of a slice through an object.

diameter: Any straight line that passes through the center of a circle and whose end points are on the circle.

force: The influence on an object that produces a change in a physical quantity. Alternate (classical physics) definition: A push or pull that causes an object with mass to accelerate.

internal force: A force caused by an external force that can be thought of as acting within the body of a material causing stress. Compression is an internal force.

load: Forces that are applied to a structure or object.

Newton (N): Unit of measure for stress. Newton = 1 (kg x m)/sec^2

Pascal (Pa): Unit of measure for stress. Pascal (pa) = 1 N/m^2

pressure: An effect that is produced when a force is applied to a surface. Examples: wind blowing on the surface of a sail, air pressure in a tire.

stress: Average amount of force per unit area.

structural component: Any member of a structure whose primary purpose is supporting the structure, such as a column or beam.

### Procedure

Background

The Introduction/Motivation section provides a basic explanation of internal forces (compression, tension, bending, torsion, and shear) to help students understand how stresses are induced in materials. Before the activity, become familiar with the basic concepts of mechanics of materials, including an understanding of force, internal force, and stress and their mathematical representations and formulations. These basic ideas can be researched online and are present in any mechanics of materials textbook. Suggested resources:

*Building Big: Forces Lab:*http://www.pbs.org/wgbh/buildingbig/lab/forces_text.html*Mechanics of Materials*, second edition, by Madhukar Vable (see Chapter 1: Stress) http://www.me.mtu.edu/~mavable/MoM2nd.htm*Mechanics of Materials*, fourth edition by Beer, Johnston and DeWolf*Mechanics of Materials*, sixth edition by R.C. Hibbeler

Before the Activity

- Gather mechanical pencils and rulers.
- Make copies of the Feel the Stress Worksheet, one per person.
- Prepare an example calculation similar to the one the students will perform to introduce them to what they will be doing. See the attached Example Stress Calculation of stress in a column.
- Have students work alone or in groups; either works well.

With the Students

1. Conduct the Introduction/Motivation section with the class. Cover the following:

- Explain the basic responsibilities of structural engineers in reference to designing components of structures and making them safe.
- Review the concepts of force and inform students that structural engineers use the term "load" to describe the forces acting on structures.
- Ask: "What kinds of loads are buildings subjected to?" Make a list on the board.
- Introduce the term "structural component" and let students know that these structural members are what engineers analyze. Let them know that this is what they will learn how to do today.
- Explain that structural components are subjected to internal forces when loads are applied to them. Define "internal force" and specifically define compression, since this type of internal force is explored in the activity. Ask students to point out some objects in the room that are in compression.
- Ask: "How do we quantify these internal forces?" The answer is stress. Define stress, and write the equation for stress on the board (stress = force / area).
- Explain why structural engineers care about stress — that analyzing stress and "loads" enables them to make sure structures are safe to use under all conditions. Engineers size structural components to minimize cost/size/weight, while still withstanding maximum internal forces.

2. Perform a sample stress calculation on the board. Refer to the attached Example Stress Calculation of stress in a column.

3. Review with students the activity procedure as described on the worksheet. (**Alternative procedure**: If safety is a concern, ask students to imagine which would be more painful, a pencil lead or eraser pushed against a finger.)

4. Pass out mechanical pencils, rulers and worksheets and direct students to begin, using the worksheets as their activity guide.

5. Conclude the activity by leading a class discussion. Review the worksheet answers with the students. Ask supplemental questions, as provided in the Assessment ("Engineering Questions") and Investigating Questions sections.

### Worksheets and Attachments

### Safety Issues

- Remind students that they should be
**gently**pressing the pencil to their finger to get an idea of what it feels like. Also, specify that no one should test anyone other than themselves. If safety is still a concern, instruct students to imagine pressing the pencil leads/erasers against their fingers.

### Troubleshooting Tips

The biggest challenge in the activity is getting students to realize the inverse relationship between stress and area. A few hints and examples help them figure this out on their own.

Example question: "What would hurt more: Someone stepping on your foot with high heels or tennis shoes?" and following with: "Which shoe has the smaller area?"

Make sure that the mechanical pencils have the lead diameter written on the pencil, since it is almost impossible to measure this with a ruler. Make sure the rulers used to measure the eraser diameters are marked in millimeters so students do not have to convert cm to mm.

### Investigating Questions

What about the mechanical pencil lead made it hurt your finger more than the eraser? (Possible answers: All the force was focused on a smaller point on my finger. The mechanical pencil lead has a smaller cross-sectional area than the eraser.)

If you were designing a second-floor room to hold boxes of books, would you locate the storage area for the books in one small corner of the room or distribute the storage area for the books over a larger area? (Answer: An engineer would consider many factors. Spreading the cases evenly over a larger area would result in less stress concentrated on any one part of the floor, so you could design the room less expensively [resulting in less costs for materials to make it stronger]. But, if you wanted to be able to put all the books in one spot so the rest of the room could be used for other purposes, or you realized that it would be hard to guarantee that people wouldn't stack all the books in one spot, an engineer could design the room structure to handle that amount of load, and it would be the safest thing to do; but it also might cost more in materials to make the structure strong enough. Costs are usually one type of constraint placed on an engineer's design. It all depends on balancing the objectives and safety considerations.)

What are some examples of loads on structures? (Possible answers: Forces on structures might include traffic, such as trains, trucks, bikes, people and cars. Other loads might be from the natural environment, such as gravity, winds, hurricanes, tornadoes, snow, earthquakes, rushing river water or standing water [say from a dam reservoir, hot tub or swimming pool]. And, another load to consider is the weight of the structure itself, the sum of all its materials!)

Suppose you were an engineer designing a steel bridge column. You analyze the column and find that it is over the stress limit for steel. What can you do to the column to lower the stress? (Possible answers: Specify a column size that is thicker across; increase the cross-sectional area. Make the column using higher-grade steel that has higher yield strength. Generally, increasing the cross-sectional area is the preferred choice, since it is less expensive than high-performance steels.)

Is decreasing the amount of force applied to an object the only way to decrease the stress on the object? (Answer: No. You can change the cross-sectional area. Increasing the area reduces the stress on the object.)

Why do engineers care about stress? (Answer: Engineers are responsible for making sure structures are safe. They do calculations to estimate the maximum expected load on each structural member, and then determine the optimum materials and design of the structural members so they can withstand the worst case scenario load. But, they also don't want to increase construction costs more than necessary, so they balance safety and efficiency factors.)

What happens to a structural member when it is stressed beyond the limit stress for the material it is made of? (Answer: It may weaken to the point at which it can no longer carry the load it was designed for, which places more load and stress on the other structural members around it, or it may completely fail and collapse.)

### Assessment

Pre-Activity Assessment

*Class Poll: *Ask for a show of hands to the following questions:

- Who thinks the pencil lead will hurt more than the eraser?
- Who thinks the only reason for this is the fact that the lead is a harder material than the eraser?

Activity Embedded Assessment

*Activity Questions: *The questions on the Feel the Stress Worksheet test the students' ideas before they perform the activity and calculations, versus what they understand afterwards. Review worksheet answers to make sure all students get the most out of the activity.

Post-Activity Assessment

*Engineering Questions: *Wrap up the activity with a discussion of worksheet answers and how the learned concept applies to structural engineering. Ask the students:

- Why do engineers care about stress so much? (Answer: For safety and efficiency reasons.)
- What happens to structural members when they are stressed beyond the limit stress of their material? (Answer: They may collapse or "fail.")
- See the Investigating Questions section for more questions/answers.

### Activity Extensions

*Moving into the Basic Mechanics of Materials:* Expand the scope of the activity by having students learn more about all of the types of forces that induce stresses: tension, compression, torsion, bending, and shear.

*Stress and Efficiency: *Explore as a class or by individual assignment the making of more efficient structural components (such as minimizing the cost, size and weight of components) by decreasing sizes or using different materials when possible, while still keeping structures safe.

*More about Loads *Have students learn the steps that engineers take to design bridges: determining the potential bridge loads, calculating the highest possible load, and calculating the amount of material needed to resist the loads. See TeachEngineering's Designing Bridges lesson and its associated activity to learn more about compression, tension, beams (girders) and columns (piers).

### Activity Scaling

- For lower grades, provide measurements and numbers that are easier for calculation.
- For upper grades, require students to measure the lead and the eraser themselves and pay close attention to units.

### Contributors

Jeffrey Mitchell### Copyright

© 2013 by Regents of the University of Colorado; original © 2009 Washington University in St. Louis### Supporting Program

GK-12 Program, School of Engineering and Applied Science, Washington University in St. Louis### Acknowledgements

This curriculum was developed with support from National Science Foundation GK-12 grant number DGE 0538541. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.

Last modified: January 12, 2019

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