Lesson Stress, Strain and Hooke's Law

Learning Objectives

After this lesson, students should be able to:

• Explain the stress and strain concepts and the relationship between them.
• Explain Hooke's law and apply it to analyze springs.
• Use Microsoft Excel® to make a simple strain plot.
• Relate stress and strain to the unit's engineering challenge.

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.

Common Core State Standards - Math

International Technology and Engineering Educators Association - Technology

State Standards

Worksheets and Attachments

Pre-Req Knowledge

A basic understanding of algebra and the ability to solve simple algebraic expressions. In addition, an understanding of the cancer detection challenge, as introduced in the previous lesson.

Introduction/Motivation

In today's lesson, we will begin to learn about Hooke's law and then we learn how to apply this proportionality to body tissue. We will learn exactly what the terms stress and strain describe as well as the relationship between them. After going through the lesson's material, I will pass out a handout with example problems. Work these to the best of your abilities independently first; then we will review the problems as a class.

After becoming familiar with using the new equations, we will explore Hooke's law in an associated activity (Applying Hooke's Law to Cancer Detection) by experimentally determining an unknown proportionality constant. After exploring Hooke's law, in the second portion of the activity, we will begin to apply what we've learned to develop a means of imaging body tissues and we will soon be able to detect malignant tumors!

You will also practice graphing prepared data to depict cancerous tissue. After we have mastered this material, we will have a quiz on stress, strain and Hooke's law. Please take careful notes and be sure to ask any questions you have about the example problems we will be working through.

Referring back to the legacy cycle which we discussed in the previous lesson, today's lesson constitutes the research and revise phase. Refer back to your initial thoughts notes and record any new information that applies to solving the challenge. Your goal today is to review, revise and expand your current knowledge! Now, let's learn how to detect cancer.

Lesson Background and Concepts for Teachers

Legacy Cycle Information

This lesson falls into the Research and Revise phase of the legacy cycle. Students begin to learn the basic concepts required for creating a strain graph to depict cancerous tissue. Following this lesson, have students revise their initial thoughts and at the conclusion of the associated activity, students should have the skills necessary to Go Public with a solution. But before Going Public, have students complete the Stress, Strain and Hooke's Law Quiz as part of the Test Your Mettle phase of the legacy cycle. The quiz serves as a formative assessment while the next lesson's Go Public phase provides a summative assessment.

Lecture Information

In the late 1600s, Robert Hooke stated that "The power of any springy body is in the same proportion with the extension." Though Hooke's law has remained valid today, the wording has been corrected, replacing power with force. The law is explained by a direct proportionality between a spring's compression or expansion and the restoring force which ensues. The relationship is given by F= -k * Δx where Δx is the distance a spring has been stretch, F is the restoring force exerted by the spring and k is the spring constant which characterizes elastic properties of the spring's material. This law is valid within the elastic limit of a linear spring, when acting along a frictionless surface.

Extending Hooke's exploration of springs, it becomes apparent that most materials act like springs with force being directly proportional to displacement. But as compared to springs, other materials possess an area which must be accounted for. Replacing force with a measure of stress and displacement with a measure of strain, the following expression may be obtained, σ = E* ε. We will now explore the measures of stress and strain.

Stress is a measure of average force per unit area, given by σ = F/A where average stress, σ, equals force, F, acting over area, A. The SI unit for stress is pascals (Pa) which is equal to 1 Newton per square meter. The Psi is an alternative unit which expresses pounds per square inch. The units of stress are equal to the units of pressure which is also a measure of force per unit area.

Stress cannot be measured directly and is therefore inferred from a measure of strain and a constant known as Young's modulus of elasticity. The relationship is given by σ = E* ε, where σ represents stress, ε represents strain and E represents Young's modulus of elasticity. Using this means of inferring stress, strain is a geometrical measure of deformation and Young's modulus is a measure used to characterize the stiffness of an elastic material. Strain does not carry a unit but the units of Young's modulus are Pa.

Strain is characterized by the ratio of total deformation or change in length to the initial length. This relationship is given by ε = Δl/l ₀ where strain, ε, is change in l divided by initial length , l ₀ .

The following problems may be worked independently and reviewed as a class, encouraging students to become more familiar with using the equations given above. Give each student a copy of the Stress, Strain and Hooke's Law Problem Set.

You are required to SHOW ALL WORK. Useful constants that are provided in a table below. (assume given constants have three significant figures (SF). Please also note that the relationships we have just discussed are given below.

Material

• steel

• Young's module: 200x10⁹ E(Pa)

2. cast iron

• Young's module: 100x10⁹ E(Pa)

3. concrete

• Young's module: 20.0x10⁹ E(Pa)

F=m*a σ=F/A ε = Δl/l₀ σ = E* ε F= -k * Δx

1. A 3340 N ball is supported vertically by a 1.90 cm diameter steel cable. Assuming the cable has a length of 10.3 m, determine the stress and the strain in the cable.
2. Consider an iron rod with a cross-sectional area of 3.81 cm2 that has a force of 66,700 N applied to it. Find the stress in the rod.
3. A concrete post with a 50.8 cm diameter is supporting a compressive load of 8910 Newtons. Determine the stress the post is bearing.
4. The concrete post in the previous problem has an initial height of 0.55 m. How much shorter is the post once the load is applied (in mm)?
5. A construction crane with a 1.90 cm diameter cable has a maximum functioning stress of 138 MPa. Find the maximum load that the crane can endure.
6. Consider Hooke's law as a simple proportionality where F is directly proportional to Δx. Therefore, we know the force stretching a spring is directly proportional to the distance the spring stretches. If 223 N stretches a spring 12.7 cm, how much stretch can we expect to result from a of 534 N?
7. Figure 1 shows a column of fatty tissue, determine the strain in each of the three regions.

   

Associated Activities

• Applying Hooke's Law to Cancer Detection - Student groups explore Hooke's law by collecting displacement data for springs with unknown spring constants by adding various masses of known weight. After answering a series of application questions, they apply their new understanding to explore a tissue of known surface area. Then then apply the pertinent relationships to depict a cancerous tumor amidst normal tissue by creating a Microsoft Excel® graph.

Vocabulary/Definitions

radiologist: A medical specialist who examines photographs of tissues, organs, bones for use in the treatment of disease.

strain: Deformation of a body or structure as a result of an applied force. Stretch beyond the proper point or limit.

stress: The physical pressure, pull or other force exerted on a system by another. A load, force, or system of forces producing a strain. The ratio of force to area.

Assessment

Post-Introduction Assessment:

Problem Set: Have students complete the Stress, Strain and Hooke's Law Problem Set in class to gauge their comprehension. The final question of the problem set and the application questions from the associated activity serve as an assessment of students' understanding of the challenge. Use these questions as a means of testing whether students are applying their acquired knowledge toward solving the engineering challenge.

Post-Lesson Assessment:

Quiz: Administer the Stress, Strain and Hooke's Law Quiz as a formative post-lesson assessment, serving as part of the Test your Mettle phase of the legacy cycle.

References

Copyright

Contributors

Supporting Program

Acknowledgements

The contents of this digital library curriculum were developed under National Science Foundation RET grant nos. 0338092 and 0742871. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.